Mutual inductance of transformer with coils of different area and length

• w1002644
In summary, determining the mutual inductance between two coils of a transformer requires considering the coupling coefficient, which measures the percentage of flux that links both coils. This coefficient becomes more important when the area and lengths of the coils are different, as all the flux may not link the other coil. Therefore, the mutual inductance can be calculated by multiplying the coupling coefficient with the standard formula for mutual inductance.
w1002644
Hello,

I am trying to derive the mutual inductance between the two coils of a transformer. I want to do this for two coils with different lengths and different cross-sectional areas (assuming that one side of the transformer core is thicker than the other). The parameters are therefore:

I1 = current in coil 1
I2 = current in coil 2
L1 = length of first coil
L2 = length of second coil
A1 = Cross sectional area of coil 1
A2 = Cross sectional area of coil 2
N1 = Number of turns in coil 1
N2 = Number of turns in coil 2

To calculate the mutual inductance, I leave the second coil open-circuited and drive a current into the first coil, so I get the flux density:

B11 = μ(N1)(I1)/L1

The total flux generated is therefore:

∅11 = (B1)(A1) = μ(N1)(I1)(A1)/L1

Assuming that all flux generated in coil 1 also links coil 2 (no flux leakage with coupling coefficient = 1) , we can say that ∅21 = ∅11.

Therefore, the voltage induced in coil 2:

V2 = (N2)d(∅21)/dt = μ(N1)(N2)(A1)/L1 d(I1)/dt

Therefore, I can concluded that the mutual inductance M21 = μ(N1)(N2)(A1)/L1

Now, repeating the steps above but with coil 1 open-circuited while driving a current I2 into the coil 2 gives the following:

B22 = μ(N2)(I2)/L2
∅22 = (B2)(A2) = μ(N2)(I2)(A2)/L2

And assuming all of ∅22 links coil 1, ∅12 = ∅22 and we get

V1 = (N1)d(∅12)/dt = μ(N1)(N2)(A2)/L2 d(I2)/dt

And this expression gives M12 = μ(N1)(N2)(A2)/L2

I know that this isn't correct because both M21 and M12 are supposed to be equal, but my derivation gives different results.

In the simple case where the coil area and lengths are the same, the above derivation does yield mutual inductances that are the same. When these are made different, I cannot seem to get the expressions for the mutual inductances to match.

Any help / comments much appreciated.

The issue here is that you are assuming that all the flux generated in one coil links the other. This is not always the case, and in fact, when the area and lengths of the coils are different, it becomes less likely that all the flux links the other coil. Therefore, when the area and lengths of the coils are different, you need to take into account the coupling coefficient, k, which is a measure of the percentage of flux that actually links both the coils. So, the mutual inductance would then be calculated as:M21 = k*μ(N1)(N2)(A1)/L1M12 = k*μ(N1)(N2)(A2)/L2where k is the coupling coefficient, which can be determined experimentally. Hope this helps!

1. What is mutual inductance?

Mutual inductance is the measure of the ability of two coils to induce a voltage in each other. It is dependent on the number of turns in each coil, the relative orientation of the coils, and the permeability of the material between the coils.

2. How does the area of the coils affect mutual inductance?

The area of the coils is directly proportional to the mutual inductance. This means that increasing the area of one of the coils will result in a higher mutual inductance and a stronger induced voltage in the other coil.

3. How does the length of the coils affect mutual inductance?

The length of the coils is inversely proportional to the mutual inductance. This means that increasing the length of one of the coils will result in a lower mutual inductance and a weaker induced voltage in the other coil.

4. Can mutual inductance be negative?

No, mutual inductance cannot be negative. It is a measure of the coupling between two coils, and a negative value would indicate a reverse coupling or a decrease in voltage instead of an increase.

5. How is mutual inductance used in transformer design?

Mutual inductance is an important factor in transformer design. It determines the ratio of the voltages between the primary and secondary coils, and can be used to control the output voltage of a transformer. By adjusting the number of turns and the area of the coils, the mutual inductance can be optimized for a desired transformer performance.

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