Question

The impedance matching network shown in the figure is to match a lossless line having characteristic impedance \( Z_0 = 50 \, \Omega \) with a load impedance \( Z_L \). A quarter-wave line having a characteristic impedance \( Z_1 = 75 \, \Omega \) is connected to \( Z_L \). Two stubs having characteristic impedance of \( 75 \, \Omega \) each are connected to this quarter-wave line. One is a short-circuited (S.C.) stub of length \( 0.25 \lambda \) connected across PQ and the other one is an open-circuited (O.C.) stub of length \( 0.5 \lambda \) connected across RS. The impedance matching is achieved when the real part of \( Z_L \) is: 

• A. 112.5 \( \Omega \)
• B. 75.0 \( \Omega \)
• C. 50.0 \( \Omega \)
• D. 33.3 \( \Omega \)

Hint:

For impedance matching using a quarter-wave line, the real part of the load impedance should be calculated based on the impedance of the quarter-wave line and the characteristic impedance of the stubs.

Answer 1

The Correct Option is A

We are given a network with a quarter-wave transformer and two stubs: a short-circuited stub and an open-circuited stub. To achieve impedance matching, the real part of the load impedance \( Z_L \) needs to be determined. The quarter-wave transformer transforms the impedance at the load \( Z_L \) by a factor of \( Z_1 \), the characteristic impedance of the quarter-wave line. Step 1: Impedance transformation due to quarter-wave line.
The relationship between the load impedance \( Z_L \) and the characteristic impedance \( Z_1 \) of the quarter-wave line is given by: \[ Z_L = Z_1 \left( \frac{Z_0}{Z_1} \right) = 75 \times 1.5 = 112.5 \, \Omega. \] Step 2: Conclusion.
For the impedance matching to be achieved, the real part of \( Z_L \) must be \( 112.5 \, \Omega \). Final Answer: \[ \boxed{112.5 \, \Omega}. \]