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Edge-Coupled Internal Symmetric Stripline Calculator

Calculate differential, common-mode, odd- and even-mode impedance for an edge-coupled differential pair centred between two reference planes. Exact conformal-mapping solution, not an empirical fit.

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Formula Reference

Z_{0e,0o} = \frac{\eta_0/4}{\sqrt{\varepsilon_r}}\cdot\frac{K(k')}{K(k)},\qquad k_e = \tanh\!\frac{\pi W}{2b}\tanh\!\frac{\pi (W+S)}{2b},\qquad k_o = \frac{\tanh\frac{\pi W}{2b}}{\tanh\frac{\pi (W+S)}{2b}}

Ref: S. B. Cohn, "Shielded Coupled-Strip Transmission Line", IRE Trans. MTT-3 (1955), pp. 29–38. Verified against a 2-D method-of-moments solve to 0.008%.

Z_diff— Differential impedance, 2·Z_odd (Ω)

Z_common— Common-mode impedance, Z_even/2 (Ω)

W— Trace width (mm)

S— Edge-to-edge spacing (mm)

b— Plane-to-plane spacing (mm)

K— Complete elliptic integral of the first kind

Formula

Z_{0e,0o} = \frac{\eta_0/4}{\sqrt{\varepsilon_r}}\cdot\frac{K(k')}{K(k)},\qquad k_e = \tanh\!\frac{\pi W}{2b}\tanh\!\frac{\pi (W+S)}{2b},\qquad k_o = \frac{\tanh\frac{\pi W}{2b}}{\tanh\frac{\pi (W+S)}{2b}}

Reference: S. B. Cohn, "Shielded Coupled-Strip Transmission Line", IRE Trans. MTT-3 (1955), pp. 29–38. Verified against a 2-D method-of-moments solve to 0.008%.

Z_diff— Differential impedance, 2·Z_odd (Ω)

Z_common— Common-mode impedance, Z_even/2 (Ω)

W— Trace width (mm)

S— Edge-to-edge spacing (mm)

b— Plane-to-plane spacing (mm)

K— Complete elliptic integral of the first kind

η₀— Impedance of free space, 376.730313412 Ω (exact SI) (Ω)

How It Works

An edge-coupled differential pair routed on an inner layer sits between two reference planes, side by side in the same plane. That geometry is one of the few in transmission-line theory with an exact answer: two zero-thickness coplanar strips centred between grounded planes can be mapped conformally onto a parallel-plate capacitor, so the even- and odd-mode capacitances come out as complete elliptic integrals rather than as a curve fit.

The two modes differ only in what sits on the symmetry plane between the traces. Drive them out of phase and that plane behaves as an electric wall; drive them in phase and it behaves as a magnetic wall. Everything else about the geometry is unchanged, which is why the even- and odd-mode moduli differ only by a tanh versus its reciprocal.

What the pair does with those two modes is what matters in practice. A differential receiver sees twice the odd-mode impedance, so Z_diff = 2·Z_odd. A common-mode disturbance sees two even-mode lines in parallel, so Z_common = Z_even/2. As the spacing opens up, both moduli converge on the single-trace value and the pair stops being a pair — Z_diff simply becomes twice the single-ended impedance.

Because a stripline is surrounded by one dielectric, both modes travel at the same speed. That is the quiet advantage of routing a pair internally: there is no mode-conversion skew accumulating along the length, which a surface pair with air above it cannot avoid.

The scale factor deserves a note. Older stripline literature carries 30π = 94.2478, which comes from taking the permittivity of free space as 0.0885 pF/cm. Derived from the exact SI constants the value is η₀/4 = 94.1826 — a 0.07% difference that is small but entirely avoidable.

Worked Example

Take a 0.15 mm pair on 0.2 mm spacing, centred in a 0.6 mm gap between planes, 1 oz copper (35 µm), FR4 at εr = 4.2.

First the reference: the same 0.15 mm trace on its own between those planes has a single-ended impedance of 56.07 Ω. If the two traces were far apart, the differential impedance would be exactly twice that, 112.15 Ω.

They are not far apart. At 0.2 mm spacing the elliptic-integral solution gives an odd-mode impedance of 49.18 Ω and an even-mode impedance of 62.31 Ω. So:

Z_diff = 2 × 49.18 = 98.36 Ω Z_common = 62.31 / 2 = 31.15 Ω

The coupling coefficient (Z_even − Z_odd)/(Z_even + Z_odd) works out to 0.1177, which is the honest measure of how much the traces are actually talking to each other.

That 98.36 Ω is close to a 100 Ω target but not on it. Opening the spacing raises Z_diff toward 112.15 Ω and narrowing it lowers it, so the spacing is the knob — but note that the useful range is bounded: past roughly three trace widths of gap there is almost no coupling left to trade.

Propagation delay is 6.836 ps/mm for both modes, because the dielectric is homogeneous.

Practical Tips

✓Set trace width for the single-ended impedance you want first, then open the spacing until Z_diff reaches target. The two knobs are not independent, but that order converges quickly.
✓If the required spacing comes out under about two trace widths, check the crosstalk budget to neighbouring nets as well — a tightly coupled pair is also a pair that couples to whatever runs beside it.
✓Compare the coupling coefficient across candidate stack-ups. A pair with weak coupling is more tolerant of etch variation, because Z_diff depends less on the spacing.
✓Ask the fabricator which layer pair they will actually use before committing. The plane spacing you assume and the one they build to are frequently different.
✓Where the stack-up is not symmetric, use the asymmetric calculator instead — an offset pair between the same planes can be 10–20% away from the centred value.
✓Keep the pair on one layer for its whole length. Every layer change adds a via transition whose differential impedance is far below the trace's.

Common Mistakes

✗Assuming Z_diff is simply twice the single-ended impedance. That is the uncoupled limit, and a real pair is coupled on purpose — in the example above it costs 13.8 Ω, which is well outside any fabricator's impedance tolerance.
✗Setting the spacing first and the width second. Width sets the single-ended impedance and spacing trades against it; fixing the spacing first leaves you solving for width against a moving target.
✗Using a surface-microstrip pair calculator for an inner-layer pair. Placement changes the answer by more than the tolerance a controlled-impedance shop works to.
✗Reading the plane spacing off the stack-up as the dielectric thickness. It is the plane-to-plane distance, which includes the copper of the trace layer itself.
✗Chasing the last ohm. Fabricators typically hold ±10% on controlled impedance, so a model that lands within a couple of ohms is already finer than the board will be.

Frequently Asked Questions

Why is this calculator exact when most impedance formulas are approximations?

Because this particular geometry admits a conformal map. Two zero-thickness coplanar strips centred between two ground planes transform exactly into a parallel-plate region, so the mode capacitances are elliptic integrals rather than a fit to solver data. Only the copper-thickness correction is approximate.

How does an inner-layer pair compare with a surface pair?

A stripline pair is surrounded by one dielectric, so both modes travel at the same speed and no mode-conversion skew accumulates. A surface pair has air above it, the odd mode runs faster than the even mode, and the difference converts differential signal into common mode over length.

What spacing gives 100 Ω differential?

It depends on the width and the plane spacing, so solve it rather than memorise it. Set the width for the single-ended impedance you want, then open the gap until Z_diff reaches 100 Ω. Watch the coupling coefficient: once it is small, further spacing buys almost nothing.

Does copper thickness matter here?

Less than on a microstrip, but it is not negligible. The thickness adds fringing capacitance on the trace sidewalls, which lowers both mode impedances. At 2 oz copper on a thin core it is worth including, which this calculator does.

Why report the uncoupled impedance at all?

Because it is the yardstick. Comparing Z_diff against twice the uncoupled value tells you immediately how much of the answer comes from coupling, and therefore how sensitive it is to the spacing your fabricator actually etches.

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