Combined Electric and Magnetic Dipoles for Mesoband Radiation, Part 2

Transcription

1 Sensor and Simulation Notes Note 53 3 May 8 Combined Eletri and Magneti Dipoles for Mesoband Radiation, Part Carl E. Baum University of New Mexio Department of Eletrial and Computer Engineering Albuquerque New Mexio 873 Abstrat This paper ontinues the disussion of the design of an antenna with ollinear eletri- and magneti-dipole moments out of phase by approximately π / (9 ). Using a high-q approximation, a swithed osillator is mathed to the antenna for radiating an osillatory pulse. ` This work was sponsored in part by the Air Fore Offie of Sientifi Researh.

2 . Introdution A reent paper introdued the onept of a balaned pair of ollinear eletri and magneti dipoles []. It was shown that this ould produe irular polarization in the far field. The alulations are based on a high-q approximation, so that narrow-band (hypoband) (CW) onepts an be used. While polarization is normally a narrow-band onept, it an be generalized to the damped sinusoidal ase [3, 7]. Mathing this antenna to a swithed osillator [], we onsider the radiation from suh a mesoband radiating soure. The details of the antenna are disussed in []. The basi geometry is indiated in Fig.. with a loop going around the eletri-dipole axis. This loop (magneti dipole) an be made in a frational- (e.g., half-) turn geometry to suppress any assoiated transverse eletri-dipole moment. z a h h e Fig.. Eletri and Magneti Dipoles

3 . Balaning Eletri and Magneti Dipoles As disussed in [] we would like to make the eletri- and magneti- dipole moments approximately balaned as m p( jω) ± j ( jω ), p( s) p ( s) z, m( s) m ( s) z two-sided Laplae transform over time, t (.) while they are π / (9 ) out of phase with eah other to give an approximately irularly polarized for field. Eah of these is related to soures as p( s) Q ( s) he Ce he V e( s) m( s) I h ( s) Ah (.) he he z, Ah Ah z s Ω + jω Laplae-transform variable or omplex frequeny This requires that V e and I h be π /(9 ) out of phase for s jω, ω π f f approximate resonane frequeny (.3) As we shall see later the resonane is moved into the left-half s-plane as s Ω + jω, Ω < (.) due to radiation losses. With st Ω t jω t p t m t e u t e e u t (.5) (), () o () () we have the ommonly used quality fator [] Q π n 3

4 n number of yles to deay to f ω Ω π Ω e (.6) For the first part of our analysis we an neglet the shift into the left half plane as a high-q approximation. Then we an orret for this shift based on energy onsiderations. So let us hoose some f and see what is required to balane the dipole moments as in (.). We are assuming that the antenna is eletrially small so that ω K << some harateristi length (.7) This allows our approximation that the fields are dominantly desribed by the dipole terms. The effetive height h e of the eletri dipole is roughly h (as in Fig..) where h is the end-to-end dimension. The resonant length is then roughly h (half wavelength). So we need λ >> h [length] (.8) In this ase we an still approximately think of an eletri dipole, but the input impedane is resistive, not apaitive. Similar onsiderations apply to the magneti dipole (loop). For a single-turn loop of radius a, eletrially small means λ >> π a irumferene (.9) not ounting the leads to the soure. In this ase there is a strong eletri-dipole term whih we need to avoid. So the restrition (.9) is important. As disussed in [] we an use a segmented loop. In partiular a half-turn (N/) loop suppresses the aforementioned resonane and the assoiated eletri-dipole moment. Reognizing our size limitations, then let us balane the dipole moments for some seleted f. We need

5 ( ω ) p j m j ± j ( ω ) ( ω ) ( ω ) p j Cehe Ve j ( ω ) ( ω ) m j Ah Ih j (.) For this we need to onsider the equivalent iruits as in Fig... L L h + _ Vs C C e A. Addition of series L and parallel C C L h L + _ Vs C e B. Addition of parallel L and series C Fig. Equivalent Ciruits for Combined Dipoles 5

6 In Fig..A we have an equivalent iruit in whih the loop indutane is in series with L, and the eletri-dipole apaitane is in parallel with C. The resonane frequeny is now [ LC] / ω k L Lh + L, C Ce + C (.) The previous formulas in [] an now be arried forward with the above substitution. Note now that the urrent I h still flows through L h as well as L, and the voltage aross C e is also aross C. The dipole moments are, however, only assoiated with L h and C e. This gives some flexibility in the design to math the dipole moments. In this ase let us adjust L h and C e to math the dipole moments for some hosen ω. We need for this A I h h( jω) V e( jω) Cehe (dipole moments) V e I h ( jω ) ( jω ) [ L L ] ω h + (indutor impedane) / Ah Lh + L ω[ Lh + L] Ch e e Ch e e Ce + C Ah C k e (dimentionless) (.) he Ce + C ω k (radian wavelength) For given h e and A h, then k must be small enough ( large enough) to ahieve the above inequality. This in turn gives onstraints on C and L. So one key to the design is to make the frequeny low enough, not only to make the antenna eletrially small, but also to ahieve the dipole mathing. Another approah is indiated in Fig..B with L in parallel with L h and C in series with ase the voltage on the eletri dipole is C e. In this C V e( jω) V s( jω) Ce + C (.3) The urrent in the loop is 6

7 ( ω ) ω ( ω ) I h j j Lh V s j (.) The resonant frequeny is [ LC] / ω L L e L + C C e C + (.5) Now we need A ( ω) h ( ω) I h j V e j Cehe (dipole moments) V s I h V e I h ( jω ) ( jω ) ( jω ) ( jω ) ω Lh (antenna indutive impedane) C ω Ce + C Lh (apaitive divider) Ah CeC Lh + L ω Lh he he Ce + C ω L Ah Lh + L k (dimensionless) he L (.6) Again for given h e and A h, then k, must be large enough ( small enough) to ahieve the above inequality. However, the antenna must be eletrially small to ahieve dipolar operation, plaing additional onstraints. So there are various tuning networks that one an use to math the dipole moments for a given resonane frequeny. The foregoing are only examples. 7

8 3. Radiated Fields and Power From [] we have the far fields ( r ) from the dipoles as (, ) r s E f r s e γ μ μ r i z p ( s) + r z m ( s) π r (, ) r s H f r s e γ r i z m ( s) r z p ( s) π r r r r θ θ + φ φ p ( s) m ( s) s γ (3.) The radiated power density at the resonane frequeny is P ( r, jω) r E f ( r, jω) H f ( r, jω) ω μ m m sin ( θ) r p( jω ) p( jω) + ( jω) ( jω) 3π r i i With the balaned-dipole ondition we have P ( r ω, ) r μ jω sin ( θ) p( jω) 6π r r (3.3) With the high-q resonane ondition (small Re( s ) ) we have the energy radiated in one yle as U tr r P ( r, jω) ds unit sphere ω μ t 3 r p( jω ) sin ( θ) dθdφ π unit sphere π tr μ p 3 ( jω ) sin ( θ) dθ π t μ rω p( jω ) 3π (3.) 8

9 tr transit line of swithed osillator (next setion) (3.) / period It is this energy loss to radiation whih will lower the Q of the osillation.. Available Energy From Osillator The stored energy in the swithed osillator is [] t U C r osv V Z tr T os transit time f T period, Z t r transit time harateristi impedane of transmission line / μos fgzw fg εos f g geometri fator (.) Z w wave impedane in osillator medium As in Fig.., the osillator is onneted to the antenna whih may inlude tuning elements (as in Fig..) to give some net L and C. We need to know the voltage aross C e (for the eletri dipole) and the urrent through L h (for the magneti dipole). A first observation onerns a nonideal aspet of the swithed osillator, namely the square-wave nature of the waveform produed on losing the swith. Suppose we remove the load (open iruit the osillator). The waveform at the output is a square wave of amplitude V with period T. Not all the energy in this osillation is at the frequeny f, but inludes omponents at 3 f, 5 f,. The total energy in all those omponents is U. Note that there is a bloking apaitor with Cb >> C, Cos (.) There is energy also stored here whih we are not onsidering for present purposes. 9

10 harateristi impedane Z I L bloking apaitor (large) V R (large) + L C V C losing swith _ Fig.. Swithed Osillator Conneted to Resonant Antenna/Load. λ After the swith loses, there are various ways to evaluate the energy in the primary resonane, depending on whih time one uses for the evaluation. If we make a Fourier analysis of the energy omponents, we an divide the energy appropriately. At t (just after the swith losure) we still have V on the enter ondutor. This gives an amplitude for the f resonane as Vs λ π os z V dz λ λ π os z dz λ λ 8 π V os z dz λ λ λ 8 λ π V sin z λ π λ V π (.3)

11 \ The energy available at f is then λ Uos Cos π os z V dz π λ tr λ V Z π λ 8 tr 8 V C osv U π Z π π (.) Comparing to (.) we an see that about 8 perent of the stored energy is available for the first resonane. 5. Q of System Allowing for the radiation losses (Setion 3) and the available energy we an establish Q from the number of yles for the osillation amplitude to deay to the amplitude as e, or energy to deay to e [, 6]. After one yle we have / Uos U U U O + Uos Uos Uos (5.) Comparing to an exponential deay gives U t U n Uos T Uos U e e n Uos T period of osillation n number of yles (5.) Q π n Applying this to the present problem gives (for the iruit in Fig..A) n Q π U os U 8 tr 3π V ω p( j ω ) π Z tr μ V ω π Z μ p( jω ) V ω Ce he π Zμ Ve

12 V C e h e π Zμ Ve Z ε V h e π Z Ce Ve (5.3) Now we an see the relative ontributions of various fators. For an osillator with free-spae onstitutive parameters we have Z Z f g (transmission-line geometri fator, dimensionless) (5.) Typial numbers might have Z a few ohms with f g of order. Even smaller numbers might be possible [5]. Large n (high Q, longer pulses) is ahieved with small f g, large (low frequenies), and small Ch. e e about π / for the iruit in Fig..A, but is inreased for the iruit in Fig..B due to the apaitive divider. V / V s is 6. Tradeoffs Negleting other (resistive) losses all the energy stored in the first osillator resonane is radiated. It is just a question of trading off amplitude and pulse width (number of yles). So we an regard Ef r n onserved parameter (6.) So the time-domain peak field is proportional to / n (or Q / ). Large fields give shorter pulse widths and onversely. In frequeny domain we an think of all the energy being radiated in a band of frequenies proportional to Q (or n ). The amplitude (proportional to square root of energy density) then is proportional to n / (or Q / ). So we have a tradeoff between spetral width and peak energy density at the enter frequeny. 7. Conluding Remarks Building on previous results [], this paper has extended the design onsiderations for ombined eletri and magneti dipoles produing approximately irular, damped sinusoidal radiation. This inludes networks to balane the dipole moments, and tradeoff between radiated field amplitudes and pulse width for a given resonane frequeny. An example alulation is given in the appendix.

13 Appendix A. Example Calulation To give the reader some feel for the interplay of the various design parameters, let us onsider a hypothetial ase. Use the equivalent iruit in Fig..A for tuning the devie. A.. Eletri-dipole parameters Let he Ce.5 m 3 pf (A.) This orresponds roughly to an antenna length of m. Using an ACD kind of design [] the feed point looks roughly like a Ω bione. We then hoose A.. Magneti-dipole parameters Choosing, as previously disussed for suppressing an eletri-dipole ontribution, N turn loop (A.) a.5 m radius ( m diameter) π Ah a.39 m b minor radius.5 a major radius 8a Lh N μ a n b.3 μh (A.3) A.3 Mathing dipole moments From (.) we have + + / L / + Ah Lh L h Ch e e Ce C + C e L Lh Ah Ce Ah C C ehe Lh he CeLh Ce.98 (A.) 3

14 By serendipity we have found a ase in whih extra indutane and/or apaitane is not required. However, one ould make the above more preise if desired. Furthermore parasiti indutane and apaitane (e.g., leads) has not been inluded. A. Resonant frequeny From (.) we have ω f λ h e ω 6 MHz π.9 m f.78 m π [ LC] /.38 9 s (A.5) The resonant half wavelength of.5 m is onsiderably larger than the approximately m length at the eletri dipole. Similar onsiderations onern this.5 m as ompared to the length of.6 m for the half loop. So the struture an be approximately onsidered as eletrially small. If a lower frequeny is desired for the given antenna dimensions, this an be aomplished by adding L and C. A.5 Swithed osillator For the swithed osillator, we have (for mathing the resonant frequeny) t r. ns (A.6) f For a free-spae dieletri this is rather long, but an be shortened by use of dieletri and/or a folded geometry. In addition we need the transmission-line harateristi impedane whih we take as Z Ω (A.7) For the present example. For energy let us take V kv t U r V 5. J Z Vs V 7 kv π t U r os V. J π Z (A.8)

15 A.6 System Q From (5.3) we have n Q Z ε V h e π π Z Ce Vs 9 (A.9) The high Q is due to the fat that we are operating below the antenna resonane (eletrially small antennas) with a low-impedane swithed osillator. This example then approahes a hypoband (narrow band) radiator with all the available energy onentrated near f. The Q an be lowered by inreasing larger V and assoiated far field. Z (and perhaps thereby allowing a A.8 Radiated fields The eletri-dipole moment at early times is approximately p Ce he Vs.9 μcm ω μ p r Ef 7 kv (from 3.) π (A.) A.9 Remarks The present is but one example. Many more an be imagined. The present example an also be saled aording to the usual rules of eletrodynami saling (e.g., multiplying linear dimensions by some onstant). Sine there are various approximations in the present analysis, one an resort to more detailed alulations and/or onstrut sale models to obtain more aurate results. 5

16 Referenes. C. E. Baum, An Equivalent-Charge Method for Defining Geometries of Dipole Antennas, Sensor and Simulation Note 7, January C. E. Baum, Combined Eletri and Magneti Dipoles for Mesoband Radiation, Sensor and Simulation Note 53, August C. E. Baum, SEM Baksattering, Interation Note 76, July C. E. Baum, Swithed osillators, Ciruit and Eletromagneti System Design Note 5, September. 5. C. E. Baum, Some Thoughts Conerning Extending the Performane of Swithed Osillators, Ciruit and Eletromagneti System Design Note 5, Marh E. A. Guillemin, Introdutory Ciruit Theory Theory, Wiley, C. E. Baum, SEM and EEM Sattering Matries and Time-Domain Satterer Polarization in the Sattering Residue Matrix, pp. 7-86, in W.-M. Boerner et al (eds.), Diret and Inverse Methods in Radar Polarimetry, Kluwer, 99. 6