Patent Publication Number: US-7911383-B2

Title: Phased array antenna system with two dimensional scanning

Description:
BACKGROUND OF THE INVENTION 
     (1) Field of the Invention 
     The present invention relates to a phased array antenna system with two dimensional scanning. It is suitable for use in all areas of technology employing scanning phased array antennas, e.g. radar, television and radio broadcasting and telecommunications, mobile cellular radio (“mobile telephones”) in particular. 
     (2) Description of the Art 
     Phased array antennas are well known: the subject is discussed in detail in for example a standard textbook well known in the art of antennas, “Microwave Scanning Antennas”, R. C. Hansen, Vol 3 Array Systems, Academic Press, NY, 1966. Such an antenna comprises an array of individual antenna elements (usually eight or more) such as dipoles or patches. The antenna has a radiation pattern incorporating a main lobe or beam and side lobes. The centre of the main lobe is the antenna&#39;s direction of maximum sensitivity in receive mode and the direction of its main output radiation beam in transmit mode. 
     It is a well known property of a phased array antenna that delaying signals received by antenna elements by a delay which varies with element distance across the array, then the antenna main radiation beam is steered or tilted towards the direction of increasing delay. The angle between main radiation beam centres corresponding to zero and non-zero variation in delay, i.e. the angle of tilt, depends on the rate of change of delay with distance across the array. Delay may be implemented equivalently by changing signal phase (hence the expression phased array, albeit a specific value of delay corresponds to different phase shifts at different frequencies). The main beam direction of the antenna pattern can therefore be altered (referred to as “beam steering”) by adjusting the phase relationship between signals fed to antenna elements. 
     A conventional technique for beam steering by adjusting the phase relationship between signals fed to antenna elements is to provide a respective variable phase, shifter or variable delay for each antenna element. This provides control of each antenna element&#39;s signal independently of other antenna elements&#39; signals. Equivalently, cascaded arrangements of variable phase shifters may be used in which each variable phase shifter provides a signal to a respective antenna element and to a respective variable phase shifter. Examples of the use of multiple variable phase shifters are disclosed by, for example, Japanese published Patent Application No. 04-320122 and U.S. Pat. Nos. 3,277,481, 4,242,352 and 5,281,974. 
     The use of variable phase shifters in numbers comparable with antenna elements is undesirable, because it greatly increases antenna design complexity and expense. A variable phase shifter is much more complex than a fixed phase shifter. This problem is particularly relevant to the case of a two dimensional phased array antenna which is required to scan in both dimensions: e.g. a phased array antenna consisting of a 64×64 array of antenna elements would require 4095 variable phase shifters and respective associated control circuitry. 
     The problem of excessive numbers of variable phase shifters has been addressed for the case of a one dimensional phased array antenna (e.g. a line of dipoles) scanned in a plane of the array dimension: the following published International Patent Applications disclose solutions to the one dimensional problem, WO 03/036756, WO 03/43127, WO 2004/036785, WO 2004/088790, WO 2004/102739 and WO 2005/048401. However, these do not scale up straightforwardly to two dimensions: for a two dimensional array of antenna elements arranged in rows and columns, using one of these prior art solutions per row or column permits scanning of all rows or columns in one dimension, but not scanning in another (orthogonal) dimension. The issue of scanning phased arrays is discussed in a standard work in the art of antennas, “Antenna Engineering Handbook”, Ed. Richard C. Johnson, McGraw Hill, 3 rd  Edition, 1993, ISBN 0-07-032381-X: see page 20-52 in particular. 
     SUMMARY OF THE INVENTION 
     It is an object of the invention to provide a phased array antenna system suitable for two dimensional scanning. 
     The present invention provides a phased array antenna system having a two dimensional array of antenna elements and a plurality of corporate feed networks, and wherein:
     a) the corporate feed networks are grouped in first and second ranks and are arranged to convert network input signals of variable phase relative to one another into network output signals phased appropriately for antenna elements of a phased array, the network output signals being in relatively greater numbers than the network input signals for corporate feed networks in at least one of the first and second ranks;   b) first rank corporate feed networks are arranged to provide network output signals as input signals to respective lines of antenna elements; and   c) second rank corporate feed networks are arranged to provide network output signals as input signals to respective lines of inputs of first rank corporate feed networks; and   d) the system includes phase difference control means for varying network input signal phasing for second rank corporate feed networks to provide control of antenna beam direction in two dimensions.   

     The invention makes possible control of antenna beam direction in two dimensions: it provides antenna array input using two ranks of corporate feed networks arranged in cascade with network input phase difference control. This provides a solution to the problem of obtaining two dimensional control of phased array antenna beam direction. 
     The phase difference control means may be arranged to:
     a) vary the phase difference between each input signal to one second rank corporate feed network and input signals to another second rank corporate feed network to provide control of antenna beam direction in a first dimension; and   b) vary both the phase difference between input signals to one second rank corporate feed network and the phase difference between input signals to another second rank corporate feed network to provide control of antenna beam direction in a second dimension.   

     The phase difference control means may be arranged to maintain:
     a) the phase difference between input signals to one second rank corporate feed network to be equal to the phase difference between input signals to another second rank corporate feed network; and   b) the phase differences between each input signal to one second rank corporate feed network and both input signals to another second rank corporate feed network to be equal.   

     In one embodiment of a system of the invention having corporate feed networks each with two inputs, the phase difference control means arranged in this way avoids cross-coupling between control of scanning in different dimensions. Here cross-coupling means that an angle of deflection of an antenna beam in one dimension is altered when an angle of deflection in another dimension is changed by scan control. 
     In another aspect, the present invention provides a phased array antenna system including a two dimensional array of antenna elements arranged in lines, and wherein:
     a) each line of antenna elements is associated with a respective first rank corporate feed network having outputs for providing signals to respective antenna elements and inputs for receiving signals of variable phase relative to one another;   b) the first rank corporate feed networks have:
       i) first inputs connected to outputs of one second rank corporate feed network;   ii) second inputs connected to outputs of another second rank corporate feed network;   
       c) the corporate feed networks provide a means for converting input signals of variable phase relative to one another into multiple output signals for phased array antenna elements, the number of output signals being relatively greater than the number of input signals; and   d) the system includes phase difference varying means for:
       i) varying the phase difference between each input signal to one second rank corporate feed network and input signals to another second rank corporate feed network to provide control of antenna beam direction in a first dimension; and   ii) varying both the phase difference between input signals to one second rank corporate feed network and the phase difference between input signals to another second rank corporate feed network to provide control of antenna beam direction in a second dimension.   
       

     Each corporate feed network may provide a means for converting input signals expressed by vectors A and B into other signal vectors given by expressions of the form p i A+q i B, where p i  and q i  are numerical factors (real or complex) in the range −1 to 1. 
     The phase difference varying means may comprise:
     a) a first variable phase shifter connected via a splitter to a second variable phase shifter and a first fixed phase shifter each connected to respective inputs of one second rank corporate feed network;   b) a second fixed phase shifter connected via a splitter to a third variable phase shifter and a third fixed phase shifter each connected to respective inputs of another second rank corporate feed network; and   c) means for ganging together operation of the second and third variable phase shifters.   

     Antenna elements may be positioned to define a curved surface such as a cylindrical, spherical or toroidal surface. 
     In an alternative aspect, the present invention provides a method of scanning a phased array antenna system having a two dimensional array of antenna elements and a plurality of corporate feed networks, and wherein:
     a) the corporate feed networks are grouped in first and second ranks and are arranged to convert network input signals of variable phase relative to one another into network output signals phased appropriately for antenna elements of a phased array, the network output signals being in relatively greater numbers than the network input signals for corporate feed networks in at least one of the first and second ranks;   b) first rank corporate feed networks are arranged to provide network output signals as input signals to respective lines of antenna elements; and   c) second rank corporate feed networks are arranged to provide network output signals as input signals to respective lines of inputs of first rank corporate feed networks; and   d) the method includes varying network input signal phasing for second rank corporate feed networks to provide control of antenna beam direction in two dimensions.   

     The step of varying network input signal phasing may comprise:
     a) varying the phase difference between each input signal to one second rank corporate feed network and input signals to another second rank corporate feed network to provide control of antenna beam direction in a first dimension; and   b) varying both the phase difference between input signals to one second rank corporate feed network and the phase difference between input signals to another second rank corporate feed network to provide control of antenna beam direction in a second dimension.   

     The step of varying network input signal phasing may include maintaining equal:
     a) the phase difference between input signals to one second rank corporate feed network and the phase difference between input signals to another second rank corporate feed network; and   b) the phase differences between each input signal to one second rank corporate feed network and both input signals to another second rank corporate feed network.   

     In a further alternative aspect, the present invention provides a method of scanning a phased array antenna system having a two dimensional array of antenna elements arranged in lines, and wherein:
     a) each line of antenna elements is associated with a respective first rank corporate feed network having outputs for providing signals to respective antenna elements and inputs for receiving signals of variable phase relative to one another;   b) the first rank corporate feed networks have:
       i) first inputs connected to outputs of one second rank corporate feed network;   ii) second inputs connected to outputs of another second rank corporate feed network;   
       c) the corporate feed networks provide a means for converting input signals of variable phase relative to one another into multiple output signals for phased array antenna elements the output signals being in relatively greater in number compared to the input signals; and the method includes:
       i) varying the phase difference between each input signal to one second rank corporate feed network and input signals to another second rank corporate feed network to provide control of antenna beam direction in a first dimension; and   ii) varying both the phase difference between input signals to one second rank corporate feed network and the phase difference between input signals to another second rank corporate feed network to provide control of antenna beam direction in a second dimension.   
       

     Each corporate feed network may provide a means for converting input signals expressed by vectors A and B into other signal vectors given by expressions of the form p i A+q i B, where p i  and q i  are numerical factors (real or complex) in the range −1 to 1. 
     The steps of varying phase difference may comprise:
     a) applying a first variable phase shift via a splitter to a second variable phase shifter and a first fixed phase shifter each connected to respective inputs of one second rank corporate feed network;   b) applying a second fixed phase shift via a splitter to a third variable phase shifter and a third fixed phase shifter each connected to respective inputs of another second rank corporate feed network; and   c) ganging together operation of the second and third variable phase shifters.   

     The method may include positioning the antenna elements to define a curved surface such as a cylindrical, spherical or toroidal surface. 
    
    
     
       DESCRIPTION OF THE FIGURES 
       In order that the invention might be more fully understood, embodiments thereof will now be described, by way of example only, with reference to the accompanying drawings, in which:— 
         FIG. 1  is a block diagram of a prior art antenna system suitable for one dimensional beam scanning; 
         FIG. 2  is a functional drawing of an embodiment of an antenna system of the invention suitable for two dimensional beam scanning; 
         FIG. 3  illustrates signal phase control circuitry for the  FIG. 2  antenna system; 
         FIG. 4  is a block diagram of a prior art antenna corporate feed network which provides two signals with variable relative phasing and which may be used in the  FIG. 2  antenna system; 
         FIG. 5  illustrates antenna element signal phasing using the corporate feed network of  FIG. 4 ; 
         FIG. 6  is a block diagram of an electrical tilt controller providing three signals with variable relative phasing and an alternative form of antenna corporate feed network which accepts input of such signals and which may be used in an antenna system of the invention; 
         FIG. 7  is a functional drawing of a further embodiment of an antenna system of the invention which incorporates the  FIG. 6  antenna corporate feed network; and 
         FIG. 8  illustrates signal phase control circuitry for the  FIG. 2  antenna system. 
     
    
    
     DESCRIPTION OF THE INVENTION 
     Referring to  FIG. 1 , a prior art phased array antenna scanning circuit is illustrated schematically and indicated generally by  10 . The circuit  10  is a generalised version of an equivalent disclosed in WO 2004/102739: it has two inputs I 1  and I 2  connected respectively to a variable delay or variable phase shifter  12  and a fixed delay or phase shifter  14 , which are in turn both connected respectively to inputs A and B of a splitter and vector combiner unit  16  having output terminals  17   1  to  17   N . The splitter and vector combiner unit  16  is referred to in the art of phased array antennas as a corporate feed network for an antenna array. The circuit  10  has a one dimensional antenna array  18 [ 1 ] consisting of a line of antenna elements  18   1  to  18   N , which are connected to the output terminals  17   1  to  17   N  respectively: here N represents any number of output terminals  17   1  etc. and antenna elements  18   1  etc., and dotted lines  20  and  22  indicate that these outputs and antenna elements may be replicated as required. 
     In operation of the circuit  10 , radio frequency (RF) input signals are fed to the inputs A and B: these signals may be obtained by splitting a single RF signal. The input signals pass to the variable and fixed phase shifters  12  and  14  respectively. The variable phase shifter  12  applies an operator-selectable phase shift or time delay, and the degree of phase shift applied here controls the angle of electrical tilt of the array  18 [ 1 ] of antenna elements  18   1  to  18   N . The fixed phase shifter  48  is not essential but convenient: it applies a fixed phase shift of half the maximum phase shift Φ M  applicable by the variable phase shifter  46 . This allows one input signal to be variable in phase in the range −Φ M /2 to +Φ M /2 relative to the other. 
     Relatively phase shifted signals pass from the variable and fixed phase shifters  12  and  14  to the splitter and vector combiner unit  16 : this unit splits the relatively phase shifted signals into component signals from which it forms various vectorial combinations to provide a respective drive signal for each individual antenna element  18   1  to  18   N . The drive signals have appropriate phasing relative to one another to provide for the antenna beam to be steerable in one dimension in response to alteration of the phase shift introduced by the variable phase shifter  12 . If the array  18 [ 1 ] of antenna elements  18   1  to  18   N  lies in a vertical plane, the antenna beam is steerable in that plane. 
     The circuit  10  may be thought of as a “few to many” signal converter or corporate feed network, since it provides for relatively few (e.g. two) signals with variable relative phase shift from inputs A and B to be converted to relatively many (e.g. twelve) antenna element drive signals with multiple variable relative shift shifts, i.e. a respective variable relative phase shift between drive signals to each adjacent pair of antenna elements  18   i  to  18   i+1  (i=1 to N−1). 
     Referring now to  FIG. 2 , there is shown a generalised block diagram representation of an embodiment of the invention, i.e. a phased array antenna system  30  providing two dimensional antenna beam scanning. Parts equivalent to those described earlier are like—referenced with changes to subscript indices as appropriate. 
     The antenna system  30  has a two dimensional planar array  18 [ 2 ] of one hundred and forty-four antenna elements  18   1,1  to  18   12,12  (only partially shown) arranged in twelve columns or vertical lines (e.g. first column or line  18   1,1  to  18   12,1 ) with twelve antenna elements (e.g.  18   1,1 ) per column. The array  18 [ 2 ] has X and Y scan directions indicated by bidirectional arrows  32  and  34 , which are respectively orthogonal and parallel to antenna element columns. Antenna elements  18   1,1  to  18   12,1  in a first column are connected to respective outputs  17   1,1  to  17   12,1  of a first splitter and vector combiner unit  16   1  in a first rank  16   1  to  16   12  of twelve such units. Likewise, antenna elements  18   1,2  to  18   12,12  in other columns (e.g. second column elements  18   1,2  to  18   12,2 ) are connected to outputs of eleven other splitter and vector combiner units  16   2  to  16   12  in the first rank respectively; i.e. antenna elements  18   1,j  to  18   12,j  in a jth column are connected to respective output terminals  17   1,j  to  17   12,j  of a jth splitter and vector combiner unit  16   j  (j=1 to 12). Connection of first rank splitter and vector combiner units  16   1  to  16   12  to antenna element columns is shown for convenience; as the system  30  could be rotated through 90° to exchange rows for columns. 
     The first rank splitter and vector combiner units  16   1  to  16   12  have respective A and B inputs A 1 /B 1  to A 12 /B 12 . The antenna system  30  has two further splitter and vector combiner units forming a second rank of such units, i.e. thirteenth and fourteenth splitter and vector combiner units  16   CD  and  16   EF  with output terminals  17   1CD  to  17   12CD  and  17   1EF  to  17   12EF  respectively. The two ranks of splitter and vector combiner units are connected in cascade as follows. The A inputs A 1  to A 12  of the first rank vector combiner units  16   1  to  16   12 , i.e. a line of upper inputs to these units, are connected to output terminals  17   1CD  to  17   12CD  of the thirteenth splitter and vector combiner unit  16   CD  respectively; similarly the B inputs B 1  to B 12  of the first rank vector splitter and vector combiner units  16   1  to  16   12 , i.e. a line of lower inputs to these units, are connected respectively to output terminals  17   1EF  to  17   12EF  of the fourteenth splitter and vector combiner unit  16   EF . The thirteenth splitter and vector combiner unit  16   CD  has inputs C and D equivalent to inputs A and B in the prior art described with reference to  FIG. 1 ; similarly the fourteenth splitter and vector combiner unit  16   EF  has likewise equivalent inputs E and F. 
     Control of scanning of the antenna array  18 [ 2 ] in two dimensions in a system of the invention requires more complex signal phasing arrangements than in the prior art, and this will now be described with reference to  FIG. 3 , in which parts equivalent to those described earlier are like-referenced.  FIG. 3  illustrates a scan control apparatus  40  for supplying signals to terminals referenced C, D, E and F, which are also the inputs C/D, E/F of the thirteenth and fourteenth splitter and vector combiner units  16   CD  and  16   EF  respectively. 
     The apparatus  40  has an RF input  42  connected to a first splitter  44 , which splits an RF input signal into two divided signals fed to a first variable delay  46  and a first fixed delay  48  respectively. Signals pass from the first variable delay  46  and the first fixed delay  48  to second and third splitters  50  and  52  respectively. In this specification delays (delay devices) and phase shifts (phase shifters) are treated as synonymous. 
     The second splitter  50  divides the signal from the first variable delay  46  into two signals which pass to a second variable delay  54  and a second fixed delay  56  respectively. Similarly, the third splitter  52  divides the signal from the first fixed delay  48  into two signals which pass to a third variable delay  58  and a third fixed delay  60  respectively. The second and third variable delays  54  and  58  are operatively ganged as indicated by dotted lines  62  so that signals reaching them are delayed for time intervals which are both variable and remain equal to one another. 
     Signals from the second fixed delay  56  and the second variable delay  54  pass respectively to the inputs C and D of the thirteenth splitter and vector combiner unit  16   CD . Similarly, signals from the third variable delay  58  and the third fixed delay  60  pass respectively to the inputs F and E of the fourteenth splitter and vector combiner unit  16   EF . 
     Connections from the second fixed delay  56  and the second variable delay  54  to the inputs C and D may be exchanged, provided that connections from the third variable delay  58  and the third fixed delay  60  to the inputs F and E are also exchanged likewise. 
     By inspection of  FIG. 3 , operation of the first variable delay  46  produces equal phase changes in signals reaching terminals C and D, and these phase changes are relative to signals reaching terminals E and F which are unaffected. Similarly, operation of the ganged second and third variable delays  54  and  58  produces equal phase changes in the signals reaching terminals D and F, and these phase changes are relative to the signals reaching terminals C and E which are unaffected. This is relevant to antenna beam scanning in two dimensions: i.e. the first variable delay  46  provides scan control in the Y direction  34  for the phased array antenna system  30 ; the ganged second and third variable delays  54  and  58  collectively provide a scan control in the X direction  32 . This will be described later in more detail. 
     Referring now to  FIG. 4 , there is shown an implementation of a prior art splitter and vector combiner circuit  16  of  FIG. 1  suitable for use with a one dimensional phased array  18 [ 1 ] with twelve antenna elements  18   1  to  18   12  arranged in a vertical line. Parts equivalent to those previously described are like-referenced. First and second splitters  70   1  and  70   2  respectively receive input signals denoted by vectors A and B: these vectors are of equal power but variable relative phase. The splitters  70   1  and  70   2  implement division into three fractions a 1 /a 2 /a 3  and b 1 /b 2 /b 3  respectively: i.e. signals a 1 A, a 2 A and a 3 A are output from splitter  70   1  and signals b 1 B, b 2 B and b 3 B from splitter  70   2 . Values for the fractions a 1 /a 2 /a 3  and b 1 /b 2 /b 3  (and also fractions c 1 /c 2 , d 1 /d 2 , e 1 /e 2 /e 3  and f 1 /f 2 /f 3  mentioned below) are given in the prior art, and can also be calculated from simple circuit and antenna phasing considerations by one of ordinary skill in the art. 
     Signals a 1 A and b 1 B pass to first and second Φ padding phase shifters  72   1  and  72   2  respectively. Here “padding” indicates a component introduced to compensate for phase shifts experienced by other signals. Signals a 2 A and b 3 B pass to I 1  and I 2  inputs of a first 180 degree hybrid directional coupler H 1  referred to as a “sum and difference hybrid” or “hybrid”. Such hybrids have the property of providing at two outputs S and D signals equal respectively to the sum and difference of signals at two inputs I 1  and I 2 . 
     Signals b 2 B and a 3 A pass to I 1  and I 2  inputs of a second hybrid H 2 . The hybrids H 1  and H 2  have difference outputs D connected as inputs to third and fourth splitters  70   3  and  70   4 , which produce two-way splitting into fractions c 1 /c 2  and d 1 /d 2  respectively. They also have sum outputs S connected to I 1  inputs of third and fourth hybrids H 3  and H 4  respectively. 
     Output signals from the first and second phase shifters  72   1  and  72   2  pass to fifth and sixth splitters  70   5  and  70   6  producing three-way splitting into fractions e 1 /e 2 /e 3  and f 1 /f 2 /f 3  respectively. Output signals from the third splitter  70   3  pass (fraction c 1 ) to an I 1  input of a fifth hybrid H 5  and (fraction c 2 ) to a third Φ padding phase shifter  72   3 . Output signals from the fourth splitter  70   4  pass (fraction d 1 ) to an I 1  input of a sixth hybrid H 6  and (fraction d 2 ) to a fourth Φ padding phase shifter  72   4 . Output signals from the fifth splitter  70   5  pass (fraction e 1 ) to an I 2  input of the fifth hybrid H 5 , (fraction e 2 ) to a fifth Φ padding phase shifter  72   5  and (fraction e 3 ) to an I 2  input of the fourth hybrid H 4 . Output signals from the sixth splitter  70   6  pass (fraction f 1 ) to an I 2  input of the sixth hybrid H 6 , (fraction f 2 ) to a sixth Φ padding phase shifter  72   6  and (fraction f 3 ) to a I 2  input of the third hybrid H 3 . Via respective fixed phase shifters  74   1  to  74   12  and terminals  17   1  to  17   12 , the antenna elements  18   1  to  18   12  receive drive signals from outputs of the third to sixth hybrids H 3  and H 6  and third to sixth phase shifters  72   3  and  72   6  as set out in the Signal Amplitude Table below. 
     
       
         
           
               
            
               
                   
               
               
                 Signal Amplitude Table 
               
            
           
           
               
               
               
            
               
                 Element 
                 Hybrid or Phase Shifter 
                 Signal Amplitude 
               
               
                   
               
               
                 18 1   
                 Hybrid H 6 , output D 
                 0.5d1(b2B − a3A) − 0.707b1f1B 
               
               
                 18 2   
                 Phase Shifter 72 4   
                 0.707d2(b2B − a3A) 
               
               
                 18 3   
                 Hybrid H 6 , output S 
                 0.5d1(b2B − a3A) + 0.707b1f1B 
               
               
                 18 4   
                 Phase Shifter 72 6   
                 b1f2B 
               
               
                 18 5   
                 Hybrid H 4 , output D 
                 0.5(b2B + a3A) − 0.707a1e3A 
               
               
                 18 6   
                 Hybrid H 4 , output S 
                 0.5(b2B + a3A) + 0.707a1e3A 
               
               
                 18 7   
                 Hybrid H 3 , output S 
                 0.5(a2A + b3B) + 0.707b1f3B 
               
               
                 18 8   
                 Hybrid H 3 , output D 
                 0.5(a2A + b3B) − 0.707b1f3B 
               
               
                 18 9   
                 Phase Shifter 72 5   
                 a1e2A 
               
               
                 18 10   
                 Hybrid H 5 , output S 
                 0.5c1(a2A − b3B) + 0.707a1e1A 
               
               
                 18 11   
                 Phase Shifter 72 4   
                 0.707c2(a2A − b3B) 
               
               
                 18 12   
                 Hybrid H 5 , output D 
                 0.5c1(a2A − b3B) − 0.707a1e1A 
               
               
                   
               
            
           
         
       
     
     Because all the terms a 1  to f 3  are fractions, all signal powers are in terms of fractions of signal vectors A and B input to the first and second splitters  70   1  and  70   2  respectively. 
     The phase shifters  72   1  to  72   6  provide compensation for the phase shift that takes place in hybrids (e.g. H 1 ). Consequently, signals or signal components that do not pass via one or more hybrids traverse two phase shifters (e.g.  72   1 ) and receive a phase shift of 2Φ before reaching antenna elements  18   3  and  18   9 . In addition, signals or signal components that pass via one hybrid traverse one phase shifter (e.g.  72   4 ) and receive a relative phase shift of Φ before reaching antenna elements (e.g.  18   2 ). 
     Referring now also to  FIG. 5 , there is shown a vector diagram for the array  18  of antenna elements  18   1  to  18   12  when the phase difference between input signal vectors A and B is 60 degrees: in this example 60 degrees is the angle at which the antenna array  18  has an optimum phase front. Drive signals for antenna elements  18   1  to  18   12  respectively are indicated in magnitude and phase by solid radius vector arrows  82   1  to  82   12  extending from a common origin O and marked to indicate signal fractions (e.g. a 1   e   2 A). Bi-directional arrows such as  86  indicate phase differences between adjacent radius vectors. 
     Components (e.g. 0.707a 1   e   1 A) of such signals are indicated by chain or dotted line vectors. Signals b 1   f   2 B and a 1   e   2 A on respective antenna elements  18   4  and  18   9  are fractions of and are in phase with input signal vectors A and B, and they are 60 degrees apart in phase as indicated by two bi-directional arrows each associated with a respective 30 degree angle marking. 
     When the phase difference between signals A and B is altered by operation of the variable phase shifter  12 , the phases of signals on individual antenna elements  18   1  to  18   12  change: this changes the direction of the antenna main lobe or beam to provide phased array beam steering. 
     Referring to  FIG. 4  once more, on comparing this drawing with  FIGS. 2 and 3 , inputs A 1  to A 12 , C and E of splitter and vector combiner units  16   1  to  16   12 ,  16   CD  and  16   EF  are equivalent to the A input to upper half first splitter  70   1 . Similarly, inputs B 1  to B 12 , D and F of splitter and vector combiner units  16   1  to  16   12 ,  16   CD  and  16   EF  are equivalent to the B input to lower half second splitter  70   2 . 
     By inspection of  FIG. 4  and the Signal Amplitude Table, the splitter and vector combiner circuit  16  is what is referred to as an antenna corporate feed network: this corporate feed network converts input signal vectors A and B into different signal vectors given by expressions of the form:
 
p i A+q i B  (1)
 
where p i  and q i  are numerical factors which (if real) take values in the range −1 to 1, and i indicates a signal supplied to an ith output  17   i . The factors p i  and q i  might alternatively be complex numbers, in which case their moduli would be in the range 0 to 1.
 
     Referring now also to  FIG. 2  once more, signal vectors C, D, E and F are now used to represent input signals at respective inputs C, D, E and F of splitter and vector combiner circuits  16   CD  and  16   EF . All the splitter and vector combiner circuits are assumed to apply the same values of p i  and q i . Applying Expression (1) above to the signal vectors C, D, E and F to express the action of the splitter and vector combiner circuits leads to the following:
         (a) the thirteenth splitter and vector combiner unit  16   CD  provides its twelve outputs  17   1CD  to  17   12CD  with signals represented by vector sums (p 1 C+q 1 D) to (p 12 C+q 12 D); i.e. the ith output  17   iCD  receives a signal (p i C+q i D), i=1 to 12;   (b) similarly, the fourteenth splitter and vector combiner unit  16   EF  provides its twelve outputs  17   1EF  to  17   12EF  with signals represented by vector sums (p 1 E+q 1 F) to (p 12 E+q 12 F); i.e. the ith output  17   iEF  receives a signal (p i E+q i F), i=1 to 12.       

     Expression (1) above is now applied to the signal vectors (p 1 C+q 1 D) and (p 1 E+q 1 F) input respectively from  17   1CD  and  17   1EF  to inputs A 1  and B 1  of the first splitter and vector combiner circuit  16   1 . This leads to the circuit  16   1  producing the following signal vectors {p 1 (p 1 C+q 1 D)+q 1 (p 1 E+q 1 F)} to {p 12 (p 1 C+q 1 D)+q 12 (p 1 E+q 1 F)}, which appear at outputs  17   1,1  to  17   12,1  connected to first column antenna elements  18   1,1  to  18   12,1  respectively. At the ith output of the first splitter and vector combiner circuit  16   1  and at the first column antenna element  18   i,1  (i=1 to 12), a signal vector given by the following expression appears:
 
p i (p 1 C+q 1 D)+q i (p 1 E+q 1 F)  (2)
 
     Varying the first variable delay  46  in  FIG. 3  varies the phases of both C and D by the same amount relative to both E and F, but does not vary the phase of either C relative to D or E relative to F. The terms (p 1 C+q 1 D) and (p 1 E+q 1 F) in parenthesis in Expression (2) are resultant vectors arising from vector addition, and they are respectively equivalent to vectors A and B in Expression (1). Varying the first variable delay  46  therefore varies the phase of a signal represented by the vector (p 1 C+q 1 D) (equivalent to vector A in Expression (1)) relative to a signal represented by the vector (p 1 E+q 1 F) (equivalent to vector B), but does not otherwise affect these signals: Expression (2) is therefore equivalent to Expression (1). With appropriate values of p i  and q i  varying along a vertical line or column of antenna elements  18   1,1  to  18   12,1  as in the prior art, Expression (1) provides for an antenna output beam to be steered in a vertical plane (Y direction  34  in the drawing) by changing the relative phase difference or delay between two signal vectors equivalent to A and B. Consequently, varying the first variable delay  46  provides beam steering in a vertical plane in the same way for the first column of antenna elements  18   1,1  to  18   12,1 . 
     Similar remarks apply to the variation of the first variable delay  46  in connection with beam steering in a vertical plane for other columns of antenna elements  18   1,j  to  18   12,j  (j=2 to 12): for the jth column the terms in parenthesis in Expression (2) become (p j C+q j D) and (p j E+q j F). However, differing values of (p j C+q j D) and (p j E+q j F) for different columns only affect signal vector resultants equivalent to vector A or B; the phase difference between (p j C+q j D) and (p j E+q j F) introduced by the first variable delay  46  equivalently affects signal vector resultants supplied to equivalently located antenna elements in different columns. Varying the first variable delay  46  therefore provides beam steering in a vertical plane in the same way for all twelve columns of antenna elements  18   1,j  to  18   12,j  (j=1 to 12), and an equivalent of Expression (2) for any of the twelve columns is given by replacing index  1  by index j as follows:
 
p i (p j C+q j D)+q i (p j E+q j F)  (3)
 
where p i  and q i  are numerical factors imposed by the first to tenth splitter and vector combiner circuits  16   1  to  16   12 , and p j  and q j  are equivalents of p i  and q i  imposed by the thirteenth and fourteenth splitter and vector combiner circuits  16   CD  and  16   EF .
 
     Turning now to the effect produced by varying the ganged second and third variable delays  54  and  58 , Expression (3) is rearranged so that index i terms appear in parenthesis and index j terms outside as follows:
 
p j (p i C+q i E)+q j (p i D+q i F)  (4)
 
     Varying the ganged second and third variable delays  54  and  58  varies the phases of both D and F by the same amount relative to both C and E, but does not vary the phase of either C relative to E or D relative to F. This therefore varies the phase of a signal represented by the vector (p i C+q i E) (equivalent to vector A in Expression (1)) relative to a signal represented by the vector (p i D+q i F) (equivalent to vector B), but does not otherwise affect these signals: just as Expression (3) therefore, Expression (4) is also equivalent to Expression (1). With appropriate values of p j  and q j  varying along an ith row (horizontal line) of antenna elements  18   i,1  to  18   i,12  (i=1 to 12), Expression (4) provides for an antenna output beam from that row to be steered in a horizontal plane (X direction  32  in the drawing) by changing the relative phase difference or delay between two signal vectors equivalent to vectors A and B. Similar remarks apply to horizontal steering of antenna output beams from all rows of antenna elements,  18   1,1  to  18   1,12 , . . .  18   i,1  to  18   i,12 , . . .  18   12,1  to  18   12,12 . 
     Consequently the two dimensional phased array antenna system  30  provides scanning of the antenna beam direction in both dimensions. 
     A more detailed treatment of the theoretical basis for the invention&#39;s two dimensional scanning of the antenna beam direction is as follows. Initially it is helpful to derive a lemma or result for use later: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             g 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   A 
                                   + 
                                   B 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             h 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   A 
                                   - 
                                   B 
                                 
                                 ) 
                               
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           = 
                           
                             
                               g 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               B 
                             
                             + 
                             
                               g 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               B 
                             
                             + 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             h 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             B 
                           
                           - 
                           
                             h 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             B 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 g 
                                 + 
                                 h 
                               
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             B 
                           
                           + 
                           
                             
                               ( 
                               
                                 g 
                                 - 
                                 h 
                               
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             B 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         g 
                                         + 
                                         h 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     B 
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         g 
                                         - 
                                         h 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     B 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ) 
                           
                           
                             1 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               A 
                               + 
                               
                                 
                                   tan 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           g 
                                           - 
                                           h 
                                         
                                         
                                           g 
                                           + 
                                           h 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         B 
                                       
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         B 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 g 
                                 2 
                               
                               + 
                               
                                 h 
                                 2 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 
                                   gh 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
                                           cos 
                                           2 
                                         
                                         ⁢ 
                                         B 
                                       
                                       - 
                                       
                                         
                                           sin 
                                           2 
                                         
                                         ⁢ 
                                         B 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                           
                             1 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               A 
                               + 
                               
                                 
                                   tan 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           g 
                                           - 
                                           h 
                                         
                                         
                                           g 
                                           + 
                                           h 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     tan 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     B 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 g 
                                 2 
                               
                               + 
                               
                                 h 
                                 2 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 gh 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 B 
                               
                             
                             ) 
                           
                           
                             1 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               A 
                               + 
                               
                                 
                                   tan 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ( 
                                       
                                         
                                           g 
                                           - 
                                           h 
                                         
                                         
                                           g 
                                           + 
                                           h 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     tan 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     B 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     With an input signal at  42  denoted by V sin ωt, the scan control apparatus  40  described with reference to  FIG. 3  provides signals V C , V D , V E  and V F  at terminals C, D, E and F, which are also like-referenced inputs of the thirteenth and fourteenth splitter and vector combiner units  16   CD  and  16   EF  respectively. The signals V C , V D , V E  and V F  are given by: 
                       V   C     =       V   2     ⁢     sin   ⁡     (       ω   ⁢           ⁢   t     +     φ   2     -     ϕ   2       )           ⁢     
     ⁢       V   D     =       V   2     ⁢     sin   ⁡     (       ω   ⁢           ⁢   t     +     φ   2     +     ϕ   2       )           ⁢     
     ⁢       V   E     =       V   2     ⁢     sin   ⁡     (       ω   ⁢           ⁢   t     -     φ   2     -     ϕ   2       )           ⁢     
     ⁢       V   F     =       V   2     ⁢     sin   ⁡     (       ω   ⁢           ⁢   t     -     φ   2     +     ϕ   2       )                   (   6   )               
where:
 
V is a constant;
 
Φ is a variable phase difference controlling antenna beam scanning in the horizontal plane indicated by X; and
 
φ is a variable phase difference controlling scan in the vertical plane indicated by Y.
 
     The numerical factor ½ multiplying V in Equations (6) arises from the fact that signals experience two splitters in cascade reducing their power to one quarter. 
     The signals V C , V D , V E  and V F  are now rewritten as: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       C 
                     
                     = 
                     
                       
                         V 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 + 
                                 
                                   φ 
                                   2 
                                 
                               
                               ] 
                             
                             - 
                             
                               ϕ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       V 
                       D 
                     
                     = 
                     
                       
                         V 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 + 
                                 
                                   φ 
                                   2 
                                 
                               
                               ] 
                             
                             + 
                             
                               ϕ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       V 
                       E 
                     
                     = 
                     
                       
                         V 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 - 
                                 
                                   φ 
                                   2 
                                 
                               
                               ] 
                             
                             - 
                             
                               ϕ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       V 
                       F 
                     
                     = 
                     
                       
                         V 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 - 
                                 
                                   φ 
                                   2 
                                 
                               
                               ] 
                             
                             + 
                             
                               ϕ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Antennas array columns are now numbered with subscript j 
     and rows with subscript i. 
     Then an input signal V Aj  to each of the A or upper inputs A 1  to A 12  of the twelve first rank splitter and vector combiner units  16   1  to  16   12  is given by: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       Aj 
                     
                     = 
                     
                       
                         
                           c 
                           j 
                         
                         ⁢ 
                         
                           V 
                           C 
                         
                       
                       + 
                       
                         
                           d 
                           j 
                         
                         ⁢ 
                         
                           V 
                           D 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       V 
                       C 
                     
                     = 
                     
                       
                         V 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 + 
                                 
                                   φ 
                                   2 
                                 
                               
                               ] 
                             
                             - 
                             
                               ϕ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       V 
                       D 
                     
                     = 
                     
                       
                         V 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 + 
                                 
                                   φ 
                                   2 
                                 
                               
                               ] 
                             
                             + 
                             
                               ϕ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       V 
                       Aj 
                     
                     = 
                     
                       
                         
                           
                             
                               c 
                               j 
                             
                             ⁢ 
                             V 
                           
                           2 
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               
                                 [ 
                                 
                                   
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   + 
                                   
                                     φ 
                                     2 
                                   
                                 
                                 ] 
                               
                               - 
                               
                                 ϕ 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             
                               d 
                               j 
                             
                             ⁢ 
                             V 
                           
                           2 
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               
                                 [ 
                                 
                                   
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   + 
                                   
                                     φ 
                                     2 
                                   
                                 
                                 ] 
                               
                               + 
                               
                                 ϕ 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     By using the lemma mentioned previously, Equations (8) can be rewritten as: 
     
       
         
           
             
               
                 
                   
                     V 
                     Aj 
                   
                   = 
                   
                     
                       V 
                       2 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             c 
                             j 
                             2 
                           
                           + 
                           
                             d 
                             j 
                             2 
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               c 
                               j 
                             
                             ⁢ 
                             
                               d 
                               j 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                           
                         
                         ) 
                       
                       
                         1 
                         2 
                       
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                               + 
                               
                                 φ 
                                 2 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         d 
                                         j 
                                       
                                       - 
                                       
                                         c 
                                         j 
                                       
                                     
                                     
                                       
                                         d 
                                         j 
                                       
                                       + 
                                       
                                         c 
                                         j 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 tan 
                                 ⁢ 
                                 
                                   ϕ 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Similarly for an input signal V Bj  to each of the B or lower inputs B 1  to B 12  of the twelve first rank splitter and vector combiner units  16   1  to  16   12  is given by: 
     
       
         
           
             
               
                 
                   
                     V 
                     Bj 
                   
                   = 
                   
                     
                       V 
                       2 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             c 
                             j 
                             2 
                           
                           + 
                           
                             d 
                             j 
                             2 
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               c 
                               j 
                             
                             ⁢ 
                             
                               d 
                               j 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                           
                         
                         ) 
                       
                       
                         1 
                         2 
                       
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                               - 
                               
                                 φ 
                                 2 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         d 
                                         j 
                                       
                                       - 
                                       
                                         c 
                                         j 
                                       
                                     
                                     
                                       
                                         d 
                                         j 
                                       
                                       + 
                                       
                                         c 
                                         j 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 tan 
                                 ⁢ 
                                 
                                   ϕ 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Rearranging brackets in Equations (9) and (10): 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       Aj 
                     
                     = 
                     
                       
                         V 
                         2 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               c 
                               j 
                               2 
                             
                             + 
                             
                               d 
                               j 
                               2 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 c 
                                 j 
                               
                               ⁢ 
                               
                                 d 
                                 j 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                             
                           
                           ) 
                         
                         
                           1 
                           2 
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 + 
                                 
                                   
                                     tan 
                                     
                                       - 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               d 
                                               j 
                                             
                                             - 
                                             
                                               c 
                                               j 
                                             
                                           
                                           
                                             
                                               d 
                                               j 
                                             
                                             + 
                                             
                                               c 
                                               j 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       tan 
                                       ⁢ 
                                       
                                         ϕ 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                             + 
                             
                               φ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       V 
                       Bj 
                     
                     = 
                     
                       
                         V 
                         2 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               c 
                               j 
                               2 
                             
                             + 
                             
                               d 
                               j 
                               2 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 c 
                                 j 
                               
                               ⁢ 
                               
                                 d 
                                 j 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                             
                           
                           ) 
                         
                         
                           1 
                           2 
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 + 
                                 
                                   
                                     tan 
                                     
                                       - 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               d 
                                               j 
                                             
                                             - 
                                             
                                               c 
                                               j 
                                             
                                           
                                           
                                             
                                               d 
                                               j 
                                             
                                             + 
                                             
                                               c 
                                               j 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       tan 
                                       ⁢ 
                                       
                                         ϕ 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                             - 
                             
                               φ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Consequently, the general antenna element  18   i,j  in the ith row and jth column receives a signal V ij  given by: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       ij 
                     
                     = 
                     
                       
                         
                           a 
                           i 
                         
                         ⁢ 
                         
                           V 
                           Aj 
                         
                       
                       + 
                       
                         
                           b 
                           i 
                         
                         ⁢ 
                         
                           V 
                           Bj 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       V 
                       ij 
                     
                     = 
                     
                       
                         
                           
                             
                               a 
                               i 
                             
                             ⁢ 
                             V 
                           
                           2 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 c 
                                 j 
                                 2 
                               
                               + 
                               
                                 d 
                                 j 
                                 2 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 
                                   c 
                                   j 
                                 
                                 ⁢ 
                                 
                                   d 
                                   j 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ϕ 
                               
                             
                             ) 
                           
                           
                             1 
                             2 
                           
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               
                                 [ 
                                 
                                   
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   + 
                                   
                                     
                                       tan 
                                       
                                         - 
                                         1 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 d 
                                                 j 
                                               
                                               - 
                                               
                                                 c 
                                                 j 
                                               
                                             
                                             
                                               
                                                 d 
                                                 j 
                                               
                                               + 
                                               
                                                 c 
                                                 j 
                                               
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         tan 
                                         ⁢ 
                                         
                                           ϕ 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                               + 
                               
                                 φ 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             
                               b 
                               i 
                             
                             ⁢ 
                             V 
                           
                           2 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 c 
                                 j 
                                 2 
                               
                               + 
                               
                                 d 
                                 j 
                                 2 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 
                                   c 
                                   j 
                                 
                                 ⁢ 
                                 
                                   d 
                                   j 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ϕ 
                               
                             
                             ) 
                           
                           
                             1 
                             2 
                           
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               
                                 [ 
                                 
                                   
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   + 
                                   
                                     
                                       tan 
                                       
                                         - 
                                         1 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 d 
                                                 j 
                                               
                                               - 
                                               
                                                 c 
                                                 j 
                                               
                                             
                                             
                                               
                                                 d 
                                                 j 
                                               
                                               + 
                                               
                                                 c 
                                                 j 
                                               
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         tan 
                                         ⁢ 
                                         
                                           ϕ 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                               - 
                               
                                 φ 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Applying the aforementioned lemma to Equation (12) produces: 
     
       
         
           
             
               
                 
                   
                     V 
                     ij 
                   
                   = 
                   
                     
                       V 
                       2 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             c 
                             j 
                             2 
                           
                           + 
                           
                             d 
                             j 
                             2 
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               c 
                               j 
                             
                             ⁢ 
                             
                               d 
                               j 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                           
                         
                         ) 
                       
                       
                         1 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             a 
                             i 
                             2 
                           
                           + 
                           
                             b 
                             i 
                             2 
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               a 
                               i 
                             
                             ⁢ 
                             
                               b 
                               i 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             φ 
                           
                         
                         ) 
                       
                       
                         1 
                         2 
                       
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             t 
                           
                           + 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         d 
                                         j 
                                       
                                       - 
                                       
                                         c 
                                         j 
                                       
                                     
                                     
                                       
                                         d 
                                         j 
                                       
                                       + 
                                       
                                         c 
                                         j 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 tan 
                                 ⁢ 
                                 
                                   ϕ 
                                   2 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         a 
                                         i 
                                       
                                       - 
                                       
                                         b 
                                         i 
                                       
                                     
                                     
                                       
                                         a 
                                         i 
                                       
                                       + 
                                       
                                         b 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 tan 
                                 ⁢ 
                                 
                                   φ 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     If θ is small then: 
     cos θ≈1, tan θ≈θ, and n tan θ≈ tan nθ≈n, 
     and hence: 
                       V   ij     =       V   2     ⁢       (       c   j   2     +     d   j   2     +     2   ⁢     c   j     ⁢     d   j         )       1   2       ⁢       (       a   i   2     +     b   i   2     +     2   ⁢           ⁢     a   i     ⁢     b   i         )       1   2       ⁢     sin   ⁡     (       ω   ⁢           ⁢   t     +       (         d   j     -     c   j           d   j     +     c   j         )     ⁢     ϕ   2       +       (         a   i     -     b   i           a   i     +     b   i         )     ⁢     φ   2         )           ⁢     
     ⁢       V   ij     =       V   2     ⁢     (       c   j     +     d   j       )     ⁢     (       a   i     +     b   i       )     ⁢     sin   ⁡     (       ω   ⁢           ⁢   t     +       (         d   j     -     c   j           d   j     +     c   j         )     ⁢     ϕ   2       +       (         a   i     -     b   i           a   i     +     b   i         )     ⁢     φ   2         )                   (   14   )               
if the spatial coordinates of antenna element  18   i,j  are x ij , y ij , where Σx ij =0=Σy ij , (i.e. the coordinates are referred to a centre in the middle of an evenly spaced rectangular array of antenna elements  18   1,1  etc.) and splitter ratios are then set so that:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             γ 
                             x 
                           
                           ⁢ 
                           
                             x 
                             ij 
                           
                         
                         = 
                         
                           2 
                           ⁢ 
                           
                             
                               
                                 d 
                                 j 
                               
                               - 
                               
                                 c 
                                 j 
                               
                             
                             
                               
                                 d 
                                 j 
                               
                               + 
                               
                                 c 
                                 j 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                         where 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           γ 
                           x 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         is 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         gearing 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ratio 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         in 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         the 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         X 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         direction 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         and 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           γ 
                           y 
                         
                         ⁢ 
                         
                           y 
                           ij 
                         
                       
                       = 
                       
                         2 
                         ⁢ 
                         
                           
                             
                               a 
                               i 
                             
                             - 
                             
                               b 
                               i 
                             
                           
                           
                             
                               a 
                               i 
                             
                             + 
                             
                               b 
                               i 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         γ 
                         y 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       gearing 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ratio 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       in 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       the 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Y 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       direction 
                     
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Then the input signal phase on antenna element  18   i,j  in the ith row and jth column is:
 
x ij γ x Φ+y ij γ y φ  (16)
 
and the antenna array  18  generates a phase front which is substantially flat and tilted in both X and Y directions.
 
     The foregoing analysis shows that the two dimensional phased array antenna system  30  provides scanning of the antenna beam direction in both dimensions. This is achieved with an example which uses two cascaded ranks of “few to many” corporate feed networks, i.e. splitter and vector combiner circuits  16   1  to  16   EF  (although it is also possible to use “few to many” corporate feed networks in one (first or second) rank coupled to another type of corporate feed network in the other (second or first) rank). A first rank of “few to many” corporate feed networks  16   1  to  16   12  (one such feed network per column) provides inputs to columns of antenna elements, and a second rank (two) of “few to many” corporate feed networks  16   CD  and  16   EF  provides inputs to the first rank, each second rank corporate feed network  16   CD  or  16   EF  providing one respective input (i.e. either Ai or Bi (i=1 to 12) but not both) to each first rank corporate feed network  16   1  to  16   12 . 
     Scanning in the dimension in which extend antenna elements connected to first rank corporate feed networks is obtained by:
         a) keeping constant both the phase difference between input signals to one second rank corporate feed network  16   CD  and the phase difference between input signals to the other second rank corporate feed network  16   EF ; and   b) varying the phase difference between each input signal to one second rank corporate feed network  16   CD  or  16   EF  and both input signals to the other second rank corporate feed network  16   EF  or  16   CD .       

     Scanning in the dimension orthogonal to that in which extend antenna elements connected to first rank corporate feed networks is obtained by:
         c) keeping constant the phase difference between each input signal to one second rank corporate feed network  16   CD  or  16   EF  and a respective input signal to the other second rank corporate feed network  16   EF  or  16   CD ,   d) varying both the phase difference between input signals to one second rank corporate feed network  16   CD  and the phase difference between input signals to the other second rank corporate feed network  16   EF .       

     Implementing b) and d) above without a) and c) produces scanning at angles inclined to both rows and columns of the antenna array  30 . Cross-coupling (as defined below) between control of scanning in the two different dimensions as above can be avoided if required by providing for the above phase control to maintain:
         a) the phase difference between input signals to one second rank corporate feed network  16   CD  to be equal to the phase difference between input signals to the other second rank corporate feed network  16   EF ; and   b) the phase differences between each input signal to one second rank corporate feed network  16   CD  or  16   EF  and both input signals to the other second rank corporate feed network  16   EF  or  16   CD  to be equal.       

     Here cross-coupling means that an angle of deflection θ x  or θ y  of the antenna beam in one (X or Y) direction or dimension is altered when an angle of deflection θ y  or θ x  in the other (Y or X) direction or dimension is changed by scan control. This may be re-expressed using Equation (16), which indicates that θ x  varies with Φ and θ y  varies with φ during normal beam scanning. If cross-coupling is to be avoided, i.e. if it is required that θ x  does not vary with φ 
             (       i   .   e   .       ∂     θ   x         ∂   φ         =   0     )         
and θ y  does not vary with Φ
 
               (       i   .   e   .       ∂     θ   y         ∂   ϕ         =   0     )     ,         
then the phase differences should be maintained equal as aforesaid. However, cross-coupling may be a useful feature in some circumstances.
 
     The example of the invention described above employed corporate feed networks  16   11  to  16   EF  each with twelve outputs for convenience of illustration: i.e. these corporate feed networks acted as “two to twelve” signal converters. Corporate feed networks may have any convenient number of outputs, and in fact in the prior art a corporate feed network with twelve outputs is preferred to give advantageous performance in a phased array system; see WO 2004/102739 previously cited. 
     The invention is not limited to a “square” antenna array  30 , i.e. an array having equal numbers of antenna elements in its rows and columns. The antenna array may for example be rectangular, i.e. it may have N antenna elements per row and M antenna elements per column, where N and M are positive integers and N≠M. Other antenna array geometries are also possible. A rectangular antenna array in particular is advantageous in applications requiring more antenna elements in the vertical dimension than in the horizontal dimension: examples of this include a mast-mounted antenna array for mobile telephones, in which beam width and extent of scan angle are required to be smaller in the vertical dimension (e.g. 15 degrees in elevation from a tower-mounted antenna array) compared to the like for the horizontal dimension (e.g. 120 degrees in azimuth). 
     The antenna array  30  is a planar array, but the invention is not limited to planar arrays. The invention may be implemented using an antenna array in which individual antenna elements have centres which are positioned to lie on or define a curved surface such as a cylindrical, spherical or toroidal surface. First rank corporate feed networks  16   1  to  16   12  connect to respective lines of antenna elements, but the lines need not be straight. The dimensions in which scanning is implemented may be orthogonal as described above, but may also be non-orthogonal: e.g. the rows of the antenna array  30  may be inclined to its columns by an angle θ which is not 90°: here again this generates cross coupling between control of angles of antenna beam deflection θ x /θ y  in different directions or dimensions. However the coupling may be counteracted by appropriate gearing of phase shifters. By combining phase shifting with various gearing one may swivel and dip an antenna beam, and indeed one may provide for the beam to sweep out a curve. 
     The embodiment of the invention described with reference to  FIGS. 1 to 5  uses a “few to many” signal converter or corporate feed network where “few” is two. It is also possible to use a “few to many” signal converter or corporate feed network where “few” is more than two. As will be described, this increases complexity but is advantageous in increasing the range of angles over which an antenna beam can be steered. 
     Referring now to  FIG. 6 , there is shown a splitter and vector combiner circuit  160  suitable for configuring one signal into three signals, two of which are variably delayed, and further configuring the three signals into eleven signals; the eleven signals are for respective antenna elements E 1 U to E 5 U, Ec and E 1 L to E 5 L of a one dimensional phased array  166  arranged in a vertical line. The circuit  160  is from GB 0622411.7 dated 10 Nov. 2006, Quintel Technology Ltd. It incorporates phase padding components (not shown) to equalize the phase shifts experienced by signals reaching antenna elements E 1 U to E 5 U, Ec and E 1 L to E 5 L after passing through it. This is known in the art and will not be described in detail (see e.g. WO 2004/102739): a signal route from an input to an antenna element incorporating hybrid couplers includes a phase shift of 180 degrees per coupler, so if the maximum number of couplers per signal route is n and the minimum is 0, a route including i couplers requires components for phase padding of 180(n−i) degrees. 
     The circuit  160  incorporates two main components, an electrical tilt controller  162  and a corporate feed  164 , the latter connected to a phased array  166  antenna. The phased array antenna  166  has eleven antenna elements, these being a central antenna element Ec, five upper antenna elements E 1 U to E 5 U disposed successively above it, and five lower antenna elements E 1 L to E 5 L disposed successively below it. 
     An RF input signal represented as a vector V is applied to an input  168  of the tilt controller  162 , in which it is split into two signal vectors c 1 .V and c 2 .V of differing amplitude by a first splitter S 1  providing voltage split ratios c 1  and c 2 . The signal vector c 2 .V is now designated as a tilt vector C, and appears at a controller output  162   c.    
     The signal vector c 1 .V is further split by a second splitter S 2  to provide first and second signal vectors c 1 .d 1 .V and c 1 .d 2 .V: the first signal vector c 1 .d 1 .V is delayed by a first variable delay T 1  to give a signal vector which is now designated as a tilt vector A and appears at a controller output  162   a ; similarly, the second signal vector c 1 .d 2 .V is delayed by a second variable delay T 2  to give a signal vector now designated as a tilt vector B and appearing at a controller output  162   b.    
     Tilt controller  162  consequently provides three antenna tilt control signals, these signals representing tilt vectors A=c 1 .d 1 .V[T 1 ], B=c 1 .d 2 .V[T 2 ] and C=c 2 .V, where [T 1 ], [T 2 ] indicate variable delays T 1 , T 2  respectively. Delays T 1  and T 2  are ganged as denoted by a dotted line  170 , which contains a −1 amplifier symbol  172  indicating change in opposite senses; i.e. T 1  increases from 0 to T when T 2  reduces from T to 0 and vice versa: here T is a prearranged maximum value of delay for both of the ganged variable delays T 1  and T 2 . Operation of a delay control  174  varies both of the ganged variable delays T 1  and T 2  in combination, and changes their respective delays by amounts which are equal in magnitude and opposite in sign as per symbol  172 , one being an increase and the other a reduction: in response to these variable delay changes, the angle of electrical tilt of the antenna array  166  also changes. 
     A third splitter S 3  with voltage split ratios e 1  and e 2  splits tilt vector C into signals e 1 .C and e 2 .C, or equivalently c 1 .e 1 .V and c 2 .e 1 .V: signal e 1 .C is designated Cc (C central) and fed as a drive signal to the central antenna element Ec (an antenna element drive signal results in radiation of that signal from the antenna element into free space). Signal e 2 .C is further split by a fourth splitter S 4  with voltage split ratios f 1  and f 2 ; this produces a signal c 2 .e 2 .f 1 .V designated Cu (C upper), and also a signal c 2 .e 2 .f 2 .V designated Cl (C lower). It is not essential that the signal Cc be not subject to delay in a variable or fixed delay device, but it is convenient to minimise circuitry and reduce design complexity and costs. Moreover, as described elsewhere herein, in practice the signal Cc is delayed or phase shifted by means not shown for phase padding purposes to compensate for delays introduced by components through which other signals pass. 
     The vectors A and Cu are used to provide drive signals to antenna elements E 1 U to E 5 L connected to the upper part of the corporate feed  164 . Fifth and sixth splitters S 5  and S 6  with voltage split ratios a 1 , a 2  and g 1 , g 2  respectively split tilt vector A into signals a 1 .A and a 2 .A, and tilt vector Cu into signals g 1 .Cu and g 2 .Cu. 
     Similarly, the vectors B and Cl are used to provide drive signals to antenna elements E 1 L to E 5 L connected to the lower part of the corporate feed  164 . Seventh and eighth splitters S 7  and S 8  with voltage split ratios b 1 , b 2  and h 1 , h 2  respectively split tilt vector B into signals b 1 .B and b 2 .B, and tilt vector Cl into signals h 1 .Cl and h 2 .Cl. 
     A ninth splitter S 9  with voltage split ratios i 1  and i 2  splits signal a 2 .A from fifth splitter S 5  into signals i 1 .a 2 .A and i 2 .a 2 .A, of which signal i 1 .a 2 .A is connected to and provides a drive signal for third upper antenna element E 3 U. A tenth splitter S 10  with voltage split ratios j 1  and j 2  splits signal b 2 .B from seventh splitter S 7  into signals j 1 .b 2 .B and j 2 .b 2 .B, of which signal j 1 .b 2 .B is connected to and provides a drive signal for third lower antenna element E 3 L. 
     The corporate feed  164  incorporates six vector combining devices HY 1  to HY 6 , each of which is a 180 degree hybrid (sum and difference hybrid) having two input terminals designated  1  and  3  and two output terminals designated  2  and  4  as shown. Signals pass from each input to both outputs: a relative phase change of 180 degrees appears between signals passing between one input-output pair as compared to the other: as indicated by the location of a character π on each hybrid, this occurs between input  1  and output  4  in hybrids HY 1  and HY 2 , and between input  3  and output  4  in hybrids HY 3  to HY 6 . Each of the hybrids HY 1  to HY 6  produces two output signals which are the vector sum and difference of its input signals. 
     The first hybrid HY 1  receives input signals a 1 .A from fifth splitter S 5  and g 2 .Cu from sixth splitter S 6 : it adds and subtracts these signals to provide their difference as input to the third hybrid HY 3  and their sum as input to the fifth hybrid HY 5 . Similarly, the second hybrid HY 2  receives input signals b 1 .B from seventh splitter S 7  and h 2 .Cl from eighth splitter S 8 : it provides these signals&#39; difference as input to the fourth hybrid HY 4  and their sum as input to the sixth hybrid HY 6 . 
     The third hybrid HY 3  receives another input signal i 2 .a 2 .A from ninth splitter S 9  in addition to that from first hybrid HY 1 , and produces sum and difference signals for output as drive signals to fourth and fifth upper antenna elements E 4 U and E 5 U respectively. 
     The fifth hybrid HY 5  receives another input signal g 1 .Cu from sixth splitter S 6  in addition to that from first hybrid HY 1 , and produces sum and difference signals for output as drive signals to first and second upper antenna elements E 1 U and E 2 U respectively. 
     The fourth hybrid HY 4  receives another input signal j 2 .b 2 .B from seventh splitter S 7  in addition to that from second hybrid HY 2 , and produces sum and difference signals for output as drive signals to fourth and fifth lower antenna elements E 4 L and E 5 L respectively. 
     The sixth hybrid HY 6  receives another input signal h 1 .Cl from eighth splitter S 8  in addition to that from second hybrid HY 2 , and produces sum and difference signals for output as drive signals to first and second lower antenna elements E 1 L and E 2 L respectively. 
     First, third and fifth hybrids HY 1 , HY 3  and HY 5  implement vector combination processes to generate signals for antenna elements E 1 U, E 2 U, E 4 U and E 5 U, and second, fourth and sixth hybrids HY 2 , HY 4  and HY 6  implement the like for antenna elements E 1 L, E 2 L, E 4 L and E 5 L. Signals for antenna elements Ec, E 3 U and E 3 L are generated by splitters without hybrids. Analysis of the signals reaching antenna elements E 1 U to E 5 U, Ec and E 1 L to E 5 L shows that their relative phasing is appropriate to collectively provide an antenna beam which is steerable in response to the tilt control  174  altering the ganged time delays T 1  and T 2  in mutually opposite senses, 
     Phasing of signal vectors or drive signals for the antenna elements Ec, E 1 U to E 5 U and E 1 L to E 5 L relative to one another is imposed by the tilt controller  162  and the corporate feed  164  in combination. This relative phasing is prearranged by choice of splitting ratios and signals for vectorial combination in hybrids: it is appropriate for phased array beam steering by control of angle of electrical tilt, which varies in response to adjustment of the two variable delays T 1  and T 2 . 
     
       
         
           
               
            
               
                   
               
               
                 Parameter Table: Splitter and Hybrid Parameters 
               
            
           
           
               
               
            
               
                 Splitter 
                 Split Ratio or Scattering Parameter 
               
            
           
           
               
               
               
               
               
            
               
                 or Hybrid 
                 Type 
                 Parameter 
                 Voltage Ratio 
                 Decibels (dB) 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 S1 
                 DBQH 
                 c1 
                 0.7045 
                 −3.04 
               
               
                   
                   
                 c2 
                 0.7097 
                 −2.98 
               
               
                 S2 
                 SDH 
                 d1, d2 
                 0.7071 
                 −3.01 
               
               
                 S3 
                 SDH 
                 e1 
                 0.6859 
                 −3.27 
               
               
                   
                   
                 e2 
                 0.7277 
                 −2.76 
               
               
                 S4 
                 SDH 
                 f1, f2 
                 0.7071 
                 −3.01 
               
               
                 S5, S7 
                 DBQH 
                 a1, b1 
                 0.5559 
                 −5.10 
               
               
                   
                   
                 a2, b2 
                 0.8313 
                 −1.61 
               
               
                 S6, S8 
                 DBQH 
                 g1, h1 
                 0.6636 
                 −3.56 
               
               
                   
                   
                 g2, h2 
                 0.7481 
                 −2.52 
               
               
                 S9, S10 
                 DBQH 
                 i1, j1 
                 0.4421 
                 −7.09 
               
               
                   
                   
                 i2, j2 
                 0.8970 
                 −0.94 
               
               
                 HY1, HY2 
                 SDH 
                 s21, s43 
                 0.7435 
                 −2.57 
               
               
                   
                   
                 s23, s41 
                 0.6688 
                 −3.49 
               
               
                 HY3, HY4 
                 SDH 
                 s21, s43 
                 0.3162 
                 −10.00 
               
               
                   
                   
                 s23, s41 
                 0.9487 
                 −0.46 
               
               
                 HY5, HY6 
                 SDH 
                 s21, s43 
                 0.3162 
                 −10.00 
               
               
                   
                   
                 s23, s41 
                 0.9487 
                 −0.46 
               
               
                   
               
            
           
         
       
     
     The splitters S 1  to S 9  and hybrids HY 1  to HY 6  provide voltage splitting ratios and input/output scattering parameters which are shown in the Parameter Table above, in which ‘DBQH’ means double box quadrature (90 degree) hybrid and ‘SDH’=sum-and-difference (180 degree) hybrid; sxy (e.g. s 21 ), where x is 2 or 4 and y is 1 or 3, represents a scattering parameter between ports x and y of each of the hybrids HY 1  to HY 6 . 
     The splitter and vector combiner circuit  160  provides an increased antenna beam tilt range of 6.5 degrees compared to 4 degrees for a comparable system, 62.5% improvement, this being for a maximum side lobe level of −18 dB relative to boresight in each case. A tilt range of 10 degrees is obtainable if the antenna beam&#39;s upper side lobe  20  can be allowed to increase to −15 dB. 
     Referring now to  FIG. 7 , there is shown a generalised block diagram representation of a further embodiment of the invention, i.e. a phased array antenna system  200  providing two dimensional scanning; the system  200  is equivalent to the system  30  described with reference to  FIG. 2  with modification to implement the three input corporate feed (or splitter and vector combiner unit)  164  shown in  FIG. 6 . Description will concentrate on differences between  FIGS. 2 and 7 . 
     The antenna system  200  has an 11×11 two dimensional planar array PA[ 2 ] of one hundred and twenty-one antenna elements A 1,1  to A 11,1  (only partially shown) arranged in eleven columns or vertical lines (e.g. first column A 1,1  to A 11,1 ) with eleven antenna elements (e.g. A 1,1 ) per column. The array PA[ 2 ] has orthogonal X and Y scan directions indicated by arrows  202  and  204 . 
     The eleven columns of the array PA[ 2 ] are connected to respective corporate feeds arranged as a first rank CF 1  to CF 11  of eleven such feeds: each of these feeds is equivalent to the three input corporate feed  164  and provides drive signal input to antenna elements in a respective column, e.g. corporate feed CF 1  has eleven outputs CF 1   1 , to CF 1   11  to provide input to antenna elements A 1,1  to A 11,1  respectively in the 1st column, with similar outputs (not shown) for other corporate feeds CF 2  etc. and columns A 1,2  to A 11,2  etc. 
     The first rank splitter and vector combiner units CF 1  to CF 11  each have three inputs for signals equivalent to those shown as tilt vectors in  FIG. 6 , or as shown A, C and B in succession vertically downwards: To avoid illustrational complexity, these inputs are shown for the first and eleventh columns only as inputs AI 1 , CI 1  and BI 1  for column one and AI 11 , CI 11  and BI 11  for column eleven. 
     The antenna system  200  has three further corporate feeds arranged as a second rank of such feeds, i.e. twelfth, thirteenth and fourteenth corporate feeds CF 12 , CF 13  and CF 14  each equivalent to any one of first rank corporate feeds CF 1  to CF 11 : twelfth corporate feed CF 12  has three input terminals D, E and F for input signals equivalent to vectors A, C and B respectively in  FIG. 6 , and thirteenth and fourteenth corporate feeds CF 13  and CF 14  each have three input terminals G to I and J to L respectively for such signals. 
     The two ranks of splitter and vector combiner units CF 1  to CF 11  and CF 12  to CF 14  are connected in cascade as follows. The A inputs such as AI 1  and AI 11  of the first rank corporate feeds CF 1  to CF 11  (i.e. a line of upper inputs to these feeds) are connected to respective output terminals (not shown) of the twelfth corporate feed CF 12 . Similarly, the C inputs such as CI 1  and CI 11  of the first rank corporate feeds CF 1  to CF 11  (i.e. a line of central inputs) are connected to respective output terminals (not shown) of the thirteenth corporate feed CF 13 . Likewise, the B inputs such as BI 1  and BI 11  of the first rank corporate feeds CF 1  to CF 11  (i.e. a line of lower inputs) are connected to respective output terminals (not shown) of the fourteenth corporate feed CF 14 . 
     Control of scanning of the antenna array A[ 2 ] in two dimensions requires more complex signal phasing arrangements than the earlier embodiment  30 . In this connection, referring now also to  FIG. 8 , a scan control apparatus  240  is shown which is for supplying signals to output terminals referenced D, E, F, G, H, I, J, K and L: these output terminals also represent the like-referenced input terminals of the twelfth, thirteenth and fourteenth corporate feeds CF 12 , CF 13  and CF 14  in the second rank. 
     The apparatus  240  consists of first, second, third and fourth tilt control units TC 1  to TC 4  of like construction and each having effect equivalent to the tilt controller  162  in  FIG. 6 . The first tilt control unit TC 1  has an input TC 1  in connected to a three way splitter SP 1 , which splits an RF input signal at TCin into three signal fractions for delay respectively at first and second variable time delays VT 11  and VT 12  and a fixed delay FT 1 . The variable time delays VT 11  and VT 12  are ganged as indicated by chain lines LX to provide delays in a like range 0 to T, but these delays vary in opposite senses as indicated by variability arrows VA 11  and VA 12  pointing in mutually orthogonal directions: i.e. first variable time delay VT 11  goes from 0 to T as second variable time delay VT 12  goes from T to 0. The ganged variable time delays VT 11  and VT 12  collectively provide an X direction scan control for the antenna system  200 . 
     Second, third and fourth tilt control units TC 2  to TC 4  have equivalent components to those of first tilt control unit TC 1 , and are like referenced with the or a first index (as the case may be) changed from 1 to 2, 3 or 4 as appropriate. 
     The three signal fractions delayed in the first tilt control unit TC 1  at first and second variable time delays VT 11  and VT 12  and a fixed delay FT 1  pass as respective input signals to the second, third and fourth tilt control units TC 2  to TC 4 . Each of the second, third and fourth tilt control units TC 2  to TC 4  splits its respective input signal into three signal fractions and delays these fractions in respective delays VTk 1 , VTk 2  and FTk (k=2, 3 or 4), two of which are variable in opposite senses to one another and the third fixed (as in the first tilt control unit TC 1 ). 
     Signal fractions delayed in the second tilt control unit TC 2  pass respectively to output terminals D, E and F; those delayed in the third tilt control unit TC 3  pass respectively to output terminals G, H and I, and those delayed in the fourth tilt control unit TC 4  pass respectively to output terminals J, K and L. The variable delays VT 21 , VT 22 , VT 31 , VT 32 , VT 41  and VT 42  of the second, third and fourth tilt control units TC 2  to TC 4  are ganged as indicated by chain lines LY and provide a Y direction scan control for the antenna system  200 . 
     Delays in signal paths between input TC 1  in to the first tilt control unit TC 1  and output terminals D to L are shown in the Delay Table below, in which Fk indicates delay at fixed delay FTq, +Tq indicates delay at variable delay VTq 1  (q=1, 2, 3 or 4) with leftward inclined arrow, and −Tq indicates delay at variable delay VTq 2  with rightward inclined arrow indicating opposite sense delay variation. 
     As has been said, output terminals D to L are also input terminals to the twelfth, thirteenth and fourteenth corporate feeds CF 12 , CF 13  and CF 14  in the second rank, which therefore receive respective groups of three input signals delayed in accordance with the Delay Table above. By a similar analysis to that given above in connection with the embodiment described with reference to  FIGS. 2 to 5 , it can be shown that operation of the X and Y direction scan controls LX and LY provides scanning of the antenna beam direction of the antenna system  200  in both X and Y (i.e. orthogonal) dimensions. 
     
       
         
           
               
            
               
                   
               
               
                 Delay Table 
               
            
           
           
               
               
               
            
               
                   
                 Output Terminal 
                 Signal Path Delay 
               
               
                   
                   
               
               
                   
                 D 
                 +T1, +T2 
               
               
                   
                 E 
                 +T1, F2  
               
               
                   
                 F 
                 +T1, −T2 
               
               
                   
                 G 
                  F1, +T3 
               
               
                   
                 H 
                 F1, F3 
               
               
                   
                 I 
                  F1, −T3 
               
               
                   
                 J 
                 −T1, +T4 
               
               
                   
                 K 
                 −T1, F4  
               
               
                   
                 L 
                 −T1, −T4 
               
               
                   
                   
               
            
           
         
       
     
     The embodiment of the invention described with reference to  FIGS. 6 to 8  uses a “few to many” signal converter or corporate feed network where “few” is three and many is “eleven”. It is also possible to use a “few to many” signal converter or corporate feed network where “few” is more than three by adding further variably delayed signals: i.e. splitters SP 1  etc. in  FIG. 8  would be modified to split into more signals and the or (as the case may be) each additional signal would be variably delayed. 
     Antenna elements (e.g. A 1,1  to A 11,11  in  FIG. 7 ) may be disposed in a square or rectangular grid as illustrated, but other element arrangements are also possible: for example, antenna elements may be in a hexagonal array i.e. at the vertices of a hexagonal grid: a hexagonal array provides minimum inter element coupling for a given number of elements and a given area of antenna array. A hexagonal array leads to alternate columns of antenna elements being staggered in position relative to respective adjacent columns by half of the spacing between adjacent elements. 
     The element array need not be fully populated: i.e. the array might be a sparse array in which elements are located at periodic positions defining a geometrical array but some array positions do not have antenna elements located there—the array has holes. This reduces the required number of elements making the antenna system cheaper: it also changes the beam shape, which is useful to provide different beam widths in azimuth and elevation 
     The perimeter of the antenna element array need not be the same shape as that indicated by element locations: e.g. if seeking equal beamwidths in azimuth and elevation, antenna elements may be used which are located on a hexagonal grid and also lying within or delimited by a circle. Alternatively, for different azimuth and elevation beamwidths, the hexagonal grid of antenna elements may lie within an ellipse. A further alternative is the hexagonal grid of antenna elements lying within a stretched hexagon. 
     The embodiments of the invention described above use like first rank and second rank corporate feeds. It is not essential for the first rank corporate feeds to be alike: they may have different numbers of outputs and be connected to differing numbers of antenna elements. They may also have different amplitude and phase weightings, in order to adjust for different element patterns resulting from differing numbers of antenna elements. In addition, the first rank corporate feeds may have differing numbers of inputs from second rank corporate feeds. If the first rank corporate feeds are not all alike, then the second rank corporate feeds may be different in consequence. 
     The invention is suitable for use in all areas of technology employing scanning phased array antennas, e.g. radar, television and radio broadcasting and telecommunications including cellular radio (“mobile telephones”). It can be used at any frequency for which appropriate components (antenna elements, corporate feed networks, phase shifters etc.) are available, and including radio, microwave, millimetric wave, near and far infrared and optical frequencies.