# TN Calculating Radiated Power and Field Strength for Conducted Power Measurements

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1 Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts TN Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts Copyright Smtch of 9

2 Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts 1 Tabl of Contnts 1 Tabl of Contnts Indx of Figurs... Introduction Antnnas and owr Dnsity Isotropic Antnnas Fr Spac ath oss Antnna Cofficints Antnna Corrction Factor Antnna Factor Fild Strngth Convrsions ffctiv Isotropic adiatd owr Convrsion frncs Indx of Figurs Figur 1: Isotropic adiator...3 Figur : On-Way Communications ink...5 Copyright Smtch 007 of 9

4 Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts oynting s thorm dfins th rlationship btwn th powr dnsity to th -fild and H-fild vctors as dfind blow: d = H Watts/m Th magnitud of th powr dnsity is thus: d = H =, whr 10π is th impdanc of fr spac or approximatly 377Ω. 10π From th abov, w can s that = 10π, and thus, H 10π = r Hnc, th rms valu of th (or lctric) fild can b calculatd from: 30 = V/m () r Th isotropic antnna is considrd to hav a powr gain, G i = 1. frring to Figur 1, abov, if an antnna with gain G T wr placd at point O, th powr rcivd at point Q would b incrasd by G T. Hnc, th fild strngth at point Q will b incrasd: 30G T T = V/m (3) r Whr T is th transmittr powr, G T is th antnna gain. Similarly, for an antnna with gain G T, w can rwrit quation (1) as: d G T T = Watts/m (4) r If G T = 1 (.g. th gain of an Isotropic antnna with gain, G i ), th radiatd powr calculatd in quation (4) is rfrrd to as th ffctiv Isotropic adiatd owr (I) and is th powr that would b radiatd from a transmittr of owr T from an idal omnidirctional sourc. -arranging (3), and with G T = 1, w can dtrmin th I in trms of th fild strngth: r T = (5) 30 As an xampl, FCC art 15.49(a) stats that th maximum fild strngth for ISM band dvics oprating in th MHz band is 50 mv/m. Sinc fild strngth is typically masurd at 3 m, w can rwrit (5) to calculat th I as follows: = 0.3 T = 0.75mW T = 1.5dBm Copyright Smtch of 9

5 Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts 4 Fr Spac ath oss Considr th on-way (or simplx) communications link illustratd blow in Figur : r Q T Figur : On-Way Communications ink Assum that a transmittr at T with transmittr powr T and antnna gain G T. Th powr dnsity, d, at th rcivr, Q is as prviously dfind in quation (4). d G T T = Watts/m r Th radiatd powr rcivd at th antnna port is a function of th ffctiv aprtur of th antnna: Gantλ A = m (6) If th gain of th rcivr antnna, G ant = G, w can calculat th powr rcivd as: = A (7) d G T TGλ = (8) ( r ) quation (8) is oftn rfrrd to as th fr spac loss quation. Whilst w can r-arrang (8) to dtrmin th thortical transmission rang, it should b noticd that in th ral world situation, factors such as rflctions, scattring, shadowing, trrain and nvironmnt will impact upon th transmission rang. Copyright Smtch of 9

6 Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts 5 Antnna Cofficints 5.1 Antnna Corrction Factor Typically laboratory basd powr masurmnts ar mad with tst quipmnt that has a charactristic impdanc of 50Ω. As has bn dfind abov, th charactristic impdanc of fr spac is (10π)Ω or 377Ω. For an antnna matchd to a load, w can r-writ quation (7) as = A (9) d If is masurd in a 50Ω load and d in a fr spac impdanc of 377Ω, thn th following rlationships apply: V = and 50 d = (10) 377 Hnc, w can r-writ quation (9): = A V (11) Not that V is in units of volts, whilst is in volts/mtr. Th ffctiv aprtur of th antnna, with units of m, convrts fr spac powr dnsity to circuit powr. Th rlationship btwn radiatd and conductd powr can b mad clarr by dfining an antnna corrction factor (ACF), which is introducd in quation (1): ACF = m - (1) V -writing (11) and substitut for ACF: ACF = 50 A By substituting quation (6) into th abov, th antnna corrction factor can b dfind as: 377 ACF = m - (13) 50 G ant λ If th frquncy, f, is in MHz, thn by substituting λ=c/f: ACF ACF = ( f, MHz) ( ) = ( )( f, MHz) Gant 1 G ant Copyright Smtch of 9

7 Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts Taking 10(log) of both sids allows th ACF to b calculatd in db: ACF( db) = 9.78dB + 0log( f, MHz) 10log( G) (14) 5. Antnna Factor From th abov, w s that th Antnna Corrction Factor links powr (in trms of V /) and powr dnsity. Most antnnas usd for fild strngth and complianc masurmnts us a cofficint to link fild strngth to rcivr voltag V dirctly, sinc V can b masurd, whilst is th quantity that is rquird by th FCC. This cofficint is a calld Antnna Factor (A F ) and is gnrally quotd as a function of frquncy (or 1/λ). A V F = = = V ACF A F = = 50 Gantλ m -1 (16) λ Gant 6 Fild Strngth Convrsions From quation (9) th powr in th load is rlatd to powr dnsity by th ffctiv antnna aprtur, A. Substituting quation (6) into this quation, w can rlat load powr dirctly to as dfind in quation 10, as follows: = λgant 377 If th frquncy, f, is in MHz and w substitut for λ=c/f: 8 (3 10 ) G = 6 ( f, MHz) (10 ) ant 377 For powr in mw and fild strngth in µv/m, and calculating th constant trms: Gant (, µ V / m) ( mw ) = ( f, MHz) (17) Taking 10(log) of both sids: 10log(, mw ) = 77.dB + 10log( G ) + 0log(, µ V / m) 0log( f, MHz) By rarranging th abov: ant ( dbµ V / m) = ( dbm) + 77.dB + 0log( f, MHz) ( db) (18) ant Copyright Smtch of 9

8 Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts Sinc is rlatd to powr masurd in a circuit (by using a spctrum analyzr) by th trms for any path gains btwn th antnna and th analyzr, w can rwrit = mas gain. quation (18) bcoms: ( dbµ V / m) = ( dbm) ( db) + 77.dB + 0log( f, MHz) ( db) (19) mas gain quation 19 is th ky quation for convrting powr masurd in a circuit (in dbm) into incidnt fild strngth in dbµv/m. 7 ffctiv Isotropic adiatd owr Convrsion To calculat th ffctiv I from th powr masurd in a circuit, from quation (7), th powr in th rcivr is rlatd to th powr dnsity and th ffctiv aprtur of th antnna: ant G T T I = A = A r r = da (0) From quation (6), th ffctiv aprtur of th antnna is th ffctiv aprtur of an isotropic antnna multiplid by th gain of th antnna (G ant = G ): Gλ A = Taking log(10) of both sids of quation (0), and if th frquncy, f, is in MHz, distanc, r, is in mtrs, and substituting for λ=c/f: = I + G ( db) log( f, MHz) 0 log( r, mtrs) (1) For th masurmnts rquird by FCC art 15, quation (1) is a convnint mthod of calculating th powr in th rcivr. Sinc masurmnts ar typically mad at a distanc of 3 mtrs, w can rwrit quation (1): = I + G ( db) log( f, MHz) () If th gain of th rcivr antnna is known and th rcivd powr has bn masurd at a distanc of r mtrs from th sourc, w can rarrang quations (1) and () to calculat th I: I( dbw ) = ( dbm) log( f, MHz) 0 log( r, mtrs) (3) If I rlativ to 1 mw is rquird: I( dbm) = ( dbm) log( f, MHz) 0 log( r, mtrs) (4) And assuming that th distanc r is 3 m as in th cas dfind for quation (): I( dbm) = ( dbm) 37 0log( f, MHz) (5) Copyright Smtch of 9

9 Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts 8 frncs F.. Connor; Antnnas dward Arnold, 197 Softwright C; F. Sandrs; Convrsion of owr Masurd in a Circuit to Incidnt Fild Strngth and Incidnt owr Dnsity, and Corrctions to Masurd mission Spctra for Non-Constant Aprtur Masurmnt Antnnas (Appndix C of NTIA port , Jan 001) ND OF DOCUMNT Copyright Smtch of 9

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