Planar Spiral Inductors: Cheap Until You Need Q

What you get for free

A planar spiral is copper you were paying for anyway. No part number, no placement, no solder joint, no tolerance stack from a supplier. For RFID antennas, wireless power coils, VCO tanks, and bias-tee chokes it is often the right answer.

A five-turn square spiral, 10 mm outer diameter, 4 mm inner, 0.3 mm traces on 0.2 mm spacing, gives about 234 nH by the Mohan expression and 236 nH by Wheeler's. Two independent approximations agreeing within 1 % is a good sign the geometry is in the range both were fitted over.

That spiral has 110 mm of trace and about 0.18 Ω of DC resistance in 1 oz copper.

The current-sheet model

Mohan's expression is the one most modern tools use. It is a current-sheet approximation with coefficients fitted per shape — square, hexagonal, octagonal, circular:

Two geometric quantities drive it. The average diameter $d_{avg} = (d_{out} + d_{in})/2$ sets the scale. The fill ratio ρ = (d_out − d_in)/(d_out + d_in) describes how much of the disc is actually spiral rather than empty middle. For the example above, d_avg = 7 mm and ρ = 0.43.

The fill ratio is the interesting parameter. A tightly wound spiral with a small hole has a high fill ratio and packs more turns into the area, but the innermost turns enclose very little flux while contributing full resistance. A hollow spiral with a large inner diameter has a low fill ratio and every turn is doing useful work.

That is why hollow spirals give better Q for the same inductance, and why the naive instinct to fill the whole area with copper is wrong.

Inductance scales as turns squared

Inductance goes as the square of turn count. Doubling the turns quadruples the inductance — but it also roughly doubles the trace length, so resistance doubles.

Q is ωL/R, so on paper more turns raise Q. In practice they do not, for two reasons that the closed-form expressions do not capture:

• Turn-to-turn capacitance grows with turn count and adjacency, dragging the self-resonant frequency down.
• Current crowding at high frequency pushes current to the inner edge of each turn, so the effective resistance rises faster than the DC figure suggests.

Both effects punish exactly the geometry that the inductance formula rewards.

Self-resonance is the real limit

Every spiral has distributed capacitance between adjacent turns and to whatever plane is below it. That capacitance resonates with the inductance, and above that frequency the structure is a capacitor, not an inductor.

The example spiral self-resonates around 68 MHz. That is a hard ceiling. You can use it at 10 MHz comfortably and at 30 MHz with care. At 100 MHz it is not an inductor at all.

This is the number to check first, before inductance and long before Q. A Q figure computed at a frequency above SRF is arithmetic without physics behind it — ωL/R keeps producing large numbers as frequency rises, but the reactance it assumes stopped being inductive. Treat any Q estimate as valid only well below SRF, and treat SRF itself as an estimate that a ground plane underneath will lower further.

What kills Q in practice

A ground plane directly beneath. It raises capacitance, lowering SRF, and it supports eddy currents that dissipate real power. If you need the spiral to perform, void the plane under it — and accept the EMC consequence of a routing gap. Skin effect. At 100 MHz the skin depth in copper is about 6.5 µm. A 35 µm thick trace is using roughly a third of its cross-section. The DC resistance figure is optimistic by a factor that grows as the square root of frequency. Substrate loss. FR-4 has a dissipation factor around 0.02. For a high-Q tank that is the dominant loss above a few hundred MHz, and no amount of copper geometry fixes it. This is where the laminate choice stops being about impedance control.

When to use one

Good fit: RFID and NFC coils, wireless power, bias chokes below SRF, VCO tanks where a Q of 20 to 40 is acceptable, and anywhere a discrete inductor's height is the problem. Bad fit: switching converter output inductors — the current and inductance are both orders of magnitude beyond what a spiral delivers. High-Q filters above a hundred MHz on FR-4. Anything needing tight inductance tolerance, since spiral inductance depends on etch accuracy and the copper thickness tolerance is typically ±10 %. Always: simulate or measure before committing. The closed-form models are good to a few percent on inductance and much rougher on Q and SRF, because those depend on the surroundings the formula knows nothing about.

The planar spiral inductor calculator reports Mohan and Wheeler side by side, plus fill ratio, trace length, DC resistance, and an SRF estimate for square, hexagonal, octagonal, and circular geometries.