The mutual inductance equation defines how a changing magnetic field in one coil induces a voltage in an adjacent coil. The foundational formula is M = k × √(L1 × L2). This relationship is the backbone of transformer design, coupled-inductor SEPIC converters, and wireless power transfer. If you are designing a flyback snubber or sizing a dual-winding inductor for a DC-DC regulator, getting this math right—and tracking your units precisely—is the difference between a stable power rail and a blown MOSFET.

The Core Mutual Inductance Equation and Symbol Definitions

Before applying the math, you must understand the physical boundaries of the formula. The mutual inductance equation assumes a linear magnetic circuit. This means the core material must not be saturated; the permeability (μ) must remain constant across your operating current range. If your core hits saturation, inductance drops non-linearly, coupling degrades, and the equation fails to predict real-world behavior.

Table 1: Mutual Inductance Symbol Definitions
Symbol Parameter Standard Unit Practical Range
M Mutual Inductance Henries (H) nH (traces) to mH (power transformers)
k Coupling Coefficient Dimensionless 0.2 (loose/RF) to 0.99 (toroidal power)
L1 Primary Self-Inductance Henries (H) μH to mH
L2 Secondary Self-Inductance Henries (H) μH to mH
V2 Induced Secondary Voltage Volts (V) mV (sensing) to kV (flyback spikes)
di1/dt Rate of Change of Primary Current Amperes/second (A/s) A/ms (linear ramps) to A/ns (switching edges)

The secondary voltage induced by a changing primary current is calculated using the derivative form: V2 = M × (di1 / dt). This is the equation that dictates your snubber diode ratings and MOSFET VDS limits in switch-mode power supplies.

Rearranged Forms for Component Sizing

On the bench, you rarely solve for M directly from scratch. Usually, you have a target mutual inductance or a known coupling coefficient from a datasheet, and you need to back-calculate a missing parameter. Here are the rearranged forms you will actually use:

  • Solving for Coupling Coefficient (k): k = M / √(L1 × L2)
    Use case: Validating a physical prototype on an LCR meter to see how much leakage inductance you have.
  • Solving for Primary Inductance (L1): L1 = M2 / (k2 × L2)
    Use case: Sizing the primary winding when the secondary requirements and core coupling are fixed.
  • Solving for Secondary Inductance (L2): L2 = M2 / (k2 × L1)
    Use case: Designing the feedback or auxiliary winding on a flyback transformer.
  • Solving for Current Slew Rate (di1/dt): di1/dt = V2 / M
    Use case: Determining the maximum allowable switching speed to keep induced voltage spikes below a component's breakdown rating.

Worked Examples with Strict Unit Tracking

The most common point of failure in magnetics design is unit mismanagement. Below are two real-world scenarios with explicit unit tracking.

Problem 1: Sizing a SEPIC Coupled Inductor

Scenario: You are designing a SEPIC converter. Your power stage requires two 10 μH inductors. You select a dual-winding component where the datasheet specifies L1 = 10 μH, L2 = 10 μH, and a coupling coefficient k = 0.95. What is the mutual inductance?

  1. Identify givens: L1 = 10 × 10-6 H, L2 = 10 × 10-6 H, k = 0.95.
  2. Apply formula: M = k × √(L1 × L2)
  3. Substitute: M = 0.95 × √((10 × 10-6) × (10 × 10-6))
  4. Multiply inside root: (10 × 10-6) × (10 × 10-6) = 100 × 10-12 H2
  5. Take square root: √(100 × 10-12) = 10 × 10-6 H (which is 10 μH)
  6. Final calculation: M = 0.95 × 10 μH = 9.5 μH

Bench Insight: Because k is less than 1.0, the remaining 0.5 μH is leakage inductance. In a SEPIC, this leakage causes high-frequency ringing at the switching node, requiring an RC snubber.

Problem 2: Calculating a Flyback Voltage Spike

Scenario: A primary MOSFET in a flyback converter turns off. The mutual inductance between primary and secondary is M = 50 μH. The primary current (i1) drops from 2.0 A to 0 A in 100 nanoseconds. What is the induced voltage spike on the secondary before the diode clamps it?

  1. Identify givens: M = 50 × 10-6 H. di1 = 2.0 A - 0 A = 2.0 A. dt = 100 ns = 100 × 10-9 s.
  2. Apply formula: V2 = M × (di1 / dt)
  3. Calculate slew rate: di1 / dt = 2.0 A / (100 × 10-9 s) = 20,000,000 A/s (or 2 × 107 A/s)
  4. Substitute: V2 = (50 × 10-6 H) × (20,000,000 A/s)
  5. Final calculation: V2 = 1,000 V

Bench Insight: This 1 kV spike explains why secondary rectifier diodes in flyback topologies must have high PIV (Peak Inverse Voltage) ratings, and why PCB layout must minimize parasitic loop inductance to prevent ringing from adding to this baseline spike.

Unit Mistakes That Break Your Calculations

Warning: The Microhenry Trap
The most frequent error in power electronics is plugging μH directly into the V = M(di/dt) equation without converting to base Henries. If you use 50 instead of 50 × 10-6, your calculated voltage will be off by a factor of one million. Always convert to base SI units (Henries, Amperes, Seconds) before calculating, then convert the result back to engineering notation.

Beyond the microhenry trap, time-base mismatches destroy calculations involving high-speed switching:

  • The Nanosecond vs. Microsecond Error: Modern GaN and SiC FETs switch in nanoseconds. If your oscilloscope reads dt in ns, but you accidentally type μs into your calculator, your predicted di/dt will be 1,000 times too low. You will underestimate your voltage spike and likely blow your snubber diode.
  • Assuming k is a Percentage: The coupling coefficient k is a ratio from 0 to 1. If a datasheet states '98% coupling', you must use 0.98 in the equation, not 98.

Decision Path: Selecting a Coupled Inductor

When your calculations dictate a specific mutual inductance, you must select a physical component. Use this decision matrix to narrow down your magnetics footprint and terminate on a specific part number.

Table 2: Coupled Inductor Selection Decision Tree
Condition / Requirement Recommended Core / Form Factor Target k Value
Iout < 1A, fsw > 1 MHz, space-constrained (IoT) 0603 or 0805 Multilayer Ceramic Coupled 0.80 - 0.90
Iout = 1A - 3A, fsw = 500 kHz, low profile (<2mm) Shielded Ferrite Drum Core (SMD) 0.95 - 0.98
Iout = 3A - 10A, fsw = 200-400 kHz, high efficiency Shielded Bobbin / T-Core (12x12mm footprint) > 0.98
Isolated Flyback, Vin > 400V, high leakage needed for ZVS Bobbin with physical winding separation (gap) 0.85 - 0.95 (Intentional leakage)

Default Recommendation: For standard non-isolated SEPIC, Zeta, or Cuk converters operating in the 2A to 5A range at 300-500 kHz, default to the Coilcraft MSD1260 series (specifically the MSD1260-103ML). It provides 10 μH per winding, handles 4.5 A RMS, and offers a tightly coupled k ≈ 0.98 in a shielded 12x12mm footprint, minimizing EMI without requiring custom magnetics.

Realistic Magnitudes and Leakage Edge Cases

Understanding what a 'normal' answer looks like prevents you from chasing ghosts on the bench. According to foundational texts on alternating current magnetics, mutual inductance scales heavily with physical geometry and core material.

  • PCB Traces: Two parallel 10-mil traces running for 1 inch on an FR4 board will exhibit a mutual inductance in the range of 1 to 5 nH. This is negligible for DC but causes severe crosstalk in 100+ MHz digital buses.
  • RF Air-Core Coils: Loosely coupled resonant coils (like in RFID or basic wireless charging) typically yield M values in the 100 nH to 1 μH range, with k values between 0.1 and 0.3.
  • Power Transformers: 50/60 Hz laminated steel or high-frequency ferrite power transformers operate with M values in the 1 mH to 100 mH range, with k values exceeding 0.99.

The Leakage Inductance Edge Case:
No physical component achieves k = 1.0. The flux that does not link both windings creates leakage inductance, calculated as Lleak = L1 × (1 - k). In a forward converter, this leakage energy has nowhere to go when the MOSFET turns off, resulting in the V2 spike calculated in Problem 2. If your calculated M implies a k of 0.99, but your bench measurement shows k = 0.92, your physical winding geometry has introduced an air gap or misalignment. Always measure L1 and L2 with the opposite winding shorted to extract the true leakage inductance before finalizing your snubber component values.