# On negative VSWR

Some calculating tools come up with a negative value of VSWR under some circumstances.

Considering the meaning of VSWR: the ratio of the voltage maximum on a long transmission line to the adjacent voltage minimum, calculated negative VSWR might seem an aberration, invalid even. Note that nothing in this definition makes VSWR a property of a dimensionless point on a line.

VSWR can be measured directly by sampling voltage along a transmission line with a voltage probe. That said, it is almost never done and VSWR is inferred from other measurements, usually point measurements.

A transmission line is free to carry waves in two directions, and the ratio of voltage to current for each of those waves is the characteristic impedance Zo.

## The Lossless line case

When a long line is terminated in some impedance Zl,  the wave components Vf and Vr, If and Ir must reconcile with Vload and Iload, the variable that accommodates this is the complex reflection coefficient Γ (see Telegrapher’s Equation). Zo of a lossless line is always a purely read number, it no imaginary part.)

If such a line is lossless, it is easy to see that as we travel back towards the source, the phase of the forward and reflected voltage components vary (the magnitude remains constant, it is a lossless line), and so at some points the voltage will be a maximum of |Vf|+|Vr| and at some other points the voltage will be a minimum of |Vf|-|Vr|.

In such a case, we can see that the ratio of Vmax to Vmin is given by (Vf+Vr)/(Vf-Vr) or (1+|Γ|)/(1-|Γ|). ρ is sometimes used to represent |Γ|, so you will also see VSWR=(1+ρ)/(1-ρ).

The last function is invertible, we can write ρ=(VSWR-1)/(VSWR+1);

## Lossy line case

In the lossy line case things are different on two important counts:

1. Zo is a complex value with a small but sometimes significant imaginary part;
2. loss means that directly measured voltage maxima and minima will not reconcile with value ρ derived at an arbitrary point, ρ varies along the line.

In some scenarios, application of the lossless line based formulas above will result in significant error, it is incumbent on the user to prove suitability of the results to the scenario.

These errors are mostly ignored until something that seems on the surface quite wrong is encountered, for example a negative VSWR result… it must be dodgy? Some propose that it can be fixed by modifying the formula to prevent a negative result… but that is taking a formula that is not valid for the scenario and doctoring the result to ‘fix’ the sign problem to make it look more pleasing but ignoring the strong hint that the formula is not suited to the scenario. It is a kind of fool’s paradise.

Note that the VSWR=f(ρ) and ρ=f(VSWR) expressions given earlier are mathematically sound even if the meaning of that VSWR seems of no use. For example, if VSWR is calculated to be -20, the calculation ρ=(-20-1)/(-20+1)=1.1 is sound and that value of ρ could occur in practice.

The VSWR formulas given in the lossless lines section above depend on that assumption and may have significant errors when applied to some scenarios of lossy lines.

## Measuring with an instrument calibrated for a purely real Zref

The specific case of negative VSWR arises from scenarios where ρ>1.

This will never happen when Zref is purely real. If you make measurements with an instrument calibrated for a purely real Zref, you will never correctly observe ρ>1. This is true even if such an instrument is inserted in a line section where ρ>1 within the line section (ie when measured wrt the actual line Zo).

## Dealing with complex Zref

If you are working with modelling tools that use complex values of Zo and Zload, and ρ is greater than 1 in the specific scenario, use Γ by all means but realise that as soon as you derive or use values for VSWR you have discarded important information and the result may have error, perhaps significant error.