﻿ Module 6.2 Standing Wave Pattern (lossless case)
Module 6.2 Standing Wave Pattern (lossless case)
Time-average power delivered to the input at z, P(z) = (mW)

### Introduction:

本模組於無損傳輸線上探討駐波現象的頻域行為與時均功率本模組我們將探討於無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：負載端發生全反射時 ( $\left|\Gamma_L\right|$ = 1 )，電壓駐波比將為無窮大 (∞)，時均功率恆為 0！
Ex.2：負載阻抗為匹配時 ( $\left|\Gamma_L\right|$ = 0 )，電壓駐波比將為1，此時的時均功率為最大！
Ex.3：當負載阻抗為開路時，於負載端可觀察到最大電壓與最小電流的現象；反之，若負載阻抗為短路時，
與負載端則可觀察到最小電壓與最大電流的現象。

### Formula:

1. $V(z)$ : Total voltage wave at z, defined by
$\left|V(z)\right|=\left|V_{0}^{+} \right|\cdot \left|1+\left|\Gamma (0)\right|e^{j(\theta -2\beta \left|z\right|)} \right| (V)$
$\Gamma (0)=\left|\Gamma (0)\right|\cdot e^{j\theta } =\frac{Z_{L} -Z_{0} }{Z_{L} +Z_{0} } =\frac{(R_{L} -Z_{0} )+jX_{L} }{(R_{L} +Z_{0} )+jX_{L} } ; Z_{L} =R_{L} +jX_{L} (\Omega)$
$\Gamma (z)=\left|\Gamma (0)\right|\cdot e^{j\theta } \cdot e^{-j2\beta \left|z\right|} =\left|\Gamma (0)\right|\cdot e^{j(\theta -2\beta \left|z\right|)}$
2. $I(z) (A)$ : Total current wave at z, defined by
$\left|I(z)\right|=\frac{\left|V_{0}^{+} \right|}{Z_{0} } \cdot \left|1-\left|\Gamma (0)\right|e^{j(\theta -2\beta \left|z\right|)} \right| (A)$
3. $P(z) (W)$ : Time-average power delivered to the input at z, defined by
$P(z)=\frac{1}{2} Re[V(z)I*(z)]=\frac{1}{2} \frac{\left|V_{0}^{+} \right|^{2} }{Z_{0} } (1-\left|\Gamma (0)\right|^{2} ) (W)$
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. $\beta$ : Phase constant, (rad/m)
5. $f$ : Operating frequency, (GHz)
6. $d$ : Length of the transmission line, value equal to 10 cm
7. $V_{0}^{+}$ : Incident voltage at z = 0, (V)
8. $Z_{0}$ : Characteristic impedance of the transmission line, $(\Omega)$