Neutron STEM Lab
Virtual engineering lab

Microstrip Geometry & Impedance

Analyze a thin PCB trace or synthesize its width for a target impedance, with effective permittivity and guided wavelength exposed.

Electrical Engineering / RF & MicrowaveintermediateValidated educational model
Learning mode

Guided laboratory

Use the checklist when you want a structured lab. Explore mode leaves the instruments unrestricted.

Explore a model

Set up your experiment

Read the equations

Start with an example, change one input, then run again. Inactive controls do not apply to the selected model. Results and validation always belong to the last completed run.

Used in analysis mode; constrained to width/height = 0.01–100.
Distance between the trace and a continuous ground plane.
unitless
Frequency-independent substrate permittivity in this model.
Real resistance or reference impedance.
Positive frequency; stored internally in hertz.

Choose an example or use the default settings, then run the experiment.

Keep this experiment

Your setup stays in this browser. A project file preserves SI inputs, display units, receiver stages, and any imported complex network samples.

Changes are saved on this device when the settings are valid.

Measurements

Instrument readings

Characteristic impedance
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Recorded from the current model setup; rerun after editing inputs.
Trace width
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Recorded from the current model setup; rerun after editing inputs.
Effective permittivity
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Recorded from the current model setup; rerun after editing inputs.
Theory

Equations and model

Fields shared between air and substrate

The zero-thickness Hammerstad–Jensen approximation estimates quasi-static impedance and effective permittivity from width/height and substrate permittivity. The empirical coefficients below depend on geometry and dielectric constant.

u=whu=\frac{w}{h}εeff=εr+12+εr−12(1+10u)−ab\varepsilon_{\mathrm{eff}}=\frac{\varepsilon_r+1}{2}+\frac{\varepsilon_r-1}{2}\left(1+\frac{10}{u}\right)^{-ab}Z0=η02πεeffln⁡(F(u)u+1+4u2)Z_0=\frac{\eta_0}{2\pi\sqrt{\varepsilon_{\mathrm{eff}}}}\ln\left(\frac{F(u)}{u}+\sqrt{1+\frac{4}{u^2}}\right)

Hammerstad–Jensen coefficients

All logarithms here are natural logarithms. These dimensionless coefficients close the impedance model above; they are not adjustable fitting controls.

a(u)=1+149ln⁡(u4+(u/52)2u4+0.432)+118.7ln⁡(1+(u/18.1)3)a(u)=1+\frac{1}{49}\ln\left(\frac{u^4+(u/52)^2}{u^4+0.432}\right)+\frac{1}{18.7}\ln\left(1+(u/18.1)^3\right)b(εr)=0.564(εr−0.9εr+3)0.053b(\varepsilon_r)=0.564\left(\frac{\varepsilon_r-0.9}{\varepsilon_r+3}\right)^{0.053}F(u)=6+(2π−6)exp⁡[−(30.666u)0.7528]F(u)=6+(2\pi-6)\exp\left[-\left(\frac{30.666}{u}\right)^{0.7528}\right]

Propagation and synthesis

The same effective permittivity determines propagation velocity and wavelength. Width synthesis is a bounded bisection of the impedance model. Dispersion, copper thickness, loss, and solder mask are omitted.

vp=cεeff,λg=vpfv_p=\frac{c}{\sqrt{\varepsilon_{\mathrm{eff}}}},\qquad \lambda_g=\frac{v_p}{f}C′=1vpZ0,L′=Z0vpC'=\frac{1}{v_pZ_0},\qquad L'=\frac{Z_0}{v_p}
Validation

Independent checks

Published reference caseNot run

Check the model against a known numerical benchmark.

Expected
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Simulated
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Error
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Tolerance
0.00001%

Run the experiment to perform this check.

Current model applicabilityNot run

Check current assumptions and report any limitations.

Expected
—
Simulated
—
Error
—
Tolerance
0%

Run the experiment to perform this check.

Engineering interpretation

Run the experiment to generate an engineering interpretation.

Assumptions and limitations
Assumptions
  • All calculations run locally; frequencies and lengths are stored in SI units.
  • The selected steady-state ideal or approximate model is appropriate to the engineering question.
  • Each experiment states its reference impedance, sign convention, and omitted effects.
Limitations
  • Model benchmarks validate the implementation, not a particular fabricated device.
  • Plots and sweeps have bounded resolution; inspect raw samples and refine a real measurement when required.
  • Guided completion requires current measurements and passing applicability checks. A warning scenario can still be useful for learning.
Local experiment export

Save your measurements and setup

Exports are generated in your browser. No account or server upload is required.

Settings JSON
Current parameters and instrument controls, including waveform, output enable, scope coupling, timebase, trigger, and cursors when present. Data labs also include the dataset, mappings, exclusions, budget, or propagation setup.
Measurements CSV
Completed instrument readings with labels, units, and descriptions.
Plot PNG
The current canvas-based scope, transient, sweep, or statistics visualization when available.
Summary
Markdown report with objectives, setup, measurements, validation, interpretation, assumptions, and graded guided concept-check results.

Ready to export the current local experiment state.

Learning objectives

What this experiment should establish

  • Analyze a reference trace.
  • Explore geometry.
  • Synthesize a target.
  • State the missing physics.
Check your understanding

Questions to answer from the experiment

  1. Why is effective permittivity generally below the substrate permittivity for microstrip?
  2. Holding substrate height and permittivity fixed, what usually happens when trace width increases?
  3. Which design detail is absent from the implemented microstrip model?
  4. A width solver converges to many decimal places. What does that establish?
Continue learning
RF & Microwave LabContinue with the other five RF experiments.AC Circuits WorkbenchReview phasors, reactance, and sinusoidal steady state.Data & Uncertainty LabAnalyze exported numerical data and measurement uncertainty.

Sources and model review

Reviewed 2026-09-17. The educational model exposes its assumptions and validation; source references do not imply external certification.