GATE 2023 EC Q.51: solution and answer

GATE 2023 · ECQ.51 · 2 marks · MCQ

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

[S] = [S_{11} S_{21} S_{12} S_{22}]

with reference to $Z_{0}$ . Two lossless transmission line sections of electrical lengths $θ_{1} = β l_{1}$ and $θ_{2} = β l_{2}$ are added to the input and output ports for measurement purposes, respectively. The S-parameters $[S^{'}]$ of the resultant two port network is

GATE 2023 EC question 51: A box labelled [S] in the centre (the original two-port). — Figure from the GATE 2023 EC paper. It is part of the question.

(A)
$[S_{11} e^{- j 2 θ_{1}} S_{21} e^{- j (θ_{1} + θ_{2})} S_{12} e^{- j (θ_{1} + θ_{2})} S_{22} e^{- j 2 θ_{2}}]$
(B)
$[S_{11} e^{j 2 θ_{1}} S_{21} e^{- j (θ_{1} + θ_{2})} S_{12} e^{- j (θ_{1} + θ_{2})} S_{22} e^{j 2 θ_{2}}]$
(C)
$[S_{11} e^{j 2 θ_{1}} S_{21} e^{j (θ_{1} + θ_{2})} S_{12} e^{j (θ_{1} + θ_{2})} S_{22} e^{j 2 θ_{2}}]$
(D)
$[S_{11} e^{- j 2 θ_{1}} S_{21} e^{j (θ_{1} + θ_{2})} S_{12} e^{j (θ_{1} + θ_{2})} S_{22} e^{- j 2 θ_{2}}]$

Answer and solution

ANSWER

(A): $[S^{'}] = [S_{11} e^{- j 2 θ_{1}} S_{21} e^{- j (θ_{1} + θ_{2})} S_{12} e^{- j (θ_{1} + θ_{2})} S_{22} e^{- j 2 θ_{2}}]$

Official answer: ASource: Official answer key, GATE 2023 EC (IIT Kanpur)

Let $a_{1}^{'}, b_{1}^{'}$ and $a_{2}^{'}, b_{2}^{'}$ be the incident and reflected wave amplitudes at the new reference planes $z_{1} = l_{1}$ and $z_{2} = l_{2}$ , and $a_{1}, b_{1}, a_{2}, b_{2}$ the ones at the original planes $z_{1} = z_{2} = 0$ . The added lines have the same characteristic impedance $Z_{0}$ as the reference impedance, so they add no reflection, only phase.
Phase delay on a matched lossless line: a wave that travels a length $l$ picks up the factor $e^{- j β l}$ . The incident wave travels from the new plane in to the old one, and the reflected wave travels from the old plane out to the new one:
$a_{1} = a_{1}^{'} e^{- j θ_{1}}, b_{1}^{'} = b_{1} e^{- j θ_{1}}, a_{2} = a_{2}^{'} e^{- j θ_{2}}, b_{2}^{'} = b_{2} e^{- j θ_{2}}$
Equivalently, $a_{1}^{'} = a_{1} e^{j θ_{1}}$ and $a_{2}^{'} = a_{2} e^{j θ_{2}}$ .
Input reflection (set $a_{2}^{'} = 0$ , which gives $a_{2} = a_{2}^{'} e^{- j θ_{2}} = 0$ ): the original network gives $b_{1} = S_{11} a_{1}$ . Then
$S_{11}^{'} = \frac{b _{1}^{'}}{a _{1}^{'}} = \frac{S _{11} a _{1} e ^{- j θ_{1}}}{a _{1} e ^{j θ_{1}}} = S_{11} e^{- j 2 θ_{1}}$
The reflected signal crosses line 1 twice: in and back out.
Forward transmission (again $a_{2}^{'} = 0$ , so $a_{2} = 0$ ): $b_{2} = S_{21} a_{1}$ . Then
$S_{21}^{'} = \frac{b _{2}^{'}}{a _{1}^{'}} = \frac{S _{21} a _{1} e ^{- j θ_{2}}}{a _{1} e ^{j θ_{1}}} = S_{21} e^{- j (θ_{1} + θ_{2})}$
The signal crosses line 1 once and line 2 once.
By the same argument with $a_{1}^{'} = 0$ (so $a_{1} = 0$ ), driving from port 2: $S_{22}^{'} = S_{22} e^{- j 2 θ_{2}}$ and $S_{12}^{'} = S_{12} e^{- j (θ_{1} + θ_{2})}$ .
In matrix form:
$[S^{'}] = [S_{11} e^{- j 2 θ_{1}} S_{21} e^{- j (θ_{1} + θ_{2})} S_{12} e^{- j (θ_{1} + θ_{2})} S_{22} e^{- j 2 θ_{2}}]$
The magnitudes are unchanged; only phase lag is added. This is option (A).

KEY IDEA

Moving the reference planes outward along matched lossless lines of electrical lengths $θ_{1}$ and $θ_{2}$ keeps every $∣ S_{ij} ∣$ the same and multiplies each entry by the phase lag of the path its signal travels: $S_{ij}^{'} = S_{ij} e^{- j (θ_{i} + θ_{j})}$ . A reflection crosses its own line twice, giving $e^{- j 2 θ_{i}}$ . A transmission crosses each line once, giving $e^{- j (θ_{1} + θ_{2})}$ .

WATCH OUT FOR

Using $e^{+ j 2 θ}$ on the reflection terms (options B and C). A positive exponent is de-embedding: it moves the reference plane in towards the network, which removes line length. Adding line only ever adds phase lag. A second mistake is to give the transmission terms a different sign from the reflection terms (option D). Every entry picks up delay, because every signal path gets longer.

From the GATE 2023 EC paper. Every question of this paper · Download the full paper · All 13 Transmission lines questions, with notes

GATE 2023 EC Q.51: solution and answer

Answer and solution

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