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GATE 2023 EC – Question 51

Electromagnetics · Transmission Lines · 2 marks · Multiple choice

The S-parameters of a two port network is given as

$$[S]=\begin{bmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{bmatrix}$$

with reference to $Z_0$. Two lossless transmission line sections of electrical lengths $\theta_1=\beta l_1$ and $\theta_2=\beta l_2$ are added to the input and output ports for measurement purposes, respectively. The S-parameters $[S']$ of the resultant two port network is

Diagram for GATE 2023 EC question 51
  1. $\begin{bmatrix}S_{11}e^{-j2\theta_1}&S_{12}e^{-j(\theta_1+\theta_2)}\\S_{21}e^{-j(\theta_1+\theta_2)}&S_{22}e^{-j2\theta_2}\end{bmatrix}$
  2. $\begin{bmatrix}S_{11}e^{j2\theta_1}&S_{12}e^{-j(\theta_1+\theta_2)}\\S_{21}e^{-j(\theta_1+\theta_2)}&S_{22}e^{j2\theta_2}\end{bmatrix}$
  3. $\begin{bmatrix}S_{11}e^{j2\theta_1}&S_{12}e^{j(\theta_1+\theta_2)}\\S_{21}e^{j(\theta_1+\theta_2)}&S_{22}e^{j2\theta_2}\end{bmatrix}$
  4. $\begin{bmatrix}S_{11}e^{-j2\theta_1}&S_{12}e^{j(\theta_1+\theta_2)}\\S_{21}e^{j(\theta_1+\theta_2)}&S_{22}e^{-j2\theta_2}\end{bmatrix}$

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Correct answer: (A) $\begin{bmatrix}S_{11}e^{-j2\theta_1}&S_{12}e^{-j(\theta_1+\theta_2)}\\S_{21}e^{-j(\theta_1+\theta_2)}&S_{22}e^{-j2\theta_2}\end{bmatrix}$

Explanation

Adding a lossless line of electrical length $\theta$ at a port moves the reference plane outward. A wave going out and returning (reflection) picks up the phase $e^{-j\theta}$ twice, so $S_{11}'=S_{11}e^{-j2\theta_1}$ and $S_{22}'=S_{22}e^{-j2\theta_2}$. A transmitted wave crosses both lines once, so $S_{21}'=S_{21}e^{-j(\theta_1+\theta_2)}=S_{12}'$ (for a reciprocal network).