3.12: Voltage Reflection Coefficient

[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.