# How to calculate mutual inductance of a rectangular loop

• endiewibowo
In summary, the conversation discusses the construction of a rogowski coil with a rectangular loop rather than a circular loop. The main question is how to calculate the mutual inductance for this variation in cross sectional area. The conversation also touches on the use of Ampere's law and Biot-Savart law in these calculations. The conclusion is that the rectangular loop can be regarded as a circular loop under certain conditions, and the results will be the same. The suggestion is to try making both coils and compare the results.
endiewibowo
I am planning to build a rogowski coil but not the conventional one, which is a circular loop.
Below is the example of a rogowski coil so you have the big picture of my question.

In the picture, the current conducting wire (with current Ip) is encircled by a big circular loop rogowski coil. The cross sectional area of the winding itself can be a circle, rectangle, or oval.
For this variations of cross sectional area, I know how to calculate the mutual inductance.

However, what if the shape of the big loop is rectangular? How to calculate the mutual inductance?

I hope you all can help me figure this out.

endiewibowo said:
However, what if the shape of the big loop is rectangular? How to calculate the mutual inductance?

Think of Amperes law, which in this case could be formulated:

circulation H⋅ds = Ip

( B = μ0*H )

What happens if you choose a rectangle as integration path instead of a circle?

Hesch said:
Think of Amperes law, which in this case could be formulated:

circulation H⋅ds = Ip

( B = μ0*H )

What happens if you choose a rectangle as integration path instead of a circle?

Let's see. Suppose I is 1kA.
So:

circulation B⋅ds = Ip0

circulation B⋅ds = 1000 A * 4π*10-7 Vs/Am = 4π*10-4

So then, how is the B ?

Last edited:
Forget B for a moment. Let's look at H when Ip = 1000A and radius for the "big loop" = R, according to your drawing in #1:

circulation H⋅ds = 1000A

Now you choose a circulation path through the "small loops", giving:

H = 1000A / (2πR). Here 2πR is the length of the rogowski coil = L ⇒ H = 1000A / L.

Now you bend the the coil into a rectangle, and you choose a circulation path following this rectangular shape. The length of the rogowski coil is still the same = L, you have just bended it. So now Hmean = 1000A / L. This means that

Hmean = H ⇒
Bmean = B ⇒
Vicircle = dψv/dt = Virect. ( Ψv = flux * turns )

According to the definition: Vi = -M × dIp/dt ⇒
Mrect = Mcircle

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Hesch said:
Forget B for a moment. Let's look at H when Ip = 1000A and radius for the "big loop" = R, according to your drawing in #1:

circulation H⋅ds = 1000A

Now you choose a circulation path through the "small loops", giving:

H = 1000A / (2πR). Here 2πR is the length of the rogowski coil = L ⇒ H = 1000A / L.

Now you bend the the coil into a rectangle, and you choose a circulation path following this rectangular shape. The length of the rogowski coil is still the same = L, you have just bended it. So now Hmean = 1000A / L. This means that

Hmean = H ⇒
Bmean = B ⇒
Vicircle = dψv/dt = Virect. ( Ψv = flux * turns )

According to the definition: Vi = -M × dIp/dt ⇒
Mrect = Mcircle

Okay, let's say Mrect = Mcircle

And suppose the small loop is rectangular as well.

To calculate M:

So:

The integration is easy since the distance x from the current wire to the coil is all the same because of circular symmetry of the big loop. However, if the big loop is rectangular, the distance x will be different for each small loop. Also, it will make a and b change all the time as well. So, how to calculate this?

Last edited:
endiewibowo said:
The integration is easy since the distance x from the current wire to the coil is all the same because of circular symmetry of the big loop. However, if the big loop is rectangular, the distance x will be different for each small loop. Also, it will make a and b change all the time as well. So, how to calculate this?

You may calculate it using Biot-Savart law, but that is comprehensive.

I think that the idea is to look at: circulation H⋅ds = 1000A. This equation is valid no matter what circulation path you choose. Now say that the distance between two windings in the rogowski coil is very small ( ds ), and the length of the big loop is constant, then the mean value of H will be the same as for rectangular/circular shape of the big loop.

Hmean = constant → Bmean = constant → Ψv = constant → Vi = constant → M = constant. ( Ψv = flux * ( number of windings ) ).

So the idea is to regard the rectangular big loop as a circular loop as you will get the same result in calculations under the conditions:

Same length of the big loop, same number of turns in the rogowski coil, same cross section area of the rogowski coil, the rogowski coil is completely surrounding the conductor (closed loop).

Last edited:
Hesch said:
You may calculate it using Biot-Savart law, but that is comprehensive.

I think that the idea is to look at: circulation H⋅ds = 1000A. This equation is valid no matter what circulation path you choose. Now say that the distance between two windings in the rogowski coil is very small ( ds ), and the length of the big loop is constant, then the mean value of H will be the same as for rectangular/circular shape of the big loop.

Hmean = constant → Bmean = constant → Ψv = constant → Vi = constant → M = constant. ( Ψv = flux * ( number of windings ) ).

So the idea is to regard the rectangular big loop as a circular loop as you will get the same result in calculations under the conditions:

Same length of the big loop, same number of turns in the rogowski coil, same cross section area of the rogowski coil, the rogowski coil is completely surrounding the conductor (closed loop).

Okay then. I will try to make both coils and compare the result. Thanks a lot.

## 1. How do I calculate the mutual inductance of a rectangular loop?

To calculate the mutual inductance of a rectangular loop, you will need to know the dimensions of the loop (length and width), the number of turns in the loop, and the current flowing through the loop. You will also need to know the distance between the rectangular loop and the other conductor. Once you have this information, you can use the formula M = μ0 * N1 * N2 * A / l, where μ0 is the permeability of free space, N1 and N2 are the number of turns in the two conductors, A is the area of the loop, and l is the distance between the two conductors.

## 2. Can I calculate mutual inductance without knowing the current?

No, the current flowing through the loop is a necessary component in the formula for calculating mutual inductance. Without knowing the current, you will not be able to accurately determine the mutual inductance.

## 3. How does the distance between the conductors affect mutual inductance?

The mutual inductance between two conductors is inversely proportional to the distance between them. This means that as the distance between the rectangular loop and the other conductor increases, the mutual inductance decreases. This is because the magnetic field produced by the current in one conductor weakens as it travels a greater distance to reach the other conductor, resulting in a lower mutual inductance.

## 4. Can mutual inductance be negative?

Yes, mutual inductance can be negative. This occurs when the two conductors have currents flowing in opposite directions, resulting in a cancellation of their magnetic fields. In this case, the mutual inductance will have a negative value in the formula.

## 5. How is mutual inductance different from self-inductance?

Mutual inductance refers to the interaction between two separate conductors, while self-inductance refers to the interaction within a single conductor. Mutual inductance is caused by the magnetic field of one conductor inducing a current in the other conductor, while self-inductance is caused by the magnetic field of a single conductor inducing a current in itself.

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