PCB trace inductance: partial and loop inductance, the formulas, why the plane matters, and a worked example
A trace is an inductor of about a nanohenry per millimetre when it stands alone, and a far smaller one when a ground plane runs beneath it. The difference is the loop: inductance belongs to the circuit the current makes, out along the trace and back along the return, and the plane puts the return right underneath. This calculator gives both figures from the trace's length, width and copper weight and the height over a plane, the reactance at a frequency, the frequency where it reaches an ohm, and the voltage a current edge drops along the trace, with the trace drawn from above with its field and from the side with its loop.
How to use the trace inductance calculator
- Type the trace's length and width and pick the copper weight in the menu row: that gives the inductance of the trace on its own.
- Add the height over a plane (the dielectric between the trace and its ground) for the loop inductance with the return path; the substrate sets εr.
- Add a frequency for the reactance and a current step with its rise time for the voltage along the trace. The Length table tab compares the common lengths.
Partial and loop inductance
Strictly, only a closed loop has an inductance: the magnetic energy lives in the area between the outgoing and the returning current. A single trace's "inductance" is its partial inductance, the share of the loop that this conductor contributes if the return is far away, and it is what the lone-trace formula gives. Bring the return close, as a plane under the trace does, and the loop shrinks to the dielectric's height; the inductance falls to a fraction. Both figures are useful: the partial one for comparing leads, vias and pieces of a path, the loop one for what a signal or a decoupling current actually meets.
The formulas
Alone: L = 0.2 l [ln(2l/(w+t)) + 0.5 + 0.2235 (w+t)/l] nH Over a plane: L = (Z₀ √εeff / c) · l
The first is Grover's formula for a flat rectangular conductor (l, w and t in millimetres), good to a few percent when the trace is much longer than it is wide. The second comes from transmission-line theory: a trace over a plane is a microstrip, and its inductance per length is its impedance times its delay, Z₀ √εeff / c, with Z₀ and εeff from the Hammerstad–Jensen formulas used in the microstrip calculator. Multiply by the length for the loop. Both ignore skin effect, which at high frequency pushes the current to the conductor's surface and trims the internal inductance a little.
Why the plane matters
A 10 mm trace alone is about 10 nH; over a plane 0.2 mm below it is about 3 nH, and the loop's magnetic field is confined under the trace rather than spread around it, so it neither radiates nor couples into its neighbours. A slot in the plane, a split between ground and power, or a layer change without a return via forces the return current to detour, opens the loop and brings the lone-trace figure back. The rules of high-speed layout, an unbroken plane beneath every fast signal and a return via beside every layer change, are all about keeping the loop small.
Where it bites
Decoupling: the trace from a capacitor's pad to the chip's pin is in series with the capacitor and sets the frequency where it stops working; keep it under a millimetre or two. Gate drives: a few centimetres of trace between a driver and a MOSFET gate ring with the gate capacitance and slow the switching. Current sense and ground: an amp switching in a nanosecond through 5 nH drops 5 V, which is why ground bounce and supply dips exist. Signal integrity: beyond a few nanohenries a lead starts to look like a transmission line, and that is a different calculation.
Your trace, step by step
- Alone: 0.2 × 10 × [ln(70.2) + 0.5 + …] = 9.52 nH, 952 pH per mm.
- Over the plane 0.20 mm below: Z₀ 59.5 Ω, εeff 3.19 → 354 pH per mm → 3.54 nH (37% of the lone figure).
- Reactance: 1 Ω at 44.9 MHz; 2.23 Ω at 100 MHz.
- Edge of 1 A in 1 ns: 3.54 V along the trace.
Worked example: 10 mm of 0.25 mm trace, 1 oz copper
Alone, with w + t = 0.285 mm, 2l/(w+t) = 70 and ln 70 = 4.25, so L = 0.2 × 10 × [4.25 + 0.5 + 0.006] = 9.52 nH: the nanohenry-per-millimetre rule. Over a plane 0.2 mm below on FR-4 the trace is a 59.5 Ω microstrip with εeff 3.19, 354 pH per millimetre, so the loop is 3.54 nH, 37% of the lone figure; on a two-layer board with the plane 1.6 mm away it would be 7.44 nH. At 100 MHz the loop's reactance is 2.23 Ω, and an amp switching in a nanosecond drops 3.54 V along it, against 9.52 V for the trace alone.
Questions
Why does the width hardly matter?
For a lone trace the width enters only through a logarithm, so quadrupling it removes about 0.28 nH per millimetre. Over a plane width matters more, because the microstrip's inductance per length falls as the trace widens, but the dielectric height matters most.
Which figure do I use for a decoupling capacitor?
The loop over the plane, for the trace from the pad to the pin, plus the via inductance and the capacitor's own; the three in series set where the capacitor stops working.
Does skin effect change it?
A little: at high frequency the current hugs the surface and the small internal inductance disappears, a few percent at most. The loop's geometry dominates.
When is a trace a transmission line instead?
When the signal's edge is shorter than about three times the trace's delay (6 ps per millimetre on FR-4): then a lumped inductance no longer describes it and the microstrip calculator's impedance does.