Squar VLF Loop Antnna, 1. m Diagonal ~ Mchanical and Elctrical Charactristics and Construction Dtails ~ Whitham D. Rv 1. Dimnsions Th loop antnna dscribd hr has a squar shap with a diagonal lngth of 1.07 m. Th loop is mad from fibrglass framwork and marin plywood bracs and consists of 64 ±1 turns of 18 AWG coatd magnt wir. Th width of a squar in trms of its diagonal lngth is W l Eq. (1.1) W = squar width l = diagonal lngth (1.07 m) Th primtr lngth of a squar is p 4 W Eq. (1.) p = primtr lngth Thrfor, for th loop in qustion W 1.07 0.853 m = 85.3 cm p 4 0.853 3.41 m Th nclosd ara of a squar is A W (0.0.853) = 0.733 m Eq. (1.3) A = nclosd ara of squar Th loop is wound with 18 AWG magnt wir, which has a coatd nominal diamtr of approximatly 0.044 in. = 1.077 mm = 0.1077 cm and uncoatd nominal diamtr of 0.0403 in. = 1.036 mm = 0.104 cm. Th total wir lngth is Lngth pt 3.4164 18.4 m T = Numbr of turns Fil: Rv_SquarLoopAntnna1.m.doc, Pag 1
Fil: Rv_SquarLoopAntnna1.m.doc, Pag
. Inductanc Four mthods ar invstigatd for calculating th loop inductanc. All mthods ar stimats and non prfctly fit th actual antnna construction but mthod 1 is th closst. Th coil cross-sction on th actual antnna is mor or lss rctangular, but two of th mthods assum a flat, singl-layr cross-sction. Th formulas usd hr ar basd on a 1937 National Burau of Standards documnt in which th CGS unit systm is usd. To avoid units convrsion rrors, all calculations givn hr us th sam units. Also includd ar actual masurmnts for comparison with calculations. Mthod 1 (Not: This mthod assums a multi-layr coil with rctangular cross-sction) Th approximat inductanc in H of a squar coil with rctangular cross-sction is givn by 1 b c 8 a b L 0.0157 a n.303 1 log y1 y Eq. (.1) 3 a 96 a d 16 a a = Avrag of inscribd and circumscribd radii, r cos N b = axial dimnsion of th coil cross-sction (cm) c = radial dimnsion of th coil cross-sction (cm) d = diagonal of th cross-sction, b c (cm) n = numbr of turns y 1 = valu from Tabl 14, pg 85 of rfrnc basd on b/c = valu from Tabl 14, pg 85 of rfrnc basd on c/b y (cm) Th following valus apply: a = 51.5 cm b =. cm c = 0.154 cm d =.3 cm n = 64 b/c = 10.3 c/b = 0.097 y 1 = 0.5898 y = 0.1317 Substituting th abov valus, th calculatd inductanc is. 0.154 851.5. L 0.015751.564.3031 log 0.5898 0.1317 351.5 9651.5.3 1651.5 1 78 H = 1.3 mh 1 Eq. 157, pg 57, Circular C74, Radio Instrumnts and Masurmnts, US Dpartmnt of Commrc, National Burau of Standards, 1937 Fil: Rv_SquarLoopAntnna1.m.doc, Pag 3
Mthod (Not: This mthod is similar to Mthod 1 but assums a singl coil layr) Th approximat inductanc in H of a singl-layr squar coil is givn by a b L 0.008 a n.303 log 0.31 0.76 a n b a 0. 008 a = width of squar masurd btwn cntrs of th cross-sction (cm) b = lngth of coil (cm) n = numbr of turns D = coil winding pitch (cm) (0.1077 cm) d = diamtr of bar wir (cm) (0.104 cm) A = function of Tabl 11 in NBS rfrnc documnt B = function of Tabl 1 in NBS rfrnc documnt A B Eq. (.) Th coil width is a = W = 85.3 cm, lngth is b. mm =. cm and th numbr of turns n = 64. A is basd on th ratio d D 0.104 0.1077 0.95 in Tabl 11 on pag 84 of th rfrnc and B is basd on th numbr of turns n = 64 in Tabl 1 on pag 84 of th rfrnc. Valus ar A = 0.506 B = 0.3 Substituting th abov valus, th calculatd inductanc is 85.3. L 0.00885.364.303log 0.31 0.76 0.008 85.364 0.506 0.3. 85.3 1 10 H = 1. mh Mthod 3 (Not: This mthod assums a rctangular cross-sction) Th approximat inductanc in H of a squar coil with rctangular cross-sction is givn by 3 a b c L b c a 0.008 a n.303 log 0.35 0.76 Eq. (.3) a b c n = width of squar masurd btwn cntrs of th cross-sction (cm) = lngth of coil (cm) = dpth of coil (cm) = numbr of turns Th coil width is a = W = 85.3 cm, lngth is b. of turns n = 64. mm =. cm, c =.154 mm = 0.154 cm and th numbr Eq. 165, pg 64, Circular C74, Radio Instrumnts and Masurmnts, US Dpartmnt of Commrc, National Burau of Standards, 1937 3 Eq. 163, pg 63, Circular C74, Radio Instrumnts and Masurmnts, US Dpartmnt of Commrc, National Burau of Standards, 1937 Fil: Rv_SquarLoopAntnna1.m.doc, Pag 4
Substituting th abov valus, th calculatd inductanc is 85.3. 0.154 L 0.00885.364.303log 0.31 0.76. 0.154 85.3 = 11 990 H = 1.0 mh Mthod 4 (Not: this mthod assums a singl-layr coil of lngth b of polygon shap) Th approximat inductanc in H of a polygon is givn by 4 0.03948 a n L K μh Eq. (.4) b a = Avrag of inscribd and circumscribd radii, r cos (cm) N Eq. (.5) r = radius of circumscribd circl (cm) N = numbr of sids (4 for a squar) n = numbr of turns b = lngth of coil, or n d (cm) d = distanc btwn turn cntrs = wir diamtr for clos spacing (cm) (0.1077 cm for 18 AWG coatd wir) K = function of a / b from tabl 10 in th rfrnc NBS documnt Th radius of a circumscribd circl for a squar is W l 1.07 r 0.6033 m = 60.33 cm Eq. (.6) and th avrag of th inscribd and circumscribd radii is a 4 60.33cos 51.5 cm Th stimatd coil lngth for singl-layr flat winding is b nd 60.1077 6.68 cm. K is basd on a / b 51.5 / 6.68 15.4 and is found in Tabl 10, pag 83 of th rfrnc NBS documnt by intrpolation, or K = 0.1498. Substituting th abov valus, th calculatd inductanc is 0.03948 51.5 64 0.1498 L 9 618 H = 9.6 mh 6.68 4 Eq. 153, pg 5, Circular C74, Radio Instrumnts and Masurmnts, US Dpartmnt of Commrc, National Burau of Standards, 1937 Fil: Rv_SquarLoopAntnna1.m.doc, Pag 5
3. Masurmnts Masurmnt dat: 10 August 018 Masurd inductanc at 15.5 C: DM4070: 1.9 mh Pak LCR45: 1.4 mh at 1 khz Pak LCR45: 13.65 mh at 15 khz Pak LCR45: 680 uh at 00 khz Masurmnt dat: Sptmbr 018 (on-sit) Masurd inductanc at 15.0 C: Kysight U1733C: 1.70 mh, Q = 16.7 at 1 khz Kysight U1733C: 1.841 mh, Q = 69.1 at 10 khz Additional masurmnts stup: Oscilloscop Siglnt SDS30X RF Signal Gnrator SG-100A 00 000 Hz 0.0 dbm 10K Slf-Rsonanc 500K Q Masurd slf-rsonanc frquncy: 46.065 khz Masurd dc rsistanc at 0.6 C: 4.535 ohms Fil: Rv_SquarLoopAntnna1.m.doc, Pag 6
Calculatd dc rsistanc: 6.386 ohms/1000 ft x 3.41 m x 64 / 0.3048 = 4.5751 ohms at 0 C Calculatd distributd capacitanc, C d, basd on slf-rsonanc: 100 pf Pak voltag at rsonanc: 0.4118 V, 6 dbv rduction: 0.4118 V x 0.707 = 0.911 V Frquncis at 6 dbv rduction ( 3 db powr): 38 173 and 55 53 Hz, Frquncy chang = 3 db powr bandwidth: 17 350 Hz Th loop originally was built in lat 009 using a 100 ft spool of 18 AWG coatd magnt wir rsulting in 96 ±1 turns. Th avrag individual winding lngth was approximatly 11.5 ft (3.43 m). Aftr construction I mad a sris of inductanc and slf-rsonant frquncy masurmnts as follows: Instrumnt: TH811D, Tmpratur: 0.6 C Tst Frquncy (Hz) Inductanc (μh) Q R (ohms) 100 7.16.494 6.8364 10 7.117.9865 6.8438 1000 7.116 3.9 7.14 10000 30.581 70 7.6 Calculatd 7.65 N/A 6.867 I masurd th slf-rsonant frquncy at 8.4 khz but this was much too low for applications as a tund loop for VLF and LF applications. I thn rmovd on turn at a tim and masurd th inductanc and slf-rsonant frquncy aftr ach turn was rmovd. I rmovd 3 turns to achiv th dsird slf-rsonant frquncy of 45.6 khz. S chart blow. 50 45 1. m Squar Loop Masurmnts ~ Turns Rmoval Inductanc (μh) or Rsonant Frquncy (khz) 40 35 30 5 0 15 10 y = 0.4506x + 31.31 Rsonant Frquncy Inductanc Linar (Rsonant Frquncy) Linar (Inductanc) y = -0.4638x + 6.348 5 0 0 5 10 15 0 5 30 35 Numbr of Turns Rmovd Fil: Rv_SquarLoopAntnna1.m.doc, Pag 7
4. Opn circuit voltag From Faraday s law of induction V d( t) Eq. (4.1) dt V = opn circuit rms voltag (V) φ(t) = magntic flux (wbr = v s ) t = tim (s) Thrfor, an inducd voltag appars across th trminals of a circuit immrsd in a changing magntic fild. If th circuit consists of an lctrically small air cor loop antnna with n turns, th voltags in th turns ar additiv. Not: An lctrically small loop antnna has circumfrnc much lss than a wavlngth. For any frquncy < 300 khz on wavlngth in fr spac is > 1,000 m, and th circumfrnc of any practical loop antnna is much smallr. d( t) V n Eq. (4.) dt Th magntic flux is rlatd to th tim varying magntic induction by ( t) B t A cos( ) B cos( t) A cos( ) Eq. (4.3) B(t) = magntic induction (tsla, T = v s / m ) B = rms magntic induction (T) A = Ara of quivalnt circular loop with radius a, ω = radian frquncy ( f, radians/s) f θ = frquncy (Hz) = angl btwn magntic fild lins and normal of loop fram (radians) a = Avrag of inscribd and circumscribd radii, r cos (m) N Eq. (4.4) r = radius of circumscribd circl, d/ (m) N = numbr of sids (4 for an squar) Th radius of a circumscribd circl for a squar is 1/ of th squar s diagonal dimnsion. For th squar loop in qustion, radius r 1.07 0.6035 m, and from q. (4.4) th avrag of inscribd and circumscribd radii is a (m) a 4 0.6035 cos 0.5151 m Fil: Rv_SquarLoopAntnna1.m.doc, Pag 8
Thrfor, for th squar loop in qustion, th quivalnt ara A a 0.5151 0.8336 m Diffrntiating Eq. (4.3) givs d( t) db( t) d cos( t) A cos( ) B A cos( ) B A cos sin t Eq. (4.5) dt dt dt Substituting Eq. (4.5) in Eq. (4.1), th opn circuit voltag across th loop trminal is cos sin V t f n A B t and th opn circuit rms voltag is V f n A Bcos Eq. (4.6) Th abov xprssion indicats th loop antnna rsponds to th magntic fild componnt (magntic induction or flux dnsity, B) of a signal and convrts it to a voltag at th antnna trminals. Th voltag at th trminals is rlatd to th lctric fild strngth, E, by V h E Eq. (4.7) h E = ffctiv antnna hight (m) = rms lctric fild strngth (V/m) Th rlationship btwn th lctric fild strngth and magntic induction is E c B Eq. (4.8) c 8 = spd of light in fr spac ( 3 10 m/s) Substituting Eq. (4.6), (4.7) and (4.8), th ffctiv hight of an air-cor loop is f n A cos n A cos h Eq. (4.9) c λ = wavlngth (m) It is sn from q. (4.7) that, for a givn lctric fild strngth, th rms voltag at th loop trminals is proportional to th ffctiv hight. Equivalntly, from q. (4.9) it is proportional to th frquncy, numbr of Fil: Rv_SquarLoopAntnna1.m.doc, Pag 9
turns and loop quivalnt ara. Not that th ffctiv hight is not rlatd to th physical hight of th loop it rlats th fild strngth to opn circuit voltag. For a givn frquncy th ffctiv hight can b incrasd by incrasing th numbr of turns or loop ara. For th cas th loop fram is paralll to th propagation dirction and th magntic fild is normal to th propagation dirction, in which cas θ = 0 dg. = 0 radians, q. (4.9) bcoms n A f h Eq. (4.10) c If th frquncy is 4 khz, th ffctiv hight of th xampl antnna is 3 n A f 640.8336 4 10 h 0.068 8 c 310 m If th fild strngth is 500 v/m at 4 khz, th opn circuit (unloadd) loop trminal voltag for th xampl loop is V h E 0.068 500 13.4 V In air and fr spac, th magntic fild strngth and magntic induction ar rlatd by B H Eq. (4.11) 0 H 0 = rms magntic fild strngth (a/m) = prmability of air or fr spac (4π x 10-7 hnry/m) Thrfor, whn th loop fram is paralll to th lin of signal propagation, th rms loop trminal voltag is maximum and in trm of th magntic fild strngth is givn by V n A f H Eq. (4.1) max 0 Fil: Rv_SquarLoopAntnna1.m.doc, Pag 10
5. Construction imags Th loop was constructd btwn 8 Novmbr and 6 Dcmbr 009. 1 squar fibrglass tub was usd for th loop fram or cross-bracs. Th loop dimnsions wr dtrmind by th lngth of th 8 ft tub I had on-hand; it was simply cut in half. Th bracs wr cut on a tabl saw with an abrasiv blad usd for cutting mtal. Th cntr of ach brac was thn notchd on th saw so thy fit togthr at th cntr. Th nds of ach brac wr cut at 45 angl on th tabl saw and lightly sandd. Th notchd nds hold th wir in plac. Fil: Rv_SquarLoopAntnna1.m.doc, Pag 11
Th two bracs fit togthr at thir cntrs. Rinforcmnts wr cut from 1/4 marin plywood with a 6 hol saw on a drill prss and thn cut into pics on a tabl saw. Aftr sanding th rinforcmnts, thy wr clampd to th bracs to nsur alignmnt and drilld with a 3/16 in brad point drill. Th fastnrs ar 18-8 stainlss stl 10-3 machin scrws, washrs and nuts. Aftr this imag was takn, th wood pics wr rmovd and paintd with urthan spar varnish. Fil: Rv_SquarLoopAntnna1.m.doc, Pag 1
Aftr parts wr prmanntly assmbld, th loop fram was clampd to a wood plat with a mandrl on thn placd on a shop tabl for winding. Th spool of magnt wir was placd on anothr mandrl clampd to an adjacnt tabl. Th starting nd of th 18 AWG coatd magnt wir was clampd to th fram and thn th fram was slowly rotatd. Compltd loop antnna with 96 turns and about 1100 ft of wir. 3 turns wr latr rmovd as discussd in th masurmnts sction laving about 70 ft on compltd antnna. Fil: Rv_SquarLoopAntnna1.m.doc, Pag 13
Viw of th wir trminations on th loop fram. Th wir is protctd by insulating slvs and tap clampd to th fram. Th trminal block was latr installd in a plastic nclosur. Th loop was hung from th shop ciling for masurmnts. This viw shows th TH811D LCR mtr usd for th inductanc and rsistanc masurmnts in 009. For slf-rsonant frquncy masurmnts in 009, a Tlulx SG-100A signal gnrator and Tktronix TDS 0B oscilloscop was usd. Th signal gnrator 50 ohm output was isolatd from th loop with 10 kohm and 500 kohm rsistors. Masurmnts wr duplicatd in 018 using this stup (but with a diffrnt oscilloscop). Fil: Rv_SquarLoopAntnna1.m.doc, Pag 14
Author - Whitham Rv is a contributing ditor for th SARA journal, Radio Astronomy. H obtaind B.S. and M.S. dgrs in Elctrical Enginring at Univrsity of Alaska Fairbanks, USA. H workd as a profssional nginr and nginring firm ownr/oprator in th airlin and tlcommunications industris for mor than 40 yars and now manufacturs lctronic quipmnt usd in radio astronomy. H has livd in Anchorag, Alaska his ntir lif. Email contact: whitrv@gmail.com Fil: Rv_SquarLoopAntnna1.m.doc, Pag 15
Author: Whitham D. Rv, Anchorag, Alaska USA Copyright: 018 W. Rv Rvision: 0.0 (Initial draft basd on UKRAA loop writ-up, 11 Jun 018) 0.1 (Substitutd actual valus for calculations, 06 Aug 018) 0. (Addd masurmnts, 11 Aug 018) 0.3 (Updatd calculations, 15 Aug 018) 0.4 (Compltd 1 st draft for distribution, 0 Aug 018) 0.5 (Addd original construction imags and data, 1 Aug 018) 0.6 (Addd opn circuit voltag calculations, 3 Aug 018) 0.7 (Addd U1733C on-sit masurmnts, 04 Sp 018) 0.8 (Minor updats to construction sction, 07 Sp 018) 0.9 (Addd wir lngth, 1 Sp 018) Word count: 355 Fil siz (byts): 81150 Fil: Rv_SquarLoopAntnna1.m.doc, Pag 16