18 August 2023

TRB 2006 - Question 2

A Very lossy $\lambda/4$ long $50\;\Omega$ transmission line is open circuited at the load end. The input impedance measured at the other end of the line is approximately.

a) $0$

b) $50\;\Omega$

c) $\infty$

d) None of these


Correct Answer: Option B

Solution:
As soon as we saw $Z_{in}$ for $\lambda/4$ transmission line, we tempt to use the formula,
$$Z_{in}=Z_0^2/Z_L$$ $$Z_{in}=50^2/\infty$$ $$Z_{in}=0\;\Omega$$

Hence, we choose the option A.

But the above formula is for lossless $\lambda/4$ transmission line. 
In our question, they have asked for very lossy transmission line.

How to approach this question?

Case 1: Lossless transmission line matched with $Z_0$
Since the transmission line is matched, there will be no standing wave and hence $Z_{in}$ will be equal to $Z_0$.

Case 2: Lossless transmission line not matched with $Z_0$
Since the transmission line is not matched, there will be reflections at the load and hence standing wave exists. Whenever standing wave is there, the impedance of the transmission line will vary in accordance voltage and current at that point.

Case 3: Very lossy transmission line not matched with $Z_0$
Since the transmission line is not matched, there will be reflections at the load and hence standing wave exists. But here, standing wave gets attenuated since the transmission line is very lossy. There will be no standing wave at input side. Hence input impedance is equal to characteristic impedance $Z_0$.

Hence the answer is Option B

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