4. Self/Mutual Inductance Solver — Theory
Inductance measures how strongly a circuit resists changes in current by creating an induced EMF.
It is defined from flux linkage:
\[
\lambda \equiv N\Phi,\qquad L \equiv \frac{\lambda}{I}
\quad\Rightarrow\quad
\varepsilon = -\,L\,\frac{dI}{dt}
\]
Self-inductance (solenoid)
For an (ideal, long) solenoid with turns density \(n=N/\ell\), cross-sectional area \(A\), and length \(\ell\):
- Inside field (Ampère’s law): \(\;B \approx \mu\,n\,I\), where \(\mu=\mu_0\mu_r\)
- Flux through one turn: \(\;\Phi \approx B A\)
- Flux linkage: \(\;\lambda = N\Phi \approx (n\ell)(\mu n I A)=\mu n^2 A\ell\,I\)
\[
L=\frac{\lambda}{I}\approx \mu\,n^2A\ell
\qquad(\text{Henry, H})
\]
Self-inductance (toroid)
For a toroid (mean magnetic path length \(\ell_m\approx 2\pi r_m\)), with \(N\) turns and cross-section area \(A\):
\[
L \approx \frac{\mu N^2 A}{\ell_m}\approx \frac{\mu N^2 A}{2\pi r_m}
\]
Mutual inductance
If coil 1 carries current \(I_1\), it produces flux that links coil 2. The mutual inductance is defined by:
\[
M \equiv \frac{\lambda_{21}}{I_1}
\qquad\text{where}\qquad
\lambda_{21}=N_2\Phi_{21}
\]
Coupling coefficient
Not all flux from one coil links the other. This is summarized by the coupling coefficient \(k\in[0,1]\):
\[
k \equiv \frac{M}{\sqrt{L_1L_2}}
\quad\Rightarrow\quad
M = k\sqrt{L_1L_2}
\]
Energy (optional)
\[
U=\frac{1}{2}LI^2
\qquad (\text{self})
\]
This calculator uses standard idealized formulas (long solenoid / toroid mean path). Real coils may differ due to fringing, leakage, and nonuniform fields.
