INTERNATIONAL JOURNAL OF NUMERICAL MODELLING : ELECTRONIC NETWORKS, DEVICES AND FIELDS,

Vol. 4, 153-162 (1991)

TIME-DOMAIN INTEGRAL EQUATIONS FOR EMP ANALYSIS

R . GOMEZ, A. SALINAS, A. RUB10 BRETONES, J . FORNIELES AND M. MARTIN

Departamento de Fisica Aplicada, Grupo de Electromagnetismo, Universidad de Granada, Facultad de Ciencias, 18071 Granada, Spain

SUMMARY Two computer programs DOTIGl and DOTIG2 were developed to calculate, in the time-domain, the interaction of transient electromagnetic pulses (EMP) with perfect electric conductor structures modelled by thin wires (DOTIGI) or patches (DOTIG2). DOTIGl uses the electric field integral equation and DOTIG2 the magnetic field integral equation. Briefly described below are the numerical procedures used to develop both programs and some results that show their characteristics.

1. INTRODUCTION

Current developments in high-resolution radar and EMP technology have generated interest in the research of radiators and scatterers of transient waveforms, i.e. pulses of very short duration whose spectral content is concentrated primarily from zero frequency through the microwave region of the spectrum. These developments can be applied to some important areas like target identification via short-pulse radar, lightning EMP and nuclear EMP analysis and interference in high-speed digital systems.

At present, numerical methods are the principal means available for studying many practical transient electromagnetic problems. These methods either involve time-harmonic analysis coupled with Fourier inversion or a direct formulation in the time-domain and its subsequent solution using a time step algorithm.

The time-domain approach may offer computational advantages and throw light on the problem in a way that is not possible using frequency-domain techniques for example when a non-linear behaviour has to be modelled.”’

One of the main problems in the study of the interaction of EMP with structures is to determine the electric current flowing on the outer skin of the structures when illuminated by these pulses. The electromagnetic penetration of the structure through apertures can be calculated using several techniques such as those based on the equivalence principle or uniqueness

Although Maxwell’s equation provides the base for studying all electromagnetic problems, depending on how the equations are manipulated to obtain the starting-point equations, we can distinguish between integral and differential formulation of the problem. The chief differences are associated with the local character of the differential operator in contrast to the global character of the Green’s function in the integral operator. The differential equation form is adopted in the following methods: finite difference time-domain (FDTD),8. ’ finite element time-domain (FETD)’” and transmission-line matrix (TLM).’ Here, Maxwell’s time-dependent curl equations are directly solved in a volume region of space which fully contains the structure to be modelled. A separate numerical wave radiation condition must be employed to truncate the computation zone in order to resolve open problems. A major advantage of this formulation is the possibility of analysing larger and more electrically complex structures than the ones that can be studied using the integral equation formulation.

The integral equation forms are adequate for the resonance region, but not for higher frequencies when the body length is several times the wavelength of the highest significant frequency of the EMP. This is because the integral equation can quickly lead to very large matrices, and because it would be necessary to store a great number of current values calculated at previous time steps.

0894-3370191 I030 153-10$05.00 0 1991 by John Wiley & Sons, Ltd.

Received 19 December 1990 Revised 25 February 1991

154 R. GOMEZ ET AL.

Two important advantages of this formulation are that the integral operator includes an explicit radiation condition and that the computation space is generally limited to the surface of the structure.6* '

This paper is limited to integral formulations and shows some characteristics and results of the computer programs DOTIGl and DOTIG2 that calculate in the time-domain the interaction of transient electromagnetic pulses with perfect electric conductor. DOTIGl uses the electric field integral equation and DOTIG2 the magnetic field integral equation. 12. l3

2. ELECTRIC FIELD INTEGRAL EQUATION (EFIE)

To calculate the electric current induced on the surfaces of complex structures, a common technique is to model the surface geometry by a net of thin wires.'"16

DOTIGl computes, in the time-domain, the currents induced on a structure modelled by interconnected wires when it is excited by an arbitrary electromagnetic signal. The method employed consists of resolving the electric field integral equation (EFIE) using the moment method. This equation for a thin wire is:6,

~a I(S',f') + __ 7 I(S'$') - cR2 as

where b and b' are tangent vectors to the wire axis of contour c(s') at positions s(r)=s and s(r ' )=s' . I(s',t') and q(s ' , t ' ) are the unknown current and charge distribution at source point s' at retarded time t' = t -R/c and E'(s,t) is the field applied to the observation point R = It--r'l. The charge q(s ' , t ' ) can be expressed in terms of I(s',t') by the equation of continuity.

Using the point-matching form of the moment method and a nine-point Lagrangian interpolation as base function, the integral equation (1) can be transformed into the group of equations (in matrix n~ ta t ion ) : ' ~

which allows us to calculate the current I j at the time ti from the elements E f of the tangential electric field scattered by currents of previous times and the elements Ej of the tangential applied electric field at the observation point and at time step t,. Matrix 2 is a matrix of interaction whose elements are time-independent. They only depend on the geometry and on the electromagnetic characteristics of the structure.

The solution of equation (1) can be achieved using a timestepping procedure. It is possible to calculate the current in the present time interval in terms of already calculated values of the current in previous times, and the value of incident field in the present time interval.h. 7 , 12, l 3

2.1. Results

The response of some wire structures were analysed in the time-domain when illuminated by a Gaussian pulse of the form:

Ei = exp{ -g2(t-rmax)2} ( 3 )

The first structure studied was the net of twelve wires that is represented in Figure 1. The electric field incident was the temporal derivative of equation ( 3 ) multiplied by g with g = lo9 s-l and t,,, = 2X10p9 s . The polarization of the field was the x-direction and it was propagating in the z-direction (normal incidence).

The second example is the stick model of the aircraft shown in Figure 2 when illuminated by a Gaussian pulse of the form (3) with g=l.2X107 s-' and t,,, = 1.8X10-7 s. The polarization of the wave was the -a direction and the angle of incidence = 30". Single stick models are very useful for estimating the natural frequencies and natural axial current modes of an aircraft.3.

In Figures 3 and 4 the current distribution is represented against time at the point P on both structures. In order to compare our results with those obtained using the numerical electromagnetic

TIME-DOMAIN INTEGRAL EQUATIONS FOR EMP ANALYSIS 155

- L -

Figure 1. Geometry of a net structure with twelve interconnected wires. L = 0.2 m. Radius of the wires a = 6.74 mm. E n = 2(t-tm,,)g exp (-g2(t-f,ax)2}f; g = 1 ~ 1 0 ~ s-' and t,,,=2x10-y s

A

Figure 2. Geometry of the stick model of an aircraft. I , = 24 m, 1, = I , = i4 = 36 rn, 1, = 6 m , /6 = I , = 1, = 12 m. (u=S6.22", P=7O.Sl0 and y=68.2". +,,,=30". Radius of all the wires a = 1 m. E'=exp {-(t-rmaJ2g2); g = l .2x107s- '

and t,,,.,,=1~8X10-7 s

code (NEC),I9 working in the frequency-domain, the time-domain results were Fourier-transformed. In Figure 5 the current magnitude at point P on the net structure was plotted against frequency; NEC results are drawn with + symbols and agree closely with the DOTIGl ones. Figure 6 shows the Fourier transform of the current on each segment of the aircraft for a wavelength X=99-93 m (continuous lines perpendicular to the structure) and with NEC (shown as points).

Finally, we show how the study in the time-domain greatly facilitates the analysis of structures with non-linear loads. These devices are often employed to reduce the late-time response, and to protect equipment from excessive voltages and currents delivered by antennas, i.e. lightning protection and EMP hardening. ''3 l8

In the case of loaded antennas, the expression of equation (2) is easily performed by including the value of the load 2'; along the antenna:

156 R. GOMEZ E T A L .

0.0008 , 0.0008

0.0007

o.ooot3 0.0005

0.0004

0.0003 - .5 0.0002

5 o.Ooo1 h

U U 0 2

-0.0001

-0.0002

-0.0003

-0.0004

-0.o005

-0.oO06

-0.0007 f I I I I I I I I I I - 7 - 7 - 0 0.4 0.8 1.2 1.6 2 2.4

( lo€-8) TIME (5)

Figure 3. Current distribution against time at point P on the net structure of Figure 1. P is situated 2 cm from the upper left corner

W

W p1

0

* N

f 0

s-i

2 0

W ; * * f I

0.01 1.07 2 -14 3.20 4.27 5.33 6.39 7.46

T IRE (s) - l o 6 Figure 4. Current distribution against time at a point of the aircraft situated 9 m from the junction on wire 1 (point P)

With this equation the analysis of non-linear loads is simple. For example, for a loaded antenna with a diode of R , R in one direction, and R2 R in the reverse direction, the value of Z'j which we must introduce into equation (4) ( R , or R2) is determined by the sign of the current, obtained from equation (2).

As a numerical example, let us consider four thin parallel conducting wires of the same length L = l m, the same radius a=0.006 73 m, and parameter of antenna 0 = 2 ln(Lla)=lO, separated by a distance d=0.5 m. The antennas are excited simultaneously at their centre by a Gaussian

TIME-DOMAIN INTEGRAL EQUATIONS FOR EMP ANALYSIS

0.0016 , 1

0.0015

0.0014

0.0013

0.0012 -. 3 0.0011 L. Q 0.001 3

2 0.0008

0.0007

2 0.0006

0.0005 c

0.0004

0.0003

0.0002

0.0001

0

's 0.0009

5

?

157

FREQUENCY IMHz) DOTlGl + NEC

Figure 5 . Current magnitude versus frequency at P point on the net structure, obtained making the Fourier transform of the time response

0 . 2 5 mA I , ,

Figure 6. Current magnitude on the aircraft for A = 99.93 m. The continuous line was calculated from the Fourier transform of the time response. The dots result from the frequency domain code NEC

pulse of voltage V(t)=exp(-gZ(t-tm,x)2), where g=3XlOY S K ' and tm,,=1-43x lopy s. In Figure 7 the solid line shows the driving point current for an antenna at the centre of the array, and the broken line shows the current for one isolated antenna.

158

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3.

R. GOMEZ E T A L .

4-

3.

0.5 m

3 -1 . 0 v)

-2'

Figure driving

I

3 0 v)

-1 .

-2'

2 m

I

7. Driving point (

point current for

Figure 8 shows currents analogous to the previous figure, when the antennas are loaded with a diode of 0 R in one direction and 5 kn in the reverse direction. In Figure 8 a considerable reduction in the late-time response can be seen, compared to the current in an array of unloaded antennas (Figure 7).

3 . MAGNETIC FIELD INTEGRAL EQUATION (MFIE)

The MFIE equation can be more suitable for calculating the unknown surface current for smooth closed surfaces. This equation can be written as:6

4 E -

-3 1 0 6 12 18 24 30 36

TIME ( n s ) Figure 8. Driving point current of a linear antenna in an array of four pulse-excited antennas when the antennas are

loaded with a diode

TIME-DOMAIN INTEGRAL EQUATIONS FOR EMP ANALYSIS 159

where J(rr , t f ) is the current distribution on the surface S at a source point r r and at the retarded time t '=t-Rlc; r' is the integration point, R = lr-r'l and H' is the incident H-field, h1 is a unitary vector normal to the surface of the scatterer. An important feature of the MFIE equation is that it presents a kernel with both derivatives and singularities of a lower order than the EFIE. As a consequence, it is possible to employ base and testing functions that are simpler than the ones required for the electric field integral equation. Basically the scheme used to develop the DOTIG2 program is the same as the one used in DOTIG1. The unknowns are the samples of J on the surface of the structure subdivided into N patches of approximately equal area.*, l 9 The current density J on each patch is considered a constant at each timestep. The time-dependence is calculated using a fourth-order Lagrangian interpolation polynomial.

Figure 9 shows two examples with two closed surfaces each that were studied using the MFIE. The upper diagram represents two spheres with different radii. In the lower diagram the second sphere is replaced by a capped cylinder. In both cases the incident magnetic field was a Gaussian pulse like the one given in equation ( 3 ) with g=6X lo8 s-l and tmax=3.5x s coming from the left side, that is, the body on the right is hidden by the one on the left.

Figures 10 and 11 present, in the solid line, the current density at the point P on the surface of the first sphere, versus time, and in the symbol line the current at the same point if the second structure were not present. The difference between both curves is due to the effect of the second body's scattered field on the first sphere.

It can be observed from these figures how time-domain procedures provide the ability to separate effects that occur at different times due to the linear relation between time and distance. In both cases there is a clear time interval between the current due to the direct wave and the

_ _

7m

-i H

t

Figure 9. Geometry of the two cases studied with the MFIE. Upper diagram, two spheres; lower diagram, sphere and capped cylinder. The geometries considered include two closed surfaces each

160 R. GOMEZ ET A L .

~~~ 0.9 , I

0.8

0.7

0.6

0.5

0.4

03

0.2

0.1

0

-0.1 km- v m ~ n ~ r n y ~

0.2 42 8.2 12.2 162 20.2

c*tlme (ml __ sphere-sphere + sphere

Figure 10. Current density at the point P on the sphere of the upper diagram of Figure 9

0.8

0.7

0.6

0.5

0.4

03

0.2

0.1

0

-0.1 km- v m ~ n ~ r n y ~

0.2 42 8.2 12.2 162 20.2

c*tlme (ml __ sphere-sphere + sphere

Figure 10. Current density at the point P on the sphere of the upper diagram of Figure 9

0.9

0.8

0.7

0.6

0.5

0.4

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02

0.1

0

-0.1

c*t/me rml - spherecyllnder + sphere

Figure 11. Current density at the point P on the sphere of the lower diagram of Figure 9

one due to the reflection from the second structure. As a consequence it is possible to isolate effects from different structures by choosing the time window so that unwanted effects cannot disturb the desired data.

TIME-DOMAIN INTEGRAL EQUATIONS FOR EMP ANALYSIS 161

4. CONCLUSIONS

This paper explains the basis for the numerical procedures employed in the DOTIG1 and DOTIG2 codes for the study in the time-domain of the interaction of arbitrary electromagnetic waves with perfect conductor structures.

The electric field integral equation, EFIE, was reduced to a form suitable for numerical computation, using the subsectional collocation form of the method of moments, and using a two- dimensional Lagrangian interpolation function of order two in each dimension.

A similar procedure was employed to resolve the magnetic field integral equation, but using a fourth-order Lagrangian interpolation function for the time variable, and using a pulse function for the space variable.

Both programs were applied to several structures, showing some important features of the solution in the time-domain.

ACKNOWLEDGEMENT

This paper was partially supported by D.G.T. through PRONTIC TIC 89-0873E.

REFERENCES

I. E. K. Miller, Time-Domain Measurements in Electromagnetics, Van Nostrand Reinhold, New York, 1986. 2. R. Mittra (Ed.), Computer Techniques for Electromagnetics, Pergamon Press, Oxford, 1973. 3. K. S. H. Lee (Ed.), EMP Interaction, Report AFLW-TR-80-402, 1980. 4. R. A. Perala, T. Rudolph and F. Eriksen, ‘Electromagnetic interaction of lightning with aircraft’, iEEE Trans.

Electromag. Compat., EMC-24, 173-203 (1982). 5. Anton G. Tijhuis, Peng Zhongqiu and A. Rubio Bretones, ‘Transient excitation of a straight thin-wire segment: a

new look at an old problem’, submitted to the Special Issue of the Proceedings of IEEE on Electromagnetics, USA. 6. A. Rubio Bretones, ‘DOTIGI, un programa para el estudio en el dominio del tiempo de la interaccion de ondas

electromagneticas con estructuras de hilo’, Ph.D. Dissertation, University of Granada, 1988. 7. A. Salinas, ‘Aplicacion de la tkcnica monopulso a una agrupacion plana de antenas lineales excitada por pulsos

electromagntticos’, Masters degree dissertation, University of Granada, 1986. 8. A. Taflove and K . Umashankar, ‘The finite-difference time-domain method for numerical modelling of electromagnetic

wave interactions’, Electromagnetics, 10, 105-106 (1990). 9. S. Gonzalez, B. Garcia, R. Gomez and K. Umashankar, ‘Effects of FDTD second order radiation boundary condition

on the convergence and accuracy of electromagnetic fields’, to be presented at Antennas and Propagation Symposium, June 1991, London, Ontario, Canada.

10. J. Sovetri and G. Costache, ‘Time domain finite element analysis of 2-D shields’, Antennas and Propagation Symposium, Dallas 1990, Vol. I , pp. 18-21, 1990. IEEE Catalog No. 90CH2776-3 Library of Congress No. 89-80729.

11. C. Christopoulos and P. Naylor, ‘Coupling between electromagnetic fields and multiconductor transmission systems using TLM’, international Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 1, 31-43, 1988.

12. A. Rubio Bretones, A. Salinas, R. Gomez and A. Perez, ‘The comparison of a time-domain numerical code (DOTIGl) with several frequency-domain codes applizd to the case of scattering from a thin-wire cross’, Applied Computational Elecfromugnetic Society (ACES), Special Issue on Computer Code Validation, 121-129 (1989).

13. A. Rubio Bretones, R. G6mez and A. Salinas, ‘DOTIGI, a time-domain numerical code for the study of the interaction of electromagnetic pulses with thin-wire structures’, Compel, The lnternational Journal for Computation and Mathematics in Electrical and Electronic Engineering, 8, 39-69 (1989).

14. J. Moore and R. Pizer, Moment Methods in Electromagnetics Techniques and Applications, Research Studies Press, John Wiley, Chichester, 1983.

15. A. C. Ludwing, ‘Wire grid modelling of surfaces’, IEEE Trans. Antennas and Propagat., 9, 1045-1048 (1987). 16. A. Rubio Bretones, A. Salinas Extremera, J . Fornieles and R. Gomez Martin, ‘Comparison of different junction

treatments in the DOTlGl code’, Seventh International Conference on ‘Antennas and Propagation’, University of York, UK, 1991.

17. R. Gomez, J . A. Morente and A. Salinas, ‘Time-domain analysis of an array of straight-wire coupled antennas’, IEE Electronics Letters, 22, 316-318 (1986).

18. R. Gomez, J. A. Morente and A. Salinas, ‘Application of the monopulse technique to a planar array of nonlinear loaded straight-wire coupled antennas’, IEEE Trans. Electromugnetic Compatibility, 29, 169-174 (1987).

19. G. J . Burke and A. J . Poggio, ‘Numerical electromagnetics code (NEC)’, Lawrence Livermore Laboratory Contract W-7405-Eng-48, USA, 1981.

20. P. D. Smith, ’Instabilities in time marching methods for scattering cause and rectification’, submitted to Electromag- nefics.

Authors’ biographies:

Rafael Gomez Martin received the M.S. degree from the University of Sevilla (Spain) in 1971 and the Ph.D. degree (cum laude) from the University of Granada (Spain) in 1974, both in Physics. Between 1971 and 1974 he was at the University of Santiago de Compostela (Spain) as a Ph.D. student. Between 1975 and 1985 he was Assistant Professor at the University of Granada where, in 1986, he became Professor and Head of the Electromagnetism Group. His current interest is in numerical methods for solving electromagnetics problems, especially in the time-domain.

162 R. GOMEZ ET AL.

Alfonso Salinas Extremera received the M.S. degree from the University of Granada (Spain) in 1986. He is currently working towards the Ph.D. degree in electromagnetic theory and applied mathematics. He is at present Research Assistant in the Department of Applied Physics at the University of Granada.

Amelia Rubio Bretones received the M.S. degree in Physics in 1985 and the Ph.D. degree (cum Zuude) in 1988, both from the University of Granada (Spain). She worked as Assistant Professor at the University of Granada from 1986 to 1989 and as Associate Professor since then. Her research has focused on the study, mainly in the time-domain, of the interaction of electromagnetic waves with structures, using numerical methods.

Jesus Fornieles Callejon received the M.S. degree in Physics from the University of Granada (Spain) in 1990. He is currently working towards the Ph.D. degree in electro- magnetic theory and applied mathematics. At present he is working in the Electromag- netism Group at Granada. The electric field integral equation in the time-domain is his current interest.

Manuel Martin Reyes received the M.S. degree in Physics from the University of Granada (Spain) in 1990. He is currently working towards the Ph.D. degree in electro- magnetic theory and applied mathematics. At present he is working in the Electromag- netism Group at Granada. The magnetic field integral equation in the time-domain is his current interest.