Question

Given that \( Z_0 = 50\Omega \), \( Z_L = (50 - j75)\Omega \), and \( P_i = 10 \) mW, find the power delivered to the load.

• A. \( 5.2 \) mW
• B. \( 6.4 \) mW
• C. \( 7.8 \) mW
• D. \( 8.6 \) mW

Hint:

In transmission line problems, power delivered is calculated using reflection coefficient, which accounts for power loss due to mismatched impedances.

Answer 1

The Correct Option is B

Step 1: Calculate the Reflection Coefficient.
\[ \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} \]
Substituting values: \[ \Gamma = \frac{(50 - j75) - 50}{(50 - j75) + 50} = \frac{-j75}{100 - j75} \]
Multiplying by the complex conjugate: \[ \Gamma = \frac{5625 - j7500}{15625} \]
\[ |\Gamma| = \sqrt{\left( \frac{5625}{15625} \right)^2 + \left( \frac{-7500}{15625} \right)^2} = 0.6 \]
Step 2: Compute Power Delivered to Load.
The formula for power delivered:
\[ P_L = P_i (1 - |\Gamma|^2) \]
\[ P_L = 10 (1 - 0.36) = 10 \times 0.64 = 6.4 \text{ mW} \]
Thus, the final power delivered is: \[ \mathbf{6.4 \text{ mW}} \]