2D Transmission Line Field Solver

by Henrik Forstén | Source code & issues

Version: 2026-10-19 (b5ea9eb)

A quasi-static and full-wave 2D field solver for transmission line analysis with adaptive mesh refinement.

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Methodology

Quasi-static solver is a finite-difference solver on a non-uniform rectilinear grid. It solves Laplace's equation for the electric potential twice, with the dielectrics and in vacuum. The two solutions give the capacitance C and the inductance L = 1/(c²C₀), from which the characteristic impedance and effective permittivity can be calculated. Dielectric loss G is integrated from the field in the lossy materials. Conductor loss is calculated from surface current on each face from the static field and weighted by the surface impedance of the metal. The static fields don't change with frequency, so dispersion is not modeled. That is strictly valid only for lines in a uniform dielectric such as stripline, but it is accurate for many practical structures.

Full-wave solver uses a triangular finite-element mesh. It solves the guided mode of Maxwell's equations as an eigenvalue problem with Nédélec elements. It captures the dispersion the quasi-static solver misses and gives more accurate results, but it's also much slower.

Full-wave conductor loss uses a magneto-quasi-static (MQS) solve. Instead of treating the metal as a surface impedance, it meshes the conductor interior and solves for the current distribution inside it. This gives accurate resistance for any ratio of skin depth to conductor thickness, including thin films where the current flows through the whole conductor.

Rectangular Waveguide

The rectangular waveguide is not TEM or quasi-TEM. It has a single conductor, so there is no static solution. It is solved as a pure eigenmode problem on the full-wave backend, and only the lowest (fundamental) mode is reported, TE₁₀ when the broad wall is wider than the narrow one. Below the cutoff frequency that mode is evanescent and only its attenuation is meaningful. Above the second cutoff the guide is over-moded and the higher-order modes it also supports are not included in the results. The Modes tab shows all the modes.

Adaptive Mesh Refinement and Certification

Both solvers start from a coarse mesh and refine it in passes. Each pass marks the cells with the largest estimated error and splits them. For differential lines, both the odd and even modes drive the refinement.

A small pass-to-pass change shows that the solution is converging slowly, not that it is close to the converged answer. When Estimate Solution Error is on, the final mesh is certified. The static fields are solved again on a uniformly refined mesh, and the remaining error is estimated by Richardson extrapolation. This extrapolation estimates the error that the current solution would have against infinitely refined mesh. Refinement continues until this estimate is below the tolerance, with a safety margin. The log reports the certified error, and a warning is shown if the node budget runs out first. The certificate covers the static quantities C and C₀, which set Z₀ and ε_eff. Conductor and dielectric loss are not certified. They converge more slowly than the field energy and their error is typically higher.

Boundaries and Ground Return

The solved region is a finite box. An open boundary truncates the static field with a zero-normal-flux condition. The automatic domain size is chosen so the truncation is small. After every solve, the field energy reaching the open walls is measured, and a warning is shown when it would shift the impedance by more than about 2%. An enclosure wall is a perfect electric conductor for the field. Its conductor loss is included, computed from the surface current on the wall.

Grounds that span the domain or reach an open edge stand for planes wider than the domain, such as the ground plane under a microstrip or a GCPW pour. These are treated as ideal returns of unlimited width. At high frequency this doesn't matter, because the return current crowds under the signal conductor. At DC and low frequency, an ideal return contributes no DC resistance, and its inductance is that of a perfect conductor plus the reactance of its metal thickness.

Grounds drawn with a finite width that don't reach the domain edge, such as custom geometry ground blocks, are solved as real conductors. Their DC and low-frequency resistance and inductance include the current spreading in them, on both solvers. Draw the ground at the actual board width when the low-frequency inductance or the DC resistance of the return path matters.

At DC the full-wave solver uses an exact DC current solve, and both solvers approach the exact DC resistance and internal inductance at low frequency.