Does superposition always apply with magnetic fields?

[htg]

Consider a solenoid which is almost toroidal, with a few turns missing (say you have 1000 turns on 99% of the circumference of the toroid) - they will form the primary winding. Let us have 10 turns on 1% of the circumference of the toroid - they will form the secondary winding. Since practically the whole flux of the primary winding goes through the secondary and most of the flux generated by the secondary winding does not go through the primary winding, we have unequal mutual inductances, against a theorem saying that they are equal.

 

[collinsmark]

Hello htg,

For simplicity, let's first consider the situation of an air core (i.e. no magnetic material in the core -- you can always repeat with a magnetic core if you wish [see below]).

Consider a solenoid which is almost toroidal, with a few turns missing (say you have 1000 turns on 99% of the circumference of the toroid) - they will form the primary winding. Let us have 10 turns on 1% of the circumference of the toroid - they will form the secondary winding. Since practically the whole flux of the primary winding goes through the secondary and most of the flux generated by the secondary winding does not go through the primary winding, we have unequal mutual inductances, against a theorem saying that they are equal.

The text in red is the incorrect assumption.

Consider a single loop of the primary winding, directly opposite the secondary winding, on the torus. The magnetic field lines of this loop extend outward from the center of the loop, and then loop back in the standard, symmetrical pattern. But since the secondary winding is relatively far away (on the other side of the torus), not many magnetic field lines from this particular loop make it into the secondary winding.

And when looking at the situation from perspective of a loop of wire in the secondary winding, the results are the same -- not many magnetic field lines originating from the particular secondary loop make it through the particular primary loop on the opposite side of the torus.

As a matter of fact, both perspectives produce the exact same answer for mutual inductance.

Now you can repeat the problem with a magnetic material core if you wish. With a magnetic core, most of the magnetic field lines of the primary are confined to the core, which pass through the secondary. However, the same is true for the secondary, since it shares the same core. Most of the secondary coil's magnetic field lines pass through the primary winding. And so again, whether you start primary winding or the secondary winding, the final mutual inductance is the same either way.

 

[htg]

You cannot analyze it by considering the field lines from a single turn - you have to apply the vector form of the principle of linear superposition. Then you will see that practically the whole flux of the primary winding goes through the secondary winding.

 

[collinsmark]

You cannot analyze it by considering the field lines from a single turn - you have to apply the vector form of the principle of linear superposition.
Then you will see that practically the whole flux of the primary winding goes through the secondary winding.

The concept of linear superposition is fundamentally based on the principle that you certainly can analyze each loop contribution separately. Where superposition applies, the total contribution of all the loops is the simple [vector] sum of the individual contributions from each loop.

If you were to analyze the whole thing precisely (without making approximations) the mutual inductance is identical, regardless if you start with the primary or secondary. You'd have to use Biot–Savart law to get a good handle on the magnetic fields, particularly concerning the secondary winding. But that would require some not so simple calculus.

If you want to use approximations, Ampere's law would work quite well for primary winding, as demonstrated here:
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/toroid.html#c2"
But won't work so well for secondary, the way you described it, being on only 1% of the toroid. If you wish to use Ampere's law to calculate B, you must choose your loop in such a way that B is constant at all parts of the loop.

But you could modify your secondary winding such that it wraps around the entire toroid symmetrically, albeit with a fewer total turns than the primary. If you do that, you can use Ampere's law to work the problem. Otherwise, you can still use Biot–Savart law and a bunch of calculus. But either way, if you use use Ampere's law for a symmetrical configuration or Biot–Savart law for a not so symmetrical configuration, you'll still get the same mutual inductance no matter which winding you start with.

 

[htg]

You may be right. I have read that in the presence of ferromagnetic materials this law can be violated. How can it be done?

 

[collinsmark]

You may be right. I have read that in the presence of ferromagnetic materials this law can be violated. How can it be done?

Well, ferromagnetic materials complicate things because the permeability can be nonlinear. In particular it can saturate, such that the flux no longer increases proportionally with the current. And hysteresis complicates things a little too.

However, if one wants to get really complicated with the math, it's still true that the mutual inductance is the same, for some given, weird fixed state of the core. But working with a linear core certainly makes things simpler.

 

[htg]

Where superposition applies, the total contribution of all the loops is the simple [vector] sum of the individual contributions from each loop.

Doesn't it ALWAYS apply?

 

[collinsmark]

Doesn't it ALWAYS apply?

Superposition always applies with magnetic fields themselves, electric fields themselves, and many other things.

But there are many relationships which it doesn't apply. For example, the energy density of an electric field or magnetic field, which increases with the square of the respective field strength.

Suppose you had two identical magnetic field sources, sharing the same space and time. You can use superposition to find the total magnetic field, which is just twice the magnetic field strength of a single source. But superposition doesn't apply to the energy density -- it increases by a factor of 4.