## A review of transformer design

In a process of designing a transformer, we often start with an approximate low frequency equivalent circuit. “Low frequency” is a relative term, it means at frequencies where each winding current phase is uniform, and the effects of distributed capacitance are insignificant.

Above is a commonly used low frequency equivalent of a transformer. Z1 and Z2 represent leakage impedances (ie the effect of magnetic flux leakage) and winding conductor loss. Zm is the magnetising impedance and represents the self inductance of the primary winding and core losses (hysteresis and eddy current losses).

### 50/60Hz power transformers

For 50/60Hz power transformers, Z1 and Z2 are mainly inductive and small (eg as would account for around 5% voltage sag under full load). Zm varies, it is large and mainly inductive for conservative designs using sufficient and good core material, and less so for designs that drive core magnetic flux into saturation.

### RF broadband transformers

For broadband RF transformers, Z1 and Z2 need to be small as they tend to be quite inductive and since inductive reactance is proportional to frequency, they tend to spoil broadband performance.

Zm shunts the input, so it spoils nominal impedance transformation (Zin=Zload/n^2) if it is relatively low. For powdered iron cores Zm is mainly inductive; and for ferrite cores Zm is a combination of inductive reactance and resistance depending on frequency and ferrite type.

Keep in mind that if Zm is sufficiently high, Im is low, and even though Zm may contain a large Rm component, Im^2*Rm may be acceptably low.

There are scores of articles on this website about ferrites, many of which show how to measure or calculate Zm from datasheets.

Proponents of powdered iron will claim that large Im does not create much loss because Rm is small, but large Im destroys broadband nominal impedance transformation (ie Insertion VSWR). Powdered iron tends to be low µ which increases leakage impedance and also destroys broadband nominal impedance transformation.

An online expert on the unsuitability of #43 for HF UNUNs discussed the stuff that masquerades as science in the name of ham radio, and gives one example which questions the exptert's opinion. Lets work through some examples, calculating and plotting two key metrics that should be considered right up front when designing an efficient broadband RF transformer with close to ideal impedance transformation (ie low InsertionVSWR).

## Analysis of a few examples

The following analyses are of expected core loss due to the magnetising impedance of the primary winding when the transformer is loaded to present an input impedance of 50+j0Ω. The magnetising impedance can be measured with only that primary winding on the core, the presence of a secondary winding, even if disconnected, may disturb the results.

Note that there is a quite wide tolerance on ferrite materials, and measured results my differ from the predictions based on published datasheets. Designs based on measurements of a single core are exposed to risks of being atypical.

Graph Y axes are not identically scaled.

### PA3HHO 2t on FT140-43

This configuration is very popular in ham radio. I am not sure who originated the design, PA3HHO's web article is a commonly cited reference.

Above is the percentage core loss when input impedance of the loaded transformer is 50+j0Ω.

Above is the Insertion VSWR caused by the magnetising impedance in shunt with 50+j0Ω. Note that this is not exactly the same configuration as for the previous chart.

### 4t on FT240-43

Above is the percentage core loss when input impedance of the loaded transformer is 50+j0Ω.

Above is the Insertion VSWR caused by the magnetising impedance in shunt with 50+j0Ω. Note that this is not exactly the same configuration as for the previous chart.

### 3t on Fair-rite 2643625002 (43)

This is a small #43 core as used in Small efficient matching transformer for an EFHW.

Above is the percentage core loss when input impedance of the loaded transformer is 50+j0Ω.

Above is the Insertion VSWR caused by the magnetising impedance in shunt with 50+j0Ω. Note that this is not exactly the same configuration as for the previous chart.

### 4t on Jaycar LO1238 (L15)

The Jaycar LO1238 is readily available in Australia, a medium size core of medium to high initial permeability (µi=1500) that seems overlooked by Australian hams in favor of harder to procure products.

Above is the percentage core loss when input impedance of the loaded transformer is 50+j0Ω.

## Other mixes

It seems many hams have a “favorite mix”, and many spurn #43, nominating others (#31, #61 often for this application).

All are possibilities that for a given core geometry and mix will require a certain minimum number of turns on the nominal 50Ω primary to meet the designer's loss and Insertion VSWR criteria. #61 is a lower loss material compared to #43, and it will require more turns to meet Insertion VSWR criteria at low frequencies, the length of the winding may limit the useful upper frequency.

## Conclusions

- The context of the article is HF broadband transformers with close to ideal nominal impedance transformation, and does not necessarily apply to other contexts.
- Three of the examples use #43 material, two of those designs have core loss less than 10% at 3.5MHz and lower on higher bands demonstrating that it is possible to design a broadband RF transformer for HF using #43 material.
- The PA3HHO example shows that insufficient turns leads to appalling core loss.
- Traditional wisdom is that higher µ cores will be even worse than #43, but the LO1238 design shows that a low cost core readily available in Australia is a worthy candidate for Australian hams.
- There is more to designing a transformer than presented here, this article describes a first analysis to screen likely candidates and find minimum primary turns for a given core to meet the design loss and InsertionVSWR criteria.
- Successful designs are almost always a compromise to meet sometimes competing / conflicting design criteria.

Read widely, and analyse critically what you read.