Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Home exercise 7.4.1

Interactive: Circular Waveguide Modes

Home exercise 7.4.1

A mode census

An air-filled circular waveguide has an inner diameter of $5.0$ cm and is fed at $f = 10$ GHz.

List all propagating modes with their cutoff frequencies.
How many independent field patterns propagate, counting both polarisations of each mode with $m \geq 1$ ?
In which frequency band does this guide carry TE $_{11}$ alone?

Cutoffs in units of f_{c,\mathrm{TE}_{11}}, blue: 10 GHz. — Cutoffs in units of $f_{c, TE_{11}}$ , blue: $10$ GHz.

Try it yourself first.

Strategy

A mode propagates when its zero $x$ is smaller than $k a$ . Here $k a = 2 π f a / c = 5.236$ , so the table of zeros of $J_{m}$ and $J_{m}^{'}$ up to $5.24$ gives the list.

$a = 2.5$ cm, $c / (2 π a) = 1.910$ GHz, $f_{c} = x \times 1.910$ GHz:

mode	$x$	$f_{c}$ [GHz]
TE $_{11}$	$1.841$	$3.52$
TM $_{01}$	$2.405$	$4.59$
TE $_{21}$	$3.054$	$5.83$
TE $_{01}$	$3.832$	$7.32$
TM $_{11}$	$3.832$	$7.32$
TE $_{31}$	$4.201$	$8.02$
TM $_{21}$	$5.136$	$9.81$

The next ones, TE $_{41}$ ( $x_{41}^{'} = 5.318$ , $10.16$ GHz) and TE $_{12}$ ( $x_{12}^{'} = 5.331$ , $10.18$ GHz), are just cut off.

Seven modes. TM $_{01}$ and TE $_{01}$ ( $m = 0$ ) have one pattern each, the five others two: $2 + 5 \times 2 = 12$ field patterns.

TE $_{11}$ alone between its cutoff and that of TM $_{01}$ : $3.52 < f < 4.59$ GHz.

Answer

7 modes, 12 field patterns at 10 GHz TE_{11} alone for 3.52 < f < 4.59 GHz

Count the bars crossed by the $10 GHz$ line below, then turn on the table of Bessel zeros and compare it with the table of part (a).

InteractiveLoading simulation…

Check

On the cutoff chart the $10$ GHz line lies at $10/3.516 = 2.844 f_{c, TE_{11}}$ . It crosses the bars of TE $_{11}$ , TM $_{01}$ , TE $_{21}$ , TE $_{01}$ , TM $_{11}$ , TE $_{31}$ and TM $_{21}$ ( $2.789$ ), but not those of TE $_{41}$ ( $2.888$ ) and TE $_{12}$ ( $2.896$ ).

Remark

A $5$ cm pipe at $10$ GHz is heavily overmoded: any bend or step converts the input mode into the others, each with its own phase velocity. For single-mode work at $10$ GHz the diameter must lie between $1.76$ and $2.30$ cm, from $8.79 GHz \cdot cm / a < 10 GHz < 11.48 GHz \cdot cm / a$ .

Home exercise 7.4.1: A mode census