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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 because it underlies many other configurations, and understanding it helps towards understanding the other configurations. Further, this balun or constructs of this balun are widely used, though often not identified as such in descriptions or circuits.
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 balun 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 5 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 common mode current I_{3} is the result of components I_{3}/2 which flow into both ends of the left hand winding of T1, and out of both ends of the right hand winding of T2. The differential mode current can be analysed by transmission line equations, and the common mode current can be analysed by lumped component circuit analysis.
Fig 3 shows the circuit rearranged to eliminate the transformers.
\[V_1=V_2 cosh(\theta) + I_2Z_0sinh(\theta) \tag{1}\] \[I_2=\frac{V_1+V_2}{4Zc}+\frac{V_2}{Z_{l_1}} \tag{2}\] Substituting (2) into (1)... \[V_1=V_2 cosh(\theta)+\left (\frac{V_1+V_2}{4Z_c}+\frac{V_2}{Z_{l_1}} \right)Z_0 sinh(\theta) \] Collecting like terms... \[V_1 \left ( 1\frac{Z_0}{4Z_c} sinh(\theta) \right )=V_2 \left( cosh(\theta)+ \left ( \frac1{4Z_c}+ \frac1{Z_{l_1}} \right ) Z_0 sinh(\theta) \right )\] \[\frac{V_2}{V_1}=\frac{1\frac{Z_0}{4Z_c} sinh(\theta) }{cosh(\theta)+ \left ( \frac1{4Z_c}+ \frac1{Z_{l_1}} \right ) Z_0 sinh(\theta)} \] 
Fig 4 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 4 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 5 shows the characteristics of Fairrite #43 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 6 shows the impedance for a 12 turn choke wound on a Fairrite 2643801002 (or 5943001001) #43 core (29mmx19mmx7.5mm). Quantity Z is the magnitude of R+jX.
Another challenge is that transmission line Zo and loss may be influenced by both skin and proximity effect. Skin effect causes the current to flow mainly on the surface of conductors, and proximity effect causes currents in opposite directions to tend to flow mainly on the adjacent surfaces of conductor that are very close together.
Fig 6a shows an estimate of Zo based on the log function, the acosh function and a curved derived from Fig 4.23 of Radio Antenna Engineering by Edmund LaPort, 1952. It would appear that LaPort's graph may be a correction to the log function that he gives in the book, and an estimate of the cosh curve. With that consideration, the acosh curve is used in the models.
A proximity resistance correction is calculated using an algorithm from the program line_zin.pas by Reg Edwards (G4FGQ).
Fig 8 shows a plot of VSWR and loss modelled for a Ruthroff 1:4 balun made with 12 turns of transmission line wound on a Fairrite 2643801002 (or 5943001001) #43 core (29mmx19mmx7.5mm). The transmission line comprises 0.5mm diameter copper conductors spaced 0.6mm centre to centre. Under matched conditions, losses are very low, 2% at worst. Estimating that the core can probably safely dissipate up to 5W, the balun is probably capable of continuous power rating of 250W.
Fig 9 shows common mode impedance. Common mode impedance is very low, which means that this type of balun does little to impede common mode current, eg on an antenna feed line connected to the balanced port.
Conclusions are:
Version  Date  Description 
1.01  12/12/2007  Initial. 
1.02  
1.03  
1.04  
1.05 
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