Tuning of High-Power Antenna Resonances by Appropriately Reactive Sources

Transcription

1 Senor and Simulation Note Note 50 Augut 005 Tuning of High-Power Antenna Reonance by Appropriately Reactive Source Carl E. Baum Univerity of New Mexico Department of Electrical and Computer Engineering Albuquerque New Mexico 873 Abtract In deigning mall high-power electromagnetic radiator (of the order of a half wavelength or o in ize) baed on witched reonant circuit, there are quetion concerning the control of the reonance frequencie. Thi paper explore ome technique for tuning thee frequencie baed on the reactive propertie of the ource.. Thi work wa ponored in part by the Air Force Office of Scientific Reearch.

2 . Introduction Begin with ome antenna with Z a( ) = Y a ( ) input impedance = Ω + jω = Laplace-tranform variable or complex frequency (.) ~ = two-ided Laplace tranform Let thi be driven, a in Fig.., by Z ( ) = Y ( ) = ource impedance (.) Thee two impedance are connected by a cloing witch which we model by V V w ( 0 ) = V 0 = charge voltage before witch cloure (.3) In time domain thi i a tep function. We hould remember that the witch doe not cloe in zero time [4], and that thi limit the performance at high frequencie. then the antenna current (at the input terminal) i jut If Z ( ) = 0 V I 0 a( ) = Y a( ) (.4) Thi typically exhibit reonant behavior at frequencie given by ( ) 0 az a a = (.5) Neglecting a = 0 the contribution of a pole i given in time domain by d t () 0 Re ( ) a Iaa t = V a Z a e u() t d (.6) = a Where the conjugate pole i now included.

3 + V w ( ) _ a ( ) Z ( ) Z Fig.. Antenna and Source 3

4 . General Conideration Now conider the influence of the ource impedance. Thi might be a imple capacitance C. However, at high frequencie Y ( ) may have more complex tructure [5]. Thi nonzero Z ( ) then combine with z ( ) to hift the reonance frequencie. a The ource impedance ha reonance given by ( ) = 0 Z (.) When combined with the antenna impedance we have new natural frequencie given by ( ) + ( ) = 0 Z a m Z m (.) Then (.6) i replaced for a ingle pole pair by d () = 0 Re ( ) + ( ) mt Iam t V m Za Z e u() t d = m (.3) So our approach i to ee how we might hift the antenna reonance in deirable direction. The factor d m Za Z d ( ) + ( ) = m (.4) can be ued (at leat for high-q reonance) a a caling factor for the trength of the reonance. 4

5 3. Ditributed Capacitive Source For preent purpoe we need a model for the ource impedance. Let u chooe an open-circuited tranmiion line a indicated in Fig. 3.. It might include a high-dielectric-contant medium with ε = εε r 0 (.4) With a tranit time t r, the capacitance (low frequency) i jut t C = r Zc Z c = characteritic impedance of tranmiion line (.5) The ource impedance i then t e r Z + ( ) = Zc = Z coth( ) t c t r (.6) e r with open-circuit reonance at ( t ) ( ω t ) inh r = 0, in r = 0 ωr t = nπ, n = 0,,, ω = n f = π tr (.7) i.e., multiple of a half wavelength. are It i intereting to ee at what frequencie the ource ha zero impedance (hort-circuit reonance). Thee ( t ) ( ω t ) coth r = 0, co r = 0 n + ω tr = π, n = 0,,, (.8) ω n + f = = π 4tr i.e., odd multiple of a quarter wavelength. One might chooe the ource then a having zero impedance at an antenna reonance o a to deliver a large voltage to the antenna. 5

6 t r tranit time ε Fig. 3. Tranmiion-Line Capacitive Source 6

7 4. Combination With Magnetic Antenna One type of electrically mall antenna i a loop of ome kind producing a magnetic-dipole moment. When operating in reonance condition there may be ome appreciable fraction of a wavelength acro the tructure [, ]. Let u model the antenna impedance (up to firt reonance of current) a Z a ( ) = + C a La L a low-frequency loop inductance (4.) Ca capacitive correction aociated with lead into loop and tray capacitance of loop tructure Note that thi neglect the radiation reitance. ource V 0 / a If the ource i modeled a a imple capacitance C, thi appear in erie with Z a when driven by the Z ( ) = Z a ( ) + = + C a + Cg La C = L = C a L + a a LaC (4.) The reonance i then at 0 = ω C mla a ω mlac ω C ω mla = mlac / / ωm = La[ Ca + C] = [ LaCa] + / C Ca (4.3) Compared to the antenna reonance [ C ] ω a = L a a (4.4) We ee that 7

8 ωm < ωa (4.5) With equality if C = 0 (or Z = C, ωm 0. ). The effect of C i to lower the reonance frequency. Note that for infinite At the ame time the trength of the reonance i changed with the factor d m Za Z d ( ) + ( ) = m = C C m + a + m a mla mla mc j = C ω C ω + a m + a + m ω ω ω C mla mla m j C = [ C + C ] [ C + C ] + + a a La a C La ωm C jl C = a + a C C + a + + a ωm C C C jl C = a + a ωm C / / j L C C = a + + a Ca Ca C / 3/ j L C C = a + Ca Ca Ca (4.6) So maller C decreae the reonant current (at the antenna port). A C (zero-impedance ource) thi reonance ha ωm 0, for which the antenna i zerowavelength long. Small C correpond to a quarter wavelength. Let u conider a higher reonance correponding to a half wavelength. 8

9 5. Tranmiion-Line Model of Loop and Source Conider the cae that both loop antenna and ource are modeled a tranmiion line a indicated in Fig. 5.. Then we have for the antenna impedanace t e Z a( ) = Zch = Z tanh ( ) t ch t (5.) + e For a zero-impedance ource we have current reonance a ( t) ( ω t) inh a = 0, in a = 0 ωat = nπ, n = 0,, (5.) ωa n fa = = π t which are multiple of a half wavelength. A pecial imple cae ha Zc = Zch (5.3) with t now the tranit time of the ource part. Thi i effectively a ingle tranmiion line of tranit time, t + t. With one end horted and the other open, the firt quarter-wave reonance i at f m ω = m = π 4 [ t + t ] (5.4) Here we ee that hortening t raie f m, conitent with the previou reult with leened ource capacitance. Here we can alo ee that a the witch approache the right end of the tranmiion line, where the current in the natural mode i weaket, the trength of the reonance i alo decreaed. The more general cae ha the reonance condition mt mt e + e Zch + Z c = mt mt + e e 0 (5.5) 9

10 hort circuit Z ch t V + _ t w ( ) Z c open circuit Fig. 5. Tranmiion-Line Model of Loop and Source 0

11 Note that for mall Z c we have Zc 0 Zch m a = jωa (5.6) a in (5.). Another pecial cae ha t = t, for which we have mt Z mt e c e + + Z ch = 0 tanh Z ( ) c mt = Zch tan Zc ( ωmt ) = Zch ωmt arctan / Zc = Zch (5.7) With additional olution baed on the periodicity of tan. The general cae (5.5) i readily olved numerically for ω m t or ω m t a a function of Zc/ Z ch and t / t. By taking the derivative of Z a a in (5.) one can alo find a perturbation olution about ω a a in Section 4.

12 6. Combination With Electric Antenna Another type of electrically mall antenna i an electric dipole of ome kind, i.e., two eparate conductor driven by ome ource between them, produding an electric dipole moment. Operated in reonance condition there may be ome appreciable fraction of a wavelength acro the tructure [3]. Let u model the antenna impedance (up to firt reonance of current) a Z a( ) = + La Ca Ca low-frequency dipole capacitance (6.) La inductive correction aociated with lead into dipole and tray inductance of dipole tructure Again thi neglect the radiation reitance. Vg / a With the ource modeled a a capacitance C, thi appear in erie with Z a when driven by the ource Z ( ) = Z a( ) + = La + + (6.) C 5 Ca C The reonance i then at 0 ωm = ωmla + ωm Ca C / = + La Ca C (6.3) Compared to the antenna reonance at [ L C ] / ωa = a a (6.4) We ee that ωm > ωa (6.5)

13 with equality if C = (or Z = 0 ). The effect of C i to raie the reonance frequency. For large C, the reonance correpond to a quarter-wave reonance related to the ource (or half wave on the two dipole conductor). For mall C the reult of (6.3) i unrealitic in that phyically thi hould go to an open-circuit or half-wave reonance related to the ource. For thi cae another model i appropriate. The trength of the reonance i changed a d m Z a( ) + Z ( ) d = m = mla + C m a C / = j + La Ca C La / j = La + Ca C (6.6) So larger C increae the reonant current (at the antenna port). 3

14 7. Tranmiion-Line Model of Electric Antenna and Source Model the electric antenna and ource a tranmiion line a indicated in Fig. 7.. Note that thi i topologically different from the loop cae ince both end are open circuited. Now the antenna impedance i t + e Za Zch Z t ch t e ( ) = = coth ( ) (7.) For a zero-impedance ource we have current reonance at ( t) ( ω t) coh a = 0, co a = 0 n + ωat = π, n = 0,,, (7.) ωa n + fa = = π 4t which are odd multiple of a quarter wavelength. For the pecial cae of Zc = Zch (7.3) we have a half-wavelgneth reonant tranmiion line of tranit time t + t. Thi give the lowet-order reonance at f m ω = m = π [ t + ] (7.4) For mall t thi become a half wavelength on each antenna conductor. However, thi alo implie a mall energy from the ource. The more general cae ha the reonance condition mt mt + e + e Zch + Z c = mt mt e e 0 (7.5) 4

15 For mall Z c we have Zc 0 Zch m a = jωa (7.6) a in (7.). Another pecial cae ha t = t, for which we have Z inh( ) c mt + inh( mt) = 0 Zch inh( mt) = 0, inh( ωmt) = 0 nπ ωmt =, n = 0,,, ωm n fm = = π 4t (7.7) The firt nonzero reonance i then when each antenna conductor i a quarter-wavelength long. Note alo that mall Z c mean more tored energy in the ource, giving a larger reonance current. The general cae (7.5) i alo readily olved numerically. 5

16 open circuit Z ch t V + _ t w ( ) Z c open circuit Fig. 7. Tranmiion-Line Model of Electric Antenna and Source 6

17 8. Concluding Remark A we can ee, judiciou choice of the frequency dependence of the ource impedance can alter the reonance frequency and reonance trength of the antenna, whether of loop or electric-dipole type. Here we have choen ome imple form of the ource impedance for illutration. More elaborate form can alo be purued. 7

18 Reference. C. E. Baum, Compact, Low-Impedance Magnetic Antenna, Senor and Simulation Note 470, December 00.. C. E. Baum, Symmetry in Low-Impedance Magnetic Antenna, Senor and Simulation Note 497, March C. E. Baum, Compact Electric Antenna, Senor and Simulation Note 500, Augut J. M. Lehr, C. E. Baum, and W. D. Prather, Fundamental Phyical Conideration for Ultrafat Spark Gap Switching, Switching Note 8, June 997; pp. -0 in E. Heyman et al (ed.), Ultra-Wideband, Short-Pule Electromagnetic 4, Kluwer Academic/Plenum Publiher, C. E. Baum, High-Dielectric-Contant Material a High-Frequency Capacitor, Energy Storage and Diipation Note, November