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Digitized by the Internet Archive 
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 http://archive.org/details/directionalchara12beck 
 
Antenna Laboratory 
 
 Technical Report No. 12 
 
 DIRECTIONAL CHARACTERISTICS 
 OF A 
 U-SHAPED SLOT ANTENNA 
 
 by 
 Richard C. Becker 
 
 30 September 1956 
 
 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 
 
ABSTRACT 
 
 An expression for the radiation pattern of U-shaped slot antennas 
 is derived, based upon an assumed distribution of magnetic current in 
 the slot. The assumed distribution is a low- frequency approximation, 
 i.e., the distribution chosen is similar to those which are usually 
 associated with antennas having dimensions which are only a fraction of 
 a wavelength in magnitude, The range of validity for this current dis- 
 tribution is determined by comparison of calculated with experimentally- 
 measured patterns of antennas which were varied in slot length from one- 
 quarter to four-fifths wavelength 
 
 Patterns have been measured using two versions of the U-slot antenna 
 an external version which is readily adapted to methods of model measure- 
 ment, and a flush-mounted version which was anticipated to be equivalent 
 in its performance and which is practical for use on high-speed aircraft 
 
 1 1 
 
ACKNOWLEDGEMENT 
 
 The author gratefully acknowledges the assistance and directive 
 guidance of Professor Paul E„ Mayes in the preparation of this report, 
 and wishes to thank Professors V„ H„ Rumsey and Raymond H. DuHamel for 
 reading the manuscript and offering suggestions which have been incor- 
 porated in the final form of this report. Appreciation is also extended 
 to R„ Trapp and members of the technician staff who performed most of 
 the pattern measurements. 
 
 111 
 
TABLE OF CONTENTS 
 
 Page 
 
 Abstract ii 
 
 Acknowledgement Hi 
 
 1„ Introduction 1 
 
 2. Derivation of an Expression for the Radiation 
 
 Pattern of the U-Slot Antenna 3 
 
 3« Calculation of Patterns for Various Antenna Dimensions 8 
 
 4„ Comparison of the Calculated and Experimental 
 
 Radiation Patterns 9 
 
 5o Conclusions 26 
 
 Bibliography 27 
 
 Distribution List 
 
 IV 
 
1. INTRODUCTION 
 
 The U shape type of slot antenna investigated in this report would 
 normally be utilized as an electrically small antenna. Under such con- 
 ditions the radiation pattern has an analytical representation which 
 can be derived using well known and straightforward techniques.. 
 
 It is the purpose of this investigation, however, to determine 
 through what range the theoretical 'low frequency" representation for 
 the current distribution on the antenna remains valid Therefore, an 
 analytical expression for the radiation pattern is derived, and patterns 
 are calculated for slot lengths which vary from one quarter to almost a 
 full wavelength. Experimental data are presented as an indication of 
 the validity of the approximation in this region of dimensional transi- 
 tion. 
 
 The derivation of the radiation pattern of the U-slot antenna is 
 presented as an extension of the methods commonly employed for .pattern 
 calculations of simple slot antennas, or current distributions, A 
 reasonable current distribution is assumed for a narrow slot and the 
 far- zone field is then calculated subject to simplifying approximations 
 which are commonly introduced in problems of this type 
 
 Figures 1 and 2 illustrate two versions of the U shaped slot an- 
 tenna which have been investigated experimentally The dimensions of 
 these antennas are such that: 
 
 / 
 I = I 
 
 h/2 = 1/10 
 
 The external version of the U-slot antenna is more easily constructed 
 and installed on models for purposes of experimentation than its 
 counterpart In addition, damage to the model surface is minimized 
 On the other hand, however, the flush-mounted slot antenna is the con- 
 figuration which is most desirable for installation on high speed air- 
 craft. 
 
Ground Plane 
 
 Figure L External U-Slot Antenna 
 
 Figure 2 Flush-Type Cavity Backed U-Slot Antenna 
 
2. DERIVATION OF AN EXPRESSION FOR THE RADIATION PATTERN OF 
 THE U-SLOT ANTENNA 
 
 For purposes of this analysis, the U-slot antenna is assumed to be 
 located in an infinite ground plane, and the radiation pattern of this 
 configuration is then calculated for an assumed distribution of electric 
 field (magnetic current) in its aperture. Figure 3 is a graphical 
 representation of the problem indicating the coordinate system chosen 
 and the slot distribution which has been assumed, Since we are 
 interested in the radiation on only one side of the infinite ground 
 plane, the slot aperture is considered to be excited by an arbitrary 
 distribution of sources in the opposite half -space. The slot then 
 functions as a diffracting aperture, and the electromagnetic field 
 on the source -free side of the plane may be determined from the 
 field distribution appearing in the slot. 
 
 An exact expression for the electric field at a point P, on the 
 
 3 
 source-free side is represented by the vector potential formula; 
 
 E(P) = A p x 
 
 e -3rpQ 
 
 n x E(Q.) — dS n 
 
 ~ " 27tr pQ 
 
 aperture 
 
 (1) 
 
 where r p Q represents the distance from P to a point , Q., in the aperture. 
 It should be pointed out that this formula is not sufficient for an 
 exact analysis , however , because tangential E in the aperture is 
 presumably unknown. 
 
 The simplified approach to the problem is presented in Fig. 4', 
 together with the notation to be used in deriving the expression for 
 the radiation pattern. The formulas for the spherical components 
 of the total distant field derivable from Eq . 1 are equivalent to 
 Eqs . 2 . The total field radiated by the U-slot is considered 
 to be the superposition of the fields produced by three simple 
 rectangular slots having been oriented to form the U-shaped configu- 
 ration. 
 
 Choose a primary coordinate system together with auxiliary 
 coordinate systems as indicated. In addition, consider the field 
 due to the i th component slot as expressed in terms of an 
 equivalent quasi-point source at the auxiliary origin, 0, , located 
 
 ■3- 
 
Plx,y,z) 
 
 £slot 
 
 ( tangent iai) = y k sin (3j(x + 2-) 
 
 2 
 
 -slot 2 ( tan Qential) = x k 
 
 -slot 3 (tangential) -y k sin (3(x + -') 
 
 Figure 3. Location of the U-Slot Antenna in an 
 
 Infinite Ground Plane, and the Assumed Electric 
 Field Distribution in its Aperture. 
 
PU,y,z) 
 (r,e,</>) 
 
 Hi s ^apert.Q * _£. 
 
 Figure 4. Jhe Magnetic Current Line Sources Wh;ch Are 
 Assumed To Represent the Radiating Effects of 
 The U-Slot Antenna, Together with Coordinate 
 Notation Used 
 
 within the respective source distribution. The expression for the total 
 field at a distant point is then given as the sum of the component 
 fields, and has the general formf 
 
 
 1 = 1 
 
 
 
 3 
 
 Eq = ) Eg. - See following page 
 
 i = l 
 
 5- 
 
-jw|j -j3r 
 e 
 
 4nr 
 
 SU.jft-^^^i 
 
 J0f • Bi 
 
 6 
 
 1=1 
 
 j<4i -j3r 
 Wv e 
 
 Jftfc 
 
 iGi e 
 
 J0P X 
 
 
 ^ 
 
 (2) 
 
 where 
 
 K = magnetic current density 
 
 p^ = vector distance from the auxiliary origin to a point, Q, in its 
 
 respective source distribution 
 r = unit vector from the primary origin to the distant field point, P. 
 R x ■ vector distance from the primary origin to the respective auxiliary 
 origin. 
 
 After performing the respective integrations, summing and combining 
 like terms, the spherical components of the field are given by 
 
 
 
 (3) 
 
 te 
 
 .1 ■ JTTe* ke 
 
 -j3rl 
 
 rcBr 
 
 7 3a . 3a 
 
 2 sin cp sin — - sin 9 sin cp cos — 
 
 \ 2 J 2 
 
 (] sin 2 9 cos 2 cp) 
 
 3a 
 
 cos — sin 
 
 2 
 
 3a 
 
 sin o coscp 
 
 . Ba fla . . 
 
 sin t) cos <p sm -— cos I— sin 6 cos 
 
 sin 3a cos cp [ |3a 
 
 — : — sin — sin 8 sin Cf 
 
 sin sin cp 1 2 
 
 [ 3a 
 
 cos — sin b cos cp 
 2 
 
 
 0a . n . \ 3a 
 
 2 ' in 'V tin j ..in 9 sin cp sin — r~ 
 
 ^1 -siu 2 8 cos 2 cp| 
 
 3a 3a 
 
 sin — cos — sin 8. cos cp 
 2 2 
 
 6 
 
P a ■ 
 
 sin 9 cos cp cos -s- sin 
 
 3a 
 
 sin fc> cos 
 
 sin 3a cos cp [ 3a . 
 
 sin ~T" sin 6 sin 
 
 sin b sin cp 
 
 P 3 : o 
 sin — sin 9 cos cp 
 
 \ 
 
 (4) 
 
 jwjiJI ke 
 
 ■J0r| 
 
 nSr 
 
 _ . [0a , . . 
 2 cos cp cos 9 sin — sm 9 sm cp 
 
 cos 
 
 (3a 
 
 (l-sin 2 9 cos 2 cp) 
 
 3a 3a . 
 
 - cos — sin — sin 9 cos cp 
 2 2 
 
 a 3a 3a 
 
 sin 9 cos cp sin - — cos — sin 9 cos cp 
 2 2 
 
 sin 3a sin cp cos 
 sin 9 sin cp 
 
 sin 
 
 P a ex P a o 
 
 — sin 9 sin cp cos — - sin 9 cos cp 
 
 2 2 
 
 + i 
 
 2 cos cp cos 9 sin 
 
 3a 3a 
 
 — sin 9 sin cp sin — — 
 2 2 
 
 (l-s:.n 2 cos 2 cp) 
 
 3a 
 
 sin — cos 
 2 
 
 3a 
 
 sin 9 cos cp 
 
 = sm 9 cos cp cos — sin — sin 
 
 2 2 
 
 cos cp 
 
 sin 3a sin cp cos 
 
 sin 
 
 03 • a ■ ■ 
 
 — sin 9 sin cp sin 
 
 3a 
 
 sin t) sm 
 
 sm 9 cos cp 
 
 (5) 
 
 Equations 3,4', and 5 thus afford a first order description of the 
 radiation pattern of the U-slot antenna, subject to the approximations 
 commonly employed to facilitate such a derivation. 
 
3 CALCULATION OF PATTERNS FOR VARIOUS ANTENNA DIMENSIONS 
 
 Several patterns have been calculated and plotted from the 
 analytical expressions for the radiation field derived in the pre- 
 vious section. Table 1 describes the parametric values for the 
 respective patterns computed and gives their simplified representations 
 The graphs in which these radiation patterns are illustrated are also 
 indicated 
 
 TABLE 1. 
 
 Figure 5: |Eg| plotted for 6 = 90°, 
 
 0° < tp < 360° 
 
 E 9 ((p) 9=90o 
 
 2 sin (*2" sin V' 
 
 sin 
 
 4 P* a 2/^ a x 
 
 sm^ "2" + cos pa sin vt cos cp) 
 
 Figure 6: \Eq\ plotted for cp = 90°, 
 
 0° < < 90° 
 
 2 Pa . (3a 
 Eq(9)cp=90o = 2 sin^ — sin (— sin 0) 
 
 Figure 7: |Eg| plotted for cp = 0°, 
 
 0° < 1 90°. 
 
 E0( e >c?=O o ' = "T Sin ^ a 
 Figure 3: lEj plotted for <p f 90° 
 
 0° < < 90° 
 
 E {p(9)<p-90 o 
 
 tin p a aojL_0_ sin (Li s in 0) 
 sin 2 
 
 Bach pattern has been calculated at seven frequencies in order to 
 resent adequately the pattern performance of this antenna in the 
 dimensional transition. At these frequencies the slot di- 
 mension, a. of Fig. 3, has the lengths: 
 
 25a, 0.3X; OAK; 0. 5A; 0.6A; 0..7X- 8A 
 
 (6) 
 
4. COMPARISON OF THE CALCULATED AND EXPERIMENTAL RADIATION 
 PATTERNS . 
 
 To obtain the desired experimental patterns, the two antenna models 
 were constructed of copper sheet and mounted on brass plates for use in a 
 test setup utilizing a ground screen 4 , The models were used as trans- 
 mitting antennas and a linearly polarized receiving antenna was used to 
 probe the radiated field. Patterns were measured at seven frequencies 
 corresponding to the slot lengths given in Eq„ 6, 
 
 Figures 5,6,7, and 8 are composite pattern plots including both 
 the experimental and calculated patterns. The experimental patterns 
 are placed on either side of the calculated pattern for ready comparison 
 of theory with experiment. The calculated patterns of Figs 5,6, and 7 
 have been plotted so that they have the same scale factor. The calculated 
 signal magnitudes of Fig. 8 would be twice as great as they appear if 
 they were plotted to the same scale as the patterns in the previous three 
 figures. It is therefore possible to observe the relative pattern magni- 
 tudes for all the calculated patterns. Comparison may then be made with 
 respect to the field components (0 and cp), and with respect to the param- 
 eters of frequency and angular position. The experimental patterns 
 afford reliable information pertaining only to the pattern structure of 
 the two versions of the U-slot antenna. 
 
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 13 
 
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 = 0.3 > 
 
 Figure G-a. E G (G) for <p 90° 
 
 L4 
 
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16- 
 
a 8 ) 
 
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 Figure 6-d. E e (6) for cp - 90 ( 
 
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 90"> 
 
 Figure 8-a. E<p(0) for rp 90° 
 
 -22- 
 
0-04 X 
 
 a* Q.§& 
 
 No Measurable Pa* 'em 
 
 Figure 8-b, EJQ) for cp =90° 
 
 ■23- 
 
-25- 
 
5= CONCLUSIONS 
 
 The pattern performance of a U-slot antenna having dimensions less 
 than a quarter wavelength can be predicted quite accurately by using the 
 proposed current distribution. This is verified by all four composite 
 pattern plots since the experimental and calculated patterns are 
 practically identical at the frequencies where the slot dimension, a, 
 is snail in terms of wavelength. Other current distributions which are 
 closely akin, such as a triangular increase of current along the 
 parallel arms and a constant-value current on the transverse arm of the 
 U-slot, would be expected to give almost identical results for antennas 
 having limited electrical length. 
 
 The purpose of this investigation has been to obtain information 
 which indicated the extent to which the proposed aperture distribution 
 remained valid in representing analytically the radiation pattern of 
 the U-s\ot antenna. It is Kern from the composite pattern plots that 
 the assumed current distribution yields radiation patterns which agree 
 quite well with those measured up to and slightly beyond a - 0.5A. 
 Figure 6- illustrates a situation in which the directional 
 i :?c .eristics of the internal and external versions; together with the 
 :ulated pattern, are essentially indistinguishable, even for slot 
 nsions of the order of a full wavelength „ Figure 7, on the other 
 tu presents a situation in which the equivalence of the flush- 
 counted and externaj antennas in -heir pattern performance is 
 iim:. ie<i to dimensions not exceeding a quarter wavelength., The dissimi- 
 larity in the actual construction and orientation with respect to the 
 mnd plane of the two modr Ls is especially pronounced at the higher 
 i < es in Fig. 7 
 
 The experimental patterns also make evident the limitations of 
 uracy for the experiment user'. The effect of the finite dimensions 
 of the gr in] plane and interaction of transmitting and receiving 
 
 Dtenna* appear.'; as raggedness on most of the patterns measured- 
 Lik« mechanical dissymmetries in the construction of the model 
 antennas become evident at the higher frequencies. 
 
 • i compel ng the computed patterns with the experimental patterns 
 lor i I is J -mounted mode J of the U-slot antenna, a general lag in 
 riatica for the calculated patterns behind the 
 
 1 ■ i quency is increased. This would seem to 
 the ' Lot Lenj f the experimental antenna is effectively 
 
 26 
 
larger in electrical length than was assumed for the theoretical 
 representation. This lag does not appear, however, when comparing the 
 performance of the external antenna and the calculated pattern. This 
 difference in correlation might be expected if one took particular 
 note of the related dimensions appearing in Figs. 1 and 2. 
 
 BIBLIOGRAPHY 
 
 Silver, S. Microwave Antenna Theory and Design, McGraw-Hill Book 
 Co., Inc., New York, N.Y. , 1949, Chapter 3. 
 
 Schelkunoff, S.A. Electromagnetic Waves, D, Van Nostrand Co., 
 New York, N.Y. , 1943, Chapter IX 
 
 Rumsey , . V.H.. HuygenS' Principle for Electromagnetic Waves, Ohio 
 State Research Foundation Report, January 25, 1954 
 
 Trapp, R.R'. Ground Screen Pattern Range, Technical Memorandum 
 
 No, 1, Antenna Section, Electrical Engineering Research Laboratory, 
 
 University of Illinois, July 10, 1955 
 
 •27 
 
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