Do Common-mode Currents Split Equally across Transmission-Line Conductors?

Are common-mode currents equal on a transmission line's conductors?  Let's look at two cases: a coaxial cable transmission line and a two-wire (e.g. twin-lead) transmission line.

Coaxial Cable Transmission Line

If the transmission line is a coaxial cable, the answer is no.

Although a coax cable consists of two conductors, the center conductor and the shield, electrically it behaves as though there are three conductors:  the center conductor, the inner-surface of the shield, and the outer-surface of the shield.

The RF signal is applied between the center conductor and the shield.  Due to physics, the signal travels on the center conductor and the inner-surface of the shield (assuming the shield has adequate thickness).  For the purpose of transporting energy from the transmitter to the antenna, the outer-surface of the coax plays no role, and the EM fields of the forward and returning currents are completely contained within the coax -- these two currents do not radiate beyond the coax.

However, the outer-surface of the coax has an undesired effect where it attaches to the antenna -- it behaves as an extra wire that is attached between one side of the antenna and the distant transmitter, running the length of the coax transmission line.  As such, it becomes a radiating part of the antenna.

The current flowing on this outer surface (and radiating RF energy) is also commonly called a "common-mode current".  Note that this common-mode current is only on the outside of the coax cable, and only it is the cause of coax cable RF emissions.

So, on a coax cable, common-mode current is only carried on the shield (its outer surface) but not on the center conductor.  The common-mode current is not distributed equally across the coax cable's conductors.

Two-wire Transmission Line

Next, consider a two-wire (e.g. twin-lead) transmission line.

A two-wire transmission line does not have the same self-shielding properties as a coax cable, and there isn't a "third" conductive element (i.e. the coax shield's outer surface) that acts as an unintentional RF radiator.

But an antenna & transmitter system can have unequal common-mode currents on a two-wire transmission line if the system is unbalanced.

Unbalance can occur if any of the following conditions exist:

1. The antenna elements are unmatched, either in length with respect to each other, or in position with respect to ground.
2. The transmitter is not balanced with respect to ground.      
3. The two wires of the transmission line are unequal in length, or one wire is closer to ground or conductive objects than the other wire.

Let's examine antenna imbalance and transmitter imbalance (skipping transmission line imbalance).  I'll use EZNEC for the antenna simulations.

The first simulation will be of a dipole that is balanced with respect to the length of its elements and their position with respect to ground.  The voltage source driving the antenna is also balanced with respect to ground in the sense that the voltage source's "midpoint" is tied to ground, so that, if one side of the transmission line is positive with respect to ground (e.g +0.5V), the other side is negative (e.g. -0.5V), and vice-versa.

This balanced voltage source is implemented with two identical in-phase voltage sources so that the total voltage swing is twice the amplitude of a single source.

In the EZNEC models, below, I'm going to accentuate any common-mode coupling paths and also minimize the effects of SWR by making the transmission line only 5 feet long.  At 3.5 MHz, this length is a very small fraction of a wavelength, nevertheless, it is still a transmission line.

Balanced Dipole Antenna and Transmitter:

In the antenna diagram below, you can see that the balanced voltage source consists of two series-connected 0.5 V sources (resulting in 1V of drive) with ground connected between them.  The dipole consists of two 50 foot elements, so it is 100 feet long.  But its height is only 5 feet above ground.

The transmission line is simply two #12 wires run in parallel with spacing of 1.2 inches (0.1 feet) between the two wires.  No thought was given to the line's characteristic impedance.  It is what it is.

The EZNEC simulation shows that for this balanced-antenna and balanced-source configuration, there are no common-mode currents -- the currents are differential only.  There is no current on the wire attached to ground.

EZNEC currents for this antenna:

Balanced Dipole, Unbalanced Transmitter:

Let's unbalance the voltage source driving the balanced antenna by removing the 0.5V source in wire 4 (as shown in the previous antenna diagram) and change the value of the remaining source from 0.5 volts to 1 volt, so that, overall, the magnitude of the driving voltage remains the same.

The dipole, itself, remains unchanged, that is, it remains balanced.

Below are the EZNEC results.  Clearly the currents in the transmission line are unbalanced, and the majority of current follows a path through wires 1, 3, 6, and 5 (the wire to ground).  This is common-mode current, and it is only on one of the transmission line's two wires (wire 3), not both.

EZNEC currents for this antenna:

By the way, the higher this antenna is raised, the smaller and smaller will be the current imbalance between the currents on the dipoles two radiating elements.  But these two currents will not be equal.

Unbalanced Dipole, Balanced Transmitter:

Let's return to a balanced voltage source driving the transmission line and change the balanced dipole to be unbalanced.  The simplest way to do this is to change the length of one of the dipole's elements, so let's change the length of the right-hand element to zero (that is, remove it).  The dipole is now a unipole.

Again, the EZNEC results show that the majority of the current flows through wires 1, 3, 6, and 5.  So again, there is significant common-mode current in this antenna system, but only on one wire (wire 3) of the two-wire transmission line.

EZNEC currents for this antenna:

Conclusions:

Although there is often an assumption that common-mode current in a cable must be equally distributed across the cable's conductors, in fact it is possible for the conductors to have different amounts of common-mode current.

Similarly, the two wires of a two-wire transmission line (e.g. twin-lead) can have mismatched amounts of common-mode current on them.

Standard Caveat:

As always, I might have made a mistake in my equations, assumptions, drawings, or interpretations.  If you see anything you believe to be in error or if anything is confusing, please feel free to contact me or comment below.

And so I should add -- this information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.