Every fast interface on a modern board — USB, HDMI, PCIe, Ethernet, SATA — sends its data as a differential pair: two traces carrying equal and opposite signals. The receiver listens to the difference between them, which rejects noise beautifully. But it only works if the pair presents the right differential impedance to the signal. USB asks for 90 ohms, HDMI and most other high-speed standards for 100 ohms. Miss that target and the edges reflect, the eye diagram closes, and a link that looked fine in simulation fails on the bench.

The catch for newcomers is that differential impedance is not a property of one trace. It emerges from how two coupled traces sit in their geometry. This article explains where that number comes from and how to design for it.

Why this calculation matters

A differential pair is a transmission line, and a transmission line only passes a signal cleanly when its impedance matches the source and load. For a pair, the relevant quantity is the differential impedance Z_diff — the impedance the line presents to the difference signal.

Standards specify it tightly. USB 2.0 high-speed wants 90 ohms differential; HDMI, DisplayPort, PCIe, and SATA cluster around 100 ohms, typically held to plus or minus 10 to 15 percent. "Controlled impedance" means the fabricator builds the stack-up so the pair lands on target — and that only happens if the trace width and spacing were chosen against the real stack-up in the first place. Get it wrong and you are paying for a board respin.

The core method

Start with a single trace. On an outer layer, a trace over a ground plane is a microstrip, and on its own it has a single-ended characteristic impedance Z_0 set by the trace width w, the dielectric height h, and the dielectric constant er. A widely used closed form (Hammerstad) for w/h greater than 1 is:

e_eff = (er + 1)/2 + (er - 1)/2 * 1 / sqrt(1 + 12 h / w)

Z_0 = 120 pi / ( sqrt(e_eff) * [ w/h + 1.393 + 0.667 * ln(w/h + 1.444) ] )

Z_0 falls as the trace gets wider and rises as the dielectric gets thicker. That single-trace number is the foundation.

Now bring a second identical trace alongside it. The two traces couple — each one's field reaches the other — and that splits the behaviour into two modes:

• Odd mode: the traces driven oppositely, which is the real differential signal. Its impedance is Z_odd.
• Even mode: the traces driven together, the common-mode case. Its impedance is Z_even.

The differential impedance is simply twice the odd-mode impedance:

Z_diff = 2 * Z_odd

Here is the key effect of coupling: bringing the traces closer lowers Z_odd below the isolated Z_0. Tighter spacing means stronger coupling means lower differential impedance. Spacing is a design knob, not an afterthought.

A worked example

Take a microstrip pair on standard FR-4: er = 4.4, dielectric height h = 1.6 mm. First size a single trace, with width w = 3.0 mm:

w/h   = 1.875
e_eff = 2.70 + 1.70 * 1/sqrt(1 + 12 x 1.6/3.0) = 3.33
Z_0   = 120 pi / ( sqrt(3.33) x 4.07 ) = 50.8 ohms

So one isolated trace is about 51 ohms. If the two traces of the pair were placed far apart — spacing several times the dielectric height — coupling would be negligible, Z_odd would stay close to Z_0, and:

Z_diff  ~  2 * Z_0  ~  102 ohms

That is already close to the 100-ohm HDMI target. But pairs are rarely routed that loosely; they are kept close so they stay matched and reject noise together. As you tighten the spacing toward the dielectric height, coupling grows, Z_odd drops below 51 ohms, and Z_diff falls below 100 ohms. To pull it back up you narrow the traces. That trade — width down, or spacing up, to raise Z_diff — is the entire routing decision, and it depends on the exact dielectric height of the layer you are on.

Common mistakes

Designing impedance without the stack-up. The width and spacing that give 100 ohms depend on the dielectric thickness of the specific layer. Get the stack-up from the fabricator first, then size the pair. Doing it the other way around means a respin.

Treating spacing as cosmetic. Intra-pair spacing directly sets the coupling, and therefore Z_diff. It is a controlled dimension, not a routing convenience.

Trusting the nominal dielectric constant. FR-4's er is not a fixed 4.4 — it varies with the resin-to-glass ratio, with frequency, and between suppliers. For a tight tolerance, use the laminate datasheet value at your operating frequency.

Breaking the reference plane. The impedance model assumes a continuous ground plane under the pair. Route over a plane split or gap and the return current has nowhere to go; the local impedance jumps and the calculation no longer applies.

Ignoring intra-pair skew. If the two traces differ in length, the signal arrives skewed and some differential energy converts to common mode — radiated noise. Length-match the pair.

Try the interactive NovaSolver calculator

Sizing a pair by hand means juggling width, spacing, height, and dielectric constant at once — easy to get wrong, tedious to iterate. The differential pair impedance calculator on NovaSolver computes the single-ended Z_0, the differential Z_diff, and the odd- and even-mode impedances of a surface microstrip pair directly from trace width, spacing, substrate height, and dielectric constant — and shows them against the USB 90-ohm and HDMI 100-ohm targets as you adjust the geometry.

Related calculators

• Coplanar waveguide — the grounded-coplanar variant, common when you need tighter field confinement.
• Stripline — for buried inner-layer traces sandwiched between two planes.
• Transmission line behaviour — to see how an impedance mismatch turns into reflections and standing waves.

Browse the full set in the RF and microwave tools hub.

Closing note

Differential pair impedance is a reminder that on a high-speed board, the geometry is the circuit. A 100-ohm pair is not a property of the copper — it is the combined result of trace width, spacing, dielectric height, and material, all working over a solid reference plane. Size the pair against the real stack-up, treat spacing as the controlled dimension it is, respect the dielectric tolerance, and keep the ground intact beneath the route. Do that and USB, HDMI, and PCIe links stop being a gamble and become ordinary, repeatable engineering.