Antenna Arrays. Contents

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1 Antenna Arras Antenna arras are forme from an assembl of raiating or receiving elements in a particular electrical an geometrical configuration. The cooperative effect of all elements in the arra permits to obtain raiation characteristics that coul not be achieve with a single element. Contents. Introuction. Two-element arras 3. N-element linear arras 4. Phase arras 5. Planar arras 6. Mutual coupling

2 . Introuction Motivation for the use of antenna arras To enlarge the imension of the antenna 4p for more irective characteristics D = A em l To be able to steer the beam in man ifferent irections (phase arras) To shape the beams for particular applications (beamforming) low-sielobe patterns Aaptive nulling

arra configuration (linear, circular, rectangular, elliptical, spherical, clinrical, conformal to a nonplanar surface,

3 Principle of function Fiels from elements in the arra interfere constructivel for certain irections ( maima) an estructivel for other irections ( nulls) S maimum Parameters that influence the overall pattern of an arra Geometrical arra configuration (linear, circular, rectangular, elliptical, spherical, clinrical, conformal to a nonplanar surface, etc) Relative spacing between elements Amplitue ecitation of elements Phase ecitation of elements Relative pattern of iniviual elements 3

4 . Two-elements arras Let us assume that two horiontal infinitesimal ipoles (length L<<l) are positione along the -ais at a istance from each other. The two ientical ipoles are fe with ientical amplitue I an phase ifference (i <ks-ee>). I I wt / / The raiation pattern in the -plane for a ipole oriente along the -ais is obtaine through a rotation See bottom of p. 8 of Wire Antennas for rigorous treatment. The epenence in 3D is in fact eual to - sin sin f which reuces to cos in the -plane ( f = 9). sin cos 4

5 Assuming no mutual coupling between the two elements, we can write the total fiel as the superposition of the fiels from both ipoles Et = E + E r r - jkr [ + /] -jkr [ - /] ki L e e aj ˆ h ì cos cos 4p r r ü ï = í + ï ý / r ï î ïþ Far-fiel approimations:»» / #: phase + / #: phase - / r» r» r r r» r + cos üï ï ïýï» r - cos ï ïþ for amplitue term for phase term / r r r ki Le 4pr -jkr - jk ( cos + ) / + jk ( cos + ) / Et = aˆ jh cos { e + e } cos[ ( k cos + ) /] / 5

6 E t -jkr ki Le = aj ˆ h cos cos[ ( k cos + ) /] 4pr single element pattern arra factor Total fiel pattern in the -plane Arra factor (far fiel) The arra factor is a function of the geometr of the arra (number of elements, relative istance) an the ecitation phase an amplitue. Simple epressions are obtaine when the elements have ientical amplitues, phases an spacing. The arra factor correspons to the pattern of the arra where the actual raiating elements are replace with isotropic point sources. Pattern multiplication (far fiel) The far-one fiel istribution of an arra of ientical elements is eual to the prouct of the fiel of a single element at the reference point (origin) an the arra factor of that arra E = E arra factor t ( single element at reference point) Note: Mutual coupling (later) is not inclue in pattern multiplication. 6

Eample : Fin the element pattern, arra factor an

-plane B Single element = Arra factor cos k cos [ ]

7 Eample : Fin the element pattern, arra factor an total fiel pattern of a two-element arra of infinitesimal ipoles with uarter-wave inter-element istance ( = l/4) an ientical phase ecitation ( = ). -plane B Single element = Arra factor cos k cos [ ] Total fiel 3D Linear = Single element in 3D: cos sin sin - f 7

8 Eample : Fin the pattern nulls for = l/4 an = -9. Solution: éæ pö ù p l p E µ cos cos kcos k ç - with = = êë è øúû l 4 ép ù E µ cos cos ( cos - ) êë 4 úû Element nulls: = = cos ( cos ) ( cos ) cos n n 9 Arra factor nulls: é p ù ép ù p n - = n - = êë 4 úû êë4 úû p p a) ( cos n - ) =+ n oes not eist 4 p p o b) ( cos n - ) =- n = 8 4 = 8

9 Eample 3: Compare the arra factors for = l/4 an the two phases = ±9. Solution: =-9 =+ 9 AF ép ù µ cos ( cos -) êë4 úû AF ép ù µ cos ( cos + ) êë4 úû constructive interference f+ = estructive interference f+ = 8 =-9 =+ 9 = l /4 f = 9 = l /4 f = 9 estructive interference - f+ =-8 constructive interference - f + = 9

10 3. N-element linear arras We now want to generalie the concepts introuce b etening the linear arra to N ientical elements along. To obtain the arra factor, we consier all elements to be isotropic sources. N 4 3 r N cos r 4 r 3 r r The elements are fe with a current ecitation with ientical amplitue I. Each succeeing element has a progressive phase relative to the preceing one. I I wt 3 Far fiel approimation: The path ifference from two ajacent elements in the far-fiel is given b cos

11 The arra factor can then be written as the superposition of all far fiel contributions N r N AF = + e + e + + e + j( k cos + ) + j ( k cos + ) + j( N - )( k cos + ) 4 3 cos r 4 r 3 r r ( ) (**) phase from path ifference ecitation phase N jn ( ) AF = å e + - Y where Y = k cos + n= * ìï = ï í ïaf e = e + e + e + + e + e ïî jy jy jn ( -) Y jn ( -) Y AF e e e e jy jy jy j3y jn ( -) Y jny ( ) ( ) ( jy ) jny ** - * AF e - = e - jn Y jn ( /) Y -jn ( /) Y e - j[ ( N-/ ) ] Y e -e [( / ) ] sin é j N- Y ( N / ) Yù AF = = e = e ë û jy j( /) Y -j( /) Y e - e -e siné( /) Yù ë û center of arra

12 If we chose the reference point in the phsical center of the arra, the arra factor reuces to æn ö sin ç Y çè AF ø = æ ö sin ç Y çè ø The maimum value of this arra factor is eual to N. Therefore, in orer to have a maimum value eual to one, the arra factor is often given in normalie form AF n = æn ö sin ç çè Y ø N æ ö sin ç Y çè ø with Y= k cos + Normalie arra factor for the uniform linear N- element arra For small values of Y, e.g. near the maimum of the arra factor AF n æn ö sin ç çè Y ø» N Y ( also for l, = )

13 Nulls an maima: Nulls are foun b setting the AF to ero, i.e. AF n = æn ö sin ç çè Y ø N æ ö sin ç Y çè ø ì æn ö sin ç Y = ï çè ø í æ ö sin Y ¹ ï çè ïî ø N Y = n = np Y= k cos + æ ö n = cos - p êp ç N ë çè øúû ì n =,, 3,... ï í ï n ¹ N, N,3 N,... ïî é l n ù - The maimum value of the AF occurs when its enominator becomes ero "" ç æ AF n ö çè ø Y= ( k cos + ) =mp = m Y= k cos + l m = cos é - (- mp) ù êë p úû m =,,,3,... The number of nulls an maima will be a function of the element spacing an the phase ecitation ifference. 3

14 Broasie arra The maimum of raiation is irecte normal to the arra ( = 9 ) N = 5 = l / = Y= ( k cos + ) = = = 9! = Elements have the same phase! En-fire arra The maimum of raiation is irecte along the ais of the arra (either = or = 8 or both) N = = l /4 =+ 8! ìïy = ( k cos + ) = k + = = = í ï ï =-k ïî! ìïy = ( k cos + ) =- k + = = 8 = 8 í ï ï = + k ïî 4

15 Arra factors for broasie arras: epenence on element spacing ( N =, = ) = l /4 = l / = l (3D: linear) 5

16 Arra factors for en-fire arras: N =, = l / 4 =-k (3 representations) =+k (3D: linear) 6

17 Arra factors for en-fire arras: Depenence on element spacing ( N = 6, =-k) = l / = l /4 = l / = 3 l /4 (3D: linear) 7

18 Broasie vs. en-fire arras: N =, = l / 4 = =-k Omniirectional pattern Directional pattern 8

19 Uniform amplitue broasie arra ( = 9 ) Uniform amplitue en-fire arra ( = ) (see p. 3) Nulls Maima æ n lö ìï n =,, 3,... ï ç è N ø ï n ¹ N, N,... ïî - n = cos í - æ lö m = ç m m = è ø cos,,,... n n,, 3,... - æ n lö ìï = = cos ï ç - í è N ø ï n ¹ N, N,... ïî - æ lö m = ç - m m = è ø cos,,,... Half-power points h - æ.39lö p cos ç ç çè pn ø l h - æ.39lö p cos ç - pn çè ø l Minor lobe maima s cos - é (s + ) lù ì s =,,... ê ï í ê ë N ú û ï p / l ïî s - cos é (s + ) lù ì s =,,... - ï í ê ë N ú û ï p / l ïî First null beamwith (FNBW) ép Q n = -cos êë - æ l ö ù ç èn øúû - æ l ö ç è N ø Q n = cos - Half-power beamwith (HPBW) First sielobe BW (FSLBW) ép - æ.39lö ù p Qh - cos ç çè pn êë øúû l ép - æ 3l ö ù p QS - cos ç çè N êë øúû l Eamples follow later. - æ.39lö p Qh cos ç - pn çè ø l - æ 3l ö p QS cos ç - N çè ø l 9

20 4. Phase arras (scanning arras) We have seen that controlling the progressive phase ecitation between the elements permits to change the irection of maimum raiation from normal (broasie) to along the arra ais (en-fire). We will show now that the maimum raiation can be steere electronicall in an irection to form a scanning arra., = =-k =? = 9 = en-fire broasie

21 To steer the main beam in irection of with 8, we nee to ajust the progressive phase ecitation between the elements Y= k cos + = k cos + = =-k cos =! n 5 N = 6 = l / = phase shifters The phase cause b path ifference is compensate b the ecitation phase for

22 Scanning arra eample N =, = l / 4 (3D: linear) = 9 = 6 = =-45 = 3 = =-77.9 =-9

23 Full-fiel simulations E-fiel amplitue in the plane Linear arra of 4 Hertian ipoles along = l/ = 9 = = 6 = -9 = 3 = (3D: linear) 3

Hansen-Wooar en-fire Arra To enhance the irectivit of an en-fire uniform arra without estroing an other characteristics, Hansen an Wooar propose that the reuire phase shift between closel space

24 Hansen-Wooar en-fire Arra To enhance the irectivit of an en-fire uniform arra without estroing an other characteristics, Hansen an Wooar propose that the reuire phase shift between closel space elements of a ver long arra shoul be æ.94ö o =- k ç + N = çè ø æ.94ö =+ k ç + = 8 N çè ø This approimation gives goo results for ver long, finite length iscrete arras with closel space elements. The above conition oes not guarantee that the maimum is along en-fire. An aitional conition efines the reuire element spacing o = l /4 = l / = N - l N 4 l» 4 for N large Uniform Hansen-Wooar arra N = =- ( k +.94/ N ) 4

25 Hansen-Wooar vs orinar en-fire arra N =, = l / 4 Hansen-Wooar en-fire arra: Higher irectivit (b.5 B) Orinar en-fire arra: Lower sielobe level (b 4 B) Hansenwooar 6.7 Orinar en-fire =- =-9 (3D: linear) 5

26 Uniform amplitue Hansen-Wooar en-fire arras towars = Same, towar = 8 Nulls n æ l ö ìï n =,, 3,... ï ç è N ø ï n ¹ N, N,... ïî - = cos + ( - n) í 8-n Maima m æ l ö ìï m =,, 3,... [ m ] ï ç è N ø ï p / l î - = cos + -( + ) í 8-m Half-power points h - æ l ö = ç - è N ø p l cos.398, N large 8-h Minor lobe maima s s,, 3,... - é sl ù ìï = = cos ï - í ê ë N ú û ï p / l ïî 8-s First null beamwith (FNBW) - æ l ö Q n = cos ç - è N ø Same Half-power beamwith (HPBW) - æ l ö ç è N ø p l Q h = cos -.398, N large Same First sielobe BW (FSLBW) - æ l ö ç è N ø p l Q S = cos - Same 6

27 Directivit of N-element linear arras Broasie Arra: U é æn ö ù é æn öù sin k cos sin k cos è ç ø çè ø ésin ù µ n =» N = æ ö N sin kcos k cos êë úû ê çè ø ú ê ë û ë úû ( ) ( AF ) é æn ö ù p sin k cos ç çè U ø = Pra = sin p N 4 ò k cos ê ë úû N - k ésin ù U =- Nk ò êë úû N k D U = U ma ésin ù p» Nk ò = êë úû Nk - p if Nk / large l Use substitution: N = kcos N =- k sin U ma = D U U ma =» Nk p Epressing D as a function of the arra length L = ( N -) p k = l æ Lö D» N = l ç + çè øl 7

28 Similarl: æ Lö Orinar en-fire arra: D» 4N = 4 l ç + çè øl Hansen-Wooar en-fire arra: æ Lö D =.85 4N =.85 4 l ç + çè øl Summar: Approimate irectivit of large closel-space arras Directivit D in terms of N Directivit D in terms of L Broasie arra Orinar en-fire arra N l 4N l æ Lö ç + çè øl æ Lö 4 ç + çè øl Hansen-Wooar en-fire arra æ Lö.85 4N.85 4 l ç + çè øl 8

29 Eample Ten isotropic elements are place along the -ais. Design a Hansen-Wooar en-fire arra with the maimum irecte towars = 8 o. Fin the following values a) esire spacing & progressive phase shift (in egrees) b) location of all the nulls (in egrees) c) first null beamwith ) irectivit Solution: N - l 9 l a) = = =.5 l ; = k + = p.5 + =.757 ra = N 4 4 N - l - ( n) b) Since in this eample = 8, n = 8- cos é + ( - n) ù = 8-cos é + - ù êë N úû êë 4.5 úû = 4.6, = 9.47, = 83.6, 3 = 56.5, = 4 5 c) - é l ù Q n = cos - êë N úû - é ù = cos - = êë 4.5úû Q n / ) D =.85 4N = l = B 9

30 Eample A linear arra is forme of isotropic elements place along the -ais with inter-element istance = l/4. Assuming uniform amplitue istribution, fin the progressive phase (in egrees), half power beamwith (in egrees), first null beamwith (in egrees), relative sie lobe level maimum (in B), an irectivit (in B) for the following arra tpes: a) broasie b) orinar en-fire c) Hansen-Wooar en-fire Solution a) broasie = ép æ.39lö ù - HPBW: Q = cos.4 h - ç pn = ê ë çè øúû ép æ l ö ù - FNBW: Q = cos 47.6 n - ç N = ê ë çè øúû ép æ - 3l ö ù FSLBW: Q = cos s - ç N = ê ë çè øúû é æ ö ù SLL: AF (at first sie lobe)» B Note: log n 3p ç 3 p ê ë çè øúû D = N = B Numerical computation gives B l ( ) -3 B Q h Q s Q n Broasie 3

99 B) l En-fire c) Hansen-Wooar æ.94ö æp ö = k + = +.94 =6.7 èç N ø èç ø Hansen-Wooar HPBW: é l ù Q = cos - -.398 = 38.

31 pl p b) Orinar en-fire = k = = = 9 l 4 é æ.39l ö ù - HPBW: Q = cos 69.5 h - ç pn = ê ë çè øúû é - l ù FNBW: Q = cos - = 6.6 n ê N ú ë û é - 3l ù FSLBW: Q = cos - = 3.84 s ê N ú ë û SLL: AF (at first sie lobe)» B n 3p D = 4N = B Numeric ( all B) l En-fire c) Hansen-Wooar æ.94ö æp ö = k + = +.94 =6.7 èç N ø èç ø Hansen-Wooar HPBW: é l ù Q = cos = 38.5 h ê N ú ë û FNBW: cos é l ù Q = - - = n ê N ú ë û FSLBW: cos é l ù Q = - - = 6.6 s ê N ú ë û SLL: AF (at first sie lobe)»-9 B n D =.85 4N = B Numericall ( B) l 3

Eample 3 Design a uniform linear scanning arra whose maimum of the arra factor is 3º from the ais of the arra ( = 3º). The esire half-power beamwith is º while the spacing between the elements is l/4.

32 Eample 3 Design a uniform linear scanning arra whose maimum of the arra factor is 3º from the ais of the arra ( = 3º). The esire half-power beamwith is º while the spacing between the elements is l/4. Determine the ecitation of the elements (amplitue an phase), length of the arra L (in wavelengths), number of elements, an irectivit (in B). Solution: Uniform amplitue istribution Progressive phase: pl =- k cos =- cos 3= l 4 From the graph: ( L ) 5 L 49.75l N l Directivit nees to be solve numericall: HPBW: D» B = 3 = N = 3

Grating Lobes When the spacing between the elements is large enough to permit

33 Grating Lobes When the spacing between the elements is large enough to permit in-phase aition of raiate fiels in more than one irection, multiple maima of eual magnitue can be forme. The principal maimum is referre to as the major lobe, the remaining as the grating lobes. To avoi grating lobes in linear broasie arras, choose < l N = = l 6 /4 N = 6 = l / N = 6 = 3 l /4 N = 6 = l N = 6 = 3 l / N = = 6 l Remark: The conition for en fire arra is: < l/ (c.f. p. 7) 33

34 Low sielobes istributions ( non uniform amplitues) The far-fiel pattern is the Fourier transform of the amplitue (aperture) istribution. Therefore, b varing the amplitues of the element ecitations, the shape of the arra factor can be influence. This is commonl use in the esign of low-sielobe istributions. Eample: N = 6 = l / = I n : variable Uniform amplitues Binomial istribution SLL: SieLobe Level SLL: B SLL: - B 34

35 Dolph-Chebshev Talor n = 4 Hamming SLL: esign parameter SLL: esign parameter SLL: fie but low Note: - Traeoff: Sielobe level vs. beamwith - The aperture efficienc of the arra is ecrease b non constant istributions 35

36 Ecitation Pattern snthesis: Variation of the amplitue an phases of the iniviual elements permits to shape the beam of the arra. Eamples: Compute with genetic algorithm ) 6 elements, linear arra 5 bit amplitue taper Amplitue control Element # Normalie far-fiel (B) Angle ( ) ) elements, linear arra with comple taper (amplitue an phase) Mask (specification) Ecitation amplitue Phase an amplitue control Ecitation phase ( ) Element # Normalie far-fiel (B) Angle ( ) 36

5. Planar arras Etening the arra in a plane (instea of just along a line) provies aitional variables which can be use to control an shape the pattern of the arra.

37 5. Planar arras Etening the arra in a plane (instea of just along a line) provies aitional variables which can be use to control an shape the pattern of the arra. Basicall, aing a imension to the arra provies control over the secon angular coorinate. The patterns of a planar arra are therefore irectional an can be use to scan the main beam of the antenna towar an point in space. Linear arra (along ) Planar arra (-plane) f 37

38 Arra factor of a planar arra Arra along Consier a linear arra of M elements in the irection M AF = å Ime AF m= = M ( - )( k sin cosf+ ) jm For uniform ecitation, we can write [ M ) Y ] [ ) Y ] sin ( / sin(/ where Y = k sin cos f+ M f Etension to D: The linear arra along is consiere as a single element (with pattern AF ) of a linear arra along Arra along AF N 38

39 Rectangular arra in the -plane N é M jm ( - )( k sincosf+ ) ù jn ( - )( k sinsinf+ ) AF = åi n Ime e êå n= ë ú m= û M jm ( - )( k sincosf+ ) jn ( - )( k sinsinf+ ) = åime åi ne m = n= AF AF N 3 4 f N M The arra factor of the rectangular arra is the prouct of the arra factors of the arras in the - an -irections. Ecitation of element mn: I mn e I I m n j é( m ) ( n ) ù ê ë úû Assume that I = I I mn m n = I = const an normalie: sin é( M / ) Y ù sin é ( N / ) Y ù AF( f, ) n = ë û ë û M sin é( / ) ù N sin é( / ) ù ë Y û ë Y û with ìy = k sin cosf+ ï í ïy = k sin sin f+ ïî 39

40 Eample of a planar arra: Isotropic elements, uniform amplitues plane plane ìï N = 7, N = 4 ï í = = l / ï = = ïî plane plane (3D: B scale) 4

41 Steering planar arras For a rectangular arra, the maima of AF n an AF n are locate at Y = k sin cos f+ = mp m =,, Y = k sin sin f+ = np n =,, The main beam is irecte towars, f when ( m =, n = ) =-k sin cosf =-k sin sin f Grating lobes will occur at ìsin cos f- sin cos f =ml/ ï í ïsin sin f- sin sin f =nl/ ïî irections where all contributions are in phase constructive interference! é - sin sin f nl/ ù f = tan êsin cos f ml/ ú ë û é - sin cos f ml/ ù é sin sin f nl/ - ù = sin = sin ê ë cosf ú û ê ë sin f ú û For a true grating lobe, both euations must be satisfie simultaneousl! 4

42 Grating lobes in rectangular arras: N = N = 5, = = (3D: linear) 4

43 Grating lobes in planar arras = = l /4 N = N = 5, = = = = l / = = 3 l /4 ( plane) = = l = = 3 l / = = l General rule to avoi grating lobes: < l/, < l/ 43

44 Beamwith To fin the beamwith of a scanne planar arra is ver ifficult. Approimate values for large arras with maimum near broasie can be foun using the results of the uniform linear arra Q : HPBW of a broasie linear arra with M elements Q : HPBW of a broasie linear arra with N elements Cross-section Q h Y h Then for a large arra with maimum near broasie, the elevation-plane beamwith (at = ) is f Q = h cos é cosf sin f ù æ ö æ ö + êèç Q ø ç Q ë è ø ú û an in the plane perpenicular to = For a suare arra with M = N an Q =Q =Q Y h = é sin f æ cosf ö ù Q Q = Y =Q : cos, h æ ö + ê ç è Q ø ç Q ë è ø ú û h 44

45 Beam soli angle W A =QhYh W = A Q Q sec é ùé æ ö ù æ ö sin f Q cosf sin f çq cos f + +ç ç èq ø ê çq ëê ûúë è ø ú û Directivit 4 p AF (, f ) AF (, f ) D = p p òò é ùé ù * ë ûë û ma * [ AF ( f, )][ AF ( f, )] sin f For large planar arras with maimum near broasie D p 3, 4» = W / ras W / egrees A A or, more accurate: D = pcos D D 45

46 Eample : Effect of scanning angle on beamwith an irectivit f = = 9 f = 3 = 9 N 5 = = l / = N = f = 6 = 9 f = 9 = 9 (3D: linear) 46

47 Eample : Design a 8 ( in the -irection an 8 in the -irection) element uniform planar arra so that the main maimum is oriente along =, = 9. For a spacing of = = /8 between the elements, fin a) the progressive phase shift between the elements in the an irections b) the irectivit of the arra c) the half-power beamwiths (in two perpenicular planes) of the arra Solution: = = l/8, M =, N = 8, =, f = 9 a) b) c) =- k sin cosf = pl =- k sin sin f =- sin=-.364 ra =-7.8 l 8 D = pcos D D ; ìï D = M = =.5 ï í l 8 ïd = N = 8 =. ïî l 8 D = p cos.5 = B é æ.39l öù - é -æ.39 8öù Q = 9 cos 9 cos - = - = 4.49 pm ç p ê ç è ø ë è øúû êë úû é æ -.39l öù é -æ.39 8öù Q = 9 cos 9 cos - = - = 5.56 pn çè 8p ê èç ø ú ëê øûú ë û (linear) 47

48 53.37 h cos cos cos9 sin 9 sin cos h 4.49 sin sin 9 cos9 cos , 4 3, 4 3, 4 D B / egrees / egrees A h h plane plane 48

49 Eample 3: Low-sielobe istributions N = N = 4, = =, = = l/ Uniform ecitation Hamming winow Linear representation B, Dnamic range: 5 B Polar, B 49

50 Eample 4: Effect of the element patterns Consier a uniform arra of horiontal ipoles oriente in irection: N = N = 5, = =, = = l = Horiontal ipoles (-irection) Arra factor (isotropic elements) Arra pattern 5

51 Arra Factor: Formula versus iniviual elements Eample : Arra with ientical amplitues an progressive phase shift between elements e e o j 3 o j o e -j o 3 e -j Eample : Arra with ifferent amplitues an same phase AF n = æ4 ö sin é o k cos + ù ç çè êë úû ø 4 æ ö o sin ék cos + ù ç êë úû çè ø jk cos AF =+ 3e + e -jk cos -jk - 3e + 5e 3 4 =+ 3e + e jk cos cos jk(.5 )cos jk(.5 )cos - 3e + 5e -jk(.5 )cos -jk(.5 )cos 5

52 Eample 3: Arra elements off the ais jk cos jk cos AF =+ e -e cos cos = a a + a = a a + a a = ( sin sin f+ cos ) ( ) r r r = a -a -a = -a a -a a = (-sin sin f-cos ) ( ) r r r ( sin sin f + cos ) - ( sin sin f + cos ) jk jk AF = e -e 5

53 Three arra eamples. Dual-polarie receiver arra for raio astronom Vivali antenna elements Each element is attache to its own receiver Transitions: Coaial to coplanar to slotline subarra Mutual coupling between elements nees to be consiere!! 53

. Conformal arras The arra elements are locate on a curve surface Avantages: Wier angle coverage possible Mounting on high-spee aircrafts Arra pattern: r r n Iˆ n E (, f) n E (, ) I e E

54 . Conformal arras The arra elements are locate on a curve surface Avantages: Wier angle coverage possible Mounting on high-spee aircrafts Arra pattern: r r n Iˆ n E (, f) n E (, ) I e E (, f) N ˆ jkrn r tot f = å n n n= irection unit vector element position element ecitation (comple) element pattern Conformal arra for LEO platform Mutual coupling nees to be consiere 54

55 3. Steerable arras ESPAR Electronicall Steerable Passive Arra Raiator Quasi maima ever 5 55

56 6. Mutual coupling Mutual coupling between the elements in the arra have been neglecte up to now. Inter-element coupling changes the current in antennas an the input impeance V I I Z L V ì V = Z I + Z I ï í ï V = Z I + Z I ïî ( Z ) V, in = = - = I Z + ZL Z Z ( Z Z ) Eample: Two element parasitic arra ìï -ZV ï í ï í ï V = Z I + Z I Z V î ïi = l ïî ì I = ï = ZI + ZI ï ZZ - Z ï ZZ - Z Z epens on spacing ( /4 results in small Z ) Z epens on length of element l > resonant length => Z has inuctive reactance (reflector) l < resonant length => Z has capacitive reactance (irector) 56

57 Yagi-Ua antenna ì = Z-, -I- + Z-,I + Z-,I +... Z-, NI V = Z, -I- + Z,I + Z,I +... Z, NIN ï í ï = ZN, -I- + ZN,I + ZN,I +... ZN, NIN ïî Since all ajustable parameters are interrelate, esigns are mainl carrie out eperimentall an/or with fiel solvers (tpical : 8- elements, gain 4 B, banwith of a few percent) N i V - main lobe li N Practical eample Directivit Bi Banwith % l i t=. To increase the banwith: lengthen reflector (low freuencies) shorten irectors (high freuencies) however: gain will be reuce b up to 5 B 57

58 Illustration: Yagi-Ua Antenna Vertical ipole + irector + 6 irectors.3l D =.76 B D» 5.4 B D» B 58

59 -D Vivali arra: Influence of mutual coupling Mutual coupling reuces return loss an increases sielobes! 59

Lecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b.

Lecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b. b. Partial erivatives Lecture b Differential operators an orthogonal coorinates Recall from our calculus courses that the erivative of a function can be efine as f ()=lim 0 or using the central ifference