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ONEWAY RADAR EQUATION / RF PROPAGATION
The oneway (transmitter to receiver) radar equation is derived in this section. This equation is most commonly used in RWR or ESM type
of applications. The following is a summary of the important equations explored in this section:
Recall
from Section 42 that 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,
[1]
If you could cover the entire spherical segment with your receiving antenna you would theoretically capture all of the transmitted energy.
You can't do this because no antenna is large enough. (A two degree segment would be about a mile and threequarters across at fifty miles from
the transmitter.)
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.
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.
This concept is shown in the following equation:
which
is known as the oneway (beacon) equation [2]
In order to maximize energy transfer between an antenna and transmitter or receiver, the antenna size should correlate λ/4. Control
of beamwidth shape may become a problem when the size of the active element exceeds several wavelengths.
The relation
between an antenna's effective capture area (A
_{e}) is:
Antenna Gain,
[3]
or: Equivalent Area, Ae =
[4]
effective aperture is in units of length squared, s
proportional to wavelength. This physically means that to maintain the gain when
doubling the frequency, the area is reduced by 1/4. This concept is illustrated in Figure 2.
If equation [4] is substituted into equation
[2], the following relationship results:
Peak Power at Receiver Input = S (or P_{R})
= [5]
is the signal calculated oneway from a transmitter to a receiver. For instance, a radar application might be to determine the signal
received by a RWR, ESM, or an ELINT receiver. It is a general purpose equation and could be The free space travel of radio waves can, of course,
be blocked, reflected, or distorted by objects in their path such
As received signal power decreases by 1/4 (6 dB). This is due to the R
^{2}
term in equation [5]. It illustrates a square on radius is decreased by 1/2, you further blow up the balloon, so the diameter or radius is doubled,
the square has quadrupled in area.
The oneway free space loss factor (
α_{1}), (sometimes
called the path loss factor) is given by the term 4
πR
^{2})(4
π/λ
^{2})
or (4
πR /λ)2. As shown in Figure 3, the loss is due to the ratio of two factors (1) the effective radiated area
of the transmit antenna, which is the surface area of a sphere (4
πR
^{2}) at that distance (R), and (2) the
effective capture
area (A
_{e}) of the receive antenna which has a gain of one. If a receiving antenna could capture
the whole surface area of the sphere, there would be no spreading loss, but a practical antenna will capture only a small part of the spherical
radiation. Space loss is calculated using isotropic antennas for both transmit and receive, so
α_{1} is
independent of the actual antenna. Using G
_{r} = 1 in equation [11] in section 31, A
_{e} = λ/
4π.
Since this term is in the denominator of
α_{1}, the higher the frequency (lower λ) the more the space loss.
Since G
_{t} and G
_{r} are part of the oneway radar equation, S (or P
_{r}) is adjusted according to actual antennas
as shown in the last portion of Figure 3. The value of the received signal (S) is:
[6]
To convert this equation to dB form, it is rewritten as:
( keep
λ and R in same units) [7]
Since λ = c / f, equation [7] can be rewritten as:
10 Log (S or P_{r}) = 10 Log(P_{t}G_{t}G_{r})  α_{1}
[8]
Where the oneway free space loss,
α_{1}, is defined as:
* [9]
The signal received equation in
dB form is: 10log (Pr or S) = 10 log P
_{t} + 10 log G
_{t} + 10 log G
_{r} 
α_{1}
[10]
The oneway free space loss,
α_{1}, can be given in terms of a variable and constant term as
follows:
[11]
The value of
f_{1} can be either in MHz or GHz as shown with commonly used units of R in the adjoining table.
Note: To avoid having to include additional terms for these calculations, always combine any transmission line loss with antenna gain.
A value for the oneway free space loss (
α_{1}) can be obtained from:
(a) The Oneway
Free Space Loss graph (Figure 4). Added accuracy can be obtained using the
Frequency Extrapolation
graph (Figure 5)
(b) The space loss nomograph (Figure 6 or 7)
(c) The formula for
α_{1}, equation [11].
FOR EXAMPLE:Find the value of the oneway free
space loss,
α_{1}, for an RF of 7.5 GHz at 100 NM.
(a) From Figure 4, find
100 NM on the Xaxis and estimate where 7.5 GHz is located between the 1
and 10 GHz lines
(note dot). Read
α_{1} as 155 dB. An alternate way would be to read the
α_{1}
at
1 GHz (138 dB) and add the frequency extrapolation value (17.5 dB for 7.5:1, dot on Figure
5) to
obtain the same 155 dB value.
(b) From the nomogram
(Figure 6), the value of
α_{1} can be read as 155 dB (Note the dashed line).
(c) From the equation 11, the precise value of
α_{1} is 155.3 dB.
Remember,
α_{1} is a free space value. If there is atmospheric attenuation because of absorption of RF due to certain molecules in the
atmosphere or weather conditions etc., the atmospheric attenuation is in addition to the space loss (refer to Section 51).
Figure 4. OneWay Free Space Loss
Figure 5. Frequency Extrapolation
Figure 6. OneWay Space Loss Nomograph For Distances Greater Than 10 Nautical
Miles
Figure 7. OneWay Space Loss Nomograph For Distances Less Than 10 Nautical
Miles
Figure 8. Visualization of OneWay Radar Equation
RWR/ESM RANGE EQUATION (OneWay)The oneway radar (signal strength) equation [5] is rearranged
to calculate the maximum range R
_{max} of RWR/ESM receivers. It occurs when the received radar signal just equals S
_{min} as
follows:
[12]
In log form:
20 log R
_{max} = 10 log P
_{t} + 10 log G
_{t}  10 log S
_{min}  20 log
f + 20 log(c/4
π) [13]
and since K
_{1} = 20 log{4
π/c times conversion units if not in m/sec, m, and Hz} (Refer to section
43 for values of K1).
10 log R
_{max} = ½[10 log P
_{t} + 10 log G
_{t}  10 log S
_{min}  20 log
f
 K
_{1}]
( keep P_{t} and S
_{min} in same units) [14]
If you want to convert back from dB, then Rmax � , where M dB is the resulting number in the brackets of equation 14.
From Section
52, Receiver Sensitivity / Noise, Smin is related to the noise factor S:
S
_{min} = (S/N)
_{min}
(NF)KT
_{o}B [15]
The
oneway RWR/ESM range equation becomes:
[16]
RWR/ESM RANGE INCREASE AS A RESULT OF A SENSITIVITY INCREASE
As shown in equation [12] S
_{min}^{1} proportional to R
_{max}.
Therefore, 10 log S
_{min} proportional
to 20 log R
_{max} and the table below results:
% Range Increase: Range + (% Range Increase) x Range
= New Range
i.e., for a 6 dB sensitivity increase, 500 miles +100% x 500 miles = 1,000 miles
Range Multiplier:
Range x Range Multiplier = New Range i.e., for a 6 dB sensitivity increase
500 miles x 2 = 1,000 miles
RWR/ESM RANGE DECREASE AS A RESULT OF A SENSITIVITY DECREASEAs shown in equation [12] S
_{min}
proportional to R
_{max}.
Therefore, 10 log S
_{min} proportional to 20 log R
_{max} and the table below results:
% Range Decrease: Range  (% Range decrease) x Range = New Range
i.e., for a 6 dB sensitivity decrease,
500 miles  50% x 500 miles = 250 miles
Range Multiplier: Range x Range Multiplier = New Range i.e., for
a 6 dB sensitivity decrease
500 miles x .5 = 250 miles
Example of OneWay Signal Strength: A 5 (or 7) GHz radar has a 70 dBm signal fed through a 5 dB loss
transmission line to an antenna that has 45 dB gain. An aircraft that is flying 31 km from the radar has an aft EW antenna with 1 dB gain and
a 5 dB line loss to the EW receiver (assume all antenna polarizations are the same).
Note: The respective transmission line losses will
be combined with antenna gains, i.e.:
5 +45 = 40 dB, 5  1 = 6 dB, 10 + 5 = 5 dB.
(1) What is the power level at the input of the EW receiver?
Answer (1): P
_{r} at the input to the EW receiver = Transmitter
power  xmt cable loss + xmt antenna gain  space loss + rcvr antenna gain  rcvr cable loss.
Space loss (from section 43) @ 5 GHz =
20 log f R + K1 = 20 log (5x31) + 92.44 = 136.25 dB.
Therefore, P
_{r} = 70 + 40  136.25  6 = 32.25 dBm @ 5 GHz (P
_{r}
= 35.17 dBm @ 7 GHz since
α_{1} = 139.17 dB)
(2) If the received signal is fed to a jammer with a gain
of 60 dB, feeding a 10 dB loss transmission line which is connected to an antenna with 5 dB gain, what is the power level from the jammer at
the input to the receiver of the 5 (or 7) GHz radar?
Answer (2): P
_{r} at the input to the radar receiver = Power at the input
to the EW receiver+ Jammer gain  jammer cable loss + jammer antenna gain  space loss + radar rcvr antenna gain  radar rcvr cable loss .
Therefore, P
_{r} = 32.25 + 60  5  136.25 + 40 = 73.5 dBm @ 5 GHz. (P
_{r} = 79.34 dBm @ 7 GHz since
α_{1} = 139.17 dB and Pt = 35.17 dBm).
This problem continues in section 44, 47, and 410.
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 
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