Standing wave ratio

The voltage standing wave ratio (VSWR or just SWR) is the ratio of maximum to minimum voltage magnitude along a Transmission line carrying both incident and reflected waves:

$S=\frac{|\tilde{V}{|}_{\max }}{|\tilde{V}{|}_{\min }}=\frac{1+|\Gamma |}{1-|\Gamma |}.$

Where $|\Gamma |$ is the magnitude of the Reflection coefficient. SWR is a single dimensionless number characterizing how mismatched the load is.

Range

• $S=1$: matched load, $|\Gamma |=0$. No reflected wave, no standing wave pattern. Voltage magnitude is constant along the line.
• $1<S<\infty$: partial reflection. Standing wave with definite maxima and minima.
• $S=\infty$: full reflection ($|\Gamma |=1$). Short, open, or purely reactive load.

SWR is real and nonnegative regardless of $\Gamma$‘s phase — it captures only the magnitude of mismatch, not the type.

Why it’s a useful metric

On a transmission line with reflection, the standing wave pattern has voltage maxima where incident and reflected waves add in phase, and minima where they oppose. The two amplitudes:

$|\tilde{V}{|}_{\max }=|V_0^{+}|(1+|\Gamma |),|\tilde{V}{|}_{\min }=|V_0^{+}|(1-|\Gamma |).$

Their ratio is independent of $|V_0^{+}|$ — a normalized, scale-invariant measure. You can measure SWR with just a voltage probe scanning the line: read the max, read the min, take the ratio. No need to know absolute amplitudes.

In terms of |Γ|

Solving $S=(1+|\Gamma |)/(1-|\Gamma |)$ for $|\Gamma |$:

$|\Gamma |=\frac{S-1}{S+1}.$

So SWR and $|\Gamma |$ encode the same information; you can convert either way.

Some reference points:

• $|\Gamma |=0\to S=1$
• $|\Gamma |=0.1\to S=1.22$
• $|\Gamma |=0.2\to S=1.50$
• $|\Gamma |=0.5\to S=3.00$
• $|\Gamma |=0.9\to S=19$
• $|\Gamma |=1.0\to S=\infty$

A 50% reflection by amplitude is SWR = 3 (severe mismatch). RF designers typically want SWR < 2, ideally < 1.5.

Standing wave geometry

The voltage magnitude along the line, in terms of distance $d$ from the load:

$|\tilde{V}(d)|=|V_0^{+}|[1+|\Gamma {|}^2+2|\Gamma |\cos (2\beta d-{\theta }_r){]}^{1/2}.$

Maxima occur where $2\beta d-{\theta }_r=2n\pi$ — incident and reflected waves are in phase. Minima occur where $2\beta d-{\theta }_r=(2n+1)\pi$ — destructive interference.

Spacing:

• Maxima are spaced $\lambda /2$ apart.
• Minima are spaced $\lambda /2$ apart.
• Each maximum is $\lambda /4$ from the nearest minimum.

So the standing wave has spatial period $\lambda /2$ — half the period of the incident or reflected waves alone. Two waves of period $\lambda$ interfering create an interference pattern of period $\lambda /2$.

First voltage max/min positions

Define $d_{\text{max}}^1$ as the position of the first voltage maximum from the load (smallest positive $d$ where max occurs). From $2\beta d_{\text{max}}-{\theta }_r=0$ (or $2\pi$ if ${\theta }_r$ is too negative):

$d_{\text{max}}^1=\{\begin{array}{ll}{\theta }_r\lambda /(4\pi )& \text{if }0\le {\theta }_r\le \pi \\ {\theta }_r\lambda /(4\pi )+\lambda /2& \text{if }-\pi \le {\theta }_r<0\end{array}$

The first minimum is $\lambda /4$ from the first maximum.

The positions of maxima and minima depend on ${\theta }_r$ (the phase of $\Gamma$), so by measuring where the first max or min sits on the line, you can extract the phase of the load reflection coefficient — and then deduce $Z_L$. This is one classical way of characterizing an unknown load.

In context

SWR is the engineering metric of matching quality. It’s measurable with a “slotted line” (1950s) or modern network analyzer. The RF community uses it daily:

• A new RF amplifier spec sheet quotes “VSWR ≤ 2:1” — meaning $|\Gamma |\le 1/3$.
• A poorly designed antenna at the wrong frequency might have SWR of 10:1 — most power reflected back to the transmitter.
• Tuners and matching networks are adjusted “until the SWR meter reads 1:1.”

The Smith chart’s “constant-SWR circles” are circles of constant $|\Gamma |$ centered at the origin — moving along the line keeps you on the same constant-SWR circle, only the phase of $\Gamma$ changes. This is a key visualization in matching design.