I have been asked to expand on the calculation of voltage magnitude and phase set out in Voltage symmetry of practical Ruthroff 4:1 baluns.

Above is Ruthroff’s equivalent circuit, Fig 3 from his paper (Ruthroff 1959). Focusing on the left hand circuit which explains the balun as a transmission line transformer (TLT), and taking the node 1 as the reference, the loaded source voltage appears at the bottom end of the combined 4R load, and transformed by the transmission line formed by the two wires of the winding, and inverted, at the top end of the combined 4R load.

It is the transformation on this transmission line that gives rise to loss of symmetry.

The complex ratio Vout/Vin is dependent on the complex reflection coefficient Gamma at both ends of the line and the line propagation constant gamma, all of which are frequency dependent complex quantities.

Vout/Vin=(1+GammaLoad)/(1+GammaSource)*e^(gamma*l)

In baluns, the real part is often close to unity, the phase is more significant.

TWLLC and family tools calculate this quantity in the long output, but I do not recall seeing it calculated in other tools. Values for Gamma may be shown in other tools, but again I have not seen gamma shown directly.

Astute readers will realise that a more correct balun could be made by including another TLT to supply the non-inverted output. By then, it looks like a Guanella 4:1 balun with the output centre tap grounded (so it behaves like a voltage balun) and has better balance… on symmetric loads.

## References / links

- Duffy, O. 2007. A model of a practical Ruthroff 1:4 balun.
- Ruthroff C. 1959. Some broad band transformers. Proceedings of the IRE.