The 1:1 Current Balun (Revised 95-11-13 and 95-11-17. Revised equations marked with (*)) This analysis - Presents an analytical circuit model of the 1:1 current balun, including finite impedance. - Gives equations for several key performance characteristics of a 1:1 current balun in typical applications. - Analyzes the effect of using a 1:1 current balun with an antenna tuner, when connected to either the tuner input or output. For information about what a balun does, see "Some Aspects of the Balun Problem" by Walt Maxwell, W2DU, QST, March, 1983, p. 38, and "Baluns: What They Do and How They Do It" in the ARRL Antenna Compendium, Volume 1, p. 157. A 1:1 current balun can be made by winding a two-conductor transmission line (e.g., twisted pair) or coaxial cable through a toroidal core or on onto a ferrite rod, or by placing ferrite cores over a twisted-pair or coax transmission line. Or, coax can be coiled to create a broadly resonant circuit for the current on the outside of the coax. The same model can be used for any of these configurations: 1:1 ideal xfmr a ---------UUUUUU---------- c b --.------UUUUUU-------.-- d | ________ | | | | | ----| Zw |----- |________| (For this entire discussion, the assumption is made that the length of the transmission line used to construct the balun is short in terms of wavelength, so it can be accurately represented by lumped elements. The analysis isn't valid if this assumption isn't true.) Zw is the winding impedance. It's the impedance that the winding would have if the winding were made of a single conductor. If a low-frequency ferrite toroidal core or beads are used, Zw is chiefly resistive; if a high- frequency ferrite is used, it's chiefly inductive, but in general it can be any combination of resistance and reactance. Without a core or coil, Zw is the "longitudinal impedance" -- the impedance of the wire itself. For coax, Zw represents the impedance to current flowing on the outside of the shield, while the "ideal transformer" models the inside of the coax. The "ideal transformer" action comes about because of the complete coupling of the fields from currents on the two conductors inside the coax. With non-coax transmission lines, the assumption is again made that the fields from the two wires completely couple. This assumption is good as long as the two conductors are very close together. The model winding impedance Zw represents the impedance to common-mode currents and can be split evenly between the two conductors or it can be placed all on either side. It makes no difference because of the action of the "ideal transformer". The "ideal transformer" is the source of the following two rules: Va - Vc = Vb - Vd and The currents in the two windings of the "ideal transformer" are equal and opposite. From the first equation can also be derived that Va - Vb = Vc - Vd. A simplified model of the balun's environment is: Is ___________ Z1 I1 --> | | ___ --> + -----a-| |-c-----|___|------. Vs | BALUN | ___ | - --.--b-| |-d-----|___|------. | |___________| <-- | | Z2 I2 | GND GND A very important point is that the two places labeled "GND" are THE SAME POINT. If Z1 and Z2 represent an antenna or antenna/feedline, the path back to the balun input must be included in values Z1 and Z2. Or, to put it another way, Z1 is the impedance measured between terminals c and b, and Z2 the impedance between d and b, with the balun disconnected. Using this simple circuit and the two balun rules, we can calculate the following: Ratio of currents in Z1 and Z2: I1/I2 = (Z2 + Zw) / Zw Balun input impedance: Vs/Is = Z1 + (Z2 || Zw) Voltage across balun winding: Vb - Vd = Vs * (Z2 || Zw) / (Z1 + (Z2 || Zw)) Ratio of coax outer shield current or twinlead "antenna current" to the total conductor current): (I1 - I2) / (I1 + I2) = (Z2 || Zw) / (2Zw - (Z2 || Zw)) (*) = Z2 / (Z2 + 2Zw) (*) where Z2 || Zw = the value of Z2 in parallel with Zw. These equations show a few interesting things. First is that as Zw gets very large, I1 and I2 become equal and the "antenna current" drops to zero. This represents the perfect current balun, and shows why we strive to maximize the balun impedance. Another interesting thing is that the current balance is dependent on only Z2 and Zw, and is completely independent of Z1. If Z2 becomes zero, the current balance is perfect regardless of Zw. If Z1 and Z2 represent an antenna or antenna and feedline, this isn't likely to happen. Remember that Z2 is the impedance from terminal d back to terminal b. Even if one end of a coaxial cable shield is connected to d and the other end to the earth, there can still be a significant impedance between the two. If the coax is a sizeable fraction of a wavelength long, the impedance can be quite high. Note that Z2 shows up only in parallel with Zw. This isn't surprising, since the two are, in fact, connected in parallel. It does lead to the observation that the currents in Z1 and Z2 can be made equal by putting another impedance Zw * Z1 / Z2 (*)from terminal c to ground. Although the connection of this extra impedance is the same as for the "tertiary" winding of a 1:1 "voltage" balun, the latter doesn't fulfill this function because of its coupling to the other windings. To use an added impedance for this purpose requires isolating it from the 1:1 current balun; it should be a separate component. If the antenna is balanced with respect to ground, the added impedance is simply equal to Zw, and another balun could be used, with its input terminals shorted together and output terminals shorted together. However, if the balun impedance is low enough that this becomes necessary to achieve balance, balun current will be large, and overheating may result. Finally, the equations show that to analyze the important elements of balun operation, we have to know Z1, Z2, and Zw -- although we can calculate the current balance by knowing only Z2 and Zw. What happens when a tuner is used? To analyze this, I modeled a simple "tuner" as a 1:n turns ratio transformer: __________ _________ --> Iin | | --> Iout | C C C in 1 C C n out | C | | __________|_|_________ Notice that the bottoms of the two windings are connected together. This is to represent the tuner's single "ground" terminal. No winding impedance is included in the model, since a tuner probably won't be a transformer in the first place, but rather some other circuit which effects an impedance transformation. This model was chosen to investigate some of the fundamental properties of baluns, which will hold regardless of the exact tuner topology. Once again, there are two rules for the model: Vout = n * Vin Iout = Iin / n Here's a representation of a typical hookup with a tuner. Is _________ ___________ Z1 I1 --> | | | | ___ --> + ------| |---a-| |-c-----|___|------. Vs | TUNER | | BALUN | ___ | - --.---| |---b-| |-d-----|___|------. | |_________| |___________| <-- | | Z2 I2 | GND GND Notice that no additional connection to "GND" is shown, since the tuner's connection to the common point called "GND" is via its lower terminal(s). Again, the rules can be applied and results calculated: Ratio of currents in Z1 and Z2: I1/I2 = (Z2 + Zw) / Zw System input impedance: Vs/Is = (Z1 + (Z2 || Zw)) / n^2 Voltage across balun winding: Vb - Vd = n * Vs * (Z2 || Zw) / (Z1 + (Z2 || Zw)) Ratio of coax outer shield current or twinlead "antenna current" to the total conductor current): (I1 - I2) / (I1 + I2) = (Z2 || Zw) / (2Zw - (Z2 || Zw)) (*) = Z2 / (Z2 + 2Zw) (*) The only changes from the no-tuner case are the input Z, which is transformed by n^2 as expected, and the voltage across the balun winding which has increased by a factor of n. Again, to achieve good current balance requires only that Zw be much greater than Z2. However, some situations requiring a tuner present a high value of Z2, making good current balance difficult to achieve. Finally, move the balun to the tuner input and do the calculations: Ratio of currents in Z1 and Z2: I1/I2 = (Z2 + Zw) / Zw System input impedance: Vs/Is = (Z1 + (Z2 || Zw)) / n^2 Voltage across balun winding: Vb - Vd = n * Vs * (Z2 || Zw) / (Z1 + (Z2 || Zw)) Ratio of coax outer shield current or twinlead "antenna current" to the total conductor current): (I1 - I2) / (I1 + I2) = (Z2 || Zw) / (2Zw - (Z2 || Zw)) (*) = Z2 / (Z2 + 2Zw) (*) The results are identical to those with the balun at the tuner output! Even the voltage across the balun winding is the same. This wasn't, to me, an obvious outcome. (And, in fact, I'd believed otherwise for years.) However, the equations have been carefully checked and are correct. In addition, an experiment was set up using a characterized balun and transformer "tuner" and the results confirm the analysis. I want to thank Tom Rausch, W8JI, for making a comment on the Internet which prodded me into doing the tuner analysis. With these models, 1:1 current balun performance should be straightforward to analyze under most operating conditions. It is hoped that this will put to rest some of the speculation surrounding these simple devices. Roy Lewallen, W7EL w7el@teleport.com October 14, 1995 Note November 13, 1995: Thanks to Frank Witt, AI1H, for spotting the error in the equations for "antenna current". The corrections don't change the conclusions. He also points out that moving the balun from the tuner output to input changes the winding-to-winding voltage, winding currents, and balun working environment (e.g. effect of stray coupling to the balun). This is true. Although the factors I've analyzed are the same with the balun in either position, other factors may not be. Choice of balun positioning or construction may involve considerations other than the ones analyzed here. If so, they should be analyzed as well and taken into account when deciding how best to construct and position the balun.