Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Related questions
Question
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{](https://content.bartleby.com/qna-images/question/eb010926-0d81-478d-9736-679f94816a5d/7b4ee2e6-d7b5-4de0-9e83-70133fc460d1/e0nsosr_processed.png)
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.
\[
\mathbf{L_s} = \begin{bmatrix}
L_{ls} + L_{ms} & -\frac{1}{2}L_{ms} & -\frac{1}{2}L_{ms} \\
-\frac{1}{2}L_{ms} & L_{ls} + L_{ms} & -\frac{1}{2}L_{ms} \](https://content.bartleby.com/qna-images/question/eb010926-0d81-478d-9736-679f94816a5d/7b4ee2e6-d7b5-4de0-9e83-70133fc460d1/kcldym5c_processed.png)
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} \
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution
Trending nowThis is a popular solution!
Step by stepSolved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, electrical-engineering and related others by exploring similar questions and additional content below.Similar questions
- 8.) Write a logical statement representing each circuit. Simplify the statement as much as possible, and draw the simplified circuit.arrow_forwardDetermine the line currents Ia, Ib and Ic.arrow_forwardThe bus admittance matrix for the power system shown in Figure 1 is given by 3-j9 −2+j6 -1+j3 0 Ybus - -2+j6 3.666-j11 -0.666+j2 −1+j3 -1+j3 -0.666+j2 3.666-j11 per unit −2+j6 0 -1+j3 −2+j6 3-j9 With the complex powers on load buses 2, 3, and 4 as shown in Figure 1. 2.1 Determine the value for V₂ that is produced by the first and second iterations of the Gauss Seidel procedure. Choose the initial guess V₂(0) = V3(0) = V4(0) = 1.0L0° per unit. (12) 2 V₁ = 1.04/0° P₁ + jQ₁ −1+j0.5 Figure 1 V₂ 0.5-10.2 0.3j0.1arrow_forward
- Can someone please explain how to convert V1=-0.4220 to poloar phasor form? (V1=0.578<0)arrow_forwardRefer to the following circuit. Apply Wye - Delta transformation or any other method, to find the voltage VAB. E 120 20°V 60Hz 2 Z₁ 0004 Z₁ 20 6500 A Z₂ 50 Zs 110 ZA 60 Z₂ 330 x Z₂ 30 90 Za 16.50 Barrow_forwardFor each matrix, let row 1 correspond to loop 1, row 2 correspond and negative values for negative voltages. Write a matrix equation that determines the loop currents. 23 V 35 2 30 V 332 20 V 19 V 10 Varrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Introductory Circuit Analysis (13th Edition)Electrical EngineeringISBN:9780133923605Author:Robert L. BoylestadPublisher:PEARSON
Delmar's Standard Textbook Of ElectricityElectrical EngineeringISBN:9781337900348Author:Stephen L. HermanPublisher:Cengage Learning
Programmable Logic ControllersElectrical EngineeringISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education 
Fundamentals of Electric CircuitsElectrical EngineeringISBN:9780078028229Author:Charles K Alexander, Matthew SadikuPublisher:McGraw-Hill Education
Electric Circuits. (11th Edition)Electrical EngineeringISBN:9780134746968Author:James W. Nilsson, Susan RiedelPublisher:PEARSON
Engineering ElectromagneticsElectrical EngineeringISBN:9780078028151Author:Hayt, William H. (william Hart), Jr, BUCK, John A.Publisher:Mcgraw-hill Education,
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:9781337900348
Author:Stephen L. Herman
Publisher:Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:9780073373843
Author:Frank D. Petruzella
Publisher:McGraw-Hill Education
Fundamentals of Electric Circuits
Electrical Engineering
ISBN:9780078028229
Author:Charles K Alexander, Matthew Sadiku
Publisher:McGraw-Hill Education
Electric Circuits. (11th Edition)
Electrical Engineering
ISBN:9780134746968
Author:James W. Nilsson, Susan Riedel
Publisher:PEARSON
Engineering Electromagnetics
Electrical Engineering
ISBN:9780078028151
Author:Hayt, William H. (william Hart), Jr, BUCK, John A.
Publisher:Mcgraw-hill Education,