Voltage Standing Wave Ratio (VSWR)

What is VSWR?

VSWR is defined as the ratio of the maximum voltage to the minimum voltage in standing wave pattern along the length of a transmission line structure. It varies from 1 to (plus) infinity and is always positive. Unless you have a piece of slotted line-test equipment this is a hard definition to use, especially since the concept of voltage in a microwave structure has many interpretations.

Sometimes VSWR is called SWR to avoid using the term voltage and to instead use the concept of power waves. This in turn leads to a mathematical definition of VSWR in terms of a reflection coefficient. A reflection coefficient is defined as the ratio of reflected wave to incident wave at a reference plane. This value varies from -1 (for a shorted load) to +1 (for an open load), and becomes 0 for matched impedance load. It is a complex number. This helps us because we can actually measure power.

The reflection coefficient, commonly denoted by the Greek letter gamma (Γ), can be calculated from the values of the complex load impedance and the transmission line characteristic impedance which in principle could also be a complex number.

Γ = (Z_l – Z₀)/(Z_l + Z₀)

The square of | Γ | is then the power of the reflected wave, the square hinting at a historical reference to voltage waves.

Now we can define VSWR (SWR) as a scalar value:

VSWR= (1 + | Γ |)/(1 – | Γ |) or in terms of s-parameters: VSWR= (1 + | S₁₁ |)/(1 – | S₁₁ |)

This is fine but what has it to do with common usage in ads and specifications. Generally, VSWR is sometimes used as a stand-in for a figure of merit for impedance matching. Sometimes this simplification of a scalar quantity and it’s restricted definition can lead to confusion in the matter of a source to load match. Most of the time there is no problem but, technically, VSWR derives from the ratio using the load impedance and the characteristic impedance of the transmission line in which the standing waves reside and not specifically to a source to load match. I prefer to think of VSWR as a figure of merit and to use the reflection coefficient whenever I am trying to solve problems.

By the way, if you think you have never experienced a standing wave personally, it’s very unlikely. Standing waves in a microwave oven are the reason that food is cooked unevenly (the turntable is a partial solution to that problem). The wavelength of the 2.45 GHz signal is about 12 centimeters, or about five inches. Nulls in the radiation (and heating) will be separated at a distance similar to wavelength.

Standing waves in nature

What’s a standing wave? Luckily there are tons of examples in nature. Any stringed instrument such as a guitar or piano makes music using standing waves. But what about a traveling wave that reflects off of an object and creates a standing wave due to constructive interference?

Let’s go to the beach. Breakers roll in off the ocean, come up on the sand, and disappear; no standing wave occurs. What’s happening? The beach is absorbing all (or at least most) of the energy, in effect it is “matched” to the wave front.

Now let’s go next door to marina where all of those expensive yachts are moored… chances are there are vertical concrete seawalls inside the marina to allow owners to bring their boats close enough so that only a small walkway is needed to get to them.

Now notice the breakwater that extends around the marina, with only a narrow opening for boats to go in and out. That’s there because the vertical walls in the marina offer near perfect reflection to moving waves (an “open circuit”).

Without the breakwater wall (which absorbs energy) huge standing waves are possible due to constructive interference, and all those boats would bob up and down like crazy corks and eventually everything would get smashed to tiny bits.

Other ways to express impedance mismatches

The reflection coefficient is what you’d read from a Smith chart. A reflection coefficient with a magnitude of zero is a perfect match, a value of one is perfect reflection.

The symbol for reflection coefficient is uppercase Greek letter gamma (). Note that the reflection coefficient is a complex value, so it includes an angle.

Unlike VSWR, the reflection coefficient can distinguish between short and open circuits. A short circuit has a value of -1 (1 at an angle of 180 degrees), while an open circuit is one at an angle of 0 degrees.

Quite often we refer to only the magnitude of the reflection coefficient. The symbol for this is the lower case Greek letter ρ.

The return loss of a load is merely the magnitude of the reflection coefficient expressed in decibels. The correct equation for return loss is:

Return loss = -20 x log [mag(Γ)]

Thus in its correct form, return loss will usually be a positive number. If it’s not, you can usually blame measurement error.

The exception to the rule is something with negative resistance, which implies that it is an active device (external DC power is converted to RF) and it is potentially unstable (it could oscillate).

Not something you have to worry about if you are just looking at coax cables!

However, many engineers often omit the minus sign and talk about “-9.5 dB return loss” for example. People that find it necessary to correct engineers who do this have underwear that is too tight.

Here are the equations that convert between VSWR, reflection coefficient (Γ) and return loss (RL) as well as mismatch loss (ML), which we will cover later). Note that |Γ| in these equations always stands for the magnitude of the complex reflection coefficient and is itself a scalar term in these equations:

Let’s end our discussion with a table of reflection VSWR, refection coefficient and return loss values (and remember that our VSWR  calculator can provide any values you need).

If you want to impress your friends, memorize as much of this table as you can. Yes, rounding off is permitted.

VSWR	Reflection coefficient	Return loss	Notes
1:1	0.00	infinity	a perfect match
1.1:1	0.05	26.44
1.2:1	0.09	20.83
1.3:1	0.13	17.69
1.4:1	0.17	15.56
1.5:1	0.20	13.98	A good rule of thumb: 1.5:1 = 14 dB
1.6:1	0.23	12.74
1.7:1	0.26	11.73
1.8:1	0.29	10.88
1.9:1	0.31	10.16	A good rule of thumb: 1.9:1 = 10 dB
2.0:1	0.33	9.54
3.0:1	0.50	6.02	A good rule of thumb: 3:1 = 6 dB
4.0:1	0.60	4.44
5.0:1	0.67	3.52
6.0:1	0.71	2.92
10:1	0.82	1.71
infinity:1	1.000	0.00	short or open circuit

Calculating VSWR from impedance mismatches

The mismatch of a load Z_L to a source Z₀ results in a reflection coefficient of:

Γ=(Z_L-Z₀)/(Z_L+Z₀)

Note that the load can be a complex (real and imaginary) impedance. If you can’t remember in which order the numerator is subtracted (did we just say “Z_L-Z₀” or Z₀-Z_L“?), you can always figure it out by remembering that a short circuit (Z_L=0) is on the left side of the Smith chart (angle = -180 degrees) which means Γ=-1 in this case, which means that the minus sign belongs in front of Z₀.

The magnitude of the reflection coefficient is given by:

ρ=mag(Γ)

For cases where Z_L is a real number,

ρ=abs((Z_L-Z₀)/(Z_L+Z₀))

Note that “abs” means “absolute value” here. VSWR can be calculated from the magnitude of the reflection coefficient:

VSWR=(1+ρ)/(1-ρ)

For cases where Z_L is real, with a little algebra you’ll see there are two cases for VSWR, calculated from load impedance:

For Z_L<Z₀: VSWR=Z₀/Z_L

For Z_L>Z₀: VSWR=Z_L/Z₀

Just remember to divide the larger impedance by the smaller impedance, because VSWR is always greater than 1. Hey, this calculation is so easy you can do it in your head!!!

Let’s look at the special case where you mix up 50 ohm parts into a 75 ohm system (or vice-versa). In either case, the resulting VSWR is 1.5:1. Yes, we did that without a calculator. While we’re at it, the reflection coefficient is:

Γ=(75-50)/(75+50)=0.2