ELECTROMAGNETIC COMPATIBILITY HANDBOOK 1. Chapter 30: Antennas

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1 ELECTROMAGNETIC COMATIBILITY HANDBOOK 1 Chaptr 30: Antnnas 30.1 Can a lump-circuit mol b us to rprsnt an lctrically-larg ipol? If ys, inicat whn. If no, why is it rprsnt by a raiation rsistanc an ractanc? 30.2 Abov what frquncy shoul twin-la transmission lins not normally b us? Stat all assumptions Driv th quation givn in this chaptr for th ohmic rsistanc of an lctrically-short ipol of lngth 2l th if th currnt istribution is uniform along its lngth: ( ) I z Io z < l = 0 z > lth Assum that th ac rsistanc pr unit lngth of th conuctors is r AC Driv th quation givn in this chaptr for th ohmic rsistanc of a ipol of lngth 2l th if th currnt istribution is scrib by ( k 1) Io z + 1 z < a a kio I ( z) = ( lth z ) a < z < l lth a 0 z > lth Assum that th ac rsistanc pr unit lngth of th conuctors is r AC Vrify that th total ohmic rsistanc is l th r AC rathr than 2l th r AC for a larg ipol with a sinusoial currnt istribution 2π I z I l z λ ( ) = o sin ( th ) whr 2l th = λ/2. Assum that th ac rsistanc pr unit lngth of th conuctors is r AC A monopol antnna us for a rmot-controll vic oprating at about 75 MHz was sign to b about 2 ft in lngth. Unfortunatly, bcaus of siz rquirmnts, th antnna must b ruc to 4 inchs. (Assum th antnna is construct of #22 AWG coppr wir.) Dtrmin th input impanc of th monopol antnna for both th 2 ft an 4 inch lngths. If th iffrnc in ths impancs is too ramatic for th transmittr, connct to th antnna, propos a mtho of rucing this impanc chang whil still rtaining th smallr siz rquirmnts. th th

2 2 ELECTROMAGNETIC COMATIBILITY HANDBOOK 30.7 A half-wav ipol oprating at a fr-spac wavlngth of 160 m is construct of #10 AWG annal coppr wir. Is th skin pth important? For a sinusoial currnt istribution on th antnna, trmin th total ohmic powr loss for th antnna if th maximum currnt amplitu along th wir is 5 A. On quartr of ach si of th wir nar th f lin is thn coat with silvr. Dtrmin th nw total powr loss for th antnna. (Slct a rasonabl silvr coating thicknss.) Why is th silvr appli nar th f lin insta of at th ns of th antnna? 30.8 It is stat that th irctivity an input impanc of a (5/8)λ vrtical monopol is gratr than a λ/4 vrtical monopol. Dtrmin whthr this is tru. Thn, compar th location of th currnt maximum for a ths two antnna. Dtrmin th total ohmic rsistanc for ach of ths antnnas assuming th ac rsistanc pr unit lngth is qual to r AC. If th groun is acting as th imag for th monopol antnna (i.., th othr si of th ipol), which antnna will hav th gratr groun losss? Dtrmin an compar th raiation rsistancs of th two antnnas. Which antnna is mor fficint? Assum that th currnt istribution on th antnna is givn by 2π I z I l z λ ( ) = o sin ( th ) Oftn raial wirs ar us to incras th ffctivnss of th groun plan for ral antnna. Th lngth of ths raials shoul b gratr for th (5/8)λ vrtical antnna than th λ/4 monopol. Why os this sm rasonabl? 30.9ES For a pic wir of spcific lngth that is lctrically short, what wir shap will provi th gratst raiation rsistanc? 30.10C Th raiation rsistanc of a half-wav ipol antnna construct of #16 AWG coppr wir is 73 + j43 Ω. A 100 ft 75 Ω coax conncts a 21 MHz transmittr (with a sourc impanc of 50 Ω) with an opn-circuit sourc voltag of 40 V to this ipol antnna. Dtrmin th powr raiat by th antnna. Dtrmin th fficincy of th antnna Th raiation rsistanc of top-loa short ipol is gratr than a nontoploa short ipol (of th sam lngth). Explain why this is tru. lacing an inuctor in sris with th antnna can also incras th raiation rsistanc. Explain why this is tru. lacing th inuctor miway along th antnna is mor fficint than nar th sourc. Explain why this is tru. [Collin, 85; Watt] Why is th capacitanc of an lctrically-short loop small? 30.13C Compar th ractanc of a λ/10 lngth straight-wir antnna using th short antnna approximation an th larg antnna quation. Assum thr iffrnt (but rasonabl) raii for th wir Why is th raiation fficincy ifficult to masur for short antnna? ropos a mtho of masuring th fficincy It is stat that th raiation rsistanc is much lss than th ohmic rsistanc for lctrically-short loop antnna. Unr what conitions is this statmnt tru?

3 ELECTROMAGNETIC COMATIBILITY HANDBOOK C For larg ipol antnna, plot th raiation rsistanc vrsus a/λ at rsonanc (whr a is th raius of th conuctor). Why is this information usful? 30.17C For larg ipol antnna, plot a/λ vrsus 2l th /λ, whr l th is th lngth corrsponing to rsonanc an a is th raius of th conuctor. Why is this information usful? On mtho to fin-tun a whip antnna is to masur th VSWR bfor an aftr touching a mtallic objct (appropriatly insulat an sparat from th iniviual prforming th tst) to th n of th antnna. If th VSWR incrass, th antnna is too long. If th VSWR crass, th antnna is too short. Explain why this rlationship is vali. Woul th siz of th mtallic objct, for a comparabl chang in VSWR, b smallr or largr as th frquncy of opration crass? 30.19C Dos th half-wav ipol hav th gratst lctric fil (magnitu) if th total lngth is limit to lss than or qual to λ? lot th fil vrsus th angl (θ) for total ipol lngths varying from λ/10 to 2λ in λ/10 incrmnts A strip-lin. suscptibility tst systm consists of a narrow strip abov a larg groun plan that is f at on n with a broaban transmittr an rsistivimpanc match at th othr n. Th lin is 1.5 m in lngth, an th istanc btwn th strip an th groun plan is 0.2 m. Th systm is us from 10 khz to 200 MHz to coupl strong fils into harnsss plac btwn th strip lin an groun plan. Th powr loss, th ratio of th loa powr to input powr, is narly 20 B at 200 MHz. If th ohmic rsistanc is rlativly small, why is th powr loss so larg at th highr frquncis? In th far fil in th cylinrical coorinat systm, trmin th quation for th irctiv gain. Stat all assumptions C lot th VSWR of a half-wav ipol (construct of #18 AWG coppr wir) that is rsonant at 28 MHz as th frquncy varis from 26 to 30 MHz. Dtrmin th banwith an Q of this antnna. Assum that th impanc of th transmission lin connct to th antnna is 75 Ω C Dtrmin th banwith of a 27 MHz half-wav ipol bas on its raiation pattrn variation. Stat all assumptions Show that th VSWR can b writtn in trms of th Q of an antnna: 1+ VSWR = 1 Qn ( Qn) ( Qn) Qn whr n is th prcnt iffrnc, xprss as a cimal (.g., 0.05 corrsponing to 5%) btwn th antnna s rsonant frquncy an frquncy whr th Q is masur. Assum that th antnna is match to th transmission lin at its cntr rsonant frquncy. lot th VSWR vrsus Q (ranging from 0.1 to 10) for n = 0.01, 0.02, 0.03, 0.04, an Us a logarithmic scal for th Q axis.

4 4 ELECTROMAGNETIC COMATIBILITY HANDBOOK Dtrmin th banwith an Q of an lctrically-small loop antnna with a cntr frquncy of 144 MHz. Assum that th loop is construct of coppr of a rasonabl gaug C Show that th irctivity of a half-wav ipol is 1.64 bginning with th basic finition provi in this chaptr: G = avg 30.27C Show that th irctivity of a full-wav ipol is 2.41 bginning with th basic finition provi in this chaptr: G = avg 30.28C Numrically vrify that for a ipol, silobs (submaximum lobs) bgin to appar for lngths gratr than on wavlngth Bginning with th basic finition for irctivity, trmin th irctivity of an antnna with th givn far fil: 2 jkr ka 2π Eφ s = ηohθ s, Hθ s = Io sinθ whr k = 2 r β Consir th four-port, linar, tim-invariant, zro-stat, passiv ntwork givn in Figur 1. With th two iffrnt currnt sourcs connct to two of th ports, th opn-circuit voltags across th rmaining two ports ar as shown. + 8 V a b c 2 A 4 A + 6 V Figur 1 Now, consir th nw xcitations shown in Figur 2. Only on opn-circuit voltag is masur. Using th Law of Rciprocity, trmin th unknown opn-circuit voltag V x.

5 ELECTROMAGNETIC COMATIBILITY HANDBOOK 5 7 A 8 V + a b c + V x 9 A Figur Hoping to moify asily th raiation pattrn of a singl ipol antnna, a stunt placs a variabl LC ntwork at th fpoint of th antnna. Will ajusting th L an C valus chang th raiation pattrn? Explain A st of antnnas is to transmit an rciv ovr th frquncy rang 30 khz to 300 MHz. Spcify a st of antnnas that will mt this critria Is a biconical antnna balanc or unbalanc? Explain S rovi on common non-emc us for ach of th broaban antnnas list S It is stat that aing two rigs to a stanar horn antnna incrass its banwith by rucing th cutoff frquncy of th ominant mo whil incrasing th cutoff frquncy of th nxt highr mo. Dtrmin th valiity of this statmnt As th gain of an antnna incrass, th main bam s with crass (th classical gain-banwith prouct). Bcaus of this, th istanc btwn th antnna an th tst sourc, r, for EMC masurmnts shoul follow th guilin r Dλ π to avoi xcssiv coupling, whr D is th irctivity of th antnna. Driv this quation an trmin its limitations. [Bronaugh; Johnson, 61]] Dtrmin th irction an th polarization (linar, circular, or lliptical) of ach of th following wavs:

6 6 ELECTROMAGNETIC COMATIBILITY HANDBOOK E = 3cos( ωt β z) aˆ 3sin ( ) ˆ x + ωt β z ay mv/m E = 3cos( ωt β y) aˆ zv/m E = 3cos( ωt β y) aˆ 2 sin ( ) ˆ x + ωt β y az mv/m E = 2 cos t x + 35 a + 3cos t x aˆ V/m ( ω β ) ˆz ( ω β ) Sktch th fils vrsus tim It is stat that for a quartr-wav long transmission lin, th magnitu of th currnt in th loa is qual to th voltag at th sourc (voltag at th input of th lin) ivi by th charactristic impanc of th lin; that is, th loa currnt (currnt at th output of th lin) is inpnnt of th loa impanc! Dtrmin whn this is tru. Why is a quartr-wav lin somtims us in a high-nsity antnna nvironmnt? [Johnson, 61] It is stat that for a half-wav long transmission lin, th magnitu of th voltag at th loa is qual to th magnitu of th voltag at th sourc (voltag at th input of th lin); that is, th loa voltag (voltag at th output of th lin) is inpnnt of th loa impanc! Dtrmin whn this is tru. Why is a half-wav lin somtims us in a high-nsity antnna nvironmnt? If th voltag at th input has a nonzro rsistanc, is this statmnt vali? [Johnson, 61] Why ar transint tsts mor maning on an antnna than swp frquncy tsts? Th currnt at th input of a rciving loop antnna that is not lctrically small is masur. This currnt is small. Dos this imply that th currnt along th loop is ngligibl an hnc th inuc currnt is small? [Wks, 64] Starting from th basic finition L 1 = I o antnna ( ) I z z show that th ffctiv lngth of a thin half-wav ipol with a sinusoial currnt istribution is λ/π E For an H-prob, trmin whthr placing a capacitor across th antnna las will incras th oprating frquncy rang of th prob A propos RF currnt prob consists of an insulat wir in th shap of a squar-figur ight as shown in Figur 3. To masur th currnt, I, in a wir, th prob is plac nar th wir as shown. It is stat that this prob is lss likly to pick up istant stray magntic fils than a simpl circular-shap prob. Dtrmin whthr thr is any valiity to this statmnt. Hint: carfully apply Lnz s law to ach loop. y

7 ELECTROMAGNETIC COMATIBILITY HANDBOOK 7 I + V OC Figur C For small valus of l th, approximat th following quivalnt lngth formula for sinusoial currnt istribution along a thin ipol of total lngth 2l th : L λ π lth = tan π λ lot th prcnt rror associat with th approximation for l th /λ valus ranging from 0.1 to C For a sinusoial currnt istribution along a thin ipol of total lngth lss than or qual to λ/2, show that th ipol s ffctiv lngth is L λ 2π lth = 1 cos π λ Thn, plot th ffctiv lngth vrsus l th using this quation an, on th sam st of axs, plot th ipol s ffctiv lngth that is bas on a mor sophisticat currnt istribution: L λ π lth = tan π λ Dtrmin th maximum lngth l th so that th rror btwn th rsults is lss than 10% Dtrmin th ffctiv lngth of a small loop antnna with N turns ach of raius a Compar th opn-circuit voltag across an E-fil prob an an H-fil prob if th lngth of th E-fil prob is qual to th circumfrnc of th loop of th H- fil prob. Which voltag is largr an unr what conitions? Assum that

8 8 ELECTROMAGNETIC COMATIBILITY HANDBOOK th probs ar masuring th fils from a plan wav. What othr practical factors shoul b consir whn comparing th two probs? Dtrmin th rlationship btwn th ffctiv lngth an ffctiv ara for a rcivr match to its antnna C lot th inuctanc vrsus numbr of turns for a singl-layr inuctor construct of 10 cm of #28 AWG aroun an air cor. Dtrmin whr th maximum inuctanc occurs. Rpat for a 5 cm lngth of wir. Assum th turns of wir ar tightly woun (but not touching) C Anothr quation us to stimat th inuctanc of a singl-layr or multilayr air-cor coil is L = 0.885kDN whr k is th inuctanc factor, N is th numbr of turns, an D is th avrag iamtr of th coil. Th inuctanc factor is a function of th ratio D to th coil lngth, l th. Th approximat valu for th inuctanc factor for svral iffrnt ratios for singl-layr an multilayr coils is givn in Tabl 1. Compar this quation for th inuctanc of singl an multilayr coils to that provi in this chaptr. [Watt] Tabl 1 D/l th k (singl layr) k (multilayr) It is stat that th maximum Q for a singl-layr air-cor inuctor occurs whn th coil lngth is qual to th coil iamtr (vn though this may not always b practical). Dtrmin th valiity of this statmnt Sktch th magntic flux nsity both insi an outsi of a currnt-carrying air-cor coil of lngth l th, th sam coil wrapp on a frrit ro of lngth l th, an th sam coil wrapp on a frrit ro of lngth 5l th. Thn, sktch th flux nsity both insi an outsi an opn-circuit air-cor coil, th sam coil wrapp on a frrit ro of th lngth l th, an th sam coil wrapp on a frrit ro of lngth 5l th whn th coil is normally incint to a uniform magntic fil. Compar th fil nsity insi th coil for all of ths cass. Is th ffctiv prmability qual to th apparnt prmability?

9 ELECTROMAGNETIC COMATIBILITY HANDBOOK Is th inuctanc of a on-loop inuctor always lss than a two-loop vrsion (closly woun) construct of th sam wir lngth, gaug, an conuctivity? 30.55E Show that if a loop is loa with an impanc that is lss than its slf ractanc, th loop s frquncy pnnc is ruc at a cost of ruc snsitivity. [Bronaugh] For irction fining systms, woul th loop or th ro antnna b mor usful? For short-rang communications (but not irct lin of sight), provi two isavantags of a vrtically orint ro antnna EC To sniff-out whr RF raiation is laking from a TV, a prob loop consisting of two thr-inch iamtr turns of insulat #20 wir is connct to th n of a pic of 50 Ω coaxial cabl. Th coax is thn connct to th 50 Ω input of an oscilloscop. Discuss th ffctivnss of this prob. What is th VSWR on th cabl? [Har] Qualitativly xplain how th voltag along a prfctly conucting antnna lmnt can vary Driv th gnral Friis quation rc in 2 ( ρ )( 1 ρ ) λo = G G 4π r t r t r whr ρ t is th rflction cofficint at th transmitting antnna an ρ r is th rflction cofficint at th rciving antnna. Show that th powr rflction cofficints ρ ρ * Zantt Z ot Zantt Z ot = = * Zantt + Zot Zantt + Zot 2 * t ρtρt * Zlr Z antr Zlr Z antr = = * Zlr + Zantr Zlr + Zantr 2 * r ρrρr ar both on whn th antnnas ar impanc match.