Richard Clark wrote:

Hi John,

Breaking magnetic lines (flux) is a commonplace of fields, motors, and
generators. A single wire that passes through a bead, torus, or core
will build a magnetic field concentrated within that structure when
the circuit is completed outside of it. The flux lines of half the
loop will penetrate the core to reach the other half of the loop. The
core breaks the magnetic line of flux. The dissymmetry of penetration
builds a magnetic field in the core.

I cannot picture what you are saying. I guess I need a picture. The
way I understand flux, is that, any current has flux wrapped around
it. Putting a core material around the current allows the flux to
increase in magnitude in proportion to the permeability of the material.

However, when the complete current loop is within the same structure,
the flux lines do not fulfill that same function.

The simplest way to say that is that two equal currents going in
opposite directions have magnetic fields that cancel well outside the
pair of conductors. It doesn't make a lot of difference if the two
currents are coaxial or side by side, as long as the flux path
surrounds both of them.

The flux lines that
do emerge from the tightly bound wires can be said to penetrate the
torus, but here the symmetry creates bucking fields, the net effect is
as though there was no core at all (except to add capacitance).

Since the magnetic fields from the two currents cancel (at sufficient
distance), there is no flux to enter a surrounding core. The only way
bucking takes place is if the two currents are side by side. Then
there is some flux leakage near the two current paths, but the fields
these create buck each other in the two halves of the core, so there
is only local flux fringing into and back out of the core.

Both models attempt to stimulate a current within the toroid, the
common mode of the single wire model above is lossy, the differential
mode of the twin line model that followed sees nothing. Superpose
these two for the coaxial solution.

As I understand common mode current, it is the net imbalance between
the current through the center conductor and the current through the
shield.

To put this to a test. Load up your rig, through a SWR meter to a
dummy load using two short connection wires (this will undoubtedly
require adapters and such to break out both paths). You should note a
1:1 indication. Place two #75 beads on ONE wire. You should note a
2:1 indication. You have just inserted 40 to 60 Ohms of additional
resistance into the circuit. Now, move the same two beads to
encompass BOTH wires. This should return the SWR meter to a 1:1
indication.

Yes, obvious to the casual observer. You can string beads all along a
coax, and if it had no common mode current in it, this will make
absolutely no difference in how the coax acts.