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Easier Microstrip Equations

OSH Park PCBs

Microstrip transmission lines are common in modern consumer devices. Most computers, cellphones, and vehicles have at least one board with copper traces, which behave as microstrips.

Ulaby's introductory electromagnetics textbook has a set of equations for the characteristic impedance of a microstrip transmission line. These equations are not shown here but can probably be found on the textbook companion site, Amanogawa. While this combination of equations is certainly a complete and accurate model for various characteristics (conductor width, dielectric properties, etc) there might be a calculation-friendly solution which is nearly accurate for most of the dimensions a student or hobbyist would encounter. Without further ado...

Trace Width
(mm)
Z0 (Ω)
Eq 2.39
Approximation
35\times(2-\ln{s})
Relative Error
(%)
0.1524 152.923 152.294 0.4
0.3048 128.055 128.034 0.0
0.4572 113.543 113.842 - 0.2
0.7620 95.385 95.963 - 0.6
1.2192 78.946 79.513 - 0.7
1.9812 62.585 62.521 0.1
3.2004 47.726 45.735 4.2
5.1816 34.841 28.871 17.1
Calculated using OSH Park standard circuit board specs:
h = 1.6mm (distance between ground plane and top traces)
Rel. Permittivity εΓ = 4.6 (@ 1MHz) for FR4 dielectric.

Intermediate values:
  • x ≈ 0.54
  • y ranges from 0.83 to 1.000
  • s ranges from 0.095 to 3.24
  • t ranges from 76.01 to 5.40 therefore (2\times\pi-6)\times e^{-t} \approx 0
  • εeff ranges from 3.02 to 3.64
Even with values of w, h, and εΓ outside of the "standard" region, many of the equations can be approximated:
\epsilon_{eff} \approx \frac{\epsilon_{\Gamma}+1}{2} + \left(\frac{10}{s} \right)^{0.5}

Z_0 \approx \frac{60}{\sqrt{\epsilon_{\Gamma}}} \times \log{(\frac{8}{s})}

Running these equations for a common circuit board (w=0.46mm, h=1.6mm, εΓ=4.6) yields a Z0 approximation of 115.80 Ω, which has a 1.99% relative error over the exhaustive calculation.
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