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Physics LibreTexts

3.13: Standing Waves

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A standing wave consists of waves moving in opposite directions. These waves add to make a distinct magnitude variation as a function of distance that does not vary in time.

To see how this can happen, first consider that an incident wave V_0^+ e^{-j\beta z}, which is traveling in the +z axis along a lossless transmission line. Associated with this wave is a reflected wave V_0^- e^{+j\beta z}=\Gamma V_0^+ e^{+j\beta z}, where \Gamma is the voltage reflection coefficient. These waves add to make the total potential

\begin{split} \widetilde{V}(z) & = V_0^+ e^{-j\beta z} + \Gamma V_0^+ e^{+j\beta z} \\ & = V_0^+ \left( e^{-j\beta z} + \Gamma e^{+j\beta z} \right) \end{split} \nonumber

The magnitude of \widetilde{V}(z) is most easily found by first finding |\widetilde{V}(z)|^2, which is:

\begin{aligned} & \tilde{V}(z) \tilde{V}^{*}(z) \\ =&\left|V_{0}^{+}\right|^{2}\left(e^{-j \beta z}+\Gamma e^{+j \beta z}\right)\left(e^{-j \beta z}+\Gamma e^{+j \beta z}\right)^{*} \\ =&\left|V_{0}^{+}\right|^{2}\left(e^{-j \beta z}+\Gamma e^{+j \beta z}\right)\left(e^{+j \beta z}+\Gamma^{*} e^{-j \beta z}\right) \\ =&\left|V_{0}^{+}\right|^{2}\left(1+|\Gamma|^{2}+\Gamma e^{+j 2 \beta z}+\Gamma^{*} e^{-j 2 \beta z}\right) \end{aligned}

Let \phi be the phase of \Gamma; i.e.,

\Gamma = \left|\Gamma\right|e^{j\phi} \nonumber

Then, continuing from the previous expression:

\begin{aligned} &\left|V_{0}^{+}\right|^{2}\left(1+|\Gamma|^{2}+|\Gamma| e^{+j(2 \beta z+\phi)}+|\Gamma| e^{-j(2 \beta z+\phi)}\right) \\ =&\left|V_{0}^{+}\right|^{2}\left(1+|\Gamma|^{2}+|\Gamma|\left[e^{+j(2 \beta z+\phi)}+e^{-j(2 \beta z+\phi)}\right]\right) \end{aligned}

The quantity in square brackets can be reduced to a cosine function using the identity

\cos\theta = \frac{1}{2}\left[e^{j\theta}+e^{-j\theta}\right] \nonumber

yielding: |V_0^+|^2 \left[ 1 + \left|\Gamma\right|^2 + 2\left|\Gamma\right| \cos\left( 2\beta z + \phi \right) \right] \nonumber

Recall that this is |\widetilde{V}(z)|^2. |\widetilde{V}(z)| is therefore the square root of the above expression:

\left|\widetilde{V}(z)\right| = |V_0^+| \sqrt{ 1 + \left|\Gamma\right|^2 + 2\left|\Gamma\right| \cos\left( 2\beta z + \phi \right) } \nonumber

Thus, we have found that the magnitude of the resulting total potential varies sinusoidally along the line. This is referred to as a standing wave because the variation of the magnitude of the phasor resulting from the interference between the incident and reflected waves does not vary with time.

We may perform a similar analysis of the current, leading to: \left|\widetilde{I}(z)\right| = \frac{|V_0^+|}{Z_0} \sqrt{ 1 + \left|\Gamma\right|^2 - 2\left|\Gamma\right| \cos\left( 2\beta z + \phi \right) } \nonumber

Again we find the result is a standing wave.

Now let us consider the outcome for a few special cases.

Matched load. When the impedance of the termination of the transmission line, Z_L, is equal to the characteristic impedance of the transmission line, Z_0, \Gamma=0 and there is no reflection. In this case, the above expressions reduce to |\widetilde{V}(z)| = |V_0^+| and |\widetilde{I}(z)| = |V_0^+|/Z_0, as expected.

Open or Short-Circuit. In this case, \Gamma=\pm1 and we find:

\left|\widetilde{V}(z)\right| = |V_0^+| \sqrt{ 2 + 2\cos\left( 2\beta z + \phi \right) } \nonumber

\left|\widetilde{I}(z)\right| = \frac{|V_0^+|}{Z_0} \sqrt{ 2 - 2\cos\left( 2\beta z + \phi \right) } \nonumber

where \phi=0 for an open circuit and \phi=\pi for a short circuit. The result for an open circuit termination is shown in Figure \PageIndex{1}(a) (potential) and \PageIndex{1}(b) (current). The result for a short circuit termination is identical except the roles of potential and current are reversed. In either case, note that voltage maxima correspond to current minima, and vice versa.

m0086_fStandingWaveOC-V.png

(a) Potential.

m0086_fStandingWaveOC-C.png

(b) Current.

Figure \PageIndex{1}: Standing wave associated with an opencircuit termination at z = 0 (incident wave arrives from left).

Also note:

The period of the standing wave is \lambda/2; i.e., one-half of a wavelength.

This can be confirmed as follows. First, note that the frequency argument of the cosine function of the standing wave is 2\beta z. This can be rewritten as 2\pi\left(\beta/\pi\right)z, so the frequency of variation is \beta/\pi and the period of the variation is \pi/\beta. Since \beta=2\pi/\lambda, we see that the period of the variation is \lambda/2. Furthermore, this is true regardless of the value of \Gamma.

Mismatched loads. A common situation is that the termination is neither perfectly-matched (\Gamma=0) nor an open/short circuit (\left|\Gamma\right|=1). Examples of the resulting standing waves are shown in Figure \PageIndex{2}.

m0086_fStandingWaves.png Figure \PageIndex{2}: Standing waves associated with loads exhibiting various reflection coefficients. In this figure the incident wave arrives from the right.

This page titled 3.13: Standing Waves is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Steven W. Ellingson (Virginia Tech Libraries' Open Education Initiative) .

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