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Unformatted text preview: Lecture 2 Lecture 2 6 6 E E C C E743 E743 3Phase Induction Machines Reference Frame Theory Part I Professor: Ali Keyhani Professor: Ali Keyhani 2 Arbitrary Reference Frame Consider stator winding of a 3phase machine Fig.1. A 2pole 3phase symmetrical induction machine. 3 Arbitrary Reference Frame Synchronous and induction machine inductances are functions of the rotor speed, therefore the coefficients of the differential equations (voltage equations) which describe the behavior of these machines are timevarying. A change of variables can be used to reduce the complexity of machine differential equations, and represent these equations in another refernce frame with constant coefficients. 4 Arbitrary Reference Frame A change of variables which formulatesa transformation of the 3phase variables of stationary circuit elements to the arbitrary reference frame may be expressed abcs s s qd f K f = [ ] , ) ( , s ds qs T s qd f f f where = f [ ] , ) ( cs bs as T abcs f f f = f , 2 1 2 1 2 1 ) 3 2 sin( ) 3 2 sin( sin ) 3 2 cos( ) 3 2 cos( cos 3 2 + + = s K ). ( ) ( + = t dt t 5 f can represent either voltage, current, or flux linkage. s indicates the variables, parameters and transformation associated with stationary circuits. represent the speed of reference frame. Arbitrary Reference Frame ( 29 . 1 ) 3 2 sin( ) 3 2 cos( 1 ) 3 2 sin( ) 3 2 cos( 1 sin cos 1 + + = s K 6 Arbitrary Reference Frame =0: Stationary reference frame. = e : synchronoulsy rotaing reference frame. = r : rotor reference frame (i.e., the reference frame is fixed on the rotor). 7 Arbitrary Reference Frame f as , f bs and f cs may be thought of as the direction of the magnetic axes of the stator windings. f qs and f ds can be considered as the direction of the magnetic axes of the new fictious windings located on qs and ds axis which are created by the change of variables. Power Equations: cs cs bs bs as as abcs i V i V i V P + + = ( 29 s s ds ds qs qs abcs s qd i V i V i V P P 2 2 3 + + = = 8 Arbitrary Reference Frame Stationary circuit variables transformed to the arbitrary reference frame. Resistive elements: For a 3phase resistive circuit, abcs s abcs i r V = ( 29 s qd s abcs i i 1 = K ( 29 s qd s abcs V V 1 = K ( 29 ( 29 s qd s s s qd s i r V 1 1 = K K ( 29 ( 29 s qd s s s s qd i r V 1 = K K s qd s s qd i r V = ( 29 ( 29 s s s s r r = 1 , K K = s s s s r r r r 9 Arbitrary Reference Frame Inductive elements: For a 3phase inductive circuit,...
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This note was uploaded on 07/17/2008 for the course ECE 743 taught by Professor Keyhani during the Spring '08 term at Ohio State.
 Spring '08
 Keyhani

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