3.13: Standing Waves
- Page ID
- 6279
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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](https://eng.libretexts.org/@api/deki/files/16225/m0086_fStandingWaveOC-V.png?revision=1&size=bestfit&width=350&height=251)
(a) Potential.
(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](https://eng.libretexts.org/@api/deki/files/16227/m0086_fStandingWaves.png?revision=1&size=bestfit&width=400&height=255)