Formula used
- M = k √(L₁ L₂) (coupling coefficient method)
- M = |ε₂| / |dI₁/dt| (induced EMF method)
- M = N₂ Φ₂₁ / I₁ (flux linkage method)
- M ≈ μ₀ μᵣ N₁ N₂ A / l (long coaxial solenoids approximation)
How to use this calculator
- Select the method that matches your available measurements.
- Enter the required values and choose an output unit.
- Click Calculate to view results above the form.
- Use the download buttons to save the computed result.
- Change inputs to compare designs, cores, or coupling.
Example data table
| Case | Method | Key inputs | Mutual inductance |
|---|---|---|---|
| 1 | k √(L1 L2) | k = 0.90, L1 = 12 mH, L2 = 8 mH | ≈ 8.82 mH |
| 2 | |ε2| / |dI/dt| | ε2 = 2.5 V, dI/dt = 50 A/s | 0.05 H |
| 3 | N2 Φ21 / I1 | N2 = 200, Φ21 = 1.2×10⁻⁴ Wb, I1 = 1.8 A | ≈ 0.0133 H |
| 4 | μ0 μr N1 N2 A / l | N1 = 400, N2 = 300, A = 6 cm², l = 12 cm, μr = 1 | ≈ 0.000754 H |
1) Mutual inductance in coupled circuits
Mutual inductance (M) describes how a changing current in one coil produces a directly linked magnetic field in a second coil. It is the core parameter behind transformers, coupled inductors, sensors, and wireless power links. Higher shared flux means larger induced voltage for the same current change.
2) Units and realistic magnitude ranges
Mutual inductance is measured in henries (H), but practical systems often fall in millihenries and microhenries. Small signal transformers may sit around 10–500 mH, while tightly coupled power inductors may be 1–50 µH. Printed coils for NFC can be a few hundred nH to a few µH.
3) Coupling coefficient and leakage flux
The coupling coefficient k (0 to 1) captures how much of the magnetic flux produced by coil 1 also links coil 2. Air‑core coils placed apart may have k ≈ 0.05–0.3, while well‑designed transformer windings on a core can reach k ≈ 0.95–0.995. Leakage flux lowers k and increases losses.
4) Using self‑inductances to estimate M
When self‑inductances L1 and L2 are known, the relation M = k·√(L1·L2) gives a fast estimate of coupling strength. This is useful for design iterations: adjust spacing, core material, or winding placement, then observe how L and k influence M. The calculator converts common inductance units automatically.
5) Induced EMF method for measured waveforms
In lab measurements, the induced EMF method uses M = |ε2| / |dI1/dt|. For example, if a 2.5 V pulse appears across coil 2 while coil 1 current ramps at 50 A/s, then M = 0.05 H. Accurate results require measuring the true current slope and using the same time window as the voltage.
6) Flux linkage method for simulations
Field‑based modeling often starts from flux linkage: M = N2·Φ21 / I1, where Φ21 is the flux through coil 2 due to current in coil 1. This is convenient when magnetic flux is obtained from simulation or a flux sensor. It also highlights linearity assumptions; if the core approaches saturation, Φ is no longer proportional to I.
7) Coaxial solenoid approximation and constants
For long coaxial solenoids sharing the same field region, an engineering approximation is M ≈ μ0·μr·N1·N2·A / l, with μ0 = 4π×10⁻7 H/m. It depends strongly on area A and turns. Because fringe fields and finite length break the ideal model, treat this option as a first‑pass estimate.
8) Practical accuracy checks and limitations
Common error sources include wrong unit conversions, neglecting resistance when inferring dI/dt, and using k values outside 0–1. Temperature and core permeability changes can shift results, especially with ferrites. Use the steps shown above the form to sanity‑check magnitudes before exporting CSV or PDF.
1) What does mutual inductance physically represent?
It represents the shared magnetic linkage between two circuits. A changing current in the first coil creates changing flux that links the second coil, producing an induced voltage proportional to M and the current rate of change.
2) Can mutual inductance be negative?
Yes. The sign depends on winding orientation and the dot convention. If the induced voltage opposes the assumed polarity, M is negative. This calculator reports magnitude in the EMF method using absolute values.
3) How is k different from M?
k is a dimensionless coupling quality from 0 to 1, while M has units of henry. For fixed L1 and L2, M increases as k increases because more flux links both coils.
4) Which method should I choose?
Use k·√(L1L2) when you know self‑inductances and coupling. Use |ε2|/|dI/dt| for test pulses. Use N2Φ/I when flux is known from simulation. Use the solenoid model for long coaxial coils.
5) Why is the solenoid option only an approximation?
It assumes a uniform field inside long solenoids and ignores fringe fields, spacing, and imperfect flux sharing. Real coils have finite length and leakage, so the result is best for preliminary sizing.
6) What units should I use for best accuracy?
Enter values in the units you measured. The calculator converts internally to henries and meters, then displays your chosen output unit. Keep significant figures consistent with your instruments.
7) What happens if the core saturates?
When a magnetic core saturates, permeability drops and flux no longer scales linearly with current. Any method assuming proportionality, especially the flux and solenoid models, can underestimate errors. Measure at the intended operating current range.