MathBench

Microstrip Impedance Calculator

Get the characteristic impedance of a PCB trace from its width, substrate height, and dielectric constant, or flip the mode and solve for the width that hits your target Z₀.

Result

—characteristic impedance
—effective εeff
—W / H ratio
—propagation speed (% of c)

How this calculator works

A microstrip is the standard PCB transmission line: a copper trace of width W above a ground plane, separated by a dielectric substrate of height H. Its characteristic impedance is set almost entirely by the W/H ratio and the substrate's dielectric constant εr. This calculator evaluates Hammerstad's closed-form equations in either direction: give it dimensions to get Z₀, or give it a target Z₀ to get the required width.

The width-solver uses the synthesis equations and then re-checks the answer through the analysis equations, so the note under the result always shows you the round-trip impedance. If they agree, the answer is self-consistent.

The formula

With u = W/H, the effective dielectric constant is:

εeff = (εr+1)/2 + (εr−1)/2 × (1 + 12/u)^(−½)   [+ 0.04(1−u)² term for u < 1]

and the impedance:

u ≤ 1:  Z₀ = 60/√εeff × ln(8/u + u/4)
u ≥ 1:  Z₀ = 120π / [√εeff × (u + 1.393 + 0.667·ln(u + 1.444))]

These are Hammerstad's equations as presented in Pozar, Microwave Engineering (§3.8). The solve-for-width mode uses the companion synthesis forms. The model assumes a thin trace and no solder mask, both worth a few percent, which is inside FR-4's own batch-to-batch εr variation.

Worked example

Say you're routing an antenna feed on 1.6 mm FR-4 (εr 4.4) with a 3 mm trace:

  1. Ratio: u = 3 ÷ 1.6 = 1.875
  2. εeff = 2.7 + 1.7 × (1 + 6.4)^(−½) = 3.325
  3. Z₀ = 120π ÷ [√3.325 × (1.875 + 1.393 + 0.667 ln 3.319)] = 377 ÷ 7.42 = 50.8 Ω

Run it backwards: asking the width mode for exactly 50 Ω on the same board returns 3.06 mm, the classic "3 mm on 1.6 mm FR-4" rule of thumb, derived.

Assumptions & tips

  • Thinner dielectric, thinner trace. On 4-layer boards the trace sits ~0.1–0.2 mm above the plane, so 50 Ω traces get narrow (~0.2–0.4 mm). Check your fab's stackup table before assuming the 2-layer numbers.
  • FR-4's εr is a range, not a number. Datasheets quote 4.2–4.7 depending on resin content, glass weave, and frequency. For anything above a few GHz or with tight tolerance, move to a controlled material (Rogers 4350B: εr 3.48 ± 0.05).
  • Keep the plane solid. The math assumes an unbroken ground plane under the whole trace. A slot or split under a 50 Ω trace turns it into an antenna and an EMC problem simultaneously.
  • Match connector geometry at the launch. The last few millimeters where the trace meets an SMA edge connector matter at high frequency: taper the trace and via-stitch the ground as the connector's datasheet shows.
  • For production boards, spec the impedance. Fabs with controlled-impedance service will adjust widths to their real stackup and test coupons. Your job is the target and tolerance (e.g., 50 Ω ± 10%), not the exact mils.

Frequently asked questions

What trace width gives 50 Ω on standard FR-4?

On 1.6 mm two-layer FR-4 (εr ≈ 4.4), a 50 Ω microstrip needs a trace about 3 mm wide, roughly twice the substrate height. On a typical 4-layer board where the trace sits 0.2 mm above the plane, the same 50 Ω needs only about 0.36 mm. The ratio of width to height is what sets the impedance, not the absolute size.

Why do RF traces need to be 50 Ω?

Fifty ohms is the convention nearly all RF connectors, cables, ICs, and test equipment are designed around, a historical compromise between the ~30 Ω of maximum power handling and ~77 Ω of minimum loss for air-dielectric coax. Matching the trace to the system impedance prevents reflections. The number itself is convention, and 75 Ω (video) or 100 Ω differential (Ethernet, USB) systems follow the same logic.

What is the effective dielectric constant?

A microstrip's field lives partly in the substrate and partly in the air above it, so the wave sees a blend of the two: the effective dielectric constant, always between 1 and εr. It sets the propagation speed and therefore electrical length: on FR-4 with εeff ≈ 3.3, signals travel at about 55 percent of the speed of light.

How accurate are these closed-form equations?

Hammerstad's equations are typically within about 1 percent of full-wave simulation for width-to-height ratios from 0.05 to 20 and εr up to about 16, tighter than the tolerance of the board material itself: FR-4's εr varies by several percent between batches and with frequency. For controlled-impedance production boards, give the fab house your target impedance and let them adjust to their measured stackup.

Does trace thickness or solder mask matter?

Both nudge the result by a few percent. Copper thickness effectively widens the trace slightly (lowering impedance about 1 to 2 Ω for 1 oz copper on thin substrates), and solder mask over the trace raises the effective dielectric constant, lowering impedance by another 1 to 3 Ω. This calculator uses the thin-trace, bare-board model. Treat the last few percent as fab-tuning territory.

Sources

  1. Equations for Microstrip Circuit Design. E. O. Hammerstad, Proceedings of the 5th European Microwave Conference, 1975, pp. 268–272. DOI 10.1109/EUMA.1975.332206. The original closed-form analysis and synthesis equations this calculator evaluates in both directions.
  2. Accurate Models for Microstrip Computer-Aided Design. E. Hammerstad and O. Jensen, 1980 IEEE MTT-S International Microwave Symposium Digest, pp. 407–409. DOI 10.1109/MWSYM.1980.1124303. The refinement of the 1975 equations that establishes the accuracy claim quoted in the FAQ, roughly one percent over the W/H range the calculator warns outside of.
  3. Microwave Engineering, 4th edition. David M. Pozar, Wiley, 2011. wiley.comSection 3.8 presents the Hammerstad equations in the exact form coded here, including the effective dielectric constant εeff and the two impedance branches either side of W/H = 1.
  4. RO4000 Series High Frequency Circuit Materials — RO4003C and RO4350B data sheet. Rogers Corporation. rogerscorp.comSource of the RO4350B figure quoted in the tips: a dielectric constant of 3.48 ± 0.05, against which FR-4's loose 4.2–4.7 spread is contrasted.
  5. IPC-2141A, Design Guide for High-Speed Controlled Impedance Circuit Boards. IPC, 2004. The industry design guide behind the closing tip on specifying a target impedance and tolerance for the fabricator.

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