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Visual calculator 03 · Planar lines

Microstrip, solved live

Your browser solves Laplace's equation on the cross-section every time you move a slider. The capacitance with and without the substrate gives εeff and Z0.

Substrate
Units

Z0 vs width: field solve against closed form

Line: Hammerstad–Jensen. Dots: every finite-difference solve you have run at this εr and t/h.

εeff vs frequency

Kirschning–Jansen dispersion: the field pulls into the substrate as frequency rises.

Check

Method

∇·(ε ∇φ) = 0 on the cross-section strip φ = 1 V, ground and far box φ = 0 half domain, symmetry wall at x = 0 graded grid, finest cell 0.004·min(w/2, h) at the edge finite-volume 5-point stencil, interface on a grid line conjugate gradients + IC(0) preconditioner C = 2 Σ g (Δφ)² · ε0 (field energy) C0 = same with εr = 1 εeff = C / C0, Z0 = 1 / (c √(C · C0))

Quasi-static (TEM) model: valid while the substrate is thin compared with the wavelength. The closed forms are Hammerstad–Jensen 1980 (static Z0 and εeff, with their thickness correction) and Kirschning–Jansen 1982 (εeff dispersion, fitted for t = 0, valid to f·h = 25 GHz·mm, 0.1 ≤ w/h ≤ 100, εr ≤ 20; the shaded part of the plot is outside that range). With t > 0 the thickness-corrected static εeff is used as its starting point.

Validation sweep · t = 0 · run in your browser

20 field solves, default and refined grids.
  1. E. Hammerstad, Ø. Jensen, "Accurate models for microstrip computer-aided design," IEEE MTT-S Int. Microwave Symp. Digest, 1980, pp. 407–409.
  2. M. Kirschning, R. H. Jansen, "Accurate model for effective dielectric constant of microstrip with validity up to millimetre-wave frequencies," Electronics Letters 18(6), 1982, pp. 272–273.
  3. M. N. O. Sadiku, Numerical Techniques in Electromagnetics, 3rd ed., CRC, 2009, Ch. 3 (finite differences for Laplace's equation, microstrip example).
  4. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, 2003, Sec. 10.3 (IC(0) preconditioning).
  5. Substrate εr: Rogers RT/duroid 5880, RO4003C and RO4350B data sheets (design Dk); FR-4 typical; 99.6% alumina; GaAs.