There is a lot of woolly thinking amongst hams about transmission line loss under mismatch, perhaps exemplified by Walt Maxwell (Maxwell 2001):

The power lost in a given line is least when the line is terminated in a resistance equal to its characteristic impedance, and as stated previously, that is called the matched-line loss. There is however an additional loss that increases with an increase in the SWR.

This article probes the folk lore with an example scenario designed to expose the failure of such thinking.

Lets consider a very short section of air insulated two wire transmission line (TL) made of 1mm copper wires separated by 10mm with three loads, Zo, 3*Zo and Zo/3 at 10MHz.

The length of line chosen is 30mm, λ/1000 at 10MHz.

For this line, length and these loads, the current along the line section is almost perfectly uniform and the distributed inductance, capacitance and insulation loss are insignificant, loss is dominated by the TL conductor resistance.

## Simple linear circuit analysis

This is such a simple scenario that it can be calculated with very good accuracy very simply using basic linear circuit analysis techniques.

### Step 1: find conductor Rrf/Rdc factor

The RF resistance of a conductor is higher than its DC resistance due to skin effect. At the spacing chosen, there is negligible proximity effect.

Let us find the factor Rrf/Rdc.

Above, we see that the RF resistance of the conductor is 12.37 times the DC resistance.

### Step 2: find RF resistance of wire

Find the resistance of the copper wire.

Above is the RF resistance of the 0.06m, the total length of both sides of the line section and the Rrf/Rdc is supplied to the calculator. Rrf=0.015876Ω.

### Step 3: calculate Zo

For a two wire air insulated line, Zo=Zo=120acosh(D/d). In this case, Zo=359Ω.

### Step 4: calculate loss with Zl=Zo

We can use a simple series circuit of the total line resistance and load resistance, and since the same current flows through both, the loss is Rtotal/Rload=(359+0.015876)/359=1.0000442, or in dB, 10*log(1.0000442)=0.000192dB. This is known as the Matched Line Loss (MLL).

### Step 4: calculate loss with Zl=Zo/3

We can use a simple series circuit of the total line resistance and load resistance, and since the same current flows through both, the loss is Rtotal/Rload=(119.7+0.015876)/119.7=1.0001327, or in dB, 10*log(1.0001327)=0.000576dB.

The loss is 3 times the matched line loss.

In TL terms, the mismatch gives rise to a reflected wave of magnitude half the forward wave (ρ=0.5), and VSWR=3.

### Step 4: calculate loss with Zl=Zo*3

We can use a simple series circuit of the total line resistance and load resistance, and since the same current flows through both, the loss is Rtotal/Rload=(1077+0.015876)/1077=1.0000147, or in dB, 10*log(1.0000147)=0.000064dB.

The loss is 1/3 of the matched line loss. Note that is in conflict with the opening quote from Maxwell: There is however an additional loss that increases with an increase in the SWR.

In TL terms, the mismatch gives rise to a reflected wave of magnitude half the forward wave (ρ=0.5), and VSWR=3.

## Wait a minute

Simple models are good… but does this pass muster, does it pass a reasonableness test?

It takes only very basic circuit analysis skills and knowledge to recognise that for a load connected by conductors of some fixed resistance with uniform current, the losses are highest for lower resistance loads.

The test scenario was designed to be capable of accurate solution using simple linear circuit analysis techniques, and the results are as expected.

## Two wire line loss calculator (TWLLC)

The scenario can be solved with greater accuracy using TLLC. Let us do just one case, Zl=Zo*3.

### Calculate loss with Zl=Zo*3

Above are the results from TWLLC, line loss is 0.0000652dB, almost the same as calculated ‘by hand' above.

TWLLC reconciles well with the basic linear circuit analysis, as it should.

## Reflections II

(Maxwell 2001) states:

Sec 17.4 Attenuation and Power Loss Now let's consider the effects of attenuation and power loss. In lines having attenuation, all power entering the line is absorbed in the load, except for that which is lost through line attenuation, regardless of the mismatch or SWR. (See Chapters 5 and 6.) Of course, the attenuation is greater when there is a load mismatch, because in addition to the attenuation of the forward power, the reflected power is also attenuated during its return to the transmatch.

…

The increase in line attenuation resulting from SWR can be determined from the graphs of Figs 1-1 and 6-1, or it may be calculated precisely from Eq 6-1.

He gives Eq 6-1 as the embodiment of his explanation without any qualification or limitations :

It can be seen that Maxwell's Eq 6-1 uses ρ the magnitude of the complex reflection coefficient for its precise calculation.

Though not stated explicitly, ρ is at the load end.

Evaluating Maxwell's formula for both mismatch cases, Zo*3 and Zo/3 gives the same result (since ρ=0.5 in both cases) of loss=0.00032dB which is 0.55 times the simple calculation for Zo/3 and 5 times the simple calculation for Zo*3.

Maxwell's equation does not reconcile with the basic linear circuit analysis, and is prone to gross error in this case.

The failure of the formula questions his whole explanation of the effect of reflections on transmission line loss.

## ARRL Antenna book 21

Like Maxwell's formula, Straw's depends on ρ the magnitude of the complex reflection coefficient and will produce the same result for both mismatch scenarios. Again, though not stated explicitly, ρ is at the load end.

Evaluating Straw's formula for both mismatch cases, Zo*3 and Zo/3 gives the same result (since ρ=0.5 in both cases) of loss=0.00032dB which is 0.55 times the simple calculation for Zo/3 and 5 times the simple calculation for Zo*3.

Straw's equation does not reconcile with the basic linear circuit analysis, and is prone to gross error in this case.

The failure of the formula questions his whole explanation of the effect of reflections on transmission line loss.

## Failure in thinking

The concept that a forward wave and reflected wave are each attenuated directly and independently is shown to be wrong by analysis of this short line scenario.

The forward and reflected waves give rise to E and I that vary along a real transmission line, and the loss is due to I^2R loss in conductors and E^2G loss in dielectric, so the loss in any incremental length of line depends on E and I at that point. The loss in any line then is the sum of the incremental losses due to varying E and I along the line.

Maxwell's rather general statement There is however an additional loss that increases with an increase in the SWR.

is proved wrong by this example.

## Conclusions

- A very simple transmission line scenario that could be solved accurately using basic linear circuit analysis was designed as a basis for evaluation of some published techniques for predicting TL loss under mismatch.
- The evaluation scenario was solved for loads Zl=Zo, Zo/3,Zo*3 using simple linear circuit analysis.
- Two Wire Line Loss Calculator (TWLLC) reconciles well with the linear circuit analysis solution.
- Reflections II did not reconcile with the linear circuit analysis solution, and showed gross error.
- ARRL Antenna book 21 did not reconcile with the linear circuit analysis solution, and showed gross error.
- The formulas proposed by (Maxwell 2001) and (Straw 2007) are embodiments of their independent forward and reverse waves attenuation explanation and do not yield correct results for the example case suggesting their explanation is flawed, that it doesn't work like they suggest.

## References

- Duffy, O. 2001. RF Two WIre Transmission Line Loss Calculator (TWLLC). VK1OD.net (offline).
- Maxwell, Walter M. 2001. Reflections II. Sacramento: Worldradio books.
- Straw, Dean ed. 2007. The ARRL Antenna Book. 21st ed. Newington: ARRL. p24-10.