Magnetic Systems: Electromechanical Energy Conversion and Magnetic Fields, Lecture notes of Engineering

Download Magnetic Systems: Electromechanical Energy Conversion and Magnetic Fields and more Lecture notes Engineering in PDF only on Docsity! Introduction to Magnetic Systems Magnetic systems are important in electrical machines, because the electromechanical energy conversion process involves the conversion of electrical energy to magnetic energy and then to mechanical energy i.e.: Figure 1: Electromechanical Energy Conversion Process Electromagnetic energy conversion depends on the production of two magnetic fields, which interact to develop torque in the machine. The two magnetic fields can be produced by: a. Electric currents in coils b. Electric currents in coils and permanent magnets Hence an analysis of the energy conversion process involves: a. Materials used to produce magnetic fields b. Development and analysis of magnetic circuits associated with the system. Before going further, it is essential to review some fundamental relationships. 1. CHARGE AND CURRENT q = Quantity of electricity or charge i = Current = Rate of flow of charge ∴ dtdqi /= A and dtiq ∫= 2. FORCE CAUSED BY MAGNETIC FIELD A particle carrying a charge ‘q’ and moving at a velocity ‘v’ in a magnetic field of strength ‘B’ is subjected to a force F, (Fig. 2) where )( BvqF rrr Λ= Electrical Energy Magnetic Energy Mechanical Energy Figure 2 Figure 3 The magnitude of the force F is given by: θsinqvBF = , where θ = angle between B and v, and F r is perpendicular to the plane in which v r and B r lies. The direction of F r is given by the right hand screw rule. The moving charge ‘q’ may form part of a current flowing in a conductor of length ‘l’. For an elementary length of conductor dl, measured in the direction of the current, Fig. 3, and where vectors ld r and B r are orthogonal, BdliBdldtdqBdtdldqdF ...)./()./.( === For the general case: )()( BldlBldiFd rrrrr Λ≡Λ= And for a conductor of length ‘L’ )()( BlLBLiF rrrrr Λ≡Λ= 4. MAGNETIC FLUX AND FLUX DENSITY Magnetic fluxΦ , is the integration of flux density B r through a closed path as in Fig. 6. Figure 6: Magnetic Flux Density through a Closed Path ∴ ∫=Φ A AdB rr . This expression is the surface integral of the scalar product of flux density and area over any surface bounded by the closed path. - Vector Ad r is normal to the elementary area dA. - Vector B r is the flux density through dA when B r is perpendicular to Ad r . BA=Φ 5. MAGNETIC FIELD INTENSITY Magnetic flux may be produced by: a. Electric Currents b. Permanent Magnets For magnetic flux produced by electric currents, there exists a physical property intermediate between current and flux density. This property is called magnetic field intensity ‘H’. The relationship is as follows: - Current i produce magnetic field intensity H - and H produces flux density B. In a vacuum, the relationship between B and H is given by: HB o rr µ= where oµ = permeability in a vacuum = 7104 −×Π H/m Vectors B r and H r are co-linear. If B r is tangential to a circle centred on the current-carrying conductor, then H r can be illustrated in the same way. Figure 7: Magnetic Field Intensity Around a Current-carrying Conductor Magnetic flux intensity H, is related to the current producing it by Ampere’s Circular Law, which states: ∫ ∫= A AdJldH rrrr .. ∫ ldH rr . = line integral around the closed path. ld r is the elemental distance in the direction of integration. ∫ A AdJ rr . = Surface integral over any surface bounded by the closed path. J r = Current density at any point in the conductor. - If the closed path is a circle of radius r, centred on the conductor. - If the closed path is normal to the flat surface which carries the current i. Then applying: ∫ ∫= A AdJldH rrrr .. irH =Π )2( ∴ riH Π= 2/ 6. INDUCED ELECTROMOTIVE FORCE AND INDUCTANCE Faraday’s Law: If the flux linking a coil changes, then an emf is induced in the coil. The flux linking the coil may be changed by: a. Moving the coil in the magnetic field b. Changing the size of the coil or c. Change the flux density N.B.: Flux linkage – flux passing through the coil. The flux linking the coil may be as a result of a current flowing through the coil due to an applied potential difference. If the current i flowing through the coil is increased by adjusting the source, the flux linking the coil will increase and consequently an emf will be induced in the coil. By Lenz’s Law, the direction of the induced emf will oppose the increase of coil current. ∴ dtde /Φ= The above equation refers to a single turn coil. For an ‘N’ turns coil. )/( dtdNe Φ= Flux Linkage Φ= Nλ ∴ dtde /λ=