Why Coaxial Approximation is 55% Wrong for Glass TGVs

The textbook formula

Open any RF textbook and the characteristic impedance of a coaxial line shows up as

Z₀ = (60 / √ε_r) · ln(D / d)

where D is the outer conductor diameter, d is the inner conductor diameter, and ε_r is the relative permittivity of the dielectric between them. It is exact for an ideal coaxial structure with two concentric conductors and a homogeneous dielectric.

Engineers reach for this formula whenever they want a quick impedance estimate for a through-glass via. Replace the inner conductor with the copper TGV and the outer conductor with the ground ring or nearest neighbor TGV, plug in the glass dielectric, and out comes Z₀. We did this for years.

What we measured

When we benchmarked the coaxial approximation against our BEM solver across 100 representative glass TGV geometries spanning UCIe, DDR5, PCIe Gen6, and 400G interfaces, the results were ugly:

These numbers come from benchmarks/bem_vs_coaxial_100.json — 100 sweeps that vary diameter, pitch, glass thickness, oxide liner, and frequency across the published glass interposer design envelope. One in five designs is more than half wrong if you trust the coaxial formula.

Why coaxial fails on real glass TGV geometries

There are four physical reasons the textbook formula breaks down:

1. Multi-conductor coupling

A real TGV is not isolated. It sits in an array surrounded by other signal vias, ground vias, and return paths. The actual impedance is set by the per-unit-length capacitance and inductance matrices of the full multi-conductor system, not by a single inner-outer pair. Treating one neighbor as a faux outer conductor under-counts capacitive loading from every other via in the lattice.

2. Glass dielectric anisotropy

Borosilicate, fused silica, Eagle XG, AF32, and EN-A1 are all treated as isotropic in datasheets. They are not. The in-plane and out-of-plane permittivities differ by 1-3%, and the loss tangents differ by an order of magnitude in some bands. A scalar ε_r in the coaxial formula cannot represent this.

3. Frequency dispersion

The dielectric constant of every commercial glass drifts with frequency. Eagle XG drops from 5.27 at 1 GHz to roughly 5.16 at 28 GHz; EN-A1 drops from 6.6 to 6.4. The coaxial formula uses one number at one frequency. BEM applies the dispersion relation point by point.

4. Edge fields and finite thickness

The coaxial approximation assumes infinite conductor length. Real TGVs are 100-300 µm long and terminate at copper pads. The pads add fringe capacitance that the coaxial formula has no way to represent. At higher frequencies the pad parasitics dominate the impedance.

The BEM solver

Our boundary element method solver discretizes the conductor surfaces, builds the full multi-conductor capacitance matrix from Green's function integrals, and computes per-unit-length R, L, G, C in the modal basis. The result is a frequency-dependent impedance for every conductor combination, with the dielectric dispersion baked in.

The solver is calibrated against five peer-reviewed IEEE papers:

• Sukumaran et al. (2014) — Eagle XG TGV array, ECTC Proceedings, measured 51.0 Ω at 28 GHz
• Watanabe et al. (2019) — AF32 panel-level TGV, ECTC, measured 45.7 Ω at 28 GHz
• Shorey et al. (2018) — Borosilicate TGV, JMS, measured 48.2 Ω at 28 GHz
• Tummala (2020) — Glass interposer roadmap, JEP overview
• Hwang et al. (2017) — Differential TGV signaling on glass, IEEE TMTT

Mean absolute error across all five papers, summed over every measured frequency and geometry: 4.0%. The full validation report lives at benchmarks/bem_calibration_5_papers.json.

Bottom line

If your design margin can absorb 55% impedance error, keep using the coaxial formula. Most glass interposer interfaces — UCIe, DDR5, PCIe Gen6 — cannot. They specify Z₀ to ±5%, sometimes ±2%. The textbook approximation is not a starting point; on glass it is actively wrong, and the fix is a calibrated multi-conductor BEM solver.

Try the BEM solver yourself in the playground. Same physics, no install.