1. Radar Range Equation

2. Objectives Calculate range for a pulsed radar system.
Calculate and interpret minimum range, unambiguous range and range resolution
Outline the derivation of the simplified radar range equation
Solve for maximum radar range using the simplified radar range equation.

3. Basic Range Range = c Dt 2 Dt = total time
elapsed for 2 – way trip
for pulse
16.5 ms
16.5 ms
4.95 km
1)       Pulsed Radar range Determination
a)       Range determination for a Pulsed Radar is based on the speed of propagation and the total travel time of the transmitted pulse and it’s return. Since the total time accounts for the distance to the contact and back to the radar receiver, 2 divide the product.
Example: How far away is a contact if it takes 33msecs for the pulse to return?
R = (3X108 m/s)(33x10-6 s) = m = km

4. Range Calculations
Shortest range at which radar can first detect a target
Range beyond which target appears as “second time around”
The min distance between 2 targets at nearly the same range that generate 2 separate returns
a)       Minimum Range = Rmin - smallest range at which radar can first detect a target.
b)       Maximum Unambiguous Range = Runambiguous - range beyond which target appear as “second time around,” echo or echo that arrives after the transmission of the next pulse. It is the maximum actual range that can be detected and displayed without ambiguity.
c)       Range Resolution = Rres The min distance between 2 targets at nearly the same range that generate 2 separate radar returns and not 1 larger return. The “range equivalent” of pulse width.

5. “Simplified” Radar Range Equation

6. Power Out
Assume pulse radiates uniformly out from antenna in all directions
Power density at any given point is
1)       DERIVATION OF THE SIMPLIFIED RADAR RANGE EQUATION
a)       Distribute the pulse Peak Power ( Ppeak ) over the surface of the sphere of radius R.
b)       Pulse radiates uniformly in all directions.
c)       Surface area is of sphere is 4pR2
a)       Power Density ( P ) for an omnidirectional antenna is:
P = Ppeak
4pR2

7. Effective Antenna Aperture
Ae is a measure of actual antenna performance
both transmit and receive
The larger the aperture, the better the radar’s performance
But no 100% efficient antenna, r is a decimal # < 1.0
a)       EFFECTIVE ANTENNA APERTURE - Ae - determines effectiveness of antenna’s ability to collect reflected energy; the larger the aperture, the better the collector, the better the range.
CALCULATE RECTANGULAR AREA OF ANTENNA: L X W
CIRCULAR AREA OF ANTENNA: PI RxR (PI R SQUARED)
A = antenna physical area
ρ = antenna [in]efficiency
Ae = antenna aperture

8. Directional Gain Omni-directional beam spreads spherically
surface area = 4R2
Directional beam
surface area = (R)(R) = R2 
Directional Gain = Maximum Radiation Intensity
Average Radiation Intensity
= Maximum power per steradian
Total power radiated isotropic
Based purely on dimensions/shape of antenna
a)       ANTENNA GAIN - ability of antenna to concentrate energy in particular direction.
b)       Directional Gain - describes antenna beam pattern (directivity); how well the beam is focused. R R R

9. Power Gain  =k / Length  = k / Width
Also simply called Gain…power out with respect to signal losses between transmitter and antenna.
Remember for radar antennas   = k/L therefore;
Includes efficiency of radar based on shape/spill over/losses
)       Power Gain - also simply called Gain; power out with respect to antenna losses and efficiency; can be given as a unit-less number or in decibels. L W
 =k / Length
 = k / Width

10. Gain Example
What is the gain of an AN/APS-116 (S-3 radar) that is operating at 9.8 GHz? The antenna is 1.07 m wide and 0.61 m high with an efficiency estimated at 77%.

11. Gain Solution G = 4Ae/2 Ae = A Gain = 4 A /k2 2 = c/ k= .88
= 3x108m/s = .03m
9.8x109/s
Gain = 4  (.77)(1.07m)(.61m) =
(.88)2 (.03m)2

12. Power Gain Combine with Power Gain (G) of a directed antenna…
a)       Combine with the Power Gain (G) of a directed antenna:
P = PpeakG
4pR2

13. Radar Cross Section

14. Reflection
Effective area that reflects the Radar Energy back to the receiver is Radar Cross Section “σ”
Returned Ray
a)       Radar pulse strikes the contact and is reflected back to the antenna. The amount of return pulse reflected is determined by the Radar Cross Section ( s ) of the contact. Therefore, the return pulse power density measured in terms of power reflected from the target is:
P = PpeakGs
4pR2
Incident Rays

15. Return Signal Spreading
Signal undergoes spherical spreading on way back to receiver
Pr = Power Returning
from target
a)       The signal undergoes spherical spreading again on its way back to the radar receiver. The power density of the return pulse at the radar antenna is now:
P = PpeakGs
(4pR2 )2

16. Received at Antenna Only a small amount of return pulse is collected
Effective Antenna Aperture Ae determines amount of energy received
a)       Only a small amount of return pulse from the contact is collected by the radar antenna and sent to the receiver. The effective Antenna Aperture ( Ae ) is used to determine the actual amount of returned pulse received. Therefore, the Power Density at the radar receiver is:
Pr = PpeakG Ae s
(4pR2 )2
A = area antenna
ρ = antenna efficiency

17. Detection When the Minimum Signal for Detection (Smin)
is equal to the power density at the receiver, target
detection will result.
Therefore,
a)       When the Minimum Signal for Detection ( Smin ) is equal to the Power Density at the radar receiver, target detection will result. Therefore,
Smin = Pr = PpeakG Ae s
(4pR2 )2

18. Solve for Maximum Range for Detection
a)      Solve for range and refer to this range as the Maximum Range ( Rmax )
Rmax = PpeakG Ae s 1/4
(4p2 ) Smin
a)       Therefore, the “simplified” Radar Range Equation is:
Units check!

19. Example: SPS-49 2D Air Search Radar
Radar Parameters
Peak Power: 500 kW
Antenna Dimensions: 8m wide x 4m high (Orange Peel Parabolic design)
Frequency: 700 MHz (approx. mid range)
Antenna Gain: 34 dB
PRF: 230pps / 800 or 1000pps
PW: 75 μs (compressed) or 2 μs (short range)
Determine horizontal and vertical beamwidths
Calculate Directional Gain (Gdir), efficiency and (Ae)
Calculate Rmax for a 1 square meter RCS and Smin of 1pW
Change peak power to 6400 kW . Determine new max range for the same target.

20. Objectives Calculate range for a pulsed radar system.
Calculate and interpret minimum range, unambiguous range and range resolution
Outline the derivation of the simplified radar range equation
Solve for maximum radar range using the simplified radar range equation.