Module 6.5 Reflection Coefficient

### Introduction:

本模組我們將探討沿著有終端負載之無損傳輸線觀察其反射係數特性，並藉由調整 toolbar 以觀察反射係數及其相位於傳輸線上之動態行為。
Ex.1：當觀察的週期拉長或將頻率( $f$ )升高時，從相位的表現可明顯發現反射係數的週期為半波長！
Ex.2：若負載阻抗為純電阻 ( $R_L$ )，當 $R_L > Z_0$ 時，負載端反射係數之相位恆為 0 度；反之，若 $R_L < Z_0$ 時，則負載端反射係數之相位恆為 180 度。
Ex.3：當頻率 ( $f$ ) 改變時，可發現輸入端的反射係數大小不會改變，僅其相位隨之變化。

### Formula:

1. $\Gamma(z)$ : Voltage reflection coefficient at z, defined by
$\Gamma(z)=\left|\Gamma_{0}\right| \cdot e^{j\theta} \cdot e^{-j2\beta|z|}$
2. $SWR$ : 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. $Z_{L}$ : 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 $Z_{L}$.
3. $\theta$ : Phase angle of the voltage reflection coefficient, (rad)
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, $(\Omega)$