5MS ntff&Sfml mm JtlMMJMaDlQJQnrUJllU JL. J IMHII L I B R.AR.Y OF THE UNIVERSITY Of ILLINOIS 621. 3C5 IJl655\e no. 3 1 -36 cop .2 The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN APR 1 8 HI76 MAR 2 8 1976 L161 — O-1096 ANTENNA LABORATORY Technical Report No. 32 THE EFFICIENCY OF EXCITATION OF A SURFACE WAVE ON A DIELECTRIC CYLINDER by James Wilbur Duncan* 25 May 1958 Contract AF33 (616)-3220 Project No. 6(7-4600) Task 40572 Sponsored by: WRIGHT AIR DEVELOPMENT CENTER Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois ♦Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering at the University of Illinois, 1958 ~JL6' 3, z). The rod is considered lossless, (J = 0, with a magnetic permeability n and permittivity £, = € € , where € is the relative dielectric constant. The medium surrounding the rod and extending to infinity is free space, with constants n and € . The electromagnetic field source is a filamentary ring of magnetic current located at the plane z = 0. The ring is of radius a, where < a < b, and is infinitesimally small in cross section. The source distribution is represented as a product of Dirac delta functions in the p and z coordinates as follows: K =^5(P-a) o(z) (2.D where (j) is a unit vector in the (j) direction, The source distribution ~K is independent of (j) (uniform), and is a unit source such that ff* a+A o+ /> 0da = J j 5(p-a) §(z) • dz dp = 1 a-A o-A a+A o+ A £(p-a) dp = 1, and j £(z) dz = 1. a-A o- A K has the dimensions of volts per square meter. A harmonic time dependence of e is selected for the source. iU r 02^ The electromagnetic field is a solution of Maxwell s equations. -iG)t Written in differential form for e time dependence, we have VX E = + i'OfjiH - K (2.2) VX H = - iCi)€ E. Taking the curl of the second equation and then substituting the first relation for VX E yields _ 2 — — - V X V X H +0) !i6 H = - iO€ K. (2.3) The only non-zero component of K is the coefficient of the unit vector . We may write the component of the vector eq„ (2.3) and obtain the non-homogenous scalar equation [- v x vx h] . + coVe ■ H . = - iO>€ £(p-a) $(z), where the (j) component of the bracketed term is indicated. The magnetic current filament generates a field having components H., E , and E , while the components E ., H , and H are equal to zero. Due to the symmetry of the source, the field is independent of

«, + (k » . t «, h dp a+A a+A i(^) dp = -1U8 J S(p-a) dp -A a-A (3.2) Assuming that h(p,£) is continuous for all p, in the limit as A— »0 and p-»a, Eq„ (3.2) reduces to = -io>8 (3.3) p=a 11 We have shown that a continuous h(p,£>) which satisfies the homogeneous equation (3.1) and whose first derivative is discontinuous by -iwg, at p = a is a solution of (2.8). The remaining boundary conditions on h(p,£>) follow from the boundary conditions imposed onllAp,z) and E (p,z) by Maxwell's equations. Refer- ring to Fig. 1, we denote the cross sectional area of the rod as region I and the space outside the rod as region II. Since tangential H is continuous at a magnetic current discontinuity, we see that Hi(p,z) is continuous at p = a for all z including the filament position z = 0. Since tangential E and H are continuous across a dielectric boundary, we note that H.(p,z) and E (p,z) must be continuous at p = b for all z. The corresponding conditions on h(p,£) are: (i) Since UAp,z) is continuous at p = a for all z, pt) = J° H (p,z) e-^ : must also be continuous at p = a, hence »(P^)] a . A = »(P.O] a+A (3.4) (ii) Similarly, since H,(p,z) is continuous at p = b for all z, h(P.O] b . A = h(p,U] b+A (3.5) (iii) From (2.5) we write l ^ E d> l 12 In region I, where - - b, the permittivity £ = £ g . In region II where p > b, £ = g . E (p,z) continuous at p = b for all z requires !5* i = 8. ?5* i ep + p H b-A b+A Taking the transform, we obtain dh 1 ,. dp + p h = e b-A dh 1 . ap + P h b+A which reduces to S r [f] - [§] ♦ [| >] C8 r -X) - o L ^ J b+A L ^ J b-A LH J b+A (3.6) by reason of (3.5) In order to determine H*(p,z) in region II, we must solve Eq. (2.8) for the corresponding h(p,£>). Inside the dielectric rod, for all p except p = a, (2.8) becomes dp h 1 dh ,,2 2 + p dp + (k l ^ - i) h - (3.7) where k, z = CO' ^O&i Outside the dielectric rod and for all p, (2.8) becomes dh ldh .2 ,2 1 N . ~2 + p dp + (k - & " "2> h = ° dp ^ ^ p (3.8) where k Q 2 =0) fi 0&() . 13 We proceed by writing general solutions h(p,£>) of (3.7) for the regions - p - a, and a - p - b. The discontinuity at p = a appears in the boundary condition (3.3). A general solution of (3.8) is written which yields h(p,£) for the region p - b. The three solutions possess six arbitrary constants. The constants are determined from the require- ments that the field be finite at p =0, that the field be regular at infinity and that it satisfy the radiation condition and the four boundary conditions (3.3), (3.4), (3.5), and (3.6). Choosing the solution h(p,£) for region II, the inversion integral (2.7) must then be evaluated to yield H.(p,z) for p > b. Equation (3.7) is recognized as a form of Bessel's differential equation. It has the general solution h(p,£) = A J^T^p) + P y^-tJ^p) where A and P are arbitrary constants, J (li p) and Y C\) p) are Bessel functions of the first and second kind, respectively, and Consider the case - p - a. Since Hj(p,z) and, therefore, h(p,£) must be finite at p = Q, we determine that P = 0, since Y (V p) is unbounded as p— >0. Hence h(p,£) = A J x Ci^p) (3,9) where 6 p £ ^=[ kl -^] 2_l/2 M For the region a - p - b, the Y function must be included, and we have where a - p - b; h(p,t,) = B J 1 Cl) 1 P) + C Y ± {]) p) (3.10) P 1/2 For the region b - p we choose to write the general solution in terms of Hankel functions of the first and second kind h(p,;> = d h 1 (1) ay)) + q h 1 (2) g)^) where ^o - K 2 - tf* It will be seen later that, in general , "U is complex; therefore we define the argument ofi) to be — arg l) - IT. As P-=»oo, H ^cp^ (2) vanishes and H ^c\P} is unbounded for the defined argument ofV ft ; therefore Q = and h(p,t,) = D H 1 U) (V o P) (3.11) where b ^ p ^o " t k o " tl - arg-u> Q - 7L In order to solve for the arbitrary constants A, B, C, and D, we apply the boundary conditions (3.3), (3.4), (3.5), and (3.6) to the 15 appropriate solutions (3.9), (3.10), and (3.11). Omitting the details, the following four equations are obtained: - (v x b) A J x ' (v x a) + (v^) B J^ (v^) + (v^) C Y^ (v^) =-x^ ± b A J 1 (v i a) - B J^i^a) - C Y (v^a) = B J^v^b) + C Y 1 (v i b) - D H^V^b) = > i b) B j' (v i b) + (v i b) C Y^ (v^) - D[£ r (v Q b) H 1 / (v Q b) + +(6 r -D Vv o b)] = (3.12) This system of equations was written in matrix form and solved by finding the inverse of the coefficient matrix. Details are given in Appendix A. The constant D was determined to be J x (va) D = 1U V (I) (I) (v x b) J (v ib ) H/ ; (v Q b) -£ r (v o b) J^b) H C ; (v Q b) (3.13) Substituting (3.13) into (3.11) we obtain h(p,?,) for the region p- b. The solution h(p,£>) may then be substituted into the inverse trans-form (2.7) to yield the integral expression for Hj(P,z) in region II. This result is . . icj^a p +0 ° J^a) H^V^) e +i ^ z d£ V P ' Z> ^ (v^J^b) Hl (1) (v b) -S r (v b) J^b) H o a) (v b) (3.14) 16 where p ^ b , v i = L k i - 1 J j V - L k - 1 J • The evaluation of (3.14) is the subject of Chapter 5. 17 4. SINGULARITIES OF THE INTEGRANT*— SOLUTION OF THE MODE EQUATION The integral for H . (p, z) is treated as a contour integral in the complex t, plane where the path of integration is along the real axis from -co to +co . In order to carry out the integration, it is first necessary to determine the singularities of the integrand. A detailed analysis of the singularities is presented in this chapter. The expression for H. (P,z) is +i0)€ ia + ^ J^a) Hl (1) (V Q P) e +iCz dC H.(p,z) = — - , t^t ^p- 211 ^ ( V l b > J o< V l b > *! < V o b > - M V „ b >V V l b >0 ( V b > -oo 1 ol 1 o rollo o (4.1) 4.1 Branch Points Consider the variable v, which appears in the arguments of J and 1 o I 2 2 J in (4.1). Since v = k - £ , then v is multiple-valued in any neighborhood of I, = + k . However, if one considers the power series expansions of J and J , one sees that the integrand is an even function of v so that the points t, = +: k are not actually branch points. The I 2 2 Hankel function arguments contain the variable v = k - t, . The Hankel function has a logarithmic singularity at v =0, and so is multiple-valued in any neighborhood of t, - + k . The points £ = + k are, therefqre, branch points of the integrand. We select branch cuts in the t, plane as shown in Fig. 2. The points + k and + k lie on the real axis with I k, I > ' k I since € , > € . The branch cuts I 1| ' I o I 1 o are defined as t = + k f i In 1. — o 18 hi t r t E « *J(x i 19 4 2 Poles Before we investigate the integrand of (4 1) for poles, it will be advantageous to consider the significance of a pole in the physical problem. Assume that the point t, = £ is a pole of the integrand. When (4.1) is evaluated as a contour integral, a residue contribution at £ = 4 must be included. Thus, Hj(P,z) will have a term of the form n x it z Ah/ A; (v p) e ° (4.2) 1 o where A = 27Ti Residue *-V 2 2 Since we assumed a lossless dielectric rod by taking k = 03 \jl€ real, we will expect (4.2) to represent a wave propagating without attenuation, which implies that t is real. If t is real and It I > |k | , o o I o | I o I then v is pure imaginary and H (v P) reduces to the K. Bessel function which decays exponentially with increasing argument. Under these conditions, (4.2) represents a surface wave of amplitude A having a radial dependence of H (v P) which propagates along the -i(CJt-£ o z) l ° rod according to e .We shall now determine if there are values of t, real which cause the denominator of (4.1) to vanish. Equating the denominator of (4.1) to zero yields J (v.b) H (1) (v b) 1 o For convenience of notation, let v b = X, and v b = X ; then (4.3) may 11 o o be written J (X,) H (1) (X ) v ol = g o o 1 W r ° Hl (1) (x) 1 o Recalling that v l = k 2 - 1 o K 2 - k > 0, 1 o y (i) Consider the case of £ real and It, | > Ik I > Ik I Then v, and v , and therefore X, and X will be pure imaginary „ Denote 1 o 1 o X, = ip and X = i£ , where p and £ are real; then (4.4) becomes 1 o J (ip) H (1) (i£) (iP) tW- H (1) (X ) H (X ) 1 o 00 where H* denotes the derivative of H o o 21 Using the definition of the Hankel functions, (4.6) may be written ^V _ . £ x W - | W A l J n (X n ) ~ r o J / (X ) + i Y / (X ) " K *'" 11 o o o o Since X is real, the left side of (4.7) is always real, while the right side is obviously complex for real X . Is it possible for the o right side to be real for some X ? The right side can be real only if the argument of the denominator is equal to that of the numerator by a multiple of nTT. Let a = arg [j^) + i Y^)] P = arg [J o (X o ) + i Y o '(X o )] Assume |3 = a + nlT, where n = + 1 , + 2 , then since a 4. , «\ tan a + tan nff tan p = tan ( a+nTT) = - — — — — =-= tan a r 1 - tan a tan nTT Y (X ) Yj{X n ) tan a = and tan [3 We obtain J (X ) J ' (X ) o o' o o Y (X ) Y > (X ) o o o o J (X ) J / (X ) o o o o J (X ) Y ' (X ) - Y (X ) J I (X ) = 0. (4.8) oooooooo However, the left side of (4.8) is the Wronskian of J , Y , which o o is non-vanishing for finite X : o W £J ,Y ? = J (X ) Y' (X ) -Y (X )J \x ) = -z§- * t o OJ oooo oooO 7TX 22 We conclude that the right side of (4„7) must always be complex. Since the left side of (4.7) can vanish, is it possible for the right side to be zero? For the right side to be zero, J (X ) and Y (X ) o o o o 9 must be zero simultaneously, Watson proves that the positive zeros of two distinct cylinder functions of the same order are interlaced; therefore, J (X ) and Y (X ) are never zero simultaneously. The same holds for -X since J (-X ) = J (X ) and Y (-X ) = Y (X ) . o o o o o o o o o Since the right side of (4.7) is always complex and never vanishes, there are no real values X. and X for which (4,6) can be satisfied. 1 o We conclude no poles exist for t, real and I £ < Ik (iii) Consider Z, real and Ik I < |£| / Ik, I ; then y is real,v is pure imaginary, and therefore X, is real and X is pure imaginary. 1 o Let X = ib , where b is positive and real, Equation (4.4) becomes i ( V J (X,) H (1) (i£) K (£) = € (-i£) -°-- = 6 e ^TT . (4.9) j o (x x ) k o (£) A graph of the functions -X . . and € £ g is presented in Fig. 3. 1 J n (X ) r K n (5 ) J o (X 1 } The function - X . > is discontinuous each time J, (X, ) vanishes 1 1 l v 1 and so an infinity of branches occur for ever increasing X . It passes through zero each time J (X, ) has a zero. The function o 1 K Q (b) € £ v (t\ approaches zero for b — »0 and increases positively for increasing £. It is evident in Fig. 3 that for every finite b, an infinity of values X exist which satisfy Eq« (4.9). In order to obtain a finite number of unique solutions of the mode equation, a second relation between X 1 and £ is needed. 2;j xT -uP \ " If L 1L 1 1 1 1 i x" 13 C (0 -^ « \ s \ \ \ > o P II ^^ fa LL LL 24 It follows from the definitions A 2 " 2 , 2 r 2 v = . k - t, o \o where £ is identically the same in v and v . ,2 Elimination of ?, yields 2 2 2 2 2 27T v/ - V / = k_- - k = (J-) (€ - 1) 1 o 1 o A. r o 2 Multiplying by b yields (v lb ) 2 - (v o b) 2 = C 2 ^) 2 (€ r -l) o o Si nee X = i£ , we obtain the second relation between X and % , which is oqq X r + i = R (4.10) where ,27Tb ) JV i = ko bjvr * HP) € The graphical solution of the mode Eq. (4.9) is illustrated in Fig. 4, which includes curves of X as a function of § obtained from Fig. 3, and the relation (4.10). Note that (4.10) is the equation of a circle of radius R with center at the origin. The multiple-valued solutions X for every £ as discussed with Fig. 3 are evident in Fig. 4. The first branch of X starts at X = 2.405 for t = and approaches X =3.83 asymptotically with increasing |. X x = 2.405 is the first zero of J^) and X = 3,83 is the first zero 25 7 i 5 A- i 1 c Mode r Solution (§, x x ) Vi 1 5 2 ^^ \ o C 5 FIGURE 4. GRAPH OF X = f(^) and X 2 + £ 2 = R 26 of J (X ) , The second branch commences at 5.52 and approaches 7,02, which corresponds to the second zeros of J (X ) and J (X ) , respectively An infinity of branches is thus established Consider the case R < 2.405. Since X ± = \| R - £ 2 ^ R, Fig, 3 shows that Eq. (4.9) cannot be satisfied for X < 2.405, that is,, no solution of the mode equation exists for R < 2.405. This is also evident in Fig. 4, Physically ; this means that the dielectric rod waveguide is below cutoff and cannot propagate a surface wave of the transverse magnetic type. Referring to Fig. 4, if 2.405 < R <5.52, a single, unique solution (£, X ) of (4.9) and (4.10) results. Thus, (4.1) has poles at t, = + t, , where t, may be determined from the solution (£ , X_ ) „ The poles occur on the real t, axis in the region | k J / \t,\ • Ik I . We see that when R = Pr— ) . |€ -1 is restricted to the range 2.405 < R < 5.52, the dielectric rod propagates a single, surface wave which is the lowest order, circularly symmetric, transverse magnetic mode. It is known i£ o Z -i£ o z as the E mode and propagates as e for z -7 o and e for z < o. As R increases, the number of solutions (£ , X ) of (4.9) and (4.10) also increases, and the dielectric rod supports an ever increasing number of higher order TM modes. The number of poles occurring between k» and k increases with R, as is shown in Fig. 5. It is interesting to note the behavior of k , k , and the poles as R increases. From the definition of v we obtain o . k 2 + (|) 2 (4.11) \| o b 27 H 1 < v c a 4J> J : : \uu\\ wwwww + i< 1 ) : if, i '\\\\ \\\\U\\\\\\ CO i 1 1 » 28 Consider R just greater than 2.405; then X ft* 2.405 and b is a very small positive quantity. Equation (4.11) shows £ is just greater than k , that is, the poles + £ first appear in the neighborhood of the branch points + k with \t,\ ~7 k , Referring to Fig. 4, as R increases, the s of any solution increases, so that £, corresponding to (£ , X ) also increases. Thus, the poles + £ move away from the origin. Each new pair of poles + t, first appear near t, = + k , Now 27T 1 I k = (r— ) , k, = J6 k , and R = k^b €- - 1, so we may think of k and k o \ 1 M r 6 \J r ' 1 as proportional to R. The distance from k to k , (k -k ) = k ( J€~-1) is also proportional to R. As R increases, k and k move away from the origin, the interval (k -k ) increases, and the number of poles between k and k increases. 4.3 Solution of the Mode Equation The object of this work is to determine the efficiency of a magnetic current filament in exciting the E _ mode on a dielectric rod. For single mode operation, R must be restricted to the range 2.405 < R < 5.52, Since R = k b I € -1 , this restriction on R is actually a limitation on k b and € , the physical constants which describe the dielectric rod r as a waveguide. Specific solutions of (4.9) and (4.10) are required in order to calculate the power transmitted in the surface wave and, ultimately, efficiency. Equations (4.9) and (4.10) were solved for the case € = 2.56, a relative dielectric constant typical for polystyrene. Six solutions (5 , X ) were obtained for selected values of k b equal to 2.2, 2.6, 3.0, 3.4, 3.8, and 4.2. Graphical solution of (4.9) and (4.10) as illustrated in Fig. 4 was considered too 29 inaccurate for the purposes of numerical computation and so (4.9) and (4.10) were solved, utilizing the University of Illinois digital computer. The solutions (£ , X ± ) are accurate to four decimal places. Details of the method used to solve the equations on the computer are given in Appendix B. Table I presents i, X , and the ratio W\ corresponding to each value of k b. The ratio X |X follows s ' g| o from the definition of v . X |X_ = g " M^© 3 (4,12) TABLE I V 2.2 2.6 3.0 3.4 3.8 4.2 5 X l x rx n g| o 5603 2.6901 .9691 1.2329 3.0043 .9036 1.9353 3.2086 .8403 2.6269 3.3366 .7913 3.2905 3 . 4204 .7560 3 . 9264 3.4788 .7305 30 5, EVALUATION OF THE CONTOUR INTEGRAL For the purposes of the following discussion, we repeat the integral solution for H-(p,z) outside the dielectric rod, which is +iG)&a p J.(va) H (1) (v n p)e +i ^> z V P ' Z) " IT J -IT) " ' (1) di; c (v i b) J o ( V» H i < v o b) " W> J i ( V> H o V> (5.1) where C is the contour along the real axis with appropriate indentations o + + * at the branch points - k and poles - b„, as shown in Fig. 2. Expression o (5.1) is evaluated by transforming from the £ plane to the complex -r plane and employing the saddle point method of integration to obtain an asymp- 10,11 totic value of the integral. The saddle point method is applicable only when certain parameters in the integrand are relatively large. It is used to obtain an approximation to the far zone radiation field. The sur- face wave field follows from the residue contribution at the pole. In the T plane the contour of integration is deformed until it passes through the saddle point of the integrand. The integrand contains an exponential function, and when the deformed contour corresponds to the path of w steep- est descent the exponential function decreases very rapidly away from the saddle point. The integrand vanishes at the ends of the path, and so the principal contribution to the integral comes from evaluation along the path of steepest descent in the very near vicinity of the saddle point. 5.1 The Radiation Field In order to apply the saddle point method of integration, we introduce 31 the transformation of variable £> = k sin t o where t = 4« + iT| (5.2) Now t = sin (£|k ) is a multiple-valued function of £, and the region of integration in the C plane transforms into a strip in the T plane bound by two curved lines which correspond to the branch cuts in the £, plane. They are defined by the equations C sin ^ coshTj =1 C sin ^ cosh T\ = ~1 1 L ^ I L (^ cos + sinh 7] = ^-2 > L cos ^ sinh r\ = ™-^ ^ ( 5 . 3) o o The contour of integration C in the ^ plane, which is defined as Im %, = with necessary detours, transforms to the path C in the t plane, as shown in Fig. 6. It follows from the definition Im £ = which corresponds to k cos 4* sinh 71 = in the t plane while maintaining C within the curved o ' 1 strip. The direction of C follows from consideration of the real part of £ along C_, where Re £ = k sin 4 1 coshTh The branch points £ = - k U o v o transform to the points t = - 7T/2 while the images of the poles - fe are the points T = +7T/2 - i cosh C /k and -f = -7T 2 + i cosh £» /k . o ' o / o o ' o' o Substituting (5.2) into the expression for v yields v = Jk 2 - C 2 - - k cos 1" „ The choice of sign here is determined by the previous definition of arg v in the £> plane. On the contour C_ , %, is real and o U when |£>| > k I we obtain v = j v I g ± *• V 2 . Since we chose to define < arg v < 77", the negative sign is excluded and we have V = V n e+ 1 ■ In ord e^ that corresponding values of f on C yield 32 33 the same argument for v the positive sign is required;, thus we establish v n = + k cos t. (5.4) o 2 Y 2 + 2 The function v, becomes v = k - fe = - k , £ - sin t. Since the 1 1 \| 1 \j^r integrand of (5.1) is an even function of v , either sign may be used and we choose 2 - sin t = k w(t) (5.5) o \J r o where for convenience of notation we let w, a function of t, represent the radical, w = £ - sin 2 T. Substitution of (5.2), (5.4), and (5,5) into (5.1) with d£ = k cos t dT yields the transformed integral in the t o plane which is ico£ a n J, (k a w) H (1) (k p cos t) e lk ° Z sinT cos T dr rr/^\ 1 lO 1 O H . (P , Z) = T-T pj v 27Tb U w J (k b w) H, (k b cos t)- £ cos t J (k bw) Bt> (kb cost) Do lo r louo C l (5.6) If p is large and k p cos T =f 0, the Hankel function may be replaced with its asymptotic representation, Then the Hankel function and exponential in (5.6) combine in the form H l (1) (k p cos T)e ik ° Z Sin T ~ e" 1 ?^ 2 > e ik o (P cos T + z sin T > o r \ 7Tk p cos t / (5.7) 34 Consider the spherical coordinate system illustrated in Fig. 7, where the FIGURE 7 CIRCULAR CYLINDRICAL COORDINATE SYSTEM (p,0,z) AND SPHERICAL COORDINATE SYSTEM (r,0,9) polar angle Q is measured from the plane z = 0. In this coordinate system we note that p = r cos 9 and z = r sin 9. Substituting for p and z in the right side of (5.7) we obtain H n (1) (k p cosT) 1 o 37T ik z sinT ^ ft -i-£- p 2 ; r cos 6 cos f- Lit k V2 ik Q r cos(-f-0) (5.8) We substitute (5.8) into (5.6) to obtain the following approximation for 35 for H. (r,9) which holds for r large and k r cos 9 # „ i/i iooe a 37T I „ , ^ H*0 for -7T+0<4 J <© and 7]<0 for < 4< < IT J (5.14) The shaded area of Fig. 6 is the region of convergence and we see that the integrand of (5.9) does vanish on C as T) approaches - oo . The relation of the path of steepest descent and the poler n will now be considered. Evaluation of (5.9) along C yields the total field s in the region outside the dielectric rod. Since the saddle point is equal to the polar angle 0, as increases towards 7T/2, the steepest descent path shifts to the right in Fig. 6. Ultimately, a residue term must be included because C will cross to the right of the pole T«. There- s fore, the total field outside the rod, which is obtained as takes on values ^0 < tt/2 s may be considered as the sum of the residue at the pole and the steepest descent contribution at the saddle point, that is, the surface wave and the radiation field . Due to the rapid decay of the exponential term away from the saddle point, it is only necessary to integrate over a short segment of C in the s near vicinity of Y= 6. 37 We proceed by setting -T - 9 = a e" 1 * .7T dX = do- e~ 17 ^ (5.15) where a is the distance along C measured from the saddle point. Consider- s ing \ f - -9| small, we may approximate 2 2 7T cos (T- ©) ,* 1 - {rf ~ 2 B) - = 1 - "Y e _i 2 (5.16) Except for the exponential term, the integrand of (5.9) is slowly varying about T 1 = @ and may be assumed as constant. We substitute (5.15) and (5.16) into (5.9) to obtain V r ' S) = «b L. [_|_] F(e) e lk o L 7T k r J +A ik^r k ro^ o dcr, (5.17) Since r is very large, the following approximation is valid J e -V- d(r « / e -V- do . = [|i] *6 (5.18) Substituting (5.18) into (5.17) and noting that W£ = k £ 1 or the far zone radiation field, which is ^o , we obtain yr,6) = . OJ 7Tb r \^o (5.19) where F(0) J (k a w) 1 o wl(k bw)R (k b cos 0)-£ cos J, (k bw) H (k b cos ©) Oolo r loOo IH and S - sin 9 r It follows from Maxwell s equations that E e (r ' e) = -"-S-> p(9) I (5.20) 5.2 The Surface Wave Field The surface wave propagating in the positive z direction is obtained by evaluating the residue of (5.1) at the pole ?> . Denote the denominator of (5.1) by G(£) so that (5.1) may be written yp.«> - w f^ (1) iCz ico^a J^a) H^ ' (v Q p) e lb 27T G (&) d£ (5.21) where G(t) = (v 1 b) J (v a li) H 1 (1) (v () b) -£ r (v Q b) J 1 (v 1 b) H (1) (v b) The integral (5.21) is the solution for the total field H,(p,z) in Region II where p > b. Let H (Pi>z) represent the surface wave portion of the field in Region II. It is given by H/X (P> z ) = 27ri Residue (1) j^a Jj^a) H x v (v Q p) e 277 G(&) (5.22) t-6. where z > 0. The pole £ follows from the solution (I , X ) of the mode equation G(£) = 0, Since G'(& o ) =^ G(&) £= 0, C is a simple pole and the residue of & »&, of the integrand at £ is given by Res iut.. J i( 5 V H (1) ,i^)eV 1 b b . (5.23) 0' + t ( T> + &r ( r^ W K o (l) + 1 1 + (6 - 1) J, (X.) K. (I) > (5.24) r 1 1 1 / Using the relation H (i £ £) = - ~ K (§^), it follows from (5.22), (5.23), and (5.24) that H0 II (P,z) =A n K x (£^) e^o z (5.25) where p > b, z > ^ (=> j, r ( 4 >+( r )] J (X l' *>< e >*<6,-» WV 6) } with and & b = J ( V ) 2 + 4 2 R 2 = Xl 2 + | 2 . It is interesting to note that the surface wave solution (5,25) exist since £> is real. This situation results from the simplifying 40 assumption that the dielectric rod is lossless. If the finite loss of the rod is included, the constant k 2 = d) 2 u £,,(1 + i^]?") is complex since i^oz (5 = 27) H

z > = A n j^) J i (x i V e 41 We obtain EAp,z) in Regions I and II from E = — - — v y H, hence P — iCoL ?)H I £ V (p ' z) = ( I3 l ) ^f = fe^ V (p ' z) (5 - 28) E p II(p,z) = ( 4^ V 1 (p,z) (529) The field components (5.25), (5,27), and (5.28), and (5.29) will be used to calculate the surface wa.e power in Chapter 6. 42 6. THE POWER INTEGRALS AND EXCITATION EFFICIENCY In the previous section, the surface wave field and radiation field generated by the magnetic current source were determined. It is now possible to calculate the corresponding powers and obtain the efficiency of the source. The efficiency with which the source delivers power to the surface wave is called the excitation efficiency of the source. Denoting efficiency by the symbol T , it is defined as S m _ W_ ; - T w s where W is the surface wave power T and W is the total power delivered by the source „ 12 Goubau has proved that, for lossless surface waveguides, the radiation and surface wave fields are orthogonal with regard to power considerations. In this case, the total energy delivered by the source is equal to the sum of the surface wave power and the radiated power. Since we are considering a lossless dielectric rod, the orthogonality condition holds and y is given by w s + w* g where W is the surface wave power and W is the radiated power. In order to determine T , we proceed with the calculation of W and W . (6.1) (6.2) 43 6,1 Radiated Power The radiated power W is obtained by integrating the time average Poynting vector over the surface of a large sphere of radius r„ W =J jp" av ■ da = j | Re ¥ X H* • da (6.3) 2 . where da = r cos 9 d 9 d [ A n j^yll V< J o 2(x i ) + J i 2 (x i> - ^ VV V x iY («.io) tor region II where p > b, W 11 is given by 27T p=o> w " = / / i ' 2 ReE P n V I%ai (6 - u > o p=b Substituting (5.29) and (5.25) into (6,11) yields p=oo f i --&*ii r p»k a '*&* p=b (6.12) he definite integral in (6.12) is also given by McLachlan, 14 btain W 11 as w " ■ <£f> A „ 2 4U 2 <6> - k x 2 (I) + I k o (?) Kl j le constant A^ is defined in Eq. (5.25), We may now substitute 5 »10), (6.13), and the value of A into (6.7) to obtain W S . After (6.13) considerable manipulation of constants one obtains the expression for the total surface wave power which is 47 where n x \ SI o o K, (I) Nr r 1 x 1 (6.14) " ( 4 )2+1 l r a X ! n 3— ■jJ (X 1 )K 1 (6) + [€ r (^) + (^J o (X 1 )K o (e) + [K o 2 (e)- Kl 2 (e) + f k o <£) ^«a/ (€ r - 1) j i (x 1 ) 1^ (I) 2 V J i where D , D , and D are the lengths defined in Fig. 12. The origin of the reflection coefficient plane is the point 0, the iconocenter is , and the center of the image circle is C. Formula (7.2) was used to calculate the efficiency for all of the experimental measurements. Before the results are presented, a brief description of the experimental apparatus and coaxial exciter will be given. 7.2 Experimental Equipment A two inch diameter polystyrene rod was selected for the experimental work. This choice followed from two main considerations. First, a two inch rod can stand on end without additional support. This method of vertical mounting avoids the usual problem of how to support a surface wave structure without disturbing the field. Second , it is easier to fabricate annular slots for a large rod than for a small rod. Table III lists the wavelength X , frequency f , the ratio X (X , and X for a two inch polystyrene rod, assuming € =2.56. Using Deschamps 7 method, one obtains the efficiency of the entire transition between the input reference plane and the output reference plane, that is, between the measuring probe and the short circuit termination. Since we are interested in measuring the dissipative 59 Unit Circle FIGURE 12 . PLOT OF THE IMAGE CIRCLE ON THE REFLECTION COEFFICIENT PLANE TABLE III k b o X o f g/ o X g centimeters megacycles sec centimeters 2 2 7 25 4132 0.969 7.03 2,6 6.14 4884 0.904 5.55 3.0 5.32 5635 0.840 4.47 3.4 4.69 6387 0.791 3.71 3.8 4.20 7138 0.756 3.17 4.2 3.80 7889 0.730 2.78 attenuation of just the annular slot, it is essential that the rest of the transition shall introduce only negligible attenuation. For the present we shall ignore the dielectric rod loss since it is very small. It follows, then, that a low loss exciter must be constructed to illuminate the annular slot. The annular slot presents a very low conductance and capacitive susceptance to the exciting waveguide. Therefore, the feed waveguide must be a low impedance line with some means of tuning out the capacitive susceptance of the slot at the ground plane position. Figure 13 shows a cross sectional view of the low impedance coaxial line which was constructed for this purpose. The inner diameter of the outer wall of the coax was 17/8 inches and the inner conductor diameter was 1 l/4 inches, which yields a 24 ohm line. A two wavelength tapered section transformed the 24 ohm line to a standard 50 ohm, type N connector. Two polystyrene rings centered the inner conductor within the cylinder. The ring located at the tapered section was one-half wavelength long, so that no impedance 61 w CP J. c c ,3 h w QJ 1 •J < 62 discontinuity would result. The ring at the ground plane position was a quarter-wavelength transformer which would match the 24 ohm line to a 9.5 ohm resistance load. A circular tuning disc or washer was placed on the end of the center conductor to provide a series inductive reactance to cancel the capacitive susceptance of the slot. The circular disc which formed the inner boundary of the annular slot was fixed to the tuning washer and the coax center conductor by a special mounting screw., Since the neighborhood of the annular slot and tuning washer is a resonant cavity, all parts were silver-plated to minimize losses,. A family of discs and rings were fabricated to permit varying the annular slot radius from 1/2 inch to 7/8 inch while maintaining the slot width constant at 1/8 inch. The 1/8 inch slot width corresponds to .067 X and .076 X at k b equal to 3.4 and 3.8, respectively, o o o Recalling that the radius of the dielectric rod is one inch, it is convenient to express the slot radius in the normalized from a/b; then the source dimension k a for any k b is given by k b (a/b)„ Six o o o ' slots were constructed for the measurements. Table IV gives the normalized slot radius a/b, and the corresponding dimension k a at the two values of k b which were used for the measurements, o A slot 1/8 inch wide and two inches long was milled in the side wall of the exciter so that the standing wave ratio could be measured. After a particular slot was mounted on the exciter, various tuning washers were tested until one was found which reduced the VSWR in the exciter to less than two when the dielectric rod was terminated with a matched load. The image circle was determined from measurements 63 TABLE IV Normalized Slot o o Radius a/b when k b=3.4 when k b^3.8 o o 0.50 1.70 1.90 0,625 2.12 2.38 0.687 2.34 2.61 0.75 2.55 2.85 0.812 2.76 3.08 0.875 2.98 3.32 made in a coaxial slotted line which was connected directly to the exciter through a type N elbow connector. A view of the slotted line, coaxial exciter , and the underside of the ground plane is given in Fig. 14. A block diagram of the waveguide apparatus used for the measurements is given in Fig. 15. When Desehamps' procedure was carried out, the dielectric rod was terminated with a six inch diameter plate, machined from 1/16 inch brass and silver plated. The ground plane, which may be seen in Fig. 10, was 60 inches square. For most of the measurements the dielectric rod was 40 centimeters long, although in some instances, the length was increased to 134 centimeters. Finally, it should be mentioned that an external probe was constructed to measure the standing wave ratio of the field along the dielectric rod. It is shown schematically in Fig. 15. It was used to measure the guide wavelength, X „ when the rod was operated at k b equal to 3.4 and 3.8. In each case, the measured guide wavelength corresponded exactly to the value predicted in Table III. A matched load was made from four tapered lengths of 300 ohm resistance cord which were 64 66 mounted symmetrically on the rod. The matched load is visible in Fig 10 The VSWR on the dielectric rod was less than 1.1 when the rod was terminated with this load, 7 3 Experimental Results When efficiency was measured as shown in Fig. 11, the result obtained was not precisely the excitation efficiency of the annular slot; instead it was the efficiency of the entire transition between the measuring probe and the short circuit plate. Although the losses in the system were small, the measured efficiency was reduced somewhat by dielectric rod attenuation and by the loss between the measuring probe and the ground plane. We shall discuss these two effects separately. First, consider the dielectric rod attenuation which caused the two port network to be terminated in a lossy short circuit. The lossy rod acts as a reflection coefficient transformer, so that the magnitude of the reflection coefficient at the ground plane, IfL is given by h ? "''" (7.3) where a is the attenuation constant of the wave on the dielectric rod and z is the length of rod between the ground plane and the short ircuit plate. It is assumed, of course, that a perfect short terminates the rod yielding a reflection coefficient of unity at that position. We wish to determine the efficiency of the slot while excluding the effect of the lossy rod. Consider Fig. 16(a), which shows [s'] , the four Pole to be measured; it is located between the slotted line and the lossy short circuit. Figure 16(b) is an alternate representation of 67 Lossy Short 51. %* I- (a) Pev-fec.+ Shovl- (b) FIGURE 16. MEASUREMENT OF A JUNCTION WITH A LOSSY SHORT CIRCUIT Fig. 16(a) in which the reflection coefficient transformer I P|has been associated with [s» ] to form the fictitious four pole [s] which is terminated with a perfect short, when one measures the four pole, assuming a perfect short, the composite four pole [s] is obtained. Assuming that | P |is known, the parameters of [S» ] are given by ] 17 S » 11 11 S ' 22 _22 12 "12 Referring to Eq. (7.2), it is evident that the efficienty,^ f , of [S> ] is related to T of [S] by T« = T (7.4) (7.5) is the slot efficiency which would be measured if the dielectric rod were lossless, and Eq. (7.5) gives the relation between the measured 6!' p efficiency T» and T ' . In order to apply (7.5) to the experimental measurements,! 1 I was obtained by measuring the efficiency of the same source with two different lengths of dielectric rod. The distance from the ground plane to the short circuit plate was 40 centimeters and 134 centimeters for the two measurements. Referring to Fig. 17, let T and f represent this efficiencies which would be measured when the short was located at the specified positions. GjirOund Plane. Die-le+vic Rod — 7 ",40m -.94n Cireu i+ FIGURE 17, ARRANGEMENT FOR MEASURING ROD ATTENUATION From (7,3) and (7.5) it follows that £-•*» (7.6) ( o where z is equal to 0,94 meters,, Substituting the measured values of T and HP into (7.6), one obtains the value of the attenuation constant a. Having determined a, it follows from (7,5) that T» = Te 2aZ (7.7) where z is now 0,40 meters, '"P ' may be thought of as the measured efficiency referred to the ground plane position. Equation (7,7) defines the adjustment which is applied to the measured efficiency, T, to account for the dielectric rod attenuation. The value of a was measured when k b was equal to 3.4 and 3.8, The results are o - ,0692, k b = 3o4 ,0716 k b = 3.8 (7.8) o o Before (7.7) is applied to the experimental measurements, the attenuation between the measuring probe and the ground plane will be discussed. The attenuation of the transition between the measuring probe and the ground plane results from losses in the type N connector and the coaxial exciter. The loss is, of course, a function of the standing wave ratio in the transition. When a particular slot is being measured , the VSWR changes with the short circuit position because the junction is lossy. Therefore, the transition loss is different for 70 each position of the short, and it is impracticable to correct every standing wave measurement performed. One may, however, establish the maximum loss, or, correspondingly,, the minimum efficiency of the transition. This was accomplished by mounting a two inch diameter, circular waveguide perpendicular to the ground plane and concentric to the annular slot, and measuring the efficiency by Deschamps' method. The circular guide was fitted with a movable shorting plunger. The annular slot excites the TM __ mode in the guide. The schematic representation of the two port junction is shown in Fig. 18. I Detector j Qround Plane Circular Waveguide FIGURE 18. ARRANGEMENT FOR MEASURING THE TRANS I ST ION EFFICIENCY 71 Assuming that the circular guide was lossless, the measured efficiency, T, is the efficiency of the transition from the slotted line to the annular slot. It is now possible to adjust the measured slot efficiency to account for both the transition loss and the dielectric rod attenuation. Let T q denote the corrected slot efficiency; it is given by 2az T TL_ Tel T T~ TT (7.9) where Tis the efficiency measured with the lossy rod present, and T T is the transition efficiency. It should be noted that T provides a maximum correction to T since, in most cases, the transition efficiency will be greater than the value Xp- This results because standing wave ratios are higher when "^ is measured than when T is determined. T was measured when k Q b was equal to 3.4 and 3.8. The results are T 0.939, T kb=3,4 o = 0.934 (7.10) b = 3.8 Substituting (7.8) and (7.10) into (7.9) yields * 1.125 T, T k b = 3.4 o = 1.132T k b = 3.8 o (7.11) The measured efficiencies were corrected according to Eqs. (7.11). tote that the correction factor is approximate. It was obtained Ln order to fix the order of magnitude of the system losses. One lay see, however, that the excitation efficiency of a slot is somewhat 72 greater than the measured value. Six slots were measured in the laboratory; the results are presented in Table V, which includes the measured efficiency °f and the corrected efficiency f for each source k a„ Measurements were performed at frequencies of 6387 megacycles/sec and 7138 megacycles/sec which result in k b equal to 3.4 and 3.8, respectively. The data of Table V have been plotted in Figs. 19 and 20 for comparison with the theoretical curves of efficiency. The very close agreement between the experimental points and the theoretical curve is evident. In all cases the measured TABLE V k b k a Measured Corrected Efficiency Efficiency T Tc 3 4 1.70 0.44 0.495 2.12 0.65 0.73 2.34 0.76 0.855 2.55 0.835 0.94 2.76 0.77 0.867 2.98 0.65 0.732 3 8 1.90 0.48 0.544 2.38 0.775 0.88 2.61 0.85 0.96 2.85 0.77 872 3.08 0,63 0.714 f 3.32 0.44 0.498 efficiency, which included the system losses, was within 10 percent of the efficiency predicted by theory. The experimental measurements verify that an excitation efficiency of approximately 95 percent may be obtained from an annular slot of dimension k a = 2.6. o 73 ! ^X / < X \ X \ v. X X 1 X 5-° >> o K c >> •H >> o O rt \ 0) 5h O \ \ 01 Si Q) cS fc A 1 S X 1 M < «S u w C\l o & Q H «3 B (0 s rvi oa *) 8 CVi 1 < * j CVJ < KH % 6 I O * § CVJ H 55 °p N ^O IO i; (O /CouQi-oij-j-g CM - 74 o fO ' / > \ 8 ^ >> o S3 •H 0) «H -H o «H O W -H (1) -P d) Vi o ft 3 03 0) Sh u 1 Xi CD o \ 1 * o 1 c\> tf 7 <0 1") € 10 >i ^ o IP cv o fo E w O It* p 3 & (V o b) Denote the coefficient matrix of (A s l) by j~A. . "1 and let the cof actor A. . be designated X. .. The solution of (A.l) is of Det [ A iJ x ll X 2! X 31 X 41 X 12 X 22 X 32 X 42 *13 X 23 X 33 X 43 \4 X 24 X 34 X 44 - ia)€ b (A, 2) which reduces to HI •Mb Det [ A u] r~ -1 \l X 12 \3 \4 Evaluating Det ["A. .1 , we obtain DGt [ A ij] = *L1 (A 22 *23 A 44 + ^23 A»4 A 42> - *11 (A 32 ^3 A 44 + ^2 ^4 A 43 ) -\2 (A 21 ^3 A 44 " ^1 ^4 A 43 ) + ^3 (A 21 ^2 A 44 " *2l ^4 A 42 ) (A3) (A, 4) since A = - A L2 and Ag = - \ , (A. 4) reduces to DGt [ A ij] " (A 13 ^1 " *ll A 23 ) (A 32 A 44 " ^4 A 42>' (A. 5) We note that (A L3 *21 " *L1 A 23 ) = (V l b) [ J l (V i a) Y l (V i a) " Y l (V i a) J l (V l a) J = (V l b) ^aT (A. 6) since the bracketed term is the Wronskian of J (v a) , Y (v a) which is | equal to [2/7T (v a)]. Substituting (A ,6) into (A, 5) we obtain 2b Det [ A ij] ■ 15 (A 32 A 44 " *34 A 42> (A.7) The cof actor X is given by From (A, 3), (A. 7) and (A. 8) we have iJJ ff (A 32 A 44 " ^4 A 42> Consider 82 14 = " Aai (A 32 A 43 -^a^a>- < A - 8 > • * h ^ * A 21 ( A 32 A 43 - A 42 A 33 ) D = " 1Cj€ l b Bet FA. .1 = + laC l* 2 , A A T" — — ■ < A ' 9 > L ijJ = (A „ A_ - A_ A ) (A 32 A 43 " A 42 ^3 ) = (V l b) LVV* T l ( V> " \ V ] = (v i b) ^bT = l (A ' 10) because of Wronskian relation. Utilizing Bessel function recurrence relations we determine that (A 32 A 44 " ^4 A 42> = (V l b) J o (V l b) *i™ (V o b) " V\> b) J l (v l b)H o (1) < V o b) (A.1D Substituting (A. 10) and (A, 11) into (A, 9) yields + ±cj€ a J (v a) >- — i_L_i (A. 12) (v ib ) J o ( Vl b) H x ( >(v b) - € r (v o b) J l(Vl b) H/ (v o b) 83 APPENDIX B The graphical solution of Eqs . (4„9) and (4.10) is illustrated in Fig. 4 of Chapter 4. Equations (4.9) and (4.10) are J Q (X) K Q (£) T[ao ' £ r (e> K]T¥) = (4.9) r 2 = (k n b) 2 (s - i) = r 2 u r (4 JO) where we omit the subscript 1 of X for convenience of notation. Referring to Fig, 4, we see that the intersection of the circle and the first branch of X = f(£) is the unique solution of (4.9) and (4.10) Consider the magnified view at the point of intersection of the curves, as shown in Fig. 21. FIGURE 21. SOLUTION OF THE MODE EQUATION 84 Denote points on the circle corresponding to values of % by X, where X = sj R 2 - i 2 from (4.10). The points X, which define the curve with positive slope, are solutions of (4.9), which may be written in the form J (X) - X j7W-V = ° (B.l) k (6) where v = £ (£) _. /gv is a constant for any selected £ and value of %. » r K (5) r We see that the points X are simply real-roots of the equation J (X) - X ( v - y = 0. Let the point of intersection, which is the exact solution of (4.9) and (4.10), have the coordinates % and X . The basic s s method for determining the point of intersection using the digital computer is illustrated in Fig. 21. It is evident that for any value £ less than § , X' > X, and when I is greater than I , x' 4 X x in 10 4 X 2 in 12 16 17 18 19 20 21 22 23 24 25 26 27 28 JO (2) 50 16L 26 115 F JO (2) J4 4 F 50 18 L 26 59 F 92 975 F L5 (3) LO (5) 40 (6) 50 523 F 7J 523 F 40 (7) 50 (6) 7J (7) 40 (7) L5 (1) 10 2 F 40 (6) 50 (6) 7J (6) 40 (6) L5 (7) LO (6) waste enter HI waste print X^/4 4 spaces 4 <£,-!) in (6) (kb/8) 2 in (7) (^[(kb)^ (£-!)] m (7) 16 in (6) 1 n2 (— >* [(kb) 2 (^-1) - e. 2 ] in A 90 50 28 L 29 26 149 F 50 (2) 30 00 2 F JO (2) 31 J4 4 F 50 31 L 32 26 59 F L5 (1) 33 LO (9) <= 36 39 L 34 L5 (1) L4 (10) 35 40 (1) 50 11 L 36 K5 F 42 11 L 37 50 12 L K5 F 38 42 12 L enter Rl , leave r-r X . ' in A 16 l clear Q form i X, ' 4 i waste print — X. ' i. | + .088 test: (^ - -^ ) advance £. i. + .01 l in (1) advance (500) address advance (510) address 22 7 L transfer control to 7 L 39 L5 42 (11) 11 L reset (500) 91 40 L5 (12) 42 12 L reset (510) 41 59 LO (2) 523 F — in A, clear Q test: . _ 4.0 - kb 8 42 < 36 L OF F 43 (1) 00 F 00 F 44 (2) 00 F 00 F 45 (3) 40 F 00 140 000 000 000 J 46 (4) 00 F 00 F 47 (5) 00 F 00 250 000 000 000 J 48 (6) 00 F 00 F 49 (7) 00 F 00 F 50 (3) 00 F 00 22 000 000 000 J 51 (9) 00 F 00 F 52 GO) 00 F 00 2 500 000 000 J 92 (11) 00 F 00 500 F (12) 00 F 00 510 F Auxiliary Subroutine 40 K5 18 F F rescue X. /4 42 8 L plant link L5 18 F 50 2 F enter V8 , leave J Q (X) • 5 50 2 L -19 and J (X) -2 in7. 26 158 F L5 18 F 10 2 F form X/16, store in 19. 40 19 F 50 6 F 79 19 F 50 (2) 66 7 F 1 J o (x) 1 f ° rm ' 16 X J]L (X) " 16 T S5 F L0 (4) JO (2) waste 22 F •19 93 TABLE B.l 275 1375 .6711 .6730 1400 .6725 .6725 - .0001, interpolating to obtain — £ (£) ( r for each £, until inter- section was detected by the method shown in Fig. 21. When intersection was detected, the computer printed out %, X, and X' on each side of crossover in the following form: k b/8 o K hi 4 i+1 K I x 4 *i+l ix'. 4 l — Y> 4 A i+1 Table B.2 presents the numerical results for each of the six values of k b. Table I was constructed from these data, o TABLE B.2 14005 .67251 .67252 14007 .67252 .67252 325 30820 .75107 .75107 30822 .75108 .75106 375 48380 .80214 .80215 48382 .80214 .80213 425 65672 .83415 .83415 65675 .83415 .83413 475 82260 .85510 .85512 82262 .85510 .85510 525 98157 .86969 .86972 98160 .86969 .86969 97 The procedure for calculating X and X" was similar to that used in the first program. The only essential difference was the addition of routine II to perform the interpolation. The tape format and order codes written for the main routine and auxiliary subroutine are as follows: Tape Format Routine N12 Routine PI 6 Routine Rl prerset parameter for HI Routine HI pre-set parameter for II Routine II Routine V8 Main Routine Auxiliary Subroutine 1 Stop - Transfer Control to Main Routine 98 Main Routine 50 21 F 50 L 26 200 F JO (1) 50 54 F 50 2 L 26 200 F 26 75 L -3-50 190 F 50 4 L 26 200 F JO (1) 50 40 F 50 6 L 26 200 F JO (1) 52 45 F 50 8 L 26 200 F read in 12 fractions, locations 21 through 32 waste read 12 fractions into 54 through 65 read 6 fractions into 190 through 195 waste read parameter constants into 40 through 44 waste -39 read integer a * 2 into 45 gg 10 11 12 13 14 15 16 17 18 19 20 21 L5 42 92 92 92 JO L5 J4 50 26 L5 40 L5 50 50 26 00 00 00 40 L5 40 10 F L5 42 F 40 12 F 45 F 17 L 707 F 131 F 515 F (1) 43 F 3 F 13 L 239 F 40 F (2) (2) 4 F 16 L 338 F F F 10 000 000 000 J (3) 41 F insert a into II entry number shift carriage return and line feed delay waste enter PI 6, print kb/8 puts initial £ into (2) current £ in A enter II, interpolates to leave Ti Y inA store interpolate in (3) X /4 in 10 X 2 /4 in 12 100 22 23 24 25 26 27 28 29 30 31 32 33 34 50 21 L 26 304 F 40 (4) L5 (2) 10 2 F L4 44 F 40 (5) 50 43 F 7J 43 F 40 (6) 50 (6) 7J (7) 40 (6) L5 (5) 10 2 F 40 (8) 50 (8) 7J (8) 40 (8) 41 F L5 (6) L0 (8) 50 32 L 26 295 F 50 (1) 00 2 F 40 (9) enter HI, leave root X. /4 in A l store X. /4 in (4) forms £ . /4 from t l t store £;/4 in (5) (kb/8) in (6) (— ) 2 [(kb) 2 (£ f l)] in (6) §./16 in (8) (i/16) 2 in (8) clear A and location ( T6" )2 t< kb > 2 <£f D - ^ 2 ] in A enter Rl , leave — X. J in A 16 i clear Q X^/4 in (9) 101 35 36 37 38 39 40 41 42 43 44 45 46 47 48 L5 (4) L0 (9) -32 41 L L5 (5) 40 (10) L5 (4) 40 (ID L5 (9) 40 (12) L5 (2) L4 (13) 40 (2) 22 15 L 92 131 F 92 515 F L5 (10) J4 5 F 50 43 L 26 239 F 92 975 F JO (1) L5 (ID J4 5 F 50 46 L 26 239 F 92 975 F JO (1) test: X./4 - X. /4 i i transfer to print § /4 in (10) X./4 in (11) X. /4 in (12) advance 6 by .0001 carriage return, line feed, delay print £ . /4 four spaces waste print X./4 four spaces waste 102 49 50 51 52 53 54 55 56 57 58 60 61 L5 (12) J4 5 F 50 49 L 26 239 F 92 131 F 92 515 F L5 (5) J4 5 F 50 52 L 26 239 F 92 975 F JO (1) L5 (4) J4 5 F 50 55 L 26 239 F 92 975 F JO (1) L5 (9) J4 5 F 50 58 L 26 239 F 59 (1) LO 43 F 36 6 L OF F 00 F print X /4 carriage return and line feed delay print i ±+1 /4 waste print X. , /4 l+l print X. + 1 /4 in A, clear Q test: 4.0 - kb 8 transfer control to read in new parameters 103 62 (1) 00 F 00 F 63 (2) 00 F 00 F 64 (3) 00 F 00 F 65 (4) 00 F 00 F 66 (5) 00 F 00 F 67 (6) 00 F 00 F 68 (7) 00 F 00 390 000 000 000 J 69 (8) 00 F 00 F 70 (9) 00 F 00 F 71 (10) 00 F 00 F 72 (11) 00 F 00 F 73 (12) 00 F 00 F 74 (13) 00 F 00 100 000 000 J 75 50 91 F 50 75 L 76 26 200 F -*— 26 4 L read 6 fractions into 91 through 96 40 18 F K5 F 42 8 L L5 18 F 50 2 F 50 2 L 26 389 F Auxilliary Subroutine rescue X. /4 plant link X/4 in A enter V8< J (X) ■J X (X) 2~ 19 in 6 2- 19 in 7 104 L5 18 F 10 2 F 40 19 F 50 6 F 79 19 F 50 (1) 66 7 F S5 F L0 (3) JO (1) -< 22 F form XA6, store in 19 F TS V x > ' 2 " 19 in A clear Q X J o (x) 16 ^(Xj.) in A X J program tape listed the function values, — £ (I) T - , g x , required by lb r K \b) the interpolation routine. It also included constants telling the computer where the table was stored, the initial value of b, X /4, X /4, 1 2i and k b/8. The parameter tape as it was read into ILLIAC is as follows: o +14217 205 +14363 079 +14509 095 +14655 256 +14801 554 +14947 990 +45745 8333 +45902 4109 +46059 0048 +46215 6141 +46372 2411 +46528 8834N +05055 518 +05089 839 +05216 642 +05343 995 +05471 883 +05600 291 105 +35300 6528 +35455 7627 +35610 9008 +35766 0675 +35921 2618 +36076 4848N +24685 093 +24837 399 +24989 766 +25142 182 +25294 660 +25447 191N +55638 3502 +55795 7729 +55953 2083 +56110 6549 +56268 1103 +56425 5788N + 555+ 66+ 68+275+ HfN + 23+74+76+ 325+25N+N +93+79+81+ 375+ 25N+N + 62+82+84+ 425+5N+N + 285+84+86+475+75N+N +92+86+88+525+75N+100N 106 APPENDIX C The integral (6.6) which was needed in order to compute the excitation efficiency is 7T/2 cos 9 F (9) d9 (6.6) where F(9) = J, (k a w) 1 o w J (k bw) H. (1) (k b cos 9) - € cos 9 J (k bw) H (1) (k b cos 9) ool o r looo and w = € - sin 2 r The program which was written to determine N utilized ILLIAC library routine E2, which computes an approximation to the integra] b 7— f (x) dx by Simpson' s rule. This routine requires that the user supply tabulated values of the integrand at an even number of equally spaced intervals. The range of integration, which was zero to 88 degrees, was subdivided into 44 intervals of two degrees each. For any k a and k b, the integrand was computed at the 45 points o o 0, 2°, 4°, 6°, ... , 86°, 88°. Since the integrand contains F(9) , which involves a division, it was necessary to scale the numerator whenever it exceeded the denominator in order to maintain | F(9) | < 1; otherwise the capacity of the ILLIAC registers would be exceeded. This 107 was done by multiplying the numerator by 1/2 as often as needed until all 45 values of F(0) were within range. Consequently, the computed integral was scaled by an integer number of multiplications by the factor 1/2. The scaled integral, I, which was computed was equal to 88 £_ n 90 2 p+l I = |8^ J <2> C ° S 9 I F(9) I 90 2 where p is the integer scale factor. R It follows from (6.6) and (Col) that N is obtained from I by For a particular choice of k b, the integral (CI) was calculated o for k a equal to 0.2, 0.4, 0.6, ... until k a took on the value k b; o o o then k b was changed to a new value, and the process was repeated. The o scaled quantities k b/8 and k a/8 were used in the computations. The o o results were printed in four columns which listed, respectively, k b/8, k a/8, p, and I. Relation (C.2) was then used to compute the o o (C2) N R corresponding value of N . Table CI presents the results as printed out by ILLIAC for k b/8 equal to 0.425, that is, k b equal to 3.4. o o The program which was written to compute the integral (C.l) was comprised of 143 order pairs. 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