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Microstrip antenna on a thick substrate with high dielectric constant problem.

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Terminator3

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Hello all! I have used calculators, such as
https://www.emtalk.com/mpacalc.php?...Lmm&Wa=6.1&W_units_list=Wmm&Rin=93.1506262859
and
https://www.everythingrf.com/rf-calculators/microstrip-patch-antenna-calculator
And some formulas from papers on patch antenna. It worked nice for 2.4GHz, and simulation gived the same center frequency.

Now trying to simulate log-periodic patch array, but at high frequencies (10 to 20GHz) i have problem to calculate center frequency using formulas/calculators. Is it true, that for thick substrates, for example 2mm and Er=4, this formulas are not very accurate? Simulation and formula sizes differs for few mm! Not even 0.1 of mm.
 

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