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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+0ejβz, which is traveling in the +z axis along a lossless transmission line. Associated with this wave is a reflected wave V0e+jβz=ΓV+0e+jβz, where Γ is the voltage reflection coefficient. These waves add to make the total potential

˜V(z)=V+0ejβz+ΓV+0e+jβz=V+0(ejβz+Γe+jβz)

The magnitude of ˜V(z) is most easily found by first finding |˜V(z)|2, which is:

˜V(z)˜V(z)=|V+0|2(ejβz+Γe+jβz)(ejβz+Γe+jβz)=|V+0|2(ejβz+Γe+jβz)(e+jβz+Γejβz)=|V+0|2(1+|Γ|2+Γe+j2βz+Γej2βz)

Let ϕ be the phase of Γ; i.e.,

Γ=|Γ|ejϕ

Then, continuing from the previous expression:

|V+0|2(1+|Γ|2+|Γ|e+j(2βz+ϕ)+|Γ|ej(2βz+ϕ))=|V+0|2(1+|Γ|2+|Γ|[e+j(2βz+ϕ)+ej(2βz+ϕ)])

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

cosθ=12[ejθ+ejθ]

yielding: |V+0|2[1+|Γ|2+2|Γ|cos(2βz+ϕ)]

Recall that this is |˜V(z)|2. |˜V(z)| is therefore the square root of the above expression:

|˜V(z)|=|V+0|1+|Γ|2+2|Γ|cos(2βz+ϕ)

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: |˜I(z)|=|V+0|Z01+|Γ|22|Γ|cos(2βz+ϕ)

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, ZL, is equal to the characteristic impedance of the transmission line, Z0, Γ=0 and there is no reflection. In this case, the above expressions reduce to |˜V(z)|=|V+0| and |˜I(z)|=|V+0|/Z0, as expected.

Open or Short-Circuit. In this case, Γ=±1 and we find:

|˜V(z)|=|V+0|2+2cos(2βz+ϕ)

|˜I(z)|=|V+0|Z022cos(2βz+ϕ)

where ϕ=0 for an open circuit and ϕ=π for a short circuit. The result for an open circuit termination is shown in Figure 3.13.1(a) (potential) and 3.13.1(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 3.13.1: Standing wave associated with an opencircuit termination at z=0 (incident wave arrives from left).

Also note:

The period of the standing wave is λ/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 2βz. This can be rewritten as 2π(β/π)z, so the frequency of variation is β/π and the period of the variation is π/β. Since β=2π/λ, we see that the period of the variation is λ/2. Furthermore, this is true regardless of the value of Γ.

Mismatched loads. A common situation is that the termination is neither perfectly-matched (Γ=0) nor an open/short circuit (|Γ|=1). Examples of the resulting standing waves are shown in Figure 3.13.2.

m0086_fStandingWaves.png Figure 3.13.2: 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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