One frequently sees discussions of coupled coils in ham fora, and the advice of the forum experts is commonly sadly lacking.

An example is the thread Impedance matching transformer where the OP is encouraged to make a transformer for 2:1 impedance transformation ratio based simply on turns ratio and a Rule Of Thumb for minimum number of turns.

Lets review a design where two windings of say 10µH and 20µH are wound on a toroidal core. With no flux leakage, the turns ratio would be 1:1.414. The model is a simple one of coupled coils and ignores self capacitance.

## 100% flux coupling

If there was no flux leakage, the mutual inductance is (10*20)^0.5=14.14µH, and we can build a three component model of the coupled coils along with the intended 100+j0Ω load.

Above the model for 100% flux coupling.

And above, the response of the network. At 7MHz, the input impedance is 48.7+j8.7Ω, not perfect, but close (VSWR=1.2).

The curve shows that this is definitely not a broadband transformer, so there is clearly more at play than simply the turns ratio and sufficient turns for the lowest frequency.

But, the reality is that there is flux leakage.

## 70% flux coupling

Flux coupling for a #2 powdered iron toroid will be well less than 100%, 70% is a more realistic figure. Lets assume again coil inductances of 10µH and 20µH.

Above, the model adjusted for 70% flux coupling factor.

Above, the results for 70% flux coupling.

The practical model does not look anything like a 2:1 impedance transformer, much less a broadband one.

It is little wonder that the constructor found that the design did not work as expected by the forum experts.

## Real world transformer

A real world transformer with turns ratio of 1:1.414 will not have inductance in the ratio of 1:2 due to the effects of flux leakage… a further factor contributing to non-ideal performance

Analysis requires measurement of the inductances (each coil and mutual inductance), and computation of the three component model.

A simple method of finding M is to measure L of two coils series aiding and series opposing, find the difference and divide it by 4 to give M.

Above is a model of a real world transformer with 13t and 19t on a T200-2 core based on measurement of the inductances.

Again, nowhere near ideal 1:2 impedance transformation.

## Conclusions

- Coupled coils are used widely in communications electronics, and very often where flux coupling is significantly less than 100%.
- Techniques for analysis where flux coupling is less than 100% are not onerous.
- The core type recommended is not suited to the specific application (near ideal impedance transformation based on the square of turns ratio).

## References

- Duffy, O. nd. Link coupling transformation ratio. VK1OD.net (offline).