Lecture 12Derivation of Equivalent Circuit from Application of γ-δFrameIn from (20) in Lecture 10, the stator current vector in the d-qframe is ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡++=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡)sin()cos(SSSSqSdttIiiϕωϕω(1) In (8) of Lecture 11, the vector is transformed to the γ-δframe as ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡SSSSSIiiϕϕδγsincos(2) Equations (1) and (2) are related by the transformation matrix as: ⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡−=⎥⎦⎤⎢⎣⎡δγωωωωSSSqSdiittttiicossinsincos(3) One can define a complex number δγSSSjiI+=~and express ]~Re[tjSSdSSaeIiiiωα===(4) In (4), one is reminded that the a-axis, the α-axis and the d-axis of the stator remain in the same position. The γ-axis, however, rotates at the speed of the stator frequency ω. Electrical engineers are familiar with ac voltages and currents expressed as complex numbers, such as SI~, associated with a synchronously rotating unit phasortjeω. This lecture shows how the equivalent circuit of the induction machine is easily derived from (5) of Lecture 11.
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