Question 9.7: A lossless transmission line is known to be shorter than 10[......
A lossless transmission line is known to be shorter than 10[m] in length. If it is open-circuited, the input impedance is given by Z^{o}_{in} = – j 81[\Omega ]. If shortcircuited, the input impedance is given by Z^{s}_{in} = j 144[\Omega ], and the first voltage maximum is observed at a distance 2.5[m] from the shorted end. Find
(a) R_{o}, (b) β, and (c) \ell .
(a) The characteristic impedance is, from Eq. (9-61a),
R_{o}= \sqrt{Z^{s}_{in} Z^{o}_{in}} . [Ω] (9-61a)
R_{o}= \sqrt{Z^{s}_{in} Z^{o}_{in}} = \sqrt{(j144)(- j 81)} = 108[\Omega ].
(b) From Eq. (9-42), the reflection coefficient of a short circuit ( Z_{L} = 0 ) is obtained as \Gamma = -1 = e^{j\pi }. Thus we obtain Φ = π.
\boxed{\Gamma = \frac{Z_{L} – Z_{o}}{Z_{L} + Z_{o}} \equiv \left|\Gamma \right| e^{j\phi }} (9-42)
Substituting Φ = π , m = 0 , and \ell^{\prime }_{max} = 2.5 [m] into Eq. (9-47a), we get
\boxed{\ell^{\prime }_{max} = \frac{1}{2\beta } (\phi + 2 m\pi )} (m = 0,1,2..) (9-47a)
β = 0.2 π. (9-62)
(c) From Eq. (9-61b) we get
\beta \ell = \tan ^{-1} \sqrt{\left|\frac{Z^{s}_{in}}{Z^{o}_{in}}\right| } + n \pi n = ( 0,1,2..) (9-61b)
\beta \ell = \tan ^{-1} \sqrt{144/81} + n\pi = 0.93 +n\pi (9-63)
Inserting Eq. (9-62) into Eq. (9-63), the line length is
\ell = (0.93 + n π)/(0.2π )
= 1.48[m], 6.48[m], 11.48[m],…
The length \ell should be longer than 2.5[m] , but shorter than 10[m] .
Therefore, we have \ell = 6.48[m].