﻿ Module 6.3 Input Impedance
Module 6.3 Input impedance

### Introduction:

本模組我們將探討有終端負載之無損傳輸線的輸入阻抗特性，並藉由調整 toolbar 以觀察輸入阻抗於該傳輸線上之動態行為與分布。
Ex.1：負載阻抗為匹配時 ( $\left|\Gamma_L\right|$ = 0 )，輸入阻抗於傳輸線上任一觀察點皆為系統阻抗 ( $Z_0$ ) ; 反之，若負載阻抗不為匹配時 ( $\left|\Gamma_L\right|$ ≠ 0 )，
輸入阻抗將會隨觀察點的移動而變化！
Ex.2：若固定負載阻抗值 ( $\left|\Gamma_L\right|$ ≠ 0 )，反射係數之大小於傳輸線上任一觀察點皆為相同，而僅有相位上的改變。
Ex.3：將觀察的週期拉長或將頻率 ( $f$ ) 升高時，從阻抗的響應分佈可明顯發現輸入阻抗的週期為半波長！

### Formula:

1. $Z_{in}(z) (\Omega)$ : Input impedance looking into the transmission line at z.
$Z_{in} \left(z\right) = Z_{0} \cdot \frac { Z_{L} + j Z_{0} { tan\beta }\left|z\right|}{Z_{0} + j Z_{L} { tan\beta }\left|z\right|} (\Omega )$
2. $\Gamma(z)$ : Voltage reflection coefficient at z, defined by
$\Gamma (z)=\left|\Gamma (0)\right|\cdot e^{j\theta } \cdot e^{-j2\beta \left|z\right|}$
3. $SWR$ : 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. $Z_{0}$ : Characteristic impedance of the transmission line, $(\Omega)$