microstrip antenna analysis

Hi guys
I'm confused with analysis of microstrip antenna
In one paper a desgin of patch antenna with the following parameters

Wsub=20mm , Lsub=30mm , h="substrate thickness"=0.8mm and Epslonr=4.4
patch dimensions are :
Wp=16 mm and Lp=8 mm excited using typical 50 Ohms microstrip line

The -10dB bandwidth simulated in paper and it was (3.5-6) GHz
My question is how to calculate the center frequency and approximated bandwidth ?
I'm caculated the proposed center frequency using the typical formula "L=Leff+2*extension length" and unfortunately it was 8.5 GHz !

Thanks in advance

Hi, Frequency1

Resonant frequency can be expressed as f0 = c/(2*L_eff*sqrt(eps_eff)), giving something near 8 GHz, in agreement with your result.
The resonant frequency would be half of that (in agreement with the paper) if the side opposite to the feed point were grounded. In that case L becomes aproximately 1/4 of wavelength instead of 1/2 wavelength (in the substrate).
Regards

Z