Closed-form expressions for the resonant frequency of rectangular microstrip antenna elements with thick substrates

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Three closed-form expressions are presented for the calculation of resonant frequencies of rectangular microstrip antenna elements. They are derived by modifying the transmission-line model and the cavity model, and by the curve-fitting method. The results of these expressions are compared with the measured resonant frequencies of antenna elements with substrates that satisfy the thickness criterion h ≥ 0.0815λ0, where λ0 is the free-space wavelength and h is the thickness of the substrate. Calculated results are in excellent agreement with measurements.