Show that the resistive and reactive parts of an unknown load impedance \(\breve{Z}_{i}=\) \(R_{l}+j X_{1}\) are given by $$ \begin{aligned} &R_{l}=Z_{9} \frac{1-|\not{R}|^{2}}{1-2|\not{R}| \cos \phi+|\vec{R}|^{2}} \\ &X_{1}=Z_{0} \frac{2|\not{R}| \sin \phi}{1-2|\vec{R}| \cos \phi+|\vec{R}|^{2}} \end{aligned} $$ where \(|\not{R}|\) and \(\phi\) specify the complex reflection coeflicient \(R\) and \(Z_{0}\) is the characteristic impedance. Note: See Prob. 1.4.3.

Question: Show that the resistive and reactive parts of an unknown load impedance \(\breve{Z}_{i}=\) \(R_{l}+jX_{1}\) can be given by the following expressions: $$R_l=\frac{Z_0\frac{1-|\not{R}|^{2}}{1-2|\not{R}| \cos \phi+|\vec{R}|^{2}}}$$ $$X_1=Z_0 \frac{2|\not{R}| \sin \phi}{1-2|\vec{R}| \cos \phi+|\vec{R}|^{2}}$$ where \(|\not{R}|\) is the reflection coefficient and \(\phi\) is its phase, and \(Z_0\) is the characteristic impedance. Answer: To show the resistive and reactive parts of an unknown load impedance depends upon the given expressions, follow four steps: 1. Understand the meaning of the reflection coefficient and its relationship with impedances: \(|\not{R}|=\frac{\breve{Z}_i-Z_0}{\breve{Z}_i+Z_0}\). 2. Rewrite the unknown load impedance as \(\breve{Z}_{i}=Z_0(1+R)\), where R is the normalized impedance, and substitute this into the reflection coefficient equation. 3. Derive the expression for \(R_l = \frac{Z_0\frac{1-|\not{R}|^{2}}{1-2|\not{R}| \cos \phi+|\vec{R}|^{2}}}\) using the obtained relationship. 4. Derive the expression for \(X_1 = Z_0 \frac{2|\not{R}| \sin \phi}{1-2|\vec{R}| \cos \phi+|\vec{R}|^{2}}\) in a similar manner. By following these steps, we can demonstrate that the resistive and reactive parts of an unknown load impedance \(\breve{Z}_{i}=\) \(R_{l}+jX_{1}\) are indeed given by the provided expressions.