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

[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:

&=
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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.