Formulae:
(1) Resonant frequency \( f_o \) for TM10 mode:
$$ f_o = \frac{c}{2 \sqrt{\epsilon_e}} \sqrt{\left(\frac{m}{L_{eff}}\right)^2 + \left(\frac{n}{W_{eff}}\right)^2} $$
Considering TM10 mode \((m=1, n=0)\), $$ f_o = \frac{c}{2L_{eff} \sqrt{\epsilon_e}} => L_{eff} = \frac{c}{2f_o \sqrt{\epsilon_e}} = \frac{\lambda}{2 \sqrt{\epsilon_e}} $$
(2) Effective length (due to the Fringing Fields) \( L_{eff} \):
$$ L_{eff} = L + 2 \Delta L $$
Where \( \Delta L \) is calculated as:
$$ \Delta L = h \times 0.412 \frac{(\epsilon_e + 0.3)(\frac{W}{h} + 0.264)}{(\epsilon_e - 0.258)(\frac{W}{h} + 0.8)} $$
(3) Effective Dielectric Constant:
$$ \epsilon_e = \frac{\epsilon_r + 1}{2} + \frac{\epsilon_r - 1}{2} \left(1 + \frac{12h}{W}\right)^{-\frac{1}{2}} < \epsilon_r $$
Due to the Fringing Fields, some of the field lines pass through the air and then through the dielectric or vice versa, hence the effective dielectric constant is calculated considering the air as well as the substrate. It is thus less than the dielectric constant of individual substrate (ϵr).
(4) Width of Patch:
$$ W = \frac{c}{2f_o \sqrt{\frac{\epsilon_r + 1}{2}}} $$
(5) Ground Plane Extension:
$$ L_g = L + 6h, \quad W_g = W + 6h $$
(6) Feed Point Location:
$$ \text{Feed point (x)} = \frac{L}{6} \text{ to } \frac{L}{4} \text{ (for most of the cases)}$$
The feed point \((x)\) is taken depending upon the characteristic impedance of the coaxial feed line, which is commonly 50 Ω or 75 Ω. At the center \((L/2)\), the Current is maximum and Voltage is minimum which implies zero impedance. Whereas at the ends, Current is minimum and Voltage is maximum which implies infinite impedance.
So, there is a point between center and the edge (along the length \(L\)) where the impedance would be 50 Ω or 75 Ω and hence a feed at that point would get matched to impedance of the coaxial line.
Where:
- \( \epsilon_r \) is the relative permittivity of the substrate material.
- \( h \) is the thickness of the substrate.
- \( W \) is the width of the microstrip patch antenna.
- \( f_o \) is the resonant frequency.
- \( L \) is the length of the microstrip patch antenna.
- \( c \) is the speed of light in vacuum.
Ref. for formulae (1), (2), (3) & (4): Constantine A. Balanis, Antenna theory: Analysis and Design, Third Edition, John Wiley & Sons, 2005