Transmission Line Power Calculation

This problem involves calculating the average power delivered to a load connected to a lossless transmission line.

Key Parameters

• Characteristic Impedance ($Z_0$): $50 \ \Omega$
• Load Impedance ($Z_L$): $(50 - j75) \ \Omega$
• Incident Power ($P_{inc}$): $10 \ mW$

Reflection Coefficient Calculation

First, calculate the reflection coefficient ($\Gamma$) at the load using the formula:

$ \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} $

Substituting the given values:

$ \Gamma = \frac{(50 - j75) \ \Omega - 50 \ \Omega}{(50 - j75) \ \Omega + 50 \ \Omega} $

$ \Gamma = \frac{-j75 \ \Omega}{100 - j75 \ \Omega} $

Magnitude Squared of Reflection Coefficient

The fraction of power delivered to the load depends on the magnitude squared of the reflection coefficient ($|\Gamma|^2$).

$ |\Gamma|^2 = \frac{|-j75|^2}{|100 - j75|^2} $

Calculate the magnitudes:

• $|-j75| = 75$
• $|100 - j75| = \sqrt{100^2 + (-75)^2} = \sqrt{10000 + 5625} = \sqrt{15625} = 125$

Now, calculate $|\Gamma|^2$:

$ |\Gamma|^2 = \frac{75^2}{125^2} = \frac{5625}{15625} = \frac{9}{25} = 0.36 $

Power Delivered to Load Calculation

For a lossless transmission line, the average power delivered to the load ($P_{load}$) is given by:

$ P_{load} = P_{inc} \times (1 - |\Gamma|^2) $

Substitute the values:

$ P_{load} = 10 \ mW \times (1 - 0.36) $

$ P_{load} = 10 \ mW \times 0.64 $

$ P_{load} = 6.4 \ mW $

Conclusion

The average power delivered to the load is $6.4 \ mW$. This value falls within the range of 6.3 to 6.5 mW.