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Differential pair impedance calculator

Differential impedance of two edge-coupled traces, on an outer layer (microstrip) or an inner layer (stripline). Also gives odd, even and common-mode impedance, and can find the trace width or gap for a target such as 90 Ω (USB) or 100 Ω (Ethernet, LVDS, PCIe).

How it is calculated

Coupled microstrip. Kirschning and Jansen's equations for the even- and odd-mode effective dielectric constants and impedances, with Jansen's thickness correction of the even and odd widths. The single-line reference uses Hammerstad and Jensen's model including copper thickness. Checked against a 2D field solver: the odd-mode impedance agrees within about 3 % for 1 oz copper (the original zero-thickness reference overestimates it by 5–10 %).

Coupled stripline. Cohn's exact conformal-mapping solution for zero-thickness coupled strips, Z0e,o = η0/(4√εr) · K(k′e,o)/K(ke,o), with ke = tanh(πw/2b)·tanh(π(w+s)/2b) and ko = tanh(πw/2b)·coth(π(w+s)/2b), corrected for thickness with Cohn's fringe-capacitance method. Offset pairs combine two centred pairs (image method) with Wadell's correction.

Definitions. Zdiff = 2·Zodd; Zcommon = Zeven/2; coupling coefficient k = (Zeven − Zodd)/(Zeven + Zodd). K(k) is the complete elliptic integral of the first kind, computed with the arithmetic-geometric mean.

Accuracy. Microstrip and centred stripline pairs: about ±3 %. Offset stripline pairs: about ±5 %. Confirm a controlled-impedance design with your manufacturer's field solver.

References

  1. M. Kirschning, R. H. Jansen, “Accurate Wide-Range Design Equations for the Frequency-Dependent Characteristic of Parallel Coupled Microstrip Lines”, IEEE Trans. MTT, vol. 32, no. 1, 1984 (corrections in vol. 33, 1985).
  2. R. H. Jansen, “High-Speed Computation of Single and Coupled Microstrip Parameters Including Dispersion, High-Order Modes, Loss and Finite Strip Thickness”, IEEE Trans. MTT, vol. 26, no. 2, 1978.
  3. S. B. Cohn, “Shielded Coupled-Strip Transmission Line”, IRE Trans. MTT, vol. 3, no. 5, 1955.
  4. B. C. Wadell, Transmission Line Design Handbook, Artech House, 1991, sections 3.6 and 3.7.

Questions

Why is the differential impedance not twice the single-ended impedance?

Coupling between the two traces lowers the odd-mode impedance. The closer the traces, the lower Zdiff is compared with twice the impedance of one trace alone.

Should I make the pair tightly or loosely coupled?

Tight coupling (gap about equal to the width) gives better noise rejection and smaller pairs. Loose coupling makes the impedance less sensitive to gap tolerance. Both work if the impedance is right and the pair stays symmetric.

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