
Properties of Modes in a Circular Waveguide


Circular
waveguides offer implementation advantages over
rectangular waveguide in that installation is much simpler when forming runs for turns and offsets 
particularly when large radii are involved  and the wind loading is less on a round crosssection, meaning towers
do not need to be as robust. Manufacturing is generally simpler, too, since only one dimension  the radius 
needs to be maintained. Applications where differential rotation is required, like a rotary joint for a radar
antenna, absolutely require a circular crosssection, so even if rectangular waveguide is used for the primary
routing, a transition to circular  and then possibly back to rectangular  is needed.
Calculations
for circular waveguide requires the application of Bessel functions, so working equations with a cheap calculator
is not going to happen. However, even spreadsheets have Bessel function (J_{n}) capability nowadays, so
determining
cutoff frequencies, field strengths, and any of the other standard
values associated with circular waveguide can be done relatively easily. The formulas below represent those
quantities most commonly needed for circular waveguides. Please see the figure at the right for variable
references.
Note: I received the following note from Brian Sequeira,
of the Johns Hopkins University Applied Physics Laboratory. "I reviewed tables on rectangular and circular
waveguides, and based on my experience of what confuses firsttime readers and what does not, I made adjustments
to notation & symbols, corrected a couple of sign errors, and put expressions in a form that make their units more
apparent." The table for circular waveguide can be viewed fullsize by clicking on the thumbnail to the right.
Brian also provided a table for rectangular waveguide.
Quantity 
TE Modes 
TM Modes 
H_{z} 

0 
E_{z} 
0 

H_{r} 


H_{ϕ} 


E_{r} 


E_{ϕ} 


β_{nm} 


Z_{h,nm} 


Z_{e,nm} 


k_{c,nm} 


λ_{c,nm} 


Power^{††} 


α^{†} 


† ^{
} 
The expression for α is not valid for
degenerate modes. 
Equations derived from "Foundations for Microwave Engineering, R.E. Collin, McGrawHill

†† Thanks to Patrick L. for finding error where
"4" in denominator should be "2." 
Values of p_{nm} for TM Modes
n 
p_{n1} 
p_{n2} 
p_{n3} 
0 
2.405 
5.520 
8.654 
1 
3.832 
7.016 
10.174 
2 
5.135 
8.417 
11.620 
Values of p'_{nm} for TE Modes
n 
p'_{n1} 
p'_{n2} 
p'_{n3} 
0 
3.832 
7.016 
10.174 
1 
1.841 
5.331 
8.536 
2 
3.054 
6.706 
9.970 



