Itroductio Circular waveguides Waveguides ca be simply described as metal pipes. Depedig o their cross sectio there are rectagular waveguides (described i separate tutorial) ad circular waveguides, which cross sectio is simply a circle. This tutorial is dedicated to basic properties of circular waveguides. For properties visualisatio the electromagetic simulatios i QuickWave software are used. All examples used here were prepared i free CAD QW-Modeller for QuickWave ad the models preparatio procedure is described i separate documets. All examples cosidered herei are icluded i the QW-Modeller ad QuickWave STUDENT Release istallatio as both, QW-Modeller ad QW-Editor projects. Table of Cotets CUTOFF FREQUENCY... 2 PROPAGATION MODES IN CIRCULAR WAVEGUIDE... 5 MODE TE11... 5 MODE TM01... 9 1
Cutoff frequecy Similarly as i the case of rectagular waveguides, propagatio i circular waveguides is determied by a cutoff frequecy. The cutoff frequecy is uique for a particular waveguide mode that is supposed to be propagatig i a waveguide of a give diameter ad determies the lower frequecy of the waveguide s operatig frequecy rage. The cutoff frequecy for circular waveguide is calculated usig the followig formula: where: v 2 f c, c, v stads for a wave velocity i a medium fillig the waveguide, c, is a cutoff phase costat which is calculated accordig to the formulae give i Table 1. Table 1 Cutoff phase costat formulae for circular waveguide modes. TE (H) mode (Trasverse Electric) c, a TM (E) mode (Trasverse Magetic), c, a where: -th root of m-th Bessel fuctio, -th root of the m-th Bessel fuctio derivative, a radius of the circular waveguide. Several Bessel fuctios ad Bessel fuctios derivatives are show i Fig. 1 ad Fig. 2. 2
Fig. 1 Bessel fuctios of the first kid Fig. 2 Derivatives of Bessel fuctios of the first kid. 3
For the egieers coveiece the values of Bessel fuctios ad Bessel fuctios derivatives are commoly give i tables (see Table 2). Table 2 Values of Bessel fuctios ad Bessel fuctios derivatives. Fuctio umber Root umber Roots of the Bessel fuctio Roots of the Bessel fuctio 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 a example, the cutoff frequecies of the TE 11 ad TM 01 modes i the circular waveguide with radius of a=10 c filled with air ca be calculated as follows: TE 11 mode: f c, TE c fc, TE11 2 a c 2 a 1,1 1,1 1.841 c 1.841 c, 879MHz 11 2 a f TE TM 01 mode: f c, TM c fc, TM 01 2 a c 2 a 0,1 0,1 2.405 c 2.405 c, 1148MHz 01 2 a f TM 4
Propagatio modes i circular waveguide Each waveguide mode is described by uique distributio of trasverse ad logitudial compoets of the electric ad magetic fields. Similarly to rectagular waveguides, two kids of waveguide modes are recogised i case of circular waveguides: TE ad TM. The waveguide mode i circular waveguide is described with m ad idexes, which stad for the field variatio i radial ad axial directios respectively. I case of circular waveguides the fudametal mode is TE 11. Mode TE 11 I this part, the distributio of trasverse ad logitudial fields compoets of TE 11 mode is ivestigated. For fields visualisatio the QuickWave software is used. The circular waveguide with radius of 10 cm ad the legth of 30 cm is cosidered. The model of such waveguide cir.qwpro ca be loaded i QW-Modeller. The cutoff frequecy of TE 11 mode i this waveguide is 0.879 GHz. The waveguide is excited at 1 GHz ad its legth is aroud half of a guide wavelegth. Ru the electromagetic simulatio with QuickWave usig Start 3). butto i Simulatio tab (Fig. Fig. 3 Simulatio tab i QW-Modeller. Press butto i 2D/3D Fields tab of QW-Simulator to ope fields visualisatio widow. The cosecutive displays show i Figs. 4-9 may be viewed by pressig butto for several times (oce for obtaiig each of the followig displays). For the visualisatio coveiece the display widows may be maximised. I case of TE 11 mode, both radial ad axial compoets of trasverse fields exists ( idexes are o-zero), resultig i the distributio of total electric ad magetic field i the waveguide s cross sectio as show i Fig. 4 ad Fig. 5 respectively. The followig pictures show the displays of electric ad magetic fields alog the waveguide. It is clearly see that there is o logitudial compoet (i the directio of wave propagatio Z directio) of the electric field ad there is logitudial magetic field oly i this case. The displays cofirm the waveguide s legth to be half of guide wavelegth sice oe wave half ca be recogised alog waveguide. 5
The fields distributio displayes are give for a radomly chose time momet. Whe the simulatio results are observed o-lie the fields variatios as the wave is passig the waveguide will be observed. Fig. 4 A distributio of electric field for TE11 mode i a cross sectio of circular waveguide (YX plae). Fig. 5 A distributio of magetic field for TE11 mode i a cross sectio of circular waveguide (YX plae). 6
Fig. 6 A distributio of electric field for TE11 mode i circular waveguide (XZ plae i the middle of the waveguide). 7
Fig. 7 A distributio of magetic field for TE11 mode i circular waveguide (XZ plae i the middle of the waveguide). Fig. 8 A distributio of electric field for TE11 mode i circular waveguide (YZ plae i the middle of the waveguide). 8
(a) (b) Fig. 9 A distributio of magetic field for TE11 mode i circular waveguide (YZ plae): i the middle of the waveguide (a) ad ear the waveguide wall (b). Mode TM 01 I this part of the tutorial the fields distributio for the TM 01 mode i a circular waveguide is preseted. As previously, the waveguide i 30 cm log ad its radius is 10 cm. The cutoff frequecy for the cosidered mode is 1.148 GHz, thus the waveguide is excited at 2 GHz, which is above the cutoff frequecy. The legth of the waveguide correspods to 1.5 of guide wavelegth. 9
The waveguide model is cotaied i cir_tm.qwpro sceario. For fields visualisatio the model ca be loaded i QW-Modeller, from where the electromagetic simulatio i QuickWave ca be ru. Press butto i 2D/3D Fields tab of QW-Simulator to ope fields visualisatio widow. The cosecutive displays show i Figs. 10-13 may be viewed by pressig butto for several times (oce for obtaiig each of the followig displays). For the visualisatio coveiece the display widows may be maximised. The TM mode meas that there is o magetic field compoet i the directio of wave propagatio. The values of m ad idexes idicate that the trasverse magetic field has oly a axial compoet (=1), thus the trasverse electric field has oly a radial compoet. Fig. 10 ad Fig. 11 show the trasverse electric ad magetic fields distributios respectively. Fig. 12 ad Fig. 13 preset the distributio of the electric ad magetic fields alog the waveguide i ZX plae (cross sectio i the middle of the waveguide). It ca be clearly see that oly the electric field has a logitudial compoet (alog the directio of wave propagatio). It is also well visible that the waveguide s legth is 1.5 of guide wavelegth sice three wave halves ca be recogised i the fields distributios i ZX plae. The fields distributios i ZY plae are the same as i ZX plae sice the TM 01 mode is characterised by a axial symmetry. Fig. 10 A distributio of electric field for TM01 mode i a cross sectio of circular waveguide (YX plae). 10
Fig. 11 A distributio of magetic field for TM01 mode i a cross sectio of circular waveguide (YX plae). Fig. 12 A distributio of electric field for TM01 mode alog the circular waveguide (ZX plae i the middle of the waveguide). 11
l Fig. 13 A distributio of magetic field for TM01 mode alog the circular waveguide (ZX plae i the middle of the waveguide). 12