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)