Transmission line impedance calculator

How it works

A PCB or package trace behaves as a transmission line once it is a sizeable part of a wavelength. Its impedance sets reflections, its effective dielectric constant sets delay, and its conductor and dielectric loss set the insertion loss. This calculator takes the trace geometry, the dielectric and the frequency, gives all of these, and solves for the width that hits a target impedance.

1. Microstrip impedance and effective permittivity. Hammerstad and Jensen's closed form gives the air-filled impedance Z01 from the width-to-height ratio u = W/h, and the effective permittivity from a blend of the dielectric constant and air. A strip thickness correction widens the strip slightly. The impedance is Z01 over the square root of the effective permittivity.Z01(u) = 60 ln( f(u)/u + sqrt(1 + (2/u)^2) ) ohm, f(u) = 6 + (2 pi − 6) exp( −(30.666/u)^0.7528 ); e_eff = (er + 1)/2 + (er − 1)/2 x (1 + 10/u)^(−a b); Z0 = Z01 / sqrt(e_eff)
2. Embedded microstrip. A cover of dielectric above the trace pulls the field into the dielectric. The effective permittivity moves from the surface value to the dielectric constant as the cover grows past about the dielectric height, and the impedance follows it. The blend was checked against field solutions.e_eff,cov = e_eff x exp(−2 c/h) + er x (1 − exp(−2 c/h)); Z0 = Z0,surface x sqrt(e_eff / e_eff,cov), with cover thickness c
3. Stripline. For a trace centred between two planes b apart, Cohn's exact solution for a thin strip uses the ratio of complete elliptic integrals, found here by the arithmetic-geometric mean. A finite thickness adds an effective width, taken from a table fitted to field solutions of rectangular traces. For an offset trace the two single-sided striplines are put in parallel and a small correction fitted to a field solution is applied.Z0 = (30 pi / sqrt(er)) x K'(k)/K(k), k = tanh( pi We / (2 b) ), We = W + extra width for thickness, b = 2 h + t; offset: Z0 = 2 Za Zb / (Za + Zb) x (1 − 0.18 g^2), g = |h1 − h2| / (h1 + h2)
4. Coplanar waveguide with a ground plane. Conformal mapping gives the impedance and effective permittivity of a centre strip with coplanar grounds over a substrate backed by a plane. The copper thickness narrows the gap by about its own thickness and lowers the effective permittivity, because the side walls hold air; both corrections are fitted to field solutions.k = a/b, k3 = tanh(pi a / 2h) / tanh(pi b / 2h), a = W/2, b = W/2 + gap; q = K'(k)K(k3) / (K(k)K'(k3)); e_eff = (1 + er q)/(1 + q); Z0 = 60 pi / (sqrt(e_eff) x (K(k)/K'(k) + K(k3)/K'(k3)))
5. Differential pairs. For edge-coupled microstrip the odd-mode impedance comes from the coupling capacitances of Garg and Bahl (parallel plate, fringing and gap terms), and the even-mode impedance is estimated as twice the single-ended value minus the odd value. For edge-coupled stripline Cohn's even and odd mode solutions are used with the effective width and a spacing narrowed by the thickness. The differential impedance is twice the odd-mode impedance.microstrip: Zodd = 1 / (c sqrt(C_odd x C_odd,air)), C_odd = Cp + Cf + Cga + Cgd; Zeven = 2 Z0 − Zodd; stripline: ke = tanh(pi We/2b) tanh(pi (We + s)/2b), ko = tanh(pi We/2b) / tanh(pi (We + s)/2b); Zdiff = 2 Zodd
6. Delay. The propagation delay per unit length depends on the effective permittivity at the frequency. For microstrip the permittivity rises with frequency by Hammerstad and Jensen's dispersion model; the other structures use their static value.delay per metre = sqrt(e_eff) / c; e_eff(f) = er − (er − e_eff) / (1 + G (f/fp)^2), fp = Z0 / (2 mu0 h)
7. Conductor loss. Wheeler's incremental inductance rule turns the impedance model into a loss: the series resistance is the surface resistance divided by the free-space impedance, times the rise in the air-line impedance when every conductor surface moves back into the metal by a small step d. Roughness multiplies the surface resistance by the Hammerstad and Jensen factor.Rs = sqrt(pi f mu0 / sigma) x Kr; Kr = 1 + (2/pi) arctan(1.4 (Delta/skin depth)^2); R = (Rs / 377) x dZair/dd; alpha_c = R / (2 Z0) Np/m; dB = 8.686 x Np
8. Dielectric loss and total. Dielectric loss follows the loss tangent and the share of the field in the dielectric. The total loss is the two added, times the length.alpha_d = (pi f tan d / c) x (er / sqrt(e_eff)) x (e_eff − 1)/(er − 1) Np/m; loss (dB) = 8.686 x (alpha_c + alpha_d) x length
9. Width for a target impedance. Impedance falls as the trace widens, so the width for a target is found by bisection between a very narrow and a very wide trace with the other dimensions held. A pair is solved for its differential impedance at the spacing entered.find W such that Z(W) = target, by halving the bracket 80 times

Sources

The methods and default figures rest on these public sources. The wording here is our own; read the source for its full text.

• Accurate Models for Microstrip Computer-Aided Design (opens in a new tab), E. Hammerstad and O. Jensen, IEEE MTT-S International Microwave Symposium Digest, pp. 407 to 409, May 1980. Microstrip impedance, effective permittivity, thickness correction, dispersion and the roughness factor.
• Shielded Coupled-Strip Transmission Line, S. B. Cohn, IRE Transactions on Microwave Theory and Techniques, vol. MTT-3, pp. 29 to 38, October 1955. Even and odd mode impedances of edge-coupled striplines, and the single stripline as the uncoupled limit.
• Microstrip Lines and Slotlines, K. C. Gupta, Ramesh Garg and I. J. Bahl, Artech House, 1979. Coupling capacitances of edge-coupled microstrip (Garg and Bahl) and the coplanar waveguide background.
• Formulas for the Skin Effect (opens in a new tab), H. A. Wheeler, Proceedings of the IRE, vol. 30, no. 9, pp. 412 to 424, September 1942. The incremental inductance rule for conductor loss.
• Coplanar Waveguide Circuits, Components, and Systems (chapter 3, conductor-backed coplanar waveguide), Rainee N. Simons, Wiley-Interscience, 2001. Quasi-static conformal-mapping expressions for the impedance and effective permittivity of coplanar waveguide with a ground plane.
• Microwave Engineering, 4th edition (stripline conductor loss and microstrip design equations), David M. Pozar, John Wiley and Sons, 2012. Used as a cross-check: the stripline conductor attenuation and the microstrip width design equations.
• RO4350B Laminates data sheet (opens in a new tab), Rogers Corporation, 2024 revision. Relative permittivity 3.48 and dissipation factor 0.0037 at 10 GHz, used in the laminate preset.