The laws of physics – common mode currents and coax

I am always suspicious when “the laws of physics” are cited to support some argument. One forum expert recently offered:

The laws of physics require that the current on an ideal coax center conductor and the current on the inside of an ideal coax braid be equal in magnitude and opposite in phase, i.e. nothing but ideal differential currents can flow inside ideal coax. Anything else would violate Maxwell’s equations. All common-mode (non-differential) current must therefore necessarily flow on the outside of the ideal coax braid.

If we consider the end of a coaxial transmission line to have just two terminals, we can define some currents for the purpose of discussion. I1 flowing out of the inner conductor terminal, and I2 flowing into the other terminal (the end of the outer conductor).

These two currents can be decomposed into differential and common mode components:

  • Id=(I1-I2)/2; and
  • Ic=(I1+I2)/2.

(Ic is the common mode component of current that flows at each of the two terminals.)

Rearranging these, we can write:

  • I1=Ic+Id; and
  • I2=Ic-Id.

So the component currents Ic and Id fully account for the current at each terminal, I1 and I2.

CoaxTEMCM-02

Back to the quote The laws of physics require that the current on an ideal coax center conductor and the current on the inside of an ideal coax braid be equal in magnitude and opposite in phase. That part is quite true, and the diagram above resolves what happens if I2 is not equal to I1, I1 flows out of the inner conductor, and I1 flows into the outer surface of the outer conductor, and a current I2-I1 flows from the outer surface of the outer conductor into the associated terminal.

The current flowing on the outer surface of the outer conductor is I2-I1 which is 2Ic by the definition given earlier, ie the current on the outer surface is purely due to common mode current, and is equal to the total common mode current (ie the sum of the common mode current components at each terminal).

But, the quote goes on to say i.e. nothing but ideal differential currents can flow inside ideal coax yet we can see from the above analysis if we are consistent about the definitions of the currents, that the current on the inner conductor being I1 which is Ic+Id does include common mode current, and for the same reason, the current on the inner surface of the outer conductor does include common mode current.