Principles of Electromagnetic Waves and Materials, 2nd Edition

Appendix 2B: Transmission Lines: Power Calculation

2B.1Transmission Lines: Power Calculation

A transmission line of length d is shown in Figure 2B.1a. The equivalent circuit is shown in Figure 2B.1b.

FIGURE 2B.1Transmission line: (a) length d and (b) equivalent input impedance.

The input impedance into the transmission line of length d with load Z_L is given by

(2B.1)$$Z_{\text{in}}=Z_0\frac{Z_{\text{L}}+\text{j}{Z}_{0}tan\left(\mathrm{\beta }d\right)}{Z_0+\text{j}{Z}_{0}tan\left(\mathrm{\beta }d\right)}=Z_0\frac{\left(1+{\mathrm{\Gamma }}_0{\text{e}}^{-\text{j}2\mathrm{\beta }d}\right)}{\left(1-{\mathrm{\Gamma }}_0{\text{e}}^{-\text{j}2\mathrm{\beta }d}\right)}.$$

The voltage at the input of the transmission line can be found as

(2B.2)$$\begin{array}{ccc}\hfill \tilde{V}\left(d\right)\hfill & =\hfill & \frac{{\tilde{V}}_{\text{g}}{Z}_{\text{in}}}{Z_g+Z_{\text{in}}}\hfill \\ =\hfill & {\tilde{V}}_0^{+}{\text{e}}^{\text{j}\mathrm{\beta }d}+{\tilde{V}}_0^{-}{\text{e}}^{-\text{j}\mathrm{\beta }d}={\tilde{V}}_0^{+}{\text{e}}^{\text{j}\mathrm{\beta }d}\left[1+{\mathrm{\Gamma }}_0{\text{e}}^{-\text{j}2\mathrm{\beta }d}\right],\hfill \end{array}$$

where ${\tilde{V}}_0^{+}$ is the voltage of the positive traveling wave and ${\tilde{V}}_0^{-}$ is the voltage of the reflected ...