I am confused about derivation of the next formula enter image description here

The question is about Cm term.

We have coupled inductors.

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The voltage across first coil is

enter image description here

And I am ok with it. I understand it.

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

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Where c is

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

I thought there would must be

enter image description here

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

  • \$\begingroup\$ 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. \$\endgroup\$
    – user103380
    Commented May 1, 2019 at 15:19
  • \$\begingroup\$ c is a constant that depends on the magnetic properties and geometry of the core. I have added this n my question just now \$\endgroup\$
    – Salekh
    Commented May 1, 2019 at 16:06
  • \$\begingroup\$ By θ I mean ϕ. It is my error \$\endgroup\$
    – Salekh
    Commented May 1, 2019 at 16:11

1 Answer 1


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:


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)




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

  • \$\begingroup\$ 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 \$\endgroup\$
    – Salekh
    Commented May 1, 2019 at 19:48
  • \$\begingroup\$ It is from my book Richard Dorf Introduction to electric circuits 9th edition. Richard Dorf is also the series editor of your CRC book. \$\endgroup\$
    – Salekh
    Commented May 1, 2019 at 20:05
  • \$\begingroup\$ 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 \$\endgroup\$
    – Voltage Spike
    Commented May 1, 2019 at 23:45
  • \$\begingroup\$ 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 \$\endgroup\$
    – Voltage Spike
    Commented May 1, 2019 at 23:46
  • \$\begingroup\$ 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 \$\endgroup\$
    – Salekh
    Commented May 2, 2019 at 10:09

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