A four-port network has the scattering matrix shown below. a. Is this network lossless? b. Is this network reciprocal? c. What is the return loss at port 1 when all the other ports are matched? d. What is the scattering matrix between ports 2 and 4 when all other ports are matched? e. What is the reflection coefficient seen at port 1 if a short circuit is placed at the terminal plane of port 3 and all other ports are matched? 0.7074–45° 0.1290° 0.70745 0.7072-45 0.707445 [s] = 0.7072–45 %3D 0.7072–45° 0.707245 0.7072–45
A four-port network has the scattering matrix shown below. a. Is this network lossless? b. Is this network reciprocal? c. What is the return loss at port 1 when all the other ports are matched? d. What is the scattering matrix between ports 2 and 4 when all other ports are matched? e. What is the reflection coefficient seen at port 1 if a short circuit is placed at the terminal plane of port 3 and all other ports are matched? 0.7074–45° 0.1290° 0.70745 0.7072-45 0.707445 [s] = 0.7072–45 %3D 0.7072–45° 0.707245 0.7072–45
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![### Scattering Matrix for a Four-Port Network
A four-port network has the scattering matrix shown below.
#### Questions:
a. **Is this network lossless?**
b. **Is this network reciprocal?**
c. **What is the return loss at port 1 when all the other ports are matched?**
d. **What is the scattering matrix between ports 2 and 4 when all other ports are matched?**
e. **What is the reflection coefficient seen at port 1 if a short circuit is placed at the terminal plane of port 3 and all other ports are matched?**
#### Scattering Matrix:
\[
[S] = \begin{bmatrix}
0.1 \angle 90^\circ & 0.707 \angle -45^\circ & 0.707 \angle 45^\circ & 0 \\
0.707 \angle -45^\circ & 0 & 0 & 0.707 \angle 45^\circ \\
0.707 \angle -45^\circ & 0 & 0 & 0.707 \angle -45^\circ \\
0 & 0.707 \angle 45^\circ & 0.707 \angle -45^\circ & 0 \\
\end{bmatrix}
\]
### Explanation:
- **Lossless Network:** A network is considered lossless if the scattering matrix is unitary. To determine this, check if the sum of the magnitudes of squares of each row (or column) equals 1.
- **Reciprocal Network:** A network is reciprocal if the scattering matrix is symmetric, meaning \( S_{ij} = S_{ji} \).
- **Return Loss at Port 1:** The return loss is calculated using the S-parameter \( S_{11} \).
- **Scattering Matrix between Ports 2 and 4:** This involves calculating the submatrix elements by eliminating the contributions from ports other than 2 and 4.
- **Reflection Coefficient at Port 1 with Short Circuit at Port 3:** Consider the new boundary condition when port 3 is short-circuited to find the reflection coefficient at port 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5a81e35a-1066-4a16-bd9f-85f7524c2bd1%2Ffa2c0b76-2627-4e9b-99ed-8ecf6544ea34%2F5z7ojc_processed.png&w=3840&q=75)
Transcribed Image Text:### Scattering Matrix for a Four-Port Network
A four-port network has the scattering matrix shown below.
#### Questions:
a. **Is this network lossless?**
b. **Is this network reciprocal?**
c. **What is the return loss at port 1 when all the other ports are matched?**
d. **What is the scattering matrix between ports 2 and 4 when all other ports are matched?**
e. **What is the reflection coefficient seen at port 1 if a short circuit is placed at the terminal plane of port 3 and all other ports are matched?**
#### Scattering Matrix:
\[
[S] = \begin{bmatrix}
0.1 \angle 90^\circ & 0.707 \angle -45^\circ & 0.707 \angle 45^\circ & 0 \\
0.707 \angle -45^\circ & 0 & 0 & 0.707 \angle 45^\circ \\
0.707 \angle -45^\circ & 0 & 0 & 0.707 \angle -45^\circ \\
0 & 0.707 \angle 45^\circ & 0.707 \angle -45^\circ & 0 \\
\end{bmatrix}
\]
### Explanation:
- **Lossless Network:** A network is considered lossless if the scattering matrix is unitary. To determine this, check if the sum of the magnitudes of squares of each row (or column) equals 1.
- **Reciprocal Network:** A network is reciprocal if the scattering matrix is symmetric, meaning \( S_{ij} = S_{ji} \).
- **Return Loss at Port 1:** The return loss is calculated using the S-parameter \( S_{11} \).
- **Scattering Matrix between Ports 2 and 4:** This involves calculating the submatrix elements by eliminating the contributions from ports other than 2 and 4.
- **Reflection Coefficient at Port 1 with Short Circuit at Port 3:** Consider the new boundary condition when port 3 is short-circuited to find the reflection coefficient at port 1.
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