Solenoid vs transformer inductance

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When they teach transformers, they give you the Ampere's law, the Faraday's law, etc. and then derive this well known formula L = μ * N^2 * A / ℓ. And everything is no nice, all these parameters are used to define some other relationships, like the concept of reluctance in magnetic circuits, etc. But it turns out that this formula may only be valid for long single layer solenoids with the coil length >> D. And for multilayer solenoids there are tons of different numeric methods. But I've never seen them discussed in the context of iron core transformers. So how come the well known relationships between Ф, B, A, H, I, E, ℓ etc. captured in various famous laws yield an unusable formula for inductance? For practical purposes, they just give you a universal transformer equation and tell you here is how you calculate the number of turns in the primary to avoid saturation. But what if I want to estimate the actual current in the primary. Then I will need to calculate L. And it turns out to be tricky. In addition to the permeability that varies with current, I am not even sure what formula for L I should use. Imagine you learn for years F=ma and other fundamental stuff that is based on /derived from other stuff and then they tell you meh these only work for some ideal model.
 

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jim hardy
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Welcome to the real world.
You'll discover most of your textbook problems were set up to solve readily without the extra complications you mention.

A transformer has a closed magnetic circuit so your formula above will work pretty well.
A solenoid or motor has an air gap
so you must calculate reluctance of each piece of the magnetic circuit
and solve it as a a system.

If you take a course in electrical machinery it should get you past your immediate hurdle

you might also look for a book titled "Inductance Calculations", it'll address weird geometries

So how come the well known relationships between Ф, B, A, H, I, E, ℓ etc. captured in various famous laws yield an unusable formula for inductance?
They get you started. That's their purpose.

Life is difficult. This is a great truth, one of the greatest truths. It is a great truth because once we truly see this truth, we transcend it. Once we truly know that life is difficult-once we truly understand and accept it-then life is no longer difficult. Because once it is accepted, the fact that life is difficult no longer matters.
- m scott peck
 
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A transformer has a closed magnetic circuit so your formula above will work pretty well.
OK, this sounds really reassuring. That would explain why just this formula is discussed in the context of iron core transformers! It just drove me crazy that one could combine the Ampere's law with the Faraday's law and rearrange things a bit, solve for this, solve for that, and finally get the expression for L, and then have doubts about validity of that formula since it is not even used in some cases. Sometimes these things just don't connect in my mind. I've seen some calculators based on the Maxwell's elliptic integral formula and they basically calculate mutual inductance between every pair of turns. And what I noticed is the more "compact" coils (shorter length more layers) are predicted to have higher inductance than longer coil with fewer layers. So that threw me off and I was afraid that in case of transformers that might also hold true and that would make the universal transformer equation less useful as the number of layers/turns per layer would have to be taken into account. But I never thought of it from the perspective of an air gap vs a closed magnetic circuit.
 
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jim hardy
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So that threw me off and I was afraid that in case of transformers that might also hold true and that would make the universal transformer equation less useful as the number of layers/turns per layer would have to be taken into account.
Things are simplified to convey the basics.
In a real transformer that's heavily loaded there are opposing mmf's that force some flux outside the iron core .
As designers push harder to minimize the amount of material in a product, those complex analyses let them eke out the last few drops of performance.
Armed with a good understanding of your basics you'll be able to handle the tedious calculations for those more complete solutions.
I forgive educators for simplifying, they have a lot to cover in a little time and humans can learn at only a finite rate.

You'll find old textbooks from prior to 1950 have good nuts&bolts explanations with graphical solutions.
 

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