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3.16: Input Impedance for Open- and Short-Circuit Terminations

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  • [m0088_Input_Impedance_for_Open_and_Short_Circuit_Terminations]

    Let us now consider the input impedance of a transmission line that is terminated in an open- or short-circuit. Such a transmission line is sometimes referred to as a stub. First, why consider such a thing? From a “lumped element” circuit theory perspective, this would not seem to have any particular application. However, the fact that this structure exhibits an input impedance that depends on length (Section [m0087_Input_Impedance_of_a_Terminated_Lossless_Transmission_Line]) enables some very useful applications.

    First, let us consider the question at hand: What is the input impedance when the transmission line is open- or short-circuited? For a short circuit, \(Z_L=0\), \(\Gamma=-1\), so we find Multiplying numerator and denominator by \(e^{+j\beta l}\) we obtain \[Z_{in}(l) = Z_0 \frac{ e^{+j\beta l} - e^{-j\beta l} }{ e^{+j\beta l} + e^{-j\beta l} }\] Now we invoke the following trigonometric identities:


    Employing these identities, we obtain: \[Z_{in}(l) = Z_0 \frac{ j2\left(\sin\beta l\right) }{ 2\left(\cos\beta l\right) }\] and finally: \[\boxed{ Z_{in}(l) = +jZ_0 \tan \beta l } \label{m0088_eZstubSC}\]

    Figure [m0088_fZin](a) shows what’s going on. As expected, \(Z_{in}=0\) when \(l=0\), since this amounts to a short circuit with no transmission line. Also, \(Z_{in}\) varies periodically with increasing length, with period \(\lambda/2\). This is precisely as expected from standing wave theory (Section [m0086_Standing_Waves]). What is of particular interest now is that as \(l\rightarrow\lambda/4\), we see \(Z_{in}\rightarrow\infty\). Remarkably, the transmission line has essentially transformed the short circuit termination into an open circuit!

    (a) Short-circuit termination (\(Z_L=0\)).

    (b) Open-circuit termination (\(Z_L\rightarrow\infty\)).

    For an open circuit termination, \(Z_L\rightarrow\infty\), \(\Gamma=+1\), and we find Following the same procedure detailed above for the short-circuit case, we find \[\boxed{ Z_{in}(l) = -jZ_0 \cot \beta l } \label{m0088_eZstubOC}\]

    Figure [m0088_fZin](b) shows the result for open-circuit termination. As expected, \(Z_{in}\rightarrow\infty\) for \(l=0\), and the same \(\lambda/2\) periodicity is observed. What is of particular interest now is that at \(l=\lambda/4\) we see \(Z_{in}=0\). In this case, the transmission line has transformed the open circuit termination into a short circuit.

    Now taking stock of what we have determined:

    The input impedance of a short- or open-circuited lossless transmission line is completely imaginary-valued and is given by Equations [m0088_eZstubSC] and [m0088_eZstubOC], respectively.

    The input impedance of a short- or open-circuited lossless transmission line alternates between open- (\(Z_{in}\rightarrow\infty\)) and short-circuit (\(Z_{in}=0\)) conditions with each \(\lambda/4\)-increase in length.