3.16: Input Impedance for Open- and Short-Circuit Terminations
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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 3.15) 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, ZL=0, Γ=−1, so we find
Zin(l)=Z01+Γe−j2βl1−Γe−j2βl=Z01−e−j2βl1+e−j2βl
Multiplying numerator and denominator by e+jβl we obtain Zin(l)=Z0e+jβl−e−jβle+jβl+e−jβl
cosθ=12[e+jθ+e−jθ]sinθ=1j2[e+jθ−e−jθ]
Employing these identities, we obtain: Zin(l)=Z0j2(sinβl)2(cosβl)
Figure 3.16.1(a) shows what’s going on. As expected, Zin=0 when l=0, since this amounts to a short circuit with no transmission line. Also, Zin varies periodically with increasing length, with period λ/2. This is precisely as expected from standing wave theory (Section 3.13). What is of particular interest now is that as l→λ/4, we see Zin→∞. Remarkably, the transmission line has essentially transformed the short circuit termination into an open circuit!
(a) Short-ciruit termination (ZL=0.
(b) Open-circuit termination (ZL→∞).
Figure 3.16.1: Input reactance (Im{Zin}) of a stub. Re{Zin} is always zero.For an open circuit termination, ZL→∞, Γ=+1, and we find
Zin(l)=Z01+Γe−j2βl1−Γe−j2βl=Z01+e−j2βl1−e−j2βl
Following the same procedure detailed above for the short-circuit case, we find Zin(l)=−jZ0cotβl
Figure 3.16.1(b) shows the result for open-circuit termination. As expected, Zin→∞ for l=0, and the same λ/2 periodicity is observed. What is of particular interest now is that at l=λ/4 we see Zin=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 ??? and ???, respectively.
The input impedance of a short- or open-circuited lossless transmission line alternates between open- (Zin→∞) and short-circuit (Zin=0) conditions with each λ/4-increase in length.