Planar Spiral Nanoinductor — On-Chip RF Inductance & Q-Factor

Coil Structure

Geometry

Turns N 5

Outer diameter 180 µm

Drive frequency 2.0 GHz

Live Readouts

Inductance L

— nH

AC resistance R

— Ω

Quality factor Q

—

Self-resonance f₀

— GHz

How it works

A planar spiral on-chip inductor is modelled with the modified Wheeler formula, the standard closed-form estimate used in RF chip design:

L = K1·μ0·N²·d_avg / (1 + K2·ρ)
d_avg = (d_out + d_in)/2
ρ     = (d_out − d_in)/(d_out + d_in)
square coil: K1=2.34, K2=2.75

A vertical solenoid (stacked 3D turns, the "3D inductor" family) instead follows the classic long-solenoid formula, trading chip area for height to recover the inductance lost when a spiral is shrunk below ~100 nm-scale trace pitch:

L = μ0·N²·A / l_coil,  A = π(d_out/2)²

Copper trace resistance rises with frequency because current crowds into a thin skin depth δ near the conductor surface:

δ = √(2ρ_Cu / (ω·μ0)),  ω = 2πf
R_ac ≈ R_dc · max(1, t / (2δ))

The quality factor Q = ωL/R_ac peaks at a moderate frequency: too low and resistive loss dominates the ratio, too high and skin-effect resistance eats the gain from a bigger ωL. A small parasitic capacitance between the innermost and outermost turns forms a self-resonant frequency f₀ = 1/(2π√(LC)), above which the "inductor" behaves capacitively instead — the practical usable ceiling for any real spiral.

Coil structure — switch between a flat on-chip spiral and a stacked vertical solenoid; same footprint, different inductance-per-area trade-off.
Turns / diameter — geometry inputs to the formulas above; more turns and larger diameter both raise L but add resistive wire length.
Drive frequency — sets ω for the skin-effect resistance and Q calculation; the moving particles on the coil represent the RF current.

Copper trace Current flow Magnetic field B

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