# Rectangular Microstrip Patch Antenna Calculator

### Input

 Resonant Frequency $$f_r$$ GHzMHzKHzHz Substrate Relative Permittivity $$\epsilon_{r}$$ Substrate Height $$h$$ millimetercentimetermeterkilometermilinch

### Output

 Patch Physical Width $$W$$ ____ millimetercentimetermeterkilometermilinch Patch Physical Length $$L$$ ____ millimetercentimetermeterkilometermilinch 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

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.

## Formula

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

$$L=\frac{c}{2f_{r}\sqrt{\epsilon_{reff}}}-0.824h \left( \frac{(\epsilon_{reff}+0.3)(\frac{W}{h}+0.264)}{(\epsilon_{reff}-0.258)(\frac{W}{h}+0.8)} \right)$$

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

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

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

$$R_{in}(y=0)=\frac{1}{2(G_1+G_{12})}$$

$$R_{in}(y=y_0)=\frac{1}{2(G_1+G_{12})}cos^2(\frac{\pi}{L}y_0)$$

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

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

 Where: $$f_r$$ = 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

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

Other Related Calculator:
- Microstrip Circular Patch Antenna Calculator

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