At (Hunt 2015) G3TXQ gave some measurements of his ‘balanced' antenna system.

Above is Hunt's equivalent circuit of his antenna system and transmitter. It is along the lines of (Schmidt nd) with different notation.

He made measurements that led to calculation of these components for the dipole which he states is visually fairly symmetric.

Above are the calculated values he gives at Leg a and Leg b. Lets assume that the values are derived from measurements made at the tuner interface with respect the the unbalanced tuner ground connection. Note that Za and Zb are not close to equal, so the antenna system is hardly symmetric or balanced.

Above is a schematic of the antenna equivalent circuit driven by equal but opposite phase voltages (ie, an ideal voltage balun).

From that we can write a set of mesh equations and solve them.

\(1=(za+zc) \cdot ia -zc \cdot ib\\

1=-zc \cdot ia+(zb+zc) \cdot ib\\

\)

This is a system of 2 linear simultaneous equations in 2 unknowns. In matrix notation:

\(\begin{vmatrix}ia\\ib \end{vmatrix}=

\begin{vmatrix}

za+zc & -zc\\

-zc & zb+zc

\end{vmatrix}^{-1} \times

\begin{vmatrix}1\\1\end{vmatrix}\\

\)

Let's do that in GNU Octave.

#equations of mesh currents #1=(za+zc)*ia -zc*ib #1=-zc*ia+(zb+zc)*ib za=15.1+79i zb=1.6-109i zc=30.7+110i A=[za+zc,-zc;-zc,zb+zc] b=[1;1] x=A\b ia=x(1) ib=x(2) ic=(ia-ib)/2 id=(ia+ib)/2 ica=abs(ic) ida=abs(id) icrel=2*abs(ic)/abs(id) #for calculator abs(ia) abs(ib) 2*abs(ic)

The console output is…

za = 15.100 + 79.000i zb = 1.6000 - 109.0000i zc = 30.700 + 110.000i A = 45.800 + 189.000i -30.700 - 110.000i -30.700 - 110.000i 32.300 + 1.000i b = 1 1 x = 0.0046179 + 0.0091411i 0.0049694 + 0.0242611i ia = 0.0046179 + 0.0091411i ib = 0.0049694 + 0.0242611i ic = -0.00017574 - 0.00756002i id = 0.0047937 + 0.0167011i ica = 0.0075621 ida = 0.017375 icrel = 0.87043 ans = 0.010241 ans = 0.024765 ans = 0.015124

The comparative statistic I will use is | 2*ic| (total common mode current) relative to |id|, it is given by the variable icrel above which has a calculated value of 0.870 for this scenario.

By way of comparison, the same scenario with a common mode choke (ie current balun) with Zcm=1500+j1500Ω, the same statistic is 0.0425. The good common mode choke results in common mode current being 26dB lower than the with ideal voltage balun alone.

Note that differential current and common mode current will almost always each be standing waves and their phase velocity may differ.

The last three values calculated are those that would be measured by a clamp on RF ammeter around the a, b and a+b wires together. These values could be plugged into Resolve measurement of I1, I2 and I12 into Ic and Ic to resolve the measurements into common mode and differential mode components.

Above, the calculator results reconcile with the results of the Octave script.

CMR (common mode reduction) (or CMRR for some) is the the magnitude of the ratio of differential current to common mode current expressed in dB.

Above, by way of comparison, the same scenario with the common mode choke (ie current balun) with Zcm=1500+j1500Ω, the same statistic is 0.0425. Adding a good current balun results in relative common mode current being 26dB lower than the ideal voltage balun alone.

You don't need complicated maths to asses an installed antenna system, measurements with a clamp on RF ammeter can be resolved into the common mode and differential mode components using the calculator.

This solution is based on Hunts measurements and equivalent circuit calculation, and the abject failure of a voltage balun on an antenna system that Hunt reported as apparently symmetric applies to this scenario, but it is not surprising as good current baluns will tend to be more effective in reduction of common mode current than voltage baluns.

## Rules of Thumb

Online experts parrot Rules of Thumb (RoT) about the most important characteristics of a common mode choke, eg:

- it is its resistance that is most important; and
- capacitive Zcm doesn't work.

Zcm (Ω) | CMR (dB) |

1500+j100 | 30.3 |

1500+j1500 | 33.45 |

100+j1500 | 30.7 |

1500-j1500 | 32.9 |

The table above shows calculated CMR for a number of alternative values for Zcm in this scenario, and the outcome does not support the RoT stated above:

- interchanging low R and low X (1500+j100Ω vs 100+j1500Ω) gives very similar CMR; and
- capacitive Zcm (1500-j1500Ω vs 1500+j1500Ω) gives very similar CMR.

## References

- Duffy, O. nd. Find three terminal equivalent circuit for an antenna system
- Hunt, S. May 2015. High performance common mode chokes in Radcom
- Schmidt, Kevin. nd. Putting a Balun and a Tuner Together: http://fermi.la.asu.edu/w9cf/articles/balun/index.html