Differential pair impedance: what Z_diff is, odd and even modes, the IPC-2141 formulas, and a worked example
USB, Ethernet, HDMI, LVDS and PCIe all send their signals as pairs: one trace carries the signal, the other its inverse, and the receiver reads the difference, which cancels noise picked up by both. The pair must present a set impedance between its traces, 90 or 100 Ω for most standards, and that impedance depends on the width of each trace, the gap between them and the dielectric below. This calculator gives Z_diff for an edge-coupled microstrip or stripline pair from those figures with the IPC-2141 coupling formulas, the odd, even and common-mode impedances, the gap and the width for a target, the delay and the wavelength, with the cross-section and its field drawn to scale.
How to use the differential pair calculator
- Choose microstrip or stripline in the menu row, with the substrate (it fills in εr) and the copper weight.
- Type the trace width, the gap between the traces' edges and the dielectric height to the plane (each plane, for a stripline).
- Add the target Zdiff your standard asks for: the calculator gives the gap at this width and the width at this gap, each usable with one click. The Gap table tab sweeps the gap.
What differential impedance is
Two traces side by side over a plane form two coupled transmission lines. Driven with opposite signals, each trace's field goes partly to the plane and partly across to the other trace, so each sees a lower impedance than it would alone, called the odd-mode impedance. The differential impedance, measured between the two traces, is twice that. The closer the traces, the more they couple and the lower Zdiff falls below twice the single-ended Z₀; far apart, the pair is just two independent lines and Zdiff = 2 Z₀.
Odd and even modes
Any pair of signals on the two traces can be split into a differential part (opposite, the odd mode) and a common part (equal, the even mode). The even mode has no field between the traces and sees a higher impedance; the two multiply to roughly the single-ended Z₀ squared. The common-mode impedance, half the even mode, matters for EMI: common-mode current on the pair radiates, which is why chokes and the receiver's common-mode rejection exist. A good pair keeps the odd mode constant along its length and gives the common mode nowhere to go.
The formulas
Microstrip: Zdiff = 2 Z₀ (1 − 0.48 e−0.96 s/h) Stripline: Zdiff = 2 Z₀ (1 − 0.347 e−2.9 s/b)
Z₀ is the single-ended impedance of one trace from the microstrip or stripline formulas, s the gap, h the dielectric height and b the plane spacing. The exponential terms are the fit in IPC-2141A (originally from National Semiconductor's LVDS notes) and are within about 10 % for gaps between a fifth and a few times the height; fabs and signal-integrity tools use a 2-D field solver for the last few percent. The even mode is estimated from Z₀² / Zodd.
Layout rules
Keep the gap constant along the whole run, route both traces together around every obstacle, and match their lengths within a small fraction of the rise time (a millimetre is about 6 ps on FR-4). Keep other traces at least three gaps away. Do not split the reference plane under the pair, and when the pair changes layers give its return current a path with a nearby via or capacitor. A gap about equal to the width is the usual start; loose coupling (a gap of two or three widths) makes Zdiff less sensitive to etch and spacing errors, tight coupling gives better noise rejection and denser routing.
Your pair, step by step
- One trace alone: Z₀ = 54.3 Ω (microstrip of 0.30 mm on 0.20 mm).
- Coupling: s/h = 1.50 → k = 0.114; Zdiff = 2 × 54.3 Ω × (1 − k) = 96.3 Ω.
- Modes: odd 48.2 Ω, even 61.3 Ω, common 30.7 Ω; 5.99 ps/mm.
- For 90.0 Ω: gap 0.21 mm at this width, or width 0.34 mm at this gap.
Worked example: a USB 2.0 pair on a 0.2 mm prepreg
USB wants 90 Ω differential. Two 0.3 mm traces with a 0.3 mm gap on 0.2 mm of FR-4 (εr 4.3) with 1 oz copper: one trace alone is 54.3 Ω; with s/h = 1.5 the coupling term is 0.114, so Zdiff = 2 × 54.3 Ω × (1 − 0.114) = 96.3 Ω, with an odd mode of 48.2 Ω and an even mode of 61.3 Ω. For exactly 90 Ω the gap would be 0.21 mm at this width, or the traces 0.34 mm wide at this gap. The pair runs at 5.99 ps/mm; the same traces buried as a stripline with 0.2 mm to each plane would be 66.2 Ω, which is why inner-layer pairs are narrower.
Questions
Why 90 Ω for USB and 100 Ω for Ethernet?
Each standard's cable: USB cable is 90 Ω differential, twisted pair 100 Ω, and the board traces match the cable so that nothing reflects at the connector. PCIe uses 85 Ω.
Does the single-ended impedance of each trace have to be 50 Ω?
No. A 100 Ω pair has an odd mode of 50 Ω, but each trace alone is a little higher, 54.3 Ω in the example; only an uncoupled pair would have 50 Ω traces.
How accurate is the formula?
About 10 %. For a controlled-impedance order, give the fab the target Zdiff and let them tune the gap and width with their field solver and real stack-up.
What about broadside-coupled pairs?
Traces stacked on adjacent layers couple differently and need another formula; most boards use edge-coupled pairs on one layer, which this calculator covers.