Understanding Reflection Coefficient Calculation

The reflection coefficient ($\Gamma$) at a load ($Z_L$) in a transmission line with characteristic impedance ($Z_0$) is given by the formula:

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

We need to match the load impedance types in List-I with their corresponding reflection coefficient values in List-II.

Matching Load Impedance with Reflection Coefficient

Case A: Short Circuit

For a short circuit, the load impedance is zero, i.e., $Z_L = 0$.

Calculating the reflection coefficient:

$ \Gamma = \frac{0 - Z_0}{0 + Z_0} = \frac{-Z_0}{Z_0} = -1 $

So, Short circuit matches with value -1 (List-II: II).

Case B: Open Circuit

For an open circuit, the load impedance is infinite, i.e., $Z_L = \infty$.

To calculate $\Gamma$ when $Z_L = \infty$, we consider the limit:

$ \Gamma = \lim_{Z_L \to \infty} \frac{Z_L - Z_0}{Z_L + Z_0} = \lim_{Z_L \to \infty} \frac{1 - Z_0/Z_L}{1 + Z_0/Z_L} = \frac{1 - 0}{1 + 0} = 1 $

So, Open circuit matches with value +1 (List-II: III).

Case C: Line Characteristic Impedance

When the load impedance equals the line characteristic impedance, $Z_L = Z_0$.

Calculating the reflection coefficient:

$ \Gamma = \frac{Z_0 - Z_0}{Z_0 + Z_0} = \frac{0}{2Z_0} = 0 $

So, Line characteristic impedance matches with value 0 (List-II: I).

Case D: 2 times Line Characteristic Impedance

Here, the load impedance is $Z_L = 2Z_0$.

Calculating the reflection coefficient:

$ \Gamma = \frac{2Z_0 - Z_0}{2Z_0 + Z_0} = \frac{Z_0}{3Z_0} = \frac{1}{3} $

So, 2 times line characteristic impedance matches with value +1/3 (List-II: IV).

Final Matching

Based on the calculations:

• A (Short circuit) matches with II (-1)
• B (Open circuit) matches with III (+1)
• C (Line characteristic impedance) matches with I (0)
• D (2 times line characteristic impedance) matches with IV (+1/3)

The correct matching is A-II, B-III, C-I, D-IV.