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2 years ago in Electrical Engineering By Madhu B

I’m designing a patch on a two-layer substrate (e.g., RO4003 over FR4). For initial hand calculations, what’s the most accurate closed-form formula for the effective dielectric constant (‑_eff)?

My stack-up has a thin, high-ε_r layer (h1, ε_r1) for the patch and a thicker, low-ε_r layer (h2, ε_r2) below. Standard microstrip formulas assume one substrate. I need a formula for ε_eff to estimate resonant frequency and feed line impedance before full-wave simulation. Is there a validated expression, perhaps using a weighted average or a modified Yamashita formula for two-layer microstrip, that provides sufficient accuracy for a first-pass design?

EpsilonEffective StackedSubstrates AntennaDesign

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By Vinay Kumar Answered 1 year ago

For a two-layer substrate, the most accurate simple formula is a modified version of the effective dielectric constant that accounts for the fraction of field in each layer:
ε_eff ≈ 1 + q1*(ε_r1 - 1) + q2*(ε_r2 - 1)
where q1 and q2 are the filling factors for each layer. For a microstrip, q1 and q2 depend on the width-to-height ratios (w/h1, w/(h1+h2)). A good first-order approximation when the top layer (under the patch) is thin is to treat the bottom layer as the main substrate for the fringing fields, but this can be off by 10-20%. For more accuracy, I use the "two-layer microstrip" model in a line calculator tool (like ADS LineCalc) as my "closed-form" solution. It's numerical but instantaneous. For true hand calculation, consult the classic papers by Yamashita & Mittra or Kowalski & Pregla, but be prepared for complex expressions.

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