Radar Range Equation
2.9 EXERCISES
1. Derive the equation
from
In these equations, θA and θB denote beamwidths in degrees and and denote beamwidths in radians. KA = 1.65.
2. A radar has a peak power of 1 MW, combined transmit and antenna losses of 1 dB, and a transmit antenna directivity of 41 dB. The radar is operating in free space so there is nothing to absorb the radiated energy. It uses a pulse with an envelope width of 1 µs.
a) Calculate the total energy on the surface of a (hypothetical) sphere with a radius of 100 km centered on the radar.
b) Repeat part a) for a sphere with a radius of 200 km centered on the radar.
c) Do your answers make sense? Explain.
3. Consider a monostatic radar with the following parameters:
• Peak transmit power at the power amp output—10 kW
• Transmit losses—1 dB
• Antenna losses—1 dB (transmit)
• Antenna losses—1 dB (receive)
• Operating frequency—6 GHz
• PRF—1,000 Hz
• Pulsewidth—100 µs
• Transmit antenna effective aperture—0.58 m2
• Receive antenna beamwidth—1.2° Az × 2.5° El (the radar has separate transmit and receive antennas positioned next to each other)
• Other losses—8 dB
• System noise temperature, Ts—1,155 K
a) Calculate the transmit antenna directivity, in dB.
b) Calculate the effective aperture, in square meters, for the receive antenna, given an antenna efficiency of 60%.
c) Calculate the ERP for the radar, in dBW.
d) Given a detection threshold of 20 dB, what is the detection range, in km, for a target with a radar cross section of 10 dBsm?
4. Consider a monostatic radar that has the following parameters:
• Peak transmit power at power amp output—100 kW
• Transmit and antenna losses—2 dB
• Operating frequency—10 GHz
• PRF—2,000 Hz
• Antenna diameter—1.5 m (circular aperture)
• Antenna efficiency—60%
• Other losses—12 dB
• Noise figure—4 dB
• The radar transmits a 10-µs rectangular pulse.
• The beam elevation angle is in the range of 1º to 5º.
a) Create a table containing all parameters necessary for the radar range equation.
Derive those parameters missing explicit values above. List as TBD those parameters with insufficient information for entering a value.
b) Calculate the unambiguous range of the radar.
c) Plot SNR, in dB, versus target range, in km, for a 6-dBsm target. Vary the range from 5 km to the radar ’s unambiguous range.
d) Given a 13-dB SNR requirement for detection, calculate the detection range, in km, for a 6-dBsm RCS target.
e) What is the maximum detection range, in km, if the minimum SNR required for detection is raised to 20 dB?
f) Calculate the antenna beamwidth, in degrees.
5. A radar generates 200 kW of peak power at the power tube and has 2 dB of loss between the power tube and the antenna. The radar is monostatic with a single antenna that has a directivity of 36 dB and a loss of 1 dB. The radar operates at a frequency of 5 GHz.
Determine the ERP, in dBW, for the radar. Determine the ERP in watts. Determine the power at the receive antenna output, in dBm, for the following conditions:
a) A 1.5-m2 RCS target at a range of 20 km b) A 20-dBsm target at a range of 100 km
6. How does doubling the range change the powers in Exercise 5? Give your answer in dB.
This problem illustrates an important rule of thumb for the radar range equation.
7. A radar with losses of 13 dB and a noise figure of 8 dB must detect targets within a search sector 360° in azimuth and from 0° to 20° in elevation. The radar must cover the
search sector in 6 s. The targets of interest have an RCS of 6 dBsm, and the radar requires 20 dB of SNR to declare a detection. The radar must have a detection range of 75 km. Calculate the average power aperture (PavgAe), in W-m2, required by the radar to satisfy the search requirements above.
8. The radar of Exercise 7 uses an antenna with fan beamwidths of 1° in azimuth and 5° in elevation. The radar operates at a frequency of 4 GHz. What average power, in kW, must the radar have? Given an antenna efficiency of 60%, calculate the approximate antenna dimensions, in m. Hint: The relative height and width of the antenna are inversely proportional to the relative beamwidths.
9. Assuming the radar of Exercise 7 uses one PRI per beam, determine the PRI for the radar. Can the radar operate unambiguously in range? Explain.
10. We typically describe the range resolution of a radar as the width of its pulses, if the radar uses unmodulated pulses. What pulsewidth does the radar of Exercise 7 require for a range resolution of 150 m? What is the peak power of the radar, in MKS units?
11. Derive (2.54).
References
[1] Barton, D. K., Radar Equations for Modern Radar, Norwood, MA: Artech House, 2013.
[2] Norton, K. A., and A. C. Omberg, “The Maximum Range of a Radar Set,” Proc. IRE, vol. 35, no. 1, Jan. 1947, pp. 4–
24. First published Feb. 1943 by U.S. Army, Office of Chief Signal Officer in the War Department, in Operational Research Group Report, ORG-P-9-1.
[3] Barton, D. K., ed., Radars, Vol. 2: The Radar Range Equation (Artech Radar Library), Dedham, MA: Artech House, 1974.
[4] Budge, M. C., Jr., “EE 619: Intro to Radar Systems.” www.ece.uah.edu/courses/material/EE619/index.htm.
[5] Erst, S. J., Receiving Systems Design, Dedham, MA: Artech House, 1984, p. 46.
[6] Skolnik, M. I., ed., Radar Handbook, New York: McGraw-Hill, 1970, p. 9–5.
[7] Skolnik, M. I., Introduction to Radar Systems, 3rd ed., New York: McGraw-Hill, 2001.
[8] Hovanessian, S. A., Radar System Design and Analysis, Norwood, MA: Artech House, 1984
[9] Stutzman, W. L., “Estimating Directivity and Gain of Antennas,” IEEE Antennas Propagat. Mag., vol. 40, no. 4, Aug.
1998, pp. 7–11.
[10] Barton, D. K., Radar System Analysis and Modeling, Norwood, MA: Artech House, 2005.
[11] Rihaczek, A. W., Principles of High-Resolution Radar, New York: McGraw-Hill, 1969, p. 64. Reprinted: Norwood, MA: Artech House, 1995, p. 64.
[12] Stutzman, W. L., and G. A. Thiele, Antenna Theory and Design, New York: Wiley & Sons, 1981.
[13] Ziemer, R. E., and W. H. Tranter, Principles of Communications, 3rd ed., Boston, MA: Houghton Mifflin, 1990.
[14] Johnson, J. B., “Thermal Agitation of Electricity in Conductors,” Phys. Rev., vol. 32, Jul. 1928, pp. 97–109.
[15] Nyquist, H., “Thermal Agitation of Electric Charge in Conductors,” Phys. Rev., vol. 32, Jul. 1928, pp. 110–113.
[16] Rohde, U. L., J. Whitaker, and T. T. N. Bucher, Communications Receivers, 2nd ed., New York: McGraw-Hill, 1997.
[17] Losee, F. A., RF Systems, Components, and Circuits Handbook, 2nd ed., Norwood, MA: Artech House, 2005.
[18] IEEE 100, The Authoritative Dictionary of IEEE Standards Terms, 7th ed., New York: IEEE, 2000.
[19] Ridenour, L. N., Radar System Engineering, vol. 1 of MIT Radiation Lab. Series, New York: McGraw-Hill, 1947.
Reprinted: Norwood, MA: Artech House (CD-ROM edition), 1999, p. 33.
[20] Uhlenbeck, G. E., and J. L. Lawson, Threshold Signals, vol. 24 of MIT Radiation Lab. Series, New York: McGraw-Hill, 1950. Reprinted: Norwood, MA: Artech House (CD-ROM edition), 1999, p. 99.
[21] Lathi, B. P., Signals, Systems and Communication, New York: Wiley & Sons, 1965, pp. 548–551.
[22] Van Voorhis, S. N., Microwave Receivers, vol. 23 of MIT Radiation Lab. Series, New York: McGraw-Hill, 1948.
Reprinted: Norwood, MA: Artech House (CD-ROM edition), 1999, p. 4.
[23] Blake, L. V., “A Guide to Basic Pulse-Radar Maximum-Range Calculation, Part 1—Equations, Definitions, and Aids to
Calculation,” Naval Research Laboratory, Washington, D.C., Rep. No. 6930, Dec. 23, 1969, p. 49. Available from DTIC as 701321.
[24] Curry, G. R., Radar System Performance Modeling, Norwood, MA: Artech House, 2001, pp. 117– 118.
[25] Mahan, E.R., and E. C. Keefer, eds., Foreign Relations of the United States, 1969–1976, vol. 32, SALT I, 1969–1976, Washington, D.C.: U.S. Government Printing Office, 2010, p. 814.
APPENDIX 2A: DERIVATION OF SEARCH SOLID ANGLE EQUATION
Figure 2A.1 Geometry for computing solid angle.
We can write the area of the small square in Figure 2A.1 as
or
To get the total area over the angles [ε1, ε2], we integrate (2A.1) and (2A.2) over these angle ranges. This yields
Performing the integral results in
Dividing by R2 yields the solid angle as
1 An assumption of this form of the radar range equation is that the radar is pulsed, not CW. For CW radars, it would be more appropriate to use the form of Section 2.3.
2 Noise figure and noise factor are often treated as synonyms, although some authors make a distinction [6]. Specifically, the term “noise figure” is used when in logarithmic form, while noise factor is used when in linear form. We will use noise figure for both the W/W and dB version in this book.
3 We assume nothing in the transmit, propagation, or receive path of the radar, up to the matched filter, distorts the rectangular pulse envelope. Clearly, this will not be the case, since a rectangular pulse has infinite bandwidth and the transmitter, environment, and receiver have finite bandwidth. As discussed in Chapters 5 and 7, we accommodate envelope distortion by including a loss factor. As a note, in practical radars, the loss due to pulse envelope distortion is usually small (< 1 dB).
4 The concept of 3-dB points should be familiar from control and signal processing theory as the standard measure used to characterize bandwidth.
5 This example is adapted from lecture notes by Dr. Stephen Gilbert.