Lecture7 - 1 Lecture 7 Matrix Manipulations in Reference...

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1 Lecture 7 Matrix Manipulations in Reference Frame Transformation Let the equations in the reference frame-1 be: 1 1 1 1 )] ( [ ] [ ] [ i M dt d dt di i R v θ + + = l (1) In order to reduce the labour of word processing, the N-tuple vectors 1 1 , i v are not underlined. The NxN resistance matrix [R] and the leakage inductance matrix ] [ l are diagonal matrices. But )] ( [ M is a full matrix. It is desired to transform (1) to the reference frame-2 using the )] ( [ ψ C matrix. 2 1 )] ( [ v C v = (2) 2 1 )] ( [ i C i = (3) Substituting (2) and (3) in (1) 2 2 2 2 )] ( )][ ( [ )] ( [ ] [ )] ( ][ [ )] ( [ i C M dt d dt i C d i C R v C + + = l (4) Pre-multiplying (4) by the inverse of )] ( [ C , 2 1 2 1 2 1 2 )] ( )][ ( [ )] ( [ )] ( [ ] [ )] ( [ )] ( ][ [ )] ( [ i C M dt d C dt i C d C i C R C v + + = l (5) The right-hand side of (5) is examined term by term. Term of 2 1 )] ( ][ [ )] ( [ i C R C Consider the case where the resistances are all equal, that is ] [ ] [ I R R R = , where R R is a scalar and ] [ I is the identity matrix. Therefore 2 2 2 1 2 1 2 1 ] [ ] [ )] ( [ )] ( [ )] ( ][ [ )] ( [ )] ( ][ [ )] ( [ i R i I R i C C R i C I C R i C R C R R R = = = = ( 6 ) One concludes that ] [ R is not changed by the transformation. Term of dt i C d C 2 1 )] ( [ ] [ )] ( [ l
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2 One follows the chain rule in the differentiation with respect to t. } )] ( [ )] ( [ ]{ [ )] ( [ )] ( [ ] [ )] ( [ 2 2 1 2 1 dt di C i dt C d C dt i C d C ψ + = l l } )] ( ][ [ )] ( [ )] ( [ ] [ )] ( [ } )] ( [ )] ( [ ]{ [ )] ( [ 2 1 2 1 2 2 1 dt di C C i C C dt d dt di C i dt C d C l l l + = + = The leakage inductance matrix ] [ l is assumed to be the product of a scalar l and an identity matrix ] [ I . Therefore 2 1 2 1 )] ( [ )] ( [ )] ( [ ] [ )] ( [ i C C dt d i C C dt d
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Lecture7 - 1 Lecture 7 Matrix Manipulations in Reference...

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