Electrical Engineers Reference Book (16th Edition), Notas de estudo de Engenharia Mecânica

Baixe Electrical Engineers Reference Book (16th Edition) e outras Notas de estudo em PDF para Engenharia Mecânica, somente na Docsity! //integras/b&h/Eer/Final_06-09-02/prelims Electrical Engineer's Reference Book //integras/b&h/Eer/Final_06-09-02/prelims Important notice Many practical techniques described in this book involve potentially dangerous applications of electricity and engineering equipment. The authors, editors and publishers cannot take responsibility for any personal, professional or financial risk involved in carrying out these techniques, or any resulting injury, accident or loss. The techniques described in this book should only be implemented by professional and fully qualified electrical engineers using their own professional judgement and due regard to health and safety issues. //integras/b&h/Eer/Final_06-09-02/prelims Contents Preface Section A ± General Principles 1 Units, Mathematics and Physical Quantities International unit system . Mathematics . Physical quantities . Physical properties . Electricity 2 Electrotechnology Nomenclature . Thermal effects . Electrochemical effects . Magnetic field effects . Electric field effects . Electromagnetic field effects . Electrical discharges 3 Network Analysis Introduction . Basic network analysis . Power-system network analysis Section B ± Materials & Processes 4 Fundamental Properties of Materials Introduction . Mechanical properties . Thermal properties . Electrically conducting materials . Magnetic materials . Dielectric materials . Optical materials . The plasma state 5 Conductors and Superconductors Conducting materials . Superconductors 6 Semiconductors, Thick and Thin-Film Microcircuits Silicon, silicon dioxide, thick- and thin-film technology . Thick- and thin-film microcircuits 7 Insulation Insulating materials . Properties and testing . Gaseous dielectrics . Liquid dielectrics . Semi-fluid and fusible materials . Varnishes, enamels, paints and lacquers . Solid dielectrics . Composite solid/liquid dielectrics . Irradiation effects . Fundamentals of dielectric theory . Polymeric insulation for high voltage outdoor applications 8 Magnetic Materials Ferromagnetics . Electrical steels including silicon steels . Soft irons and relay steels . Ferrites . Nickel±iron alloys . Iron±cobalt alloys . Permanent magnet materials 9 Electroheat and Materials Processing Introduction . Direct resistance heating . Indirect resistance heating . Electric ovens and furnaces . Induction heating . Metal melting . Dielectric heating . Ultraviolet processes . Plasma torches . Semiconductor plasma processing . Lasers 10 Welding and Soldering Arc welding . Resistance welding . Fuses . Contacts . Special alloys . Solders . Rare and precious metals . Temperature- sensitive bimetals . Nuclear-reactor materials . Amorphous materials Section C ± Control 11 Electrical Measurement Introduction . Terminology . The role of measurement traceability in product quality . National and international measurement standards . Direct-acting analogue measuring instruments . Integrating (energy) metering . Electronic instrumentation . Oscilloscopes . Potentiometers and bridges . Measuring and protection transformers . Magnetic measurements . Transducers . Data recording 12 Industrial Instrumentation Introduction . Temperature . Flow . Pressure . Level transducers . Position transducers . Velocity and acceleration . Strain gauges, loadcells and weighing . Fieldbus systems . Installation notes 13 Control Systems Introduction . Laplace transforms and the transfer function . Block diagrams . Feedback . Generally desirable and acceptable behaviour . Stability . Classification of system and static accuracy. Transient behaviour . Root-locus method . Frequency-response methods . State-space description . Sampled-data systems . Some necessary mathematical preliminaries . Sampler and zero-order hold . Block diagrams . Closed-loop systems . Stability . Example . Dead-beat response . Simulation . Multivariable control . Dealing with non linear elements . //integras/b&h/Eer/Final_06-09-02/prelims Disturbances . Ratio control . Transit delays . Stability . Industrial controllers . Digital control algorithms . Auto-tuners . Practical tuning methods 14 Digital Control Systems Introduction . Logic families . Combinational logic . Storage . Timers and monostables . Arithmetic circuits . Counters and shift registers . Sequencing and event driven logic . Analog interfacing . Practical considerations . Data sheet notations 15 Microprocessors Introduction . Structured design of programmable logic systems . Microprogrammable systems . Programmable systems . Processor instruction sets . Program structures . Reduced instruction set computers (RISC) . Software design . Embedded systems 16 Programmable Controllers Introduction . The programmable controller . Programming methods . Numerics . Distributed systems and fieldbus . Graphics . Software engineering . Safety Section D ± Power Electronics and Drives 17 Power Semiconductor Devices Junction diodes . Bipolar power transistors and Darlingtons . Thyristors . Schottky barrier diodes . MOSFET . The insulated gate bipolar transistor (IGBT) 18 Electronic Power Conversion Electronic power conversion principles . Switch-mode power supplies . D.c/a.c. conversion . A.c./d.c. conversion . A.c./a.c. conversion . Resonant techniques . Modular systems . Further reading 19 Electrical Machine Drives Introduction . Fundamental control requirements for electrical machines . Drive power circuits . Drive control . Applications and drive selection . Electromagnetic compatibility 20 Motors and Actuators Energy conversion . Electromagnetic devices . Industrial rotary and linear motors Section E ± Environment 21 Lighting Light and vision . Quantities and units . Photometric concepts . Lighting design technology . Lamps . Lighting design . Design techniques . Lighting applications 22 Environmental Control Introduction . Environmental comfort . Energy requirements . Heating and warm-air systems . Control . Energy conservation . Interfaces and associated data 23 Electromagnetic Compatibility Introduction . Common terms . The EMC model . EMC requirements . Product design . Device selection . Printed circuit boards . Interfaces . Power supplies and power-line filters . Signal line filters . Enclosure design . Interface cable connections . Golden rules for effective design for EMC . System design . Buildings . Conformity assessment . EMC testing and measurements . Management plans 24 Health and Safety The scope of electrical safety considerations . The nature of electrical injuries . Failure of electrical equipment 25 Hazardous Area Technology A brief UK history . General certification requirements . Gas group and temperature class . Explosion protection concepts . ATEX certification . Global view . Useful websites Section F ± Power Generation 26 Prime Movers Steam generating plant . Steam turbine plant . Gas turbine plant . Hydroelectric plant . Diesel-engine plant 27 Alternative Energy Sources Introduction . Solar . Marine energy . Hydro . Wind . Geothermal energy. Biofuels . Direct conversion . Fuel cells . Heat pumps 28 Alternating Current Generators Introduction . Airgap flux and open-circuit e.m.f. . Alternating current windings . Coils and insulation . Temperature rise . Output equation . Armature reaction . Reactances and time constants . Steady-state operation . Synchronising . Operating charts . On-load excitation . Sudden three phase short circuit . Excitation systems . Turbogenerators . Generator±transformer connection . Hydrogenerators . Salient-pole generators other than hydrogenerators . Synchronous compensators . Induction generators . Standards 29 Batteries Introduction . Cells and batteries . Primary cells . Secondary cells and batteries . Battery applications . Anodising . Electrodeposition . Hydrogen and oxygen electrolysis Section G ± Transmission and Distribution 30 Overhead Lines General . Conductors and earth wires . Conductor fittings . Electrical characteristics . Insulators . Supports . Lightning . Loadings //integras/b&h/Eer/Final_06-09-02/prelims 31 Cables Introduction . Cable components . General wiring cables and flexible cords . Supply distribution cables . Transmission cables . Current-carrying capacity . Jointing and accessories . Cable fault location 32 HVDC Introduction . Applications of HVDC . Principles of HVDC converters . Transmission arrangements . Converter station design . Insulation co-ordination of HVDC converter stations . HVDC thyristor valves . Design of harmonic filters for HVDC converters . Reactive power considerations . Control of HVDC . A.c. system damping controls . Interaction between a.c. and d.c. systems . Multiterminal HVDC systems . Future trends 33 Power Transformers Introduction . Magnetic circuit . Windings and insulation . Connections . Three-winding transformers . Quadrature booster transformers . On-load tap changing . Cooling . Fittings . Parallel operation . Auto-transformers . Special types . Testing . Maintenance . Surge protection . Purchasing specifications 34 Switchgear Circuit-switching devices . Materials . Primary-circuit- protection devices . LV switchgear . HV secondary distribution switchgear . HV primary distribution switchgear . HV transmission switchgear . Generator switchgear . Switching conditions . Switchgear testing . Diagnostic monitoring . Electromagnetic compatibility . Future developments 35 Protection Overcurrent and earth leakage protection . Application of protective systems . Testing and commissioning . Overvoltage protection 36 Electromagnetic Transients Introduction . Basic concepts of transient analysis . Protection of system and equipment against transient overvoltage . Power system simulators . Waveforms associated with the electromagnetic transient phenomena 37 Optical Fibres in Power Systems Introduction . Optical fibre fundamentals . Optical fibre cables . British and International Standards . Optical fibre telemetry on overhead power lines . Power equipment monitoring with optical fibre sensors 38 Installation Layout . Regulations and specifications . High-voltage supplies . Fault currents . Substations . Wiring systems . Lighting and small power . Floor trunking . Stand-by and emergency supplies . Special buildings . Low-voltage switchgear and protection . Transformers . Power-factor correction . Earthing . Inspection and testing Section H ± Power Systems 39 Power System Planning The changing electricity supply industry (ESI) . Nature of an electrical power system . Types of generating plant and characteristics . Security and reliability of a power system . Revenue collection . Environmental sustainable planning 40 Power System Operation and Control Introduction . Objectives and requirements . System description . Data acquisition and telemetering . Decentralised control: excitation systems and control characteristics of synchronous machines . Decentralised control: electronic turbine controllers . Decentralised control: substation automation . Decentralised control: pulse controllers for voltage control with tap-changing transformers. Centralised control . System operation . System control in liberalised electricity markets . Distribution automation and demand side management . Reliability considerations for system control 41 Reactive Power Plant and FACTS Controllers Introduction . Basic concepts . Variations of voltage with load . The management of vars . The development of FACTS controllers . Shunt compensation . Series compensation . Controllers with shunt and series components . Special aspects of var compensation . Future prospects 42 Electricity Economics and Trading Introduction . Summary of electricity pricing principles . Electricity markets . Market models . Reactive market 43 Power Quality Introduction . Definition of power quality terms . Sources of problems . Effects of power quality problems . Measuring power quality . Amelioration of power quality problems . Power quality codes and standards Section I ± Sectors of Electricity Use 44 Road Transport Electrical equipment of road transport vehicles . Light rail transit . Battery vehicles . Road traffic control and information systems 45 Railways Railway electrification . Diesel-electric traction . Systems, EMC and standards . Railway signalling and control 46 Ships Introduction . Regulations . Conditions of service . D.c. installations . A.c. installations . Earthing . Machines //integras/b&h/Eer/Final_06-09-02/prelims Electrical Engineer's Reference BookÐonline edition As this book goes to press an online electronic version is also in preparation. The online edition will feature . the complete text of the book . access to the latest revisions (a rolling chapter-by-chapter revision will take place between print editions) . additional material not included in the print version To find out more, please visit the Electrical Engineer's Reference Book web page: http://www.bh.com/newness?isbn=0750646373 or send an e-mail to newnes@elsevier.com //integras/b&h/Eer/Final_06-09-02/part Section A General Principles //integras/b&h/Eer/Final_06-09-02/part //integras/b&h/eer/Final_06-09-02/eerc001 This reference section provides (a) a statement of the International System (SI) of Units, with conversion factors; (b) basic mathematical functions, series and tables; and (c) some physical properties of materials. 1.1 International unit system The International System of Units (SI) is a metric system giving a fully coherent set of units for science, technology and engineering, involving no conversion factors. The starting point is the selection and definition of a minimum set of inde- pendent `base' units. From these, `derived' units are obtained by forming products or quotients in various combinations, again without numerical factors. For convenience, certain combinations are given shortened names. A single SI unit of energy (joule ˆ( kilogram metre-squared per second-squared) is, for example, applied to energy of any kind, whether it be kinetic, potential, electrical, thermal, chemical . . . , thus unify- ing usage throughout science and technology. The SI system has seven base units, and two supplement- ary units of angle. Combinations of these are derived for all other units. Each physical quantity has a quantity symbol (e.g. m for mass, P for power) that represents it in physical equations, and a unit symbol (e.g. kg for kilogram, W for watt) to indicate its SI unit of measure. 1.1.1 Base units Definitions of the seven base units have been laid down in the following terms. The quantity symbol is given in italic, the unit symbol (with its standard abbreviation) in roman type. As measurements become more precise, changes are occasionally made in the definitions. Length: l, metre (m) The metre was defined in 1983 as the length of the path travelled by light in a vacuum during a time interval of 1/299 792 458 of a second. Mass: m, kilogram (kg) The mass of the international prototype (a block of platinum preserved at the International Bureau of Weights and Measures, SeÁ vres). Time: t, second (s) The duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. Electric current: i, ampere (A) The current which, main- tained in two straight parallel conductors of infinite length, of negligible circular cross-section and 1 m apart in vacuum, pro- duces a force equal to 2 ( 10�7 newton per metre of length. Thermodynamic temperature: T, kelvin (K) The fraction 1/273.16 of the thermodynamic (absolute) temperature of the triple point of water. Luminous intensity: I, candela (cd) The luminous intensity in the perpendicular direction of a surface of 1/600 000 m2 of a black body at the temperature of freezing platinum under a pressure of 101 325 newton per square metre. Amount of substance: Q, mole (mol) The amount of sub- stance of a system which contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. The elementary entity must be specified and may be an atom, a molecule, an ion, an electron . . . , or a specified group of such entities. 1.1.2 Supplementary units Plane angle: , & . . . , radian (rad) The plane angle between two radii of a circle which cut off on the circumfer- ence of the circle an arc of length equal to the radius. Solid angle: , steradian (sr) The solid angle which, having its vertex at the centre of a sphere, cuts off an area of the surface of the sphere equal to a square having sides equal to the radius. International unit system 1/3 1.1.3 Notes Temperature At zero K, bodies possess no thermal energy. Specified points (273.16 and 373.16 K) define the Celsius (centigrade) scale (0 and 100C). In terms of intervals, 1C ˆ( 1 K. In terms of levels, a scale Celsius temperature & corresponds to (&‡ 273.16) K. Force The SI unit is the newton (N). A force of 1 N endows a mass of 1 kg with an acceleration of 1 m/s2. Weight The weight of a mass depends on gravitational effect. The standard weight of a mass of 1 kg at the surface of the earth is 9.807 N. 1.1.4 Derived units All physical quantities have units derived from the base and supplementary SI units, and some of them have been given names for convenience in use. A tabulation of those of inter- est in electrical technology is appended to the list in Table 1.1. Table 1.1 SI base, supplementary and derived units Quantity Unit Derivation Unit name symbol Length metre Mass kilogram Time second Electric current ampere Thermodynamic temperature kelvin Luminous intensity candela Amount of mole substance Plane angle radian Solid angle steradian Force newton Pressure, stress pascal Energy joule Power watt Electric charge, flux coulomb Magnetic flux weber Electric potential volt Magnetic flux density tesla Resistance ohm Inductance henry Capacitance farad Conductance siemens Frequency hertz Luminous flux lumen Illuminance lux Radiation activity becquerel Absorbed dose gray Mass density kilogram per cubic metre Dynamic viscosity pascal-second Concentration mole per cubic m kg s A K cd mol rad sr kg m/s2 N N/m2 Pa N m, W s J J/s W A s C V s Wb J/C V s Wb/m2 T V/A Wb/A, V s/A H C/V, A s/V F A/V S �1 Hz cd sr lm lm/m2 lx s �1 Bq J/kg Gy kg/m3 Pa s mol/ 3metre m Linear velocity metre per second m/s Linear metre per second- m/s2 acceleration squared Angular velocity radian per second rad/s cont'd //integras/b&h/eer/Final_06-09-02/eerc001 1/4 Units, mathematics and physical quantities Table 1.1 (continued ) Quantity Unit Derivation Unit name symbol Angular radian per second- acceleration squared rad/s2 Torque newton metre N m Electric field strength volt per metre V/m Magnetic field strength ampere per metre A/m Current density ampere per square metre A/m2 Resistivity ohm metre m Conductivity siemens per metre S/m Permeability henry per metre H/m Permittivity farad per metre F/m Thermal capacity joule per kelvin J/K Specific heat joule per kilogram capacity kelvin J/(kg K) Thermal watt per metre conductivity kelvin W/(m K) Luminance candela per square metre cd/m2 Decimal multiples and submultiples of SI units are indi- cated by prefix letters as listed in Table 1.2. Thus, kA is the unit symbol for kiloampere, and mF that for microfarad. There is a preference in technology for steps of 103. Prefixes for the kilogram are expressed in terms of the gram: thus, 1000 kg ˆ 1 Mg, not 1 kkg. Table 1.2 Decimal prefixes 1.1.5 Auxiliary units Some quantities are still used in special fields (such as vacuum physics, irradiation, etc.) having non-SI units. Some of these are given in Table 1.3 with their SI equivalents. 1.1.6 Conversion factors Imperial and other non-SI units still in use are listed in Table 1.4, expressed in the most convenient multiples or sub- multiples of the basic SI unit [ ] under classified headings. 1.1.7 CGS electrostatic and electromagnetic units Although obsolescent, electrostatic and electromagnetic units (e.s.u., e.m.u.) appear in older works of reference. Neither system is `rationalised', nor are the two mutually compatible. In e.s.u. the electric space constant is "&0 ˆ 1, in e.m.u. the magnetic space constant is 0 ˆ 1; but the SI units take account of the fact that 1/H("&00) is the velocity of electromagnetic wave propagation in free space. Table 1.5 lists SI units with the equivalent number n of e.s.u. and e.m.u. Where these lack names, they are expressed as SI unit names with the prefix `st' (`electrostatic') for e.s.u. and `ab' (`absolute') for e.m.u. Thus, 1 V corresponds to 10�2/3 stV and to 108 abV, so that 1 stV ˆ 300 V and 1 abV ˆ 10�8V. 1.2 Mathematics Mathematical symbolism is set out in Table 1.6. This sub- section gives trigonometric and hyperbolic relations, series (including Fourier series for a number of common wave forms), binary enumeration and a list of common deriva- tives and integrals. 1018 exa E 1015 peta P 1012 tera T 109 giga G 106 mega M 103 kilo k 102 hecto h 101 deca da 10�1 deci d 10�3 milli m 10�6 micro & 10�9 nano n 10�12 pico p 10�15 femto f 10�18 atto a 10�2 centi c Table 1.3 Auxiliary units Quantity Symbol SI Quantity Symbol SI Angle Mass degree () /180 rad tonne t 1000 kg minute (0) Ð Ð second (0 0) Ð Ð Nucleonics, Radiation becquerel Bq 1.0 s �1 Area gray Gy 1.0 J/kg are a 100 m 2 curie Ci 3.7  1010 Bq hectare ha 0.01 km2 rad rd 0.01 Gy barn barn 10�28 m 2 roentgen R 2.6  10�4 C/kg Energy Pressure erg erg 0.1 mJ bar b 100 kPa calorie cal 4.186 J torr Torr 133.3 Pa electron-volt eV 0.160 aJ Time gauss-oersted Ga Oe 7.96 mJ/m3 minute min 60 s Force hour h 3600 s dyne dyn 10 mN day d 86 400 s Length AÊ ngstrom AÊ 0.1 mm Volume litre 1 or L 1.0 dm3 //integras/b&h/eer/Final_06-09-02/eerc001 Mathematics 1/5 Table 1.4 Conversion factors Length [m] Density [kg/m, kg/m3] 1 mil 25.40 mm 1 lb/in 17.86 kg/m 1 in 25.40 mm 1 lb/ft 1.488 kg/m 1 ft 1 yd 1 fathom 1 mile 0.3048 m 0.9144 m 1.829 m 1.6093 km 1 lb/yd 1 lb/in3 1 lb/ft3 1 ton/yd3 0.496 kg/m 27.68 Mg/m3 16.02 kg/m3 1329 kg/m3 1 nautical mile 1.852 km Area [m2] 1 circular mil 1 in 2 1 ft2 1 yd2 1 acre 1 mile2 Volume [m3] 1 in 3 1 ft3 1 yd3 1 UKgal 506.7 mm 2 645.2 mm 2 0.0929 m 2 0.8361 m 2 4047 m 2 2.590 km2 16.39 cm 3 0.0283 m 3 0.7646 m 3 4.546 dm3 Flow rate [kg/s, m 3/s] 1 lb/h 1 ton/h 1 lb/s 1 ft3/h 1 ft3/s 1 gal/h 1 gal/min 1 gal/s Force [N], Pressure [Pa] 1 dyn 1 kgf 1 ozf 0.1260 g/s 0.2822 kg/s 0.4536 kg/s 7.866 cm 3/s 0.0283 m 3/s 1.263 cm 3/s 75.77 cm 3/s 4.546 dm 3/s 10.0 mN 9.807 N 0.278 N 1 lbf 4.445 N Velocity [m/s, rad/s] Acceleration [m/s2, rad/s 2] 1 ft/min 1 in/s 1 ft/s 1 mile/h 1 knot 1 deg/s 5.080 mm/s 25.40 mm/s 0.3048 m/s 0.4470 m/s 0.5144 m/s 17.45 mrad/s 1 tonf 1 dyn/cm2 1 lbf/ft2 1 lbf/in2 1 tonf/ft2 1 tonf/in2 1 kgf/m2 1 kgf/cm2 9.964 kN 0.10 Pa 47.88 Pa 6.895 kPa 107.2 kPa 15.44 MPa 9.807 Pa 98.07 kPa 1 rev/min 0.1047 rad/s 1 mmHg 133.3 Pa 1 rev/s 1 ft/s2 1 mile/h per s 6.283 rad/s 0.3048 m/s2 0.4470 m/s2 1 inHg 1 inH2O 1 ftH2O 3.386 kPa 149.1 Pa 2.989 kPa Mass [kg] Torque [N m] 1 oz 28.35 g 1 ozf in 7.062 nN m 1 lb 0.454 kg 1 lbf in 0.113 N m 1 slug 14.59 kg 1 lbf ft 1.356 N m 1 cwt 50.80 kg 1 tonf ft 3.307 kN m 1 UKton 1016 kg 1 kgf m 9.806 N m Energy [J], Power [W] 1 ft lbf 1 m kgf 1 Btu 1 therm 1 hp h 1 kW h 1.356 J 9.807 J 1055 J 105.5 kJ 2.685 MJ 3.60 MJ Inertia [kg m 2] Momentum [kg m/s, kg m 2/s] 1 oz in2 1 lb in2 1 lb ft2 1 slug ft2 1 ton ft2 0.018 g m 2 0.293 g m 2 0.0421 kg m 2 1.355 kg m 2 94.30 kg m 2 1 Btu/h 1 ft lbf/s 0.293 W 1.356 W 1 lb ft/s 1 lb ft2/s 0.138 kg m/s 0.042 kg m 2/s 1 m kgf/s 9.807 W 1 hp 745.9 W Viscosity [Pa s, m 2/s] Thermal quantities [W, J, kg, K] 1 W/in2 1 Btu/(ft2 h) 1 Btu/(ft3 h) 1 Btu/(ft h F) 1 ft lbf/lb 1.550 kW/m2 3.155 W/m2 10.35 W/m3 1.731 W/(m K) 2.989 J/kg 1 poise 1 kgf s/m2 1 lbf s/ft2 1 lbf h/ft2 1 stokes 1 in 2/s 1 ft2/s 9.807 Pa s 9.807 Pa s 47.88 Pa s 172.4 kPa s 1.0 cm 2/s 6.452 cm 2/s 929.0 cm 2/s 1 Btu/lb 1 Btu/ft3 1 ft lbf/(lb F) 1 Btu/(lb F) 1 Btu/(ft3 F) 2326 J/kg 37.26 KJ/m3 5.380 J/(kg K) 4.187 kJ/(kg K) 67.07 kJ/m 3 K Illumination [cd, lm] 1 lm/ft2 1 cd/ft2 1 cd/in2 10.76 lm/m2 10.76 cd/m2 1550 cd/m2 //integras/b&h/eer/Final_06-09-02/eerc001 1/8 Units, mathematics and physical quantities Figure 1.3 Hyperbolic relations If u is a quadrature (`imaginary') number jv, then 3 4 exp…jv† ˆ 1  jv � v 2 =2! jv =3!‡ v =4! . . . because j2 ˆ�1, j3 ˆ�j1, j4 ˆ‡ 1, etc. Figure 1.2 (right) shows the summation of the first five terms for exp(j1), i.e. exp…j1† ˆ 1 ‡ j1 � 1=2 � j1=6 ‡ 1=24 a complex or expression converging to a point P. The length OP is unity and the angle of OP to the datum axis is, in fact, 1 rad. In general, exp(jv) is equivalent to a shift by €v rad. It follows that exp(jv) ˆ cos v  j sin v, and that exp…jv† ‡ exp…�jv† ˆ 2 cos v exp…jv† � exp…�jv† ˆ j2 sin v For a complex number (u ‡ jv), then exp…u ‡ jv† ˆ exp…u†
exp…jv† ˆ exp…u†
€v Hyperbolic functions A point P on a rectangular hyper- bola (x/a)2�( (y/a)2 ˆ 1 defines the hyperbolic `sector' area 2Sh ˆ 1a ln[(x/a � (y/a)] shown shaded in Figure 1.3 (left). By 2 analogy with &ˆ 2Sc/h2 for the trigonometrical angle , the hyperbolic entity (not an angle in the ordinary sense) is u ˆ 2Sh/a 2, where a is the major semi-axis. Then the hyperbolic functions of u for point P are: sinh u ˆ y=a cosech u ˆ a=y cosh u ˆ x=a sech u ˆ a=x tanh u ˆ y=x coth u ˆ x=y Figure 1.2 Exponential relations The principal relations yield the curves shown in the diagram (right) for values of u between 0 and 3. For higher values sinh u approaches cosh u, and tanh u becomes asymptotic to 1. Inspection shows that cosh(�u) ˆ cosh u, sinh(�u) ˆ�sinh u and cosh2 u� sinh2 u ˆ 1. The hyperbolic functions can also be expressed in the exponential form through the series 4 6cosh u ˆ 1 ‡ u 2 =2!‡ u =4!‡ u =6!‡


( 5 7sinh u ˆ u ‡ u 3 =3!‡ u =5!‡ u =7!‡


( so that cosh u ˆ 1 ‰exp…u† ‡ exp…�u†Š( sinh u ˆ 1 ‰exp…u† � exp…�u†Š2 2 cosh u ‡ sinh u ˆ exp…u†( cosh u � sinh u ˆ exp…�u†( Other relations are: sinh u ‡ sinh v ˆ 2 sinh 1 …u ‡ v†
cosh 1 …u � v†2 2 cosh u ‡ cosh v ˆ 2 cosh 1 …u ‡ v†
cosh 1 …u � v†2 2 cosh u � cosh v ˆ 2 sinh 1 …u ‡ v†
sinh 1 …u � v†2 2 sinh…u  v† ˆ sinh u
cosh v  cosh u
sinh v cosh…u  v† ˆ cosh u
cosh v  sinh u
sinh v tanh…u  v† ˆ …tanh u  tanh v†=…1  tanh u
tanh v†( //integras/b&h/eer/Final_06-09-02/eerc001 Mathematics 1/9 Table 1.8 Exponential and hyperbolic functions u exp(u) exp(�u) sinh u cosh u tanh u 0.0 1.0 1.0 0.0 1.0 0.0 0.1 1.1052 0.9048 0.1092 1.0050 0.0997 0.2 1.2214 0.8187 0.2013 1.0201 0.1974 0.3 1.3499 0.7408 0.3045 1.0453 0.2913 0.4 1.4918 0.6703 0.4108 1.0811 0.3799 0.5 1.6487 0.6065 0.5211 1.1276 0.4621 0.6 1.8221 0.5488 0.6367 1.1855 0.5370 0.7 2.0138 0.4966 0.7586 1.2552 0.6044 0.8 2.2255 0.4493 0.8881 1.3374 0.6640 0.9 2.4596 0.4066 1.0265 1.4331 0.7163 1.0 2.7183 0.3679 1.1752 1.5431 0.7616 1.2 3.320 0.3012 1.5095 1.8107 0.8337 1.4 4.055 0.2466 1.9043 2.1509 0.8854 1.6 4.953 0.2019 2.376 2.577 0.9217 1.8 6.050 0.1653 2.942 3.107 0.9468 2.0 7.389 0.1353 3.627 3.762 0.9640 2.303 10.00 0.100 4.950 5.049 0.9802 2.5 12.18 0.0821 6.050 6.132 0.9866 2.75 15.64 0.0639 7.789 7.853 0.9919 3.0 20.09 0.0498 10.02 10.07 0.9951 3.5 33.12 0.0302 16.54 16.57 0.9982 4.0 54.60 0.0183 27.29 27.31 0.9993 4.5 90.02 0.0111 45.00 45.01 0.9998 4.605 100.0 0.0100 49.77 49.80 0.9999 5.0 148.4 0.0067 74.20 74.21 0.9999 5.5 244.7 0.0041 122.3 cosh u # tanh u # 6.0 403.4 0.0025 201.7 ˆ sinh u ˆ 1.0 6.908 1000 0.0010 500 ˆ 1 2 exp(u) sinh…u  jv† ˆ …sinh u
cos v†  j…cosh u
sin v†( cosh…u  jv† ˆ …cosh u
cos v†  j…sinh u
sin v†(„ d…sinh u†=du ˆ( cosh u sinh u
du ˆ( cosh u „ d…cosh u†=du ˆ( sinh u cosh u
du ˆ( sinh u Exponential and hyperbolic functions of u between zero and 6.908 are listed in Table 1.8. Many calculators can give such values directly. 1.2.3 Bessel functions Problems in a wide range of technology (e.g. in eddy currents, frequency modulation, etc.) can be set in the form of the Bessel equation   2d2 y 1 dy n‡
‡( 1 �( y ˆ( 0 2dx2 x dx x and its solutions are called Bessel functions of order n. For n ˆ 0 the solution is 4 =22 6 =22
42J0…x† ˆ (1 � …x 2 =22† ‡ …x
42† � …x
62† ‡


( and for n ˆ 1, 2, 3 . . .  # n 2 4x x x Jn…x† ˆ ( 1 �( ‡( �


( 2nn! 2…2n ‡ 2†( 2
4…2n ‡ 2†…2n ‡ 4†( Table 1.9 gives values of Jn(x) for various values of n and x. 1.2.4 Series Factorials In several of the following the factorial (n!) of integral numbers appears. For n between 2 and 10 these are 2! ˆ( 2 1/2! ˆ 0.5 3! ˆ( 6 1/3! ˆ 0.1667 4! ˆ( 24 1/4! ˆ 0.417  10�1 5! ˆ( 120 1/5! ˆ 0.833  10�2 6! ˆ( 720 1/6! ˆ 0.139  10�2 7! ˆ( 5 040 1/7! ˆ 0.198  10�3 8! ˆ( 40 320 1/8! ˆ 0.248  10�4 9! ˆ( 362 880 1/9! ˆ 0.276  10�5 10! ˆ 3 628 800 1/10! ˆ 0.276  10�6 Progression Arithmetic a ‡ (a ‡ d) ‡ (a ‡ 2d) ‡


‡ [a ‡ (n � 1)d] ˆ 1 n (sum of 1st and nth terms) 2 nGeometric a ‡ ar ‡ ar 2 ‡


‡ arn�1 ˆ a(1�r )/(1�r) Trigonometric See Section 1.2.1. Exponential and hyperbolic See Section 1.2.2. Binomial n…n � 1†…n � 2†(…1  x†n ˆ( 1  nx ‡( n…n � 1†( x 2 ( x 3 ‡


( 2! 3! n! ‡ …�1†r xr ‡


( r!…n � r†! n …a  x†n ˆ( an‰1  …x=a†Š //integras/b&h/eer/Final_06-09-02/eerc001 1/10 Units, mathematics and physical quantities Binomial coefficients n!/[r! (n�r)!] are tabulated: Term r ˆ( 0 1 2 3 4 5 6 7 8 9 10 n ˆ 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 7 1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 1 9 1 9 36 84 126 126 84 36 9 1 10 1 10 45 120 210 252 210 120 45 10 1 Power If there is a power series for a function f(h), it is given by ii†( iii†f …h† ˆ f …0† ‡ hf …i†…0† ‡ …h2 =2!†f … …0† ‡ …h3 =3!†f … …0† ‡


( ‡ …hr =r!†f …r†…0† ‡


( …Maclaurin†( …ii†f …x ‡ h† ˆ f …x† ‡ hf …i†…x† ‡ …h2 =2!†f …x† ‡


( ‡ …hr =r!†f …r†…x† ‡


( …Taylor†( Permutation, combination nPr ˆ n…n � 1†…n � 2†…n � 3† . . . …n � r ‡ 1† ˆ n!=…n � r†! nCr ˆ …1=r!†‰n…n�1†…n�2†…n�3† . . . …n�r ‡ 1†Š ˆ n!=r!…n�r†! Bessel See Section 1.2.3. Fourier See Section 1.2.5. 1.2.5 Fourier series A univalued periodic wave form f() of period 2& is repre- sented by a summation in general of sine and cosine waves of fundamental period 2& and of integral harmonic orders n (ˆ 2, 3, 4, . . .) as f …† ˆ c0 ‡ a1 cos &‡ a2 cos 2&‡


‡ an cos n&‡


( ‡ b1 sin &‡ b2 sin 2&‡


‡ bn sin n&‡


( The mean value of f() over a full period 2& is 1 …#2& c0 ˆ( f …†
d& 2& 0 and the harmonic-component amplitudes a and b are 1 …#2& 1 …#2& an ˆ( f …†
cos n&
d;& bn ˆ( f …†
sin n&
d& & 0 & 0 Table 1.10 gives for a number of typical wave forms the harmonic series in square brackets, preceded by the mean value c0 where it is not zero. 1.2.6 Derivatives and integrals Some basic forms are listed in Table 1.11. Entries in a given column are the integrals of those in the column to its left and the derivatives of those to its right. Constants of integration are omitted. 1.2.7 Laplace transforms Laplace transformation is a method of deriving the response of a system to any stimulus. The system has a basic equation of behaviour, and the stimulus is a pulse, step, sine wave or other variable with time. Such a response involves integration: the Laplace transform method removes integration difficulties, as tables are available for the direct solution of a great variety of problems. The pro- cess is analogous to evaluation (for example) of y ˆ 2.13.6 by transformation into a logarithmic form log y ˆ 3.6  log(2.1), and a subsequent inverse transformation back into arithmetic by use of a table of antilogarithms. The Laplace transform (L.t.) of a time-varying function f(t) is …#1( L‰ f …t†Š ˆ F…s† ˆ ( exp…�st†
f …t†
dt 0 and the inverse transformation of F(s) to give f(t) is L�1‰F…s†Š ˆ f …t† ˆ lim 1 …#‡j!& exp…st†
F…s†
ds 2 �j!& The process, illustrated by the response of a current i(t) in an electrical network of impedance z to a voltage v(t) applied at t ˆ 0, is to write down the transform equation I…s† ˆ V…s†=Z…s†( where I(s) is the L.t. of the current i(t), V(s) is the L.t. of the voltage v(t), and Z(s) is the operational impedance. Z(s) is obtained from the network resistance R, inductance L and capacitance C by leaving R unchanged but replacing L by Ls and C by 1/Cs. The process is equivalent to writing the network impedance for a steady state frequency !& and then replacing j!& by s. V(s) and Z(s) are polynomials in s: the quotient V(s)/Z(s) is reduced algebraically to a form recog- nisable in the transform table. The resulting current/time relation i(t) is read out: it contains the complete solution. However, if at t ˆ 0 the network has initial energy (i.e. if currents flow in inductors or charges are stored in capa- citors), the equation becomes I…s† ˆ ‰V…s† ‡U…s†Š=Z…s†( where U(s) contains such terms as LI0 and (1/s)V0 for the inductors or capacitors at t ˆ 0. A number of useful transform pairs is listed in Table 1.12. 1.2.8 Binary numeration A number N in decimal notation can be represented by an ordered set of binary digits an, an�2, . . . , a2, a1, a0 such that N ˆ 2nan ‡ 2n�1 an�1 ‡


‡ 2a1 ‡ a0 Decimal 1 2 3 4 5 6 7 8 9 10 100 Binary 1 10 11 100 101 110 111 1000 1001 1010 1100100 //integras/b&h/eer/Final_06-09-02/eerc001 Mathematics 1/13 Table 1.10 (continued ) Wave form Series  # 1 2 & sin & cos 2& cos 4& cos 6& Rectified sine (half-wave): a ‡ a �( �( �( �


(  & 4 1
3 3
5 5
7  # 2 4 cos 2& cos 4& cos 6& cos 8& Rectified sine (full-wave): a � a ‡( ‡( ‡( ‡


(  & 1
3 3
5 5
7 7
9  # m & 2m & cos m& cos 2m& cos 3m& Rectified sine (m-phase): a sin ‡ a sin �( ‡( �


( & m & m m2 � 1 4m2 � 1 9m2 � 1  # & 2 sin &
cos & sin 2&
cos 2& sin 3&
cos 3& Rectangular pulse train: a ‡ a ‡( ‡( ‡


(  & 1 2 3  # & 2& cos & cos 2& cos 3& a ‡ a ‡( ‡( ‡


( for & ( &  & 1 2 3 "# 1 1 1 & 4 sin2 …2 †( sin2 2…( †( sin2 3…( †2Triangular pulse train: a ‡a cos‡( 2 cos2‡( cos3‡


Š ( 2& & 1 4 9  & a ‡a ‰cos‡cos2‡cos 3‡


Š ( for && 2 & where the as have the values either 1 or 0. Thus, if N ˆ 19, 19 ˆ 16 ‡ 2 ‡ 1 ˆ (24)1 ‡ (23)0 ‡ (22)0 ‡ (21)1 ‡ (20)1 ˆ 10011 in binary notation. The rules of addition and multiplication are 0 ‡ 0 ˆ 0, 0 ‡ 1 ˆ 1, 1 ‡ 1 ˆ 10; 00 ˆ 0, 01 ˆ 0, 11 ˆ 1 1.2.9 Power ratio In communication networks the powers P1 and P2 at two specified points may differ widely as the result of ampli- fication or attenuation. The power ratio P1/P2 is more convenient in logarithmic terms. Neper [Np] This is the natural logarithm of a voltage or current ratio, given by a ˆ( ln…V1=V2 †( or a ˆ( ln…I1=I2† Np If the voltages are applied to, or the currents flow in, identical impedances, then the power ratio is a ˆ( ln…V1=V2 †2 ˆ( 2 ln…V1=V2†( and similarly for current. Decibel [dB] The power gain is given by the common logarithm lg(P1/P2) in bel [B], or most commonly by A ˆ 10 log(P1/P2) decibel [dB]. With again the proviso that the powers are developed in identical impedances, the power gain is A ˆ( 10 log…P1 =P2† ˆ (10 log…V1 =V2†2 ˆ( 20 log…V1 =V2† dB Table 1.13 gives the power ratio corresponding to a gain A (in dB) and the related identical-impedance voltage (or current) ratios. Approximately, 3 dB corresponds to a power ratio of 2, and 6 dB to a power ratio of 4. The decibel equivalent of 1 Np is 8.69 dB. 1.2.10 Matrices and vectors 1.2.10.1 Definitions If a11, a12, a13, a14 . . . is a set of elements, then the rectangu- lar array 2 3# a14 . . . a1na11 a12 a13 7a24 . . . a2n 6# a21 a22 a23 6 7A ˆ( 4 5# am1 am2 am3 am4 . . . amn arranged in m rows and n columns is called an (m  n) matrix. If m ˆ n then A is n-square. //integras/b&h/eer/Final_06-09-02/eerc001 1/14 Units, mathematics and physical quantities Table 1.11 Derivatives and integrals „ d[ f(x)]/dx f(x) f(x)
( dx 1 x n�1 n nx x (n =�1) �1/x 2 1/x 1/x ln x exp x exp x cos x sin x �sin x cos x 2 sec x tan x �cosec x
( cot x cosec x sec x
( tan x sec x �cosec2 x cot x 21/ H(a 2�x ) arcsin(x/a) 2�1/H(a 2�x ) arccos(x/a) 2a/(a 2 ‡( x ) arctan(x/a) 2�a/x H(x 2�a ) arccosec(x/a) 2a/x H(x �a 2) arcsec(x/a) 2�a/(a 2 ‡( x ) arccot(x/a) cosh x sinh x sinh x cosh x sech2 x tanh x �cosech x
( coth x cosech x �sech x
( tanh x sech x �cosech2 x coth x 1/H(x 2 ‡ 1) arsinh x 1/H(x 2�1) arcosh x 1/(1�x 2) artanh x �1/x H(x 2 ‡( 1) arcosech x �1/x H(1�x 2) arsech x 1/(1�x 2) arcoth x dv du u(x)
( v(x)u ‡( v dx dx 1 du u dv u…x† v dx �( v2 dx v…x† r exp(xa) ( sin(!x ‡&‡( ) exp(ax) ( sin(!x ‡( ) 1 2 x2 n ‡ 1x /(n ‡ 1) ln x x ln x�x exp x �cos x sin x ln(sec x) ln ( tan 1 x)2 ln(sec x ‡ tan x) ln(sin x) 2x arcsin(x/a) ‡H(a 2�x ) 2x arccos(x/a)�H(a 2�x ) 1x arctan(x=a) �( 2 a ln (a2+x2) 2x arccosec(x/a) ‡ a ln | x ‡( H(x 2�a ) | 2x arcsec(x/a)�a ln | x ‡H(x 2�a ) | 2x arccot(x/a) ‡ 1 a ln (a +x2)2 cosh x sinh x ln(cosh x) �ln(tanh 1 x†2 2 arctan (exp x) ln(sinh x) x arsinh x�H(1 ‡( x 2) x arcosh x�H(x 2�1) x artanh x ‡ 1 2 ln (1 �( x2) x arcosech x ‡ arsinh x x arsech x ‡ arcsin x x arcoth x ‡ 1 2 ln (x2 �( 1) …# du uv �( v dv dv Ð (1/r)exp(ax)sin(!x ‡�) 2r ˆ(H(!2 ‡( a ) &ˆ( arctan (!/a) An ordered set of elements x ˆ( [x1, x2, x3 . . . xn] is called 1.2.10.3 Rules of operation an n-vector. (i) Associativity A ‡( (B ‡( C) ˆ( (A ‡( B) ‡( C, An (n ( 1) matrix is called a column vector and a (1 ( n) A(BC) ˆ( (AB)C ˆ(ABC. matrix a row vector. (ii) Distributivity A(B ‡C) ˆ(AB ‡AC, (B ‡C)A ˆ(BA ‡CA. 1.2.10.2 Basic operations (iii) Identity If U is the (n ( n) matrix (ij), i, j ˆ( 1 . . . n, If A ˆ( (ars), B ˆ( (brs), where ij ˆ( 1 if i ˆ( j and 0 otherwise, then U is the (i) Sum C ˆ(A ‡B is defined by crs ˆ( ars ‡ brs, for diagonal unit matrix and A U ˆ(A. (iv) Inverse If the product U ˆ(AB exists, then B ˆ(A�1,r ˆ( 1 . . . m; s ˆ( 1 . . . n. the inverse matrix of A. If both inverses A�1 and B�1 (ii) Product If A is an (m ( q) matrix and B is a (q ( n) A�1exist, then (A B)�1 ˆ(B�1 .matrix, then the product C ˆ(AB is an (m ( n) matrix defined by (crs) ˆ(p arp bps, p ˆ( 1 . . . q; r ˆ( 1 . . . m; (v) Transposition The transpose of A is written as AT s ˆ( 1 . . . n. If AB ˆ(BA then A and B are said to commute. and is the matrix whose rows are the columns (iii) Matrix-vector product If x ˆ( [x1 . . . xn], then b ˆ(Ax is of A. If the product C ˆ(AB exists then defined by (br) ˆ(p arp xp, p ˆ( 1 . . . n; r ˆ( 1 . . . m. C T ˆ( (AB)T ˆ( BTAT . (iv) Multiplication of a matrix by a (scalar) element If k is (vi) Conjugate For A ˆ( (ars), the congugate of A is an element then C ˆ( kA ˆ(Ak is defined by (crs) ˆ( k(ars). denoted by A* ˆ( (ars*). (v) Equality If A ˆ(B, then (aij) ˆ( (bij), for i ˆ( 1 . . . n; (vii) Orthogonality Matrix A is orthogonal if AAT ˆ( U.j ˆ( 1 . . . m. //integras/b&h/eer/Final_06-09-02/eerc001 & �( & Mathematics 1/15 Table 1.12 Laplace transforms Definition f( t ) from t ˆ( 0+ F(s) ˆ( L‰ f (t)Š ˆ ( „1( 0�( f (t)
( exp (�st)
( dt Sum First derivative nth derivative Definite integral Shift by T Periodic function (period T ) Initial value Final value af1(t)+bf2(t) (d/dt) f (t) n(dn/dt ) f (t) „ T f (t)
( dt 0� f(t�T ) f(t) f(t), t!0+ f(t), t!1( aF1(s)+bF2(s) sF(s)�f(0�) n n�2s F(s)�s n�1f(0�)� s f (1)(0�)�
(
(
�f (n�1)(0�) 1 F(s) s exp(�sT )
(F(s) 1 …#T exp (�sT)
( f (t)
( dt 1 �( exp ( �( sT ) 0�( sF(s), s!1( sF(s), s!0 Description f(t) F(s) f(t) to base t 1. Unit impulse (t) 2. Unit step H(t) 3. Delayed step H(t�T ) 4. Rectangular pulse (duration T ) H(t)�H(t�T ) 5. Unit ramp t 6. Delayed ramp (t�T )H(t�T ) 7. nth-order ramp tn 8. Exponential decay exp(� t) 9. Exponential rise 1�exp(� t) 10. Exponential ( t t exp(� t) 11. Exponential ( tn tn exp(� t) 12. Difference of exponentials exp(� t)�exp(� t) 1 1 s exp (�st) s 1 �( exp (�sT ) s 1 s2 exp (�sT) 2s n! sn‡1 1 s ‡( & & s(s ‡( ) 1 (s ‡( )2 n! (s ‡( )n‡1 (s ‡( )(s ‡( ) cont'd //integras/b&h/eer/Final_06-09-02/eerc001 1/18 Units, mathematics and physical quantities definition, but may be described, as an aid to an intuitive appreciation. Energy is the capacity for `action' or work. Work is the measure of the change in energy state. State is the measure of the energy condition of a system. System is the ordered arrangement of related physical entities or processes, represented by a model. Mode is a description or mathematical formulation of the system to determine its behaviour. Behaviour describes (verbally or mathematically) the energy processes involved in changes of state. Energy storage occurs if the work done on a system is recoverable in its original form. Energy conversion takes place when related changes of state concern energy in a different form, the process sometimes being reversible. Energy dissipation is an irreversible conversion into heat. Energy transmission and radiation are forms of energy transport in which there is a finite propagation time. W In a physical system there is an identifiable energy input i and output Wo. The system itself may store energy Ws and dissipate energy W. The energy conservation principle states that Wi ˆWs ‡W ‡Wo Comparable statements can be made for energy changes
w and for energy rates (i.e. powers), giving
wi ˆ
ws ‡
w ‡
wo and pi ˆ ps ‡ p ‡ po 1.3.1.1 Analogues In some cases the mathematical formulation of a system model resembles that of a model in a completely different physical system: the two systems are then analogues. Consider linear and rotary displacements in a simple mechanical system with the conditions in an electric circuit, with the following nomenclature: A mechanical element (such as a spring) of compliance k (which describes the displacement per unit force and is the inverse of the stiffness) has a displacement l ˆ kf when a force f is applied. At a final force f1 the potential energy stored is W=1 kf1 2. For the rotary case, &ˆ kM and2 W =1 kM1 2. In the electric circuit with a pure capacitance 2 C, to which a p.d. v is applied, the charge is q ˆCv and the 1electric energy stored at v1 is W=2 Cv 2 f 1. Use is made of these correspondences in mechanical problems (e.g. of vibration) when the parameters can be con- sidered to be `lumped'. An ideal transformer, in which the primary m.m.f. in ampere-turns i1N1 is equal to the second- ary m.m.f. i2N2 has as analogue the simple lever, in which a force f1 at a point distant l1 from the fulcrum corresponds to 2 at l2 such that f1l1 ˆ f2l2. A simple series circuit is described by the equation v ˆ L(di/dt) ‡Ri ‡ q/C or, with i written as dq/dt, 2v ˆL(d2 q/dt ) ‡R(dq/dt) ‡ (1/C)q A corresponding mechanical system of mass, compliance and viscous friction (proportional to velocity) in which for a displacement l the inertial force is m(du/dt), the compli- ance force is l/k and the friction force is ru, has a total force f ˆ m…d2l=dt2† ‡ r…dl=dt† ‡ …1=k†l Thus the two systems are expressed in identical mathemat- ical form. 1.3.1.2 Fields Several physical problems are concerned with `fields' having stream-line properties. The eddyless flow of a liquid, the cur- rent in a conducting medium, the flow of heat from a high- to a low-temperature region, are fields in which representative lines can be drawn to indicate at any point the direction of f m force [N] mass [kg] M J torque [N m] inertia [kg m 2] r friction [N s/m] r friction [N m s/rad] k compliance [m/N] k compliance [rad/N m] l displacement [m] & displacement [rad] u velocity [m/s] !& angular velocity [rad/s] v voltage [V] L inductance [H] R resistance [ ] C capacitance [F] q charge [C] i current [A] The force necessary to maintain a uniform linear velocity u against a viscous frictional resistance r is f ˆ ur; the power is p ˆ fu ˆ u 2 r and the energy expended over a distance l is W ˆ fut ˆ u 2rt, since l ˆ ut. These are, respectively, the ana- logues of v ˆ iR, p ˆ vi ˆ i2R and W ˆ vit ˆ i2Rt for the corresponding electrical system. For a constant angular velocity in a rotary mechanical system, M ˆ!r, p ˆM!ˆ!2 r and W ˆ!2rt, since &ˆ!t. If a mass is given an acceleration du/dt, the force required is f ˆm(du/dt) and the stored kinetic energy at velocity u1 2is W =1 mu1. For rotary acceleration, M ˆ J(d!/dt) and 2 W =1 J!2 1. Analogously the application of a voltage v to a 2 pure inductor L produces an increase of current at the rate di/dt such that v ˆL(di/dt) and the magnetic energy stored 1at current i1 is W=2 Li 2. the flow there. Other lines, orthogonal to the flow lines, con- nect points in the field having equal potential. Along these equipotential lines there is no tendency for flow to take place. Static electric fields between charged conductors (having equipotential surfaces) are of interest in problems of insula- tion stressing. Magnetic fields, which in air-gaps may be assumed to cross between high-permeability ferromagnetic surfaces that are substantially equipotentials, may be studied in the course of investigations into flux distribution in machines. All the fields mentioned above satisfy Laplacian equations of the form …@2 V=@x 2† ‡ …@2V=@y 2† ‡ …@2V=@z 2† ˆ 0 The solution for a physical field of given geometry will apply to other Laplacian fields of similar geometry, e.g. System Potential Flux Medium current flow voltage V current I conductivity & heat flow temperature & heat q thermal conductivity & electric field voltage V electric flux Q permittivity "& magnetic field m.m.f. F magnetic flux & permeability & //integras/b&h/eer/Final_06-09-02/eerc001 Physical quantities 1/19 The ratio I/V for the first system would give the effective conductance G; correspondingly for the other systems, q/& gives the thermal conductance, Q/V gives the capacitance and /F gives the permeance, so that if measurements are made in one system the results are applicable to all the others. It is usual to treat problems as two-dimensional where possible. Several field-mapping techniques have been devised, generally electrical because of the greater convenience and precision of electrical measurements. For two-dimensional problems, conductive methods include high-resistivity paper sheers, square-mesh `nets' of resistors and electrolytic tanks. The tank is especially adaptable to three-dimensional cases of axial symmetry. In the electrolytic tank a weak electrolyte, such as ordinary tap-water, provides the conducting medium. A scale model of the electrode system is set into the liquid. A low-voltage supply at some frequency between 50 Hz and 1 kHz is connected to the electrodes so that current flows through the electrolyte between them. A probe, adjustable in the horizontal plane and with its tip dipping vertically into the electrolyte, enables the potential field to be plotted. Electrode models are constructed from some suitable insulant (wood, paraffin wax, Bakelite, etc.), the electrode outlines being defined by a highly conductive material such as brass or copper. The metal is silver-plated to improve conductivity and reduce polarisation. Three-dimensional cases with axial symmetry are simulated by tilting the tank and using the surface of the electrolyte as a radial plane of the system. The conducting-sheet analogue substitutes a sheet of resistive material (usually `teledeltos' paper with silver- painted electrodes) for the electrolyte. The method is not readily adaptable to three-dimensional plots, but is quick and inexpensive in time and material. The mesh or resistor-net analogue replaces a conductive continuum by a square mesh of equal resistors, the potential measurements being made at the nodes. Where the bound- aries are simple, and where the `grain size' is sufficiently small, good results are obtained. As there are no polarisation troubles, direct voltage supply can be used. If the resistors are made adjustable, the net can be adapted to cases of inhomo- geneity, as when plotting a magnetic field in which perme- ability is dependent on flux density. Three-dimensional plots are made by arranging plane meshes in layers; the nodes are now the junctions of six instead of four resistors. A stretched elastic membrane, depressed or elevated in appropriate regions, will accommodate itself smoothly to the differences in level: the height of the membrane everywhere can be shown to be in conformity with a two-dimensional Laplace equation. Using a rubber sheet as a membrane, the path of electrons in an electric field between electrodes in a vacuum can be investigated by the analogous paths of rolling bearing-balls. Many other useful analogues have been devised, some for the rapid solution of mathematical processes. Recently considerable development has been made in point-by-point computer solutions for the more compli- cated field patterns in three-dimensional space. 1.3.2 Structure of matter Material substances, whether solid, liquid or gaseous, are conceived as composed of very large numbers of molecules. A molecule is the smallest portion of any substance which cannot be further subdivided without losing its characteristic material properties. In all states of matter molecules are in a state of rapid continuous motion. In a solid the molecules are relatively closely `packed' and the molecules, although rapidly moving, maintain a fixed mean position. Attractive forces between molecules account for the tendency of the solid to retain its shape. In a liquid the molecules are less closely packed and there is a weaker cohesion between them, so that they can wander about with some freedom within the liquid, which consequently takes up the shape of the vessel in which it is contained. The molecules in a gas are still more mobile, and are relatively far apart. The cohesive force is very small, and the gas is enabled freely to contract and expand. The usual effect of heat is to increase the intensity and speed of molecular activity so that `collisions' between molecules occur more often; the average spaces between the molecules increase, so that the substance attempts to expand, producing internal pressure if the expansion is resisted. Molecules are capable of further subdivision, but the resulting particles, called atoms, no longer have the same properties as the molecules from which they came. An atom is the smallest portion of matter than can enter into chemical combination or be chemically separated, but it cannot gener- ally maintain a separate existence except in the few special cases where a single atom forms a molecule. A molecule may consist of one, two or more (sometimes many more) atoms of various kinds. A substance whose molecules are composed entirely of atoms of the same kind is called an element. Where atoms of two or more kinds are present, the molecule is that of a chemical compound. At present over 100 elements are recognised (Table 1.14: the atomic mass number A is relative to 1/12 of the mass of an element of carbon-12). If the element symbols are arranged in a table in ascend- ing order of atomic number, and in columns (`groups') and rows (`periods') with due regard to associated similarities, Table 1.15 is obtained. Metallic elements are found on the left, non-metals on the right. Some of the correspondences that emerge are: Group 1a: Alkali metals (Li 3, Na 11, K 19, Rb 37, Cs 55, Fr 87) 2a: Alkaline earths (Be 4, Mg 12, Ca 20, Sr 38, Ba 56, Ra 88) 1b: Copper group (Cu 29, Ag 47, Au 79) 6b: Chromium group (Cr 24, Mo 42, W 74) 7a: Halogens (F 9, Cl 17, Br 35, I 53, At 85) 0: Rare gases (He 2, Ne 10, Ar 18, Kr 36, Xe 54, Rn 86) 3a±6a: Semiconductors (B 5, Si 16, Ge 32, As 33, Sb 51, Te 52) In some cases a horizontal relation obtains as in the transition series (Sc 21 . . . Ni 28) and the heavy-atom rare earth and actinide series. The explanation lies in the struc- ture of the atom. 1.3.2.1 Atomic structure The original Bohr model of the hydrogen atom was a central nucleus containing almost the whole mass of the atom, and a single electron orbiting around it. Electrons, as small particles of negative electric charge, were discovered at the end of the nineteenth century, bringing to light the complex structure of atoms. The hydrogen nucleus is a proton, a mass having a charge equal to that of an electron, but positive. Extended to all elements, each has a nucleus comprising mass particles, some (protons) with a positive charge, others (neutrons) with no charge. The atomic mass number A is the total number of protons and neutrons in the nucleus; the atomic number Z is the number of positive charges, and the normal number of orbital electrons. The nuclear structure is not known, and the forces that bind the protons against their mutual attraction are conjectural. //integras/b&h/eer/Final_06-09-02/eerc001 1/20 Units, mathematics and physical quantities The hydrogen atom (Figure 1.4) has one proton (Z ˆ( 1) and one electron in an orbit formerly called the K shell. Helium (Z ˆ( 2) has two protons, the two electrons occupy- ing the K shell which, by the Pauli exclusion principle, can- not have more than two. The next element in order is lithium (Z ˆ( 3), the third electron in an outer L shell. With elements of increasing atomic number, the electrons are added to the L shell until it holds a maximum of 8, the surplus then occupying the M shell to a maximum of 18. The number of `valence' electrons (those in the outermost shell) determines the physical and chemical properties of the element. Those with completed outer shells are `stable'. Isotopes An element is often found to be a mixture of atoms with the same chemical property but different atomic masses: these are isotopes. The isotopes of an element must have the same number of electrons and protons, but differ in the number of neutrons, accounting for the non-integral average mass numbers. For example, neon comprises 90.4% of mass number 20, with 0.6% of 21 and 9.0% of mass number 22, giving a resultant mass number of 20.18. Energy states Atoms may be in various energy states. Thus, the filament of an incadescent lamp may emit light when excited by an electric current but not when the current is switched off. Heat energy is the kinetic energy of the atoms of a heated body. The more vigorous impact of atoms may not always shift the atom as a whole, but may shift an electron from one orbit to another of higher energy level within the atom. This position is not normally stable, and the electron gives up its momentarily acquired potential energy by falling back to its original level, releasing the energy as a light quantum or photon. Ionisation Among the electrons of an atom, those of the outermost shell are unique in that, on account of all the electron charges on the shells between them and the nucleus, they are the most loosely bound and most easily removable. In a variety of ways it is possible so to excite an atom that one of the outer electrons is torn away, leaving the atom ionised or converted for the time into an ion with an effect- ive positive charge due to the unbalanced electrical state it has acquired. Ionisation may occur due to impact by other fast-moving particles, by irradiation with rays of suitable wavelength and by the application of intense electric fields. 1.3.2.2 Wave mechanics The fundamental laws of optics can be explained without regard to the nature of light as an electromagnetic wave phenomenon, and photoelectricity emphasises its nature as a stream or ray of corpuscles. The phenomena of diffraction or interference can only be explained on the wave concept. Wave mechanics correlates the two apparently conflicting ideas into a wider concept of `waves of matter'. Electrons, atoms and even molecules participate in this duality, in that their effects appear sometimes as corpuscular, sometimes as of a wave nature. Streams of electrons behave in a corpus- cular fashion in photoemission, but in certain circumstances show the diffraction effects familiar in wave action. Considerations of particle mechanics led de Broglie to write several theoretic papers (1922±1926) on the parallel- ism between the dynamics of a particle and geometrical optics, and suggested that it was necessary to admit that classical dynamics could not interpret phenomena involving energy quanta. Wave mechanics was established by SchroÈ dinger in 1926 on de Broglie's conceptions. When electrons interact with matter, they exhibit wave properties: in the free state they act like particles. Light has a similar duality, as already noted. The hypothesis of de Broglie is that a particle of mass m and velocity u has wave Table 1.14 Elements (Z, atomic number; A, atomic mass; KLMNOPQ, electron shells) Z Name and symbol A Shells K L 1 Hydrogen H 1.008 1 Ð 2 Helium He 4.002 2 Ð 3 Lithium Li 6.94 2 1 4 Beryllium Be 9.02 2 2 5 Boron B 10.82 2 3 6 Carbon C 12 2 4 7 Nitrogen N 14.01 2 5 8 Oxygen O 16.00 2 6 9 Fluorine F 19.00 2 7 10 Neon Ne 20.18 2 8 KL M N 11 Sodium Na 22.99 10 1 Ð 12 Magnesium Mg 24.32 10 2 Ð 13 Aluminium Al 26.97 10 3 Ð 14 Silicon Si 28.06 10 4 Ð 15 Phosphorus P 31.02 10 5 Ð 16 Sulphur S 32.06 10 6 Ð 17 Chlorine Cl 35.46 10 7 Ð 18 Argon Ar 39.94 10 8 Ð 19 Potassium K 39.09 10 8 1 20 Calcium Ca 40.08 10 8 2 21 Scandium Sc 45.10 10 9 2 22 Titanium Ti 47.90 10 10 2 23 Vanadium V 0.95 10 11 2 24 Chromium Cr 52.01 10 13 1 25 Manganese Mn 54.93 10 13 2 26 Iron Fe 55.84 10 14 2 27 Cobalt Co 58.94 10 15 2 28 Nickel Ni 58.69 10 16 2 29 Copper Cu 63.57 10 18 1 30 Zinc Zn 65.38 10 18 2 31 Gallium Ga 69.72 10 18 3 32 Germanium Ge 72.60 10 18 4 33 Arsenic As 74.91 10 18 5 34 Selenium Se 78.96 10 18 6 35 Bromine Br 79.91 10 18 7 36 Krypton Kr 83.70 10 18 8 KLM N O 37 Rubidium Rb 85.44 28 8 1 38 Strontium Sr 87.63 28 8 2 39 Yttrium Y 88.92 28 9 2 40 Zirconium Zr 91.22 28 10 2 41 Niobium Nb 92.91 28 12 1 42 Molybdenum Mo 96.0 28 13 1 43 Technetium Tc 99.0 28 14 1 44 Ruthenium Ru 101.7 28 15 1 45 Rhodium Rh 102.9 28 16 1 46 Palladium Pd 106.7 28 18 Ð 47 Silver Ag 107.9 28 18 1 48 Cadmium Cd 112.4 28 18 2 49 Indium In 114.8 28 18 3 50 Tin Sn 118.7 28 18 4 51 Antimony Sb 121.8 28 18 5 52 Tellurium Te 127.6 28 18 6 53 Iodine I 126.9 28 18 7 54 Xenon Xe 131.3 28 18 8 KLM N O P 55 Caesium Cs 132.9 28 18 8 1 56 Barium Ba 137.4 28 18 8 2 cont'd //integras/b&h/eer/Final_06-09-02/eerc001 Physical quantities 1/23 Table 1.16 Physical properties of metals Approximate general properties at normal temperatures: & density [kg/m3] k thermal conductivity [W/(m K)] E elastic modulus [GPa] Tm melting point [K] e linear expansivity [mm/(m K)] & resistivity [n m] c specific heat capacity [kJ/(kg K)] & resistance±temperature coefficient [m /( K)] Metal & E e c k Tm & & Pure metals 4 Beryllium 1840 300 120 1700 170 1560 33 9.0 11 Sodium 970 Ð 71 710 130 370 47 5.5 12 Magnesium 1740 44 26 1020 170 920 46 3.8 13 Aluminium 2700 70 24 900 220 930 27 4.2 19 Potassium 860 Ð 83 750 130 340 67 5.4 20 Calcium 1550 Ð 22 650 96 1120 43 4.2 24 Chromium 7100 25 8.5 450 43 2170 130 3.0 26 Iron 7860 220 12 450 75 1810 105 6.5 27 Cobalt 8800 210 13 420 70 1770 65 6.2 28 Nickel 8900 200 13 450 70 1730 78 6.5 29 Copper 8930 120 16 390 390 1360 17 4.3 30 Zinc 7100 93 26 390 110 690 62 4.1 42 Molybdenum 10 200 Ð 5 260 140 2890 56 4.3 47 Silver 10 500 79 19 230 420 1230 16 3.9 48 Cadmium 8640 60 32 230 92 590 75 4.0 50 Tin 7300 55 27 230 65 500 115 4.3 73 Tantalum 16 600 190 6.5 140 54 3270 155 3.1 74 Tungsten 19 300 360 4 130 170 3650 55 4.9 78 Platinum 21 500 165 9 130 70 2050 106 3.9 79 Gold 19 300 80 14 130 300 1340 23 3.6 80 Mercury 13 550 Ð 180 140 10 230 960 0.9 82 Lead 11 300 15 29 130 35 600 210 4.1 83 Bismuth 9800 32 13 120 9 540 1190 4.3 92 Uranium 18 700 13 Ð 120 Ð 1410 220 2.1 Alloys Brass (60 Cu, 40 Zn) 8500 100 21 380 120 1170 60 2.0 Bronze (90 Cu, 10 Sn) 8900 100 19 380 46 1280 Ð Ð Constantan 8900 110 15 410 22 1540 450 0.05 Invar (64 Fe, 36 Ni) 8100 145 2 500 16 1720 100 2.0 Iron, soft (0.2 C) 7600 220 12 460 60 1800 140 Ð Iron cast (3.5 C, 2.5 Si) 7300 100 12 460 60 1450 Ð Ð Manganin 8500 130 16 410 22 1270 430 0.02 Steel (0.85 C) 7800 200 12 480 50 1630 180 Ð Electron emission A metal may be regarded as a potential `well' of depth �V relative to its surface, so that an electron in the lowest energy state has (at absolute zero temperature) the energy W ˆ(Ve (of the order 10 eV): other electrons occupy levels up to a height "* (5±8 eV) from the bottom of the `well'. Before an electron can escape from the surface it must be endowed with an energy not less than &ˆ( W�"*, called the work function. Emission occurs by surface irradiation (e.g. with light) of frequency v if the energy quantum hv of the radiation is at least equal to . The threshold of photoelectric emission is therefore with radiation at a frequency not less than v ˆ(/h. Emission takes place at high temperatures if, put simply, the kinetic energy of electrons normal to the surface is great enough to jump the potential step W. This leads to an expression for the emission current i in terms of temperature T, a constant A and the thermionic work function : i ˆ( AT2 exp…�=kT †( Electron emission is also the result of the application of a high electric field intensity (of the order 1±10 GV/m) to a metal surface; also when the surface is bombarded with electrons or ions of sufficient kinetic energy, giving the effect of secondary emission. Crystals When atoms are brought together to form a crystal, their individual sharp and well-defined energy levels merge into energy bands. These bands may overlap, or there may be gaps in the energy levels available, depending on the lattice spacing and interatomic bonding. Conduction can take place only by electron migration into an empty or partly filled band; filled bands are not available. If an elec- tron acquires a small amount of energy from the externally applied electric field, and can move into an available empty level, it can then contribute to the conduction process. 1.3.2.5 Insulators In this case the `distance' (or energy increase
w in electron- volts) is too large for moderate electric applied fields to endow electrons with sufficient energy, so the material remains an insulator. High temperatures, however, may //integras/b&h/eer/Final_06-09-02/eerc001 1/24 Units, mathematics and physical quantities Table 1.17 Physical properties of non-metals Approximate general properties: & density [kg/m3] Tm melting point [K] e linear expansivity [mm/(m K)] & resistivity [M m] c specific heat capacity [kJ/(kg K)] r relative permittivity [�] k thermal conductivity [W/(m K)] Material & e c k Tm & "&r Asbestos (packed) 580 Ð 0.84 0.19 Ð Ð 3 Bakelite 1300 30 0.92 0.20 Ð 0.1 7 Concrete (dry) Diamond 2000 3510 10 1.3 0.92 0.49 1.70 165 Ð 4000 Ð 107 Ð Ð Glass 2500 8 0.84 0.93 Ð 106 8 Graphite Marble 2250 2700 2 12 0.69 0.88 160 3 3800 Ð 10�11 103 Ð 8.5 Mica 2800 3 0.88 0.5 Ð 108 7 Nylon Paper Paraffin wax 1140 900 890 100 Ð 110 1.7 Ð 2.9 0.3 0.18 0.26 Ð Ð Ð Ð 104 109 Ð 2 2 Perspex 1200 80 1.5 1.9 Ð 1014 3 Polythene Porcelain 930 2400 180 3.5 2.2 0.8 0.3 1.0 Ð 1900 Ð 106 2.3 6 Quartz (fused) Rubber 2200 1250 0.4 Ð 0.75 1.5 0.22 0.15 2000 Ð 1014 107 3.8 3 Silicon 2300 7 0.75 Ð 1690 0.1 2.7 Table 1.18 Physical properties of liquids Average values at 20C (293 K): & density [kg/m3] k thermal conductivity [W/(m K)] v e viscosity [mPa s] cubic expansivity [10�3/K] Tm Tb melting point [K] boiling point [K] c specific heat capacity [kJ/(kg K)] "&r relative permittivity [�] Liquid & v e c k Tm Tb "&r Acetone (CH3)2CO 792 0.3 1.43 2.2 0.18 178 329 22 Benzine C6H6 881 0.7 1.15 1.7 0.14 279 353 2.3 Carbon disulphide CS2 1260 0.4 1.22 1.0 0.14 161 319 2.6 Carbon tetrachloride CCl4 1600 1.0 1.22 0.8 0.10 250 350 2.2 Ether (C2H5)2O 716 0.2 1.62 2.3 0.14 157 308 4.3 Glycerol C3H5(OH)3 1270 1500 0.50 2.4 0.28 291 563 56 Methanol CH3OH 793 0.6 1.20 1.2 0.21 175 338 32 Oil Ð 850 85 0.75 1.6 0.17 Ð Ð 3.0 Sulphuric acid H2SO4 1850 28 0.56 1.4 Ð 284 599 Ð Turpentine C10H16 840 1.5 0.10 1.8 0.15 263 453 2.3 Water H2O 1000 1.0 0.18 4.2 0.60 273 373 81 result in sufficient thermal agitation to permit electrons to `jump the gap'. 1.3.2.6 Semiconductors Intrinsic semiconductors (i.e. materials between the good conductors and the good insulators) have a small spacing of about 1 eV between their permitted bands, which affords a low conductivity, strongly dependent on temperature and of the order of one-millionth that of a conductor. Impurity semiconductors have their low conductivity raised by the presence of minute quantities of foreign atoms (e.g. 1 in 108) or by deformations in the crystal struc- ture. The impurities `donate' electrons of energy level that can be raised into a conduction band (n-type); or they can attract an electron from a filled band to leave a `hole', or electron deficiency, the movement of which corresponds to the movement of a positive charge (p-type). 1.3.2.7 Magnetism Modern magnetic theory is very complex, with ramifica- tions in several branches of physics. Magnetic phenomena are associated with moving charges. Electrons, considered as particles, are assumed to possess an axial spin, which gives them the effect of a minute current turn or of a small permanent magnet, called a Bohr magneton. The gyro- scopic effect of electron spin develops a precession when a magnetic field is applied. If the precession effect exceeds the spin effect, the external applied magnetic field produces less //integras/b&h/eer/Final_06-09-02/eerc001 Physical quantities 1/25 Table 1.19 Physical properties of gases Values at 0C (273 K) and atmospheric pressure: c & density [kg/m3] k thermal conductivity [m W/(m K)] v viscosity [mPa s] Tm melting point [K] p specific heat capacity [kJ/(kg K)] Tb boiling point [K] cp/cv ratio between specific heat capacity at constant pressure and at constant volume Gas & v cp cp/cv k Tm Tb Air Ammonia Carbon dioxide Carbon monoxide Chlorine Deuterium Ethane Fluorine Helium Hydrogen Hydrogen chloride Krypton Methane Neon Nitrogen Oxygen Ozone Propane Sulphur dioxide Xenon O O N H F Ð NH3 CO2 CO Cl2 D C2H6 2 He 2 HCl Kr CH4 Ne 2 2 3 C3H8 SO2 Xe 1.293 0.771 1.977 1.250 3.214 0.180 1.356 1.695 0.178 0.090 1.639 3.740 0.717 0.900 1.251 1.429 2.220 2.020 2.926 5.890 17.0 9.3 13.9 16.4 12.3 Ð 8.6 Ð 18.6 8.5 13.8 23.3 10.2 29.8 16.7 19.4 Ð 7.5 11.7 22.6 1.00 2.06 0.82 1.05 0.49 Ð 1.72 0.75 5.1 14.3 0.81 Ð 2.21 1.03 1.04 0.92 Ð 1.53 0.64 Ð 1.40 1.32 1.31 1.40 1.36 1.73 1.22 Ð 1.66 1.41 1.41 1.68 1.31 1.64 1.40 1.40 1.29 1.13 1.27 1.66 24 22 14 23 7.6 Ð 18 Ð 144 174 Ð 8.7 30 46 24 25 Ð 15 8.4 5.2 Ð 195 216* 68 171 18 89 50 1.0 14 161 116 90 24 63 55 80 83 200 161 Ð 240 194 81 239 23 184 85 4.3 20 189 121 112 27 77 90 161 231 263 165 *At pressure of 5 atm. magnetisation than it would in free space, and the material of which the electron is a constituent part is diamagnetic. If the spin effect exceeds that due to precession, the material is paramagnetic. The spin effect may, in certain cases, be very large, and high magnetisations are produced by an external field: such materials are ferromagnetic. An iron atom has, in the n ˆ( 4 shell (N), electrons that give it conductive properties. The K, L and N shells have equal numbers of electrons possessing opposite spin direc- tions, so cancelling. But shell M contains 9 electrons spin- ning in one direction and 5 in the other, leaving 4 net magnetons. Cobalt has 3, and nickel 2. In a solid metal further cancellation occurs and the average number of unbalanced magnetons is: Fe, 2.2; Co, 1.7; Ni, 0.6. In an iron crystal the magnetic axes of the atoms are aligned, unless upset by excessive thermal agitation. (At 770C for Fe, the Curie point, the directions become random and ferromagnetism is lost.) A single Fe crystal magnetises most easily along a cube edge of the structure. It does not exhibit spontaneous magnetisation like a per- manent magnet, however, because a crystal is divided into a large number of domains in which the various magnetic directions of the atoms form closed paths. But if a crystal is exposed to an external applied magnetic field, (a) the elec- tron spin axes remain initially unchanged, but those domains having axes in the favourable direction grow at the expense of the others (domain wall displacement); and (b) for higher field intensities the spin axes orientate into the direction of the applied field. If wall movement makes a domain acquire more internal energy, then the movement will relax again when the exter- nal field is removed. But if wall movement results in loss of energy, the movement is non-reversibleÐi.e. it needs Table 1.20 Characteristic temperatures Temperature T [kelvin] corresponds to c ˆ(T � 273.15 [degree Celsius] and to f ˆ( c (9/5)�32 [degree Fahrenheit]. Condition T c f Absolute zero 0 �273.15 �459.7 Boiling point of oxygen 90.18 �182.97 �297.3 Zero of Fahrenheit scale 255.4 �17.78 0 Melting point of ice 273.15 0 32.0 Triple point of water 273.16 0.01 32.02 Maximum density of water 277.13 3.98 39.16 `Normal' ambient 293.15 20 68 Boiling point of water 373.15 100 212 Boiling point of sulphur 717.8 444.6 832 Freezing point of silver 1234 962 1762 Freezing point of gold 1336 1064 1945 external force to reverse it. This accounts for hysteresis and remanence phenomena. The closed-circuit self-magnetisation of a domain gives it a mechanical strain. When the magnetisation directions of individual domains are changed by an external field, the strain directions alter too, so that an assembly of domains will tend to lengthen or shorten. Thus, readjustments in the crystal lattice occur, with deformations (e.g. 20 parts in 106) in one direction. This is the phenomenon of magnetostriction. The practical art of magnetics consists in control of mag- netic properties by alloying, heat treatment and mechanical working to produce variants of crystal structure and conse- quent magnetic characteristics. //integras/b&h/eer/Final_06-09-02/eerc001 1/28 Units, mathematics and physical quantities Figure 1.8 Conduction in low-pressure gas always present due to stray radiations (light, etc.). The elec- trons produced attach themselves to gas atoms and the sets of positive and negative ions drift in opposite directions. At very low gas pressures the electrons produced by ionisa- tion have a much longer free path before they collide with a molecule, and so have scope to attain high velocities. Their motional energy may be enough to shockionise neutral atoms, resulting in a great enrichment of the electron stream and an increased current flow. The current may build up to high values if the effect becomes cumulative, and eventually conduction may be effected through a spark or arc. In a vacuum conduction can be considered as purely electronic, in that any electrons present (there can be no molecular matter present in a perfect vacuum) are moved in accordance with the force exerted on them by an applied electric field. The number of electrons is small, and although high speeds may be reached, the conduction is generally measurable only in milli- or microamperes. Some of the effects are illustrated in Figure 1.8, represent- ing part of a vessel containing a gas or vapour at low pres- sure. At the bottom is an electrode, the cathode, from the surface of which electrons are emitted, generally by heating the cathode material. At the top is a second electrode, the anode, and an electric field is established between the electro- des. The field causes electrons emitted from the cathode to move upward. In their passage to the anode these electrons will encounter gas molecules. If conditions are suitable, the gas atoms are ionised, becoming in effect positive charges associated with the nuclear mass. Thereafter the current is increased by the detached electrons moving upwards and by the positive ions moving more slowly downwards. In certain devices (such as the mercury arc rectifier) the impact of ions on the cathode surface maintains its emission. The impact of electrons on the anode may be energetic enough to cause the secondary emission of electrons from the anode surface. If the gas molecules are excluded and a vacuum is established, the conduction becomes purely electronic. 1.5.2.5 Insulators If an electric field is applied to a perfect insulator, whether solid, liquid or gaseous, the electric field affects the atoms by producing a kind of `stretching' or `rotation' which displaces the electrical centres of negative and positive in opposite directions. This polarisation of the dielectric insu- lating material may be considered as taking place in the manner indicated in Figure 1.9. Before the electric field is Figure 1.9 Polarisation and breakdown in insulator applied, the atoms of the insulator are neutral and unstrained; as the potential difference is raised the electric field exerts opposite mechanical forces on the negative and positive charges and the atoms become more and more highly strained (Figure 1.9(a)). On the left face the atoms will all present their negative charges at the surface: on the right face, their positive charges. These surface polarisations are such as to account for the effect known as permittivity. The small displacement of the atomic electric charges con- stitutes a polarisation current. Figure 1.9(b) shows that, for excessive electric field strength, conduction can take place, resulting in insulation breakdown. The electrical properties of metallic conductors and of insulating materials are listed in Tables 1.22 and 1.23. 1.5.2.6 Convection current Charges can be moved mechanically, on belts, water-drops, dust and mist particles, and by beams of high-speed electrons (as in a cathode ray oscilloscope). Such movement, indepen- dent of an electric field, is termed a convection current. 1.5.3 Charges in acceleration Reference has been made to the emission of energy (photons) when an electron falls from an energy level to a lower one. Radiation has both a particle and a wave nature, the latter associated with energy propagation through empty space and through transparent media. 1.5.3.1 Maxwell equations Faraday postulated the concept of the field to account for `action at a distance' between charges and between magnets. Maxwell (1873) systematised this concept in the form of electromagnetic field equations. These refer to media in bulk. They naturally have no direct relation to the elec- tronic nature of conduction, but deal with the fluxes of elec- tric, magnetic and conduction fields, their flux densities, and the bulk material properties (permittivity ", permeabil- ity & and conductivity ) of the media in which the fields exist. To the work of Faraday. AmpeÁ re and Gauss, Maxwell added the concept of displacement current. Displacement current Around an electric field that changes with time there is evidence of a magnetic field. By analogy with the magnetic field around a conduction current, the rate of change of an electric field may be represented by the presence of a displacement current. The concept is applicable to an electric circuit containing a capacitor: there is a conduction current ic in the external circuit but not between the electrodes of the capacitor. The capacitor, however, must be acquiring or losing charge and its electric field must be changing. If the rate of change is represented by a displacement current id ˆ( ic, not only is the magnetic field accounted for, but also there now exists a `continuity' of current around the circuit. Displacement current is present in any material medium, conducting or insulating, whenever there is present an //integras/b&h/eer/Final_06-09-02/eerc001 Electricity 1/29 Table 1.22 Electrical properties of conductors Typical approximate values at 293 K (20 C): g conductivity relative to I.S.A.C. [%] & resistivity [n m] & resistance±temperature coefficient [m /( K)] Material g  & International standard annealed copper (ISAC) Copper annealed hard-drawn Brass (60/40) cast rolled Bronze Phosphor-bronze Cadmium-copper, hard-drawn Copper-clad steel, hard-drawn Aluminium cast hard-drawn duralumin Iron wrought cast grey white malleable nomag Steel 0.1% C 0.4% C core 1% Si 2% Si 4% Si wire galvanised 45 ton 80 ton Resistance alloys* 80 Ni, 20 Cr 59 Ni, 16 Cr, 25 Fe 37 Ni, 18 Cr, 2 Si, 43 Fe 45 Ni, 54 Cu 20 Ni, 80 Cu 15 Ni, 62 Cu, 22 Zn 4 Ni, 84 Cu, 12 Mn Gold Lead Mercury Molybdenum Nickel Platinum Silver annealed hard-drawn Tantalum Tungsten Zinc 100 17.2 3.93 99 17.3 3.90 97 17.7 3.85 23 75 1.6 19 90 1.6 48 36 1.65 29±14 6±12 1.0 82±93 21±18 4.0 30±40 57±43 3.75 66 26 3.90 62 28 3.90 36 47 Ð 16 107 5.5 2.5 700 Ð 1.7 1000 2.0 5.9 300 Ð 1.1 1600 4.5 8.6 200 4.2 11 160 4.2 10 170 Ð 4.9 350 Ð 3.1 550 Ð 12 140 4.4 10 170 3.4 8 215 3.4 (1) 1.65 1090 0.1 (2) 1.62 1100 0.2 (3) 1.89 1080 0.26 (4) 3.6 490 0.04 (5) 6.6 260 0.29 (6) 5.0 340 0.25 (7) 3.6 480 0.0 73 23.6 3.0 7.8 220 4.0 1.8 955 0.7 30 57 4.0 12.6 136 5.0 14.7 117 3.9 109 15.8 4.0 98.5 17.5 4.0 11.1 155 3.1 31 56 4.5 28 62 4.0 *Resistance alloys: (1) furnaces, radiant elements; (2) electric irons, tubular heaters; (3) furnace elements; (4) control resistors; (5) cupro; (6) German silver, platinoid; (7) Manganin. electric field that changes with time. There is a displacement current along a copper conductor carrying an alternating current, but the conduction current is vastly greater even at very high frequencies. In poor conductors and in insulating materials the displacement current is comparable to (or greater than) the conduction current if the frequency is high enough. In free space and in a perfect insulator only displacement current is concerned. Equations The following symbols are used, the SI unit of each appended. The permeability and permittivity are absolute values (&ˆ(r0, "&ˆ( "&r "&0). Potentials and fluxes are scalar quantities: field strength and flux density, also surface and path-length elements, are vectorial. Field Electric Magnetic Conduction Potential V [V] F [A] V [V] Field strength E [V/m] H [A/m] E [V/m] Flux Flux density Q [C] D [C/m2] & [Wb] B [T] I [A] J [A/m2] Material property "& [F/m] & [H/m] & [S/m] The total electric flux emerging from a charge ‡Q or entering a charge �Q is equal to Q. The integral of the elec- tric flux density D over a closed surface s enveloping the charge is …# D
( ds ˆ( Q …1:1†( s If the surface has no enclosed charge, the integral is zero. This is the Gauss law. The magnetomotive force F, or the line integral of the magnetic field strength H around a closed path l, is equal to the current enclosed, i.e. …# H
( dl ˆ( F ˆ( ic ‡( id …1:2†( o This is the AmpeÁ re law with the addition of displacement current. The Faraday law states that, around any closed path l encircling a magnetic flux & that changes with time, there is an electric field, and the line integral of the electric field strength E around the path is …# E
( dl ˆ( e ˆ �…d=dt† …1:3†( o Magnetic flux is a solenoidal quantity, i.e. it comprises a structure of closed loops; over any closed surface s in a mag- netic field as much flux leaves the surface as enters it. The sur- face integral of the flux density B is therefore always zero, i.e. …# B
( ds ˆ( 0 …1:4†( s To these four laws are added the constitutive equations, which relate the flux densities to the properties of the media in which the fields are established. The first two are, respectively, electric and magnetic field relations; the third relates conduction current density to the voltage gradient in a conducting medium; the fourth is a statement of the dis- placement current density resulting from a time rate of change of the electric flux density. The relations are D ˆ( "E; B ˆ( H; Jc ˆ( E; Jd ˆ( @D=@t //integras/b&h/eer/Final_06-09-02/eerc001 1/30 Units, mathematics and physical quantities Table 1.23 Electrical properties of insulating materials Typical approximate values (see also Section 1.4): "&r relative permittivity E electric strength [MV/m] tan & loss tangent & maximum working temperature [C] k G thermal conductivity density [mW/(m K)] [kg/m3] Material "&r E tan & & k G Air at n.t.p. 1.0 3 Ð Ð 25 1.3 Alcohol 26 Ð Ð Ð 180 790 Asbestos 2 2 Ð 400 80 3000 paper 2 2 Ð 250 250 1200 Bakelite moulding 4 6 0.03 130 Ð 1600 paper 5 15 0.03 100 270 1300 Bitumen pure 2.7 1.6 Ð 50 150 1200 vulcanised 4.5 5 Ð 100 200 1250 Cellulose film 5.8 28 Ð Ð Ð 800 Cotton fabric dry Ð 0.5 Ð 95 80 Ð impregnated Ð 2 Ð 95 250 Ð Ebonite 2.8 50 0.005 80 150 1400 Fabric tape, impregnated 5 17 0.1 95 240 Ð Glass flint 6.6 6 Ð Ð 1100 4500 crown 4.8 6 0.02 Ð 600 2200 toughened 5.3 9 0.003 Ð Ð Ð Gutta-percha 4.5 Ð 0.02 Ð 200 980 Marble 7 2 0.03 Ð 2600 2700 Mica 6 40 0.02 750 600 2800 Micanite Ð 15 Ð 125 150 2200 Oil transformer 2.3 Ð Ð 85 160 870 castor 4.7 8 Ð Ð Ð 970 Paper dry 2.2 5 0.007 90 130 820 impregnated 3.2 15 0.06 90 140 1100 Porcelain 5.7 15 0.008 1000 1000 2400 Pressboard 6.2 7 Ð 95 170 1100 Quartz fused 3.5 13 0.002 1000 1200 2200 crystalline 4.4 Ð Ð Ð Ð 2700 Rubber pure 2.6 18 0.005 50 100 930 vulcanised 4 10 0.01 70 250 1500 moulding 4 10 Ð 70 Ð Ð Resin 3 Ð Ð Ð Ð 1100 Shellac 3 11 Ð 75 250 1000 paper 5.5 11 0.05 80 Ð 1350 Silica, fused 3.6 14 Ð Ð Ð Ð Silk Ð Ð Ð 95 60 1200 Slate Ð 0.5 Ð Ð 2000 2800 Steatite Ð 0.6 Ð 1500 2000 2600 Sulphur 4 Ð 0.0003 100 220 2000 Water 70 Ð Ð Ð 570 1000 Wax (paraffin) 2.2 12 0.0003 35 270 860 In electrotechnology concerned with direct or low- frequency currents, the Maxwell equations are rarely used in the form given above. Equation (1.2), for example, appears as the number of amperes (or ampere-turns) required to produce in an area a the specified magnetic flux &ˆBa ˆHa. Equation (1.3) in the form e ˆ�(d/dt) gives the e.m.f. in a transformer primary or secondary turn. The concept of the `magnetic circuit' embodies Equation (1.4). But when dealing with such field phenomena as the eddy currents in massive conductors, radio propagation or the transfer of energy along a transmission line, the Maxwell equations are the basis of analysis. //integras/b&h/Eer/Final_06-09-02/eerc002 2 Electrotechnology 2.1 Nomenclature 2/3 2.1.1 2/3 2.1.2 2/4 2.2 2/6 2.2.1 Resistance 2/6 2.2.2 2/8 2.3 2/10 2.3.1 Electrolysis 2/10 2.3.2 Cells 2/11 2/12 2.4.1 2/12 2.4.2 2/14 2.4.3 2/15 2.4.4 Inductance 2/17 2.5 2/19 2.5.1 Electrostatics 2/20 2.5.2 Capacitance 2/20 2.5.3 2/22 2/22 2.6 2/23 2.6.1 2/23 2.6.2 2/23 2.6.3 2/23 2.7 2/25 2.7.1 Introduction 2/25 2.7.2 2/26 2.7.3 2/27 2/28 M G Say PhD, MSc, CEng, ACGI, DIC, FIEE, FRSE Heriot-Watt University (Sections 2.1±2.6) G R Jones PhD, DSc, CEng, FIEE, MInst P University of Liverpool (Section 2.7) Contents Circuit phenomena Electrotechnical terms Thermal effects Heating and cooling Electrochemical effects 2.4 Magnetic field effects Magnetic circuit Magnetomechanical effects Electromagnetic induction Electric field effects Dielectric breakdown 2.5.4 Electromechanical effects Electromagnetic field effects Movement of charged particles Free space propagation Transmission line propagation Electrical discharges Types of discharge Discharge±network interaction 2.7.4 Discharge applications //integras/b&h/Eer/Final_06-09-02/eerc002 //integras/b&h/Eer/Final_06-09-02/eerc002 Nomenclature 2/3 Electrotechnology concerns the electrophysical and allied principles applied to practical electrical engineering. A com- pletely general approach is not feasible, and many separate ad hoc technologies have been developed using simplified and delimited areas adequate for particular applications. In establishing a technology it is necessary to consider whether the relevant applications can be dealt with (a) in macroscopic terms of physical qualities of materials in bulk (as with metallic conduction or static magnetic fields); or (b) in microscopic terms involving the microstructure of materials as an essential feature (as in domain theory); or (c) in molecu- lar, atomic or subatomic terms (as in nuclear reaction and semi-conduction). There is no rigid line of demarcation, and certain technologies must cope with two or more such sub- divisions at once. Electrotechnology thus tends to become an assembly of more or less discrete (and sometimes apparently unrelated) areas in which methods of treatment differ widely. To a considerable extent (but not completely), the items of plant with which technical electrical engineering dealsÐ generators, motors, feeders, capacitors, etc.Ðcan be repre- sented by equivalent circuits or networks energised by an electrical source. For the great majority of cases within the purview of `heavy electrical engineering' (that is, generation, transmis- sion and utilisation for power purposes, as distinct from telecommunications), a source of electrical energy is con- sidered to produce a current in a conducting circuit by reason of an electromotive force acting against a property of the circuit called impedance. The behaviour of the circuit is described in terms of the energy fed into the circuit by the source, and the nature of the conversion, dissipation or stor- age of this energy in the several circuit components. Electrical phenomena, however, are only in part asso- ciated with conducting circuits. The generalised basis is one of magnetic and electrical fields in free space or in material media. The fundamental starting point is the conception contained in Maxwell's electromagnetic equations (Section 1.5.3), and in this respect the voltage and currents in a circuit are only representative of the fundamental field phenomena within a restricted range. Fortunately, this range embraces very nearly the whole of `heavy' electrical engineering practice. The necessity for a more comprehensive viewpoint makes itself apparent in connection with problems of long-line transmission; and when the technique of ultra- high-frequency work is reached, it is necessary to give up the familiar circuit ideas in favour of a whole-hearted application of field principles. 2.1 Nomenclature 2.1.1 Circuit phenomena Figure 2.1 shows in a simplified form a hypothetical circuit with a variety of electrical energy sources and a representative selection of devices in which the energy received from the source is converted into other forms, or stored, or both. The forms of variation of the current or voltage are shown in Figure 2.2. In an actual circuit the current may change in a quite arbitrary fashion as indicated at (a): it may rise or fall, or reverse its direction, depending on chance or control. Such random variation is inconveniently difficult to deal with, and engineers prefer to simplify the conditions as much as poss- ible. For example (Figure 2.2(b)), the current may be assumed to be rigidly constant, in which case it is termed a direct cur- rent. If the current be deemed to reverse cyclically according to a sine function, it becomes a sinusoidal alternating current (c). Less simple waveforms, such as (d), may be dealt with by Figure 2.1 Typical circuit devices. G, source generator; R, resistor; A, arc; B, battery; P, plating bath; M, motor; L, inductor; C, capacitor; I, insulator Figure 2.2 Modes of current (or voltage) variation application of Fourier's theorem, thus making it possible to calculate a great range of practical casesÐsuch as those involving rectifiersÐin which the sinusoidal waveform assumption is inapplicable. The cases shown in (b), (c) and (d) are known as steady states, the current (or voltage) being assumed established for a considerable time before the circuit is investigated. But since the electric circuit is capable of stor- ing energy, a change in the circuit may alter the conditions so as to cause a redistribution of circuit energy. This occurs with a circulation of transient current. An example of a simple oscillatory transient is shown in Figure 2.2(e). The calculation of circuits in which direct currents flow is comparatively straightforward. For sine wave alternating current circuits an algebra has been developed by means of which problems can be reduced to a technique very similar to that of d.c. circuits. Where non-sinusoidal waveforms are concerned, the treatment is based on the analysis of the current and voltage waves into fundamental and harmonic sine waves, the standard sine wave method being applied to the fundamental and to each of the harmonics. In the case of transients, a more searching investigation may be necessary, but there are a number of common modes in which transients usually occur, and (so long as the circuit is relatively simple) it may be possible to select the appro- priate mode by inspection. Circuit parametersÐresistance, inductance and capaci- tanceÐmay or may not be constant. If they are not, approximation, linearising or step-by-step computation is necessary. 2.1.1.1 Electromotive-force sources Any device that develops an electromotive force (e.m.f.) capable of sustaining a current in an electric circuit must be associated with some mode of energy conversion into the electrical from some different form. The modes are (1) mechanical/electromagnetic, (2) mechanical/electrostatic, (3) chemical, (4) thermal, and (5) photoelectric. //integras/b&h/Eer/Final_06-09-02/eerc002 2/6 Electrotechnology the magnetic field intensity between the points, except in the presence of electric currents. Magnetic space constant: The permeability of free space. Magnetising force: The same as magnetic field strength. Magnetomotive force: Along any path, the line integral of the magnetic field strength along that path. If the path is closed, the line integral is equal to the total magnetising current in ampere-turns. Paramagnetic: Having a permeability greater than that of free space. Period: The time taken by one complete cycle of a wave- form. Permeability: The ratio of the magnetic flux density in a medium or material at a point to the magnetic field strength at the point. The absolute permeability is the product of the relative permeability and the permeability of free space (magnetic space constant). Permeance: The ratio between the magnetic flux in a mag- netic circuit and the magnetomotive force. The reciprocal of reluctance. Permittivity: The ratio between the electric flux density in a medium or material at a point and the electric field strength at the point. The absolute permittivity is the pro- duct of the relative permittivity and the permittivity of free space (electric space constant). Phase angle: The angle between the phasors that repre- sent two alternating quantities of sinusoidal waveform and the same frequency. Phasor: A sinusoidally varying quantity represented in the form of a complex number. Polarisation: The change of the electrical state of an insulat- ing material under the influence of an electric field, such that each small element becomes an electric dipole or doublet. Potential: The electrical state at a point with respect to potential zero (normally taken as that of the earth). It is measured by the work done in transferring unit charge from pontential zero to the point. Potential difference: A difference between the electrical states existing at two points tending to cause a movement of positive charges from one point to the other. It is meas- ured by the work done in transferring unit charge from one point to the other. Potential gradient: The potential difference per unit length in the direction in which it is a maximum. Power: The rate of transfer, storage, conversion or dis- sipation of energy. In sinusoidal alternating current circuits the active power is the mean rate of energy conversion; the reactive power is the peak rate of circulation of stored energy; the apparent power is the product of r.m.s. values of voltage and current. Power factor: The ratio between active power and appar- ent power. In sinusoidal alternating current circuits the power factor is cos , where  is the phase angle between voltage and current waveforms. Quantity: The product of the current and the time during which it flows. Reactance: In sinusoidal alternating current circuits, the quantity !L or 1/!C, where L is the inductance, C is the capacitance and ! is the angular frequency. Reactor: A device having reactance as a chief property; it may be an inductor or a capacitor. A nuclear reactor is a device in which energy is generated by a process of nuclear fission. Reluctance: The ratio between the magnetomotive force acting around a magnetic circuit and the resulting magnetic flux. The reciprocal of permeance. Remanence: The remanent flux density obtained when the initial magnetisation reaches the saturation value for the material. Remanent flux density: The magnetic flux density remain- ing in a material when, after initial magnetisation, the mag- netising force is reduced to zero. Residual magnetism: The magnetism remaining in a mater- ial after the magnetising force has been removed. Resistance: That property of a material by virtue of which it resists the flow of charge through it, causing a dissipation of energy as heat. It is equal to the constant potential differ- ence divided by the current produced thereby when the material has no e.m.f. acting within it. Resistivity: The resistance between opposite faces of a unit cube of a given material. Resistor: A device having resistance as a chief property. Susceptance: The reciprocal of reactance. Time constant: The characteristic time describing the duration of a transient phenomenon. Voltage: The same as potential difference. Voltage gradient: The same as potential gradient. Waveform: The graph of successive instantaneous values of a time-varying physical quantity. 2.2 Thermal effects 2.2.1 Resistance That property of an electric circuit which determines for a given current the rate at which electrical energy is converted into heat is termed resistance. A device whose chief property is resistance is a resistor, or, if variable, a rheostat. A current I flowing in a resistance R develops heat at the rate P ˆ( I 2R joule/second or watts a relation expressing Joule's law. 2.2.1.1 Voltage applied to a resistor In the absence of any energy storage effects (a physically unrealisable condition), the current in a resistor of value R is I when the voltage across it is V, in accordance with the relation I ˆ(V/R. If a steady p.d. V be suddenly applied to a resistor R, the current instantaneously assumes the value given, and energy is expended at the rate P ˆ( I2R watts, continuously. No transient occurs. If a constant frequency, constant amplitude sine wave voltage v is applied, the current i is at every instant given by i ˆ( v/R, and in consequence the current has also a sine waveform, provided that the resistance is linear. The instantaneous rate of energy dissipation depends on the instantaneous current: it is p ˆ( vi ˆ( i2R. Should the applied voltage be non-sinusoidal, the current has (under the restriction mentioned) an exactly similar waveform. The three cases are illustrated in Figure 2.3. Figure 2.3 Voltage applied to a pure resistor //integras/b&h/Eer/Final_06-09-02/eerc002 Thermal effects 2/7 In the case of alternating waveform, the average rate of energy dissipation is given by P ˆ( I2R, where I is the root- mean-square current value. 2.2.1.2 Voltage±current characteristics For a given resistor R carrying a constant current I, the p.d. is V ˆ( IR. The ratio R ˆ(V/I may or may not be invariable. In some cases it is sufficient to assume a degree of con- stancy, and calculation is generally made on this assump- tion. Where the variations of resistance are too great to make the assumption reasonably valid, it is necessary to resort to less simple analysis or to graphical methods. A constant resistance is manifested by a constant ratio between the voltage across it and the current through it, and by a straight-line graphical relation between I and V (Figure 2.4(a)), where R ˆ(V/I ˆ( cot . This case is typical of metallic resistance wires at constant temperature. Certain circuits exhibit non-linear current±voltage rela- tions (Figure 2.4(b)). The non-linearity may be symmetrical or asymmetrical, in accordance with whether the conduction characteristics are the same or different for the two current- flow directions. Rectifiers are an important class of non- linear, asymmetrical resistors. r A hypothetical device having the current±voltage charac- teristic shown in Figure 2.4(c) has, at an operating condition represented by the point P, a current Id and a p.d. Vd. The ratio Rd ˆ(Vd/Id is its d.c. resistance for the given condition. If a small alternating voltage
va be applied under the same condition (i.e. superimposed on the p.d. Vd), the current will fluctuate by
ia and the ratio ra ˆ(
va/
ia is the a.c. or incremental resistance at P. The d.c. resistance is also obtainable from Rd ˆ( cot , and the a.c. resistance from a ˆ( cot . In the region of which Q is a representative point, the a.c. resistance is negative, indicating that the device is capable of giving a small output of a.c. power, derived from its greater d.c. input. It remains in sum an energy dissipator, but some of the energy is returnable under suitable conditions of operation. 2.2.1.3 D.c. or ohmic resistance: linear resistors The d.c. or ohmic resistance of linear resistors (a category confined principally to metallic conductors) is a function of the dimensions of the conducting path and of the resistivity of the material from which the conductor is made. A wire of length l, cross-section a and resistivity  has, at constant given temperature, a resistance R ˆ( l=a ohms where , l and a are in a consistent system of dimensions (e.g. l in metres, a in square metres,  in ohms per 1 m length and 1m2 cross-sectionÐgenerally contracted to ohm-metres). The expression above, though widely applicable, is true only on the assumption that the current is uniformly distributed over the cross-section of the conductor and flows in paths parallel to the boundary walls. If this assumption is inadmissible, it is necessary to resort to integration or the use of current-flow lines. Figure 2.5 summarises the expressions for the resistance of certain arrangements and shapes of conductors. Resistivity The resistivity of conductors depends on their composition, physical condition (e.g. dampness in the case of non-metals), alloying, manufacturing and heat treat- ment, chemical purity, mechanical working and ageing. The resistance-temperature coefficient describes the rate of change of resistivity with temperature. It is practically 0.004 /C at 20C for copper. Most pure metals have a resistivity that rises with temperature. Some alloys have a very small coefficient. Carbon is notable in that its resistiv- ity decreases markedly with temperature rise, while uranium dioxide has a resistivity which falls in the ratio 50:1 over a range of a few hundred degrees. Table 2.2 lists the resistivity  and the resistance-temperature coefficient for a number of representative materials. The effect of temperature is assessed in accordance with the expressions R1 ˆ( R0…l ‡( 1†; R2 =R1 ˆ …1 ‡( 2†=…1 ‡( 1†; or R2 ˆ( R1‰1 ‡( …2 �( 1†Š( where R0, R1 and R2 are the resistances at temperatures 0, 1 and 2, and is the resistance-temperature coefficient at 0C. 2.2.1.4 Liquid conductors The variations of resistance of a given aqueous solution of an electrolyte with temperature follow the approximate rule: R ˆ( R0 =…1 ‡( 0:03†( where  is the temperature in degrees Celsius. The conduc- tivity (or reciprocal of resistivity) varies widely with the per- centage strength of the solution. For low concentrations the variation is that given in Table 2.2. 2.2.1.5 Frequency effects The resistance of a given conductor is affected by the frequency of the current carried by it. The simplest example is that of an isolated wire of circular cross-section. The inductance of the central parts of the conductor is greater than that of the outside skin because of the additional flux linkages due to the internal magnetic flux lines. The impedance of the central parts is consequently greater, and the current flows mainly at and near the surface of the conductor, where the impedance is least. The useful cross-section of the conductor is less than the actual area, and the effective resistance is consequently higher. This is called the skin effect. An analogous phenom- enon, the proximity effect, is due to mutual inductance between conductors arranged closely parallel to one another. Figure 2.4 Current±voltage characteristics Figure 2.5 Resistance in particular cases //integras/b&h/Eer/Final_06-09-02/eerc002 2/8 Electrotechnology Table 2.2 Conductivity of aqueous solutions* (mS/cm) Concentration (%) a b c d e f g h, j k l, m 1 4 0 18 12 10 10 8 6 4 3 3 2 72 35 23 20 20 16 12 8 6 6 3 102 51 34 30 30 24 18 12 9 8 4 130 65 44 39 39 32 23 16 11 10 5 79 55 4 8 4 7 39 28 20 13 11 7.5 110 79 69 67 54 39 29 18 16 10 99 90 85 69 31 22 20 *(a) NaOH, caustic soda; (b) NH4Cl, sal ammoniac; (c) NaCl, common salt; (d) NaNO3, Chilean saltpetre; (e) CaCl2, calcium chloride; (f) ZnCl2, zinc chloride; (g) NaHCO3, baking soda; (h) Na2CO3, soda ash; (j) Na2SO4, Glauber's salt; (k) Al2(SO4)3K2SO4, alum; (l) CuSO4, blue vitriol; (m) ZnSO4, white vitriol. The effects depend on conductor size, frequency f of the current, resistivity  and permeability  of the material. For a circular conductor of diameter d the increase of effective resistance is proportional to d2f/. At power frequencies and for small conductors the effect is negligible. It may, however, be necessary to investigate the skin and proximity effects in the case of large conductors such as bus-bars. 2.2.1.6 Non-linear resistors Prominent among non-linear resistors are electric arcs; also silicon carbide and similar materials. Arcs An electric arc constitutes a conductor of somewhat vague dimensions utilising electronic and ionic conduction in a gas. It is strongly affected by physical conditions of tempera- ture, gas pressure and cooling. In air at normal pressure a d.c. arc between copper electrodes has the voltage±current rela- tion given approximately by V ˆ 30 ‡ 10/I ‡ l [1 ‡ 3/I]103 for a current I in an arc length l metres. The expression is roughly equivalent to 10 V/cm for large currents and high voltages. The current density varies between 1 and 1000 A/mm2, being greater for large currents because of the pinch effect. See also Section 2.7. Silicon carbide Conducting pieces of this material have a current±voltage relation expressed approximately by I ˆKVx , where x is usually between 3 and 5. For rising voltage the current increases very rapidly, making silicon carbide devices suitable for circuit protection and the dis- charge of excess transmission-line surge energy. 2.2.2 Heating and cooling The heating of any body such as a resistor or a conducting circuit having inherent resistance is a function of the losses within it that are developed as heat. (This includes core and dielectric as well as ohmic I2R losses, but the effective value of R may be extended to cover such additional losses.) The cooling is a function of the facilities for heat dissipation to outside media such as air, oil or solids, by radiation, con- duction and convection. 2.2.2.1 Rapid heating If the time of heating is short, the cooling may be ignored, the temperature reached being dependent only on the rate of development of heat and the thermal capacity. If p is the heat development per second in joules (i.e. the power in watts), G the mass of the heated body in kilograms and c its specific heat in joules per kilogram per kelvin, then …# Gc
d ˆ p
dt; giving  ˆ …1=Gc†( p
dt For steady heating, the temperature rise is p/Gc in Kelvin per second. Standard annealed copper is frequently used for the wind- ings and connections of electrical equipment. Its density is G ˆ 8900 kg/m3 and its resistivity at 20C is 0.017 m -m; at 75C it is 0.021 m -m. A conductor worked at a current density J (in amperes per square metre) has a specific loss (watts per kilogram) of J2/8900. If J ˆ 2.75 MA/m2 (or 2.75 A/mm2), the specific loss at 75C is 17.8 W/ kg, and its rate of self-heating is 17.8/375 ˆ 0.048C/s. 2.2.2.2 Continuous heating Under prolonged steady heating a body will reach a tem- perature rise above the ambient medium of m ˆ p/A, where A is the cooling surface area and  the specific heat dissipation (joules per second per square metre of surface per degree Celsius temperature rise above ambient). The expression is based on the assumption, roughly true for moderate temperature rises, that the rate of heat emission is proportional to the temperature rise. The specific heat dissipation  is compounded of the effects of radiation, conduction and convection. Radiation The heat radiated by a surface depends on the absolute temperature T (given by T ˆ ‡ 273, where  is the Celsius temperature), and on its character (surface smooth- ness or roughness, colour, etc.). The Stefan law of heat radiation is pr ˆ 5:7eT 4  10�8 watts per square metre where e is the coefficient of radiant emission, always less than unity, except for the perfect `black body' surface, for which e ˆ 1. The radiation from a body is independent of the temperature of the medium in which it is situated. The process of radiation of a body to an exterior surface is accompanied by a re-absorption of part of the energy when re-radiated by that surface. For a small spherical radiating body inside a large and/or black spherical cavity, the radiated power is given by the Stefan-Boltzmann law: pr ˆ 5:7e1 ‰T4 � T 4 Š10�8 watts per square metre1 2 where T1 and e1 refer to the body and T2 to the cavity. The emission of radiant heat from a perfect black body surface is independent of the roughness or corrugation of the surface. If e < 1, however, there is some increase of radiation if the surface is rough. //integras/b&h/Eer/Final_06-09-02/eerc002 ‡( �( ‡( �( ‡( �( ‡( �( Electrochemical effects 2/11 Table 2.4 Electrochemical equivalents z (mg/C) circuit e.m.f. at 20C is about 1.018 30 V, and the e.m.f./ temperature coefficient is of the order of � 0.04mV/C. Element Valency z Element Valency z H 1 0.010 45 Zn Li 1 0.071 92 As Be 2 0.046 74 Se O 2 0.082 90 Br F 1 0.196 89 Sr Na 1 0.238 31 Pd Mg 2 0.126 01 Ag Al 3 0.093 16 Cd Si 4 0.072 69 Sn S 2 0.166 11 Sn S 4 0.083 06 Sb S 6 0.055 37 Te Cl 1 0.367 43 I K 1 0.405 14 Cs Ca 2 0.207 67 Ba Ti 4 0.124 09 Ce V 5 0.105 60 Ta Cr 3 0.179 65 W Cr 6 0.089 83 Pt Mn 2 0.284 61 Au Fe 1 0.578 65 Au Fe 2 0.289 33 Hg Fe 3 0.192 88 Hg Co 2 0.305 39 Tl Ni 2 0.304 09 Pb Cu 1 0.658 76 Bi Cu 2 0.329 38 Th 2 0.338 76 2.3.2.3 Secondary cells 3 0.258 76 In the lead±acid storage cell or accumulator, lead peroxide 4 0.204 56 reacts with sulphuric acid to produce a positive charge at 1 0.828 15 the anode. At the cathode metallic lead reacts with the acid 2 0.454 04 to produce a negative charge. The lead at both electrodes 2 combines with the sulphate ions to produce the poorly 4 0.276 4 1 1.117 93 2 0.582 44 2 0.615 03 4 0.307 51 3 0.420 59 4 0.330 60 1 1.315 23 1 1.377 31 2 0.711 71 3 0.484 04 5 0.374 88 6 0.317 65 4 0.505 78 1 2.043 52 3 0.681 17 1 2.078 86 2 1.039 42 1 2.118 03 2 1.073 63 3 0.721 93 4 0.601 35 soluble lead sulphate. The action is described as Charged PbO2 ‡( 2H2SO4 ‡( Pb Discharged Brown Strong acid Grey ˆ( PbSO4 ‡( 2H2O ‡( PbSO4 Sulphurate Weak acid Sulphate Both electrode reactions are reversible, so that the initial con- ditions may be restored by means of a `charging current'. In the alkaline cell, nickel hydrate replaces lead peroxide at the anode, and either iron or cadmium replaces lead at the cathode. The electrolyte is potassium hydroxide. The reactions are complex, but the following gives a general indication: Charged Discharged 2Ni…OH†3 ‡KOH ‡ Fe ˆ( 2Ni OH †2 ‡KOH ‡( Fe…OH†2…( or 2Ni…OH†3 ‡KOH ‡Cd ˆ( 2Ni…OH†3 ‡KOH ‡Cd…OH†2 2.3.2.4 Fuel cell Whereas a storage battery cell contains all the substances in the electrochemical oxidation±reduction reactions involved and has, therefore, a limited capacity, a fuel cell is supplied with its reactants externally and operates continuously as long as it is supplied with fuel. A practical fuel cell for direct conversion into electrical energy is the hydrogen±oxygen cell (Figure 2.8). Microporous electrodes serve to bring the gases into intimate contact with the electrolyte (potassium hydroxide) and to provide the cell terminals. The hydrogen and oxygen reactants are fed continuously into the cell from externally, and electrical energy is available on demand. At the fuel (H2) electrode, H2 molecules split into hydrogen atoms in the presence of a catalyst, and these combine with OH�( ions from the electrolyte, forming H2O and releasing electrons e. At the oxygen electrode, the oxygen molecules (O2) combine (also in the presence of a catalyst) with water molecules from the electrolyte and with pairs of electrons arriving at the electrode through the external load from the fuel electrode. Perhydroxyl ions (O2H � ) and hydroxyl ions (OH� ) are produced: the latter enter the electrolyte, while the more resistant O2H �( ions, with special catalysts, can be Figure 2.8 Fuel cell 2.3.2 Cells 2.3.2.1 Primary cells An elementary cell comprising electrodes of copper (positive) and zinc (negative) in sulphuric acid develops a p.d. between copper and zinc. If a circuit is completed between the electro- des, a current will flow, which acts in the electrolyte to decom- pose the acid, and causes a production of hydrogen gas round the copper, setting up an e.m.f. of polarisation in opposition to the original cell e.m.f. The latter therefore falls considerably. In practical primary cells the effect is avoided by the use of a depolariser. The most widely used primary cell is the Leclanche . It comprises a zinc and a carbon electrode in a solution of ammonium chloride, NH4Cl. When current flows, zinc chloride, ZnCl, is formed, releasing electrical energy. The NH4 positive ions travel to the carbon electrode (positive), which is packed in a mixture of manganese dioxide and carbon as depolariser. The NH4 ions are split up into NH3 (ammonia gas) and H, which is oxidised by the MnO2 to become water. The removal of the hydrogen prevents polarisation, provided that the current taken from the cell is small and intermittent. The wet form of Leclanche cell is not portable. The dry cell has a paste electrolyte and is suitable for continuous moder- ate discharge rates. It is exhausted by use or by ageing and drying up of the paste. The `shelf life' is limited. The inert cell is very similar in construction to the dry cell, but is assembled in the dry state, and is activated when required by moistening the active materials. In each case the cell e.m.f. is about 1.5 V. 2.3.2.2 Standard cell The Weston normal cell has a positive element of mercury, a negative element of cadmium, and an electrolyte of cadmium sulphate with mercurous sulphate as depolariser. The open- //integras/b&h/Eer/Final_06-09-02/eerc002 2/12 Electrotechnology reduced to OH� ions and oxygen. The overall process can be summarised as: Fuel electrode H2 ‡ 2OH� ˆ 2OH2O ‡ 2e 1Oxygen electrode 2 O2 ‡H2O ‡ 2e ˆ 2OH�( 1Net reaction H2 ‡ 2 O2 ! 2e flow ! H2O In a complete reaction 2 kg hydrogen and 16 kg oxygen combine chemically (not explosively) to form 18 kg of water with the release of 400 MJ of electrical energy. For each kiloamp/hour the cell produces 0.33 1 of water, which must not be allowed unduly to weaken the electrolyte. The open-circuit e.m.f. is 1.1 V, while the terminal voltage is about 0.9 V, with a delivery of 1 kA/m2 of plate area. 2.4 Magnetic field effects The space surrounding permanent magnets and electric cir- cuits carrying currents attains a peculiar state in which a number of phenomena occur. The state is described by saying that the space is threaded by a magnetic field of flux. The field is mapped by an arrangement of lines of induction giving the strength and direction of the flux. Figure 2.9 gives a rough indication of the flux pattern for three simple cases of magnetic field due to a current. The diagrams show the conventions of polarity, direction of flux and direction of current adopted. Magnetic lines of induction form closed loops in a magnetic circuit linked by the circuit current wholly or in part. 2.4.1 Magnetic circuit By analogy with the electric circuit, the magnetic flux pro- duced by a given current in a magnetic circuit is found from the magnetomotive force (m.m.f.) and the circuit reluctance. The m.m.f. produced by a coil of N turns carrying a current I is F ˆNI ampere-turns. This is expended over any closed path linking the current I. At a given point in a magnetic field in free space the m.m.f. per unit length or magnetising force H gives rise to a magnetic flux density B0 ˆ0H, where 0 ˆ 4/107. If the medium in which the field exists has a relative permeability r, the flux density established is B ˆ rB0 ˆ r0H ˆ H The summation of H
dl round any path linking an N-turn circuit carrying current I is the total m.m.f. F. If the distribu- tion of H is known, the magnetic flux density B or B0 can be Figure 2.9 Magnetic fields Figure 2.10 Magnetic circuits found for all points in the field, and a knowledge of the area a of the magnetic path gives F ˆBa, the total magnetic flux. Only in a few cases of great geometrical simplicity can the flux due to a given system of currents be found precisely. Among these are the following. Long straight isolated wire (Figure 2.10(a)): This is not strictly a realisable case, but the results are useful. Assume a current of 1 A. The m.m.f. around any closed linking path is therefore 1 A-t. Experiment shows that the magnetic field is symmetrical about, and concentric with, the axis of the wire. Around a closed path of radius x metres there will be a uniform distribution of m.m.f. so that H ˆ F =2x ˆ 1=2x…A-t=m†( Consequently, in free space the flux density (T ) at radius x is B0 ˆ 0H ˆ 0=2x In a medium of constant permeability ˆr0 the flux density is B ˆrB0. There will be magnetic flux following closed circular paths within the cross-section of the wire itself: at any radius x the m.m.f. is F ˆ (x/r)2 because the circular path links only that part of the (uniformly distribu- ted) current within the path. The magnetising force is H ˆF/2x ˆ x/2r 2 and the corresponding flux density in a non-magnetic conductor is B0 ˆ 0H ˆ 0x=2r 2 and r times as much if the conductor material has a relative permeability r. The expressions above are for a conductor current of 1 A. Concentric conductors (Figure 2.10(b)): Here only the inner conductor contributes the magnetic flux in the space between the conductors and in itself, because all such flux can link only the inner current. The flux distribution is found exactly as in the previous case, but can now be summed in defined limits. If the outer conductor is suffi- ciently thin radially, the flux in the interconductor space, per metre axial length of the system, is …#R 0 0 R F ˆ( dx ˆ( ln 2x 2 rr Toroid (Figure 2.10(c)): This represents the closest approach to a perfectly symmetrical magnetic circuit, in which the m.m.f. is distributed evenly round the magnetic path and the m.m.f. per metre H corresponds at all points exactly to the flux density existing at those points. The magnetic flux is therefore wholly confined to the path. Let the mean radius of //integras/b&h/Eer/Final_06-09-02/eerc002 Magnetic field effects 2/13 the toroid be R and its cross-sectional area be A. Then, with N uniformly distributed turns carrying a current I and a toroid core of permeability , F ˆ( NI ; H ˆ F =2R; B ˆ H; F ˆ( FA=2R This applies approximately to a long solenoid of length l, replacing R by 1/2. The permeability will usually be 0. Composite magnetic circuit containing iron (Figure 2.10(d)): For simplicity practical composite magnetic circuits are arbitrarily divided into parts along which the flux density is deemed constant. For each part F ˆ( Hl ˆ Bl= ˆ BlA=A ˆ( FS where S ˆ l/A is the reluctance. Its reciprocal  ˆ 1/S ˆA/l is the permeance. The expression F ˆFS resembles E ˆ IR for a simple d.c. circuit and is therefore sometimes called the magnetic Ohm's law. The total excitation for the magnetic circuit is F ˆ( H1l1 ‡H2l2 ‡H3l3 ‡


( for a series of parts of length l1, l2 . . . , along which magnetic field intensities of H1, H2 . . . (A-t/m) are necessary. For free space, air and non-magnetic materials, r ˆ 1 and B0 ˆ0H, so that H ˆB0/0 ^ 800 000 B0. This means that an excita- tion F ˆ 800 000 A-t is required to establish unit magnetic flux density (1 T) over a length l ˆ 1 m. For ferromagnetic materials it is usual to employ B±H graphs (magnetisation curves) for the determination of the excitation required, because such materials exhibit a saturation phenomenon. Typical B±H curves are given in Figures 2.11 and 2.12. 2.4.1.1 Permeability  Certain diamagnetic materials have a relative permeability slightly less than that of vacuum. Thus, bismuth has r ˆ 0.9999. Other materials have r slightly greater than unity: these are called paramagnetic. Iron, nickel, cobalt, steels, Heusler alloy (61% Cu, 27% Mn, 13% Al) and a number of others of great metallurgical interest have ferro- magnetic properties, in which the flux density is not directly proportional to the magnetising force but which under suit- able conditions are strongly magnetic. The more usual con- structional materials employed in the magnetic circuits of electrical machinery may have peak permeabilities in the neighbourhood of 5000±10 000. A group of nickel±iron alloys, including mumetal (73% Ni, 22% Fe, 5% Cu), Figure 2.12 Magnetisation and permeability curves permalloy `C' (77.4% Ni, 13.3% Fe, 3.7% Mo, 5% Cu) and hypernik (50% Ni, 50% Fe), show much higher permeabil- ities at low densities (Figure 2.12). Permeabilities depend on exact chemical composition, heat treatment, mechanical stress and temperature conditions, as well as on the flux density. Values of r exceeding 5  105 can be achieved. 2.4.1.2 Core losses A ferromagnetic core subjected to cycles of magnetisation, whether alternating (reversing), rotating or pulsating, exhibits hysteresis. Figure 2.13 shows the cycle B±H relation typical of this phenomenon. The significant quantities remanent flux density and coercive force are also shown. The area of the hysteresis loop figure is a measure of the energy loss in the cycle per unit volume of material. An empirical expression for the hysteresis loss in a core taken through f cycles of magnetisation per second is ph ˆ( kh fBx watts per unit mass or volume m Here Bm is the maximum induction reached and kh is the hysteretic constant depending on the molecular quality and structure of the core metal. The exponent x may lie between 1.5 and 2.3. It is often taken as 2. A further cause of loss in the same circumstances is the eddy current loss, due to the I2R losses of induced currents. It can be shown to be pe ˆ ket2F 2B2 watts per unit mass or volume the constant ke depending on the resistivity of the metal and t being its thickness, the material being laminated to decrease the induced e.m.f. per lamina and to increase the resistance of the path in which the eddy currents flow. In practice, curves of loss per kilogram or per cubic metre for various flux densities are employed, the curves being constructed from the results of Figure 2.11 Magnetisation curves Figure 2.13 Hysteresis //integras/b&h/Eer/Final_06-09-02/eerc002 2/16 Electrotechnology Flux change: This law has the basic form e ˆ �N…dF=dt†( and is applicable where a circuit of constant shape links a changing magnetic flux. Flux cutting: Where a conductor of length l moves at speed u at right angles to a uniform magnetic field of density B, the e.m.f. induced in the conductor is e ˆ Blu This can be applied to the motion of conductors in constant magnetic fields and when sliding contacts are involved. Linkage change: Where coils move in changing fluxes, and both flux-pulsation and flux-cutting processes occur, the general expression. e ˆ �d =dt must be used, with variation of the linkage expressed as the result of both processes. 2.4.3.4 Constant linkage principle The linkage of a closed circuit cannot be changed instant- aneously, because this would imply an instantaneous change of associated magnetic energy, i.e. the momentary appear- ance of infinite power. It can be shown that the linkage of a closed circuit of zero resistance and no internal source can- not be changed at all. The latter concept is embodied in the following theorem. Constant linkage theorem The linkage of a closed passive circuit of zero resistance is a constant. External attempts to change the linkage are opposed by induced currents that effectively prevent any net change of linkage. The theorem is very helpful in dealing with transients in highly inductive circuits such as those of transformers, synchronous generators, etc. 2.4.3.5 Ideal transformer An ideal transformer comprises two resistanceless coils embracing a common magnetic circuit of infinite permeabil- ity and zero core loss (Figure 2.18). The coils produce no leakage flux: i.e. the whole flux of the magnetic circuit com- pletely links both coils. When the primary coil is energised by an alternating voltage V1, a corresponding flux of peak value Fm is developed, inducing in the N1-turn primary coil an e.m.f. E1 ˆV1. At the same time an e.m.f. E2 is induced in the N2-turn secondary coil. If the terminals of the second- ary coil are connected to a load taking a current I2, the primary coil must accept a balancing current I1 such that I1N1 ˆ I2N2, as the core requires zero excitation. The operating conditions are therefore N1=N2 ˆ E1=E2 ˆ I2=I1; and E1I1 ˆ E2 I2 The secondary load impedance Z2 ˆ E2/I2 is reflected into the primary to give the impedance Z1 ˆ E1/I1 such that Figure 2.18 Ideal transformer Z1 ˆ …N1=N2†2Z2 A practical power transformer differs from the ideal in that its core is not infinitely permeable and demands an excita- tion N1I0 ˆN1I1 �N2I2; the primary and secondary coils have both resistance and magnetic leakage; and core losses occur. By treating these effects separately, a practical trans- former may be considered as an ideal transformer con- nected into an external network to account for the defects. 2.4.3.6 Electromagnetic machines An electromagnetic machine links an electrical energy sys- tem to a mechanical one, by providing a reversible means of energy flow between them in the common or `mutual' mag- netic flux linking stator and rotor. Energy is stored in the field and released as work. A current-carrying conductor in the field is subjected to a mechanical force and, in moving, does work and generates a counter e.m.f. Thus the force± motion product is converted to or from the voltage±current product representing electrical power. The energy-rate balance equations relating the mechanical power pe, and the energy stored in the magnetic field wf, are: Motor: pe ˆ pm ˆ dwf =dt Generator: pm ˆ pe ‡ dwf =dt The mechanical power term must account for changes in stored kinetic energy, which occur whenever the speed of the machine and its coupled mechanical loads alter. Reluctance motors The force between magnetised surfaces (Figure 2.15(b)) can be applied to rotary machines (Figure 2.19(a)). The armature tends to align itself with the field axis, developing a reluctance torque. The principle is applied to miniature rotating-contact d.c. motors and synchronous clock motors. Machines with armature windings Consider a machine rotating with constant angular velocity !r and developing a torque M. The mechanical power is pm ˆM!r: the electrical power is pe ˆ ei, where e is the counter e.m.f. due to the reaction of the mutual magnetic field. Then ei ˆM!r ‡ dwf/dt at every instant. If the armature conductor a in Figure 2.19(b) is running in a non-time-varying flux of local density B, the e.m.f. is entirely rotational and equal to er ˆBlu ˆBl!rR. The tangential force on the conductor is f ˆBli and the torque is M ˆBliR. Thus, eri ˆM!r because dwf/dt ˆ 0. This case applies to constant flux (d.c., three-phase synchronous and induction) machines. If the armature in Figure 2.19(b) is given two conductors a and b they can be connected to form a turn. Provided the turn is of full pitch, the torques will always be additive. More turns in series form a winding. The total flux in the machine results from the m.m.f.s of all current-carrying conductors, whether on stator or rotor, but the torque arises from that component of the total flux at right angles to the m.m.f. axis of the armature winding. Armature windings (Figure 2.19(c)) may be of the commu- tator or phase (tapped) types. The former is closed on itself, and current is led into and out of the winding by fixed brushes which include between them a constant number of conductors in each armature current path. The armature m.m.f. coincides always with the brush axis. Phase windings have separate external connections. If the winding is on the rotor, its current and m.m.f. rotate with it and the external connections must be made through slip-rings. Two (or three) such windings with two-phase (or three-phase) currents can produce a resultant m.m.f. that rotates with respect to the windings. //integras/b&h/Eer/Final_06-09-02/eerc002 Figure 2.19 Electromagnetic machines Torque Figure 2.19(d) shows a commutator winding arranged for maximum torque: i.e. the m.m.f. axis of the winding is displaced electrically /2 from the field pole cen- tres. If the armature has a radius R and a core length l, the flux has a constant uniform density B, and there are Z con- ductors in the 2p pole pitches each carrying the current I, the torque is BRlIZ/2p. This applies to a d.c. machine. It also gives the mean torque of a single-phase commutator machine if B and I are r.m.s. values and the factor cos  is introduced for any time phase angle between them. The torque of a phase winding can be derived from Figure 2.19(e). The flux density is assumed to be distributed sinu- soidally, and reckoned from the pole centre to be Bm cos . The current in the phase winding produces the m.m.f. Fa, having an axis displaced by angle  from the pole centre. The total torque is then ˆ 1M ˆ BmFalR sin  2 FFa sin  per pole pair. This case applies directly to the three-phase synchronous and induction machines. Types of machine For unidirectional torque, the axes of the pole centres and armature m.m.f. must remain fixed relative to one another. Maximum torque is obtained if these axes are at right angles. The machine is technically better if the field flux and armature m.m.f. do not fluctuate with time (i.e. they are d.c. values): if they do alternate, it is preferable that they be co-phasal. Workable machines can be built with (1) concentrated (`field') or (2) phase windings on one member, with (A) commutator or (B) phase windings on the other. It is basic- ally immaterial which function is assigned to stator and which to rotor, but for practical convenience a commutator winding normally rotates. The list of chief types below gives the type of winding (1, 2, A, B) and current supply (d or a), with the stator first: D.C. machine, 1d/Ad: The arrangement is that of Figure 2.19(d). A commutator and brushes are necessary for the rotor. Single-phase commutator machine, 1a/Aa: The physical arrangement is the same as that of the d.c. machine. The field flux alternates, so that the rotor m.m.f. must also alter- nate at the same frequency and preferably in time phase. Series connection of stator and rotor gives this condition. Synchronous machine, Ba/1d: The rotor carries a concen- trated d.c. winding, so the rotor m.m.f. must rotate with it at corresponding (synchronous) speed, requiring a.c. (normally three-phase) supply. The machine may be inverted (1d/Ba). Magnetic field effects 2/17 Induction machine, 2a/Ba (Figure 2.19(e)): The polyphase stator winding produces a rotating field of angular velocity !1. The rotor runs with a slip s, i.e. at a speed !1(1 � s). The torque is maintained unidirectional by currents induced in the rotor winding at frequency s!1. With d.c. supplied to the rotor (2a/Bd) the rotor m.m.f. is fixed relatively to the windings and unidirectional torque is maintained only at synchronous speed (s ˆ 0). All electromagnetic machines are variants of the above. 2.4.3.7 Magnetohydrodynamic generator Magnetohydrodynamics (m.h.d.) concerns the interaction between a conducting fluid in motion and a magnetic field. If a fast-moving gas at high temperature (and therefore ionised) passes across a magnetic field, an electric field is developed across the gaseous stream exactly as if it were a metallic conductor, in accordance with Faraday's law. The electric field gives rise to a p.d. between electrodes flanking the stream, and a current may be made to flow in an external circuit connected to the electrodes. The m.h.d. generator offers a direct conversion between heat and electrical energy. 2.4.3.8 Hall effect If a flat conductor carrying a current I is placed in a magnetic field of density B in a direction normal to it (Figure 2.20), then an electric field is set up across the width of the conduc- tor. This is the Hall effect, the generation of an e.m.f. by the movement of conduction electrons through the magnetic field. The Hall e.m.f. (normally a few microvolts) is picked off by tappings applied to the conductor edges, for the meas- urement of I or for indication of high-frequency powers. 2.4.4 Inductance The e.m.f. induced in an electric circuit by change of flux linkage may be the result of changing the circuit's own cur- rent. A magnetic field always links a current-carrying circuit, and the linkage is (under certain restrictions) proportional to the current. When the current changes, the linkage also changes and an e.m.f. called the e.m.f. of self-induction is induced. If the linkage due to a current i in the circuit is ˆFN ˆLi, the e.m.f. induced by a change of current is e ˆ �d =dt† ˆ �N…dF=dt† ˆ �L…di=dt†( L is a coefficient giving the linkage per ampere: it is called the coefficient of self-induction, or, more usually, the induct- ance. The unit is the henry, and in consequence of its rela- tion to linkage, induced e.m.f., and stored magnetic energy, it can be defined as follows. A circuit has unit inductance (1 H) if: (a) the energy stored in the associated magnetic field is 1 2 J when the current is 1 A; (b) the induced e.m.f. is 1 V when the current is changed at Figure 2.20 Hall effect //integras/b&h/Eer/Final_06-09-02/eerc002 2/18 Electrotechnology the rate 1 A/s; or (c) the flux linkage is 1 Wb-t when the current is 1 A. 2.4.4.1 Voltage applied to an inductor I Let an inductor (i.e. an inductive coil or circuit) devoid of resistance and capacitance be connected to a supply of con- stant potential difference V, and let the inductance be L. By definition (b) above, a current will be initiated, growing at such a rate that the e.m.f. induced will counterbalance the applied voltage V. The current must rise uniformly at V/L amperes per second, as shown in Figure 2.21(a), so long as the applied p.d. is maintained. Simultaneously the circuit develops a growing linked flux and stores a growing amount of magnetic energy. After a time t1 the current reaches 1 ˆ (V/L)t1, and has absorbed a store of energy at voltage V and average current I1/2, i.e. ˆ 1 VI1t1 ˆ 1 LI2 joules2 W1 ˆ V
1 I1
t1 2 2 1 since V ˆ I1L/t1. If now the supply is removed but the circuit remains closed, there is no way of converting the stored energy, which remains constant. The current therefore continues to circulate indefinitely at value I1. Suppose that V is applied for a time t1, then reversed for an equal time interval, and so on, repeatedly. The resulting current is shown in Figure 2.21(b). During the first period t1 the current rises uniformly to I1 ˆ (V/L)t1 and the stored 2energy is then 1LI1 . On reversing the applied voltage the 2 current performs the same rate of change, but negatively so as to reduce the current magnitude. After t1 it is zero and so is the stored energy, which has all been returned to the sup- ply from which it came. If the applied voltage is sinusoidal and alternates at fre- quency f, such that v ˆ vm cos 2ft ˆ vm cos !t, and is switched on at instant t ˆ 0 when v ˆ vm, the current begins to rise at rate vm/L (Figure 2.21(c)); but the immediate reduc- tion and subsequent reversal of the applied voltage require corresponding changes in the rate of rise or fall of the cur- rent. As v ˆL(di/dt) at every instant, the current is therefore …# i ˆ( v dt ˆ( vm sin !t L !L The peak current reached is im ˆ vm/!L and the r.m.s. current is I ˆV/!L ˆV/XL, where XL ˆ!L ˆ 2fL is the inductive reactance. Should the applied voltage be switched on at a voltage zero (Figure 2.21(d)), the application of the same argument results in a sine-shaped current, unidirectional but pulsat- ing, reaching the peak value 2im ˆ 2vm/!L, or twice that in the symmetrical case above. This is termed the doubling effect. Compare with Figure 2.21(b). 2.4.4.2 Calculation of inductance To calculate inductance in a given case (a problem capable of reasonably exact solution only in cases of considerable geometrical simplicity), the approach is from the standpoint of definition (c). The calculation involves estimating the magnetic field produced by a current of 1 A, summing the linkage FN produced by this field with the circuit, and writ- ing the inductance as L ˆFN. The cases illustrated in Figure 2.10 and 2.22 give the following results. Long straight isolated conductor (Figure 2.10(a)) The mag- netising force in a circular path concentric with the conduc- tor and of radius x is H ˆF/2x ˆ 1/2x; this gives rise to a circuital flux density B0 ˆ0H ˆ0/2x. Summing the link- age from the radius r of the conductor to a distance s gives ˆ …0=2† ln…s=r†( weber-turn per metre of conductor length. If s is infinite, so is the linkage and therefore the inductance: but in practice it is not possible so to isolate the conductor. There is a magnetic flux following closed circular paths within the conductor, the density being Bi ˆx/2r 2 at radius x. The effective linkage is the product of the flux by that proportion of the conductor actually enclosed, giving /8 per metre length. It follows that the internal linkage produces a contribution Li ˆ/8 henry/metre, regardless of the conductor diameter on the assumption that the cur- rent is uniformly distributed. The absolute permeability  of the conductor material has a considerable effect on the internal inductance. Concentric cylindrical conductors (Figure 2.10(b)) The inductance of a metre length of concentric cable carrying equal currents oppositely directed in the two parts is due to the flux in the space between the central and the tubular conductor set up by the inner current alone, since the cur- rent in the outer conductor cannot set up internal flux. Summing the linkages and adding the internal linkage of the inner conductor: L ˆ …=8† ‡ …0 =2† ln…R=r† henry=metre Parallel conductors (Figure 2.22) Between two conductors (a) carrying the same current in opposite directions, the link- age is found by summing the flux produced by conductor Figure 2.21 Voltage applied to a pure inductor Figure 2.22 Parallel conductors //integras/b&h/Eer/Final_06-09-02/eerc002 Electric field effects 2/21 electric flux is 1 C. From the field pattern the electric flux density D at any point is found. Then the electric field strength at the point is E ˆD/, where ˆ r0 is the absolute permittivity of the insulating medium in which the electric flux is established. Integration of E over any path from one electrode to the other gives the p.d. V, whence the capacitance is C ˆ 1/V. Parallel plates (Figure 2.27(a)) The electric flux density is uniform except near the edges. By use of a guard ring main- tained at the potential of the plate that it surrounds, the capacitance of the inner part is calculable on the reasonable assumption of uniform field conditions. With a charge of 1 C on each plate, and plates of area S spaced a apart, the electric flux density is D ˆ 1/S, the electric field intensity is E ˆD/ˆ 1/S, the potential difference is V ˆEa ˆ a/S, and the capacitance is therefore C ˆ q=V ˆ …S=a†( A case of interest is that of a parallel plate arrangement (Figure 2.27(b)), with two dielectric materials, of thickness a1 and a2 and absolute permittivity 1 and 2, respectively. The voltage gradient is inversely proportional to the permittivity, so that E11 ˆE22. The field pattern makes it evident that the difference in polarisation produces an interface charge, but in terms of the charge qc on the plates themselves the electric flux density is constant throughout. The total voltage between the plates is V ˆV1 ‡( V2 ˆE1a1 ‡E2a2, from which the total capacitance can be obtained. Concentric cylinders (Figure 2.27(c)) With a charge of 1 C per metre length, the electric flux density at radius x is 1/2x, whence Ex ˆ 1/2x. Integrating for the p.d. gives V ˆ …1=2† ln…R=r†( The capacitance is consequently C ˆ 2= ln…R=r† farad=metre The electric field strength (voltage gradient) E is inversely proportional to the radius, over which it is distributed hyperbolically. The maximum gradient occurs at the surface of the inner conductor and amounts to Em ˆ V=r ln…R=r†( At any other radius x, Ex ˆEm(r/x). For a given p.d. V and gradient Em there is one value of r to give minimum overall radius R: this is r ˆ V=Em and R ˆ 2:72r For the cylindrical capacitor (d ) with two dielectrics, of permittivity 1 between radii r and  and 2 between  and R, the maximum gradients are related by Em11r ˆEm22. Parallel cylinders (Figure 2.27(e)) The calculation leads to the value C ˆ = ln…a=r† farad=metre for the capacitance between the conductors, provided that a 4 r. It can be considered as composed of two series-con- nected capacitors each of C0 ˆ 2= ln…a=r† farad=metre C0 being the line-to-neutral capacitance. A three-phase line has a line-to-neutral capacitance identical with C0, the inter- pretation of the spacing a for transposed asymmetrical lines being the same as for their inductance. The voltage gradient of a two-wire line is shown in Figure 2.27(e). If a 4 r, the gradient in the immediate vicinity of a wire may be taken as due to the charge thereon, the further wire having little effect: consequently, Em ˆ V=r ln…a=r†( is the voltage gradient at a conductor surface. 2.5.2.3 Connection of capacitors If a bank of capacitors of capacitance C1, C2, C3 . . . , be connected in parallel and raised in combination each to the p.d. V, the total charge is the sum of the individual charges VC1, VC2, VC3 . . . , whence the total combined capacitance is C ˆ C1 ‡ C2 ‡ C3 ‡


( With a series connection, the same displacement current occurs in each capacitor and the p.d. V across the series assembly is the sum of the individual p.d.s: V ˆ V1 ‡ V2 ‡ V3 ‡


( ˆ q‰…1=C1† ‡ …1=C2† ‡ …1=C3† ‡


Š ( ˆ q=C so that the combined capacitance is obtained from C ˆ 1=‰…1=C1† ‡ …1=C2† ‡ …1=C3 † ‡


Š ( 2.5.2.4 Voltage applied to a capacitor The basis for determining the conditions in a circuit con- taining a capacitor to which a voltage is applied is that the p.d. v across the capacitor is related definitely by its capaci- tance C to the charge q displaced on its plates: q ˆCv. Let a direct voltage V be suddenly applied to a circuit devoid of all characteristic parameters except that of cap- acitance C. At the instant of its application, the capacitor must accept a charge q ˆCv, resulting in an infinitely large current flowing for a vanishingly short time. The energy stored is W ˆ 1 2 Vq ˆ 1 CV2 joules. If the voltage is raised or 2 lowered uniformly, the charge must correspondingly change, by a constant charging or discharging current flow- ing during the change (Figure 2.28(a)). Figure 2.27 Capacitance and voltage gradient Figure 2.28 Voltage applied to a pure capacitor //integras/b&h/Eer/Final_06-09-02/eerc002 2/22 Electrotechnology q If the applied voltage is sinusoidal, as in (b), such that v ˆ vm cos 2ft ˆ vm cos !t, the same argument leads to the requirement that the charge is q ˆ qm cos !t, where m ˆCvm. Then the current is i ˆ dq/dt, i.e. i ˆ �!Cvm sin !t with a peak im ˆ!Cvm and an r.m.s. value I ˆ!CV ˆV/Xc, where Xc ˆ 1/!C is the capacitive reactance. 2.5.3 Dielectric breakdown A dielectric material must possess: (a) a high insulation resistivity to avoid leakage conduction, which dissipates the capacitor energy in heat; (b) a permittivity suitable for the purposeÐhigh for capacitors and low for insulation generally; and (c) a high electric strength to withstand large voltage gradients, so that only thin material is required. It is rarely possible to secure optimum properties in one and the same material. A practical dielectric will break down (i.e. fail to insulate) when the voltage gradient exceeds the value that the mater- ial can withstand. The breakdown mechanism is complex. 2.5.3.1 Gases With gaseous dielectrics (e.g. air and hydrogen), ions are always present, on account of light, heat, sparking, etc. These are set in motion, making additional ionisation, which may be cumulative, causing glow discharge, sparking or arcing unless the field strength is below a critical value. A field strength of the order of 3 MV/m is a limiting value for gases at normal temperature and pressure. The dielectric strength increases with the gas pressure. The polarisation in gases is small, on account of the com- paratively large distances between molecules. Consequently, the relative permittivity is not very different from unity. 2.5.3.2 Liquids When very pure, liquids may behave like gases. Usually, however, impurities are present. A small proportion of the molecules forms positive or negative ions, and foreign par- ticles in suspension (fibres, dust, water, droplets) are prone to align themselves into semiconducting filaments: heating produces vapour, and gaseous breakdown may be initiated. Water, because of its exceptionally high permittivity, is especially deleterious in liquids such as oil. 2.5.3.3 Solids Solid dielectrics are rarely homogeneous, and are often hygroscopic. Local space charges may appear, producing absorption effects; filament conducting paths may be present; and local heating (with consequent deterioration) may occur. Breakdown depends on many factors, especially thermal ones, and is a function of the time of application of the p.d. 2.5.3.4 Conduction and absorption Solid dielectrics in particular, and to some degree liquids also, show conduction and absorption effects. Conduction appears to be mainly ionic in nature. Absorption is an apparent storing of charge within the dielectric. When a capacitor is charged, an initial quantity is displaced on its plates due to the geometric capacitance. If the p.d. is maintained, the charge gradually grows, owing to absorptive capacitance, probably a result of the slow orientation of permanent dipolar molecules. The current finally settles down to a small constant value, owing to conduction. Absorptive charge leaks out gradually when a capacitor is discharged, a phenomenon observable particularly in cables after a d.c. charge followed by momentary discharge. 2.5.3.5 Grading The electric fields set up when high voltages are applied to electrical insulators are accompanied by voltage gradients in various parts thereof. In many cases the gradients are any- thing but uniform: there is frequently some region where the field is intense, the voltage gradient severe and the dielectric stress high. Such regions may impose a controlling and limit- ing influence on the insulation design and on the working voltage. The process of securing improved dielectric operating conditions is called grading. The chief methods available are: (1) The avoidance of sharp corners in conductors, near which the gradient is always high. (2) The application of high-permittivity materials to those parts of the dielectric structure where the stress tends to be high, on the principle that the stress is inversely pro- portional to the permittivity: it is, of course, necessary to correlate the method with the dielectric strength of the material to be employed. (3) The use of intersheath conductors maintained at a suit- able intermediate potential so as to throw less stress on those parts which would otherwise be subjected to the more intense voltage gradients. Examples of (1) are commonly observed in high-voltage apparatus working in air, where large rounded conductors are employed and all edges are given a large radius. The application of (2) is restricted by the fact that the choice of materials in any given case is closely circumscribed by the mechanical, chemical and thermal properties necessary. Method (3) is employed in capacitor bushings, in which the intersheaths have potentials adjusted by correlation of their dimensions. 2.5.4 Electromechanical effects Figure 2.29 summarises the mechanical force effects observ- able in the electric field. In (a), (b) and (c) are sketched the field patterns for cases already mentioned in connection with the laws of electrostatics. The surface charges developed on high- materials are instrumental in producing the forces indicated in (d ). Finally, (e) shows the forces on pieces of dielectric material immersed in a gaseous or liquid insulator and subjected to a non-uniform electric field. The force Figure 2.29 Electromechanical forces //integras/b&h/Eer/Final_06-09-02/eerc002 Electromagnetic field effects 2/23 direction depends upon whether the piece has a higher or lower permittivity than the dielectric medium in which it lies. Thus, pieces of high permittivity are urged towards regions of higher electric field strength. 2.6 Electromagnetic field effects Electromagnetic field effects occur when electric charges undergo acceleration. The effects may be negligible if the rate of change of velocity is small (e.g. if the operating frequency is low), but other conditions are also significant, and in certain cases effects can be significant even at power frequencies. 2.6.1 Movement of charged particles Particles of small mass, such as electrons and protons, can be accelerated in vacuum to very high speeds. Static electric field The force developed on a particle of mass m carrying a positive charge q and lying in an electric field of intensity (or gradient) E is f ˆ qE in the direction of E, i.e. from a high-potential to a low-potential region (Figure 2.30(a)). (If the charge is negative, the direction of the force is reversed.) The acceleration of the particle is a ˆ f/m ˆE(q/m); and if it starts from rest its velocity after 2time t, is u ˆ at ˆE(q/m)t. The kinetic energy 1mu imparted2 is equal to the change of potential energy Vq, where V is the p.d. between the starting and finishing points in the electric field. Hence, the velocity attained from rest is u ˆ p‰2V…q=m†Š( For an electron (q ˆ� 1.6  10� 19 C, m0 ˆ 0.91  10� 30 kg) falling through a p.d. of 1 V the velocity is 600 km/s and the kinetic energy is w ˆVq ˆ 1.60  10� 19 J, often called an electron-volt, 1 eV. If V ˆ 2.5 kV, then u ˆ 30 000 km/s; but the speed cannot be indefinitely raised by increasing V, for as u approaches c ˆ 300 000 km/s, the free-space electromagnetic wave velo- city, the effective mass of the particle begins to acquire a rapid relativistic increase to m ˆ m0=‰1 � …u=c†2Š( compared with its `rest mass' m0. Static magnetic field A charge q moving at velocity u is a current i ˆ qu, and is therefore subject to a force if it moves across a magnetic field. The force is at right angles to u and to B, the magnetic flux density, and in the simple case of Figure 2.30(b) we have f ˆ quB ˆma ˆmu 2/R, the particle being constrained by the force to move in a circular path of radius R ˆ (u/B)(m/q). For an electron R ˆ 5.7  10� 22 (u/B). Combined electric and magnetic fields The two effects described above are superimposed. Thus, if the E and B fields are coaxial, the motion of the particle is helical. The influence of static (or quasi-static) fields on charged particles is applied in cathode ray oscilloscopes and accelerator machines. 2.6.2 Free space propagation In Section 1.5.3 the Maxwell equations are applied to pro- pagation of a plane electromagnetic wave in free space. It is shown that basic relations hold between the velocity u of propagation, the electric and magnetic field components E and H, and the electric and magnetic space constants 0 and 0. The relations are: u ˆ 1=p…00† ' 3  108 metre=second is the free space propagation velocity. The electric and magnetic properties of space impose a relation between E (in volts per metre) and H (in amps per metre) given by E=H ˆ p…0=0† ˆ 377 : called the intrinsic impedance of space. Furthermore, the energy densities of the electric and magnetic components are the same, i.e. 1 0 E 2 ˆ 1 0H2 2 2 Propagation in power engineering is not (at present) by space waves but by guided waves, a conducting system being used to direct the electromagnetic energy more effect- ively in a specified path. The field pattern is modified (although it is still substantially transverse), but the essen- tial physical propagation remains unchanged. Such a guide is called a transmission line, and the fields are normally specified in terms of the inductance and capacitance pro- perties of the line configuration, with an effective impedance z0 ˆ p…L=C† differing from 377 . 2.6.3 Transmission line propagation (see also Section 36) If the two wires of a long transmission line, originally dead, are suddenly connected to a supply of p.d. v, an energy wave advances along the line towards the further end at velocity u (Figure 2.31). The wave is characterised by the fact that the advance of the voltage charges the line capacitance, for which an advancing current is needed: and the advance of the current establishes a magnetic field against a counter- e.m.f., requiring the voltage for maintaining the advance. Thus, current and voltage are propagated simultaneously. Let losses be neglected, and L and C be the inductance and capacitance per unit length of line. In a brief time interval dt, the waves advance by a distance u
dt. The voltage is established across a capacitance Cu
dt and the rate of charge, or current, is i ˆ vCu
dt=dt ˆ vCu Figure 2.30 Motion of charged particles Figure 2.31 Transmission-line field //integras/b&h/Eer/Final_06-09-02/eerc002 2/26 Electrotechnology discharge. The discharge is maintained by the creation and movement of ions and electrons which in many discharges constitute a plasma (see Section 10.8). Such discharges may be produced between two electrodes which form part of an electrical network and across which a sufficient potential difference exists to ionise the insulation. Alternatively, discharges may be electromagnetically induced, for instance by strong radiofrequency fields. Electrical discharges may be characterised for electrical network applications in terms of the current and voltage values needed for their occurrence (Figure 2.34). 2.7.2 Types of discharge Discharges have historically been subdivided into two categor- ies namely self-sustaining and non-self-sustaining discharges. The transition between the two forms (which constitutes the electrical breakdown of the gas) is sudden and occurs through the formation of a spark. Non-self-sustaining discharges occur at relatively low currents (10�8 A) (region 0A, Figure 2.34) of which Townsend discharges are a particular type. The form of the current±voltage characteristic in this region is governed firstly by a current increase caused by primary electrons ionising the gas by collision to produce secondary electrons, and subsequently by the positive ions formed in this process gaining sufficient energy to produce further ionisation. Such discharges may be induced by irradiating the gas in between two electrodes to produce the initial ionisation. They are non-sustaining because the current flow ceases as soon as the ionising radiation is removed. When the voltage across the electrodes reaches a critical value Vs (Figure 2.34), current level ( 10�5 A, the current increases rapidly via a spark to form a self-sustaining discharge. The sparking potential Vs for ideal operating con- ditions (uniform electric field) varies with the product of gas pressure (p) and electrode separation (d ) according to Paschen's law (Figure 2.35). There is a critical value of pd for which the breakdown voltage Vs is a minimum. The self-sustaining discharge following breakdown may be either a glow or arc discharge (regions B±C and D±F, respectively, in Figure 2.34) depending on the discharge path and the nature of the connected electric circuit. The region between Vs and B (Figure 2.34) is known as a `normal' glow discharge and is characterised by the potential Figure 2.34 Current±voltage characteristic for electrical discharges. 0A, Townsend discharge; B, normal glow; C, abnormal glow; DF, arc; Vs, spark; Vo�Io, load line Figure 2.35 Breakdown voltage as a function of the pressure±electrode separation product (Paschen's law) difference across the discharge being nearly independent of current, extending to at least 10�3 A if not several amperes. For higher currents the voltage increases to form the `abnor- mal ' glow discharge (region C, Figure 2.34). The glow discharge is manifest as a diffusely luminous plasma extending across the discharge volume but may consist of alternate light and dark regions extending from the cathode in the order: Aston dark space, cathode glow, cathode dark space, negative glow, Faraday dark space, positive glow (which is extensive in volume), the anode glow and the anode dark space (Figure 2.36). The glowing regions correspond to ionisation and excitation processes being particularly active and their occurrence and extent depends on the particular operating conditions. The voltage across the glow discharge consists of two major components: the cathode fall and the positive column (Figure 2.36a). Most of the voltage drop occurs across the cathode fall. For sharply curved surfaces (e.g. wires) and long elec- trode separations the gas near the surface breaks down at a voltage less than Vs to form a local glow discharge known as a corona. The electric arc is a self-sustained discharge requiring only a low voltage for its sustenance and capable of causing currents from typically 10� 1 A to above 105 A to flow. A major difference between arc and glow discharges is that the current density at the cathode of the arc is greater than that at the glow cathode (Figure 2.37). The implication is that the electron emission process for the arc is different from that of the glow and is often thermionic in nature. At higher gas pressures both the anode and cathode of the arc may be at the boiling temperature of the electrode material. Materials having high boiling points (e.g. carbon and tungsten) have lower cathode current densities (ca. 5 ( 102 A/cm2) than materials with lower boiling points (e.g. copper and iron; ca. 5 ( 103A/cm2). The arc voltage is the sum of three distinct components (Figure 2.36(b)): the cathode and anode falls and the positive column. Cathode and anode fall voltages are each typically about 10 V. Short arcs are governed by the electrode fall regions, whereas longer arcs are dominated by the positive column. //integras/b&h/Eer/Final_06-09-02/eerc002 Figure 2.36 Voltage distributions between discharge electrodes: (a) glow; (b) arc Figure 2.37 Cathode current density distinction between glow and arc discharges At low pressures the arc may be luminously diffuse and the plasma is not in thermal equilibrium. The temperature of the gas atoms and ions is seldom more than a few hundred degrees Celsius whereas the temperature of the electrons may be as high as 4 ( 104K (Figure 2.38). At atmospheric pressure and above, the arc discharge is manifest as a constricted, highly luminous core surrounded by a more diffusively luminous aureole. The arc plasma column is generally, although not exclusively, in thermal equilibrium. The arc core is typically at temperatures in the range 5 ( 103 to 30 ( 103 K so that the gas is completely disso- Electrical discharges 2/27 Figure 2.38 Typical electron and gas temperature variations with pressure for arc plasma ciated and highly ionised. Conversely, the temperature of the aureole spans the range over which dissociation and chemical reactions occur (ca. 2 ( 103 to 5 ( 103K). The current voltage characteristic of the long electric arc is governed by the electric power (VI) dissipated in the arc column to overcome thermal losses. At lower current levels (10� 1 to 102 A) the are column is governed by thermal con- duction losses leading to a negative gradient for the current- voltage characteristic. At higher current levels radiation becomes the dominant loss mechanism yielding a positive gradient characteristic (Figure 2.34). Thermal losses and hence electric power dissipation increase with arc length (e.g. longer electrode separation), gas pressure, convection (e.g. arcs in supersonic flows) and radiation (e.g. entrained metal vapours). As a result the arc voltage at a given current is also increased, causing the discharge characteristic (Figure 2.34) to be displaced parallel to the voltage axis. 2.7.3 Discharge±network interaction For quasi-steady situations the interaction between an elec- trical discharge and the interconnected network is governed by the intersection of the load line VoIo (Figure 2.34) V ˆ( Vo �( iR (where R is the series-network resistance and Vo is the source voltage) and the current±voltage characteristic of the discharge. The negative gradient of the low current arc branch of the discharge characteristic produces a negative incremental resistance which makes operation at point D (Figure 2.34) unstable whereas operation at point E or B is stable. In practice the operating point is determined by the manner in which the discharge is initiated. Initiation by electrode separation (e.g. circuit breaker contact opening) or by fuse rupture leads to arc operation at E. However, if the discharge is initiated by reducing the series resistance R gradually so that the load line is rotated about Vo, opera- tion as a glow discharge at B may be maintained. If the cathode is heated to provide a large supply of electrons a transition from B to E may occur. A variation of the source voltage Vo causes the points of intersection of the load line and discharge characteristic to //integras/b&h/Eer/Final_06-09-02/eerc002 2/28 Electrotechnology vary so that new points of stability are produced. When the voltage falls below a value, which makes the load line tan- gential to the negative characteristic, the arc is, in principle, not sustainable. However, in practice, the thermal inertia of the arc plasma may maintain ionisation and so delay even- tual arc extinction. The behaviour of electric arcs in a.c. networks is gov- erned by the competing effects of the thermal inertia of the arc column (due to the thermal capacity of the arc plasma) and the electrical inertia of the network (produced by circuit inductance and manifest as a phase difference between current and voltage). The current±voltage characteristic of the discharge changes from the quasi-steady (d.c.) form of Figure 2.34 (corresponding to arc inertia being considerably less than the electrical inertia) via an intermediate form (when the thermal and electrical inertias are comparable) to an approximately resistive form (when the thermal inertia is considerably greater than the electrical inertia) (Figure 2.39). 2.7.4 Discharge applications Electrical discharges occur in a number of engineering situa- tions either as limiting or as essential operating features of systems and devices. Spark discharges are used in applications which utilise their transitional nature. These include spark gaps for pro- tecting equipment against high frequency, high voltage transients and as rapid acting make switches for high power test equipment or pulsed power applications. They are also used for spark erosion in machining materials to high tolerances. Glow discharges are utilised in a variety of lamps, in gas lasers, in the processing of semiconductor materials and for the surface hardening of materials (e.g. nitriding). Operational problems in all cases involve maintaining the discharge against extinction during the low current part of the driving a.c. at one extreme and preventing transition Figure 2.39 Arc current±voltage characteristics for a.c. conditions having different thermal/electrical inertia ratios: (1) thermal ( electrical inertia (d.c. case); (2) thermal ^ electrical inertia; (3) thermal 4 electrical inertia (e.g. high frequency, resistive case) into an arcing mode (which could lead to destructive thermal overload) at the other extreme. Glow discharge lamps either rely on short discharge gaps in which all the light is produced from the negative glow covering the cathode (e.g. neon indicator lamps) or long discharge gaps in which all the light comes from the positive column confined in a long tube (e.g. neon advertising lights). In materials processing the glow is used to provide the required active ionic species for surface treating the material which forms a cathodic electrode. Both etching of surface layers and deposition of complex layers can be achieved with important applications for the production of integ- rated circuits for the electronics industry. Metallic surfaces (e.g. titanium steel) may be hardened by nitriding in glow discharges. Corona on high voltage transmission lines constitute a continuous power loss which for long-distance transmis- sion may be substantial and economically undesirable. Furthermore, such corona can cause a deterioration of insulating materials through the combined action of the discharges (ion bombardment) and the effect of chemical compounds (e.g. ozone and nitrogen oxides) formed in the discharge on the surface. Arc discharges are used in high pressure lamps, gas lasers, welding, and arc heaters and also occur in current-interrup- tion devices. The distinction between the needs of the two classes of applications is that for lamps and heaters the arc needs to be stably sustained whereas for circuit interruption the arc needs to be extinguished in a controlled manner. The implication is that, when for the former applications an a.c. supply is used, the arc thermal inertia needs to be relatively long compared with the electrical inertia of the network (e.g. to minimise lamp flicker). For circuit- interruption applications the opposite is required in order to accelerate arc extinction and provide efficient current interruption. Such applications require that a number of different current waveforms (Figure 2.40) should be inter- ruptable in a controlled manner. High voltage a.c. networks need to be interrupted as the current passes naturally through zero to avoid excessive transient voltages being produced by the inductive nature of such networks. Low voltage, domestic type networks benefit from interruption via the current limiting action of a rapidly lengthening arc. High voltage d.c. network interruption relies upon the arc producing controlled instabilities to force the current artificially to zero. The arc discharges which are utilised for these applica- tions are configured in a number of different ways. Some basic forms are shown in Figure 2.41. These may be divided into two major categories corresponding to the symmetry of the arc. Axisymmetrical arcs include those which are free burning vertically (so that symmetry is maintained by buoyancy forces) wall stabilised arcs, ablation stabilised arcs (which essentially represent arcs in fuses) and axial convection controlled arcs (which are used for both gas heating, welding and high voltage circuit interruption). Non-axisymmetric arcs include the crossflow arc, the linearly driven electromagnetic arc (which has potential for the electromagnetic drive of projectiles or by driving the arc into deionising plates for circuit interruption), the rotary driven electromagnetic arc (which may be configured either between ring electrodes and used for circuit interruption, or helically and used both for gas heating and circuit inter- ruption) and the spiral arc with wall stabilisation. //integras/b&h/eer/Final_06-09-02/eerc003 3 Network Analysis (Sections 3.3.1±3.3.5) 3.1 Introduction 3/3 3.2 3/3 3.2.1 3/3 3.2.2 3/4 3.2.3 3/4 3.2.4 3/5 3.2.5 Two-ports 3/6 3.2.6 3.2.7 3/10 3.2.8 3/10 3.2.9 3/10 3.2.10 3/14 3.2.11 3/15 3.2.12 3/17 3.2.13 3/18 3.2.14 3/19 3.2.15 3/22 3.2.16 Non-linearity 3/26 3.3 3/28 3.3.1 Conventions 3/28 3.3.2 3/29 3.3.3 3/31 3.3.4 3/31 3.3.5 3/34 3.3.6 M G Say PhD, MSc, CEng, FRSE, FIEE, FIERE, ACGI, DIC Formerly of Heriot-Watt University M A Laughton BASc, PhD, DSc(Eng), FREng, FIEE Formerly of Queen Mary & Westfield College, University of London Contents Basic network analysis Network elements Network laws Network solution Network theorems Network topology 3/7 Steady-state d.c. networks Steady-state a.c. networks Sinusoidal alternating quantities Non-sinusoidal alternating quantities Three-phase systems Symmetrical components Line transmission Network transients System functions Power-system network analysis Load-flow analysis Fault-level analysis System-fault analysis Phase co-ordinate analysis Network power limits and stability 3/42 //integras/b&h/eer/Final_06-09-02/eerc003 //integras/b&h/eer/Final_06-09-02/eerc003 Basic network analysis 3/3 3.1 Introduction In an electrical network, electrical energy is conveyed from sources to an array of interconnected branches in which energy is converted, dissipated or stored. Each branch has a charac- teristic voltage±current relation that defines its parameters. The analysis of networks is concerned with the solution of source and branch currents and voltages in a given network configuration. Basic and general network concepts are dis- cussed in Section 3.2. Section 3.3 is concerned with the special techniques applied in the analysis of power-system networks. 3.2 Basic network analysis 3.2.1 Network elements Given the sources (generators, batteries, thermocouples, etc.), the network configuration and its branch parameters, then the network solution proceeds through network equa- tions set up in accordance with the Kirchhoff laws. 3.2.1.1 Sources In most cases a source can be represented as in Figure 3.1(a) by an electromotive force (e.m.f.) E0 acting through an internal series impedance Z0 and supplying an external `load' Z with a current I at a terminal voltage V. This is the Helmholtz±Thevenin equivalent voltage generator. As regards the load voltage V and current I, the source could equally well be represented by the Helmholtz±Norton equivalent current generator in Figure 3.1(b), comprising a source current I0 shunted by an internal admittance Y0 which is effectively in parallel with the load of admittance Y. Comparing the two forms for the same load current I and terminal voltage V in a load of impedance Z or admittance Y ˆ 1/Z, we have: Voltage generator Current generator V ˆ E0 � IZ0 I ˆ I0 � VY0 I ˆ …E0 � V†=Z0 V ˆ …I0 � I†=Y0 ˆ E0=Z0 � V=Z0 ˆ I0=Y0 � I =Y0 ˆ I0 � VY0 ˆ E0 � IZ0 These are identical provided that I0 ˆE0/Z0 and Y0 ˆ 1/Z0. The identity applies only to the load terminals, for internally the sources have quite different operating conditions. The two forms are duals. Sources with Z0 ˆ 0 and Y0 ˆ 0 (so that V ˆE0 and I ˆ I0) are termed ideal generators. 3.2.1.2 Parameters When a real physical network is set up by interconnecting sources and loads by conducting wires and cables, all parts (including the connections) have associated electric and magnetic fields. A resistor, for example, has resistance as the Figure 3.1 (a) Voltage and (b) current sources Figure 3.2 Pure parameters prime property, but the passage of a current implies a mag- netic field, while the potential difference (p.d.) across the resistor implies an electric field, both fields being present in and around the resistor. In the equivalent circuit drawn to represent the physical one it is usual to lump together the significant resistances into a limited number of lumped resistances. Similarly, electric-field effects are represented by lumped capacitance and magnetic-field effects by lumped inductance. The equivalent circuit then behaves like the physical prototype if it is so constructed as to include all significant effects. The lumped parameters can now be considered to be free from `residuals' and pure in the sense that simple laws of behaviour apply. These are indicated in Figure 3.2. (a) Resistance For a pure resistance R carrying an instanta- neous current i, the p.d. is v ˆRi and the rate of heat production is p ˆ vi ˆRi2. Alternatively, if the conduc- tance G ˆ 1/R is used, then i ˆGv and p ˆ vi ˆGv2. There is a constant relation v ˆ Ri ˆ v=G; i ˆ Gv ˆ v=R; p ˆ Ri2 ˆ Gv2 (b) Inductance With a self-inductance L, the magnetic linkage is Li, and the source voltage is required only when the linkage changes, i.e. v ˆ d(Li)/dt ˆL(di/dt). An inductor stores in its magnetic field the energy w ˆ 1Li2. The behaviour equations are 2 „ v ˆ L…di=dt†; i ˆ …1=L† v dt; w ˆ 1 Li2 2 Two inductances L1 and L2 with a common magnetic field have a mutual inductance L12 ˆL21 such that an e.m.f. is induced in one when current changes in the other: e1 ˆ L12…di2=dt†; e2 ˆ L21…di1=dt†( The direction of the e.m.f.s depends on the change (increase or decrease) of current and on the `sense' in which the inductors are wound. The `dot convention' for establishing the sense is to place a dot at one end of the symbol for L1, and a dot at that end of L2 which has the same polarity as the dotted end of L1 for a given change in the common flux. (c) Capacitance The stored charge q is proportional to the p.d. such that q ˆCv. When v is changed, a charge must enter or leave at the rate i ˆ dq/dt ˆC(dv/dt). The electric-field energy in a charged capacitor is w ˆ 1Cv2. Thus „ i ˆ C…dv=dt†; v ˆ …1=C† i dt; w ˆ 1 Cv2 2 2 //integras/b&h/eer/Final_06-09-02/eerc003 3/6 Network analysis Figure 3.6 The Millman theorem 3.2.4.6 Millman (Figure 3.6) The Millman theorem is also known as the parallel-generator theorem. The common terminal voltage of a number of sources connected in parallel to a common load of impedance Z is V ˆ IscZp, where Isc is the sum of the short-circuit currents of the individual source branches and Zp is the effective impedance of all the branches in parallel, including the load Z. If E1 and E2 are the e.m.f.s of two sources with internal impedances Z1 and Z2 connected in parallel to supply a load Z, and if I1 and I2 are the currents contributed by these sources to the load Z, then their common terminal voltage V must be V ˆ …I1 ‡ I2†Z ˆ ‰…E1 � V†=Z1 ‡ …E2 � V†=Z2 ŠZ whence V…1=Z ‡ 1=Z1 ‡ 1=Z2 † ˆ E1=Z1 ‡ E2=Z2 The term in parentheses on the left-hand side of the equation is the effective admittance of all the branches in parallel. The right-hand side of the equation is the sum of the individual source short-circuit currents, totalling Isc. Thus V ˆ IscZp. The theorem holds for any number of sources. 3.2.4.7 Helmholtz±Thevenin (Figure 3.7) The current in any branch Z of a network is the same as if that branch were connected to a voltage source of e.m.f. E0 and internal impedance Z0, where E0 is the p.d. appearing across the branch terminals when they are open-circuited and Z0 is the impedance of the network looking into the branch terminals with all sources represented by their internal impedance. In Figure 3.7, the network has a branch AB in which it is required to find the current. The branch impedance Z is removed, and a p.d. E0 appears across AB. With all sources replaced by their internal impedance, the network presents the impedance Z0 to AB. The current in Z when it is replaced into the original network is I ˆ E0=…Z0 ‡ Z†( The whole network apart from the branch AB has been replaced by an equivalent voltage source, resulting in the simplified condition of Figure 3.1(a). 3.2.4.8 Helmholtz±Norton The Helmholtz±Norton theorem is the dual of the Helm- holtz±Thevenin theorem. The voltage across any branch Y of a network is the same as if that branch were connected to a current source I0 with internal shunt admittance Y0, where I0 is Figure 3.7 The Helmholtz±Thevenin theorem the current between the branch terminals when short circuited and Y0 is the admittance of the network looking into the branch terminals with all sources represented by their internal admittance. Then across the terminals AB in Figure 3.7 the voltage is V ˆ I0 =…Y0 ‡ Y †( Thus the whole network apart from the branch AB has been replaced by an equivalent current source, i.e. the system in Figure 3.1(b). 3.2.5 Two-ports In power and signal transmission, input voltage and current at one port (i.e. one terminal-pair) yield voltage and current at another port of the interconnecting network. Thus in Figure 3.8a voltage source at the input port 1 delivers to the passive network a voltage V1 and a current I1. The corres- ponding values at the output port 2 are V2 and I2. 3.2.5.1 Lacour According to the theorem originated by Lacour, any passive linear network between two ports can be replaced by a two-mesh or T network, and in general no simpler form can be found. Such a result is obtained by iterative star±delta conver- sion to give the T equivalent; by one more star±delta conver- sion the -equivalent is obtained (Figure 3.9). In general, the equivalent networks are asymmetric; in some cases, however, they are symmetric. It can be shown that a passive two-port has the input and output voltages and currents related by V1 ˆ AV2 ‡ BI2 and I1 ˆ CV2 ‡DI2 where ABCD are the general two-port parameters, constants for a given frequency and with AD �BC ˆ 1. The conven- tions for voltage polarity and current direction are those given in Figure 3.8. Figure 3.8 Two-port network //integras/b&h/eer/Final_06-09-02/eerc003 Figure 3.9 T and  two-ports 3.2.5.2 T network Consider the asymmetric T in Figure 3.9. Application of the Kirchhoff laws gives I1 ˆ I2 ‡ …V2 ‡ I2Z2†Y ˆ V2 Y ‡ I2 …1 ‡ YZ2†( V1 ˆ V2…1 ‡ YZ1† ‡ I2…Z1 ‡ Z2 ‡ Z1Z2Y†( Hence in terms of the series and parallel branch components A ˆ 1 ‡ YZ1 B ˆ Z1 ‡ Z2 ‡ Z1 Z2Y C ˆ Y D ˆ 1 ‡ YZ2 Multiplication shows that AD �BC ˆ 1. For the symmetric T with Z1 ˆZ2 ˆ 1Z,2 A ˆ 1 ‡ 1 YZ ˆ D; B ˆ Z ‡ 1 YZ2; C ˆ Y2 4 3.2.5.3  network In a similar way, the general parameters for the asymmetric case are A ˆ 1 ‡Y2Z; B ˆZ; C ˆY1 ‡Y2 ‡Y1Y2Z; D ˆ 1 ‡Y1 Z which reduce with symmetry to A ˆ 1 ‡ 1 YZ ˆ D; B ˆ Z; C ˆ Y ‡ 1 Y2Z2 4 The values of the ABCD parameters, in matrix form,  # A B C D are set out in Table 3.1 for a number of common cases. 3.2.5.4 Characteristic impedance If the output terminals of a two-port are closed through an impedance V2/I2 ˆZ0, and if the input impedance V1/I1 is then also Z0, the quantity Z0 is the characteristic impedance. Consider a symmetrical two-port (A=D) so terminated: if V1/I1 is to be Z0 we have V1 V2A ‡ I2B V2 …A ‡ B=Z0†( A ‡ B=Z0ˆ ˆ( ˆ Z0 I1 V2 C ‡ I2A I2…A ‡ CZ0 †( A ‡ CZ0 which is Z0 for B/Z0 ˆCZ0. Thus the characteristic impedance is Z0 ˆH(B/C). The same result is obtainable from the input impedances with the output terminals first open circuited (I2 ˆ 0) giving Zoc, then short circuited (V2 ˆ 0) giving Zsc: thus ˆ pZoc ˆ A=C; Zsc ˆ B=A; Z0 ˆ p…ZocZsc …B=C†( Basic network analysis 3/7 3.2.5.5 Propagation coefficient The parameters ABCD are functions of frequency, and Z0 is a complex operator. For the Z0 termination of a sym- metrical two-port (for which A2�BC ˆ 1) the input/output voltage or current ratio is V1 =V2 ˆ I1 =I2 ˆ A ‡p…BC† ˆ A ‡p…A2 � 1†( ˆ exp… † ˆ exp… ‡ j †( The magnitude of V1 exceeds that of V2 by the factor exp( ) and leads it by the angle , where is the attenuation coeffi- cient, is the phase coefficient and the combination ˆ ‡ j is the propagation coefficient. 3.2.5.6 Alternative two-port parameters I There are other ways of expressing two-port relationships. For generality, both terminal voltages are taken as applied and both currents are input currents. With this convention it is necessary to write �I2 for I2 in the general parameters so far discussed. The mesh-current and node-voltage methods (Section 3.2.4) give V1 ˆ I1z11+I2z12, etc., and I1 ˆ( V1y11+ V2y12, etc., respectively. A further method relates V1 and I2 to 1 and V2 by hybrid (impedance and admittance) parameters. The four relationships are then obtained as follows: General Impedance # # # # # # # #   # V1 ˆ( A B V2 V1 ˆ( z11 z12 I1 I1 C D �I2 V2 z21 z22 I2 Admittance Hybrid   # # # # # # # # # I1 ˆ( y11 y12 V1 V1 ˆ( h11 h12 I1 I2 y21 y22 V2 I2 h21 h22 V2 h If the networks are passive, then z12 ˆ z21, y12 ˆ y21 and 12 ˆ�h21. If, in addition, the networks are symmetrical, then A ˆD, z11 ˆ z22 and y11 ˆ y22. If the networks are active (i.e. they contain sources), then reciprocity does not apply and there is no necessary relation between the terms of the 2  2 matrix. 3.2.6 Network topology In multibranch networks the solution process is aided by representing the network as a graph of nodes and inter- connections. The topology is the scheme of interconnec- tions. A network is planar if it can be drawn on a closed spherical (or plane) surface without cross-overs. A non-planar network cannot be so drawn: a single cross-over can be elimin- ated if the network is drawn on a more complicated surface resembling a doughnut, and more cross-overs require closed surfaces with more holes. The nomenclature employed in topology is as follows. Graph A diagram of the network showing all the nodes, with each branch represented by a plain line. Tree Any arrangement of branches that connects all nodes together without forming loops. A tree branch is one branch of such a tree. Link A branch that, added to a tree, completes a closed loop. Tie set A loop of branches with one a link and the others tree branches. Cut set A set of branches comprising one tree branch, the other branches being tree links. A cut set dissociates two main portions of a network in such a way that replacing any one element destroys the dissociation. //integras/b&h/eer/Final_06-09-02/eerc003 3/8 Network analysis Table 3.1 General ABCD two-port parameters Network Matrix Network Matrix Direct connection Loaded network  #   1 0 A ‡( BY0 B 0 1 C ‡(DY0 D Cross-connection Shunted network  #  #�1 0 A B 0 �1 C ‡( AY1 D ‡( BY1 Series impedance  # 1 Z 0 1 Shunt admittance  # 1 0 Y 1 L network  # 1 ‡( YZ Z Y 1 L network  # 1 Z Y 1 ‡( YZ T network # 1 ‡( YZ1 Z1 ‡( Z2 ‡( YZ1Z2 Y 1 ‡( YZ2 Symmetrical T network # 1 ‡ YZ/2 Z(1 ‡( YZ/4) Y 1 ‡( YZ/2  network # 1 ‡( Y2 Z Y1 ‡( Y2 ‡( Y1Y2 Z Symmetrical  network # 1 ‡( YZ/2 Z Y (1 ‡( YZ/4) 1 ‡( YZ/2 Cascaded networks # A1A2 ‡( B1C2 A1B2 ‡( B1D2 A2C1 ‡( C2D1 B2C1 ‡(D1D2 Mutual inductance  # 0 �j!L12 �1/j!L12 Mutual inductance  # 0 �j!L12 1/j!L12 0 Ideal transformer  # N1/N2 0 0 N2/N1 # # # Z 1 ‡( Y1Z # # cont'd 0 //integras/b&h/eer/Final_06-09-02/eerc003 Basic network analysis 3/11 Figure 3.14 Phasors subtraction of r.m.s. values are performed as if the lines were co-planar vector forces in mechanics. Physically, how- ever, the lines are not vectors: they substitute for scalar quan- tities, alternating sinusoidally with time. They are termed phasors. Certain associated quantities, such as impedance, admittance and apparent power, can also be represented by directed lines, but as they are not sinusoids they are termed complexors or complex operators. Both phasors and complex- ors can be dealt with by application of the theory of complex numbers. The definitions concerned are listed below. Complexor A generic term for a non-vector quantity expressed as a complex number. Phasor A complexor (e.g. voltage or current) derived from a time-varying sinusoidal quantity and expressed as a complex number. Complex operator A complexor derived for the ratio of two phasors (e.g. impedance and admittance); or a complexor which, operating on a phasor, gives another phasor (e.g. V ˆ IZ, in which V and I are phasors, but Z is a complex operator). 3.2.9.1 Complexor algebra The four arithmetic processes for complexors are applications of the theory of complex numbers. Complexor a in Figure 3.15 can be expressed by its magnitude a and its angle  with respect to an arbitrary `datum' direction (here taken as horizontal) as the simple polar form a ˆ a € . Alternatively it can be written as a ˆ p+jq, the rectangular form, in terms of its projection p on the datum and q on a quadrature axis at right angles thereto: q (as a scalar magnitude along the datum) is rotated counter-clockwise by angle 1  rad (90) by the 2 operator j. Two successive operations by j (written as j2) give a rotation of  rad (180), making the original +q into �q, in effect a multiplication by �1. Three operations (j3) give �jq and four give ‡q. Thus any complexor can be located in the complex datum±quadrature plane. Further obvious forms are the trigonometric, a ˆ a(cos ‡ j sin ), and the exponen- tial, a ˆ a exp(j). Summarising, the four descriptions are: Polar : a ˆ a €  Rectangular : a ˆ p ‡ jq Exponential : a ˆ a exp…j†( Trigonometric : a ˆ a…cos  ‡ j sin †( 2where a ˆH(p +q 2) and ˆ arctan(q/p). Consider complexors a ˆ p ‡ jq ˆ a €( and b ˆ r ‡ js ˆ b € . The basic manipulations are: Addition a ‡ b ˆ …p ‡ r† ‡ j…q ‡ s†( Subtraction a � b ˆ …p � r† ‡ j…q � s†( Multiplication The exponential and polar forms are more direct than the rectangular or trigonometric: ab ˆ …pr � qs† ‡ j…qr ‡ ps†( ˆ( ab exp‰j… ‡ †Š ˆ ab € … ‡ †( Division Here also the angular forms are preferred: a=b ˆ ‰…pr ‡ qs† ‡ j…qr � ps†Š=…r 2 ‡ s 2†( ˆ …a=b† exp‰j… � †Š ˆ …a=b† € (… � †( Conjugate The conjugate of a complexor a ˆ p ‡ jq ˆ a € is a* ˆ p � jq ˆ a € (� ), the quadrature com- ponent (and therefore the angle) being reversed. Then ab* ˆ ab € … � †( a*b ˆ ab € … � †( a*a ˆ aa* ˆ a 2 ˆ p 2 ‡ q 2 The last expression is used to `rationalise' the denominator in the complexor division process. 3.2.9.2 Impedance and admittance operators Sinusoidal voltages and currents can be represented by phasors in the expressions V ˆ IZ ˆ I/Y and I ˆVY ˆV/Z. Current and voltage phasors are related by multiplication or division with the complex operators Z and Y. Series resist- ance R and reactance jX can be arranged as a right-angled triangle of hypotenuse Z ˆH(R2 ‡X2) and the angle between Z and R is ˆ arctan(X/R). The relation between Z and Y for the same series network elements with Z ˆR ‡ jX is 1 1 R � jX ˆ( R � jX Y ˆ ˆ( ˆ( Z R ‡ jX …R ‡ jX†…R � jX†( R2 ‡ X2 ˆ R=Z2 � j…X=Z2† ˆ G � jB where G and B are defined in terms of R, X and Z. The series components R and X become parallel branches in Y, one a pure conductance, the other a pure susceptance. Further, a positive-angled impedance has, as inverse equivalent, a negative-angled admittance (Figure 3.16). The impedance and phase angle of a number of circuit combinations are given in Table 3.3. Figure 3.15 Complexors Figure 3.16 Impedance and admittance triangles //integras/b&h/eer/Final_06-09-02/eerc003 3/12 Network analysis Table 3.3 Impedance of network elements at angular frequency ! (rad/s) Impedance: Z ˆR+jX ˆ |Z| €  |Z| ˆH(R2+X2) ˆ arctan(X/R) Admittance: Y ˆ 1/Z ˆ |Y| € (�) |Y|ˆH[(R/Z2)2+(X/Z2)2] ˆ�arctan(X/R) Z: R 1/j!C j!L (C1+C2)/j!C1C2 j!(L1+2L12) R+j!L R+1/j!C : 0 �/2 ‡ /2 �/2 ‡/2 arctan(!L/R) �arctan(j!CR) Z: j(!L � 1/!C) R+j(!L � 1/!C) !LR !L ‡ jR R2 ‡ !2 L2 R 1 � j!CR 1 ‡ !2C2 R2 j!L 1 � !2LC : /2 arctan[(!L � 1/!C)/R] arctan(R/!L) �arctan(!CR)  /2 Z : 1=R � j(!C � 1/!L) (1=R)2 ‡ (!C � 1/!L)2 R ‡ j!‰L(1 � !2LC) � CR2 Š( (1 � !2LC)2 ‡ !2C2R2 A ‡ jB (R ‡ r)2 ‡ (!L � 1/!C)2 A ˆRr(R+r)+!2L2 r+R/!2C2 B ˆ!Lr2�R2/!C�(L/C) (!L�1/!C) : �arctan[R(!C � 1/!L)] arctan {![L(1 �!2 LC) �CR2]/R} arctan (B/A) Resonance conditions for LC networks numbered 1±6 above, for !ˆ!0 ˆ 1/H(LC): (1) |Z| ˆ 0, ˆ 0; (2) |Z| ˆR, ˆ 0; (3) |Z| ˆ1, ˆ 0; (4) |Z| ˆR, !ˆ 0; (5) |Z| ˆL/CR, ˆ�arctan(!CR) for R 5 !L; (6) |Z| ˆR (const.) for R ˆ r ˆH(L/C) Impedance and admittance loci If the characteristics of a device or a circuit can be expressed in terms of an equivalent circuit in which the impedances and/or admittances vary according to some law, then the current taken for a given applied voltage (or the voltage for a given current) can be obtained graphically by use of an admittance or impedance locus diagram. In Figure 3.17(a), let OP represent an impedance Z ˆ( R ‡ jX and OQ the corresponding admittance Y ˆG � jB. Point Q is obtained from P by finding first the geometric inverse point Q0( such that OQ0 ˆ 1/OP to scale, and then reflecting OQ0( across the datum line to give OQ and thus a reversed angle �, a process termed complexor inversion. If Z has successive values Z1, Z2, . . . , on the impedance locus, the corresponding admittances Y1, Y2, . . . , lie on the admittance locus. The inversion process may be point-by-point, but in many cases certain propositions can reduce the labour: (1) Inverse of a straight lineÐthe geometric inverse of a straight line AB about a point O not on the line is a circle passing through O with its centre M on the perpendicular OC from O to AB (Figure 3.17(b)). Then A0( is the geometric inverse of A, B0( of B, etc.; also, A is the inverse of A0, B of B0, etc. Figure 3.17 Inversion (2) Inverse of a circleÐfrom the foregoing, the geometric inverse of a circle about a point O on its circumference is a straight line. If, however, O is not on the circumference, the inverse is a second circle between the same tangents; but the distances OM and OM0( from the origin O to the centres M and M0( of the circles are not inverses of each other. //integras/b&h/eer/Final_06-09-02/eerc003 Basic network analysis 3/13 The choice of scales arises in the inversion process: for example, the inverse of an impedance Z ˆ 50 € 70( is Y ˆ 0.02 € (�70) S. It is usually possible to decide on a scale by taking a salient feature (such as a circle diameter) as a basis. 3.2.9.3 Power The instantaneous power delivered to a load is the product of the instantaneous voltage v and current i. Let v ˆ vm sin !t and i ˆ im sin(!t �) as in Figure 3.18(a); then the instanta- neous power is p ˆ 1 vmim‰cos � cos…2!t � †Š2 This is a quantity fluctuating at angular frequency 2! with, in general, excursions into negative power (i.e. that returned by the load to the source). Over an integral number of periods the mean power is P ˆ 1 vm im cos  ˆ VI cos 2 where V and I are r.m.s. values. Now resolve i into the active and reactive components ip ˆ …im cos † sin !t and iq ˆ …im sin † sin…!t � 1 †2 as in Figure 3.18(b); then the instantaneous power can be written p ˆ …vm…im cos † sin2 !t � vm …im sin † sin !t cos !t Over a whole number of periods the average of the first term is P ˆ 1 vm im cos  ˆ VI cos 2 giving the average rate of energy transfer from source to load. The second term is a double-frequency sinusoid of average value zero, the energy flow changing direction rhythmically between source and load at a peak rate Q ˆ 1 vmim sin  ˆ VI sin 2 The power conditions thus summarise to the following: Active power P The mean of the instantaneous power over an integral number of periods giving the mean rate of energy transfer from source to load in watts (W). Reactive power Q The maximum rate of energy interchange between source and load in reactive volt-amperes (var). Apparent power S The product of the r.m.s. voltage and current in volt-amperes (V-A). Both P and Q represent real power. The apparent power S is not a power at all, but is an arbitrary product VI. Nevertheless, because of the way in which P and Q are defined, we can write 2 2 2P2 ‡Q2 ˆ …VI † …cos  ‡ sin2 † ˆ …VI † whence S=H(P2+Q2), a convenient combination of mean active power with peak power circulation. Complex power The active and reactive powers can be determined for voltage and current phasors by S ˆ P  jQ ˆ VI * or S ˆ V*I using the conjugate of either I or V. Power factor The ratio of active to apparent power, P/S=cos  for sinusoidal conditions. 3.2.9.4 Resonance A condition of resonance occurs when the load contains two forms of energy-storing element (L and C) such that, at the frequency of operation, the two energies are equal. The reactive power requirements are then satisfied internally, as the inductor releases energy at the rate that the capacitor requires it. The source supplies only the active power demand of the energy-dissipating load components, the load externally appearing to be purely resistive. Acceptor resonance The series RLC circuit in Figure 3.19(a) has, at angular frequency !, the impedance Z ˆR ‡ jX, where X is !L�1/!C, which for !ˆ!0 ˆ1/H(LC) is zero. The impedance is then ZˆR and the input current has a maximum I0 ˆV/R, conditions of acceptor resonance. Internally, large voltages appear across the reactive com- ponents, viz. VL ˆ I0!L ˆ V!0…L=R† and VC ˆ I0…1=!0C† ˆ V=!0 …CR†( The terms L/R and 1/CR are the time constants of the reactive elements, and !0L/R is the Q value of a practical inductor of Figure 3.18 Active and reactive power Figure 3.19 Resonance //integras/b&h/Eer/Final_06-09-02/prelims Electrical Engineer's Reference Book //integras/b&h/Eer/Final_06-09-02/prelims Important notice Many practical techniques described in this book involve potentially dangerous applications of electricity and engineering equipment. The authors, editors and publishers cannot take responsibility for any personal, professional or financial risk involved in carrying out these techniques, or any resulting injury, accident or loss. The techniques described in this book should only be implemented by professional and fully qualified electrical engineers using their own professional judgement and due regard to health and safety issues. //integras/b&h/Eer/Final_06-09-02/prelims Contents Preface Section A ± General Principles 1 Units, Mathematics and Physical Quantities International unit system . Mathematics . Physical quantities . Physical properties . Electricity 2 Electrotechnology Nomenclature . Thermal effects . Electrochemical effects . Magnetic field effects . Electric field effects . Electromagnetic field effects . Electrical discharges 3 Network Analysis Introduction . Basic network analysis . Power-system network analysis Section B ± Materials & Processes 4 Fundamental Properties of Materials Introduction . Mechanical properties . Thermal properties . Electrically conducting materials . Magnetic materials . Dielectric materials . Optical materials . The plasma state 5 Conductors and Superconductors Conducting materials . Superconductors 6 Semiconductors, Thick and Thin-Film Microcircuits Silicon, silicon dioxide, thick- and thin-film technology . Thick- and thin-film microcircuits 7 Insulation Insulating materials . Properties and testing . Gaseous dielectrics . Liquid dielectrics . Semi-fluid and fusible materials . Varnishes, enamels, paints and lacquers . Solid dielectrics . Composite solid/liquid dielectrics . Irradiation effects . Fundamentals of dielectric theory . Polymeric insulation for high voltage outdoor applications 8 Magnetic Materials Ferromagnetics . Electrical steels including silicon steels . Soft irons and relay steels . Ferrites . Nickel±iron alloys . Iron±cobalt alloys . Permanent magnet materials 9 Electroheat and Materials Processing Introduction . Direct resistance heating . Indirect resistance heating . Electric ovens and furnaces . Induction heating . Metal melting . Dielectric heating . Ultraviolet processes . Plasma torches . Semiconductor plasma processing . Lasers 10 Welding and Soldering Arc welding . Resistance welding . Fuses . Contacts . Special alloys . Solders . Rare and precious metals . Temperature- sensitive bimetals . Nuclear-reactor materials . Amorphous materials Section C ± Control 11 Electrical Measurement Introduction . Terminology . The role of measurement traceability in product quality . National and international measurement standards . Direct-acting analogue measuring instruments . Integrating (energy) metering . Electronic instrumentation . Oscilloscopes . Potentiometers and bridges . Measuring and protection transformers . Magnetic measurements . Transducers . Data recording 12 Industrial Instrumentation Introduction . Temperature . Flow . Pressure . Level transducers . Position transducers . Velocity and acceleration . Strain gauges, loadcells and weighing . Fieldbus systems . Installation notes 13 Control Systems Introduction . Laplace transforms and the transfer function . Block diagrams . Feedback . Generally desirable and acceptable behaviour . Stability . Classification of system and static accuracy. Transient behaviour . Root-locus method . Frequency-response methods . State-space description . Sampled-data systems . Some necessary mathematical preliminaries . Sampler and zero-order hold . Block diagrams . Closed-loop systems . Stability . Example . Dead-beat response . Simulation . Multivariable control . Dealing with non linear elements . //integras/b&h/Eer/Final_06-09-02/prelims Disturbances . Ratio control . Transit delays . Stability . Industrial controllers . Digital control algorithms . Auto-tuners . Practical tuning methods 14 Digital Control Systems Introduction . Logic families . Combinational logic . Storage . Timers and monostables . Arithmetic circuits . Counters and shift registers . Sequencing and event driven logic . Analog interfacing . Practical considerations . Data sheet notations 15 Microprocessors Introduction . Structured design of programmable logic systems . Microprogrammable systems . Programmable systems . Processor instruction sets . Program structures . Reduced instruction set computers (RISC) . Software design . Embedded systems 16 Programmable Controllers Introduction . The programmable controller . Programming methods . Numerics . Distributed systems and fieldbus . Graphics . Software engineering . Safety Section D ± Power Electronics and Drives 17 Power Semiconductor Devices Junction diodes . Bipolar power transistors and Darlingtons . Thyristors . Schottky barrier diodes . MOSFET . The insulated gate bipolar transistor (IGBT) 18 Electronic Power Conversion Electronic power conversion principles . Switch-mode power supplies . D.c/a.c. conversion . A.c./d.c. conversion . A.c./a.c. conversion . Resonant techniques . Modular systems . Further reading 19 Electrical Machine Drives Introduction . Fundamental control requirements for electrical machines . Drive power circuits . Drive control . Applications and drive selection . Electromagnetic compatibility 20 Motors and Actuators Energy conversion . Electromagnetic devices . Industrial rotary and linear motors Section E ± Environment 21 Lighting Light and vision . Quantities and units . Photometric concepts . Lighting design technology . Lamps . Lighting design . Design techniques . Lighting applications 22 Environmental Control Introduction . Environmental comfort . Energy requirements . Heating and warm-air systems . Control . Energy conservation . Interfaces and associated data 23 Electromagnetic Compatibility Introduction . Common terms . The EMC model . EMC requirements . Product design . Device selection . Printed circuit boards . Interfaces . Power supplies and power-line filters . Signal line filters . Enclosure design . Interface cable connections . Golden rules for effective design for EMC . System design . Buildings . Conformity assessment . EMC testing and measurements . Management plans 24 Health and Safety The scope of electrical safety considerations . The nature of electrical injuries . Failure of electrical equipment 25 Hazardous Area Technology A brief UK history . General certification requirements . Gas group and temperature class . Explosion protection concepts . ATEX certification . Global view . Useful websites Section F ± Power Generation 26 Prime Movers Steam generating plant . Steam turbine plant . Gas turbine plant . Hydroelectric plant . Diesel-engine plant 27 Alternative Energy Sources Introduction . Solar . Marine energy . Hydro . Wind . Geothermal energy. Biofuels . Direct conversion . Fuel cells . Heat pumps 28 Alternating Current Generators Introduction . Airgap flux and open-circuit e.m.f. . Alternating current windings . Coils and insulation . Temperature rise . Output equation . Armature reaction . Reactances and time constants . Steady-state operation . Synchronising . Operating charts . On-load excitation . Sudden three phase short circuit . Excitation systems . Turbogenerators . Generator±transformer connection . Hydrogenerators . Salient-pole generators other than hydrogenerators . Synchronous compensators . Induction generators . Standards 29 Batteries Introduction . Cells and batteries . Primary cells . Secondary cells and batteries . Battery applications . Anodising . Electrodeposition . Hydrogen and oxygen electrolysis Section G ± Transmission and Distribution 30 Overhead Lines General . Conductors and earth wires . Conductor fittings . Electrical characteristics . Insulators . Supports . Lightning . Loadings //integras/b&h/Eer/Final_06-09-02/prelims 31 Cables Introduction . Cable components . General wiring cables and flexible cords . Supply distribution cables . Transmission cables . Current-carrying capacity . Jointing and accessories . Cable fault location 32 HVDC Introduction . Applications of HVDC . Principles of HVDC converters . Transmission arrangements . Converter station design . Insulation co-ordination of HVDC converter stations . HVDC thyristor valves . Design of harmonic filters for HVDC converters . Reactive power considerations . Control of HVDC . A.c. system damping controls . Interaction between a.c. and d.c. systems . Multiterminal HVDC systems . Future trends 33 Power Transformers Introduction . Magnetic circuit . Windings and insulation . Connections . Three-winding transformers . Quadrature booster transformers . On-load tap changing . Cooling . Fittings . Parallel operation . Auto-transformers . Special types . Testing . Maintenance . Surge protection . Purchasing specifications 34 Switchgear Circuit-switching devices . Materials . Primary-circuit- protection devices . LV switchgear . HV secondary distribution switchgear . HV primary distribution switchgear . HV transmission switchgear . Generator switchgear . Switching conditions . Switchgear testing . Diagnostic monitoring . Electromagnetic compatibility . Future developments 35 Protection Overcurrent and earth leakage protection . Application of protective systems . Testing and commissioning . Overvoltage protection 36 Electromagnetic Transients Introduction . Basic concepts of transient analysis . Protection of system and equipment against transient overvoltage . Power system simulators . Waveforms associated with the electromagnetic transient phenomena 37 Optical Fibres in Power Systems Introduction . Optical fibre fundamentals . Optical fibre cables . British and International Standards . Optical fibre telemetry on overhead power lines . Power equipment monitoring with optical fibre sensors 38 Installation Layout . Regulations and specifications . High-voltage supplies . Fault currents . Substations . Wiring systems . Lighting and small power . Floor trunking . Stand-by and emergency supplies . Special buildings . Low-voltage switchgear and protection . Transformers . Power-factor correction . Earthing . Inspection and testing Section H ± Power Systems 39 Power System Planning The changing electricity supply industry (ESI) . Nature of an electrical power system . Types of generating plant and characteristics . Security and reliability of a power system . Revenue collection . Environmental sustainable planning 40 Power System Operation and Control Introduction . Objectives and requirements . System description . Data acquisition and telemetering . Decentralised control: excitation systems and control characteristics of synchronous machines . Decentralised control: electronic turbine controllers . Decentralised control: substation automation . Decentralised control: pulse controllers for voltage control with tap-changing transformers. Centralised control . System operation . System control in liberalised electricity markets . Distribution automation and demand side management . Reliability considerations for system control 41 Reactive Power Plant and FACTS Controllers Introduction . Basic concepts . Variations of voltage with load . The management of vars . The development of FACTS controllers . Shunt compensation . Series compensation . Controllers with shunt and series components . Special aspects of var compensation . Future prospects 42 Electricity Economics and Trading Introduction . Summary of electricity pricing principles . Electricity markets . Market models . Reactive market 43 Power Quality Introduction . Definition of power quality terms . Sources of problems . Effects of power quality problems . Measuring power quality . Amelioration of power quality problems . Power quality codes and standards Section I ± Sectors of Electricity Use 44 Road Transport Electrical equipment of road transport vehicles . Light rail transit . Battery vehicles . Road traffic control and information systems 45 Railways Railway electrification . Diesel-electric traction . Systems, EMC and standards . Railway signalling and control 46 Ships Introduction . Regulations . Conditions of service . D.c. installations . A.c. installations . Earthing . Machines //integras/b&h/Eer/Final_06-09-02/prelims Electrical Engineer's Reference BookÐonline edition As this book goes to press an online electronic version is also in preparation. The online edition will feature . the complete text of the book . access to the latest revisions (a rolling chapter-by-chapter revision will take place between print editions) . additional material not included in the print version To find out more, please visit the Electrical Engineer's Reference Book web page: http://www.bh.com/newness?isbn=0750646373 or send an e-mail to newnes@elsevier.com //integras/b&h/Eer/Final_06-09-02/part Section A General Principles //integras/b&h/Eer/Final_06-09-02/part //integras/b&h/eer/Final_06-09-02/eerc001 This reference section provides (a) a statement of the International System (SI) of Units, with conversion factors; (b) basic mathematical functions, series and tables; and (c) some physical properties of materials. 1.1 International unit system The International System of Units (SI) is a metric system giving a fully coherent set of units for science, technology and engineering, involving no conversion factors. The starting point is the selection and definition of a minimum set of inde- pendent `base' units. From these, `derived' units are obtained by forming products or quotients in various combinations, again without numerical factors. For convenience, certain combinations are given shortened names. A single SI unit of energy (joule ˆ( kilogram metre-squared per second-squared) is, for example, applied to energy of any kind, whether it be kinetic, potential, electrical, thermal, chemical . . . , thus unify- ing usage throughout science and technology. The SI system has seven base units, and two supplement- ary units of angle. Combinations of these are derived for all other units. Each physical quantity has a quantity symbol (e.g. m for mass, P for power) that represents it in physical equations, and a unit symbol (e.g. kg for kilogram, W for watt) to indicate its SI unit of measure. 1.1.1 Base units Definitions of the seven base units have been laid down in the following terms. The quantity symbol is given in italic, the unit symbol (with its standard abbreviation) in roman type. As measurements become more precise, changes are occasionally made in the definitions. Length: l, metre (m) The metre was defined in 1983 as the length of the path travelled by light in a vacuum during a time interval of 1/299 792 458 of a second. Mass: m, kilogram (kg) The mass of the international prototype (a block of platinum preserved at the International Bureau of Weights and Measures, SeÁ vres). Time: t, second (s) The duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. Electric current: i, ampere (A) The current which, main- tained in two straight parallel conductors of infinite length, of negligible circular cross-section and 1 m apart in vacuum, pro- duces a force equal to 2 ( 10�7 newton per metre of length. Thermodynamic temperature: T, kelvin (K) The fraction 1/273.16 of the thermodynamic (absolute) temperature of the triple point of water. Luminous intensity: I, candela (cd) The luminous intensity in the perpendicular direction of a surface of 1/600 000 m2 of a black body at the temperature of freezing platinum under a pressure of 101 325 newton per square metre. Amount of substance: Q, mole (mol) The amount of sub- stance of a system which contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. The elementary entity must be specified and may be an atom, a molecule, an ion, an electron . . . , or a specified group of such entities. 1.1.2 Supplementary units Plane angle: , & . . . , radian (rad) The plane angle between two radii of a circle which cut off on the circumfer- ence of the circle an arc of length equal to the radius. Solid angle: , steradian (sr) The solid angle which, having its vertex at the centre of a sphere, cuts off an area of the surface of the sphere equal to a square having sides equal to the radius. International unit system 1/3 1.1.3 Notes Temperature At zero K, bodies possess no thermal energy. Specified points (273.16 and 373.16 K) define the Celsius (centigrade) scale (0 and 100C). In terms of intervals, 1C ˆ( 1 K. In terms of levels, a scale Celsius temperature & corresponds to (&‡ 273.16) K. Force The SI unit is the newton (N). A force of 1 N endows a mass of 1 kg with an acceleration of 1 m/s2. Weight The weight of a mass depends on gravitational effect. The standard weight of a mass of 1 kg at the surface of the earth is 9.807 N. 1.1.4 Derived units All physical quantities have units derived from the base and supplementary SI units, and some of them have been given names for convenience in use. A tabulation of those of inter- est in electrical technology is appended to the list in Table 1.1. Table 1.1 SI base, supplementary and derived units Quantity Unit Derivation Unit name symbol Length metre Mass kilogram Time second Electric current ampere Thermodynamic temperature kelvin Luminous intensity candela Amount of mole substance Plane angle radian Solid angle steradian Force newton Pressure, stress pascal Energy joule Power watt Electric charge, flux coulomb Magnetic flux weber Electric potential volt Magnetic flux density tesla Resistance ohm Inductance henry Capacitance farad Conductance siemens Frequency hertz Luminous flux lumen Illuminance lux Radiation activity becquerel Absorbed dose gray Mass density kilogram per cubic metre Dynamic viscosity pascal-second Concentration mole per cubic m kg s A K cd mol rad sr kg m/s2 N N/m2 Pa N m, W s J J/s W A s C V s Wb J/C V s Wb/m2 T V/A Wb/A, V s/A H C/V, A s/V F A/V S �1 Hz cd sr lm lm/m2 lx s �1 Bq J/kg Gy kg/m3 Pa s mol/ 3metre m Linear velocity metre per second m/s Linear metre per second- m/s2 acceleration squared Angular velocity radian per second rad/s cont'd //integras/b&h/eer/Final_06-09-02/eerc001 1/4 Units, mathematics and physical quantities Table 1.1 (continued ) Quantity Unit Derivation Unit name symbol Angular radian per second- acceleration squared rad/s2 Torque newton metre N m Electric field strength volt per metre V/m Magnetic field strength ampere per metre A/m Current density ampere per square metre A/m2 Resistivity ohm metre m Conductivity siemens per metre S/m Permeability henry per metre H/m Permittivity farad per metre F/m Thermal capacity joule per kelvin J/K Specific heat joule per kilogram capacity kelvin J/(kg K) Thermal watt per metre conductivity kelvin W/(m K) Luminance candela per square metre cd/m2 Decimal multiples and submultiples of SI units are indi- cated by prefix letters as listed in Table 1.2. Thus, kA is the unit symbol for kiloampere, and mF that for microfarad. There is a preference in technology for steps of 103. Prefixes for the kilogram are expressed in terms of the gram: thus, 1000 kg ˆ 1 Mg, not 1 kkg. Table 1.2 Decimal prefixes 1.1.5 Auxiliary units Some quantities are still used in special fields (such as vacuum physics, irradiation, etc.) having non-SI units. Some of these are given in Table 1.3 with their SI equivalents. 1.1.6 Conversion factors Imperial and other non-SI units still in use are listed in Table 1.4, expressed in the most convenient multiples or sub- multiples of the basic SI unit [ ] under classified headings. 1.1.7 CGS electrostatic and electromagnetic units Although obsolescent, electrostatic and electromagnetic units (e.s.u., e.m.u.) appear in older works of reference. Neither system is `rationalised', nor are the two mutually compatible. In e.s.u. the electric space constant is "&0 ˆ 1, in e.m.u. the magnetic space constant is 0 ˆ 1; but the SI units take account of the fact that 1/H("&00) is the velocity of electromagnetic wave propagation in free space. Table 1.5 lists SI units with the equivalent number n of e.s.u. and e.m.u. Where these lack names, they are expressed as SI unit names with the prefix `st' (`electrostatic') for e.s.u. and `ab' (`absolute') for e.m.u. Thus, 1 V corresponds to 10�2/3 stV and to 108 abV, so that 1 stV ˆ 300 V and 1 abV ˆ 10�8V. 1.2 Mathematics Mathematical symbolism is set out in Table 1.6. This sub- section gives trigonometric and hyperbolic relations, series (including Fourier series for a number of common wave forms), binary enumeration and a list of common deriva- tives and integrals. 1018 exa E 1015 peta P 1012 tera T 109 giga G 106 mega M 103 kilo k 102 hecto h 101 deca da 10�1 deci d 10�3 milli m 10�6 micro & 10�9 nano n 10�12 pico p 10�15 femto f 10�18 atto a 10�2 centi c Table 1.3 Auxiliary units Quantity Symbol SI Quantity Symbol SI Angle Mass degree () /180 rad tonne t 1000 kg minute (0) Ð Ð second (0 0) Ð Ð Nucleonics, Radiation becquerel Bq 1.0 s �1 Area gray Gy 1.0 J/kg are a 100 m 2 curie Ci 3.7  1010 Bq hectare ha 0.01 km2 rad rd 0.01 Gy barn barn 10�28 m 2 roentgen R 2.6  10�4 C/kg Energy Pressure erg erg 0.1 mJ bar b 100 kPa calorie cal 4.186 J torr Torr 133.3 Pa electron-volt eV 0.160 aJ Time gauss-oersted Ga Oe 7.96 mJ/m3 minute min 60 s Force hour h 3600 s dyne dyn 10 mN day d 86 400 s Length AÊ ngstrom AÊ 0.1 mm Volume litre 1 or L 1.0 dm3 //integras/b&h/eer/Final_06-09-02/eerc001 Mathematics 1/5 Table 1.4 Conversion factors Length [m] Density [kg/m, kg/m3] 1 mil 25.40 mm 1 lb/in 17.86 kg/m 1 in 25.40 mm 1 lb/ft 1.488 kg/m 1 ft 1 yd 1 fathom 1 mile 0.3048 m 0.9144 m 1.829 m 1.6093 km 1 lb/yd 1 lb/in3 1 lb/ft3 1 ton/yd3 0.496 kg/m 27.68 Mg/m3 16.02 kg/m3 1329 kg/m3 1 nautical mile 1.852 km Area [m2] 1 circular mil 1 in 2 1 ft2 1 yd2 1 acre 1 mile2 Volume [m3] 1 in 3 1 ft3 1 yd3 1 UKgal 506.7 mm 2 645.2 mm 2 0.0929 m 2 0.8361 m 2 4047 m 2 2.590 km2 16.39 cm 3 0.0283 m 3 0.7646 m 3 4.546 dm3 Flow rate [kg/s, m 3/s] 1 lb/h 1 ton/h 1 lb/s 1 ft3/h 1 ft3/s 1 gal/h 1 gal/min 1 gal/s Force [N], Pressure [Pa] 1 dyn 1 kgf 1 ozf 0.1260 g/s 0.2822 kg/s 0.4536 kg/s 7.866 cm 3/s 0.0283 m 3/s 1.263 cm 3/s 75.77 cm 3/s 4.546 dm 3/s 10.0 mN 9.807 N 0.278 N 1 lbf 4.445 N Velocity [m/s, rad/s] Acceleration [m/s2, rad/s 2] 1 ft/min 1 in/s 1 ft/s 1 mile/h 1 knot 1 deg/s 5.080 mm/s 25.40 mm/s 0.3048 m/s 0.4470 m/s 0.5144 m/s 17.45 mrad/s 1 tonf 1 dyn/cm2 1 lbf/ft2 1 lbf/in2 1 tonf/ft2 1 tonf/in2 1 kgf/m2 1 kgf/cm2 9.964 kN 0.10 Pa 47.88 Pa 6.895 kPa 107.2 kPa 15.44 MPa 9.807 Pa 98.07 kPa 1 rev/min 0.1047 rad/s 1 mmHg 133.3 Pa 1 rev/s 1 ft/s2 1 mile/h per s 6.283 rad/s 0.3048 m/s2 0.4470 m/s2 1 inHg 1 inH2O 1 ftH2O 3.386 kPa 149.1 Pa 2.989 kPa Mass [kg] Torque [N m] 1 oz 28.35 g 1 ozf in 7.062 nN m 1 lb 0.454 kg 1 lbf in 0.113 N m 1 slug 14.59 kg 1 lbf ft 1.356 N m 1 cwt 50.80 kg 1 tonf ft 3.307 kN m 1 UKton 1016 kg 1 kgf m 9.806 N m Energy [J], Power [W] 1 ft lbf 1 m kgf 1 Btu 1 therm 1 hp h 1 kW h 1.356 J 9.807 J 1055 J 105.5 kJ 2.685 MJ 3.60 MJ Inertia [kg m 2] Momentum [kg m/s, kg m 2/s] 1 oz in2 1 lb in2 1 lb ft2 1 slug ft2 1 ton ft2 0.018 g m 2 0.293 g m 2 0.0421 kg m 2 1.355 kg m 2 94.30 kg m 2 1 Btu/h 1 ft lbf/s 0.293 W 1.356 W 1 lb ft/s 1 lb ft2/s 0.138 kg m/s 0.042 kg m 2/s 1 m kgf/s 9.807 W 1 hp 745.9 W Viscosity [Pa s, m 2/s] Thermal quantities [W, J, kg, K] 1 W/in2 1 Btu/(ft2 h) 1 Btu/(ft3 h) 1 Btu/(ft h F) 1 ft lbf/lb 1.550 kW/m2 3.155 W/m2 10.35 W/m3 1.731 W/(m K) 2.989 J/kg 1 poise 1 kgf s/m2 1 lbf s/ft2 1 lbf h/ft2 1 stokes 1 in 2/s 1 ft2/s 9.807 Pa s 9.807 Pa s 47.88 Pa s 172.4 kPa s 1.0 cm 2/s 6.452 cm 2/s 929.0 cm 2/s 1 Btu/lb 1 Btu/ft3 1 ft lbf/(lb F) 1 Btu/(lb F) 1 Btu/(ft3 F) 2326 J/kg 37.26 KJ/m3 5.380 J/(kg K) 4.187 kJ/(kg K) 67.07 kJ/m 3 K Illumination [cd, lm] 1 lm/ft2 1 cd/ft2 1 cd/in2 10.76 lm/m2 10.76 cd/m2 1550 cd/m2 //integras/b&h/eer/Final_06-09-02/eerc001 1/8 Units, mathematics and physical quantities Figure 1.3 Hyperbolic relations If u is a quadrature (`imaginary') number jv, then 3 4 exp…jv† ˆ 1  jv � v 2 =2! jv =3!‡ v =4! . . . because j2 ˆ�1, j3 ˆ�j1, j4 ˆ‡ 1, etc. Figure 1.2 (right) shows the summation of the first five terms for exp(j1), i.e. exp…j1† ˆ 1 ‡ j1 � 1=2 � j1=6 ‡ 1=24 a complex or expression converging to a point P. The length OP is unity and the angle of OP to the datum axis is, in fact, 1 rad. In general, exp(jv) is equivalent to a shift by €v rad. It follows that exp(jv) ˆ cos v  j sin v, and that exp…jv† ‡ exp…�jv† ˆ 2 cos v exp…jv† � exp…�jv† ˆ j2 sin v For a complex number (u ‡ jv), then exp…u ‡ jv† ˆ exp…u†
exp…jv† ˆ exp…u†
€v Hyperbolic functions A point P on a rectangular hyper- bola (x/a)2�( (y/a)2 ˆ 1 defines the hyperbolic `sector' area 2Sh ˆ 1a ln[(x/a � (y/a)] shown shaded in Figure 1.3 (left). By 2 analogy with &ˆ 2Sc/h2 for the trigonometrical angle , the hyperbolic entity (not an angle in the ordinary sense) is u ˆ 2Sh/a 2, where a is the major semi-axis. Then the hyperbolic functions of u for point P are: sinh u ˆ y=a cosech u ˆ a=y cosh u ˆ x=a sech u ˆ a=x tanh u ˆ y=x coth u ˆ x=y Figure 1.2 Exponential relations The principal relations yield the curves shown in the diagram (right) for values of u between 0 and 3. For higher values sinh u approaches cosh u, and tanh u becomes asymptotic to 1. Inspection shows that cosh(�u) ˆ cosh u, sinh(�u) ˆ�sinh u and cosh2 u� sinh2 u ˆ 1. The hyperbolic functions can also be expressed in the exponential form through the series 4 6cosh u ˆ 1 ‡ u 2 =2!‡ u =4!‡ u =6!‡


( 5 7sinh u ˆ u ‡ u 3 =3!‡ u =5!‡ u =7!‡


( so that cosh u ˆ 1 ‰exp…u† ‡ exp…�u†Š( sinh u ˆ 1 ‰exp…u† � exp…�u†Š2 2 cosh u ‡ sinh u ˆ exp…u†( cosh u � sinh u ˆ exp…�u†( Other relations are: sinh u ‡ sinh v ˆ 2 sinh 1 …u ‡ v†
cosh 1 …u � v†2 2 cosh u ‡ cosh v ˆ 2 cosh 1 …u ‡ v†
cosh 1 …u � v†2 2 cosh u � cosh v ˆ 2 sinh 1 …u ‡ v†
sinh 1 …u � v†2 2 sinh…u  v† ˆ sinh u
cosh v  cosh u
sinh v cosh…u  v† ˆ cosh u
cosh v  sinh u
sinh v tanh…u  v† ˆ …tanh u  tanh v†=…1  tanh u
tanh v†( //integras/b&h/eer/Final_06-09-02/eerc001 Mathematics 1/9 Table 1.8 Exponential and hyperbolic functions u exp(u) exp(�u) sinh u cosh u tanh u 0.0 1.0 1.0 0.0 1.0 0.0 0.1 1.1052 0.9048 0.1092 1.0050 0.0997 0.2 1.2214 0.8187 0.2013 1.0201 0.1974 0.3 1.3499 0.7408 0.3045 1.0453 0.2913 0.4 1.4918 0.6703 0.4108 1.0811 0.3799 0.5 1.6487 0.6065 0.5211 1.1276 0.4621 0.6 1.8221 0.5488 0.6367 1.1855 0.5370 0.7 2.0138 0.4966 0.7586 1.2552 0.6044 0.8 2.2255 0.4493 0.8881 1.3374 0.6640 0.9 2.4596 0.4066 1.0265 1.4331 0.7163 1.0 2.7183 0.3679 1.1752 1.5431 0.7616 1.2 3.320 0.3012 1.5095 1.8107 0.8337 1.4 4.055 0.2466 1.9043 2.1509 0.8854 1.6 4.953 0.2019 2.376 2.577 0.9217 1.8 6.050 0.1653 2.942 3.107 0.9468 2.0 7.389 0.1353 3.627 3.762 0.9640 2.303 10.00 0.100 4.950 5.049 0.9802 2.5 12.18 0.0821 6.050 6.132 0.9866 2.75 15.64 0.0639 7.789 7.853 0.9919 3.0 20.09 0.0498 10.02 10.07 0.9951 3.5 33.12 0.0302 16.54 16.57 0.9982 4.0 54.60 0.0183 27.29 27.31 0.9993 4.5 90.02 0.0111 45.00 45.01 0.9998 4.605 100.0 0.0100 49.77 49.80 0.9999 5.0 148.4 0.0067 74.20 74.21 0.9999 5.5 244.7 0.0041 122.3 cosh u # tanh u # 6.0 403.4 0.0025 201.7 ˆ sinh u ˆ 1.0 6.908 1000 0.0010 500 ˆ 1 2 exp(u) sinh…u  jv† ˆ …sinh u
cos v†  j…cosh u
sin v†( cosh…u  jv† ˆ …cosh u
cos v†  j…sinh u
sin v†(„ d…sinh u†=du ˆ( cosh u sinh u
du ˆ( cosh u „ d…cosh u†=du ˆ( sinh u cosh u
du ˆ( sinh u Exponential and hyperbolic functions of u between zero and 6.908 are listed in Table 1.8. Many calculators can give such values directly. 1.2.3 Bessel functions Problems in a wide range of technology (e.g. in eddy currents, frequency modulation, etc.) can be set in the form of the Bessel equation   2d2 y 1 dy n‡
‡( 1 �( y ˆ( 0 2dx2 x dx x and its solutions are called Bessel functions of order n. For n ˆ 0 the solution is 4 =22 6 =22
42J0…x† ˆ (1 � …x 2 =22† ‡ …x
42† � …x
62† ‡


( and for n ˆ 1, 2, 3 . . .  # n 2 4x x x Jn…x† ˆ ( 1 �( ‡( �


( 2nn! 2…2n ‡ 2†( 2
4…2n ‡ 2†…2n ‡ 4†( Table 1.9 gives values of Jn(x) for various values of n and x. 1.2.4 Series Factorials In several of the following the factorial (n!) of integral numbers appears. For n between 2 and 10 these are 2! ˆ( 2 1/2! ˆ 0.5 3! ˆ( 6 1/3! ˆ 0.1667 4! ˆ( 24 1/4! ˆ 0.417  10�1 5! ˆ( 120 1/5! ˆ 0.833  10�2 6! ˆ( 720 1/6! ˆ 0.139  10�2 7! ˆ( 5 040 1/7! ˆ 0.198  10�3 8! ˆ( 40 320 1/8! ˆ 0.248  10�4 9! ˆ( 362 880 1/9! ˆ 0.276  10�5 10! ˆ 3 628 800 1/10! ˆ 0.276  10�6 Progression Arithmetic a ‡ (a ‡ d) ‡ (a ‡ 2d) ‡


‡ [a ‡ (n � 1)d] ˆ 1 n (sum of 1st and nth terms) 2 nGeometric a ‡ ar ‡ ar 2 ‡


‡ arn�1 ˆ a(1�r )/(1�r) Trigonometric See Section 1.2.1. Exponential and hyperbolic See Section 1.2.2. Binomial n…n � 1†…n � 2†(…1  x†n ˆ( 1  nx ‡( n…n � 1†( x 2 ( x 3 ‡


( 2! 3! n! ‡ …�1†r xr ‡


( r!…n � r†! n …a  x†n ˆ( an‰1  …x=a†Š //integras/b&h/eer/Final_06-09-02/eerc001 1/10 Units, mathematics and physical quantities Binomial coefficients n!/[r! (n�r)!] are tabulated: Term r ˆ( 0 1 2 3 4 5 6 7 8 9 10 n ˆ 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 7 1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 1 9 1 9 36 84 126 126 84 36 9 1 10 1 10 45 120 210 252 210 120 45 10 1 Power If there is a power series for a function f(h), it is given by ii†( iii†f …h† ˆ f …0† ‡ hf …i†…0† ‡ …h2 =2!†f … …0† ‡ …h3 =3!†f … …0† ‡


( ‡ …hr =r!†f …r†…0† ‡


( …Maclaurin†( …ii†f …x ‡ h† ˆ f …x† ‡ hf …i†…x† ‡ …h2 =2!†f …x† ‡


( ‡ …hr =r!†f …r†…x† ‡


( …Taylor†( Permutation, combination nPr ˆ n…n � 1†…n � 2†…n � 3† . . . …n � r ‡ 1† ˆ n!=…n � r†! nCr ˆ …1=r!†‰n…n�1†…n�2†…n�3† . . . …n�r ‡ 1†Š ˆ n!=r!…n�r†! Bessel See Section 1.2.3. Fourier See Section 1.2.5. 1.2.5 Fourier series A univalued periodic wave form f() of period 2& is repre- sented by a summation in general of sine and cosine waves of fundamental period 2& and of integral harmonic orders n (ˆ 2, 3, 4, . . .) as f …† ˆ c0 ‡ a1 cos &‡ a2 cos 2&‡


‡ an cos n&‡


( ‡ b1 sin &‡ b2 sin 2&‡


‡ bn sin n&‡


( The mean value of f() over a full period 2& is 1 …#2& c0 ˆ( f …†
d& 2& 0 and the harmonic-component amplitudes a and b are 1 …#2& 1 …#2& an ˆ( f …†
cos n&
d;& bn ˆ( f …†
sin n&
d& & 0 & 0 Table 1.10 gives for a number of typical wave forms the harmonic series in square brackets, preceded by the mean value c0 where it is not zero. 1.2.6 Derivatives and integrals Some basic forms are listed in Table 1.11. Entries in a given column are the integrals of those in the column to its left and the derivatives of those to its right. Constants of integration are omitted. 1.2.7 Laplace transforms Laplace transformation is a method of deriving the response of a system to any stimulus. The system has a basic equation of behaviour, and the stimulus is a pulse, step, sine wave or other variable with time. Such a response involves integration: the Laplace transform method removes integration difficulties, as tables are available for the direct solution of a great variety of problems. The pro- cess is analogous to evaluation (for example) of y ˆ 2.13.6 by transformation into a logarithmic form log y ˆ 3.6  log(2.1), and a subsequent inverse transformation back into arithmetic by use of a table of antilogarithms. The Laplace transform (L.t.) of a time-varying function f(t) is …#1( L‰ f …t†Š ˆ F…s† ˆ ( exp…�st†
f …t†
dt 0 and the inverse transformation of F(s) to give f(t) is L�1‰F…s†Š ˆ f …t† ˆ lim 1 …#‡j!& exp…st†
F…s†
ds 2 �j!& The process, illustrated by the response of a current i(t) in an electrical network of impedance z to a voltage v(t) applied at t ˆ 0, is to write down the transform equation I…s† ˆ V…s†=Z…s†( where I(s) is the L.t. of the current i(t), V(s) is the L.t. of the voltage v(t), and Z(s) is the operational impedance. Z(s) is obtained from the network resistance R, inductance L and capacitance C by leaving R unchanged but replacing L by Ls and C by 1/Cs. The process is equivalent to writing the network impedance for a steady state frequency !& and then replacing j!& by s. V(s) and Z(s) are polynomials in s: the quotient V(s)/Z(s) is reduced algebraically to a form recog- nisable in the transform table. The resulting current/time relation i(t) is read out: it contains the complete solution. However, if at t ˆ 0 the network has initial energy (i.e. if currents flow in inductors or charges are stored in capa- citors), the equation becomes I…s† ˆ ‰V…s† ‡U…s†Š=Z…s†( where U(s) contains such terms as LI0 and (1/s)V0 for the inductors or capacitors at t ˆ 0. A number of useful transform pairs is listed in Table 1.12. 1.2.8 Binary numeration A number N in decimal notation can be represented by an ordered set of binary digits an, an�2, . . . , a2, a1, a0 such that N ˆ 2nan ‡ 2n�1 an�1 ‡


‡ 2a1 ‡ a0 Decimal 1 2 3 4 5 6 7 8 9 10 100 Binary 1 10 11 100 101 110 111 1000 1001 1010 1100100 //integras/b&h/eer/Final_06-09-02/eerc001 Mathematics 1/13 Table 1.10 (continued ) Wave form Series  # 1 2 & sin & cos 2& cos 4& cos 6& Rectified sine (half-wave): a ‡ a �( �( �( �


(  & 4 1
3 3
5 5
7  # 2 4 cos 2& cos 4& cos 6& cos 8& Rectified sine (full-wave): a � a ‡( ‡( ‡( ‡


(  & 1
3 3
5 5
7 7
9  # m & 2m & cos m& cos 2m& cos 3m& Rectified sine (m-phase): a sin ‡ a sin �( ‡( �


( & m & m m2 � 1 4m2 � 1 9m2 � 1  # & 2 sin &
cos & sin 2&
cos 2& sin 3&
cos 3& Rectangular pulse train: a ‡ a ‡( ‡( ‡


(  & 1 2 3  # & 2& cos & cos 2& cos 3& a ‡ a ‡( ‡( ‡


( for & ( &  & 1 2 3 "# 1 1 1 & 4 sin2 …2 †( sin2 2…( †( sin2 3…( †2Triangular pulse train: a ‡a cos‡( 2 cos2‡( cos3‡


Š ( 2& & 1 4 9  & a ‡a ‰cos‡cos2‡cos 3‡


Š ( for && 2 & where the as have the values either 1 or 0. Thus, if N ˆ 19, 19 ˆ 16 ‡ 2 ‡ 1 ˆ (24)1 ‡ (23)0 ‡ (22)0 ‡ (21)1 ‡ (20)1 ˆ 10011 in binary notation. The rules of addition and multiplication are 0 ‡ 0 ˆ 0, 0 ‡ 1 ˆ 1, 1 ‡ 1 ˆ 10; 00 ˆ 0, 01 ˆ 0, 11 ˆ 1 1.2.9 Power ratio In communication networks the powers P1 and P2 at two specified points may differ widely as the result of ampli- fication or attenuation. The power ratio P1/P2 is more convenient in logarithmic terms. Neper [Np] This is the natural logarithm of a voltage or current ratio, given by a ˆ( ln…V1=V2 †( or a ˆ( ln…I1=I2† Np If the voltages are applied to, or the currents flow in, identical impedances, then the power ratio is a ˆ( ln…V1=V2 †2 ˆ( 2 ln…V1=V2†( and similarly for current. Decibel [dB] The power gain is given by the common logarithm lg(P1/P2) in bel [B], or most commonly by A ˆ 10 log(P1/P2) decibel [dB]. With again the proviso that the powers are developed in identical impedances, the power gain is A ˆ( 10 log…P1 =P2† ˆ (10 log…V1 =V2†2 ˆ( 20 log…V1 =V2† dB Table 1.13 gives the power ratio corresponding to a gain A (in dB) and the related identical-impedance voltage (or current) ratios. Approximately, 3 dB corresponds to a power ratio of 2, and 6 dB to a power ratio of 4. The decibel equivalent of 1 Np is 8.69 dB. 1.2.10 Matrices and vectors 1.2.10.1 Definitions If a11, a12, a13, a14 . . . is a set of elements, then the rectangu- lar array 2 3# a14 . . . a1na11 a12 a13 7a24 . . . a2n 6# a21 a22 a23 6 7A ˆ( 4 5# am1 am2 am3 am4 . . . amn arranged in m rows and n columns is called an (m  n) matrix. If m ˆ n then A is n-square. //integras/b&h/eer/Final_06-09-02/eerc001 1/14 Units, mathematics and physical quantities Table 1.11 Derivatives and integrals „ d[ f(x)]/dx f(x) f(x)
( dx 1 x n�1 n nx x (n =�1) �1/x 2 1/x 1/x ln x exp x exp x cos x sin x �sin x cos x 2 sec x tan x �cosec x
( cot x cosec x sec x
( tan x sec x �cosec2 x cot x 21/ H(a 2�x ) arcsin(x/a) 2�1/H(a 2�x ) arccos(x/a) 2a/(a 2 ‡( x ) arctan(x/a) 2�a/x H(x 2�a ) arccosec(x/a) 2a/x H(x �a 2) arcsec(x/a) 2�a/(a 2 ‡( x ) arccot(x/a) cosh x sinh x sinh x cosh x sech2 x tanh x �cosech x
( coth x cosech x �sech x
( tanh x sech x �cosech2 x coth x 1/H(x 2 ‡ 1) arsinh x 1/H(x 2�1) arcosh x 1/(1�x 2) artanh x �1/x H(x 2 ‡( 1) arcosech x �1/x H(1�x 2) arsech x 1/(1�x 2) arcoth x dv du u(x)
( v(x)u ‡( v dx dx 1 du u dv u…x† v dx �( v2 dx v…x† r exp(xa) ( sin(!x ‡&‡( ) exp(ax) ( sin(!x ‡( ) 1 2 x2 n ‡ 1x /(n ‡ 1) ln x x ln x�x exp x �cos x sin x ln(sec x) ln ( tan 1 x)2 ln(sec x ‡ tan x) ln(sin x) 2x arcsin(x/a) ‡H(a 2�x ) 2x arccos(x/a)�H(a 2�x ) 1x arctan(x=a) �( 2 a ln (a2+x2) 2x arccosec(x/a) ‡ a ln | x ‡( H(x 2�a ) | 2x arcsec(x/a)�a ln | x ‡H(x 2�a ) | 2x arccot(x/a) ‡ 1 a ln (a +x2)2 cosh x sinh x ln(cosh x) �ln(tanh 1 x†2 2 arctan (exp x) ln(sinh x) x arsinh x�H(1 ‡( x 2) x arcosh x�H(x 2�1) x artanh x ‡ 1 2 ln (1 �( x2) x arcosech x ‡ arsinh x x arsech x ‡ arcsin x x arcoth x ‡ 1 2 ln (x2 �( 1) …# du uv �( v dv dv Ð (1/r)exp(ax)sin(!x ‡�) 2r ˆ(H(!2 ‡( a ) &ˆ( arctan (!/a) An ordered set of elements x ˆ( [x1, x2, x3 . . . xn] is called 1.2.10.3 Rules of operation an n-vector. (i) Associativity A ‡( (B ‡( C) ˆ( (A ‡( B) ‡( C, An (n ( 1) matrix is called a column vector and a (1 ( n) A(BC) ˆ( (AB)C ˆ(ABC. matrix a row vector. (ii) Distributivity A(B ‡C) ˆ(AB ‡AC, (B ‡C)A ˆ(BA ‡CA. 1.2.10.2 Basic operations (iii) Identity If U is the (n ( n) matrix (ij), i, j ˆ( 1 . . . n, If A ˆ( (ars), B ˆ( (brs), where ij ˆ( 1 if i ˆ( j and 0 otherwise, then U is the (i) Sum C ˆ(A ‡B is defined by crs ˆ( ars ‡ brs, for diagonal unit matrix and A U ˆ(A. (iv) Inverse If the product U ˆ(AB exists, then B ˆ(A�1,r ˆ( 1 . . . m; s ˆ( 1 . . . n. the inverse matrix of A. If both inverses A�1 and B�1 (ii) Product If A is an (m ( q) matrix and B is a (q ( n) A�1exist, then (A B)�1 ˆ(B�1 .matrix, then the product C ˆ(AB is an (m ( n) matrix defined by (crs) ˆ(p arp bps, p ˆ( 1 . . . q; r ˆ( 1 . . . m; (v) Transposition The transpose of A is written as AT s ˆ( 1 . . . n. If AB ˆ(BA then A and B are said to commute. and is the matrix whose rows are the columns (iii) Matrix-vector product If x ˆ( [x1 . . . xn], then b ˆ(Ax is of A. If the product C ˆ(AB exists then defined by (br) ˆ(p arp xp, p ˆ( 1 . . . n; r ˆ( 1 . . . m. C T ˆ( (AB)T ˆ( BTAT . (iv) Multiplication of a matrix by a (scalar) element If k is (vi) Conjugate For A ˆ( (ars), the congugate of A is an element then C ˆ( kA ˆ(Ak is defined by (crs) ˆ( k(ars). denoted by A* ˆ( (ars*). (v) Equality If A ˆ(B, then (aij) ˆ( (bij), for i ˆ( 1 . . . n; (vii) Orthogonality Matrix A is orthogonal if AAT ˆ( U.j ˆ( 1 . . . m. //integras/b&h/eer/Final_06-09-02/eerc001 & �( & Mathematics 1/15 Table 1.12 Laplace transforms Definition f( t ) from t ˆ( 0+ F(s) ˆ( L‰ f (t)Š ˆ ( „1( 0�( f (t)
( exp (�st)
( dt Sum First derivative nth derivative Definite integral Shift by T Periodic function (period T ) Initial value Final value af1(t)+bf2(t) (d/dt) f (t) n(dn/dt ) f (t) „ T f (t)
( dt 0� f(t�T ) f(t) f(t), t!0+ f(t), t!1( aF1(s)+bF2(s) sF(s)�f(0�) n n�2s F(s)�s n�1f(0�)� s f (1)(0�)�
(
(
�f (n�1)(0�) 1 F(s) s exp(�sT )
(F(s) 1 …#T exp (�sT)
( f (t)
( dt 1 �( exp ( �( sT ) 0�( sF(s), s!1( sF(s), s!0 Description f(t) F(s) f(t) to base t 1. Unit impulse (t) 2. Unit step H(t) 3. Delayed step H(t�T ) 4. Rectangular pulse (duration T ) H(t)�H(t�T ) 5. Unit ramp t 6. Delayed ramp (t�T )H(t�T ) 7. nth-order ramp tn 8. Exponential decay exp(� t) 9. Exponential rise 1�exp(� t) 10. Exponential ( t t exp(� t) 11. Exponential ( tn tn exp(� t) 12. Difference of exponentials exp(� t)�exp(� t) 1 1 s exp (�st) s 1 �( exp (�sT ) s 1 s2 exp (�sT) 2s n! sn‡1 1 s ‡( & & s(s ‡( ) 1 (s ‡( )2 n! (s ‡( )n‡1 (s ‡( )(s ‡( ) cont'd //integras/b&h/eer/Final_06-09-02/eerc001 1/18 Units, mathematics and physical quantities definition, but may be described, as an aid to an intuitive appreciation. Energy is the capacity for `action' or work. Work is the measure of the change in energy state. State is the measure of the energy condition of a system. System is the ordered arrangement of related physical entities or processes, represented by a model. Mode is a description or mathematical formulation of the system to determine its behaviour. Behaviour describes (verbally or mathematically) the energy processes involved in changes of state. Energy storage occurs if the work done on a system is recoverable in its original form. Energy conversion takes place when related changes of state concern energy in a different form, the process sometimes being reversible. Energy dissipation is an irreversible conversion into heat. Energy transmission and radiation are forms of energy transport in which there is a finite propagation time. W In a physical system there is an identifiable energy input i and output Wo. The system itself may store energy Ws and dissipate energy W. The energy conservation principle states that Wi ˆWs ‡W ‡Wo Comparable statements can be made for energy changes
w and for energy rates (i.e. powers), giving
wi ˆ
ws ‡
w ‡
wo and pi ˆ ps ‡ p ‡ po 1.3.1.1 Analogues In some cases the mathematical formulation of a system model resembles that of a model in a completely different physical system: the two systems are then analogues. Consider linear and rotary displacements in a simple mechanical system with the conditions in an electric circuit, with the following nomenclature: A mechanical element (such as a spring) of compliance k (which describes the displacement per unit force and is the inverse of the stiffness) has a displacement l ˆ kf when a force f is applied. At a final force f1 the potential energy stored is W=1 kf1 2. For the rotary case, &ˆ kM and2 W =1 kM1 2. In the electric circuit with a pure capacitance 2 C, to which a p.d. v is applied, the charge is q ˆCv and the 1electric energy stored at v1 is W=2 Cv 2 f 1. Use is made of these correspondences in mechanical problems (e.g. of vibration) when the parameters can be con- sidered to be `lumped'. An ideal transformer, in which the primary m.m.f. in ampere-turns i1N1 is equal to the second- ary m.m.f. i2N2 has as analogue the simple lever, in which a force f1 at a point distant l1 from the fulcrum corresponds to 2 at l2 such that f1l1 ˆ f2l2. A simple series circuit is described by the equation v ˆ L(di/dt) ‡Ri ‡ q/C or, with i written as dq/dt, 2v ˆL(d2 q/dt ) ‡R(dq/dt) ‡ (1/C)q A corresponding mechanical system of mass, compliance and viscous friction (proportional to velocity) in which for a displacement l the inertial force is m(du/dt), the compli- ance force is l/k and the friction force is ru, has a total force f ˆ m…d2l=dt2† ‡ r…dl=dt† ‡ …1=k†l Thus the two systems are expressed in identical mathemat- ical form. 1.3.1.2 Fields Several physical problems are concerned with `fields' having stream-line properties. The eddyless flow of a liquid, the cur- rent in a conducting medium, the flow of heat from a high- to a low-temperature region, are fields in which representative lines can be drawn to indicate at any point the direction of f m force [N] mass [kg] M J torque [N m] inertia [kg m 2] r friction [N s/m] r friction [N m s/rad] k compliance [m/N] k compliance [rad/N m] l displacement [m] & displacement [rad] u velocity [m/s] !& angular velocity [rad/s] v voltage [V] L inductance [H] R resistance [ ] C capacitance [F] q charge [C] i current [A] The force necessary to maintain a uniform linear velocity u against a viscous frictional resistance r is f ˆ ur; the power is p ˆ fu ˆ u 2 r and the energy expended over a distance l is W ˆ fut ˆ u 2rt, since l ˆ ut. These are, respectively, the ana- logues of v ˆ iR, p ˆ vi ˆ i2R and W ˆ vit ˆ i2Rt for the corresponding electrical system. For a constant angular velocity in a rotary mechanical system, M ˆ!r, p ˆM!ˆ!2 r and W ˆ!2rt, since &ˆ!t. If a mass is given an acceleration du/dt, the force required is f ˆm(du/dt) and the stored kinetic energy at velocity u1 2is W =1 mu1. For rotary acceleration, M ˆ J(d!/dt) and 2 W =1 J!2 1. Analogously the application of a voltage v to a 2 pure inductor L produces an increase of current at the rate di/dt such that v ˆL(di/dt) and the magnetic energy stored 1at current i1 is W=2 Li 2. the flow there. Other lines, orthogonal to the flow lines, con- nect points in the field having equal potential. Along these equipotential lines there is no tendency for flow to take place. Static electric fields between charged conductors (having equipotential surfaces) are of interest in problems of insula- tion stressing. Magnetic fields, which in air-gaps may be assumed to cross between high-permeability ferromagnetic surfaces that are substantially equipotentials, may be studied in the course of investigations into flux distribution in machines. All the fields mentioned above satisfy Laplacian equations of the form …@2 V=@x 2† ‡ …@2V=@y 2† ‡ …@2V=@z 2† ˆ 0 The solution for a physical field of given geometry will apply to other Laplacian fields of similar geometry, e.g. System Potential Flux Medium current flow voltage V current I conductivity & heat flow temperature & heat q thermal conductivity & electric field voltage V electric flux Q permittivity "& magnetic field m.m.f. F magnetic flux & permeability & //integras/b&h/eer/Final_06-09-02/eerc001 Physical quantities 1/19 The ratio I/V for the first system would give the effective conductance G; correspondingly for the other systems, q/& gives the thermal conductance, Q/V gives the capacitance and /F gives the permeance, so that if measurements are made in one system the results are applicable to all the others. It is usual to treat problems as two-dimensional where possible. Several field-mapping techniques have been devised, generally electrical because of the greater convenience and precision of electrical measurements. For two-dimensional problems, conductive methods include high-resistivity paper sheers, square-mesh `nets' of resistors and electrolytic tanks. The tank is especially adaptable to three-dimensional cases of axial symmetry. In the electrolytic tank a weak electrolyte, such as ordinary tap-water, provides the conducting medium. A scale model of the electrode system is set into the liquid. A low-voltage supply at some frequency between 50 Hz and 1 kHz is connected to the electrodes so that current flows through the electrolyte between them. A probe, adjustable in the horizontal plane and with its tip dipping vertically into the electrolyte, enables the potential field to be plotted. Electrode models are constructed from some suitable insulant (wood, paraffin wax, Bakelite, etc.), the electrode outlines being defined by a highly conductive material such as brass or copper. The metal is silver-plated to improve conductivity and reduce polarisation. Three-dimensional cases with axial symmetry are simulated by tilting the tank and using the surface of the electrolyte as a radial plane of the system. The conducting-sheet analogue substitutes a sheet of resistive material (usually `teledeltos' paper with silver- painted electrodes) for the electrolyte. The method is not readily adaptable to three-dimensional plots, but is quick and inexpensive in time and material. The mesh or resistor-net analogue replaces a conductive continuum by a square mesh of equal resistors, the potential measurements being made at the nodes. Where the bound- aries are simple, and where the `grain size' is sufficiently small, good results are obtained. As there are no polarisation troubles, direct voltage supply can be used. If the resistors are made adjustable, the net can be adapted to cases of inhomo- geneity, as when plotting a magnetic field in which perme- ability is dependent on flux density. Three-dimensional plots are made by arranging plane meshes in layers; the nodes are now the junctions of six instead of four resistors. A stretched elastic membrane, depressed or elevated in appropriate regions, will accommodate itself smoothly to the differences in level: the height of the membrane everywhere can be shown to be in conformity with a two-dimensional Laplace equation. Using a rubber sheet as a membrane, the path of electrons in an electric field between electrodes in a vacuum can be investigated by the analogous paths of rolling bearing-balls. Many other useful analogues have been devised, some for the rapid solution of mathematical processes. Recently considerable development has been made in point-by-point computer solutions for the more compli- cated field patterns in three-dimensional space. 1.3.2 Structure of matter Material substances, whether solid, liquid or gaseous, are conceived as composed of very large numbers of molecules. A molecule is the smallest portion of any substance which cannot be further subdivided without losing its characteristic material properties. In all states of matter molecules are in a state of rapid continuous motion. In a solid the molecules are relatively closely `packed' and the molecules, although rapidly moving, maintain a fixed mean position. Attractive forces between molecules account for the tendency of the solid to retain its shape. In a liquid the molecules are less closely packed and there is a weaker cohesion between them, so that they can wander about with some freedom within the liquid, which consequently takes up the shape of the vessel in which it is contained. The molecules in a gas are still more mobile, and are relatively far apart. The cohesive force is very small, and the gas is enabled freely to contract and expand. The usual effect of heat is to increase the intensity and speed of molecular activity so that `collisions' between molecules occur more often; the average spaces between the molecules increase, so that the substance attempts to expand, producing internal pressure if the expansion is resisted. Molecules are capable of further subdivision, but the resulting particles, called atoms, no longer have the same properties as the molecules from which they came. An atom is the smallest portion of matter than can enter into chemical combination or be chemically separated, but it cannot gener- ally maintain a separate existence except in the few special cases where a single atom forms a molecule. A molecule may consist of one, two or more (sometimes many more) atoms of various kinds. A substance whose molecules are composed entirely of atoms of the same kind is called an element. Where atoms of two or more kinds are present, the molecule is that of a chemical compound. At present over 100 elements are recognised (Table 1.14: the atomic mass number A is relative to 1/12 of the mass of an element of carbon-12). If the element symbols are arranged in a table in ascend- ing order of atomic number, and in columns (`groups') and rows (`periods') with due regard to associated similarities, Table 1.15 is obtained. Metallic elements are found on the left, non-metals on the right. Some of the correspondences that emerge are: Group 1a: Alkali metals (Li 3, Na 11, K 19, Rb 37, Cs 55, Fr 87) 2a: Alkaline earths (Be 4, Mg 12, Ca 20, Sr 38, Ba 56, Ra 88) 1b: Copper group (Cu 29, Ag 47, Au 79) 6b: Chromium group (Cr 24, Mo 42, W 74) 7a: Halogens (F 9, Cl 17, Br 35, I 53, At 85) 0: Rare gases (He 2, Ne 10, Ar 18, Kr 36, Xe 54, Rn 86) 3a±6a: Semiconductors (B 5, Si 16, Ge 32, As 33, Sb 51, Te 52) In some cases a horizontal relation obtains as in the transition series (Sc 21 . . . Ni 28) and the heavy-atom rare earth and actinide series. The explanation lies in the struc- ture of the atom. 1.3.2.1 Atomic structure The original Bohr model of the hydrogen atom was a central nucleus containing almost the whole mass of the atom, and a single electron orbiting around it. Electrons, as small particles of negative electric charge, were discovered at the end of the nineteenth century, bringing to light the complex structure of atoms. The hydrogen nucleus is a proton, a mass having a charge equal to that of an electron, but positive. Extended to all elements, each has a nucleus comprising mass particles, some (protons) with a positive charge, others (neutrons) with no charge. The atomic mass number A is the total number of protons and neutrons in the nucleus; the atomic number Z is the number of positive charges, and the normal number of orbital electrons. The nuclear structure is not known, and the forces that bind the protons against their mutual attraction are conjectural. //integras/b&h/eer/Final_06-09-02/eerc001 1/20 Units, mathematics and physical quantities The hydrogen atom (Figure 1.4) has one proton (Z ˆ( 1) and one electron in an orbit formerly called the K shell. Helium (Z ˆ( 2) has two protons, the two electrons occupy- ing the K shell which, by the Pauli exclusion principle, can- not have more than two. The next element in order is lithium (Z ˆ( 3), the third electron in an outer L shell. With elements of increasing atomic number, the electrons are added to the L shell until it holds a maximum of 8, the surplus then occupying the M shell to a maximum of 18. The number of `valence' electrons (those in the outermost shell) determines the physical and chemical properties of the element. Those with completed outer shells are `stable'. Isotopes An element is often found to be a mixture of atoms with the same chemical property but different atomic masses: these are isotopes. The isotopes of an element must have the same number of electrons and protons, but differ in the number of neutrons, accounting for the non-integral average mass numbers. For example, neon comprises 90.4% of mass number 20, with 0.6% of 21 and 9.0% of mass number 22, giving a resultant mass number of 20.18. Energy states Atoms may be in various energy states. Thus, the filament of an incadescent lamp may emit light when excited by an electric current but not when the current is switched off. Heat energy is the kinetic energy of the atoms of a heated body. The more vigorous impact of atoms may not always shift the atom as a whole, but may shift an electron from one orbit to another of higher energy level within the atom. This position is not normally stable, and the electron gives up its momentarily acquired potential energy by falling back to its original level, releasing the energy as a light quantum or photon. Ionisation Among the electrons of an atom, those of the outermost shell are unique in that, on account of all the electron charges on the shells between them and the nucleus, they are the most loosely bound and most easily removable. In a variety of ways it is possible so to excite an atom that one of the outer electrons is torn away, leaving the atom ionised or converted for the time into an ion with an effect- ive positive charge due to the unbalanced electrical state it has acquired. Ionisation may occur due to impact by other fast-moving particles, by irradiation with rays of suitable wavelength and by the application of intense electric fields. 1.3.2.2 Wave mechanics The fundamental laws of optics can be explained without regard to the nature of light as an electromagnetic wave phenomenon, and photoelectricity emphasises its nature as a stream or ray of corpuscles. The phenomena of diffraction or interference can only be explained on the wave concept. Wave mechanics correlates the two apparently conflicting ideas into a wider concept of `waves of matter'. Electrons, atoms and even molecules participate in this duality, in that their effects appear sometimes as corpuscular, sometimes as of a wave nature. Streams of electrons behave in a corpus- cular fashion in photoemission, but in certain circumstances show the diffraction effects familiar in wave action. Considerations of particle mechanics led de Broglie to write several theoretic papers (1922±1926) on the parallel- ism between the dynamics of a particle and geometrical optics, and suggested that it was necessary to admit that classical dynamics could not interpret phenomena involving energy quanta. Wave mechanics was established by SchroÈ dinger in 1926 on de Broglie's conceptions. When electrons interact with matter, they exhibit wave properties: in the free state they act like particles. Light has a similar duality, as already noted. The hypothesis of de Broglie is that a particle of mass m and velocity u has wave Table 1.14 Elements (Z, atomic number; A, atomic mass; KLMNOPQ, electron shells) Z Name and symbol A Shells K L 1 Hydrogen H 1.008 1 Ð 2 Helium He 4.002 2 Ð 3 Lithium Li 6.94 2 1 4 Beryllium Be 9.02 2 2 5 Boron B 10.82 2 3 6 Carbon C 12 2 4 7 Nitrogen N 14.01 2 5 8 Oxygen O 16.00 2 6 9 Fluorine F 19.00 2 7 10 Neon Ne 20.18 2 8 KL M N 11 Sodium Na 22.99 10 1 Ð 12 Magnesium Mg 24.32 10 2 Ð 13 Aluminium Al 26.97 10 3 Ð 14 Silicon Si 28.06 10 4 Ð 15 Phosphorus P 31.02 10 5 Ð 16 Sulphur S 32.06 10 6 Ð 17 Chlorine Cl 35.46 10 7 Ð 18 Argon Ar 39.94 10 8 Ð 19 Potassium K 39.09 10 8 1 20 Calcium Ca 40.08 10 8 2 21 Scandium Sc 45.10 10 9 2 22 Titanium Ti 47.90 10 10 2 23 Vanadium V 0.95 10 11 2 24 Chromium Cr 52.01 10 13 1 25 Manganese Mn 54.93 10 13 2 26 Iron Fe 55.84 10 14 2 27 Cobalt Co 58.94 10 15 2 28 Nickel Ni 58.69 10 16 2 29 Copper Cu 63.57 10 18 1 30 Zinc Zn 65.38 10 18 2 31 Gallium Ga 69.72 10 18 3 32 Germanium Ge 72.60 10 18 4 33 Arsenic As 74.91 10 18 5 34 Selenium Se 78.96 10 18 6 35 Bromine Br 79.91 10 18 7 36 Krypton Kr 83.70 10 18 8 KLM N O 37 Rubidium Rb 85.44 28 8 1 38 Strontium Sr 87.63 28 8 2 39 Yttrium Y 88.92 28 9 2 40 Zirconium Zr 91.22 28 10 2 41 Niobium Nb 92.91 28 12 1 42 Molybdenum Mo 96.0 28 13 1 43 Technetium Tc 99.0 28 14 1 44 Ruthenium Ru 101.7 28 15 1 45 Rhodium Rh 102.9 28 16 1 46 Palladium Pd 106.7 28 18 Ð 47 Silver Ag 107.9 28 18 1 48 Cadmium Cd 112.4 28 18 2 49 Indium In 114.8 28 18 3 50 Tin Sn 118.7 28 18 4 51 Antimony Sb 121.8 28 18 5 52 Tellurium Te 127.6 28 18 6 53 Iodine I 126.9 28 18 7 54 Xenon Xe 131.3 28 18 8 KLM N O P 55 Caesium Cs 132.9 28 18 8 1 56 Barium Ba 137.4 28 18 8 2 cont'd //integras/b&h/eer/Final_06-09-02/eerc001 Physical quantities 1/23 Table 1.16 Physical properties of metals Approximate general properties at normal temperatures: & density [kg/m3] k thermal conductivity [W/(m K)] E elastic modulus [GPa] Tm melting point [K] e linear expansivity [mm/(m K)] & resistivity [n m] c specific heat capacity [kJ/(kg K)] & resistance±temperature coefficient [m /( K)] Metal & E e c k Tm & & Pure metals 4 Beryllium 1840 300 120 1700 170 1560 33 9.0 11 Sodium 970 Ð 71 710 130 370 47 5.5 12 Magnesium 1740 44 26 1020 170 920 46 3.8 13 Aluminium 2700 70 24 900 220 930 27 4.2 19 Potassium 860 Ð 83 750 130 340 67 5.4 20 Calcium 1550 Ð 22 650 96 1120 43 4.2 24 Chromium 7100 25 8.5 450 43 2170 130 3.0 26 Iron 7860 220 12 450 75 1810 105 6.5 27 Cobalt 8800 210 13 420 70 1770 65 6.2 28 Nickel 8900 200 13 450 70 1730 78 6.5 29 Copper 8930 120 16 390 390 1360 17 4.3 30 Zinc 7100 93 26 390 110 690 62 4.1 42 Molybdenum 10 200 Ð 5 260 140 2890 56 4.3 47 Silver 10 500 79 19 230 420 1230 16 3.9 48 Cadmium 8640 60 32 230 92 590 75 4.0 50 Tin 7300 55 27 230 65 500 115 4.3 73 Tantalum 16 600 190 6.5 140 54 3270 155 3.1 74 Tungsten 19 300 360 4 130 170 3650 55 4.9 78 Platinum 21 500 165 9 130 70 2050 106 3.9 79 Gold 19 300 80 14 130 300 1340 23 3.6 80 Mercury 13 550 Ð 180 140 10 230 960 0.9 82 Lead 11 300 15 29 130 35 600 210 4.1 83 Bismuth 9800 32 13 120 9 540 1190 4.3 92 Uranium 18 700 13 Ð 120 Ð 1410 220 2.1 Alloys Brass (60 Cu, 40 Zn) 8500 100 21 380 120 1170 60 2.0 Bronze (90 Cu, 10 Sn) 8900 100 19 380 46 1280 Ð Ð Constantan 8900 110 15 410 22 1540 450 0.05 Invar (64 Fe, 36 Ni) 8100 145 2 500 16 1720 100 2.0 Iron, soft (0.2 C) 7600 220 12 460 60 1800 140 Ð Iron cast (3.5 C, 2.5 Si) 7300 100 12 460 60 1450 Ð Ð Manganin 8500 130 16 410 22 1270 430 0.02 Steel (0.85 C) 7800 200 12 480 50 1630 180 Ð Electron emission A metal may be regarded as a potential `well' of depth �V relative to its surface, so that an electron in the lowest energy state has (at absolute zero temperature) the energy W ˆ(Ve (of the order 10 eV): other electrons occupy levels up to a height "* (5±8 eV) from the bottom of the `well'. Before an electron can escape from the surface it must be endowed with an energy not less than &ˆ( W�"*, called the work function. Emission occurs by surface irradiation (e.g. with light) of frequency v if the energy quantum hv of the radiation is at least equal to . The threshold of photoelectric emission is therefore with radiation at a frequency not less than v ˆ(/h. Emission takes place at high temperatures if, put simply, the kinetic energy of electrons normal to the surface is great enough to jump the potential step W. This leads to an expression for the emission current i in terms of temperature T, a constant A and the thermionic work function : i ˆ( AT2 exp…�=kT †( Electron emission is also the result of the application of a high electric field intensity (of the order 1±10 GV/m) to a metal surface; also when the surface is bombarded with electrons or ions of sufficient kinetic energy, giving the effect of secondary emission. Crystals When atoms are brought together to form a crystal, their individual sharp and well-defined energy levels merge into energy bands. These bands may overlap, or there may be gaps in the energy levels available, depending on the lattice spacing and interatomic bonding. Conduction can take place only by electron migration into an empty or partly filled band; filled bands are not available. If an elec- tron acquires a small amount of energy from the externally applied electric field, and can move into an available empty level, it can then contribute to the conduction process. 1.3.2.5 Insulators In this case the `distance' (or energy increase
w in electron- volts) is too large for moderate electric applied fields to endow electrons with sufficient energy, so the material remains an insulator. High temperatures, however, may //integras/b&h/eer/Final_06-09-02/eerc001 1/24 Units, mathematics and physical quantities Table 1.17 Physical properties of non-metals Approximate general properties: & density [kg/m3] Tm melting point [K] e linear expansivity [mm/(m K)] & resistivity [M m] c specific heat capacity [kJ/(kg K)] r relative permittivity [�] k thermal conductivity [W/(m K)] Material & e c k Tm & "&r Asbestos (packed) 580 Ð 0.84 0.19 Ð Ð 3 Bakelite 1300 30 0.92 0.20 Ð 0.1 7 Concrete (dry) Diamond 2000 3510 10 1.3 0.92 0.49 1.70 165 Ð 4000 Ð 107 Ð Ð Glass 2500 8 0.84 0.93 Ð 106 8 Graphite Marble 2250 2700 2 12 0.69 0.88 160 3 3800 Ð 10�11 103 Ð 8.5 Mica 2800 3 0.88 0.5 Ð 108 7 Nylon Paper Paraffin wax 1140 900 890 100 Ð 110 1.7 Ð 2.9 0.3 0.18 0.26 Ð Ð Ð Ð 104 109 Ð 2 2 Perspex 1200 80 1.5 1.9 Ð 1014 3 Polythene Porcelain 930 2400 180 3.5 2.2 0.8 0.3 1.0 Ð 1900 Ð 106 2.3 6 Quartz (fused) Rubber 2200 1250 0.4 Ð 0.75 1.5 0.22 0.15 2000 Ð 1014 107 3.8 3 Silicon 2300 7 0.75 Ð 1690 0.1 2.7 Table 1.18 Physical properties of liquids Average values at 20C (293 K): & density [kg/m3] k thermal conductivity [W/(m K)] v e viscosity [mPa s] cubic expansivity [10�3/K] Tm Tb melting point [K] boiling point [K] c specific heat capacity [kJ/(kg K)] "&r relative permittivity [�] Liquid & v e c k Tm Tb "&r Acetone (CH3)2CO 792 0.3 1.43 2.2 0.18 178 329 22 Benzine C6H6 881 0.7 1.15 1.7 0.14 279 353 2.3 Carbon disulphide CS2 1260 0.4 1.22 1.0 0.14 161 319 2.6 Carbon tetrachloride CCl4 1600 1.0 1.22 0.8 0.10 250 350 2.2 Ether (C2H5)2O 716 0.2 1.62 2.3 0.14 157 308 4.3 Glycerol C3H5(OH)3 1270 1500 0.50 2.4 0.28 291 563 56 Methanol CH3OH 793 0.6 1.20 1.2 0.21 175 338 32 Oil Ð 850 85 0.75 1.6 0.17 Ð Ð 3.0 Sulphuric acid H2SO4 1850 28 0.56 1.4 Ð 284 599 Ð Turpentine C10H16 840 1.5 0.10 1.8 0.15 263 453 2.3 Water H2O 1000 1.0 0.18 4.2 0.60 273 373 81 result in sufficient thermal agitation to permit electrons to `jump the gap'. 1.3.2.6 Semiconductors Intrinsic semiconductors (i.e. materials between the good conductors and the good insulators) have a small spacing of about 1 eV between their permitted bands, which affords a low conductivity, strongly dependent on temperature and of the order of one-millionth that of a conductor. Impurity semiconductors have their low conductivity raised by the presence of minute quantities of foreign atoms (e.g. 1 in 108) or by deformations in the crystal struc- ture. The impurities `donate' electrons of energy level that can be raised into a conduction band (n-type); or they can attract an electron from a filled band to leave a `hole', or electron deficiency, the movement of which corresponds to the movement of a positive charge (p-type). 1.3.2.7 Magnetism Modern magnetic theory is very complex, with ramifica- tions in several branches of physics. Magnetic phenomena are associated with moving charges. Electrons, considered as particles, are assumed to possess an axial spin, which gives them the effect of a minute current turn or of a small permanent magnet, called a Bohr magneton. The gyro- scopic effect of electron spin develops a precession when a magnetic field is applied. If the precession effect exceeds the spin effect, the external applied magnetic field produces less //integras/b&h/eer/Final_06-09-02/eerc001 Physical quantities 1/25 Table 1.19 Physical properties of gases Values at 0C (273 K) and atmospheric pressure: c & density [kg/m3] k thermal conductivity [m W/(m K)] v viscosity [mPa s] Tm melting point [K] p specific heat capacity [kJ/(kg K)] Tb boiling point [K] cp/cv ratio between specific heat capacity at constant pressure and at constant volume Gas & v cp cp/cv k Tm Tb Air Ammonia Carbon dioxide Carbon monoxide Chlorine Deuterium Ethane Fluorine Helium Hydrogen Hydrogen chloride Krypton Methane Neon Nitrogen Oxygen Ozone Propane Sulphur dioxide Xenon O O N H F Ð NH3 CO2 CO Cl2 D C2H6 2 He 2 HCl Kr CH4 Ne 2 2 3 C3H8 SO2 Xe 1.293 0.771 1.977 1.250 3.214 0.180 1.356 1.695 0.178 0.090 1.639 3.740 0.717 0.900 1.251 1.429 2.220 2.020 2.926 5.890 17.0 9.3 13.9 16.4 12.3 Ð 8.6 Ð 18.6 8.5 13.8 23.3 10.2 29.8 16.7 19.4 Ð 7.5 11.7 22.6 1.00 2.06 0.82 1.05 0.49 Ð 1.72 0.75 5.1 14.3 0.81 Ð 2.21 1.03 1.04 0.92 Ð 1.53 0.64 Ð 1.40 1.32 1.31 1.40 1.36 1.73 1.22 Ð 1.66 1.41 1.41 1.68 1.31 1.64 1.40 1.40 1.29 1.13 1.27 1.66 24 22 14 23 7.6 Ð 18 Ð 144 174 Ð 8.7 30 46 24 25 Ð 15 8.4 5.2 Ð 195 216* 68 171 18 89 50 1.0 14 161 116 90 24 63 55 80 83 200 161 Ð 240 194 81 239 23 184 85 4.3 20 189 121 112 27 77 90 161 231 263 165 *At pressure of 5 atm. magnetisation than it would in free space, and the material of which the electron is a constituent part is diamagnetic. If the spin effect exceeds that due to precession, the material is paramagnetic. The spin effect may, in certain cases, be very large, and high magnetisations are produced by an external field: such materials are ferromagnetic. An iron atom has, in the n ˆ( 4 shell (N), electrons that give it conductive properties. The K, L and N shells have equal numbers of electrons possessing opposite spin direc- tions, so cancelling. But shell M contains 9 electrons spin- ning in one direction and 5 in the other, leaving 4 net magnetons. Cobalt has 3, and nickel 2. In a solid metal further cancellation occurs and the average number of unbalanced magnetons is: Fe, 2.2; Co, 1.7; Ni, 0.6. In an iron crystal the magnetic axes of the atoms are aligned, unless upset by excessive thermal agitation. (At 770C for Fe, the Curie point, the directions become random and ferromagnetism is lost.) A single Fe crystal magnetises most easily along a cube edge of the structure. It does not exhibit spontaneous magnetisation like a per- manent magnet, however, because a crystal is divided into a large number of domains in which the various magnetic directions of the atoms form closed paths. But if a crystal is exposed to an external applied magnetic field, (a) the elec- tron spin axes remain initially unchanged, but those domains having axes in the favourable direction grow at the expense of the others (domain wall displacement); and (b) for higher field intensities the spin axes orientate into the direction of the applied field. If wall movement makes a domain acquire more internal energy, then the movement will relax again when the exter- nal field is removed. But if wall movement results in loss of energy, the movement is non-reversibleÐi.e. it needs Table 1.20 Characteristic temperatures Temperature T [kelvin] corresponds to c ˆ(T � 273.15 [degree Celsius] and to f ˆ( c (9/5)�32 [degree Fahrenheit]. Condition T c f Absolute zero 0 �273.15 �459.7 Boiling point of oxygen 90.18 �182.97 �297.3 Zero of Fahrenheit scale 255.4 �17.78 0 Melting point of ice 273.15 0 32.0 Triple point of water 273.16 0.01 32.02 Maximum density of water 277.13 3.98 39.16 `Normal' ambient 293.15 20 68 Boiling point of water 373.15 100 212 Boiling point of sulphur 717.8 444.6 832 Freezing point of silver 1234 962 1762 Freezing point of gold 1336 1064 1945 external force to reverse it. This accounts for hysteresis and remanence phenomena. The closed-circuit self-magnetisation of a domain gives it a mechanical strain. When the magnetisation directions of individual domains are changed by an external field, the strain directions alter too, so that an assembly of domains will tend to lengthen or shorten. Thus, readjustments in the crystal lattice occur, with deformations (e.g. 20 parts in 106) in one direction. This is the phenomenon of magnetostriction. The practical art of magnetics consists in control of mag- netic properties by alloying, heat treatment and mechanical working to produce variants of crystal structure and conse- quent magnetic characteristics. //integras/b&h/eer/Final_06-09-02/eerc001 1/28 Units, mathematics and physical quantities Figure 1.8 Conduction in low-pressure gas always present due to stray radiations (light, etc.). The elec- trons produced attach themselves to gas atoms and the sets of positive and negative ions drift in opposite directions. At very low gas pressures the electrons produced by ionisa- tion have a much longer free path before they collide with a molecule, and so have scope to attain high velocities. Their motional energy may be enough to shockionise neutral atoms, resulting in a great enrichment of the electron stream and an increased current flow. The current may build up to high values if the effect becomes cumulative, and eventually conduction may be effected through a spark or arc. In a vacuum conduction can be considered as purely electronic, in that any electrons present (there can be no molecular matter present in a perfect vacuum) are moved in accordance with the force exerted on them by an applied electric field. The number of electrons is small, and although high speeds may be reached, the conduction is generally measurable only in milli- or microamperes. Some of the effects are illustrated in Figure 1.8, represent- ing part of a vessel containing a gas or vapour at low pres- sure. At the bottom is an electrode, the cathode, from the surface of which electrons are emitted, generally by heating the cathode material. At the top is a second electrode, the anode, and an electric field is established between the electro- des. The field causes electrons emitted from the cathode to move upward. In their passage to the anode these electrons will encounter gas molecules. If conditions are suitable, the gas atoms are ionised, becoming in effect positive charges associated with the nuclear mass. Thereafter the current is increased by the detached electrons moving upwards and by the positive ions moving more slowly downwards. In certain devices (such as the mercury arc rectifier) the impact of ions on the cathode surface maintains its emission. The impact of electrons on the anode may be energetic enough to cause the secondary emission of electrons from the anode surface. If the gas molecules are excluded and a vacuum is established, the conduction becomes purely electronic. 1.5.2.5 Insulators If an electric field is applied to a perfect insulator, whether solid, liquid or gaseous, the electric field affects the atoms by producing a kind of `stretching' or `rotation' which displaces the electrical centres of negative and positive in opposite directions. This polarisation of the dielectric insu- lating material may be considered as taking place in the manner indicated in Figure 1.9. Before the electric field is Figure 1.9 Polarisation and breakdown in insulator applied, the atoms of the insulator are neutral and unstrained; as the potential difference is raised the electric field exerts opposite mechanical forces on the negative and positive charges and the atoms become more and more highly strained (Figure 1.9(a)). On the left face the atoms will all present their negative charges at the surface: on the right face, their positive charges. These surface polarisations are such as to account for the effect known as permittivity. The small displacement of the atomic electric charges con- stitutes a polarisation current. Figure 1.9(b) shows that, for excessive electric field strength, conduction can take place, resulting in insulation breakdown. The electrical properties of metallic conductors and of insulating materials are listed in Tables 1.22 and 1.23. 1.5.2.6 Convection current Charges can be moved mechanically, on belts, water-drops, dust and mist particles, and by beams of high-speed electrons (as in a cathode ray oscilloscope). Such movement, indepen- dent of an electric field, is termed a convection current. 1.5.3 Charges in acceleration Reference has been made to the emission of energy (photons) when an electron falls from an energy level to a lower one. Radiation has both a particle and a wave nature, the latter associated with energy propagation through empty space and through transparent media. 1.5.3.1 Maxwell equations Faraday postulated the concept of the field to account for `action at a distance' between charges and between magnets. Maxwell (1873) systematised this concept in the form of electromagnetic field equations. These refer to media in bulk. They naturally have no direct relation to the elec- tronic nature of conduction, but deal with the fluxes of elec- tric, magnetic and conduction fields, their flux densities, and the bulk material properties (permittivity ", permeabil- ity & and conductivity ) of the media in which the fields exist. To the work of Faraday. AmpeÁ re and Gauss, Maxwell added the concept of displacement current. Displacement current Around an electric field that changes with time there is evidence of a magnetic field. By analogy with the magnetic field around a conduction current, the rate of change of an electric field may be represented by the presence of a displacement current. The concept is applicable to an electric circuit containing a capacitor: there is a conduction current ic in the external circuit but not between the electrodes of the capacitor. The capacitor, however, must be acquiring or losing charge and its electric field must be changing. If the rate of change is represented by a displacement current id ˆ( ic, not only is the magnetic field accounted for, but also there now exists a `continuity' of current around the circuit. Displacement current is present in any material medium, conducting or insulating, whenever there is present an //integras/b&h/eer/Final_06-09-02/eerc001 Electricity 1/29 Table 1.22 Electrical properties of conductors Typical approximate values at 293 K (20 C): g conductivity relative to I.S.A.C. [%] & resistivity [n m] & resistance±temperature coefficient [m /( K)] Material g  & International standard annealed copper (ISAC) Copper annealed hard-drawn Brass (60/40) cast rolled Bronze Phosphor-bronze Cadmium-copper, hard-drawn Copper-clad steel, hard-drawn Aluminium cast hard-drawn duralumin Iron wrought cast grey white malleable nomag Steel 0.1% C 0.4% C core 1% Si 2% Si 4% Si wire galvanised 45 ton 80 ton Resistance alloys* 80 Ni, 20 Cr 59 Ni, 16 Cr, 25 Fe 37 Ni, 18 Cr, 2 Si, 43 Fe 45 Ni, 54 Cu 20 Ni, 80 Cu 15 Ni, 62 Cu, 22 Zn 4 Ni, 84 Cu, 12 Mn Gold Lead Mercury Molybdenum Nickel Platinum Silver annealed hard-drawn Tantalum Tungsten Zinc 100 17.2 3.93 99 17.3 3.90 97 17.7 3.85 23 75 1.6 19 90 1.6 48 36 1.65 29±14 6±12 1.0 82±93 21±18 4.0 30±40 57±43 3.75 66 26 3.90 62 28 3.90 36 47 Ð 16 107 5.5 2.5 700 Ð 1.7 1000 2.0 5.9 300 Ð 1.1 1600 4.5 8.6 200 4.2 11 160 4.2 10 170 Ð 4.9 350 Ð 3.1 550 Ð 12 140 4.4 10 170 3.4 8 215 3.4 (1) 1.65 1090 0.1 (2) 1.62 1100 0.2 (3) 1.89 1080 0.26 (4) 3.6 490 0.04 (5) 6.6 260 0.29 (6) 5.0 340 0.25 (7) 3.6 480 0.0 73 23.6 3.0 7.8 220 4.0 1.8 955 0.7 30 57 4.0 12.6 136 5.0 14.7 117 3.9 109 15.8 4.0 98.5 17.5 4.0 11.1 155 3.1 31 56 4.5 28 62 4.0 *Resistance alloys: (1) furnaces, radiant elements; (2) electric irons, tubular heaters; (3) furnace elements; (4) control resistors; (5) cupro; (6) German silver, platinoid; (7) Manganin. electric field that changes with time. There is a displacement current along a copper conductor carrying an alternating current, but the conduction current is vastly greater even at very high frequencies. In poor conductors and in insulating materials the displacement current is comparable to (or greater than) the conduction current if the frequency is high enough. In free space and in a perfect insulator only displacement current is concerned. Equations The following symbols are used, the SI unit of each appended. The permeability and permittivity are absolute values (&ˆ(r0, "&ˆ( "&r "&0). Potentials and fluxes are scalar quantities: field strength and flux density, also surface and path-length elements, are vectorial. Field Electric Magnetic Conduction Potential V [V] F [A] V [V] Field strength E [V/m] H [A/m] E [V/m] Flux Flux density Q [C] D [C/m2] & [Wb] B [T] I [A] J [A/m2] Material property "& [F/m] & [H/m] & [S/m] The total electric flux emerging from a charge ‡Q or entering a charge �Q is equal to Q. The integral of the elec- tric flux density D over a closed surface s enveloping the charge is …# D
( ds ˆ( Q …1:1†( s If the surface has no enclosed charge, the integral is zero. This is the Gauss law. The magnetomotive force F, or the line integral of the magnetic field strength H around a closed path l, is equal to the current enclosed, i.e. …# H
( dl ˆ( F ˆ( ic ‡( id …1:2†( o This is the AmpeÁ re law with the addition of displacement current. The Faraday law states that, around any closed path l encircling a magnetic flux & that changes with time, there is an electric field, and the line integral of the electric field strength E around the path is …# E
( dl ˆ( e ˆ �…d=dt† …1:3†( o Magnetic flux is a solenoidal quantity, i.e. it comprises a structure of closed loops; over any closed surface s in a mag- netic field as much flux leaves the surface as enters it. The sur- face integral of the flux density B is therefore always zero, i.e. …# B
( ds ˆ( 0 …1:4†( s To these four laws are added the constitutive equations, which relate the flux densities to the properties of the media in which the fields are established. The first two are, respectively, electric and magnetic field relations; the third relates conduction current density to the voltage gradient in a conducting medium; the fourth is a statement of the dis- placement current density resulting from a time rate of change of the electric flux density. The relations are D ˆ( "E; B ˆ( H; Jc ˆ( E; Jd ˆ( @D=@t //integras/b&h/eer/Final_06-09-02/eerc001 1/30 Units, mathematics and physical quantities Table 1.23 Electrical properties of insulating materials Typical approximate values (see also Section 1.4): "&r relative permittivity E electric strength [MV/m] tan & loss tangent & maximum working temperature [C] k G thermal conductivity density [mW/(m K)] [kg/m3] Material "&r E tan & & k G Air at n.t.p. 1.0 3 Ð Ð 25 1.3 Alcohol 26 Ð Ð Ð 180 790 Asbestos 2 2 Ð 400 80 3000 paper 2 2 Ð 250 250 1200 Bakelite moulding 4 6 0.03 130 Ð 1600 paper 5 15 0.03 100 270 1300 Bitumen pure 2.7 1.6 Ð 50 150 1200 vulcanised 4.5 5 Ð 100 200 1250 Cellulose film 5.8 28 Ð Ð Ð 800 Cotton fabric dry Ð 0.5 Ð 95 80 Ð impregnated Ð 2 Ð 95 250 Ð Ebonite 2.8 50 0.005 80 150 1400 Fabric tape, impregnated 5 17 0.1 95 240 Ð Glass flint 6.6 6 Ð Ð 1100 4500 crown 4.8 6 0.02 Ð 600 2200 toughened 5.3 9 0.003 Ð Ð Ð Gutta-percha 4.5 Ð 0.02 Ð 200 980 Marble 7 2 0.03 Ð 2600 2700 Mica 6 40 0.02 750 600 2800 Micanite Ð 15 Ð 125 150 2200 Oil transformer 2.3 Ð Ð 85 160 870 castor 4.7 8 Ð Ð Ð 970 Paper dry 2.2 5 0.007 90 130 820 impregnated 3.2 15 0.06 90 140 1100 Porcelain 5.7 15 0.008 1000 1000 2400 Pressboard 6.2 7 Ð 95 170 1100 Quartz fused 3.5 13 0.002 1000 1200 2200 crystalline 4.4 Ð Ð Ð Ð 2700 Rubber pure 2.6 18 0.005 50 100 930 vulcanised 4 10 0.01 70 250 1500 moulding 4 10 Ð 70 Ð Ð Resin 3 Ð Ð Ð Ð 1100 Shellac 3 11 Ð 75 250 1000 paper 5.5 11 0.05 80 Ð 1350 Silica, fused 3.6 14 Ð Ð Ð Ð Silk Ð Ð Ð 95 60 1200 Slate Ð 0.5 Ð Ð 2000 2800 Steatite Ð 0.6 Ð 1500 2000 2600 Sulphur 4 Ð 0.0003 100 220 2000 Water 70 Ð Ð Ð 570 1000 Wax (paraffin) 2.2 12 0.0003 35 270 860 In electrotechnology concerned with direct or low- frequency currents, the Maxwell equations are rarely used in the form given above. Equation (1.2), for example, appears as the number of amperes (or ampere-turns) required to produce in an area a the specified magnetic flux &ˆBa ˆHa. Equation (1.3) in the form e ˆ�(d/dt) gives the e.m.f. in a transformer primary or secondary turn. The concept of the `magnetic circuit' embodies Equation (1.4). But when dealing with such field phenomena as the eddy currents in massive conductors, radio propagation or the transfer of energy along a transmission line, the Maxwell equations are the basis of analysis. //integras/b&h/Eer/Final_06-09-02/eerc002 2 Electrotechnology 2.1 Nomenclature 2/3 2.1.1 2/3 2.1.2 2/4 2.2 2/6 2.2.1 Resistance 2/6 2.2.2 2/8 2.3 2/10 2.3.1 Electrolysis 2/10 2.3.2 Cells 2/11 2/12 2.4.1 2/12 2.4.2 2/14 2.4.3 2/15 2.4.4 Inductance 2/17 2.5 2/19 2.5.1 Electrostatics 2/20 2.5.2 Capacitance 2/20 2.5.3 2/22 2/22 2.6 2/23 2.6.1 2/23 2.6.2 2/23 2.6.3 2/23 2.7 2/25 2.7.1 Introduction 2/25 2.7.2 2/26 2.7.3 2/27 2/28 M G Say PhD, MSc, CEng, ACGI, DIC, FIEE, FRSE Heriot-Watt University (Sections 2.1±2.6) G R Jones PhD, DSc, CEng, FIEE, MInst P University of Liverpool (Section 2.7) Contents Circuit phenomena Electrotechnical terms Thermal effects Heating and cooling Electrochemical effects 2.4 Magnetic field effects Magnetic circuit Magnetomechanical effects Electromagnetic induction Electric field effects Dielectric breakdown 2.5.4 Electromechanical effects Electromagnetic field effects Movement of charged particles Free space propagation Transmission line propagation Electrical discharges Types of discharge Discharge±network interaction 2.7.4 Discharge applications //integras/b&h/Eer/Final_06-09-02/eerc002 //integras/b&h/Eer/Final_06-09-02/eerc002 Nomenclature 2/3 Electrotechnology concerns the electrophysical and allied principles applied to practical electrical engineering. A com- pletely general approach is not feasible, and many separate ad hoc technologies have been developed using simplified and delimited areas adequate for particular applications. In establishing a technology it is necessary to consider whether the relevant applications can be dealt with (a) in macroscopic terms of physical qualities of materials in bulk (as with metallic conduction or static magnetic fields); or (b) in microscopic terms involving the microstructure of materials as an essential feature (as in domain theory); or (c) in molecu- lar, atomic or subatomic terms (as in nuclear reaction and semi-conduction). There is no rigid line of demarcation, and certain technologies must cope with two or more such sub- divisions at once. Electrotechnology thus tends to become an assembly of more or less discrete (and sometimes apparently unrelated) areas in which methods of treatment differ widely. To a considerable extent (but not completely), the items of plant with which technical electrical engineering dealsÐ generators, motors, feeders, capacitors, etc.Ðcan be repre- sented by equivalent circuits or networks energised by an electrical source. For the great majority of cases within the purview of `heavy electrical engineering' (that is, generation, transmis- sion and utilisation for power purposes, as distinct from telecommunications), a source of electrical energy is con- sidered to produce a current in a conducting circuit by reason of an electromotive force acting against a property of the circuit called impedance. The behaviour of the circuit is described in terms of the energy fed into the circuit by the source, and the nature of the conversion, dissipation or stor- age of this energy in the several circuit components. Electrical phenomena, however, are only in part asso- ciated with conducting circuits. The generalised basis is one of magnetic and electrical fields in free space or in material media. The fundamental starting point is the conception contained in Maxwell's electromagnetic equations (Section 1.5.3), and in this respect the voltage and currents in a circuit are only representative of the fundamental field phenomena within a restricted range. Fortunately, this range embraces very nearly the whole of `heavy' electrical engineering practice. The necessity for a more comprehensive viewpoint makes itself apparent in connection with problems of long-line transmission; and when the technique of ultra- high-frequency work is reached, it is necessary to give up the familiar circuit ideas in favour of a whole-hearted application of field principles. 2.1 Nomenclature 2.1.1 Circuit phenomena Figure 2.1 shows in a simplified form a hypothetical circuit with a variety of electrical energy sources and a representative selection of devices in which the energy received from the source is converted into other forms, or stored, or both. The forms of variation of the current or voltage are shown in Figure 2.2. In an actual circuit the current may change in a quite arbitrary fashion as indicated at (a): it may rise or fall, or reverse its direction, depending on chance or control. Such random variation is inconveniently difficult to deal with, and engineers prefer to simplify the conditions as much as poss- ible. For example (Figure 2.2(b)), the current may be assumed to be rigidly constant, in which case it is termed a direct cur- rent. If the current be deemed to reverse cyclically according to a sine function, it becomes a sinusoidal alternating current (c). Less simple waveforms, such as (d), may be dealt with by Figure 2.1 Typical circuit devices. G, source generator; R, resistor; A, arc; B, battery; P, plating bath; M, motor; L, inductor; C, capacitor; I, insulator Figure 2.2 Modes of current (or voltage) variation application of Fourier's theorem, thus making it possible to calculate a great range of practical casesÐsuch as those involving rectifiersÐin which the sinusoidal waveform assumption is inapplicable. The cases shown in (b), (c) and (d) are known as steady states, the current (or voltage) being assumed established for a considerable time before the circuit is investigated. But since the electric circuit is capable of stor- ing energy, a change in the circuit may alter the conditions so as to cause a redistribution of circuit energy. This occurs with a circulation of transient current. An example of a simple oscillatory transient is shown in Figure 2.2(e). The calculation of circuits in which direct currents flow is comparatively straightforward. For sine wave alternating current circuits an algebra has been developed by means of which problems can be reduced to a technique very similar to that of d.c. circuits. Where non-sinusoidal waveforms are concerned, the treatment is based on the analysis of the current and voltage waves into fundamental and harmonic sine waves, the standard sine wave method being applied to the fundamental and to each of the harmonics. In the case of transients, a more searching investigation may be necessary, but there are a number of common modes in which transients usually occur, and (so long as the circuit is relatively simple) it may be possible to select the appro- priate mode by inspection. Circuit parametersÐresistance, inductance and capaci- tanceÐmay or may not be constant. If they are not, approximation, linearising or step-by-step computation is necessary. 2.1.1.1 Electromotive-force sources Any device that develops an electromotive force (e.m.f.) capable of sustaining a current in an electric circuit must be associated with some mode of energy conversion into the electrical from some different form. The modes are (1) mechanical/electromagnetic, (2) mechanical/electrostatic, (3) chemical, (4) thermal, and (5) photoelectric. //integras/b&h/Eer/Final_06-09-02/eerc002 2/6 Electrotechnology the magnetic field intensity between the points, except in the presence of electric currents. Magnetic space constant: The permeability of free space. Magnetising force: The same as magnetic field strength. Magnetomotive force: Along any path, the line integral of the magnetic field strength along that path. If the path is closed, the line integral is equal to the total magnetising current in ampere-turns. Paramagnetic: Having a permeability greater than that of free space. Period: The time taken by one complete cycle of a wave- form. Permeability: The ratio of the magnetic flux density in a medium or material at a point to the magnetic field strength at the point. The absolute permeability is the product of the relative permeability and the permeability of free space (magnetic space constant). Permeance: The ratio between the magnetic flux in a mag- netic circuit and the magnetomotive force. The reciprocal of reluctance. Permittivity: The ratio between the electric flux density in a medium or material at a point and the electric field strength at the point. The absolute permittivity is the pro- duct of the relative permittivity and the permittivity of free space (electric space constant). Phase angle: The angle between the phasors that repre- sent two alternating quantities of sinusoidal waveform and the same frequency. Phasor: A sinusoidally varying quantity represented in the form of a complex number. Polarisation: The change of the electrical state of an insulat- ing material under the influence of an electric field, such that each small element becomes an electric dipole or doublet. Potential: The electrical state at a point with respect to potential zero (normally taken as that of the earth). It is measured by the work done in transferring unit charge from pontential zero to the point. Potential difference: A difference between the electrical states existing at two points tending to cause a movement of positive charges from one point to the other. It is meas- ured by the work done in transferring unit charge from one point to the other. Potential gradient: The potential difference per unit length in the direction in which it is a maximum. Power: The rate of transfer, storage, conversion or dis- sipation of energy. In sinusoidal alternating current circuits the active power is the mean rate of energy conversion; the reactive power is the peak rate of circulation of stored energy; the apparent power is the product of r.m.s. values of voltage and current. Power factor: The ratio between active power and appar- ent power. In sinusoidal alternating current circuits the power factor is cos , where  is the phase angle between voltage and current waveforms. Quantity: The product of the current and the time during which it flows. Reactance: In sinusoidal alternating current circuits, the quantity !L or 1/!C, where L is the inductance, C is the capacitance and ! is the angular frequency. Reactor: A device having reactance as a chief property; it may be an inductor or a capacitor. A nuclear reactor is a device in which energy is generated by a process of nuclear fission. Reluctance: The ratio between the magnetomotive force acting around a magnetic circuit and the resulting magnetic flux. The reciprocal of permeance. Remanence: The remanent flux density obtained when the initial magnetisation reaches the saturation value for the material. Remanent flux density: The magnetic flux density remain- ing in a material when, after initial magnetisation, the mag- netising force is reduced to zero. Residual magnetism: The magnetism remaining in a mater- ial after the magnetising force has been removed. Resistance: That property of a material by virtue of which it resists the flow of charge through it, causing a dissipation of energy as heat. It is equal to the constant potential differ- ence divided by the current produced thereby when the material has no e.m.f. acting within it. Resistivity: The resistance between opposite faces of a unit cube of a given material. Resistor: A device having resistance as a chief property. Susceptance: The reciprocal of reactance. Time constant: The characteristic time describing the duration of a transient phenomenon. Voltage: The same as potential difference. Voltage gradient: The same as potential gradient. Waveform: The graph of successive instantaneous values of a time-varying physical quantity. 2.2 Thermal effects 2.2.1 Resistance That property of an electric circuit which determines for a given current the rate at which electrical energy is converted into heat is termed resistance. A device whose chief property is resistance is a resistor, or, if variable, a rheostat. A current I flowing in a resistance R develops heat at the rate P ˆ( I 2R joule/second or watts a relation expressing Joule's law. 2.2.1.1 Voltage applied to a resistor In the absence of any energy storage effects (a physically unrealisable condition), the current in a resistor of value R is I when the voltage across it is V, in accordance with the relation I ˆ(V/R. If a steady p.d. V be suddenly applied to a resistor R, the current instantaneously assumes the value given, and energy is expended at the rate P ˆ( I2R watts, continuously. No transient occurs. If a constant frequency, constant amplitude sine wave voltage v is applied, the current i is at every instant given by i ˆ( v/R, and in consequence the current has also a sine waveform, provided that the resistance is linear. The instantaneous rate of energy dissipation depends on the instantaneous current: it is p ˆ( vi ˆ( i2R. Should the applied voltage be non-sinusoidal, the current has (under the restriction mentioned) an exactly similar waveform. The three cases are illustrated in Figure 2.3. Figure 2.3 Voltage applied to a pure resistor //integras/b&h/Eer/Final_06-09-02/eerc002 Thermal effects 2/7 In the case of alternating waveform, the average rate of energy dissipation is given by P ˆ( I2R, where I is the root- mean-square current value. 2.2.1.2 Voltage±current characteristics For a given resistor R carrying a constant current I, the p.d. is V ˆ( IR. The ratio R ˆ(V/I may or may not be invariable. In some cases it is sufficient to assume a degree of con- stancy, and calculation is generally made on this assump- tion. Where the variations of resistance are too great to make the assumption reasonably valid, it is necessary to resort to less simple analysis or to graphical methods. A constant resistance is manifested by a constant ratio between the voltage across it and the current through it, and by a straight-line graphical relation between I and V (Figure 2.4(a)), where R ˆ(V/I ˆ( cot . This case is typical of metallic resistance wires at constant temperature. Certain circuits exhibit non-linear current±voltage rela- tions (Figure 2.4(b)). The non-linearity may be symmetrical or asymmetrical, in accordance with whether the conduction characteristics are the same or different for the two current- flow directions. Rectifiers are an important class of non- linear, asymmetrical resistors. r A hypothetical device having the current±voltage charac- teristic shown in Figure 2.4(c) has, at an operating condition represented by the point P, a current Id and a p.d. Vd. The ratio Rd ˆ(Vd/Id is its d.c. resistance for the given condition. If a small alternating voltage
va be applied under the same condition (i.e. superimposed on the p.d. Vd), the current will fluctuate by
ia and the ratio ra ˆ(
va/
ia is the a.c. or incremental resistance at P. The d.c. resistance is also obtainable from Rd ˆ( cot , and the a.c. resistance from a ˆ( cot . In the region of which Q is a representative point, the a.c. resistance is negative, indicating that the device is capable of giving a small output of a.c. power, derived from its greater d.c. input. It remains in sum an energy dissipator, but some of the energy is returnable under suitable conditions of operation. 2.2.1.3 D.c. or ohmic resistance: linear resistors The d.c. or ohmic resistance of linear resistors (a category confined principally to metallic conductors) is a function of the dimensions of the conducting path and of the resistivity of the material from which the conductor is made. A wire of length l, cross-section a and resistivity  has, at constant given temperature, a resistance R ˆ( l=a ohms where , l and a are in a consistent system of dimensions (e.g. l in metres, a in square metres,  in ohms per 1 m length and 1m2 cross-sectionÐgenerally contracted to ohm-metres). The expression above, though widely applicable, is true only on the assumption that the current is uniformly distributed over the cross-section of the conductor and flows in paths parallel to the boundary walls. If this assumption is inadmissible, it is necessary to resort to integration or the use of current-flow lines. Figure 2.5 summarises the expressions for the resistance of certain arrangements and shapes of conductors. Resistivity The resistivity of conductors depends on their composition, physical condition (e.g. dampness in the case of non-metals), alloying, manufacturing and heat treat- ment, chemical purity, mechanical working and ageing. The resistance-temperature coefficient describes the rate of change of resistivity with temperature. It is practically 0.004 /C at 20C for copper. Most pure metals have a resistivity that rises with temperature. Some alloys have a very small coefficient. Carbon is notable in that its resistiv- ity decreases markedly with temperature rise, while uranium dioxide has a resistivity which falls in the ratio 50:1 over a range of a few hundred degrees. Table 2.2 lists the resistivity  and the resistance-temperature coefficient for a number of representative materials. The effect of temperature is assessed in accordance with the expressions R1 ˆ( R0…l ‡( 1†; R2 =R1 ˆ …1 ‡( 2†=…1 ‡( 1†; or R2 ˆ( R1‰1 ‡( …2 �( 1†Š( where R0, R1 and R2 are the resistances at temperatures 0, 1 and 2, and is the resistance-temperature coefficient at 0C. 2.2.1.4 Liquid conductors The variations of resistance of a given aqueous solution of an electrolyte with temperature follow the approximate rule: R ˆ( R0 =…1 ‡( 0:03†( where  is the temperature in degrees Celsius. The conduc- tivity (or reciprocal of resistivity) varies widely with the per- centage strength of the solution. For low concentrations the variation is that given in Table 2.2. 2.2.1.5 Frequency effects The resistance of a given conductor is affected by the frequency of the current carried by it. The simplest example is that of an isolated wire of circular cross-section. The inductance of the central parts of the conductor is greater than that of the outside skin because of the additional flux linkages due to the internal magnetic flux lines. The impedance of the central parts is consequently greater, and the current flows mainly at and near the surface of the conductor, where the impedance is least. The useful cross-section of the conductor is less than the actual area, and the effective resistance is consequently higher. This is called the skin effect. An analogous phenom- enon, the proximity effect, is due to mutual inductance between conductors arranged closely parallel to one another. Figure 2.4 Current±voltage characteristics Figure 2.5 Resistance in particular cases //integras/b&h/Eer/Final_06-09-02/eerc002 2/8 Electrotechnology Table 2.2 Conductivity of aqueous solutions* (mS/cm) Concentration (%) a b c d e f g h, j k l, m 1 4 0 18 12 10 10 8 6 4 3 3 2 72 35 23 20 20 16 12 8 6 6 3 102 51 34 30 30 24 18 12 9 8 4 130 65 44 39 39 32 23 16 11 10 5 79 55 4 8 4 7 39 28 20 13 11 7.5 110 79 69 67 54 39 29 18 16 10 99 90 85 69 31 22 20 *(a) NaOH, caustic soda; (b) NH4Cl, sal ammoniac; (c) NaCl, common salt; (d) NaNO3, Chilean saltpetre; (e) CaCl2, calcium chloride; (f) ZnCl2, zinc chloride; (g) NaHCO3, baking soda; (h) Na2CO3, soda ash; (j) Na2SO4, Glauber's salt; (k) Al2(SO4)3K2SO4, alum; (l) CuSO4, blue vitriol; (m) ZnSO4, white vitriol. The effects depend on conductor size, frequency f of the current, resistivity  and permeability  of the material. For a circular conductor of diameter d the increase of effective resistance is proportional to d2f/. At power frequencies and for small conductors the effect is negligible. It may, however, be necessary to investigate the skin and proximity effects in the case of large conductors such as bus-bars. 2.2.1.6 Non-linear resistors Prominent among non-linear resistors are electric arcs; also silicon carbide and similar materials. Arcs An electric arc constitutes a conductor of somewhat vague dimensions utilising electronic and ionic conduction in a gas. It is strongly affected by physical conditions of tempera- ture, gas pressure and cooling. In air at normal pressure a d.c. arc between copper electrodes has the voltage±current rela- tion given approximately by V ˆ 30 ‡ 10/I ‡ l [1 ‡ 3/I]103 for a current I in an arc length l metres. The expression is roughly equivalent to 10 V/cm for large currents and high voltages. The current density varies between 1 and 1000 A/mm2, being greater for large currents because of the pinch effect. See also Section 2.7. Silicon carbide Conducting pieces of this material have a current±voltage relation expressed approximately by I ˆKVx , where x is usually between 3 and 5. For rising voltage the current increases very rapidly, making silicon carbide devices suitable for circuit protection and the dis- charge of excess transmission-line surge energy. 2.2.2 Heating and cooling The heating of any body such as a resistor or a conducting circuit having inherent resistance is a function of the losses within it that are developed as heat. (This includes core and dielectric as well as ohmic I2R losses, but the effective value of R may be extended to cover such additional losses.) The cooling is a function of the facilities for heat dissipation to outside media such as air, oil or solids, by radiation, con- duction and convection. 2.2.2.1 Rapid heating If the time of heating is short, the cooling may be ignored, the temperature reached being dependent only on the rate of development of heat and the thermal capacity. If p is the heat development per second in joules (i.e. the power in watts), G the mass of the heated body in kilograms and c its specific heat in joules per kilogram per kelvin, then …# Gc
d ˆ p
dt; giving  ˆ …1=Gc†( p
dt For steady heating, the temperature rise is p/Gc in Kelvin per second. Standard annealed copper is frequently used for the wind- ings and connections of electrical equipment. Its density is G ˆ 8900 kg/m3 and its resistivity at 20C is 0.017 m -m; at 75C it is 0.021 m -m. A conductor worked at a current density J (in amperes per square metre) has a specific loss (watts per kilogram) of J2/8900. If J ˆ 2.75 MA/m2 (or 2.75 A/mm2), the specific loss at 75C is 17.8 W/ kg, and its rate of self-heating is 17.8/375 ˆ 0.048C/s. 2.2.2.2 Continuous heating Under prolonged steady heating a body will reach a tem- perature rise above the ambient medium of m ˆ p/A, where A is the cooling surface area and  the specific heat dissipation (joules per second per square metre of surface per degree Celsius temperature rise above ambient). The expression is based on the assumption, roughly true for moderate temperature rises, that the rate of heat emission is proportional to the temperature rise. The specific heat dissipation  is compounded of the effects of radiation, conduction and convection. Radiation The heat radiated by a surface depends on the absolute temperature T (given by T ˆ ‡ 273, where  is the Celsius temperature), and on its character (surface smooth- ness or roughness, colour, etc.). The Stefan law of heat radiation is pr ˆ 5:7eT 4  10�8 watts per square metre where e is the coefficient of radiant emission, always less than unity, except for the perfect `black body' surface, for which e ˆ 1. The radiation from a body is independent of the temperature of the medium in which it is situated. The process of radiation of a body to an exterior surface is accompanied by a re-absorption of part of the energy when re-radiated by that surface. For a small spherical radiating body inside a large and/or black spherical cavity, the radiated power is given by the Stefan-Boltzmann law: pr ˆ 5:7e1 ‰T4 � T 4 Š10�8 watts per square metre1 2 where T1 and e1 refer to the body and T2 to the cavity. The emission of radiant heat from a perfect black body surface is independent of the roughness or corrugation of the surface. If e < 1, however, there is some increase of radiation if the surface is rough. //integras/b&h/Eer/Final_06-09-02/eerc002 ‡( �( ‡( �( ‡( �( ‡( �( Electrochemical effects 2/11 Table 2.4 Electrochemical equivalents z (mg/C) circuit e.m.f. at 20C is about 1.018 30 V, and the e.m.f./ temperature coefficient is of the order of � 0.04mV/C. Element Valency z Element Valency z H 1 0.010 45 Zn Li 1 0.071 92 As Be 2 0.046 74 Se O 2 0.082 90 Br F 1 0.196 89 Sr Na 1 0.238 31 Pd Mg 2 0.126 01 Ag Al 3 0.093 16 Cd Si 4 0.072 69 Sn S 2 0.166 11 Sn S 4 0.083 06 Sb S 6 0.055 37 Te Cl 1 0.367 43 I K 1 0.405 14 Cs Ca 2 0.207 67 Ba Ti 4 0.124 09 Ce V 5 0.105 60 Ta Cr 3 0.179 65 W Cr 6 0.089 83 Pt Mn 2 0.284 61 Au Fe 1 0.578 65 Au Fe 2 0.289 33 Hg Fe 3 0.192 88 Hg Co 2 0.305 39 Tl Ni 2 0.304 09 Pb Cu 1 0.658 76 Bi Cu 2 0.329 38 Th 2 0.338 76 2.3.2.3 Secondary cells 3 0.258 76 In the lead±acid storage cell or accumulator, lead peroxide 4 0.204 56 reacts with sulphuric acid to produce a positive charge at 1 0.828 15 the anode. At the cathode metallic lead reacts with the acid 2 0.454 04 to produce a negative charge. The lead at both electrodes 2 combines with the sulphate ions to produce the poorly 4 0.276 4 1 1.117 93 2 0.582 44 2 0.615 03 4 0.307 51 3 0.420 59 4 0.330 60 1 1.315 23 1 1.377 31 2 0.711 71 3 0.484 04 5 0.374 88 6 0.317 65 4 0.505 78 1 2.043 52 3 0.681 17 1 2.078 86 2 1.039 42 1 2.118 03 2 1.073 63 3 0.721 93 4 0.601 35 soluble lead sulphate. The action is described as Charged PbO2 ‡( 2H2SO4 ‡( Pb Discharged Brown Strong acid Grey ˆ( PbSO4 ‡( 2H2O ‡( PbSO4 Sulphurate Weak acid Sulphate Both electrode reactions are reversible, so that the initial con- ditions may be restored by means of a `charging current'. In the alkaline cell, nickel hydrate replaces lead peroxide at the anode, and either iron or cadmium replaces lead at the cathode. The electrolyte is potassium hydroxide. The reactions are complex, but the following gives a general indication: Charged Discharged 2Ni…OH†3 ‡KOH ‡ Fe ˆ( 2Ni OH †2 ‡KOH ‡( Fe…OH†2…( or 2Ni…OH†3 ‡KOH ‡Cd ˆ( 2Ni…OH†3 ‡KOH ‡Cd…OH†2 2.3.2.4 Fuel cell Whereas a storage battery cell contains all the substances in the electrochemical oxidation±reduction reactions involved and has, therefore, a limited capacity, a fuel cell is supplied with its reactants externally and operates continuously as long as it is supplied with fuel. A practical fuel cell for direct conversion into electrical energy is the hydrogen±oxygen cell (Figure 2.8). Microporous electrodes serve to bring the gases into intimate contact with the electrolyte (potassium hydroxide) and to provide the cell terminals. The hydrogen and oxygen reactants are fed continuously into the cell from externally, and electrical energy is available on demand. At the fuel (H2) electrode, H2 molecules split into hydrogen atoms in the presence of a catalyst, and these combine with OH�( ions from the electrolyte, forming H2O and releasing electrons e. At the oxygen electrode, the oxygen molecules (O2) combine (also in the presence of a catalyst) with water molecules from the electrolyte and with pairs of electrons arriving at the electrode through the external load from the fuel electrode. Perhydroxyl ions (O2H � ) and hydroxyl ions (OH� ) are produced: the latter enter the electrolyte, while the more resistant O2H �( ions, with special catalysts, can be Figure 2.8 Fuel cell 2.3.2 Cells 2.3.2.1 Primary cells An elementary cell comprising electrodes of copper (positive) and zinc (negative) in sulphuric acid develops a p.d. between copper and zinc. If a circuit is completed between the electro- des, a current will flow, which acts in the electrolyte to decom- pose the acid, and causes a production of hydrogen gas round the copper, setting up an e.m.f. of polarisation in opposition to the original cell e.m.f. The latter therefore falls considerably. In practical primary cells the effect is avoided by the use of a depolariser. The most widely used primary cell is the Leclanche . It comprises a zinc and a carbon electrode in a solution of ammonium chloride, NH4Cl. When current flows, zinc chloride, ZnCl, is formed, releasing electrical energy. The NH4 positive ions travel to the carbon electrode (positive), which is packed in a mixture of manganese dioxide and carbon as depolariser. The NH4 ions are split up into NH3 (ammonia gas) and H, which is oxidised by the MnO2 to become water. The removal of the hydrogen prevents polarisation, provided that the current taken from the cell is small and intermittent. The wet form of Leclanche cell is not portable. The dry cell has a paste electrolyte and is suitable for continuous moder- ate discharge rates. It is exhausted by use or by ageing and drying up of the paste. The `shelf life' is limited. The inert cell is very similar in construction to the dry cell, but is assembled in the dry state, and is activated when required by moistening the active materials. In each case the cell e.m.f. is about 1.5 V. 2.3.2.2 Standard cell The Weston normal cell has a positive element of mercury, a negative element of cadmium, and an electrolyte of cadmium sulphate with mercurous sulphate as depolariser. The open- //integras/b&h/Eer/Final_06-09-02/eerc002 2/12 Electrotechnology reduced to OH� ions and oxygen. The overall process can be summarised as: Fuel electrode H2 ‡ 2OH� ˆ 2OH2O ‡ 2e 1Oxygen electrode 2 O2 ‡H2O ‡ 2e ˆ 2OH�( 1Net reaction H2 ‡ 2 O2 ! 2e flow ! H2O In a complete reaction 2 kg hydrogen and 16 kg oxygen combine chemically (not explosively) to form 18 kg of water with the release of 400 MJ of electrical energy. For each kiloamp/hour the cell produces 0.33 1 of water, which must not be allowed unduly to weaken the electrolyte. The open-circuit e.m.f. is 1.1 V, while the terminal voltage is about 0.9 V, with a delivery of 1 kA/m2 of plate area. 2.4 Magnetic field effects The space surrounding permanent magnets and electric cir- cuits carrying currents attains a peculiar state in which a number of phenomena occur. The state is described by saying that the space is threaded by a magnetic field of flux. The field is mapped by an arrangement of lines of induction giving the strength and direction of the flux. Figure 2.9 gives a rough indication of the flux pattern for three simple cases of magnetic field due to a current. The diagrams show the conventions of polarity, direction of flux and direction of current adopted. Magnetic lines of induction form closed loops in a magnetic circuit linked by the circuit current wholly or in part. 2.4.1 Magnetic circuit By analogy with the electric circuit, the magnetic flux pro- duced by a given current in a magnetic circuit is found from the magnetomotive force (m.m.f.) and the circuit reluctance. The m.m.f. produced by a coil of N turns carrying a current I is F ˆNI ampere-turns. This is expended over any closed path linking the current I. At a given point in a magnetic field in free space the m.m.f. per unit length or magnetising force H gives rise to a magnetic flux density B0 ˆ0H, where 0 ˆ 4/107. If the medium in which the field exists has a relative permeability r, the flux density established is B ˆ rB0 ˆ r0H ˆ H The summation of H
dl round any path linking an N-turn circuit carrying current I is the total m.m.f. F. If the distribu- tion of H is known, the magnetic flux density B or B0 can be Figure 2.9 Magnetic fields Figure 2.10 Magnetic circuits found for all points in the field, and a knowledge of the area a of the magnetic path gives F ˆBa, the total magnetic flux. Only in a few cases of great geometrical simplicity can the flux due to a given system of currents be found precisely. Among these are the following. Long straight isolated wire (Figure 2.10(a)): This is not strictly a realisable case, but the results are useful. Assume a current of 1 A. The m.m.f. around any closed linking path is therefore 1 A-t. Experiment shows that the magnetic field is symmetrical about, and concentric with, the axis of the wire. Around a closed path of radius x metres there will be a uniform distribution of m.m.f. so that H ˆ F =2x ˆ 1=2x…A-t=m†( Consequently, in free space the flux density (T ) at radius x is B0 ˆ 0H ˆ 0=2x In a medium of constant permeability ˆr0 the flux density is B ˆrB0. There will be magnetic flux following closed circular paths within the cross-section of the wire itself: at any radius x the m.m.f. is F ˆ (x/r)2 because the circular path links only that part of the (uniformly distribu- ted) current within the path. The magnetising force is H ˆF/2x ˆ x/2r 2 and the corresponding flux density in a non-magnetic conductor is B0 ˆ 0H ˆ 0x=2r 2 and r times as much if the conductor material has a relative permeability r. The expressions above are for a conductor current of 1 A. Concentric conductors (Figure 2.10(b)): Here only the inner conductor contributes the magnetic flux in the space between the conductors and in itself, because all such flux can link only the inner current. The flux distribution is found exactly as in the previous case, but can now be summed in defined limits. If the outer conductor is suffi- ciently thin radially, the flux in the interconductor space, per metre axial length of the system, is …#R 0 0 R F ˆ( dx ˆ( ln 2x 2 rr Toroid (Figure 2.10(c)): This represents the closest approach to a perfectly symmetrical magnetic circuit, in which the m.m.f. is distributed evenly round the magnetic path and the m.m.f. per metre H corresponds at all points exactly to the flux density existing at those points. The magnetic flux is therefore wholly confined to the path. Let the mean radius of //integras/b&h/Eer/Final_06-09-02/eerc002 Magnetic field effects 2/13 the toroid be R and its cross-sectional area be A. Then, with N uniformly distributed turns carrying a current I and a toroid core of permeability , F ˆ( NI ; H ˆ F =2R; B ˆ H; F ˆ( FA=2R This applies approximately to a long solenoid of length l, replacing R by 1/2. The permeability will usually be 0. Composite magnetic circuit containing iron (Figure 2.10(d)): For simplicity practical composite magnetic circuits are arbitrarily divided into parts along which the flux density is deemed constant. For each part F ˆ( Hl ˆ Bl= ˆ BlA=A ˆ( FS where S ˆ l/A is the reluctance. Its reciprocal  ˆ 1/S ˆA/l is the permeance. The expression F ˆFS resembles E ˆ IR for a simple d.c. circuit and is therefore sometimes called the magnetic Ohm's law. The total excitation for the magnetic circuit is F ˆ( H1l1 ‡H2l2 ‡H3l3 ‡


( for a series of parts of length l1, l2 . . . , along which magnetic field intensities of H1, H2 . . . (A-t/m) are necessary. For free space, air and non-magnetic materials, r ˆ 1 and B0 ˆ0H, so that H ˆB0/0 ^ 800 000 B0. This means that an excita- tion F ˆ 800 000 A-t is required to establish unit magnetic flux density (1 T) over a length l ˆ 1 m. For ferromagnetic materials it is usual to employ B±H graphs (magnetisation curves) for the determination of the excitation required, because such materials exhibit a saturation phenomenon. Typical B±H curves are given in Figures 2.11 and 2.12. 2.4.1.1 Permeability  Certain diamagnetic materials have a relative permeability slightly less than that of vacuum. Thus, bismuth has r ˆ 0.9999. Other materials have r slightly greater than unity: these are called paramagnetic. Iron, nickel, cobalt, steels, Heusler alloy (61% Cu, 27% Mn, 13% Al) and a number of others of great metallurgical interest have ferro- magnetic properties, in which the flux density is not directly proportional to the magnetising force but which under suit- able conditions are strongly magnetic. The more usual con- structional materials employed in the magnetic circuits of electrical machinery may have peak permeabilities in the neighbourhood of 5000±10 000. A group of nickel±iron alloys, including mumetal (73% Ni, 22% Fe, 5% Cu), Figure 2.12 Magnetisation and permeability curves permalloy `C' (77.4% Ni, 13.3% Fe, 3.7% Mo, 5% Cu) and hypernik (50% Ni, 50% Fe), show much higher permeabil- ities at low densities (Figure 2.12). Permeabilities depend on exact chemical composition, heat treatment, mechanical stress and temperature conditions, as well as on the flux density. Values of r exceeding 5  105 can be achieved. 2.4.1.2 Core losses A ferromagnetic core subjected to cycles of magnetisation, whether alternating (reversing), rotating or pulsating, exhibits hysteresis. Figure 2.13 shows the cycle B±H relation typical of this phenomenon. The significant quantities remanent flux density and coercive force are also shown. The area of the hysteresis loop figure is a measure of the energy loss in the cycle per unit volume of material. An empirical expression for the hysteresis loss in a core taken through f cycles of magnetisation per second is ph ˆ( kh fBx watts per unit mass or volume m Here Bm is the maximum induction reached and kh is the hysteretic constant depending on the molecular quality and structure of the core metal. The exponent x may lie between 1.5 and 2.3. It is often taken as 2. A further cause of loss in the same circumstances is the eddy current loss, due to the I2R losses of induced currents. It can be shown to be pe ˆ ket2F 2B2 watts per unit mass or volume the constant ke depending on the resistivity of the metal and t being its thickness, the material being laminated to decrease the induced e.m.f. per lamina and to increase the resistance of the path in which the eddy currents flow. In practice, curves of loss per kilogram or per cubic metre for various flux densities are employed, the curves being constructed from the results of Figure 2.11 Magnetisation curves Figure 2.13 Hysteresis //integras/b&h/Eer/Final_06-09-02/eerc002 2/16 Electrotechnology Flux change: This law has the basic form e ˆ �N…dF=dt†( and is applicable where a circuit of constant shape links a changing magnetic flux. Flux cutting: Where a conductor of length l moves at speed u at right angles to a uniform magnetic field of density B, the e.m.f. induced in the conductor is e ˆ Blu This can be applied to the motion of conductors in constant magnetic fields and when sliding contacts are involved. Linkage change: Where coils move in changing fluxes, and both flux-pulsation and flux-cutting processes occur, the general expression. e ˆ �d =dt must be used, with variation of the linkage expressed as the result of both processes. 2.4.3.4 Constant linkage principle The linkage of a closed circuit cannot be changed instant- aneously, because this would imply an instantaneous change of associated magnetic energy, i.e. the momentary appear- ance of infinite power. It can be shown that the linkage of a closed circuit of zero resistance and no internal source can- not be changed at all. The latter concept is embodied in the following theorem. Constant linkage theorem The linkage of a closed passive circuit of zero resistance is a constant. External attempts to change the linkage are opposed by induced currents that effectively prevent any net change of linkage. The theorem is very helpful in dealing with transients in highly inductive circuits such as those of transformers, synchronous generators, etc. 2.4.3.5 Ideal transformer An ideal transformer comprises two resistanceless coils embracing a common magnetic circuit of infinite permeabil- ity and zero core loss (Figure 2.18). The coils produce no leakage flux: i.e. the whole flux of the magnetic circuit com- pletely links both coils. When the primary coil is energised by an alternating voltage V1, a corresponding flux of peak value Fm is developed, inducing in the N1-turn primary coil an e.m.f. E1 ˆV1. At the same time an e.m.f. E2 is induced in the N2-turn secondary coil. If the terminals of the second- ary coil are connected to a load taking a current I2, the primary coil must accept a balancing current I1 such that I1N1 ˆ I2N2, as the core requires zero excitation. The operating conditions are therefore N1=N2 ˆ E1=E2 ˆ I2=I1; and E1I1 ˆ E2 I2 The secondary load impedance Z2 ˆ E2/I2 is reflected into the primary to give the impedance Z1 ˆ E1/I1 such that Figure 2.18 Ideal transformer Z1 ˆ …N1=N2†2Z2 A practical power transformer differs from the ideal in that its core is not infinitely permeable and demands an excita- tion N1I0 ˆN1I1 �N2I2; the primary and secondary coils have both resistance and magnetic leakage; and core losses occur. By treating these effects separately, a practical trans- former may be considered as an ideal transformer con- nected into an external network to account for the defects. 2.4.3.6 Electromagnetic machines An electromagnetic machine links an electrical energy sys- tem to a mechanical one, by providing a reversible means of energy flow between them in the common or `mutual' mag- netic flux linking stator and rotor. Energy is stored in the field and released as work. A current-carrying conductor in the field is subjected to a mechanical force and, in moving, does work and generates a counter e.m.f. Thus the force± motion product is converted to or from the voltage±current product representing electrical power. The energy-rate balance equations relating the mechanical power pe, and the energy stored in the magnetic field wf, are: Motor: pe ˆ pm ˆ dwf =dt Generator: pm ˆ pe ‡ dwf =dt The mechanical power term must account for changes in stored kinetic energy, which occur whenever the speed of the machine and its coupled mechanical loads alter. Reluctance motors The force between magnetised surfaces (Figure 2.15(b)) can be applied to rotary machines (Figure 2.19(a)). The armature tends to align itself with the field axis, developing a reluctance torque. The principle is applied to miniature rotating-contact d.c. motors and synchronous clock motors. Machines with armature windings Consider a machine rotating with constant angular velocity !r and developing a torque M. The mechanical power is pm ˆM!r: the electrical power is pe ˆ ei, where e is the counter e.m.f. due to the reaction of the mutual magnetic field. Then ei ˆM!r ‡ dwf/dt at every instant. If the armature conductor a in Figure 2.19(b) is running in a non-time-varying flux of local density B, the e.m.f. is entirely rotational and equal to er ˆBlu ˆBl!rR. The tangential force on the conductor is f ˆBli and the torque is M ˆBliR. Thus, eri ˆM!r because dwf/dt ˆ 0. This case applies to constant flux (d.c., three-phase synchronous and induction) machines. If the armature in Figure 2.19(b) is given two conductors a and b they can be connected to form a turn. Provided the turn is of full pitch, the torques will always be additive. More turns in series form a winding. The total flux in the machine results from the m.m.f.s of all current-carrying conductors, whether on stator or rotor, but the torque arises from that component of the total flux at right angles to the m.m.f. axis of the armature winding. Armature windings (Figure 2.19(c)) may be of the commu- tator or phase (tapped) types. The former is closed on itself, and current is led into and out of the winding by fixed brushes which include between them a constant number of conductors in each armature current path. The armature m.m.f. coincides always with the brush axis. Phase windings have separate external connections. If the winding is on the rotor, its current and m.m.f. rotate with it and the external connections must be made through slip-rings. Two (or three) such windings with two-phase (or three-phase) currents can produce a resultant m.m.f. that rotates with respect to the windings. //integras/b&h/Eer/Final_06-09-02/eerc002 Figure 2.19 Electromagnetic machines Torque Figure 2.19(d) shows a commutator winding arranged for maximum torque: i.e. the m.m.f. axis of the winding is displaced electrically /2 from the field pole cen- tres. If the armature has a radius R and a core length l, the flux has a constant uniform density B, and there are Z con- ductors in the 2p pole pitches each carrying the current I, the torque is BRlIZ/2p. This applies to a d.c. machine. It also gives the mean torque of a single-phase commutator machine if B and I are r.m.s. values and the factor cos  is introduced for any time phase angle between them. The torque of a phase winding can be derived from Figure 2.19(e). The flux density is assumed to be distributed sinu- soidally, and reckoned from the pole centre to be Bm cos . The current in the phase winding produces the m.m.f. Fa, having an axis displaced by angle  from the pole centre. The total torque is then ˆ 1M ˆ BmFalR sin  2 FFa sin  per pole pair. This case applies directly to the three-phase synchronous and induction machines. Types of machine For unidirectional torque, the axes of the pole centres and armature m.m.f. must remain fixed relative to one another. Maximum torque is obtained if these axes are at right angles. The machine is technically better if the field flux and armature m.m.f. do not fluctuate with time (i.e. they are d.c. values): if they do alternate, it is preferable that they be co-phasal. Workable machines can be built with (1) concentrated (`field') or (2) phase windings on one member, with (A) commutator or (B) phase windings on the other. It is basic- ally immaterial which function is assigned to stator and which to rotor, but for practical convenience a commutator winding normally rotates. The list of chief types below gives the type of winding (1, 2, A, B) and current supply (d or a), with the stator first: D.C. machine, 1d/Ad: The arrangement is that of Figure 2.19(d). A commutator and brushes are necessary for the rotor. Single-phase commutator machine, 1a/Aa: The physical arrangement is the same as that of the d.c. machine. The field flux alternates, so that the rotor m.m.f. must also alter- nate at the same frequency and preferably in time phase. Series connection of stator and rotor gives this condition. Synchronous machine, Ba/1d: The rotor carries a concen- trated d.c. winding, so the rotor m.m.f. must rotate with it at corresponding (synchronous) speed, requiring a.c. (normally three-phase) supply. The machine may be inverted (1d/Ba). Magnetic field effects 2/17 Induction machine, 2a/Ba (Figure 2.19(e)): The polyphase stator winding produces a rotating field of angular velocity !1. The rotor runs with a slip s, i.e. at a speed !1(1 � s). The torque is maintained unidirectional by currents induced in the rotor winding at frequency s!1. With d.c. supplied to the rotor (2a/Bd) the rotor m.m.f. is fixed relatively to the windings and unidirectional torque is maintained only at synchronous speed (s ˆ 0). All electromagnetic machines are variants of the above. 2.4.3.7 Magnetohydrodynamic generator Magnetohydrodynamics (m.h.d.) concerns the interaction between a conducting fluid in motion and a magnetic field. If a fast-moving gas at high temperature (and therefore ionised) passes across a magnetic field, an electric field is developed across the gaseous stream exactly as if it were a metallic conductor, in accordance with Faraday's law. The electric field gives rise to a p.d. between electrodes flanking the stream, and a current may be made to flow in an external circuit connected to the electrodes. The m.h.d. generator offers a direct conversion between heat and electrical energy. 2.4.3.8 Hall effect If a flat conductor carrying a current I is placed in a magnetic field of density B in a direction normal to it (Figure 2.20), then an electric field is set up across the width of the conduc- tor. This is the Hall effect, the generation of an e.m.f. by the movement of conduction electrons through the magnetic field. The Hall e.m.f. (normally a few microvolts) is picked off by tappings applied to the conductor edges, for the meas- urement of I or for indication of high-frequency powers. 2.4.4 Inductance The e.m.f. induced in an electric circuit by change of flux linkage may be the result of changing the circuit's own cur- rent. A magnetic field always links a current-carrying circuit, and the linkage is (under certain restrictions) proportional to the current. When the current changes, the linkage also changes and an e.m.f. called the e.m.f. of self-induction is induced. If the linkage due to a current i in the circuit is ˆFN ˆLi, the e.m.f. induced by a change of current is e ˆ �d =dt† ˆ �N…dF=dt† ˆ �L…di=dt†( L is a coefficient giving the linkage per ampere: it is called the coefficient of self-induction, or, more usually, the induct- ance. The unit is the henry, and in consequence of its rela- tion to linkage, induced e.m.f., and stored magnetic energy, it can be defined as follows. A circuit has unit inductance (1 H) if: (a) the energy stored in the associated magnetic field is 1 2 J when the current is 1 A; (b) the induced e.m.f. is 1 V when the current is changed at Figure 2.20 Hall effect //integras/b&h/Eer/Final_06-09-02/eerc002 2/18 Electrotechnology the rate 1 A/s; or (c) the flux linkage is 1 Wb-t when the current is 1 A. 2.4.4.1 Voltage applied to an inductor I Let an inductor (i.e. an inductive coil or circuit) devoid of resistance and capacitance be connected to a supply of con- stant potential difference V, and let the inductance be L. By definition (b) above, a current will be initiated, growing at such a rate that the e.m.f. induced will counterbalance the applied voltage V. The current must rise uniformly at V/L amperes per second, as shown in Figure 2.21(a), so long as the applied p.d. is maintained. Simultaneously the circuit develops a growing linked flux and stores a growing amount of magnetic energy. After a time t1 the current reaches 1 ˆ (V/L)t1, and has absorbed a store of energy at voltage V and average current I1/2, i.e. ˆ 1 VI1t1 ˆ 1 LI2 joules2 W1 ˆ V
1 I1
t1 2 2 1 since V ˆ I1L/t1. If now the supply is removed but the circuit remains closed, there is no way of converting the stored energy, which remains constant. The current therefore continues to circulate indefinitely at value I1. Suppose that V is applied for a time t1, then reversed for an equal time interval, and so on, repeatedly. The resulting current is shown in Figure 2.21(b). During the first period t1 the current rises uniformly to I1 ˆ (V/L)t1 and the stored 2energy is then 1LI1 . On reversing the applied voltage the 2 current performs the same rate of change, but negatively so as to reduce the current magnitude. After t1 it is zero and so is the stored energy, which has all been returned to the sup- ply from which it came. If the applied voltage is sinusoidal and alternates at fre- quency f, such that v ˆ vm cos 2ft ˆ vm cos !t, and is switched on at instant t ˆ 0 when v ˆ vm, the current begins to rise at rate vm/L (Figure 2.21(c)); but the immediate reduc- tion and subsequent reversal of the applied voltage require corresponding changes in the rate of rise or fall of the cur- rent. As v ˆL(di/dt) at every instant, the current is therefore …# i ˆ( v dt ˆ( vm sin !t L !L The peak current reached is im ˆ vm/!L and the r.m.s. current is I ˆV/!L ˆV/XL, where XL ˆ!L ˆ 2fL is the inductive reactance. Should the applied voltage be switched on at a voltage zero (Figure 2.21(d)), the application of the same argument results in a sine-shaped current, unidirectional but pulsat- ing, reaching the peak value 2im ˆ 2vm/!L, or twice that in the symmetrical case above. This is termed the doubling effect. Compare with Figure 2.21(b). 2.4.4.2 Calculation of inductance To calculate inductance in a given case (a problem capable of reasonably exact solution only in cases of considerable geometrical simplicity), the approach is from the standpoint of definition (c). The calculation involves estimating the magnetic field produced by a current of 1 A, summing the linkage FN produced by this field with the circuit, and writ- ing the inductance as L ˆFN. The cases illustrated in Figure 2.10 and 2.22 give the following results. Long straight isolated conductor (Figure 2.10(a)) The mag- netising force in a circular path concentric with the conduc- tor and of radius x is H ˆF/2x ˆ 1/2x; this gives rise to a circuital flux density B0 ˆ0H ˆ0/2x. Summing the link- age from the radius r of the conductor to a distance s gives ˆ …0=2† ln…s=r†( weber-turn per metre of conductor length. If s is infinite, so is the linkage and therefore the inductance: but in practice it is not possible so to isolate the conductor. There is a magnetic flux following closed circular paths within the conductor, the density being Bi ˆx/2r 2 at radius x. The effective linkage is the product of the flux by that proportion of the conductor actually enclosed, giving /8 per metre length. It follows that the internal linkage produces a contribution Li ˆ/8 henry/metre, regardless of the conductor diameter on the assumption that the cur- rent is uniformly distributed. The absolute permeability  of the conductor material has a considerable effect on the internal inductance. Concentric cylindrical conductors (Figure 2.10(b)) The inductance of a metre length of concentric cable carrying equal currents oppositely directed in the two parts is due to the flux in the space between the central and the tubular conductor set up by the inner current alone, since the cur- rent in the outer conductor cannot set up internal flux. Summing the linkages and adding the internal linkage of the inner conductor: L ˆ …=8† ‡ …0 =2† ln…R=r† henry=metre Parallel conductors (Figure 2.22) Between two conductors (a) carrying the same current in opposite directions, the link- age is found by summing the flux produced by conductor Figure 2.21 Voltage applied to a pure inductor Figure 2.22 Parallel conductors //integras/b&h/Eer/Final_06-09-02/eerc002 Electric field effects 2/21 electric flux is 1 C. From the field pattern the electric flux density D at any point is found. Then the electric field strength at the point is E ˆD/, where ˆ r0 is the absolute permittivity of the insulating medium in which the electric flux is established. Integration of E over any path from one electrode to the other gives the p.d. V, whence the capacitance is C ˆ 1/V. Parallel plates (Figure 2.27(a)) The electric flux density is uniform except near the edges. By use of a guard ring main- tained at the potential of the plate that it surrounds, the capacitance of the inner part is calculable on the reasonable assumption of uniform field conditions. With a charge of 1 C on each plate, and plates of area S spaced a apart, the electric flux density is D ˆ 1/S, the electric field intensity is E ˆD/ˆ 1/S, the potential difference is V ˆEa ˆ a/S, and the capacitance is therefore C ˆ q=V ˆ …S=a†( A case of interest is that of a parallel plate arrangement (Figure 2.27(b)), with two dielectric materials, of thickness a1 and a2 and absolute permittivity 1 and 2, respectively. The voltage gradient is inversely proportional to the permittivity, so that E11 ˆE22. The field pattern makes it evident that the difference in polarisation produces an interface charge, but in terms of the charge qc on the plates themselves the electric flux density is constant throughout. The total voltage between the plates is V ˆV1 ‡( V2 ˆE1a1 ‡E2a2, from which the total capacitance can be obtained. Concentric cylinders (Figure 2.27(c)) With a charge of 1 C per metre length, the electric flux density at radius x is 1/2x, whence Ex ˆ 1/2x. Integrating for the p.d. gives V ˆ …1=2† ln…R=r†( The capacitance is consequently C ˆ 2= ln…R=r† farad=metre The electric field strength (voltage gradient) E is inversely proportional to the radius, over which it is distributed hyperbolically. The maximum gradient occurs at the surface of the inner conductor and amounts to Em ˆ V=r ln…R=r†( At any other radius x, Ex ˆEm(r/x). For a given p.d. V and gradient Em there is one value of r to give minimum overall radius R: this is r ˆ V=Em and R ˆ 2:72r For the cylindrical capacitor (d ) with two dielectrics, of permittivity 1 between radii r and  and 2 between  and R, the maximum gradients are related by Em11r ˆEm22. Parallel cylinders (Figure 2.27(e)) The calculation leads to the value C ˆ = ln…a=r† farad=metre for the capacitance between the conductors, provided that a 4 r. It can be considered as composed of two series-con- nected capacitors each of C0 ˆ 2= ln…a=r† farad=metre C0 being the line-to-neutral capacitance. A three-phase line has a line-to-neutral capacitance identical with C0, the inter- pretation of the spacing a for transposed asymmetrical lines being the same as for their inductance. The voltage gradient of a two-wire line is shown in Figure 2.27(e). If a 4 r, the gradient in the immediate vicinity of a wire may be taken as due to the charge thereon, the further wire having little effect: consequently, Em ˆ V=r ln…a=r†( is the voltage gradient at a conductor surface. 2.5.2.3 Connection of capacitors If a bank of capacitors of capacitance C1, C2, C3 . . . , be connected in parallel and raised in combination each to the p.d. V, the total charge is the sum of the individual charges VC1, VC2, VC3 . . . , whence the total combined capacitance is C ˆ C1 ‡ C2 ‡ C3 ‡


( With a series connection, the same displacement current occurs in each capacitor and the p.d. V across the series assembly is the sum of the individual p.d.s: V ˆ V1 ‡ V2 ‡ V3 ‡


( ˆ q‰…1=C1† ‡ …1=C2† ‡ …1=C3† ‡


Š ( ˆ q=C so that the combined capacitance is obtained from C ˆ 1=‰…1=C1† ‡ …1=C2† ‡ …1=C3 † ‡


Š ( 2.5.2.4 Voltage applied to a capacitor The basis for determining the conditions in a circuit con- taining a capacitor to which a voltage is applied is that the p.d. v across the capacitor is related definitely by its capaci- tance C to the charge q displaced on its plates: q ˆCv. Let a direct voltage V be suddenly applied to a circuit devoid of all characteristic parameters except that of cap- acitance C. At the instant of its application, the capacitor must accept a charge q ˆCv, resulting in an infinitely large current flowing for a vanishingly short time. The energy stored is W ˆ 1 2 Vq ˆ 1 CV2 joules. If the voltage is raised or 2 lowered uniformly, the charge must correspondingly change, by a constant charging or discharging current flow- ing during the change (Figure 2.28(a)). Figure 2.27 Capacitance and voltage gradient Figure 2.28 Voltage applied to a pure capacitor //integras/b&h/Eer/Final_06-09-02/eerc002 2/22 Electrotechnology q If the applied voltage is sinusoidal, as in (b), such that v ˆ vm cos 2ft ˆ vm cos !t, the same argument leads to the requirement that the charge is q ˆ qm cos !t, where m ˆCvm. Then the current is i ˆ dq/dt, i.e. i ˆ �!Cvm sin !t with a peak im ˆ!Cvm and an r.m.s. value I ˆ!CV ˆV/Xc, where Xc ˆ 1/!C is the capacitive reactance. 2.5.3 Dielectric breakdown A dielectric material must possess: (a) a high insulation resistivity to avoid leakage conduction, which dissipates the capacitor energy in heat; (b) a permittivity suitable for the purposeÐhigh for capacitors and low for insulation generally; and (c) a high electric strength to withstand large voltage gradients, so that only thin material is required. It is rarely possible to secure optimum properties in one and the same material. A practical dielectric will break down (i.e. fail to insulate) when the voltage gradient exceeds the value that the mater- ial can withstand. The breakdown mechanism is complex. 2.5.3.1 Gases With gaseous dielectrics (e.g. air and hydrogen), ions are always present, on account of light, heat, sparking, etc. These are set in motion, making additional ionisation, which may be cumulative, causing glow discharge, sparking or arcing unless the field strength is below a critical value. A field strength of the order of 3 MV/m is a limiting value for gases at normal temperature and pressure. The dielectric strength increases with the gas pressure. The polarisation in gases is small, on account of the com- paratively large distances between molecules. Consequently, the relative permittivity is not very different from unity. 2.5.3.2 Liquids When very pure, liquids may behave like gases. Usually, however, impurities are present. A small proportion of the molecules forms positive or negative ions, and foreign par- ticles in suspension (fibres, dust, water, droplets) are prone to align themselves into semiconducting filaments: heating produces vapour, and gaseous breakdown may be initiated. Water, because of its exceptionally high permittivity, is especially deleterious in liquids such as oil. 2.5.3.3 Solids Solid dielectrics are rarely homogeneous, and are often hygroscopic. Local space charges may appear, producing absorption effects; filament conducting paths may be present; and local heating (with consequent deterioration) may occur. Breakdown depends on many factors, especially thermal ones, and is a function of the time of application of the p.d. 2.5.3.4 Conduction and absorption Solid dielectrics in particular, and to some degree liquids also, show conduction and absorption effects. Conduction appears to be mainly ionic in nature. Absorption is an apparent storing of charge within the dielectric. When a capacitor is charged, an initial quantity is displaced on its plates due to the geometric capacitance. If the p.d. is maintained, the charge gradually grows, owing to absorptive capacitance, probably a result of the slow orientation of permanent dipolar molecules. The current finally settles down to a small constant value, owing to conduction. Absorptive charge leaks out gradually when a capacitor is discharged, a phenomenon observable particularly in cables after a d.c. charge followed by momentary discharge. 2.5.3.5 Grading The electric fields set up when high voltages are applied to electrical insulators are accompanied by voltage gradients in various parts thereof. In many cases the gradients are any- thing but uniform: there is frequently some region where the field is intense, the voltage gradient severe and the dielectric stress high. Such regions may impose a controlling and limit- ing influence on the insulation design and on the working voltage. The process of securing improved dielectric operating conditions is called grading. The chief methods available are: (1) The avoidance of sharp corners in conductors, near which the gradient is always high. (2) The application of high-permittivity materials to those parts of the dielectric structure where the stress tends to be high, on the principle that the stress is inversely pro- portional to the permittivity: it is, of course, necessary to correlate the method with the dielectric strength of the material to be employed. (3) The use of intersheath conductors maintained at a suit- able intermediate potential so as to throw less stress on those parts which would otherwise be subjected to the more intense voltage gradients. Examples of (1) are commonly observed in high-voltage apparatus working in air, where large rounded conductors are employed and all edges are given a large radius. The application of (2) is restricted by the fact that the choice of materials in any given case is closely circumscribed by the mechanical, chemical and thermal properties necessary. Method (3) is employed in capacitor bushings, in which the intersheaths have potentials adjusted by correlation of their dimensions. 2.5.4 Electromechanical effects Figure 2.29 summarises the mechanical force effects observ- able in the electric field. In (a), (b) and (c) are sketched the field patterns for cases already mentioned in connection with the laws of electrostatics. The surface charges developed on high- materials are instrumental in producing the forces indicated in (d ). Finally, (e) shows the forces on pieces of dielectric material immersed in a gaseous or liquid insulator and subjected to a non-uniform electric field. The force Figure 2.29 Electromechanical forces //integras/b&h/Eer/Final_06-09-02/eerc002 Electromagnetic field effects 2/23 direction depends upon whether the piece has a higher or lower permittivity than the dielectric medium in which it lies. Thus, pieces of high permittivity are urged towards regions of higher electric field strength. 2.6 Electromagnetic field effects Electromagnetic field effects occur when electric charges undergo acceleration. The effects may be negligible if the rate of change of velocity is small (e.g. if the operating frequency is low), but other conditions are also significant, and in certain cases effects can be significant even at power frequencies. 2.6.1 Movement of charged particles Particles of small mass, such as electrons and protons, can be accelerated in vacuum to very high speeds. Static electric field The force developed on a particle of mass m carrying a positive charge q and lying in an electric field of intensity (or gradient) E is f ˆ qE in the direction of E, i.e. from a high-potential to a low-potential region (Figure 2.30(a)). (If the charge is negative, the direction of the force is reversed.) The acceleration of the particle is a ˆ f/m ˆE(q/m); and if it starts from rest its velocity after 2time t, is u ˆ at ˆE(q/m)t. The kinetic energy 1mu imparted2 is equal to the change of potential energy Vq, where V is the p.d. between the starting and finishing points in the electric field. Hence, the velocity attained from rest is u ˆ p‰2V…q=m†Š( For an electron (q ˆ� 1.6  10� 19 C, m0 ˆ 0.91  10� 30 kg) falling through a p.d. of 1 V the velocity is 600 km/s and the kinetic energy is w ˆVq ˆ 1.60  10� 19 J, often called an electron-volt, 1 eV. If V ˆ 2.5 kV, then u ˆ 30 000 km/s; but the speed cannot be indefinitely raised by increasing V, for as u approaches c ˆ 300 000 km/s, the free-space electromagnetic wave velo- city, the effective mass of the particle begins to acquire a rapid relativistic increase to m ˆ m0=‰1 � …u=c†2Š( compared with its `rest mass' m0. Static magnetic field A charge q moving at velocity u is a current i ˆ qu, and is therefore subject to a force if it moves across a magnetic field. The force is at right angles to u and to B, the magnetic flux density, and in the simple case of Figure 2.30(b) we have f ˆ quB ˆma ˆmu 2/R, the particle being constrained by the force to move in a circular path of radius R ˆ (u/B)(m/q). For an electron R ˆ 5.7  10� 22 (u/B). Combined electric and magnetic fields The two effects described above are superimposed. Thus, if the E and B fields are coaxial, the motion of the particle is helical. The influence of static (or quasi-static) fields on charged particles is applied in cathode ray oscilloscopes and accelerator machines. 2.6.2 Free space propagation In Section 1.5.3 the Maxwell equations are applied to pro- pagation of a plane electromagnetic wave in free space. It is shown that basic relations hold between the velocity u of propagation, the electric and magnetic field components E and H, and the electric and magnetic space constants 0 and 0. The relations are: u ˆ 1=p…00† ' 3  108 metre=second is the free space propagation velocity. The electric and magnetic properties of space impose a relation between E (in volts per metre) and H (in amps per metre) given by E=H ˆ p…0=0† ˆ 377 : called the intrinsic impedance of space. Furthermore, the energy densities of the electric and magnetic components are the same, i.e. 1 0 E 2 ˆ 1 0H2 2 2 Propagation in power engineering is not (at present) by space waves but by guided waves, a conducting system being used to direct the electromagnetic energy more effect- ively in a specified path. The field pattern is modified (although it is still substantially transverse), but the essen- tial physical propagation remains unchanged. Such a guide is called a transmission line, and the fields are normally specified in terms of the inductance and capacitance pro- perties of the line configuration, with an effective impedance z0 ˆ p…L=C† differing from 377 . 2.6.3 Transmission line propagation (see also Section 36) If the two wires of a long transmission line, originally dead, are suddenly connected to a supply of p.d. v, an energy wave advances along the line towards the further end at velocity u (Figure 2.31). The wave is characterised by the fact that the advance of the voltage charges the line capacitance, for which an advancing current is needed: and the advance of the current establishes a magnetic field against a counter- e.m.f., requiring the voltage for maintaining the advance. Thus, current and voltage are propagated simultaneously. Let losses be neglected, and L and C be the inductance and capacitance per unit length of line. In a brief time interval dt, the waves advance by a distance u
dt. The voltage is established across a capacitance Cu
dt and the rate of charge, or current, is i ˆ vCu
dt=dt ˆ vCu Figure 2.30 Motion of charged particles Figure 2.31 Transmission-line field //integras/b&h/Eer/Final_06-09-02/eerc002 2/26 Electrotechnology discharge. The discharge is maintained by the creation and movement of ions and electrons which in many discharges constitute a plasma (see Section 10.8). Such discharges may be produced between two electrodes which form part of an electrical network and across which a sufficient potential difference exists to ionise the insulation. Alternatively, discharges may be electromagnetically induced, for instance by strong radiofrequency fields. Electrical discharges may be characterised for electrical network applications in terms of the current and voltage values needed for their occurrence (Figure 2.34). 2.7.2 Types of discharge Discharges have historically been subdivided into two categor- ies namely self-sustaining and non-self-sustaining discharges. The transition between the two forms (which constitutes the electrical breakdown of the gas) is sudden and occurs through the formation of a spark. Non-self-sustaining discharges occur at relatively low currents (10�8 A) (region 0A, Figure 2.34) of which Townsend discharges are a particular type. The form of the current±voltage characteristic in this region is governed firstly by a current increase caused by primary electrons ionising the gas by collision to produce secondary electrons, and subsequently by the positive ions formed in this process gaining sufficient energy to produce further ionisation. Such discharges may be induced by irradiating the gas in between two electrodes to produce the initial ionisation. They are non-sustaining because the current flow ceases as soon as the ionising radiation is removed. When the voltage across the electrodes reaches a critical value Vs (Figure 2.34), current level ( 10�5 A, the current increases rapidly via a spark to form a self-sustaining discharge. The sparking potential Vs for ideal operating con- ditions (uniform electric field) varies with the product of gas pressure (p) and electrode separation (d ) according to Paschen's law (Figure 2.35). There is a critical value of pd for which the breakdown voltage Vs is a minimum. The self-sustaining discharge following breakdown may be either a glow or arc discharge (regions B±C and D±F, respectively, in Figure 2.34) depending on the discharge path and the nature of the connected electric circuit. The region between Vs and B (Figure 2.34) is known as a `normal' glow discharge and is characterised by the potential Figure 2.34 Current±voltage characteristic for electrical discharges. 0A, Townsend discharge; B, normal glow; C, abnormal glow; DF, arc; Vs, spark; Vo�Io, load line Figure 2.35 Breakdown voltage as a function of the pressure±electrode separation product (Paschen's law) difference across the discharge being nearly independent of current, extending to at least 10�3 A if not several amperes. For higher currents the voltage increases to form the `abnor- mal ' glow discharge (region C, Figure 2.34). The glow discharge is manifest as a diffusely luminous plasma extending across the discharge volume but may consist of alternate light and dark regions extending from the cathode in the order: Aston dark space, cathode glow, cathode dark space, negative glow, Faraday dark space, positive glow (which is extensive in volume), the anode glow and the anode dark space (Figure 2.36). The glowing regions correspond to ionisation and excitation processes being particularly active and their occurrence and extent depends on the particular operating conditions. The voltage across the glow discharge consists of two major components: the cathode fall and the positive column (Figure 2.36a). Most of the voltage drop occurs across the cathode fall. For sharply curved surfaces (e.g. wires) and long elec- trode separations the gas near the surface breaks down at a voltage less than Vs to form a local glow discharge known as a corona. The electric arc is a self-sustained discharge requiring only a low voltage for its sustenance and capable of causing currents from typically 10� 1 A to above 105 A to flow. A major difference between arc and glow discharges is that the current density at the cathode of the arc is greater than that at the glow cathode (Figure 2.37). The implication is that the electron emission process for the arc is different from that of the glow and is often thermionic in nature. At higher gas pressures both the anode and cathode of the arc may be at the boiling temperature of the electrode material. Materials having high boiling points (e.g. carbon and tungsten) have lower cathode current densities (ca. 5 ( 102 A/cm2) than materials with lower boiling points (e.g. copper and iron; ca. 5 ( 103A/cm2). The arc voltage is the sum of three distinct components (Figure 2.36(b)): the cathode and anode falls and the positive column. Cathode and anode fall voltages are each typically about 10 V. Short arcs are governed by the electrode fall regions, whereas longer arcs are dominated by the positive column. //integras/b&h/Eer/Final_06-09-02/eerc002 Figure 2.36 Voltage distributions between discharge electrodes: (a) glow; (b) arc Figure 2.37 Cathode current density distinction between glow and arc discharges At low pressures the arc may be luminously diffuse and the plasma is not in thermal equilibrium. The temperature of the gas atoms and ions is seldom more than a few hundred degrees Celsius whereas the temperature of the electrons may be as high as 4 ( 104K (Figure 2.38). At atmospheric pressure and above, the arc discharge is manifest as a constricted, highly luminous core surrounded by a more diffusively luminous aureole. The arc plasma column is generally, although not exclusively, in thermal equilibrium. The arc core is typically at temperatures in the range 5 ( 103 to 30 ( 103 K so that the gas is completely disso- Electrical discharges 2/27 Figure 2.38 Typical electron and gas temperature variations with pressure for arc plasma ciated and highly ionised. Conversely, the temperature of the aureole spans the range over which dissociation and chemical reactions occur (ca. 2 ( 103 to 5 ( 103K). The current voltage characteristic of the long electric arc is governed by the electric power (VI) dissipated in the arc column to overcome thermal losses. At lower current levels (10� 1 to 102 A) the are column is governed by thermal con- duction losses leading to a negative gradient for the current- voltage characteristic. At higher current levels radiation becomes the dominant loss mechanism yielding a positive gradient characteristic (Figure 2.34). Thermal losses and hence electric power dissipation increase with arc length (e.g. longer electrode separation), gas pressure, convection (e.g. arcs in supersonic flows) and radiation (e.g. entrained metal vapours). As a result the arc voltage at a given current is also increased, causing the discharge characteristic (Figure 2.34) to be displaced parallel to the voltage axis. 2.7.3 Discharge±network interaction For quasi-steady situations the interaction between an elec- trical discharge and the interconnected network is governed by the intersection of the load line VoIo (Figure 2.34) V ˆ( Vo �( iR (where R is the series-network resistance and Vo is the source voltage) and the current±voltage characteristic of the discharge. The negative gradient of the low current arc branch of the discharge characteristic produces a negative incremental resistance which makes operation at point D (Figure 2.34) unstable whereas operation at point E or B is stable. In practice the operating point is determined by the manner in which the discharge is initiated. Initiation by electrode separation (e.g. circuit breaker contact opening) or by fuse rupture leads to arc operation at E. However, if the discharge is initiated by reducing the series resistance R gradually so that the load line is rotated about Vo, opera- tion as a glow discharge at B may be maintained. If the cathode is heated to provide a large supply of electrons a transition from B to E may occur. A variation of the source voltage Vo causes the points of intersection of the load line and discharge characteristic to //integras/b&h/Eer/Final_06-09-02/eerc002 2/28 Electrotechnology vary so that new points of stability are produced. When the voltage falls below a value, which makes the load line tan- gential to the negative characteristic, the arc is, in principle, not sustainable. However, in practice, the thermal inertia of the arc plasma may maintain ionisation and so delay even- tual arc extinction. The behaviour of electric arcs in a.c. networks is gov- erned by the competing effects of the thermal inertia of the arc column (due to the thermal capacity of the arc plasma) and the electrical inertia of the network (produced by circuit inductance and manifest as a phase difference between current and voltage). The current±voltage characteristic of the discharge changes from the quasi-steady (d.c.) form of Figure 2.34 (corresponding to arc inertia being considerably less than the electrical inertia) via an intermediate form (when the thermal and electrical inertias are comparable) to an approximately resistive form (when the thermal inertia is considerably greater than the electrical inertia) (Figure 2.39). 2.7.4 Discharge applications Electrical discharges occur in a number of engineering situa- tions either as limiting or as essential operating features of systems and devices. Spark discharges are used in applications which utilise their transitional nature. These include spark gaps for pro- tecting equipment against high frequency, high voltage transients and as rapid acting make switches for high power test equipment or pulsed power applications. They are also used for spark erosion in machining materials to high tolerances. Glow discharges are utilised in a variety of lamps, in gas lasers, in the processing of semiconductor materials and for the surface hardening of materials (e.g. nitriding). Operational problems in all cases involve maintaining the discharge against extinction during the low current part of the driving a.c. at one extreme and preventing transition Figure 2.39 Arc current±voltage characteristics for a.c. conditions having different thermal/electrical inertia ratios: (1) thermal ( electrical inertia (d.c. case); (2) thermal ^ electrical inertia; (3) thermal 4 electrical inertia (e.g. high frequency, resistive case) into an arcing mode (which could lead to destructive thermal overload) at the other extreme. Glow discharge lamps either rely on short discharge gaps in which all the light is produced from the negative glow covering the cathode (e.g. neon indicator lamps) or long discharge gaps in which all the light comes from the positive column confined in a long tube (e.g. neon advertising lights). In materials processing the glow is used to provide the required active ionic species for surface treating the material which forms a cathodic electrode. Both etching of surface layers and deposition of complex layers can be achieved with important applications for the production of integ- rated circuits for the electronics industry. Metallic surfaces (e.g. titanium steel) may be hardened by nitriding in glow discharges. Corona on high voltage transmission lines constitute a continuous power loss which for long-distance transmis- sion may be substantial and economically undesirable. Furthermore, such corona can cause a deterioration of insulating materials through the combined action of the discharges (ion bombardment) and the effect of chemical compounds (e.g. ozone and nitrogen oxides) formed in the discharge on the surface. Arc discharges are used in high pressure lamps, gas lasers, welding, and arc heaters and also occur in current-interrup- tion devices. The distinction between the needs of the two classes of applications is that for lamps and heaters the arc needs to be stably sustained whereas for circuit interruption the arc needs to be extinguished in a controlled manner. The implication is that, when for the former applications an a.c. supply is used, the arc thermal inertia needs to be relatively long compared with the electrical inertia of the network (e.g. to minimise lamp flicker). For circuit- interruption applications the opposite is required in order to accelerate arc extinction and provide efficient current interruption. Such applications require that a number of different current waveforms (Figure 2.40) should be inter- ruptable in a controlled manner. High voltage a.c. networks need to be interrupted as the current passes naturally through zero to avoid excessive transient voltages being produced by the inductive nature of such networks. Low voltage, domestic type networks benefit from interruption via the current limiting action of a rapidly lengthening arc. High voltage d.c. network interruption relies upon the arc producing controlled instabilities to force the current artificially to zero. The arc discharges which are utilised for these applica- tions are configured in a number of different ways. Some basic forms are shown in Figure 2.41. These may be divided into two major categories corresponding to the symmetry of the arc. Axisymmetrical arcs include those which are free burning vertically (so that symmetry is maintained by buoyancy forces) wall stabilised arcs, ablation stabilised arcs (which essentially represent arcs in fuses) and axial convection controlled arcs (which are used for both gas heating, welding and high voltage circuit interruption). Non-axisymmetric arcs include the crossflow arc, the linearly driven electromagnetic arc (which has potential for the electromagnetic drive of projectiles or by driving the arc into deionising plates for circuit interruption), the rotary driven electromagnetic arc (which may be configured either between ring electrodes and used for circuit interruption, or helically and used both for gas heating and circuit inter- ruption) and the spiral arc with wall stabilisation. //integras/b&h/eer/Final_06-09-02/eerc003 3 Network Analysis (Sections 3.3.1±3.3.5) 3.1 Introduction 3/3 3.2 3/3 3.2.1 3/3 3.2.2 3/4 3.2.3 3/4 3.2.4 3/5 3.2.5 Two-ports 3/6 3.2.6 3.2.7 3/10 3.2.8 3/10 3.2.9 3/10 3.2.10 3/14 3.2.11 3/15 3.2.12 3/17 3.2.13 3/18 3.2.14 3/19 3.2.15 3/22 3.2.16 Non-linearity 3/26 3.3 3/28 3.3.1 Conventions 3/28 3.3.2 3/29 3.3.3 3/31 3.3.4 3/31 3.3.5 3/34 3.3.6 M G Say PhD, MSc, CEng, FRSE, FIEE, FIERE, ACGI, DIC Formerly of Heriot-Watt University M A Laughton BASc, PhD, DSc(Eng), FREng, FIEE Formerly of Queen Mary & Westfield College, University of London Contents Basic network analysis Network elements Network laws Network solution Network theorems Network topology 3/7 Steady-state d.c. networks Steady-state a.c. networks Sinusoidal alternating quantities Non-sinusoidal alternating quantities Three-phase systems Symmetrical components Line transmission Network transients System functions Power-system network analysis Load-flow analysis Fault-level analysis System-fault analysis Phase co-ordinate analysis Network power limits and stability 3/42 //integras/b&h/eer/Final_06-09-02/eerc003 //integras/b&h/eer/Final_06-09-02/eerc003 Basic network analysis 3/3 3.1 Introduction In an electrical network, electrical energy is conveyed from sources to an array of interconnected branches in which energy is converted, dissipated or stored. Each branch has a charac- teristic voltage±current relation that defines its parameters. The analysis of networks is concerned with the solution of source and branch currents and voltages in a given network configuration. Basic and general network concepts are dis- cussed in Section 3.2. Section 3.3 is concerned with the special techniques applied in the analysis of power-system networks. 3.2 Basic network analysis 3.2.1 Network elements Given the sources (generators, batteries, thermocouples, etc.), the network configuration and its branch parameters, then the network solution proceeds through network equa- tions set up in accordance with the Kirchhoff laws. 3.2.1.1 Sources In most cases a source can be represented as in Figure 3.1(a) by an electromotive force (e.m.f.) E0 acting through an internal series impedance Z0 and supplying an external `load' Z with a current I at a terminal voltage V. This is the Helmholtz±Thevenin equivalent voltage generator. As regards the load voltage V and current I, the source could equally well be represented by the Helmholtz±Norton equivalent current generator in Figure 3.1(b), comprising a source current I0 shunted by an internal admittance Y0 which is effectively in parallel with the load of admittance Y. Comparing the two forms for the same load current I and terminal voltage V in a load of impedance Z or admittance Y ˆ 1/Z, we have: Voltage generator Current generator V ˆ E0 � IZ0 I ˆ I0 � VY0 I ˆ …E0 � V†=Z0 V ˆ …I0 � I†=Y0 ˆ E0=Z0 � V=Z0 ˆ I0=Y0 � I =Y0 ˆ I0 � VY0 ˆ E0 � IZ0 These are identical provided that I0 ˆE0/Z0 and Y0 ˆ 1/Z0. The identity applies only to the load terminals, for internally the sources have quite different operating conditions. The two forms are duals. Sources with Z0 ˆ 0 and Y0 ˆ 0 (so that V ˆE0 and I ˆ I0) are termed ideal generators. 3.2.1.2 Parameters When a real physical network is set up by interconnecting sources and loads by conducting wires and cables, all parts (including the connections) have associated electric and magnetic fields. A resistor, for example, has resistance as the Figure 3.1 (a) Voltage and (b) current sources Figure 3.2 Pure parameters prime property, but the passage of a current implies a mag- netic field, while the potential difference (p.d.) across the resistor implies an electric field, both fields being present in and around the resistor. In the equivalent circuit drawn to represent the physical one it is usual to lump together the significant resistances into a limited number of lumped resistances. Similarly, electric-field effects are represented by lumped capacitance and magnetic-field effects by lumped inductance. The equivalent circuit then behaves like the physical prototype if it is so constructed as to include all significant effects. The lumped parameters can now be considered to be free from `residuals' and pure in the sense that simple laws of behaviour apply. These are indicated in Figure 3.2. (a) Resistance For a pure resistance R carrying an instanta- neous current i, the p.d. is v ˆRi and the rate of heat production is p ˆ vi ˆRi2. Alternatively, if the conduc- tance G ˆ 1/R is used, then i ˆGv and p ˆ vi ˆGv2. There is a constant relation v ˆ Ri ˆ v=G; i ˆ Gv ˆ v=R; p ˆ Ri2 ˆ Gv2 (b) Inductance With a self-inductance L, the magnetic linkage is Li, and the source voltage is required only when the linkage changes, i.e. v ˆ d(Li)/dt ˆL(di/dt). An inductor stores in its magnetic field the energy w ˆ 1Li2. The behaviour equations are 2 „ v ˆ L…di=dt†; i ˆ …1=L† v dt; w ˆ 1 Li2 2 Two inductances L1 and L2 with a common magnetic field have a mutual inductance L12 ˆL21 such that an e.m.f. is induced in one when current changes in the other: e1 ˆ L12…di2=dt†; e2 ˆ L21…di1=dt†( The direction of the e.m.f.s depends on the change (increase or decrease) of current and on the `sense' in which the inductors are wound. The `dot convention' for establishing the sense is to place a dot at one end of the symbol for L1, and a dot at that end of L2 which has the same polarity as the dotted end of L1 for a given change in the common flux. (c) Capacitance The stored charge q is proportional to the p.d. such that q ˆCv. When v is changed, a charge must enter or leave at the rate i ˆ dq/dt ˆC(dv/dt). The electric-field energy in a charged capacitor is w ˆ 1Cv2. Thus „ i ˆ C…dv=dt†; v ˆ …1=C† i dt; w ˆ 1 Cv2 2 2 //integras/b&h/eer/Final_06-09-02/eerc003 3/6 Network analysis Figure 3.6 The Millman theorem 3.2.4.6 Millman (Figure 3.6) The Millman theorem is also known as the parallel-generator theorem. The common terminal voltage of a number of sources connected in parallel to a common load of impedance Z is V ˆ IscZp, where Isc is the sum of the short-circuit currents of the individual source branches and Zp is the effective impedance of all the branches in parallel, including the load Z. If E1 and E2 are the e.m.f.s of two sources with internal impedances Z1 and Z2 connected in parallel to supply a load Z, and if I1 and I2 are the currents contributed by these sources to the load Z, then their common terminal voltage V must be V ˆ …I1 ‡ I2†Z ˆ ‰…E1 � V†=Z1 ‡ …E2 � V†=Z2 ŠZ whence V…1=Z ‡ 1=Z1 ‡ 1=Z2 † ˆ E1=Z1 ‡ E2=Z2 The term in parentheses on the left-hand side of the equation is the effective admittance of all the branches in parallel. The right-hand side of the equation is the sum of the individual source short-circuit currents, totalling Isc. Thus V ˆ IscZp. The theorem holds for any number of sources. 3.2.4.7 Helmholtz±Thevenin (Figure 3.7) The current in any branch Z of a network is the same as if that branch were connected to a voltage source of e.m.f. E0 and internal impedance Z0, where E0 is the p.d. appearing across the branch terminals when they are open-circuited and Z0 is the impedance of the network looking into the branch terminals with all sources represented by their internal impedance. In Figure 3.7, the network has a branch AB in which it is required to find the current. The branch impedance Z is removed, and a p.d. E0 appears across AB. With all sources replaced by their internal impedance, the network presents the impedance Z0 to AB. The current in Z when it is replaced into the original network is I ˆ E0=…Z0 ‡ Z†( The whole network apart from the branch AB has been replaced by an equivalent voltage source, resulting in the simplified condition of Figure 3.1(a). 3.2.4.8 Helmholtz±Norton The Helmholtz±Norton theorem is the dual of the Helm- holtz±Thevenin theorem. The voltage across any branch Y of a network is the same as if that branch were connected to a current source I0 with internal shunt admittance Y0, where I0 is Figure 3.7 The Helmholtz±Thevenin theorem the current between the branch terminals when short circuited and Y0 is the admittance of the network looking into the branch terminals with all sources represented by their internal admittance. Then across the terminals AB in Figure 3.7 the voltage is V ˆ I0 =…Y0 ‡ Y †( Thus the whole network apart from the branch AB has been replaced by an equivalent current source, i.e. the system in Figure 3.1(b). 3.2.5 Two-ports In power and signal transmission, input voltage and current at one port (i.e. one terminal-pair) yield voltage and current at another port of the interconnecting network. Thus in Figure 3.8a voltage source at the input port 1 delivers to the passive network a voltage V1 and a current I1. The corres- ponding values at the output port 2 are V2 and I2. 3.2.5.1 Lacour According to the theorem originated by Lacour, any passive linear network between two ports can be replaced by a two-mesh or T network, and in general no simpler form can be found. Such a result is obtained by iterative star±delta conver- sion to give the T equivalent; by one more star±delta conver- sion the -equivalent is obtained (Figure 3.9). In general, the equivalent networks are asymmetric; in some cases, however, they are symmetric. It can be shown that a passive two-port has the input and output voltages and currents related by V1 ˆ AV2 ‡ BI2 and I1 ˆ CV2 ‡DI2 where ABCD are the general two-port parameters, constants for a given frequency and with AD �BC ˆ 1. The conven- tions for voltage polarity and current direction are those given in Figure 3.8. Figure 3.8 Two-port network //integras/b&h/eer/Final_06-09-02/eerc003 Figure 3.9 T and  two-ports 3.2.5.2 T network Consider the asymmetric T in Figure 3.9. Application of the Kirchhoff laws gives I1 ˆ I2 ‡ …V2 ‡ I2Z2†Y ˆ V2 Y ‡ I2 …1 ‡ YZ2†( V1 ˆ V2…1 ‡ YZ1† ‡ I2…Z1 ‡ Z2 ‡ Z1Z2Y†( Hence in terms of the series and parallel branch components A ˆ 1 ‡ YZ1 B ˆ Z1 ‡ Z2 ‡ Z1 Z2Y C ˆ Y D ˆ 1 ‡ YZ2 Multiplication shows that AD �BC ˆ 1. For the symmetric T with Z1 ˆZ2 ˆ 1Z,2 A ˆ 1 ‡ 1 YZ ˆ D; B ˆ Z ‡ 1 YZ2; C ˆ Y2 4 3.2.5.3  network In a similar way, the general parameters for the asymmetric case are A ˆ 1 ‡Y2Z; B ˆZ; C ˆY1 ‡Y2 ‡Y1Y2Z; D ˆ 1 ‡Y1 Z which reduce with symmetry to A ˆ 1 ‡ 1 YZ ˆ D; B ˆ Z; C ˆ Y ‡ 1 Y2Z2 4 The values of the ABCD parameters, in matrix form,  # A B C D are set out in Table 3.1 for a number of common cases. 3.2.5.4 Characteristic impedance If the output terminals of a two-port are closed through an impedance V2/I2 ˆZ0, and if the input impedance V1/I1 is then also Z0, the quantity Z0 is the characteristic impedance. Consider a symmetrical two-port (A=D) so terminated: if V1/I1 is to be Z0 we have V1 V2A ‡ I2B V2 …A ‡ B=Z0†( A ‡ B=Z0ˆ ˆ( ˆ Z0 I1 V2 C ‡ I2A I2…A ‡ CZ0 †( A ‡ CZ0 which is Z0 for B/Z0 ˆCZ0. Thus the characteristic impedance is Z0 ˆH(B/C). The same result is obtainable from the input impedances with the output terminals first open circuited (I2 ˆ 0) giving Zoc, then short circuited (V2 ˆ 0) giving Zsc: thus ˆ pZoc ˆ A=C; Zsc ˆ B=A; Z0 ˆ p…ZocZsc …B=C†( Basic network analysis 3/7 3.2.5.5 Propagation coefficient The parameters ABCD are functions of frequency, and Z0 is a complex operator. For the Z0 termination of a sym- metrical two-port (for which A2�BC ˆ 1) the input/output voltage or current ratio is V1 =V2 ˆ I1 =I2 ˆ A ‡p…BC† ˆ A ‡p…A2 � 1†( ˆ exp… † ˆ exp… ‡ j †( The magnitude of V1 exceeds that of V2 by the factor exp( ) and leads it by the angle , where is the attenuation coeffi- cient, is the phase coefficient and the combination ˆ ‡ j is the propagation coefficient. 3.2.5.6 Alternative two-port parameters I There are other ways of expressing two-port relationships. For generality, both terminal voltages are taken as applied and both currents are input currents. With this convention it is necessary to write �I2 for I2 in the general parameters so far discussed. The mesh-current and node-voltage methods (Section 3.2.4) give V1 ˆ I1z11+I2z12, etc., and I1 ˆ( V1y11+ V2y12, etc., respectively. A further method relates V1 and I2 to 1 and V2 by hybrid (impedance and admittance) parameters. The four relationships are then obtained as follows: General Impedance # # # # # # # #   # V1 ˆ( A B V2 V1 ˆ( z11 z12 I1 I1 C D �I2 V2 z21 z22 I2 Admittance Hybrid   # # # # # # # # # I1 ˆ( y11 y12 V1 V1 ˆ( h11 h12 I1 I2 y21 y22 V2 I2 h21 h22 V2 h If the networks are passive, then z12 ˆ z21, y12 ˆ y21 and 12 ˆ�h21. If, in addition, the networks are symmetrical, then A ˆD, z11 ˆ z22 and y11 ˆ y22. If the networks are active (i.e. they contain sources), then reciprocity does not apply and there is no necessary relation between the terms of the 2  2 matrix. 3.2.6 Network topology In multibranch networks the solution process is aided by representing the network as a graph of nodes and inter- connections. The topology is the scheme of interconnec- tions. A network is planar if it can be drawn on a closed spherical (or plane) surface without cross-overs. A non-planar network cannot be so drawn: a single cross-over can be elimin- ated if the network is drawn on a more complicated surface resembling a doughnut, and more cross-overs require closed surfaces with more holes. The nomenclature employed in topology is as follows. Graph A diagram of the network showing all the nodes, with each branch represented by a plain line. Tree Any arrangement of branches that connects all nodes together without forming loops. A tree branch is one branch of such a tree. Link A branch that, added to a tree, completes a closed loop. Tie set A loop of branches with one a link and the others tree branches. Cut set A set of branches comprising one tree branch, the other branches being tree links. A cut set dissociates two main portions of a network in such a way that replacing any one element destroys the dissociation. //integras/b&h/eer/Final_06-09-02/eerc003 3/8 Network analysis Table 3.1 General ABCD two-port parameters Network Matrix Network Matrix Direct connection Loaded network  #   1 0 A ‡( BY0 B 0 1 C ‡(DY0 D Cross-connection Shunted network  #  #�1 0 A B 0 �1 C ‡( AY1 D ‡( BY1 Series impedance  # 1 Z 0 1 Shunt admittance  # 1 0 Y 1 L network  # 1 ‡( YZ Z Y 1 L network  # 1 Z Y 1 ‡( YZ T network # 1 ‡( YZ1 Z1 ‡( Z2 ‡( YZ1Z2 Y 1 ‡( YZ2 Symmetrical T network # 1 ‡ YZ/2 Z(1 ‡( YZ/4) Y 1 ‡( YZ/2  network # 1 ‡( Y2 Z Y1 ‡( Y2 ‡( Y1Y2 Z Symmetrical  network # 1 ‡( YZ/2 Z Y (1 ‡( YZ/4) 1 ‡( YZ/2 Cascaded networks # A1A2 ‡( B1C2 A1B2 ‡( B1D2 A2C1 ‡( C2D1 B2C1 ‡(D1D2 Mutual inductance  # 0 �j!L12 �1/j!L12 Mutual inductance  # 0 �j!L12 1/j!L12 0 Ideal transformer  # N1/N2 0 0 N2/N1 # # # Z 1 ‡( Y1Z # # cont'd 0 //integras/b&h/eer/Final_06-09-02/eerc003 Basic network analysis 3/11 Figure 3.14 Phasors subtraction of r.m.s. values are performed as if the lines were co-planar vector forces in mechanics. Physically, how- ever, the lines are not vectors: they substitute for scalar quan- tities, alternating sinusoidally with time. They are termed phasors. Certain associated quantities, such as impedance, admittance and apparent power, can also be represented by directed lines, but as they are not sinusoids they are termed complexors or complex operators. Both phasors and complex- ors can be dealt with by application of the theory of complex numbers. The definitions concerned are listed below. Complexor A generic term for a non-vector quantity expressed as a complex number. Phasor A complexor (e.g. voltage or current) derived from a time-varying sinusoidal quantity and expressed as a complex number. Complex operator A complexor derived for the ratio of two phasors (e.g. impedance and admittance); or a complexor which, operating on a phasor, gives another phasor (e.g. V ˆ IZ, in which V and I are phasors, but Z is a complex operator). 3.2.9.1 Complexor algebra The four arithmetic processes for complexors are applications of the theory of complex numbers. Complexor a in Figure 3.15 can be expressed by its magnitude a and its angle  with respect to an arbitrary `datum' direction (here taken as horizontal) as the simple polar form a ˆ a € . Alternatively it can be written as a ˆ p+jq, the rectangular form, in terms of its projection p on the datum and q on a quadrature axis at right angles thereto: q (as a scalar magnitude along the datum) is rotated counter-clockwise by angle 1  rad (90) by the 2 operator j. Two successive operations by j (written as j2) give a rotation of  rad (180), making the original +q into �q, in effect a multiplication by �1. Three operations (j3) give �jq and four give ‡q. Thus any complexor can be located in the complex datum±quadrature plane. Further obvious forms are the trigonometric, a ˆ a(cos ‡ j sin ), and the exponen- tial, a ˆ a exp(j). Summarising, the four descriptions are: Polar : a ˆ a €  Rectangular : a ˆ p ‡ jq Exponential : a ˆ a exp…j†( Trigonometric : a ˆ a…cos  ‡ j sin †( 2where a ˆH(p +q 2) and ˆ arctan(q/p). Consider complexors a ˆ p ‡ jq ˆ a €( and b ˆ r ‡ js ˆ b € . The basic manipulations are: Addition a ‡ b ˆ …p ‡ r† ‡ j…q ‡ s†( Subtraction a � b ˆ …p � r† ‡ j…q � s†( Multiplication The exponential and polar forms are more direct than the rectangular or trigonometric: ab ˆ …pr � qs† ‡ j…qr ‡ ps†( ˆ( ab exp‰j… ‡ †Š ˆ ab € … ‡ †( Division Here also the angular forms are preferred: a=b ˆ ‰…pr ‡ qs† ‡ j…qr � ps†Š=…r 2 ‡ s 2†( ˆ …a=b† exp‰j… � †Š ˆ …a=b† € (… � †( Conjugate The conjugate of a complexor a ˆ p ‡ jq ˆ a € is a* ˆ p � jq ˆ a € (� ), the quadrature com- ponent (and therefore the angle) being reversed. Then ab* ˆ ab € … � †( a*b ˆ ab € … � †( a*a ˆ aa* ˆ a 2 ˆ p 2 ‡ q 2 The last expression is used to `rationalise' the denominator in the complexor division process. 3.2.9.2 Impedance and admittance operators Sinusoidal voltages and currents can be represented by phasors in the expressions V ˆ IZ ˆ I/Y and I ˆVY ˆV/Z. Current and voltage phasors are related by multiplication or division with the complex operators Z and Y. Series resist- ance R and reactance jX can be arranged as a right-angled triangle of hypotenuse Z ˆH(R2 ‡X2) and the angle between Z and R is ˆ arctan(X/R). The relation between Z and Y for the same series network elements with Z ˆR ‡ jX is 1 1 R � jX ˆ( R � jX Y ˆ ˆ( ˆ( Z R ‡ jX …R ‡ jX†…R � jX†( R2 ‡ X2 ˆ R=Z2 � j…X=Z2† ˆ G � jB where G and B are defined in terms of R, X and Z. The series components R and X become parallel branches in Y, one a pure conductance, the other a pure susceptance. Further, a positive-angled impedance has, as inverse equivalent, a negative-angled admittance (Figure 3.16). The impedance and phase angle of a number of circuit combinations are given in Table 3.3. Figure 3.15 Complexors Figure 3.16 Impedance and admittance triangles //integras/b&h/eer/Final_06-09-02/eerc003 3/12 Network analysis Table 3.3 Impedance of network elements at angular frequency ! (rad/s) Impedance: Z ˆR+jX ˆ |Z| €  |Z| ˆH(R2+X2) ˆ arctan(X/R) Admittance: Y ˆ 1/Z ˆ |Y| € (�) |Y|ˆH[(R/Z2)2+(X/Z2)2] ˆ�arctan(X/R) Z: R 1/j!C j!L (C1+C2)/j!C1C2 j!(L1+2L12) R+j!L R+1/j!C : 0 �/2 ‡ /2 �/2 ‡/2 arctan(!L/R) �arctan(j!CR) Z: j(!L � 1/!C) R+j(!L � 1/!C) !LR !L ‡ jR R2 ‡ !2 L2 R 1 � j!CR 1 ‡ !2C2 R2 j!L 1 � !2LC : /2 arctan[(!L � 1/!C)/R] arctan(R/!L) �arctan(!CR)  /2 Z : 1=R � j(!C � 1/!L) (1=R)2 ‡ (!C � 1/!L)2 R ‡ j!‰L(1 � !2LC) � CR2 Š( (1 � !2LC)2 ‡ !2C2R2 A ‡ jB (R ‡ r)2 ‡ (!L � 1/!C)2 A ˆRr(R+r)+!2L2 r+R/!2C2 B ˆ!Lr2�R2/!C�(L/C) (!L�1/!C) : �arctan[R(!C � 1/!L)] arctan {![L(1 �!2 LC) �CR2]/R} arctan (B/A) Resonance conditions for LC networks numbered 1±6 above, for !ˆ!0 ˆ 1/H(LC): (1) |Z| ˆ 0, ˆ 0; (2) |Z| ˆR, ˆ 0; (3) |Z| ˆ1, ˆ 0; (4) |Z| ˆR, !ˆ 0; (5) |Z| ˆL/CR, ˆ�arctan(!CR) for R 5 !L; (6) |Z| ˆR (const.) for R ˆ r ˆH(L/C) Impedance and admittance loci If the characteristics of a device or a circuit can be expressed in terms of an equivalent circuit in which the impedances and/or admittances vary according to some law, then the current taken for a given applied voltage (or the voltage for a given current) can be obtained graphically by use of an admittance or impedance locus diagram. In Figure 3.17(a), let OP represent an impedance Z ˆ( R ‡ jX and OQ the corresponding admittance Y ˆG � jB. Point Q is obtained from P by finding first the geometric inverse point Q0( such that OQ0 ˆ 1/OP to scale, and then reflecting OQ0( across the datum line to give OQ and thus a reversed angle �, a process termed complexor inversion. If Z has successive values Z1, Z2, . . . , on the impedance locus, the corresponding admittances Y1, Y2, . . . , lie on the admittance locus. The inversion process may be point-by-point, but in many cases certain propositions can reduce the labour: (1) Inverse of a straight lineÐthe geometric inverse of a straight line AB about a point O not on the line is a circle passing through O with its centre M on the perpendicular OC from O to AB (Figure 3.17(b)). Then A0( is the geometric inverse of A, B0( of B, etc.; also, A is the inverse of A0, B of B0, etc. Figure 3.17 Inversion (2) Inverse of a circleÐfrom the foregoing, the geometric inverse of a circle about a point O on its circumference is a straight line. If, however, O is not on the circumference, the inverse is a second circle between the same tangents; but the distances OM and OM0( from the origin O to the centres M and M0( of the circles are not inverses of each other. //integras/b&h/eer/Final_06-09-02/eerc003 Basic network analysis 3/13 The choice of scales arises in the inversion process: for example, the inverse of an impedance Z ˆ 50 € 70( is Y ˆ 0.02 € (�70) S. It is usually possible to decide on a scale by taking a salient feature (such as a circle diameter) as a basis. 3.2.9.3 Power The instantaneous power delivered to a load is the product of the instantaneous voltage v and current i. Let v ˆ vm sin !t and i ˆ im sin(!t �) as in Figure 3.18(a); then the instanta- neous power is p ˆ 1 vmim‰cos � cos…2!t � †Š2 This is a quantity fluctuating at angular frequency 2! with, in general, excursions into negative power (i.e. that returned by the load to the source). Over an integral number of periods the mean power is P ˆ 1 vm im cos  ˆ VI cos 2 where V and I are r.m.s. values. Now resolve i into the active and reactive components ip ˆ …im cos † sin !t and iq ˆ …im sin † sin…!t � 1 †2 as in Figure 3.18(b); then the instantaneous power can be written p ˆ …vm…im cos † sin2 !t � vm …im sin † sin !t cos !t Over a whole number of periods the average of the first term is P ˆ 1 vm im cos  ˆ VI cos 2 giving the average rate of energy transfer from source to load. The second term is a double-frequency sinusoid of average value zero, the energy flow changing direction rhythmically between source and load at a peak rate Q ˆ 1 vmim sin  ˆ VI sin 2 The power conditions thus summarise to the following: Active power P The mean of the instantaneous power over an integral number of periods giving the mean rate of energy transfer from source to load in watts (W). Reactive power Q The maximum rate of energy interchange between source and load in reactive volt-amperes (var). Apparent power S The product of the r.m.s. voltage and current in volt-amperes (V-A). Both P and Q represent real power. The apparent power S is not a power at all, but is an arbitrary product VI. Nevertheless, because of the way in which P and Q are defined, we can write 2 2 2P2 ‡Q2 ˆ …VI † …cos  ‡ sin2 † ˆ …VI † whence S=H(P2+Q2), a convenient combination of mean active power with peak power circulation. Complex power The active and reactive powers can be determined for voltage and current phasors by S ˆ P  jQ ˆ VI * or S ˆ V*I using the conjugate of either I or V. Power factor The ratio of active to apparent power, P/S=cos  for sinusoidal conditions. 3.2.9.4 Resonance A condition of resonance occurs when the load contains two forms of energy-storing element (L and C) such that, at the frequency of operation, the two energies are equal. The reactive power requirements are then satisfied internally, as the inductor releases energy at the rate that the capacitor requires it. The source supplies only the active power demand of the energy-dissipating load components, the load externally appearing to be purely resistive. Acceptor resonance The series RLC circuit in Figure 3.19(a) has, at angular frequency !, the impedance Z ˆR ‡ jX, where X is !L�1/!C, which for !ˆ!0 ˆ1/H(LC) is zero. The impedance is then ZˆR and the input current has a maximum I0 ˆV/R, conditions of acceptor resonance. Internally, large voltages appear across the reactive com- ponents, viz. VL ˆ I0!L ˆ V!0…L=R† and VC ˆ I0…1=!0C† ˆ V=!0 …CR†( The terms L/R and 1/CR are the time constants of the reactive elements, and !0L/R is the Q value of a practical inductor of Figure 3.18 Active and reactive power Figure 3.19 Resonance //integras/b&h/eer/Final_06-09-02/eerc003 3/16 Network analysis 3.2.11.3 Delta The line-to-line e.m.f. is that of the phase across which the lines are connected. The line current is the difference of the currents in the phases forming the line junction, so that the relations for symmetric loading are E1 ˆEph and I1 ˆH3 Iph. 3.2.11.4 Interconnected star A line-to-neutral e.m.f. comprises contributions from suc- cessive half-phases and sums to 1 p 3 of a complete phase 2 e.m.f. The line-to-line e.m.f. is 1 1 times the magnitude of 2 a complete phase e.m.f. and the line current is numerically equal to the phase current. 3.2.11.5 Power The total power delivered to or absorbed by a polyphase system, be it symmetric and balanced or not, is the algebraic sum of the individual phase powers. Consider an m-phase system with instantaneous line currents i1, i2, . . . , im, the alge- braic sum of which is zero by the Kirchhoff node law. Let the voltages of the input (or output) terminals, with reference to a common point X, be v1 � vx, v2 � vX, . . . , vm � vx; then the instantaneous powers will be (v1 � vx)i1, (v2 � vx)i2, . . . , (vm �( vx)im, which together sum to the total instantaneous power p. There is no restriction on the choice of X; it can be any of the terminals, say M. In this case vm � vx=vm � vm=0, and the power summation has only m�1 terms. The average power over a full period T is, therefore, …#T P ˆ …1=T†( ‰…v1 � vm†i1 ‡


‡ …vm�1 � vm†im�1Š dt 0 The first term of the sum in brackets represents the indication of a wattmeter with i1 in its current circuit and v1 � vm across its volt circuit, i.e. connected between terminals 1 and M. It follows that three wattmeters can measure the power in a three-phase four-wire system, and two in a three-phase three- wire system. Some of the common cases are listed below. (1) Three-phase, four-wire, load unbalancedÐThe connec- tions are shown in Figure 3.23(a). Wattmeters W1, W2 Figure 3.23 Three-phase power measurement and W3 measure the phase powers separately. The total power is the sum of the indications: P ˆ P1 ‡ P2 ‡ P3 (2) Three-phase, four-wire, load balancedÐwith the connec- tions shown in Figure 3.23(a), all the meters read the same. Two of the wattmeters can be omitted and the reading of the remaining instrument multiplied by 3. (3) Three-phase, three-wire, load unbalancedÐtwo watt- meters are connected with their current circuits in any pair of lines, as in Figure 3.23(b). The total power is the algebraic sum of the readings, regardless of waveform. A two-element wattmeter summates the power automa- tically; with separate instruments, one will tend to read reversed under certain conditions, given below. (4) Three-phase, three-wire, load balancedÐwith sinusoidal voltage and current the conditions in Figure 3.23(c) obtain. Wattmeters W1 and W2 indicate powers P1 and P2 where P1 ˆ VabIa cos (30 ‡ ) ˆ V1I1 cos (30 ‡ ) P2 ˆ Vcb Ic cos (30 � ) ˆ V1I1 cos (30 � ) The total active power P=P1+P2 is therefore P ˆ V1I1[ cos (30 ‡ ) ‡ cos (30 � )] ˆ p3V1I1 cos  where cos  is the phase power factor. The algebraic difference is P1�P2 ˆV1I1 sin , whence the reactive power is given by Q ˆ p3V1 I1 sin  ˆ p3(P1 � P2) and the phase angle can be obtained from ˆ arctan (Q/P). For ˆ 0 (unity power factor) both wattmeters read alike; for ˆ 60( (power factor 0.5 lag) W1 reads zero; and for lower lagging power factors W1 tends to read backwards. The active power of a single phase has a double- frequency pulsation (Figure 3.18). For the asymmetric two-phase system under balanced conditions and a phase dis- placement of 90, and for all symmetric systems with m ˆ 3 or more, the total power is constant. 3.2.11.6 Harmonics Considering a symmetrical balanced system of three-phase non-sinusoidal voltages, and omitting phase displacements (which are in the context not significant), let the voltage of phase A be va ˆ v1 sin !t ‡ v2 sin 2!t ‡ v3 sin 3!t ‡


( Writing !t � 2  and !t � 4 , respectively, for phases B and 3 3 C, and simplifying, we obtain va ˆ v1 sin !t ‡ v2 sin 2!t ‡ v3 sin 3!t ‡


( vb ˆ v1 sin …!t � 2† ‡ v2 sin 2…!t � 4† ‡ v3 sin 3!t ‡


(3 3 vc ˆ v1 sin …!t � 4† ‡ v2 sin 2…!t � 23 † ‡ v3 sin 3!t ‡


(3 The fundamentals have a normal 2/3 rad (120) phase rela- tion in the sequence ABC, as also do the 4th, 7th, 10th, . . . , harmonics. The 2nd (and 5th, 8th, 11th, . . . ) harmonics have the 2/3 rad phase relation but of reversed sequence ACB. The triplen harmonics (those of the order of a multiple of 3) are, however, co-phasal and form a zero-sequence set. The relation V1 ˆH3 Vph in a three-phase star-connected system is applicable only for sine waveforms. If harmonics are present, the line- and phase-voltage waveforms differ because of the effective phase angle and sequence of the har- monic components. The nth harmonic voltages to neutral in two successive phases AB are vn sin n!t and vn sin n(!t � 2 ), 3 //integras/b&h/eer/Final_06-09-02/eerc003 Basic network analysis 3/17 and between the corresponding line terminals the nth harmonic voltage is 2vn sin n( 1 ). For triplen harmonics 3 this is zero; hence no triplens are present in balanced line voltages because, in the associated phases, their components are equal and in opposition. In a balanced delta connection, again no triplens are present between lines: the delta forms a closed circuit to triplen circulating currents, the impedance drop of which absorbs the harmonic e.m.f.s. 3.2.12 Symmetrical components Figure 3.21(a) shows the sine waves and phasors of a balanced symmetric three-phase system of e.m.f.s of sequence ABC. The magnitudes are equal and the phase displacements are 2/3 rad. In Figure 3.21(b), the asym- metric sine waveforms have also the sequence ABC, but they are of different magnitudes and have the phase dis- placements , and . Problems of asymmetry occur in the unbalanced loading of a.c. machines and in fault conditions on power networks. While a solution is possible by the Kirchhoff laws, the method of symmetrical components greatly simplifies analysis. Any set of asymmetric three-phase e.m.f.s or currents can be resolved into a summation of three sets of symmetrical components, respectively of positive phase-sequence (p.p.s.) ABC, negative phase-sequence (n.p.s.) ACB, and zero phase-sequence (z.p.s.). Use is made of the operator , resembling the 90( operator j (Section 3.2.9.1) but implying a counter-clockwise rotation of 2/3 rad (120). Thus ˆ 1 € 120 ˆ 1…�1 ‡ jp3†2 2 ˆ 1 € 240 ˆ 1…�1 � jp3†2 3 ˆ 1 € 360 ˆ 1 ‡ j0 1 ‡ ‡ 2 ˆ 0 A symmetric three-phase system has only p.p.s. components Ea ˆ Ea‡; Eb ˆ 2 Ea‡; Ec ˆ Ea‡( whereas an asymmetric system (Figure 3.24) comprises the three sets z:p:s: : Ea0; Eb0 ˆ Ea0; Ec0 ˆ Ea0 p:p:s: : Ea‡; Eb‡ ˆ 2Ea‡; Ec‡ ˆ Ea‡( n:p:s: : Ea�; Eb� ˆ Ea�; Ec� ˆ 2Ea�( where the subscripts 0, + and � designate the z.p.s., p.p.s. and n.p.s. components, respectively. The p.p.s. and the n.p.s. components sum individually to zero. Therefore, if the originating phasors Ea, Eb, Ec also sum to zero there are no z.p.s. components; if they do not, their residual is the sum of the three z.p.s. components. The asymmetrical phasors have now been reduced to the sum of three sets of symmetrical components: Ea ˆ Ea0 ‡ Ea‡ ‡ Ea�( Eb ˆ Eb0 ‡ Eb‡ ‡ Eb�( Ec ˆ Ec0 ‡ Ec‡ ‡ Ec�( The components are evaluated from the arbitrary identities Ea ˆ Z ‡ P ‡N Eb ˆ Z ‡ 2P ‡ N Ec ˆ Z ‡ P ‡ 2N where Z ˆ …Ea ‡ Eb ‡ Ec†=3 Figure 3.24 Symmetrical components P ˆ …Ea ‡ Eb ‡ 2Ec†=3 N ˆ …Ea ‡ 2Eb ‡ Ec†=3 Figure 3.24 is drawn for an asymmetric system with voltages Ea ˆ 200, Eb ˆ 100 and Ec ˆ 400 V, and phase- displacement angles ˆ 90(, ˆ 120( and ˆ 150(. In phasor terms, Ea ˆ 200 € 0 ˆ 200 ‡ j0 V Eb ˆ 100 € …�90† ˆ 0 � j100 V Ec ˆ 400 € 150 ˆ �346 ‡ j200 V Then Z ˆ …200 � j100 � 346 ‡ j200†=3 ˆ �49 ‡ j33 ˆ Ea0 P ˆ …200 ‡ 87 ‡ j50 ‡ 347 ‡ j200†=3 ˆ 211 ‡ j83 ˆ Ea‡( N ˆ …200 � 87 ‡ j50 � j400†=3 ˆ 38 � j117 ˆ Ea�( The summation Ea0 ‡Ea ‡ ‡Ea�ˆ 200 ‡ j0 ˆEa. The p.p.s and n.p.s. components of Eb and Ec are readily obtained. 3.2.12.1 Power In linear networks there is no interference between currents of different sequences. Thus p.p.s. voltages produce only p.p.s. currents, etc. The total power is therefore P ˆ Pa ‡ Pb ‡ Pc ˆ 3…V0I0 cos 0 ‡ V‡I‡ cos ‡ ‡ V�I� cos �†( This is equivalent to the more obvious summation of phase powers P ˆ VaIa cos a ‡ VbIb cos b ‡ VcIc cos c Symmetrical-component techniques are useful in the ana- lysis of power-system networks with faults or unbalanced //integras/b&h/eer/Final_06-09-02/eerc003 3/18 Network analysis loads: an example is given in Section 3.3.4. Machine perfor- mance is also affected when the machine is supplied from an asymmetric voltage system: thus in a three-phase induction motor the n.p.s. components set up a torque in opposition to that of the (normal) p.p.s. voltages. 3.2.13 Line transmission Networks of small physical dimensions and operated at low frequency are usually considered to have a zero propagation time; a current started in a closed circuit appears at every point in the circuit simultaneously. In extended circuits, such as long transmission lines, the propagation time is sig- nificant and cannot properly be ignored. The basics of energy propagation on an ideal loss-free line are discussed in another section. Propagation takes place as a voltage wave v accompanied by a current wave i such that v/i ˆ z0 (the surge impedance) travelling at speed u. Both z0 and u are functions of the line configuration, the electric and magnetic space constants 0 and 0, and the relative permit- tivity and permeability of the medium surrounding the line conductors. At the receiving end of a line of finite length, an abrupt change of the electromagnetic-field pattern (and there- fore of the ratio v/i) is imposed by the discontinuity unless the receiving-end load is z0, a termination called the natural load in a power line and a matching impedance in a telecommuni- cation line. For a non-matching termination, wave reflection takes place with an electromagnetic wave running back towards the sending end. After many successive reflections of rapidly diminishing amplitude, the system settles down to a steady state determined by the sending-end voltage, the receiving-end load impedance and the line parameters. 3.2.13.1 A.c. power transmission The steady-state condition considered is the transfer of a constant balanced apparent power per phase from a genera- tor at the sending end (s) to a load at the receiving end (r) by a sinusoidal voltage and current at a frequency f ˆ!/2. The line has uniformly distributed parameters: a conductor resistance r and a loop inductance L effectively in series, and an insulation conductance g and capacitance C in shunt, all per phase and per unit length. The series impe- dance, shunt admittance and propagation coefficient per unit length are z ˆ r ‡ j!L, y ˆ g ‡ j!C and ˆ p(yz), respectively. For a line of length l the overall parameters are zl ˆZ, yl ˆY and lp(yz) ˆ p(YZ) ˆ l. The solution for the receiving-end terminal conditions is in terms ofp (YZ) and its hyperbolic functions as a two-port: Vs ˆ VrA ‡ I rB ˆ Vr cosh…pYZ† ‡ Irz0 sinh…pYZ†( Is ˆ VrC ‡ I rD ˆ Vr…1=z0† sinh…pYZ† ‡ Ir cosh…pYZ†( Usingp the hyperbolic series (Section 1.2.2) and writing z0 ˆ( (Z=Y ), we obtain for a symmetrical line A ˆ 1 ‡ YZ=2 ‡ …YZ†2 =24 ‡


ˆ D B ˆ Z‰1 ‡ YZ=6 ‡ …YZ†2 =120 ‡


Š ( C ˆ Y ‰1 ‡ YZ=6 ‡ …YZ†2 =120 ‡


Š ( The significance of the higher powers of YZ depends on: (i) the line configuration, (ii) the properties of the ambient medium, and (iii) the physical length of the line in terms of the wavelength ˆ u/f. For air-insulated overhead lines the inductance is large and the capacitance small: the propaga- tion velocity approximates to u ˆ 3105 km/s (corresponding to a wavelength ˆ 6000 km at 50 Hz), with a natural load z0 Figure 3.25 Transmission-line phasor diagram of the order of 400±500 . Cable lines have a low inductance and a large capacitance: the permittivity of the dielectric material and the presence of armouring and sheathing result in a propagation velocity around 200 km/s, a surge impe- dance below 100 , and the possibility (in high-voltage cables) that the charging current may be comparable with the load current if the cable length exceeds 25±30 km. V For balanced three-phase power transmission, the general equations are applied for the line-to-neutral voltage, line current and phase power factor. Phasor diagrams for the load and no-load (Ir ˆ 0) receiving-end conditions for an overhead-line transmission are shown in Figure 3.25, with Vr as datum. On no load, Vs ˆVrA, and as A has a magnitude less than unity and a small positive angle , the phasor VrA is smaller than Vr and leads it by angle : thus r >Vs, the Ferranti effect. For the loaded condition, IrB is added to VrA to give Vs. Similarly VrC is added to IrD to obtain Is. Q The product VrIr ˆ Ir(Vs�VrA) is the receiving-end com- plex apparent power Sr. Let Vs lead Vr by angle ; then the receiving-end load has the active and reactive powers Pr and r given by Pr ˆ …VsVr =B† cos…� † � …Vr2A=B† cos… � †( Qr…Vs Vr =B† sin…� † ‡ …Vr2A=B† sin… � †( where and are the angles in the complexors A and B. The importance of B (roughly the overall series impedance) is clear. Line chart Operating charts for a transmission circuit can be drawn to relate graphically Vs, Vr, Pr and Qr, using the appropriate overall ABCD parameters (e.g. with terminal transformers included). Receiving-end chart A receiving-end chart gives active and reactive power at the receiving end for Vr constant (Figure 3.26(a)). The co-ordinates (x, y) and the radius (r) of the constant-voltage circles are x ˆ �V2 …A=B† cos… � †(r y ˆ �Vr2 …A=B† sin… � †( r ˆ VsVr =B where A and B are scalar magnitudes, and and the angles in A and B. For a given Vr the chart comprises a family of concentric circles, each corresponding to a parti- cular Vs. If a given receiving-end load is located by its active and reactive power components, Vs is obtained from the corresponding Vs circle. //integras/b&h/eer/Final_06-09-02/eerc003 Figure 3.29 Transients in a capacitive±resistive circuit As this must be V/R at t ˆ 0‡, then k ˆV/R, as shown in Figure 3.29(a). In Figure 3.29(b) the initiation of a CR circuit with a sine voltage is shown. Summary for an RC circuit The transient current is a decaying exponential k exp(�t/T). The initial current is determined by the voltage difference between the voltage applied by the source and that of the capacitor. (In Figure 3.29 the capacitor is in each case uncharged.) If this p.d. is V0, then the initial current is V0/R. Double-energy system A typical case is that of a series RLC circuit. The transient form is obtained from L(di/dt) ‡Ri ‡ q/C ˆ 0, differentiated to d2i=dt2 ‡ …R=L†…di=dt† ‡ …1=LC†i ˆ 0 Thus  2 ‡ (R/L)‡ 1/LC ˆ 0 is the required equation, with the roots R # R2 1 1=2 1; 2 ˆ � ( �( 2L 4L2 LC The resulting transient depends on the sign of the quantity in parentheses, i.e. on whether R/2L is greater or less than 1/H(LC). Four waveforms are shown in Figure 3.30. (1) Roots real: R > 2H(L/C). The transient current is uni- directional and results from two simple exponential curves with different rates of decay. Basic network analysis 3/21 (2) Roots equal: R ˆ 2H(L/C). This has more mathematical than physical interest, but it marks the boundary between unidirectional and oscillatory transient current. (3) Roots complex: R < 2H(L/C). The roots take the form �  j!n, and the transient current oscillates with the interchange of magnetic and electric energies respect- ively in L and C; but the oscillation amplitude decays by reason of dissipation in R. With R ˆ 0 the oscillation persists without decay at the undamped natural fre- quency !n ˆ 1/H(LC). Pulse drive The response of networks to single pulses (or to trains of such pulses) is an important aspect of data transmission. An ideal pulse has a rectangular waveform of duration (`width') tp. It can be considered as the resultant of two opposing step functions displaced in time by tp as in Figure 3.31(a). In practice a pulse cannot rise and fall instantaneously, and often the amplitude is not constant (Figure 3.31(b)). Ambiguity in the precise position of the peak value Vp makes it necessary to define the rise time as the interval between the levels 0.1 Vp and 0.9 Vp. The tilt is the difference between Vp and the value at the start of the trail- ing edge, expressed as a fraction of Vp. The response of the output network to a voltage pulse depends on the network characteristics (in particular its time- constant T ) and the pulse width tp. Consider an ideal input Figure 3.30 Double-energy transient forms Figure 3.31 Pulse drive //integras/b&h/eer/Final_06-09-02/eerc003 3/22 Network analysis voltage Vi of rectangular waveform applied to an ideal low- pass series network (Figure 3.31(c)), the output being the voltage v0 across the capacitor C. Writing p for d/dt, then v0 ˆ( 1=pC ˆ( 1 1 ˆ( Vi R ‡ 1=pC 1 ‡ pCR 1 ‡ pT where T ˆCR is the network time-constant. This represents an exponential growth v0 ˆVi[1 � exp(�t/T)] over the interval tp. The trailing edge is an exponential decay, with t reckoned from the start of the trailing edge. Three typical responses are shown. For CR 5 tp the output voltage reaches Vi; for CR > tp the rise is slow and does not reach Vi; for CR @ tp the rise is almost linear, the final value is small and the response is a measure of the time-integral of Vi. With C and R interchanged as in Figure 3.31(d ) to give a high-pass network, the whole of Vi appears across R at the leading edge, falling as C charges. Following the input pulse there is a reversed v0 during the discharge of the capacitor. The output/input voltage relation is given by v0 R ˆ( pCR ˆ( pT ˆ( Vi R ‡ 1=pC 1 ‡ pCR 1 ‡ pT For CR @ tp the response shows a tilt; for CR 5 tp the capacitor charges rapidly and the output v0 comprises positive- and negative-going spikes that give a measure of the time-differential of Vi. 3.2.14.3 Laplace transform method Application of the Laplace transforms is the most usual method of solving transient problems. The basic features of the Laplace transform are set out in Section 1.2.7 and Table 3.4, which gives transform pairs. The advantages of the method are that: (1) any stimulus, including discontinu- ous and pulse forms, can be handled, (2) the solution is complete with both steady-state and transient components, (3) the initial conditions are introduced at the start, and (4) formal mathematical processes are avoided. Consider the system in Figure 3.28(a). The applied direct voltage V has the Laplace transform V(s) ˆV/s; the opera- tional impedance of the circuit is Z(s) ˆR ‡Ls. Then the Laplace transform of the current is V…s†( V 1 V 1 I …s† ˆ ˆ( ˆ( Z…s†( s R ‡ Ls L s…s ‡ R=L†( The term V/L is a constant unaffected by transformation. The term in s is almost of the form a/s(s ‡ a). So, if we write V a I …s† ˆ ( aL s…s ‡ a†( where a ˆR/L ˆ 1/T, the inverse Laplace transform gives i…t† ˆ …V=aL†‰1 � exp…�at†Š ˆ …V=R†‰1 � exp…�t=T†Š( which is the complete solution. More complex problems require the development of partial fractions to derive recognisable transforms which are then individually inverse-transformed to give the terms in the solution of i(t). 3.2.15 System functions It is characteristic of linear constant-coefficient systems that their operational solution involves three parts: (i) the excita- tion or stimulus, (ii) the output or response and (iii) the system function. Thus in the relation I(s) ˆV(s)/Z(s) for the current in Z resulting from the application of V, 1/Z(s) is the system function relating voltage to current. For the simple Figure 3.32 System functions electrical system shown in Figure 3.32(a) the system function Y(s) relating V(s) to I(s) in I(s) ˆV(s)Y(s) is Y (s)= 1/(R ‡ Ls ‡ 1/Cs). Different functions could relate the capacitor charge or the magnetic linkage in the inductor to the transform V(s) of the stimulus v(t). The mechanical analogue (Figure 3.32(b)) of this elec- trical system, as indicated in Section 1.3.1, has a system trans- fer function to relate force f(t) to velocity u(t) of the mass m and one end of the spring of compliance k in the presence of viscous friction of coefficient r. Then F(s) and U(s) are the transforms of f(t) and u(t), and the operational `mechanical impedance' has the terms ms, 1/ks and r. In general, an input i(s) and an output o(s) are related by a system trans- fer function KG(s) (Figure 3.32(c)), where K is a numerical or a dimensional quantity to include amplification or the value of some physical quantity (such as admittance). The transform of the integro-differential equation of variation with time is expressed by the term G(s). The system is then represented by the block diagram in Figure 3.32(c); i.e. o(s)/i(s) ˆKG(s). A number of typical system transfer functions for rela- tively simple systems are given in Table 3.4. The output of one system may be used as the input to another. Provided that the two do not interact (i.e. the individual transfer functions are not modified by the connection) the overall system function is the product [K1G1(s)] ( [K2G2(s)] of the individual functions. If the systems are paralleled and their outputs are additively combined, the overall function is their sum. 3.2.15.1 Closed-loop systems In Figure 3.32, parts (a), (b) and (c) are open-loop systems. However, the output can be made to modify the input by feedback through a network Kf Gf(s) as in (d ). The signal f …s† ˆ ‰Kf Gf …s†Šo …s†( is combined with i(s) to give the modified input. For positive feedback, the resultant input is (s) ˆ i(s) ‡( f(s), and the effect is usually to produce instability and oscillation. //integras/b&h/eer/Final_06-09-02/eerc003 Basic network analysis 3/23 Table 3.4 System transfer functions [the relation f2(t)/f1(t ) of output to input quantity in terms of the Laplace transform F2(s)/F1(s)] System Transfer function Z2(s)1 Electrical network V2(s) ˆ( V1(s) Z1(s) ‡( Z2(s) 2 Electrical network Z1(s)Z2(s)V2(s) ˆ( I1(s) Z1(s) ‡( Z2(s) 3 Feedback amplifier Z2(s)V2(s) ˆ( V1(s) Z1(s) 4 Second-order system 12(s) ˆ( T ˆ p(J/K) 1(s) 1 ‡( 2csT ‡( s2T 2 c ˆ( F =2p…JK†( V2…s†( ka T ˆ p…M=K†5 Accelerometer ˆ( A1…s†( 1 ‡( 2csT ‡( s2T 2 c ˆ( F =2p…MK†( V2…s†6 Permanent-magnet generator ˆ( ke !1 …s†( V2…s†( ke!7 Separately excited generator ˆ( T ˆ( L=R V1…s†( R…1 ‡( sT†( 2…s†( Ke8 Motor: armature control ˆ( Ke ˆ( ke =…FR ‡( k2e †(V1…s†( s…1 ‡( sT †( T ˆ( JR=…FR ‡( k2 †e 2…s†( keˆ( T1 ˆ( J=F ,9 Motor: field control V1…s†( s…1 ‡ sT1†…1 ‡ sT2†( T2 ˆ( L=R v voltage L inductance i current ke e.m.f. coefficient Z impedance c damping coefficient R, r resistance T time-constant For negative feedback, the resultant input is the difference (s) ˆ( i(s) � f(s), an `error' signal. With the main system KG(s) now relating  and o, the output/input relation is o…s†( KG…s†(ˆ( i…s†( 1 ‡ ‰KG…s†Š‰Kf Gf …s†Š( Suppose that there is unity feedback KfGf (s) ˆ( 1, then if KG (s) is large o…s†=i…s† ˆ (KG…s†=‰1 ‡( KG…s†Š( '( 1  angular displacement M mass ! angular velocity J inertia a acceleration F viscous friction coefficient ka acceleration coefficient K stiffness and the output closely follows the input in magnitude and wave shape, a condition sought in servo-mechanisms and feedback controls. 3.2.15.2 System performance In general, a system function takes the form numerator/ denominator, each a polynomial in s, relating response to input stimulus. Two forms are