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Rectangular & Circular Waveguide: Equations, Fields, & f_{co} Calculator 
The following equations and images describe electromagnetic waves inside both rectangular waveguide and circular
(round) waveguides. Oval waveguide equations are not included due to the mathematical complexity.
Click here for a transmission
lines & waveguide presentation.
A Cavity Resonance Calculator is included in
Espresso Engineering Workbook™ for FREE.
Rectangular Waveguide Cutoff Frequency
The lower cutoff frequency (or wavelength) for a particular mode in rectangular
waveguide is determined by the following equations (note that the length, x, has no bearing
on the cutoff frequency):

Cutoff Frequency Calculator
This example is for TE_{1,0} (the mode with the lowest cutoff frequency)
in WR284 waveguide (commonly used for Sband radar systems). It has a width of
2.840" (7.214 cm) and a height of
1.340"(3.404 cm).
Espresso Engineering Workbook™

where 
a = b = m = n = ε
= µ = 
Inside width (m), longest dimension Inside height (m), shortest dimension Number of ½wavelength
variations of fields in the "a" direction Number of ½wavelength variations of fields in the "b" direction
Permittivity (8.854187817E12 for free space) Permeability
(4πE7 for
free space) 

The TE_{10} mode is the dominant mode of a rectangular waveguide with a>b, since it has the lowest
attenuation of all modes. Either m or n can be zero, but not both.
End View (TE_{10})
Side View (TE_{10})
Top View (TE_{10})
____ Electric field lines p
_ _ _ Magnetic field lines

For TM modes, m=0 and n=0 are not possible, thus, TM_{11} is the lowest possible TM mode.
End View (TM_{11})
Side View (TM_{11})
____ Electric field lines
_ _ _ Magnetic field lines


The lower cutoff frequency (or wavelength) for a particular TE mode in circular waveguide is determined by
the following equation:
,
where p'_{mn} is
0 
3.832 
7.016 
10.174 
1 
1.841 
5.331 
8.536 
2 
3.054 
6.706 
9.970 

The lower cutoff frequency (or wavelength) for a particular TM mode in circular waveguide is determined by
the following equation:
(m), where p_{mn} is
0 
2.405 
5.520 
8.654 
1 
3.832 
7.016 
10.174 
2 
5.135 
8.417 
11.620 

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