BSTJ 40: 1. January 1961: Normal Modes and Mode Conversion in Helix Waveguide. (Unger, H.G.)

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Normal Modes and Mode Conversion
in Helix Waveguide

By H.4. UNGER

Helis woseguie, compu of ctvety word slat copper ere ect
vat enh aw acre oe martin sort, raat ev eee raves
eth fon lms, Aerhanonltwfertona, such a errs eet dara

toe vnplng bere Rive tri aes et att ee
Vgc the tonoyssnn Fa sai Pais wc for a paste

tac appfinsan, «frst wed eft thet mininaoo he fae
Uagradation, Une mnie charters ud Bast ephing a

tres be ruse Bn atten a tony te rote af & ace
Span soning mii Bese fons

I penn es boo ep fr mutiny Siding
by ime mpousiaton, St fou the name oats Sate chat
amavis be vhavotorils een a ted ta all yet sue of

we the pe! ent ll wee of Stren. The sepvarneton

avntcice ace (eton af ell impacts ta
Vinipnaton of motes Btempnsnne sosegaide, The Ty moaen of hit
avepniie nth 27 50% hace oly 0 Tattd atevetion. Phos ete
titer tie desig of made flooe. Movefacuriny simpenfietions ine
een te every By aa dsp fhe wh dupa. ern
‘Sonatave wih dee eareatintxnace J paced yagi tenor,
Dt ie conti Go the arage Tae ie nite in @ Rel vege
toith ey Pre wet pea,

omg vate

oie wove esl wound from salted eopper wirw dow
cee wth on aleneptve or rove vee i 9 goa Usman mc
{Tis freer ete sven! a ong rnnnieation, ave
eve ca be big el 2 ale er, to negotiate ben om

Tetras, lo serve ae the reais Be prope
"Asin medal wave, the hee ral electric waves deena
stony nis fron oni if pelee: heli wove. Any curv
25 ame meu aveeioe eines goUNNAt, sAseau LORE

tre of the guide as, femal of the ems ston or devition of
"he wing ei le nd ver pel es oe nl degre
‘he transsson chartoretin«

Tn a pevfert elix waveguide cioularclecris wavs propagate a
iturbed. Leapetione esti coupling betwen etcalar eee waves
tnd other modes. Power is leet by onversion to umanted made and
reconversion start any month transmission characte.

Ts onder ta censtel mice enivesion and remaverinn in ix we
side with practes imperfection, and tleo to deren. heli waveguides
for mode filters and intentional beads, che unmaated mode character
iste and unyaated mode couplng must be invested. eis calcu
latinas have rested inn euruteriie ein hil ply de
ermine The propectow uf hele woguidomuae? wre le in exit
xpress fe varvan coupling ecient? ?* Nemeriew) evaluatone
Bf these equaciens have been veny informative. Thy wer, hewn,
ot compete enragh ln reveal all Une newaned rele popestin onl
end not serve ae &. kee foe els wave deg in every pple

‘he neues of a nore extant xe vmod ration lic wave
side eqonriene il ho prewated bene Ln few typi exp sese
results will ke applied to nex waweguice detign probleme, Mist the
cequutione whiet deseo wuve oropusution im pesfect anal impefee:
Teli wae wile Fed

bets waveguide (Hg. 1) wil cll peo whe the helix fue
‘rah eeelareylinder al i won with ioe and nif pith

ie mathenitical model which then replace st ie am anisorgopi in

pedance sheet at radius a conducting pesfectty in cincumferential dire
‘ton but with a wal impedance Zin aialdizeetion. Tho Z rplaces the
jek srrounding the Helix aks ini eran Ui Gite se of
hele wines ‘The elerlewngnetin Geld omsponen in ended eae
dinate system (ra) ate hon subjen at — a Uo the boundary cone
‘tions

@

@

Solutions of Maxwell's equations in eylindies! gonna are Bese
[SOLMUAL MODE AN MOR. CONTERSTON 2st

ig 1 This navies hound rondo

otons fy de li rigonometie Funetios of the anal ad
sspoaetal fayetnns of the aa. dst

Ell
22) bon

where pis the asisuthal oor of the wae. "Phe asi! propagation eon
ant aed taal propane ane vate wi the 28-
Ini pongacer: eon tu we of Ie wera ing te weweguide

(acu ”

Won te bonndary onditions (2) aud (2 ae igposed on the solutions
(2) af Mayle rquations the fllowing ebaraeteristi egal reals

atta)
Jad - oe
PETA Ae

‘Values a > Year atnty she eharsentic equation (3 are the preps:
‘iow eonstones of norms modes uf th por: ix waco. Ths
Alserite wave propasstion in pert ele ease completely
256 MULE ATEN MeeRICRE JOURN, TANTARY 1961

Fave pation in ampere e's sence a Lee decribed
hay eenvalinn telegraphs oqo In a perfect bel waves
‘veri! mode w of emplitude , Ry propagates independency ‘rom sl
‘ther re

Moy

= Ally wo

Impentome eau incewetion Hote ene oa ne weave wp
sles nee nals coupled

WB

am

B= Leake

"The coupling coetTicients ace determines by the kind and sizeof the im
pesfetion, But they are alo tama dependent on the wall impedance

Fr irenlarvlccre wave applications, only coupling petneen these
and ofaer eves Ie of interest. Coupling coeconts of typical imper
‘retin in alix waveguide willbe leted. Tho subserine m wil reer
[5 he Tyg wave; ew oe te ans of tas souoled moda A normal
sito felon

TEI

%

Brig —nyyte de
wre net

with

w

0)

a Cure

“There i ony coupling tern siculae netic mes and mes of
fase aaimchel coder in a curve bee meee
= END thE Tyo toy an

oe Me Ve RL an geal IR

one ia he ning of ryan,
XOMIAT WOBRE AND MULE CONELIOS nt

2 Deformation of th Coos Seton
‘Phe redive oof «deformed guide of muni mains # oan he wetten
a = ah + Ey 0 ped 2)

Back componel wil erase coupling bet sseule elects modes
‘and mae of eaeacel ord p

ig ym he
ne ean

Hie Wong!

bay an

a Teo

Ie poriet oi wage te ge Velanen a eli wite and the
rom sls i stall enough © be regia se veo. Than egular

sietng hr angen bowie
¥ = Diane oy

ae mnt ee coin ae of altel ade
gg = Mtb 8)

"The prmpazation votsLant of normal aedes in Helis ssid 2, 9
4, sl £3), only split yin a a Fonet on of freqnoury ard wae
‘aie pmentoetere Pie you ito find be eoumple sels o£ eae
“oeuetal ane compas oy am

"Wit p+ ean be elvis! frm 3, The, form oe fhequancy
fang guide nase the saaritsie eyuation deny © a a Foneson
mee

Feb) 0. a6,
oe Z = Ute suanasoristieonuation degra into
i — 0, = 0, a

Ue sms of whi ongesponil be TNT snd IE wence ees
mela wre,

‘tasting fo the nose rool of (17 for Z — 0, the wlio of (15)
fo helix waveruie erie tram by enedlfmerosing the wal ime
26) amp mata, sreymAromemteAT, aHNKt, JANCANOY HT

prota. Tf fy al Zy a on soon of (6), then an apie
alae fr the statin ab 2, = Ze} 32 given by

3 Reb,
ey, i)

ka, Za]

A bertae appraxiniatin ie Zand by Newon's fem:
Hb
Ariz

oo

Tne farther maprovement, che proses (10) ean be raped 0 any dee
ine anoracy.

"Te fs eel he alsrting pnt fare mest ol tthe ashen
ing value of wall impedanes, Pur che numeri evuluston, the well
inpedanco wae rated ta the impedance ef tre epuce Zy = V/s

Zaye con

‘The sototons were traced along Hines of constant phase © of Z. The
inecompct bo wae varied and kopt sufiiently sual to insure continuity
che proce,

he evalstton we prugmmied ry Vi. ©. Te Realty for wstomutie
esectinn on on IBM TO4 Daca Processnu Sete

p characershis cgi sno. evalesie far a wall pean
swith passive pays solu alen mp ANN ss AN Te
‘Sons ese tod whi fr neo wal aptace tr Uae Following
rmotallie wer gide mndes:

"Ta, YMa THe, TM, THe, TMs
Tha, TMa, THe, TMs,
Ty, TMa, TBs, TB,

For sue speria! wall impedance phases che evaluations were extended

oer many more modes. A value of @/h — 17 ws essued eorespond:

ing to a eeator Fequeney: of Fae jones 6 a 98 he egy bd

forthe inch inside dmc wns yr.

Tn mumnerial resi were ald uted to cateTase fram the eenaration

constant, the propagation eostunt i ile nal ad enone pale
igs. 2 though 0 ure pllled fm Use cri, These gras hi
Sunast ons a azobu Cow OS ot

ig ently ne eile of

conten ines of constant pe sd constens em
impedauce drown in the compies plane o propagation eonstant 7. The
fee on Hefei hax beetles hy he Ey phase Gaz = 2:805
fd represents the iference ia phe cenwiaal between Ml, and te
lle mod

Tar lingam for a purtculue value of, peeing the respective

lite p of the wal
QML me ania sxereg rnewNteAL sSAt, 2AKART 19

ures for wall impedanoes with complex phase fan ea int the
‘plane, Some retum {o the imaginary assy others continue mone and
‘ore ous ta ever ineteasing vatues ofthe actemtation constant.

The propegstion conetant ir & mnitivalund Sunetion of ube wall
pedance. Forany one wall padanee value thre xe a9 may dir
ates ofthe propagasion ennstant ce there ave points of end wall ho

Fig 5 Prongguioa conta » = ap +t + ate in atc wos of

wal Eaprdanceouioar ny how 1 eeede apres pe gle
Sonus MoDus sn" sO, CONFEBSION 285

(G+ 48 In Re: ware

raotion costent caress

weet With respect nny field component

le lsignton of metallic waveguide
i Nevertbles, the mode designation
red hele vei, for that
20) mus wou evaTEW geomNtest socRNac, saNCARE IHL

atten, on any’ heterogeneous warexuide, when she yplane i divided
ftp 20 aut in ach ragien the propagation constant ie singesvaled
Tneion of the erica guice parameter. In the present case tie vall
Ipsos the exieal paras.

"The diving les i oe ple wil be araneh outs nthe Z-plene
‘Tiny ert the inte gor of Urauhes of 7 iors wach ebber, cash
ancl rrrespandiag co 8 Heix wiveguide male. The ranch cute of
“aul! conaeet the branch yin the Z-plane. "The branch pints
(ery im ho plane are male pins of 2 in The plane. Bran ras
If ehould therefore eo thmungh thy snlle its Ziv the plan
‘A taany branches mill hein ontcs at the cade point ae ie the onder
Of the eadle point. From iopection of the diagrams all saddle points
fre found to of rose! utes Heel, oly two branches of are
Sh eonfact at Those sede pints and only one dividing ine or branch
mas env ag ach,

"The ressiing path ef the brace ews ie arbitrary: They should
onvvenionty folow a course ih never ruta vonlour es of wopaat
hase nf che wall impedacee gal eds either in innit oro she foe
the pint faite wall sedans

er example, she brane et le TE aa TM startin Fig 2
at the eoresponding pint of infil wall ease cw neni
fontour ine & = 40°) orig fron Ty from the conto line (
435°) coming fom TMy. Somewbere in the +-plane the dividing ‘ine
hte w saddle poi of Z— J). Hevond this saddle point the branch
feats continued according to tae same vule, always separating contour
Tis f ennstant phare which originated ac diffrent pointe of zero wall

‘Farh sch rion, bounded by the dans wl the brah bs broken
Tins inthe diagrame) is ow designated by the zotalle waveguide mode
Ieee within, The sera medes of hex wares are then defined
snide, anda further asexual er ns
ese.

"Ta mode desigmation is bls wwvepuile az bo defined in fea
worn rs: Aue i li waveghide of fnite sl Smpedace fe
Heated ith the mele sa vege mode into shies it degenerates
‘vue the wll isp plate hepl-constant and the wall aspedance
split we ro

‘Modes in any Deleraaseitt waveguide ean enttespnstingly be ide
‘id with metallic wvoyuide unto oe the eiical parasite are
stbjested tn the rope Tinting moves AM erseal parameters should
Te heptrorstant exes Rha ane whieh in it mit ebatge tke patil