Análise de Máquinas de Indução Trifásicas em Referência abc

Máquina de Indução Trifásica 2 abc Reference Frame Fig 1 A twopole 3phase symmetrical induction machine 3 abc Reference Frame Fig 1 A twopole 3phase symmetrical induction machine 4 abc Reference Frame Winding arrangement for a 2pole 3phase wyeconnected symmetrical induction machine is shown in Fig1 Stator windings are identical sinusoidally distributed windings displaced by120 with Ns equivalent turns and resistance rs Consider the case when rotor windings are also three identical sinusoidally distributed windings displaced by120 with Nr equivalent turns and resistance rr 5 In abc reference frame voltage equations can be written as s denotes variables and parameters associated with the stator circuits r denotes variables and parameters associated with the rotor circuits abc Reference Frame cr br ar T abcr cs bs as T abcs abcr r abcr abcr abcs s abcs abcs f f f f f f f f p r i V p r i V 6 Lls and Lms are respectively the leakage and magnetizing inductance of the stator windings Llr and Lmr are respectively the leakage and magnetizing inductance of the rotor windings abc Reference Frame abcr abcs r T sr sr s abcr abcs i i L L L L 2 1 2 1 2 1 2 1 2 1 2 1 ms ls ms ms ms ms ls ms ms ms ms ls s L L L L L L L L L L L L L mr lr mr mr mr mr lr mr mr mr mr lr r L L L L L L L L L L L L 2 1 2 1 2 1 2 1 2 1 2 1 L where 7 Lsr is the amplitude of the mutual inductances between stator and rotor windings A majority of induction machines are not equipped with coilwound rotor windings instead the current flows in copper or aluminum bars which are uniformly distributed in a common ring at each end of the rotor This type of rotor is referred to as a squirrelcage rotor abc Reference Frame cos 3 2 cos 3 2 cos 3 2 cos cos 3 2 cos 3 2 cos 3 2 cos cos r r r r r r r r r ms rs sr L L L 8 Rotor variables can be referred to the stator windings by appropriate turns ratio abc Reference Frame sr r s ms abcr r s abcr abcr r s abcr abcr s r abcr L N N L N N N V N V N i N i 2 cos 3 2 cos 3 2 cos 3 2 cos cos 3 2 cos 3 2 cos 3 2 cos cos r r r r r r r r r ms sr r s sr L N N L L 9 Also abc Reference Frame r s r r ms s r mr N N L N N L L L 2 2 ms lr ms ms ms ms lr ms ms ms mr lr r L L L L L L L L L L L L 2 1 2 1 2 1 2 1 2 1 2 1 L where lr r s lr L N N L 2 10 Flux linkage may be expressed as Voltage equations expressed in terms of machine variables referred to the stator windings may be written as abc Reference Frame abcr abcs r T sr sr s abcr abcs i i L L L L abcr abcs r r T sr sr s s abcr abcs i i p p p p V V L r L L L r where r r s r r N N r 2 11 Energy stored in the coupling field may be written as Voltage equations expressed in terms of machine variables referred to the stator windings may be written as Torque Equation in Machine Variables abcr lr r T abcr abcr sr T abcs abcs ls s T abcs f c i i i i i i W W 2 1 2 1 L I L L L I L where I identity matrix r r j c r j e P W i T i 2 12 Since Ls and Lr are functions of r the above equation for the electromagnetic torque yields The torque and rotor speed are related by Torque Equation in Machine Variables r br ar cs ar cr bs cr br as r ar br cr cs cr ar br bs cr br ar as ms abcr sr r T abcs e i i i i i i i i i i i i i i i i i i i i i L P i p i T cos 2 3 sin 2 1 2 1 2 1 2 1 2 1 2 1 2 2 L L r e T J P p T 2 13 In the analysis of induction machines it is desirable to transform the variables associated with the symmetrical rotor windings to the arbitrary reference frame Equations of Transformation for Rotor Circuit cr br ar T abcr r dr qr T r qd abcr r r qd f f f f f f f f f f 0 0 0 K 2 1 2 1 2 1 3 2 sin 3 2 sin sin 3 2 cos 3 2 cos cos 3 2 Kr 14 Equations of Transformation for Rotor Circuit 1 3 2 sin 3 2 cos 1 3 2 sin 3 2 cos 1 sin cos 1 r K where r See Fig2 0 0 r t r r t dt r subscript indicates the variable parameters and transformation associated with rotating circuits Fig 2 Axis of 2pole 3phase Symmetrical machine 15 Voltage Equations in Arbitrary Reference Frame Variables For twopole 3phase symmetrical induction abcr r abcr abcr abcs s abcs abcs p r i V p r i V r abcr abcs T sr abcr abcr sr abcs s abcs L i i L i L i L r qd s abcr qd r r abcr s qd s abcs qd s s abcs i i V V i i V V 0 0 0 0 K K K K 16 Voltage Equations in Arbitrary Reference Frame Variables Fig1 Axis of 2pole 3phase symmetrical induction 17 Voltage Equations in Arbitrary Reference Frame Variables Using the above transformation equations we can transform the voltage equations to an arbitrary reference frame rotating at speed of Flux linkage equations in abc reference frame can be transformed to qd axes using Ks and Kr transformation matrices qd r qdr r r qd r r qd qd s qds s qd s s qd p r i V p r i V 0 0 0 0 0 0 0 0 qr dr T qdr qs ds T qds where 18 Voltage Equations in Arbitrary Reference Frame Variables r qd s qd r r r s sr r r sr s s s s r qd s qd i i 0 0 1 1 1 1 0 0 K L K K L K K K L L K K where ms ls ls ls s s s L M M L M L M L 2 3 0 0 0 0 0 0 1 L K K ms lr lr lr r r r L M M L M L M L 2 3 0 0 0 0 0 0 1 K L K 19 Voltage equations written in expanded form can be expressed as Voltage Equations in Arbitrary Reference Frame Variables M M M s T sr r r sr s 0 0 0 0 0 0 1 1 K L K K L K s s s s ds qs s ds ds qs ds s qs qs p r i V p r i V p r i V 0 0 0 r r r r dr qr r r dr dr qr dr r r qr qr p r i V p r i V p r i V 0 0 0 20 Flux linkage equations are Since machine and power system parameters are nearly always given in ohms or percent or per unit of a base impedance it is convenient to express the voltage and flux linkage equations in terms of reactances rather than inductances Voltage Equations in Arbitrary Reference Frame Variables s ls s dr ds ls ds ds qr qs ls qs qs i L i M i i L i M i i L 0 0 r lr r dr ds lr dr dr qr qs lr qr qr i L i M i i L i M i i L 0 0 21 Let Then Voltage Equations in Arbitrary Reference Frame Variables b s b s s s ds b qs b s ds ds qs ds b s qs qs p r i V p r i V p r i V 0 0 0 r b r r r dr b qr b r r dr dr qr b dr b r r qr qr p r i V p r i V p r i V 0 0 0 22 And flux linkages become flux linkages per second with the units of volts Fig 2 presents the arbitrary reference frame equivalent circuits for a 3phase symmetrical induction machine Voltage Equations in Arbitrary Reference Frame Variables s ls s dr ds m ls ds ds qr qs m ls qs qs i X i i X i X i i X i X 0 0 r lr r dr ds m lr dr dr qr qs m lr qr qr i X i i X i X i i X i X 0 0 23 Fig 2 A 2pole 3phase symmetrical induction machine Voltage Equations in Arbitrary Reference Frame Variables 24 Fig 2 A 2pole 3phase symmetrical induction machine Voltage Equations in Arbitrary Reference Frame Variables 25 Fig 3 Equivalent circuits of a 3phase symmetrical induction machine with rotating qd axis at speed of Voltage Equations in Arbitrary Reference Frame Variables 26 Fig 3 Equivalent circuits of a 3phase symmetrical induction machine with rotating qd axis at speed of Voltage Equations in Arbitrary Reference Frame Variables 27 Fig 3 Equivalent circuits of a 3phase symmetrical induction machine with rotating qd axis at speed of Voltage Equations in Arbitrary Reference Frame Variables 28 Electromagnetic torque in terms of arbitrary reference frame variables may be obtained by substituting the equations of transformation in After some work we will have the following Torque Equation in Arbitrary Reference Frame Variables qd r r sr r T qd s s abcr sr r T abcs e i L i P i L P i T 0 1 0 1 2 2 K K 2 2 3 ds qr qs dr e i i P M i i T 29 Where Te is positive for motor action Other expressions for the electromagnetic torque of an induction machine are Torque Equation in Arbitrary Reference Frame Variables 2 2 3 dr qr qr dr e i i P T 2 2 3 qs ds ds qs em i i P T 1 2 2 3 dr qr dr qr b e i i P T 30 qd Axis Voltage Equations qd axis voltage equations can be written by inspection of Fig1 And flux linkage equations are s s s s ds qs s ds ds qs ds s qs qs p r i V p r i V p r i V 0 0 0 r r r r dr qr r r dr dr qr dr r r qr qr p r i V p r i V p r i V 0 0 0 s ls s dr ds M ls ds ds qr qs M ls qs qs i L i i L i L i i L i L 0 0 r lr r dr ds M lr dr dr qr qs M lr qr qr i L i i L i L i i L i L 0 0 31 qd Axis Voltage Equations Fig1 Axis of 2pole 3phase symmetrical induction 32 qd Axis Voltage Equations Fig 2 A 2pole 3phase symmetrical induction machine 33 qd Axis Voltage Equations Fig 2 A 2pole 3phase symmetrical induction machine 34 qd Axis Voltage Equations Fig 3 Equivalent circuits of a 3phase symmetrical induction machine with rotating qd axis at speed of 35 qd Axis Voltage Equations Fig 3 Equivalent circuits of a 3phase symmetrical induction machine with rotating qd axis at speed of 36 qd Axis Voltage Equations Fig 3 Equivalent circuits of a 3phase symmetrical induction machine with rotating qd axis at speed of 37 When 0 stationary reference frame voltage equations become When r rotor reference frame which is also referred to as Parks transformation voltage equations become s s s s ds s ds ds qs s qs qs p r i V p r i V p r i V 0 0 0 r r r r dr qr r r dr dr qr dr r r qr qr p r i V p r i V p r i V 0 0 0 qd Axis Voltage Equations s s s s ds qs r s ds ds qs ds r s qs qs p r i V p r i V p r i V 0 0 0 r r r r dr r dr dr qr r qr qr p r i V p r i V p r i V 0 0 0 38 qd Axis Voltage Equations When e synchronously rotating reference frame the voltage equations become Synchronously rotating reference frame is used when incorporating the dynamic characteristics od an induction machine into a digital computer program used to study the transient and dynamic stability of large power systems Synchronously rotating reference frame is also used in variable frequency study of induction machines s s s s ds qs e s ds ds qs ds e s qs qs p r i V p r i V p r i V 0 0 0 r r r r dr qr r e r dr dr qr dr r e r qr qr p r i V p r i V p r i V 0 0 0 39 Per Unit System Machine data Stator line frequency f Hz Output horse power HP hp Line to line voltage V rms V Pole number P Base values ohms P V 2 3 I V Z amp peak 3V 2P I peak volts 3 2V V watts 746 hp P b 2 b b b b b b b b b 40 Per Unit System Base electrical angular velocity b2fb radsec Base mechanical angular velocity brm2fbP2 3V rms I rms 2 V I 3 P m N ω P 2 P ω P T b b b b b b b brm b b b s b s s s b qs ds b s ds ds b ds qs b s qs qs p r i V p r i V p r i V 0 0 0 r b r r r dr b qr b r r dr dr qr b dr b r r qr qr p r i V p r i V p r i V 0 0 0 Let 41 Per Unit System Define Machine voltage equations in pu can be written as b qs qs p u qs b qs qs p u qs b qs qs qs p u V I i i i V V V V b b ds b qs b b qs b s b qs V V p I i Z r V V s b s b qs ds b s ds ds b ds qs b s qs qs p V p r i V p r i V 0 0 r b r dr b qr b r r dr dr qr b dr b r r qr qr p V p r i V p r i V 0 0 42 Per Unit System Torque base qs ds ds qs i i b e b b b b b b ω 1 2 P 2 3 T 2 V I 3 2ω P ω P 2 P T qs ds qs ds b ds b qs b qs b ds b b b qs ds qs ds b b e e pu i i I i V I i V 2 V I 3 2ω P i i ω 1 2 P 2 3 T T T 43 Per Unit System Per unit equation of motion is Define quantity b b b r b b L b e ω P 1 2 P ω ω dt d P2 J ω T T T T b r b 2 2 b L pu e pu ω ω dt d P 2 P J ω T T power base stored kinetic energy at ω H b 44 Per Unit System b 2 2 b b 2 brm P 2 P 2 J ω 1 P 2 J ω 1 H Therefore 2H P 2 P ω J b 2 2 b J kgm2 Pb watts b radsec 45 The quantity H is called the inertia constant and has units of seconds Per Unit System b r pu r b r L pu e pu ω ω ω t in sec ω ω dt 2H d T T KVA n Wk 0237 10 H KVA n J 548 10 H 2 2 6 2 6 n rpm KVA base KVA J kgm2 Wk2 Lbft2 46 Per Unit System Example HP 115 Poles 4 Vrated 210rmsphase f 50 Hz rated J 100 Lbft2 rs 0016 rr 0001 Xls 00706 Xlr 00903 Xm 28413 Pb 746115 85800 watts Vb 2102 297 Volts peak Ib 23PbVb 193 Amppeak Zb VbIb 297193 154 Base values 47 Per Unit System Base angular frequency b 250 314 radsec Torque base Tb PbP2b 547 Nm 0045 154 00706 Z X X 002 154 0031 Z r r 00103 154 0016 Z r r b ls ls b r r b s s 48 Per Unit System sec 0607 858 1500 100 023110 H 1845 154 28413 Z X X 00587 154 00903 Z X X 2 6 b M M b lr lr