Module 4.4 Bouncing Signal in a T.-Line System w Shunt/Series Discontinuity

Formula:

1. \(\Gamma_S\): Source reflection coefficient
    \(\Gamma_S = \frac{Z_S-Z_{01}}{Z_S+Z_{01}}\)
2. \(\Gamma_L\) : Load reflection coefficient
    \(\Gamma_L = \frac{Z_L-Z_{03}}{Z_L+Z_{03}}\)
3. In the condition of Series :
    \(\Gamma_{13} = \frac{R+Z_{03}-Z_{01}}{R+Z_{03}+Z_{01}}\);    \(\tau_{13} = (1+\Gamma_{13}) \times \frac{Z_{03}}{R+Z_{03}}\)
    \(\Gamma_{31} = \frac{R+Z_{01}-Z_{03}}{R+Z_{01}+Z_{03}}\);    \(\tau_{31} = (1+\Gamma_{31}) \times \frac{Z_{01}}{R+Z_{01}}\)
4. In the condition of Shunt :
    \(Z_{1} = \frac{1}{R^{-1}+Z_{01}^{-1}}\);    \(Z_{3} = \frac{1}{R^{-1}+Z_{03}^{-1}}\)
    \(\Gamma_{13} = \frac{Z_{3}-Z_{01}}{Z_{3}+Z_{01}}\);      \(\tau_{13} = 1+\Gamma_{13}\)
    \(\Gamma_{31} = \frac{Z_{1}-Z_{03}}{Z_{1}+Z_{03}}\);      \(\tau_{31} = 1+\Gamma_{31}\)
5. \(V(z,t)\)
    \(V_1^+ = \frac{Z_{01}}{Z_{01}+Z_S}\);               \(V_1^- = \frac{Z_{01}}{Z_{01}+Z_S} \times \tau_{13} \times \Gamma_L \times \tau_{31}\)
    \(V_2^+ = \frac{Z_{01}}{Z_{01}+Z_S} \times \tau_{13}\);    \(V_2^{++} = \frac{Z_{01}}{Z_{01}+Z_S} \times \Gamma_{13} \times \Gamma_S \times \tau_{13}\)

Parameters:

1. \(Z_S\) : Source impedance (Ω)
2. \(Z_L\) : Load impendance (Ω)
3. \(Z_{01}, Z_{03}\) : Characteristic impendance of the transmission lines (Ω)
4. \(L_1, L_3\) : Length of the transmission lines (m)
5. \(R\) : Series/shunt resistance at discontinuity (Ω)
6. \(V_p\) :Speed of electromagnetic wave in the transmission line (m/s)