# Mutual inductance

I am confused about derivation of the next formula

The question is about Cm term.

We have coupled inductors.

The voltage across first coil is

And I am ok with it. I understand it.

I know that current I1 causes a flux in the magnetic core

Where c is

But I cannot understand the first formula. How do we get Cm term?

I thought there would must be

My misunderstanding does not relate to the nature of Cm or C1 terms. I do not understand how do we get it mathematically.

• I'm assuming that $c$ is the equivalent of $\lambda$ in some sources, which indicates the flux linkage between the coils. Am I correct? While $v,i,N,$ and $\phi$ are very common nomenclature for mutual inductance, $c$ and $\theta$ are not so you may have to explain what that those indicate.
– user103380
Commented May 1, 2019 at 15:19
• c is a constant that depends on the magnetic properties and geometry of the core. I have added this n my question just now Commented May 1, 2019 at 16:06
• By θ I mean ϕ. It is my error Commented May 1, 2019 at 16:11

I'm not sure what text your pulling this from, but just looking at the first equation you have:

$$V_2 = C_MN_1N_2\frac{di_1}{dt}= M\frac{di_1}{dt}$$

which means that from the equations listed above at the time of writing:

$$M=C_MN_1N_2$$

In other literature (such as this article on mutual inductance or my CRC book) M is the mutual inductance ratio k is the coupling factor which describes the leakage out of each inductor (not all of the magnetic flux lines flow through the other side of the transformer)

$$M=k\sqrt{L_1L_2}$$

or

$$M=\sqrt{k_1L_1k_2L_2}$$

In a problem k or C_M are usually given to you, or they come from another leakage parameter because inductors are not perfect. For iedal transformers, these parameters are 1. For non ideal transformers, between 1 and 0

• Yes. But why Cm and not C1? It was my question. I wrote that my question does not relate to the nature or meaning of C. How did they get Cm instead of C1. That is the question Commented May 1, 2019 at 19:48
• It is from my book Richard Dorf Introduction to electric circuits 9th edition. Richard Dorf is also the series editor of your CRC book. Commented May 1, 2019 at 20:05
• You can see that in the equations above with the k values, k is the leakage parameter for both, k1 and k2 apply to reach inductor individually. For Cm, you'll have a parameter for both and a C1 and C2 for each of the individual inductors Commented May 1, 2019 at 23:45
• You also said I do not understand how to get "it" what it are you referring to, in cases like this it helps to be specific Commented May 1, 2019 at 23:46
• C is permeance. By "it" I mean " how do we get cm term in the equation for V2 (voltage at the second coil)". Just substitute the formula for ϕ in the V2's equatuon and you will get the same result. I acked about mathematical deriveation. Read my question simply Commented May 2, 2019 at 10:09