Indian Journa l or Pure & Applied Physics

Vo l. 37 . November 1999. pp. X4X-X52

Effect of surface roughness on electromagnetic propagation through waveguides

S K Popalghat;i. Ajay Chauclhari h & P B Patil h

"Department or Physics . J E S College. J,ilna 43 1 203

hDepartment of Phys ics. Dr BAM Uni vers ity. Aurangahad 431 O()4

Rece ived 20 M ay 1999: rev ised 27 Jul y 1999: accepted X September 19l)<)

The effect o f surface roughness on electromagnetic propagat ion throug h rectangu lar waveguide is stud ied using Fi r, ite Ele illent

Method. The eITect of surface roughness on TE to. TE20 mode cutoff frequencies and passbands is studied. The surface roug hness

on the wa ll s of the waveguide has been gcnerated by using the RND funct ion in Turbo Basic. Thc random va riation in cutorf

frequencics 1'0 1' T E lo. T E20 mode is observed. The passband decreases w ith increase in surface roughness.

Introduction The surface roughness of waveguide wa ll s remains a

subject of interest, for studying the different aspects of wave propagati on. The surface roughness of waveguide i ~ a practica l rea lity. lt may occur due to various reasons, such as polishing of irregul ar waveguide structure, un­even surface coating or co rruga ti on of surface whil e fa hri cating the waveguides . The probl em of roughness lllay also arise due to ex pos ure of wavegu ide to environ­lllcnt. The environmental corrosion increases with age­in g wh ich results in increasing roughness of waveguide wal ls. So, it becomes very important to study the effec t of surface rough ness on elec tromagnet ic wave propaga­ti on.

The effect of rough surface is also noted in the prop"­gati on of acoust ic waves as well. T he dependence of axial wave number on corrugation of acoustic duct is discussed by Rayleigh I. The effect of corrugation and surface roughness of waveguide on electromagneti c propagation is studied by different workers using vari­ous methods. Lennart and Lundqvi st2 in ves ti gated infi ­nite corrugated waveguides by use of the null fi eld approach. The refl ec ti on and tran smissi on coe ffi c ient s are determined by lllode lllatching and reported th at as 'orrugati un increases, the stopband broadens. Rozzi ('/

(II ' studi ed th e losses in curv ed d ie lec tri c rid ge w<lvegui de /"0 1' va ri ous va lues of depth of corrugat ion. The corru ga ti on studi ed by them was due to fabricati on. Dispers ion characterist ics were calcul ated by Markkll

\ - Illail address: halllll<lll r <!!) hoIl14. vsnl.nel.in

and Oksanen.J fo r multide pth co rru gated diel e tri c loaded wavegu ides. Benali el ({IS stuclied the scatterin g of e lec tromagneti c waves from conducti ng di elec tric rough surfaces. Erkin el a/' (' had studi ed the radiati on angle and radiati on effi c iency of millimete r wave in corru gated ferrite slab structure whi ch h a~ signifi ca nt importance in hi gh reso lution radar.

In the work menti oned above the surface roughness is represented by some mathemati cal model or <;ome peri od ic functi on such as sinusoidal , tri angul ar or rec­tangul ar function and are studi ed by Rayleigh' s met hod. perturbation method, Besse l series method or in ver\c Fourier tran sform method.

Bu t corrugation or roughness of surface is random in nature so it s mathemat ica l mode l is not a real repre­sentation. Approaching to a close realit y the authors have considered the roughness as a random functi on. The surface with heights and depths are di stributed randomly. The magnitude of surface height and depth (surface amplitude) is a l ~o considered randoml y wi thin some specified limit. A specified limiting magnitude of surface height and depth is considered as a measure o /" surface roughness in that C~l.~e .

To deal wit h these irregul ar and randoll1 geometry o/" waveguides of rough surface, finite Ele ment l'vle thod with variati onal principle is the appropri ate choicc7 The roughness of waveguide surface is defined at the nodcs on the boundary elements, located on the waveguide surface by a rand om fun cti on. The rough sur face i ~

simulated by computer by using RND func ti on in Turho Basic. The TE lo and TE2() modes are work ed out by

r

POPALGHAT 1'1 ai.: EM PROPAGATION THROUGH WAVEGUlDES X4lJ

considering propagati on through waveguide as an eigenva lue problem. As the nature of surface is random, the TEJI) and TE20 modes of propagation are estimated for same measure of roughness in fifteen different sa lll­piing events for better ana lys is.

2 Statement of the Problem Consider a homogeneous rectangular waveguide

with air as a dielec tri c medium inside. The wa lls of the wavegui de are perfec tl y conducting havi ng random roughn ess. The boundari es i. e. th e wal ls of th e waveguide appear straight lines i r measure of roughness is zero, otherwise it appears as zigzag lines on proper magnifi cat ion.

The cross-section of waveguide in X-Y plane of Cartes ian coordinate system is considered as a problem

domain Q , with zigzag line boundari es. The nodes ap­pearing on domain boundary lines parallel to X-axi s, have va ryin g Y - coordinate randoml y and nodes appear­ing on bOllndary lines paralle l to Y-axis ha ve X-coordi ­nate varyin g rand oml y within a spec ifi ed limit of magnitude of surface amplitude, whi ch vary above and below the boundary surface. This limit defin es the meas­ure of rou ghness of the surface.

3 Variational Formulation The electric and magnetic fi e ld s inside the waveguide

will sati sfy the Ma xwe ll 's equati ons. With the time

dependence as ex p( iron , the Max well 's curl equations can be combined to give

V x (£-1 V x H ) = K2 H .. . ( I )

') ? where K- = ro-~ltl£tl

Here ro is the angular frequency, £" the permitti vity

and ~lo the permeability of free space. £ is the relati ve permitti vity of the med ium, in side the waveguide . S ince

there is air med ium inside the waveguide, so £ = I . The magnetic fi e ld H must sa tisfy the suitable bou nd­

ary condition at the conductin g boundari es, given by

(II x V X H )Ptl'lIld: l1 ) = () ... (2)

since the tangenti al co mponent of the e lectri c fie ld is 7.e ro on conductor boundari es .

Thus the problem of electromagnetic propaga ti on under considerati on invo lves so lvin g of Eq. ( I ) sub­jected to condition of Eq. (2).

To avo id the spurious solutions, the magnetic fi e ld vector formul at ionK is used for the so lution of the prob­lem.

The ex pression fo r the fun ctional n is :

n = ~ J I ( V x H " ) . ( £- 1 \7 x H ) -

K 2 H ' · H + ( V . H ' ) . ( V . H) 18 Q .. . (3)

The stationary character of n requires 8n = O.

The first variation in n is given by:

8 I f r ( V x H ' ) . ( £- 1 V x H ) -

K 2 H ' . H + ( V . H " ) . ( V . H ) Id Q = () ... (4)

The domain i.e. cross secti on of wa veguide is di videe.! . I I . I t' I 'J 10 TI t' Int o rectangu ar e ements wit 1 'our noc es · . le unc-ti onal over each element is:

n" = ~ r f l ( V x H "' ) . ( £- 1 V x H " ) d Q ' -

f K 2 H " . . H " d Q " + J( V . H ". ) . ( V . He ) d Q I

.. . (:'))

where the unknown vector He has three component s H,. H y ancl H 7 . Each unknown component H " H , and H , is

defined in terms of the nodal values H I,' , H :' and H ~'

respectively. The linear mapping function is used , whi ch needs

four nodes per rectangul ar element.

Functi onal n° can be written as n<' = ( 112) I I WIT I S" I {H" } - K 2 I H " IT rT'1 {He I I

... (6)

The fun cti onal for the who le region Q is given hy:

n = ( 1/2) I {H I r I S I I H I - K 2 I HIT IT I II-I I I ... (7 )

The conditi on on /o, IHI = () leads to the foll ow ing matri x equation

lSI {H} _K2ITI {HI =0 ... (8)

Eq. (7) is the matri x equati on to be so lved fo r eigen-

values K" = A and eigenvectors i.e. H fi e ld compo-nents.

4 Numerical Calculations Consider a rectangul ar waveguide with cross-sec ti on

of length a = 2.4 cm and breadth h = 1.2 cm. The systemati c analysis of the electromagneti c pro p~lg, lli ()1l

fo r diffe re nt measures o f rand o m rough ness o r wavegui de surface is carried out. The roughness of surface is varied from + 0.04 cm ( 0 + O. 16 cm in t he step of + 0.02 cm.

Cross-secti on of th is waveguide is di vided i nlo 128

rectangular elements with four nodes per e lement. The total number of nodes in the geometry are I S3 Ilodes,

out of which 48 nodes are the boundary nodes. on which

850 INDIAN J PURE APPL PHYS, VOL 37, NOVEMBER 1 999

boundary conditions are specified. For each node there are three unknown field components . Therefore the ma­trix problem to be solved for the system, consists of

matrices of the order 459 x 459. As these matrices are symmetric, only half matrices

are stored in the memory . The sky l ine storage has been used for storing these matrices. The subspace i teration method I I has been used, to solve this eigenvalue prob­lem, which consists of the fol lowing three steps.

( I ) Estab l ishment of q starting iteration vectors, q > p, where fJ is the number of eigenvalues and vectors to be calculated.

(2) Use of s imultaneous inverse i teration on the q vectors and Ritz analysis to extract the best eigenvalue and eigenvector approximations from the q i terat ion vectors.

(3) Use of the strum sequence check to verify whether some eigenvalues are missed in the set that i s calculated.

Roughness of waveguide surface is developed in the computer programme by us ing the function RND in the Turbo Basic . The RND function generates the random number between 0 and I . The roughness measure i s decided by using the value generated by RND funct ion

and the maxi mum roughness value, i .e . ± 0.04 to ± 0. 1 6 cm.

S ince the surface roughness is considered as a random function i n d i str ibut ion of surface ampl i tude, the waveguide with same measure of surface roughness is analyzed for 1 5 different samples. For example, the roughness of + 0.04 cm is developed for 1 5 t imes and each t ime the cut-off frequency for TEI O and TE20 modes are worked out. The analysis is repeated for 8 different

surface roughness measures, v iz. ± 0.04 cm. , ± 0.06 cm. ,

± 0.08 cm. , ± O. I cm. , ± 0. 1 2 cm. , ± 0. 1 4 cm. and ± 0. 1 6 cm.

5 Results

The TEl o mode frequency varies randomly from 6. 1 348 GHz to 6.4532 GHz. The random variation of TElo mode frequency increases with measure of surface roughness. (TEI O mode frequency is 6.2579 GHz for smooth surface of X-band rectangular waveguide with zero measure roughness) . The variat ion in TE l o mode cut-off frequency for d ifferent roughness measures is given in Table I .

The TE20 mode frequency varies randomly about standard rectangular X-band waveguide TE20 mode fre-

Table 1 - Variation of TEIO mode cut-off frequency with surface roughness for 1 5 different random variations

Sr. No

2

4

7

9

1 0

\ I 1 2

1 1

1 4

1 5

± O.O4

6 .2663

6.2735

6 .2 1 70

6.2755

6.2604

6 .2292

6 .244X

6. 2793

6.25 1 3

6.2382

6.254 1

6 .2 1 49

6.2508

6 .2675

6.27 6 1

±O 06

6.2737

6.278 1

6.3099

6. 1 960

6.305 1

6. 1 866

6 .2422

6 .293 1

6 .2568

6.2045

6.2048

6.3255

6.2723

6 .2 1 09

6.26 1 3

fin GHz for surface roughness i n cm

± 0.08 ± O. I O ± 0. 1 2 ± 0. 1 4 ± 0. 1 6

6.2998 6.2864 6. 1 9 10 6.2938 6.3343

6.3384 6.3047 6 .329 1 6 .2495 6.2624

6.3083 6.3 1 08 6.2768 6.3242 6.4 1 42

6.2879 6.3677 6 .2 1 1 1 6 .1895 6 .2971

6.2699 6 .2628 6 .2855 6.2626 6.2797

6.2402 6.2 1 42 6 .3049 6.3948 6.4532

6 .2672 6.352 1 6 .3 1 70 6.3354 6.20 1 2

6 .2026 6. 1 348 6.3 1 23 63800 6. 1 835

6 .2692 6.3373 6. 1 49 1 6.2853 6.3664

6. 1 966 6.2279 6. 1 8 1 3 6.3 1 1 9 6. 1 8 1 5

6.2 1 1 6 6.2 1 40 6.34 1 3 6 . 3087 6.3246

6. 2320 6.2886 6 . 1 6 1 1 6 .3870 6 .3M\9

6.3074 6. 1 647 6.2562 6. 3292 6.323X

6.29 1 5 6.2866 6.3348 6.2018 6.3 1 86

6.3 1 52 6.2 1 95 6.2853 6.3 1 99 6.2 1 1 4

p-

r-

POPALGH AT t' l al.: EM PROPAGATION THROUGH WAVEGUIDES

Tah le 2 - Variati on of TE2o mode cut-oil frequency with surface roughness for 15 different ranuolll variati ons

Sr. No f in GHz for surface roughness in cm

± 0 .04 ±()O6 ± 0 .08

12.5849 12.6 174 12.6 147

2 12.5555 12.52 11 12.5646

:1 12.5109 12.5922 12.65 I I

4 12.5907 12. 570:1 12.4288

:; 12.6002 12. 6317 12.555 1

(1 12.4998 12.5 I 74 12.4989

7 12.55 10 12. 6 148 12.6630

8 12.5534 12.58R7 12.5400

9 12.5398 12.6649 12.5894

10 12.5520 12.4598 12.5322

II 12.6270 /2. 6 13 1 12 .52 13

12 12.4999 12.5620 12.5642

13 12 .5 723 12.6 140 12 .()930

14 12.565') 12.5270 12.(1552

15 12.5938 12.52 11 12.748 1

quency 12.5805 GHz. The vari ati on of TE20 is observed

from 12.2563 GHz to 12.8952 GHz randomly . The mode

va ri at ion inc reases with measure o f surface roughness.

The variati on in T E20 mode c ut-off frequency for diffe r­

ent roughness measures is g iven in Table 2. The TE to and TE20 propagat in g mode in waveguides

decides bandwidth of the waveguide. The e ffect ive

bandw idth is reduced due to random vari ati on obse rved

in TE lo and TE21l mode frequenc ies of the waveguides

with wa ll s hav ing surface of rando m roughness . The

useful band width is reduced as compared to standard

X-band rectangular waveguide of smooth po li shed sur­

face (6.3206 GHz). The e ffecti ve bandw idth decreases

with inc rease in surface roughness, the minimum band­

w idth of 5.86 15 GH z is obse rved fo r hi gher measure of

surface roughness . The ma ximum va ri ation of T E 1o,

T E20 mode cut -o il freq uenc ies and bandw idth is g iven

in Tab le 3.

Conclusions The surface rQughness affects the cut -off frequency

values and the bandwidth . More the ro ughness of the

surface. less will be the bandwidth.

± O. IO ± 0. 12 ± 0. 14 ± () . 16

12.4637 12.5352 12.557:1 12.7072

12. 5716 12.6842 12.5220 12.5995

12.5969 12.5725 12.72 19 12.4167

12.7202 12.5 169 12.8952 I 2.()94 I

12.4075 12.837 1 12.4422 12. 3475

12. 6677 12.3654 12. 5099 12.47:17

12.5785 1'2 1\ / (12 12.5263 12.524 1

12.4872 12.7267 12.R935 12. :1659

12.5962 12.366 1 12.54:10 12.593(1

12.44 17 12.6227 12.7585 12.5665

12.6670 12.6638 12.5481 12.7374

12.6087 12.5455 12.70 I 0 12.613 2

12.368 1 12.5239 12.6428 12.7243

12.6040 12.72% 12.4566 12.7953

12.5673 12.5520 12. 6323 12 .6%5

Table 3 - Maximum value of cut-o tT frequency for T Elo(max) and

minimum value of cut-off frequency for T E20 mode and hand widt h

(B.W. ) w ith surface roughness

Sr. Roughness fi n G Hz Certain B.W.

No measure in inGHI.

Clll

T Elo T E20 (max.) (min .)

± O.GO 6.2599 12.5805 6.32()6

2 ± 0.04 6.2793 12 .4998 6.2205

3 ± O.O6 6.3255 12.4598 6. 1343

4 ± 0.08 6.3384 12.4288 6.()904

5 ± 0. 10 6.3677 12.368 I (1 O()O4

6 ± 0 .12 6.3413 12.3654 6 0241

7 ±0. 14 6.3948 12.2563 5.8615

8 ± 0 .16 6.4532 12.3475 5.8\)43

INDIAN.I PU RE APPL PHYS. VOL :'7, NOVEMBER 1999

Acknowledgement The financial support from the Uni ve rsity Grants

Commission, New Delhi , is gratefull y ack nowledged,

References I Ib ylcigh L. The l iI, 'oJ'." o( ,1'Ol/lIri , (Maclllill 'lIl. Ncw York ).

. '

..j

In9,

Lcnn,m S & Lundqvisl G. lI~r:/:' Tml/s 0 11 Micrml'{{ l'l' Th eory

T,'(-/1. :l(l ( I tJXX) 2X ,

ROlli T, Cceri G. Chiarlllcc F, Lco R D & Orll1onclrnyd R F . 11:'1:'1-: Tmlls Oil Mi('}'oll '(/\ '" Th('or." (Jll d T{'( -fI. 36 ( I tJXX ) oX ,

j\ \;lrkku & Ok sancn I. ICCI:' Tmlls ()II Jl4i( '/'()II '{/\'(' Th l'() I'." ({ li d

]'1'l'h, 37 ( I ()XtJ ) I tJ I.

5

7

8

Bcn,ili A. Chanc\c7.Un J & Fonlaine .l . ILL/-: Tro llS ()1I /111I"II II({

(Jill! Proflng(Jl io ll . 40 ( ItJ92) 141

Erkin S. Chang S. M ;1hcri H & TUSlllllllll i M . II:LC lrollS 1111

Micrml '({I'(' Th('on' alld T('ch. 30 ( 198X) 56X,

Leo Young. Adl'w/c('s ill M icrOlvat'('s. Vol. 7, (Acadcmic Prcss. Ncw York). 1971 ,

HaY;1la K. Koshiba M. Eguchi M & Suzuki M.IEL'E h ullS Oil

Microw(/\'(' Th ('on' allil Tech. :'4 (1986) 1120,

9 Rcddy .l N. All !lI lmillll'lioll IO/illile (,!(,III (,III II/( Ihod. (McGI'aw Hill. NC'.'J York ). 199 1,

10 Ak in.l E. Aprlicalion and imrlcmclllalion or rin it e! cic ment Illcthod, (Acadcmic Pres" Ncw York). I tJSX ,

I I Bathe K .I & Wil son E L. NIIIII('rim/ II /{'{/wds in/ill il(, (''' 'II /{'II/

({I/(J /rsis. (Prentice Hall. Ind i;I). 1987,