Determine flux in air gap (Magnetic Core)

[ElectroViruz]

1. Homework Statement
As shown above

Homework Equations

*Not too sure if they are relevant. Pretty sure there's more. We have textbooks from the library as references but we still can't quite figure it out :/
Φ = B_cA_c = B_gA_g

B = Φ/A_c = (Ni/R)/A_c

R = l_c/(A_c.μ)

The Attempt at a Solution

Attempt has been very little. Me and my classmates aren't a smart bunch. So we haven't gotten really far. There were several arguments on how we should handle the flux directions as well.

Our attempt:

To find Φ (which is flux in core first),
Hence using Φ = BA, we find A first in the question Where A = 40cm² = 0.004m²

To find B, we used:
B = Φ/A_c = (Ni/R)/A_c

So to find R, we used:
R = l_c/(A_c.μ) where we used μ_r=2500 and μ₀ to get μ for the R equation.

So we used l_c which is 1.2m (use the entire core length right ?). We arrived at
R= 95480.6 At/Wb

To get B we need Ni which is the At, but we don't know how to deal with the two coils and the flux.

We've resorted to look for answers online and we hope someone could help us with this. At this point we don't even know if our attempts are appropriate.

We appreciate any help given. Thank you.

 

[cnh1995]

To get B we need Ni which is the At, but we don't know how to deal with the two coils and the flux.

Use the right hand rule to find the direction of flux from each coil and compute the total mmf accordingly. Replace the two coils with a single coil supplying the resultant mmf.

 

[ElectroViruz]

Use the right hand rule to find the direction of flux from each coil and compute the total mmf accordingly.

But isn't the flux going against each other at the corner ? Left coil flux goes up, and the top coil goes to the left.
How do we deal with that ?

 

[cnh1995]

But isn't the flux going against each other at the corner ? Left coil flux goes up, and the top coil goes to the left.
How do we deal with that ?

Subtract the smaller mmf from the larger mmf. Isn't it like connecting two batteries in series opposition?

 

[ElectroViruz]

Subtract the smaller mmf from the larger mmf. Isn't it like connecting two batteries in series opposition?

So we'd get the net flux of 2000AT which is only going in one direction clockwise right ?
Some of my buds were saying it's supposed to be parallel because both ends of the coil connects to each other.

 

[cnh1995]

So we'd get the net flux **mmf** of 2000AT

Some of my buds were saying it's supposed to be parallel because both ends of the coil connects to each other.

I don't think so. The coils are wound on the same flux path, i.e. they are in series.

 

[ElectroViruz]

I don't think so. The coils are wound on the same mmf path, i.e. they are in series.

When the question says neglect fringing and flux leakage, it's safe to say that Φ = B_cA_c = B_gA_g right ? So can we also say that A_c = A_g ?

The problem now is, to get the MMF, I require the R_c and R_g, and R_g requires A_g that the question does not provide.

Otherwise, how do you suggest that I find A_g (or R_g for this matter) ? :)

 

[cnh1995]

So can we also say that Ac = Ag ?

Yes.
You can first find reluctances of all the sections and compute the equivalent reluctance using series-parallel reduction (just like in electrical circuits). Then you can find the total flux using mmf and net reluctance. Using flux division (just like current division in electrical circuits), you can determine the flux through the air gap.

 

[ElectroViruz]

Yes.
You can first find reluctances of all the sections and compute the equivalent reluctance using series-parallel reduction (just like in electrical circuits). Then you can find the total flux using mmf and net reluctance. Using flux division (just like current division in electrical circuits), you can determine the flux through the air gap.

I really appreciate your help, cn1995. I apologise if this question seem effortless to solve, but I am having difficulties with it.
I'm doing my mechanical degree first year but I have electric subjects as well.

Can you help me verify if
R_CORE= 95480.6 1/H or 39789 1/H
?

I noticed that difference between me and my friends' answers were that we used different length.
Following what you said: to replace the two coils with a single coil supplying the resultant mmf.
Hence when calculating the R_CORE, I immediately used the length 1.2m which gave me 95480.6 1/H

However my friends did a very complex way, such as splitting the resistance (they did it the electrical circuit way where the core reluctance was the resistance) into several parts and summing them up.
Hence the length they used were 0.5m instead (their workings were too messy for me to understand how they got there).
They got 39789 1/H.

They based their calculations on someone else's anyway so they didn't know how to explain.

 

[cnh1995]

However my friends did a very complex way, such as splitting the resistance (they did it the electrical circuit way where the core reluctance was the resistance) into several parts and summing them up.
Hence the length they used were 0.5m instead (their workings were too messy for me to understand how they got there).
They got 39789 1/H.

Well, you should go with the electrical circuits way. You can't use the length of the core directly because the flux is not same throughout the length. It is getting divided in the middle portion. I also got the air gap reluctance as 397888 At/Wb.

It's really late here in India now, so I'd better be off to bed. Will look at this tomorrow. Good luck!

 

[ElectroViruz]

Well, you should go with the electrical circuits way. You can't use the length of the core directly because the flux is not same throughout the length. It is getting divided in the middle portion. I also got the air gap reluctance as 397888 At/Wb.

It's really late here in India now, so I'd better be off to bed. Will look at this tomorrow. Good luck!

Alright. Thank you so much once again

 

[cnh1995]

First, identify all possible flux paths.
There are three paths:
1)The left C section (length=1.202m)
2)The middle limb (again has two iron paths and one air gap)
3The right C (reverse C) section (length=1.202m).
Calculate reluctances of these paths.

Think on how they are interconnected in terms of series or parallel fashion. Form an electrical equivalent circuit for simplicity. Compute the equivalent reluctance.

Then you can find the total flux using mmf and net reluctance. Using flux division (just like current division in electrical circuits), you can determine the flux through the air gap.

 

[ElectroViruz]

First, identify all possible flux paths.
There are three paths:
1)The left C section (length=1.202m)
2)The middle limb (again has two iron paths and one air gap)
3The right C (reverse C) section (length=1.202m).
Calculate reluctances of these paths.

Think on how they are interconnected in terms of series or parallel fashion. Form an electrical equivalent circuit for simplicity. Compute the equivalent reluctance.

Thank you !
I was able to get the answer :)
Here's what I did

*After finding the RC1, RC2, RC3 and Rgap
Φ3 is the flux that goes through the air gap

Using KVL:
Loop 1: -2000 + (RC2 + Rgap) Φ3 + RC1Φ1 = 0
15915.49 Φ3 + 397887.36 Φ3 + 39788.74 Φ1 = 2000
39788.74 Φ1 + 413802.85 Φ3 = 2000 — ①

Loop 2: (- Rgap - RC2) Φ3 + RC3Φ2 = 0
- 397887.36 Φ3 – 15915.49 Φ3 + 39788.74 (Φ1 – Φ3) = 0
- 397887.36 Φ3 – 15915.49 Φ3 – 39788.74 Φ3 + 39788.74 Φ1
- 453591.59 Φ3 + 39788.74 Φ1 = 0 — ②

Solve ① and ② simultaneously:
39788.74 Φ1 + 413802.85 Φ3 = 2000
39788.74 Φ1 - 453591.59 Φ3 = 0

Therefore, Φ1= 0.0262 Wb
Φ3 = 0.0231 Wb

Answer = 2.31mWbHowever, is it wrong to use mesh analysis for this ? Using mesh analysis, I got 3mWb. So my friends suggested I used KVL instead.
Also is there a simpler way to calculate this question ?

 

[cnh1995]

However, is it wrong to use mesh analysis for this ? Using mesh analysis, I got 3mWb.

Do you mean nodal analysis? Because mesh analysis involves use of KVL.

Also is there a simpler way to calculate this question ?

Calculate reluctances of these paths.

Think on how they are interconnected in terms of series or parallel fashion. Form an electrical equivalent circuit for simplicity. Compute the equivalent reluctance.

Then you can find the total flux using mmf and net reluctance. Using flux division (just like current division in electrical circuits), you can determine the flux through the air gap.

This way, you need not keep track of signs and solve for multiple unknowns. So, this method may be simpler.

 

[ElectroViruz]

Do you mean nodal analysis? Because mesh analysis involves use of KVL.

Hmm I must've miscalculated when doing mesh then. I did mesh but my answer was 3mWb. I'm supposed to get the same answer as the way I did KVL right ?

This way, you need not keep track of signs and solve for multiple unknowns. So, this method may be simpler.

I see. Thanks again man :)

 

[cnh1995]

I see. Thanks again man :)

You're welcome!