ONE-WAY RADAR EQUATION / RF PROPAGATION The one-way (transmitter to receiver) radar equation is derived in this section. This equation is most commonly used in RWR or ESM type of applications. The following is a summary of the important equations explored in this section: ONE-WAY RADAR EQUATION Peak Power at 4BA 2 PtGtAe e G8 ' ' and Antenna Gain, G ' or: Equivalent Area, A ' Receiver Input, Pr (or S) PDAe 2 e 4BR 2 8 4B So the one-way radar equation is : ( P G G 82 2 Values of K1 (in dB) ' t t r ' c ' c S (orPr) Pt Gt Gr (Note: 8 ) Range f in MHz f in GHz (4BR)2 (4BfR)2 f 1 1 (units) K = K = * keep 8, c, and R in the same units 1 1 NM 37.8 97.8 On reducing to log form this becomes: km 32.45 92.45 m -27.55 32.45 10log Pr = 10log Pt + 10log Gt + 10log Gr - 20log f R + 20log (c/4B) yd -28.33 31.67 or in simplified terms: ft -37.87 22.13 ______________________ 10log Pr = 10log Pt + 10log Gt + 10log Gr - "1 (in dB) Where: "1 = one-way free space loss = 20log (f1R) + K1 (in dB) Note: Losses due to antenna and: K1 = 20log [(4B/c)(Conversion factors if units if not in m/sec, m, and Hz)] polarization and atmospheric Note: To avoid having to include additional terms for these calculations, absorption (Sections 3-2 & 5-1) always combine any transmission line loss with antenna gain are not included in any of these equations. Recall from Section 4-2 that the power density at a distant point from a radar with an antenna gain of Gt Same Antenna is the power density from an isotropic antenna Capture Area multiplied by the radar antenna gain. P G Power density from radar, ' t t [1] PD 4BR 2 If you could cover the entire spherical segment with your receiving antenna you would theoretically capture all of the transmitted energy. You can't do this Range 1 Range 2 because no antenna is large enough. (A two degree Received Signal Received Signal segment would be about a mile and three-quarters across at fifty miles from the transmitter.) Figure 1. Power Density vs. Range A receiving antenna captures a portion of this power determined by it's effective capture Area (Ae). The received power available at the antenna terminals is the power density times the effective capture area (Ae) of the receiving antenna. For a given receiver antenna size the capture area is constant no matter how far it is from the transmitter, as illustrated in Figure 1. This concept is shown in the following equation: 4-3.1 PtGtAe PR (or S) = P e = which is known as the one-way (beacon) equation 4BR 2 In order to maximize energy transfer between an antenna and transmitter or receiver, the antenna size shoul correlate 8/4. Control o beamwidth shape may become a problem when the size of the active element exceeds several wavelengths. Th relation between an antenna's effectiv capture area (Ae is: 4BA Antenna Gain, G ' e 82 G82 or: Equivalent Area, A ' [4] e 4B Lower Frequency Higher Frequency Antenna Has Antenna Has Larger Area Smaller Area effective aperture is in units of length Low Frequency Higher Frequency squared, s Antenna Area Antenna Area proport Received Signal Received Signal wavelength. This physically means that to maintain the gain when doubling the frequency, the area i Figure 2. Capture Area vs Frequency reduced by 1/4. This concept is illustrated in Figure 2. If equation [4] is substituted into equation [2], the following relationship results: P G G 82 P G G 82 ' ' t t r ' t t r Peak Power at Receiver Input S (or PR) [5] (4B)2R 2 (4BR)2 is the signal calculated one-way from a transmitter to a receiver. For instance, a radar application might be to rmine the signal received by a RWR, ESM, or an ELINT receiver. It is a general purpose equation and could be The free space travel of radio waves can, of course, be blocked, reflected, or distorted by objects in their path such As received signal power decreases by 1/4 (6 dB). This is due to the ONE WAY SIGNAL STRENGTH (S) 2 term in equation [5]. S 2R S decreases by 6 dB 6 dB when the distance doubles illust a (1/4 pwr) R square on radius is decreased by 1/2, 6 dB R (4x pwr) S increases by 6 dB you further blow up the balloon, so the diameter or radius i when the distance is half doubled, the square has quadrupled in area. S 0.5 R 4-3.2 The one-way free space loss factor ("1), PHYSICAL CONCEPT - One-way Space Loss (sometimes called the path loss factor) is given by the term 2 2 2 TRANSMITTER (4BR )(4B/8 ) or (4BR /8) . As shown in Figure 3, the RECEIVER G = 1 loss is due to the ratio of two factors (1) the effective r radiated area of the transmit antenna, which is the surface Pt 2 S ( or P r ) area of a sphere (4BR ) at that distance (R), and (2) the G t = 1 effective capture area (Ae) of the receive antenna which has a gain of one. If a receiving antenna could capture the EQUIVALENT CIRCUIT - One-way Space Loss whole surface area of the sphere, there would be no TRANSMITTER spreading loss, but a practical antenna will capture only a RECEIVER small part of the spherical radiation. Space loss is P calculated using isotropic antennas for both transmit and t " , TRANSMITTER TO RECEIVER 1 ONE-WAY SPACE LOSS S ( or P ) receive, so "1 is independent of the actual antenna. Using r 2 Gr = 1 in equation [11] in section 3-1, Ae = 8 /4B. Since EQUIVALENT CIRCUIT - One-Way Space Loss with Actual Antennas this term is in the denominator of "1, the higher the TRANSMITTER frequency (lower 8) the more the space loss. Since G and RECEIVER t Gt G G are part of the one-way radar equation, S (or P ) is r r r P adjusted according to actual antennas as shown in the last t " 1 S ( or P ) portion of Figure 3. The value of the received signal (S) is: XMT RECEIVE r ANTENNA ANTENNA GAIN GAIN P G G 82 2 ' t t r ' 8 Figure 3. Concept of One-Way Space Loss S (or PR) PtGtGr [6] (4BR)2 (4BR)2 To convert this equation to dB form, it is rewritten as: ( 8 10log(S orP ) ' 10log(P G G ) % 20log (( keep 8 and R in same units) [7] r t t r 4BR Since 8 = c / f, equation [7] can be rewritten as: 10 Log (S or Pr) = 10 Log(PtGtGr) - "1 [8] 4BfR Where the one-way free space loss, "1, is defined as: " ' 20Log * [9] 1 c The signal received equation in dB form is: 10log (Pr or S) = 10log Pt + 10log Gt + 10log Gr - "1 [10] The one-way free space loss, "1, can be given in terms of a variable and constant term as follows: ( 4BfR " ' 20Log ' 20Log f R % K (in dB) [11] 1 c 1 1 The value of f1 can be either in MHz or GHz as shown with commonly used units of R in the adjoining table. Values of K1 (dB) Range f 1 in MHz f 1 in GHz 4B where K ' 20Log @(Conversion units if not in m/sec, m, and Hz) (units) K 1 = K 1 = 1 c NM 37.8 97.8 km 32.45 92.45 Note: To avoid having to include additional terms for these m -27.55 32.45 calculations, always combine any transmission line loss with antenna gain. yd -28.33 31.67 ft -37.87 22.13 4-3.3 A value for the one-way free space loss ("1) can be obtained from: (a) The One-way Free Space Loss graph (Figure 4). Added accuracy can be obtained using the Frequency Extrapolation graph (Figure 5) (b) The space loss nomograph (Figure 6 or 7) (c) The formula for "1, equation [11]. FOR EXAMPLE: Find the value of the one-way free space loss, "1, for an RF of 7.5 GHz at 100 NM. (a) From Figure 4, find 100 NM on the X-axis and estimate where 7.5 GHz is located between the 1 and 10 GHz lines (note dot). Read "1 as 155 dB. An alternate way would be to read the "1 at 1 GHz (138 dB) and add the frequency extrapolation value (17.5 dB for 7.5:1, dot on Figure 5) to obtain the same 155 dB value. (b) From the nomogram (Figure 6), the value of "1 can be read as 155 dB (Note the dashed line). (c) From the equation 11, the precise value of "1 is 155.3 dB. Remember, "1 is a free space value. If there is atmospheric attenuation because of absorption of RF due to certain molecules in the atmosphere or weather conditions etc., the atmospheric attenuation is in addition to the space loss (refer to Section 5-1). 180 = 20 Log fR + 37.8 dB 1 100 GHz 160 f in MHz & R in NM Point From Example 10 GHz 140 1 GHz 120 100 MHz 100 10 MHz 80 1 MHz 60 0.1 0.2 0.3 0.5 1.0 2 3 5 10 20 30 50 100 200 300 RANGE (NM) Figure 4.