INEL 5606 Dr. Sandra Cruz-Pol ECE, UPRM Circular Waveguides INEL 5606 Dr. Sandra Cruz-Pol ECE, UPRM http://www.rfwireless-world.com/Terminology/Rectangular-waveguide-vs-Circular-waveguide.html http://www.miwv.com/

Circular Waveguides In 1897 Lord Rayleigh performed the first theoretical analysis of a wave in a circular waveguide By Vanessaezekowitz - SVG version by Vanessa Ezekowitz, based on this PNG, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=4520294

Use Cylindrical coordinates

From Maxwell Eqs. we can derive for Ez and Hz, all other components: TEM waves not supported where And we have assumed waves travel to +z , so

TE Modes: (Ez=0) Solving the wave equation for Hz: Use Separation of Variables: In cylindrical coordinates:

Since Hz must be periodic: Since they are both equal to a constant Kc, we can separate by variables: Since Hz must be periodic: kf must be an integer

Jn=Bessel function of the 1st kind Yn=Bessel function of the 2nd Kind This is general solution for Bessel’s Equation! This is Bessel’s Equation! Jn=Bessel function of the 1st kind Yn=Bessel function of the 2nd Kind n is the order

Bessel’s Functions Jn Yn Similar to Sine and Cosine but amplitude goes down w/argument Bessel functions are the radial part of the modes of vibration of a circular drum and circular antennas! Jn Yn n is the order

Bessel Function of the 1st Kind

So we are left with: We need to satisfy: Therefore, we need: derivative Therefore, we need:

Substituting The Cutoff frequency is: Note we have A and B, which depend on excited power.

TE11 is the dominant mode Due to symmetry of guide, we can rotate the axis of the coordinate system so that either A or B are zero:

TM Modes: (Hz=0) Solving the wave equation for Ez: Use Separation of Variables: In cylindrical coordinates: Following similar procedure as for TE, now for TM we obtain:

Bessel

The Propagation Constant The cutoff frequency:

TM fields and impedance

Cylindrical Geometry

Modes of Propagation https://www.youtube.com/watch?v=kp33ZprO0Ck

Dominant Mode

http://www. rfcafe. com/references/electrical/circular-waveguide-modes http://www.rfcafe.com/references/electrical/circular-waveguide-modes.htm

Conventional sizes d=2a

Advantages Circular polarization waves and virtually any other type of polarization can be propagated thru it. Circular waveguides offer implementation advantages over rectangular waveguide in that installation is much simpler when forming runs for turns and offsets. Manufacturing is generally simpler, too. http://www.rfcafe.com/references/electrical/circular-waveguide-modes.htm