3- (2002) Thin Wire Representation in Finite Difference Time Domain Surge Simulation

840 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 17, NO. 3, JULY 2002

Thin Wire Representation in Finite DifferenceTime Domain Surge Simulation

Taku Noda, Member, IEEE, and Shigeru Yokoyama, Fellow, IEEE

AbstractSimulation of very fast surge phenomena in athree-dimensional (3-D) structure requires a method based onMaxwells equations, such as the finite difference time domain(FDTD) method or the method of moments (MoM), because cir-cuit-equation-based methods cannot handle the phenomena. Thispaper presents a method of thin wire representation for the FDTDmethod which is suitable for the 3-D surge simulation. The thinwire representation is indispensable to simulate electromagneticsurges on wires or steel flames of which the radius is smaller than adiscretized space step used in the FDTD simulation. Comparisonsbetween calculated and laboratory test results are presented toshow the accuracy of the proposed thin wire representation. Thedevelopment of a general surge analysis program based on theFDTD method is also described in the present paper.

Index TermsElectromagnetic transient analysis, finitedifference time domain (FDTD) methods, Maxwell equations,simulation, surges, wire.

I. INTRODUCTION

CONVENTIONAL surge problems have successfullybeen solved by the circuit theory, where transmissionlines consisting of wires parallel to the earth surface are mod-eled by distributed-parameter circuit elements and the othercomponents by lumped-parameter circuit elements [1]. Thedistributed-parameter circuit theory assumes the plane-wavepropagation that is a reasonable and accurate approximationfor the transmission lines, and this assumption enables thehandling of the electromagnetic wave propagation within thecircuit theory. On the other hand, very fast surge phenomena ina three-dimensional (3-D) structure, which includes surge prop-agation in a transmission tower and in a tall building, cannotbe approximated by the plane-wave propagation. Thus, thosephenomena cannot be dealt with by the circuit theory but needto be solved by Maxwells equations as an electromagnetic fieldproblem. Nowadays, the surge propagation in a transmissiontower needs to be analyzed for economical insulation design.Furthermore, in a tall building, it is also important to assessthe interference of lightning surges with information devicesinside the building.

In order to solve the very fast surge phenomena in a 3-Dstructure as an electromagnetic field problem, the finite dif-ference time domain (FDTD) method [2], [3] and the methodof moments (MoM) [4], [5] are currently available as practical

Manuscript received August 15, 2001.T. Noda is with the Department of Electrical Insulation, Central Research

Institute of Electric Power Industry (CRIEPI), Tokyo 201-8511, Japan (e-mail:[email protected]).

S. Yokoyama is with Central Research Institute of Electric PowerIndustry (CRIEPI), Komae-shi, Tokyo 201-8511, Japan (e-mail:[email protected]).

Publisher Item Identifier S 0885-8977(02)05920-4.

choices. In surge simulations, accurate modeling of a thin wireis necessary to represent transmission wires and steel frames ofa building. Furthermore, an imperfectly conducting medium isrequired to be accurately modeled to represent currents in theearth. Comparing the theories of FDTD and MoM, the formeris more advantageous to handle 3-D currents in an imperfectlyconducting medium such as earth soil without any difficulty,even if the medium is nonhomogeneous. This is reported in de-tail in [6]. On the other hand, the latter is more advantageous toaccurately represent the thin wire.

This paper proposes a method to accurately represent thethin wire in the FDTD simulation (thin wire is definedas a conductive wire of which the radius is smaller thanthe size of a discretized cell used in FDTD). The proposedmethod corrects adjacent electric and magnetic fields alonga thin wire according to its radius taking into account thediscretization error of FDTD. This correction gives accuratesurge impedance of the thin wire, which is very importantfor surge simulations. In this regard, the proposed method isdifferent from a method by Umashankar et al. [7] which isused for the calculation of fields scattered by a thin wire.The Umashankar method corrects only magnetic fields withoutconsidering the discretization error. By utilizing the proposedthin wire representation, FDTD is able to model both thethin wire and the earth currents accurately, although MoMcannot handle the earth currents except in simple configurations[8]. This paper first describes the proposed method, and thencomparisons between calculated and laboratory test results areshown to validate the method.

II. REVIEW OF THE FDTD METHOD

A. FormulationThere exist several different formulations of FDTD method.

In order to precisely describe the proposed method of thin wirerepresentation, the formulation used in this paper is brieflyreviewed here. Assuming neither anisotropic nor dispersivemedium in the space of interest, the Maxwell equations in theCartesian coordinates are

(1)and (2)

whereelectric field;magnetic field;charge density;permittivity;

0885-8977/02$17.00 2002 IEEE

NODA AND YOKOYAMA: THIN WIRE REPRESENTATION IN FDTD SURGE SIMULATION 841

permeability;conductivity.

The space of interest is a rectangular-parallelepiped, and it isdiscretized by a small length (referred to as the space stephereafter) in all the directions. As a result, the space is filledwith cubes of which the sides are , and each cube is called acell. Fig. 1 shows the cell with the configuration of electric andmagnetic fields that are considered to be constant within the cell.In (1), the derivatives with respect to , , and are replaced bya central difference formula

(3)

and the derivatives with respect to time are replaced by

(4)

where denotes a component of or . Assumingthat the electric fields are calculated at time steps

and the magnetic fields atby turns, we finally

obtain (5)(10), shown at the bottom of the page (an approx-imation is employed in thederivation). denotes component electricfield at position , , , and at

(5)

(6)

(7)

(8)

(9)

(10)


Fig. 1. Configuration of electric and magnetic fields in cell.

time , and the other components are expressed in thesame manner. Coefficients , , and are given by

(11)

Equations (5)(11) are the FDTD formulas of the Maxwellequations [2], [3]. Although (2) is not explicitly formulated, itis proven that (5)(11) automatically satisfies (2) [3].

B. Time Step and Space StepEquations (5)(10) are considered as numerical integration,

and stable integration is performed if the following condition issatisfied (Courants condition) [3]

(12)

On the other hand, the grid dispersion error is minimized whenthe above relation is an equality. Thus, the following formula isused in all calculations in this paper to determine time stepby user defined space step :

(13)

is a small positive value specified by a user in order to preventinstability of the numerical integration due to round-off error in(5)(10).

III. PROPOSED THIN WIRE REPRESENTATION

If the space step were chosen to be small enough to representthe shape of wires cross section, an accurate representationwould be possible. However, it requires impractical compu-tational resources at this moment. The thin wire is definedas a conductive wire of which the radius is smaller than thesize of a cell in the FDTD simulation. In antenna simula-tions, the thin wire is mainly used to represent an antennaelementthe most important part. In surge simulations, it isalso important to represent wires (phase and ground wires ofa transmission/distribution line) and steel frames of a buildingalong which surges propagate. Umashankar et al. proposeda method of thin wire representation by correcting the adja-cent magnetic fields of the wire according to its radius [7],and [9] reports that the method is valid for the calculationof radiated fields by an antenna. However, the Umashankarmethod cannot give accurate surge impedance, as shown inSection V-B of this paper.

Fig. 2. Thin wire and configuration of adjacent electric and magnetic fields.

A. Modification of Permittivity and PermeabilityThe proposed method of thin wire representation that cor-

rects both the adjacent electric and magnetic fields of the wireaccording to its radius gives accurate surge impedance. The cor-rection of the fields is carried out by equivalently modifyingthe permittivity and permeability of the adjacent cells. Fig. 2(a)shows a wire with radius placed in the direction, and thepermittivity and permeability of the space are and . Fig. 2(b)shows the cross section of the wire with the adjacent electricfields, and Fig. 2(c) with the adjacent magnetic fields. In theFDTD method, a wire is, in principle, represented by forcingthe electric fields along the center line of the wire to be zero,and s are forced to be zero in this case. Calculated electricand magnetic fields around the wire are in a certain distribution(including the effects of space and time discretization), and thedistribution coincides with a real one around a wire with radius

. In other words, is considered to be the radius of whichthe real distribution of electric and magnetic fields around thewire is the same as one obtained by the FDTD method by simplyforcing electric fields along a line to be zero, and is called theintrinsic radius in this paper (the value of is evaluated later).Therefore, in order to represent the desired radius , permittivity

to calculate the adjacent electric fields , , ,[see Fig. 2(d)] is multiplied by a correction factor , and alsopermeability to calculate the adjacent magnetic fields ,

, , [see Fig. 2(e)] is divided by the same factor ,based on the fact that forcing zero electric fields along the wireautomatically gives the intrinsic radius . The correction factor

is, of course, a function of .


Fig. 3. Electric field around thin wire.

B. Correction Factor and Intrinsic RadiusThe correction factor is determined so that the four adjacent

electric fields of Fig. 2(b) are equal to those of Fig. 2(d),and also so that the four adjacent magnetic fields of Fig. 2(c)are equal to those of Fig. 2(e). Because the distance betweenthe wire and boundary is , the distance is theoreticallyshort enough to regard the electric field perpendicular to thewire surface as inversely proportional to the distance from thecenter of the wire in the region between the wire surface and

. Therefore, is an approximate equipotential surface withrespect to the wire, and can be determined by equating thecapacitance (between the wire and ) of Fig. 2(b) and (d).Assuming that is a cylinder with radius for simplicity,the following equation must be satisfied

(14)

Thus, the correction factor is obtained as

(15)

The modified permittivity of Fig. 2(d) gives the same capac-itance value as Fig. 2(b) with desired radius and with originalpermittivity . In the same manner, it can also be derived that themodified permeability of Fig. 2(e) gives the same induc-tance value as Fig. 2(c) with desired radius and with originalpermeability . The above theory is based on that the electricand magnetic fields are electrostatic and magnetostatic respec-tively in the vicinity of the wire (within boundary ).

Next, we evaluate the value of the intrinsic radius . Fig. 3(a)shows a thin wire in an FDTD calculation, and the electric fieldsalong the thin wire are simply forced to be zero without thepermittivity and permeability corrections described previously.Fig. 3(b) is a current waveform converging into a constant value

, and this current is flowed in the thin wire in the FDTD cal-culation. We take notice of electric field strength andof which the direction is perpendicular to the thin wire. Theactual FDTD calculation gives after the currentreaches to sufficiently, in the case that is normalized tounity, as given in Appendix A. It should be noted that the value

also takes into account the effects of the discretiza-tion, i.e., the finite difference formulation, with respect to timeand space. Because the current reaches the constant value ,

the vicinity electric field perpendicular to the thin wire is an-alytically given as in inverse proportion to distance from thecenter of the wire

(16)

This is also normalized as . Fig. 3(c) shows the curveof (16) and electric fields , , and calculated by theFDTD calculation as shown by circles. The circles farther than

agree well with the curve (even in farther region which is notshown in the figure). Because represents the electric field inthe range between and (the origin of is at thecenter of the thin wire), the potential difference betweenand obtained by the FDTD calculation is thatcorresponds to the area enclosed by a broken line in Fig. 3(c). Onthe other hand, the analytical expression (16) gives the potentialdifference in the following form:

(17)

Equating the above expression to gives

(18)This is the value of the intrinsic radius of the FDTD thin wirerepresentation. Substituting (18) into (15) gives the final form ofthe correction factor

(19)

The proposed thin wire representation is summarized as follows.1) Preceding an FDTD calculation itself, the correction

factor of each thin wire is calculated due to (19).2) In the FDTD calculation due to (5)(11), electric fields

around each thin wire are calculated using the modifiedpermittivity .

3) In the same manner, magnetic fields around each thinwire are calculated using the modified permeability

.

C. Theoretical Comparison with the Umashankar MethodThe Umashankar method is based on the concept of the

subcell [3], [7]. According to the subcell concept, theintrinsic radius of the FDTD thin wire representation is

which is different from what we


obtained in (18). The subcell concept applies Faradays elec-tromagnetic induction law to a region smaller than , wherethe effects of the discretization with respect to time and spaceare significant. However, the subcell concept does not takeinto account the discretization effects. On the other hand, theproposed value of the intrinsic radius given in (18) takes intoaccount the discretization effects as mentioned above.

Another difference is that the Umashankar method modifiesmagnetic fields only. This difference may come from the differ-ence of purposes. The main purpose of the Umashankar methodis to accurately simulate electromagnetic fields scattered by thinwires, where the surge impedance is unimportant [7], [9]. Onthe other hand, the proposed method modifies both the elec-tric and magnetic fields in order to precisely simulate the surgeimpedance.

IV. GENERAL SURGE ANALYSIS PROGRAMThe development of a general surge analysis program, named

Virtual Surge Test Lab. (VSTL), which is based on the FDTDmethod and the proposed thin wire representation method, isbriefly described in this section. Sections IV-AE feature thedeveloped program.

A. Treatment of BoundariesEach boundary of the space of interest can independently

be defined as a perfectly conducting plane or an absorbingplane. The perfectly conducting plane can easily be representedby forcing the tangential components of electric fields at theboundary to be zero. The second-order Liaos method is usedto represent the absorbing plane, because it is more accurateand requires less memory compared with other methods [10].An open space can be assumed by applying the absorbing planeto all the boundaries of the space of interest.

B. Imperfectly Conducting EarthThe goal of surge analysis is usually to find the solution of

surge propagation in a 3-D skeleton structure above an imper-fectly conducting earth. In the FDTD calculation, the represen-tation of the imperfectly conducting earth with resistivity canbe achieved by simply setting the value of in (11) to inthe region defined as the earth soil.

C. Rectangular-Parallelepiped ConductorsThe geometrical shape of most power equipments can be

represented by a combination of several rectangular-paral-lelepiped objects. The rectangular-parallelepiped conductor issimply modeled by forcing the tangential electric fields on itssurface to be zero.

D. Localized Voltage and Current SourcesUnlike the static electric fields, the transient electric fields

do not satisfy . Thus, in the analysis of transientfields, the voltage or the voltage difference do not make sensein general. However, if we take notice of an electric fieldcomponent of a cell, the voltage difference across a side ofthe cell can reasonably be defined as , becausewaves of which the wave length is shorter thando not present in the FDTD calculation due to the bandwidth

Fig. 4. Calculation procedure of developed program.

limitation of . Based on this fact, a localized voltage sourcewith and without its internal resistance can be modeled in theFDTD calculation as in [3].

In the case of a current source, because current itself is ageneral quantity even in the transient fields, a localized currentsource with and without its internal resistance can also be mod-eled as in [3].

E. Calculation Procedure and OutputThe flowchart of the calculation procedure of the developed

program is shown in Fig. 4. The output of the program includesthe waveform of localized voltage differences and currentintensities at a specified position in a specified direction, andan animation of electric or magnetic field distribution in anarbitrary section is also included. The visualization of theanimation is carried out with the help of MATLAB.

V. SIMULATION RESULTSA. Horizontal Conductor System

Fig. 5 shows a horizontal conductor system, one of themost fundamental elements of the surge analysis, where a thinwire conductor with radius and length m is placed


Fig. 5. Conductor arrangement (horizontal conductor system).

Fig. 6. Calculated and measured waveforms (horizontal conductor system):(a) waveform of voltage source used in the simulation was obtained as piecewiselinear approximation of measured open voltage; (b) and (c) comparison betweenmeasured and calculated waveforms.

above a copper plate at height . The horizontal conductoris excited by a pulse generator (PG) of which the internalresistance is 50 , and connected via a vertical lead wire. Inthis configuration, voltage and current waveforms at PG weremeasured, and an FDTD simulation was also carried out. Inthe simulation, the dimensions of the analysis space were 2 m,6 m, and 2 m in the , , and directions, respectively, and thespace step was 5 cm. The time step was determined by (13)with . All the six boundaries were treated as thesecond-order Liaos absorbing boundary, and the resistivity oftwo-cell layers at the bottom of the analysis space was set to

simulating the copper plate. PG was modeledas a -direction voltage source with its internal resistance 50in series, of which the waveform was given by a piecewiselinear approximation of its measured open voltage as shownin Fig. 6(a). The radii of the horizontal conductor and thevertical lead wire were taken into account by the proposedmethod. Fig. 6(b) and (c) show the measured and calculatedwaveforms in the case of cm and cm, and thecalculated waveforms agree well with measured ones. Fig. 7 isthe electric field distribution at ns on the conductor plane (a snapshot of its animation visualized by MATLAB).

Fig. 7. Electric field strength at t = 20 ns (horizontal conductor system);electric field strength in [V/m] corresponding to gray scale at the bottom; unitof vertical and horizontal axes is in cells (s = 5 cm).

Fig. 8. Comparison between calculated and measured surge impedance;different density (gray scale) indicates different conductor height.

This shows an instance that the reflected wave is about to goback toward the sending end.

B. Accuracy of Surge ImpedanceFig. 8 shows a comparison of surge impedance between

measured and calculated values with varying and . Thecalculated values are obtained both by the proposed method andby Umashankars method. The surge impedance was definedas the average value between ns. It is obvious thatthe proposed method is far more accurate than Umashankarsmethod.

C. Vertical Conductor SystemThe modeling of a vertical conductor is important as a basis of

transmission tower modeling. Fig. 9 shows a vertical conductorsystem consisting of four cylindrical pipes each of which the ra-dius is 16.5 mm. This is the same configuration in which a mea-surement was carried out in [11]. The vertical conductors areexcited by a PG through a current lead wire, and the tower-topvoltage is defined as the voltage between the tower top and avoltage measuring wire. In the simulation, the dimensions of


Fig. 9. Conductor arrangement (vertical conductor system).

Fig. 10. Calculated and measured waveforms (vertical conductor system).

the analysis space were 9.09 m all in the , , and directionswith space step cm. The time step was determinedwith , and all the six boundaries were treated as thesecond-order Liaos absorbing boundary. The thickness and theresistivity of the earth were set to 3.03 m and 1.69 10 m.PG was modeled by a current source, with internal resistance5 k , of which the waveform was given in Fig. 10(a). Fig. 10(b)and (c) show waveforms of tower-top voltage and current, andthe calculated results agree well with measured ones. Undera different condition that four 1.01-m vertical grounding elec-trodes are attached to the tower feet in the earth and the earthresistivity is set to 100 m, another simulation was carried out.Fig. 11 shows the magnetic field distribution at ns onthe tower plane, when a part of the incoming wave reflectsat the earths surface and the remaining part penetrates into theearth.

D. Computation TimeIt may be believed that the FDTD method is a time-consuming

method. However, the progress of computers in terms of speed

Fig. 11. Magnetic field strength at t = 35:5 ns (vertical conductor system);expressed by gray scale gradation from dark black = 0 A/m to clear white= 0.1 A/m; unit of axes is in cells (s = 10:1 cm).

and memory is considerable, and even a personal computercan be used for the FDTD calculations. In fact, the simulationspresented in this paper were performed by a personal computerwith Pentium III 600 MHz CPU and 256 MB RAM. Thecomputation time for the horizontal conductor case is 2 minand 53 s, and that for the vertical conductor case is 8 minand 35 s.

VI. CONCLUSIONS

In this paper, a method of thin wire representation in theFDTD calculation was developed, and it was shown by acomparison with a laboratory test result that the new methodgives more accurate surge impedance than previously proposedUmashankars method. This paper also described the devel-opment of a general surge analysis program using the FDTDmethod incorporating the new thin wire representation method.Two conductor systems, a horizontal conductor system anda vertical conductor system, were analyzed by the developedprogram, and its accuracy was validated by comparisons be-tween the simulation results and corresponding laboratory testresults.

APPENDIXGENERAL VALIDITY OF

Although was obtained with a particular valueof , is generally valid regardless of values ofas long as (13) is used to determine because of the followingreason: A thin wire is placed in the space where . In suchspace, and always appear in the form of in theFDTD formulas (5)(11), and the ratio is always fixedto be by (13). Thus, any practical value of gives

as long as is determined by (13).

ACKNOWLEDGMENT

The authors wish to thank R. Yonezawa, Tokyo Universityof Agriculture and Technology, and H. Arai, CRIEPI, for theircontributions and Drs. T. Shindo, Y. Sunaga, and K. Tanabe,CRIEPI, for their valuable discussions.


REFERENCES[1] H. W. Dommel, Digital computer solution of electromagnetic transients

in single- and multi-phase networks, IEEE Trans. Power App. Syst., vol.PAS-88, pp. 388399, Apr. 1969.

[2] K. S. Yee, Numerical solution of initial boundary value problems in-volving Maxwells equations in isotropic media, IEEE Trans. AntennasPropagat., vol. AP-14, pp. 302307, May 1966.

[3] K. S. Kunz and R. J. Luebbers, The Finite Difference Time DomainMethod for Electromagnetics. Boca Raton, FL: CRC, 1993.

[4] R. F. Harrington, Field Computation by Moment Methods. New York:Macmillan, 1968.

[5] G. J. Burke and A. J. Poggio, Numerical Electromagnetics Code(NEC)Method of Moments: Lawrence Livermore Laboratory, 1981.

[6] K. Tanabe, Novel method for analyzing the transient behavior ofgrounding systems based on the finite-difference time-domain method,in Proc. IEEE Power Engineering Society Winter Meeting, vol. 3, 2001,pp. 11281132.

[7] K. R. Umashankar et al., Calculation and experimental validation ofinduced currents on coupled wires in an arbitrary shaped cavity, IEEETrans. Antennas Propagat., vol. AP-35, pp. 12481248, Nov. 1987.

[8] G. J. Burke and E. K. Miller, Modeling antennas near to and pene-trating a lossy interface, IEEE Trans. Antennas Propagat., vol. AP-32,pp. 10401049, Oct. 1984.

[9] T. Kashiwa, S. Tanaka, and I. Fukai, Time domain analysis ofYagi-Uda antennas using the FDTD method, IEICE Trans. Commun.,vol. J76-B-II, pp. 872872, Nov. 1993.

[10] Z. P. Liao, H. L. Wong, B.-P. Yang, and Y.-F. Yuan, A transmittingboundary for transient wave analysis, Science Sinica, Series A, vol. 27,no. 10, pp. 10631063, 1984.

[11] T. Hara et al., Transmission tower model for surge analysis, in Proc.H3 IEE Japan Power and Energy Conf., 1991, Paper no. II-270.

Taku Noda (S94M97) was born in Osaka,Japan, on July 4, 1969. He received the B.S., M.S.,and Ph.D. degrees in engineering from DoshishaUniversity, Kyoto, Japan, in 1992, 1994, and 1997,respectively.

He was with DEI Simulation Software, Neskowin,OR, in 1994, and was a Consultant at the BonnevillePower Administration (BPA), Portland, OR, in 1995.In 1997, he joined the Central Research Institute ofElectric Power Industry (CRIEPI), Tokyo, Japan,where he holds the position of Research Scientist.

Since January 2001, he has been a Visiting Scientist at the University ofToronto, Toronto, ON, Canada. His research interests include transient analysisof power systems.

Dr. Noda is a Member of IEE of Japan.

Shigeru Yokoyama (M83SM91F96) was bornin Miyagi, Japan, on March 5, 1947. He received theB.S. and Ph.D. degrees in engineering from the Uni-versity of Tokyo, Tokyo, Japan, in 1969 and 1986,respectively.

In 1969, he joined the Central Research Institute ofElectric Power Industry (CRIPEPI), Tokyo, where hecurrently holds the position of Associate Vice Presi-dent. His research interests include lightning protec-tion and the insulation coordination of transmissionand distribution lines.

Dr. Yokoyama is one of the Vice Presidents of IEE of Japan.

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3- (2002) Thin Wire Representation in Finite Difference Time Domain Surge Simulation