Admittance of a Wedge Excited Co-axial Antenna With a Plasma Sheath Chi-fu Den October 1966'Technical Report 5825-9-T On NASA Grant NsG-472 This work was submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan Prepared for National Aeronautics and Space Administration Langley Research Center Langley Field, Virginia 23365

C 7~ _.J S,

ABSTRACT Cylinder with a wedge and a coaxial shell with an axial slot make up the antenna. A stationary expression for the admittance is obtained when the antenna is enclosed by a plasma sheath. The basis of the admittance calculation is the electric field of the wedge aperture derived from a solution of two coupled integral equations. The calculations are carried out for the parameter ranges: The radii of the cylinder and shell are, in wavelength, from 0. 05/?r to 2/7r and from 0.055/7r to 2. 2/i, respectively. The plasma sheath thickness is from 0 to 2. 5/r. The plasma frequency to signal frequency ratio, w /w, is from 0 to 5. Collision frequency to signal frequency ratio, v/u, is 0; 0. 01; 0. 1, and 0. 5. The angular width of the wedge slot and the shell slot are the same and equal to 0. 06 radians. The results indicate: For wp/w > 1, conductance and susceptance depend weakly on the plasma sheath thickness. For w p/w > 1 and v/w = 0, conductance decreases exponentially when either the sheath thickness or w /w increases. Susp ceptance depends primarily on _o/w and inappreciably on the sheath thickness. An increase of v/w increases the conductance but modifies the susceptance only slightly. The coaxial slotted shell behaves as an ideal voltage transformer in the equivalent antenna circuit. iii

ACKNOWLEDGMENT The candidate wishes to thank the members of his doctoral committee for the encouragement during this study. He is particularly indebted to Prof. Andrejs Olte, his committe chairman who brought the present problem to his attention and provided continuous guidance throughtout the course of this study. He is also thankful to Mrs. Katherine McWilliams and Miss Deirdre E. Nolan for typing the manuscript. This research was supported by NASA under Grant NsG-472 with Langley Research Center, Virginia. The numerical computations were supported in part by the Department of Electrical Engineering, The University of Michigan, v

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS ix CHAPTER I INTRODUCTION 1 1. Survey of Previous Work 1 2. Problem to be Investigated 4 3. Outline of the Report 5 CHAPTER II INTEGRAL EQUATIONS AND THE TERMINAL ADMITTANCE OF THE WEDGE WAVEGUIDE 6 1. Introduction 6 2. Field Expressions 9 3. Formulation of the Integral Equations 12 4. Terminal Admittance of the Wedge Waveguide 21 CHAPTER III SOLUTION OF INTEGRAL EQUATIONS (2-3-31) 24 1. Introduction 24 2. Noble's Scheme 29 3. Solution of (3-1-9) 31 4. Solution of (3-1-10) 34 5. Discussion 39 6. Solution for Narrow Slot 45 CHAPTER IV APPROXIMATE SOLUTION OF INTEGRAL EQUATION (2-3-32) 48 1. Introduction 48 2. Reduction of Integral Equation (2-3-32) 49 3. Approximate Solutions for e(?r) and E() 56 CHAPTER V SLOT VOLTAGES AND TERMINAL ADMITTANCE OF THE WEDGE WAVEGUIDE FOR NARROW SLOTS 60 1. Introduction 60 2. Voltages of the Wedge Aperture and Shell Slot 61 3. Terminal Admittance of the Wedge Waveguide When 0' 0 << 1. 62 CHAPTER VI EQUIVALENT CIRCUIT FOR A COAXIAL ANTENNA WITH A PLASMA SHEATH 75 1. Introduction 75 2. An Equivalent Circuit of the Coaxial Antenna 76 3. Physical Significance of the Circuit Components 79 CHAPTER VII NUMERICAL RESULTS AND CONCLUSIONS 1. Introduction 86 2. Numerical Results 87 3. Conclusion 95 vii

TABLE OF CONTENTS (continued) APPENDIXES A-1 PROOF OF THE STATIONARY PROPERTY OF y(a) 100 A-2 SOLUTION OF INTEGRAL EQS. (3-3-2)AND(3-3-3) 102 A-3 SOLUTION OF INTEGRAL EQS. (3-4-5) AND (3-4-6) 108 A-4 PROPERTIES OF X 117 qp (0) (0) A-5 INTEGRATION OF A AND B 123 ma m A-6 DIFFERENTIATION OF W(0) 124 A-7 PROPERTIES OF THE INTEGRALS (4-3-6)TO(4-3-9) 127 00 A-8 SUMMATION OF THE SERIES 5q (Q(e)/Q())2/n AND n=1 J (nA 0)/a 129 A-9 TERMINAL ADMITTANCE OF A WEDGE WAVEGUIDE IN A PERFECTLY CONDUCTING CYLINDER 132 A-10 POYNTING'S ENERGY THEOREM IN THE PLASMA SHEATH AND THE FREE SPACE 135 BIBLIOGRAPHY 138 viii

LIST OF ILLUSTRATIONS Figure Page 1-1 CROSS-SECTION OF A TYPICAL SLOTTED CYLINDER WITH A PLASMA SHEATH 2 1-2 RELATIVE POSITION OF SHELL SLOT AND MAGNETIC LINE SOURCE 3 1-3 CROSS-SECTION OF A WEDGED CYLINDER, SLOTTED SHELL AND PLASMA SHEATH 4 2-1 POSITIVE VOLTAGE AND CURRENT OF WEDGE GUIDE 8 6-1 EQUIVALENT CIRCUIT FOR THE COAXIAL LINE 78 7-1 (a) NORMALIZED CONDUCTANCE AND (b) SUSCEPTANCE VERSUS 9 WITH NO PLASMA SHEATH AND (koa; k0b) AS THE PARAMETER88 7-2 (a) CONDUCTANCE AND (b) SUSCEPTANCE WITH A COLLISIONLESS PLASMA SHEATH VERSUS 0 FOR k a=1.0, k0b =1. 1, kc = 1.2 AND wp/w AS THE PARAMETER 89 7-3 (a) CONDUCTANCE AND (b) SUSCEPTANCE VERSUS 9 WITH ka = 1.0, k b = 1.1, kc = 1.2, w /w = 1.5 AND v/w AS THE PARAME TE 90 7-4 (a) CONDUCTANCE AND (b) SUSCEPTANCE WITH NO PLASMA SHEATH VERSUS k (b-a) FOR k a=l.O, AND 0 AS THE PARAMETER 9 2 7-5 (a) CONDUCTANCE AND (b) SUSCEPTANCE WITH COLLISIONLESS PLASMA SBEATH VERSUS k (c -b) FOR k a= 1.0, k0b=l.l, 8=00, AND w /w AS TE PARAMETE~% 93 7-6 (a) CONDUCTANCE AND (b) SUSCEPTANCE WITH A PLASMA SHEATH VERSUS k.(c -b) FOR k a = 1.0, k b =1., =0 0 v/ =0.1 AND p /w AS THE PARPMETER 94 p 7-7 (a) CONDUCTANCE AND (b) SUSCEPTANCE WITH NO PLASMA SHEATH VERSUS k0a, kob/k0a -1.1 AND 9 AS THE PARAMETER 95 7-8 (a) CONDUCTANCE AND (b) SUSCEPTANCE WITH PLASMA SHEATH VERSUS o /w FOR k a=1.0, kob=1.1, kOc1=.3 AND v/w AS THE PARAME TER 96 A-9-1 PERFECTLY CONDUCTING CYLINDER SLOTTED BY A WEDGE 132 ix

CHAPTER I INTRODUCTION 1. Survey of Previous Work: In the course of re-entry, a space vehicle travels through the upper atmosphere with hypersonic speed, thus a highly ionized non-uniform plasma layer is generated. This plasma layer encloses the body of the vehicle, therefore it tends to block the radio contact between the vehicle and the outside stations. In the last few years, this problem has attracted the attention of a number of investigators. Hodara (1963) calculated the radiation pattern of a slot on an infinitely conducting plane covered with a homogeneous, but anisotropic plasma layer. In his approach, he first assumed a reasonable aperture field and then obtained the far field. He did not consider the slot admittance. To obtain the slot admittance, one has to know the field in the slot much more accurately than is required for the far field calculations. Galejs (1963) considered a slot on an infinitely conducting plane, backed by a rectangular cavity and excited by a current generator. He formulated a stationary expression for the slot admittance. In his more recent papers (1964, 1965a, 1965b), he applied the same technique to evaluate the slot admittance when the conducting plane is covered with a homogeneous plasma layer. A. T. Villeneuve (1965) considered a problem which involves a rectangular waveguide terminated on an infinitely conducting plane coated with a plasma layer. He employed the reaction concept to derive a stationary form for the terminal admittance of the waveguide. Both Galejs and Villeneuve limited their calculation to the case of signal frequency w greater than plasma frequency p 1

2 Unless the aperture size and free space wavelength are much smaller than the size of the vehicle, we could not use the plane geometry to approximate the surface of the vehicle; otherwise a geometry closer to reality should be considered. Some authors choose to consider the circular cylindrical geometry. A typical geometrical configuration is shown in Fig. 1-1 Plasma Layer FIG. 1-1: CROSS-SECTION OF A TYPICAL SLOTTED CYLINDER WITH A PLASMA SHEATH. where the slot A may be either axial slot or circumferencial slot. The existing work for the above configuration almost entirely is concerned with evaluating the radiation pattern of the slot when the plasma layer is assumed to be of the following: (a) homogeneous and isotropic (Knop 1961, Sengupta 1964) (b) homogeneous and anisotropic (Chen and Cheng, 1965) (c) isotropic but'inhomogeneous (Rusch 1964, Swift 1964,'Taylor 1961)

3 The last case is of particular interest to us. Rusch and Swift assumed that the density of plasma varies continuously according to a specified function of the radial variable r. Taylor (Rotman and Meltz, 1961) considered the plasma sheath to be stepwise inhomogeneous. The inner step is very thin and highly overdense in comparison with the wavelength. Therefore he regarded this sublayer as metal-like sheet. This metal-like sheet is then followed by a comparatively thick dielectric-like sublayer. To prevent a short circuit on the antenna, a dielectric layer is placed between the metal surface of the vehicle and the metal-like sublayer. He also pointed out that the radio communication blackout is due to the metal-like sublayer. Olte (1965) in a recent paper considered a conducting cylinder enclosed by a slotted coaxial metal shell with an axial slot which represents the metallike plasma sheath. The electromagnetic field is excited by an axial magnetic line source on the cylinder (Fig. 1-2). He calculated the power radiated through the shell for different combinations of the cylinder size, shell size, and the separation angle e between the line source A and the shell slot. Perfectly Conducting Shell Perfectly Conducting Cylinder FIG. 1-2: RELATIVE POSITION OF SHELL SLOT AND MAGNETIC LINE SOURCE.

4 2. Problem to be Investigated Although the radiation problem of a slot on a cylinder with a plasma sheath has been treated by many authors, few of them have been concerned with the admittance. The prime purpose of this report is to partly fill this gap. The geometrical configuration we consider is shown below: Free D C B A 0 Plasma Layer Perfectly conducting Coaxial Region shell with a slot Perfectly conducting cylinder with a wedge FIG. 1-3: CROSS-SECTION OFA WEDGED CYLINDER, SLOTTED SHELL, AND PLASMA SHEATH. where A is a circular cylinder with a wedge of width 200 B is a dielectric with Mr = 1 Er = 1, C is a uniform dielectric-like plasma layer, D is the r r free space region, E is a circular conducting shell with an axial slot, a, b, and c are the radii of the cylinder, circular shell and the outer boundary of the plasma layer respectively, 0 represents the center to center angle between the shell slot and the wedge slot. If we assume a magnetic line source at the apex of the wedge, then the electromagnetic energy radiated from the line

5 source is guided by the wedge to the coaxial region and then through the shell slot and the plasma sheath to the free space. Therefore we may regard the wedge as the antenna feeding line and we proceed to calculate the terminal admittance of the wedge waveguide. 3. Outline of the Report In the next chapter, we first assume the source strength to be VO volts and then write down the fields in the form of infinite series for the wedge guide, the coaxial region, the plasma, and the free space. From the continuity of the tangential electromagnetic fields in the two apertures, we formulate two coupled integral equations with n-directed electric field in the wedge aperture and shell slot as the unknown functions. From these expressions we formulate the terminal admittance of the wedge waveguide which is proved to be stationary with respect to the variations of the wedge aperture field. In chapter III and chapter IV, we present the methods and the solutions of the coupled integral equations. Upon employing these solutions, we obtain in chapter V the explicit expressions for the voltages of the two slots and the terminal admittance of the wedge waveguide when both slots are narrow. Parallel to the stationary formulation of the terminal admittance of the wedge waveguide, in chapter VI, we formulate this admittance in an alternate form. This new formulation is not stationary, but provides a basis for the discussion of the contribution of different regions to the terminal admittance. From this formulation, we construct an equivalent circuit. In chapter VII, we present the numerical values of the terminal conductance and terminal susceptances computed from the expression of the admittance obtained in chapter V. Finally, we draw some brief conclusions for this report. In order to maintain the main sequence of thought, we leave some of the detailed derivations to the apendices A-1 through A-10.

6 CHAPTER II INTEGRAL EQUATIONS AND THE TERMINAL ADMITTANCE OF THE WEDGE WAVEGUIDE 1. Introduction: The geometrical configuration which we choose to consider suggests us to employ the cylindrical coordinates, of which the z-axis is aligned with the axis of the cylinder and p is measured in a counter-clockwise direction from the center of the slot of the shell. Because the antenna is excited by an axial magnetic line source, only the following field components exist: H axial magnetic field, E0 circumferencial electric field, E radial electric field. r By superscripts I, II, III and IV, we will denote the wedge waveguide, the coaxial region, the plasma sheath, and the free space, respectively. Since tangential electromagnetic field must be continuous across the wedgeguide aperture, the shell slot, and the tangential electric field must vanish on the perfectly conducting cylindrical walls one obtains at r = a: E E ( e - e o -2-1-<e+e) EI0; 0 0+0B (2r+e-80) (2-1-2) HI = HIz; - 0 < < +0, (2-1-3) z z 0

atr =b: Ep Et- M E(0); - PO < 0 (2-1-4) =; l:l (2-1-5) Hz Hz.; <o ppo (2-1-6) at r =c E E -< r 1 r (2-1-7) H = HIV;4.r. (2-1-8) Z Z In the last part of this chapter we use the forgoing relations to formulate two coupled integral equations with the wedge guide aperture field e () and the shell slot field E (0) as the unknown functions. One may consider the wedge guide as a transmission line with TEM wave as the transmission mode. We consider a section of wedged-waveguide of length L, in which the transmission line voltage and current are governed by the following equations (Montgomery, 1948) d V(r) d r - j k Z (r) I (r) (2-1-9) d I(r) dr =r - j k0 Y(r) Vo(r) (2-1-10) where Y0(r) (r) = 20' (2-1-11) 0 Z.0(r) 2 0. r V~o

The positive directions for the current and voltage are shown in Fig. 2-1 Xr) V(r) FIG. 2-1: POSITIVE VOLTAGE AND CURRENT OF WEDGE GUIDE The solution of differential equations (2-1-9) and (2-1-10) are easily obtained as I(r) = - [A'H(2) (k0r) + B'H) (k0r) (2-1-12 V(r) = 1 r A'H(2) (k r) + B'H0 (kr)0 (2-1-13) v j (r) LOO j g r "0 where the primes indicate the derivatives of the Hankel function. If one defines the normalized admittance at a cross-section r as y(r) = 1r) 1 (2-1-14)

then from previous two equations one obtains AH 2)(k0r) + BH (1)(k0r) y(r) = j (2)' (1) (2-1-15) A'H (kOJ)+B H 0 (k r) The normalized terminal admittance of the wedge waveguide is y(a). Thus our problem is to find the constants A' and B'. 2. Field Expressions: The field expressions (Stratton, 1941) in the wedge waveguide are (2-2-1) EI = w 0 k2 [A (k r)+ B(H (k r) + BH 0 (k r) cos A(- + z= k0 A0H o0 n (0V n (2-2-2) v0A nrir (k0r) sin H (k(r +. (2-2-3) n r n10 2n 0 0 Vr)= jn0r260AoH J (kor), +Bi n0H( - +], (2-2-5) v(~):~~~~ - 0~k~o oH o (2 —5

10 I! One may specify the value of A0, or B0, or their linear combination. We choose to specify the voltage at the apex of wedge, i. e. to fix the strength of the magnetic line source. Let limj jWPok r2O0LAOHO O(k r)+B H (ko) = V~ (2-2-6) r-H ) (k0 r) + 0 H (1)0 (0 After taking the limit, the result is r, V A B=- 0 (2-2-7) 0 0 4% (0% In the coaxial region: H =k ~ [AnJn (k r)+Bn Nn( 0r), (2-2-a HII 02 C_ [A Jn(kor)+B N (kPr)]e-jnl (2-2-n) n= -oD n = oo n=- 0 1I FA J(kr) + Bn Nn (k r)1 e jn~ (2-2-10) n = -Co In the plasma sheath: n - Hz =kl [D (k r) + E N (k r e2-2-11) n= -Co

11 n= oa noo E = j;0 k L J n (kIlr)+ EnN(k1 r)]e (2-2-12) n= -00 n = 00 Em r= r T D n Jn(kr)+E N (k r)e (2-2-13) n= -oD where (P)2 kl 1 k v.=k0 ~ (2-2-14) 1+j0 If we let k= kr k j ki (2-2-15) then F2i2 2 V 1 1 1 )+1) 2 + 2( k = { ( 1 I)+ + v 2 (2-2-16) 2 2 2 11/ 1+() 1+(-) /2 k, }12 i 2 [ 1+(v)2 ( +v2 i)] 2 +(v)2 (2-2-17) In the free space: n=aoD HIV = k E C (k) (=o eCH-r (2-2-18) z 0 k0 n n nl = -00

12 n=00 E jc ~k k Cn H2n (kor) e (2-2-19) 11= -00 n = o00 IV 0 )2' = Z n C H )(kr) e J (2-2-20) r r n n n = -oo 3. Formulation of the integral equations Upon substituting the required EO expansions in the boundary conditions at r = a, r a b and r = C, i. e. in (2-1-1), (2-1-4) and (2-1-7) respectively, one obtains (2)' nr + 8 A(O) (1)' ka) +o __ 0 jiAok AoH(o) (koa) + B0 H0 (k a) + J' (ko a)co s ( 0E n=l 20 (2-3-1) n=100 lo0k0 AnJ n(k 0a)+ B nNn(k0a))e J= () (2-3-2) n = -ao n= OD IJwA O (A Jt(kb)+B N I(kb ) e no=E() (2-3-3) n= o00 n=00 j klk 2 (Dn Jn (klb) + E N I(k b) ) e j n = E(), (2-3-4) n= -00

and nl=Gj jw0k 1 (DnJn(k1c) + EnNn(k1c))en = -o n = o j1 OC H2 n (k c) e- (2-3-5) n = -o00 Applying the orthogonality of the circular functions to eqns. (2-3-1) through (2-3-5), there results: Q+o 1 (2)1 40 LA A0H() (koa)+Bo0H O (k0a) = 2jOopoko d (2-3-6) ~~~~90 n + 80 1A lE(' c (('-+cos 0+0) fn OLot/40koJ' (k0oa) e-e 20 O 0 n > 1, (2-3-7) A Jn(k a)+B N (k a) = 1' n' ~1n nOd n( n3k-b) ~ n(ka)+B~ 0 n n j (0k') e dots (2-3-8) D J (k b)+B N (k b) = E() ejn d' 23

14 and lE(D J (klc) + E N (kc)) = C H (kc) (2-3-11) Combining Eqs. (2-2-7) with (2-3-6), (2-3-8) with (2-3-9),and (2-3-10) with (2-3-11), one has three sets of two variable simultaneous algebraic equations. Their solutions are A R' V0 H (ka) + ( AO 80 4'0 e~ko j E (o0 d 00 8e0 J (k0a) 4j Ok0 J(k a) (2-3-12) irV0 H()'(k0a) 1 1 0 ( Bo 8 % WHO + 4 j %,j,,o k (0') d', (2-3-13) 8 Jo0 (k0a) jO(k0a) A, (Nn(koa) P -Nn(k~b)c) * (2-3-14) n J(k b) Nn(koa)- Jn(koa) N(kb) (2-3-14) I { v Tr B'.'..' (- Jn(k0a) pn+J (k0b) qn) J (k0b) Nn (ka)- Jn (ka) Nn (kb) (2-3-15) 1 n (2)1 (1, D [J (kC) N (kb)- (kc N(kb) Nn (k C (2-3-16)

[ n(1)n(1 c) Nl) -Jn(1) n(k1C )]n n n (2-3-17) where 1 1, n (2-3-18) pn 2rjWk E(0') ejn dp' (2-3-18) n 27jrc i sko -o E (2jw k) e' do'. (2-3-19) We substitute the required H expansions in (2-1-3), (2-1-6), and (2-1-8), themagnetic boundary conditions at r =- a, r = 1, and r = c respectively, and obtain 0) 0) +A __(k0a)cos n A H (2) (k a) + BO H() (k a) + An Jn X (ko) 2a -0 0 n=1 200 n=00 ~(An n (koa) + B N (ka) ) e -j n n = -oo 8-80 < p 88 0, (2-3-20) n=co n= o (A Jn(k b)+ BnNn(k b) ) e =k (DJn (klb) + E N (klb)) e jn, n= -oo n= -OD ~~- ~PO,~ 4 ~ P ~ P~ t.(2-3-21)

16 and n=OD n=co k2 (DJn(k1C))+E NkC = e CnH((k0c) e1n n=-oD n=-OD - r A ~ X.W (2-3-22) Upon using (2-3-22) we eliminate Cn in the expressions for Dn and En and obtain D= -k[Nn(k 1c)2klC) - (kc)Pn n [ 1 nc) 1 ) JNkb)Nn (k1 b1nOC[Jn(k1)N (k1b)-J(k b)N'(k b (2-3-23) N (k1c) H(k0c) -kH (k c) Jn(klc)] n.n n k) b)J(k b)N (k, ] k k)N b) -J'(k b)N'(k c (2-3-24) If we s1ubstitute (2-3-14), (2-3-15), (2-3-23) and (2-3-24) in (2-3-21), we have.,.k.,Jnk,,, Pn e u —o:Jk(k b)Nc(k a) - J-(k0a)Nn(k0b) n Jn(k0b)Nn(k0b) - Jn(k0b) Nn(k0b) -n jn p J'a Nt~b %.te n= n=-ooJn(k0b)Nn(k0a) - JIn(ka) Nn(kIb) n -oo (2-3-25) where

17 2)'(k c) J (Substitutib)N (k2-3-18)-J (2-3c)N (kand the iJ (kN -J (c)N b) n -I2) 2)' N -J;lsb)~lsc) 2 {J(1jc)N'(kb) -J (bb) N I (k(b)N (k)c - _ N O -n n n nbklc H %c)n (2-3-26) Substituting (2-3-18), (2-3-19), and the identity Jn(kb)Nn(kb) - Jn(kb)Nn(kb) = 2 (2-3-27) in (2-3-25), we obtain n2 (k0c) J n(kcb)Nn(k0a) - Nn(kOb)Jn(k0a) 0 _> 7 (,-,. )7 E (~')e-jn(~-~ do' _ (2)' J(k ) n=- n (k Jn(k b)Nn(ka)- Jn(k0a)N (kob) 0 2 1 A' nI o e-Jn(}-)d~' rk b I ",' 1, do Jn = n(kob)Nn(k0a) - J(k (ka)Nn(k0b) O0 - ~o < P < 0o' (2-3-28)

18 Upon interchanging the summation with integration on the left-hand -ide, one obtains D —H(n2)kc k)N(k) J (kb)N(k)- n (k)Nn(kb) 0 n=^O H n (k c) Jn(kob)Nn k -) J (kaNPN (k n=oD +0 rk0 b 2 n=- JI(kJb)No (k)N(k()ei-J i' N c - < < ~P0' ~ (2-3-29) We then substitute (2-3-7) and (2-3-12) through (2-3-15) in (2-3-20), rearrange terms, and using the Wronskian of the Hankel functions Hn (ka Hn ((kka H (koa) H n - rOa) arrive at0 arrive at

19 1 Vo 2 j eowoko aJo (koa) J (k a) 2 %wo ko ~nlr (Oa) 2% 2 2% 1 n=oD Nn(kob)Jn(koa) - J n(kob)Nn(koa) 0 -jn(O-p) 2 no0jn n 0 0 ~0 0 Jn(k~b)Nn(k0a) - Jl(kOa)b N (kOb)d Or < < O +0 (2-3-30) where E = 1 for n = O and 2 for n f 0. We note that in (2-3-29) and on the right handside of (2-3-30) the series are summed on n from -a to o. If we employ the relations Z (r) = (-1) Z (r) n -n Zn (r) = (-1)n Zt (r) n -n where Z (r) and Z (r) denote the cylindrical function and its derivative; n n

20 one may simplify (2-3-29) and (2-3-30) respectively to E(pt)~ n (kdc) Jn(kob) N (k&) -J(k )N(kB) -O n =0 iE; (%bc) J(kcP) Nn(kg) - J(ka)Nn(kob) n:J n Onn n A k o Jn(kob) o N(k -J(k&) - o (2-3-31) and ~nr (k0a) =_ 0 ~. 2e 0~ D 200 nr ^ nf 180 0 +%, (k Nn -J(k b) N (k a) 0A n=0 n J(k0b)Nn(ka) -J(k n(ka)Nn(k b) Vo 1 2 ~oC " - - + 2 E(0')cos n(0.-4,), a o(koa) %a' (kNk - J(ka)NkJ) n =0.nCf n nU u e-00 < 0 e+0. (2-3-32)

21 Equations (2-3-31) and (2-3-32) are dual integral equations. Thus the boundary value problem is reduced to the dual integral equations. In chapter III, we regard P (p') as a known quantity and solve integral equation (2-3-31). Then we insert this result in (2-3-32) to eliminate E(0'). An approximate solution for "(0') is obtained for narrow wedge and narrow shell slot. The kernels of the last two integral equations will be studied respectively in chapter m and chapter IV; it is shown that these kernels have a logarithmic singularity when,' — p. 4. Terminal Admittance of the Wedge Waveguide Using (2-3-6), the first two terms on the left hand side of (2-3-20) can be written as ( kaB 1'2) 1) _.' (2)(k~a) + B0 ~O(k ) = 1 0 B (ka)d AO HO)(k a) + B;0(k&) 2jO0 wo ko 2'H()''(2)' E(p) do', 0~ AO0 I (ko) + BH(\ ) (kg),_ Oo but A0 H( 2)(k0a) + BoH( (k0a) J' (2) )' = y(a) A0H 0 (ka) + B0 ( and therefore (2) (1) (2-1) A0H 0 (k0a)+B I 0 (k a) y(a).= o (2-4-1) % 0 0 0 0 2 e w% 0,0 0

22 Upon substituting (2-4-1) in (2-3-20), we have,+0 O0 y(a) E (o') do+ J (ka) cos -(0 + +) 20woko( n nr o0 -80 n= 1 280 n=oo = I (An Jn(k0a) + BnNn(kb))e (2-4-2) n= -ao If one inserts (2-3-7), (2-3-14) and (2-3-15) into the last expression, one obtains J N(k 0a) (n =0 A k I I0 ((o n ( )d _ 0a0 n2(~}') (k a) 20o 0 Nn(k0b)Jn(k0a)- J(k0b )Nn(k0a) _n=0 a Jn(kob)Nn(koa) - Jn(koa)Nn(kob) 0 ko a I 1(2-4-3) (2-4-3)

23 Multiplying (2-4-3) by E(0) do and integrating from e -00 to 9+00, we have A!~~~~~~ nQ 0 n zn=1 206 2~ % +0~. J ( k)Nn(kob) -Jn(kb)Nn(kob) +0 +j,, I ka - J do E(c E(O')cos n(O-0')dp' n=O Jn(kob)Nn(k) - Jn(k)N (kob) d ) =0O J (k~b) N, d]~ (. (~) E(]')cos n ((k-N') d]J'. (2-4-4) One may note from (2-1-14) and (2-1-15) that y(a) is a dimensionless quantity. If the value of the terminal admittance Y(a) of the wedge waveguide is of interest, then by virtue of (2-1-11) and (2-1-14), we have Y(a) = 2 j y(a) (2-4-5) In A-1, assuming the solution of integral equation (2-3-31) is obtainable, the stationary property of (2-4-4) with respect to a small variation of E (~) is established. Thus in order to use (2-4-5) to calculate the terminal admittance Y(a), one needs to solve first the integral equation (2-3-31).

CHAPTER m SOLUTION OF INTEGRAL EQUATIONS (2-3-31) 1. Introduction The p-directed electric field in the shell slot E (p) may be considered as the sum of E (p) and E(0), respectively, symmetric and antisymmetric part with regard to p = 0 (Olte, 1965). Since \ E(p)sinndp = 0 and EO (p) cos n d = 0 One may split (2-3-31) into two integral equations::Ee(fI)~Ef~....n c,' ~,...., | eCos ncos n' dp' Ee(). nO H''k(c) Jn(kb)N N(k) - J (ka) N(k0b)J 0 0J(kb)NcJ(kb) -n(k JO k b) 0 nE I I E(01~ c no, dot(3-1-1) xO n n n n 8+ 2480

25 W EO(k c J (kNb)Nn(kJa) -J(kj)N(k) n=~ " O l n I n' i n i n s Uo nO (koC) Jn(kJa)NJ(kOa)-J'koa)Nk(kb)n OD 2 sinn _ _ 2 — -- sin n | E(p')sin n' do' 0n= 1 Jn(k0b)Nn(k0a) -Jn(k0a)Nn(k0b) 0(3-1-2) For large value of n, one finds 1 _(b)2n 1+ c 2 (2) b 2n H (k C) k) -0b (b) 2 n ___... c -2 n (2)' n b2n k (3-1-3) H (k c) I n O c +n -b )2n ( (k c and a 2n Jn(k0b)Nn(k0a) - Jn(k0a)Nn(k0b) kb 1 + ()2n....... 0....n Q. b (3-_ 14) J (k0b)Nn(k 0a) -J (k0a)N(k0b) n 1( As n - oD (2) Jk) ka kb H (k0c) J (k0b)N (k 0a)-Jn(k0Nn(k0b) k0 b 2 vn... l I t 0 - n(1 k n2))(kc) J (kb)N J() -J'kna)Nn(kb)n

26 Thus if we define H(0 (k0c) J0(k0b)N0(k0a) - J0(k0a)N0(k0b) -'r; I., (3-1-5) T 02 nH~ J(koc) J (k b)N (k a) - J0(ka)N(kb) H(2) H(n )(kC) J(kb) N(ka) - Jn(koa)Nn(kob) knb k7b l 2g(lb)N)c -J kLb)Nn-(ka,.J n0 aNk, n -C n+l)N(k (k)Nn+l(kob) I n n 2) - nk16 then the series which represents the kernels of integral equations (3-1-1) and (3-1-2)respectively become nH(k0c) Jn(kOb)Nn(k0a) -J(ka)N (kb ki b {b2)l n l~c(kNc) -J (k Nb) (c)-N lb)n(kc'' ('''', n no n=O \H (k0a) Jn(kOb)Nn(koa) Jn(ka)N(k - =, cosn cosnp'-2('+1E) n (3-1-7) n=0 n= 1

27 n l (k0c) Jn(k0b)Nn(k0a) - Jn(k0a)N (k0b) in _ _i n 0G - sin n +sinn i n(k 0c) Jn(k )Nn(k0a) - J(kOa)N(k0b) E2C~ma) bEop~a~~.zjlr~5*,~. sin nf sin nt' 2 Tnr sin no sin no' -2 (i1E+%) k b^ sinnf 8 n (3-1-8) n=l n=l From (3-1-3) and (3-1-4), it is easily found that as n — )co (2) b)2n Hn(kc) 2k0b 2 c V 2) + 12 — - 2 E k0b n (2)'(n) n H( n(k0c) and J (kob) N(koa) - J(koa) N(kb) a2 n, -2k0b Jn(k0b)N (k0a) - J (k0a)N (k0b) Therefore the series E kn n=l converges, and thus n T Cos ncos n n=O Co E n sinnp sin n' n_=1

28 converge uniformly in the region - )0 <, p S -0' Hence if one replaces the infiniite series with the finite sums N.E T CO n cos nO' n=0 N Tn sin no sin no' n=l then the error over the square region - 0 0 p, p' < P0 is less than a constant, independent of (p, *') Upon substituting (3-1-7) and (3-1-8) in (3-1-1) and (3-1-2) respectively and replacing the uniformly convergent series by their finite sum, we obtain | OEe(d~,,'- e n 0 =- e cos nCos nII I n=O n(kBN n (koa)-Jn(k a)Nn (kob) - (3-1-9) E(0p) sin ng sin n0l O'T sin no sin no dN n2 sinkb nn' n=1 nk)N(koa) -Jn(kOa)N (kob)-

29 These are the two integral equations to be solved in this chapter. 2. Noble's Scheme In this section, we reproduce a scheme due to Noble (Langer, 1962) to solve an integral equation of the form N E(x U )[ K(00) + l(0) )]0 do G(p0N).(3-2-1) A n=O If one knew the solutions of the auxidliary integral equations F(p') K(d, p') do' = G(0) (3-2-2) A and fn(0') K(O, p') do'' n(). n = O 12... N, (3-2-3) then upon substituting (3-2-3) in (3-2-1), we have E(') K(I, p')dp' + E(p) (, f(~) K(IA, p)d' do' = G(0) ~~A A n=O A (3-2-4) Let a - E(b') n(p') d~' I (3-2-5) Then (3-2-4) reduced to (~') + an fn(~') ]K(,~') d' = G(~) (3-2-6) nO

30 Upon comparing this equation with (3-2-2 we have N E(0')= F (0') f-nf(O') (3-2-7) n=0 Multiplying both side of (3-2-7) byfm(9') and integrating over the interval A < p', B, we arrive at 1m(,'19 E(O') do I+ d n(0) ~m(' 0 dot' m = 0, 1, 2,... N We let A fn(P') m("' ) do' (3-2-8) Bn1 F(p ) (0) d't (3-2-9) A then N a + A a uB., m 1, 2,... N (3-2-10) m n mnan m n=0 From (3-2-10), we are able to determine a.

31 3. Solution of (3-1-9) Comparing (3-1-9) with (3-2-1), we observe that E(0') = Ee0') K(p, p'O) = cos n cos n' n=1 (3-3-1) ~ (09 = fnTncoo nP' cos nO I,,1 ~ 2(1 +i2) kO G() = 1 OD E coSi(np)r(e) 1r( 1+ IC) c Jn(kob)Nn(k0a) - J(k0a)N (k0b) where E(Oj cos n p do' 0o and the auxiliary integral equations are e?,) cosn0cosn0 d, 1 = n n =1 n( 1(k p2 J (ko)Nn(k) Jn(k a)N (k P) 00 B o,~ (3-3-2) and f~e) cos no cond = 1 n n d =Coo n, n5 0, 1 2 N n.I~~;2(1+-k ( k0b Po)0 -P<b, Po *(3-3-3)

32 These two integral equations can be solved by employing Schwinger's transformation (Lewin, 1951) which is defined in (A-2-4). For detailed derivations of the solutions, the reader may refer to A-2; here we only state the results: e coo A (e) OD In e) COS 1 2 0 (csc 2 ~O='p)=2 S co 0 0; (co[mcos ( c[ c 2cos-coeo4.+;2)(k d-Cos 2 lec- = 2 -<Po 0, (3-3-4) fgw) 2 On + mn -1 1' 2(1+k")kJ os -co t Onc + mcos (csc -8-cpot -i n 2(1+ 2)k8 2Io0-c~8^o 4hncsc~ 0 < 0 O<' r(3-3-5) where E r(e)X.~e) En p (e) (3-3-6) p=O J (k b)N (k a) - J(k a) Np(k b) p0 p0 p 0 p0 (e) =2 2 p mp (3-3-7) p=m Jp(kob)Np(koa) -Jp(kOa)Np(kob) and Xm jcoOs(mlcs[pCo (Co s 2 s 2 sin co] d (3-3-8) (cos ~~~..ii

33 Therefore the solution of equation (3-1-9) is: -cosIe m e)I 2e 20 E(p)=.2 | 0 os[mcos 1(c2 cosO-coIc2 F X2m 1 20 2 2 4br n csc2In the last equation, the coefficients 0(e) can be determined by solving the ae)+ a (e)A(e) = B(e), m=O, 1,2,... N, (3-3-10) m2 / n mn m where (e m) e~f'(n m ) cos mp d, S-c =emm 5 F cos~mpd <, (3-3-12) To express Amn and B( explicitly, wesubstitute (3-3- in (3-3-11) and (3-3-4) in (3-3-12) and obtain (e)= _ X= ( A(e) em'm Xo m XqnXqm (3-3-13) mn 2 b0 (mIn on4,21 Oncsc A t — - A. A.'I'VIIQ -Q I.12

34 m rn o2 B tt+|?>tid>2 { I.+ t qS( X } (3-3-14)' (l2) (k b) 4,2 1 n esc 2 = 4. Solution of (3-1-10) To start this section, we introduce a transformation W (0) = E0o(') d' (3-4-1) -.0 for.the oddpart of the unknown slot field, E0(). Since for p' < 80, E(p') behaves as |7Rv, _r,1 when I "I —- 0o but is otherwise continuous, therefore the integral3 E0(0') dO exists and the function W(0) is defined at every point inside the closed interval - p0 < b < while its first derivative exists in the corresponding open interval. Thus one has Eo(0) = a od -f co < p < co - 0g(- do Also because EO() is an odd function of P, W(O) is symmetric with respect to. One may set w(- p 0)= and it then follows that W(TO) = 0 Thus integrating by parts

35 ~o~~~~~~~~~~ E0(') sin n do = sin nW() - W() cos n' do' % -~0 "-~o~ - = W(-)coo no' do' (3-4-2)'Po Let 9+0 to) -s ecp~~~~~sillnb'dp'. ~~~(3-4-3) r'(0)- N(') sin not do' I.343 n 9-0 0% Applying (3-4-2) and (3-4-3) to the integral equation (3-1-10), we obtain 6,I0 I N inn~cs b W(t') sin no cos not n nd {~1(1+k:kk0b =1 n n=1 OD ~r (O)sin no 2 n__ (3-4-4) v I b) n=1 J (k0b)N (k0a) - Jn(k0a)N (k0b) Last equation has the same form as (3-2-1), 1. e., E(o') W(1') K(O, 0') = sinncosno' n=1 ~.(nt) = n r COB,. (0): sinn (1+k2) (k0b) = - 2( 0 sin na' (1+ (k~b) 2n 1J(k0b)Nn(k0a)- Jn(k0a)N(k0b)

36 The auxiliary integral equations for (3-4-4) are: er,~~~~~~20 r 0)sinno F0 sinn0 C n0'd0' 2 2 J' (k (ka)-J'(kOa)N'(ka) "-0 < -0~ 0 VO'(3-4-5) -0 (1+k 2)(k b) 0 <0<0 O0 (3-4-6) We may again apply the Schwinger's transformation to the last two integral equations and obtain the solutions for F (0) and f (0). Similar to 0 n last section, we leave the detailed derivations to A-3 and state the results as: (0) 0 < 00 0 (+k )(irk0b) m1 (3-4-7) (0)(0) Ju I(0)U ( fn() _2 co 2 /Os p - cos p,, m~ U (), I <0 00 <0 (1+k )(r.k b) (3-4-8) where a - ( 1 L X + )' (3-4-9) b() p,p 2 2n - L2m, b0) _ 1'ij-)- C2X ) nodd, (3-4-10a) p = 0, 2o 4, p m PM- 1p p

37 b() _ 1X ) even, (3-4-10b) nm 7r = 5 Xm-lp _m+lp U ( m-p) e p- (m -p)oos p -(C 2cos0-cot2 ) -0U0: 0: * 0' (3-4-11) and r (0) L m (3-4-12) P m=nlp+3,... J(kob)N'(koa) -J (koa)N'(kob) Thus the solution of integral equation (3-4-4) is (i+k) )(7rk b)2 aUIn W(9) = 2 ~cos~ —cs0O {a0 U m() N,0In +rk~Obn n b(0) (0)nn U()(3-4-13) + 7rkob an b nmUm( 3-4-13 In the above expression, the coefficients a() can be found by solving the simultaneous equations a(0)+n= A (0) (0) (0) a O)+ A() ( = B) m 1,2.... N (3-4-14) m I rnnn r where A(0) m co=s(rmn )(0) dO' (3-4-15) mn m

38 B(m) S i m7m cos(m0')F(0) (0')dO'. (3-4-16) m m 0 -00 We perform the integrations in Appendix 5, and obtain AM = i m p b(0) (cos(pir)X - X ) (3-4-17) mn (0 Om pm (1+k )(7rk0b) p 0 (0) si_ (0) B s - m~2 2 a0 a(cos(pir)X, - X (3-4-18) m (1+2k )(wk0b) p = The antisymmetric part of the 0-directed electric field in the shell slot can be obtained by differentiating W (0), i.e., Eo(0) = d W 00 ~o We carry out the differentiation in Appendix 6, and state that sin a E __(0) __ 2 E - (+k )(7rk0b) Vos0 - cos00 {lcos(0 2) [ b (O)m U(ba (0) Cos a u ()+ 7~ l (0) k(l a b]u + (1 + cos0) [m am )mVm(0) + irk0b N (r0 lb0mV (0) ] - 00 < 0 < 0 (3-4-19)

39 where Vm(~) =+2 aam 11 2 + 2 [p _ 0 (0) = + 2osos (csc cos - cot2 ) m odd (3-4-20) rn- 1 r 1 2' 2 0n1 c2 Cos pCos (csc — cos0 - cot j). m even 5. Discussion In the first section of this chapter, we obtained a pair of integral equations (3-1-9) and (3-1-10) from (3-1-1) and (3-1-2) respectively by truncating the uniformly convergent series. Therefore (3-3-9) and (3-4-19) are the approximate solutions of (3-1-1) and (3-1-2), respectively. The accuracy of these approximate solutions depend largely on the value of N. But N + 1 and N., respectively, are the degrees of freedom of the simultaneous systems (3-3-10) and (3-4-14). We may encounter the usual difficulties of solving a large simultaneous system of algebraic equations. We attempt to reduce this difficulty here. Expression (3-3-9) suggests a transformation z(e)'( ae)x (3-5-1) m a mn' n =m Upon substituting this transformation in (3-3-9), one has E 1 = Cos 0(e) OMS(e) Ee(~) = ~ 20 20 + m co m cos1 (csc os 0 - cot2 ) m= ~i -1 2 2O2~22 m1

40 rk b [ 1 (e) N (e) 1 200 20 47r 2Incsc m 4ir tncsc- 0 - < z c 0m cos (csc * (3-5-2) If we multiply (3-3-10) by Xpme and sum on m, with the knowledge of (3-5-1), we have N (e) =ommOmpm (e)N N (e) 2q Q X X )z (e) 0 z O q( + "2( b b4tncs-(1r l+k2kob 8r (1+k )(k0b)lncs 2 0 1 = ~ 2 m Om pm r q12 q mb p qm PM Upon multiplying (3-5-3) with the factor R defined as 47r (1+k )k0b RpN L N(3-5-4) m Om one obtains a. new system of equations (e) sOP (e) _ 4 ] iP k bncs qp q 0 2 S~~pO S mN5 - g SOD + = 7k-b mp

where N f T X X M m mm qm pm S =. (3-5-6) m Om If we express the simultaneous system in a form of a matrix equation, we have a ze)]= b], q, p = O 1,... N. (3-5-7) where [a denotes a square matrix of order N + 1, while z(e)] and bp] denote the column matrix of the same order. Comparing (3-5-7) with (3-5-5), we obtain SOp ap= R:6 - ~P * 21ncsc 2 a =R6 - 4qS, q 0 (3-5-8) qP qP qp where = 1, f q = p qp = o if qfp and 4e) ao mS(e) b = Sp+8 Zm S (3-5-9) P — o P m 1k0b mp 7rk blncsc 2 Sqp plays an important role in further reducing the matrix equation (3-5-7). In the following paragraph, we state some of the properties of S~ qp

42 In Appendix 4, we show that for any 0 Xqm = 0 when q > m. Upon employing this property of Xqm, we may conclude that Sqp is symmetric with respect to its subscript index q and p, i.e., S = S. (3-5-10) qp pq We recall that Tm is defined by (3-1-6). It depends on k0a, k0b, k0c and k. For m>m (mo 2kOb<N), 7 behaves as - ()+( )2 m m c b for1 [ 1 I a)2m P for p=, 0 and as + (-) for =O. For m<kb, t m m(m -1) b o 0 the T values are large and may be oscillating in sign. From the (A-4-5) m property of Xj k which is also explicitly accounted for as far as the p subscript is concerned in the definition of Sqp in (3-5-6),we see that as p increases, the sum maldng up Sqp consists of terms involving Tm for which m> p. But the Tm terms decrease rapidly once m> mO and thus since lXi ki < 27r we see that Sqp will decrease rapidly once p > mo. Because of (3-5-10), the same behavior is exhibited also on the q subscript of Sqp The properties of Sqp are further modified if we consider the angular width of the slot 2 00. From the discussion in Appendix 4,, it is clear that for 00 sufficiently small, there is a number j such that X j k0 k<m0 jk O The net effect of this is that the magnitude of Sqp is further reduced as either q-subscript or p-subscript increases. In view of this discussion as can be seen from Eq. (3-5-8), the matrix [ap] can be reduced in size. We indicate the size of this reduced matrix by N'. In fact, for a very narrow slot, we only need to consider in the matrix the first element a00' i.e., N' = 0.

43 If the slot is very wide, ( = r - A, A is very small) then kXjke 2r6jk and S - (Tq/T0)6 qp, p O.1, 2,.... N. Therefore the matrix aqp] becomes a diagonal one, i. e., the problem becomes a separable one. The fact that Sqp in (3-5-8) is multiplied by q does not change the order of magnitude of our arguments. For the odd part of the 0-directed electric field E0(0) in the slotgiven by (3-4-20), we introduce a transformation (0) N (0) (0) z = - a. (3-5-11) p I m mp Then E2(0) = a a [cos 2 n(9+m(1+cos0) Vm(0)J (l+k X7rk0b) ~cososi 0 m I = (0)~ 2 +' rk b Cs -0U m(2 ) + m(l+cos)V ()m( (3-5-12) where Um(O) and Vm(0) are given by(3-4-11) and (3-4-20), respectively. Upon using the transformation (3-5-11), (3-4-14) can be reduced to a new simultaneous system of algebraic equations. We express this new system in a matrix form [a] ]b],,p 12 (3-5-13) where ap] is a square matrix of degree N, z(O)] and b'] are the column matrix of the same degree. The elements of [a'q and bP] are, respectively, a' = R' + sin(-2) [To cos q - T ] qqp 2 Oqsin2 )(TopC ~s T qq

44 and b'= sin2 a() [T cosmir-T (3-5-15) p 2 m op mp In the above equations, we define (1 + k2) (rk0b)2 N (3-5-16).m T X b(0) m lm ml and N(0). mT X b() m qrn mp T = mNp (3-5-17) qp N s T m7 X b(0) IIm lm ml We observe that in (3-5-13) Tp plays the same role as S in (3-5-7). X = 0 if q > m, therefore T becomes qm qp mt X bt(0) ~m: m qm mp T mq (3-5-18) qp N (0) mT X1 b nm lm Mr We note that from (3-4-10), lb (0) < m, thus in the numerator of (3-5-17) m 2 mP is at most multiplied by m. But since Tm decreases rapidly for increasing m when m > m0 the m behavior will prevail over m. Therefore the magnitude of T will decrease rapidly as either p or q exceeds mO. The effect of the slot width enters into T in a similar manner as for S qp qp

45 Based on the above discussion, we conclude that [ajp can be reduced in size for a narrow slot. We denote this reduced size by N". For a very narrow slot, we may choose N" = 1, i.e., we only need to consider the first element all 1 For a very wide slot, because of the property of Xj ks Tq = 6 Matrix [at] is thus reduced to a diagonal form. 6. Solution for Narrow Slot In this section, we shall extract the solutions for the narrow slot from the general solutions (3-5-2) and (3-5-12). This is a case of some practical importance. In the later chapters, we use these results to attain an approximate solution for the 0-directed electric field in the wedge slot and to have an explicit form of the terminal admittance of the wedge waveguide. If the slot width 200 is so narrow that we may apply the approximate relations X - 1-O(N02) op2 X1p O0(N0o) to (3-5-2) and (3-5-12), then we may neglect all terms of order O(N00 ). The results are: _ _ _ _ _Cos - 7rk b E (0) T 2_ 2 ____S_ (e)+ 0 Z(e) 4ir (1+k )(3-6-1) 0 2 -00 <0 <0 0 (3-6-1) and ff (a {0) + irk bz ()) sin E (0 01 2 2 (1 +< < (k3-6-2) { cos2(:9)Ui(0) + (1 + cos 0) v1(0)1 * -00 < 0 < 0n X 362

46 Since U1(0) = -1, V1(0) = 1, (Eqs. (3-4-11) and (3-4-20)) (3-6-2) can be further reduced to Y2(a(0 + 7rkbz () sin. cos0 E(0) - 2'0 0. * (3-6-3) E0 (0) s k~~-2 )2 ____3 (1 + k )(irk0b) ycos0 cos0 0 The unknown factors z0(e) and z (0) in the last two equations can be obtained by choosing N = 0 in (3-5-7) and N = 1 in (3-5-13), i.e., ~~S ~sh~(e) 00 (e) 0S (R- " ) Z)o - "0 SOo (3-6-4) 21n csc - rk0b In csc 2 and 2 ~o (0) sin (0) 1 2 sna T2 T (3-6-5) (1+k )(7rk0b) From (3-6-5), it is seen that z is of 0(0 ), therefore we may neglect (0) (0) 1 z1 in comparison with a1 in (3-6-3). Upon substituting (3-6-4) in (3-6-1) and reorganiz ing terms, we have (e) cos 2 -2 0 22 8 7r2(1 +k2)(kb)in csc — 2 - T m m Om - 0, < <~00. (3-6-6)

47 Employing the approximate formulas 2 C0os y 1siny —y, y<< 1, the even part and odd part of the slot electric field can be further simplified to (e) 1 E () et.(0) a0 0 < 0<50 * (3-6-7) and a 0 f 1 0 4<.. (3-6-8) E0( 7(1+k )(kb)2 022

CHAPTER IV APPROXIMATE SOLUTION OF INTEGRAL EQUATION (2-3-32) 1. Introduction: If we interchange summation with integration and reorganize the terms on the left hand side of (2-3-32), we have 0+90 Ci O k0 %a 1 Co s no (0-+( )Co n(01-0+ )+Cos n(0- 0)1 Jn (koa) J(ko.a) o0 Jo0%a)Nokob)-J%6 NO(kOa) + 2%o kOa J' (k a)+ Jr (kb)Nn6(a)Jkob) n Jn0( - n 2%o 2 0 1 o2 Non(0-d0 a J(koa)+ 7rka 7. J c(kob)N'(k a)-J((ka)N(b) | n(0)cosn(-0 O+ (4-1-1) From the recurrence relation of cylindrical functions Zp+ 1(Z) = pZ'(z)- zZ (z), (4-1-2) p+1 ~p p one may easily show that J n(kOa) J n 1) J 7k a) 0 00(k0a) 2 00 ka 2kaO (O 21..n.. _ 0 - (4-1-3) J'ns (kOa) It nJ' (oa 280 0 20 48

49 and J (k a)N(k b) -J (kb)N(koa) koa koa J l(k a)N'(kob)-J'(k b)N (k a) nO nO nO nO - 1 nO n O n-1 ~O J'(k b)NI (k a) -J'(k a)Na0ub) "- I J' ( b)N'(k a)-J'(k a)N'(k b) (4-1-4) For 0 > > k a in (4-1-3) and n>> ka, kob in (4-1-4) the last two equations 2%0 0 behave, respectively, as - and s+ n ()n Therefore the second and the third series of the kernel of integral Eq. (4-1-1) are uniformly convergent on a square interval 0 - g< 0.p0' < +0 o while the first series has a logarithmic singularity when 0' - 0. Thus the chance of solving (4-1-1) depends largely on whether or not one can solve the integral equation F(T')n [cos 0r)cos + cosnrcos 0nr' dq7' = G(0) (4-1-5) Unfortunately Schwinger's transformation is not applicable to this integral equation. Therefore, to solve (4-1-1), a new transformation of some form is required. If both the wedge slot width 200 and shell slot width 20o are much smaller than unity, we may substitute (3-6-7) and (3-6-8) in (2-3-32) and then employ Galeldkin's method(Kantorovich, 1958) to obtain an approximate solution for the integral equation (2-3-32). 2. Reduction of Integral Eq. (2-3-32): In Eq. (2-3-32), the variables 0 and 0'! are referred to the center of the shell slot, while the unknown function is the 0-directed electric field in the wedge aperture. It is more convenient to express (2-3-32) as function of a new set of variables rl and rn' defined as

50 7 = 07 — = 0'-e. (4-2-1) It is seen that from (4-2-1), r1 and r7' are referred to the center of the wedge aperture. Upon substituting (4-2-1) in (2-3-32), we obtain J (k0a) _ 0 J 00 20 0 OD J (koa)Nn (kob)-J'(kob)N (ko a) 0 =OO n nO n 0 1 1 0 CA - - a bJNa+ko)-Jn (k a)N' b' (tncosn(n-rj9d a JI(k a) rkAa n0 J b)N (k a)-JIk N0 NO(4-2-2) In (4-2-2) we may regard E(vi) as the sum of a symmetric part E (r) and antisymmetric part E 0(rl); thus E(n) = e (n) + E0(77) (4-2-3) Since cos 20 (rl+ 00) is an even function of nr when n is even, and an odd function of nr when n is odd, we have 7e(rl) cos2 (Or+ O)dr7 l= 0, n = 1, 3 5,. (4-2-4) e~~~~~~~~ 20"

51 A ax 0#2OB2~(t+ g~= n=0 2, 4,... (4-2-5) 0 0 0 ^(4-2-) 0 Eo(r)cosndrl = 0. (4-2-7) -00 If we substitute (4-2-3) into (4-2-2), and use the relations (4-2-4) through (4-2-7) we may obtain two equations: gn a(koa) 00 _n_-, ~acos (n ) 4 fl(7')cos (.7' + 6 d. n= ( 1319 0o 0 co J (ko a)N'(k b) - Jn(kob)N'(k a) ~0 O Ji OE)N'(k a)- Jk a )N(k +-n J(k0b)NJ cos nrl _ )cosrl' drl O0 C OD E - (e) 0 1 + 2 O0 nn -- _ +2 8n=0)

52 and J 2n- 1(k0a) 0 2n- 1 2n- 1 J'2n-1.0 s (n O) Eo(rl') 2C0 ( 0'+ 0)d' 200 + Jn Kn ) n l n 0 n O sin nr 0 o( )sinn7 dw Ir J I(k0b)Nn(k0a) Jn(ka)Nn (kb)) *9 where (e) = E (r)cos nrdtr, (4-2-10) and (0) E(rl) sin n rd r (4-2-11) In the preceeding chapter we expressed the shell slot field as function of 0, therefore to perform the last two integrals, it is more convenient to go back to the 0 variable. If we employ (4-2-1), then (4-2-10) and (4-2-11) respectively, become 00 y (e) = cos0 nO Ee(0')cos n0' d0' -0 o t sinne 0 Eo(4')sin-) te (4-2-n 12) -0 0

53 and 7n(0) = cos n 09 E0(0') sin n 0' d 0 - sin n 8O Ee(0')cos n d. (4-2-13) -0 If we limit ourselves to the case that the angular width of the shell slot 200 is much smaller than unity, then substituting (3-6-7) and (3-6-8) in (4-2-12) and (4-2-13), respectively, and neglecting the term of 0(0 2), we obtain _> (e ) ~ 4S (e) J0(n 0)cos n O (e) e 0 0 (4-2-14) n 2 2 0 kob [87r (1+k')k0b In A- T en XOm ] and 4S (e) (n0 )sinne (O)r~.... 0 O (4-2-15) n k0b [87r2(1+k2 )k0bnI0 - NTn c nX0m As was stated in the introduction of the present chapter, we confine ourselves to the case that the angular width of the wedge slot is much smaller than unity. Therefore i- > > 1, and the first series on the left hand side of 0(4-2-8) andpectively, can (4-2-8) and (4-2-9), respectively, can be approximated by

54 J (koa) no 0 0 Ft7 (k'a) 0os e 0+ 0d 2n J. (k a) cos ~0 (r+ 0 0) 500 0 O 0 oo2ka 0 n7r A 3 - co-COS (77+ ) I E (?')cos —(r'+e0 )d +( 0 )0 n = 1 - -00 0 e 00 0 0 (4-2-16) and 2n-1 (k0a) %02k 2n- 1 2n - 2,, I co 28- -.o(o+ H) Eo(,')cos 2 l0+) d n= 1'n-1 (2n-) 2 1 200 0 20 W -00 0.~ 2k a 2n-1 0 0~ COS 2~0 n - 1 80 S O A 2n - 1 T =1 (211l 7 ( 1)) E2(')cos 26 (r'+ O)dr'+O(/03) -0 (4-2-17) If one inserts (4-2-14) and (4-2-15) into (4-2-8) and (4-2-9), respectively, and introduces the notations (1) 1 Vn J'(k b)N(ka)-J (ka)N' kb) (4-2-18) n nO nO nO v(2) J (k a)N0(k0b) -J (kob)No(k0a) o J (kob)N0 (k a)-J (k a)N0(k b) (4-2-19) v (2) ka J0 J (k a)N(k b)-J(k b)N l(k a) (2) n- 0 n 0 n n- 0 n(kOb IVnO a On n

55 one arrives at 8 ~~~~~~~~~8o Jo(koa) 0 0 oD 2ka 0 0 0 0 n,,dt+ A n~r 00 n ".~2~cos!(ri1+ 0 E O(il')cos ( 0'+ 0 )d r' J'I (k a)i ei 7r' 0o u e 00 0 0 0 J -e 0 0 o D 2k a e0o O0 0 0 An' (2) "cos n El)cosnr d l+ -E v COSn E (i)cosnrl dr n ~ 3 n e -00 0 0 4 S (e) o 20O -00( 1~10 a J'(ka) 9 (4-2-e20) and a2 2k a n 0 40 2 n obl2)n n n On V_0 a) (4-2- 20) and - 2k oa 2n-,' 0 (2)- 1) o'%7~l 0 i -0 0 00 2 sin n 4E 0sinn0(e dn' + Oin n7 n9sin neinn 0 n n 0 n n 03 0 -8o 4S(e) k+ a N 2)sn n in n n 0 0rk 2 -0-2 2 2 ~Nn n"OUOI~'UVI~U~ k b [87r (1+k2)kO bsnt E TXs i 0 n n O( O~~~~~~ (4-2-21)

56 where So(e) is given by (3-3-6) and can be rewritten as s(e) = n en(1)c osne0n e0 df -2 E ~ sin os19O E si (n') cos' n(4-d2-2 -e0 In the next section, from the last three equations, we will find the solutions for E (ri) and E (r). e 0 3. Approximate Solutions for E (r) and EO(rl): e 0 We will apply qalerkin's method (Kantorovich, 1958) to find the approximate solutions for Ee(rl) and 0(7). This method requires us to choose the forms of Ee (n) and EO(f) in advance and then to determine the arbitrary constant for each field by substituting back in the integral equations. Since the electromagnetic fields in the vicinity of a perfectly conducting right angle edge behave as 3 r (R. E. Collin, 1960) where r is the distance from the field point to the edge. Thus e() and 0() may take the forms A(e) -E(r7) =, (4-3-1) E0(7) 3 (4-3-2) The remaining problem is to determine the constants A (e) and A40).

57 We first substitute (4-3-1) and (4-3-2) in (4-2-20) and (4-2-21). Then multplying (4-2-20) with 1/(00 2 72)1/3 and (4-2-21) with /(002 ) 1/3 and integrating with respect to r7 from -00 to 00, we have a) (e) 2 00o o 2ka ((e))2 2ka (e) 2 V (2)(e A(e) (Q ) + -(P + (Q ) + v } 0 ~ (ka O nn n n n n o n V n 20 n= (e) 0 1 (e) S -Q 7rk a v2 2 N 2 0 a JO(k 0) 0 k0b [ (2(1 +2)k I K 2] 0 (4-3-3) 2k0 a (0) 2))2 O { 0(P 2)2 0 (Q())2 (2 Q(0))2 A 2n - 1 n n n n 0 o a + n 0 () 0 (4-3-4) koa2 -2 2 N k0b 8r1 + k )k0b n 29 E- n and (e) (e) ODv(1)i (e).0 (0) = A0 ~nva cos(n6O nQ -2 A0 Vn sidn2)K n (4-3-5) nO n n n n where nI cos_- rWdrl p(e)_ (4-3-6) 3 3 2 -eo -

58 2n - 1 80 7 sin 20 -! (O) 0, =.... dn (4-3-7) n 3 2 S40 iS7Integrals (4-3-6) through (4-3-9) are discussed in A-7. It is shown in A-7 that 1 1.(e) 93 2 26 n 0 () ) J16(n 1 1 (4-3-10) (e) 3 2 2 6 Q) = eo 3) ne 1 1/6(neo) while P()/P() and Q()/Q() areatleastof 0(9) and 0(9 )2 respectively. From (4-3-3) through (4-3-5), one can easily obtain that (e) (e)' 0'' A(0 a (4-3-11) and 2 (1 (e) O (1) (0) (e) [ E0nvn (no)Qn cosnej[ a v n o(no0 s) A() ob,, V a0 n 2(1+k )kObn e (o (4-3-12)

59 where h - - 2Je)2 o(ka (.e)2 0Q(e 2/u A= J0(k0a) (Q0 e)(k) a (P /n+ 2koa (Q(n /n OD1 ((e) 2 ( 1)j (n, (e) os no) OD) (2) (e))2 4 n 0 )nCOSn) + 2 v 2 ~0 n nO (4-3-13) In Eq. (4-3-11) and (4-3-12) we use the approximately equal sign because in (4-3-13) we have neglected the terms of O(0 8/3) and replaced X0n with 2rJ0(n 00) on account of (A-4-16) and for the convenience of computation. One also notices from (4-3-10) that the series (Q(e))2/n in (4-3-13) n1 n converges very slowly when 00 < < 1. Fortunately, under this condition, it is found in A-8 that (Qn )'/n = (Q0 I n (2/00) + 0.050532/3. (4-3-14) n -

V SLOT VOLTAGES AND TERMINAL ADMITTANCE OF THE WEDGE WAVEGUIDE FOR NARROW SLOTS 1. Introduction In this chapter we will obtain explicitly the three important physical quantities: the wedge slot voltage, the shell slot voltage and the terminal admittance of the wedge waveguide when the angular width of the shell slot and the wedge aperture are very small in comparison with unity. The voltages of the wedge aperture and of the shell slot are defined, respectively, as 0 and V = -\ bE(0)d0 (5-1-2) Since e-o 90 0 and S bE0(0)d0 = 0 60

61 consequently, we arrive at 0 V ( aE (rn)dr) (5-1-3) w e, O V = - bE (0) d. (5-1-4) -00 In deriving the explicit form of the terminal admittance, we neglect all terms whose magnitude are of 0(002), or O(00 ),or less in comparison with the magnitude of S (e) 2. Voltages of the Wedge Aperture and the Shell Slot From (5-1-3), (5-1-4), (3-6-7) and (4-3-1), it is obvious that V V [Q e)J 2/A, (5-2-1) and ~V E ~ Rv(1 sn)cos(n9)Jo(n )Q s 2 (5-2-2) - (e) 2'-2 2 N 2 w rk0a Q0 2(1+k )k0bin- - En7nJ0 (n) where A is given by (4-3-13). It is seen from (4-3-13) that V is only weakly dependent on the slot separation angle e, plasma sheath and the coaxial spacing, except when J0(k0a) is close to a zero. For this exception one can show for 00 << 1 that V - Vo/JO(koa). (5-2-3)

62 Thus for the radiation problem, one may even regard the narrow wedge aperture as a constant voltage source. On the other hand the voltage of the shell slot depends not only on V but also on w, (k0b-k0a), and plasma sheath. 3. Terminal Admittance of the Wedge Waveguide When 00, 00 < < 1 The terminal admittance shown in (2-4-4) consists of three series. We will consider these series in the next few paragraphs. The first series is Jn (k0a) 0nr 1 = 2j J1 nr (k0a) 2 -) - (00-e+%)d0 n m220e - 8+ 00 0 which can be separated into two series, i. e., J (k0a) i A 2 (r 20nd 2j'I (ko a) E (ri')cos (' +- 00 0 0 2n- 1 (k a) E(kf)0i') sin 20 (rl' + 0)dn=1 2n-1 0 280 since 0' and rl' are related by (4-2-1). Since 80 < < 1, similar to (4-2-16) and (4-2-17), we employ the approximate farmulas J' (k0a) n

63 J 2n- 1 (k0a) 20 80 2k0a 2n- 1 (ka) r 2n-1 280 then using (4-3-1) and(4-3-2). we have i 2jk0a 0 (A (P(e))2/n+(A() As we indicate in A-7, the magnitude of Pn e)/P(e) is at least of 0(80), therefore one may neglect the second term and arrive at 0 (e)2 co 2k0 (e)2 I O A(A ) _ (P ))2. (5-3-1) The second series of (2-4-4) by change of variable (4-2-1) and dsing (4-2-3) can be yvritten in the form o O J (k a)N'(kOb)-J'(kOb)Nn(k a) r I - 0' E nO en(i)cosn dr) 2 = n r nJ I(kOeb)N a)-J IOka)N I(k b) L [ f 00 2o + ( E(r) sin nr drl)2 -0 If one substitutes (4-3-1) and (4-3-2) in last equation, and neglects the terms of O (80), one has (A(0e)2 nJ(k0a)NI(k0b) - J (kb)Nn(k0a) (e))2 i ne j O (A ) 2 ir 0 0 en Jn(k b)Nn(k a)-J'(k a)N?(k b) ( Usi(4-14),n reduce n n n form, i.e. Using (4-1-4). we can reduce 12 in a more convenient form, i.e.,

64 2 = (AO ) v (Q(e))2+ 2koa (Q /n (5-3-2) n=0 n ni n=1 The last series of (2-4-4) is 13 r 7rkaE [ I (0)cos E(')cos cosn'd0' 9%-o -o O+ % 00 +\ E (0) sinn 0d0 Eo(0) sinn0d] (5-3-3) 0 -00 where v (1) is given by (4-2-18). Again if we make use of (4-2-1) and (4-2-3) and consider E(rl) as the sum of the even function A(e) - and the odd function A(o)or/(9 - 72) 1/3 it is obvious that 0 0 O+ 00 A =. (e) (e) (0) (0) E(0)cosn~d0 = A0 Qn cosne - A Qnsin n (5-3-4) o-Oo and e+O0 01%0(0) Q(O) (e) (e) E (E0)sinn0d0 = A0 Qn cos n0+A0 Qn sinnO. (5-3-5) 0 - 0 0 From (3-6-7) and (3-6-8), respectively, 0 lo ~ 4"(e)J (n0) Ee(0)cosn0d0 22N (5-3-6)'0 k ob 8r2(l+k2)kobln(2/0o)- n n On

65 and 00 a(0)o E(0))sinn0d = 2 ( ) (5-3-7) ~0 7ir~(1~+k )(k b) Upon substituting (5-3-4) through (5-3-7) in (5-3-3) and neglecting the terms of o(02 ), (O (02) and O(000o) (e)(e) (e) s(e) ~~ n n n cos(nex (n0)Qn.02 On\O I3 - -j N (5-3-8) 3 Ir irkoa 2 2 2 N 22 7r kob 2(1+k )(kob)ln -O n n=O ( 0 where S (e) is given by (4-3-5). Therefore, inserting (4-3-5) in the last equation, one obtains 4 (e)E e (e) 2 00 2 22r (A 0 e)n- n vJ0(no0) Qn cos n!.EQrkoa e 2kb - b2(1+k 2)kob I n 2.-T 2 (5-3-1) -, 0 W, J0 ( y(a) 12e j ja ) (Po)Q (2) (e)4 2 2 E v i0n n 0 n=0 OD (2) (e) 2k 0ab + E V (Q ) ))N la (5-3-10) n n r n n -2 2 2 2(1 +k )k bblnw — E n inJ (no.)

66 Thus substituting (5-3-10) in (2-4-5), we obtain an explicit formula for the terminal admittance Y(a) of the wedge waveguide. For the convenience in presenting data, in Eq. (2-4-5), we choose L = a and attain the resulting form of i O ae)/Q(e) ~ (e) (e)/ 2 Y(a) W 1J — 2kO ( u / )/+ + L 2(1) n& 1n'(e) (e) 2 a 2 Lr aenn *~ ( (e)/ (0 )cos ne a __{ (4) ]k. bn2 a ra N n=OO( /( 0 n=O The last term inside the brace can be written as where the voltage ratio VS/Vw is given by (5-2-2). In view of (3-1-6), we further introduce the notations (3) Jo((k b)N(a) - J)(k a)N (k b) V3) 00 00 0_0 00'~ JIJ0(k b)N0(k a) -J(k a)N0(k b) k b J+ l(kb)N'(k a)-J (k0a)N (k b) n(3) 0 n+1 0 n 0 n 0 n+1 0 V nJ(k b)Nu(k a)-J'(k a)N'(kb);, nO0 n 0 nO0 nO0

67 and ir = 7r 0 0,:.2kj' -lN'(klbc)-J'( kcC)N l(kb)]' j(2: in(kb)N(c)-N i(kcJ -(klc) 7r kn n n (2)(2) H~!0~~~~H (k0c) (5-3-13) Then 70 and T become n n H(2)'k tance, ten from (5-3-11) through (5-3-14) one arrives at n 0klvi (3) 4.42- b..*Im[a. n 7r' )*(n.( J0(n.0)- 2(m ) k0b n(2/0) n 15n b'IIV ]Vw~2 n: n Hn)(koC) 2=n n (1) 000 (5-3-15)

68 B - 1- {2ke)OD. (e (e) )2 () ( (e) 2 0a ) 2 /n+ (Qn /Q ))2n 27r % 0 n a1 0 O n 01 + (2)Q(e) /Q(e))2 a- (3 ) 2k n20 =V n (Qn /Q 0 b 2 n v j V 2 N HM (kc)k ~ A2E Re E e(7r') (1) J(n 2 2k bln(2/00) wb i2n n 0o0 n n H1 (k0oC) (5-3-16) where Im and Re are the abreviations of "real part" and "imaginary part" respectively. Since (5-3-15) and (5-3-16) are so complicated that in general one can hardly obtain any information before actually performing the numerical computations. However, in a certain special cases, some properties of the conductance G and susceptance B can be read from the expressions. Case a: In this case, we assume no plasma sheath, i.e. we let c -- b and k -> 1. From (2-3-26), it can be shown that r n — 1. Since H (k b) nO0 2 H( l)(kb) 0 n O then (5-3-15) is easily reduced to a form e 2 ~N 1 J2(no) Iw2 2 G J a 1 n (5-3-17) 2 b 2_ _ J(n01)2 w 0 n~~

69 If we divide (5-3-17) by (A-9-4), then we obtaina formula of similar form as Eq. (38) in one of Olte's recent papers (1965). This coincidence is a physical consequence because from the circuit point of view, if V is the terminal 2 voltage, G is input conductance then it is well known that the power P = GV It is interesting to note that as 0 — O, G decreases as -1/ln00 and B tends to an expression independent of D0 and 0, B ~g )/Q( /n +) / n 1 0 (2(e) (e) (e) 2 +2ka a n kn (Qne) /QO1 (5-3-18) The susceptance given by last equation is the terminal susceptance of the wedge waveguide for the case of no shell slot, i. e. a continuous shell shrouds the cylindrical antenna. Case b: In the present case, we assume that I k 1 < 1.0, kbb -k0a<<k a, k0b and k a = m, a positive integer.. In this report, we limit ourselves to k a < 5. Since the series'C'O (2) (e) (e) 2 O 1),A (e) (e) Since the series ev(Q /Q ) and E EV no)(Q /Q )cosnO n0 nn n 0 n n n 0 n=O n=O (the numerator of V /Vw, (5-2-2)) converge absolutely, we may truncate them N (3) 2 at Mth term and then these truncated series as well as the ~ n a J0O(n00) will be proved to be dominated by their respective runth terms. If kob- k a < < ko a k b, it is found that nJ (ka)N'(kkab)- J'n(kob)N (k b) n- 0n 20) n 0 n- 0 k a irk a 0( a 0~~~~~~ 0~ ~ )nka-'~~) nlka]+..

70 J (k b)N' (k a)-J' (k a)N (k b) n+1 0 n 0 n 0 n+1 0 -n - -k ( (b a) J +l(kOb)N"(k b) -J"n(kb)N N (kOb) +. k0b rkb n+ n n 0 0 J' (k b) N' (k a) -J' (ka) N' (k b) n0 nO O 0 nO 2k (b-a) )2 (1 b-a)2 rk0 a Lk0a ] 2 L 0 n O nO n O where n > 1. Thus if one made use of the above results, it is clear that (e) (2) (Q(e) / (e)4 2 1 m )2 Ok b-k a nn n 0 2 2 F (e) [ ] n=~ k0 k2a) k 0 m (5-3-19) (e) M (1) ((e) (e) 4) sm ZE V( J (n0o)(Q )/Q0 ) cos n 24(e) O[k0 b -k a]} 0 0 n 0k (b - a) Q 0 0 m (5-3-20) and N (3)2 4 2 (mO } (n 0)( 2 2 k b m + (k b-k ) (5-3-21) 0n n n 0 k 2( a) 2 k 0b F ( b- 0a2 0 where F mJ' (k a)N' (k a) - J' (k a)N"' (k a) (5-3-22) m m (0) O m O mb 0

71 If one substitutes (5-3-19) through (5-3-22) in (5-3-16) and (5-2-2), one obtains 0X 4 f2 2{ (Q e))2 k (b -a) co 2k0a Q (e) B ce ( 1 0 4 2 m )2 + n n 2 2 [0 k 2(b- a)2 Fk a (e) 4 (e) b 2 V 2 42 k (ba) 1 J2(MO)+ 0 ( 2k ain(2/0) jv2LdrF ka 0 0 0 0] in 0i v 2 2o(b-a' 2 r +a. 2 s. - kU (2n0) -2( (12)I kbib n (2/ 0)]} ( k0a= m (5-3-23) and s, 2 (J0(m0'e' cos meAQ0'F Vie ka |2(m(0) k b 2 a() k0a =m (5-3-24) Since ikl < 1 and the angular width of the wedge aperture and shell slot in practical case are small but finite, Eqs. (5-3-23) and (5-3-24) can be further reduced to

72 V bQ (e) V Q(e) JCos(m) +C k0(b-a)] kOam (5-3-25) w aQ0 0 BY 4 ) b os2me+0[kO(b-a)1} kga = m. (5-3-26) If we substitute (5-3-25) in (5-3-15), we obtain Q (e) b m )2c2b 2 Q0a (e) mM Q0 J0(m0 ) (1) N H( (kc) I { E n(7rn ) J O (n0 - 2(k) k bIn(2/00)j n o + [ko(b-a)], ka= m. (5-3-27) It is interesting to note that in the present case, both G and B depends strongly on k a, k (b - a) and O but only G depends on the plasma sheath and shell slot width 200. Case C: If we keep the radii k a and k b constant, 00 and O0 small but finite, 1=0, v/w = 0 and wp/w >> 1, then

73 P)2 e and H(2) (koc) 2 k b n H(2)1 (k c) C n n 0 C H (k c)cos Pk (c-b)-H (kO sin Pk (c -b-. n (k' Co 2( o 1C) H( ()(k c)cos ko(c -b)+H (kc) sin k0(c -b [ n o0 2w2 2rk _e n (2 (5-3-29) H (=0hik0c Re I E (7r ) mJ (no) n n 2 0)' H H bn H (k C) n W 2 Nkb Co N 2 Co J(no )- c J (no (5-3-30) n 0 0 ) n2o = n

74 On the right hand-side of (5-3-30), one notes that from (A-8-12), the first term is approximately equal to -2( P)2kObIn(2/0) Hence, upon employing (5-3-29) and (5-3-30), V /Vwl 2 G and B respectively become V 2 s w 2 (-)2 (CS/C) (5-3-31) w p n0 (1) (e) J(e) C E V (Q /* )(n cos nO - (5-3-32) 5 7rk0a n n 0 0 N -t e n (ne) (5-3-33) n=0 2 n ((wa ka FN J(no 1 22 -2-Pk (c-b) G ( e 0 ( ) an s2w2e ( 0w p to th ca 7r k bc n=0 n Ho(1 (k5C- ) 0 n C (5-3-34) kOa c (e) (e))2 (e) 2 1 OD (2) (e) (e) 2 B C --- (P /Q /n+ (Q /Q /n+ neV (Q Q 7r 0 1n n n 0 2k0a nn n 0 N (3) 2 01n (no)+ 2kr0no0, + bin20 -(C /2k b) Cj 25 P Ct P Equation (5-3-34) and (5-3-35) show that for an overdense plasma sheath, if we ignore the collision effects, the terminal conductance G decreases with a factor (w/w ) 2 e 2k 0 (cp/ and the terminal susceptance B approaches to the case of no shell slot as shown in (5-3-18 ).

VI EQUIVALENT CIRCUIT FOR A COAXIAL ANTENNA WITH A PLASMA SHEATH 1. Introduction: In chapter II, we derived the general stationary form for the terminal admittance of the wedge waveguide. Upon using the results from chapter III and chapter IV for 00 < < 1 we finally arrived at an explicit formula for this admittance in chapter V. It is clear that this admittance is a function of the following factors: k0a, k0b, k0c, 0, Wp/O', v/W, 00, and 00. If one can find some explicit expressions to indicate the individual role of each of the above factors in Y(a), then one knows all details of the coaxial antenna. Unfortunately, this is practically impossible. However, it is also valuable to know the individual influence of the wedge region, the coaxial region, plasma sheath and free space on Y(a), respectively. If we refer to the normalized stationary form of the terminal admittance y (a), (2-4-4), it is found that one can hardly identify the individual influence of each of the above four regions on y(a). Thus we turn to seek some other way to formulate the normalized terminal admittance of the wedge waveguide so that the effects of the above four regions can be discussed. In section 2 of this chapter, we furnish a new formulation of y(a) which allows one to propose an equivalent circuit for the antenna. In section 3, we also discuss the physical significance of each circuit component of the equivalent circuit. However, the new formulation of y(a) is not stationary with respect to the functional variation of the wedge aperture field and therefore, as long as the exact solution for the wedge aperture field is not found, the stationary formulation of y (a), (2-4-4), is still important in producing the numerical results. 75

76 2. An Equivalent Circuit of the Coaxial Antenna: Upon multiplying (2-4-3) with E (0)d0 and integrating from 0-00 to 0 +00 we have i 02 -2_o y(a) 0 (0)d0\.OfIi {2O J a)j(n=lk a) e ( )cos (0-0+%)d0 0 0 0 J (ka)N'(kb)-J'(kb)N (ka) +0 0+ 0E0 J (k b)N'(k) J ) n 0 n (cosn(-d' n= n 0 nn 0 n ~0 % e r 2rka n vn n do E (0) E (0 ) cos n (0 01) d0) (6-2-1) is not stationary. Fromkthe definition (2-4-5)of Y(a), taking L = a and by virtue of integral Eq. (2-3-31), we write (6-2-1) in the form 0 2 JY(a) jBb j( c c(B )2 (6-2-2) +Ja) = j w el cC2

77 where e+0 2 E 0(0) os2(0 -0+%)d0 On=~ 29 20 Iw BW =,/A;;C 0 (6-2-3) 0+0o 0+0o k d0 E"(0) E(0')cos n(0-0')d0 _ 1 N 0 J (k a)N'(kOb)-J'(k b)N(kO a) 0-0 0-0 ~B =-AI —T n O nO n nO ca O2 nJ0(kOb)N%(kOa)'-J(kOa)Nn (kb) IT I 2 0 n = 0 n 0 n 0 n 0 n T (6-2-4) B 2 a 2_r Jn k7b)NI(k~a)-J%0a)N(%0b) 2MO O l0o n ~ nO n) nO (6-2-5) H do E (0) E (0' )cos n (0 - 0') d' co 1 H(1)(kc) - -0 pf a 27rn n n o IT I 2 Ho(kc) ITlj (6-2-6) e+0 T = 1 e(0) do (6-2-7) w-8

78 0 T E (0)d0. (6-2-8) and |T | x 1 (6-2-9) ITwI Equation (6-2-2) suggests an equivalent terminal circuit of the wedge transmission line as shown in Fig. 6-1. I: a I IV o Yo(r)= Yo(a)~ jB B B Yf I I I Wedge Transmission Ideal Transformer I Wedge Transmission Line I FIG. 6-1: EQUIVALENT CIRCUIT FOR THE COAXIAL ANTENNA It is seen from (6-2-3) through (6-2-9) that Bw B 1' B2 are real quantities while Y is a complex quantity, therefore the first three pf circuit components are susceptances and last one is an admittance. One may also note that the only 0 dependent circuit component is the transformer turn

79 ratio 1. In the next section, we will discuss the physical significance of these circuit components in detail. 3. Physical Significance of the Circuit Components: In (6-2-3), we note that B depends only on k0a, 90 and e (*), the wedge aperture field. Furthermore, if we let Er E and H represent the electric field and magnetic field in the wedge region minus the TEM mode field, then the integral a a 9+ 0 s,Eh, h h, h h-,- ], 2 dz drC e(E E + E E H d 0 0 e - a in view of Eqs. (2-2-1) through (2-2-3), can be reduced to a form exactly the same as shown on the right hand-side of (6-2-3) which defines the susceptance B. Therefore, we may regard Bw as the susceptance due to the higher order mode fields in the wedge region. For narrow angular width of the wedge or small k0a, B can be reduced to 0 W Bw = LWC + O(k0a0)2] (6-3-1) where e+9 2 E 0a co0)cos1- 9+%)d00)d2 C (0)Cos (- e+2 )do 1T (6-3-2) 0 It is seen from (6-3-1) that the dominant part in square bracket is the capacitor C which from (6-3-2), depends upon the radius of the cylinder and the angular width of the wedge aperture.

80 In Eq. (6- 2-4), if we make use of (4-2-19), we may separate the right hand-side into two series and obtain 0+0 0+00 doe (0~) ( E (0) cos n (0 - 0') d0' -WC +n 1 0 ZE 0 0 (6-3-3) where d0 E (0) (0)cos n ( ) d0 E a a 0-_ 0 -0a C = 0 0 (6-3-4) When the angular width of the wedge is very small, we may employ the wedge aperture field (4-3-1) and then because of (A-8-12), we have Eea CCl 1 ln(2/0). (6-3-5) The second term of (6-3-3), because of v, (4-2-19), converges rapidly. Ccl may be considered as the capacitance due to the frin age fields of the wedge aperture. The same fringe capacitance can be found when the circular shell and plasma sheath are not present. Schelkunoff (1952) in deriving the terminal admittance of the biconical antenna also found a capacitance which has a logarithmic singularity as the cone angle f0 el The next circuit component to be discussed is B2* Since c2'

81 J (k b)NI(k a) -J (kOa)Nn(k b) kb (3) nO nO no nO 0 (3) = + v nn>1 J'(k b)N'(k a)-J'(k a)N'(k b) n n where v(3) is given by (5-3-12), we may write (6-2-5) in a form "d 0 E'(0) E (0') cos n( - 0')d0' B C + b 1V (3) -o - c2 c2 a2i (TJ 2a' 2'"'~ n ITj2 (6-3- 6) where b d0E (0) 5 E (0')cos n(0 - 0')d0' C n b e~b 1 j0j 20.(6-3-7) c2n a r 1 IT 2 Similar to C cl we regard Cc2 as the capacitance due to the fringing field of the shell slot in the coaxial region. For narrow shell slot, upon substituting (3-6-7) in (6-3-7) and making use of (A-8-12), we obtain b ~0b C 2 a r n(2/0) Hence, as the angular width of the shell slot approaches zero, the capacitance Cc2 also has a logarithmic singularity. The series in (6-3-6) converges rapidly and for narrow shell slot 2 00 it is weakly dependent on 00. Since in the plasma sheath, there is also a fringe field neighboring to the shell slot, we expect that this fringe field will contribute to a capacitance. To investigate the nature of this capacitance, we turn our attention to Ypf,

82 -2 k kb (6-2-6). Comparing (5-3-14) with (3-1-6), we observe that rn = n' - n for n / 0 where Ir' is given by (5-3-13). Therefore Ypf can be reduced to 0 (kn 0 YpfJ (12) Aft* ~~(6 — 38) IT| and A d0E*() E(0c) cos n(0 d - O') do' n1 U 1T4 2 Afr a n n (6-3-9) We see from (6-3- 9 )that for 20 < < 1, we find Afr ()2 2 2 (6-3-10) fr a In (6-3-8), we may regard the first term on the right side as the admittance associated with shell slot fringe field in the plasma sheath. From the defining equation of the dielectric constant k of the plasma, (2-2-14), we can show that the above fringe admittance is composed of three parallel branches and can be written as w a_2 1._ j -A —(k )* A = (C3 2 )+G (6-3-11)

83 where E0a C A- (6-3-12) c3 7T fr L ra 2 1+ (v/w)2 c~L (6-3-13) c3 7r k0a p/)2 p fr vcEa (w 1w) 2 A G = fr (6-3-14) c3 7r 1 + (v/w)2 C is a capacitance due to fringing fields, L 3 is an inductance due to the plasma, and G 3 is a conductance which accounts for the power dissipatedby the shell slot fringe field in the plasma sheath. As t00 0 it is seen from (6-3-12) through (6-3-14) that Cc3 and G 3 have logarithmic singularity while Lc3 approaches zero. Since Afr is only weakly dependent on the plasma sheath, thus Cc3 is also weakly dependent on the plasma constants. L is inversely proportional to (wp/w)2 and G increases as (wp/w) 2 Thus c3 p c3 p increasing the plasma density tends to short out the shell slot. The real part of the second term on the right hand-side of (6-3- 8) may be associated with the power radiated into the free space and the power loss in the plasma sheath by other than the fringe field of the slot. The imaginary part of this term may be related to the stored energy in the plasma sheath and the free space with the slot fringe field excluded. To investigate the connections between Ypf and the stored energies, the power loss in the plasma sheath,and the power radiated into the free space, it is more convenient to start with (6-2-6). From (A-10-9 ), it can be easily shown that 2P + j 4w (WIII_ WE ) + 2P +; 4uw( HT W - WIV )=b IT I Y (6-3-15)

84 where P = time averaged power radiated into the free space pIII time averaged power loss in the plasma sheath W H = time averaged energy stored in the magnetic field WE = time averaged energy stored in the electric field and the superscripts III and IV denote the plasma sheath and the free space, respectively. If we define the radiation conductance Gr and plasma conductance G, respectively, as 2P G r' (6-3-16) 2P G =2 2 (6-3- 17) P b2 ITs2 then G + G Re Y (6-3- i8) r p pf The imaginary part of Ypfo from (6-3-15) is a susceptance which accounts for the difference of the time averaged stored energies in the magnetic field and electric field exterior to the conducting shell. It can be visualized that [C3 -1/(w2L3)] and G3 area part of the susceptanceandconductance representedby Im Ypf and G respectively. In (A-10-10) we derived the expression for Pr From this equation and (6-3-16), when Lp > > o, one can show that G p r

85 -x k0(c -b) decreases as the factor e if x k (c -b)> > 1 where f =+1 E (v21 1/2) 1/2 X p + -+ I1+4() The coaxial region not only behaves as a reactive element, but also couples the two slots. In the equivalent circuit, we indicate this coupling effect by a transformer of turn ratio I as defined in (6-2-9). Since these two slots are separated by an angle 0, I will be a circuit constant in Fig. 6-1 that depends upon the separation angle 0. Apparently, I also depends on the radii k0a and k b. However, since the shell slot opens into the plasma sheath, I is also modified by the plasma constants. To have an idea as to how I depends upon these factors, the reader may refer to (5-2-2) in which the angular width of wedge aperture and shell slot are assumed very narrow.

VII NUMERICAL RESULTS AND CONCLUSIONS 1. Introduction: In this chapter, we present the numerical results based on computations from (5-3-11). From this equation we note that Y(a) is a function of k a, k0b, k0c, 0, w /W and v/w; thus in presenting data, we successively choose 0o k (b-a), k0(c-b), k0a and wp/w as the abscissas. The computations were performed on a digital computer 7090 for 00 and 00 equal to 0.03 radians. Since the method of solution of the integral Eqs. (2-3-31) and (2-3-32) given in chapter III and chapter I V, respectively, is primarily a low frequency approximation, we limit k a in the computations to the interval 0.1 < k0a < 4. 3. In (5-3-11), we sum the series (Pn /Qo ))2/n to 250 terms and the series E (Q(e)/Q) )/n by the n=l n=l (1) method shown in A-8. The factor vn enters into the series defining the numerator of the last term inside the brace of (5-3-11); this series we sum to M terms. The number M is determined by two conditions: a) in the last terms preceeding the Mth term, vn(1) decreasesmonotonically, b):M)I/ |V < 10-6 where v(1) is the largest among (). Since (2) (1) v, T decrease faster than v as n becomes large, we sum the n n n (2) (e) (e) 2 series n vn (Qn /Q)) to Mth term and for the finite sum N 2 Cn~TnJ0(no0) we set N = M. 86

87 In the following sections, we first present and discuss the numerical results, then we summarize what we have done in this report. Finally, we make some brief conclusions based on the theoretical discussions and numerical results. 2. Numerical Results: In Fig. 7-1 (a) and (b), we plot the normalized conductance G/G' and the normalized susceptance B/B' as a function of e for the no plasma sheath case. In this figure, the radii k a and k b are the parameters; G' and B', respectively, are the terminal conductance and susceptance of the wedge waveguide without the conducting shell and plasma sheath. Their formulas are (A-9-4) and (A-9-5), respectively. G' and B' depend only upon the radius k0a and the wedge width 2e0. In Fig. 7-1 (a) and (b), four different values of the radius k a are used. We tabulate the corresponding values of G' and B' in Table VII-1 for reference. k0a G mhos B',i mhos 0.2 1.33 x 10-4 8.72 x 10-4 1.0 1.04x 103 3.59 x 103 1.8 2.08 x 10-3 5.69 x 10 3 4.3 5.35x 103 10.5 x10 3 TABLE VII-1: TERMINAL CONDUCTANCE AND SUSCEPTANCE WITHOUT CONDUCTING SHELL. WEDGE WIDTH 0.06 RADIANS. In Fige 7-2(a) and (b), we plot the terminal conductance and the terminal susceptance versus separation angle e, with w /w as the parameter for the collision-free plasma sheath. The radii k0a, k0b, and k0c are kept at 1.0, 1.1, and 1.2, respectively. In Fig. 7-3 (a) and (b), we repeat Fig. 7-2, except v/w is used as the parameter and wp/w = 1. 5. It is seen p

aii hI~V viad aHL SV (q 0: I0 a(INY HLVtHS VWSerIcd ON HILIM SfasuEA AIDNVLJYa SflS (q) aNV aI3NVLOf1nNOo QaZII.W oN (B):'-L'DIa 008L o091[ O1I 006 009 o0~ 00 I I,I', I I I - (q) 9-./;( \o 1) 01~/' ('(.g'o') - o!09 0091_ OZ 006 009 0 0S'l / OSI,I (O q al I= =. 0 (I I OO) O 7 X / \ ---------- -(-zFD) 80 0~ 0

,II3zLAVcd MaHL SV M/ m aNY Z'" = D 0'1'I =q0''o= 1 tO 0 S1S aoa A sa HLV VHS VW Id SSahqNOISIrHOD V HLiA aDNvLd3DSfS (q) (NV 3DNv3 L(iNO) (B):Z-L.DI QIX01(q), OlXG_:0 / *0 CD 0=T - 0 0ol oOZ 06 009 O (ma 0 0 0 0 0 \ -9-'' / 0~o (o) 68

90 -2 10 P v 0,o20~~~~~~~~~~~~. -4 o L ---— *-. E -, -0.01t \ / A 0;0.10.5 0.05 0.1 =0 0 o 304-6 PARAMETER B,

91 from the last three figures that a) conductance versus 0 curve and susceptance versus O curve for the case k a = 1.0 and k b = 1.1 are neatly checked with Eqs. (5-3-26) and (5-3-27) except w /w = 5; b) when v/w increases, the conductance also increases while the susceptance is practically not affected. In Fig. 7-4 (a) and (b) we plot the terminal conductance and the terminal susceptance against the width of coaxial region, k0(b - a), for the no plasma sheath case, the separation angle 9 as the parameter, and k a = 1.0. One may observe that for small k (b-a), when 0=00 and 1800 the conductance and the susceptance are approximately equal to G' and B' for k a = 1.0; when 0= 900, G becomes very small while B becomes a large inductive susceptance. If we refer back to (5-3-26) and (5-3-27), a similar result can be observed. For a large value of k (b - a), the conductance is small while the susceptance approaches a positive constant, i. e., a capacitive susceptance. Furthermore, one may note that the conductance versus k0(b - a) curves shown in Fig. 7-4 (a) are maximum when k0 (b - a) - 0. 4. In Fig. 7-5(a) and (b), we plot G and B versus the plasma sheath thickness k0(b-c) with w /w as the parameter and k a = 1.0, k b = 1.1, = 00, v/w = 0. In Fig. 7-6(a) and (b), we repeat the last figure except for v /w = 0.1. From the last two figures one may observe that: a) When w /w = 0. 5, the conductance and the susceptance are weakly dependent on the sheath thickness for the cases v w = 0 and v /w = 0. 1. Galejs (1964) in a paper on the admittance of a slot in a perfectly conducting plate covered with a plasma sheath showedthatthe slot conductance and susceptance are practically independent of the thickness of the plasma sheath when w /w < 1. His numerical P results are not accurate for a thin sheath. b) When w 1/w = 1.5 and the sheath thickness k0(c -b) exceeds 1.5, further increasing the sheath thickness will decrease the conductance exponentially for v/t = 0, but makes it approach a constant for v w = 0. 1. For k (c - b) > 1. 5 the susceptance is essentially independent of k0(c -b) and the collision frequency. c) When k0(c -b)

HI3aL ThV1IVd 2H~ SV & GSNV ~i' =u0 0 OSI: (g-q)0I SfSiS:[A HLV aHS VISVIdc ON HLIM A33NvLdo3SflS (q) (INV a3DNVOL3fINO (e):'~-L,'IDI (q) ~06 0lIXj- (D-q) OM 01 9~(0~)~01 g_.' /'/o 06 (D-q) 01 0 1 g6

o 0=eT T =q' 0'T =B u HOI (q- o)j S StIatA HHLaVlHS ~VWSVrId SSariNOISIrrIOD HULIM 3NVTLdJLDSSS (q) aGN aNVLOfnclNo (o ):'-L'lIa (0o:'g) (q) c, =r (o =:'o:- -) - (q_) - OM1 9 9(o' 9) (D) (')l (100-00 1) 0o \ dn ~ 6 E6

94 -2 10 V -0.1 (a) - 1. 5 (). ++, ++ ++ 0 - -C I10 l-4 (a) -s5 ko (c-b) 10 I. I I. I, 0 I 2 3 4 5 6 -3 — -— P0.5 4. = 5xO w — =0.1 0 1.0 az \"~~~~~ ko (c-b) 1 2 3 4 5 6 o\v \ I I I I I I t -3 + -IxlO lO (b) ++\ ++ 1.5 FIG. 7-6: (a) CONDUCTANCE AND (b) SUSCEPTANCE WITH A PLASMA SHEATH VERSUS k (c-b) FOR koa l.O, =l. 000 v/w = 0.1, AND w /t AS THE PARAMETER

'JLSaIIVWVd aHJL SV 0 CINV I' = 0)t/q 0I' 0 SflSWSIA HLV3HS VISVrIcI ON HLIMA 3DoNVj3DSfnS (q) CINV aDNVLDfn(INOJ (D ):L-L'Ij U 1~, / InoyrI)AA, o o.~..~~~,0., o oxG, 0-t_ S~ 0_ _ S'_Z 0- __ O S0 0O,. Jf W g' i',* U'' *",'._,z - *-,-*-~-*' or***,Tr**(a) / C3 Ino'i,'~ 0'O --'U

96 o io I._, I, I. - I, I, I 0 2 3 4 5 61 0p/ I10 -3 CL 0 -4 10 (a) cno 0. 5 -3xlo - (b) o, o0. FIG. 7-8: (a) CONDUCTANCE AND (b) SUSCEPTANCE WITH PLASMA SHEATH VERSUS w /w FOR k0a=1.0, k b=1.1, kc =1.3, AND v/w AS THE PARkMETER

97 approaches zero, the conductance and the susceptance for all cases approach respectively to 1. 18 x 10 mhos and 6. 26 x 10 mhos, the magnitude of the terminal conductance and the terminal susceptance for the case without plasma sheath. We did not plot the case w/w = 1.0, V/w = 0 because when the plasma frequency is equal to the radio frequency, the effect of the collision can not be neglected. In Fig. 7-7(a) and (b), we keep k0b/k0a = 1.1 and have no plasma sheath. G and B are plotted against k0a with e as the parameter. The primary purpose of this figure is to show the effect of the radio frequency on G and B for a constant cylinder radius a. We note that as k0a increases, the conductance peaks at k0a - 0.43, 1. 3, 2.2, 3. 13, etc. The susceptance peaks almost at the same values of k0a as G. In Fig. 7-8(a) and (b), we plot G and B as function of w p/w with v/w as the parameter. The values of k0a, k0b and k0c are chosen as 1, 1. 1, and 1.3, respectively. One notes that for large values of wp/w, G decreases exponentially with further increasing of uw/w when v/w =0 and approaches to a constant value when v /w 0. The susceptance, on the other hand, for large p/w, is approximately a straight line with a negative slope. The effect of v/w is to shift the straight line upward. The susceptance in this region of w /W is inductive. In Fig. 7-8 (a) we plotted G versus p W /w for the cases v /w=O, 0.1, and 0. 5 and in Fig. 7-8(b), B versus w p/w for the same parameters. Notice that v/w =0, and 0.1 curves for B are not distinguishable on the graph. 3. Conclusion: The antennaproblem encountered in this report is basically a boundary value problem. To attack such a problem, we first express the electromagnetic fields in the wedge region, the coaxial region, the plasma sheath and the free space in a series whose coefficients are in terms of the 0directed electric field, E (i) and E (0), in the wedge aperture and the

98 shell slot, respectively. Then upon applying the boundary conditions, we formulate two coupled integral equations in which P (0) and E(0) are the unknown functions. Both integral equations are of the first kind of the Fredholm type if one of the slots fields is assumed known. Only in one, however, the magnetic current source is present. This we will call the inhomogeneous equation; the other one - the homogeneous equation, for the purpose of present discussion. Thus the boundary value problem is reduced to the problem of solving these two coupled integral equations. However, for practical purposes, we may regard the wedge region as a transmission line loaded at the cylinder surface by a terminal admittance. The knowledge of the terminal admittance is fundamentally importantin s tudying the behavior of an antenna. For this purpose, from the inhomogeneous integral equation, we formulated two different expressions for the terminal admittance. On the assumption that the solution of the homogeneous integral equation mentioned above is obtainable, one of the above two expressions for the terminal admittance is proved to be stationary with respect to the functional variation of P (0). An analytical solution of the homogeneous integral equation in a series form has been found for the low frequency region. This solution depends on the radii k0a, k0b, k0c; W /Wo v/Wo and the angular width of the shell slot 20O. For narrow shell slot, the series which represents the solution converges rapidly. The other form of the terminal admittance of the wedge waveguide is not found stationary with respect to the functiomal variation of E (0). However, this new form of the terminal admittance gives us some physical insight about the antenna via an equivalent circuit. When the angular width of the wedge aperture and shell slot are very narrow, from the stationary form of the terminal admittance, we obtained an explicit expression for the terminal admittance. Based on this explicit form, in some special cases, we were able to discuss the behavior of the terminal admittance theoretically. From the above discussions and the

99 numerical results presented in the preceeding section of this chapter we may briefly conclude: a) The slotted circular shell functions as a tuning element and a matching transformer. Therefore a suitable choice of the width of the coaxial region and the slot separation angle 0 will result in more power radiated into the free space than by the wedge cylinder alone. b) The frequency response of the conductance and susceptance of the coaxial antenna peak repeatedly at different frequencies, with narrow bandwidth in comparison with the wedge cylinder. c) When wp/ < 1, the plasma sheath thickness k0(c -b) has little p effect on the conductance and susceptance. When w/ w > 1, and plasma p collisions are neglected, for large sheath thickness, the conductance decreases exponentially while the susceptance approaches to a constant which depends on the ratio w 1w. If the collisions are not negligible, we p observe that the behavior of the susceptance is not changed but the conductance approaches to a constant depending upon v/w. d) For a fixed operating frequency and plasma sheath thickness, when w /w< 1, the collision term v/w has little effect on the susceptance, p but increases the magnitude of the conductance. For large w /w, further increasing the plasma density will have the same effect on the conductance as the increasing of plasma sheath thickness, but will make the terminal susceptance decreases continuously to the case of unslotted conducting shell.

APPENDIX A-1 PROOF OF THE STATIONARY PROPERTY OF y(a) To start the proof we take the first variation of Eq. (2-4-4); the result is e+ + 6+6 + 6 y(a)[ E (0)d] + 2y(a)' 6 E (0)d0o E (01t' nJr (koa) 0+9o j4~ Jlnr ~I~aO O 0. O < 6 ~ 29 ao J (k a)N'(k b)-J'(k b)N(k a) E ~0 0Co 2- on Jn(kb)N'(k0)_ J'(kob)N'(k b) d- J( )t Et )cosn(k-0,)d0, ~2r 2rka /ODnn 1E d0E0 o c E(0' )cos n(0 - 0')d0''r- k a n n e+6% +, d06E(0) E (0) cosn(00)d0]. (A-i-) - O From integral Eq. (2-3-31) one can show that 100

101 e+ eo rk2a r Env doP(0)l 6 E (0) cos n ( - 0)d0, rk0a Or EVNe S d06E(0) E(0')cosn(0 - 0)dd'. (A-1-2) n n Upon applying (A-1-2) to (A-l-l) and moving the second term on the left hand side of (A-l-l) to the right hand side it is seen from (2-4-3) that the right hand side is zero, i.e. 6 y(a) = O One thus concludes that a first variation in the aperture field of the wedge gives a second variation of the terminal admittance of the wedge waveguide.

APPENDIX A-2 SOLUTION OF INTEGRAL EQS. (3-3-2) and (3-3-3) In this appendix, we will employ the Schwinger transformation (Lewin, 1951) and use the trigonometric series (Schmeidler, 1955) to solve integral Eqs. (3-3-2) and (3-3-3). Since C1 n0 cos = 12 cos 0' - cos0 (A-2-1) n =12 (3-3-2) and (3-3-3) become, respectively, (e) 1 ~0 (e)(p,)ln2 cos0'-coso do' = 1 2ern cosn0 (A-2-2) 2(l+k2 )(kob)2 Jt(k b)N'(ka) -J(k a)N (k b) and f(e) (0) In 2 cos 0'- cos 0 d' 0 c. (A-2-3).2. cosno (A-2-3) (l+k )(k0b) 102

103 We then introduce a transformation due to Schwinger, i.e., cos~ = cos2 + sin2 0~cos s. (A-2-4) It is obvious that one may map the region -00 < 0 < 00 into the region -7r < s < 7r with the transformation (A-2-4) but not in one to one correspondence. Thus we further introduce the restrictions -r < s < 0 corresponding to -00 < 0 < O O < s < 7r corresponding to 0 < 0 < 00 to the transformation (A-2-4). In this report, whenever the Schwinger transformation is mentioned, these two restrictions as well as (A-2-4) are implied. Upon applying the Schwinger transformation to (A-2-2) and (A-2-3), one obtains 0 - (e)t (e)(t ) D C c cos mt dt a ncsc2 - F0 dt + (t) dt 2 O0 dt 0 dt m (e) -1 200 200 1 1 n c-sn cos (Cos + sins2 coss (A-2-5) -2 2 nji(k b)N (k a-)-J(k a)N? (k b) ir(l+k )(k0b) n n nO n O nO and 0J0 (~ ~e) dt (e)(t) dJ' cos mscosmt ncsc2 T 3 in — dt + n dt m1 m = _ 1 *2 cosFncos -1(2 2 Bi 2 )1 (A-2-6) =.:2_'.'cscs(o1~+k )kbL~-os

104 where (eO (S) - FOe) [fFIA(S)1 (A-2-7) (e)() - f(e) [(S) (A-2-8) for -0O < 0 < 00 and -Jr < s < J The free terms of Eqs. (A-2- 5) and (A-2-6), respectively, can be expanded into Fourier series; thus, (e) - 2 00 200 1 cO E <ecoslmcos (cos + sin c2 coss) (e) m~mr L~ 2 2 (e)~ a cos p s m= = I(k b)N' (koa)-J'(koa)N' (kob) P=O (A-2-9) -1 z2~0 2 0 ) ] (e) cos ncos (cos + sin 2- S coss bp s (A-2-10) c 2p=0 np with er /, (e) (e) 1 mm Om (A-2-1 0m0 mO mO mO m0O r(e)x a(e) -2 m PM (A-2-12) P - J Im(k0b)N' (k0a) -J (k0a)NI (k0b) (e) 1 be = X (A-2-13) nO 2r On (e) 1 b(e 1X (A-2-14) np 7r pn

105 where 7r The properties of XPM will be investigated in A-4, however for the present pm discussion, we note that X 0= for p>m (A-2-16) pm If we let 00 yFe)(t)tt (e) + a(e)cospt, (A-2-17) e (t) dt (e) + t(e) cospt (A-2-18) n dt nO np for -r < t <; upon substituting (A-2-9), (A-2-1(), (A-2-17) and (A-2-18) in (A-2-5) and (A-2-6), respectively, and employing the expressions (A-2-11) through (A-2-14), we obtain (e) (e)= 1 1 m m Om O (+k)( b) 2 0 mO Jm(k b)Nm(kOa)-Jm ma)N (kOb) 0 4r tncsc (A-2-19)

106 (e)_ 1 2p mE (e)x A20 p (1+2 ) 2r (kb)2J (k a) Jm(kab)Nr n(ka) (ka)(k0b) r (1+k)(k b) m=p 0 a) mOmO (e) 1 X0 2(1 +k )(k0b) 4ir2Incsc (e) =1 P (A-2-22) 2(1+k2)(k0b) 72 p If one differentiates the Schwinger transformation with respect to 0, one has dt c dt =21EGOBp < V < 2 (A-2-23) Thus from (A-2-7) and (A-2-17), we obtain - cos -L f: r (e) -1 2 20 (e)(~ _ 2'02 F )(0) = G il,__ ( (e)+ E a( )cosp cos (csc 2 cos 0 -cot 2)] p=l -00 < 0 < 0 0'(A-2-24) and from (A-2-8) and (A-2-18), fc s02 (e) n (e) -1 20O2 0)]} f( )(0) =...... nO + p lnP cos (cc cos (csc - cot -0 < < 0 n=0, 1,2,... N, (A-2-25)

107 If one defines co ~ r(e) 0b m J'k0b)Na) m0 (A-2-26) pm =O Jm (k b)Nm (k a)-Jm (k a)N' (k b) then F(e)(0) and f(e)(0) becomes 0 n (e) Cos - S(e) c (e) F ) (0) 2(+k)(k b)2 { pL ns 2. cos[p COS-l(csc2'O cos 0 cot2 -_ < 0 < p (A-2-28) cos x n pX f(e)(0)f X 2 On e pn n 2(l +~20)(kob) 4 r0 2In csc 2 _ 2 (csCoscos (s2 -cot ~ -0k < 0 < 00 (A-2-29)

APPENDIX A-3 SOLUTION OF INTEGRAL EQ. (3-4-5) and (3-4-6) Same as in A-2 we employ the Schwinger transformation and the trigonometrical series method to solve integral Eqs. (3-4-5) and (3-4-6). If we differentiate the well known formula (A-2-1) with respect to 0 for all 0, 0' < 0o except 0 = 0', we have sinm C0os M0' C O- sin 0 (A-3-1) M =! Upon substituting (A-3-1) into (3-4-5) and (3-4-6), one obtains, respectively, c (0)-o (0) 0~1 FO)(p') d 1 + m sin m 910 c00 o - r-+ )(k0b)2 J'(k0b)N(k0a)-J (k0a)Nm(k0b) i (A-3-2) and i.n 1 sinno 2 CoS2 coLco3 s ( 2b n=1,2, N. (A-3-3) Since ~si~ n o- 1 = B emcosm0 for n odd, (A-3-4) m =0, 2, 4,... 108

109 the free term of (A-3-2) can be written as <)o r~ b)Nm sin n EL cosp, (A-3-5) J I ~k b)NI (k a)-J' (k a)N' (k b) sin mikm 0 mO, m-= mO po where L = vp2m + 2m - 1 (A-3-6) m=. p+2m-1 p+2m-1 (1) 1 (A-3-7) n n IJ(k b)N 0n(k0a) - J'(k0a)N (k0b) Upon substituting (A-3-5) in (A-3-2) and (A-3-4) in (A-3-3), we have o0 F(0)(0')dot 1 _ 2 -of ir(1+k2)(k;b)2~EL cospp= (A-3-8) -0 -i Cos 01- - Cos 0 ( + k )(k0b) po p n00 f (0)(o d0' 2 n - 1 - =....... CO spOP n even (1 +k2)(k0b) P....._ 1 E cospO, n = odd (A-3-9) (1+k )(k0b) p=0,2,4.. P We apply Schwinger's transformation (A-2-4) to (A-3-8) and (A-3-9) andlet Fn )(t) =Fo) FA(t] (A-3-10)

{(ot) - f() [0(t)0 (A-3-11) for - < 0 and -7r < t < ir, and obtain respectively ~1 dt 1 -1 20 200 00_ o4jd2 2p=E L co cos (cos + sin - cos is 277 t 7r(1+k )(kb) p p sin _(cos t-cos s) = (A-3-12) 2 co Cos. + in cos + ( l (l+k)k(kbb)p0 4 L 22 ] n = even, n = odd. (A-3-13) In order to generate convenient expansions we multiply both sides of the last three integral equations by sins. The free terms of these new integral equations can be expanded by Fourier series, i.e. n-l~7~ L co6~ -1 28O -i E pL cospcos (cos 2+sin 2-coss)I sins= s a )sinms (A-3-14) n p (co 2+ 2 mD mo 1 2coscos (cos *W+sin - coss ) sins bn sin ms n = even (A-3-15)

111 and n - 1 S -1 240 z2 0 s (0)sinms p e PCOS C os +Cos s+s) s co ss sinms n = odd (A-3-16) Since sin ms sins cos cos (os 2+ sin2 coss ds - X -X 2 m-l,p m+l, p we obtain am) PLPXmi -p2 LPXm+i p)' (A-3-i7) a LX 2- (e L X (A-3-17) m rpm- p p m-l, pp -+l p () 1 n -i n- b -- 2 ) n =odd, (A-3-18) nm 27r p m-i p M+ p b0) (0) We note that b =0 when m> n. Because nmn sin s 00 sin sC sin m s cosmt 2(cps t - coss ) m=l (A-3-12) and (A-3-13) become

112 7rO i 20Oi _(o~) d~ot~ z2 (0) dt) Z.sin m s cos mt dt =.-. a sinms, (A-3-20) -r ml 7r (1 +k )(k0b) m1 7 [ 200 (200) dot OD sin'- n n (0) )(t) d't sinms cos mtdt 2 b inms mn d (1+k )(k0b)m 1 nm n = 1,2,... N (A-3-21) Now we let 4Ot) dO' (A-3-22) ) dt CO + C1 cost+ C2 os 2t +... (A-3-22) -(0) do' n (t) + dd cos t + d cos 2t +.. (A-3-23) n dt no ni n2 Upon substituting (A-3-22) and (A-3-23) in (A-3-20) and (A-3-21), the Fourier coefficients C and d are found as m nm 200 (0) sin2 a C =. (A-3-24) m r (1 + -2) (k b) 2 __ d 2 b( m> 1 (A-3-25) nm (1+ ki2) (kb) nm

113 Hence 2 00 (0) -(0) dO' sin -2 a2 1csm - ir < t < ir (A-3-26) sin - n (0 (t) dt= dd 2 5 7 b Cos mt n dt nO -2 cos nmt r(+k k0b) m= -Ir < t < 7T (A-3-27) We recall that co0' A =cn + Si200 2 00 and therefore sin 2sin t dt - - -oIt is obvious eal' tha*~It -do > 0 as t - r dt

114 Therefore, -(O) do' F( (t) -a o dt and hence from (A-3-26) and (A-3-27), respectively, we have sin -- r (le+k Xkb) m; m 0 2 sin - n (0) =d 2 - n r (A-3-29) no'2(1 +k 2) kgb nm Thus F 0)(t) and f (t) are 0 n11 -0) sin 2 dt 00 F (t) -i2) 2f am (cos mir - cos mt) 0 2 )2 d m r 1+k )(k0b) m=l -r < t < lr (A-3-30) sin 2 __ -0(t) s.....2 dt b(0) (cos mir - cos mt) n 2 d nm 7(1+k k0b d- m=l -7r < t < ir (A-3-31)

115 For the convenience of further investigation we introduce a new function m (t) s mr - cos mt e(- )m - P( - ) osp t (A-3-32) rn1 + cost p If we let U (0) denote the function Um(t) in 0 interval, then () - 1 1)m -p -1 200 2 0o] rU nE p=rnec(-L) P(mcos(csc cos 0 - cot2 ) (A-3-33) Thus Eqs. (A-3-30) and (A-3-31) become 2 0o (Q) sin -2 dt (0) - F (t) — (1 + cos t) a U t) i (1+k )(k0b) rn1 -ir < t < w, (A-3-34) and 2 00 sin -- n (0) sin 2 dt b (0) (t) (1+k )lrk7b k (1+ cost) n.. -7r < t < 7r (A-3-35) From the Schwinger's transformation, we obtain sin2 0 dt - c 0 (A-3-36) si2 (1+ cost) = 2 cos 2 COS - cos 0A

116 Therefore, OD (o) (0) co" cos(08 - Cosd s a( u m(0) F (0) 1-2 2 x2 v m (l++k )(7rk b) m m -00 -~ <- 0 c 00(A-3-37) (0) O) co() O b-m U m(0) I (l+k (irk0 b) 0-0 < _: (A-3-38)

APPENDIX A-4 PROPERTIES OF X qP In Eq. (3-3-8) we put 2 2 coss=x, cos =b and sin = a (A-4-1) then X becomes qP Xqp 2 cos(qcos x)cos[pcos (b+ax) dx (A-4-2) Tchebychev polynomial is defined as T (x) = cos qcos x]. TO(x) 1 therefore one may rewrite (A-4-2) in the form T (x)T (b+ax) X =2 Pq p dx (A-4-3) -1 — x It is obvious that for any 00 X = 2 r (A-4-4) T (b+ax) is a p-th order polynomial of (b+ax), while T (x) Is a m - th order polynomial of x, therefore 117

118 T (b+ax) = m a T (x) n mm If we multiply both sides of that equation with 2 Tq (x)/ - and integrate from x=-1 to x=1, we find T (x)Tq(x) X =2 m dx 1 l-x But T (x) T (x) dx,~m q = 27 1 m q $ I, mq0 and therefore we conclude that for any 00 X = 0 if q > p (A-4-5) DuHamel (1953), Salzer (1956), Brown (1957) and others in their works on radiation pattern of antenna arrays also studied integral (A-4-3). By different approaches, they carried out the integration and arrived at a tedious formula E q = 2r(2b~naq ~ 1(_ 1 r[ (+n-r) (q+n-r-1) r 4b2) in/2]-rq2jr 2 [n/2] ci-2j ~ (q+n 2r) 4a 2(A-4-6) q+ 2j 4 2

119 where (q) denotes the binomial coefficient rq and y] denotes r r' (q - r):, and [y] denotes the largest integer not exceeding y. For the convenience of further discussion, we list X for n=O to n=4: q, q+n e X = 2raq q q,q e X = 27r(q+l)aq(2b) q qq+2 2 Qq q q+3 27r(q+3)aq[2b)j(q+ 1) (l+2 (2b)2 (q+2)a2 2 +a(bq J(q+) (q+ 2)(q+ 3) (q+) (q2) 4 q q,q+41 4 2 ~ 121. 21 +(q+2)(,+3 _(q+2) a2 (2b) +.. a a4 (A-4-7) For n -- o, one may evaluate the integral (A-4-2) by the method of stationary phase: cos (n~o + qO X me 27 cos qr (n+q (A-4-8) q,q+n 2irntan Combining the informations given by (A-4-7) and (A-4-8), we may state the following behavior of X q+n: For any q * 0, as n increases from 0, X increases gradually from r aq to its first maximum and then q, q+n repeatedly swings from negative maximum to positive maximum with a decreasing amplitude. For q = 0, as n increases from 0, X0n decreases

120 gradually from 2 r to a negative maximum, then swings up and down with a gradually reducing amplitude. In the case q * 0, suppose the first maximum of X occurs at n= M0 then it is seen that the increasing of q reduces q, q+n the value of X as well as increases the value of M. q, q Now we consider the special case of narrow slot. The necessary p values for the narrow slot satisfy the condition p002 < < 1. If in p[p2os] [2] r -2 p1 p-r P-r-1p2r cO Cos [pcosz = (-l)r2Pr[ ) _ ) (A-4-9) 2 00 2 o00 we replace z by cos 2 + sin -2 cost and make use of 2 2 200 - 2 (cos _ + in2 osp 2r 1- (p 2r) (1 cs t) (A-4-10) we have -1 2 m0 20 p cos (cos -+ sin2 cos t - (_l)r2p-2r-1[2Pr) (p-r- )] r r 02 02 4 g(p)+ 4 g(p)cost (A-4-11) where g(P) = (-1) 2P2( )] ~ (p-2r) (A-4-12) = r

121 We note that g(p) < p. Upon substituting (A-4-11) in (A-4-2) and employing the identity [p/2] (Pr 1)] 1 (-1)r 2P2r-l[2(Pr-r -l we attain X~p - 27r[ - -'-g(p)1, (A-4-13) 2 XoP line 2 7rL1 -4 P) (A-4-13) l XT 0 g(p) (A-4-14) lp 4 and X X 0 q > 2 (A-4-15) qP There is an alternate approach to find X0p for 00 < < 1. From Schwinger's transformation, for q =0 we may rewrite (A-4-2) as 00 2 cos A cosp 0 XOp S_ cbs -c~. d0 2 2 do= 2irJ0(pf) (A-4-16) %00 - ~o

122 If the angular width of the shell slot is very wide, we may let 00 = - A, where A is much smaller than unity, then we obtain 200 200 2 cos 2 + sin -cost = sin + cos cos t Now we let b = sin 2 a =cos 2 2 Salzer (1956) showed that T (ax+b)=i l2] (1)r 2p-2r [2(n-r) (n-r-1)] [(p -q)/2-r r [(p 2-(q.+2j)p-2r a q+2j p-q-2r-2j * T (x) (A-4-17) q If pA < < 1, (A-4-17) can be reduced to 2 T (ax+b)- T (x)+ p T (x) (A-4-18) p p 2 p-i Therefore X ir, p fO P, P (p+ 1)a2 pp+l 2 and X - 0, for n> 2 pp+n

APPENDIX A-5 INTEGRATION OF A(0) AND B(0) mn m Using the Schwinger transformation we obtain from (3-4-15) and (3-4-16), respectively, (0) _ s" 17 - 2 0 2 00 0) do AM= m S(cos +sin -cost) (t) -- dt (A-5-1) mn =O 2 2 a dt and (0) 1 200 20) 0-(0) Mm MT C (cos,2 +sin 2 st) (t) dt (A-5-2) Upon substituting (A-3-34) and (A-3-35) in (A-5-1) and (A-5-2), we have, respectively 2 o00 (0) MT Sin (0) A = m 2 b() (cospp Xpm) (A-5-3) m(+k 2) r k0b up PM 200 r( m 20 a )(c(s0) - Xm) (A-5-4) (1+k ) rk0b p1 p Om pm 123

APPENDIX A-6 DIFFERENTIATION OF W(0) Since ~[c ~~~B. a(2cos0+l-coso)sin-A F [Os 0 YCOS 0 - COs.. 0=-o )in0(A-6-1) we find from (3-4-13) 2 dW (0) 1 (2cos0+ 1 - cos0)sin 0 (l+k )(rk0b) -do =a - o0 (o) N (0) n (0) *+1aosm0 Um(0) + 7r k0b a=n +rkbLu=m (0)I (A-6-2) m = dt (A-6-3) and d m( ) sint msinmt + cos mar - cos mt (A-6-4) +In() I dt 1+cost sint 1+cos d 124

125 If we define (t) sinmt V (t) -s r < t < 7r m sin t m- 1 = 1+ 2 cospt, m=odd m-l = 2 cospt, m= even (A-6-5) p=l and Mr- 100 200 V (p) =1+ 2= cosp[cosl(csc2 cos0-cot ] m=odd m = = 2 cosp[c (sos(csc2 0 - cot2 )] m =even (A-6-6) then we have dU (t) m _ sint [m V(t)t U (t)1 (A-6-7) dt 1 + cos t m -L From the Schwinger's transformation, one can show that dt = 1 sin 0(A-6-8) d~~-~. sint sin2 Thus it is seen from (A-6-3), (A-6-7) and (A-6-8) that dc*U I(0) A -9sin =- sinO F m V (0)-UT (0) 1 (A-6-9)

126 Hence (A-6-2) can be written as 2WS( n Asin 2 N (1+k )(rkob) c os - a m(0)+7rk ob a)b(OUm( ma7CX~O ( ( b7n (00) [m = 1 m - u 3 nmn -0 < 0 < d00.(A-6-10)

APPENDIX A-7 PROPERTIES OF THE INTEGRALS (4-3-6) TO (4-3-9) If we let x = r/e0 then from (4-3-6) to (4-3-9) we obtain, respectively, 1 1 0(e) 3= cos nrx dx (A-7-1) n 0 3 1 1 (e) 3 s cosne0x dx Qn = 00 3 (A-7-2) - - 4 1 i 2n-1 sin n7rdx P(0) = 3 2 1 (A-7-3) n: 0 3 -n Q(0) = 0 3 dx (A-7-4) It is well known that ^1 1V IFF (V+l (l -xcoszxdx =2 2()V J (z) (A-7-5) 2 z V 0 where r (v+ ) is the gamma function with v+ - as its argument. Therefore 2 ~~~~2 for n > 1, P and Q ) respectively, can be written as n1 n 127

128 1 1 p(e) 3 ( 2 6 e =e wrVrO)(2) J 1 (nir), (A-7-6) n 0 3 nr(A-7-6) 6 1 1 Q(e) =0 3<. (2)(2) 6J (nr) (A-7-7) n O';"r3 pnir 1 6 It is difficult to express P(0) and Q(0) in terms of any classical n n (0) (e) functions. However, it is rather obvious that (/Pe is at least of 0(0 ) and Q( /QOe) is at least of the order 0 2 when 0 << 1. 0 n 0 0 0

APPENDIX A-8 SUMMATION OF THE SERIES (Q(e)/Q(e)2 /n AND i J0 (n00)/n n=l Letting = 00 cos a we transform (4-3- 8) into 1 7 3 1 (e) sin (A-8-1) n Q sin acosn cos adda (A-8-1) n0 and hence 2 r 1 (Q ) /n = Q sin aco(n0 cosa)da nl 0n=l 0, 1 * sin 3 cos(n0o C0 )d * (A-8-2) Upon interchanging the summation and integration, we arrive at 2 n 1 n 1 l(Q(e)2/n = da sin 3 sin3 cos (n co0 0 oD cos(ne coso)cos(ne cosi3) _z~ 0.. d, (A-8-3) 129

130 But cos (n e cos a) cos (n 8 cos B 1 n = 2 An2jcos(e0cosa) -cos(0cos )I n= (A-8-4) For e0 << 1, we have 0 cos 0cosea — 1 - - (l+cos 2a) (A-8-5) cos ecost - 1 - -4 (1 + cos 2I) and (A-8-4) becomes co cos (n 0 cos a) cos (n 0 cos3) 2 1 -n Qn - 2 n2 cos 23 - cos 2a = 12 + t cos2acos23 (A-8-6) 0 n= 1 Upon substituting (A-8-6) in (A-8-3) one has 2 f T 1 1 ( 2 3 23 2 1 (Q(e)2 2 n 0in d) n=o " 1 i 1 + ( sin a cos 2nc da)2 (A-8-7) n IV0

131 One can see from (A-8-1) that 1 7r 1 Q e) = % 3 sin3 ac da (A-8-8) and because the series on the right-hand side of (A-8-7) converges very fast to 0.05053 0 2/3 the series 0' (. (e) (e)2 2 _(Q ) /n = In( )+ 0.0195 (A-8-9) Since 7r J0(n00) = cos(n0cos e)do, (A-8-10) 0 we have 7r 7r 0 J (n 2 cos (n00cos) dc c s (n00cosI3)do3 n=l n=l n7r 0 0 (A-8-11) Following the same steps from (A-8-3) to (A-8-9), we obtain 2 0 (n 0)/n In () (A-8-12) n= 0

APPENDIX A-9 TERMINAL ADMITTANCE OF A WEDGE WAVEGUIDE IN A PERFECTLY CONDUCTING CYLINDER The geometry is as shown in Fig. A-9-1. 0 II I \ ~ /,?- l FIG. A-9-1: PERFECTLY CONDUCTING CYLINDER SLOTTED BY A WEDGE. The perfectly conducting circular cylinder body is of radius a. The width of the wedge is 2 80. If we put a magnetic line source at the apex of the wedge, then the source excites EM fields in the free space (region II) as well as inside the wedge (region I). In Fig. 1-3 if we let 8 = 0, 00 > 80 and c -> b -a, then we obtain the same geometry as shown Fig. A-9-1. If y' (a) is the normalized wedge terminal admittance defined by (2-1-15) where r =a, it can be shown by a similar procedure as in chapter II that (0 yit(a) =2: J (k0a) 0o ) 2 (%.) l E(~')oos 132

133 0o c13.(0')cos n 0'' -0 H(2)(k a) 7r ) | g )2 (A-9-1) -o where 2 (0') is the tangential electric field in the wedge aperture. This is a stationary expression with respect to e (0). Hence, when 80 < < 1, one may let A E(0') = w 2 and since J (k0a) 0 0, 0 k0a ~J'(k) - for e0 << 1, n> 0 J' (ka) 7r n 0 nir 0 0 Eq. (A-9-1) becomes r 2k a ~ OD H(2)(k a) ~(a) j (Q 2 T(P(e ) /n - 37 n 0 (e). (A-9-2) y'(a r() 2ka 2 C H(2)ka) n Q0 H"n (k0a) If Y'(a) is the terminal admittance of a section of the wedge waveguide of length a meters, then

134 Y' (a) 2' (a) (A-9-3) It is straight forward to write down the conductance G' and susceptance B' from (A-9-3) as (e) __ 0 n n 2 G' = 2 n E 2 ( A)2 ( mhos (A-9-4) r2ka LO n=O l+x Q N'(k a) 0 n 0 nO0 ODc { ((e) B' = 0 2ka 1 (-e n Q(e) (e) Nn(ka) +XnYn n 2 n ) ( (e) ka mhos (A-9-5) n=O n 1 + x 2 Q N' (k a) n 0 nO where' (koa)/N (koa) n nu nn Yn Jn(k0a)/Nn(k0a)

APPENDIX A-10 POYNTING'S ENERGY THEOREM IN THE PLASMA SHEATH AND THE FREE SPACE We consider a volume V enclosed by a surface S in which the electromagnetic fields are of periodic time variation.. The Poynting's theorem for this volume then is -i,(ExH )n ds = 4jw(W -W )+ 2P (A-10-1) H E where n is the outward normal of S and WH = time-averaged stored magnetic energy in V 1 P - -* ~= H H dv (A-10-2) V W = time-averaged stored electric energy in V -4 e E E dv, (A-10-3) V P = time-averaged dissipated power in V a E ~ E dv (A-10-4) V 135

136 As in the main text, we choose to consider a section of the coaxial antenna of length a meters in z-direction and apply (A-10-1) to the plasma sheath and free space of this section respectively; we have 7r 77 2Pro + j4w(W III a E I = H cd0 H E 0 Z Z -o r =b o r ==c (A-10-5) 7r 7r IV +j4w(WIV - = a E cd -a EI HI rd| H E 0az H c -7r r=c -= r 7 (A-10-6) It is obvious that there is no dissipated power in the free space, thus PIV = 0. Furthermore, if Pr denotes the power radiated by this section of the antenna, then P EIV rd (A-10-7) 82 s~-?rEVr -> co Since at r = c, E =E and HIII= HIV, one may combine (A-10-5) through 0 0 z z (A-10-7) together and obtain III III III IV IV IH In 2PII +j4 w(WI _WIE )+ j4w(W -W )+2P E H III (A-10-8) H E H E rb -7r Upon substituting (2-2-11) and (2-2-12) in the right hand-side of (A-10-8) and carrying out the integration, we have 2P IV +2PI n) + 4 (W - 4(W - WI ) r H E H E

137 ~ab,~co H ()(kOC)' n 0 i~ n-O H(n (k0c) 00 00 * E d0E (pE0)cos n (0 - 01')dp'. (A-10-9) Similarly, if we substitute (2-2-18) and (2-2-19) in (A-10-7), we attain a formula as shown, ac f 2 3 nosn(-')d' Pdr = E ( ( E ( 40')cosn(p- 0')d d' r T -f Pt C n -A A nn -00 -00 (A-10-10) where A -k H(0) (kCc) [J(k c)Nn(klb)- J'(kb)N(kc)] n n 0 nInbnbnli H 0)(k ) J n(klc)Nn(k b)- JI(klb)N'(klc)3 (A-.-l) n 0 LnI n 1 n 1 n j For no plasma sheath case, we let c — > b, k -1 and obtain straight forwardly from (A-10-10) that Pr 4 rkb) n- O (22 d p0E -"(0) E (E(0')cosn( -')dO' Vr 4: 0 O k0bn=:H'(k0b), 72 d = 2E n ~-0 0-0 0 (A-10-12)

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