Coupled Inductor
How Coupled Windings Share One Core
When two windings occupy the same core, the flux generated by one partly links the other. The total flux linkage of winding 1 becomes the sum of its self term L1i1 and a mutual term Mi2, and similarly for winding 2. The mutual inductance M captures how strongly the windings interact and is bounded by the geometric mean of the two self-inductances. Designers express the strength of that interaction through the coupling coefficient k: a value near 1 means almost all the flux is shared (the tight-coupling regime of a transformer equivalent), while moderate k values of 0.5 to 0.9 are intentional in many power-converter coupled inductors so that leakage inductance can perform useful filtering work.
The polarity of the windings, shown by dot convention, decides whether the windings add or oppose. In a multiphase buck regulator the windings are connected inverse-coupled, so the DC fluxes from each phase partly cancel in the shared core leg, which relieves DC bias and lets a smaller core avoid saturation. Because the phase ripple currents are interleaved out of step, the mutual term raises the effective steady-state (ripple) inductance each phase sees, cutting per-phase ripple, while the transient inductance during a load step stays low for fast current slew. The net result is comparable dynamic response with substantially smaller core volume than two independent inductors would require.
Core selection follows from the energy that must be stored in the gap. Gapped ferrite, powdered iron, and distributed-gap composite cores all see use; the gap length sets the inductance and prevents saturation under DC bias. High-frequency operation raises winding loss through skin effect and proximity effect, so litz wire or planar windings are common above several hundred kHz. The magnetizing branch, governed by the core's magnetizing inductance, must be high enough that magnetizing current stays a small fraction of the load current.
Governing Equations
k = M / √(L1 × L2) (0 ≤ k ≤ 1)
Mutual Inductance from Series Tests:
M = (Laid − Lopp) / 4
where Laid = L1 + L2 + 2M, Lopp = L1 + L2 − 2M
Leakage Inductance:
Lleak ≈ L1 × (1 − k)
Coupled Winding Voltage:
v1 = L1(di1/dt) ± M(di2/dt)
Example: L1 = L2 = 4.7 μH, k = 0.5 → M = 2.35 μH, Lleak ≈ 2.35 μH. The ± sign follows the dot convention; inverse coupling uses the minus term.
Coupled Inductor vs. Discrete and Transformer Magnetics
| Property | Coupled Inductor | Discrete Inductors | Transformer |
|---|---|---|---|
| Typical coupling k | 0.5 to 0.95 | 0 (separate cores) | 0.98 to 0.999 |
| Air gap | Yes (energy storage) | Yes | Minimal or none |
| DC bias per winding | High | High | Low / balanced |
| Primary benefit | Ripple cancel, smaller core | Simplest, independent | Isolation, voltage scaling |
| Leakage role | Useful filtering / parasitic | Not applicable | Parasitic (spikes) |
| Common topologies | SEPIC, Cuk, multiphase buck | Single buck/boost | Forward, push-pull, LLC |
| Footprint vs. discrete | 20 to 50% smaller | Baseline | Comparable |
Frequently Asked Questions
How does a coupled inductor differ from a transformer?
Both put multiple windings on a shared core, but a transformer targets near-unity coupling (k > 0.99), minimizes stored energy, and carries little DC bias, so it uses little or no air gap. A coupled inductor deliberately stores energy in a gap and carries large DC current per winding, often at moderate k of 0.5 to 0.95. A flyback "transformer" is really a coupled inductor: it stores energy in the gap during the on time and releases it during the off time.
Why do coupled inductors reduce ripple in multiphase converters?
The phases switch out of step, so their ripple currents are partly out of phase. Connecting the windings inverse-coupled makes the mutual term oppose the steady-state DC flux while reinforcing transient response. The phase then sees a high steady-state inductance that cuts ripple, plus a low transient inductance for fast load steps. A two-phase k = 0.5 design can cut output ripple 30 to 50% versus two discrete inductors of equal footprint.
How do I measure the coupling coefficient k on the bench?
Measure each winding's self-inductance L1 and L2 on an LCR meter, then measure series-aiding (Laid = L1 + L2 + 2M) and series-opposing (Lopp = L1 + L2 − 2M). Compute M = (Laid − Lopp) / 4, then k = M / √(L1L2). Leakage inductance is roughly L1(1 − k), the part of the flux that does not link the other winding.
What sets the leakage inductance and why does it matter?
Leakage inductance is the flux that links only its own winding and not the other; it scales roughly with L1(1 − k) and depends on winding geometry, interleaving, and spacing. In flyback and forward converters it causes voltage spikes across the switch at turn-off, demanding snubbers or active clamps. In SEPIC and Cuk converters a controlled amount of leakage can usefully shape the ripple, so designers tune it rather than simply minimize it.