Electronic Warfare and Radar Systems Engineering Handbook  Power Density  
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POWER DENSITY
Radio Frequency (RF) propagation is defined as the travel of electromagnetic waves through or along a medium. For RF propagation
between approximately 100 MHz and 10 GHz, radio waves travel very much as they do in free space and travel in a direct line of sight. There
is a very slight difference in the dielectric constants of space and air. The dielectric constant of space is one. The dielectric constant of
air at sea level is 1.000536. In all but the highest precision calculations, the slight difference is neglected.
From chapter 3, Antennas,
an isotropic radiator is a theoretical, lossless, omnidirectional (spherical) antenna. That is, it radiates uniformly in all directions. The
power of a transmitter that is radiated from an isotropic antenna will have a uniform power density (power per unit area) in all directions.
The power density at any distance from an isotropic antenna is simply the transmitter power divided by the surface area of a sphere (4πR^{2}) at that distance. The surface area of the sphere increases by the square of the radius, therefore the
power density, P_{D}, (watts/square meter) decreases by the square of the radius.
[1]
P_{t} is either peak or average power depending on how PD is to be specified.
Radars use directional
antennas to channel most of the radiated power in a particular direction. The Gain (G) of an antenna is the ratio of power radiated in the desired
direction as compared to the power radiated from an isotropic antenna, or:
The power density at a distant point from a radar with an antenna gain of G_{t} is the power density from an isotropic antenna
multiplied by the radar antenna gain.
Power density from radar, [2]
Pt is either peak or average power depending
on how PD is to be specified. Another commonly used term is effective radiated power (ERP), and is defined as: ERP = P_{t}
G_{t}
A receiving antenna captures a portion of this power determined by it's effective capture Area (A_{e}).
The received power available at the antenna terminals is the power density times the effective capture area (A_{e}) of the receiving
antenna.
e.g. If the power density at a specified range is one microwatt per square meter and the antenna's effective capture area is one square meter
then the power captured by the antenna is one microwatt. 
For a given receiver antenna size the capture area is constant no matter how far it is from the transmitter, as illustrated in Figure
1. Also notice from Figure 1 that the received signal power decreases by 1/4 (6 dB) as the distance doubles. This is due to the R2 term in the
denominator of equation [2].
Sample Power Density Calculation  Far Field (Refer to Section 35 for the definition of near field and far
field)
Calculate the power density at 100 feet for 100 watts transmitted through an antenna with a gain of 10.
Given: P_{t}
= 100 watts G_{t} = 10 (dimensionless ratio)
R = 100 ft
This equation produces power density in watts per square range unit.
For safety (radiation hazard) and EMI calculations, power density is usually expressed in milliwatts per square cm. That's nothing more
than converting the power and range to the proper units.
100 watts = 1 x 10^{2} watts = 1 x 10^{5} mW
100 feet
= 30.4785 meters = 3047.85 cm.
However, antenna gain is almost always given in dB, not as a ratio. It's then often easier to express ERP in dBm.
ERP (dBm) = P_{t} (dBm) + G_{t} (dB) = 50 + 10 = 60 dBm
To reduce calculations, the graph in Figure 2 can be used. It gives ERP in dBm, range in feet and power density in mW/cm^{2}. Follow
the scale A line for an ERP of 60 dBm to the point where it intersects the 100 foot range scale. Read the power density directly from the Ascale
xaxis as 0.0086 mW/cm^{2} (confirming our earlier calculations).
Figure 2. Power Density vs Range and ERP
Example 2
When antenna gain and power (or ERP) are given in dB and dBm, it's necessary to convert back to ratios in order to perform the calculation
given in equation [2]. Use the same values as in example 1 except for antenna gain.
Suppose the antenna gain is given as 15 dB: G_{t}
(dB) = 10 Log (G_{t})
Follow the 65 dBm (extrapolated) ERP line and verify this result on the Ascale Xaxis.
Example
3  Sample Real Life Problem
Assume we are trying to determine if a jammer will damage the circuitry of a missile
carried onboard an aircraft and we cannot perform an actual measurement. Refer to the diagram at the right.
Given the following:
Jammer power: 500 W (P_{t} = 500)
Jammer line loss and antenna gain:
3 dB (G_{t} = 2)
Missile antenna
diameter: 10 in
Missile antenna gain: Unknown
Missile limiter protection (maximum antenna power input): 20 dBm (100mW) average
and peak.
The power density at the missile antenna caused by the jammer is computed as follows:
The maximum input power actually received by the missile is either:
P_{r} = P_{D} A_{e}
(if effective antenna area is known) or
P_{r} = P_{D} G_{m}λ / 4π
(if missile antenna gain is known)
To cover the case where the missile antenna gain is not known, first assume an aperture efficiency
of 0.7 for the missile antenna (typical). Then:
P_{r} = P_{D} Aη = 8.56 W/m^{2} (π)[
(10/2 in)(.0254 m/in)]^{2} (0.7) = 0.3 watts
Depending upon missile antenna efficiency, we can see that the power received will
be about 3 times the maximum allowable and that either better limiter circuitry may be required in the missile or a new location is needed for
the missile or jammer. Of course if the antenna efficiency is 0.23 or less, then the power will not damage the missile's receiver.
If
the missile gain were known to be 25 dB, then a more accurate calculation could be performed. Using the given gain of the missile (25 dB= numeric
gain of 316), and assuming operation at 10 GHz (λ = .03m) P_{r} = P_{D} G_{m}λ^{2 }/ 4π = 8.56 W/m (316)(.03) / 4π = .19 watts (still double the allowable tolerance
Table of Contents for Electronics Warfare and Radar Engineering Handbook
Introduction 
Abbreviations  Decibel  Duty
Cycle  Doppler Shift  Radar Horizon / Line
of Sight  Propagation Time / Resolution  Modulation
 Transforms / Wavelets  Antenna Introduction
/ Basics  Polarization  Radiation Patterns 
Frequency / Phase Effects of Antennas 
Antenna Near Field  Radiation Hazards 
Power Density  OneWay Radar Equation / RF Propagation
 TwoWay Radar Equation (Monostatic) 
Alternate TwoWay Radar Equation 
TwoWay Radar Equation (Bistatic) 
Jamming to Signal (J/S) Ratio  Constant Power [Saturated] Jamming
 Support Jamming  Radar Cross Section (RCS) 
Emission Control (EMCON)  RF Atmospheric
Absorption / Ducting  Receiver Sensitivity / Noise 
Receiver Types and Characteristics 
General Radar Display Types 
IFF  Identification  Friend or Foe  Receiver
Tests  Signal Sorting Methods and Direction Finding 
Voltage Standing Wave Ratio (VSWR) / Reflection Coefficient / Return
Loss / Mismatch Loss  Microwave Coaxial Connectors 
Power Dividers/Combiner and Directional Couplers 
Attenuators / Filters / DC Blocks 
Terminations / Dummy Loads  Circulators
and Diplexers  Mixers and Frequency Discriminators 
Detectors  Microwave Measurements 
Microwave Waveguides and Coaxial Cable 
ElectroOptics  Laser Safety 
Mach Number and Airspeed vs. Altitude Mach Number 
EMP/ Aircraft Dimensions  Data Busses  RS232 Interface
 RS422 Balanced Voltage Interface  RS485 Interface 
IEEE488 Interface Bus (HPIB/GPIB)  MILSTD1553 &
1773 Data Bus  This HTML version may be printed but not reproduced on websites.







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