Module 5.6 Wave Propagation on a Transmission Line (Lossy case)

Introduction:

    In this module, we discuss the time-domain behavior of the incident, reflected and standing waves on a lossy transmission line. Observe the
    dynamic behavior and superposition of incident and reflected waves by adjusting the range sliders.
    Ex.1：As the attenuation constant ( \(\alpha\) ) gradually increases, the amplitudes of incident and reflected waves decrease. Furthermore, amplitude of the
    combined wave becomes smaller, or even vanishes, as it approaches to the load.
    Ex.2：When the attenuation constant ( \(\alpha\) ) is not zero, the incident and reflected waves have different amplitudes so that a pure standing wave cannot
    be generated.

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)