Mutual Inductance Interactive Calculator

Two coils placed near each other will exchange energy through their shared magnetic field — but how much? That depends on their individual inductances, their geometry, and how tightly their flux links together. Use this Mutual Inductance Interactive Calculator to calculate mutual inductance (M), coupling coefficient (k), induced voltage, mutual flux, secondary inductance, or coaxial solenoid M using inputs like self-inductance values, turns counts, current change rate, and core geometry. Getting this right matters in transformer design, wireless EV charging, and RFID antenna engineering — undershoot the coupling and your system wastes power or fails to start; overshoot and you're dealing with voltage transients you didn't plan for. This page includes the key formulas, a worked transformer example, engineering theory, real-world application scenarios, and an FAQ.

What is Mutual Inductance?

Mutual inductance is a measure of how strongly two coils are electromagnetically linked — specifically, how much voltage one coil induces in the other when its current changes. It is measured in henries (H). The stronger the magnetic coupling between the coils, the higher the mutual inductance.

Simple Explanation

Think of 2 coils sitting near each other. When current flows through the first coil, it creates a magnetic field — like an invisible bubble of energy around the wire. If that bubble reaches the second coil, it pushes a voltage into it, even though nothing is physically touching. Mutual inductance is just a number that tells you how effective that invisible push is: a high number means strong coupling, a low number means most of the magnetic energy misses the second coil entirely.

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Diagram

Mutual Inductance Calculator

How to Use This Calculator

1. Select your calculation mode from the dropdown — choose what you want to solve for (M, k, induced voltage, flux, secondary inductance, or coaxial solenoid M).
2. Enter the required input values that appear for your selected mode — these may include self-inductance values (H), coupling coefficient, turns counts, current change rate (A/s), or solenoid geometry.
3. Check your units — inductance in henries, current in amps, area in m², length in metres.
4. Click Calculate to see your result.

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Equations & Formulas

Use the formula below to calculate mutual inductance from the coupling coefficient and coil self-inductances.

Use the formula below to calculate induced voltage using Faraday's Law.

Use the formula below to calculate mutual flux linkage through the secondary coil.

Use the formula below to calculate mutual inductance for coaxial solenoids.

Use the formula below to calculate coupling coefficient from measured inductances.

Simple Example

Two coils with k = 0.5, L₁ = 0.04 H, and L₂ = 0.04 H.

M = k × √(L₁ × L₂) = 0.5 × √(0.04 × 0.04) = 0.5 × 0.04 = 0.02 H

If primary current then changes at 100 A/s, the induced secondary voltage is V₂ = −0.02 × 100 = −2 V.

Theory & Engineering Applications

Fundamental Electromagnetic Coupling Principles

Mutual inductance quantifies the magnetic coupling between two circuits through their shared magnetic flux. When alternating or time-varying current flows through a primary coil, it generates a changing magnetic field that induces electromotive force (EMF) in any nearby secondary coil. This phenomenon, governed by Faraday's law of electromagnetic induction, forms the operational basis for transformers, inductive coupling systems, and proximity sensors across electrical engineering disciplines.

The coupling coefficient k represents the fraction of magnetic flux from the primary coil that successfully links the secondary coil. In practice, k rarely equals unity due to geometric constraints, air gaps, core material limitations, and flux leakage paths. A non-obvious limitation arises in high-frequency applications: above certain frequencies determined by coil geometry and core material properties, skin effect and proximity effect redistribute current density within conductors, effectively reducing the mutual inductance from its low-frequency value by 15-30% in typical power electronics transformers operating above 100 kHz.

Core Material Selection and Permeability Effects

The permeability μ of the magnetic core material dramatically influences mutual inductance magnitude. Air-core transformers use μ₀ = 4π × 10⁻⁷ H/m (permeability of free space), yielding relatively weak coupling suitable for radio-frequency tuned circuits. Ferrite cores with relative permeabilities between 1,500 and 15,000 increase mutual inductance proportionally, enabling compact high-frequency transformer designs. Silicon steel laminations (μᵣ ≈ 4,000) dominate power-frequency transformers where eddy current losses must be minimized through lamination thickness optimization.

A critical non-textbook consideration: core saturation fundamentally limits mutual inductance validity. When magnetic flux density exceeds the core's saturation threshold (typically 1.2-1.8 Tesla for silicon steel, 0.3-0.5 Tesla for ferrites), permeability collapses nonlinearly, and mutual inductance becomes current-dependent rather than constant. This nonlinearity generates harmonic distortion in transformer secondary voltages and reduces power transfer efficiency by 5-20% depending on operating point.

Wireless Power Transfer Engineering

Modern inductive charging systems for electric vehicles and consumer electronics exploit resonant mutual inductance where both primary and secondary circuits are tuned to the same frequency. The quality factor Q and coupling coefficient k determine maximum achievable efficiency through the figure of merit k²Q. Automotive wireless charging pads operating at 85 kHz typically achieve k = 0.15-0.25 due to ground clearance constraints (150-250 mm air gaps), necessitating Q values above 200 to maintain 90% efficiency. Precise coil alignment mechanisms compensate for lateral misalignment, which degrades coupling coefficient by approximately 8% per centimeter of horizontal displacement in typical pad geometries.

Worked Example: Transformer Design Calculation

Problem: An engineer designs a step-down transformer for a switched-mode power supply. The primary coil has 847 turns with measured self-inductance L₁ = 3.76 mH. The secondary coil has 124 turns with L₂ = 82.3 μH. Measurements using an LCR meter show mutual inductance M = 467 μH. Calculate the coupling coefficient, determine coupling quality, predict the secondary voltage when primary current changes at 2,350 A/s, and calculate the flux through each secondary turn when primary current is 1.83 A.

Solution:

Step 1: Calculate coupling coefficient

Using k = M / √(L₁L₂):

L₁ = 3.76 mH = 3.76 × 10⁻³ H
L₂ = 82.3 μH = 82.3 × 10⁻⁶ H
M = 467 μH = 467 × 10⁻⁶ H
√(L₁L₂) = √(3.76 × 10⁻³ × 82.3 × 10⁻⁶) = √(3.094 × 10⁻⁷) = 5.562 × 10⁻⁴ H
k = 467 × 10⁻⁶ / 5.562 × 10⁻⁴ = 0.840

Step 2: Assess coupling quality

With k = 0.840, this transformer exhibits strong coupling (k greater than 0.7). This indicates minimal flux leakage and efficient magnetic circuit design, typical of ferrite-core high-frequency transformers with close winding coupling.

Step 3: Calculate induced voltage

Using V₂ = -M(dI₁/dt):

dI₁/dt = 2,350 A/s
V₂ = -(467 × 10⁻⁶) × 2,350 = -1.097 V
The magnitude of induced voltage is 1.10 V (negative sign indicates polarity per Lenz's law)

Step 4: Calculate flux per secondary turn

Using Φ₂₁ = MI₁/N₂:

I₁ = 1.83 A
N₂ = 124 turns
Φ₂₁ = (467 × 10⁻⁶ × 1.83) / 124 = 6.891 × 10⁻⁶ Wb = 6.89 μWb

Engineering Interpretation: The coupling coefficient of 0.840 confirms excellent magnetic coupling, justifying the transformer design for high-efficiency operation. The induced voltage calculation demonstrates that during rapid current transitions (as occur in switching converters), significant voltage transients appear across the secondary, requiring appropriate snubber or clamping circuits. The flux calculation reveals that each secondary turn intercepts 6.89 μWb, which can be used to verify core saturation margins by comparing against the core's cross-sectional area and maximum flux density specification.

Automotive Sensor Applications

Inductive position sensors in automotive throttle bodies, camshaft positioning systems, and ride height measurement employ mutual inductance variation with target proximity. A primary coil driven at 50-500 kHz creates an alternating magnetic field. Eddy currents induced in nearby metallic targets alter the effective mutual inductance between primary and secondary coils, producing amplitude or phase changes in the secondary voltage proportional to target distance. These sensors operate reliably across -40°C to +150°C, immune to contamination from oil, dirt, and moisture that would compromise optical or capacitive technologies.

For further exploration of electromagnetic design principles and related engineering calculations, visit the complete calculator library covering inductance, magnetic circuits, and electromagnetic compatibility topics.

Practical Applications

Frequently Asked Questions