Polyphase Systems-Power Electronics-Handout, Exercises of Power Electronics

Download Polyphase Systems-Power Electronics-Handout and more Exercises Power Electronics in PDF only on Docsity! Polyphase Systems Introduction An ac generator designed to develop a single sinusoidal voltage for each rotation of the shaft (rotor) is referred to as a single-phase ac generator. If the number of coils on the rotor is increase in a specified manner, the result is a Polyphase ac generator, which develops more than one ac phase voltage per rotation of the rotor. In general, three-phase systems are preferred over single-phase systems for the transmission of power for many reasons, including the following: 1. Thinner conductor can be used to transmit the same kVA at the same voltage, which reduces the amount of copper required (typically about 25% less) and in turn reduces construction maintenance costs. 2. The lighter lines are easier to install, and the supporting structure can be less massive and farther apart. 3. Three-phase equipment and motors have preferred running and starting characteristics compared to single-phase systems because of a more even flow of power to the transducer than can be delivered with a single-phase supply. 4. In general, larger motors are three phase because they are essentially self-starting and do not require a special design or additional starting circuitry. docsity.com cies were chosen primarily because they ean be generated by 2 rela- oe t stable mechani cal design that is sensitive to the size Benerating systems and the demand that must be met during peak periods, On aircraft and ships the demand levels permit the use of a 400-Hz line frequency. : The three-phase system is used by almost all commercial electric generators. This docs aot mean that single-phase and two-phase gener- ating systems are obsolete. Most small emergency generators, such as the gasoline type, are one-phase generating systems. The two-phase , system is commonly used in servomechanisms, which are self-correct- ing contre! systems capable of detecting and adjusting their own opera- tion. Servomechanisms are used in ships and aircraft to keep them on course automatically, or, in simpler devices such as a thermostatic cir- cuit, to regulate heat output. In many cases, however, where single- phase and two-phase inputs are required, they are supplied by one and two phases of a three-phase generating system rather than generated independently. - ; The number of phase voltages that can be produced by a polyphase generator is not limited to three. Any number of phases can be obtained by spacing the windings for each phase at the proper ‘angular position around the rotor. Some electrical systems operate more efficiently if more than three phases are used. One such system involves the process of rectification, which is used to convert an alternating output to one having an average, or dc, value. The greater the number of phases, the smoother the dc output of the system. docsity.com  )30(cos2 0 AN AB E E 2 3 2  AN AB E E ANAB EE 3 docsity.com ANBNAB EEE  BNANAB EEE  00 1200 )()(  BNrmsANrmsAB EEE )120sin120(cos 00)()( jEEE BNrmsANrmsAB  ) 2 3 2 1 ()()( jEEE BNrmsANrmsAB  ) 2 3 2 1 ()()( jEEE ANrmsANrmsAB  ][ )()( BNrmsANrms EE  ) 2 3 2 3 ()( jEE ANrmsAB  docsity.com ) 2 1 2 3 (3 )( jEE ANrmsAB  00 )( 0 )( 3003303  ANrmsABANrmsAB EEEE ……………………………………………………………………………………………… BNCNBC EEE  CNBNBC EEE  00 240120 )()(  CNrmsBNrmsBC EEE )240sin240(cos)120sin120(cos 00)( 00 )( jEjEE CNrmsBNrmsBC  ) 2 3 2 1 () 2 3 2 1 ( )()(   jEjEE CNrmsBNrmsBC ) 2 3 2 1 () 2 3 2 1 ( )()(   jEjEE BNrmsBNrmsBC ][ )()( CNrmsBNrms EE  ) 2 32 ()( jEE BNrmsBC  )(3 )( jEE BNrmsBC  0903 )(  BNrmsBC EE 00 )( 0 )( 301203903  BNrmsBCBNrmsBC EEEE ……………………………………………………………………………………………… CNANCA EEE  ANCNCA EEE  00 0240 )()(  ANrmsCNrmsCA EEE docsity.com Line Voltages (ABC Sequence) )(00)( referenceEE ABrmsAB  0120)(  BCrmsBC EE 0240)(  CArmsCA EE Determining the phase sequence from the line voltages of a three-phase generator ACB Phase Sequence (Y-Connected Generator) Phase Voltages (ACB Sequence) )(00)( referenceEE ANrmsAN  0120)(  CNrmsCN EE 0240)(  BNrmsBN EE docsity.com Determining the phase sequence from the phase voltages of a three-phase generator Line Voltages (ACB Sequence) )(00)( referenceEE ABrmsAB  0120)(  CArmsCA EE 0240)(  BCrmsBC EE docsity.com Determining the phase sequence from the line voltages of a three-phase generator docsity.com 00 120240 )()(  CBrmsACrmsCc III )120sin120(cos)240sin240(cos 00)( 00 )( jIjII CBrmsACrmsCc  ) 2 3 2 1 () 2 3 2 1 ( )()( jIjII CBrmsACrmsCc    ) 2 3 2 1 () 2 3 2 1 ( )()( jIjII ACrmsACrmsCc    ][ )()( CBrmsACrms II  ) 2 32 ()( jII ACrmsCc  )(3 )( jII ACrmsCc  00 )( 0 )( 3024032703  ACrmsCcACrmsCc IIII ……………………………………………………………………………………………… 0 )( 303  BArmsAa II 0 )( 30 AarmsAa II )()( 3 BArmsAarms II  )30sin()30sin(2 0)( 0 )(  tItIi AamAarmsAa  ……………………………………………………………………………………………… 0 )( 1503  CBrmsBb II 0 )( 150 BbrmsBb II )()( 3 CBrmsBbrms II  )150sin()150sin(2 0)( 0 )(  tItIi BbmBbrmsBb  ……………………………………………………………………………………………… docsity.com 0 )( 2703  ACrmsCc II 0 )( 270 CcrmsCc II )()( 3 ACrmsCcrms II  )270sin()270sin(2 0)( 0 )(  tItIi CcmCcrmsCc  ………………………………………………………………………………………….... )30(cos2 0 BA Aa I I 2 3 2  BA Aa I I BAAa II 3 docsity.com 23.8 ‘PHASE SEQUENCE (4-CONNECTED GENERATOR) Even though the line and phase voltages of a A-connected system are the same, it is standard practice to describe the phase sequence in terms of the line voltages. The method used is the same as that desctibeil for the line voltages of the Y-connected generator. For example, the phasor diagram of the line voltages for a phase sequence ABC is shown in Fig. 23.19. In drawing such a diagram, one must take care to have the sequence of the first and second subscripts the same. In phasor notation, Ey = Exp 20° Esc = Ege 2—120° Ec, = Ecy £120° docsity.com ) 2 3 2 3 ()( jEE CNrmsCA  ][ )()( CNrmsANrms EE  ) 2 1 2 3 (3 )( jEE CNrmsCA  00 )( 0 )( 3012031503  CNrmsCACNrmsCA EEEE ……………………………………………………………………………………………… ACB Phase sequence ACAaBA III  ACBAAa III  00 1200 )()(  ACrmsBArmsAa III )120sin120(cos 00)()( jIII ACrmsBArmsAa  ) 2 3 2 1 ()()( jIII ACrmsBArmsAa  ) 2 3 2 1 ()()( jIII BArmsBArmsAa  ][ )()( ACrmsBArms II  ) 2 3 2 3 ()( jII BArmsAa  ) 2 1 2 3 (3 )( jII BArmsAa  00 )( 0 )( 3003303  BArmsAaBArmsAa IIEI ……………………………………………………………………………………………… docsity.com BABbCB III  BACBBb III  00 0240 )()(  BArmsCBrmsBb III )( 00 )( )240sin240(cos BArmsCBrmsBb IjII  )()( )2 3 2 1 ( BArmsCBrmsBb IjII    )()( )2 3 2 1 ( CBrmsCBrmsBb IjII  ][ )()( BArmsCBrms II  ) 2 3 2 3 ()( jII CBrmsBb  ) 2 1 2 3 (3 )( jII CBrmsBb  00 )( 0 )( 3024032103  CbrmsBbCBrmsBb IIII ……………………………………………………………………………………………… CBCcAC III  CBACCc III  00 240120 )()(  CBrmsACrmsCc III )240sin240(cos)120sin120(cos 00)( 00 )( jIjII CBrmsACrmsCc  ) 2 3 2 1 () 2 3 2 1 ( )()(   jIjII CBrmsACrmsCc ) 2 3 2 1 () 2 3 2 1 ( )()( jIjII ACrmsACrmsCc  docsity.com ][ )()( CBrmsACrms II  ) 2 32 ()( jII ACrmsCc  )(3 )( jII ACrmsCc  00 )( 0 )( 301203903  ACrmsCcACrmsCc IIII ……………………………………………………………………………………………… The Rotating Magnetic Field Before we have looked at how if two magnetic fields are present in a machine, then a torque will be created which will tend to line up the two magnetic fields. If one magnetic field is produced by the stator of an ac machine and the other by the rotor, then a torque will be induced in the rotor which will cause the rotor to turn and align itself with the stator magnetic field. If there were some way to make the stator magnetic field rotate, then the induced torque in the rotor would cause it to ‘chase’ the stator magnetic field. How do we make the stator magnetic field to rotate? Fundamental principle – a 3-phase set of currents, each of equal magnitude and differing in phase by 120º, flows in a 3-phase winding, then it will produce a rotating magnetic field of constant magnitude. The rotating magnetic field concept is illustrated below – empty stator containing 3 coils 120º apart. It is a 2-pole winding (one north and one south). docsity.com )] 2 3 2 1 ( 2 3 [)] 2 3 2 1 ( 2 3 [ jBjBB MMnet  ] 4 3 4 3 [] 4 3 4 3 [ jBjBB MMnet  ] 4 3 4 3 4 3 4 3 [ jjBB Mnet  ] 2 3 [ jBB Mnet  0905.1)(5.1  MMnet BjBB The resulting magnetic flux is as shown below: At time 090t TBB Maa 00  TBTBB MMbb 000 1205.0120)30(sin   TBTBB MMcc 0000 2405.0240)24090(sin   docsity.com The total magnetic field from all three coils added together will be cc B bb B aa BB net  TBBBB MMMnet 000 2405.01205.00  TjBjBBB MMMnet ] 2 3 2 1 [5.0] 2 3 2 1 [5.0  TjBjBBB MMMnet ] 4 3 4 1 [] 4 3 4 1 [  TBBB MMnet 5.0 TBB Mnet 005.1  The resulting magnetic flux is as shown below: Proof of Rotating Magnetic Field Concept At any time t, the magnetic field will have the same magnitude 1.5 BM and it will continue to rotate at angular velocity . Proof: docsity.com vectorunithorizontalx ˆ vectorunitverticaly ˆ Refer again to the stator in Figure 4.1. x direction is to the right and y direction is upward. Assume that we represent the direction of the magnetic field densities in the form of: To find the total magnetic flux density in the stator, simply add vector ally the three component magnetic fields and determine their sum. We know that the flux densities equations are: TtBtB Maa 00sin)(    TxtBtB Maa ˆ)(sin)(   TtBtB Mbb 00 120)120(sin)(    TttBtB Mbb 000 120)120sincos120cos(sin)(    TttBtB Mbb 0120)cos 2 3 sin 2 1 ()(    TyxttBtB Mbb )ˆ 2 3 ˆ 2 1 )(cos 2 3 sin 2 1 ()(    TyttBxttBtB MMbb ˆ)cos 4 3 sin 4 3 (ˆ)cos 4 3 sin 4 1 ()(    TtBtB Mcc 00 240)240(sin)(    TttBtB Mcc 000 240)240sincos240cos(sin)(    TttBtB Mcc 0240)cos 2 3 sin 2 1 ()(    TyxttBtB Mcc )ˆ 2 3 ˆ 2 1 )(cos 2 3 sin 2 1 ()(    TyttBxttBtB MMcc ˆ)cos 4 3 sin 4 3 (ˆ)cos 4 3 sin 4 1 ()(    cc B bb B aa BB net  docsity.com Mechanical Cycle: One mechanical rotation around the stator surface. Electrical Cycle: N pole-S pole-N pole Electrical Frequency (Speed): ef (In Hz or cycle per second) e (In radians per second) Mechanical Speed of Magnetic Field Rotation: mf (In revolution per second) m (In radians per second) The Relation between Electrical Frequency (Speed) and the Mechanical Speed of Magnetic Field Rotation Relation between ef and mf for two Pole Stator We know that there is only one electrical cycle (N-S-N) in two pole stator in one mechanical rotation around the stator surface as shown in Fig. 1 Fig. 1 docsity.com mfef  (Two poles) Relation between ef and mf for four Pole Stator We know that there are two electrical cycles (N-S-N and N-S-N) in four pole stator in one mechanical rotation around the stator surface as shown in Fig. 2 Fig. 2 mfef 2 (Four poles) mf P ef 2  (General relation) Similarly me   (Two poles) me  2 (Four poles) m P e  2  (General relation) me   (Two poles) me  2 (Four poles) m P e  2  (General relation) docsity.com Mechanical Speed of Magnetic Field Rotation in “rpm” The mechanical speed of magnetic field rotation in “rpm” is denoted by mn . m fmn 60 We know that: mf P ef 2  P e f mf 2  Now again we have: m fmn 60 P e f mn 2 60 P e f mn 120  docsity.com