ONE-WAY RADAR EQUATION / RF PROPAGATION The one-way (tansmitte to eceive) ada equation is deived in this section. This equation is most commonly used in RWR o ESM type of applications. The following is a summay of the impotant equations exploed in this section: ONE-WAY RADAR EQUATION Peak Powe at P (o S) ' P D A e ' P G A t t e and Antenna Gain, G ' 4BA e o: Equivalent Aea, A 4BR 8 e ' Receive Input, G8 4B So the one-way ada equation is : S (op ) ' P G t 8 c ' P (4BR) t G ( (Note: 8' c Values of K (in db) ) Range f (4BfR) f in MHz f in GHz (units) K * keep 8, c, and R in the same units = K = NM 37.8 97.8 On educing to log fom this becomes: km 3.45 9.45 0log P = 0log + 0log + 0log - 0log f R + 0log (c/4b) m -7.55 3.45 yd -8.33 3.67 o in simplified tems: ft -37.87.3 0log P = 0log + 0log + 0log -" (in db) Whee: " = one-way fee space loss = 0log (f R) + K (in db) and: K = 0log [(4B/c)(Convesion factos if units if not in m/sec, m, and Hz)] Note: To avoid having to include additional tems fo these calculations, always combine any tansmission line loss with antenna gain Note: Losses due to antenna polaization and atmospheic absoption (Sections 3- & 5-) ae not included in any of these equations. Recall fom Section 4- that the powe density at a distant point fom a ada with an antenna gain of is the powe density fom an isotopic antenna multiplied by the ada antenna gain. Same Antenna Captue Aea Powe density fom ada, P D ' [] 4BR If you could cove the entie spheical segment with you eceiving antenna you would theoetically captue all of the tansmitted enegy. You can't do this because no antenna is lage enough. (A two degee segment would be about a mile and thee-quates acoss at fifty miles fom the tansmitte.) Range Range Received Signal Received Signal Figue. Powe Density vs. Range A eceiving antenna captues a potion of this powe detemined by it's effective captue Aea (A ). The eceived e powe available at the antenna teminals is the powe density times the effective captue aea (A ) of the eceiving antenna. e Fo a given eceive antenna size the captue aea is constant no matte how fa it is fom the tansmitte, as illustated in Figue. This concept is shown in the following equation: 4-3.
Antenna Gain, A P e R (o S) = P e = which is known as the one-way (beacon) equation 4BR In ode to maximize enegy tansfe between an antenna and tansmitte o eceive, the antenna size shoul coelate 8/4. Contol o beamwidth shape may become a poblem when the size of the active element exceeds seveal wavelengths. captue aea (A e is: Th elation between an antenna's effectiv G ' 4BA e 8 o: Equivalent Aea, A e ' G8 4B effective apetue is in units of length squaed, s popot wavelength. This physically means that to maintain the gain when doubling the fequency, the aea i educed by /4. This concept is illustated in Figue. [4] Low Fequency Antenna Aea Received Signal Lowe Fequency Antenna Has Lage Aea Highe Fequency Antenna Aea Received Signal Figue. Captue Aea vs Fequency Highe Fequency Antenna Has Smalle Aea If equation [4] is substituted into equation [], the following elationship esults: to Peak Powe at Receive Input ' S (o P R ) ' 8 ' 8 (4B) R (4BR) is the signal calculated one-way fom a tansmitte to a eceive. Fo instance, a ada application might be mine the signal eceived by a RWR, ESM, o an ELINT eceive. It is a geneal pupose equation and could be [5] The fee space tavel of adio waves can, of couse, be blocked, eflected, o distoted by objects in thei path such As eceived signal powe deceases by /4 (6 db). This is due to the tem in equation [5]. illust a squae on adius is deceased by /, you futhe blow up the balloon, so the diamete o adius i doubled, the squae has quadupled in aea. ONE WAY SIGNAL STRENGTH (S) S S deceases by 6 db R 6 db (/4 pw) when the distance doubles R 6 db (4x pw) S S inceases by 6 db when the distance is half R 0.5 R 4-3.
The one-way fee space loss facto (" ), (sometimes called the path loss facto) is given by the tem (4BR )(4B/8 ) o (4BR /8). As shown in Figue 3, the loss is due to the atio of two factos () the effective adiated aea of the tansmit antenna, which is the suface aea of a sphee (4BR ) at that distance (R), and () the effective captue aea (A e) of the eceive antenna which has a gain of one. If a eceiving antenna could captue the whole suface aea of the sphee, thee would be no speading loss, but a pactical antenna will captue only a small pat of the spheical adiation. Space loss is calculated using isotopic antennas fo both tansmit and eceive, so " is independent of the actual antenna. Using G = in equation [] in section 3-, A e = 8 /4B. Since this tem is in the denominato of ", the highe the fequency (lowe 8) the moe the space loss. Since Gt and G ae pat of the one-way ada equation, S (o P ) is adjusted accoding to actual antennas as shown in the last potion of Figue 3. The value of the eceived signal (S) is: S (o P R ) ' 8 (4BR) ' 8 (4BR) [6] PHYSICAL CONCEPT - One-way Space Loss TRANSMITTER G = t EQUIVALENT CIRCUIT - One-way Space Loss TRANSMITTER EQUIVALENT CIRCUIT - One-Way Space Loss with Actual Antennas TRANSMITTER XMT ANTENNA GAIN " G = RECEIVE ANTENNA GAIN RECEIVER S ( o P ) RECEIVER ", TRANSMITTER TO RECEIVER ONE-WAY SPACE LOSS S ( o P ) Figue 3. Concept of One-Way Space Loss RECEIVER S ( o P ) To convet this equation to db fom, it is ewitten as: 0log(S op ) ' 0log( ) % 0log 8 4BR Since 8 = c / f, equation [7] can be ewitten as: ( (( keep 8 and R in same units) [7] 0 Log (S o P ) = 0 Log(P G G ) - " [8] t t Whee the one-way fee space loss, ", is defined as: " ' 0Log 4BfR * [9] c The signal eceived equation in db fom is: 0log (P o S) = 0log P + 0log G + 0log G - " [0] t t The one-way fee space loss, ", can be given in tems of a vaiable and constant tem as follows: " ' 0Log 4BfR c ( ' 0Log f R % K (in db) [] The value of f can be eithe in MHz o GHz as shown with commonly used units of R in the adjoining table. whee K ' 0Log 4B c @(Convesion units if not in m/sec, m, and Hz) Note: To avoid having to include additional tems fo these calculations, always combine any tansmission line loss with antenna gain. Values of K (db) Range f in MHz f in GHz (units) K = K = NM 37.8 97.8 km 3.45 9.45 m -7.55 3.45 yd -8.33 3.67 ft -37.87.3 4-3.3
A value fo the one-way fee space loss (" ) can be obtained fom: (a) The One-way Fee Space Loss gaph (Figue 4). Added accuacy can be obtained using the Fequency Extapolation gaph (Figue 5) 80 (b) The space loss nomogaph (Figue 6 o 7) (c) The fomula fo ", equation []. FOR EXAMPLE: Find the value of the one-way fee space loss, ", fo an RF of 7.5 GHz at 00 NM. (a) Fom Figue 4, find 00 NM on the X-axis and estimate whee 7.5 GHz is located between the and 0 GHz lines (note dot). Read " as 55 db. An altenate way would be to ead the " at GHz (38 db) and add the fequency extapolation value (7.5 db fo 7.5:, dot on Figue 5) to obtain the same 55 db value. (b) Fom the nomogam (Figue 6), the value of " can be ead as 55 db (Note the dashed line). (c) Fom the equation, the pecise value of " is 55.3 db. Remembe, " is a fee space value. If thee is atmospheic attenuation because of absoption of RF due to cetain molecules in the atmosphee o weathe conditions etc., the atmospheic attenuation is in addition to the space loss (efe to Section 5-). 60 40 = 0 Log fr + 37.8 db 00 GHz f in MHz & R in NM Point Fom Example 0 GHz 0 GHz 00 00 MHz 80 0 MHz MHz 60 0. 0. 0.3 0.5.0 3 5 0 0 30 50 00 00 300 RANGE (NM) Figue 4. One-Way Fee Space Loss 4-3.4
db 0 FOR USE WITH ONE-WAY FREE SPACE LOSS GRAPH 8 6 Point Fom Example 4 0 8 6 4 0 3 4 5 6 8 0 n DELTA FREQUENCY (f ) [ whee: F = (f ) x 0 ] Figue 5. Fequency Extapolation Figue 6. One-Way Space Loss Nomogaph Fo Distances Geate Than 0 Nautical Miles 4-3.5
Figue 7. One-Way Space Loss Nomogaph Fo Distances Less Than 0 Nautical Miles ERP NOTE: Dawing not to scale P T Space Loss Appoaching Receive Note: In the example on page 4-3.6, the eceive antenna gain is negative vs positive. If powe is actually measued in this egion, it is stated in eithe powe density (mw/cm ) o field intensity (V/m) P R RWR / ESM Receive 0 log + 0 log - " + 0 log = 0 log P SIGNAL POSITION IN SPACE Figue 8. Visualization of One-Way Rada Equation Figue 8 is the visualization of the losses occuing in one-way ada equation. Note: To avoid having to include additional tems, always combine any tansmission line loss with antenna gain. Losses due to antenna polaization and atmospheic absoption also need to be included. 4-3.6
RWR/ESM RANGE EQUATION (One-Way) The one-way ada (signal stength) equation [5] is eaanged to calculate the maximum ange R max of RWR/ESM eceives. It occus when the eceived ada signal just equals S as follows: R max 8 (4B) S min o c (4Bf) S min o A e 4BS min min [] In log fom: 0log R = 0log P + 0log G - 0log S - 0log f + 0log(c/4B) [3] max t t min and since K = 0log{4B/c times convesion units if not in m/sec, m, and Hz} (Refe to section 4-3 fo values of K ). 0log R = ½[ 0log P + 0log G - 0log S - 0log f - K ] ( keep P and S in same units) [4] max t t min t min 0 MdB If you want to convet back fom db, then R max, whee M db is the esulting numbe in the backets of equation 4. 0 Fom Section 5-, Receive Sensitivity / Noise, S is elated to the noise facto S: S = (S/N) (NF)KT B [5] min min min o The one-way RWR/ESM ange equation becomes: R max 8 (4B) (S/N) min (NF)KT o B o c (4Bf) (S/N) min (NF)KT o B o A e 4B(S/N) min (NF)KT o B [6] RWR/ESM RANGE INCREASE AS A RESULT OF A SENSITIVITY INCREASE As shown in equation [] S - % R min max Theefoe, -0 log S min % 0 logr max and the table below esults: % Range Incease: Range + (% Range Incease) x Range = New Range i.e., fo a 6 db sensitivity incease, 500 miles +00% x 500 miles =,000 miles Range Multiplie: Range x Range Multiplie = New Range i.e., fo a 6 db sensitivity incease 500 miles x =,000 miles db Sensitivity % Range Range db Sensitivity % Range Range Incease Incease Multiplie Incease Incease Multiplie + 0.5 6.06 0 6 3.6.0. 55 3.55.5 9.9 98 3.98 6.6 3 347 4.47 3 4.4 4 40 5.0 4 58.58 5 46 5.6 5 78.78 6 53 6.3 6 00.0 7 608 7.08 7 4.4 8 694 7.94 8 5.5 9 79 8.9 9 8.8 0 900 0.0 4-3.7
RWR/ESM RANGE DECREASE AS A RESULT OF A SENSITIVITY DECREASE As shown in equation [] S - % R min max Theefoe, -0 log S min % 0 logr max and the table below esults: % Range Decease: Range - (% Range decease) x Range = New Range i.e., fo a 6 db sensitivity decease, 500 miles - 50% x 500 miles = 50 miles Range Multiplie: Range x Range Multiplie = New Range i.e., fo a 6 db sensitivity decease 500 miles x.5 = 50 miles db Sensitivity % Range Range db Sensitivity % Range Range Decease Decease Multiplie Decease Decease Multiplie - 0.5 6 0.94-0 68 0.3 -.0 0.89-7 0.8 -.5 6 0.84-75 0.5-0.79-3 78 0. - 3 9 0.7-4 80 0.0-4 37 0.63-5 8 0.8-5 44 0.56-6 84 0.6-6 50 0.50-7 86 0.4-7 56 0.44-8 87 0.3-8 60 0.4-9 89 0. - 9 65 0.35-0 90 0.0 Example of One-Way Signal Stength: A 5 (o 7) GHz ada has a 70 dbm signal fed though a 5 db loss tansmission line to an antenna that has 45 db gain. An aicaft that is flying 3 km fom the ada has an aft EW antenna with - db gain and a 5 db line loss to the EW eceive (assume all antenna polaizations ae the same). Note: The espective tansmission line losses will be combined with antenna gains, i.e.: -5 +45 = 40 db, -5 - = -6 db, -0 + 5 = -5 db. () What is the powe level at the input of the EW eceive? Answe (): P at the input to the EW eceive = Tansmitte powe - xmt cable loss + xmt antenna gain - space loss + cv antenna gain - cv cable loss. Space loss (fom section 4-3) @ 5 GHz = 0 log f R + K = 0 log (5x3) + 9.44 = 36.5 db. Theefoe, P = 70 + 40-36.5-6 = -3.5 dbm @ 5 GHz (P = -35.7 dbm @ 7 GHz since " = 39.7 db) () If the eceived signal is fed to a jamme with a gain of 60 db, feeding a 0 db loss tansmission line which is connected to an antenna with 5 db gain, what is the powe level fom the jamme at the input to the eceive of the 5 (o 7) GHz ada? Answe (): P at the input to the ada eceive = Powe at the input to the EW eceive+ Jamme gain - jamme cable loss + jamme antenna gain - space loss + ada cv antenna gain - ada cv cable loss. Theefoe, P = -3.5 + 60-5 - 36.5 + 40 = -73.5 dbm @ 5 GHz. (P = -79.34 dbm @ 7 GHz since " = 39.7 db and = -35.7 dbm). This poblem continues in section 4-4, 4-7, and 4-0. 4-3.8