Time Domain Modeling of Nonlinear Loads - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. "-31,

121

NO. 1 , JANUARY 1983

Time Domain Modeling of Nonlinear Loads

Abstract-The behavior of an antenna or scatterer when loaded with a nonlinearelementcanbechangedgreatlyfrom that observed under linear conditions. In some cases, the nonlinearity causes effects such as the intermodulation products arising from the nonlinear mixing of two frequencies. In other cases, the nonlinearity maybe exploited, for example, to reducelate-timeringing on apulse-excitedantenna. A procedure for treating general nonlinear loads is described and illustrated. This procedure is applied using a computer to several specific types of nonlinearities.Thetreatmentwithintheframeworkofthethin-wire is developed. As approximation to theelectricfieldintegralequation such, the treatment canbe applied to the large class of objects modeledby wires. The nonlinear load types that are considered include those with piecewise-linear voltage-current curves (e&, oneor more diodes), a load with a time varying resistance (which permits modulating the scattered fields), and a general nonlinear load represented by specified voltagecurrent functions.

INTRODUCTION

where (< t )

incident electric fieldobservation at point ;at time t , s' and i' unit vectors parallel to C(?) at ?and C light, of velocity PO freepermeability of space Z(s', t') and q(s', t') currentandcharge,respectively,at t' = (t point s' andretardedtime R/C), where R = ;1 - 7'1 and C(?) = structure contour, assumed for now to be a perfectly conducting wire. Einc

?'$

~

Vector quantities are denoted by an arrowover the quantity. Solving (1) numerically by the method of moments (using, in this case, point matching and a nine-point Lagrangian interpolation scheme for the current at space' points protect equipment fromexcessive voltages or currents delivered by antennas, i.e., lightning protection and EMP hardening. Here, we will examine the response of nonlinearelementsthatlimit(orclip)thecurrentwith a series load. A dipole 10 m long and 2 mm in diameter was divided into49segments. A plane wave illuminatedthedipolefrom broadside with the electric field parallel to the dipole. The time history of the electric field was a typical EMP waveform: E(t) = 5.25 X lo4 [exp (- 4.00 X 106r) -exp (- 4.76 X 108t)] V/m. A 50 R load and a current-limiting nonlinear element were placed inseries atthecenter of thedipole. Fig. 4 showsthecurrent through the nonlinear element for clipping at a level of 25 A (01 1250 V across the 5 0 4 load). (A current of about 430-A peak is delivered in the absence of the nonlinear element.) The voltage

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LAh'DT e t a [ . : TIME DOMAIN MODELING

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Fig. 4. EMP response of a dipole antenna loaded by a nonlinear element that limits the current at the center of the antennato 25 A.

TIME

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15.1

Electric field radiated broadside from the antenna of Fig.

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TIME (ns]

Fig. 5 .

Fig. 3.

Electric field radiated by the example of Fig. resistance of 400 !2.

1 but with a reverse

Voltage across the nonlinear element of Fig. 4.

across the nonlinear element is shown in Fig. 5 . In this example, the initial current charges up the dipole, and the nonlinear elementbecomesahigh resistance whenlimitingthecurrent (a This results in a voltage that keeps maximum of about 16 000 a>. thenonlinearelement in acurrent-limitingmodefor several cycles of thenaturalresonantfrequency of theantenna.The power capability of the element must be at least 10 hlW and will dissipate about 0.04 J. Fig. 6 shows the current resulting if the limiting is raised to 50 A. The voltageacross thiselement is shown in Fig. 7. In this case, the power capability of the element must be about 18 MW, but only about 0.03 J are dissipated in the nonlinear element. Because about 0.1 J will be delivered to the

124

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-31, NO. 1, JANUARY 1983 60

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Fig. 8 . Electric far field scattered from a dipole antenna illuminated a 16-MHz sinusoidalwaveandloaded by aresistancevaryingina sinusoidal fashion at 4 MHz.

by

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Fig. 7.

Fig. 9. Spectrum

Voltage across the nonlinear element of Fig. 6 .

TIME-DEPENDENT LOADS The use of a time-varying load has some interesting possibilities, including a shifting of the apparent frequency of the scattered field to provide a “false” Doppler shift. To study this possibility, a linear dipole 9 m long and 0.2 m in radius was divided into nine segments and illuminated from broadside with a sinusoidal wave of 16 MHz and 1-V/m peak electric field parallel to the dipole. A time-varying resistance was placed at the center of the dipole with a value

+ sin (at)]a,

40

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80

IFREQUENCY [MHz)

unprotected 5042 load,muchoftheenergycollectedbythe dipole is reradiated and not dumped in the protection device.

R(t) = 500[1

20

(10)

of the waveform of Fig. 8.

where w = 2nf and f is frequency. The resulting backscattered field is greatly altered from the no-load case and is shown in Fig. 8 for a resistance modulation frequency f of 4 MHz. The spectrum of this field is shown in Fig. 9. Significant energy has been placed in sidebands with frequencies offset in multiples of 4 MHz from the 16 MHz incident wave. Miller and Landt [ 111 presented similarresults, butfora resistancevaryinginasquare-wave fashion between 0 and I000 i-2.The results are very similar, but slightly more energy is transferred to the higher sidebands in the square-wave case. Fig. 10 shows the backscattered field strength if the resistance modulating frequency f i s 5 MHz and thus not locked to the frequency of the incident wave. In this case, the beat-frequency pattern is more complicated than that of Fig. 8. The spectrum of the backscattered field is shown in Fig. 11, and

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LANDT er al.: TIME DOMAIN MODELING

Nonlinearities that can be modeled include arbitrary V-I relationships exhibited by diodes, square-law detectors and limiters, and time-varying loads. We have presented examples of these types. Applications include analysis of nonlinear behavior resulting from large amplitude excitation (e.g., that caused by EMP and lightning); the influence of protective devices like spark gaps and electrical surge arresters (ESA’s); intermodulationresultingfrominadvertent nonlinearities (i.e.: the “rusty bolt” effect); pulse shaping through the use of diode loads; and spectrum spreading and modification using time-varying loads.

REFERENCES

0

Fig. 10.

0.4

0.8 TIME Ips)

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0.12

Electric far field of the example of Fig. 8 for a resistance varying at 5 MHz. 15

FREQUENCY (MHz]

Fig. 11. Spectrum of the waveform of Fig. 10.

againsignificant amountsofenergy have been placed in sidebands, now offset at multiples at 5 MHz from the frequency of theincident wave. It is alsointeresting tonotethatpractical schemescanbeused t o producetime-varyingresistance,and several of these (including usingfield effect transistors) require little power and are completely quiet electromagnetically, only acting as passive (but nonlinear) reflectors.

CONCLUSION The analysis on nonlinear effects is readilyachievable in the timedomain. A straightforwardmodificationof a thin-wire time-domaincode(WT-MBA/LLLlB) [7] provides a computationally efficient way to analyze various types of nonlinear loads on wire-type structures.

D. E. Merewether. ”Transient currents induced on a metallic body of revolution by an electromagnetic pulse,” IEEE Trans. Electromagn. Comp., vol. EMC-13, pp. 4 1 4 4 , May 1971. D. E. Merewether and T. F. Ezell, “The interactionofcylindrical posts and radiation-induced electric field pulses in ionizedmedia,’‘ IEEE Trans. h’uc. Sci., vol. NS-21, pp. 4-13, Oct. 1974. H. K . Schuman, “Time-domain scattering from a nonlinearly loaded wire,” IEEE Trans. Antennas Propagat., vol.AP-22,pp. 611-613, July 1974. T. K . Liu and F. M. Tesche, “Analysis of antennas and scatterers with nonlinear loads,” IEEE Trans. Antennas Propagat., vol. AP-24, pp. 131-139,Mar.1976. of nonlinearly T. K. Sarkar and D. D. Weiner, “Scattering analysis loaded antennas,” IEEE Trans. Antennas Propagat., vol. AP-24, pp. 125-131,Mar.1976. J. A. Landt, “Network loadingof thin-wire antennas and scatters in the time domain,” Radio Sci., vol. 16, no. 6, pp. 1241-1247, Nov.-Dec. 1981. J . A. Landt, E. K . Miller, and M. Van Blaricum, “WT-MBA/LLLIB: A computer program for the time-domain electromagnetic response of thin-wire structures,” Lawrence Livermore Nat. Lab., Livermore, CA, Rep. UCRL-51585, May 1974. E. K. Miller, A. J . Poggio, and G.J. Burke, “An integro-differential equation technique for the time-domain analysisof thin wire structhres: I. The numerical method,” J . Comput. Phys., vol. 12, pp. 24-28, May 1973. J . A.Landt,“Thetransientresponseofthin-wireantennasand scatterers loaded with circuits and nonlinear elements,” Los Alamos Nat. Lab., Los Alamos, NM, Rep. LA-8242-MS, Feb. 1980. ofthe E. K. Millerand J. A.Landt,“Short-pulsecharacteristics vol. AP-25, conical spiral antenna,” IEEE Trans. Antennas Propagat., pp. 621-626, Sept. 1977. -, “Directtime-domaintechniquesfortransientradiationand scatteringfromwires,” Proc. IEEE, vol.68,pp.1396-1423,Nov. 1980.

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-31, NO. 1 , JANUARY 1 9 8 3

E d m u n d K. Miller (S’60-M’6&SM’70-SM.S I) was born in Milwaukee, W1. in 1935. He received the B.S. degree with honors in electrical engineering in 1957 from Michigan Technological University,the M.S. degreeinnuclearengineering in 1958 and the M.S. and Ph.D. degrees in electrical engineering in 1961and1965.respectively,all from the University of Michigan, Ann Arbor. During 1958-1959 he taught in the Physics Department of MichiganTechnologicalUniversity. From1958to1965 he waswiththeRadiation Laboratory of the University of Michigan where he was involved in both experimental and analytical research concerning plasmaelectromagnetic wave interaction. He further pursued this line of research to with the High Altitude Research Laboratory of the university from 1966 1968 in the specific area of antenna probes used for ionospheric diagnostics. I n 1968hejoined M B Associates,SanRamon. CA. wereheworked on integral-equation methods for modeling antennas and scattering problems in both the time and frequency domains. His involvement in numerical techniques has continued at the Lawrence Livermore National Laboratory. Livermore. CA, nhich he joined in 197 I . and was instrumental in initiation of the to fostermore Computer Code Seasletter whichthelaboratorypublished effective dissemination and use of computer codes. .He is currently Division at theLanrenceLivermore LcadcroftheEngineeringResearchDivision Laboratory. Dr. h,liller is amember of EtaKappa Ku, TauBetaPi.SigmaXi.the American Physical Society, and Commissions A,B and C of the International

ScientificRadioUnion.Hehaslecturedwidely on computermethods in electromagnetics, has taught at several short courses on the subject, and is a past IEEE Distinguished Lecturer for the Antennas and Propagation Society.

Frederick J. Deadrick received the B.E.E. degree in electricalengineeringfromtheUniversityof Minnesota, Minneapolis, in 1962 the and M.S.E.E.degreeinelectricalengineering/comof California. putersciencefromtheUni%ersity Davis. in 1972. Hehas been employed by the Lawrence LivermoreKationalLaboratory.Livermore,CA.since 1962. I n 1971 he joined the Engineering Research Division‘s Electromagnetic Systems Research Group to develop numerical modeling techniques for antenna analysis and EMP damage assessments. Included in the electromagneticswork was the development of a time-domain EM modeling teat facility at the Laboratoryfor EMPmeasurements.He currently is the Lead Electrical Engineer for the plasma diagnostics group on the Mirror Fusion Test Facility.