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Clive Ruthroff described several baluns and ununs in his article 'Some BroadBand Transformers' published in 1959.
One particular configuration, the 1:4 balun is of particular interest as it is promoted as part of the feed system of a multiband unloaded vertical.
Ruthroff's own explanation, and it seems many of those who came after him deal with the balun by simplification to equivalent circuits at low and high frequencies. Whilst Ruthroff's article predicts insertion loss, it does so for an equivalent circuit that does not include any balun loss elements (eg conductor loss, magnetic core loss).
This article proposes a lossy transmission line model of a practical Ruthroff 1:4 unun that is effective for all frequencies within and immediately adjacent to the pass band. The model is of a two wire transmission line wound around a toroidal ferrite core, and includes:
Such a model can be used for exploring not only variations in design such as different core materials, wire thickness and spacing, number of turns, etc, but exploration of loads other than the nominally matched balanced load.
Fig 1 is Ruthroff's equivalent circuit, Fig 3 from his paper. (The current designations are not used further in this article.
Fig 2 shows the proposed equivalent circuit. The transformers T1 and T2 are ideal transformers that are introduced for the purpose of separating the common mode and differential currents. The common mode current is routed through the common mode choke impedance Z_{c}, and the differential current flows in the transmission line TL. The differential mode current can be analysed by transmission line equations, and the common mode current can be analysed by lumped component circuit analysis.
\[V_1=V_2 cosh(\theta) + I_2Z_0sinh(\theta) \tag{1}\] \[I_2=\frac{V_1+V_2}{Z_l}+\frac{V_1+V_2}{4Z_c}\] \[I_2=(V_1+V_2) \left (\frac1{Z_l}+\frac1{4Z_c} \right) \tag{2}\] Substituting (2) into (1)... \[V_1=V_2 cosh(\theta)+(V_1+V_2)\left ( \frac1{Z_l}+\frac1{4Z_c} \right) Z_0 sinh(\theta)\] \[\frac{V_2}{V_1}=\frac{1\left ( \frac1{Z_l}+\frac1{4Z_c} \right) Z_0 sinh(\theta) }{cosh(\theta)+\left ( \frac1{Z_l}+\frac1{4Z_c} \right) Z_0 sinh(\theta)}\] 
Fig 3 is a solution to the proposed equivalent circuit. The quantity θ is the product of γ, the transmission line complex propagation coefficient, and transmission line length. All other currents, voltages, and powers can be derived from the V_{2}/V_{1} relationship. Note that quantities in Fig 3 may be complex values, and many are frequency dependent.
The choke impedance depends on the ferrite core, and its ferrite permeability and loss are frequency dependent.
Fig 4 shows the characteristics of Fairrite #52 mix. The choke impedance cannot simply be modelled as a fixed idealised inductor. The model needs to calculate the equivalent series inductive reactance and loss resistance at each frequency of interest. Choke impedance Z_{c}=j*2*π*f*n^2*A_{l}/µ_{ilf}*(µ'jµ'') where A_{l} is inductance for 1 turn at µ_{r}=µ_{ilf}, and µ_{ilf} is the initial permeability at low frequency.
Fig 5 shows the choke impedance for a 12 turn bifilar winding on an FT24052 core.
Fig 6 shows the VSWR(50) and Loss of the prototype unun with a 200+j0Ω load. If the unun could safely dissipate a maximum of 20W, the continuous power rating implied by the loss curve is 1700W.
Fig 7 shows the VSWR(50) and Loss of the prototype unun with a 1000+j0Ω load.
Fig 8 shows input impedance with a 1000+j0Ω load. Note that 'ideal' transformation is achieved in a very small band from about 3.5MHz to about 6.5MHz. In the expectation that these devices would be used in a system with an ATU, nonideal transformation ratio is not usually a big issue.
Fig 7 showed a large increase in loss with the higher impedance load. If the unun could safely dissipate a maximum of 20W, the continuous power rating implied by the loss curve is lower at 500W, see Fig 9.
Power handling is very much affected by the complex load impedance and frequency, and power handling is often an important issue.
Watch this space... a work in progress!!!
To Do:
Conclusions are:
The Ruthroff 1:4 unun made with a bifilar winding on a ferrite core can be modelled as a Transmission Line Transformer.
Power handling is very much affected by the complex load impedance and frequency, and power handling is often an important issue
Transformation ratio is varies with load and frequency. In the expectation that these devices would be used in a system with an ATU, nonideal transformation ratio is not usually a big issue.
Term  Meaning 
ATU  Antenna Tuning Unit, an impedance transforming device. 
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