Lecture11

# Lecture11 - Lecture 11 Transformation from d-q frame to γ...

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Unformatted text preview: Lecture 11 Transformation from d-q frame to γ- δ frame In Lecture 10, one notes that in the induction machine the magnetic fluxes produced by the currents of the stator and the rotor rotate around the air-gap at the same angular velocity ω , the synchronous velocity of the supply frequency. The d-q frame axes of the stator and the rotor coincide and are stationary. Their voltages and currents are at the same frequency ω . The objective of transforming to the γ- δ frame is to view the magnetic fluxes as stationary. This is by rotating the axes frame at the same angular velocity ω . The γ- δ frame axes of the stator and the rotor coincide because they have been coinciding in the d-q frame. The voltages and currents associated with the rotating γ- δ frame windings are constant voltages and currents. Having the voltages and currents time-invariant is an important reason for the transformation. The rotational transformation, which is applied to both the stator and the rotor d-q windings, is ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = t t t t t C dq ω ω ω ω ω γδ cos sin sin cos )] ( [ (1) The complete transformation is: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ R S dq dq R dq S dq e e C C e e γδ γδ γδ γδ ω ω )] ( [ )] ( [ (2) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ R S dq dq R dq S dq i i C C i i γδ γδ γδ γδ ω ω )] ( [ )] ( [ (3) Substituting (2) and (3) in the equations in the d-q frame: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + + + + − + + − + + + + = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 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 ) ( ) ( ) ( ) ( ) ( ) ( l l l l l l ω ω ω ω (4) the resultant γ- δ frame equations are: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢...
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Lecture11 - Lecture 11 Transformation from d-q frame to γ...

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