RF Cascade Workbook

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- Two-Way Radar Equation (Monostatic) -

**TWO-WAY RADAR EQUATION (MONOSTATIC)**

In this section the radar equation is derived from the one-way equation (transmitter to receiver) which is then extended to the two-way 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 co-located). Note the similarity of Figure 1 to Figure 3 in Section 4-3. Transmitted power, transmitting and receiving antenna gains, and the one-way free space loss are the same as those described in Section 4-3. 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 4-3, One-Way Radar Equation / RF Propagation, the power in the receiver is:

[1]

From equation [3] in Section 4-3:

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 4-11. 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 re-radiation of the power captured by an antenna of area σ (the RCS). Therefore, the effective capture area (A

[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

[5]

If like terms are cancelled, the two-way radar equation results. The peak power at the radar receiver input is:

[6]

* Note: λ=c/

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 one-way free space loss factor defined in Section 4-3. 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

From Section 4-3, 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

[12]

The "Target Gain Factor" (G

If frequency is given in MHz and RCS (σ) is in m

[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 two-way radar equation, the one-way free space loss factor (α

If the transmission loss in Figure 1 from P

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

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 two-way 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 Two-Way Radar 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

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 5-2, Receiver Sensitivity / Noise, S

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

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 2-11). A CW signal requires a very narrow bandwidth (approximately 100 Hz). Therefore, receiver noise is very low and good sensitivity results (see Section 5-2). If the radar pulse is narrow, the receiver filter bandwidth must be increased for a match (see Section 5-2), 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

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 line-of-sight/radar horizon (see Section 2-9) well before a typical target faded below S

Two-Way 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

Answer:

Starting with: 10 log S = 10 log P

We know that: α

and that: G

(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

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.

TWO-WAY RADAR RANGE INCREASE AS A RESULT OF A SENSITIVITY INCREASE

As shown in equation [17] S

% 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

TWO-WAY RADAR RANGE DECREASE AS A RESULT OF A SENSITIVITY DECREASE

As shown in equation [17] S

% 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 | One-Way Radar Equation / RF Propagation
| Two-Way Radar Equation (Monostatic) |
Alternate Two-Way Radar Equation |
Two-Way 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 |
Electro-Optics | Laser Safety |
Mach Number and Airspeed vs. Altitude Mach Number |
EMP/ Aircraft Dimensions | Data Busses | RS-232 Interface
| RS-422 Balanced Voltage Interface | RS-485 Interface |
IEEE-488 Interface Bus (HP-IB/GP-IB) | MIL-STD-1553 &
1773 Data Bus |

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