no. ?,o 
 
 tof 
 
 .2- 
 
 C6HKKKCE ROOM 
 
 'pGJNGERIW 
 ANTENNA LABORATORY 
 Technical Report No. 20 
 
 /OF ILLINOIS 
 
 U88ANA, ILLINOIS 
 
 FREQUENCY INDEPENDENT ANTENNAS 
 
 MWHHBE ROOM 
 
 TH-'-Ui, COL i 
 
 by 
 V. H. RUMSEY 
 
 25 October 1957 
 
 Contract No. AF 33 (6161-32 20 
 Project No. 6(7-4600) Task 40572 
 WRIGHT AIR DEVELOPMENT CENTER 
 
 ELECTRICAL ENGINEERING RESEARCH LABORATORY 
 
 ENGINEERING EXPERIMENT STATION 
 
 UNIVERSITY OF ILLINOIS 
 
 URBANA, ILLINOIS 
 
 
 36Q 
 
 
Antenna Laboratory 
 
 Technical Report No. 20 
 
 FREQUENCY INDEPENDENT ANTENNAS 
 
 by 
 V.H. Rumsey 
 
 25 October 1957 
 
 Contract AF33( 616) -3230 
 
 Project No. 6(7-4600) Task 40572 
 
 WRIGHT AIR DEVELOPMENT CENTER 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
ii 
 
 sin <£ 
 
 CONTENTS 
 
 
 Page 
 
 Abstract iii 
 
 Acknowledgement iv 
 
 1. Antennas Specified by Angles 1 
 
 2 . The General Approach 3 
 
 3. The Pattern 5 
 
 4. The Impedance 7 
 
 5. Pseudo-Frequency- Independent Antennas 9 
 
 Bibliography 11 
 
 Appendix Surfaces for Which a Rotation is Equivalent to an 
 
 Expansion 12 
 
 Distribution List 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/frequencyindepen20rums 
 
iii 
 ABSTRACT 
 
 There is a class of antennas whose pattern as well as impedance is 
 practically independent of frequency for all frequencies above a certain 
 value. The general formula for their shape is r = e F(#) where 
 r ; 0,f> are the usual spherical coordinates, a and <p are constants and F(9) 
 is any function of 0. Assuming a to be positive,^ ranges from -«° to 
 sqijpe finite value which determines the low frequency limit. For such 
 antennas, a change of frequency is equivalent to a rotation of the antenna 
 about 0=0. It appears that the pattern converges to the characteristic 
 pattern as the frequency is raised, if a is not infinite, and that the 
 impedance converges to the characteristic impedance for all ^. 
 
iv 
 ACKNOWLEDGEMENT 
 
 This work was started in connection with a Signal Corps Contract 
 at Ohio State University and carried on under an Air Force Contract at 
 the University of Illinois. 
 
1. ANTENNAS SPECIFIED BY ANGLES 
 
 To introduce the idea of frequency independent antennas, let us consider 
 the familiar problem of modelling an antenna to a different wavelength. As 
 we know ; if all the dimensions of a lossless antenna are increased by some 
 factor K, the pattern and impedance remain fixed if the operating wavelength 
 is also increased by the factor K. In other words, the performance of the 
 antenna is independent of frequency if its dimensions measured in wavelengths 
 are held constant. It follows that if the shape of the antenna were such 
 that it could be specified entirely by angles, its performance would be 
 independent of frequency. The infinite biconical antenna is the most 
 familiar example: it is specified by the angles of the two cones and the 
 angle between their axes. However, there is an infinite variety of shapes 
 which are completely specified by angles, and these form the starting 
 point for the design of frequency independent antennas. They must all 
 extend to infinity (because if they did not they would have at least one 
 characteristic length) , and therefore they do not immediately lead to 
 practical designs. The key problem is therefore to determine how rapidly, 
 if at all, the performance of the finite structure converges to that of 
 the infinite structure. Let us, however, defer consideration of this 
 question until we have explored some applications of the "angle method" 
 to familiar antennas. 
 
 The problem of a typical directional antenna can be illustrated by 
 considering a unipole in front of a plane reflector as shown in Fig. 1: 
 it is specified by the lengths ABCDdjfc. One of the major limitations on 
 its pattern and impedance bandwidth is represented by the distance D from 
 the unipole to the reflector. By changing the reflector and unipole to 
 coapical cones ; all dimensions except B and jL are replaced by angles. This 
 
 simple application of the angle method does indeed give significant 
 
 2 
 improvement of the impedance and pattern bandwidth. The impedance is 
 
 practically constant above a certain frequency, but unfortunately the 
 
 pattern is not. This appears to be typical of most conventional antennas 
 
 designed according to the angle method. However, there is a whole class 
 
 of unconventional antennas whose pattern, as well as impedance, remains 
 
FIGURE 1 
 
 Common Point of 
 Intersection 
 
 FIGURE 2 
 
practically constant above a certain frequency. In view of the fact that 
 all experience suggests that the pattern of any antenna develops more and 
 sharper lobes as the frequency is increased, this result is indeed remark- 
 able. 
 
 2. THE GENERAL APPROACH 
 
 Let us consider, then, the entire class of antenna shapes which are 
 specified by angles. To begin with, let us restrict the discussion to 
 plane sheet antennas; that is, we consider all plane curves which remain 
 essentially the same when scaled to a different unit of length. Such 
 curves can be used to determine the shape of a plane sheet antenna, by 
 taking the input terminals at the common point of intersection of four 
 curves, as illustrated in Fig. 2. Obviously we do not want to have to 
 move the terminals when we change frequency, so we add the restriction 
 that the location of the terminals remain fixed. The fact that a typical 
 curve remains essentially unchanged by a change of scale implies that the 
 new curve can be made to coincide with the old one by translation and 
 rotation. Since a translation is eliminated by the requirement that the 
 terminals stay fixed, the problem is to determine all curves such that a 
 change of scale is equivalent to a rotation. This can be stated symbolically 
 in the form 
 
 Kr(^) = r(<P+ C) (1) 
 
 where r(<p) denotes the radius r as a function of the polar angle (p , K is 
 the scale change and C the angle of rotation to which it is equivalent. 
 Thus K depends on C, but K and C are independent of <P (and r) . 
 
 (2) 
 
 and K ~~^b' " X Q<p" ~' (3) 
 
 9r(<p+ C) dr(<P+ C) 3r( V+ C) , ., 
 
 But jn; = TTTH .„x ■ = 5Tff3 ( 4 ) 
 
 (5) 
 (6) 
 
 ] 
 
 rli| 
 
 }) dC " 
 
 be 
 
 
 
 K 
 
 dr(<P) 
 d!^ 
 
 d r ( ( P+ C) 
 3<P 
 
 
 9r(<p + 
 Be 
 
 Cj 
 
 dr(<P 
 d(«P 
 
 + C) 9r(<P+ C) 
 +C) " d<p 
 
 
 r 
 
 yY) dC 
 dr 
 
 v dr(p) 
 
 ! ar 
 
 
where a is independent of V: 
 
 It follows from (6) that 
 
 1 dK 
 K dC 
 
 (7) 
 
 a.(p . „ 
 
 r = r e where r is a constant. (8) 
 
 o o 
 
 Let r = e ^ ° where (Q is a constant (9) 
 
 o o 
 
 or a<& = In r . (10) 
 
 T o o 
 
 Then r = e aC ? + ^ (11) 
 
 or <£+<j? = In r 1/a . (12) 
 
 We recognize (11) or (12) as the formula for an equiangular spiral: it 
 
 contains the two parameters, a, which represents the rate of expansion, and 
 
 (D , which represents the orientation. Thus the shape of all plane sheet 
 ' o 
 
 frequency independent antennas must be defined by equiangular spirals. 
 
 (Note that when — = the spiral degenerates into the straight line @ = ~P .) 
 a o 
 
 Theoretically it might appear that we could obtain four curves such as those 
 
 shown in Fig. 2 by selecting four different combinations of a and p , but 
 
 o 
 
 it is easily verified that unless a is the same for all, such curves overlap 
 
 at infinitesimal values of r, thus placing a short circuit across the 
 
 terminals. It is therefore necessary to choose four different <£> with the 
 
 o 
 
 same a . 
 
 The general problem is to find all surfaces which have the property 
 
 that a change in the unit of length is equivalent to a certain rotation. 
 
 Then if we construct a metal antenna whose surface is one of these surfaces, 
 
 its performance will be the same at all wavelengths except for a rotation of 
 
 the coordinate system. This problem is analyzed in the appendix. The solution 
 
 can be written in the form „,,-„, ^ \ 
 
 r = e a (? + ?°> F(e) (13) 
 
 where in principle F(0) can be any function of Q. The shapes represented 
 
 by (13) can be very complicated because in general an increase of 2?T t in p 
 
 does not give the same r: as <£> ranges from - oo to <x> the surface weaves 
 
 around through all space. Figure 3 illustrates a simple example which 
 
gives a practical antenna design. Figure 4 illustrates the case where 
 In F(0) is periodic in 9 with period 27Ta . This gives a simple surface 
 like a screw thread which is uniformly expanded in proportion to the 
 distance from the origin: an increase of 27T in <j? is equivalent to moving 
 
 one turn along the screw. The plane antennas considered in the paper by 
 
 3 
 R.H. DuHamel and D.E. Isbell represent a cross section through the axis 
 
 of this kind of antenna. 
 
 3. THE PATTERN 
 
 To examine the question of pattern convergence, we make use of a 
 characteristic of the field which depends on the fact that a rotation is 
 equivalent to an expansion. Since we are dealing with the frequency 
 independent mode, the analysis of the field can be simplified by consider- 
 ing the static or DC case. For this case it has been shown by P. E. Mast' 
 
 that for the plane antenna the field is a function of the two variables 
 
 -a<p 
 S = re and 9, rather than the three variables r6 ( <p. (Note that S is 
 
 constant on any equiangular spiral in the family characterized by the 
 parameter a) . This result implies that the direction of the current on 
 the antenna is along the lines S = constant; i.e., the currents follow a 
 spiral flow. The simplest way to analyze the pattern is to make use of 
 the intuitive idea of coupling between adjacent turns of this spiral flow. 
 Assuming, to begin with, that the AC current distribution is like the DC 
 distribution traveling with the free space phase velocity, we see that a 
 "resonance 1 will build up in the current distribution where the mean 
 circumference of the spiral is about one wavelength. This resonance will 
 be highly damped by radiation so that most of the current is dissipated in 
 the first resonance. The pattern due to the first resonance is therefore 
 almost the same as the pattern due to the infinite structure. Admittedly 
 this argument is weak, but it is not altogether worthless because it does 
 give a simple way of estimating the pattern which agrees satisfactorily 
 with measurements on finite antennas. It shows that the condition for 
 pattern convergence is that the current flow be spiral rather than recti- 
 linear^ i.e., that the parameter a should not be infinite. 
 
& 
 
 F(0) 
 
 e^ F(6) 
 
 FIGURE 3 
 
 In F(6) 
 
 FIGURE 4 
 
These theoretical conjectures are borne out in practice. The case 
 
 a = oo represents the biconical type of antenna for which the pattern does 
 
 i 
 not converge. Various cases for a ^ oo have been investigated by J.D. Dyson 
 
 5 
 and R.L. Carrel, and convincing evidence of the constancy of the pattern 
 
 over a 20:1 frequency range has been obtained. Note that a change of 
 
 frequency is equivalent to a rotation. The pattern at one frequency, f , 
 
 is the same as the pattern at another frequency, f , if the coordinate 
 
 1 f l 
 system is rotated about the 9=0 axis through an angle — In ~~: the 
 
 a ± 2 
 
 pattern scans around the © = axis at a rate which depends on a. If 
 
 measurements are made in a fixed plane, the pattern in general varies 
 
 with frequency, and the frequency independent nature of the complete spheri- 
 cal pattern may be missed in such measurements. 
 
 4. THE IMPEDANCE 
 
 The convergence of the input impedance to a constant value as the 
 frequency is increased is familiar in connection with the biconical antenna. 
 
 It has been confirmed in all cases of the general type represented by (13) 
 
 4,5 
 which have been investigated. The determination of the characteristic 
 
 impedance is an important problem in connection with frequency independent 
 
 antennas. Schelkunoff 's well-known formula for the characteristic impedance 
 
 of two coaxial cones is 
 
 A B 
 377 In (tan - cot g) 
 
 where A and B are the cone angles measured from the common axis. The 
 
 6 
 formula for two inclined cones is 
 
 377 
 
 tanh 
 
 tan 
 
 % 
 
 tan 
 
 + tanh 
 
 tan 
 
 % 
 
 where ip , Cf> and ^ are defined in Fig. 5. The formula for a symmetrical 
 plane sheet antenna consisting of two triangles with a common apex at the 
 
 terminals is 
 
 189 
 
 K(cos 40 
 K(sin ip) 
 
\ 
 
 X 
 
 **< 
 
 Cone 
 
 FIGURE 5 
 
where K represents the elliptic integral defined by 
 
 .1 
 
 dt 
 
 (1 - t*) (1 - x^t*) 
 
 and V represents the half angle of the triangular strips measured from 
 the common axis. 
 
 In connection with the impedance, we should note an interesting 
 property which was pointed out by Mushiake in one of the Tohoku University 
 reports. It is that the impedance of any plane sheet antenna whose shape 
 is the same as the shape of its complement (except for a trivial change of 
 coordinates) is independent of frequency and equal to 60-77 = 189 ohms. The 
 complementary antenna is defined as the portion of a metal plane which is 
 not covered by the original antenna: when the antenna and its complement 
 are fitted together they completely cover the whole plane without over- 
 lapping. The constant impedance of a "self -complementary" antenna follows 
 
 Q 
 
 from the relation 
 
 Z ][ Z 2 = (607T) 2 
 
 between the impedance Z of the antenna and the impedance Z of its comple- 
 ment. Figure 6 gives some examples of self -complementary shapes. 
 
 5. PSEUDO-FREQUENCY- INDEPENDENT ANTENNAS 
 
 The idea of a pseudo-frequency-independent antenna is illustrated by 
 the horn antenna shown in Fig. 7. It consists of metal sheet perforated 
 by holes of uniformly expanding size: any hole is exactly like its neigh- 
 bor on the left except for a fixed expansion. The idea is that the effective 
 size of the horn remains roughly independent of wavelength because the metal 
 sheet becomes approximately transparent once the holes become greater than 
 about half a wavelength square. More precisely, it can be seen that if the 
 horn started from a point and extended to infinity, it would "look" the same 
 to any two wavelengths whose ratio was equal to the expansion factor. Some 
 
 interesting examples of pseudo-frequency— independent antennas are described 
 
 3 
 
 in the paper by R H. DuHamel and D.E. Isbell . 
 
Terminals 
 
 10 
 
 FIGURE 6 
 
 FIGURE 7 
 
11 
 
 BIBLIOGRAPHY 
 
 1. S.A. Schelkunoff, Electromagnetic Waves, D. Van Nostrand Co., Inc., 
 New York, 1943. 
 
 2. Ohio State University Research Foundation Reports 510-4 and 510-5, 
 May 1953. 
 
 3. R.H. DuHamel and D.E. Isbell, "Broadband Logarithmically Periodic 
 Antenna Structures," University of Illinois Electrical Engineering 
 Technical Report No. 1£, Contract AF33(616)-3220, May 1, 1957. 
 
 4. J.D. Dyson, "The Equiangular Spiral Antenna," USAF Antenna Symposium, 
 University of Illinois, October 1955. 
 
 5. R.L. Carrel, "Conical Spiral Antenna," University of Illinois Electrical 
 Engineering Technical Report No. 22, Contract AF33( 616) -3220, 
 
 May 25, 1957. 
 
 6. V.H. Rumsey, "The Characteristic Impedance of Two Inclined Cones," 
 IRE Trans . on . Antennas . To be published. 
 
 7. R.L. Carrel, "The Characteristic Impedance of the Fin Antenna of Infinite 
 Length," University of Illinois Electrical Engineering Technical Report 
 No. 16, Contract AF33(616)-3220, January 1957. 
 
 8. H.G. Booker, "Slot Aerials and Their Relation to Complementary Wire 
 Aerials, (Babinet's Principle) ," JIEE, Part IIIA, 1946. 
 
12 
 
 APPENDIX 
 
 SURFACES FOR WHICH A ROTATION IS EQUIVALENT TO AN EXPANSION 
 
 A surface can be represented by the equation 
 
 r = f(0/P) (14) 
 
 where r, and tp are the conventional spherical coordinates. The 
 condition that a uniform expansion be equivalent to a rotation can be 
 written as 
 
 K f(0 ; P) = f(e;P') or Kr = r* (15) 
 
 where K is the expansion factor and (0|^P') represents the direction after rota- 
 tion of the line represented by (0^), eg., for a rotation about the z axis 
 (0=0), 1 = 0, and <P' = <P + c. For a general rotation, we express the 
 connection in the following way, in order to have the advantage of work- 
 ing with linear transformations . We define 
 
 e x = 0; 2 =<p (16) 
 
 P 1 = sin e i cos 2 ; P 2 = sin 1 sin £ ; p g = cos X (17) 
 
 a = li 
 
 \ P 3/ (18) 
 
 Thus Q_ represents the rectangular coordinates of the point (0 ,0 ) on 
 
 1 A 
 
 the unit sphere: /0 s is similarly defined with respect to *, ' . 
 
 4 
 Then the connection between 0.0 and © f o can be expressed as 
 
 T 2 i > ^ 
 
 iQ.' = T p_ (19) 
 
 where T is a (3 X 3) matrix independent of e 1 > e 2 ,d l'®2' ' To ex P ress T 
 
 we introduce 
 
 / cos c sin c 0\ 
 J (c) = -sin c cos c oj 
 
 1 1/ (20) 
 
13 
 
 It is easily verified that«/(c) /D represents a rotation through the angle c 
 about the z axis. Now the general rotation can be taken as a rotation c 
 about the z axis followed by a rotation c about the y axis followed by a 
 rotation c about the z axis. In effect, the latter two rotations define 
 the direction of the axis of rotation, and the first defines the angle of 
 rotation about that axis. Since 
 
 z \ 
 
 /0 it 
 
 /x 
 
 
 
 X - 
 
 = 10 
 
 y 
 
 =•• W Q., 
 
 where 
 
 i y / 
 
 \0 1 0/ 
 [0 l\ 
 
 \ ml 
 
 
 
 u 
 
 = 10 
 
 \0 1 0/ 
 
 
 
 
 (21) 
 
 and UU = 1 (22) 
 
 (U denotes the transpose of U) , it is easily verified that these three 
 rotations are represented by 
 
 T =/( Cl ) fy (c 2 ) u/(c 3 ). (23) 
 
 To find the solutions of (15) we note that it must be valid for all 
 9 ,QJ& S&' 1 and all expansions K and the corresponding rotations specified 
 by c 1 > c ^c 3 - Thu s 
 
 dr d r ' dr' 9e i* 
 
 K W. a©, a©.' a©. ^ ; 
 
 i i j i 
 
 with the understanding that summation is implied over any repeated suffix, 
 such as j in (24) . The matrix M and the vector a are defined by 
 
 M. . = w^ (25) 
 
 Then (24) becomes 
 
 Also from (15) 
 
 8r 8r' 
 
 Q i " 85. i «l' = W ■ (26) 
 
 K a = M a' (27) 
 
 3K 8r' 8r' ^j , N 
 
 i i j i 
 
14 
 
 The matrix S and the vector (3 are defined by 
 
 s ij - ST <29) 
 
 (*i - 5r ■ (30) 
 
 i 
 Then (28) becomes 
 
 r p = S a' . (31) 
 
 Our aim now is to eliminate a_ from (27) and (31) and thus obtain a set 
 
 of differential equations for r in terms of and . This requires that 
 
 1 £ 
 
 M and S be expressed in terms of and , which is accomplished as follows: 
 
 From (19) 
 
 W = T W or "Se^ = T ">J 3e~ * (32) 
 
 i i i i 
 
 H-=Hv^o r ^- = ^-^ . (33, 
 
 i J i i j i 
 
 ap • , %,' 
 
 Define A. . = ^ ; A ±J = -ggj- • (34) 
 
 Then (32) and (33) give 
 
 T A = A ' M or A T = M A* . (35) 
 
 Bp ' dp ' be.' 
 
 y *7 
 
 ^=W7^ 
 
 8p ' 8t 
 
 and jgf- = ^ P^ (37) 
 
 i i 
 
 whence S A = N (38) 
 
 3t. 
 
 where N ±j = ^- 0^ 
 
 (39) 
 
15 
 
 * 
 Note that (35) and (39) comprise 15 equations for the 16 elements of M A 
 
 and S. In addition we have A ' = from (34) and (17)^ so that in principle 
 
 we have enough equations to proceed.. 
 
 At this point j we can simplify the analysis by choosing the coordinate 
 
 system so that the axis of rotation is the z axis, i.e. ; 
 
 Cj = 0) c 2 = o ; c 3 = c (40) 
 
 and T =^(c) . (41) 
 
 d 
 
 This does not. involve any less of generality provided -g — is carried out 
 before (40). It is easily verified that (41) is equivalent to 
 
 e i = 8 1 ; @ 2 = ®2 + ° ' (42) 
 
 Hence from (25) 
 
 M = 1 (43) 
 
 /- (27) gives K a = a/ „ (44) 
 
 Substitution in (31) gives 
 
 r J3 = K S a (45) 
 
 which can be expressed more concisely by defining 
 
 if{- In r and k = In K * (46) 
 
 Then, from the definitions of a and (3 in (26) and (30), (45) can be 
 expressed as 
 
 ¥<r ■ S ij it • (47) 
 
 i j 
 To evaluate S, we can rearrange (38) as follows 
 
 o it 1/ ~* e 
 
 S A A =NA 
 (note that S is 3 X 2 , A is 2 X 3 and N is 3 X 3) 
 
 N A (A A ) (48) 
 
 where ( ) denotes the inverse matrix, 
 
From (17) and (42) 
 
 p ' = sin 9 cos(0 -c) 
 
 16 
 
 sin 9 sin(9 -c) 
 1 £ 
 
 P 3 8 = cos e x . 
 
 (49) 
 
 ,*pfrpm (34) 
 
 cos 9 cos(9 -c) cos 9 sin(9 -c) 
 1 ^ 1 ^ 
 
 -sin 9 sin(9 -c) sin 9 cos(9 -c) 
 
 - sin 9, 
 
 /.(A'A 8 )" 1 
 
 .'MA'A')" 1 
 
 A'A' = 
 
 \o 
 
 1 
 
 
 2 
 
 sin 9, 
 
 cosec 6., 
 
 cos 9 cos(9 -c) 
 
 = cos 9 sin(9 -c) 
 
 sin(9 2 -c) 
 
 sin 9 
 
 cos(e 2 " c ) 
 sin 9, 
 
 (50) 
 (51) 
 
 (52) 
 
 ■sin 9, 
 
 
 
 (53) 
 
 To evaluate N from (39) we use the general formula for T given in (23) 
 Thus 
 
 3t 
 
 ^ = c/( Cl ) U o/(c 2 ) U /(c 3 ) 
 
 o^'(0) IJC7 7 (0) Uc7(c) 
 
 from (40) 
 
 -sin c cos c 
 -cos c -sin c 
 
 (54) 
 
 dT 
 
 "3c\ 
 
 J (0) uV'(O) Uo^(c) 
 
 (55) 
 
17 
 
 3T rf 1 
 
 ■g — = / (c) = / -sin c cos c 
 
 | -cos c -sin c 
 
 0/ (56) 
 
 Thus from (39) 
 
 N = / sin sin(0 -c) -sin cos(6 -c) 
 
 i. A 1 A 
 
 cos 9 sin e cos(0 -c) 
 
 sin 9 sin(6 -c) -sine cos(0 -c) 
 
 (57) 
 
 From (48), (53) and (57) 
 
 From (47) and (58) 
 
 ■cos(0 -c) cot 9 sin(9 -c) 
 
 a k _ _ ace 
 
 ^7 = -^ 
 
 (58) 
 
 (59) 
 
 ■^ = -cos(e 2 -c) -g^- + cot B 1 sin(6 2 -c) -g^ (60) 
 
 2 1 2 
 
 a k _ a*e 
 
 3 2 
 
 (61) 
 
 Now the left hand sides of (59), (60) and (61) are independent of ©,©„• 
 From (61) 
 
 fa = D 6 2 + f(6 ) (62) 
 
 where D is constant, and f(6 ) any function of 9 . Substitution of (62) 
 in (60) gives 
 
 -jsr cos(e.-c) = const + D cot 9 n sin(6 -c) (63) 
 do 2 1 2, 
 
 which evidently has no non-trivial solution since both sides must match 
 for all and 6 . This implies that the rotation must be of a more 
 
18 
 
 restricted kind than the general rotation represented by c , c , c . If 
 we consider a reduction to two degrees of f reedom ; we have 
 
 c = s^ome function of c ,c = g(c , c_) (64) 
 
 and k = h(e r; c a ,c 3 ) = Kc^, Cg) (65) 
 
 a k = a_i ■■ a_h a g a h 
 "5^ ac 2 - a Cl JT 2 + ~b~T 2 • 
 
 (66) 
 
 a k a t v 
 
 Similarly, for -g — and -g — . Thus in place of (59), (60) and (61) we 
 
 C 3 °i 
 have now two equations which, so far as 9 and e are concerned, are :.' 
 
 formed by taking linear combinations of corresponding sides of (59) 
 
 with (60) and (61): 
 
 9 k e£ . , are ^ a . , Q , Be 
 
 ac~ = A ^e" " cos ( e 2 ~ c) de + cot e i sin < e 2 ~ c) W (67) 
 
 a k 
 
 3c" 
 
 B W~ ' < 68 > 
 
 3 2 
 
 Again there is no non-trivial solution. The rotation therefore has 
 only one degree of freedom. Therefore T is a function of only one 
 variable which we shall call t. Now since T T = 1 
 
 |T| = Determinant T = 1 (69) 
 
 •'• 4^=0. (70) 
 
 dt 
 
 But (70) is the condition that there be some Q_ and p ' satisfying 
 
 dp' Jm 
 
 -sr = £ e» - ° • <71) 
 
 dt dt Ha 
 
 „\ /2 ' is independent of t. The rotation therefore has a fixed axis 
 
 represented by Q_ ' = Q. . Thus (59), (60) and (61) reduce to (61), 
 
 giving the solution (62) where f(© ) is an arbitrary function. From 
 (61) we also obtain 
 
 k = - Dc . (72) 
 
19 
 
 Thus we conclude that the general formula for a surface which is 
 transformed into itself by a rotation and an expansion is 
 
 In r = Dp+ f(0) (73) 
 
 where r ; 9 ; *P are conventional spherical coordinates and the expansion 
 factor K is related to the rotation angle c by 
 
 In K = k = - Dc (74) 
 
 which follows directly from (61). Alternatively (73) can be expressed 
 in the form 
 
 r = e r F(9) . (75) 
 
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 Bell Aircraft Corporation 
 ATIWs Ut Jo Do Shantz . 
 Buffalo 5 New York 
 WF Contract Wo33(038)°14169 
 
DISTRIBUTION LIST (CONT ) AF33( 616) - 3220 
 One copy each unless otherwise indicated 
 
 page 2 
 
 Chief 
 
 Bureau of Ships 
 
 Department of the Navy 
 
 ATTN; Code 838D, L E, Shoemaker 
 
 Washington 25„ D C 
 
 Director 
 
 Naval Research Laboratory 
 
 ATTN, Dr J I Bohnert 
 
 Anacostia 
 
 Washington 25„ D„C 
 
 National Bureau of Standards 
 Department of Commerce 
 ATTN? Dr A G McNish 
 Washington 25, DX, 
 
 Director 
 
 UoSo Navy Electronics Lab 
 
 Point Loma 
 
 San Diego 52„ California 
 
 Commander 
 
 USA White Sands Signal Agency 
 
 White Sands Proving Command 
 
 Aim SIGWS-FC 02 
 
 White Sands, N.M.. 
 
 Consolidated Vultee Aircraft Corp 
 
 Fort Worth Division 
 
 ATTN- CJ. Curnutt 
 
 Fort Worth, Texas 
 
 M/F Contract AF33(038)=21117 
 
 Textron American^ Inc„ Div, 
 Dalmo Victor Company 
 ATTN: Mr, Glen Walters 
 1414 El Camino Real 
 San Carlos, California 
 M/F Contract AF33(038)=30525 
 
 Dome & Margolin 
 
 30 Sylvester Street 
 
 Westbury 
 
 Long Is land ,j New York 
 
 M/F Contract AF33( 616) -2037 , 
 
 Douglas Aircraft Company „. Inc<> 
 
 Long Beach Plant 
 
 ATTN J„Co Buckwalter 
 
 Long Beach l s California 
 M/F Contract AF33( 600) -25669 
 
 Boeing Airplane Company 
 
 ATTN; F Bushman 
 
 7755 Marginal Way 
 
 Seattle, Washington 
 
 M/F Contract AF33(038)~31096 
 
 Chance-Vought Aircraft Division 
 United Aircraft Corporation 
 ATTNs Mr F„N Kickerman 
 TIRUs BuAer Representative 
 Dallas, Texas 
 
 Consoiidated-Vultee Aircraft Corp 
 
 ATTN? Mr R E, 'loner 
 
 Po0 Box 1950 
 
 San Diego 12, California 
 
 M/F Contract AF33( 600 )« 26530 
 
 Grumman Aircraft Engineering Corp 
 
 ATTN; J,S C Erickson 5 
 
 Asst, Chief; Avionics Dept. 
 
 Bethpage 
 
 Long Is land £ New York 
 
 M/F Contract NOa(s) 51=118 
 
 Hallicrafters Corporation 
 
 ATTN Norman Foot 
 
 440 W„ 5th Avenue 
 
 Chicago Illinois 
 
 M/F Contract AF33(600)=26117 
 
 Hoffman Laboratories t Inc 
 ATTN? S, Varian 
 Los Angeles California 
 M/F Contract AF33(600) = 17529 
 
 Hughes Aircraft Corporation 
 
 Division of Hughes Tool Company 
 
 ATTN; D„ Adcock 
 
 Florence Avenue at Teale 
 
 Culver City, California 
 
 M/F Contract AF33(600)«27615 
 
 Illinois University of 
 
 Head, Department of Elec Eng, 
 
 ATTN: Dr EC Jordan 
 
 Urbana Illinois 
 
 Johns Hopkins University 
 Radiation Laboratory 
 ATTNs Librarian 
 1315 St„ Paul Street 
 Baltimore 2* Maryland 
 M/F Contract AF33(616)<=>68 
 
DISTRIBUTION LIST (CONT ) AF33(616) 3220 
 
 PM* 3 
 
 One copy each unless otherwise indicated 
 
 Electronics Research p Inc 
 
 2300 N; New York Avenue 
 
 P Oo Box 327 
 
 Evansville 4„ Indiana 
 
 M/F Contract AF33(616)o2113 
 
 Glenn L Martin Company 
 
 Baltimore 3„ Maryland 
 
 M/F Contract AF33(600) =21703 
 
 McDonnell Aircraft Corporation 
 ATTN? Engineering Library 
 Lambert Municipal Airport 
 St„ Louis 21 f Missouri 
 M/F Contract AF33( 600) -87 43 
 
 Michigan!, University of 
 Aeronautical Research Center 
 ATTN? Dr, L. Cutrona 
 Willow Run Airport 
 Ypailantij Michigan 
 M/F Contract Af33(038)-21573 
 
 Massachusetts Institute of Tsch a 
 ATTN g . Profo H J Zimmermami 
 
 Research Leb of Electronics 
 Cambridge Massachusetts 
 M/F Contract AF33(616)=2107 
 
 North American Aviation,, Inct, 
 Aerophysiea Laboratory 
 ATINg Dr JoAo Marsh 
 12214 Lakewood Boulevard 
 Downey,, California 
 M/F Contract AF33C 038) -18319 
 
 North American Aviation 5 Inc* 
 Los Angeles International Airport 
 ATTNs Mr u Dave Mason 
 
 Engineering Data Section 
 Los Angeles 45 California 
 M/F Contract AF33(038) 18319 
 
 Northrop Aircraft Incorporated 
 ATTN? Northrop Library 
 Depto 2135 
 
 Hawthorne,, California 
 
 M/F Contract AF33(6Q0)« 22313 
 
 Radioplane Company 
 
 Van Nuys California 
 
 M/F Contract AF33C 600) -23893 
 
 Lockheed Aircraft Corporation 
 ATINs C.Lo Johnson 
 P,0 Box 55 
 Burbankp California 
 M/F NOa(S)«52=763 
 
 Raytheon Manufacturing Company 
 
 ATIT^ Robert Borts 
 
 Wayiand Laboratory , Wayland s Mass 
 
 Republic Aviation Corporation 
 ATTNi Engineering Library 
 Farmingdale 
 
 Long Island, New York 
 
 M/F Contract AF33(038) 14810 
 
 Sperry Gyroscope Company 
 
 ATIWg Mr, Bo Berkowitx 
 
 Great Neek 
 
 Long Island, New York 
 
 M/F Contract AF3 3(038) -14324 
 
 Temco Aircraft Corp 
 ATTOs Mr,, George Cramer 
 Garland, Texas 
 (Contract AF33( 600) -21714 
 
 Farnsworth Electronics Co* 
 
 ATINs George Giffin 
 
 Ft- Wayne s Indiana 
 
 Marked! For Con s AF33( 600) =25523 
 
 North American Aviation, Inc» 
 4300 E Fifth Ave 
 ColusnWsj Ohio 
 ATTN! Mr. James D! Leonard 
 Contract/ NO(as> 54-323 
 
 Stanford Research Institute 
 Aircraft Radiation Systems Lab, 
 ATOSg Dr. J„V. N„ Granger 
 Stanford California 
 M/F CoBatract AF33C 600) -25669 
 
 Weafciaaghossse Electric Corporation 
 
 Aiir Arm Division 
 
 ATOi Mffo P, Do Newhouser 
 
 D»v<el©pfl»®rat Engineering 
 Friendship Airp@fffe Maryland 
 Contort AF33C 60 0)o 27852 
 
page 4 
 
 DISTRIBUTION LIST (CONT. ) 
 
 AF33(616)-3220 
 
 One copy each unless otherwise indicated 
 
 Ohio State Univ. Research Foundation 
 ATTN: Dr. T.C Tice 
 
 310 Administration Bldg. 
 
 Ohio State University 
 Columbus J.0, Ohio 
 M/F Contract AF33(616)-3353 
 
 Air Force Development Field 
 
 Representative 
 
 ATTN: Capt. Carl B. Ausfahl 
 
 Code 1010 
 Naval Research Laboratory 
 Washington 25, D.C. 
 
 Chief of Naval Research 
 Department of the Navy 
 ATTN: Mr. Harry Harrison 
 
 Code 427, Room 2604 
 
 Bldg. T-3 
 
 Washington 25, D.C. 
 
 Beech Aircraft Corporation 
 ATTN: Chief Engineer 
 6600 E. Central Avenue 
 Wichita 1 , Kansas 
 M/F Contract AF33( 600) -20910 
 
 Land-Air Incorporated 
 Cheyenne Division 
 ATTN: Mr. R.J. Klessig 
 
 Chief Engineer 
 Cheyenne ) Wyoming 
 M/F Contract AF33(600)-22964 
 
 Director, National Security Agency 
 RADE 1 GM, ATTN: Lt . Manning 
 Washington 25, D.C. 
 
 Mel par, Inc. 
 3000 Arlington Blvd. 
 Fails Church, Virginia 
 ATTN: K.S. Kelleher 
 
 Naval Air Missile Test Center 
 Point Mugu, California 
 ATTN: Antenna Section 
 
 Fairchild Engine & Airplane Corp. 
 Fairchild Airplane Division 
 ATTN: L. Fahnestock 
 Hagerstown, Maryland 
 M/F Contract AF33(038)-18499 
 
 Federal Telecommunications Lab. 
 
 ATTN: Mr. A. Kandoian 
 
 500 Washington Avenue 
 
 Nutley 10, New Jersey 
 
 M/F Contract AF33( 616) -3071 
 
 Ryan Aeronautical Company 
 
 Lindbergh Drive 
 
 San Diego 12, California 
 
 M/F Contract W-33(038)-ac-21370 
 
 Republic Aviation Corporation 
 ATTN: Mr. Thatcher 
 Hicksville, Long Island, New York 
 M/F Contract AF18( 600) -1602 
 
 General Electric Co. 
 
 French Road 
 
 Utica, New York 
 
 ATTN: Mr. Grimm, LMEED 
 
 M/F Contract AF33(600)-30632 
 
page 5 
 
 DISTRIBUTION LIST (CONT. ) 
 
 AF33(616)-3220 
 
 One copy each unless otherwise indicated 
 
 Stanford Research Institute 
 
 Southern California Laboratories 
 
 ATTN: Document Librarian 
 
 820 Mission Street 
 
 South Pasadena, California 
 
 Contract AF19(604)-1296 
 
 Prof. J.R. Whinnery 
 Dept. of Electrical Engineering 
 University of California 
 Berkeley, California 
 
 Professor Morris Kline 
 Mathematics Research Group 
 New York University 
 45 Astor Place 
 New York, N.Y. 
 
 Prof. A. A. Oliner 
 Microwave Research Institute 
 Polytechnic Institute of Brooklyn 
 55 Johnson Street - Third Floor 
 Brooklyn, New York 
 
 Dr. C.H. Papas 
 
 Dept . of Electrical Engineering 
 California Institute of Technology 
 Pasadena, California 
 
 Electronics Research Laboratory 
 Stanford University 
 Stanford, California 
 ATTN: Dr. F.E. Terman 
 
 Radio Corporation of America 
 R.C.A. Laboratories Division 
 Princeton, New Jersey 
 ATTN: Librarian 
 
 Electrical Engineering Res . Lab. 
 University of Texas 
 Box 8026, University Station 
 Austin, Texas 
 
 Dr. Robert Hansen 
 8356 Chase Avenue 
 Los Angeles 45, California 
 
 Technical Library 
 
 Bell Telephone Laboratories 
 
 463 West St. 
 
 New York 14, N.Y. 
 
 Dr. R.E. Beam 
 Microwave Laboratory 
 Northwestern University 
 Evanston, Illinois 
 
 Department of Electrical Engineering 
 Cornell University 
 Ithaca, New York 
 ATTN: Dr. H.G. Booker 
 
 Applied Physics Laboratory 
 Johns Hopkins University 
 8621 Georgia Avenue 
 Silver Spring, Maryland 
 
 Exchange and Gift Division 
 The Library of Congress 
 Washington 25, D.C. 
 
 Ennis Kuhlman 
 
 c/o McDonnell Aircraft 
 
 P.O. Box 516 
 
 Lambert Municipal Airport 
 
 St. Louis 21, Mo. 
 
 Physical Science Lab. 
 New Mexico College of A and MA 
 State College, New Mexico 
 ATTN: R. Dressel 
 
 Technical Reports Collection 
 
 303 A Pierce Hall 
 
 Harvard University 
 
 Cambridge 38, Mass. 
 
 ATTN: Mrs. M.L. Cox, Librarian 
 
 Dr. R.H. DuHamel 
 Collins Radio Company 
 Cedar Rapids, Iowa 
 
 Dr. R. F. Hyneman 
 
 8725 Yorktown Avenue 
 
 Los Angeles 45, California 
 
UNIVERSITY OF ILLINOIS-URBANA 
 
 Q.621.365IL655TE C002 
 
 TECHNICAL REPORT$URBANA 
 20 1957 
 
 iiiiiiiiigiiiiiiNiiviniiiii 
 
 IH^KERSITY OF IUW*Q£ 
 UBRARY