a Matlab package for phased array beam shape inspection
a Matlab package for phased array beam shape inspection a Matlab package for phased array beam shape inspection
42 A FIGURES TO CHAPTERS 2–7 I A E T ∝ G T (û)I A û V (B) A (a) Transmission from A. V (A) B û E R ∝ I B I B (b) Transmission from B. Figure 8: Antenna directional gain in reception. A small test antenna, e.g. a short dipole, is at a fixed distance from the antenna of interest, always oriented perpendicularly to the direction unit vector û. We envision û but not the distance to be varied by moving the test antenna, while always keeping it perpendicular to û, and inquiry the dependency on û of the voltage V (A) B at A caused by the current at B, the situation shown in panel (b). We first consider the transmission from A as in panel (a). Total transmitted power is proportional to IA 2 , so by definition of the directional amplitude gain G T (û), the far field E T in direction û is proportional to G T (û)I A . On the other hand, the induced voltage V (B) A to I A and G T . We write this as V (B) depends linearly on E T , so that also V (B) A A is proportional = Z AB I A , where Z AB = a G T (û) and the factor a that does not depend on û. By reciprocity, the impedance Z AB gives also the coupling from B to A in panel (b): V (A) B = a G T (û) I B . The (linear) relation between ‖E R ‖ and I B cannot depend on û, so we must also have V (A) B = b G T (û) ‖E R ‖, with some b that is independent of û. That is, all directional dependency in V (A) B comes via G T (û). This shows that the gain pattern in reception is equal to the gain pattern in transmission. To find the constants of proportionality a and b requires knowledge of the actual radiation fields associated with a short dipole. With standard assumptions of polarization and impedance matching in the receiving system, this leads to the general result A eff = (G T (û)) 2 λ 2 /4π for the effective area of an receiving antenna.
43 E 0 E m û d m P xy ∆ m P u Figure 9: Computing the time delay/the phasing factor for an array element . We are interested in computing the far field of an antenna array in the direction û, when the array elements are at the positions R m . The plane P u is normal to û. For a monochromatic plane wave, the relative phases of the fields E m are equal to the elements’ phase distances 2π∆ m /λ from this plane. The time distance is T m = ∆ m /c, independent of the wave length. 1 0.9 0.8 0.7 M = 5 M = 10 M = 50 |diric(Ψ, (x,M)| M)| 0.6 0.5 0.4 0.3 0.2 0.1 ∼ 2 3π ∼ 1 M 0 0 0.2 0.4 0.6 0.8 1 Ψ/2π x / 2 π Figure 10: Magnitude of the Dirichlet kernel for three values of the M-parameter. The value of the largest local maxima, located at about Ψ ≈ 3π/M and Ψ ≈ 2π − 3π/M, and the smallest local maximum, is indicated. The largest maximum has value of about 0.212 (-13.5 dB in terms of power), roughly independent of M for any larger M, while the smallest maximum, near Ψ ≈ π, goes towards zero as 1/M. The kernel has zeros at the points Ψ = 2πn/M.
- Page 1 and 2: E3ANT A Matlab Package for Phased A
- Page 3 and 4: List of Figures 1. Beam steering. .
- Page 5 and 6: 5 2. Time steering and phase steeri
- Page 7 and 8: 2.2 Monochromatic signals—time-st
- Page 9 and 10: 3.1 Transmission 9 lobes have equal
- Page 11 and 12: 3.2 Reception 11 current I goes via
- Page 13 and 14: 3.2 Reception 13 equations. But fir
- Page 15 and 16: 15 “losing sensitivity” (due) t
- Page 17 and 18: 4.2 Array factor as 2-D discrete Fo
- Page 19 and 20: 4.6 Parseval’s theorem 19 closed
- Page 21 and 22: 4.9 The power integral for an array
- Page 23 and 24: 4.12 Computing the antenna directiv
- Page 25 and 26: 4.14 2-D beam cross section 25 or m
- Page 27 and 28: 5.1 Timing accuracy versus pointing
- Page 29 and 30: 6.1 Method 29 excitation amplitudes
- Page 31 and 32: 31 electric field and the scatterin
- Page 33 and 34: 33 Table 1: Predicting SNR for the
- Page 35 and 36: 35 s(t, r; û) E p û û 0 0 R m
- Page 37 and 38: 37 D=1.0; θ 0 = 60, θ g = −8, 6
- Page 39 and 40: 39 (a) (b) (c) (d) Figure 5: Random
- Page 41: 41 A + Z 2 I A Z 1 B + Z 3 V V (B)
- Page 45 and 46: 45 80 60 40 20 θ 0 (°) 0 −20
- Page 47 and 48: 47 10 Beam width (°) 9 8 7 6 5 4 M
- Page 49 and 50: 49 Y X Figure 17: A 3D view of an a
- Page 51 and 52: 51 45 40 Directivity (dBi) 35 30 25
- Page 53 and 54: 53 ARRAY 50×20 1.50×1.50 δx=0.0
- Page 55 and 56: 55 ARRAY 50×20 1.50×1.50 δx=0.0
- Page 57 and 58: 57 ∆X z E p χ P Q ∆Z z r 2 R 2
- Page 59 and 60: 0.02 59 12×4 1.4×1.4 φ 0 =0.0°
- Page 61 and 62: 61 V eff for a long pulse (km 3 ) V
- Page 63 and 64: 63 V eff for a long pulse (km 3 ) V
- Page 65 and 66: 65 B. Pointing geometry The functio
- Page 67 and 68: 67 KIRUNA Lat=67.86 Lon=20.44 Hgt=0
- Page 69: 69 C. Matlab functions There are tw
43<br />
E 0<br />
E m<br />
û<br />
d m<br />
P xy<br />
∆ m<br />
P u<br />
Figure 9: Computing the time delay/the phasing factor <strong>for</strong> an <strong>array</strong> element . We are<br />
interested in computing the far field of an antenna <strong>array</strong> in the direction û,<br />
when the <strong>array</strong> elements are at the positions R m . The plane P u is normal<br />
to û. For a monochromatic plane wave, the relative phases of the fields E m<br />
are equal to the elements’ phase distances 2π∆ m /λ from this plane. The time<br />
distance is T m = ∆ m /c, independent of the wave length.<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
M = 5<br />
M = 10<br />
M = 50<br />
|diric(Ψ, (x,M)| M)|<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
∼ 2<br />
3π<br />
∼ 1 M<br />
0<br />
0 0.2 0.4 0.6 0.8 1<br />
Ψ/2π x / 2 π<br />
Figure 10: Magnitude of the Dirichlet kernel <strong>for</strong> three values of the M-parameter. The<br />
value of the largest local maxima, located at about Ψ ≈ 3π/M and Ψ ≈ 2π −<br />
3π/M, and the smallest local maximum, is indicated. The largest maximum<br />
has value of about 0.212 (-13.5 dB in terms of power), roughly independent of<br />
M <strong>for</strong> any larger M, while the smallest maximum, near Ψ ≈ π, goes towards<br />
zero as 1/M. The kernel has zeros at the points Ψ = 2πn/M.
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