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.