2D Transmission Line Field Solver

A browser-based quasi-static 2D field solver for transmission line analysis. Computes characteristic impedance, effective permittivity, RLGC parameters, losses, and S-parameters.

Try it online: https://hforsten.com/field_solver.html

Features

• Transmission Line Types: Microstrip, stripline, GCPW (single-ended and differential)
• Electric Field Solving: Calculates characteristic impedance
• Full RLGC Extraction: Resistance, inductance, capacitance, conductance per unit length
• Loss Modeling: Conductor losses (skin effect, surface roughness) and dielectric losses
• S-Parameter Export: Touchstone .s2p and .s4p file generation
• Visualization: 2D potential plots, E-field streamlines, frequency-dependent plots
• Adaptive Meshing: Automatic mesh refinement for accurate field solutions

Quick Start

1. Open src/field_solver.html in a browser

Solution Flow

1. Geometry Setup: Define conductors and dielectric regions. Only rectangles are supported currently.
2. Mesh Generation: Create non-uniform coarse grid
3. Laplace Solve: Solve ∇²V = 0. Refine mesh until solution converges
4. Parameter Extraction:
   • Capacitance from field energy
   • Losses from perturbation method
   • RLGC extraction
5. Frequency Sweep: Repeat for multiple frequencies
6. S-Parameters: Convert RLGC to S-parameters via ABCD matrix

Validity

• Designed for microstrip, stripline, and coplanar waveguide structures commonly used in PCB RF and high-speed digital designs.
• Provides good results when the transmission line supports TEM or quasi-TEM propagation, which covers most practical PCB geometries below the onset of higher-order modes.
• Results have been checked against EM solver and actual measurement data with different geometries and transmission line types, showing close agreement with small error in typical use cases (tests folder).
• Accurate from RF through microwave and high-speed digital frequencies where return currents are confined and skin effect is significant.
• Sufficient for Most Practical Designs. Suitable for impedance control, loss estimation, and S-parameter generation in the vast majority of PCB transmission line applications.

Limitations

• Higher-order modes, dispersion, and cutoff behavior are not modeled. Structures that support non-TEM modes (e.g., waveguides) are outside the solver’s validity range.
• Current is modeled at the surface for AC and DC resistance is blended smoothly at low frequencies. Full 2D/3D current density inside conductors is not solved, which can reduce accuracy at frequencies where skin depth is comparable to conductor thickness.
• At DC and low frequencies (~<1 MHz), return current spreads over the ground plane. Since the solver infers inductance from capacitance, partial inductance and finite ground width effects may be inaccurate.
• Results apply to uniform, infinitely long transmission lines. Bends can often be approximated to behave similarly to straight line if they curve smoothly.
• Radiation is not modeled.

Common Tasks

Run the Web App

# Open in browser
open src/field_solver.html

Run Tests

node tests/test_vs_hfss.js

Build WASM Solver

WebAssembly (WASM) module is used for high-performance sparse matrix solving. This module is built using Emscripten and the Eigen C++ library. Compiled js and wasm is already included. Compiling is only needed if changes to WASM module are necessary.

Prerequisites

Emscripten Compiler (emcc)

Install the Emscripten SDK:

# Clone the emsdk repository
git clone https://github.com/emscripten-core/emsdk.git
cd emsdk

# Install and activate the latest SDK
./emsdk install latest
./emsdk activate latest

# Add to PATH (add this to your .bashrc or .zshrc for permanent use)
source ./emsdk_env.sh

Verify installation:

emcc --version

Eigen Library

The Eigen library is included as a git submodule. Initialize it:

git submodule update --init --recursive

Build Steps

cd src/wasm_solver
make

This compiles solver.cpp using Emscripten and generates:

• solver.js - JavaScript wrapper for the WASM module
• solver.wasm - Compiled WebAssembly binary