Planar Spiral Inductor

Inductance of square, hexagonal, octagonal and circular PCB spiral inductors from the Mohan–Hershenson–Boyd–Lee expressions, with DC resistance and a Q estimate.

Method, formulas and references

Inductance

The spiral is described by the number of turns n, the trace width w, the spacing s and the outer diameter d_out (across flats for the polygons). The inner diameter follows from these:

(1)d_in = d_out − 2n w − 2(n − 1)s, d_avg = (d_out + d_in)/2, ρ = (d_out − d_in)/(d_out + d_in)

Mohan et al. give two simple expressions for the DC inductance. The first is a modified Wheeler formula:

(2)L_mw = K₁ μ₀ n² d_avg / (1 + K₂ ρ)

The second comes from a current-sheet approximation:

(3)L_gmd = μ₀ n² d_avg c₁/2 · [ ln(c₂/ρ) + c₃ρ + c₄ρ² ]

Layout	K1	K2	c1	c2	c3	c4
Square	2.34	2.75	1.27	2.07	0.18	0.13
Hexagonal	2.33	3.82	1.09	2.23	0.00	0.17
Octagonal	2.25	3.55	1.07	2.29	0.00	0.19
Circular	—	—	1.00	2.46	0.00	0.20

The coefficients are Tables I and II of the paper, checked against the published PDF. The authors report typical errors of 2–3 % against the ASITIC field solver. The current-sheet error rises as s/w grows, with a maximum of 8 % for s ≤ 3w. The fit covered d_in from 0.1 to 0.9 d_out. The expressions were derived for on-chip spirals, but they contain only length ratios and ignore conductor thickness, so they scale to PCB dimensions. A copper plane close under the spiral carries eddy currents and lowers the inductance. The expressions do not include this effect, so keep planes away from the coil or model it with a field solver.

Resistance and Q

Turn k has a centre-line diameter of d_out − w − 2k(w + s). The average over all turns is exactly d_avg, so the trace length is n · P · d_avg. Here P = N tan(π/N) is the perimeter of a regular N-gon per unit across-flats diameter: 4 for a square, 3.464 for a hexagon, 3.314 for an octagon and π for a circle. The leads to the centre are not included. Resistance is ρ_Cul/(w t) with ρ_Cu = 1.7241 µΩ·cm at 20 °C. The quality factor Q = ωL/R_DC is optimistic. Skin and proximity effects raise the resistance well above its DC value at MHz frequencies, and the self-resonance of the coil is not modelled.

References

S. S. Mohan, M. del Mar Hershenson, S. P. Boyd, T. H. Lee, “Simple Accurate Expressions for Planar Spiral Inductances”, IEEE J. Solid-State Circuits, vol. 34, no. 10, pp. 1419–1424, Oct. 1999.
H. A. Wheeler, “Simple inductance formulas for radio coils”, Proc. IRE, vol. 16, no. 10, pp. 1398–1400, Oct. 1928.

Related guides

Planar Inductor Design: PCB Spiral Inductance, Resistance and Q

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