(Terman 1955) gives a meaning for the term SWR (or VSWR).

The character of the voltage (or current) distribution on a transmission line can be conveniently described in terms of the ratio of the maximum amplitude to minimum amplitude possessed by the distribution. This quantity is termed the standing wave ratio (often abbreviated SWR)…

Standing-wave ratio=S=Emax/Emin

Note that the use of capital E implies the magnitude of voltage, so Emax/Emin must always be a positive number.

## Lossless line example

Let’s look at an example of a 5Ω load on a line with Zo=50+j0Ω at 0.1MHz.

The standing wave is observable, the expression VSWR=Emax/Emin seems straight forward enough. The voltage along the line could be sampled and VSWR determined, seems all very practical.

VSWR is often expressed in terms of ρ, the magnitude of the complex reflection coefficient Γ.

VSWR=(1+ρ)/(1-ρ)

Above is a plot of ρ and VSWR for the example, ρ and VSWR are both uniform along the line, and VSWR reconciles with (1+ρ)/(1-ρ).

This is the world of lossless lines, the world of most text books, but it is not the real world. The real world introduces some complications that make these simple definitions less adequate.

## Lossy line with same load

Let’s look at a more practical example:

- Zload=5+j0Ω
- Nominal Zo=50Ω;
- vf=0.66;
- Conductor loss 0.005dB/m;
- Dielectric loss 0.00003dB/m;
- f=0.1MHz.

The line parameters are similar to RG58A/U, the characteristics of a line of that size will be very similar.

Note that actual Zo at 0.1MHz is not 50+j0Ω, it is 50.80-j8.87Ω, a consequence of the line loss.

Above is the voltage and current distribution on the lossy line.

Immediately it is obvious that the simple definition of VSWR=Emax/Emin is a problem as line loss also affects the observed values of Emax and Emin. The problem is that direct measurement of VSWR essentially depends on observation of Emax and Emin at different points and is exposed to influence by line loss. Whilst it may be possible to extrapolate Emax and Emin from more local samples, it is error prone and is not really direct measurement of Emax and Emin.

ρ is a property of a point on the line, and so we can calculate a ‘virtual’ VSWR at a point from ρ at that point.

The above plot shows the variation in ρ due to mismatch and line loss.

Also shown is VSWR calculated from ρ.

That all seems pretty sane, a possible solution, but is it?

## Lossy line with load ρ > 1

Let’s look at an interesting example that gives rise to ρ>1.

- Zload=5+j50Ω
- Nominal Zo=50Ω;
- vf=0.66;
- Conductor loss 0.005dB/m;
- Dielectric loss 0.00003dB/m;
- f=0.1MHz.

Above is the voltage and current standing wave distribution, focussing on the 300m nearest to the load. It might not be obvious from the chart, but the VSWR cannot be determined by direct measurement of Emax and Emin for the same reasons as in the last case… so we hope to again calculate VSWR from ρ.

Above, the plot of ρ which passes through 1.0 and VSWR(ρ) is singular at that point, it is undefined at that point. For ρ>1.0, VSWR(ρ) using the traditional formula given above will be negative… which is in conflict with the definition as Emax/Emin being a positive number.

One could change VSWR(ρ) to always produce a positive number, but it is probably better to allow the negative result as the function has a simple inverse, ie ρ=(VSWR-1)/(VSWR+1).

Forcing a positive result is a fudge that only masks the underlying problem and ruins the inverse function

## Does ρ>1 imply ‘reflected power’ greater than ‘forward power’

No, it does not.

The common notion that ‘forward power’=Vf^2/Zo and ‘reflected power’=(ρ*Vf)^2/Zo where Zo is assumed to be purely real is an approximation that is invalid when Xo≠0 as in the lossy line cases above. Any argument about conservation of energy based invalid assumptions is itself invalid.

## Summary

- It is possible to have ρ>1.
- VSWR calculated from ρ is singular where ρ=1, and will yield negative VSWR where ρ>1.
- It may be unwise to ‘correct’ the VSWR(ρ) formula to give positive results as there is no longer the simple inverse function ρ=(VSWR-1)/(VSWR+1).

## References / links

- Stannard, G.E. Jan 1967, Calculation of power on a transmission line In
*Proceedings of the IEEE*, vol.55, no.1, pp.132-132, Jan. 1967. - Terman 1955. Electronic and Radio Engineering: McGraw-Hill New York.
- Vernon, R and Seshadri, S. Jan 1969. Reflection coefficient and reflected power on a lossy transmission line In Proc IEEE, Jan 1969.