The article Antennas – disturbing the thing being measured – open wire lines #2 did a simple analysis of current flows in the model scenario using ideal voltage balun drive with a current balun.
It was mentioned that solution of the lumped values network is only a first approximation and not as good as the NEC solution which properly models the coupled conductors, and their mutual effect on the distribution of currents in the system.
This article reports the NEC current solution decomposed into differential mode and common mode components.
Ideal voltage balun drive
Above is a zoomed in view of the feed point with a balanced pair of voltage sources feeding the line against the ground electrode.
The current in the lower segments of the feed line is obtained.
i1:
i2:
The same was done after a load of 1130+j1657Ω was inserted in the segment to the ground electrode.
Lets use the Python console to perform the complex maths to obtain the common mode (Ic) and differential (Id) current components, and the ratio of Ic to Id.
import os
os.system('cls')
import math
import cmath
#ideal voltage balun drive
i1=-0.3-0.84j
i2=0.64+1.01j
id=(i1-i2)/2
ic=(i1+i2)/2
cmath.polar(id)
cmath.polar(ic)
20*math.log10(cmath.polar(ic)[0]/cmath.polar(id)[0])
#current balun Zcm=1130+j1657Ω
i1=-0.4-1.11j
i2=0.38+1.13j
id=(i1-i2)/2
ic=(i1+i2)/2
cmath.polar(id)
cmath.polar(ic)
20*math.log10(cmath.polar(ic)[0]/cmath.polar(id)[0])
… and with the results…
>>> >>> import math >>> import cmath >>> >>> #ideal voltage balun drive ... i1=-0.3-0.84j >>> i2=0.64+1.01j >>> id=(i1-i2)/2 >>> ic=(i1+i2)/2 >>> cmath.polar(id) (1.0375572273373648, -2.040909362946856) >>> cmath.polar(ic) (0.19006577808748215, 0.4636476090008062) >>> 20*math.log10(cmath.polar(ic)[0]/cmath.polar(id)[0]) -14.742162640762551 >>> >>> #current balun Zcm=1130+j1657Ω ... i1=-0.4-1.11j >>> i2=0.38+1.13j >>> id=(i1-i2)/2 >>> ic=(i1+i2)/2 >>> cmath.polar(id) (1.185959527134042, -1.9058794244856845) >>> cmath.polar(ic) (0.014142135623730885, 2.3561944901923506) >>> 20*math.log10(cmath.polar(ic)[0]/cmath.polar(id)[0]) -38.47109740837243
So, with current balun, the common mode component of current at this point in each conductor is 38.7dB less than the differential component, 23.8dB lower than the ideal voltage balun drive. The current balun has significantly reduced common mode current at this point.
The NEC result is somewhat different to the linear circuit analysis of inserting the balun into the antenna equivalent circuit (itself derived from the NEC model) and illustrates the problem of simple linear circuit analysis to coupled conductors problems, and the utter futility of most back of the envelope ‘ohms law’ analyses of these type of problems presented by online experts.