Design of a Certain Class of Broad Band Dipole Antennas

Transcription

1 Sensor and Simulation Notes Note 57 Idus Martiae 5 Design of a Certain Class of Broad Band Dipole Antennas I L Gallon 4, St Katerine s Avenue, Bridport, Dorset, DT6 3DE, UK ilandpamgallon@tiscali.co.uk and D V Giri Pro-Tec, -C Orcard Court, Alamo, CA 9457, USA Dept. of E. C. E., University of New Meico, Albuquerque, NM, USA Abstract Antennas are regarded as transmission lines wit space dependent parameters. Te frequency response is generally scale dependent. However, imposing frequency independence on te transmission line equations leads to a scale independent geometry.

2 . Introduction Bandwidt can be an important attribute of an antenna, if it is required to perform over a range of frequencies. Oter performance parameters of any antenna can be summarized by an acronym BRIDGE [] denoting Beamwidt, Radiation pattern, Input impedance, Directivity, Gain and Effective area. In tis note, we focus on te bandwidt and investigate a way of improving te bandwidt. en we talk of te bandwidt of an antenna, we need to distinguis between two types wic we will call i) C bandwidt and ii) transient bandwidt. For instance, wen te so called Frequency Independent Antennas came on te scene in 958 [], it is noted tat tis class of antennas ave a large C bandwidt. A log periodic antenna works for a large range of frequencies, if tese frequency signals are applied at different times. It is igly dispersive and does not work if one applies a transient pulse wic contains many frequencies at te same time. In oter words, a log-periodic antenna as a large C bandwidt and no transient bandwidt to speak of. If we apply a pulse to a log periodic antenna, different frequency components get radiated from different parts of te antenna and reac an observer at a distance at sligtly different times resulting in a significant loss of pulse fidelity. Essentially te pase centre of te log periodic antenna is not stationary. Pulsed antennas ave received a lot of attention in recent years and a good eample is te Impulse Radiating Antenna [3-5]. Suc an antenna is TEM wave-fed and ence non-dispersive. It as a large transient bandwidt and can also be used at single C frequency over a wide ( to ) frequency band. Tere eist a number of metods for increasing te bandwidt of antennas. Increasing te diameter of te elements of a dipole [6], assembling arrays of non-resonant antennas [7], specifying te antenna sape by angles [], varying te impedance wit position so as to inibit reflections [8]. Te first tree approaces ave relatively severe limitations, approac only producing a modest range, approac results in a frequency dependent direction, and approac 3 requires an infinite scale. It is te last approac tat is developed ere. It is found tat finite structures eist tat support outgoing waves wit no reflections.. Te Approac An antenna is modelled as a transmission line wit position dependent parameters, and a wave solution is sougt tat maintains te wave sape as it progresses. Te transmission line (TL) equations are V I L t () I V C t () were L and C are te inductance and capacitance per unit lengt of line. Letting te inductance be a function of position Eliminating te current I from te TL equations, we obtain L L () (3) V L V L V L C t (4)

3 noting tat L C (5) c for a line wit dielectric ε. e now look for a solution representing an arbitrary travelling voltage wave wit varying amplitude V V ()f( ct) (6) Inserting tis function into te differential equation d V d f dl V L d V is independent of t and so we must ave dv dl d L d ( ct) f ( ct) (7) d V dl dv d L d d (8) dv dl V d L d (9) An arbitrary coice of te inductance variation will not in general be compatible wit bot te above equations. Eliminating V tere results dl d dl L d d L d () Te solution of tis equation, for a line of lengt is Substituting tis result into te first order equation for V and integrating L L () e must ave or V () V C L c (3) C C (4) c L it tis variation of parameters wit position we ave a frequency independent transformer were te impedance is, for a lengt X, 3

4 Z out L C L C X Z in X (5) If te transformer is left open circuit, we ave an antenna. Not only is it frequency independent, it is also scale independent. Practically tere will be limitations; in particular we cannot epect te igest frequency to be greater tan tat indicated by te dimensions of te feed. Te low frequency limit is more difficult to define. Matcing te antenna to Z for a 5 Ω input impedance X (6) Te dependence of te impedance on te ratio of te dimensions means tat te set of antenna designs is infinite. A few of te more interesting designs are considered below. 3. TEM Horn Antennas A TEM orn antenna can be considered first as a triangular plate transmission line α β Figure. A TEM orn antenna viewed as a triangular plate transmission line Te impedance of tis line is given approimately by Z Z (7) were Z is te impedance of free space. As te impedance depends only on angle it is independent of frequency, provided te plates etend to infinity. If we now epress as a function of measured along te mid plane, and put α tan (8) β cos 4

5 e ten ave β tan (9) β β β tan cos Z sin Z Z () α α tan tan Reflections from te ends of finite planes limit te bandwidt. It is to be epected tat for (β <π) tere will be some enanced directionality to te radiated field, but tis as not been investigated. Assuming a moderate amount of directionality is of benefit, we coose β π/. Ten, taking te initial impedance as 5Ω giving α 377 tan.67 5 () o α 38.9 () e ten impose te widt variation, leading to a finite line α tan.67 (3) Tere is a significant error, of order %, in te approimation due to ignoring te fringe fields. Te approimation may be improved, and is investigated later. Note tat te antenna lengt can be reduced by taking it suc tat te end impedance as some ig some ig value, 377Ω being an obvious coice. Note tat tis as te wrong dimensions to matc free space! Te need to so modify designs is te requirement for ecessively large or small dimensions. Setting ZZ we require leading to β sin α tan X For 5Ω and Ω and βπ/,. 485,.636. ( β/) sin α tan (4) (5) 5

6 6 3. Fringe Field Correction An approimation for te fringe field of a parallel plate transmission line due to Palmer [9] is ln π ε C' o π (6) Palmer compares te above approimation wit te eact solution from te Scwartz-Cristoffel transformation for values of (/) in te limited range from to. Te approimation in equation (6) is reasonable for.5 < (/) <. Unfortunately, as te widt approaces zero te formula diverges. Including a function of / tat varies from zero to one in te argument of te logaritmic term removes te divergence and as little effect away from zero, were [] sows te approimation is accurate. e ten ave π ln π ε C' f (7) Similarly Tere is a furter correction term for tick plates due to Yang [] tat as been ignored. Te epression for te impedance ten becomes e ten require o π ln π Z Z (3) π ln π π f ln π (3) Setting β tan u and (3) were κ is some suitable number, (8) ln π L o π µ f (9) π f ln π Z Z o { } f κu ep

7 u πu ( ln[ ep( κu) π u] ) A (33) e ten ave u u A π ( ln[ ep( κu) π ]) (34) were A is given by Z A lnπ (35) π Z π For β and Z() 5 and Ω, Α 7.54 and 3.77 (36) 5 A grapical solution sown in figure, ten gives 6.6 and. 567 respectively and κ. It was found tat te curves canged very slowly over equation (56) over te range of.-5 for κ. From te definition of u we ave For β π / () u tan ( β/) (37) u (38) Plate Profile (5 Om) Corrected for Fringe Field Plate Profile ( Om) Corrected for Fringe Field t id.5 a lfw H i dt lfw.5 a H Position Position (a) 5 Om case (b) Om case (/ ) Figure. Plate profile wit fringe field correction (/ ) 7

8 Comparing te corrected wit te uncorrected profiles te alf widt maimum is in error by ~7% for te 5Ω line and ~33% for te Ω line, wile te lengt is in error by~ 5% and ~ 4% respectively. Tis error is discussed later. At tis point, it is relevant to review some related publised work in te literature on tis topic. Numerical evaluation of te caracteristic impedance of a TEM orn of finite lengt as been etensively calculated and tabulated [], using te metod of stereograpic transformation. Conformal transformation tecniques are available for two-dimensional problems, and a stereograpic projection is needed for a tree-dimensional problem. A TEM orn as a sperical wave front wic is tree-dimensional. One approac uses te metod of terminating te TEM orn in its caracteristic impedance to avoid reflections [3-5]. Tere are several ways to terminate a TEM orn. Figure 3 illustrates some eamples of back-termination of te TEM orns. Suc a back termination minimizes te back radiation at low frequencies. At low frequencies, (wavelengts large compared to orn dimensions), te TEM orn is caracterized by a pair of electric ( p ) and magnetic dipole ( m ). Te resultant radiation is in te direction ( p m ) wic is ortogonal to bot dipole moments. If te TEM orn is terminated at te eit aperture, te resultant radiation at lowfrequencies is in te backward direction. Back termination reverses te direction of te magnetic dipole moment and results in te low-frequencies going out in te bore sigt direction. Figure 3. Back- terminated TEM orn [reproduced ere from [5] 8

9 Bandwidt of a TEM orn is enanced by bot lowering te low frequency cut off and increasing te igfrequency cutoff. It is also recalled tat a TEM orn, unlike a dipole antenna is an aperture type of antenna. Tis means te radiated field from te aperture of a TEM orn is a time derivative of te aperture field. Tis was eperimentally demonstrated in [6] were tey considered a TEM orn as a transmission line and attempted, witout muc success, to apply te integral of a double eponential pulse at te input in order to radiate a double eponential pulse. Te second is an eperimental approac [7] along te lines suggested in tis note, by controlling te impedance as a function of position. In [7], tey use a FDTD numerical metod to analyze tree different types of TEM orns. ) A constant impedance TEM orn wit no cange of impedance as a function of position ) A saped TEM orn wit a resistive termination at te open end 3) A TEM orn were te plates ave a resistance varying continuously along its lengt In all tree cases (sown in Figure 4) te reflected voltage from te open end is estimated and measured. It is clear tat tis reflection degrades te performance of te antenna in terms of bandwidt and radiated waveform features. At te ape of te orn, te caracteristic impedance is set equal to tat of te feeding line, typically 5 Oms. One can ten set te impedance at te aperture to be 377 Oms like we considered before. Te idea is tat tere would be very little reflection from te open end if te impedance of te line is 377 Oms at te end. In fact, tey find tis to be not te case. Tis as been called te TIT (travelling wave, impedance taper) TEM orn. Te impedance as a function of position is varied in a special manner. Resistive cards were placed over te plates at te very end to furter reduce te reflections. Te ecitation voltage was a Gaussian voltage pulse and te reflected voltage is measured in te feed coa. Te peak of reflected voltage (in time domain) from te open end was measured to be about 4% of te incident voltage. So, arranging for 377 Om impedance at te open end appears to be an erroneous concept, probably because 377 Oms is te impedance of a uniform plane wave in free space and tat is not wat we ave at te open end of a TEM orn. Slager et al [7] also found tat te reflections reduced from 4% to 7%, if tey introduced resistive seets at te end in addition to te matcing to 377 Oms. Eample of te TEM orn were it is continuously loaded by a triangular resistive profile r() r ), < < ( / ) ( (39) Is in accordance wit u-king profile described in [8] in te contet of a cylindrical dipole antenna. Here r() is te resistance per unit lengt at te ape and it gradually increases to near infinity ( or very large values) at te open end. Here te idea is tat te current flowing on te plate essentially decays out so tat tere is noting to reflect back, tus eliminating te resonance of te orn plates. In conclusion one migt say tat proper loading can improve te bandwidt of te TEM orn and tere are tecniques available to lower te low-frequency cutoff and increase te ig-frequency cut off. However, tere is no escaping te fact tat te radiated field of an aperture antenna suc as te TEM orn is a time derivative of te aperture field. Te feed point considerations are also critical. Application of ig voltages at te feed requires space to avoid breakdown at te feed, wic figts te ig frequency cut off. Larger electrode spacing near te feed necessarily lowers te ig-frequency performance of te antenna. Tis is te perennial problem of ig voltages and ig frequencies. 9

10 (A) Unloaded and unterminated TEM orn (B) Saped and loaded to matc 377Oms/square (C) Continuously loaded using u-king [8] loading profile Figure 4. Tree types of TEM orns considered in [7]

11 4. Modified Plate Antenna A triangular plate antenna can be modified by bending te plates at a certain lengt into a parallel formation wile at te same time modifying te sape to comply wit te requirement for a frequency independent response Figure 5. Notional modification of a TEM orn Assuming tat te impedance of te triangular portion of te plates is Z() frequency independence is imposed on te parallel section. Setting at te beginning of te parallel section, te impedance is given by Z() Z() Z o () (4) Solving for () (4) Half widt of te parallel section is estimated and plotted in Figure 6. Parallel Section Half idt /() Figure 6. Calculated alf widt of te parallel section as a function of position Suc an antenna was fabricated and sown in Figure 7.

12 Ω launc region Ω load Figure 7. Potograp of a modified TEM orn antenna Provisional measurements made wit te modified TEM orn antenna of Figure 7 indicated a wide bandwidt. One can attempt a Fringe Field Correction for te modified TEM orn antenna of Figure 5 and 7. Te impedance of te tapered section is given by Z Z o Setting π u, we ten ave ln π Z o π ln π (4) u πu u() π u ( ln[ π u] ) ( ln[ π u ]) (43) yields ()/ 6.6 for a 5Ω line and.567 for te Ω line. Calculating as a function of u and plotting u as a function of te following profiles of Figures 8 and 9 are obtained. ()/ 5om () / / l Figure 8. Profile of te widt as a function of position

13 Rescaled Fringe Correction Parallel Section Half idt ()/() /() / /l/.8833 ; /.8833 / Figure 9. Rescaled widt profiles as a function of position Te above comparison wit te rescaled corrected profile and te uncorrected profile accounts for te broad band beaviour of te model based on te uncorrected design. A furter modification is to etend te triangular plates by a parallell pair of plates of constant widt, and ten to increase te separation wit position. Te geometry, as before, of te triangular plate is given by Figure. Setting βπ/, and te impedance equal to 5Ω Coosing.3, and tis determines a 377 tan o α 58.8 Z m Z m a tan. 77 (44) (45) (46) (47) Te etension of te plates will take te form illustrated below in Figure. Similar considerations of saping te plates were also addressed in [-3]. 3

14 .5 Antenna Profile Upper plate Figure. Flared Strip Line Antenna Te main section of te antenna is constructed as a strip line starting wit a separation of.4m, increasing according to L o (48) were L is an arbitrary lengt. e can matc tis antenna to free space by truncating at X were ZZ o. e ten ave X L.636 (49) e are free to coose X, and so to maintain a compact antenna, coose X.3m, and we ave L. 3/ m (5) 5. Te Coaial Frequency Independent Antenna Te impedance of a coaial line is Zo b Z ln π a Imposing frequency independence we make b dependent on position and set (5) were ( ). Solving for Z Zo b ln π a Z (5) 4

15 () a b b ep ln a b a (53) Alternatively we can make te inner radius variable. e ten ave, wit ( ) yielding g b b epln a ( ) g ( ) a (54) (55) Te ratio b/a is.3 for 5Ω and 5.94 for Ω. Te electrode profiles are sown plotted in Figure. Coaial Electrode Profiles 5 Om Log Radius Outer Radius Inner Radius Position Figure. Electrode profiles of a coaial transmission line antenna Only varying te outer electrode rapidly leads to an inconveniently large radius, wile only varying te inner leads to an impossibly small radius. Truncating te antenna suc tat te end impedance is Z sets /.636, resulting in an outer radius of ~34 Initial Radius. it an inner radius of mm te maimum outer radius becomes mm, nominally seen in Figure. Te dimensions can be made more practical by allowing bot radii to become -dependent. For eample, we may take α (56) 5

16 Figure. Coaial Antenna terminated at 377 Om were α as yet to be cosen. In Figure 3 below.4 α 3.49 giving te angle as ~74 o.7.7m epansion section.4m 5 Ω cable Figure 3. Profile of Central Electrode for te case of α

17 Tis profile may be truncated for eample at 4.5 cm were te impedance is 377 Ω. It is not known but weter it is better to truncate tan ave te full lengt at a greater diameter tan specified. Coosing an eponential orn, b (/ ) epα g(/ ) epα a (57) Te central electrode profile is given in Figure 4. Eponential Coa Antenna Central Electrode cm Relative Radius Relative Lengt Central Electrode Profile Figure 4. Electrode Profiles for 5 Ω /377 Ω coaial frequency independent antenna 6. Te Vertical 'Monopole' Frequency Independent Antenna Te input impedance of a cone over a ground plane sown in Figure 5 is given by Z π Z θ ln cot (58) θ Figure 5. Mono-cone 7

18 To consider te variation of impedance wit eig t, te initial impedance is set at Z θ ln cot Z π Epressing Z in terms of te constants in equation (59), substituting into equation (58) and imposing frequency independence (59) Solving tis for θ Setting θ ln cot θ H ln cot θ cot θ cot r tan( θ) H r θ H tancot cot H H Te profile of a 5 Om frequency independent monopole antenna is sown in Figure 6. (6) (6) (6) Ω -.5 Figure 6. Profile of te 5Ω frequency independent monopole antenna Te antenna can again be truncated at /H.636 corresponding to 377Ω for eample. Tis is a limiting case of te co-aial antenna, wit te difference tat te feed is not part of te design. Again in te case of a monocone antenna, using te u-king resistive loading profile several large antennas ave been built for Nuclear Electromagnetic pulse (NEMP) simulation [9-]. One of te largest suc structure is te EMPRESS II HEMP simulator sown in Figure 7 [3]. EMPRES II (wic is about 4m in eigt) uses a u-king resistive profile described in [8] to avoid reflections from te end of te monocone. Te saping of te monocone can be an alternate metod to continuously loading te monocone. It is anticipated in tis case, te continuous loading to eliminate all reflections from te end of te monocone is superior to saping te cone to acieve 377 Om impedance at te end of te cone. 8

19 Figure 7. EMPRESS II HEMP Simulator [reproduced from 4], wic is basically a resistively loaded monocone antenna above a ground plane 7. References [Refs Re [] D. V. Giri and F. M. Tesce, Energy Patterns of te Prototype Impulse Radiating Antenna (IRA), Sensor and Simulation Note 55, 5 February. Tis reference can be downloaded from [] V. Rumsey, Frequency Independent Antennas, IRE International Convention Record, volume 5, pp 4-8, -5 Marc 966. [3] C. E. Baum, Radiation of Impulse-Like aveforms, Sensor and Simulation Note 3, 5 November 989. Tis reference can be downloaded from [4] D. V. Giri et al., Design, Fabrication and Testing of a Paraboloidal Reflector Antenna and Pulser System for Impulse-Like aveforms, Invited Paper, IEEE Transactions on Plasma Science, Volume 5, Number, pp 38-36, April 997. [5] D. V. Giri, Hig-Power Electromagnetic Radiators, Nonletal eapons and Oter Applications, publised by Harvard University press, 4. [6] D.. Fry, F. K. Goward, Aerials for Centimetre avelengts, Cambridge University Press, United Kingdom,95 [7] S. Silver, Microwave Antenna Teory and Design, Radiation Laboratory Series, 949. [8] E Hallen, Electromagnetic Teory, Capman & Hall, st U.S. Edition, December 96 [9] H. B. Palmer, "Capacitance of a parallel-plate capacitor by te Scwartz-Cristoffel transformation," Trans. AIEE, Vol. 56, pp. 363, Marc 97. 9

20 [] V. Leus, D. Elata, Fringing Field effect in electrostatic actuators Tecnion, Haifa, Israel, Tecnical Report ETR-4-, May 4, available at: ttp://meeng.tecnion.ac.il/researc/treports/4/etr-4-.pdf [] H. Yang, "Microgyroscope and Microdynamics," P. D. Dissertation, December,. [] F. C. Yang and K. S. H. Lee, Impedance of a two-conical-plate transmission line, Sensor and Simulation Note, November 976. Tis reference can be downloaded from [3] C. E. Baum, Low-frequency compensated TEM orn, Sensor and Simulation Note 377, 8 January 99. Tis reference can be downloaded from [4] M. H. Vogel, Design of te low-frequency compensation of an etreme-bandwidt TEM orn and lens IRA, Sensor and Simulation Note 39, 9 April 996. Tis reference can be downloaded from [5] D. V. Giri, H. Lackner, G. Francescetti, J. Tatoian V. Carboni and J. Ler, Design, Frabrication and Testing of a Timed Array of TEM Hrray for Beam Steering, Sensor and Simulation Note 469, May. Tis reference can be downloaded from [6] Y. ang, Y. Cen and Q. ang, Application of a TEM orn antenna in radiating NEMP Simulator, 7 t International Conference on Applied Electrostatics (ICAES-), Journal of Pysics Conference Series 48 (3), avaialble at [7] K. Slager, G. S. Smit and J. G. Maloney, Accurate Analysis of TEM Horn Antennas for Pulse Radiation, IEEE Transactions on Electromagnetic Compatibility, volume 38, Number 3, August 996. [8] T. T. u and R.. P. King, A cylindrical antenna wit non-reflecting resistive loading, IEEE Transactions on Antennas and Propagation, Volume AP-3, pp , May 965. [9] C. E. Baum, Resistively loaded Radiating Dipole Based on a Transmission-Line Model, Sensor and Simulation Note 8, 7 April 969. Tis reference can be downloaded from [] D. V. Giri, Time-Domain Radiated Fields of Resistively Loaded Biconical Antenna Based on a Transmission-Line Model, Sensor and Simulation Note 366, April 994. Tis reference can be downloaded from [] D. V. Giri, Impedance Matri Caracterization of an Incremental Lengt of a Periodic Array of ave Launcers, Sensor and Simulation Note 36, April 989. Tis reference can be downloaded from [] C. E. Baum, Canonical Eamples for Hig-Frequency Propagation on Unit Cell of ave-launcer Array, Sesnsor and Simulation Note 37, 9 April 989. Tis reference can be downloaded from [3] D. V. Giri, A Family of Canonical Eamples for Hig-Frequncy Propagation on Unit Cell of ave- Launcer Array, Sensor and Simulaiton Note 38, 5 June 989. Tis reference can be downloaded from [4] International ElectroTecncial Commission, Testing and Measuremnt Tecnciques- Hig Altitude Electronagnetic Pulse (HEMP) Simulator Compendium, IEC 6-4-3,.