# 3.15: Input Impedance of a Terminated Lossless Transmission Line

- Page ID
- 6281

[m0087_Input_Impedance_of_a_Terminated_Lossless_Transmission_Line]

Consider Figure [m0087_fZinTerminatedTL], which shows a lossless transmission line being driven from the left and which is terminated by an impedance \(Z_L\) on the right. If \(Z_L\) is equal to the characteristic impedance \(Z_0\) of the transmission line, then the input impedance \(Z_{in}\) will be equal to \(Z_L\). Otherwise \(Z_{in}\) depends on both \(Z_L\) and the characteristics of the transmission line. In this section, we determine a general expression for \(Z_{in}\) in terms of \(Z_L\), \(Z_0\), the phase propagation constant \(\beta\), and the length \(l\) of the line.

Using the coordinate system indicated in Figure [m0087_fZinTerminatedTL], the interface between source and transmission line is located at \(z=-l\). Impedance is defined at the ratio of potential to current, so: \[Z_{in}(l) \triangleq \frac{\widetilde{V}(z=-l)}{\widetilde{I}(z=-l)}\] Now employing expressions for \(\widetilde{V}(z)\) and \(\widetilde{I}(z)\) from Section [m0086_Standing_Waves] with \(z=-l\), we find: Multiplying both numerator and denominator by \(e^{-j\beta l}\): \[\boxed{ Z_{in}(l) = Z_0 \frac{ 1 + \Gamma e^{-j2\beta l} }{ 1 - \Gamma e^{-j2\beta l} } } \label{m0087_eZin1}\] Recall that \(\Gamma\) in the above expression is: \[\Gamma = \frac{ Z_L-Z_0 }{ Z_L+Z_0} \label{m0087_eGamma}\]

Summarizing:

Equation [m0087_eZin1] is the input impedance of a lossless transmission line having characteristic impedance \(Z_0\) and which is terminated into a load \(Z_L\). The result also depends on the length and phase propagation constant of the line.

Note that \(Z_{in}(l)\) is periodic in \(l\). Since the argument of the complex exponential factors is \(2\beta l\), the frequency at which \(Z_{in}(l)\) varies is \(\beta/\pi\); and since \(\beta=2\pi/\lambda\), the associated period is \(\lambda/2\). This is very useful to keep in mind because it means that all possible values of \(Z_{in}(l)\) are achieved by varying \(l\) over \(\lambda/2\). In other words, changing \(l\) by more than \(\lambda/2\) results in an impedance which could have been obtained by a smaller change in \(l\). Summarizing to underscore this important idea:

The input impedance of a terminated lossless transmission line is periodic in the length of the transmission line, with period \(\lambda/2\).

Not surprisingly, \(\lambda/2\) is also the period of the standing wave (Section [m0086_Standing_Waves]). This is because – once again – the variation with length is due to the interference of incident and reflected waves.

Also worth noting is that Equation [m0087_eZin1] can be written entirely in terms of \(Z_L\) and \(Z_0\), since \(\Gamma\) depends only on these two parameters. Here’s that version of the expression: \[Z_{in}(l) = Z_0 \left[ \frac{ Z_L + jZ_0\tan\beta l }{ Z_0 + jZ_L\tan\beta l } \right] \label{m0087_eZin2}\] This expression can be derived by substituting Equation [m0087_eGamma] into Equation [m0087_eZin1] and is left as an exercise for the student.

Finally, note that the argument \(\beta l\) appearing Equations [m0087_eZin1] and [m0087_eZin2] has units of radians and is referred to as *electrical length*. Electrical length can be interpreted as physical length expressed with respect to wavelength and has the advantage that analysis can be made independent of frequency.