Transmission Line Impedance Calculation

Understanding the behavior of transmission lines, especially when they are terminated with a load different from their characteristic impedance, is crucial in electrical and communication engineering. This scenario leads to the formation of standing waves, causing the impedance to vary along the length of the line. Our goal is to determine the minimum impedance that can be measured on such a line.

Line Parameters

Let's first list the given parameters for this transmission line problem:

• Characteristic Impedance (\(Z_0\)): This is an inherent property of the transmission line, representing the impedance it would have if it were infinitely long. For this problem, \(Z_0 = 50{\rm{\;\Omega }}\).
• Load Resistance (\(Z_L\)): This is the resistance at the end of the transmission line, terminating the line. Here, \(Z_L = 100{\rm{\;\Omega }}\).

Since the load resistance (\(100{\rm{\;\Omega }}\)) is not equal to the characteristic impedance (\(50{\rm{\;\Omega }}\)), there is an impedance mismatch, and standing waves will be present on the line.

Standing Wave Ratio (SWR) Determination

The Standing Wave Ratio (SWR) is a dimensionless quantity that quantifies the impedance match of a load to a transmission line. It is a measure of the ratio of the maximum to the minimum voltage or current on a line that contains standing waves. For a purely resistive load, the SWR is calculated as the ratio of the larger impedance to the smaller impedance, ensuring SWR ≥ 1:

If \(Z_L > Z_0\), then \[SWR = \frac{Z_L}{Z_0}\]

If \(Z_0 > Z_L\), then \[SWR = \frac{Z_0}{Z_L}\]

In this specific case, \(Z_L = 100{\rm{\;\Omega }}\) and \(Z_0 = 50{\rm{\;\Omega }}\). Since \(Z_L > Z_0\), we apply the first formula:

\[SWR = \frac{100{\rm{\;\Omega }}}{50{\rm{\;\Omega }}}\]

\[SWR = 2\]

A SWR of 2 indicates that the peak voltage (or current) is twice the minimum voltage (or current) on the line.

Minimum Impedance Calculation

The impedance along a mismatched transmission line varies. The minimum impedance (\(Z_{min}\)) occurs at locations where the voltage is at its minimum and the current is at its maximum. It can be calculated using the characteristic impedance (\(Z_0\)) and the Standing Wave Ratio (SWR) with the following formula:

\[Z_{min} = \frac{Z_0}{SWR}\]

Now, we substitute the values we have:

\[Z_{min} = \frac{50{\rm{\;\Omega }}}{2}\]

\[Z_{min} = 25{\rm{\;\Omega }}\]

Thus, the minimum impedance measured on the transmission line is \(25{\rm{\;\Omega }}\). This value is less than the characteristic impedance because the load resistance is higher than the characteristic impedance, resulting in a reflection that creates points of minimum impedance.