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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.
