The primary mutual inductance formula is M = k√(L₁L₂). This equation defines the magnetic linkage between two coils, where M is the mutual inductance in Henrys, k is the dimensionless coupling coefficient (0 to 1), and L₁ and L₂ are the self-inductances of the individual coils in Henrys. On the bench, this formula dictates everything from flyback voltage spikes in relay circuits to power transfer efficiency in SEPIC converters.
The Core Mutual Inductance Formula and Symbol Definitions
While M = k√(L₁L₂) is the most practical form for component selection, the foundational physics definition relies on magnetic flux linkage. Both forms are required for a complete understanding of coupled magnetic circuits. Below is the definitive symbol table for both the coupling coefficient form and the flux linkage form (M = N₂Φ₁₂ / I₁).
| Symbol | Parameter | SI Unit | Practical Bench Unit | Definition & Constraints |
|---|---|---|---|---|
| M | Mutual Inductance | Henry (H) | µH or mH | The ratio of induced voltage in coil 2 to the rate of current change in coil 1. |
| k | Coupling Coefficient | Dimensionless | 0.0 to 1.0 | Fraction of magnetic flux from L₁ that intersects L₂. 1.0 is perfect coupling (impossible in reality). |
| L₁, L₂ | Self-Inductance | Henry (H) | µH or mH | Inductance of each coil measured independently with the other coil open-circuited. |
| N₂ | Turns of Coil 2 | Dimensionless | Integer | Total number of wire wraps in the secondary/induced winding. |
| Φ₁₂ | Mutual Flux | Weber (Wb) | µWb | The specific magnetic flux generated by coil 1 that physically passes through the cross-section of coil 2. |
| I₁ | Primary Current | Ampere (A) | mA or A | DC or instantaneous AC current flowing through coil 1 generating the magnetic field. |
Rearranged Forms and the Time-Domain Voltage Equation
Designing with coupled inductors requires isolating specific variables. Below are the algebraically rearranged forms of the core formula, followed by the time-domain derivative equation that actually causes MOSFETs to fail on the bench.
Rearranged Coupling Forms
- Solve for coupling coefficient (k): k = M / √(L₁L₂)
- Solve for primary inductance (L₁): L₁ = M² / (k²L₂)
- Solve for secondary inductance (L₂): L₂ = M² / (k²L₁)
- Solve for mutual flux (Φ₁₂): Φ₁₂ = (M × I₁) / N₂
The Time-Domain Induction Equation
The static formula defines the component, but the dynamic formula defines the circuit behavior. The voltage induced in coil 2 (V₂) is governed by:
V₂ = M(di₁/dt)
- Solve for M: M = V₂ / (di₁/dt)
- Solve for slew rate (di₁/dt): di₁/dt = V₂ / M
Worked Examples with Strict Unit Tracking
Abstract formulas fail when unit prefixes are mismanaged. These two solved problems track every prefix conversion explicitly.
Example 1: Calculating M for a SEPIC Coupled Inductor
Scenario: You are designing a SEPIC DC-DC converter using a dual-winding coupled inductor. The datasheet specifies L₁ = 47 µH, L₂ = 47 µH, and a typical coupling coefficient k = 0.96. Find M.
- Identify given values: L₁ = 47 µH, L₂ = 47 µH, k = 0.96.
- Select formula: M = k × √(L₁ × L₂)
- Substitute values (keeping units in µH for simplicity, as √(µH × µH) = µH):
M = 0.96 × √(47 µH × 47 µH) - Solve the square root:
√(2209 µH²) = 47 µH - Final multiplication:
M = 0.96 × 47 µH = 45.12 µH
Example 2: Calculating Flyback Voltage Spike in a Relay Coil
Scenario: A microcontroller switches off a relay driver. The relay coil acts as a secondary winding coupled to the collapsing field of the primary circuit (or simply consider the self-inductance flyback, but here we model the induced spike on an adjacent sniffer coil used for zero-crossing detection). Let M between the relay coil and the sniffer coil be 15 mH. The relay current (I₁) drops from 120 mA to 0 A in 4 µs. Find the induced voltage V₂.
- Convert to base SI units:
M = 15 mH = 0.015 H
di = 120 mA - 0 mA = 0.12 A
dt = 4 µs = 0.000004 s (4 × 10⁻⁶ s) - Calculate current slew rate (di/dt):
di/dt = 0.12 A / 0.000004 s = 30,000 A/s - Apply time-domain formula:
V₂ = M × (di/dt) - Substitute and solve:
V₂ = 0.015 H × 30,000 A/s = 450 V
Assumptions, Unit Traps, and Realistic Magnitudes
The mutual inductance formula is not a universal law; it operates under strict physical assumptions. Violating these assumptions or falling for common unit traps will yield simulations that fail on the physical bench.
When the Formula Applies (and When It Breaks)
- Linear B-H Curve: The formula assumes the core material operates in its linear region. If the core saturates (e.g., exceeding the I_sat rating of a ferrite core), permeability drops, and L₁, L₂, and M all collapse non-linearly.
- Constant Geometry: It assumes fixed physical spacing. If the coils vibrate or move (like in a loose transformer), k becomes a function of time, inducing microphonic noise.
- Sinusoidal or Monotonic DC: For complex waveforms, you must apply the derivative form (V = M di/dt) at every instantaneous point, not just use RMS averages.
Unit Mistakes That Break the Math
- The Prefix Mismatch: Multiplying L₁ in mH by L₂ in µH under the square root without converting to base Henrys. This yields a mathematically valid but physically meaningless number. Always convert both to Henrys (or both to µH) before multiplying.
- The Time-Squared Trap: In di/dt, engineers often write 'µs' but calculate as 'ms', introducing a 1000x error in the induced voltage calculation. Always write out the scientific notation (e.g., 10⁻⁶) for time.
Realistic Answer Magnitudes
If your calculated M falls outside these ranges, check your inputs:
- PCB Trace Crosstalk: 1 nH to 50 nH (nanohenries).
- RF / Signal Transformers: 1 µH to 500 µH (microhenries).
- Power Coupled Inductors (SEPIC/Ćuk): 10 µH to 10 mH.
- Grid-Scale Power Transformers: 1 H to 500 H (Henries).
Decision Path: Specifying and Measuring Coupled Inductors
Use this decision tree to determine how to handle mutual inductance in your specific design scenario. Follow the if-then logic to arrive at a concrete component or measurement strategy.
| Design Requirement | If-Then Decision Logic | Concrete Action / Part Pick |
|---|---|---|
| Need high power transfer efficiency in a DC-DC converter (k > 0.95) | If tight coupling is required, discrete inductors will fail due to leakage flux. You must use a single-core dual-winding component. | Select the Coilcraft MSD1260-473ML (47µH, k≈0.99, 2.3A I_sat) or equivalent shielded coupled inductor. |
| Need to measure unknown M of a salvaged transformer on the bench | If you lack a specialized mutual inductance fixture, use the series-aiding/opposing method with a standard LCR meter. | Measure L_aiding and L_opposing at 10 kHz. Calculate M = (L_aiding - L_opposing) / 4. |
| Designing a snubber for a relay/inductive kickback scenario | If calculating V_spike yields a voltage > 2x your MOSFET V_ds rating, the calculated M or di/dt is too high for the current switching speed. | Slow down the gate turn-off time (increase R_gate) or add a 1N4148 flyback diode directly across the primary coil. |
| Need loose coupling for a proximity sensor or wireless power (k < 0.5) | If physical air gaps are mandatory, ferrite cores will saturate or detune. Air-core or planar PCB coils are required. | Design a planar spiral PCB coil (e.g., 4-layer FR4, 20 turns, 0.2mm trace width) and simulate k in Ansys Maxwell. |
For 90% of hobbyist and commercial power supply designs requiring a known, stable mutual inductance, the default recommendation is to abandon discrete winding and specify a factory-matched coupled inductor like the Coilcraft MSD1260 series. The guaranteed k value and integrated magnetic shielding eliminate the parasitic variables that make the raw mutual inductance formula so difficult to apply to hand-wound bench prototypes.
For deeper theoretical background on magnetic coupling, refer to the All About Circuits chapter on Mutual Inductance, and for physical component specifications, consult the Coilcraft Coupled Inductor catalog.
