Planar phased array fan beam scanning system
Nelson Apr. s, 1975 PLANAR PHASED ARRAY FAN BEAM SCANNING SYSTEM 3,646,558 2/1972 Campanella 343/100 SA Primary E.\&#39;aminerT. H. Tubbesing Inventor: Everett A. Nelson, Whitesboro, N.Y.  
 [73] Assignee: General Electric Company, Utica. [57] ABSTRACT Array element phase commands used to scan the [22] Fil d; A 9, 1973 beam of a planar phased array are conventionally obtained by adding to the beam shaping phases, phases 349334 obtained by sampling a linear (planar) function of position on the array surface. When this technique is ap- [52] U.S. Cl. 343/100 SA; 343/854 plied to phase s p a a the a a quires [51] Int. Cl. H0lq 3/26 an increasingly rl d r arped shape as the scan [58] Field of Search 343/100 SA angle is increased- To offset this n y. e ca ning phase commands are obtained by sampling a [56] Referen e Cit d function which becomes increasingly nonlinear UNITED STATES PATENTS (warped) as scan angle is increased. 3,482,244 l2/l969 Gadenne 343/100 SA 1 Claim, 11 Drawing Figures 23 i sin a 30&#39; 4&#39; ,40 38 f 3/5; t; esus/2&#39;&#34; L 2 24 z,  
 i 49 g DR sins, 361 42 Foe 24 M I v sme, Us M I 23 l A I 345 L 24 25 /k -[r x,,)+ksin,fx 60 I I i D 24 25 23 on sm 1&#39;. I r i kx sirw. I l l I &#39;62 e3 .26 31 I Ass: 33 n ELEMENT PATENTEBAPR 81s SAIU 2 [IF 4 x FlG.3b.  
 Yx) kx cos 4,  
 CONSTANT 9 ARC BEAM SCANNED Ad, IN xz PLANE EDGE OF ELEMENT PATTERN CONSTANT 1} ARC 5 2 2 2 2 4 4 4 2 4 2 2 m 2 m 0 n I w 2 .llllllll l l l l l l I l .ll |.|l| 2 x n LOG C MAN FOLD f g 0 n y x 1% Z I n a u 0 a .m 2 S 5 m 4 4 r k M, 4 n M ..N 0 D X EN 0% W 4, l BA n n n m X MM I l\ CM 3 2 0 mm m k M n P V H I l l l I l I l l l I l l ll. I I l l l I 4).. o rill l I I l l l I l l l l l l I II I L 5 I w v x M s a a a L T w rfw m d, LL N mms m .m .m MW M w 8 SW 3 S s A E L E PLANAR PHASED ARRAY FAN BEAM SCANNING SYSTEM BACKGROUND OF THE INVENTION This invention relates to phased array technology radio wave propagation and more particularly to directive scanning of a phased array beam.  
  A planar phased array consists of a plurality of array elements, the radiation from which adds and cancels to provide a beam. The beam scanned by the phased array is determined by the phase commands provided to each element. The present invention is particularly applicable to planar phased arrays having large apertures in terms of radiated wave length.&#39;Fan beam scanning is used, for example, in airborne radar for ground mapping. While the present invention is discussed in many aspects in terms of fan beam scanning, it should be realized that this is only by way of exemplification, and the present invention is also applicable, for example, to pencil beam scanning and other forms of shaped beam scanning in accordance with the teachings of the present invention.  
  In conventional prior systems, a linear function has been utilized for scanning a fan beam. The linear function is sampled, and added to the phase command for each array element. While the ideal fan beam would have a generally planar three dimensional shape, the fan beam of a conventionally scanned planar phased array will in general have a conical shape. Consequently, for a ground mapping beam scanned in the conventional way, the intersection of the beam with the ground is a hyperbolic arc, rather than the desired straight line. The intersection only approaches a straight line asymptotically at longer ranges. Deviation of the hyperbolic are from the straight line is greater at short ranges, and increases rapidly with scan angle. This effect is commonly known as beam curling. The result of beam curling is gain reduction and directional error. The analogy may be drawn of an operator looking at a radar display of a curled beam and the person looking at an object at the bottom of the swimming pool. To both, the object from which returns are received are at a place other than where it appears to be. At moderate to large scan angles, the short range portion of the cathode ray tube ground map display of returns from a conventionally scanned planar phased array will be both distorted and washed out.  
 SUMMARY OF THE INVENTION It is therefore an object of the present invention to provide a planar phased array scanning system in which beam curling is minimized.  
  It is also an object of the present invention to scan a planar phased array by sampling of a warped function rather than a linear function.  
  It is another object of the present invention to provide a method and means for scanning a planar phased array to provide a fan beam that is more nearly planar rather than conical in shape, whereby a beam directed from the air toward the ground has an intersection with the ground that is a straight line.  
  It is yet another object of the present invention to provide a method and means for scanning a planar phased array whereby gain variation with scan angle is minimized.  
  It is a general object of the present invention to pro vide a method and means for scanning a planar phased array in which distortion and directional error is minimized.  
  Briefly stated, in accordance with the present invention, there is provided a means of scanning a planar phased array beam in which scanning phases are obtained by sampling a warped function. The warped function is determined from the derivative of the phase function used to obtain the desired beam shape. The amount of warping is increased as scan angle increases in order to compensate for the increasing tendency of the scanned beam to curl.  
 BRIEF DESCRIPTION OF THE DRAWINGS The means by which the foregoing objects and features of novelty are achieved are pointed out with particularity in the claims forming the concluding portion of the specification. The invention, both as to its organization and manner of operation, may be further understood by reference to the following description taken in connection with the following drawings.  
 Of the drawings:  
  FIG. 1 is illustrative of a sphere in the far field of the aperture of a planar phased array illustrating various forms of antenna beams;  
  FIGS. 2 and 3 are geometrical illustrations useful in understanding the theory of operation of the present invention;  
  FIG. 4 is a block diagramatic representation of one means for implementing the present invention; and  
  FIGS. 5 and 6 are charts useful in understanding the operation of the present invention.  
 DESCRIPTION OF THE PREFERRED EMBODIMENT Referring now to FIG. 1, there is illustrated a sphere in the far field of the aperture of a planar phased array at an origin 0. The far field is defined by stating that lines from any point on the aperture to a point in the far field are substantially parallel. The x axis is vertical, and the y and z axes are horizontal. The origin 0 may be viewed as the position of a planar phased array in an aircraft having a heading along the z axis and flying in the y-z plane. In this conceptualization, the ground may be represented as a y&#39;z&#39; plane parallel to the y-z plane.  
  A fan beam is a beam having a major lobe with a highly elliptical cross section. For ground mapping purposes, the major diameter of the ellipse is large with respect to the minor diameter. For purposes of the present exemplification, the fan beam is described in terms of the major diameter. Due to resolution of a radar system and conventional shaping of a fan beam, it is proper to approximate the fan beam as having a cross section being a line. In FIG. 1, the curve 1 represents the intersection of an unscanned fan beam with the far field sphere. All lines drawn to the beam lie in a plane, here the x-z plane. The projection of the curve 1 on the y&#39;z plane lies along the z axis. It is desired to scan the beam through an azimuth angle 6 and maintain the same beam shape, namely a scanned beam 2 intersecting the far field sphere along a great circle and having an intersection 3 with the y&#39;z plane which is a straight line. In this manner,&#39;no directional error appears in the return wave. This form of accurate scanning is achieved with mechanical scanning since the entire antenna rotates. However, to scan a phased array, phase commands must be provided to provide a resultant beam which is scanned while the antenna itself does not rotate about the x axis.  
  An explanation of the relation of the mathematical function utilized to the beam produced and of beam steering by element phasing is found in Allen, .I. L. Array Antennas: New Applications For An Old Technique, I.E.E. Spectrum, Nov. 1964, pp ll5l30. The conventional form of phased array scanning by sampling a linear function produces a scanned beam 6. The beam 2 comprises an arc which at all portions is of constant azimuth for a given angle 0. However, the conventionally scanned beam 6 for a given angle 6 is not of constant azimuth, but is at a constant angle to the y axis. All lines drawn to the curve 6 from the origin form the side of the cone. Therefore, the beam 6 is said to be curled. The intersection 7 of the beam 6 with the yz plane is a hyperbolic arc. Deviation of intersection 7 from intersection 3 is indicative of directional error of the beam 6. The directional error is at its greatest value at short ranges from the x axis and also increases rapidly with increasing values of 9.  
  In order to illustrate this problem, the equations by which conventional scanning is expressed are exam ined. It has been the convention to assume that the phase scanning. of the beam of a planar aperture of an arry implies the superposition on the aperture of a phase distribution that is a linear function of the position on the aperture. It is an inherent characteristic of such a phase scanned beam to tend to retain its shape in cosine space independent of scan angle. This leads to the above described distortion, as, for example, of a fan-shaped beam known in the art as a csc cos beam frequency used for ground mapping (where is the dip angle. or angle below the horizontal plane). This examination is made with reference to FIG. 2 in which FIG. 2a is a spherical plot illustrating direction angles and the radial unit vector TJ FIG. 2b is a plot of cos a vs. cos or, (defined below) illustrating cosine space; and FIG. 2c is a plot of a sphere in far field of the aperture, similar to FIG. 1.  
  Consider an array of N identical element radiators in the .\&#39;-y plane (FIG. 2a). The array is large enough so that edge effects may be assumed negligible. Let the lo-.  
 cation of an element n be given by F Ecr -ti y&#34; (l) and let the relative current at a reference point be A =|A,,|,. tlm, where 17 is a unit vector and 1!; is a phase function. Then. the field intensity in the radial direction 11,- is given by N FUI =f(il 2 A ei n&#39; r (2) where k=2 1r and f(fi, is the field intensity of a single element in the array environment when A,,=1 for that element. The summation in equation (2) is called the array factor. In fact, it is the radiation pattern of an array of isotropic elements. When f(fi is very broad (as it almost always is) the shape of the radiation pattern of a large aperture is usually determined primarily by the array factor. The subscript o is used below to indicate a direction or angle in which a beam is steered. HE.) is a maximum when the contribution of all the elements add in phase. Therefore, let  
 From equation 1 ERIE- m) X&#34; (cos a, cos a y,, (cos a cos Therefore, the array factor F,, of equation (4) can be written,  
 F (cos (1 cos a It should be noted that, in the space defined by FIG. 2b, the shape of F is independant of (cos a cos a This property of planar apertures is not restricted to pencil beams.  
 Let &#39;y(.\-) be a non linear function of .r such that when ground map beam is formed in the .r-z plane. It is assumed that the desired shaped beam extends from 5 to 45 below the y-z (horizontal) plane and has all of its power in a region very close to the .r-z plane. This corresponds to or, and 01,, The intersection of the beam with a sphere in the far field of the aperture is indicated by the heavy black arc in FIG. 20.  
  Conventionally, such a beam has been scanned by superimposing on the aperture the linear phase function [ti -fi so that the element phase commands are now obtained from The array factor then becomes F (cos a cos m When beam scanning of a shaped beam is defined in this manner, it can be seen from equation (12) that the shape of F is still independent of (cos cos in cosine space (FIG. 2b). This means the beam will retain its shape on constant direction cosine arc. At moderate to large scan angles (the definition in numerical terms of moderate to large depending on the particular application), this retention of shape constitutes beam warping. Restrict 7, to the y-z plane. Then by: yn cos an!) 7 ii) and equation (10) becomes F,,(cos a cos a,,)  
  The present invention comprises means for providing a non-linear phasing function based on the derivative of the conventional beam function. In this manner, the dimensional error due to change in slope as the intersection 7 of FIG. 1 approaches the x axis is minimized. The theory of operation of the present invention is set forth below in order to demonstrate the manner in which beam curling is substantially eliminated. FIG. 3 is utilized in this explanation. FIG. 3a is a representation or a circular, planar array aperture; FIG. 3b is a plot of a phase function utilized and FIG. 3c is a plot of a sphere in the far field of an aperture, similar to FIG. 1.  
  If the intersection with ground (i.e., the y&#39;z&#39; plane of FIG. 1) of a ground map beam formed by a planar phased array is to be a straight line through the x axis, the intersection of the beam with the far field sphere must be an arc of constant 6. A scanning procedure that approximates this desired result will be developed.  
  The pattern of a large (in wave lengths) planar array can be approximated by that of a large continuous distribution of electromagnetic sources. Therefore, since it is also more convenient to do so, the phasing function for scanning a ground map beam will first be developed for a continuous aperture. Then the element phase commands, 1b,, as given by equation (3) or equation (9), can be approximated by a continuous function ofaperture position. It will be designated 111(5) Thus, for the continuous aperture, equation (3) can be replaced by (1 where prepresents the location of any point on the aperture, rather than the discrete element locations. Adding a&#39; constant to the right side of equation (13) will have no effect on the beam. This will be done for later convenience and equation 13) will be rewritten as follows:  
  The circular aperture illustrated in FIG. 3a will be used as an example. Let &#39;y (.r) be the function required to obtain the desired unscanned ground map beam in the .\&#39;z plane. To scan this beam in the conventional manner, equation (9) can be rewritten for application to a continuous aperture as w (3) no k 7;- Hr.  
 or, using the notation of equations (2) and (5) (F) &#39;y(.\&#39;) k.r coS a ky cos a,  
  For the moment, let&#39;cos a =0 (a Thus, the beam is scanned in the x-z plane, only. The intersection of this beam with the far field sphere is indicated by the dark arc in the x-z plane of the FIG. 30. Now, suppose a is held constant and the beam is scanned away from the .\&#39;z plane. The intersection of the beam with the far field sphere is now indicated by the dark constant a, are of FIG. 30. The dotted arc represents the desired beam intersection which is a constant 6 are (See FIG.  
  Divide the aperture of FIG. 3a into incremental strips such as the one shown at x=.r,. On each incremental strip, approximate the x dependent terms of equation (16) by a straight line function.  
 Replacing &#39;y (.r) +kx cos a in equation 16) by B,- (x), the phase function for strip i becomes w.- (i) B.- o) ky cos (18) which is a linear function of position on the strip.  
  Now, suppose that the power radiated by each incremental strip of the aperture is directed to a given segment of the radiation pattern on a one for one basis. For instance, suppose that all the power radiating from strip i passes through the beam cross section segment indicated by Au in FIG. 30. Holding a constant as the beam is scanned away from the plane, the segment A121,,- moves along a constant a, arc. However, if each beam strip can be independently phased. then each beam segment can be placed on the desired arc, instead of the constant a arc. Thus, the desired beam intersection could be obtained. Since 111, (5) is separable in .r and y, each beam segment can be steered to the constant 6 arc, 0=0,,, if the proper value of a, or a, is used in equation (18) for each aperture strip. That is each strip must be scanned by applying the equation.  
  l!- (7 ,3,- (x) ky cos a,  
 Considering FIGS. 1 and 2a it can be seen that cos a, cos 4),- sin 0., sin 4), cos 01,,  
 Therefore, it is necessary to find 01,,- for each strip. Each strip can be scanned by application of equation (14). Let  
 ll),- (5) kx cos 04,,- ky cos 01 B From equation (19) and (21) it can be seen that ,8,- (x) k.\- cos a B Comparing equations (17) and (22) cos a,,-=l/k m,-  
 Therefore,  
 cos ib sin a V k m 7 Substituting equations (20) and (24) into equation (19) yields W Bi y Sin n k2 &#34;H of course, in the limit as A x,- 0  
 m,- d/d.\&#39; [3,- (x) When x x Bi (X) &#39;y(.\-) kx cos a &#39;y(.\&#34;) kx sin 4),,  
 and  
 m,- &#39;y(.\-) k sin d where y&#39;(.\&#39;) =d/dx &#39;y(.\&#39;): Thus, in the limit equation (25) can be rewritten 41(3) y(.\&#39;) +kx sin (1)., +v sin 0,, V k [&#39;y(.\&#39;) +k sin This is the phase function desired to minimize the curl of a phase scanned fan beam. To apply equation (29) to a planar array, simply sample the function of each element position thus.  
  In order to compare the phase function of equation (30) to the conventional phase function equation (9) may be rewritten in terms of functions of azimuth and elevation as follows:  
  Utilizing the relationship cos =l-sin q5, equation (3 I) may be rewritten as follows:  
  By comparing the radicals of equations (32) and (30). it is seen that the apparatus of the present invention provides a correction to the scanning function dependent upon -y&#39;, the slope thereoffConsequently, the apparatus of the present invention operates to provide improved scanning of the radar beam.  
  Referring now to FIG. 4 there is illustrated in block diagrammatic form a phased array system constructed in accordance with the present invention. An array control computer 20 which is well-known in prior systems provides beam direction angles based on beam direction requirements and aircraft attitude to a beam steering computer 21 coupled thereto. The beam steering computer corresponds to the prior art computer providing the linear phasing commands described above. Outputs from the beam steering computer 21 are provided to a logic manifold 23, which provides phase commands to phase shift drivers 24. The logic manifold 23 comprises prior art interface circuitry between the beam steering computer 21 and the phase shifter drivers 24 which provide the phase commands to the antenna elements 25. Radio frequency energy is provided to the antenna elements 25 at an input port 26 from a well-known radio frequency source (not shown). One form of array may include two thousand ant&#39;enna elements 25. Only four are illustrated for simplicity.  
  The array control computer provides signals comprising digital words representing trigonometric functions of the beam direction angles which are characterized in the present exemplification as being provided at output terminals 30-36. Focusing phase command digital words are provided at an output terminal 32, and commands for enabling selected elements are provided at an output terminal 33. These commands are all provided in prior systems of the type described.  
  The beam steering computer 21 may take many forms in accordance with the present invention. In the present exemplification, the beam steering computer 21 is illustrated as including a pencil beam phase command generator 40 which is included in prior beam steering computers of the type described. A ganged electronic switch 41 is provided and connected such that in one position the output terminal is connected to the pencil beam phase command generator 40 and in its other position, the output terminals 31, 34, and 36 are connected to a plurality of other generators 42 through 45. In other words, in operation, for providing a pencil beam, a prior art method of scanning is used and for generating a fan beam, the present invention is utilized. Therefore, in FIG. 4, the switch 41 is illustrated in its second position. When the switch 41 is in its first position, the digital words representing trigonometric functions of the beam direction angles are coupled to the pencil beam phase command generator 40 which provides an output to an adder 48. Focusing phase commands are provided to the adder 48 from the focusing phase command generator 49, which has an input coupled to the output terminal 32. Pencil beam phase command words are coupled to the logic manifold 24 from the adder 48.  
  In accordance with the present invention, the switch 41 is coupled in its second position, and the output terminal 36 is coupled to an azimuth scan command generator 42, terminal 34 to a non-linear function generator 43, terminal 35 to an elevation scan generator 44, and terminal 31 to a linear scan generator 45. The azimuth scan generator 42 provides the digital word indicative of y sin 6,, for determining the relative phase of one array element 25 to another in a horizontal direction in order to achieve azimuth scanning. The nonlinear function generator 43 provides a digital word representing Vk [&#39;y&#39;(x,,)+k sin 6,] which is indicative of the derivative of the fan beam command function in order to compensate for beam curling in accordance with the above-described derivation. The outputs of the generators 42 and 43 are provided a multiplier 60. The generator 44 provides a linear command in the form of a digital word representing kx, sin (1),, determining relative phase of one array element 25 to the next in a vertical direction for determining the angle of depression of the fan beam. The linear scan generator 45 provides the basic fan beam function y(x,,) in accordance with the teachings of the above-described publication. Outputs of the generators 44 and 45 are coupled to an adder 61. In order to provide the command signal, the outputs of the multiplier 60 and adder 61 are coupled to an adder 62 which provides an input to an adder 63 for combining the steering command with a focusing command coupled to the adder 63 from the focusing phase command generator 49. The output of the adder 63 is connected to the logic manifold 24, whereby phase commands in accordance with the present invention are coupled to the array elements 25.  
  In operation, phase commands are provided to the elements 25 in accordance with the above-described derivation. Each of the generators 42-45 is constructed in a well-known manner and it-may utilize handbook circuitry. Other circuitry for providing the desired phase commands will suggest itself to those skilled in the art.  
  In operation, phase commands are provided to the array elements 25 in accordance with the above derivation. FIGS. a and 5b are respectively illustrative of typical prior art and present invention plots of beam amplitude versus azimuth angle. In FIGS. 5a and 5b, the solid line curves represent cross-sections of the beam scanned at different azimuth angles, and for one elevation angle, dotted lines connect corresponding corss-sections of a beam scanned to different azimuth angles. FIG. 5a shows the ground map beam in the .\&#39;z plane (FIG. 1) and in five scanned positions. For each beam position. five cross-sections of the ground map beam are shown. Ideally the cross-section peaks should line up for each beam azimuth position. For the conventionally scanned case of FIG. 5a, every beam position shows significant beam skew, while in FIG. 5b, beam skew becomes severe only at the heel (i.e., for larger values of 4)) of the beam and only for the extreme angle. Further. gain loss at the heel of the beam in FIG. 5b is almost negligible, while comparatively large in FIG. 5a, as indicated by the slope of the dotted lines drawn in FIGS. 51! and 5b.  
  FIG. 6 is a plot of the intersection with the ground of beams of arrays scanned conventionally and in accordance with the present invention, for an aircraft at a given altitude. The y and z axes of FIG. 6 correspond to those of FIG. I. The prior scan is illustrated in dashed lines while the scan of the present invention is illustrated in solid lines. For the exemplification of FIG. 6, the aircraft is at an altitude of eight miles. Deviation of the beam from a straight line going to the origin of the plot of FIG. 6 is minimized in accordance with the present invention. Consequently, positional distortion is minimized, and the return beam produced in accordance with the present invention may be used in conjunction with a conventional radar display rather than a display requiring compensation for a positional distortion.  
  What is thus provided according to the present invention are a method and apparatus for scanning a phased array to produce a unidirectional beam of substantially constant azimuth by sampling a non-linear function for the phase command to compensate for beam curling which is produced by sampling a linear phase command.  
  What is claimed as new and desired to besecured by Letters Patent is:  
  1. In a beam steering computer for connection between an array control computer and a logic manifold in a phased array radar system, the combination comprising:  
 a first function generator connected to the array control computer for generating a first signal indicative of a scanning function for producing a selected shape beam to comprise a term in a phase command;  
 a second function generator controlled by the array control computer for roducin a second signal indicataive of Vk [&#39;y&#39;(.r,,)+/ sin 5,1 where &#39;y&#39; is the derivative of the scanning function, k is 21r/A, A is the wavelength of the phased array radar system output, x is the x plane location of a phased array element, x axis being vertically disposed and (75,, is a value of aximuth angle;  
 horizontal and vertical sweep command generators constructed and arranged to produce horizontal and vertical sweep phase signals; and  
 signal integration means for combining said first signal with said vertical sweep signal and said second signal with said horizontal sweep signal and for passing said combined signals as a phasing command to said logic manifold whereby a planar, vertically&#39; disposed phased array beam is generated and scanned over an azimuth range without distortion of said planar configuration.  
 * l= =l= l=