3.15: Input Impedance of a Terminated Lossless Transmission Line
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Consider Figure \(\PageIndex{1}\), 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.
Figure \(\PageIndex{1}\): A transmission line driven by a source on the left and terminated by an impedance \(Z_L\) at \(z = 0\) on the rightUsing the coordinate system indicated in Figure \(\PageIndex{1}\), the interface between source and transmission line is located at \(z=-l\). Impedance is defined at the ratio of potential to current, so:
\[Z_{in}(l) \triangleq \frac{\widetilde{V}(z=-l)}{\widetilde{I}(z=-l)} \nonumber \]
Now employing expressions for \(\widetilde{V}(z)\) and \(\widetilde{I}(z)\) from Section 3.13 with \(z=-l\), we find:
\begin{aligned}
Z_{i n}(l) &=\frac{V_{0}^{+}\left(e^{+j \beta l}+\Gamma e^{-j \beta l}\right)}{V_{0}^{+}\left(e^{+j \beta l}-\Gamma e^{-j \beta l}\right) / Z_{0}} \\
&=Z_{0} \frac{e^{+j \beta l}+\Gamma e^{-j \beta l}}{e^{+j \beta l}-\Gamma e^{-j \beta l}}
\end{aligned}
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 \ref{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 3.13). 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 \ref{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 \ref{m0087_eGamma} into Equation \ref{m0087_eZin1} and is left as an exercise for the student.
Finally, note that the argument \(\beta l\) appearing Equations \ref{m0087_eZin1} and \ref{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.