I have the problem of a few independent current loops (wires) that on a portion of the circuit are close to one another so there will be inductive coupling between them.

Each wire, on the portion of interest, has its self inductance and the mutual inductance will add to that because the currents have the same direction in this case (aiding inductors).

The problem is that there are many wires and any single wire will have partial inductances, with every other wire. The formula for partial inductance is:

enter image description here

Can the sum of the coupling coefficients exceed 1 ? That is, can wire nr. 1 have a coupling coefficient of 0.6 to wire nr.2 , 0.4 to wire nr. 3 and 0.2 to wire nr. 4?

If the sum of the coupling coefficients of a wire to the rest can't exceed 1, then for this case of aiding inductors and same self-inductance for all wires, the maximum equivalent inductance (self partial inductance + mutual partial inductance) for one of the wires can't exceed two times the self inductance.

So, one could avoid calculating all the partial inductances and just use two times the self inductance as an upper limit of the inductance that can be on that portion.

Is this true?

EDIT: it's obvious now that the above assumption is not correct. I think this can be explained with the relative permeability of the materials, which is almost 1 for both copper and common insulators. Basically, the magnetic field goes through wires as if they weren't there.

In this case, could one calculate the mutual inductances for two wires as if the rest weren't even there? That would be a useful approximation, which I hope is the correct one now.

  • \$\begingroup\$ Are you referring to the calculation of self-inductance of a multi-turn coil, or Mutual inductance between primary and secondary? Please add a picture/illustration. \$\endgroup\$
    – Pojj
    Commented Feb 27, 2019 at 7:41
  • \$\begingroup\$ independent current loops, I've edited the post. \$\endgroup\$ Commented Feb 27, 2019 at 11:28
  • \$\begingroup\$ If they are independent loops, yes you can calculate mutual inductance independently. But if you are connecting them in series/parallel, then you have to consider the cumulative effect all mutual inductances to calculate self-inductance... see here or this paper - if you have access \$\endgroup\$
    – Pojj
    Commented Feb 28, 2019 at 12:19

2 Answers 2


The coupling coefficient presents how big part of the flux of one winding goes through another winding. Think a transformer which has several secondary windings. It would be useless if all coupling coefficients between the primary and the secondaries couldn't be nearly 1.


Think about three closely coupled coils. Imagine that they were coupled 1:1:1. That means that the coupling from coil 1 to coil 2 is 1 and, from coil 1 to coil 3 is 1.

Can the sum of the coupling coefficients exceed 1 ?

The example above would be 2 for coil 1.

If ten closely coupled coils were made, the sum of couplings (not at all relevant to any design that I'm aware of) is 9. It's a nonsense number (or indicator of anything) but, if it helps you in some way so be it.


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