The system and the transmitter
2.8 Quantitative scanner analysis
2.8.1 Elevation performance, marine and VTS slotted arrays
Elevation sidelobe performance is not usually stated in datasheets, but can be inferred from the following analysis, which is based on the worst case of uniform illumination.
From the geometry, off-axis voltage, V, relative to axial voltage VmSiX is y-sinxy
v — nnax-
X
Hence, off-axis gain is
G = ( - — j Gmax = — - Gmax numerically, (2.10a) where v is the elevation angle from axis (rad) [—1 < v < 1], 0 the beamwidth between half-power points (per datasheet) (rad), G the numerical gain at angle v,
^max the numerical gain on axis (per datasheet), gQ\ the loss of gain at angle v, x = Cv/ko where C is a constant ^2.78312, a the aperture (m) and w the aperture fill factor.
Actual gain, G, should be used when calculating performance against targets lying well out of the scanner equatorial plane, replacing Gmax by a term Gm a x/gei; the loss recurs on the receive leg. Like all losses in this book, gQ\ > 1 numerically.
gel= G m a X „ or gel = Gm a x- G d B . (2.10b)
G numerically
By definition, when v = ± 0 / 2 , G = Gm a x/2, gQ\ = 2 and loss is 3 dB. By sub- stitution in Eq. (2.7a), here sin2jc = 0.5JC2, from which x ~ dil.392v/(0/2) = 2.78312v/0. That is, C = 2.78312 ~ ^7T3/2. Substituting in Eq. (2.7a) at angle v
Gmax f 2.783v/0 I2 .
= pel ^ numerically
G 6 |_ sin (2.783 v/0) J y or
gci - 20 log r - ^ f o ^ i
d B- (
2-
ioc>
|_sin(2.783v/0) J
Nulls occur when the sine term is zero. Here, where n is an integer,
2.78312- = ±7T,±27r,...,ibur. (2.11a)
0
At the first nulls, v = ±1.12880. Here V/Vmax = Oandjc = jr. But here sin O = ko/a.
If a > Ao, sin O ~ O. Substituting,
2.78312X0 Wk0
Tt = , soa = . (2.11b) aq O
For the fully filled case considered here, w ~ 0.886 = ^^/TT and efficiency D = (0.886)2 = 0.785, depressed below unity by the power lost to sidelobes.
Sidelobe peaks occur when the sine term is ± 1 . The principal sidelobe is when n = LAt peaks:
2.78312- = ± — , ± ^ L , . . . and v ~ ±1.690, ± 2 . 8 2 0 , . . . . (2.11c) 0 2 2
Substituting in Eq. (2.10c), gain at principal sidelobe peaks = 0.045. That is 4.5 per cent of the radiated power is lost to each of the four sidelobes (two each in azimuth and elevation); a few per cent is also lost to the secondary sidelobes, together accounting for the value of D obtained above.
At the -2OdB points, sin (2.78312v/0) = 0.1 (2.78312v/0). Assuming uniform illumination, the width is just over twice the half-power beamwidth:
v_20dB = ±1.0250. (2.1Id) Figure 2.29 shows calculated radiation patterns of uniformly illuminated apertures having elevation beamwidths typical of ships' scanners. Actual patterns may differ
Figure2.29 Elevation patterns, ships' scanners. For beamwidths 20° and 25°, assuming uniform illumination. One-way transmission. Roll or pitch reduces the effective gain on target elevation
Down
Axis Elevation, degrees.
Horizontal (if no roll or pitch) Main beams
Heavy line: 25° beamwidth Light line: 20° beamwidth Rectangular beam approximation (25° beamwidth)
- 3 dB (half power) points Beamwidth 20°
First sidelobe peaks ~ 13.3 dB below main beam peaks Sidelobes
Beamwidth 25°
Gain, G - Gmax, dBi
(Symmetrical about axis)
slightly because illumination may not be quite uniform. Gain remains reasonably constant within the half-power points, then falls rapidly.
When assuming a rectangular beamshape (sometimes called square) in perfor- mance calculations:
G = Gmax (i.e. ge\ = 1) if - 0 < v < 0, otherwise
G = O (i.e. gel = 0) numerically. (2.1Ie) Effective aperture height, b, can be inferred from published beamwidth assuming the aperture is fully filled:
b = I v ^ = 0.886^. (2.12)
2 0 0 Substituting for typical beamwidths:
• at 3 GHz, X0 = 0.1 m: b = 0.254 m (20°), 0.203 m (25°);
• at 9 GHz, X0 = 0.032 m: b = 0.081 m (20°), 0.065 m (25°).
Using flares, actual scanner heights exceed these apertures because illumination is not fully uniform and window sealing flanges have to be accommodated. As noted earlier, polyrod designs may trade front to back length for height, permitting sharp height reduction.
Severe roll or pitch swings targets lying near to the roll or pitch plane through the scanner elevation polar diagram. Assuming uniform aperture illumination in the elevation plane and sinusoidal roll motion, Eq. (2.10c) can be used to calculate gain reduction throughout the roll or pitch cycle. If the roll is sinusoidal, reduction is 1 dB one-way when the roll component in the target plane is 0 peak-peak.
2.8.2 Inverse cosecant squared VTS scanners
In the cosec2 elevation region, for half-power beamwidth 0 rad:
/ cosec v \
voltage at v rad off axis = ) numerically. (2.13a) V cosec 0 / 2 /
At the half-power point, again v = 0 / 2 , and gain loss gQ\ = 2. Putting cosec (angle) = 1/ sin (angle), in the cosec2 region:
gain relative to half-power point gain = I — J numerically; (2.13b) V sin 0 / 2 /
• 1 / v ux G™ x 2(sin.x)2 z ^ i o x
gain loss (positive number) gei = = — TT-^T numerically. (2.13c) G (sin0/2)z
Figure 2.30 Elevation patterns, VTS scanners. For uniformly illuminated and inverse cosec squared beams, each having beamwidth 4°. Although in practice the inverse cosec2 pattern may contain ripples, it avoids the deep nulls at —4.5° and —9°. It illuminates short-range targets more uniformly, at cost of slightly lower Gmaxfor a given aperture The expression for the one-way elevation off-axis loss relative to Gmax of an inverse cosecant squared scanner in both its regions, taking upward angles as positive, is:
<t> fsin(2.78312v/0)]
I f v > - - , ft, = 20hg [
2?8312y/(/)J,
otherwise
The first term represents (in dB) the region of uniform illumination (Eq. (2.1Oc)), the second being the cosec2 region (Eq. (2.13c)). Figure 2.30 compares elevation beamshapes of uniformly illuminated and inverse cosecant squared large VTS scan- ners, the latter shown depressed 1° for clarity. The pattern locus lies a few decibels below the sidelobe peaks of a same-size uniformly illuminated scanner, because of the energy transferred to the former null angles. In practice the cosecant squared law is not always closely followed; when calculating short-range performance actual gain/angle values should be substituted if known. Above the lower half-power point both types have the same nose shape.
An inverse cosec2 scanner receives approximately constant echo power from a surface target at the shorter ranges for thee following reason. Assuming the scanner
(Sin x)lx pattern above lower —3 dB point Power gain, G, dBi
Nominal shape of cosec2 beam Feint line Gain reduction at A, g dB
Inverse cosec2 Heavy line Beam axis shown depressed 1°
Cosec2 law usually maintained to -10°
without sidelobe nulls
Gain ill-defined beyond
Beamwidths (j> 4° between - 3 dB points
Sidelobes (Sinjc)/x, uniform illumination
Pattern symmetrical about axis
Down
Second null -9.0° First null -4.5°
Horizontal
First null 4.5°
Up Elevation
Figure 2.31 Typical azimuth patterns. Typical 9 GHz band marine radar scanners.
At 3GHz, gains for the same apertures would be 5dB lower and beamwidths 3 times wider. Large aperture is needed for high gain.
Linear gain scaling would make the patterns look much thinner is mounted at height H above a flat Earth, from the geometry
R = HcOSQCV. (2.14) Neglecting multipath effects, received echo power oc G2/R4. Substituting G oc cosec2 v, echo power oc (cosec2 v)2/(Hcosec v)4 oc I / / /4, irrespective of R.
This incidentally shows that at short range doubling scanner height reduces signal by 12 dB because v is higher and gain lower.
2.8.3 Azimuth radiation pattern
When considering operation of any type of scanner with target off-axis by angle v, and having half-power beamwidth O9 loss factor gaz is defined similarly to ge\. If the target is off beam in both planes, loss factor is gazge\ numerically.
Figure 2.31 shows azimuth radiation patterns of two of the typical ships' scanners listed in Table 2.3, with main beams and close-in sidelobes. When a rectangular beam is assumed, following Eq. (2.11c):
G = Gmax (i.e. gaz = OdB) if-°-<v<°-, (2.15) otherwise
G = 0(i.e.ga z = ocdB). (2.16)
Idealised rectangular beamshapes used in performance calculations
Power gain, dBi
Heavy and dashed lines
Gain dB First sidelobes
Beam axis Azimuth Half-power points, G1113x