Transmission Line Calculations: Reflection Coefficient and SWR

The problem requires calculating the voltage reflection coefficient ($\Gamma$) and the Standing Wave Ratio (SWR) for a transmission line with a given characteristic impedance ($Z_0$) connected to a specific load impedance ($Z_L$).

Calculate Voltage Reflection Coefficient ($\Gamma$)

The voltage reflection coefficient ($\Gamma$) quantifies the ratio of the reflected voltage wave to the incident voltage wave at the load. For a low-loss line, it is calculated using the load impedance ($Z_L$) and the characteristic impedance ($Z_0$) with the following formula:

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

Given:

• Characteristic Impedance, $Z_0 = 50 \Omega$
• Load Impedance, $Z_L = 100 \Omega$

Substitute the values into the formula:

$ \Gamma = \frac{100 \Omega - 50 \Omega}{100 \Omega + 50 \Omega} = \frac{50 \Omega}{150 \Omega} = \frac{1}{3} $

The voltage reflection coefficient is $1/3$.

Calculate Standing Wave Ratio (SWR)

The Standing Wave Ratio (SWR), also known as the Voltage Standing Wave Ratio (VSWR), measures the ratio of the maximum amplitude to the minimum amplitude of the standing wave on the line. It is related to the magnitude of the reflection coefficient ($|\Gamma|$). The formula is:

$ SWR = \frac{1 + |\Gamma|}{1 - |\Gamma|} $

Using the calculated reflection coefficient $\Gamma = 1/3$:

$ |\Gamma| = \left| \frac{1}{3} \right| = \frac{1}{3} $

Now, calculate the SWR:

$ SWR = \frac{1 + \frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{3}{3} + \frac{1}{3}}{\frac{3}{3} - \frac{1}{3}} = \frac{\frac{4}{3}}{\frac{2}{3}} = \frac{4}{3} \times \frac{3}{2} = 2 $

The Standing Wave Ratio (SWR) is 2.

Final Answer

The voltage reflection coefficient ($\Gamma$) is $1/3$, and the Standing Wave Ratio (SWR) is 2. Therefore, the correct option is ($1/3$ and 2).