﻿ Module 6.7 Standing Wave Pattern (lossy case)
Module 6.7 Standing Wave Pattern (lossy case)
Time-average power delivered to the input at z, P(z) = (mW)

### Introduction:

本模組我們將探討於有損傳輸線上駐波現象的頻域行為與時均功率，並藉由調整 toolbar 以觀察電壓駐波比 ( $SWR$ ) 之動態行為
及該傳輸線上之電壓與電流分布。
Ex.1：在負載端未發生全反射時 ( $\left|\Gamma_L\right|$ < 1 )，將衰減係數 ( $\alpha$ ) 逐漸升高，可發現時均功率將會隨之下降！
Ex.2：衰減係數 ( $\alpha$ ) 逐漸增加時，於負載端所觀察到的電壓與電流會隨之逐漸降低。

### Formula:

1. $V(z)$ : Total voltage wave at z, defined by
$\left|V(z)\right|=\left|V_{0}^{+}~{\cdot}~e^{\alpha \left|z\right|} \right|\cdot \left|1+\left|\Gamma (0)\right|~{\cdot}~e^{j(\theta -2\beta \left|z\right|)}~{\cdot}~e^{-2 \alpha \left|z\right|} \right| (V)$
$\Gamma (0)=\left|\Gamma (0)\right|\cdot e^{j\theta } =\frac{Z_{L} -Z_{0} }{Z_{L} +Z_{0} } =\frac{(R_{L} -Z_{0} )+jX_{L} }{(R_{L} +Z_{0} )+jX_{L} } ; Z_{L} =R_{L} +jX_{L} (\Omega)$
$\Gamma (z)=\left|\Gamma (0)\right|\cdot e^{j\theta } \cdot e^{-j2\beta \left|z\right|}~{\cdot}~e^{-2 \alpha \left|z\right|}$
2. $I(z) (A)$ : Total current wave at z, defined by
$\left|I(z)\right|=\frac{\left|V_{0}^{+} ~{\cdot}~e^{\alpha \left|z\right|} \right|}{Z_{0} } \cdot \left|1-\left|\Gamma (0)\right|~{\cdot}~e^{j(\theta -2\beta \left|z\right|)}~{\cdot}~e^{-2 \alpha \left|z\right|} \right| (A)$
3. $P(z) (W)$ : Time-average power delivered to the input at z, defined by
$P(z)=\frac{1}{2} Re[V(z)I*(z)]=\frac{1}{2} \frac{\left|V_{0}^{+} \right|^{2} }{Z_{0} } (e^{2 \alpha \left|z\right|}-\left|\Gamma (0)\right|^{2}~{\cdot}~e^{-2 \alpha \left|z\right|} ) (W)$
4. $SWR$ : Voltage standing wave ratio, defined by
$SWR=\frac{V_{\max } }{V_{\min } } =\frac{1+\left|\Gamma (0)\right|}{1-\left|\Gamma (0)\right|}$

### Parameters:

1. $\left|\Gamma (0)\right|$ : Voltage reflection coefficient at z = 0, defined by
$\Gamma (0)=\left|\Gamma (0)\right|\cdot e^{j\theta } =\frac{Z_{L} -Z_{0} }{Z_{L} +Z_{0} }$
2. $\theta$ : Phase angle of the voltage reflection coefficient, (rad)
3. $Z_{L}$ : Load impedance, defined by
$Z_{L} =R_{L} +jX_{L} =Z_{0} \cdot \frac{1+\Gamma (0)}{1-\Gamma (0)} (\Omega)$
a. $R_{L}$ : Real part of $Z_{L}$,
b. $X_{L}$ : Imaginary part of $Z_{L}$.
4. $\alpha$ : Attenuation constant, (Np/m)
5. $\beta$ : Phase constant, (rad/m)
6. $f$ : Operating frequency, (GHz)
7. $d$ : Length of the transmission line, value equal to 10 cm
8. $V_{0}^{+}$ : Incident voltage at z = 0, (V)
9. $Z_{0}$ : Characteristic impedance of the transmission line, $(\Omega)$