In an social media discussion about loss of EFHW transformers under mismatch conditions, one of the gathered experts said:

It doesn't even have to be highly complex Z. Just presenting an impedance other than 2450 sends the loss through the roof. The back to back transformer test is misleading unless the antenna presents something very close to 2450 on each band for which it is used.

giving this graphic to quote someone else's work in support.

Interpreting this graphic is fraught with risks, the author obviously does not understand and accept / follow the conventional meaning of term loss.

Nevertheless, the author has set up a Spectrum Analyser (SA) and Tracking Generator (TG) to pass a signal through a 1:49 transformer then an impedance matching attenuator (2650? and 30Ω load cases on the transformer) and factored in the loss in those attenuators by setting the SA reference level so that the display reads the network output power relative to input (ie Gain).

Remember that the conventional meanings are:

- Gain=Power
_{out}/Power_{in}which can be expressed as Gain_{dB}=10log(Gain); - Loss=Power
_{in}/Power_{out}which can be expressed as Loss_{dB}=10log(Loss); and - Loss
_{dB}=-Gain_{dB}.

A further caveat is that transmitters are not necessarily well represented as a Thevenin source, so measurements using such sources (VNA, SA with TG) and application of linear circuit theory are not necessarily applicable.

Let's skip the hand waving sends the loss through the roof

and talk in numbers, it is the beginning of understanding. A quotation from Lord Kelvin:

When you can measure what you are speaking about, and express it in numbers, you know something about it. But when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind. It may be the beginning of knowledge but you have scarcely in your thoughts advanced to the state of science.

It does appear that he is trying to prove by measurement the 50Ω MismatchLoss of a 1:49 transformer with 30+j0Ω load.

## Ideal 1:49 transformer with 30+j0Ω load – theoretical

Let's calculate the case of a 2450Ω Thevenin source with load of 30+j0Ω.

The quantity 1/(1-|s|^2) is the MismatchLoss, equivalent to InsertionLoss in this ideal transformer scenario, is 13.2dB.

Let's calculate the equivalent case of a 50Ω Thevenin source with load of (30+j0)/49=0.6122+j0Ω.

The quantity 1/(1-|s|^2) is the MismatchLoss, equivalent to InsertionLoss in this ideal transformer scenario, is 13.2dB.

## Real 1:49 transformer with 30+j0Ω load – calibrated model

Lets look at the Simsmith model documented at An improved simple Simsmith model for exploration of a common EFHW transformer designs (v1.03) for a FT240-43 with 3:14t with compensation at 7.1MHz.

A calibrated model means that the model parameters were adjusted to reconcile with measurements of a real transformer with nominal load and short circuit load.

This is not a particularly good design, but very popular among hams.

### Nominal load (2450+j0Ω)

The model is setup so that under matched conditions, power in the load is 0dBm (see right hand side).

The power developed in the nominal load 2450+j0Ω is -0.44dBm, 0.44dBm less than the matched case, and so InsertionLoss is 0.44dBm.

I should mention that this model assumes copper loss is insignificant, a reasonable assumption for the EFHW application. Copper loss may be significant if a transformer designed for 50:2450Ω (ie wire appropriate to the current) is used with a very low impedance load at much higher current.

### Grossly mismatched load of 30+j0Ω

The mismatch case used in the quoted graphic is 30+j0Ω, an extreme mismatch.

The power developed in the 30+j0Ω load is -13.09dBm, 13.09dBm less than the matched case, and so InsertionLoss is 13.09dBm. The VSWR at the source is 74, and extreme mismatch.

Note that this is almost equal to the theoretical value calculated earlier of 13.2dBm, but it 0.1dB less due to the real transformer and compensation capacitor effects and losses.

We can see that the power into the Ccomp element is -12.77dBm, this is the power from the source. We are not obtaining the 0dBm available from the source because of severe mismatch. This 12.77dB MismatchLoss makes up most of the InsertionLoss of 13.09dB. The remaining 0.32dB is (Transmission) Loss, it accounts for input power converted to heat in core loss (in this model).

So returning the to quote:

It doesn't even have to be highly complex Z. Just presenting an impedance other than 2450 sends the loss through the roof. The back to back transformer test is misleading unless the antenna presents something very close to 2450 on each band for which it is used.

and the example given of a 30+j0Ω on the 1:49 transformer (an extreme mismatch), one should expect with a good transformer, that InsertionLoss will be around 13dB which is mainly due to extreme mismatch rather than transformer losses. Something similar would be observed by connecting a 0.6Ω load to a 50Ω source.

Beware of hand waving on social media, there might not be much sound science behind it.

What about a complex load other than 2450+j0Ω? The same techniques can be used to estimate the InsertionLoss (again assuming a Thevenin source).

In finishing, let me repeat the caveat: transmitters are not necessarily well represented as a Thevenin source, so measurements using such sources (VNA, SA with TG) and application of linear circuit theory are not necessarily applicable.

## Conclusions

If the quoted graphic author's calibration and measurements of his real transformer were correct, his measured InsertionLoss of 12.8dB is very close to the theoretical expectation of an ideal 1:49 transformer (13.2dB), it is actually 0.6dB lower InsertionLoss, and does not suggest this is a ‘lossy transformer' per se, just wrongly applied.

In the world of ham radio, ALL antennas ‘work' (whatever that means)… just some ‘work' better than others. If you use s 1:49 transformer to feed a quarter wave, you might expect poor performance (as in the 30Ω example worked here).