Computing Radar Characteristics

References:
Introduction to Airborne Radar, Second Edition
Detecting and Classifying Low Probability of Intercept Radar by Phillip E. Pace

Step 1: Pick a Mode of Operation

Pulsed Operation: In this mode, the radar alternates between transmit and receive modes. This eliminates problems involving the transmitter interfering with the receiver. Range measurement is also simplified.

Continuous Wave: In this mode, the radar is continuously radiating and listening for the echoes. In ground and shipborne CW systems, where space is not a problem (usually); the transmitter and receiver are usually two separate antennas spaced far enough from each other to isolate most interference. In airborne systems, where space is limited, a single antenna must be used usually, leading to noise leakage between the transmitter and the receiver, degrading the performance of the system.

Step 2: Pick a Wavelength.

You can consult this page to find out more about wavelengths.

Picking a wavelength is a very important step, and each wavelength has it's own advantages and disadvantages.

Size of Antenna:

Higher wavelengths allow a smaller antenna to have the same gain.

For example, at X Band (3 cm); you need an antenna 45” in diameter to get a gain of 40 dB; while at S Band (10 cm); you need an antenna 150” in diameter to get the same gain of 40 dB.

Atmospheric Attenuation

From the above; you might wonder why people still use S-Band, despite it's huge antenna sizes required, and it's low resolution.

The reason comes from the fact that as the radar frequency rises, so does it's absorption by the atmosphere. Rough figures for atmospheric attenuation are below:

Band

Loss Factor
(divide signal strength by)

P-Band (1.5 meter)

1.13

L-Band (30 cm)

1.91

S-Band (10 cm)

2.18

X-Band (3 cm)

3.55

As you can see from the table above; a P-Band radar would lose less of it's energy to atmospheric attenuation at long ranges; while X-Band would be severely affected.

Weather Attenuation

As wavelengths start to approach that of rain drops; radar clutter increases – at X-Band, a rainstorm can reflect a substantial amount of radiation – for this reason, X-Band is a very popular choice for airborne weather radar.

At K-Band (1.24 cm or 24 GHz); the frequency is close enough to the absorption frequency of water vapor in the atmosphere (22.2 GHz); that the range of a 1.24 cm radar is virtually negligible. It was for this reason that the Ku (K-Under) and Ka (K-Above) bands were chosen; to avoid this effect.

Space Uses

Of course, in outer space, atmospheric attenuation doesn't apply; making X-Band or smaller very attractive. However, there is a reason to carry a decent S-Band radar on your space dreadnought – frequency diversity and defeating stealth coatings. The RCS of a target can differ by several orders, depending on what wavelength you are targeting it with.

Step 2: Compute Antenna Gain

Antenna Gain is computed through the following formula:

Gain = 4π ((AntennaArea * AntennaEfficiency) / Wavelength2)

Where:

Gain: Antenna gain at the center of the mainlobe.

Wavelength: The length of the radar's wavelength.

AntennaArea= The area of the antenna's aperture.

AntennaEfficiency = The efficiency factor of the antenna.

NOTE: Both AntennaArea and Wavelength must be in the same measurement units; e.g. if the antenna area is in square meters, the wavelength must be in meters as well.

Typical Antenna Efficiency Factors

1

Uniformly illuminated aperture
(Practical impossibility)

0.6 to 0.8

Planar Arrays

As low as 0.45

Parabolic Reflectors



EXAMPLE: What is the gain of a 60 cm diameter circular antenna operating in the 3-cm wavelength with an average efficiency factor of 70%?

Basic math gives us a surface area of 2,827.43 square cm for the antenna.

4π ((2.827.43 cm2 * 0.7) / 32) = 2,763.49

Converting the answer to dB gives us:

10 * log10(2,763.49) = 34.41 dB

An antenna gain of 34.41 dB.

Computing Range Resolution:

PulseLength = 500 * PulseWidth

Where:

PulseWidth is in Microseconds

EXAMPLE: What is the range resolution of a radar with a pulse width of 0.1 microseconds?

500 * 0.1 microseconds = 50 feet

As a rule of thumb, shapes can be recognized at 1/5th to 1/20th of their major dimension, e.g. a 100 foot diameter oil tank needs a radar resolution of around 5 to 20 feet to be recognized easily.

Typical Targets in Ground Mapping Mode and Resolutions Needed

Target

Resolution

Pulse Width

Bandwidth Needed

Coastlines, large cities, mountains.

500 ft

1 μs

1 MHz

Major highways

60-100 ft

0.12 μs to 0.2 μs

5 to 8.33 MHz

Road Map Details, streets, large buildings, small airfields

30-50 ft

0.06 μs to 0.1 μs

10 to 16.67 MHz

Vehicles, small buildings

5-10 ft

0.01 μs to 0.02 μs

50 to 100 MHz

Computing Duty Factor for Pulsed Radars

DutyFactor = (AveragePower / PeakPower)

Computing Duty Factor (Alternate Method)

DutyFactor = (PulseWidth / PRI)

Where:

DutyFactor: Percentage of time the radar is transmitting
PulseWidth: Pulse Width in microseconds
PRI: Pulse Repetition Intervals in microseconds.

EXAMPLE: What is the Duty factor of a radar with a pulse width of 0.75 microseconds and a PRI of 200 microseconds?

(0.75 / 200) = 0.00375

Thus, the duty factor is 0.00375, or 0.375%.

Computing Bandwidth

In order for the majority of power in radar pulses to be passed along, the 3-dB bandwidth must be on the order of:

BandwidthNeeded = (1 / PulseWidth)

Where:

BandwidthNeeded is in megahertz (MHz).
PulseWidth is in microseconds.

EXAMPLE: What is the bandwidth needed at 3-dB for the majority of power to be passed through for a 0.5 microsecond pulse width?

(1 / 0.5 microseconds) = 2 MHz

As a rule of thumb, for it to work well, the bandwidth required must be between 3 and 10 percent of the radar's operating frequency; e.g a 400 MHz radar would have 12 to 40 MHz of bandwidth available; limiting it's effective pulse width to the range between 0.025 μs and 0.083 μs.

Computing the Power Density of a CW Radar from an Isotropic Antenna:

PD = AP / (4 * π * R2)

Where:

PD = Power Density in Watts/m2
A
P = Average Power of the Radar in Watts
R = Range in Meters

Computing the Power Density of a CW Radar from an Directive Antenna:

PD = (AP * AG * L1) / (4 * π * R2)

Where:

PD = Power Density in Watts/m2
AP = Average Power of the Radar in Watts
AG = Antenna Gain
L1 = One Way Atmospheric Transmission Factor
R = Range in Meters

Computing Pulse Repetition Interval (PRI)

NOTE: PRI is also known as the interpulse period.

PRI = (1 / PRF)

Where

PRI = Pulse Repetition Interval of radar in seconds.
PRF = Pulse Repetition Frequency of Radar in Hertz (Hz).

EXAMPLE: What is the PRI of a radar with a PRF of 500 Hertz?

(1 / 500 Hz) = 0.002 Seconds

0.002 seconds is equal to 2,000 microseconds.

Computing the Maximum Unambiguous Range for Pulsed Radar

NOTE: Beyond this distance, a target can be erroneously identified as being much closer than it actually is due to pulse interference.

MUR = SpeedLight / ( 2 * PRF)

Computing Average Transmitted Power

PowerAverage = PowerPeak * DutyFactor

Where:

PowerAverage: Average Power of the Radar in Watts.

PowerPeak: Peak Power of the Radar in Watts.

DutyFactor: The Percentage of time the Radar is transmitting.

EXAMPLE: What is the Average Power of a radar with a peak power of 12 kilowatts and a duty factor of 6 percent?

12,000 watts * 0.06 = 720 watts

Thus, average power is 720 watts

Computing Duty Factor

DutyFactor = Pulse Width / Interpulse Period

Where:

DutyFactor: The Percentage of time the Radar is transmitting.

Pulse Width: Measured in Microseconds.

Interpulse Period: Measured in Microseconds.

EXAMPLE: What is the Duty Factor of a radar with a pulse width of 50 microseconds and an interpulse period of 500 microseconds?

50 / 500 = 0.1

Thus, the duty factor is 10%

Computing the 3-dB Beamwidth for a Uniformly Illuminated Synthetic Aperture array:

These use the speed of the vehicle along with mathematical trickery to produce the equivalent of an array antenna that is much longer than the actual physical dimensions of the array itself.

3dBBW = 0.44 * (Wavelength / AntennaLength)

Where:

3dBBW = The 3 dB beamwidth of the radar in radians.
Wavelength = The wavelength of the radar
AntennaLength = The length of the antenna.

NOTE: Both Wavelength and AntennaLength have to be the same unit of measurement (meters or centimeters, etc).

Computing the 3-dB Beamwidth for a SLAR array

3dBBW = Wavelength / AntennaLength

Where:

3dBBW = The 3 dB beamwidth of the radar in radians.
Wavelength = The wavelength of the radar
AntennaLength = The length of the antenna.

NOTE: Both Wavelength and AntennaLength have to be the same unit of measurement (meters or centimeters, etc).

Computing the Azimuth Resolution for a SLAR Array

ResolutionAzimuth = (Wavelength * Range) / LengthAntenna

NOTE: Units for wavelength, range, and antenna length must be the same. Also, due to atmospheric attenuation, the minimum wavelength that is practical for SLARs is around 3 cm.:

EXAMPLE: What is the Azimuth Resolution of a 10 foot long SLAR array operating at a wavelength of 0.1 foot (3.04 cm or ~10 GHz) at 50 nautical miles (300,000 feet)?

(0.1 * 300,000) / 10 = 3,000 feet

Computing the Azimuth Resolution for a Uniformly Illuminated Synthetic Aperture array:

AR = (Wavelength / ( 2 x AntennaLength) ) * R

The Maximum Acceptable Spacing of Radiating Elements in an AESA

MaxSpacing = (Wavelength / (1 + sin MaxLookAngle))

Where:

MaxLookAngle: How wide the radar beam is in degrees. For use in this formula, degrees MUST be converted to radians.

EXAMPLE: What is the maximum acceptable spacing of radiating elements in a 3 cm wavelength AESA with a maximum look angle of 60 degrees?

(3 cm / (1 + sin 1.05 radians)) = (3 cm / 1.87) = 1.61 cm

Thus, the maximum acceptable spacing of each element is 1.61 cm away from each other to avoid grating lobes, which rob the radar of performance.