Exercise 1-4 The Radar Equation EXERCISE OBJECTIVE When you have completed this exercise, you will be familiar with the different parameters in the radar equation, and with the interaction between these parameters in a radar system. DISCUSSION OUTLINE The Discussion of this exercise covers the following points: Derivation of the radar equation Use of the radar equation DISCUSSION OF FUNDAMENTALS One of the most important parameters of a radar system is the maximum detection range. This range is ultimately determined by the signal-to-noise ratio required by the receiver. Many different factors, however, affect the power of the received echo signal, the most important of which are explained below. Transmitted power: the power of the received signal is proportional to the average transmitted power. Target range: the received signal power diminishes rapidly as the target range is increased. This is mainly due to the spreading over a greater and greater area of the transmitted waves as they travel away from the antenna. This spreading reduces the power density of the waves. The power density of electromagnetic waves is equal to the emitted power per unit cross-sectional area normal to the direction of propagation: (1-7) For example, if 100 watts were radiated uniformly over a surface of one square meter, the power density would be equal to 100 W/1 m 2 = 100 W/m 2. If the same 100 W were spread uniformly over an area of 1000 m 2, the power density would be 100 W/1000 m 2 = 0.1 W/m 2. Antenna gain: the directivity of a transmitting antenna is a measure of how much the antenna concentrates the transmitted signal into a narrow beam. The greater the directivity, the more power will be directed towards the target, and the greater the received signal power. The antenna gain is equal to the directivity times an efficiency factor, since some losses are always present in a practical antenna. Festo Didactic 38542-00 69
Ex. 1-4 The Radar Equation Discussion of Fundamentals Target radar cross section: the target intercepts a portion of the transmitted signal and reflects it in various directions. How much of the signal is intercepted, how well the target reflects radar waves, and how much of the reflected signal is actually directed back towards the radar antenna determines the size of the target, as seen by the radar. The measure of this size is called the radar cross section. The radar cross section, which has units of area, varies from one target to another, and even from one orientation to another of the same target. Antenna effective area: the signal reflected by the target spreads as it travels back towards the radar. The greater the effective area of the receiving antenna, the more power will be intercepted, and the greater the power of the received signal will be. As an example, if the power density at a receiving antenna were equal to 10 W/m 2, and the effective area of the antenna were 0.5 m 2, the power received by the antenna would be Derivation of the radar equation Imagine that a lossless, isotropic antenna (one which radiates equally in all directions) is at the centre of a large sphere of radius, as shown in Figure 1-48. The power density over the surface of the sphere would be uniform, and equal to the total power transmitted divided by the surface area of the sphere. The power density at a distance from an isotropic antenna is therefore equal to the transmitted power divided by the surface area of a sphere of radius : where is the average power output of the radar transmitter. is the distance from the antenna. (1-8) Figure 1-48. Isotropic antenna at the centre of a sphere of radius R. 70 Festo Didactic 38542-00
Ex. 1-4 The Radar Equation Discussion of Fundamentals Note that the power density is proportional to. This is because the power of a signal is spread over an increasingly large area as it travels away from the target. The gain of an antenna is the ratio of the power radiated by the antenna in the direction of maximum radiation, to the power that would have been radiated by a lossless, isotropic antenna. The power density at a distance from a directive antenna with a gain of is equal to the power density from an isotropic antenna times the gain : (1-9) A target situated at a distance from the antenna intercepts a portion of the radiated power and reflects it in various directions. The amount of power reflected back towards the radar is equal to the power density at the target times the radar cross section of the target: (1-10) The reflected signal spreads as it travels the distance back towards the radar. The power density of the reflected signal when it reaches the radar antenna is equal to the power reflected by the target in the direction of the radar divided by (the area of a sphere of radius ): (1-11) The antenna captures a portion of the echo signal. The power of the signal received by the radar antenna is equal to the power density at the antenna times the effective area of the antenna: (1-12) The power of the received echo signal drops very rapidly as the range is increased. According to Equation (1-12), the received power is proportional to. This means that doubling the target range causes the received power to be decreased by a factor of 16. The relationship between target range and the relative power of the received signal is illustrated in Figure 1-49. The relative received signal power for a certain target at a range of 1 km is taken to be 0 db. This figure shows that increasing the target range from 1 km to 1.2 km reduces the received signal power by a factor of two ( 3dB). Festo Didactic 38542-00 71
Ex. 1-4 The Radar Equation Discussion of Fundamentals Figure 1-49. Received power as a function of target range. As previously stated, the maximum detection range is ultimately determined by the minimum signal-to-noise ratio required by the receiver. For a given noise level at the input of the receiver, the minimum signal-to-noise ratio depends on the minimum detectable signal power. A signal weaker than would be covered by noise and would probably not be detected. A target is at the maximum range at which it can be detected when the power of the received echo signal is equal to the minimum detectable signal power. By substituting for and for in Equation (1-12), and rearranging, we obtain the classical form of the radar equation: (1-13) where is the maximum range at which a target can be detected. is the average radar transmitter power. is the transmitting gain of the antenna. is the radar cross section of the target. is the receiving effective area of the antenna. is the minimum detectable signal power. 72 Festo Didactic 38542-00
Outline Use of the radar equation The radar equation can be used to estimate the maximum range at which a target can be detected. It should be noted that this classical form of the equation neglects many losses such as atmospheric absorption, system degradation in the field, etc. The actual maximum range may be only half that predicted by the classical form of the equation. The radar equation shows the relationship between the parameters which affect the operation of the radar. It shows that for long ranges, the transmitter power must be high, the antenna must be large and have a high gain, and the receiver must be able to detect weak signals. Increasing the transmitter power by a certain factor yields a relatively small increase in maximum range. For example, doubling power increases maximum range by a factor of 2 1/4 = 1.19, an increase of approximately 20%. Antenna theory states that the gain and the effective area of an antenna are directly related. When the same antenna is used for both transmission and reception, it is convenient to express the equivalent area in terms of gain, using the relation: (1-14) where is the wavelength of the radar signal. The radar equation then becomes: PROCEDURE OUTLINE The Procedure is divided into the following sections: Setting up a basic pulsed radar Effect of transmitted power on maximum range Effect of target range on received power Effect of radar cross section on received power Effect of target material on radar cross section Effect of antenna parameters on received power PROCEDURE Setting up a basic pulsed radar In this section, you will set up a basic pulsed radar and calibrate the A-scope display. The block diagram of the system you will use is shown in Figure 1-51. Festo Didactic 38542-00 73
a In this exercise, you are often asked to set the target range so that the amplitude of the target blip observed on the A-scope display is positive and maximum. With time, however, the amplitude of the target blip may vary. This is due to the RF OSCILLATOR of the Radar Transmitter which undergoes a slight frequency drift with temperature. To reduce drift to a minimum, it is preferable to let the Radar Training System warm up for at least half an hour before beginning this exercise. If the amplitude of the target blip still varies significantly, slightly readjust the target range as required. 1. The main elements of the Radar Training System, that is the antenna and its pedestal, the target table and the training modules, must be set up properly before beginning this exercise. Refer to Appendix B of this manual for setting up the Radar Training System, if this is not done yet. Set up the modules on the Power Supply / Antenna Motor Driver as shown in Figure 1-50. On the Radar Transmitter, make sure that the RF POWER switch is in the STANDBY position. On the Antenna Controller, make sure that the MANual ANTENNA ROTATION MODE is selected and that the SPEED control is in the 0 position. Set the POWER switch of the Power Supply to the I (on) position, and then those of the other modules. Figure 1-50. Module Arrangement. 2. Figure 1-51 shows the block diagram of the basic pulsed radar that can be obtained using the Radar Training System. Connect the modules according to this block diagram. a You should use a medium-length (approximately 75 cm) SMA cable to connect the RF OSCILLATOR OUTPUT of the Radar Transmitter to the LOCAL OSCILLATOR INPUT of the Radar Receiver. This is because a short SMA cable will be necessary later in this exercise to modify the pulsed radar. 74 Festo Didactic 38542-00
Figure 1-51. Block diagram of the basic pulsed radar. Festo Didactic 38542-00 75
3. Make the following adjustments: On the Radar Transmitter RF OSCILLATOR FREQUENCY... CAL. PULSE GENERATOR PULSE WIDTH... 1 ns On the Radar Synchronizer PRF MODE... SINGLE PRF... 216 Hz 4. Refer to Appendix C of this manual to calibrate the A-scope display so that its origin is as near as possible to the antenna horn (0 m), and its range span is equal to 3.6 m. Once you have finished the calibration, the display on the oscilloscope should resemble Figure 1-52. Figure 1-52. Calibrated A-scope display of a fixed target located at the origin. 5. Place the target table so that the grid is located approximately 2.1 m from the antenna horn, as shown in Figure 1-53. Make sure that the metal rail of the target table remains aligned with the antenna beam axis while moving the target table. a In this exercise, you are often asked to vary the position of the target table or the orientation of the target while the RF power is on. This requires standing near or in front of the antenna. This practice could be very dangerous with a full-scale radar and should normally be avoided. However, the low radiation levels of the Radar Training System allow these manipulations to be carried out safely. On the oscilloscope, set the Y-channel sensitivity to 0.1 V/DIV. 76 Festo Didactic 38542-00
On the Target Controller, use the Y-axis POSITION control to vary the target range by a few millimeters so that the peak voltage of the target blip is positive and maximum. Slightly vary the orientation of both the target and the antenna in order to maximize the peak voltage of the blip. Determine the target range using the A-scope display. Target range m Figure 1-53. Position of the target table. Effect of transmitted power on maximum range In this section, you will attenuate the transmitted power and measure the maximum range of the pulsed radar. You will then attenuate the transmitted power by an additional 3 db, and again measure the maximum range of the pulsed radar. This will allow you to verify the relationship between the maximum range and the transmitted power. 6. On the Radar Transmitter, place the RF POWER switch in the STANDBY position. Using a short SMA cable (approximately 40 cm), connect a 6-dB attenuator in series with the cable connected to the PULSED RF OUTPUT of the Radar Transmitter. Place the RF POWER switch in the ON position. On the Target Controller, use the Y-axis POSITION control to vary the target range by a few millimeters so that the peak voltage of the target blip is positive and maximum. Notice that the target blip on the A-scope display is appearing at a range which is greater than that measured in the previous step. This is due to the connection of the short SMA cable and 6-dB attenuator in series with the PULSED RF OUTPUT of the Radar Transmitter. This causes a delay in the reception of the RF echo signal which changes the origin calibration of the A- scope display. Festo Didactic 38542-00 77
On the Dual-Channel Sampler, slightly readjust the ORIGIN control so that the target blip on the A-scope display appears at the range measured in the previous step. This compensates for the delay introduced by the added SMA cable and attenuator. On the oscilloscope, set the Y-channel sensitivity to 50 mv/div. On the Target Controller, use the Y-axis POSITION control to place the target at the far end of the target table, then vary the target range by a few millimeters so that the peak voltage of the target blip is positive and maximum. The display on the oscilloscope screen should resemble Figure 1-54. Figure 1-54. A-scope display of a weak target blip. Determine the peak voltage of the target blip using the A-scope display. For the purpose of this exercise, the power of this signal is arbitrarily considered to be the minimum detectable signal. ( ) mv Determine the target range using the A-scope display. This is considered to be the maximum range of the pulsed radar, for this particular target, when the transmitted signal is attenuated 6 db. ( 6 db) m 7. Using the radar equation, predict the maximum range that the pulsed radar should have if the transmitted power were to be reduced by half. 78 Festo Didactic 38542-00
On the Radar Transmitter, place the RF POWER switch in the STANDBY position. Using a U-shaped, semi-rigid SMA cable, connect a 3-dB attenuator in series with the 6-dB attenuator and cable connected to the PULSED RF OUTPUT of the Radar Transmitter. This will reduce by half the power transmitted by the pulsed radar, with respect to that transmitted in the previous step. Place the RF POWER switch in the ON position. On the Target Controller, use the Y-axis POSITION control to vary the target range by a few millimeters so that the peak voltage of the target blip is positive and maximum. Notice that the target blip on the A-scope display is appearing anew at a range which is greater than that measured in the previous step. This is because the 3-dB attenuator adds to the delay introduced by the short SMA cable and 6-dB attenuator. On the Dual-Channel Sampler, slightly readjust the ORIGIN control so that the target blip on the A-scope display appears at the range measured in the previous step. This compensates for the additional delay introduced by the 3-dB attenuator. On the Target Controller, use the Y-axis POSITION control to decrease the target range until the peak voltage of the target blip is again equal to ( ). Determine the target range using the A-scope display. Since the peak voltage of the target blip is equal to the minimum detectable signal, this range is considered to be the maximum range of the pulsed radar for this particular target, when the transmitted signal is attenuated 9 db. ( 9 db) m Compare the measured maximum range with the predicted maximum range. What does this confirm? Effect of target range on received power In this section, you will verify the relationship between the received power and the target range, by placing the same target at two different ranges and measuring the amplitudes of the target blips. 8. On the Radar Transmitter, place the RF POWER switch in the STANDBY position. Remove both attenuators connected at the PULSED RF OUTPUT, then reconnect the cable going to the RF INPUT of the Rotating-Antenna Pedestal to the PULSED RF OUTPUT. Place the RF POWER switch of the Radar Transmitter in the ON position. Festo Didactic 38542-00 79
On the oscilloscope, set the Y-channel sensitivity to 0.1 V/DIV. a For the rest of this exercise, adjust the Y-channel sensitivity of the oscilloscope as required. 9. On the Target Controller, use the Y-axis POSITION control to place the target at the near end of the target table, then slowly vary the target range by a few millimeters so that the peak voltage of the target blip is positive and maximum. The target range is now approximately 2.1 m. However, the target blip is appearing at a shorter range on the A-scope display because the attenuator and the SMA cable have been removed. On the Dual-Channel Sampler, slightly readjust the ORIGIN control so that the target blip on the A-scope display appears at a range of approximately 2.1 m. Slightly vary the orientation of the target in order to maximize the peak voltage of the target blip. Determine the peak voltage of the target blip using the A-scope display. for the small metal plate target (2.1 m) mv 10. Knowing that the power received Pr from any object is proportional to 1/R 4, predict the peak voltage of the target blip that should be obtained if the target were moved to 3.0 m from the antenna horn. On the Target Controller, use the Y-axis POSITION control to place the target at the far end of the target table, then slowly vary the target range by a few millimeters so that the peak voltage of the target blip is positive and maximum. The target range is now approximately equal to 3.0 m. Determine the peak voltage of the target blip using the A-scope display. for the small metal plate target (3.0 m) mv Compare the measured peak voltage with the predicted peak voltage when the target range is equal to 3.0 m. What does this confirm? 80 Festo Didactic 38542-00
Effect of radar cross section on received power In this section, you will verify the relationship between the received power and the radar cross section of targets, by using two targets with different radar cross sections placed at the same range. 11. Replace the small metal plate target with the large metal plate target. The radar cross section of this target is sixteen times greater than that of the small metal plate target (See Appendix D for more details on radar cross sections). Knowing that the power received Pr from a target is proportional to the radar cross section, predict the peak voltage that should be obtained with the large metal plate target. Use the peak voltage measured in the previous step to do your calculation. Set the orientation of the large metal plate target so that the peak voltage of the target blip is maximized. On the Target Controller, use the Y-axis POSITION control to slowly vary the target range by a few millimeters to ensure that the peak voltage of the target blip is maximized. Determine the peak voltage of the target blip using the A-scope display. for the large metal plate target (3.0 m) mv Compare the measured peak voltage with the predicted peak voltage for the large metal plate target. What does this confirm? Effect of target material on radar cross section In this section, you will observe the effect of the target material on the radar cross section, by using two targets of identical shape and size, at the same range, but made of different materials. 12. Replace the large metal plate target with the large plexiglass plate target. Make sure that the plexiglass target is oriented so that it is perpendicular to the metal rail of the target table. On the Target Controller, use the Y-axis POSITION control to slowly vary the target range by a few millimeters so that the peak voltage of the target blip is positive and maximum. Slightly vary the orientation of the plexiglass target to ensure that the peak voltage of the target blip is maximized. Festo Didactic 38542-00 81
Determine the peak voltage of the target blip using the A-scope display. for the large plexiglass plate target (3.0 m) mv Does the target material have any effect on the radar cross section of the target? Explain. Effect of antenna parameters on received power In this section, you will verify the relationship between the received power and the antenna parameters, by using the Radar Antenna and a horn antenna. 13. On the Radar Transmitter, place the RF POWER switch in the STANDBY position. Replace the large plexiglass plate target with the large metal plate target. Make sure that the metal target is oriented so that it is perpendicular to the metal rail of the target table. Loosen the screw that secures the Radar Antenna to the Rotating-Antenna Pedestal and remove the Radar Antenna. Assemble the horn, the waveguide-to-sma coaxial adaptor and the support pin of the horn as shown in Figure 1-55. Install the horn assembly on the fixed mast provided with the target table. Figure 1-55. Horn assembly. Move the Rotating-Antenna Pedestal by approximately 45 cm without changing the distance which separates it from the grid of the target table. Place the horn at approximately 2.5 m from the grid of the target table, as shown in Figure 1-56. 82 Festo Didactic 38542-00
Figure 1-56. Position of the horn assembly. Using a medium-length SMA cable and the mechanical adaptor provided with the Connection Leads and Accessories kit, connect the horn assembly to the Rotating-Antenna Pedestal as shown in Figure 1-57. Figure 1-57. Connection of the horn assembly to the Rotating-Antenna Pedestal. 14. On the Radar Transmitter, place the RF POWER switch in the ON position. Figure 1-58 shows an example of what you might observe on the oscilloscope screen. There may be undesired blips on the A-scope display from various reflecting objects in the laboratory because the beamwidth of the horn is larger than that of the Radar Antenna. Festo Didactic 38542-00 83
a If the signal waveform is not vertically centred on the oscilloscope screen, readjust the I-CHANNEL DC OFFSET control. Figure 1-58. A-scope display of various undesired blips. On the Target Controller, use the Y-axis POSITION control to vary the target range in order to find the target blip on the A-scope display. Adjust the target range so that the target blip is not affected by undesired blips, then slightly vary the target range so that the peak voltage of the target blip is positive and maximum. Slightly vary the orientation of both the target and the horn in order to maximize the peak voltage of the target blip. Figure 1-59 shows an example of what you might observe on the oscilloscope screen. Figure 1-59. A-scope display of a weak target blip along with various weak undesired blips. 84 Festo Didactic 38542-00
Ex. 1-4 The Radar Equation Conclusion Determine the peak voltage of the target blip using the A-scope display. (horn) mv 15. On the Radar Transmitter, place the RF POWER switch in the STANDBY position. Disconnect the horn assembly from the Rotating-Antenna Pedestal. Mount the Radar Antenna on the Rotating-Antenna Pedestal, then place it at the position shown in Figure 1-53. The antenna beam axis should be aligned with the target. The gain of the Radar Antenna is approximately 13 db greater than that of the horn. Using Equation (1-12) and Equation (1-14) and the peak voltage measured in the previous step, predict the peak voltage that should be obtained with the Radar Antenna. On the Radar Transmitter, place the RF POWER switch in the ON position. On the Target Controller, use the Y-axis POSITION control to slowly vary the target range by a few millimeters so that the peak voltage of the target blip is positive and maximum. Slightly vary the orientation of the antenna beam axis to maximize the peak voltage of the target blip. Determine the peak voltage of the target blip using the A-scope display. (antenna) mv Compare the measured peak voltage with the predicted peak voltage for the Radar Antenna. What does this confirm? 16. On the Radar Transmitter, make sure that the RF POWER switch is in the STANDBY position. The RF POWER STANDBY LED should be lit. Place all POWER switches in the O (off) position and disconnect all cables. CONCLUSION In this exercise, you verified the relationship between the transmitted power and the maximum range of the radar. You found that the maximum range decreases as the transmitted power decreases. Festo Didactic 38542-00 85
Ex. 1-4 The Radar Equation Review Questions You also verified the relationship between the received power and the following parameters: the target range, the radar cross section of targets and the antenna parameters. You found that the received power decreases very rapidly when the target range increases. You also found that the received power is directly proportional to the radar cross section of the target, and to the gain and effective aperture of the antenna. Finally, you observed that the target material influences the radar cross section. REVIEW QUESTIONS 1. Explain the term radar cross section. 2. What is the relationship between the power of the received echo signal and the target range? 3. Explain how the radar equation is used. 4. What conditions are necessary in order for the radar to detect targets at long ranges? 5. Does increasing transmitter power of a radar cause a large increase in maximum range? Explain. 86 Festo Didactic 38542-00