Determining the inductance of a coil can be necessary for the
restoration of vintage radios or for understanding their design. There are no
analytical formulas giving the exact value of the inductance. That is why at the beginning of the 20th
century, the first approximate calculation tools based on tables or empirical formulas
appeared and gave satisfactory results for common needs. The arrival of
digital calculators has enabled the rapid, precise and adapted determination for the most
different winding shapes of mutual inductance and thus self-inductance.
fig.1 2 loops
A loop \(B1\) with current \(i_1\) creates a flux that passes through it. If it is near another
loop \(B2\), a part of the flux \(\phi_1\) passes through \(B2\). A current \(i_2\) then appears which
depends on \(i_1\), the distance \(r\) between the loops, their shapes, etc.
This current \(i_2\) creates a flux \(\phi_2\). A part of the flux \(\phi_2\) passes through the first loop \(B1\). So this one is crossed by 2 fluxes \(\phi_1\) and \(\phi_2\). The flux \(\phi_1\) depends on the self-inductance \(L1\) of loop B1 and the portion of flux from loop \(B2\) depends on the inductance \(L2\) but also on the shape and distance of this second loop. We then introduce a coefficient \(M_{12}\) called mutual inductance.
Two loops in series very far apart have an inductance that is the sum of the self-inductances of each loop. When the loops are close, the interaction between them involves mutual inductance.
This current \(i_2\) creates a flux \(\phi_2\). A part of the flux \(\phi_2\) passes through the first loop \(B1\). So this one is crossed by 2 fluxes \(\phi_1\) and \(\phi_2\). The flux \(\phi_1\) depends on the self-inductance \(L1\) of loop B1 and the portion of flux from loop \(B2\) depends on the inductance \(L2\) but also on the shape and distance of this second loop. We then introduce a coefficient \(M_{12}\) called mutual inductance.
Two loops in series very far apart have an inductance that is the sum of the self-inductances of each loop. When the loops are close, the interaction between them involves mutual inductance.
fig.2 2 coaxial loops
Neumann's formula
Mutual inductance can be calculated using Neumann's formula.\(\displaystyle M=\frac {\mu_0}{4\pi} \iint \frac {\overrightarrow {ds} \overrightarrow {ds'}}{r} \)
Here \(ds\) and \(ds'\) are elementary segment vectors of the loops and r is the distance separating them. \(a\) and \(A\) are the radii of the coaxial loops occupying 2 parallel planes separated by \(b\).
\(\displaystyle M_{12}=\frac {\mu_0}{4\pi} \iint \frac {\cos \epsilon}{r} \)
For example, in the case of 2 coaxial turns placed in 2 parallel planes, Neumann's formula can be written as follows.
This classic integral can be written using 2 elliptic integrals.
$$\begin{cases} r=\sqrt{A^2+a^2+b^2-2Aa\cos {(\varphi-\varphi')}}\\ \epsilon=\varphi - \varphi' \mbox{ , } ds=ad\varphi \mbox{ , } ds'=Ad\varphi' \end{cases} $$
Again, relying on the assumption of concentric turns, we can write the distance \(r\) separating the 2 elements \(ds\) and \(ds'\) as a function of the respective radii \(a\), \(A\) and the angle \(\epsilon\) .
By making the changes of integration variables opposite, we can write Neumann's formula in the form given below. $$\displaystyle M_{12}=\frac {\mu_0}{4\pi}\int_0^{2\pi}\!\!\!\!\!\int_0^{2\pi}\frac{Aa\cos {(\varphi-\varphi')}d\varphi d\varphi'}{\sqrt{A^2+a^2+b^2-2Aa\cos {(\varphi-\varphi')}}}$$ This classic integral can be expressed using the functions \(K\) and \(E\).
\(K\) and \(E\) are complete elliptic integrals of the first and second kind respectively.
$$\displaystyle M_{12}=-\mu_0\sqrt{Aa}\left\lbrack(k-\frac{2}{k})K+-\frac{2}{k}E\right\rbrack\frac{}{} \mbox{ ; }k=\frac{2\sqrt{Aa}}{\sqrt{(A+a)^2+b^2}}$$
At this stage, \(M_{12}\) still has a non-approximate form. We introduce a variable \(k\). It remains to determine \(K\) and \(E\)
\(K\) is a complete elliptic integral of the first kind $$\displaystyle K=F(k,\pi/2)=F(k)=\int_0^{\pi/2}\frac{d\varphi}{\sqrt{1-k^2\sin^2\varphi}d\varphi}$$ \(E\) is a complete elliptic integral of the second kind $$\displaystyle E=E(k,\pi/2)=E(k)=\int_0^{\pi/2}\sqrt{1-k^2\sin^2\varphi}d\varphi$$
In the case of a coil composed of \(n\) turns and applying the principle of superposition, we can sum the Mij as shown opposite. The number of terms in the sum is not \(n^2\) but \(2^n - 1\). $$\displaystyle M_{total}=\sum_{i=1}^{n_1}\sum_{j=1}^{n_1}M_{ij}$$
$$\begin{cases} r=\sqrt{A^2+a^2+b^2-2Aa\cos {(\varphi-\varphi')}}\\ \epsilon=\varphi - \varphi' \mbox{ , } ds=ad\varphi \mbox{ , } ds'=Ad\varphi' \end{cases} $$
Again, relying on the assumption of concentric turns, we can write the distance \(r\) separating the 2 elements \(ds\) and \(ds'\) as a function of the respective radii \(a\), \(A\) and the angle \(\epsilon\) .
By making the changes of integration variables opposite, we can write Neumann's formula in the form given below. $$\displaystyle M_{12}=\frac {\mu_0}{4\pi}\int_0^{2\pi}\!\!\!\!\!\int_0^{2\pi}\frac{Aa\cos {(\varphi-\varphi')}d\varphi d\varphi'}{\sqrt{A^2+a^2+b^2-2Aa\cos {(\varphi-\varphi')}}}$$ This classic integral can be expressed using the functions \(K\) and \(E\).
\(K\) and \(E\) are complete elliptic integrals of the first and second kind respectively.
$$\displaystyle M_{12}=-\mu_0\sqrt{Aa}\left\lbrack(k-\frac{2}{k})K+-\frac{2}{k}E\right\rbrack\frac{}{} \mbox{ ; }k=\frac{2\sqrt{Aa}}{\sqrt{(A+a)^2+b^2}}$$
At this stage, \(M_{12}\) still has a non-approximate form. We introduce a variable \(k\). It remains to determine \(K\) and \(E\)
\(K\) is a complete elliptic integral of the first kind $$\displaystyle K=F(k,\pi/2)=F(k)=\int_0^{\pi/2}\frac{d\varphi}{\sqrt{1-k^2\sin^2\varphi}d\varphi}$$ \(E\) is a complete elliptic integral of the second kind $$\displaystyle E=E(k,\pi/2)=E(k)=\int_0^{\pi/2}\sqrt{1-k^2\sin^2\varphi}d\varphi$$
In the case of a coil composed of \(n\) turns and applying the principle of superposition, we can sum the Mij as shown opposite. The number of terms in the sum is not \(n^2\) but \(2^n - 1\). $$\displaystyle M_{total}=\sum_{i=1}^{n_1}\sum_{j=1}^{n_1}M_{ij}$$
