Cantenna mathematics

cantenna.md

Waveguide theory

• http://www.antenna-theory.com/tutorial/waveguides/waveguide.php

Cutoff frequency

Circular waveguide cutoff frequency equation for TE11 mode

fc = 1.8412 c / 2 pi a = 1.8412 c / pi D

Where:

• fc = circular waveguide cutoff frequency in Hz
• c = speed of light within the waveguide in metres per second
• a = the internal radius for the circular waveguide in metres

http://www.radio-electronics.com/info/antennas/waveguide/cutoff-frequency.php

λc = c / fc = 2 pi a / 1.8412 = pi D / 1.8412

Which means that:

λf < λc = 1.706 D
D = 0.586 λc > 0.586 λf = (1.8412 / pi) λf

Wifi is in the frequency range 2.412-2.484GHz, which is 12.077-12.438cm.

which for wifi (λf =~ 12.285cm, f =~ 2.422GHz) means:

D > 0.538 * 12.438 = 7.288 cm

For the TM01 the constant is 2.4049. We don't want this so the sign is the other way:

D < (2.4049 / pi) * 12.077 = 9.245 cm

2.4049 / pi = 0.766

Constants are zeros of the derivative of Bessel functions.

http://www.tuks.nl/pdf/Reference_Material/Circular_Waveguides.pdf

Guide wavelength

Guide wavelength is defined as the distance between two equal phase planes along the waveguide. The guide wavelength is a function of operating wavelength (or frequency) and the lower cutoff wavelength, and is always longer than the wavelength would be in free-space. Here's the equation for guide wavelength:

λG = λf / \sqrt{1- \frac{λf/λc}^2}

λf is the freespace wavelength, i.e. the wavelength you want to build your antenna for.

http://www.microwaves101.com/encyclopedias/waveguide-mathematics#guide

This is derived from the more general equation

1/λf^2 = 1/λG^2 + 1/λc^2

Application to cans

Writing λc in terms of the diameter we get:

λc = pi D / 1.8412

λc^2 = (pi/1.8412)^2 D^2 = 2.911 D^2

Which leads to the formula for λG found on http://www.vias.org/wirelessnetw/wndw_06_08_04.html

λG = λf / \sqrt{1- \frac{λf^2}{2.911 D^2}}

The length is apparently not as important, since it mostly affects the directionality. Most sites recommend at least 0.75 λG.

Calculators

• http://www.wikarekare.org/Antenna/WaveguideCan.html (metric)
• http://www.csgnetwork.com/antennawncalc.html (imperial)

Parts

1. N-Type Female chassis mount connector.
2. Copper wire (12 gauge?) about 3cm
3. Cable to connect to wireless card
4. Can
5. Assembly stuff

What I have

Can Diameter = 7.8cm = 0.078m

λf = 0.1227 m
λc = 0.078*2*pi/1.8412 = 0.2661 m
λG = λf / \sqrt{1- \frac{λf/λc}^2} = 0.3168 m