Transmission Line Input Impedance Calculation

This problem involves finding the input impedance ($Z_{in}$) of a dissipation-less transmission line given its characteristic impedance ($Z_0$), load impedance ($Z_L$), frequency ($f$), and length ($l$). The result needs to match one of the provided options, which appear to represent the magnitude of the impedance.

Given Parameters

• Characteristic Impedance: $Z_0 = 200 \, \Omega$
• Load Impedance: $Z_L = (100 + j100) \, \Omega$
• Frequency: $f = 300 \, \text{MHz} = 3 \times 10^8 \, \text{Hz}$
• Line Length: $l = 25 \, \text{cm} = 0.25 \, \text{m}$
• Speed of Light (in assumed lossless medium): $c \approx 3 \times 10^8 \, \text{m/s}$

Calculate Wavelength and Electrical Length

1. Calculate the phase constant ($\beta$): $ \beta = \frac{2 \pi f}{c} = \frac{2 \pi (3 \times 10^8 \, \text{Hz})}{3 \times 10^8 \, \text{m/s}} = 2 \pi \, \text{rad/m} $
2. Calculate the electrical length ($\beta l$): $ \beta l = (2 \pi \, \text{rad/m}) \times (0.25 \, \text{m}) = \frac{\pi}{2} \, \text{rad} $
3. Interpretation: An electrical length of $\frac{\pi}{2}$ radians corresponds to a quarter of a wavelength ($\lambda/4$).

Input Impedance Formula for Quarter-Wave Line

For a dissipation-less transmission line with length $l = \lambda/4$, the input impedance can be calculated using the relationship:

This formula arises from the general transmission line formula when $\tan(\beta l)$ approaches infinity.

Calculate Input Impedance ($Z_{in}$)

1. Substitute the values into the formula: $ Z_{in} = \frac{(200 \, \Omega)^2}{(100 + j100) \, \Omega} = \frac{40000}{100(1 + j)} \, \Omega $
2. Simplify the expression: $ Z_{in} = \frac{400}{1 + j} \, \Omega $
3. Rationalize the denominator: $ Z_{in} = \frac{400}{1 + j} \times \frac{1 - j}{1 - j} \, \Omega = \frac{400(1 - j)}{1^2 - (j)^2} \, \Omega = \frac{400(1 - j)}{1 - (-1)} \, \Omega = \frac{400(1 - j)}{2} \, \Omega $
4. Final complex impedance: $ Z_{in} = 200(1 - j) \, \Omega = (200 - j200) \, \Omega $

Determine Magnitude for Option Matching

The options provided are real numbers, suggesting the magnitude of the input impedance is required.

• Calculate the magnitude: $ |Z_{in}| = |200 - j200| \, \Omega = \sqrt{200^2 + (-200)^2} \, \Omega $ $ |Z_{in}| = \sqrt{40000 + 40000} \, \Omega = \sqrt{80000} \, \Omega $ $ |Z_{in}| = \sqrt{40000 \times 2} \, \Omega = 200\sqrt{2} \, \Omega \approx 282.84 \, \Omega $

The calculated magnitude, approximately 282.84 ohms, closely matches Option B.