Skip to main content
Engineering LibreTexts

3.13: Standing Waves

  • Page ID
    6279
  • 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

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

    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.,

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

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

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

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

    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: \[\left|\widetilde{I}(z)\right| = \frac{|V_0^+|}{Z_0} \sqrt{ 1 + \left|\Gamma\right|^2 - 2\left|\Gamma\right| \cos\left( 2\beta z + \phi \right) }\]

    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:

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

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

    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.

    Contributors and Attributions