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3.12: Voltage Reflection Coefficient

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    6278
  • [m0084_Voltage_Reflection_Coefficient]

    We now consider the scenario shown in Figure [m0084_fWaveIncidentOnZL]. Here a wave arriving from the left along a lossless transmission line having characteristic impedance \(Z_0\) arrives at a termination located at \(z=0\). The impedance looking into the termination is \(Z_L\), which may be real-, imaginary-, or complex-valued. The questions are: Under what circumstances is a reflection – i.e., a leftward traveling wave – expected, and what precisely is that wave?

    The potential and current of the incident wave are related by the constant value of \(Z_0\). Similarly, the potential and current of the reflected wave are related by \(Z_0\). Therefore, it suffices to consider either potential or current. Choosing potential, we may express the incident wave as \[\widetilde{V}^+(z) = V_0^+ e^{-j\beta z}\] where \(V_0^+\) is determined by the source of the wave, and so is effectively a “given.” Any reflected wave must have the form \[\widetilde{V}^-(z) = V_0^- e^{+j\beta z}\] Therefore, the problem is solved by determining the value of \(V_0^-\) given \(V_0^+\), \(Z_0\), and \(Z_L\).

    Considering the situation at \(z=0\), note that by definition we have \[Z_L \triangleq \frac{\widetilde{V}_L}{\widetilde{I}_L} \label{m0084_eZL}\] where \(\widetilde{V}_L\) and \(\widetilde{I}_L\) are the potential across and current through the termination, respectively. Also, the potential and current on either side of the \(z=0\) interface must be equal. Thus, where \(\widetilde{I}^+(z)\) and \(\widetilde{I}^-(z)\) are the currents associated with \(\widetilde{V}^+(z)\) and \(\widetilde{V}^-(z)\), respectively. Since the voltage and current are related by \(Z_0\), Equation [m0084_eI1] may be rewritten as follows: \[\frac{\widetilde{V}^+(0)}{Z_0} - \frac{\widetilde{V}^-(0)}{Z_0} = \widetilde{I}_L \label{m0084_eI2}\]

    Evaluating the left sides of Equations [m0084_eV1] and [m0084_eI2] at \(z=0\), we find: Substituting these expressions into Equation [m0084_eZL] we obtain: \[Z_L = \frac{V_0^+ + V_0^-}{V_0^+/Z_0 - V_0^-/Z_0}\] Solving for \(V_0^-\) we obtain \[V_0^- = \frac{Z_L-Z_0}{Z_L+Z_0}~V_0^+\] Thus, the answer to the question posed earlier is that \[V_0^- = \Gamma V_0^+ ~~\mbox{, where}\] \[\boxed{ \Gamma \triangleq \frac{Z_L-Z_0}{Z_L+Z_0} } \label{m0084_eGamma}\]

    The quantity \(\Gamma\) is known as the voltage reflection coefficient. Note that when \(Z_L=Z_0\), \(\Gamma=0\) and therefore \(V_0^-=0\). In other words,

    If the terminating impedance is equal to the characteristic impedance of the transmission line, then there is no reflection.

    If, on the other hand, \(Z_L \neq Z_0\), then \(\left|\Gamma\right|>0\), \(V_0^-= \Gamma V_0^+\), and a leftward-traveling reflected wave exists.

    Since \(Z_L\) may be real-, imaginary-, or complex-valued, \(\Gamma\) too may be real-, imaginary-, or complex-valued. Therefore, \(V_0^-\) may be different from \(V_0^+\) in magnitude, sign, or phase.

    Note also that \(\Gamma\) is not the ratio of \(I_0^-\) to \(I_0^+\). The ratio of the current coefficients is actually \(-\Gamma\). It is quite simple to show this with a simple modification to the above procedure and is left as an exercise for the student.

    Summarizing:

    The voltage reflection coefficient \(\Gamma\), given by Equation [m0084_eGamma], determines the magnitude and phase of the reflected wave given the incident wave, the characteristic impedance of the transmission line, and the terminating impedance.

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