Question 2.4: Evaluate the input admittance of a short line in short and t......
Evaluate the input admittance of a short line in short and the input impedance of a short line in open. Let the p.u.l. impedance and admittance be Z and Y, respectively, and the line length l.
The line complex propagation constant and characteristic impedance are:
\gamma=\sqrt{ Z Y }, \quad Z_0=\sqrt{\frac{ Z }{ Y }},
while the input impedance of a line in open, for l → 0, can be expanded as:
\begin{aligned} Z_{\text {in,open }} & =Z_0 \text{ coth }(\gamma l) \approx Z_0\left(\frac{1}{\gamma l}+\frac{1}{3} \gamma l\right) \\ & =\sqrt{\frac{ Z }{ Y }} \frac{1}{\sqrt{ Z Y } l}+\frac{1}{3} \sqrt{\frac{ Z }{ Y }} \sqrt{ Z Y } l=\frac{1}{ Y l}+\frac{1}{3} Z l, \end{aligned}
i.e., the series of an impedance equal to 1/ (Yl) and of an impedance equal to Zl/3. Notice that the first term is dominant unless the p.u.l. admittances and impedances are one purely real and the other purely imaginary. Consider, e.g., an RC line in open where:
Z = R , \quad Y = j \omega C,
then:
Z_{ \text{in,open} } \approx \frac{1}{ j \omega C l}+\frac{1}{3}Rl
is the series of a resistor and a capacitor. Similarly, for the admittance of a short line in short we have:
\begin{aligned} Y_{\text {in,short }} & =Y_0 \text{ coth }(\gamma l) \approx Y_0\left(\frac{1}{\gamma l}+\frac{1}{3} \gamma l\right) \\ & =\sqrt{\frac{ Y }{ Z }} \frac{1}{\sqrt{ Z Y} l}+\frac{1}{3} \sqrt{\frac{ Y }{ Z }} \sqrt{ Z Y } l=\frac{1}{ Z l}+\frac{1}{3} Y l,\qquad \text{(2.25)} \end{aligned}
i.e., the parallel of an admittance equal to 1/ (Zl) and of an admittance equal to Yl/3.