IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 3, AUGUST 1994 20 1

Calculation of Radiated Electromagnetic Fields from Cables Using Time-Domain Simulation

David W. P. Thomas, Christos Christopoulos, Member, ZEEE, and Elisete T. Pereira

Abstruct- Radiated electromagnetic fields are produced by currents in cables or transmission lines interconnecting various circuits. An elegant method of computing the resultant electro- magnetic field, produced by several radiating current elements, is given. The current in each radiating cable is first found from a time-domain simulation algorithm and this may be a steady- state or transient current. The radiated field is then calculated by assuming a radiating transmission line can be treated as a chain of short radiating dipoles. The problems associated with the calculation of the near-zone term at low frequencies and the overall response near the radiator are clarified. The proposed technique is fully evaluated and compared with other methods.

I. INTRODUCTION

HE FIELDS generated by transmission lines such as T tracks on a printed-circuit board make a major con- tribution to the emitted spectrum of electromagnetic (EM) radiation. Electromagnetic compatibility (EMC) regulations specify maximum permissible radiated fields at various dis- tances from a device, e.g., 3, 10, and 30 m. At low frequencies and for distances of the order of a few meters the measurement point is almost invariably in the near field of the radiator. At such close range, electrostatic and inductive contributions to the response can be very significant. Formulas used to calculate directly fields generated from transmission lines are based on approximations which are good enough for antenna work but which should be improved for the purpose of EMC studies. The objective of this paper is to study the effect of commonly made approximations in the calculation of the electrostatic and inductive terms at low frequencies and near radiating elements. It will be shown that at the 3-m range significant errors are present in the calculations which may affect the certainty of EMC compliance. Improvements in the estimation of near- zone contributions and in the formulas used for near-field calculations are described in the next two sections.

II. RADIATION FROM DIPOLE ELEMENTS Radiating sections in an electrical circuit are considered

to be made up of a number of ideal radiating dipoles, short enough for the current to be approximated as constant along the length of each dipole. The total radiated electromagnetic field is the sum of all the contributions from each constituent dipole element. For a single short dipole element of length Z in free space (permittivity EO and permeability PO) with the

Manuscript received May 21, 1993; revised February 1, 1994. The authors are with the Department of Electrical and Electronic Engineer-

ing, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. IEEE Log Number 9401879.

co-ordinate system given in Fig. 1, the radiated fields can be obtained in the frequency domain as described in [l].

For time-dependent transient studies, however, radiated fields are more conveniently expressed and calculated in the time domain [2], [3] and for systems involving several radiating elements the Cartesian co-ordinate system is more convenient for the summation of the field contributions [4]. The expressions for the fields then become [4]

t 3 I ( r - T / C ) d r

7-3 Ez(t) = - + O

1 1 aI( t - T / C )

C2T a(t - T / C ) + -

1 1 aI( t - T / C )

C2T a(t - T / C ) + -

t

r / c ) d r

7-3 &(t) = - ~ ” + ~

47T€oT2 1 CT2

1 1 aq t - T / C )

C2T a(t - T / C ) + - (3)

H z ( t ) = - -qt - T / C ) + - 1 a r ( t - T / C ) ] (4) -lY 47T [ T 3 c T 2 a(t - T / C )

] ( 5 ) 1 a l ( t - T / C )

CT2 d( t - T / C ) H&) = - -qt - T / C ) + -

41r zx [’ T3

where t - T / C is the retarded time [(I, I is the current amplitude, and T is the distance of the /observation point at location (z, y, 2) from the dipole at the qrigin (0, 0, 0).

0018-9375/94$04.00 0 1994 IEEE

~~

202 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 3, AUGUST 1994

E, A

X

Fig. 1. A Hertzian dipole and the coordinate systems.

Therefore, from knowledge of the current in an isolated radiating dipole at the retarded time t - T / C , the radiated electromagnetic field can be found at time t at a field point P a distance T from the dipole center. The terms in (1)-(3) that are a function of charge (J I(t - T / C ) dt) lead to the near- zone field (this is an electrostatic field for dc). The terms in (1)-(6) which are a function of the current represent what is usually termed the induction fields and the terms involving the derivative of the dipole currents give the radiated fields.

These formulas have been used to determine the radiation field of isolated radiators such as the lightning channel [3]. They have also been used to determine the field due to transmission lines such as parallel tracks in printed-circuit boards, ribbon cables, etc. [6]. A typical configuration used in the latter application is shown in Fig. 2(a). In using these formulas the radiating properties of this circuit are assumed to be described by a collection of short dipole radiators as shown in Fig. 2(b). The length of each segment S1 to Sk is chosen to be short enough to permit the assumption that the current along each section is constant. Currents 11 to 1; which are in general time-varying are obtained by analytical or, more often, numerical means. For the examples presented in this paper the current in the time domain is obtained for each segment by simulation using the TLM method [2], [4]. In computing the electromagnetic field it is important to clarify the validity of the model shown in Fig. 2(b) and the assumptions inherent in (1)-(6). These aspects of the simulation are addressed below.

One difficulty with the model shown in Fig. 2(b) is that the fields due to the line terminations are not included so large uncompensated charges will appear at the extremities A, B and A’, B’. These charges, which do not appear in the original circuit shown in Fig. 2(a), establish a strong near-zone field and result in substantial errors at low frequencies. This point does not appear to have been emphasised enough in previously reported work [5], [6] and, as a result, low-frequency near fields have been overestimated. It should be pointed out that this problem does not arise when calculating the radiated field from isolated radiators, such as the lightning channel [3], as in such cases the solution is found by integrating the current over the entire current-carrying region.

It is possible to obtain the correct field solution by including the fields due to the terminations. It has been found that in such calculations phase information is very important and that the

A’S; Si s‘, 8‘ 9; +,+ +- - - - - -+

11 12 I’n (b)

Fig. 2. (a) A simple two-wire transmission line system. (b) The segmentation of a two-wire transmission line for field calculations.

terminations AA’ and BB’ have to be modeled in detail with at least six current segments. This may lead to too fine a time step and too long a computation time. One solution to the problem of the unnaturally large near-zone term due to the charge at the extremities, is to subtract from the fields obtained from (1H6) the electrostatic field of a dipole with charges placed at A-B and A’-B’. This correction is not fully effective as the near- zone term obtained from (1H6) is not accurate enough. This is particularly true near the radiators where the approximations in calculating the field of the Hertzian dipole give sufficient errors to make this kind of correction inadequate. Another, computationally intensive, solution is to obtain a continuous current waveform by interpolation and to calculate the field due to the entire closed circuit as suggested in [7].

An elegant alternative solution is to omit the electrostatic terms in (1H6) altogether and calculate the electrostatic field directly from the charge at the junctions of the constituent dipole elements (Qj,j+l). The charge at the junctions is given by the requirement of continuity

where Ij is the current in the j dipole element and Ij+l is the current in the adjacent j + 1 dipole element.

The electrostatic fields are then given by the well-known expressions for the electrostatic field of a point charge [l]

where T is now the distance from the dipole junction to the observation point P and z, y, z are the co-ordinates of P with respect to the dipole junction.

THOMAS et al.: CALCULATION OF RADIATED ELECTROMAGNETIC FIELDS

Using (7)-(10) provides reasonable accuracy for studies where the spacial resolution AR and the time resolution At are related by

AR = c h t . (1 1)

These fields are then summed for all the junctions of the radiating dipole elements and superimposed on the induction and radiation electric fields to give the total electric field amplitude. The charge at the extremities of the radiating body is taken to be approximately zero. This approach has been used to obtain the results presented in the next section.

When a perfectly conducting ground plane is present the method of images may be used fl]. The fields from the image dipole are then calculated in the same manner to that of the actual dipole, using (1HlO) where T is now the distance from the image dipole to the field point P.

III. RESULTS In the computation of the electromagnetic fields, using the

TLM model of the electrical circuit [2], all distances are rounded off to the nearest segment length 1. This does not introduce any significant errors as the segment lengths are significantly smaller than the minimum wavelength in the transient. Time and retarded time are also discretized in time steps At. This also does not introduce any significant errors as the maximum frequency in the transient is very much less than the inverse of the time step.

The field contributions from each dipole are delayed by the retarded time factor TIC associated with each dipole and then summed. Note that the origin of the axes used in (1)-(6) is at the center of each dipole element and the origin of the axes used in (7)-(10) is at the ends of the dipole elements. Since the Cartesian co-ordinate system is used, however, the field contribution in each axis due to each element can be simply added without further complication.

In order to demonstrate the precision and range of ap- plication of the technique, a number of simulations were performed including predictions of electromagnetic radiation from a lightning return stroke [3] and the results obtained were in good agreement with the analytic solutions. Results are presented below for the case of radiation from a two-wire transmission line.

The electromagnetic fields due to the differential current in a two-wire transmission line are calculated. The configuration used is as shown in Fig. 3. A two-wire transmission line of length I consists of two identical parallel wires of separation s. Paul and Bush [6] have derived analytic solutions for the electric fields parallel to the transmission line at a point a distance D away as shown in Fig. 3. For wavelengths very much greater than the transmission-line length, the analytic solution for the electric-field amplitude, given by Paul and Bush, is

where V, is the source voltage, R, is the source impedance, RI is the load impedance, and @ is the wavenumber (27~/X).

203

"s

Fig. 3. Geometry of a simple two-wire transmission-line system for which remote fields at a distance D are calculated. The observation point is on the plane of the line.

The terms proportional to p2 and are the radiation and the induction terms, respectively, and the constant term is the predicted near-zone field amplitude.

In deriving (12), Paul and Bush [6] assumed that the two- wire transmission line behaves as two independent isolated dipoles and that their physical separation significantly affected the phase only of the field contribution from each wire. It will be shown that these two assumptions lead to the near- zone term being overestimated and the other terms being underestimated.

For higher frequencies, the electric field can be approxi- mated by the radiation electric field term which is given in 161.

where R, is the line characteristic impedance, p~ is the traveling-wave voltage reflection coefficient at the load, and p s is the traveling-wave voltage reflection coefficient at the source.

Equation (13) allows for the variation in the current ampli- tude along the transmission line and can be compared with the electric field predicted assuming the transmission line behaves as a loop antenna, which is given in [6]

where A is the area and I is the loop current. Fig. 4(a) shows the results obtained in [6] for frequencies

below 100 MHz (using (12)) and Fig. 4(b) shows the results obtained using the computational scheme described here. Fig. 5 shows the results for frequencies above 100 MHz. The results shown are for a two-wire line 0.5 m in length and a 0.006-m separation with the observation point 3 m away. The transmission line was split into 100 line sections of length 0.005 m. This required a time step of 16.667 ps which gave a phase resolution of 0.005/A. This resolution was chosen as a good compromise between the need to describe the line length and the wire separation and did not affect significantly the

204 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY. VOL. 36, NO. 3, AUGUST 1994

-;i‘50

Electric field (D=3.0 metres)

60 r ,’

R a d i a t i o n <* -, , , , ‘ -

0 ‘ I 2

1 0 2 3 4 5 6 7 8 9

10’ Frequency (MHz)

(a)

Electric f ield (D=3.0 metres)

2 102

3 k 5 6 7 8 9 -30 I

1 0‘ Frequency (MHz)

(b)

Fig. 4. Predicted electric-field magnitudes at a distance 3 m from the transmission line for frequencies between 10 and 100 MHz.. (a) Field magnitude given by Paul and Bush [6]. (b) Computed field magnitude using the proposed algorithm.

Electric Field (D =3.0 metres) radiotion term only

c o m p u t e d r e s u l t s ’ ::r , , , , -i’, , , , , ,

Pau l & Bush 30

20 3 I 5 6 7 8 9

1 0 2

frequency (Muz)

Fig. 5. Predicted electric-field magnitudes at a distance 3 m from the transmission line for frequencies between 100 M H z and 1.0 GHz.

accuracy of the results. The computation was carried out for a period long enough for a steady state to be established along the transmission-line length. The current in the “return line” segments was assumed to be equal and opposite to the current in the corresponding “line” segments of the two-line system.

It can be seen that at frequencies above 100 MHz (Fig. 5) there is very good agreement between computed electric fields using the scheme described here and those predicted by the analytical solution (13) given by Paul and Bush

[6]. At frequencies below 100 MHz (Fig. 4), there is a marked difference between the analytic solutions and the computed solutions. Examination of the results shows that the disagreement is due to the assumptions used in obtaining the analytic solutions.

The first assumption, used by Paul and Bush [6], is that at low frequencies the two lines can be treated as two in- dependent isolated dipoles. This leads to a large oscillating charge at the line ends giving rise to the near-zone term. At low frequencies the voltage is approximately uniform along the line giving a uniform charge distribution. For a uniform charge distribution there is no near-zone field parallel to the transmission line. This is apparent from the low near-zone fields calculated using the scheme described here as shown in Fig. 4(b). At higher frequencies, the near-zone field does approach the value predicted by Paul and Bush but this field is still significantly less than the radiation electric field and can thus be neglected.

The second assumption, used by Paul and Bush [6], is that the electric field amplitude observed is essentially due to the phase difference (p . s) between the fields produced by the two wires. At low frequencies, however, this phase difference becomes less important and the amplitude variation due to the 1 / D and the 1/D2 terms assumes greater importance. If the correction to the amplitude term is taken into account then the induction and radiation field terms are greater (by approximately 5 dB) than those predicted by Paul and Bush [61.

At low frequencies, the expression for the electric fields should, therefore, be of the form

] (15)

1 3opl e - j P D ~ - - I P ( D + S )

I E ’ = (R, + Rz) [T - D + s 301 e - ~ P D e - ~ P ( D + ~ )

+ (R, + Rz) [+F + j ( D + s ) ~

where the terms proportional to the wavenumber /3 give the radiation component and the other terms are the induction components. Thus the first two terms are the radiation field terms and the last two terms are the induction terms. From (15) the amplitude of the induction field can be approximated by

d m (16)

and the amplitude of the radiation field can be approximated by

3011 P i n d l

30p11 &G-p@p. (17) IErd’ D ( D + s )

The results obtained using (16) and (17) for the two-wire transmission-line system under study are given in Fig. 6. This figure shows very good agreement with the computed electric fields given in Fig. 4(b).

IV. CONCLUSION A computational scheme has been described which is suit-

able for calculating the electromagnetic fields produced by transient currents in an electrical circuit. Radiating sections of

THOMAS et al.: CALCULATION OF RADIATED ELECTROMAGNETIC FIELDS 205

Electric Field (D = 3.0 metres)

5 0 1 Radiation -,

-20

I 2

102 3 I 5 6 7 0 9

-30 ‘ 10’

Frequency (MH2)

Fig. 6. Results obtained for the electric-field amplitude at a distance 3 m from the transmission line using (15) and (16) which include phase and attenuation with distance.

the electrical circuit are assumed to behave as ideal radiating antennas with a transient current given from a time-domain simulation of the circuit. In this example, the simulation procedure used a TLM [2], [4] algorithm to derive the transient current although any other time-domain transient simulation algorithm can also be used.

A simple elegant method of estimating the electrostatic term was proposed which removes errors due to approximations in the field computations and in the model. Examination of the results shows good agreement with analytic solutions.

From a study of the computed results obtained an improved estimate of low-frequency fields can be made. The computa- tional scheme proposed here is a simple but effective method for electromagnetic compatibility studies. It compliments the analytical solution of simple systems and its generality enables solutions to be easily found for conditions where analytical results are difficult to obtain.

REFERENCES

111 J. D. Kraus, Electromagnetics, 3rd ed. New York: McGraw-Hill, 1984. [21 P. B. Johns and M. O’Brien, “Use of the transmission line modelling

(TLM) method to solve non-linear lumped networks,” Radio Electron. Eng., vol. 50, no. 12, pp. 59-70, Jan./Feb. 1980.

[3] M. Rubinstein and M. A. Uman, “Methods for calculating the elec- tromagnetic fields from a known source distribution: application to

lightning,” IEEE Trans Electromagn. Compat. , vol. 31, no. 2, pp.

[4] E. T. Pereira, D. W. P. Thomas, A. F. Howe, and C. Christopoulos, “Computation of electromagnetic switching transients in a substation,” in IEE Int. Conf on Computation in Electromagnetics (Nov. 25-27, 1991), IEE London, Conf. Pub. 350, pp. 331-334.

[5] J. Kujalowicz, “The field of interference around a domestic power network for short distance,” in 8th Zurich Symp. on Electromagnetic Compatibility, 1989, paper 8a-c5, pp. 87-89.

[6] C. R. Paul and D. R. Bush, “Radiated fields of interconnected cables,” in Int. Conf on Electromagnetic Compatibility (University of Surrey, Sept. 18-21, 1984), E R E Pub. 60, pp. 259-264.

[7] E. K. Miller, A. J. Poggio, and G. J. Burke, “An integro-differential equation technique for time-domain analysis of thin wire structures,” J. Compufat. Phys., vol. 12, pp. 24-48, 1973.

183-189, 1989.

David W. P. Thomas, was bom in Padstow, UK, on May 5,1959. He received the B.Sc. degree in physics from Imperial College of Science and Technology, London, UK, the M.Phil. degree in space physics from Sheffield University, Sheffield, UK, and the Ph.D. degree from Nottingham University, Nottingham, UK, in 1981, 1987, and 1990, respectively.

In 1984 he joined the Department of Electrical and Electronic Engineering at the University of Nottingham as a Research Assistant, where he is now a Lecturer.

Christos Christopoulos (M92) was born in Patras, Greece, on September 17, 1946. He received the Diploma in Electrical and Mechanical Engineering degree from the National Technical University of Athens, Athens, Greece, in 1969, and the M.Sc. and D. Phil. degrees from the University of Sussex in 1970 and 1974, respectively.

In 1974 he joined the Arc Research Project at the University of Liverpool, Liverpool, UK, and spent two years working on vacuum arcs and breakdown while on attachment to the UKAEA Culham Laboratories. In 1976 he joined the University of Durham as a Senior Demonstrator in Electrical Engineering Science. In October 1978 he joined the Department of Electrical and Electronic Engineering, University of Nottingham. His research interests are in Electrical Discharges and Plasmas, Electromagnetic Compatibility, Electromagnetics, and Protection and Simulation of Power Networks.

Elisete T. Pereira was born in Brazil, on January 23, 1956. She received the B.Sc. degree in electrical engineering from Universidade Fed. de Santa Cata- rina, Brazil, the M.Sc. degree in control of power systems from the University of Waterloo, Waterloo, Ont., Canada, and the Ph.D. degree from Nottingham University, Nottingham, UK, in 1979, 1982, and 1993, respectively.

She is now Professor of Electrical Engineering in the Department of Electrical Engineering, PURB-Universidade Regional de Blumenau, Brazil.