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 )] ( [ θ 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 ψ = (2) 2 1 )] ( [ i C i ψ = (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 ψ θ ψ ψ ψ + + = l (4) Pre-multiplying (4) by the inverse of )] ( [ ψ C , 2 1 2 1 2 1 2 )] ( )][ ( [ )] ( [ )] ( [ ] [ )] ( [ )] ( ][ [ )] ( [ i C M dt d C dt i C d C i C R C v ψ θ ψ ψ ψ ψ ψ + + = l (5) The right-hand side of (5) is examined term by term. Term of 2 1 )] ( ][ [ )] ( [ i C R C ψ ψ 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 = = = = ψ ψ ψ ψ ψ ψ (6) One concludes that ] [ R is not changed by the transformation. Term of dt i C d C 2 1 )] ( [ ] [ )] ( [ ψ ψ l
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