Lecture10

# Lecture10 - Lecture 10 Transformation from frame to d-q...

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Lecture 10 Transformation from α - β frame to d-q frame Fig. 10-(a) α - β Frame Fig. 10-(b) d-q Frame In the α - β frame (setting aside the zero-sequence) the stator quantities are [] = = t t V e e e line S S S ω αβ sin cos and + + = = ) sin( ) cos( S S S S S S t t I i i i ϕ and the rotor quantities are + + = = ) sin( ) cos( VR R VR R R R R R t t V e e e and + + = = ) sin( ) cos( iR R iR R R S S R t t I i i i The frame transformation consists replacing the fictitious rotor - windings to an equally fictitious rotor windings whose axes are always aligned with the stationary stator α - β windings. The common axes are called the d-q windings. After the transformation, the voltage and current vectors are for the stator: s axis s ra x i s x i s m θ sq-axis sd-axis

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[] = Sq Sd S dq e e e and = Sq Sd S dq i i i (1) and for the rotor = Rq Rd R dq e e e and = Sq Sd R dq i i i (2) In this transformation, the stator quantities are not transformed. Therefore, the stator quantities are transformed by the 2x2 identity matrix [I]. The rotor quantities are transformed by: = m m m m m dq n n n n n C θ αχ cos sin sin cos )] ( [ (3) The complete transformation is: = R dq S dq m dq R S e e n C I e e )] ( [ 0 0 ] [ αβ (4) = R dq S dq m dq R S i i n C I i i )] ( [ 0 0 ] [ (5) The equations in the d-q frame is: + + + + + + + + + + = Rq Rd Sq Sd R R m R m m R R R m S S S S Rq Rd Sq Sd i i i i p M R n M Mp Mn n M p M R Mn Mp Mp p M R Mp p M R e e e e ) ( ) ( ) ( ) ( 0 ) ( 0 0 0 ) ( l l l l l l ω (6) In order to appreciate how (6) has been derived, consider the mutual induction matrix (14) from Lecture 9:
= M n M M M n M n M n M n

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Lecture10 - Lecture 10 Transformation from frame to d-q...

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