3.1 Cutoff frequency
Similarly as in the case of rectangular waveguides, propagation in circular waveguides is determined by a cutoff frequency. The cutoff frequency is unique for a particular waveguide mode that is supposed to be propagating in a waveguide of a given diameter and determines the lower frequency of the waveguide’s operating frequency range.
The cutoff frequency for circular waveguide is calculated using the following formula:
where:
stands for a wave velocity in a medium filling the waveguide, bc, m,n is a cutoff phase constant which is calculated according to the formulae given below:
TE (H) mode (Transverse Electric): 
TM (E) mode (Transverse Magnetic): 
where:
– n-th root of m-th Bessel function,
– n-th root of the m-th Bessel function derivative ,
a– radius of the circular waveguide.
Several Bessel functions and Bessel functions derivatives are shown in Fig. 1 and Fig. 2.
Fig. 1 Bessel functions of the first kind
Fig. 2 Derivatives of Bessel functions of the first kind.
For the engineers’ convenience the values of Bessel functions and Bessel functions derivatives are commonly given in tables (see Table 1).
Table 1 Values of Bessel functions and Bessel functions derivatives.
|
Function number
|
Root number
|
Roots of the Bessel function
|
Roots of the Bessel function derivatives
|
|
0 |
1 |
2,405 |
3,832 |
|
0 |
2 |
5,520 |
7,016 |
|
0 |
3 |
8,654 |
10,173 |
|
1 |
1 |
3,832 |
1,841 |
|
1 |
2 |
7,016 |
5,331 |
|
2 |
1 |
5,136 |
3,054 |
|
2 |
2 |
8,417 |
6,706 |
|
3 |
1 |
6,380 |
4,201 |
As an example, the cutoff frequencies of the TE11 and TM01 modes in the circular waveguide with radius of a=10 cm, filled with air can be calculated as follows:
TE11 mode:
TM01 mode:
