Rectangular Microstrip Patch Antenna Calculator


Resonant Frequency \(f_r \)
Substrate Relative Permittivity \(\epsilon_{r}\)
Substrate Height \(h\)


Patch Physical Width \(W \) ____
Patch Physical Length \(L \) ____
Effective Length \(L_{eff}\) ____ centimeter
Input Impedance at Edge\((y = 0)\) \(R_{in}\) ____ ohm
50 ohm Feed Position \(y_{0}\) ____ centimeter
Single Slot Conductance \(G_{1}\) ____ mho
Mutual Conductance \(G_{12}\) ____ mho
Directivity D ____ dBi

Rectangular Patch Antenna

The dimensions of rectangular patch antenna is calculated for the lowest resonant frequency \(TM_{010} \)(dominant) mode. The ground plane is assumed to be infinite and all the metallic surfaces are lossless.


\(W = \frac{c}{2f_{r}}\sqrt{\frac{2}{\epsilon_{r}+1}}\)


\(\epsilon_{reff}=\frac{\epsilon_{r}+1}{2}+\frac{\epsilon_{r}-1}{2}\left [ \frac{1}{\sqrt{1+12\left (\frac{h}{W} \right )}}\right ]\)

\(G_{1}=\frac{1}{120\pi^2}\int_{0}^{\pi}\left [ \frac{sin\left ( \frac{K_{0}W}{2}cos\theta \right )}{cos\theta} \right ]^2sin^3\theta d\theta\)

\(G_{12}=\frac{1}{120\pi^{2} }\int_{0}^{\pi}\left [ \frac{sin\left ( \frac{K_{0}W}{2}cos\theta \right )}{cos\theta} \right ]^2J_{0}(K_{0}Lsin\theta)sin^3\theta d\theta\)



\(I_{1}=\int_{0}^{\pi}\int_{0}^{\pi}\left [\frac{sin\left ( \frac{K_{0}W}{2}cos\theta \right )}{cos\theta} \right ]^2 sin^3\theta cos^2( \frac{K_{0}L_e}{2}sin\theta sin\phi) d\theta d\phi\)

\(D=\left ( \frac{2\pi W}{\lambda_0} \right )^2\frac{\pi}{I_1}\)

\(f_r \) = Resonant Frequency of Rectangular Patch Antenna
\(\epsilon_{r}\) = Substrate Relative Permittivity
\(h\) = Substrate Height
\(c \) = Velocity of Light
\(L \) = Physical Length of Patch
\(W \) = Physical Width of Patch
\(\epsilon_{reff} \) = Effective Relative Permittivity
\(K_{0}\) = Free-Space Phase Constant \(\frac{2\pi}{\lambda_0}\)
\(\lambda_0\) = Free-Space Wavelength
\(G_{1}\) = Single Slot Conductance
\(G_{12}\) = Mutual Conductance
\(R_{in}(y=0)\) = Input impedance of resonant patch at edge
\(R_{in}(y=y_0)\) = Input impedance of resonant patch at \(y=y_0\)
\(J_{0}\) = Bessel function of the first kind of order zero
\(D\) = Directivity of Patch Antenna

- [1] Balanis, C.A. (2016). Antenna Theory: Analysis and Design. 4th ed. Hoboken, New Jersey Wiley, pp.788–814. Chapter 14.2 Rectangular Patch.

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