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TWOWAY RADAR EQUATION (MONOSTATIC)
In this section the radar equation is derived from the oneway equation (transmitter to receiver) which is then extended to the twoway
radar equation. The following is a summary of the important equations to be derived here:
Peak power at the radar receiver input
is:
Figure 1 illustrates the physical concept and equivalent circuit for a target being illuminated by a monostatic
radar (transmitter and receiver colocated). Note the similarity of Figure 1 to Figure 3 in Section 43. Transmitted power, transmitting and
receiving antenna gains, and the oneway free space loss are the same as those described in Section 43. The physical arrangement of the elements
is different, of course, but otherwise the only difference is the addition of the equivalent gain of the target RCS factor.
From
Section 43, OneWay Radar Equation / RF Propagation, the power in the receiver is:
[1]
From equation [3] in Section 43: Antenna Gain, G = [2] Similar to a receiving antenna, a radar target
also intercepts a portion of the power, but reflects (reradiates) it in the direction of the radar. The amount of power reflected toward the
radar is determined by the Radar Cross Section (RCS) of the target. RCS is a characteristic of the target that represents its size as seen by
the radar and has the dimensions of area (σ) as shown in Section 411. RCS area is not the same as physical area. But, for a radar target, the
power reflected in the radar's direction is equivalent to reradiation of the power captured by an antenna of area σ (the RCS). Therefore, the
effective capture area (A_{e}) of the receiving antenna is replaced by the RCS (σ).
[3] so we now have: [4]
The equation for the power reflected in the radar's direction is the same as equation [1] except that P_{t} G_{t}, which
was the original transmitted power, is replaced with the reflected signal power from the target, from equation [4]. This gives:
[5]
If like
terms are cancelled, the twoway radar equation results. The peak power at the radar receiver input is:
[6]
* Note: λ=c/f and F = RCS. Keep λ or c, σ, and R in the same units.
On reducing equation [6] to log form we have: 10 log P_{r} = 10 log P_{t} + 10 log G_{t} + 10 log G_{r} + 10 log σ  20 log
f  40 log R  30 log 4π + 20 log c [7]
Target Gain Factor
If Equation [5] terms are rearranged instead of cancelled, a recognizable form results: [8]
In log form:
[9]
The fourth and sixth terms can each be recognized as α, where α is the oneway
free space loss factor defined in Section 43. The fifth term containing RCS (σ) is the only new factor, and it is the "Target Gain Factor".
In simplified terms the equation becomes: 10 log [S (or P_{r})] = 10 log P_{t} + 10 log
G_{t} + 10 log G_{r} + G_{σ}  2α_{1} (in dB)
[10]
Where α_{1} and G_{σ} are as follows: From Section 43, equation [11], the space loss
in dB is given by: [11]
* Keep c
and R in the same units. The table of values for K_{1} is again presented here for completeness. The constant, K_{1}, in the
table includes a range and frequency unit conversion factor. While it's understood that RCS is the antenna aperture area equivalent to an isotropically
radiated target return signal, the target gain factor represents a gain, as shown in the equivalent circuit of Figure 1. The Target Gain Factor
expressed in dB is G_{σ} as shown in equation [12].
[12]
The "Target Gain Factor" (G_{σ}) is a composite of RCS, frequency, and dimension conversion factors and is called by various
names: "Gain of RCS", "Equivalent Gain of RCS", "Gain of Target Cross Section", and in dB form "Gainsub Sigma".
If frequency is given
in MHz and RCS (σ) is in m^{2}, the formula for G is:
[13]
or: Gα = 10 log σ + 20 log f_{1}  38.54
(in dB) [14]
For this example,
the constant K2 is 38.54 dB. This value of K2 plus K2 for other area units and frequency multiplier values are summarized in the adjoining
table.
In the twoway radar equation, the oneway free space loss factor (α_{1}) is used twice, once
for the radar transmitter to target path and once for the target to radar receiver path. The radar illustrated in Figure 1 is monostatic so
the two path losses are the same and the values of the two α_{1}'s are the same.
If the transmission
loss in Figure 1 from P_{t} to G_{t} equals the loss from G_{r} to P_{r} , and G_{r} = G_{t},
then equation [10] can be written as:
10 log [S or P_{r}] = 10 log P_{t} + 20 log G_{tr}  2α_{1}
+ Gσ (in dB) [15]
The space loss factor (α_{1}) and the target gain factor (Gσ) include all the necessary unit conversions
so that they can be used directly with the most common units. Because the factors are given in dB form, they are more convenient to use and
allow calculation without a calculator when the factors are read from a chart or nomograph.
Most radars are monostatic. That is, the
radar transmitting and receiving antennas are literally the same antenna. There are some radars that are considered "monostatic" but have separate
transmitting and receiving antennas that are colocated. In that case, equation [10] could require two different antenna gain factors as originally
derived:
10 log [S or P_{r}] = 10 log P_{t} + 10 log G_{t} + 10 log G_{r}  2α_{1} + Gσ (in dB) [16]
Note: To avoid having to include additional terms for these calculations, always combine any transmission line loss with antenna gain.
Figure 2 is the visualization of the path losses occurring with the twoway radar equation. Note: to avoid having to include additional
terms, always combine any transmission line loss with antenna gain. Losses due to antenna polarization and atmospheric absorption also need
to be included.
Figure 2. Visualization of TwoWay Radar Equation
RADAR RANGE EQUATION (TwoWay Equation)
The Radar Equation is often called the "Radar Range Equation". The
Radar Range Equation is simply the Radar Equation rewritten to solve for maximum Range. The maximum radar range (R_{max}) is the distance
beyond which the target can no longer be detected and correctly processed. It occurs when the received echo signal just equals S_{min}.
The Radar Range Equation is then: [17]
The first equation, of the three above, is given in Log form by:
[18]
As shown previously, Since K1 = 20log [(4B/c) times conversion units if not in m/sec, m, and Hz], we have:
[19]
If you want
to convert back from dB, then
Where M dB is the resulting
number within the brackets of equation 19.
From Section 52, Receiver Sensitivity / Noise, S_{min} is related to
the noise factors by:
S_{min} = (S/N)_{min}(NF)kT_{0}B [20]
The Radar Range Equation for a tracking radar (target continuously in the antenna beam) becomes:
[21]
Pt in equations [17], [19], and [21] is the peak power of a CW or pulse signal. For pulse signals these equations assume the radar pulse
is square. If not, there is less power since Pt is actually the average power within the pulse width of the radar signal. Equations [17] and
[19] relate the maximum detection range to S_{min}, the minimum signal which can be detected and processed (the receiver sensitivity).
The bandwidth (B) in equations [20] and [21] is directly related to S_{min}. B is approximately equal to 1/PW. Thus a wider pulse width
means a narrower receiver bandwidth which lowers S_{min}, assuming no integration.
One cannot arbitrarily change the receiver
bandwidth, since it has to match the transmitted signal. The "widest pulse width" occurs when the signal approaches a CW signal (see Section
211). A CW signal requires a very narrow bandwidth (approximately 100 Hz). Therefore, receiver noise is very low and good sensitivity results
(see Section 52). If the radar pulse is narrow, the receiver filter bandwidth must be increased for a match (see Section 52), i.e. a 1 μs
pulse requires a bandwidth of approximately 1 MHz. This increases receiver noise and decreases sensitivity.
If the radar transmitter
can increase its PRF (decreasing PRI) and its receiver performs integration over time, an increase in PRF can permit the receiver to "pull"
coherent signals out of the noise thus reducing S/N_{min} thereby increasing 44.6 the detection range. Note that a PRF increase may
limit the maximum range due to the creation of overlapping return echoes (see Section 210).
There are also other factors that limit
the maximum practical detection range. With a scanning radar, there is loss if the receiver integration time exceeds the radar's time on target.
Many radars would be range limited by lineofsight/radar horizon (see Section 29) well before a typical target faded below S_{min}.
Range can also be reduced by losses due to antenna polarization and atmospheric absorption (see Sections 32 and 51).
TwoWay Radar Equation (Example)
Assume that a 5 GHz radar has a 70 dBm (10 kilowatt) signal fed through
a 5 dB loss transmission line to a transmit/receive antenna that has 45 dB gain. An aircraft that is flying 31 km from the radar has an RCS
of 9 m^{2}. What is the signal level at the input to the radar receiver? (There is an additional loss due to any antenna polarization
mismatch but that loss will not be addressed in this problem). This problem continues in Sections 43, 47, and 410.
Answer:
Starting with: 10 log S = 10 log P_{t} + 10 log G_{t} + 10 log G_{r} + G_{σ}  2α_{1}
(in dB)
We know that: α_{1} = 20l og f R + K1 = 20 log (5x31) + 92.44 = 136.25 dB
and that: G_{σ} = 10 log σ + 20 log f_{1} + K2 = 10 log 9 + 20 log 5 + 21.46 = 44.98 dB (see Table 1)
(Note: The aircraft transmission line losses (5 dB) will be combined with the antenna gain (45 dB) for both receive and transmit paths
of the radar)
So, substituting in we have: 10 log S = 70 + 40 + 40 + 44.98  2(136.25) = 77.52 dBm @ 5 GHz
The answer changes
to 80.44 dBm if the tracking radar operates at 7 GHz provided the antenna gains and the aircraft RCS are the same at both frequencies.
α_{1} = 20 log (7x31) + 92.44 = 139.17 dB, G_{σ} = 10 log 9 + 20 log 7 + 21.46 = 47.9 dB (see
Table 1)
10 log S = 70 + 40 + 40 + 47.9  2(139.17) = 80.44 dBm @ 7 GHz
Table 1. Values of the Target Gain Factor (G_{σ}) in dB for Various Values of Frequency and RCS
Note: Shaded values were used in the examples.
TWOWAY RADAR RANGE INCREASE AS A RESULT OF A SENSITIVITY
INCREASE
As shown in equation [17] S_{min}^{1} ≈ R_{max}^{4} Therefore, 10 log S_{min}
≈ 40 log R_{max} and the table below results:
% Range Increase: Range + (% Range Increase) x Range = New Range i.e., for a
12 dB sensitivity increase, 500 miles +100% x 500 miles = 1,000 miles
Range Multiplier: Range x Range Multiplier = New Range i.e., for
a 12 dB sensitivity increase 500 miles x 2 = 1,000 miles
Table 2. Effects of Sensitivity Increase
TWOWAY RADAR RANGE DECREASE AS A RESULT OF A SENSITIVITY DECREASE
As shown in equation
[17] S_{min}^{1} ≈ R_{max}^{4} Therefore, 10 log S_{min} ≈ 40 log R_{max} and the table
below results:
% Range Decrease: Range  (% Range Decrease) x Range = New Range i.e., for a 12 dB sensitivity
decrease, 500 miles  50% x 500 miles = 250 miles
Range Multiplier: Range x Range Multiplier = New Range
i.e., for a 12 dB sensitivity decrease 500 miles x 0.5 = 250 miles
Table 3. Effects of Sensitivity Decrease
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.
