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)