Traveling and Standing Waves

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Standing Waves

In the previous section, we introduced the voltage reflection coefficient that relates the forward to reflected voltage phasor.

$\begin{eqnarray} \tilde{V}(z)= \tilde{V}_0^+ (e^{-j \beta z} + \Gamma_L e^{j \beta z } ) \label{eq:vtlfin} \\ I(z)= \frac{\tilde{V}_0^+}{Z_0} (e^{-j \beta z} - \Gamma_L e^{j \beta z} ) \label{eq:itlfin} \end{eqnarray}$

Let us look at the physical meaning of these variables.

(a): $\Gamma _L$ is the voltage reflection coefficient at the load (z=0),
(b): $z$ is the axis in the direction of wave propagation,
(c): $\beta$ is the phase constant,
(d): $Z_0$ is the impedance of the transmission line.
(e): $\tilde {V}_0^+$ is the phasor of the forward-going wave at the load,
(f): $\Gamma _L \tilde {V}_0^+$ is the phasor of the reflected going wave at the load,
(g): $\tilde {V}_0^+ e^{-j \beta z}$ is a forward voltage anywhere on the line,
(h): $\Gamma _L \tilde {V}_0^+ e^{j \beta z}$ is a reflected voltage anywhere on the line,
(i): $\tilde {V}(z)$ is the total voltage phasor on the line, the sum of forward and reflected voltage.
(j): $\tilde {I}_0^+$ is the phasor of the forward-going current at the load,
(k): $-\Gamma _L \frac {\tilde {V}_0^+}{Z_0}$ is the phasor of the reflected current at the load,
(l): $\tilde {I}_0^+ e^{-j \beta z}$ is the phasor of a forward current anywhere on the line,
(m): $-\Gamma _L \frac {\tilde {V}_0^+}{Z_0} e^{j \beta z}$ is the phasor of a reflected current anywhere on the line,
(n): $\tilde {I}(z)$ is the phasor of the total current on the line, the sum of forward and reflected voltage.

The magnitude of a complex number can be found as $|z|=\sqrt {z z^*}$ .Therefore the magnitude of the voltage anywhere on the line is $|\tilde {V}(z)|=\sqrt {\tilde {V}(z) \tilde {V}(z)^*}$ . We can simplify this equation as shown in Figure eq:SWsw1.

$\begin{eqnarray} |\tilde{V}(z)|=\sqrt{\tilde{V}(z) \tilde{V}(z)^*} \nonumber \\ |\tilde{V}(z)|=\sqrt{ (\tilde{V}_0^+)^2 (e^{-j \beta z} - |\Gamma_L| e^{j \beta z + \Theta_r} ) \tilde{V}_0^+ (e^{j \beta z} - |\Gamma_L| e^{-(j \beta z + \angle \Gamma_L)} )} \nonumber \\ |\tilde{V}(z)|= \tilde{V}_0^+ \sqrt{(e^{-j \beta z} - |\Gamma_L| e^{j \beta z + |\angle \Gamma_L} ) (e^{j \beta z} - |\Gamma_L| e^{-(j \beta z + \angle \Gamma_L)} )} \nonumber \\ |\tilde{V}(z)|= \tilde{V}_0^+ \sqrt{1+ |\Gamma_L| e^{-(2j \beta z + |\angle \Gamma_L)} + |\Gamma_L| e^{j 2 \beta z + \angle \Gamma_L} +|\Gamma_L|^2 )} \nonumber \\ |\tilde{V}(z)|= \tilde{V}_0^+ \sqrt{1+ |\Gamma_L|^2 + |\Gamma_L| ( e^{-(2j \beta z + \angle \Gamma_L)} \angle \Gamma_L + e^{(j 2 \beta z + \angle \Gamma_L)} )} \nonumber \\ |\tilde{V}(z)|= \tilde{V}_0^+ \sqrt{1+ |\Gamma_L|^2 + 2 |\Gamma_L| \cos(2 \beta z +\angle \Gamma_L)} \label{eq:SWsw1} \end{eqnarray}$

The Equation eq:SWsw1 is written in terms of $z$ . We set up the load at $z=0$ , and the generator at $z=-l$ . The positions of the maximums and minimum total voltage on the line will be at some position at the negative part of z-axis $z_{max}=-l_{max}$ , and the minimums will be at $z_{max}=-l_{min}$ .

The magnitude of the total voltage on the transmission line is given by Eq.eq:SWsw1. We will now visualize how the magnitude of the voltage looks on the transmission line.

Find the magnitude of the total voltage anywhere on the transmission line if $\Gamma _L=0$ .

Let us start from a simple case when the voltage reflection coefficient on the transmission line is $\Gamma _L=0$ and draw the magnitude of the total voltage as a function of z. Equation eq:SWsw1 shows the magnitude of the total voltage anywhere on the line is equal to the magnitude of the voltage at the load $|\tilde {V}(z)|= |\tilde {V}_0^+|$ . The magnitude of the voltage is constant everywhere on the transmission line, and so the line is called ”flat,” and it represents a single, forward traveling wave from the generator to the load. The magnitude is the green line in Figure fig:SWflatline. To see the movie of this transmission line, go to the class web page under Instructional Videos. Forward voltage is shown in red, reflected voltage in pink, and the magnitude of the voltage is green.

Figure 1: Flat line.

Find the magnitude of the total voltage anywhere on the transmission line if $\Gamma _L=0.5 e^{j0}$ .

Let’s look at another case, $\Gamma _L=0.5$ and $\angle \Gamma _L=0$ . Equation eq:SWswc1 represents the magnitude of the voltage on the transmission line, and Figure fig:SWreflcoeffvid shows in green how this function looks on a transmission line. This case is shown in Figure fig:SWreflcoeffvid.

$\begin{eqnarray} |\tilde{V}(z)|=\tilde{V}_0^+ \sqrt{\frac{5}{4}+ cos{2 \beta z} }\label{eq:SWswc1} \end{eqnarray}$

Figure 2: Voltage on a transmission line with reflection coefficient magnitude 0.5, and zero phase.

The function in Equation eq:SWswc1 is at its maximum when $cos(2 \beta z+\angle \Gamma _L)=1$ or $z=\frac {k}{2} \lambda$ , and the function value is $\tilde {V}(z)=1.5\tilde {V}_0^+$ . It is at its minimum when $cos(2 \beta z+ \angle \Gamma _L)=-1$ or $z=\frac {2 k +1}{4} \lambda$ and the function value is $\tilde {V}(z)=0.5 \tilde {V}_0^+$

The function that we see looks like a cosine with an average value of $\tilde {V}_0^+$ , but it is not a cosine. The minimums of the function are sharper than the maximums.

Observe waves in the app below.

Is the black wave moving left or right?

left right

Is the black wave a forward, reflected or the sum of the two?

Forward Reflected The sum of the two

Find the magnitude of the total voltage anywhere on the transmission line if $\Gamma _L=1$ .

Another case we will look at is when the reflection coefficient is at its maximum of $\Gamma _L =1$ . The function is shown if Figure fig:SWStanding. In this case, we have a pure standing wave on a transmission line.

Figure 3: Shorted Transmission Line.

Observe waves in the app below.

Is the black wave moving left or right? The wave is not moving left or right, it is standing in place, so it is called a standing wave. Is the black wave a forward, reflected or the sum of the two?

Forward Reflected The sum of the two

Find the magnitude of the total voltage anywhere on the transmission line for any $\Gamma _L$ , and the position of voltage maximums on the line.

The magnitude of the total voltage on the line is given in Equationeq:SWsw1. In general the voltage maximums will occur when the cosine function is at its maximum $cos(2 \beta z+\angle \Gamma _L)=1$ . In this case, the maximum value of the magnitude of total voltage on the line is shown in Equation eq:SWMaxValue.

$\begin{eqnarray} |\tilde{V}(z)_{max}|=|\tilde{V}_0^+|\sqrt{1+|\Gamma_L|^2+2 |\Gamma_L|} \nonumber \\ |\tilde{V}(z)_{max}| =|\tilde{V}_0^+|\sqrt{(1+|\Gamma_L|)^2} \nonumber \\ |\tilde{V}(z)_{max}| =|\tilde{V}_0^+|(1+|\Gamma_L|) \label{eq:SWMaxValue} \end{eqnarray}$

The Equation eq:SWmaximums shows position of voltage maximums $z_{max}$ on the line.

$\begin{eqnarray} cos(-2 \beta l_{max}+ \angle \Gamma_L)=1 \nonumber \\ cos(2 \beta l_{max}- \angle \Gamma_L)=1 \nonumber \\ 2 \beta z_{max} - \angle \Gamma_L = 2 n \pi \nonumber \\ z_{max} =\frac{ 2 n \pi +\angle \Gamma_L }{ 2 \beta} \nonumber \\ z_{max}= \lambda \frac{ 2 n \pi +\angle \Gamma_L }{ 4 \pi} \label{eq:SWmaximums} \end{eqnarray}$

In general the voltage minimums will occur when $cos(2 \beta z)=-1$ .

$\begin{eqnarray} |\tilde{V}(z)_{min}|=|\tilde{V}_0^+|\sqrt{1+|\Gamma_L|^2-2 |\Gamma_L|} \nonumber \\ |\tilde{V}(z)_{min}| =|\tilde{V}_0^+|\sqrt{(1-|\Gamma_L|)^2} \nonumber \\ |\tilde{V}(z)_{min}| =|\tilde{V}_0^+| (1-|\Gamma_L|) \end{eqnarray}$

The Equation eq:SWminimums shows the position of voltage minimums on the line.

$\begin{eqnarray} cos(2 \beta z_{max}-\angle \Gamma_L)=-1 \nonumber \\ 2 \beta z_{max} + \angle \Gamma_L = (2 n+1) \pi \nonumber \\ z_{max} =\frac{ (2 n+1) \pi +\angle \Gamma_L }{ 2 \beta} \nonumber \\ z_{max}=\lambda \frac{ (2 n+1) \pi +\angle \Gamma_L }{ 4 \pi} \label{eq:SWminimums} \end{eqnarray}$

Voltage Standing Wave Ratio (VSWR) - pron: ”vee-s-uh-are”

The ratio of voltage minimum on the line over the voltage maximum is called the Voltage Standing Wave Ratio (VSWR) or just Standing Wave Ratio (SWR).

$\begin{eqnarray} SWR=\frac{\tilde{V}(z)_{max}}{ \tilde{V}(z)_{min}} \nonumber \\ SWR=\frac{1+|\Gamma_L|}{1-|\Gamma_L|} \end{eqnarray}$

Note that the SWR is always equal or greater than 1.