Mutual Inductance Calculator

Calculate mutual inductance, coupling coefficient, and induced EMF between two coupled inductors.

Perfect coupling k=1, Air core: 0.1-0.5
For induced EMF calculation
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How One Coil Induces Voltage in Another

Mutual inductance measures how effectively a changing current in one circuit creates magnetic flux linkage in a second circuit. If current I1 changes, the linked flux through coil 2 changes and Faraday’s law produces an induced emf ε2=−M dI1/dt. The minus sign expresses Lenz’s law: the induced effect opposes the change that created it.

For two inductors with self-inductances L1 and L2, M=k√(L1L2), where the coupling coefficient 0≤k&le1 describes how much of one coil’s magnetic flux links the other. Geometry, spacing, orientation, and magnetic core material strongly influence k.

M=k√(L1L2),   |ε|=M|dI/dt|
SymbolMeaningWhy it appears / units
MMutual inductanceH; strength of magnetic coupling between two circuits.
kCoupling coefficientDimensionless, from 0 for no coupling to 1 for ideal complete coupling.
dI/dtCurrent change rateA/s; faster current changes induce larger voltage.

A large M does not by itself guarantee high efficiency; winding resistance, core losses, leakage flux, and frequency effects also matter. In transformer analysis, dot convention determines the relative polarity of induced voltages.

Mutual inductance cannot exceed √(L1L2) for passive coupled coils. The coupling coefficient k=M/√(L1L2) should lie between 0 and 1. A computed k above 1 indicates inconsistent inductance data or a unit-conversion error.

Worked Examples

Example 1: Power transformer: L1=100mH, L2=400mH, k=0.99
M=0.99×√(0.1×0.4)=0.99×0.2
Result: M=198mH, turns ratio=1:2
Step-up transformer
Example 2: Wireless charging coils: k=0.5, L1=L2=10mH
M=0.5×10=5mH
Result: Loosely coupled — typical wireless charger
k<0.6 typical for wireless power transfer
Example 3: Coupled inductors
L1=4mH, L2=9mH, k=0.5
Result: M=3mH
The geometric mean gives 6mH, and 50% coupling reduces the mutual inductance to 3mH.
Example 4: Induced emf
M=20mH, dI/dt=50A/s
Result: |ε|=1.0V
A faster-changing primary current increases induced voltage even when M is unchanged.

Common Mistakes

⚠️
Allowing k to exceed 1

For passive coupled inductors, the magnitude of coupling coefficient cannot exceed 1. Values above 1 violate M²≤L₁L₂.

⚠️
Ignoring the sign from Lenz’s law

Magnitude calculations may omit the sign, but circuit polarity depends on winding orientation and the dot convention.

⚠️
Using current instead of current-change rate

Induced emf depends on dI/dt. A steady DC current can establish flux but produces no continuing transformer emf after transients settle.

Frequently Asked Questions

What is coupling coefficient k?
k=1: perfect coupling (all flux links both coils, ideal transformer). k=0: no coupling. Real transformers: k=0.95-0.999. Wireless chargers: k=0.3-0.8 depending on coil alignment and separation.
Mutual inductance in circuit analysis?
Coupled inductors add complexity: Z_in = jωL₁ + ω²M²/(jωL₂+Z_load). Dot convention shows polarity of induced EMF. Series-aiding: L_total = L₁+L₂+2M. Series-opposing: L_total = L₁+L₂-2M.
What determines mutual inductance physically?
Mutual inductance depends on turn counts, coil geometry, spacing, orientation, permeability of the magnetic path, and how much flux from one coil links the other.
Can mutual inductance be negative?
A signed mutual term can be negative depending on chosen current directions and dot convention. The coupling magnitude M is usually quoted as positive, while signs are handled in circuit equations.
Why does a transformer need changing current?
Faraday’s law requires changing magnetic flux. A steady DC primary current eventually produces constant flux, so after the switching transient there is no sustained induced secondary voltage.
How are M and k related?
The relation M=k√(L₁L₂) normalizes coupling against the two self-inductances. It lets very different coil sizes be compared on a common scale from zero to ideal coupling.
Can mutual inductance be negative?
The physical coupling magnitude M is commonly reported as nonnegative, while the sign of induced voltage depends on winding orientation and the dot convention. Reversing one winding changes the relative polarity. Circuit equations therefore may contain +M or −M terms even though the tabulated coupling magnitude is positive.

Formula Explorer connections

Interpretation: This relationship connects magnetic fields, moving charge, flux, induction or electromagnetic material response. Assumption: Specify field direction and sign convention. Uniform fields, linear materials, negligible edge effects or sinusoidal steady state may be assumed.

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