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MECA Electronics

Properties of Modes in a Circular Waveguide

Circular waveguide formula variables - RF CafeCircular 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 cross-section, 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 cross-section, so even if rectangular waveguide is used for the primary routing, a transition to circular - and then possibly back to rectangular - is needed.

Tabel of Time-Harmonic Mode Properties in a Circular Waveguide, by Brian Sequeira - RF CafeCalculations 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 (Jn) 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 first-time 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 full-size by clicking on the thumbnail to the right. Brian also provided a table for rectangular waveguide.

QuantityTE ModesTM Modes
HzRF Cafe: Circular waveguide equations0
Ez0RF Cafe: Circular waveguide equations
HrRF Cafe: Circular waveguide equationsRF Cafe: Circular waveguide equations
HϕRF Cafe: Circular waveguide equationsRF Cafe: Circular waveguide equations
ErRF Cafe: Circular waveguide equationsRF Cafe: Circular waveguide equations
EϕRF Cafe: Circular waveguide equationsRF Cafe: Circular waveguide equations
βnmRF Cafe: Circular waveguide equationsRF Cafe: Circular waveguide equations
Zh,nmRF Cafe: Circular waveguide equations 
Ze,nm RF Cafe: Circular waveguide equations
kc,nmRF Cafe: Circular waveguide equationsRF Cafe: Circular waveguide equations
λc,nmRF Cafe: Circular waveguide equationsRF Cafe: Circular waveguide equations
Power††RF Cafe: Circular waveguide equationsRF Cafe: Circular waveguide equations
αRF Cafe: Circular waveguide equationsRF Cafe: Circular waveguide equations
RF Cafe: Circular waveguide equations
The expression for α is not valid for degenerate modes.
Equations derived from "Foundations for Microwave Engineering, R.E. Collin, McGraw-Hill
†† Thanks to Patrick L. for finding error where "4" in denominator should be "2."

Values of pnm for TM Modes

npn1pn2pn3
02.4055.5208.654
13.8327.01610.174
25.1358.41711.620

Values of p'nm for TE Modes

np'n1p'n2p'n3
03.8327.01610.174
11.8415.3318.536
23.0546.7069.970

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
- NEETS - Waveguide Theory and Application
- EWHBK, Microwave Waveguide and Coaxial Cable

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