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Rectangular & Circular Waveguide: Equations, Fields, & fco 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 RF Cafe Calculator 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):

Rectangular Waveguide - RF Cafe

Rectangular Waveguide Cutoff Frequency equation - RF Cafe

Rectangular Waveguide Cutoff Wavelength equation - RF Cafe

Rectangular Waveguide TEm,n Mode
Cutoff Frequency Calculator

This example is for TE1,0 (the mode with the lowest cutoff frequency) in WR284 waveguide (commonly used for S-band radar systems). It has a width of 2.840" (7.214 cm) and a height of 1.340"(3.404 cm).

where a =
b =
m =
n =
ε =
µ =
Inside width (m), longest dimension
Inside height (m), shortest dimension
Number of ½-wavelength variations of fields in the "a" direction
Number of ½-wavelength variations of fields in the "b" direction
Permittivity (8.854187817E-12 for free space)
Permeability (4πE-7 for free space)

TE (Transverse Electric) Mode

The TE10 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.


Waveguide equation formula drawing fields End View TE - RF Cafe
End View (TE10)

 

Waveguide equation formula drawing fields Side View TE - RF Cafe

Side View (TE10)

 

Waveguide equation formula drawing fields Top View TE - RF Cafe

Top View (TE10)

____ Electric field lines
p    _ _ _ Magnetic field lines

TM (Transverse Magnetic) Mode

For TM modes, m=0 and n=0 are not possible, thus, TM11 is the lowest possible TM mode.

Waveguide equation formula drawing fields End View TM - RF Cafe
End View (TM11)

 

Waveguide equation formula drawing fields Side View TM - RF Cafe
Side View (TM11)

____ Electric field lines
   _ _ _ Magnetic field lines

TE (Transverse Electric) Mode Circular Waveguide - RF Cafe

The lower cutoff frequency (or wavelength) for a particular TE mode in circular waveguide is determined by the following equation: Circular Waveguide Cutoff Wavelength equation - RF Cafe, where p'mn is

m p'm1 p'm2 p'm3
0 3.832 7.016 10.174
1 1.841 5.331 8.536
2 3.054 6.706 9.970

TM (Transverse Magnetic) Mode

The lower cutoff frequency (or wavelength) for a particular TM mode in circular waveguide is determined by the following equation: Waveguide equation formula drawing fields (m), where pmn is

m pm1 pm2 pm3
0 2.405 5.520 8.654
1 3.832 7.016 10.174
2 5.135 8.417 11.620

Related Pages on RF Cafe

- Properties of Modes in a Rectangular Waveguide
- Properties of Modes in a Circular Waveguide
- Waveguide & Flange Selection Guide
- Rectangular & Circular Waveguide: Equations & Fields
- Rectangular waveguide TE1,0 cutoff frequency calculator.
- Waveguide Component Vendors
- Waveguide Design Resources
- NEETS - Waveguide Theory and Application
- EWHBK, Microwave Waveguide and Coaxial Cable

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