11

RadarRadarThe radar range equationThe radar range equation

Prof. N.V.S.N. SarmaProf. N.V.S.N. Sarma

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OutlineOutline

1. Basic radar range equation2. Developing the radar range equation3. Design impacts4. Receiver sensitivity5. Radar cross-section6. Low observability7. Exercises

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1.1. Basic radar range equationBasic radar range equation

• There are many different versions of the radar range equation.

• We will use, and fully derive, the one presented below.

4

min3

22

)4( SGPR t

Max πσλ

=

44

1.11.1 Components of the equationComponents of the equation

• Rmax – the maximum range of the radar• Pt – average power of the transmitter• G – gain of the transmit/receive antenna• λ – wavelength of the operating frequency• σ – radar cross-section of the target• Smin – minimum detectable signal power

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1.21.2 Units of the equationUnits of the equation

4

min3

22

)4( SGPR t

Max πσλ

=

mW

mmWRofunits Max == 422

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2. 2. Developing radar range equationDeveloping radar range equation

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2.12.1 Transmitted powerTransmitted power

• Recall from the previous lecture that the average transmitted power is a function of peak pulse power and the pulse duration:

PRFTwhere

TP

PP pp

peakavet

1, ===τ

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2.22.2 Power density at target Power density at target [4][4]

• Recall that power density decreases as a function of distance traveled:

24 RGPRrangeatdensitypower t

π=

99

2.32.3 Reflected powerReflected power

• The amount of power reflected back from a target is a function of the power density at the target and the target’s radar cross-section, σ:

σπ

⋅= 24 RGPreflecteddensitypower t

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2.42.4 Power density of echo at antennaPower density of echo at antenna

• The power density of the returned signal, echo, again spreads as it travels back towards the radar receive antenna.

22 44 RRGPantennaatreceiveddensitypower t

πσ

π⋅=

1111

2.52.5 Power of echo at receiverPower of echo at receiver**

• The antenna captures only a portion of the echoed power density as a function of the receive antenna’s effective aperture:

πλ

πσλ

πσ

4

,)4()4(

,

2

43

22

42

GAthatrecalling

RGPA

RGPPreceiveratpower

e

te

tr

=

=⋅=

* In this equation the receiver is assumed to be all radar receive chain components except the antenna.

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2.5.1 Relative power received 2.5.1 Relative power received αα rangerange

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2.62.6 Minimum detectable signal powerMinimum detectable signal power

• Therefore a radar system is capable of detecting targets as long as the received echo power is greater than or equal to the minimum detectable signal power of the receive chain:

4

min3

22

maxmin )4(,

SGPRSPfor t

r πσλ

==

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3.3. Radar design impactsRadar design impacts• A careful study of the radar range equation

provides further insight as to the effect of several radar design decisions.

• In general the equation tells us that for a radar to have a long range, the transmitter must be high power, the antenna must be large and have high gain, and the receiver must be very sensitive.

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3.13.1 Power, PPower, Ptt

• Increase in transmitter power yields a surprisingly small increase in radar range, since range increases by the inverse fourth power.– For example, a doubling of transmitter peak power

results increases radar range by only 19%,

19.124 ≈

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3.23.2 TimeTime--onon--target, target, ττ/T/Tpp

• The average power transmitted can also be increased by increasing the pulse duty cycle, sometimes referred to as the “time-on-target”.

• A combined doubling of the pulse width and doubling of the transmitter peak power will give a fourfold increase in average transmitted power, and ~41% increase in radar range.

41.144 ≈

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3.33.3 GGain, Gain, G

• Antenna gain is a major consideration in the design of the radar system.– For a parabolic dish, doubling the antenna size

(diameter) will yield a fourfold increase in gain and a doubling of radar range.

4 44 2max

2)2/(

DorGRand

DorAGdishaFor p

αα

αα

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3.43.4 Receiver sensitivity, SReceiver sensitivity, Sminmin

• Similar to that of transmitter power, increases in receiver sensitivity yield relatively small increases in radar range.– Only 19% range increase for a halving of sensitivity,

and at the expense of false alarms.• Receiver design is a complex subject beyond

the scope of this course, see §3.5.3.• Simplistically, the smaller the radar pulse width,

the larger the required receiver bandwidth and the larger the receiver noise floor.

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3.4.1 3.4.1 Receiver bandwidthReceiver bandwidth

2020

3.4.2 3.4.2 SignalSignal--toto--noisenoise

2121

3.4.3 3.4.3 Receiver thresholdReceiver threshold

2222

4.4. Radar crossRadar cross--section, section, σσ• The radar cross-section of a target is a measure

of its size as seen by a radar, expressed as an area, m2.

• It is a complex function of the geometric cross-section of the target at the incident angle of the radar signal, as well as the directivity and reflectivity of the target.

• The RCS is a characteristic of the target, not the radar.

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4.1.1 4.1.1 RCS of a RCS of a metalmetal plateplate

• Large RCS, but decreases rapidly as the incident angle deviates from the normal.

2

224λ

πσ ba=

2424

4.1.2 RCS of a 4.1.2 RCS of a metalmetal spheresphere

• Small RCS, but is independent of incident angle.

2rπσ =

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4.1.3 RCS of a 4.1.3 RCS of a metalmetal cylindercylinder

• RCS can be quite small or fairly large depending on orientation.

endthefrom

viewedas

r

ra

,4

,2

2

43

2

λπσ

λπσ

=

=

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5.5. Low ObservabilityLow Observability• From the previous discussion on the radar

cross-section of targets, it should be obvious that determining the radar cross-section of an airplane is a complicated task.

• The art of designing an aircraft to specifically have a low RCS is known as low observability, or more commonly known as “stealth”.

• Stealth is a relatively new technology,– even full RCS prediction is only 2 decades old.

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5.25.2 Aircraft high RCS areas Aircraft high RCS areas [1][1]

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5.35.3 Low observability design areas Low observability design areas [1][1]

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5.4 Comparative RCS 5.4 Comparative RCS [1][1]

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6.6. InIn--class exercisesclass exercises

3131

6.16.1 Quick response exercise # 1Quick response exercise # 1

• Think carefully about the derivation of the radar range equation just presented. Is there a potentially significant loss component missing?– Hint: recall the simple link equation from your

very early lectures.

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6.26.2 Quick response exercise # 2Quick response exercise # 2

• Why have designers of stealth aircraft sought to blend the physical transitions / features of the aircraft?

• Will reduction in your aircraft RCS alone make you invisible to the enemy?– How else might they find you?

3333

6.36.3 Radar range equation calculationRadar range equation calculation

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6.36.3 Radar range equation calculationRadar range equation calculation

• A specific Air Search Radar has the following technical specifications:– Operating frequency 2900-3100 MHz– Transmitter peak power 60-2200 kW– PRF 161-1366 Hz, and pulse widths of 9 / 3 μsec– Phased array antenna with a gain of 38.5 dB

• For its published maximum range of 250 miles for a nominal target such as the F-18, what is the receiver chain sensitivity in dBm?

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ReferencesReferences

1) Moir & Seabridge, Military Avionics Systems, American Institute of Aeronautics & Astronautics, 2006. [Sections 2.6 & 2.7]

2) David Adamy, EW101 - A First Course in Electronic Warfare, Artech House, 2000. [Chapters 3,4 & 6]

3) George W. Stimson, Introduction to Airborne Radar, Second Edition, SciTch Publishing, 1998.

4) Principles of Radar Systems, student laboratory manual, 38542-00, Lab-Volt (Quebec) Ltd, 2006.

5) John C. Vaquer, US Navy Surface Officer Warfare School Documents, Combat Systems Engineering : Radar, http://www.fas.org/man/dod-101/navy/docs/swos/cmd/fun12/12-1/sld001.htm

6) Mark A. Hicks, "Clip art licensed from the Clip Art Gallery on DiscoverySchool.com"