mathematical_model

Mathematical model of HyperloopUPV H10 antenna

When considering different types of antennas for the hyperloop vehicle, the microstrip antenna stands out as a very good option overall. The antenna should be able to fit in the vehicle without being too big, reach 5.8 GHz and be directional. The microstrip antenna accomplishes all of the requirements with a very small size and a good directivity.

Our antenna is a microstrip rectangular antenna with direct feeding and an impedance matching strip. It consists of a ground plane, a substrate plane and the metal layer with the rectangular shape and the feeding.

Selecting the frequency of the antenna

Our desired frequency is 5.8 GHz, as it allows us to use WiFi 5 (IEEE 802.11ac) and also not have as many interferences as it would have around 5 GHz. Also the desired bandwidth is higher than 20 MHz, so we looked for channels with a bandwidth of 40 MHz or higher.

With all that we selected our objective frequency at 5.775 GHz, which is the center frequency of the WiFi band 155 with 80 MHz bandwidth.

Formulas used to calculate the dimensions of the antenna

0. Constants used

Parameter	Value	Description
Substrate Height ($h$)	0.813 mm	Thickness of the dielectric substrate
Relative Permittivity ($\varepsilon_r$)	3.38	Dielectric constant of the substrate
Desired Impedance ($Z$)	50 Ω	Desired impedance for the matching strip
Matching Line Length ($L_m$)	8.3405 mm	Length of the impedance matching strip
Feed Line Width ($W_f$)	1.8351 mm	Width of the feedline
Feed Line Length ($L_f$)	7.9445 mm	Length of the feedline
Centre Frequency ($f_r$)	5.775 GHz	Operating resonant frequency
Speed of Light ($c_0$)	299.792458 m/s	Speed of light in vacuum

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1. Patch Width (W)

The width of the patch is determined by the operating frequency and substrate properties:

$$ W = \frac{c}{2 f_r \sqrt{\frac{\varepsilon_r + 1}{2}}} $$

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2. Effective Permittivity ($\varepsilon_{\text{eff}}$)

The effective permittivity accounts for the fringing effect at the edges of the patch:

$$ \varepsilon_{\text{eff}} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2} \left(1 + 12\frac{h}{W}\right)^{-\frac{1}{2}} $$

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3. Patch Length Extension ($\Delta L$)

Due to fringing fields, the effective electrical length is slightly longer than the physical length. The extension is given by:

$$ \Delta L = 0.412h \frac{\left(\varepsilon_{\text{eff}} + 0.3\right)\left(\frac{W}{h} + 0.264\right)}{\left(\varepsilon_{\text{eff}} - 0.258\right)\left(\frac{W}{h} + 0.8\right)} $$

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4. Patch Length (L)

The length of the patch is adjusted using the effective permittivity and fringing effects:

$$ L = \frac{c}{2 f_r \sqrt{\varepsilon_{\text{eff}}}} - 2\Delta L $$

This determines the final patch length for proper resonance.

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5. Characteristic Impedance of the matching strip ($Z_f$)

The microstrip matching strip width is calculated to match the standard 50 Ω impedance:

$$ Z_m = \frac{60}{\sqrt{\varepsilon_{\text{eff}}}} \ln\left(\frac{8h}{W_m} + 0.25 \frac{W_m}{h}\right) $$

This ensures minimum reflection at the feeding point.

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6. Ground Plane Dimensions

To minimize edge diffraction effects, the ground plane should be at least twice the size of the patch, in our case:

$$ W_g = 2W,\ L_g = 3L $$

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Summary

After all the required calculations, the final dimensions are:

Parameter	Value	Description
Substrate Height ($h$)	0.813 mm	Thickness of the dielectric substrate
Relative Permittivity ($\varepsilon_r$)	3.38	Dielectric constant of the substrate
Patch Width ($W$)	17.4531 mm	Width of the radiating patch
Patch Length ($L$)	13.4209 mm	Length of the radiating patch
Matching Line Width ($W_m$)	0.3765 mm	Width of the impedance matching strip
Matching Line Length ($L_m$)	8.3405 mm	Length of the impedance matching strip
Ground Plane Width ($W_g$)	34.9062 mm	Width of the ground plane
Ground Plane Length ($L_g$)	40.3629 mm	Length of the ground plane
Feed Line Width ($W_f$)	1.8351 mm	Width of the feedline
Feed Line Length ($L_f$)	7.9445 mm	Length of the feedline
Centre Frequency ($f_r$)	5.775 GHz	Operating resonant frequency
Speed of Light ($c_0$)	299.792458 m/s	Speed of light in vacuum