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Differential Via Calculator

Calculate differential impedance and insertion loss for a via pair, including the stub resonance that dominates a via transition long before conductor and dielectric loss matter.

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Formula

Zodd=η0εr(Cself+2Cm),Cselfε=2πln⁡(D/d),Cmε=πarccosh(p/d),fres=c4LstubεrZ_{odd} = \frac{\eta_0}{\sqrt{\varepsilon_r}\left(C_{self} + 2C_m\right)},\qquad \frac{C_{self}}{\varepsilon} = \frac{2\pi}{\ln(D/d)},\qquad \frac{C_m}{\varepsilon} = \frac{\pi}{\mathrm{arccosh}(p/d)},\qquad f_{res} = \frac{c}{4 L_{stub}\sqrt{\varepsilon_r}}

Reference: Two-wire and coaxial impedances are standard results; the stub loading follows from S₂₁ of a shunt admittance. A via transition is three-dimensional, so this is a screening model — see the linked full-wave tool.

Z_diff— Differential impedance of the via pair (Ω)
d— Finished barrel diameter (mm)
D— Antipad diameter (mm)
p— Via pitch, centre to centre (mm)
L_stub— Unused barrel below the exit layer (mm)
η₀— Impedance of free space, 376.730313412 Ω (exact SI) (Ω)

How It Works

A via transition is the least controlled part of an otherwise controlled-impedance channel. The trace is held to a few percent; the via is whatever the drill and the antipad happen to produce. This calculator gives the two numbers that decide whether that matters: the differential impedance of the pair, and the insertion loss it costs across a band.

The impedance comes from one consistent problem decomposed into modes. Each barrel has a coaxial capacitance to its own antipad, and the two barrels have a mutual capacitance to each other. The even mode puts both barrels at the same potential, so the mutual capacitance carries no charge and the even-mode impedance is simply the isolated coaxial value. The odd mode charges it, so the odd-mode impedance is always lower. That ordering holds by construction rather than by coincidence, and both modes converge as the pitch opens.

The loss is dominated by something that has nothing to do with the via's own resistance. Below the layer where the signal leaves, the barrel carries no signal but is still connected — an open-circuited stub hanging off the through path. A shunt admittance across a line produces a transmission notch when its length reaches a quarter wavelength, and for a stub in a typical board that happens at a frequency well below where conductor and dielectric loss become interesting. Conductor and dielectric loss through 1.6 mm of barrel amount to a fraction of a decibel even at 20 GHz; the stub can take tens.

That is why backdrilling exists. Removing the stub removes the notch, and what is left is the small loss the via itself costs.

One limit is worth stating plainly. The mutual term assumes the two barrels share a single clearance opening, which is what a differential via pair is. Space them further apart than the antipad and plane copper comes between them, screening the coupling — the calculator says so rather than pretending the two-wire term still applies.

This is a lumped screening model, not a field solution. A real via transition includes pad capacitance, plane-cavity resonance and the launch discontinuity, none of which are here. Use it to decide whether a geometry deserves a full-wave simulation, then run one.

Worked Example

Take a 0.25 mm finished barrel, 0.9 mm antipad, 0.8 mm pitch, in a 1.6 mm board of FR4 at εr = 4.2 with tan δ = 0.02. The signal enters at the top and leaves at 0.4 mm depth. Band of interest: 0.1 to 20 GHz.

Impedance first. The coaxial capacitance of one barrel to its antipad gives an even-mode impedance of 37.48 Ω, which is also what a single via would present on its own. Adding the mutual capacitance to the odd mode brings it down to 22.05 Ω, so:

Z_diff = 2 × 22.05 = 44.10 Ω Z_common = 37.48 / 2 = 18.74 Ω

That 44 Ω is the point of the exercise. A 100 Ω differential channel passing through a 44 Ω via sees a substantial discontinuity, and no amount of trace tuning fixes it.

Now the stub. The signal exits at 0.4 mm in a 1.6 mm board, leaving 1.2 mm of unused barrel. Its quarter-wave resonance lands at 30.48 GHz — above the 20 GHz band edge, so the notch itself is outside the band, but its skirt is not.

Across 0.1 to 20 GHz the worst insertion loss is 2.266 dB, at the top of the band. Of that, dielectric loss through the barrel contributes 0.119 dB and conductor loss 0.030 dB. The remaining ~2.1 dB is the stub. The via's own material loss is under a sixth of a decibel; everything else is the piece of copper that is not carrying signal.

Set the exit depth to 1.6 mm — a fully backdrilled or bottom-exiting via — and the stub disappears along with almost all of the loss.

Practical Tips

  • ✓Compare the worst in-band insertion loss with and without the stub before deciding whether backdrilling is worth its cost. If the resonance is far above the band, it may not be.
  • ✓Move the transition to a deeper layer if you can — it shortens the stub for free and moves the resonance up.
  • ✓Look at the differential impedance alongside the trace's. The mismatch, not the absolute value, is what the channel sees.
  • ✓Antipad diameter is the most effective impedance knob here, and the one your fabricator is most willing to change.
  • ✓Where the antipads overlap into a single oblong opening, treat the common-mode figure as optimistic — the calculator flags this.
  • ✓Use the linked full-wave S-parameter tool for any transition that this model puts close to a limit.

Common Mistakes

  • ✗Budgeting via loss from conductor and dielectric loss alone. Those are the small terms. The stub is the large one, and it is resonant rather than gradual.
  • ✗Assuming a via is electrically short because it is physically short. A 1.2 mm stub in FR4 is a quarter wavelength at 30 GHz, and its effect is felt long before that.
  • ✗Treating differential via impedance as close to the trace impedance. A via pair is typically well under half the differential impedance of the trace it interrupts.
  • ✗Spacing the vias apart to raise the differential impedance without checking whether the antipads still overlap. Once plane copper comes between them, the model no longer applies and the calculator says so.
  • ✗Enlarging the antipad without limit. It raises the impedance toward the trace value, but it also removes plane copper, which is a return-path problem of its own.
  • ✗Taking this model as a substitute for a full-wave simulation on a channel that is genuinely marginal.

Frequently Asked Questions

Because it is resonant. Conductor and dielectric loss in a 1.6 mm barrel are a small fraction of a decibel even at 20 GHz, while a quarter-wave stub puts a deep notch exactly where it resonates. A via that transitions on an upper layer of a thick board leaves a long stub and can notch inside your band.
It is a screening model. The impedances are standard results for the idealised geometry, but a real via transition includes pad capacitance, plane-cavity effects and the launch discontinuity, none of which are here. Use it to decide what to simulate.
It removes the stub, which removes the notch. Set the exit depth equal to the board thickness to see the case with no stub at all — the remaining insertion loss is what the via itself costs, and it is usually small.
Because in the even mode both barrels are at the same potential, so no charge flows into the capacitance between them. The mutual term drops out entirely and each barrel behaves as if the other were not there.
Plane copper comes between the two barrels and screens them from each other. The two-wire mutual term no longer describes the geometry, so the calculator flags it — the reported coupling will be too strong and the differential impedance too low.

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