I have been trying my best to solve a problem that has been haunting and taunting me for the past couple days. The questions poses a scenario of two inductors wound around the same iron core. One inductor possesses $800$ turns and is connected to power. The other inductor possesses $500$ turns and is not connected to power. A set of data is given to plot the magnetic field in units of kilolines to current (this is an old book). It then asks the student to find the self inductance of the $800$ turn coil and the mutual inductance between the coil. The question asks me to plot the self-inductance of the $800$ turn coil and the mutual inductance between the coils at each point.

Here is an example of the data set:

  • $I=0.1\;\mathrm{A}$ in $800$ turn coil, $B_{800 \mathrm{turns}}= 13\;\mathrm{kilolines}$, $B_{500\mathrm{turns}}= 13\;\mathrm{kilolines}$
  • $I=0.4\;\mathrm{A}$ in $800$ turn coil, $B_{800 \mathrm{turns}}=49 \;\mathrm{kilolines}$, $B_{500\mathrm{turns}}=47\;\mathrm{kilolines}$.

I know $B = \mu_0 n I$ where $\mu_0$ = magnetic permissivity costant??? $4 \pi 10^{-7}\;\frac{\mathrm{Vs}}{\mathrm{Am}}$. Also inductance is the magnetic field differential $\mathrm{d}B$ divided by the current differential $\mathrm{d}I$. I can find the slope between the points to find the inductance for the $800$ turn coil, but I do no know how I would separate the mutual inductance from the self-inductance between the plots for each coil. Could someone please lend a hand? Thank you kindly =)

  • $\begingroup$ If I get it right, these are DC currents in two cases? So there is no $\mathrm{d}i/\mathrm{d}t$? $\endgroup$ – mikuszefski Jun 7 '17 at 6:13
  • $\begingroup$ Yes, I believe so. $\endgroup$ – Truth_Seeker24 Jun 10 '17 at 19:28

There are several ways to express the inductance. The one you are looking for is probably (see here) $$L=\frac{N\Phi}{I},$$ with inductance $L$, flux $\Phi$, current $I$ and winding number $N$. From this one might already guess the formula for the mutual inductance $M$, $$M=\frac{N_2\Phi_{1 2}}{I_1},$$ where $N_2$ is thewinding number of coil 2, $\Phi_{1 2}$ the flux of coil 1 in coil 2, and $I_1$ the current in coil 1. For more details have a look, e.g., here.

  • $\begingroup$ I appreciate the comment. Sorry for the long wait. I'll think about it and let you know if I have any more questions. $\endgroup$ – Truth_Seeker24 Jun 10 '17 at 19:35
  • $\begingroup$ I think I understand how to solve this problem. The second coil does not have current flowing through it, since the current is steady in coil 1, but it does possess some magnetic field lines from coil 1. In that case, I would find the slope of the second coil first to find the mutual inductance. Since, there is no changing current, the inductance of coil 1 would just be estimated as the slope between two points in the inductor. I cannot find flux because I don't have enough information. Am I on the right track? $\endgroup$ – Truth_Seeker24 Jun 10 '17 at 19:44
  • $\begingroup$ @Truth_Seeker24 I guess one educational purpose of the question is: be aware that the flux of coil 1 in coil 2 can be field dependent, e.g. by non-liniearities of the core.So you are just applying the above formulae in two cases. Due to the definition of flux, inductance, etc. you can actually calculate it from DC values. And concerning the amount of information: If you check the units carefully it is not $B$ what is given $[B]=\mathrm{Wb}/\mathrm{m}^2$ but the flux. $13\;\mathrm{kilolines}=0.00013\;\mathrm{Wb}$. $\endgroup$ – mikuszefski Jun 12 '17 at 5:45

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