Input Impedance of Short-Circuited Transmission Lines

Understanding the behavior of transmission lines is crucial in electrical engineering, especially when dealing with high-frequency signals. The input impedance of a transmission line depends on its length, its characteristic impedance, and the load connected at its end. For a specific case, such as a short-circuited line that is a quarter-wavelength long and lossless, we can determine its input impedance.

Formula for Transmission Line Input Impedance

The general formula for the input impedance \(Z_{in}\) of a lossless transmission line with a length \(l\), characteristic impedance \(Z_0\), and load impedance \(Z_L\) is given by:

\[Z_{in} = Z_0 \frac{Z_L + j Z_0 \tan(\beta l)}{Z_0 + j Z_L \tan(\beta l)}\]

Where:

• \(Z_{in}\) is the input impedance looking into the line.
• \(Z_0\) is the characteristic impedance of the transmission line.
• \(Z_L\) is the load impedance connected at the far end.
• \(\beta\) is the phase constant, typically \(\beta = \frac{2\pi}{\lambda}\), where \(\lambda\) is the wavelength.
• \(l\) is the physical length of the transmission line.
• \(j\) is the imaginary unit.

Analyzing a Short-Circuited Quarter-Wavelength Line

Let's apply the formula to the specific conditions given in the question:

1. Short-Circuited Line: For a short-circuited transmission line, the load impedance \(Z_L\) at the end of the line is zero.

   Thus, \(Z_L = 0\).

2. Quarter-Wavelength Line: The length of the transmission line \(l\) is a quarter wavelength.

   Thus, \(l = \frac{\lambda}{4}\).

3. Lossless Transmission Line: A lossless line implies no resistive losses, meaning the characteristic impedance \(Z_0\) is purely resistive, and there is no attenuation.

Step-by-Step Calculation of Input Impedance

Now, substitute the conditions \(Z_L = 0\) and \(l = \frac{\lambda}{4}\) into the input impedance formula:

First, calculate the term \(\beta l\):

\[\beta l = \left(\frac{2\pi}{\lambda}\right) \times \left(\frac{\lambda}{4}\right) = \frac{2\pi}{4} = \frac{\pi}{2}\]

Next, substitute \(Z_L = 0\) and \(\beta l = \frac{\pi}{2}\) into the input impedance formula:

\[Z_{in} = Z_0 \frac{0 + j Z_0 \tan(\frac{\pi}{2})}{Z_0 + j (0) \tan(\frac{\pi}{2})}\]

Simplify the expression:

\[Z_{in} = Z_0 \frac{j Z_0 \tan(\frac{\pi}{2})}{Z_0}\]

We know that \(\tan(\frac{\pi}{2})\) (or \(\tan(90^\circ)\)) approaches infinity (\(\infty\)).

Therefore, substituting this into the equation:

\[Z_{in} = Z_0 \frac{j Z_0 \times \infty}{Z_0}\]

As the numerator approaches infinity, the input impedance \(Z_{in}\) also approaches infinity.

Thus, a short-circuited lossless transmission line quarter wavelength behaves like an open circuit at its input. This property is fundamental to the design of various microwave components, such as resonant stubs and filters.