Module 6.6 Wave Propagation on a Transmission Line (lossy case)

### Introduction:

本模組我們將探討有損傳輸線之入射波、反射波與駐波之時域行為，除藉由調整 toolbar 以觀察入射波與反射波之動態行為與合成。
Ex.1：將衰減係數 ( $\alpha$ ) 逐漸升高，入射波與反射波振幅將會隨之下降，且越靠近負載端時，合成波振幅會變得越小，甚至消失為 0。
Ex.2：衰減係數 ( $\alpha$ ) 不為 0 時，因入射波與反射波振幅不再相等，故完全駐波已無法被觀察到。

### Formula:

1. $V_{total}(z,t)(V)$ : Total voltage waveform in time domain, defined by
$V_{total}(z,t) = \left|V_0^+(z)\right|cos(2 \pi ft-\frac{2 \pi f}{c}z+\varphi^+)$$e^{- \alpha z}$
$+ \left|V_0^-(z)\right|cos(2 \pi ft+\frac{2 \pi f}{c}z+\varphi^+)$$e^{\alpha z}$ (V)
where $\left|V_0^{+\prime}\right| = \left|V_0^+\right|e^{\alpha d}$
2. $V_{incident}(z,t)(V)$ : Incident voltage waveform in time domain, defined by
$V_{incident}(z,t) = \left|V_0^+(z)\right|cos(2 \pi ft-\frac{2 \pi f}{c}z+\varphi^+)$$e^{- \alpha z}$
3. $V_{reflected}(z,t)(V)$ : Reflected voltage waveform in time domain, defined by
$V_{reflected}(z,t) = \left|V_0^-(z)\right|cos(2 \pi ft+\frac{2 \pi f}{c}z+\varphi^-)$$e^{\alpha z}$

### Parameters:

1. $\left|V_0^+\right|$ : Amplitude of the incident voltage wave at z=0, (V)
2. $\left|V_0^-\right|$ : Amplitude of the reflected voltage wave at z=0, (V)
3. $\varphi^+$ : Phase angle of complex voltage $V_0^+$, (rad)
4. $\varphi^-$ : Phase angle of complex voltage $V_0^-$, (rad)
5. $\alpha$ : Attenuation constant, (Np/m)
6. $\beta$ : Phase constant, $\beta$ = $\frac{w}{c}$ = $\frac{2 \pi f}{c}$, (rad/m)
7. $f$ : Operating frequency, (GHz)
8. $d$ : Length of the transmission line, value equal to 10 (cm)
9. $c$ : Light speed, value equal to $3x10^8$ (m/s)