Question 9.6: A lossless line of characteristic resistance Ro is terminate......
A lossless line of characteristic resistance R_{o} is terminated in a load impedance Z_{L}. The waves on the line have a phase constant β. At a distance \ell ^{\prime} from the load toward the generator, the input impedance looking toward the load is defined as Z_{in} (\ell^{\prime }) \equiv V (\ell^{\prime }) / I(\ell^{\prime }) , and the reflection coefficient is defined as \Gamma (\ell^{\prime }) = V^{-} (\ell^{\prime }) / V^{+}(\ell^{\prime }). Find
(a) expression for \Gamma (\ell^{\prime }),
(b), \Gamma (\ell^{\prime }) at the locations of V_{max} and V_{min}, and
(c) Z_{in} (\ell^{\prime }) at the locations of V_{max} and V_{min}.
(d) Show that S = Z_{in} (\ell^{\prime }) / R_{o} at the location of V_{max}.
(a) At a point with z = z_{1} on the line, or at a distance \ell^{\prime } = \ell – z_{1} from the load, the forward voltage wave is given by the first term on the right-hand side of Eq. (9-46):
V (z) = V^{+}_{o} e^{- j \beta z} \left[ 1 + \left|\Gamma \right| e^{j (\phi – 2\beta \ell^{\prime })} \right] (9 – 46)
V^{+} (z_{1}) = V^{+}_{o} e^{- j \beta z_{1}}At the same point on the line, the backward voltage wave is given by the second term on the right-hand side of Eq. (9-46):
V^{-} (z_{1}) = V^{+}_{o} e^{- j \beta z_{1}} \left|\Gamma \right| e ^{j(\phi – 2\beta \ell^{\prime })}The reflection coefficient at a distance \ell^{\prime } from the load is therefore
\Gamma (\ell^{\prime }) = \frac{ V^{-} (z_{1})}{ V^{+} (z_{1})} = \left|\Gamma \right| e ^{j(\phi – 2\beta \ell^{\prime })} (9-54)
(b) From Eq. (9-46), we have a voltage maximum if the relation e ^{j(\phi – 2\beta \ell^{\prime })} = 1 is satisfied, and a voltage minimum if e ^{j(\phi – 2\beta \ell^{\prime })} = -1. Therefore, from Eq. (9- 54), we obtain
\Gamma (\ell^{\prime }) = \left|\Gamma \right| : at the location of V_{max} (9-55a) \\ \Gamma (\ell^{\prime }) = – \left|\Gamma \right| : at the location of V_{min} (9-55b)We note that \Gamma (\ell^{\prime }) is purely real at the locations of V_{max} and V_{min}.
(c) From Eq. (9-43), the input impedance at a distance \ell^{\prime } from the load is written as
were we used γ = jβ and Z_{o} = R_{o}.
Noting that e ^{j(\phi – 2\beta \ell^{\prime })} = 1 for the voltage maximum and e ^{j(\phi – 2\beta \ell^{\prime })} = -1 for the voltage minimum, from Eq. (9-56) we obtain
Z_{in} (\ell^{\prime })= R_{o}\frac{1 + \left|\Gamma \right| }{1 – \left|\Gamma \right|} : at the location of V_{max} (9-57a) \\ Z_{in} (\ell^{\prime })= R_{o}\frac{1 – \left|\Gamma \right| }{1 + \left|\Gamma \right|} : at the location of V_{min} (9-57b)We note that Z_{in} (\ell^{\prime }) is purely real at the locations of V_{max} and V_{min}.
(d) At the location of V_{max}, dividing both sides of Eq. (9-57a) by R_{o}we obtain
\frac{Z_{in} (\ell^{\prime })}{ R_{o}} = \frac{1 + \left|\Gamma \right| }{1 – \left|\Gamma \right|} = S (9-58)