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2n cos e + 3 2n cos e cos e 3 2 sin 0 3 sin 0 3 2n sin 0+ 3 K, 1 1 1 cose sine 1 (K,)- =| cos| 0 sin 0- 3 1 3 %3D cos 0+ sin 0+ 3 3 Lis + Lms Lms - Lms 2 L, = 1 - Lms s + Lms 1 Lms -- --Lms Lis + Lms Lis + LM K,L,(K,)-' = Lis + LM Lis

2n cos e + 3 2n cos e cos e 3 2 sin 0 3 sin 0 3 2n sin 0+ 3 K, 1 1 1 cose sine 1 (K,)- =| cos| 0 sin 0- 3 1 3 %3D cos 0+ sin 0+ 3 3 Lis + Lms Lms - Lms 2 L, = 1 - Lms s + Lms 1 Lms -- --Lms Lis + Lms Lis + LM K,L,(K,)-' = Lis + LM Lis

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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 mutual leakage terms for flux linkage 

Ks, Ks^-1 , Ls

If I multiply Ks by it inverse I should get the intentity matrix, then If I muliply by Ls matrix I should get the Ls matrix right.

I did the matrix but It doenst not become the indenty matrix multpy by Ls and I also I have no Idea how the anwer would Lis at the last term of the last matrix on the first picture. 

## Matrix Multiplication and Transformation ### Original Matrix Components The matrices involved here represent transformations using trigonometric functions and constants. They appear to be part of calculations in electrical engineering, possibly related to inductance and angular relationships. #### Components 1. **Trigonometric Functions:** - \( \cos(\Theta) \) - \( \sin(\Theta) \) - \( \cos(\Theta - \frac{2\pi}{3}) \) - \( \sin(\Theta - \frac{2\pi}{3}) \) - \( \cos(\Theta + \frac{2\pi}{3}) \) - \( \sin(\Theta + \frac{2\pi}{3}) \) 2. **Inductance Components:** - \(L_{is} + L_{ms}\) - \(-\frac{1}{2}L_{ms}\) ### Initial Matrix The initial interaction is shown through two separate matrices being multiplied: - **First Matrix:** \[ \begin{bmatrix} \cos(\Theta) & \cos(\Theta - \frac{2\pi}{3}) & \cos(\Theta + \frac{2\pi}{3}) \\ \sin(\Theta) & \sin(\Theta - \frac{2\pi}{3}) & \sin(\Theta + \frac{2\pi}{3}) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \] - **Second Matrix:** \[ \begin{bmatrix} L_{is} + L_{ms} & -\frac{1}{2}L_{ms} & -\frac{1}{2}L_{ms} \\ -\frac{1}{2}L_{ms} & L_{is} + L_{ms} & -\frac{1}{2}L_{ms} \\ -\frac{1}{2}L_{ms} & -\frac{1}{2}L_{ms} & L_{is} + L_{ms} \end{bmatrix} \] ### Resulting Matrix The multiplication of these matrices results in a new matrix represented in MatrixForm: \[ \left( \begin{
Transcribed Image Text:## Matrix Multiplication and Transformation ### Original Matrix Components The matrices involved here represent transformations using trigonometric functions and constants. They appear to be part of calculations in electrical engineering, possibly related to inductance and angular relationships. #### Components 1. **Trigonometric Functions:** - \( \cos(\Theta) \) - \( \sin(\Theta) \) - \( \cos(\Theta - \frac{2\pi}{3}) \) - \( \sin(\Theta - \frac{2\pi}{3}) \) - \( \cos(\Theta + \frac{2\pi}{3}) \) - \( \sin(\Theta + \frac{2\pi}{3}) \) 2. **Inductance Components:** - \(L_{is} + L_{ms}\) - \(-\frac{1}{2}L_{ms}\) ### Initial Matrix The initial interaction is shown through two separate matrices being multiplied: - **First Matrix:** \[ \begin{bmatrix} \cos(\Theta) & \cos(\Theta - \frac{2\pi}{3}) & \cos(\Theta + \frac{2\pi}{3}) \\ \sin(\Theta) & \sin(\Theta - \frac{2\pi}{3}) & \sin(\Theta + \frac{2\pi}{3}) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \] - **Second Matrix:** \[ \begin{bmatrix} L_{is} + L_{ms} & -\frac{1}{2}L_{ms} & -\frac{1}{2}L_{ms} \\ -\frac{1}{2}L_{ms} & L_{is} + L_{ms} & -\frac{1}{2}L_{ms} \\ -\frac{1}{2}L_{ms} & -\frac{1}{2}L_{ms} & L_{is} + L_{ms} \end{bmatrix} \] ### Resulting Matrix The multiplication of these matrices results in a new matrix represented in MatrixForm: \[ \left( \begin{
### Transformation Matrices and Inductance Matrix in Electrical Engineering #### Clarke Transformation Matrix (\( \mathbf{K_s} \)) The Clarke transformation matrix, \( \mathbf{K_s} \), is used to transform three-phase coordinates into two-phase orthogonal coordinates. This matrix is defined as: \[ \mathbf{K_s} = \frac{2}{3} \begin{bmatrix} \cos \theta & \cos \left( \theta - \frac{2\pi}{3} \right) & \cos \left( \theta + \frac{2\pi}{3} \right) \\ \sin \theta & \sin \left( \theta - \frac{2\pi}{3} \right) & \sin \left( \theta + \frac{2\pi}{3} \right) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \] #### Inverse Clarke Transformation (\( (\mathbf{K_s})^{-1} \)) The inverse of the Clarke transformation matrix converts two-phase orthogonal coordinates back to three-phase coordinates. \[ (\mathbf{K_s})^{-1} = \begin{bmatrix} \cos \theta & \sin \theta & 1 \\ \cos \left( \theta - \frac{2\pi}{3} \right) & \sin \left( \theta - \frac{2\pi}{3} \right) & 1 \\ \cos \left( \theta + \frac{2\pi}{3} \right) & \sin \left( \theta + \frac{2\pi}{3} \right) & 1 \end{bmatrix} \] #### Stator Self and Mutual Inductance Matrix (\( \mathbf{L_s} \)) This matrix represents the self and mutual inductances in a three-phase system. \[ \mathbf{L_s} = \begin{bmatrix} L_{ls} + L_{ms} & -\frac{1}{2}L_{ms} & -\frac{1}{2}L_{ms} \\ -\frac{1}{2}L_{ms} & L_{ls} + L_{ms} & -\frac{1}{2}L_{ms} \
Transcribed Image Text:### Transformation Matrices and Inductance Matrix in Electrical Engineering #### Clarke Transformation Matrix (\( \mathbf{K_s} \)) The Clarke transformation matrix, \( \mathbf{K_s} \), is used to transform three-phase coordinates into two-phase orthogonal coordinates. This matrix is defined as: \[ \mathbf{K_s} = \frac{2}{3} \begin{bmatrix} \cos \theta & \cos \left( \theta - \frac{2\pi}{3} \right) & \cos \left( \theta + \frac{2\pi}{3} \right) \\ \sin \theta & \sin \left( \theta - \frac{2\pi}{3} \right) & \sin \left( \theta + \frac{2\pi}{3} \right) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \] #### Inverse Clarke Transformation (\( (\mathbf{K_s})^{-1} \)) The inverse of the Clarke transformation matrix converts two-phase orthogonal coordinates back to three-phase coordinates. \[ (\mathbf{K_s})^{-1} = \begin{bmatrix} \cos \theta & \sin \theta & 1 \\ \cos \left( \theta - \frac{2\pi}{3} \right) & \sin \left( \theta - \frac{2\pi}{3} \right) & 1 \\ \cos \left( \theta + \frac{2\pi}{3} \right) & \sin \left( \theta + \frac{2\pi}{3} \right) & 1 \end{bmatrix} \] #### Stator Self and Mutual Inductance Matrix (\( \mathbf{L_s} \)) This matrix represents the self and mutual inductances in a three-phase system. \[ \mathbf{L_s} = \begin{bmatrix} L_{ls} + L_{ms} & -\frac{1}{2}L_{ms} & -\frac{1}{2}L_{ms} \\ -\frac{1}{2}L_{ms} & L_{ls} + L_{ms} & -\frac{1}{2}L_{ms} \
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