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### Easier Microstrip Equations

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) |
Z_{0} (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

ε_{Γ} = 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
- ε
_{eff} ranges from 3.02 to 3.64

Even with values of w, h, and

ε_{Γ} 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,

ε_{Γ}=4.6)
yields a

Z_{0} approximation of 115.80 Ω, which has a 1.99% relative error over
the exhaustive calculation.

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