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So I just provided the answer to this Stackexchange question and I noticed something kinda cool and interesting in the final equation:

electronics.stackexchange.com/

The equation, for those of you on a Latex capable instance is:

\( k \cdot \frac{N_p}{N_s} \cdot \sqrt{\frac{L_s}{L_p}} = R_\phi\)

In this case R, the flux linking ratio, must be a value between 0 and 1 and represents the percentage of flux that is mutual between two coupled inductors (a transformer).

Similarly k is Coefficient of Coupling, which is also a ratio between 0 and 1. It is a different measure of how well coupled two inductors are.

What I find interesting about the equation is that all the other variables can be anything from 0 to infinity, so at first glance you would think that it would be possible to have a scenario where either R or k are greater than 1. But in reality because the number of windings of the primary and secondary have a reciprocal relationship to that of the square root of the ratio of the inductances, in reality any real world values that you could plug in here would actually never be able to produce a contradiction.

If you look at the equation to calculate the inductance of an inductor based on its number of turns this becomes obvious. for simplicity if we assume the diameter if the inductor and the wire thickness are all the same then the relationship is basically the following:

\( L = N^2 \cdot C \)

Where L is the inductance, N is the number of turns, and C is some constant. So essentially the square of the number of turns of the inductor is linearly proportional to its inductance. Going back to the equation I was talking about this basically counters the sqrt function in the equation and it becomes obvious why any real world values would always satisfy the equation correctly.

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@freemo Another way you could look at this is on the basis of conservation of energy in the context of Faraday's law. Consider the following thought experiment: two inductors are nested within one another with one connected to a battery, and the other forming a closed loop. The switch on the first is closed inducing a magnetic field in the second. A sort of "cascade" effect occurs when you think about this in terms of Faraday's Law. The first coil produces a magnetic field, but it does not experience an external one at the instant of closing the switch. The second in turn will experience an external magnetic field from the first, but also an induced magnetic field due to the current induced by the external field. This splitting is built into Faraday's Law. Faraday's Law doesn't care which magnetic field is which, just that there is a total magnetic field. Ultimately, the induced component must be smaller than the external component and opposite in direction. If it was larger, then the response in the next "time step" to the first coil would have an increase in current in the direction that was creating the original magnetic field. This would cascade to infinite current in both inductors and clearly conservation of energy is violated. Analogously you could consider the two coils like paddles in pong where either the choice is that the bounce the ball back and forth dampening and slowing it with each collision, or they swing harder and harder giving the ball an infinite velocity. As a result, your flux linking ratio MUST be less than 1. Otherwise energy conservation is violated. All of this is a consequence of Lenz's "Law" or the notion of "nature abhorring changes in flux" but really is embedded into Faraday's law via the negative sign and careful adherence to sign conventions when applying the right hand rule.

@wuphysics87 You should model your example of a shorted out secondary transformer, you might be surprised the results you get.. As I recall it wont actually cause a feedback in the way you suggest as the secondary coil isnt driving the field in the first place.

Want me to model it for you to show you?

@freemo The feedback is a conceptualization of the way you couple the inductors together. Changing magnetic flux induces current. Which in turn produces a magnetic field from the current that is induced. Which is seen as a changing magnetic flux by the inductor producing the original magnetic field. The point is a matter of the interpretation of how you apply Faraday's Law and what your "choice" is when applying sign convention. One choice violates conservation of energy. Which makes it incorrect. The other results in the scaling relationship that you saw in your calculation.

@wuphysics87 the thing is if you had an ideal inductor that was short circuited coupled with another inductor, I dont think it would actually have any feedback effect on the driving inductor as you suggest, specifically because in the ideal world there is no resistance. If you shorted a coupled inductor in the real world it would of course have an effect but only due tot he resistive losses on the coupled side.

But its been a while since i ran the math on a transformer.

@wuphysics87 Nevermind your right, i realized it the second i started setting up the equation. The mutual inductance virtual inductor is modeled in each circuit half and dependent on the other circuits current. Which at the very least represents what you were saying as "feedback". Still im going to work through the whole math for the fun of it then share.

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