Differential Via

Differential, odd-, even- and common-mode impedance of a via pair through a reference plane, from a 2D field solution of the barrels in their antipad, plus the stub resonance and the delay through the board.

Method, formulas and references

Model

Where the via pair passes through a reference plane, the barrels and the edge of the antipad form a shielded two-conductor line. The tool solves Laplace's equation for this cross-section on a fine grid, with curved-boundary correction, and derives the odd- and even-mode impedances in the same way as the trace field solver:

(1)Z_diff = 2 Z_odd, Z_comm = Z_even / 2

The solver is validated against the exact coaxial result Z = (60/√εr)·ln(D/d) for a single via, and against the twin-wire limit for a pair in a very large clearance. Between planes a real via also couples to the nearest planes and to its return vias, so treat the value as the impedance of the plane crossings. For a full-length model, including pads and stubs, a 3D solver is required.

(2)f_stub = c / (4 · L_stub · √ε_r)

The unused part of the barrel below the exit layer is an open stub. It notches the channel at its quarter-wave frequency. Back-drilling removes it.

References

E. Bogatin, Signal and Power Integrity – Simplified, 3rd ed.
H. Johnson, M. Graham, High-Speed Signal Propagation: Advanced Black Magic, 2003.

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