I have been trying to find a formula to calculate the impedance of an embedded microstrip where the substrate has a different dielectric constant to the superstrate (so the stack up is conducting plane – dielectric substrate – trace – dielectric superstrate) – with no luck so far.
Everyone only treats the case where substrate and superstrate are identical.
So my question is: is there such a formula and where can I find it?
Best Answer
As your microstrip dimensions are substrate 1 mm thick, superstrate 10 mm thick, you could reasonably approximate your superstrate thickness to infinity. The upper fringeing fields are only going to be significant on the order of the substrate thickness.
This now leaves you with an approximate microstrip geometry, with different media above and below the line. If you scale both dielectrics so the upper is equivalent to air, then you can use existing approximate formulae for microstrip with air above and a specific substrate below. Increasing both er's back to their original values will increase the capacitance by that scaling factor, so reduce the impedance by the square root of the scale.
It's been a while since I used microstrip approximation formulae in anger, so they might only be valid for er_sub >> 1, or er_sub > 1. You don't say what the ratio of your dielectrics is. If they don't meet the range of applicability for microstrip formulae, then that's not going to work. However I've just tried a couple of online microstrip calculators, and neither complains for er ratios even below 1. That still may not mean the formulae are valid for that range, just that the calculators have not implemented catches for this unexpected range of use.