DIELECTRICAL ROD* THEORY OF OPEN RESONATOR WITH AN …

THEORY OF OPEN RESONATOR WITH AN AXIAL

DIELECTRICAL ROD*

By N. NARASIMHAN, V. C. ANANTHAN, AND S. K. CHATTERHE

(Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore-12 India)

[Received : November 22, 19711

1. ABSTRACT

The Q factor of a microwave resonator consisting of an uniform circular

dielectric rod excited by HE n and Ecn modes and terminated at both ends by

circular metal plates has been derived. The constant percentage power contour

for both the modes show that most of the energy is located within the resonator

and very little is lost by radiation outside the resonator.

2. INTRODUCTION

Theoretical and experimental studies of microwave resonator, closed and open types and resonator consisting of an axial uniform and corrugated metal rod have been made and reported elsewhere, by chatterjee, et.all' 1 . The object of the present paper is to derivp an expression for the Q-factor of a resonator consisting of an uniform circular dielectric rod and excited in HEn and Eot modes and terminated at both ends by circular metal plates of diameter much larger than the diameter of the dielectric rod. All the other sides of the resonator are open. rt is also the object to determiln the constant percentage power contours round the rod and show that the loss of the energy by radiation is insignificantly small when the diameter of the end-plates are much larger compared to the wavelength of excitation.

3. FIELD COMPONENTS

The field components inside and outside the dielectric rod when it is excited by //En mode are as follows (See Fig. 1).

• The project is suppor ted by PL-480, Contract No. E-262-69(N), dated August 30, 1969.

64

Theory of Open Resonator with an Axial Dielectric Rod

For p c a Medium 1 (Inside the rod) :

p

Ep1 = — B Mk, p)+ bk _—_ 1 J:(1c 1 P)1 sin es- Or &tic,

+ Jai( P] ccs &IP'

P PctJ

i (k p )1 sin çli et a

/3k 1 b 1 (AP I P)+ — p)1 cos e -nit

B p

B _ 1EL- (k P) k i (k, P) il sin rifl' P w Pv

65

66 N. NARASIMHAN, V. C. ANANTHAN AND S. K. CHATTERJEE

si Bi(ic?tho m o) .1 1 (1c p)] cos # rig: (11

For P eza Medium 2 (outside the rod) :

1SU C 2 #

11 ) (k2 + (k 2 P) sinkrig' C we e

n id, n 1 ;2= — C [ k2 Hi lir (k2 r ) + C

1 P jrs — — •••■•••••.■ 1 ( I (r. r sr COS iciS Erg ,

C C p I

) 2

i

Ea c k2

= Hp) (k2 F)i sin q; el1321 Cc puce

C 1 1 H 1,2= C

2 ll ix 1 (k2 + —

% (k, p)] cos # es 01.

1.0,140

(k 2 ir” I sin# tr Jff P w mo

H= — [(k2uto,u0 Hp) (k 2 p)] cos so e-mx . • [21

4. BOUNDARY CONDITIONS

In order to derive the characteristic equation propagation constant, the following boundary conditions interface (3 to a) between the dielectric and air.

and determine the are applied at the

Ez = Ez2 ;

Hz1 = Hz2

E3 1 = ge2 ;

H42

[3]

5, CHARACTERISTIC EQUATION

By using the proper field components and appropriate boundary conditions, the following characteristic equation is obtained.

I 1 n(xl) _ 1 HPV (x2) 1[c1 .1:(x1 ) J1 (x 1 )

. I x2

HP° (x2)1 L x 1 Merl ) x2 11 3" ) (x2) i [ x t Hp ) (x2 )1

= (4-4) (4 - €, 1 xTra- [4]

Theory of Open Resonator with an Axial Dielectric Rod 67

Where, xi a,

CII =E1/€0 ( 2 7r a )2 xl +(x2 1j)

2 (en = 1)

Ao Is]

6. EXCITATION CONSTANTS

The excitation contants B, C, b and c are related as follows.

H 0) (X ) 2

Ji (xt )

Ht(1 ' (x2)

4-4 c i 4 (x1 ) _ co

x14 [ x

i tii (x i ) x2 Hp (x2) 1 Him(x2)

C 13 c o

C com e

2 2 X 1 - 2 1 e [ '1 2 2 X X2 .K i

.111 (x 1 ) co

J1 (xi ) x2

H im' (X2) I

The above relations are obtained from the field components and using boundary conditions.

7. STANDING WAVES

The standing waves formed along the rod due to reflections taking place at both ends of the rod are represented on a vector b Isis as follows.

En tr-- Er 4. E, His = + Hz —

Ept s= Ep +

I-/PS = lip s+ Hp

E 4, s tr- E 4, 4, 4- E3 _

where,

+E7 H, f = —

EP-

Ii p+= + Hp

E4 E4 ic +113_ 181


The field components of the standing waves are therefore, for P c a.

1 b Epis

13k i j (ki p)} sin sin 132 + co E ' B J i(k P)

p

b Eeit e —2 B[k l (k i _ (k, p)] cosq sin 13:

B w E 1

Ens = Bi bn ------ sin cos f3z B Ito( '

b 1 1101 ,2e2BE—lik J (k i p)-f- Jai p) cos cos13::

.0442 0 B p

—2 Bi

1

13 Jak i P) (k, p) sioçbcos flz

P tomo B

k 2 11 2 BH— p)1 cos 96 sin As;

iwiao 191

The components in the region outside the rod Pa can be written similarly.

8. RESONANT WAVE

The field cornponets Ep and E3 at z--0 and z= I satisfy the conditions E3 =O and El, =0 which lead to sin flz-= 0 which requires filt=nrn where, m is a positive integer (1, 2, 3 . . .) indicating the number of half cycle variation of the field components in the z—direction. The components of the resonant waves are obtained substituting P=(wn//) in the equations for slanding waves.

For p 28 j (lc p)4. b /3k1 P)] sin sin (7) • B w f

Enter I

mit • b — t p )1 cos* Eor = —2B lk i i: (k P) 8— 73 /3 wet

P71 ) E r —2B [± k 2 ii(ici pd sin

B btu E


, b M Hp s r = 2 B[k-L- J i (k I P) I-

I --- i (k i P)1 cos cos' ---) z

w M o p

13 1131 ,--= - 2 1

.11 (k i P)-1- k i f it (k i p)] sin cos (ii_i_LT) z p come 1

ks 11, 1 ,- -2B1. -2--- J i (k i P)1 cos (I) sin (71Ln ) z [10]

j(0,42 0 /

C fl/ c 2 -2C [a-1 Him (k 2 0+ __ __si Nor (k P)} sin I) sin ("1 z

P c wco 1 2 /

it . „ - — — — __ i ,.._2 . , _ __ 7., sin k _ c 13 1 E32, = — 2C 1k 2 H (1)f (k Pl+ Ho ) tic Hems) s' (mn )2 C coE 0 P 1

. c k 2

Ez2r = 2C -- --1. Hp) (lc?) inck cos P-7 )z C Puce, /

f3k Ha2r =--2C1

co--- Hi" (k 2 p)+ 11( 1 )(k2 p) }cosy", cos ( m z

,u0 C p

114, 2s — 2C [-- 1 Hi" ) (k 2 p) + k up)" (k2 ni sin 4) cos ( m z T. 1 P w

H22r k2_ 112 rn -2C _ 10) (k 2 P) cos 4, sin — z 01) ../wmo

If the field inside the resonator is completely described by equations [1O] and [11], the mode of oscillation is pure ffEn . lithe frequency of excitation or the distance between the end plates is so adjusted that the above conditions are satisfi.:d, the cavity formed by the structure with the two end plates is said to be in resonance

9. MAGNETIC ENERGY STORED

The energy stored in the magnetic field consists of two parts Wm , , in

the field inside the rod and Wm2 in the field outside the rod (W42). The

total enery Wm stored in the magnetic field is therefore.

WA,I -i- Wm 2

a 2 -n

122° j I p.0 3=0

pdp dod:

d 2.n 1, la f 1 11212

2

pg-a z-o

pdp dodz [12]

•

70 N. NARAS1MHAN, V. C. ANANTHAN AND S. K. CHATTERJEE

where,

I Hi r = 1 ;it r + i Hid. r + i Hilt 1 1

I H21 2 4= 1 11 P2r12 ÷ 1 1102,1 2 4- 1 Har 1 2 [131

The energy stored in the two media are : a

WM1 AA A1/4 It I 4B { y P (k P)1 2 d P

2 2 ( ith c I P

S a

J I b\2 1 ft/ (ic pvi2 dp 2b ki j(k

i „ , ) I- (k i p) dp j f7 1` "J B w mo

p =0 P 0

a a

+ (ky va,p)rd, f b k2 p [41 1 (k dp

mo P

F"ACO Polo

a 2 2 a

+2b ./: (Le i p) mk, p) dp

it.it p pn2 dP

mo w

pv0 p=0

which results in wm 1 =itmo B p i if 13 V + Lb e V . I k i VI

I lkwihoi k B 1 k (Ak itt 0 if 1 2 \

+ [./i (k i a)]2>1 — i( w mo 13— — ----b

B

+

(

A)' ik i a Jo (k i a) J1 (ki a)}1 [14] wm o

WM2 n Me C12i(1—)2 '

wm. ./22.) 1( { (k22d)2 woo , (k2 dip

+ [Hp) (k2 (1 )12) (k2 C)2 „ 2 IEHO li (k2 a)? i l) (k2 a)12})

(k, Off 2

(k2 d — Vi i0) co Me C k

± t(k2d)Hi o (k2d)Hp ) (k2d)—(k2a)HP ) (k201110) (k2a)}1 wmo

[15)


10. POWER Loss

The total power lo dielectric rod, power lost outside the resonatar, i.e. ;

;t ti is composed of the power lost Pd in the in the end plates Po and power lost by radiation PR

Pr Pd +Pe+ PR

[16]

If the diameters 4 d of the end plates is mach larger than the diameter (2a) of the rod, the loss by radiation P R outside the resonator can be ignored compared to the other two terms.

The power lost in the dielectric rod is given by 2it a

f I a t 1E1 2 pdp dO [17] 4,-o p =0

where a., = to E l tan 8 and

1E1 2 ---- 1 E t 1 E r +1 E zi

Substituting the values of the field components, equation [17] becomes

a

P d tc Cr 7t B2 1. ---1 P ak t p)? dp je b 1625 Bwc 4

)2

pn0

a

± 2b P k l J;(k t p).1 1 (lc i P)dP +kP [ti P)1 2 dP . B coE

P -0 P

a 2 I 2b ,

( Bbatijc, ) P)1 2 dP + it (Kt tkt Piar

p-o ptQ

a

(—

bki --) p 2 p, (k, p)12 JP)] acoE,

p-'0

e n/ or, 11 2/ Oc t a) 2 I bg ( b ki a Vt po (ki

\s 1 2 [ I 2 B we I I b lc? a 2

4- (k a)} 2 --. ,bLY (kAll Bb_c01-3E :1 (a ) Ad E t

+ Po (k a) .1 1 (k 1 a)} ("9-7-14:1 )1)

[18]

72 N. NARASIMHAN, V. C. ANANTHAN AND S. K. CHATTERHT

The power lost in the two end plates is

it el p.2 x

2s/2 at

e Pati

Vrt

.1 I H uta' p dp • d [191

0-0 where,

a i,==conductivity of the end plates

I Hula 1 2 = I Hp2 12 + I lin 12 •

Substituting the value of the field components, equation [191 after integration yields.

e2 fr,t4 e\ f113 kA2 ick2 \21 d2 (,) n2 0 k2 er e/ [ l‘tomoi cc) ll EH° (42d)11— 4k

2 a)] 2;

+ {( 114)2 + (cL!)2 + (limo *)2} l ied 2 [H' (k2d))2 Ta2 :Mc" (k21,0]21.1

(201

I 1. Q FACTOR

The quality factor Q of the resonator is defined as

WMI + WM2 Wm, + WM2 4.0 • to [21]

Plots! Pe + Pd

Substituting appropriate expressions for the maximum energy stored in the magnetic field and the total power lost, the expression for Q becomes

2“02.1€0)pie+(c 2113 2)yi Q—

(0E,, tan 8) (U+ V ) + (C 2/B 2) (1/€ 0 1) (t ,u 0120 e)" W [221

where

drz-1

is

) 4 (

2 b 2 k i 2 Oc i a) 2 [ vici Op 4- { j i (k i a)} 2 1 (c:110 (C; /TO

f g 2 2 i(--- b

g .1 1 (k, 012 1 + A 1 } {(1; I a) Jo(k. I a) Mk I a)} 40,u0 B w mo


( k 2 ) 2 t j(k 2 d) 2 [Ho„ )(k2dn , )(k,d)? \ A-4 0 t

(k202 iruco:)(k2a)]2-i- Hc 1) (k2a)1 2 Il - 1 1-

k [1-1 (1 1 (k 2 a)J2}

)

.1 (k 2d ) H (01 )(k 2d ) 11 (1 1) (k 2d)

,wmo

(k 2 a) H 101 ) (k 2 a) H (1 1 ) (k 2a)}

1 u (k 1 a) 2 bfi \2 1 WI a)

2) [J o(kia)1 2 \13w€ 1 1 I 2 kthoc i

g ) 2 (k i 011 1 b kfL \ 2 1 V nE iiJ i (k 1 a)32}1— (1 — + 2 Bcoc i i

(bkiii {J0 (k1 a) (k, a) } I bk? qt. \

2 ‘Bw€ ) 1( 1 a \ 13(0€ I /

w ft k 2 )2 (ck2y} d 2 rtin 2

ikcom o —c— (k2 d -ar [1-1c 1) (k2

( 3 k 2 \2 ± ic k \2 + 1 c y} d 2 (1) ‘-e k 67-20 - [Hi (k2 d)J2

a2 [WI (k2 a)12 1

12. EVALUATION OF k i AND k 2

The values of radial propagation constants k i and k2 are found from the Values of x i and x2 which are determined by solvicig the characteristic

equation (4) with the aid of equation (5). The variation of radial propa-

gation constants with (2a/A 0) is shown in Fig. 2. Fig. 3 shows the variation

of axial phase constant with (2a/A0) which is determined from the relation

between k and g.

13. EVALUATION OF Q '

The values of Q of the resonator as a function of (2a/A ) , L, tan 8 , and A calculated from equation (22) with proper values of

k i , k2 and f3 are ploted in figures 4 8 respectively.

y/ (( )6 \ 2 c 2 \ ciTA70) Z,"

2 )

{±. \12 e

w ma P U2d

74 N. NARASIIIIHAN t V. C. ANANTHAN AND S. K. CHATTERME

K 2A

2.0

K

0.5

0 h a

al" •

•

Mix ftsei

Pro. 2

s I

2

z

2 5

ae

4

30


FIG. 3

14. FIELD COMPONENTS Ewe MODE

The field components inside (pls.. a) and outside the rod (Pa) are respectively

Medium 1: P ca

r, Bkile

6' 1 k r tic pl ext. ( —113z)

WE I

Ezi zan (ki P)exP ( —i13z) poe t

H el = Bk i J i (k i p) exp efiez) . . , [24]

4

4100

IWO

1600

4

76 N. NARASIMHAN, V. C. ANANTkiAN AND S. K. CHATTERJEE

now aw• 10

0 0.4

1.0

uvhoi. -rem

Fm. 4

0 a 0.47t0 Cat A a 1.14 cm Cr • 2.11i

4030

2400

2200

11100

140C

$000

30A

Theory of Open Resonator with an Axial Dielectric Rod

I 450^

44

77

41(

401 I I - . . .

Tan 6 —or LCG v1011) ---+P

Flo. 6

FIG. 7

21/4 (cm) —IP-

FIG. 8

78 N. NAR.ASIMHAN, V. C. ANANTHAN AND S. K. CHATTERJEE

•.. Medium 2: Pr-a

Ck 2 fit (1 ) En c---- H o (k2P) exp -j gz)

wco

Ck 2 , 1 % Ez2 2- lc (k2P ) eXP Jaz)

.00€ 0

C k H ( s ) (k. P) exp gz) 11.2' - 2 2

15. STANDING WAVES

In this case, the standing waves are represented by

, E =E + E Ps P — P-

Ep _

where,

114 + r [271

16. RESONANT WAVES

Due to the vanishing of the tangential components of the electric field at the two end plates (zic 0 and s= I), $=n n/1, where n is a positive integer. The field components of the resonant waves are

Medium 1: pas. a

k fit (n it Z E B (k i p) sin --- ply as

WE )

Inicz) Lflr B 2 J0 (k, p) cos

jape,

Pt 7C Z) +17 2 Bk i J 1 (k i pi cos -----

(28j

Medium 2: a

k B E, 2,= 2 C 117 ) (k2 9) sin "

we u

k2r as 2 C k 2 2_ HU

j see °

I

( . II TT 21)

(k 7 ') bill — I

14 42,== 2 C k2 p) cos [29]


17. MAXIMUM ENERGY STORED IN MAGNETIC FIELD

The total maximum energy stored in the magnetic field is

where, the magnetic energy stored in medium 1 (wmi ) and in medium 2 (Wm2 ) are respectively

1 li mn d v WM1

, 1

2 "

4

a 2en t

WM 1 re. Ail si I I 4 B2 lc? f .1 1 (ki P) 12 cos 2 (n in r ) P dP d 0 dr I

P=0 + -so 2=-0

--= 7T ,110 B 2 k ? i a2 10 (k , a) 2 +J 1 (k t a)2 - [ ---2 .10 (k t a) J 1 a)

k 1 a [ 30)

[

Whi2 =--- 7C Mo C2 kl I di {[ HT (k 2 d) 1 2 + [ Hit) (IC2 d ) 2

2 k 2 d

HS' ) (k 2 d) F1 (1" (k 2 d)}

a2 (k 2 a) ] 2 HT ) (k2 a) i2 2

(k2 a)FIV ) (ki 0)} [311 k 2a

Therefore, the total maximum energy stored in the magnetic field is

2 Wm = W Mo I a 2 B 2 [k {{1 0(k a)} 2 + t (k Q 11 2 icy; jo (1( 1 a) J 1 (Ic e a)}

- C2 ki ( d---2-1[11 (4 ) (k 2 d )1 2 + [HY (k 2 d )1 2 ._-2

B 2 (22 k l a

- /Wm (k 2 a)12 + [HI" (k 21)]2 2 FI,V ) (k 2a) 1-1 (1" (k 2a) Di

( 0 k2a [32]

18. POWER Loss IN THE RESONATOR Foi — MODE

The total power loss in the resonator is 1. 31 )

P Pd + P,

80 N. NARASINHAN, V. C. ANANTHAN AND S. K. CHATTERIEE

neglecting the loss by radiation outside the resonator. Where, the power loss in the two end plates is

d Vu ( come )1/2

Pe = 2 x H,..1 2 pdp 2 V2 ere

p=a 41 wJ

[341

where 1 Iltani '1 142 I > a e and IA, are the conductivity and permeability respectively of the end plates and d is the radius of the end plate

Substituting H 412 in the integrand, eqation [34] becomes

Pe a it /CI C2 (---("49 112 id2 ([14 (01) (k2 d)}2 +H' (k2 d) 2 2 c

— W." (k 2 a) WI " (k2 d)) k 2 d "

2 a2 ({}-1" ) (k 01 2 + 114 (1 1) (k2 a)} 2— ---- Wo n (k2 a) HY ) (k 2 a))1 [35] 0 2 k2 a

The power loss in the dielectric rod is

27 a

Pel = is is IE12 PdP do i361 4)=0 p =0

where, o 1 =wc 1 tan s and 1E12=lEp1 12 + lEzi

Substituting Etan in the integrand, equation [36] reduces to

7r 1 B 2 k 2 a 2 Pd 2 (0 2 1 E2 [ ( 00 (k1 PI (ki 4 )} 2i (fl 2 —k?)

—2 P2 J0 (k 1 a) .1 1 (k 1 [36a] k a

19. ' Q' FACTOR: E0 1 — MODE

Q. 6, (iyhdP) gives

(m0/€0) [R I +(C/B) 2 R21 [37] Q f( to € ri tan 8)/21 [R31+ (qv (1pE0)(w7,720,y12 [R4 1

where, R,, R2 , R3 and R4 are given by the following expressions.

R I = a2 4{[J0 (lc, a)]2 + p, (k, 2 (k, a) it (k i a)} k a


2 k [12 H (01) (k2 d)]2 + [H (1 1) (k2 (/)] 2/727/ H ‘0" (1/ 2 d) H' ) (k2 d)}

– al[H (01 ) (k 2 a)? + [Hil l ) (k. 2 a)? – 7c--a2 1-1 (01) (k2 a) Hy ) (k 2 a)111

2 R3 - (—Lk ) 2 f {P0 (k 1 a)? + [J I (k 1 a)] 1 + {oh _k;)} -- ____ J o (k I a) J i (k t al k a I

Re- id idlill 2d)1 2 + [WI " (k 2 d)] 2 rd2 H (0 1) (k2 d) litt 1) (k2 d)}

2 a 2 I[H 40 1) (k2 a)) 2 + [H (1 1 ) (k 2 a)] 2 - 2

H (1 ) (k2 a) H' ) (k 2 a)} °

The value of CM is given by the following expression

Jo (c i a) 2 (C/B) 2 ----

E rl H (0 1 ) (k 2 a)

The variation of Q for E0 mode as a function of 2a/A0 , In tan 8 and a, are shown in Fig. 9-13.

20. CONSTANT PERCENTAGE POWER CONTOURS

In calculating the Q-factor of the resonator, radiation loss outside the resonator has been assumed to be negligible. Th is is justified if most of the energy is located inside the resonator, i.e., within a radius ' c/' from the axis of the resonator where ' d ' is the radius of the end plates. This can be determined by calculating the constant percentage power contour.

The amount of relative power flow outside the guide can be represented from the constant percentage power contours round the guide which are determined as follows.

representing P

to the total

If P =fe l , r2 , r3 • • • r„ represent the radii or the circles

the contours inside which constant powers P, 1 , Pa , Pz3 • • • along the rod are located, then the ratio of powers with respect power P,„ is

Pit iPin Wt at Part at p

=-- W3 at P--er3

• • •

Paant - Wn at P r,1

t 4 norx

82 N. NARASIMHAN, V. C. ANA/kiTHAN AND S. K. CHATTERIEE

nOi

*oo

s2o<

4.40(

IWO

bles""■,..„..eammaiismsab. sm.__ I

CI 1. lo

FIG. 9

f0 3

CS s 1.21 Cin

A • 3-14 Cm E a • 2. 56 Cin

_

_ _ ..

FIG. 10

$o

3.75

5000

4000

0

3000

200C

100C


0 .05

9600

9550 4 -t

1.27 Cm

50.0 Crn

2.56 3.14 Cm

5.0 5.2 5-4

• tog (re)

FIG. 12

An,

Tan 6 Ftc. 11

FIG. 13

84 N. NARASIMHAN, V. C. ANANTHAN AND S. K. CHATTERHE

where,

P SP' +P°, zi z,

Pz2 Pzi2

Pa r PzI3 4- PS

contained within a radius pwri •

contained within a radius p s=r2 •

contained within a radius p=r 3 •

•

•

Pais Fin+ in contained within a radius

r 1 <r2 < r3 • • • <r,, . The values of Pzi , P22, Pal • • • Pi„, mined from equation [46] for HE mode and for the E 0 —mode by the integrals

f p=a

by P=ti

f , pca

0=rt

f • plies

4 . Paarn f

Paa

respectively.

20.1 Power Contour for H E Mode

The power flow in the radial, azimuthal and longitudinal directions are

Pp r + Re ff [ E4 H: H:1 Pti el z [39a]

Pe -4 .Re ff E, FI: — H:] dP dz [3914

P,=4 Re fsf [E, H 4*. - Wp ] pdp [39c)

The power launched in the dielectric rod will be transmitted in the longitudinal direction, where there is no radiation. But if some power is lost by radiation, the power will not only be transmitted in the longitudinal direction but also in the radial and azimuthal directions. In order to determine how much power is lost by radiation, the power flow Pp and P3 in the p and 0 directions respectively are also calculated.

P the total power contained within a contour of radius rn and are deter.

replacing

Theory of Open Resonator with an Axial Dielectric Rod 85 The power flow inside pi and P° in the three directions are

a Iv 2

-r B B * [—B

( I a Jo(k p) (ic k dP E

P 0

n PI * ( 1 b 2 1 ) Jovci p) aki p)dp

ice) /20 = 0

C

7C FY I 1 b2 \ f 1_ j (lc dp

it° Vuo B 2 Et JP ti P = 0

a I' (1— *1 ) 1.-1 1J I R I P)} 2 dp B w 2 sue € 1 P

P -= 0

1c 2 ( 1 2

b2 1 I eq- B 2 Mo

PI Jo(k i P)} 2 [40]

iv k2( 1 — 2 1 ) H (01 ) (k2P) ts.) M 0 0

tma

Ifil ) (k2P) dP

n .1 2 k2 1 its) Mo

+ C2 .) f H (01 )(k2P) C 2 CO

p ta

Hy)(k

2P) LIP

a

.Y2 ( ± ) .1 .[Hy ) 2 012 dP ito /20 C €13 P

p a

a .... ce I

°Y1 \ j C

—{1-1 (11) (k2 P)} 2 dP to 2 mo co/ P

P ca

7T 2 ( + 2 k 2

2jco

1 C2

12 0 C p iii(01 )(k2 p)} 2 dp

[in I P k 3 Iv = j CC* — ---? 2 w

_

In calculating # and P° the and the limit of z is from

n ••••■■

/ 14( I •■■•■•■•

2 40 mo

86 N. NARASINMAN, V. C. ANANTIIAN AND S. K. CHATTERJEE

a BB* I- b 2i,91k? J. 1 Pi (Ic a p)}. dP

P4 B (0 2 mo E ) P Pro

a b 2 lfl1k? Jo (k p) j i (k i p) dP B ts2 m o c i

pn0

a

1 k? (1 + b2 1B2

) pi (k i P)1 2 Tt; P

p

[421

CC it 2 032 hi ± (k2 p)} 2 ti p j 4 C (0 2 iti o c o

C 2i $2k 2 HS" (k2 p) wit(k p) dp

c (0 2 duo c o

. P=t1

__ 114(1 j_ 1)11 1 u bn

"1" 11.11 1) (A2 r)} 2 [43]

w mO C.2

Eo P pa

In calculating and P:, the limit of z has been taken from z=0 to z=1 where 4 1 ' is the length of the dielectric rod pf and P: has been evaluated for 0=n/4 as P3 =0 for 41=0, 7c/2, "/C 3w/2, and 2n.

pin jBB*1-71 1 - _ -L) (k i p) I 2 w mo B2 E l

/ b2 1 )

2 to \ 47:0 -/-31 T (lc ) p)} [441

1 c- 1 - _. H (1) (k2 p) my) (k 2 P) 0 (ito C2 E 0

c2

E.

1 — — )111 (1 1 ) (k, roll [45] cz

limit of 0 has been taken from 0 =0 to 96 =-- 2 7C

2 -- 0 to z=1. pi and P° have been evaluated for


p a, The axial propagation constant "Y =ji3 , and ($i /32 = p) has been obtained from the solution of the characteristic equation which gives 1( 1 and k 2 and from equation relating /3 to the radial constants k.

The power flow /) and P for different values of 20. have been calculated for perspex rod (En i 2.56) and the results are tabulated below.

TABLE I

Power flow inside and outside the rod in the z-direction

201x 0 P z (Watts/Sq. em.)

0 P

c (Watts/Sq. cm.)

i o PccP + P

(Watts/‘. an.) .

0.8 0.01363 181 2 0.02 1308 181 2 0.0149 181 2

0.6 0.01349 181 2 0.01698 1131 2 0.0305 181 2

0.4 0.0 2 9965 1B I 2 0.0448 1 B 1 2 0.0548 1B l 2 0.3 0.02 7806 1 B 1 2 0.0490 I 13J 2 0.0569 1131 2

TABLE 2

Power flow inside and outside•the rod in the radial (p) direction

2a1X0 P1 Watts/Sq. cm. Watts/Sq. cm.

pc, Watts per sq. cm .

0.8 j 0.049086 I 11 2 j 0.03 6389 1 I B 1 2 j 0.03 7298 1 i B1 2

0.6 -j 0.0'6965 1 1B12 j 0.0'2418 1 1131 2 j 0.03 1722 1 113 1 2

0. 4 -j 0.02 2031 I 1 B 1 2 j 0.02 1387 1 1 B1 2 -j 0.036440 1 1B1 2

0.3 -j 0.02 3466 1 ! 131 2

-J 0.02500 1 1 8 1 2 -j 0.02847 I 1 B1 2 -

TABLE 3

Power flower inside and outsidi

2a/X0 Pi cts Watts/cm 2

0.8 - j 0.024620 1 1 1 2 -j

0.6 -j 0.02 7099 118 1 2 -j

0.4 -j 0.02 8581 1 1B12 -1

01 -j 0.0 2 6279 1 jB 2 ej _

he guide in the azimuthn

pa 4,

Wa tts/cm 2

03 7518 1 1 B 1 2 -j

0 2 2197 1 I B

02134 1 1131 2 j

0107 1 jB 2 -j

d direction

PciSil 4+ PO, Watts/cm'

0.025371 1 I B1 2

0.02 9296 I 1B V

0.02(29 I 1131 2

0.0169 1 1612

88 N. NARAS1MHAN,

Values of

2a/x0

V. C ANANTHAN AND

TABLE 4

k i , k, and for the rod excited in H

S. K. CHATTERWE

Ell mode=3.2 ems.

f3 lc ' -jks ..

0.30 2.46 0 43 2.00 0.40 2.42 0.54 2.04 0.50 2.15 1.20 2.32 0.60 1.92 1.58 2.52 0.70 1.71 1.80 2.66 0.80 1.56 1.93 2.75 090 1.41 2.04 2.83 1.00 1.28 2.12 2.89 1.50 0.90 2.31 3.04 2.00 0.69 2.38 3.08

The power flow P 0 and P 4 are reactive and hence the relative power flowing inside the guide is expressed as a percentage as follows

w- Piz i17).6- 100 [46]

The rusults of computation of pil IP. as a function of 200 for c, 1 =2 56 and as a function of 2a/A 0 =0.8 at A=3.2 cm. for the ti E l 1 mode are shown in Figures 14 and 15, respectively. Figure 16 represents graphically the relation between W% and P=r 1 , r2 , ... rn for different values of (2a/A 0) and e r,=2.56. From Fig. 17, the radii of the constant percentage power contours for different values of (2a/A 0) are ploted.

20.2 Power contour for E o mode

The power flow inside and outside the dieletric rod excited in E 0 mode has been calculated following the above method,

• #703 2 ,Ba2 ki p. e - ----(Joifr i a)? +Lir i (lc i a)? - 2 10(k1 a) jal a) ) ......_ 2 wc o c r, 1 k i a [471

PP = Re it` P2k I P (di {[Wo' 1 (k 2d )12 + [Hy )(k2d)12 _ 2Froi )k2d) I-1 (1 1) (k2d))

2 wc o k 2d

( ., 21-IV ) (k 2a) H il l )(k 2a) _ (72 i t 4 tH (i t ) ( k2tois _ A .., ...li■I=1■111SIM■e e eabw•■•

k 20


where B and D are related by the following relation

B k1 ‘ 1 2 1 j°(k l a) 1 „ 1 k 2 [ H col (k2 a )

which is obtained by using the appropriate field components and boundary

'condition. The total power flow is therefore,

B 2p k i ,12 — 2 Ja i a) J i (k i a) a 2 [Jo (k l a)? + [

2(0E 0 E,1 kia

1 (lc: _) 10 (k 19)

(k2a) IC

} d2([11 ' )(k 2 d)12 [H (11)(k211)11 Cr 1 I Won

H (1 ) (k 2d ) H cl ) (k 2d _ a2 En (01 )(1(201 2 + [H(11)(k2a)]2

_ k 2d

2 H (01 ) (k2a) H (1 1 ) (k 2a))1 k 2a

Figures 18 and 19 respectively represent W% versus p and constant percentage power contours.

21. CONCLUSIONS

The following interesting points emerge as a result of the above theoretical investigations.

(i) For the same value of 2a/A0 , and E ri , more power is concentrated near the rod in the case of HE mode compared to the Ec, mode excitation.

(ii) The power flow inside the rod for HE mode excitation is more compared to that in the case of E0 mode (12) excitation.

(iii) For the same value of L, 2a/A0 , E r 1 ae and tan 8 the Q factor of the resonator is greater in the case of E 0 —mode than in the case of HE mode excitation. This is justified due to the fact that in the case of HE mode though the loss in the end plates due to finite cre

is less than Eo mode due to less radial field spread, but the loss due to tan 3 in the case of HE —mode is comparatively much mole than in the case of E0 —mode due to higher concentration of power

inside the rod. (iv) The nature of variation of' the Q— factor with wavelength of

excitation is practically the same in both the cases of excitation.

' ft may therefore be concluded that a dielectric rod excited in HE mode will act as a better surface wave guide than when excited in E c, mode provided

the dielectric rod is made of material having low value of loss tangent.

.01

Si

20


CVA0

FIG. 14

•

100

95 ci/A 0 ota

4.

60 Ne 1 4 3 4 S 6 i

Mt ATM DIELEciRt4 CONSTANT $ € r OF THE &RD

Fm. 15

100

90

80

70

60

50

40

2(

Theory of Open Resonator with an Axial Dielectric. Rod 91

C.) 1 2 3

RADIUS or THE ARIA 114 CI

FIG. 16

92 N. NARASIMHAN, V. C. A NANTHAN AND S. K. CHATTERHE

• r •2.56

3.0 907.

I . 5

eoX

soz\

i.0

0-5

0 • • it o.2 0.4

_

Etc. 17

RADIUS OF 'THE ROD'e

(1) 1.27 cm

(2) 1.4287 cm

(3) l• St175 cem 1 (4) 1.9050 Cm .

(S) 2.2225 Cm i

(4) 2.54 Cm .

S.

)1- ( 3 )


4.3

2.0 3.0

RADiuS OF THE AREA tt4 Cm

FIG . I S

0

a

0.6 a ' I' 4 0.5 1.0 1-2


22. ACKNOWLEDGEMENTS

The investigator-in-charge (S. K. Chatterjee) is indebted to the Dr. S. Dhawan, Director, Indian Institute of Science for providing nec essary facilities for the work. He is also grateful to University Grants Commission, New Delhi for permission to conduct investiga7ions on the project. He expresses his grateful thanks to Dr. James R. Wait, Monitor, for encouragement and to the U. S. Department of Commerce for providing PL -480 funds, and to N 0.A„ U.S.A.

23. REFERENCES

1. Chatterjee. S. K. • . . . J. Bt. Ins/n. Radio Engrs., 1953, 13, 475.

2. and lacharia, K. P. .. Radio and Electrontc Eigineer 1968, 36. 111.

3. ____ • . . . I. Indian Inst. Sc!. 1952, 34. 99.

4. _ • • . . Ibid. 1953, 35, 59.

5. _ _____ • • • • J. Instn. Telecommun. Engrs. 1965, 11, 407.

6. (Miss) Prabhavatbi, A. S , Chatterjee, S. K.

.. J. Indian Inst. Sri., (under publication)

7. Chartterjee, S. K., • . . • J. Instn, Telecommun. Engrs. 1965, 11, 528.

8. _ 7 . . . . J Indian Inst. Sci., 1952, 34, 43.

9. _ and Chat terjee, i , .. Mid. 1968. 50, 345.

10. and et. al., • • • • Ibid. 1971, 53, 63.

11. and Chatterjee, R. . . Proc. JERE 1966, 4, 53.

12. Shankara K. N., Chatterjee S. K., under publication.

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