1. Module 2. Principles of work, key parameters of radio location systems Topic 2.2. Range of radar Lecture 2.2.1. RADAR EQUATION.

2. The Radar Equation The radar equation represents the physical dependences of the transmit power, that is the wave propagation up to the receiving of the echo-signals. The power P E returning to the receiving antenna is given by the radar equation, depending on the transmitted power P S, the slant range R, and the reflecting characteristics of the aim (described as the radar cross- section σ). At known sensibility of the radar receiver the radar equation determines the achieved by a given radar set theoretically maximum range. Furthermore one can assess the performance of the radar set with the radar equation.

3. Nondirectional power density diminishes as geometric spreading of the beam.

4. Argumentation/Derivation First we assume, that electromagnetic waves propagate under ideal conditions, i.e. without dispersion. If high-frequency energy is emitted by an isotropic radiator, than the energy propagate uniformly in all directions. Areas with the same power density therefore form spheres (A= 4 π R²) around the radiator. The same amount of energy spreads out on an incremented spherical surface at an incremented spherical radius. That means: the power density on the surface of a sphere is inversely proportional to the to the square of the radius of the sphere.isotropic radiator

5. So we get the equation to calculate the Nondirectional Power Density S u S u = P s /4 · π · R 1 2, in W/ m 2 P S = transmitted power [W] S u = nondirectional power density R 1 = Range antenna - aim [m]

6. Since a spherical segment emits equal radiation in all direction (at constant transmit power), if the power radiated is redistributed to provide more radiation in one direction, then this results an increase of the power density in direction of the radiation. This effect is called antenna gain. This gain is obtained by directional radiation of the power.

7. So, from the definition, the directional power density is: S g = S u · G S g = directional power density S u = nondirectional power density G = antenna gain

8. Of course in reality radar antennas aren't “partially radiating” isotropic radiators. Radar antennas must have a small beam width and an antenna gain up to 30 or 40 dB. (e.g. parabolic dish antenna or phased array antenna).parabolic dish antenna phased array antenna

9. The target detection isn't only dependent on the power density at the target position, but also on how much power is reflected in the direction of the radar. In order to determine the useful reflected power, it is necessary to know the radar cross section σ. This quantity depends on several factors. But it is true to say that a bigger area reflects more power than a smaller area.radar cross section σ

10. That means: A Jumbo jet offers more radar cross section than a sporting aircraft at same flight situation. Beyond this the reflecting area depends on design, surface composition and materials used.

11. With this in mind we can say: The reflected power P r at the radar depends on the power density S u, the antenna gain G, and the variable radar cross section σ: P r = (P s /4 · π · R 1 2 )· G · σ in [W] P r = reflected power [W] σ = radar cross section [m 2 ] R 1 = range, distance antenna - aim [m]

12. Simplified a target can be regarded as a radiator in turn due to the reflected power. In this case the reflected power P r is the emitted power. Since the echoes encounter the same conditions as the transmitted power, the power density yielded at the receiver S e is given by: S e = P r /4 · π · R 2 2, in W /m 2 S e = power density at receiving place P r = reflected power [W] R 2 = range aim - antenna [m]

13. Connection between equations

14. At the radar antenna the received power P E is dependent on the power density at the receiving site s e and the effective antenna aperture A W. P E = S e · A W P E = received power [W] A W = effective antenna aperture [m²] The effective antenna aperture arises from the fact that an antenna suffer from losses, therefore, the received power at the antenna is not equal to the input power. As a rule, the efficiency of the antenna is around 0.6 to 0.7 (Efficiency K a ).

15. Applied to the geometric antenna area, the effective antenna aperture is: A W = A · K a A W = effective antenna aperture [m²] A = geometric antenna area [m²] K a = efficiency

16. The power received, P E is then calculated:

17. The transmitted and reflected waves have been seen separately. The next step is to consider both transmitted and reflected power: Since R 2 (aim - antenna) is the distance R 1 (antenna - aim) then,

21. Another equation, which will not be derived here, describes the antenna gain G in terms of the wavelength λ. G = 4 · π· A · K a /λ 2 Solving for A, antenna area, and replacing A, after simplification it yields: P e = P s · G 2 · σ · λ 2 /(4 · π) 3 · R 4, in [W]

22. Solving for range R, we obtain the classic radar equation:

23. All quantities that influence the wave propagation of radar signals were taken into account at this equation. Before we attempt to use the radar equation in the practice for example to determine the efficiency of radar sets, some further considerations are necessary. For a given radar equipment most sizes (P s, G, λ) can be regarded as constant since they are only variable parameters in very small ranges. The radar cross section, on the other hand, varies heavily but for practical purposes we will assume 1 m².

24. The smallest received power that can be detected by the radar is called P Emin. Smaller powers than P Emin aren't usable since they are lost in the noise of the receiver. The minimum power is detect at the maximum range R max as seen from the equation.noise

25. An application of this radar equation easily visualize how the performance of the radar sets influence the range.

26. Influences on the Maximum Range of a Radar Set All considerations, when calculating the radar equation, were made assuming that the electromagnetic waves propagate under ideal conditions without disturbing influences. In the practice a number of losses should be considered since they reduce the effectiveness of the radar considerably.number of losses

27. First the radar equation is extended by including the loss factor L ges.

28. This factor includes the following losses: - L D = internal attenuation factors of the radar set on the transmitting and receiving paths - L f = fluctuation losses during the reflection - L Atm = atmospheric losses during propagation of the electromagnetic waves to and from the target High frequency components, such as waveguides, filters and also a radome, generate internal losses. For a given radar set this loss is relatively constant and also easily measured. Atmospheric attenuation and reflections at the Earth's surface are permanent influences.fluctuation losseswaveguidesradome Atmospheric attenuation

29. Transmitters Power The more transmitted power, the more power of range, but: Note this fourth root! To double the maximum range you must increase the transmitted power 16-fold! The inversion of this argument is also permissible: if the transmit power is reduced by 1/16 (e.g. failures into two of 32 transmitter modules), then the change on the maximum range of the radar set is negligible in the practice: <2%

33. Noise The value of the MDS echo depends on the Signal-to-Noise-Ratio, defined as the ratio of the signal energy to the noise energy. All radars, as with all electronic equipment, must operate in the presence of noise. The main source of noise is termed thermal noise and is due to agitation of electrons caused by heat.

34. The noise can arise from - received atmospheric or cosmic noise - receiver noise - generated internally in the radar receiver. The overall receiver sensitivity is directly related to the noise figure of the radar receiver. It becomes clear, that a low noise figure receiver is accomplished by a good design in the very front- end components. An aspect to a very low noise figure receiver is achieved through minimizing the noise factor of the very first block. This component usual is characterized by a low noise figure with high gain. This is the reason for the often used denomination, low noise preamplifier (LNA).

38. False Alarm Rate A false alarm is “an erroneous radar target detection decision caused by noise or other interfering signals exceeding the detection threshold”. In general, it is an indication of the presence of a radar target when there is no valid target. The False Alarm Rate (FAR) is calculated using the following formula: FAR= false targets per PRT /number of range cells

39. False alarms are generated when thermal noise exceeds a pre-set threshold level, by the presence of spurious signals (either internal to the radar receiver or from sources external to the radar), or by equipment malfunction. A false alarm may be manifested as a momentary blip on a cathode ray tube (CRT) display, a digital signal processor output, an audio signal, or by all of these means. If the detection threshold is set too high, there will be very few false alarms, but the signal-to-noise ratio required will inhibit detection of valid targets. If the threshold is set too low, the large number of false alarms will mask detection of valid targets.

40. Probability of Detection The received and demodulated echo signal is processed by threshold logic. This threshold shall be balanced so that as of certain amplitude wanted signals being able to pass and noise will be removed. Since high noise exists in the mixed signal tops which lie in the range of small wanted signals the optimized threshold level shall be a compromise. Wanted signals shall on the one hand reach the indication as of minimal amplitude; on the other hand the false alarm rate may not increase. P= (detected targets /all possible blibs) 100% The system must detect, with greater than or equal to 80% probability at a defined range, a one square meter radar cross section.

41. Influence of the Earth's Surface An extended, but less frequent used form of the radar equation considers additional terms, like the Earth's surface but does not classify receiver sensitivity and atmospheric absorption.

43. Radar Reflections from Flat Ground The trigonometric representation shows the influence of the Earth's surface. The Earth plane surrounding a radar antenna has a significant impact on the vertical polar diagram. The combination of the direct and re-reflected ground echo changes the transmitting and receiving patterns of the antenna. This is substantial in the VHF range and decreases with increasing frequency. For the detection of targets at low heights, a reflection at the Earth's surface is necessary. This is possible only if the ripples of the area within the first Fresnel zone do not exceed the value 0.001 R (i.e.: Within a radius of 1000 m no obstacle may be larger than 1 m!).Fresnel zone

44. Detour of ground reflections

45. Specialized Radars at lower (VHF-) frequency band make use of the reflections at the Earth's surface and lobing to maximize cover at low levels. At higher frequencies these reflections are more disturbing. The following picture shows the lobe structure caused by ground reflections. Normally this is highly undesirable as it introduces intermittent cover as aircraft fly through the lobes. The technique has been used in ATC ground mounted radars to extend the range but is only successful at low frequencies where the broad lobe structure permits adequate cover at higher elevations.VHF-

46. A vertical pattern diagram with influences of ground reflections

47. Raising the height of the antenna has the effect of making the lobbing pattern finer. A fine grained lobing structure is often filled in by irregularities in the ground plane. Specifically, if the ground plane deviates from a flat surface then the reinforcement and destruction pattern resulting from the ground reflections breaks down. Avoidance of lobe effects is one of the prime considerations when selecting a radar location and the height of the antenna.

49. Every radarsystem has got miscellaneous losses. Some of these are preventible, or at least reducible by a well designed radar. Some losses can even minimized by maintenance. But unfortunately most of these losses are inevitable. The sum of losses in Table 1 is declared very hard width the value of 21.1 Decibels. Well designed radars have a fairer loss of about 13 to 15 Decibels mostly.Decibels

50. Atmospheric Losses These are losses due to atmospheric absorption by the atmosphere. They are dependent upon the radar operating frequency, the range to the target and the elevation angle of the target relative to the radar. These losses are insignificant at low frequencies less than 3 Gigahertzes by clear weather condition.atmospheric absorption

51. Beamshape Loss This loss term accounts for the fact that, as the beam scans across the target, the signal amplitudes of the pulses coherently or non- coherently integrated varies. Because of the, the full integration gain of the integrator can’t be realized. From the Skolnik Radar Handbook typical values are: - 1.6 dB for a scanning, fan beam radar - 3.2 dB for a thinner beam, scanning radar - 3.2 dB for a phased array radar wherein the beams of a search sector overlap at the 3-dB beam positions.

52. For phase array radars the beam doesn’t move continuously (in most cases) but in discrete steps. This means that the phased array radar may not point the beam directly at the target. This means, in turn, that the antenna gain used in the radar range equation will not be its maximum value. As with the other cases, this phenomena is accommodated through the inclusion of a loss term called, in this case, beam shape loss.

53. Beam width factor The azimuth beam width of a radar antenna has not the same value in all elevation angles. This is summarized in an additional loss factor. Fluctuation Loss This relatively high loss is a result of the fluttering in the values of radar cross section. The gaps are frequency depending! In order to overcome some of the target size fluctuations many radars use two or more different illumination frequencies. Frequency diversity typically uses two transmitters operating in tandem to illuminate the target with two separate frequencies.radar cross section

54. Miscellaneous Signal Processing Loss If the radar uses an MTI with a staggered PRF waveform, and a good MTI and PRF stagger design, it will suffer up to 3 dB signal processing loss. Transmit Line Losses Typically associated with the wave guides and other components between the power amplifier and the antenna. These are typically 1 to 2 dB in a well-designed radar.

55. Receive Losses Typically associated with the wave guides and other components between the antenna and RF amplifier. These are also typically 1 to 2 dB for a well-designed radar. If the noise figure is referenced to the antenna terminals, receive losses are included in the noise figure.

56. Sensitivity of the Receiver While evaluating the minimal received power we follow a different procedure: It's also under the 4th root, but in the denominator. Well, a reduction of the minimal received power of the receiver gets an increase of the maximum range. For every receiver there is a certain receiving power as of which the receiver can work at all. This smallest workable received power is frequently often called MDS - Minimum Discernible Signal in radar technology. Typical radar values of the MDS echo lie in the range of -104 dBm to -110 dBm.MDS echo

57. Antenna Gain The antenna gain is squared under the 4th root (Remember: the same antenna is used during transmission and reception). If one quadruples the antenna gain, it will double the maximum range. Here is a concrete example from VHF- radar technology: Sometimes the P-12 (yagi antennae array: G = 69) was mounted at the antenna of the P-14 (same frequency, parabolic dish antenna: G = 900). This combination was often mentioned jocularly to “P–13”.P-12P-14parabolic dish antenna

58. Frequency-diversity Radar In order to overcome some of the target size fluctuations many radars use two or more different illumination frequencies. Frequency diversity typically uses two transmitters operating in tandem to illuminate the target with two separate frequencies like shown in the following picture.

61. By putting aside the power doubling achieved with two transmitters at constant frequencies, the maximum range through frequency diversity mode can never be better due losses caused by fluctuation (3 to 8 dB).