Casual Microstrip Design 8

I think this will be the last in this unintentional series on microstrip design. I just wanted to add a bit on dispersion to the earlier posts. Continuing with the formulae found in Chapter 7 of (Edwards and Steer 2016), defining improved expressions for dispersion at higher frequencies. The design formula by Kirschning and Jansen¹ covers a much wider range of permittivities, aspect ratios and frequencies.

Leading up to the main \(\varepsilon_{eff}(f)\) expression, we have to present several preliminary formulae, where frequency, F, is in GHz, and thickness, H, is in cm. The first four are, \[\begin{align*} P_1 & = 0.27488 + [0.6315 + 0.525 / (1 + 0.157 F H ) ^{20}] (w/h) \\ & - 0.065683 exp(-8.7513 w/h) \\ P_2 & = 0.33622 [1- exp(-0.03442 \varepsilon_r)] \\ P_3 & = 0.0363 exp(-4.6 w/h) (1-exp[-(F H / 3.87)^{4.97} ]) \\ P_4 & = 1 + 2.751 (1- exp[-(\varepsilon_r / 15.916)^8]) \end{align*}\] These are used as input to this formula, \[\begin{equation} P(F) = P_1 P_2 [(0.1844 + P_3 P_4) 10 F H]^{1.5763} \end{equation}\] Finally we get to the basic formula, \[\begin{equation} \varepsilon_{eff}(f) = \varepsilon_r - \frac{\varepsilon_r - \varepsilon_{eff}}{1+ P(F)} \end{equation}\] An accuracy of better than 0.6% is claimed to 60 GHz, but only tested to 30 GHz. The validity ranges are, \[\begin{equation} \begin{split} 1 \leq \varepsilon_r \leq 20 \\ 0.1 \leq w/h \leq 100 \\ 0 \leq h/\lambda_o \leq 0.13 \\ \end{split} \end{equation}\]

So, what does that really look like? The below charts use a relative permittivity, \(\varepsilon_r=9.8\) and substrate, h=1.6. The deviation/difference over an impedance range is shown here, for 10 GHz and 30 GHz.

The \(\varepsilon_{eff}\) and \(\varepsilon_{eff}(f)\) are mostly impedance driven, more so than frequency. However, when frequency goes higher, the \(\varepsilon_{eff}(f)\) value rises, showing the dispersion effects at the higher frequencies; and at very high frequencies, where \(F_r \geq 3000\) GHz, approaches \(\varepsilon_r\).

That about wraps up these thoughts on dispersion effects at higher frequencies, and covers the areas I have been exploring. These posts are a great way to have the information available as I push on in my investigations of the microwave region and filter design in microstrip. Future investigations may include attempting to make prototype printed circuit boards (PCBs) using a CNC router. One last thing, I have compiled this series into a downloadable PDF file, found here

Have a great day, and may the Lord Jesus Bless your efforts and keep you safe!

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References

Edwards, Terry C., and Michael B. Steer. 2016. Foundations for Microstrip Circuit Design, 4th Edition. West Sussex, United Kingdom: Wiley & Sons, Ltd.

Footnotes

1. M. Kirschning and R. Jansen, “Accurate model for effective dielectric constant of microstrip with validity up to millimeter-wave freuencies.” Electronics Letters, vol. 18, no. 6, pp. 272-273, 1982.