Question 6.10: On the simplified Smith chart, locate the normalized impedan...

The real and the imaginary parts of the reflection coefficient in (6.42) can be computed by multiplying the numerator and the denominator by the complex conjugate of the denominator. We obtain

\mathcal{R}=\mathcal{R}_{r}+j \mathcal{R}_{i}=\frac{Z_{L}-1}{Z_{L}+1}          (6.42)

\mathcal{R} = \mathcal{R} _{r}+j \mathcal{R} _{i}=\frac{z_{L}-1}{z_{L}+1}=\frac{(r-1)+j x}{(r+1)+j x} \times \frac{(r+1)-j x}{(r+1)-j x}=\frac{r^{2}-1+x^{2}}{(r+1)^{2}+x^{2}}+j \frac{2 x}{(r+1)^{2}+x^{2}}