Nikolova John H. Dunlavy Jr., US Patent 13,588,905: Wide range tunable transmitting loop antenna, 1967.

Transcription

1 LECTURE 1: Loop Antennas (Radiation paametes of a small loop. Cicula loop of constant cuent. Equivalent cicuit of the loop antenna. The small loop as a eceiving antenna. Feite loops.) Equation Section 1 1. Intoduction Loop antennas featue simplicity, low cost and vesatility. They may have vaious shapes: cicula, tiangula, squae, elliptical, etc. They ae widely used in communication links up to the micowave bands (up to 3 GHz). They ae also used as electomagnetic (EM) field pobes in the micowave bands. Electically small loops ae widely used as compact tansmitting and eceiving antennas in the low MHz ange (3 MHz to 30 MHz, o wavelengths of about 10 m to 100 m). Loop antennas ae usually classified as electically small ( C < λ /3) and electically lage (C λ ). Hee, C denotes the loop s cicumfeence. The small loops of a single tun have small adiation esistance (< 1 Ω) usually compaable to thei loss esistance. Thei adiation esistance, howeve, can be impoved by adding moe tuns. Also, the small loops ae naowband. Typical bandwidths ae less than 1%. Howeve, cleve impedance matching can povide low-eflection tansition fom a coaxial cable to a loop antenna with a tuning fequency ange as high as 1:10. 1 Moeove, in the HF and VHF bands whee the loop diametes ae on the ode of a half a mete to seveal metes, the loop can be made of lage-diamete tubing o coaxial cable, o wide coppe tape, which can dastically educe the loss. Fig. 1: Altenative constuctions fo shielded Faaday loop. [fom Leigh Tune VK5KLT, An oveview of the undeestimated magnetic loop HF antenna, coutesy of D. James R. La Fieda (N6MV)] 1 John H. Dunlavy J., US Patent 13,588,905: Wide ange tunable tansmitting loop antenna, Nikolova 016 1

2 The small loops, egadless of thei shape, have a fa-field patten vey simila to that of a small electic dipole nomal to the plane of the loop. This is expected because they ae equivalent to a magnetic dipole. Note, howeve, that the field polaization is othogonal to that of the electic dipole. As the cicumfeence of the loop inceases, the patten maximum shifts towads the loop s nomal, and when C λ, the maximum of the patten is along the loop s nomal.. Radiation Chaacteistics of a Small Loop A small loop is by definition a loop of constant cuent. Its adius satisfies λ a <, (1.1) 6π o, equivalently, C < λ /3. The limit (1.1) is mathematically deived late in this Lectue fom the fist-ode appoximation of the Bessel function of the fist ode J1( x ) in the geneal solution fo a loop of constant cuent. Actually, to make sue that the cuent has nea-constant distibution along the loop, a tighte limit must be imposed: a < 0.03λ, (1.) o, C < λ /5. A good appoximate model of a small loop is povided by the infinitesimal loop (o the infinitesimal magnetic dipole). The expessions fo the field components of an infinitesimal loop of electic cuent of aea A wee aleady deived in Lectue 3. Hee, we give only the fafield components of the loop, the axis of which is along z: e jβ Eϕ = ηβ ( IA) sinθ, (1.3) 4π e jβ Hθ = β ( IA) sinθ. (1.4) 4π It is obvious that the fa-field patten, E ϕ ( θ) = sinθ, (1.5) is identical to that of a z-diected infinitesimal electic dipole although the polaization is othogonal. The powe patten is identical to that of the infinitesimal electic dipole: Nikolova 016

3 Radiated powe: F( θ) = sin θ. (1.6) Radiation esistance: In fee space, η = 10π Ω, and 1 ϕ sin Π= E θdθdϕ, η 1 ηβ 1π ds ( IA) Π= 4. (1.7) 8 3 A = R η π 3 R λ. (1.8). (1.9) 31171( A/ λ ) Equation (1.9) gives the adiation esistance of a single loop. If the loop antenna has N tuns, then the adiation esistance inceases with a facto of N (because the adiated powe inceases as I ): R 8 3 A = η π N 3 λ. (1.10) The elation in (1.10) povides a handy mechanism to incease R and the adiated powe Π. Unfotunately, the losses of the loop antenna also incease (although only as N ) and this may esult in low efficiency. The diectivity is the same as that of an infinitesimal dipole: Umax D0 = 4π = 1.5. (1.11) Π ad 3. Cicula Loop of Constant Cuent Geneal Solution So fa, we have assumed that the loop is of infinitesimal adius a, which allows the use of the expessions fo the infinitesimal magnetic dipole. Now, we deive the fa field of a cicula loop, which might not be necessaily vey small, but still has constant cuent distibution. This deivation illustates the geneal Nikolova 016 3

4 loop-antenna analysis as the appoach is used in the solutions to cicula loop poblems of nonunifom distibutions, too. The cicula loop can be divided into an infinite numbe of infinitesimal cuent elements. With efeence to the figue below, the position of a cuent element in the xy plane is chaacteized by 0 ϕ < 360 and θ = 90. The position of the obsevation point P is defined by ( θϕ, ). The fa-field appoximations ae R acos ψ, fo the phase tem, 1 1, fo the amplitude tem. R (1.1) In geneal, the solution fo A does not depend on ϕ because of the cylindical symmety of the poblem. Hee, we set ϕ = 0. The angle between the position vecto of the souce point Q and that of the obsevation point P is detemined as cos ψ = ˆ ˆ = ( xˆsin θcosϕ + yˆsin θsinϕ + zˆcos θ) ( xˆcos ϕ + y ˆsin ϕ ), z P x a θ ψ ϕ ϕ Q I 0 R y cosψ = sinθcosϕ. (1.13) Now the vecto potential integal can be solved fo the fa zone: µ e A(, θϕ, ) = I d 4π jβ( asinθcos ϕ ) 0 l (1.14) C Nikolova 016 4

5 whee dl = φ ˆ adϕ is the linea element of the loop contou. The cuent element changes its diection along the loop and its contibution depends on the angle between its diection and the espective A component. Since all cuent elements ae diected along ˆφ, we conclude that the vecto potential has only A ϕ component, i.e., A = φ, ˆ whee A ϕ j π µ e β A(,, ) ˆ (,, ) ( Ia sin cos 0 ) ( ˆ ˆ ) ejβa θ ϕ ϕ = = d Since θϕ φa θϕ φφ ϕ. (1.15) 4π the vecto potential is φˆ φˆ = ( xˆcos ϕ + yˆsin ϕ) ( xˆcos ϕ + yˆsin ϕ ) = = cosϕcosϕ + sinϕsinϕ = = cos( ϕ ϕ ) = cos ϕ, ϕ = 0 j π µ e β A(,0) ( Ia sin cos 0 ) cos ejβa θ ϕ ϕ = d Nikolova (1.16) θ ϕ ϕ 4π, (1.17) j π π µ e β A ( ) ( I sin cos sin cos 0a) cos ejβa θ ϕ d cos ejβa θ ϕ ϕ θ = ϕ ϕ + ϕ dϕ 4π 0 π. We apply the following substitution in the second integal: ϕ = ϕ + π. Then, j π π µ I0ae β A ( ) cos ejβasinθcosϕ d cos e jβasinθcosϕ ϕ θ = ϕ ϕ ϕ dϕ 4π 0 0. (1.18) The integals in (1.18) can be expessed in tems of Bessel functions, which ae defined as π cos( n ϕ ) ejz cosϕd ϕ = π jnj n ( z). (1.19) 0 Hee, Jn( z ) is the Bessel function of the fist kind of ode n. Fom (1.18) and (1.19), it follows that µ e jβ Aϕ ( θ) = ( Ia 0 ) π j J1( βasinθ) J1( βasinθ) 4π. (1.0)

6 Since equation (1.0) educes to J ( z) = ( 1) n J ( z), (1.1) n n µ e jβ Aϕ ( θ) = j ( Ia 0 ) J1( βasin θ). (1.) J1(x) x The fa-zone fields ae deived as e jβ Eϕ ( θ) = βη( Ia 0 ) J1( βasin θ), (1.3) E e jβ ϕ Hθ ( θ) = = β( Ia 0 ) J1( βasin θ). η The pattens of constant-cuent loops obtained fom (1.3) ae shown below: Nikolova 016 6

7 [Balanis] The small-loop field solution in (1.3)-(1.4) is actually a fist-ode appoximation of the solution in (1.3). This becomes obvious when the Bessel function is expanded in seies as 1 1 J1( βasin θ) ( βasin θ) ( βasin θ) 16 = 3 +. (1.4) Nikolova 016 7

8 Fo small values of the agument ( β a < 1/3), the fist-ode appoximation is acceptable, i.e., 1 J1( βasin θ) ( βasin θ). (1.5) The substitution of (1.5) in (1.3) yields (1.3)-(1.4). It can be shown that the maximum of the patten given by (1.3) is in the diection θ = 90 fo all loops, which have cicumfeence C < 1.84λ. Radiated powe and adiation esistance We substitute the E ϕ expession (1.3) in 1 Π= Eϕ sin θdθdϕ, η which yields ( ) π ωµ ( ) ( sin )sin 0 1 4η 0 ds Π= I A J βa θ θdθ. (1.6) Hee, A= π a is the loop s aea. The integal in (1.6) does not have a closed fom solution. Often, the following tansfomation is applied: π β a 1 J1 ( βasin θ)sin θdθ = J( x) dx βa. (1.7) 0 0 The second integal in (1.7) does not have a closed fom solution eithe but it can be appoximated with a highly convegent seies: β a J ( x) dx = J ( βa). (1.8) 0 m+ 3 m= 0 The adiation esistance is obtained as ( ) π ωµ 1 Π R = = A J ( βasin θ)sinθdθ I η. (1.9) 0 0 The adiation esistance of small loops is vey small. Fo example, fo λ /100 < a < λ / 30 the adiation esistance vaies fom Ω to 0.5 Ω. Nikolova 016 8

9 This is often less than the loss esistance of the loop. That is why small loop antennas ae constucted with multiple tuns and on feomagnetic coes. Such loop antennas have lage inductive eactance, which is compensated by a capacito. This is convenient in naowband eceives, whee the antenna itself is a vey efficient filte (togethe with the tuning capacito), which can be tuned fo diffeent fequency bands. Low-loss capacitos must be used to pevent futhe incease in the loss. 4. Cicula Loop of Nonunifom Cuent When the loop adius becomes lage than 0.λ, the constant-cuent assumption does not hold. A common assumption is the cosine distibution.,3 Lindsay, J., 4 consides the cicula loop to be a defomation of a shoted paallelwie line. If I s is the cuent magnitude at the shoted end, i.e., the point opposite to the feed point whee ϕ = π, then I( α) = I cosh( γaα) (1.30) s whee α = π ϕ is the angle with espect to the shoted end, γ is the line popagation constant and a is the loop adius. If we assume loss-fee tansmission-line model, then γ = jβ and cosh( γaα) = cos( βaα). Fo a loop in open space, β is assumed to be the fee-space wave numbe ( β= ω µε 0 0 ). The cosine distibution is not vey accuate, especially close to the teminals, and this has a negative impact on the accuacy of the computed input impedance. That is why the cuent is often epesented by a Fouie seies: 5,6 I( ϕ ) = I + I cos( nϕ ). (1.31) 0 N n= 1 Hee, ϕ is measued fom the feed point. This way, the deivative of the cuent distibution with espect to ϕ at ϕ = π (the point diametically opposite to the feed point) is always zeo. This imposes the equiement fo a symmetical cuent distibution on both sides of the diamete fom ϕ = 0 to ϕ = π. The n E.A. Wolff, Antenna Analysis, Wiley, New Yok, A. Richtscheid, Calculation of the adiation esistance of loop antennas with sinusoidal cuent distibution, IEEE Tans. Antennas Popagat., Nov. 1976, pp J. E. Lindsay, J., A cicula loop antenna with non-unifom cuent distibution, IRE Tans. Antennas Popagat., vol. AP-8, No. 4, July 1960, pp H. C. Pocklington, Electical oscillations in wie, in Cambidge Phil. Soc. Poc., vol. 9, 1897, pp J. E. Stoe, Input impedance of cicula loop antennas, Am. Inst. Elect. Eng. Tans., vol. 75, Nov Nikolova 016 9

10 complete analysis of this geneal case will be left out, and only some impotant esults will be given. When the cicumfeence of the loop appoaches λ, the maximum of the adiation patten shifts exactly along the loop s nomal. Then, the input esistance of the antenna is also good (about 50 to 70 Ω). The maximum diectivity occus when C 1.4λ but then the input impedance is too lage. The input esistance and eactance of the lage cicula loop ae given below. Nikolova

11 (Note: typo in autho s name, ead as J. E. Stoe) Nikolova

12 The lage cicula loop is vey simila in its pefomance to the lage squae loop. An appoximate solution of vey good accuacy fo the squae-loop antenna can be found in W.L. Stutzman and G.A. Thiele, Antenna Theoy and Design, nd Ed., John Wiley & Sons, New Yok, Thee, it is assumed that the total antenna loop is exactly one wavelength and has a cosine cuent distibution along the loop s wie. y x λ 4 The pincipal plane pattens obtained though the cosine-cuent assumption (solid line) and using numeical methods (dash line) ae shown below: Nikolova 016 1

13 5. Equivalent Cicuit of a Loop Antenna R C L A L i Z in Z in R l C R l R L A L i - esonance capacito - loss esistance of the loop antenna - adiation esistance - inductance of the loop - inductance of the loop conducto (wie) (a) Loss esistance Usually, it is assumed that the loss esistance of loosely wound loop equals the high-fequency loss esistance of a staight wie of the same length as the loop and of the same cuent distibution. In the case of a unifom cuent distibution, the high-fequency esistance is calculated as l π fµ Rhf = Rs, Rs =, Ω (1.3) p σ whee l is the length of the wie, and p is the peimete of the wie s coss-section. We ae not concened with the cuent distibution now because it can be always taken into account in the same way as it is done fo the dipole/monopole antennas. Howeve, anothe impotant phenomenon has to be taken into account, namely the poximity effect. Nikolova

14 J1 J When the spacing between the tuns of the wound wie is vey small, the loss esistance due to the poximity effect is lage than that due to the skin effect. The following fomula is used to calculate exactly the loss esistance of a loop with N tuns, wie adius b, and tun sepaation c: Na Rp Rl = Rs 1 b + R (1.33) 0 whee Rs, Ω, is the suface esistance (see (1.3)), Rp, Ω / m, is the ohmic esistance pe unit length due to the poximity effect, NR R0 = s, Ω /m, is the ohmic esistance pe unit length due to the skin πb effect. a c b Nikolova

15 The atio Rp / R 0 has been calculated fo diffeent elative spacings c/ b, fo loops with N 8 in: G.N. Smith, The poximity effect in systems of paallel conductos, J. Appl. Phys., vol. 43, No. 5, May 197, pp The esults ae shown below: is (b) Loop inductance The inductance of a single cicula loop of adius a made of wie of adius b L a µ = ln b H. (1.34) cic 8 A 1 a Nikolova

16 The inductance of a squae loop with sides a and wie adius b is calculated as sq a a LA1 = µ ln π b H. (1.35) The inductance of a multi-tun coil is obtained fom the inductance of a singletun loop multiplied by N, whee N is the numbe of tuns. The inductance of the wie itself (intenal inductance) is vey small and is often neglected. It can be shown that the HF self-inductance pe unit length of a staight wie of cylindical coss-section is µ a 4ac + 3c + 4c ln( a/ c) Lint = 8 π ( a c) H/m, (1.36) whee c= a δ and δ is the skin depth. To obtain the total intenal inductance of the wie, simply multiply L int by the oveall length of the wie used to constuct the multi-tun loop antenna. (c) Tuning capacito The susceptance of the capacito B must be chosen to eliminate the susceptance of the loop. Assume that the equivalent admittance of the loop is whee Rin = R + Rl, X = jω( L + L ). in The following tansfomation holds: whee A int Y in 1 1 = = Z R + jx in in in (1.37) Yin = Gin + jbin (1.38) Nikolova

17 G The susceptance of the capacito is B in in = R Rin + X in in X in = R + X B in in,. (1.39) = ωc. (1.40) Fo esonance to occu at f 0 = ω 0 / ( π) when the capacito is in paallel with the loop, the condition must be fulfilled. Theefoe, B π fc 0 = R C = = B (1.41) in X in + X in in 1 X in π f R X ( in + in ) Unde esonance, the input impedance Z in becomes in in in, (1.4). (1.43) 1 1 R in + Xin Zin = Rin = = =, (1.44) G G R X in Zin = Rin +, Ω. (1.45) R in 5. The Small Loop as a Receiving Antenna The small loop antennas have the following featues: 1) high adiation esistance povided multi-tun feite-coe constuctions ae used; ) high losses, theefoe, low adiation efficiency; 3) simple constuction, small size and weight. Small loops ae usually not used as tansmitting antennas due to thei low efficiency e cd. Howeve, they ae much pefeed as eceiving antennas in AM Nikolova

18 adio-eceives because of thei high signal-to-noise atio (they can be easily tuned to fom a vey high-q esonant cicuit), thei small size and low cost. Loops ae constucted as magnetic field pobes to measue magnetic flux densities. At highe fequencies (UHF and micowave), loops ae used to measue the EM field intensity. In this case, feite ods ae not used. Since the loop is a typical linealy polaized antenna, it has to be oiented popely to optimize eception. The optimal case is a linealy polaized wave with the H-field aligned with the loop s axis. z i H ψ a 0 θi i E y optimal incidence ϕ i x V oc The open-cicuit voltage at the loop teminals is induced by the time-vaying magnetic flux though the loop: Hee, Ψ m is the magnetic flux, Wb; V j j j H a oc = ωψ m = ωbs = ωµ z π, (1.46) H = Hi cosψ sinθ. (1.47) z i Nikolova

19 ( i, i) θ ϕ ae the angles specifying the diection of incidence; ψ is the angle between the H i vecto and the plane of incidence. Finally, the open-cicuit voltage can be expessed as Voc = jωµ SH icosψ sinθ i i = jβse cosψ sinθi. (1.48) Hee, S = π a denotes the aea of the loop, and β = ω µε is the phase constant. V is maximum fo θ = 90 and ψ = 0. oc 6. Feite Loops i The adiation esistance and adiation efficiency can be aised by inseting a feite coe, which has high magnetic pemeability in the opeating fequency band. Lage magnetic pemeability µ = µµ 0 means lage magnetic flux Ψ m, and theefoe lage induced voltage V oc. The adiation esistance of a small loop was aleady deived in (1.10) to include the numbe of tuns, and it was shown that it inceases as N. Now the magnetic popeties of the loop will be included in the expession fo R. The magnetic popeties of a feite coe depend not only on the elative magnetic pemeability µ of the mateial it is made of but also on its geomety. The incease in the magnetic flux is then moe ealistically epesented by the effective elative pemeability (effective magnetic constant) µ. We show next eff that the adiation esistance of a feite-coe loop is ( µ ) eff times lage than the adiation esistance of the ai-coe loop of the same geomety. When we calculated the fa fields of a small loop, we used the equivalence between an electic cuent loop and a magnetic cuent element: jωµ ( IA) = Iml. (1.49) Fom (1.49) it is obvious that the equivalent magnetic cuent is popotional to µ. The field magnitudes ae popotional to I m, and theefoe they ae popotional to µ as well. This means that the adiated powe Π ad is popotional to µ, and theefoe the adiation esistance inceases as ( µ ) eff. Finally, we can expess the adiation esistance as R 8 3 A = η0 π Nµ eff 3 λ. (1.50) Nikolova

20 Hee, A= π a is the loop aea, and η 0 = µ 0 / ε0 is the intinsic impedance of vacuum. An equivalent fom of (1.50) is R C 0 π ( Nµ ) λ eff 4 (1.51) whee we have used the appoximate expession η 0 10π and C is the cicumfeence of the loop, C= π a. Some notes ae made below with egad to the popeties of feite coes: The effective magnetic constant of a feite coe is always less than the magnetic constant of the feomagnetic mateial it is made of, i.e., µ < µ eff. Tooidal coes have the highest µ, and feite-stick coes have the eff lowest µ. eff The effective magnetic constant is fequency dependent. One has to be caeful when picking the ight coe fo the application at hand. The magnetic losses of feomagnetic mateials incease with fequency. At vey high (micowave) fequencies, the magnetic losses ae vey high. They have to be calculated and epesented in the equivalent cicuit of the antenna as a shunt conductance G. m Nikolova 016 0