EE: Mutual Inductances in Magnetic Circuit

[ckpage86]

Homework Statement

The symmetric magnetic circuit has three windings. Winding A and B each have N turns and are wound on the two bottom legs of the core. The core dimensions are indicated in the figure.

b.) Find the mutual inductances between the three pairs of windings.

figure : http://i844.photobucket.com/albums/ab2/cpage86/133ece321hmwk1_zps0b1b4f42.png

Homework Equations

$\varphi$ = $\Im$/$\Re$

Inductance = $\lambda$/current

$\lambda$ = N$\varphi$

The Attempt at a Solution

In the problem I went through and found the self-inductances for part a.), but cannot get the mutual inductances. I have tried about 15 times, and according to the solution I am close, but still not 100% correct.

I have tried finding the flux in each leg and determining the flux linkages of the coils, but I keep ending up with a huge mess of papers with never-ending equations.

I have tried just utilizing the flux through the gap.

I have tried to blindly guess which reluctances are necessary to solve this.

This class just started this week, and I have read the book. I have gone back and forth through the chapter and the only example that is close to this one is of a square loop core with a single gap and two windings. In that particular example they disregarded the core permeability and completed the necessary work in a few steps; which is pretty misleading not helpful.

Part of my work is here:

http://i844.photobucket.com/albums/ab2/cpage86/fe163d3b-b6a3-420f-b12c-ccc26c537ff5_zps8ee39d5d.jpg

According to the book I am pretty close, but the "la" term in the numerator is not supposed to be there. Other than that, my solution is right on.

I mainly am hoping to gain some insights on how to go about setting this up and solving for the mutual inductances. If you have any advice or wisdom to drop on me I am all ears.

 

[rude man]

Sorry, can't read your work so I'll start from scratch. Sorry if much or all of this is redundant:

First, do you see that there are only two mutual inductances to be computed? I mean, there are 6 (M1a, Ma1, M2a, Ma2, M12, M21) but 4 are redundant by symmetry and mutual inductance theory.

So let us choose to compute M1a and M12:

Are you familiar with magnetomotive force (mmf) and reluctance R in addition to the usual H, B, flux ψ and current i? What equation relates mmf to reluctance R and flux ψ? where ψ = BA, A = cross-sectional area.

Then, what is the defintion of mutual inductance between coil a and coil 1, assuming coil a is excited by a current ia?

The method is to equate mmf to the sum of magnetic potential drops (mpd) around the closed magnetic path.

 

[ckpage86]

Rude man, Thanks for the reply, I've got some good news. While dining at Arby's yesterday on my lunch break I had a breakthrough and finally solved that bad boy.

The feeling of relief was huge.

:)

 

[rude man]

Rude man, Thanks for the reply, I've got some good news. While dining at Arby's yesterday on my lunch break I had a breakthrough and finally solved that bad boy.

The feeling of relief was huge.

:)

Good going!

 

[rezeew]

Hello, it sounds like you have put a lot of effort into trying to solve this problem and have made some progress. It can be challenging to solve problems involving mutual inductances, but I am confident that with some guidance you will be able to find the correct solution. Here are a few tips that may help you:

1. First, make sure you understand the concept of mutual inductance. It is the measure of the amount of magnetic flux that is linked between two coils. In other words, when current flows through one coil, it creates a magnetic field that induces a voltage in the other coil. This voltage is directly proportional to the mutual inductance.

2. Next, try to visualize the magnetic circuit in your mind. This can help you understand how the flux is distributed throughout the circuit and how it links the different coils.

3. It may be helpful to start by finding the total flux through the entire circuit. This can be done by summing up the individual fluxes through each leg of the core. Keep in mind that the total flux is equal to the sum of the fluxes through the individual legs, since magnetic flux is a conserved quantity.

4. Once you have the total flux, you can use it to find the mutual inductances between each pair of coils. Remember that mutual inductance is given by the ratio of flux linkage to current. So, for example, the mutual inductance between winding A and winding B would be equal to the flux through winding B divided by the current in winding A.

5. Finally, don't be afraid to break the problem down into smaller parts or to try different approaches. Sometimes, solving a problem like this requires some trial and error.

I hope these tips help you in solving this problem. Good luck!