&-+?! OPY ANTENNA LABORATORY Technical Report No. 28 PHASE VELOCITIES IN RECTANGULAR WAVEGUIDE PARTIALLY FILLED WITH DIELECTRIC . by WALTER L. WEEKS 20 December 1957 Contract No. AF33(616)-3220 Project No. 6(7-4600) Task 40572 WRIGHT AIR DEVELOPMENT CENTER ELECTRICAL ENGINEERING RESEARCH LABORATORY ENGINEERING EXPERIMENT STATION UNIVERSITY OF ILLINOIS URBANA, ILLINOIS ANTENNA LABORATORY Technical Report No. 28 PHASE VELOCITIES IN RECTANGULAR WAVEGUIDE PARTIALLY FILLED WITH DIELECTRIC by Walter L. Weeks 20 December 1957 Contract AF33 (616) -3220 Project No. 6(7-4600) Task 40572 WRIGHT AIR DEVELOPMENT CENTER Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois Digitized by the Internet Archive in 2013 http://archive.org/details/phasevelocitiesi28week ii CONTENTS Page Abstract iii List of Symbols iv 1. Introduction 1 2. Field Theory 3 3 7 63 63 64 2.1 PM Modes (H = 0) 2.2 PE Modes (E y m y 0) 3. Computation 4. Note Added in Proof 5, Acknowl edgement References Distribution List iii ABSTRACT A compilation is made of the phase velocity of the dominant mode of propagation in a rectangular waveguide which is partially loaded with a dielectric, in the case that the dielectric interface is parallel to the broad wall of the waveguide. Curves are presented which show the phase velocity ratio (c/v) as a function of the wavelength to guide width ratio (— ) for seven values of dielectric constant in the range from 1.6 to 13.7; cut-off information on three other important modes is also given. The roots of the transcendental equation for the propagation constants are also tabulated. Enough of the field theory is presented to allow intelligent application of the data. iv LIST OF SYMBOLS E electric field intensity H magnetic field intensity a waveguide width b waveguide height c speed of light in free space d thickness of dielectric material j imaginary unit k propagation constant in free space (2ir/X) k propagation constant in x direction in waveguide k propagation constant in y direction in waveguide k propagation constant in z direction in waveguide phase velocity in waveguide y } unit vectors in coordinate directions € permittivity of dielectric X free space wavelength (j. permeability of dielectric to radian frequency (radians/sec) 1 . INTRODUCTION There is considerable current interest in the production of guided electromagnetic waves having phase velocities equal to or less than the speed of light in free space (for example, in the design of traveling wave antennas and of devices involving electron- traveling -wave inter- actions)) . A convenient way to obtain such phase velocities is to partially load a rectangular waveguide with a dielectric material. In antenna work particularly, because of the field configurations, it is usually desirable to place the dielectric interface parallel to the broad wall of the waveguide, as indicated in Fig. 1. This problem has FIGURE 1. PARTIALLY DIELECTRIC LOADED WAVEGUIDE SHOWING COORDINATE SYSTEM AND DIMENSION DESIGNATIONS 1-3 been considered by others, and there is published information on some 3 of the cut-off frequencies, but (since in this case there is no convenient relationship between the cut-off frequencies and the propagation constants) there has been little detailed information available on the phase velocities as a function of waveguide geometry and dielectric material. This report presents such information. Main consideration is given to the dominant (hybrid) mode under the assumption of zero energy dissipation, but cut-off data for the next three higher order modes are included. Results are presented for most of the common solid dielectric materials which have a small loss tangent. The University of Illinois digital computer (ILLIAC) was employed, first to solve the transcendental equations, and then to calculate the c/v ratios. The results were spot checked with desk calcu- lators. 2. FIELD THEORY The elimination of one or the other of the field variables from the Maxwell equations with harmonic time variation results in the equation yxvxB- w 2 |J.eB= (1) where EJ is either E or H. The mathematical problem is thus to find solutions of the vector Helmholtz equation (1) which satisfy the boundary conditions and Maxwell's equations. It is convenient to represent the fields in terms 4 of a pair of scalars. The customary representation, in which the scalars are related to the longitudinal components of the fields, with the consequent representation in terms of TM and TE modes, is not particularly appropriate since ordinarily the simplest field configuration which can propagate in a waveguide as shown in Fig. 1 is neither TM nor TE. Consequently, we will employ the alternative representation in which the modes are separated into those for which there is no y-component of magnetic field (H = 0) and those for which there is no y-component of electric field (E =0). For convenience, we will designate these modes as PM (for parallel magnetic, i.e., H is parallel to the dielectric interface) and PE (E is parallel to the dielectric interface) . 2.1 PM Modes (H = 0) We first look for solutions such that H = Vx f $ (2) (carat symbol designates unit vector) . In this case we find from Eqs (1) and (2) the equation V X(V X V X fy - w U. € fy) = (3) from which it follows that V x- Vxfy - w 2 |i€fy = vu, (4) where U is an arbitrary scalar. If we take U = $f/9y we find that f must satisfy the equation V 2 f + w 2 jjl€ f = 0. (5) We are looking for the fields which propagate in the z direction, hence we let k z f = F(x,y)e Z (6) and find the equation d 2 F a 2 F 2 2 ox dy To proceed, we divide the waveguide into two regions— in each of which the values of € is constant. The solution consists of those functions F and F which satisfy (7) and the following boundary conditions. 1 2 A d± ■X s • H = ( = - -£-) at x = and x = a (8) ^ ' Sj_ - & • I 2 at y = d (9) ^■H 1 =^.H 2 aty = d & • H = ||) (10) o * ■ £ = ° <= j^e ^5^) at y = and y = b (11) £ * E. = £ • E at y = d (12) ™1 ^ 2 ^ . E = (= — ( — | f co 2 |jL€f)) at x = and x = a (13) jC0£ d Y z a 2 * ' S = ° (= ~m W _) at x = and x = a J y and at y = and y = b (14) z ° EjS^- E 2 at y = d. (15) It is clear from (9), for example, that the propagation constant k z must be the same in both regions. Assuming a product solution of (7) gives solutions of the type F = sin (k X +(f) ) cosh (k y + 0) (16) and applying the conditions (8), (9), and (14) gives the functions for the regions as follows: F, = A, sin — cosh k y (17) 11a y 1 F = A sin -^ cosh k (y-b) . (18) * * a y 2 Then to satisfy (10) we must have the equation A, cosh k d = A„ cosh k (d-b) (19) 1 y l 2 y 2 and to satisfy (15) we must have — A, k sinh k d * ~ A„k sinh k (d-b). (20) € 1 1 y l y l € 2 2y 2 y 2 These latter two equations combine to give the transcendental equation k k y y 2-i tanh k d = ^- 2 - tanh k (d-b) (21) 1 y l 2 y 2 which can be used to solve for the propagation constants since by (7) (22) - k 2 + k 2 + (co 2 n € + k ) * x y r z so that 2 2 _ , 2 2 _ ,„ „ k + w ll. € = k + co w € (23) y H. 1 y ^2 2 If we put |i,, = |x and let medium 2 be vacuum, then 2 2 27T 2 \ - X + ( X> <*r " » < 24 > where € is the relative permittivity of medium 1. \ The phase velocity ratio — = — — can thus be found by solving (21) subject to (24) and using (22) to find v V r 2a A careful examination discloses that equation (21) has no solutions if k is pure real, but there is no such limitation on k ; for the dominant y i y 2 mode (a>b), k is real . y 2 From Eq„ (2), with (6), (17) and (18), we find the expressions for the magnetic field components k z H = - Ak sin — cosh k y e Z (26) \ z a y 1 " y k z . nfl" n7Tx , , z ,„„. H ■= A — cos cosh k y e (27) z x a a y x with similar expressions for region two, except with the cosh argument changed to k (y-b) . y 2 From the Maxwell equation E = ^— (VX H), we find that the electric field components are as follows: ^ A nU , nffx . , , z ,„„ v E = ,-?-«: — k cos sinh k y e (28) *! J"^ a Yi a y^ _\ r.n7T.2 2~] n — cosh k y e Z (29) y-, o w € , I a z I a y x A -, « k z 1 n ^x , . , z ,„„. E = . , ■ _ ■ ■ k k sin sinh k ye (30) Z l J 1 y l Z a y l with an obvious change of subscripts and argument to characterize the fields in region 2, 2.2 PE Modes (E = 0) To delineate the PE modes, we proceed in a fashion analogous to that above under PM modes. Thus we look for solutions (E = 0) for which y E=VX gy\ (31) As before / we look for solutions of the type for which k z g = G(x,y) e Z (32) and we find the equation ^§ + ^§ + (coV * k 2 ) G = 0. (33) 8x 2 8y 2 Z The boundary conditions (8) to (14), (except for the parentheses) apply with the additional condition, a 2 1 - g 2 y • H = ( = - 1 — (— ■ - + w u£ g) at y = and y = b. (34) J W M- a y 2 The conditions on g and G can be found by interchanging the roles of E and H, (In particular — dz ' ox <> . w - 1 9 g O . w _ x JX\ - jqj. ox 9y' - jcojj. 9z 9y Application of these boundary conditions gives the result _ nJTx . , G, = B, cos sinh k y i i a y t G„ = B„ cos sinh k (y-b) 2 2 a y 2 and the transcendental equation |a- tanh k d |i tanh, k (d-b) 1 y l 2 y 2 k k' y i y 2 The field components are found in straightforward fashion to be k z „ , n7Tx . . , z E = - B, k cos sinh k ye x x 1 z a y x k z „ „ nil nTTx . , , z E = - B, — sxn sinh k y e z ± 1 a a y;L B , k z 1 nTT J nTTx , , z H = -rfr sin cosh, k y e x 1 jo^x y-. 1 , ,n7J\2 , 2. nflx . . , z H = -: ( ( — ) - k ) cos sinh k y e y l J °^l a z a ^i -B, «■ k z • In, n^x v. , z k k cos cosh k y e z 1 ja^j, z ^ a y. Note that n = gives a permissible PE solution, and these modes are also TE. 3. COMPUTATION The computational difficulty lies in the fact that the quantities k and k which appear in the equations of the foregoing section depend ^1 ^2 upon the parameters 6 , b, and d in such a way that a change in any one of these requires a new numerical solution to the transcendental equation (21). Thus, any extensive tabulation of results make the use of a high speed computing machine almost imperative. The availability of the ILLIAC made the present compilation feasible. In selecting the parameters for calculation, an effort was made to obtain values for those situations most likely to be of interest. Thus, data are available for seven values of dielectric constant in the range from 1.6 to 13.7, with fillings (d/b) ranging from 10 to 90 percent, and waveguide aspect ratios varying from 0.1 to 1.0. Selections from these data are presented here in graphical form. The results from the digital computer were printed out with eight significant figures; however, for most applications such accuracy is neither warranted nor realistic. The program was set up to instruct the computer to calculate c/v ratios in small steps of \/2a until it reached a value for which (c/v) was negative. Consequently, the cut-off dimension lies between the last calculated point and the next higher regular step. Table 1 is a facsimile of the form of the data as obtained from the computer. It is convenient to distinguish the modes of propagation by a pair of numerical subscripts. The first subscript will specify the root of the transcendental equation which appears in the solution (roots numbered from the smallest) p while the second subscript specifies the integer n in the argument of the trigonometric functions (nif/a). Thus, the mode PM. is the dominant mode if a>b. The c/v ratios for this mode are plotted in detail in the following figures. The cut-off dimensions for the PE mode and the PM mode were calculated by the ILLIAC and are plotted on the curves for values of d/b = .2, .4, .5, .6, .8. The symbols are as follows: a circle (O) for the PE mode, and a triangle (A) for the PM mode. The cut-off dimension for the PM mode was obtained (less accurately) by extrapolation. 1» This information also appears in the graphs, with the square symbol (D) 10 designating the PM mode,, The cut-off dimensions for the modes analogous to these latter three modes in the limiting case of d/b equal to zero or one are as follows: X V^r"" at cut-off PE ( O ) -£- = - V*T' at cut-off 10 2a a v r b / f -r a Vl + (b/a) : > 2^iii/7 ^.2 Figures 2 through 43 contain the phase velocity information plotted so as to be immediately useful in the design problem in which the object is to prescribe the dimensions in a waveguide for single mode propagation when the phase velocity at a given frequency is specified. An indication of other ways of presenting the same information to increase its utility is given in Figs. 44 through 48. For example, Fig. 44 shows the variation of phase velocity with dielectric constant for particular waveguide geometries. To facilitate interpolation between the d/b and b/a ratios used in the calculation, the data can be presented as in Figs, 45 through 48. Tables 2 through 8 give the roots of the transcendental equation (21) for different dielectric constants and waveguide geometries. The results can be used directly in Eq.(25) to make specific calculations for particular designs. 11 FIGURE 2. PHASE VELOCITY RATIO OF TiiE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. f- £y -.\.b ex. r.25 i A o .2. >247 .254- .4 .266 .274- .5 ,278 .287 .6 .2.89 .299 .8 .302 ,315 .3 ■ s ^ V ^ 53 V \ V .2. t ix \ \ s \ 1 VN \ V. \ ^ s\ ^ \ S \\ \ \ \ \ \ V \ \\ \ \ \ \ \ V \ \ \ \ V \ \ \ 3 \\ \ \ r \ A\ \ \ t \ H I \ \ A \ \ A \ \ 1.0 V.2. 12 FIGURE 3. PHASE VELOCITY RATIO OF THE PM„ FIGURE 1. BEE WAVEGUIDE AS SHOWN IN 1H ^r = 1.6 b /a = .4- •s cut-off A /2 _^ .1 .360 ,4-06 .4 .408 .436 .5 .425 .4-53 .6 .440 .478 .8 .4feO .501 .5 N^ fi V V .4 \ N .2 N t\ N> V s \ N \> \ \ \ Oi \\ V \ \ \ \ \ \ \ s \ \ \ A \ \ \ \ V \\ \ \ \ V- \ \ \ V A \ \ a \\ \ r \ A \ \ t \ \ \ \ 4 \ i i \ .5 .8 • 3 j±_ 1.0 2a M 13 FIGURE 4. PHASE VELOCITY RATIO OF THE PM U FIGURE 1. MODE IN THE WAVEGUIDE AS SHOWN IN 1.2 i-> £» -=1.6 b/ .r.4S .8 oof-o-ff \ a d /b A .2 .421 .4- .451 • 5 .469 .6 -484 .$ .507 .£ \ ^ •5, ^ V V s .40 S ^ X \ \ q .2' V N ^ ^ ^ "s \\ \N \ V C \ \ \r \ \ v •7 \ v \ \ \ \ \ .6 \ A \ \ \ V \ \ \ \ > \ V \ \ \ V \\ \ \ J \\ \ • 3 r \\ \ \ t \ H \ \ .2 3 \ ,\ 1 i \ .3 .8 .3 \ 10 1.1 2-a. 14 FIGURE 5. PHASE VELOCITY RATIO OF THE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. 1.2 1.0 .8 V .4 £=< "-^^ £ „=■ \.G >V„ = • S .3 co1-of{ Via d /b A .2. .4-5 9 .4- .432 .5 .5IO .€. .5 27 • 8 .55! ■& Ik $ N X V .4 ^ s/^ \ \ .u *< ^N N k v \ \\ \ v \ \\ \ \ \ V 5 \ \ \ V \ A \ \ i \ ' \ \ \ V v A \ \ 3 \ \\ I \ r \ \ \ \ x \ \ \ \ A \ \ n .5 ,€> .7 2a i.o 1. 2. 15 FIGURE 6. PHASE VELOCITY RATIO OF THE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. 1.2 LO .9 • 7 .5 .2 *•' £.y. :|.fe .8 X, b/ a =.T5 .6 .5 sN .4 x: sN V s ^ ^ N .2 y V X N N tN \ \ V \ \ \ v_ \ \ \ \ \ \ \ \ \ \ A \ \ \ \ ( \ \ \ \ \ \ \\ \ \ \ ~^ V \ \ \ r^ \ \ \ \ j^ 1 1 \ \ \ \ \ \ ,3 _^ 1.0 2a 16 FIGURE 7, PHASE VELOCITY RATIO OF THE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. 17 FIGURE 8, PHASE VELOCITY RATIO OF THE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. 8r~- 2. 1 b / A -- 25 cy1-off A /2- .358 .344- d 4r ,9 "N • 7\ ^ N?" S: N < t> ^ xi >s \] ^ Sn o\; \ A \ \ V \ \ A \ti [ \ \ \ Y r \ \ \ \ r ■] 1 i i .4 .S ^ I.0 1. 2 1. 4 18 FIGURE 9 1.6 PHASE VELOCITY RATIO OF THE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. *»• ,3^ ^^^ * N ^ ^ t r -- 2.\ ■ A •4v x"N & ^ v \ \ b /a= " 2 >fc ^ > N^ ^ S: s^ 0\ ^s 5x sN <^ \ A ^ k\ V \ \ \ \ \v ^\\ V \ V \\ \ \ V \ \ .4- s x i.o i. a 1.4 19 FIGURE 10. PHASE VELOCITY RATIO OF THE PM n MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. 20 FIGURE 11. PHASE VELOCITY RATIO OF THE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. 21 FIGURE 12, PHASE VELOCITY RATIO OF THE PM^ MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. 22 FIGURE 13. PHASE VELOCITY RATIO OF THE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1, 23 FIGURE 14.. PHASE VELOCITY RATIO OF THE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. d/b=l. .* 2.34 A Z'zA 52S 51(* • 9l ^ o .2 .2: \ \ \ \ V \\\ \ \ \ \ ft \ \ \ \ \ \ \ \\ \ \ 1 1 r V 1 I 2.CJ 25 FIGURE 16. PHASE VELOCITY RATIO OF THE P1VL MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE J. . 26 FIGURE 17 PHASE VELOCITY RATIO OF THE PM^ MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. 27 FIGURE 18. PHASE VELOCITY RATIO OF THE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1„ 1.6 d/b= l '^.54 1 ^ >s s---^ 1-4 .s 1 .4 sN ^ ^ \ N .3 <\ ^ \ \ .2 c/v .1 V O ^ \ .& \ \ \ \\ \ \ \ \\ \ \\ \\ \ \ y \ \\ I \\ \ \ \ \ \ \ \ o ] I.O 1-0. i.^i- !•£> 284 .6 .4S6 .4-14- .8 .47^ .45^> b /a = ■ 2* *-' .& k \ ^ .6, ■S\ A- \ s \ •Z. s >l \ \ o \ V \ \ v \ \ \ \ >L 1 \ \ r \ I \ i .4. .6 1.0 . 1.1 1.4 1-6 1.8 it 30 FIGURE 21 PHASE VELOCITY RATIO OF THE PM, , MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1 11 iH 1.8 .8 I • fe, £ r = 3.78 1.6 .£< N X ^,=.4 .4, 1.4 ^ \ \ \ 1.2 c. \ \ \ \ V •* jv N \ \ V 1.0 \ C \ \ <- ^ V \ \ •8 \ \ \ \ \ • 6 \ \ \ •4- \ A \ ■ 2 \ \ \ 1 .4 .& •8 1.0 1.1. 1.4 1.6 1-6 h. 31 FIGURE 22 PHASE VELOCITY RATIO OF THE PM, , MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1 11 f-l .6 N .6 .5, £_ r = 3,78 Ay .2 \ \ \ ^ \ \ \ \ \ \ \\ \ \ 1 1 .4 1.0 l-l 2. 1-8 .8 S^ *"-1 ■6 N s= a. y -^»-re> 1.6 .4i N N N \ b /«-~ • 5 1.4- 1.2 • 2 ( 1.0 V \ • 8 \ v .& \ \ ,4 V \ \ .2 \ \ ' 1 \ \ I O ■ 1 1 .4 1.0 1-2 Jo. 1.4- i.fe i-e 33 FIGURE 24 PHASE VELOCITY RATIO OF THE PM n MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. 2.0 1-8 1.6 1.4- 1.2 1-0 • 8 *•' •G •5 ^? ^ .41 ^ ^ ^ s £,.iS.7S •1 ^ \ b / a -.75 .2, \ s s; 1 V \ \ \ \ V \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ J \ I \ 1 1.0 t. 2. X 2a. t-4- I.G 1.6 34 FIGURE 25 PHASE VELOCITY RATIO OF THE PM n MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. 2.0 1.8 l.fe 1.4- 1.2 1.0 *2 ^g; >.. ^*** .4 ^ i, >>, d^.= 2>.7S ^ = i .2, \ \ \ \ ^ \ \ \ \ o \ \ \ \ \ \ v L_ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i \ \ \ \ ■9— e ) -0- ^-L .6 .8 r.O 1-2. 1.4- l.g 1.8 2-0 A 35 FIGURE 26 PHASE VELOCITY RATIO OF THE PMn MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. & t r ~- 5. IS <:uT- off A / 2a d /b o A .2 .290 ,2B5 ,4. .424- .408 .5 .484- -458 .<& .5"bl .497 v .S .588 .548 -8 ( .6 .5, V \ \ \ < •4 ( \ \ •2, V \ \ \ \ \ \ V \ \ \ \ \ 1, \ \ [ \ \ \ ) \ \ \ \ \ \ 1.0 l.i - 1.4 2a ~* 16 2.0 36 FIGURE 27 PHASE VELOCITY RATIO OF THE PM n MODE IN THE 2.4 WAVEGUIDE AS SHOWN IN FIGURE-- 1. b ' 2.1 .8 s N 2.0 i .6 i \ !^4 75. 1.8 .5. v /c \ l.G .4 > \ 1.4 \ \ \ s. V 1.2 \ \ 1.0 .2< •8 ,6 \ \ \ .4 \ \ \ i 1 \ i Ui A. ( ' 4 2a i-6 2.0 37 WAVEGUIDE AS SHOWN IN FIGURE 1. 2.4 2.2 2.0 1-8 1.6 1.4 i-l i-o .8 #-> ,e< ,&• £. •■5, \ \ V h r - -3 i ^> \ \ V -y .5' .4, \ N \ .2 v \ \ \ \ c \ \ X \ \ \ \ \ \ \ \ \ k \ V ^ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k \ \ \ \ \ \ \ \ \ \ \ 1.2. ^ U4 1.6 1.8 2.0 40 FIGURE 31. PHASE VELOCITY RATIO OF THE PM MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. J..*k b i .4. ^ ^ \ ^ $ ix s \ <^ $ s ^ \ rv 6- / 1 ^ .2 \ Q > 'il- > \j i-S \ \ \ \ \ \ D \ \ 1. 4- \ \ \ \ \t \ \ \ 1.2. \ \ \ \ t. \ \ 8 \ \ \ \ \ \ \ i \ v \ \ \ .4- \ ] \ | \ .2 \ \ \ — e ) ■e— 1.0 1-2. 1.4- * I.& 1.6 2..0 2a .s) ^ 6s -- \o ^ ^ V <=3 & \ \ X N X \ \ \ *? — o- \ X X 1 i \ \ 1 ' l,o \,A IS Z.(=> 46 FIGURE 37 PHASE VELOCITY RATIO OF THE PM n MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. cH WAVEGUIDE AS SHOWN IN FIGURE 1. 47 fl £ r = lZ>. 7 -8< % = .15 .6 i .3, A ■N \ w A V l) .4* [\ \ A \ \ \ \ V \ \ v \ .2, \ \ \ \ \ \ v V \ \ \ 1 1.0 J.4. .8 ^ 2.2 2.6 3.0 3.4 48 FIGURE 39 PHASE VELOCITY RATIO OF THE PM X1 MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1„ 40 3.6 f' .8 •6, £r - I2>.7 3.2 .5 h /a - A .4 2.8 \ 24 c \ 2.0 c v \ i \ \ 1.6 t \ •^ ^ [ \ y \ 1-2 \ \ \ t \ \ \ t •8 6 \ \ \ ) \ v i .4- \ \ 1.0 1.4 ,.8 ^_ 2.2 2-a 2.6 3.0 3.4- 49 FIGURE 40 PHASE VELOCITY RATIO OF THE PU^ MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. *' .8 •6 £ r = 13,. 7 •5 •4i b /a = .4-5 \ .2, { \ S { \ \ \ \ \ \ \ \ \ ■ \ \ \ \ \ r \ \ .6 1.0 1.4- I.e. . 2.2 2.6 3.0 3-4- 25l FIGURE 41 PHASE VELOCITY RATIO OF THE PM X1 MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE L 50 A A ^-1 3.£ 6 ^ s <^r- 13.7 3.2 .4. b /* r.S Z»8 \ c v •2« \ \ \ \ 1 j \ J..0 \ \ \ \ \ \ 1 v !•€> \ \ \ \ \ \>2. I \ \ I \ \ \ ♦o \ \ 1 \ 1 \ \ .4 \ \ 1.0 1.4- I.Q ^ 2-Z 2.6 3.0 3.4 Tex. 51 FIGURE 42 PHASE VELOCITY RATIO OF THE PM lx MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1 . ih ^ ^ &r= ?3.T 5a- 75 ^s K & V \ 6 V \ \ \ v \ \ 3 \ \ \ i \ \ i \ \ \ \ \ \ \ - .& 1.0 \.4 I.B X 2.2. 2..G 3.0 3.4- 52 FIGURE 43 PHASE VELOCITY RATIO OF THE PM 1X MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1. £r = IB. 7 b /a = L0 ^-o« A /j2 a tb o .2. .15-1 .4- -25M .5 .231 .<& .324 .8 .362 "\ V s ^ & £s \ ^ C^ \ \ ^8 \ 6 \4- \ 5 k \.2 \ \ \ \ \ \ i No \ \ \ \ i .£> 1.0 1.4 i-S % 2.2 2.6 3.0 3.4- 2^ FIGURE 44 PHASE* VELOCITY RATIO OF THE PM 1]L MODE IN THE WAVEGUIDE AS SHOWN J.N FIGURE!, \0 U \Z. \3 V4 54 FIGURE 45 PHASE VELOCITY RATIO OF THE PM-q MODE IN THE WAVEGUIDE AS SHOWN IN FIGURE 1= 2« .5/^ 1 2<3 ■l.o 2*3 UC - - fA nO IA CM CM NO XA CM O-rH -=f_=r (A On rH CO On On ON fA nO CM crv C^ II CVJ CVJ r- O OO CM IA r-i O vO CO o JU\ CO -3 c On O CVJ On VO O c--co r-i O On CO r-i O f- o oTi> O- fA CO "LA On nO On C^ • • CJO rH vO 0\NKM>- OO O CO "LA _3 CO -J fA OCOOOCrHCCO-Lr\ vO ONfAC—OvONrHONfA 1— 1 JHON03M30 fAIA fA rH (n-n CM -3 -3 -3 CM rH O roOvnH HlA-3 OCOfA-^IfAr>-Oaj-3 »3CVJ rH -3 On CO CM CO CM CO Ov O rH. rH rH (vn, CU J 0\in..OJ O fV CO nO On O CM CO _3 (Tn. On On NO II CV _3 rH OlAJH OS Wcc n-3 C JroH Ijn.CO OrncovO A'Ot— CVjAr>-rHr>NO-3COfA coi>-Or--o-Oc--0 "lArHl^rHC-NO OCV1 vOWVOo Jf^K IN-CJ. t>-CMr<-vrH CNfAfAOfA c COvO O CM CO On CA rH CM o IA -3 IN O f~ CM H Jr>JvOf> Jf\ r>- rHfA v O OnCO A'fAOON i>- O rf\J(nOnw WU\aJOOOQrnH VC C CA NO t— O CA IT, A OIAO CM-J-IAfACOlA t>T> rH rH O O\0O [N-lJ>O0 wh o o a r>- no -3 CM CM H O O CO IN- "IA. CM fACMrH <-> OOnCOnO-3 rH rH rH rH rH rH rH rH rH rH rH H rH rH rH rH co _3_3 _3co CV NO CM o\cvjc^corH_3 oca nOJHo\COO\J(MO NOCMONfAOrHlACMO r^lA^OcoOooH on C^OrHrrMAlArHOv IA I— On CM O fA CO CM "LA O H lf\ CO ro r>f> H rr, -3 nO H r^CN^OCO O uv co no tA c- 1 no aj co rH _3 -3 nO CM CM NO t>- NO Osno cano r- fA-^rvrvNO vOir\oorHc^-i>-r--co OrrMNCClAC CIA CO OO _TJ On vO fA fA O "LA MDrHOrHCMfAfArHOV GO ck IA CN H \Tv l> H NO O "IA -J rH "LA CO rH "LAOO'LACMOnO-ACO CVlH^vO J\fl C— rH On I>nO\0 fA VO OCOCMCM i -Zjr^-CV'CViCOr^t^-O OnCO O CO ~ « ) rH_^-LTV t>-CO\ACfANOl>-ONO r— t >> v5 CM IA nO [N-COCOCO On IN- ON r— • CV CVJ fA (v-\ rA r— ' rrvuvvrv. nqno o-r^r-^ -3 uv no r>- i>- r>- i>- 1> - co _3:_3_3-3-_3-_3:_3.-3" fArA-3-3-3-^T-3-3 nr^ntnnnrAf^fA ACMCJCMCMCMCMCMCM -a • *|c3 IA vO r— CO On O i— 1 CVJ lAvO t>-CO r>OH W lANOr— COONOr—'CMrA "Lr\NOl>-COONOrHc\jrA rH rH rH rH rH H rH rH r-i r-i r-i r-i r-i r-i II ,D|ro rH CM T3IX1 NO in- CO On • CM rH rH -3 O- l>- fA II 0- -3 II fA rH rH IA IA CM ■ fA rH O CM fAlA \J C On IA O- O CO O CO fA C^ CM On On -Cj CO o CM rH IA o|> CVJ fA IA NO [>- rH « • • nO fA rH CM rH rH O IA -3 ON CO o OCOlAroO W IA fACO CO CO fA nUMSCvicovO rA CM -3 IA On O CM o- H NO nH JOCO m co cvj nTj r- o _d i>- _3 -CT -3- CO nO Jf^WJfJNWN CVJ NO IA -fA NO "LA rA Ov -3 -3 r-i _^ OO CV NO t— O ON CV: CM nO "IA rH nO CO C~- rH CnJ fAlA rH fA II OnH fA.O O OlAOnCM-3 -3 _3 vO -3 -3 "LA rH rH (A _tT CO fA nO (TV On C— rH f> IN A fA fA CM O NO "LA -H fAMD CVJ -3 IA On A CO _3 NO CO O _3 [^- vO A CA O r-i fAlACO O- CO fA C rH NO -^ NO [^ CO r— rH O CO rH rH CA-3 rfMAK CO nO CM rH vO CO On NO NO rH rH On CO "LA CO O rAlAJ On fA -3" CO O On NO IA C CVJlAM^vO C\jH o?> On CO IN- nO -3 CM On co CO nO IA fA O Ov cc o- vO -3 C On On CO NO VTV CM rH 0\COr-N03 rH rH* rH rH "LAlA o- o evj -^J OO ax On Cnj co IA _3 CM -3 |>- CO CO NO CVJ nQ IA \A fA CM fAlA r-i NO NO r-i _^t ; fA rH "LA NO NO -J rH •o rH -3 nO -3 -3 CM CVJ CVJ IA1A C rH cv n\0 q-,nO r- Ox co r>- O i>- cvi niAnvOco r>-\co • \ UMAHIACMJA On vO CM -3 NO fA On OO [^ UV CM IA -3 OOHHMA3 ^N "LA CVJ rH "LA rH fA -3 i_r\co O -3nO f> CM CN CM fA On r>- r>- CM rrN,NO \A O t^r> H r> J cncc t< rH CO CV -^ nO r— fA NO _3 NO fA CO IA rH t~-VArolA "LT\ O. -3" VA CO r-i CO CM "LA CO p"\ fS F>- CM OOHHHH r-\NO CO On O O NO IA CM -3 "LA O 1A-3 On CM"LT\nO On O CM O "LA CO rH >>* _cr ^j_a--a- -3 -3 rH rH rH rH CVJ CVJ r>COK On On On fA -CT "IA trv nO nO nO C-- O rH CM CM CM fA NO NO NO NO NO NO vO nO O vO nO nO IA IA "LA \A IA "LA "LA VA U^ IA "LIMA "LA -3 "LA IA IA "LA "LA Vf \ <«|c3 VnnO c^co On O "LA nO 0- CO On O UAvO r>cc OvO "UV nO D- OO O O rH IAnO r>-CO On O r-i 1 r-i * rH rH r-i r-i r-i r-i -d.|x> rH cvj fA -3 "LA 59 TABLE II. VALUES OF (k X)/j WHICH SATISFY EQUATION (21), RELATIVE DIELECTRIC CONSTANT € 1 = 1.60 V 1.60 X C k Yl X)/j b d b - 2 .h .5 .6 .8 1.00 h.ohoi 2.8501 2.U309 2.10U6 1.5995 1.20 U. 2212 3.1712 2.7U12 2.387U 1.7985 1.60 1.3797 3.5781 3.1705 2.79U1 2.0769 2.00 U.14JU02 3.7791 3.^080 3.03U6 2.2L3U 2. IjO h.h693 3.8832 3.5398 3.1757 2.3h5l 3.00 h.)t912 3 .9629 3.6b50 3.2931 2.1i338 li.00 lx.5071 lu0207 3.723U 3.3837 2.5059 5.oo U.51U2 h.0h6l 3.7583 3.k2k9 2.5399 6.1a0 U.5189 1.0632 3.7819 3.h531 2.5638 8.00 1.5215 h.0726 3.7950 3.1688 2.5773 8.80 3.7990 3.1j737 2.5815 60 NO -4 CO rH WW n w\o m ota ! lAOvHON^OmOW H Jtr\NtAO\ o CMnOCM-OcoOnOOni-ICM CN-d "LfMJA o T-Pv CO i— 1 _^f \/\ C^ N c^-co COOOCO COCO • ••••••••••• r-l i— 1 CM CM CM CM CM CM CM CM CM CM CM CM ^OIA^OOO O rr\ rH CN On l>- rH On CM O m r^ c<-\ r- 1 l>- nO H J1AOIA cO"LT\ CO NnO-4coocco-4c^OMHf^ -criA • r^-CMvOr-i niA^O c^ r- c^ooco coco rH cm CMp^o^o^c^mr^ c> o~, c> mm On rH t^to JoHvOJHr^t- Uv-=T no nr- jco cv r-coiAcMAH nOca i>- HvO rlCO N W (^^(^1 T-f"\ l>- O (H CM e OlAOvO On cm mr^-S-ct-zJ-LPv UM-Tv cm cm r«-\r^o^-^_3-d--ci_zf^J-ct -3-ct \Q Jr^JvflHcooD 0\O(^^\0\0 co MHnosCN^HjvOOn H_J\0 O 0_CfLf\rHl3\OCOO nO CM nO HCM o roO\-4MlA[^co OOnOOO oo cm cm o^-^r-^-jj-zj-^r-^-^-^iA uyia mco OlA^N O -^CC On CO -CtO\ lAt^lACJlAf^aHvO Win 1 i-HIA o U\ HHcofr\OOoo r^iAt^oa 1 t-HrH rH T)U> ° r — 3onOw wnrr\r\r^ 1 _=J_3 CM II cm «r> c~\ -3 uma in i_r> \a uma "LfNlPv r-l rH CM CM -^ CO -d nO m rH l>- On ravO o~\ r- d O On UN, t— IOr v -l>--3' 1 -ct \0(nH-01AO\W\00(>OH I mm • WO^OW-^lAvO^OOKN 1 o-p- • ••••••■•••| r^-^r-^nj>XAlA\A*LfMfMA'Lf\ •LAl-iA CMtY\,ONmcr\CMONf— OOnOn OnnO OCM 0\OOCHU\1AHH\00 1 c\ oc • oo • ••••••••••| J JlAUMAlAlAl^.lAUMA nOnO r— 1 "LT\ C~- nO MD OO CM rH NO CM UACM -dCO NfMvOvOOOlAO^H f^ 1 C^GC CVJ •LT\t-HlAO.\OoOi-HrHCMCV | CM CM • H r>Osr- It— 1 r-H CM CM CV CM CM 1 CM CM • ••••••••••| • • UMAXfNvO^OvOvO^OvOvOvO nOnO O nQ CO nO r~wO — 3C i>- nt-- P-On cMo\0\aco~c^ccooc I r-HrH rH CM CM • _=j-=r • •*... •••••! ^OvO\00\OvOvO\OvQ\0\0 nOnO r>~\ c- O r^,vO COOOOCOOO o o X|.Q Or^vO-JOOOOOCOO o o • • iHr-HrHCMr<>-J"U>vOl>-COO\CMvOO r-l t— 1 CM T3U> • \A CMl>-00'Lrvo-COCOQOOr^\C>'-'l>- (^JaOHooHWUMMONJ OCJ\O00WJnJHlAr-O\C0(^J i— I CMCM CM mmmmmmmmmm Ovf^UNrrMACM WHvOOWOrlt- CM r-H tT\ O CM nO CO a. 0\ O t> W N OO t^^OOCNOO J-_^'Lr\CO_d'CO'UAr>~\CO "LA _d CM CO 1A -4 On -d C 3 P- 1A nO 1A O OfAC^-fD O CO CO -d r— On mtANO _Cjr-lc^I>-CM[^COOsOOOr-lt-liH CM moA-3"LfMA'LfMAsOvO v O v -OvOO t— lONlAONOlAmCMCM-dCV CM rH CO NONOrHmOCMCMCMCOr 3 rH NO CM (T\(^J JHIAcn\Q O CM _zj r>- CO On COsO m -d On CM m-^JlATAlAlAIAlA CM nJlAlAO\00^)\0\OvOvO\0 \A^O f>COOJOwO O^AJO\ c^-cjf^iHvO 0\r>Hr>HH _3;nO CM U\ On rH O 1A C- On O _d f^ r- It— I -4 r> OO co QO oo On tr l -3\A\0'>0\00\00 , OvO , On r-t l r— cm I cv m I On O- I • • NO NO c-nno CwhOco h^o wr- COCO 0>>-r--ON_ZfNOrHl>-CM NOc^OOaooN-^tNOcocooN r^mrlco 0\OrlHrlHH -^jianonOno r~~ t>- p- r>- c^ r— mlA I NO CO I o o I CM CM I • • rlNP.CvJ0OHr>rt\H\Ot s - CM rH mr-CM-d"tANOCMrHr-o\rH I t^oc ONOAmONlAOxrHCMCVCMrn I C\(T\ t-\OOWrr\f^444 4-4 I _a -3 UN,NOt--c'^r-t>-c v -r>-r>-r--r— r— c*- (T\(\j OJ CM r- 1 ONl^f" I m O m lAt^ -3 0lr\J\AW^)COO\0\C I HH CA O nO O r— I CM CM CM CM CM m I en tT\ P-xIAVAnOnOnOnOnOnOnOnO I ^O\0 m c^ O rnO OOOOOOO OOOO O cAnO -d O O O O CM O OOOO rH r— I r— I CM r~\ ■ 61 TABLE V. VALUES OF (k \)/j WHICH SATISFY EQUATION (21) RELATIVE DIELECTRIC CONSTANT € = 3.78 € 1 = 3.78 X d b b .2 .1* o5 .6 .8 1.067 7.1015 3.9063 3.1721 2.6687 2.0221 1.60 9.0P27 5.6157 a .6.115 3.9023 2.9588 2.00 9.6732 6.71*22 5.6037 h.7665 3.6073 2. hO 9.8900 7.6709 6.1*881 5.5568 1*.1952 3.00 10.0116 8.5859 7.5310 6.5532 1*. 91*28 1*.00 10.0801* 9.1971 8.1*713 7 .6626 5.8191 5.00 10.1066 9.1*025 8.81*511 8.11*01 6.3328 6.00 9.h9l*0 9 oi55 8.3951* 6.6316 6.1*0 10.1229 9.5169 9.0580 8.1*608 6.711*8 8.00 10.1316 9.5728 9.1608 8.6205 6.9311* 9.60 9.6008 9.2118 8.700h 7.01*7)* TABLE VI. VALUES OF (k \)/j WHICH SATISFY EQUATION (21). RELATIVE DIELECTRIC CONSTANT € J = 5.75 € 1 = 5.75 X d b b .2 J* .5 .6 .8 1.20 8.5898 h.5297 3.65U* 3.0618 2.3105 1.60 10.8068 5.91*81* I*c8l76 l*.ol*5l 3 .0561 2.00 12.2759 7.3065 5.9h72 5.0059 3.7828 2.1*0 12.9082 8.5829 7.031*1* 5.9381 1*.L81*7 3.00 13.1906 10.2502 8.51*87 7.2611* 5J4762 l*.oo 13.3123 11.9092 10.5215 9.1399 6.9082 5.00 13.3519 12.1*61*9 11.5680 10.1*122 7.9975 6.00 12.6695 12.0200 11.1126 8.71*80 6.1*0 13.3750 12.7160 12.1231 11.2876 8.9703 8.00 13.3867 12.8212 12.3513 11.6886 9.5613 9.60 i — — 12.8699 12.1*532 11.8699 9.871*7 62 TABLE VII. VALUES OF (k \)/j WHICH SATISFY EQUATION (21) RELATIVI 2 DIELECTRIC CONSTANT € = 10.0 € - 1 ~ 10.0 X d b b .2 ok .5 .6 .8 1.20 9.09U6 k 06363 3o72l6 3.1082 2.337U 1.60 11.9152 6.1U53 ii.9397 14.1289 3ol075 2.00 11.5080 7.6327 6. 1153 5. Hill* 3 08718 2oU0 I6o6205 9.09l*a 7.3368 6.1UI4.6 a. 6288 3.00 18.0837 11.2235 9.0920 7.6283 5.7i*6l4 UoOO 18.U618 lU.U5h6 11.8839 10.0186 7.5)il7 5.00 18.5381 16.6706 lJUo3?6U 12.22U2 9.2125 6.00 17.539a 16.0369 IU.0790 10.7010 6 hO 18.575a 17.695b 16.U682 lii.6766 11.2319 8 o 00 18.5926 17.9823 17.3075 16.1831 12.9273 9.60 13,0875 17.5921 16.796U 13.9 811 ll o 20 17.7216 17.071^3 D*.5972 TABLE VIII. VALUES OF (k X)/j WHICH SATISFY EQUATION (21). RELATIVE DIELECTRIC CONSTANT € = 13.7 13.7 X d b .2 .1* .5 .6 .8 1.20 9.2268 a. 6657 3.71*02 £.9733 3.1210 a. 1521 2.3U* 7 1.60 12.1878 6.1991 3.1209 2.00 15.0U66 7.7200 6.1991 5.1781 3.8830 2. hO 17.7190 9.2?68 7.1168 6.1990 a.6633 3.00 20.7905 11.1*51*9 9.226U 7.7190 5.8083 1**00 -21.9512 15.0381 12.1811 10.2132 7.6836 S.00 22. 08 hi I8.2h05 15.0013 12.6295 9.a989 6.00 20.3523 17.5165 ia.9018 11.22a5 6.U0 22.1369 20.7837 I8.3608 l5.7ai*5 11.8811 8.00 22.1591 21.1*1*96 20.3759 is.a5o6 ia.2ao5 9.60 21.63)49 21.0071 i9.eaa8 i6.oai5 11.20 21.2U53 20.a336 17.2h25 63 4. NOTE ADDED IN PROOF In looking over the curves in proof, it was apparent that some accuracy was lost in the reproduction of the graphs. For example, in Figs. 2-43, the c/v curves for the cases d/b = and d/b = 1 (drawn in for reference) are circles centered at the origin with radii unity (in units of X/2a) and Je , respectively. These circles were accurately v r drawn on the original graphs; therefore, an estimate of the error in reproduction can be made by drawing these circles on the graphs and comparing. If great accuracy is required, the values in Tables II-VIII can be used with equation (25) (substitute the negative of the square 2 of the table entry for the quantity (k X) in equation (25)). y i 5. ACKNOWLEDGEMENT The author was helped and encouraged by discussions with P„E. Mayes and some unpublished notes of V.H. Rumsey. A special acknowledgement is due to Mrs. Judy Blankfield, who set up the ILLIAC program, and to Mr. G. Mochel, who plotted most of the graphs. 64 REFERENCES 1. Pincherle, L. Phys, Rev, 86 , pg , 118, September 1944. 2. Marcuvitz, N„ Waveguide Handbook, McGraw-Hill Book Co., Inc., New York 3. Van Bladel, J. and Biggins, T»J M J. Ap_p_l . Phys . , 22, pg 324 , March 1951 „ 4. After the method described in: Morse, P. and Feshbach, B ;„ , Methods of Theor etical Physics , McGraw-Hill Book Co., Inc., New York, 1953, Chap. XIII. DISTRIBUTION LIST FOR REPORTS ISSUED UNDER CONTRACT AF33 (616) -3220 One copy each unless otherwise indicated Armed Services Technical Information Agency Knott Building 4th & Main Streets 4 and 1 repro. Dayton 2, Ohio ATTN: TIC-SC (Excluding Top Secret and Restricted Data) (Reference AFR 205-43) Commander Wright Air Development Center Wright-Patterson Air Force Base Ohio 3 copies ATTN: WCLRS-6, Mr. W.J. Portune Commander Wright Air Development Center Wright-Patterson Air Force Base Ohio ATTN: WCLN, Mr. N. 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