# Phased Array China Lake

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```					Planar Element Lattices
Coverage                                    FOV = field-of-view (scan volume)

Angular = Antenna FOV
PT =Transmit Power
1                            G = Antenna Gain
 PtG   2   2   4                             =Target cross-section

4 3Smin 
Range                                                     Smin = Min Detectable Signal
             

Resolution                              B = bandwidth
c = speed of light
2c
Range =
B
 3dB
Angle =
S
km    2 
ψ3dB = 3dB beamwidth
km = monopulse error slope                         SA-12
N
(S/N) = signal-to-noise ratio
Directivity vs. Angle
Directivity, D(,), describes the angular variation in an
antenna’s ability to transmit or receive plane waves

Isotopic
Directivity vs Angle from attay Normal
directivity                       30
pattern
20

10

Directivity in dB
0

                                                     -10

-20

-30

-40
-90 -80 -70 -60 -50 -40 -30 -20 -10     0   10    20   30      40   50   60   70   80   90

0°                                                                        Angle in Degree
Maximum Directivity
Dmax 
I
Dmax  max
Pr
4
where Pr is the total power radiatedby the antenna

D
Therefore, max is a ratio of the maximum power intensityradiatedby
total power
Expression for Maximum
Directivity
 4A
Dmax   
2
A
where         is the area of the antennnaaperturein square wavelengths
 2

and is the apertureefficiency factor

d
If the power is uniformly distribute across the array and there are no
amplitude or phase errors,  1.0, otherwise  1.0
Gain vs. Directivity
Imax
G0          D0  L
Pin
4
where Pin is the total power into the antenna and L is loss in the antenna

• Directivity is a ratio the power radiated by the antenna
with respect to an isotropic antenna with the same
• Gain is a ratio of the power radiated by a antenna with
respect to an lossless isotropic antenna with the same
transmit power
• The gain of an antenna is the directivity minus the
antenna losses
Beamwidth (Angular Resolution)
40

3 dB or 1/2 power beamwidth, ψ
k
30
Gain (or Directivity) (dBi)

Beamwidth                                              
L ,
20
where k is a constant depending on
the aperture taper,and
10                                                                    L is the length of the aperture

0                                                         For uniform linear or rectangular
aperture, k=50.6°

-10

-20
-90           -75   -60   -45   -30   -15   0   15   30   45   60    75   90

Angle (degrees)
Beamwidth & Directivity
4Lx Ly
Dmax                 , and
Pattern of a 48                                               2

element array                                                 k

Pattern of a 16                   L
element array
 Dmax  x  y  4

x y

Ly

Lx

The directivity of and antenna is inversely related the area of its main beam.
Thus, you cannot have wide beamwidth and high directivity at the same time
Aperture Projection,
Element Pattern, and Scan Loss
       When an array forms a beam in
it projects a smaller aperture:

* The maximum directivity(Dmax=4A/2)
is reduced by cos

* The main beam (ψ=k/L) is broadened in the
direction of scan by 1/cos

Therefore, the element pattern will
roll off by at least cos. More typical
loss is cos1.5

angle
Element Pattern & Array Factor
2
j        nx sin   s 
E (  )  Ee   e         

N
Array Factor
Element Pattern

Array Factor

Element Pattern

Sine Space Coordinate System
u  sin  cos                    w                               u and v are the x and y

v  sin  sin 
components of a unit vector,k,
pointing in the direction (,)

w  cos
k

sin   u2  v 2                                              v
y
w  1 u  v2     2


u

x
2
j        mx uus ny  v v s 
Eu, v   E e u, v  a mne          

M   N
Grating Lobe Locations for a
Rectangular Lattice
2
j        mx uus ny  v v s 
Eu, v   E e u, v  a mne           

M   N

Eu, v   E e u, v  a mne  j2 mx uus ny vvs 
M   N

AF u, v    a mne  j2 mx uus ny vvs                         The x,y coordinate
M   N
system has been
                          1 
AF u, v   AF u 
1                                                      normalized by wavelength
, v   AF u, v     
     x                  y 

AF u, v  is periodic with period
1      1
and
x     y
This equation tell us
where all the grating
p                 q                                                  lobes are!
u  us    and v  v s 
x                 y
where p  0,1,2,... and q  0,1,2,...
Grating Lobe Diagram for a
Rectangular Lattice
u  sin  cos
v  sin  sin 
w  cos
w  1  u2  v 2
sin   u2  v 2
at   90 , u2  v 2  1

2
p           q 
2

if  us        vs      1
      x          y 
This is the equation for a set of
for p or q  0,1,2,...          unit circles in sine space each
centered at a nominal grating lobe
Grating Lobe Diagram for a
Triangular Lattice
u  sin  cos
v  sin  sin 
w  cos
w  1  u2  v 2
sin   u2  v 2
at   90 , u2  v 2  1    Centers of grating
lobe circles for a
triangular lattice
maxima are set at :
p                  q
u - us       and v - v s 
2 x               2 y
for p or q  0,1,2,...and p  q is even
Grating Lobe Locations for a
Rectangular Lattice                            - j
2p
{m DX ( u - us ) - n DY (v - v s )}
E (u , v ) = Eelement (u , v )å     åa      mn   e          l

M    N

E (u , v ) = Eelement (u , v )å     åa      mn   e - j 2p {mDx (u - us ) - nDy (v - v s )}
M    N

AF (u , v ) = å    åa  mn   e - j 2p {mDx (u - us ) - nDy (v - v s )}
M   N
The x,y coordinate
system has been
1                     1                                                   normalized by wavelength
AF (u , v ) = AF (u +       , v ) = AF (u , v + )
Dx                    Dy

1      1
\ AF (u , v ) is periodic with period                             and
Dx     Dy
This equation tell us
where all the grating
p               q                                                                             lobes are!
u-us =  and v-v s =
Dx              Dy
where p = 0, ±1, ±2,.. and q = 0, ±1, ±2,..
Grating Lobe Diagram for a
Rectangular Lattice
u = sin q cos f
v = sin q sin f
w = cos q
w = 1 - u2 - v 2
sin q = u 2 + v 2
at q = 90°, u 2 + v 2 = 1

p 2         q
if (us +  ) + (v s + ) 2 = 1
Dx           Dy
This is the equation for a set of
for p or q ¹ 0                                unit circles in sine space each
centered at a nominal grating lobe
The grating lobe will be on the unit circle
Grating Lobe Diagram for a
Triangular Lattice
AF (u , v ) = é1 + e { ( s ) ( s )} ù
- j 2 p Dx u - u +Dy v - v
ë                                û
å
×    åa  mn e
- j 2 p { 2 mDx (u - us )+2 nDy (v - v s )}

M   N

where maxima are at
p
u - us =       p = 0, ±1, ±2,...
2 Dx
q
v - vs =       q = 0, ±1, ±2,...
2 Dy
and p + q is even                                             Grating lobe circles
for a triangular
lattice
Grating Lobe Migration with
Beam Scanning
Kx0 = us
Ky0 = vs   When the array is scanned, the
grating lobes (GLs) scan the same
amount in the same direction

When the main beam is scanned to a
location that crosses a GL circle, a GL
migrates into real space on the
opposite side of the unit circle

This diagram will show you:
• Where you can scan without GLs
• How many GLs there are
• Where they are
Real GL
Array Element Pattern

Cosine element pattern is only an approximation

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