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Full changelogEngine 1.18.3

Controlled Impedance & Delay

PCB & Copper

Characteristic impedance for six transmission-line types, propagation delay, and the critical length.

Inputs

Board profile

Fills Dielectric constant from your board profile.

FR-4 ≈ 4.2–4.6 at 1 GHz, falling with frequency · Rogers 4350 ≈ 3.48 · PTFE ≈ 2.2

Results

Characteristic impedance Z0

Not judged: no target impedance in this mode. The geometry is inside the model's validity range.
48.9Ω
Show working
  1. Geometry ratiow / h = 0.2 mm / 0.11 mm = 1.818
  2. Effective permittivityε_eff at w/h = 1.82, ε_r = 4.2, t = 35 µm, microstrip (Hammerstad-Jensen) = 3.009
  3. Characteristic impedanceZ₀ = f(w/h, ε_eff), microstrip (Hammerstad-Jensen) = 48.9 Ω
Effective εr

Hammerstad-Jensen
3.009
All results (2)
Inductance
0.283nH/mm
Capacitance
0.118pF/mm
Z0 vs width

Accuracy

Source
  • Hammerstad and Jensen (1980)
  • Cohn (1954)

Hammerstad and Jensen, Accurate Models for Microstrip Computer-Aided Design (1980) the equation

Cohn, Characteristic Impedance of the Shielded-Strip Transmission Line (1954) the equation

Valid range
  • Dielectric constant 1 to 20

Dielectric constant The closed-form fits were derived for ordinary laminate permittivities. Hammerstad and Jensen, 1980

Precision
Closed-form fits to the exact static solution.

Within their range, ε_r (quoted at one frequency, not yours) and etch tolerance move the answer more than model error does. Use the tolerance analysis to see the band.

Most of the uncertainty comes from Dielectric constant. Tighten that first.

Verified against

3 independent cases. See the working.

This is a design aid. The engineer remains responsible for the design and for checking the standard itself.

Parameter sweep

Vary one input over a range and see the answer and verdict at each step, as a table and a curve.Pro

Worst-case corners

Put a tolerance on each input and get the worst-case band around the answer.Pro

Give at least one input a tolerance above zero.

Cross-section

The principle

Every trace over a reference plane is a transmission line. Whether you must treat it as one depends on the rise time, not the clock: if the round trip along the trace is comparable with the edge, reflections distort the waveform.

Microstrip (Hammerstad-Jensen, 1980)
Z0=η02πεeff  ln⁡ ⁣(f(u)u+1+(2u)2),u=whZ_0 = \frac{\eta_0}{2\pi\sqrt{\varepsilon_{eff}}}\;\ln\!\left(\frac{f(u)}{u} + \sqrt{1 + \left(\frac{2}{u}\right)^{2}}\right), \qquad u = \frac{w}{h}
f(u) = 6 + (2π − 6)·exp(−(30.666/u)^0.7528), with the paper’s thickness correction folded into u. Outside 0.1 ≤ w/h ≤ 10 you get a warning: a conservative range set here, not one quoted from the paper.
Effective dielectric constant (microstrip)
εeff=εr+12+εr−12(1+10u)−a(u) b(εr)\varepsilon_{eff} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2}\left(1 + \frac{10}{u}\right)^{-a(u)\,b(\varepsilon_r)}
a(u) and b(εr) are the paper’s fitted exponents. Embedded microstrip moves this toward εr with burial depth: εeff,em = εeff·e^(−2h₂/h) + εr·(1 − e^(−2h₂/h)).
Stripline (Cohn, exact at t = 0)
Z0=30πεr  K(k)K(k′),k=sech ⁣(πw2b)Z_0 = \frac{30\pi}{\sqrt{\varepsilon_r}}\;\frac{K(k)}{K(k')}, \qquad k = \t{sech}\!\left(\frac{\pi w}{2 b}\right)
K is the complete elliptic integral. Copper thickness enters through Wheeler’s effective-width correction; b is the plane-to-plane spacing.
Grounded coplanar waveguide
Z0=60πεeff  1K(k1)K(k1′)+K(k3)K(k3′),k1=ww+2sZ_0 = \frac{60\pi}{\sqrt{\varepsilon_{eff}}}\;\frac{1}{\tfrac{K(k_1)}{K(k_1')} + \tfrac{K(k_3)}{K(k_3')}}, \qquad k_1 = \frac{w}{w + 2s}
k₃ = tanh(πw/4h)/tanh(π(w+2s)/4h). Conformal mapping per Ghione-Naldi; the model assumes zero copper thickness.
Differential (edge-coupled, IPC-2141)
Zdiff≈2Z0(1−0.48 e−0.96 s/h)  microstripZ_{diff} \approx 2 Z_0\left(1 - 0.48\,e^{-0.96\,s/h}\right)\;\t{microstrip}
2Z₀(1 − 0.347·e^(−2.9 s/b)) for stripline. These empirical factors are the weakest step, about ±10 %.
Propagation delay
tpd=εeffc0≈3.34 εeff  ps/mm=84.7 εeff  ps/incht_{pd} = \frac{\sqrt{\varepsilon_{eff}}}{c_0} \approx 3.34\,\sqrt{\varepsilon_{eff}}\;\t{ps/mm} = 84.7\,\sqrt{\varepsilon_{eff}}\;\t{ps/inch}
Line parameters
Z0=LCtpd=LCZ_0 = \sqrt{\frac{L}{C}} \qquad t_{pd} = \sqrt{LC}
Critical length
lcrit=tr2 tpd(electrically long)ldesign=tr6 tpd(safe)l_{crit} = \frac{t_r}{2\,t_{pd}} \quad(\t{electrically long}) \qquad l_{design} = \frac{t_r}{6\,t_{pd}} \quad(\t{safe})
Signal bandwidth from rise time
fknee≈0.35trf_{knee} \approx \frac{0.35}{t_r}
  • wwtrace width
  • hhdielectric height to the reference plane (b for stripline)
  • h2h_2dielectric above an embedded trace
  • ttcopper thickness
  • sspair spacing, or the CPW gap to the coplanar ground
  • trt_r10–90 % rise time of the driver
  • η0\eta_0impedance of free space, 376.73 Ω
More detail

Layout rules

  • Never route across a plane split. The return current cannot follow, and the loop area grows. It is a common cause of EMC failures.
  • Add a return via next to any signal via that changes reference plane, or a decoupling capacitor if the planes are at different potentials.
  • Match delay, not length. Microstrip and stripline on FR-4 differ, about 5.8 against 6.9 ps/mm.
Sanity check on FR-4 (εr = 4.2). About 0.19 mm for 50 Ω microstrip over a 0.11 mm prepreg. Solder mask, trapezoidal etch and the frequency dependence of εr move the real value, so take the fabricator's stack-up as the final answer.
References: E. Hammerstad and Ø. Jensen, Accurate Models for Microstrip Computer-Aided Design, IEEE MTT-S 1980; S. B. Cohn, Characteristic Impedance of the Shielded-Strip Transmission Line, IRE Trans. MTT, 1954; G. Ghione and C. Naldi, Electronics Letters, 1984; B. C. Wadell, Transmission Line Design Handbook, Artech House, 1991; IPC-2141A; IPC-2251. For anything above ~2 GHz use a 2-D field solver and your fabricator's measured stack-up.

Engine version ⁨1.18.3⁩