Module 5.1 Wave Propagation on a Transmission Line (Lossless Case)

Introduction:

    In this module, we discuss the time-domain behavior of the incident, reflected and standing waves on a lossless transmission line.
    Observe the dynamic behavior and superposition of the incident and reflected waves by adjusting the range sliders.
    Ex.1：When the load impedance is matched ( \(\left|\Gamma_L\right|\) = 0 ), the amplitude of the reflected wave is zero and the incident wave is
    identical to the combined wave.
    Ex.2：When the frequency is lower, the transmission-line effects become smaller, that is, the variations of amplitude and phase
    along the transmission line are smaller.
    Ex.3：When \(\left|\Gamma_L\right|\) = 1, the incident and reflected waves have the same amplitude, and the combined wave is a standing wave.

Formula:

1. \(V_{total}(z,t)(V)\) : Total voltage waveform in time domain, defined by
    \(V_{total}(z,t) = \left|V_0^+(z)\right|cos(\omega t-\beta z+\varphi^+)\)
                        \(+ \left|V_0^-(z)\right|cos(\omega t+\beta z+\varphi^-)\)
2. \(V_{incident}(z,t)(V)\) : Incident voltage waveform in time domain, defined by
    \(V_{incident}(z,t) = \left|V_0^+(z)\right|cos(\omega t-\beta z+\varphi^+)\)
3. \(V_{reflected}(z,t)(V)\) : Reflected voltage waveform in time domain, defined by
    \(V_{reflected}(z,t) = \left|V_0^-(z)\right|cos(\omega t+\beta z+\varphi^-)\)

Parameters:

1. \(\left|V_0^+\right|\) : Amplitude of the incident voltage wave at z=0, (V)
2. \(\varphi^+\) : Phase angle of the complex voltage \(V_0^+\), (rad)
3. \(\left|\Gamma_L\right|\) : Amplitude of the reflection coefficient at z=0
4. \(\theta\) : Phase angle of the reflection coefficient \(\Gamma_L\), (rad)
5. \(V_0^-\) : Reflected voltage wave at z=0, (V)
              \(V_0^- = V_0^+ \Gamma_L = \left|V_0^-\right|\angle\varphi^-\)
6. \(\beta\) : Phase cnstant, (rad/m)
7. \(f\) : Operating frequency, (GHz)
8. \(d\) : Length of the transmission line, value equal to 10 (cm)