application of the partial element equivalent circuit method to analysis of transient potential...

726 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 3, AUGUST 2011

Application of the Partial Element Equivalent CircuitMethod to Analysis of Transient Potential Rises in

Grounding SystemsPeerawut Yutthagowith, Student Member, IEEE, Akihiro Ametani, Life Fellow, IEEE,

Naoto Nagaoka, Member, IEEE, and Yoshihiro Baba, Member, IEEE

Abstract—This paper presents calculations of lightning transientpotential rises in grounding systems. A partial element equivalentcircuit (PEEC) method, adopting a modified image method, is em-ployed in this paper. The modified image method in this paper hastwo options either including or neglecting images of conductioncurrents along conductors for calculating series impedances. Theeffect of retardation in the PEEC method is also investigated. Com-parisons of simulation results by the proposed method with those bythe method of moments, the finite-difference time-domain method,and experimental results collected from the literature show that thePEEC method with the modified image method is quite effective inthe evaluation of transient potential rise in a grounding system.

Index Terms—Grounding systems, modified image method, par-tial element equivalent circuit (PEEC) method, transient potentialrise.

I. INTRODUCTION

AGROUNDING system is generally made of buried con-ductors in the form of vertical and horizontal rods and/or

grid networks. The grounding system is used to ensure the safetyof persons, to prevent damage to the electrical installation, and todissipate fault currents. Moreover, it is employed as a referencevoltage for electrical and electronic systems.

The transient and steady-state characteristics of groundingsystems have been studied by an experiment [1], by using cir-cuit theory [2], [3], transmission line approaches (see, e.g.,[4]–[11]), and numerical electromagnetic methods based onthe method of moments (MoM) [12]–[15], the finite-elementmethod (FEM), the finite-difference time-domain (FDTD)method [11], [16]–[18], and the hybrid electromagnetic model(HEM) method [19], [20]. The accuracy of the MoM and theFDTD methods was verified by comparing both with experi-mental results and with other reference simulation results andgood agreement was found [17].

The partial element equivalent circuit (PEEC) method [21] isderived from Maxwell’s equations and provides a full-wave so-lution to them. The method is applied to both time domain [22]

Manuscript received October 19, 2009; revised March 19, 2010 and May26, 2010; accepted July 25, 2010. Date of publication March 24, 2011; date ofcurrent version August 18, 2011.

The authors are with the Department of Electrical Engineering, DoshishaUniversity, Kyoto 610-0321, Japan (e-mail: [email protected];[email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEMC.2010.2077676

and frequency domain. A major difference from the MoM isthat the problem under consideration is transformed into a cir-cuit domain, where circuit analysis techniques can be applied.Despite the fact that the PEEC method has an exact field theo-retical basis, it was not primarily developed for the computationof electromagnetic fields. Instead, a circuit designer has a toolat hand to analyze parasitic effects in connecting structures ofcircuits without leaving the familiar area of network theory. Adetailed description of the approach can be found in [21]. Ini-tially, the time retardation effect was neglected. This methodwas intensively developed further in the 1990s when retarda-tion, external field excitation, and the treatment of dielectricmaterials were investigated [23]. The derivation of the PEECmethod for the thin wire structure described in Appendix A isclosely related to the MoM derivation and that referring to re-lated papers of Harrington [24] and Visacro [20] would help thereader understand the derivation more easily.

One of the main advantages of the PEEC method over com-monly used full-wave approaches such as the MoM and theFDTD method is that it directly yields the scalar potential alongthe grounding system without requiring any post processing.In the MoM and the FDTD method, the potential is generallycalculated by the integration of the electric field along a definedpath. Moreover, it can incorporate electrical components basedon circuit theory, such as resistive, inductive, and capacitive(RLC) elements, transmission lines, cables, transformers, andswitches.

For the problem relating to lightning effects, the PEECmethod has been employed to model lightning protection sys-tem, for coupling to coaxial cable, and to calculate lightning-induced voltage [25]–[27]. To the best of our knowledge, thisis the first time that the PEEC method considering or ne-glecting retardation is applied to analyzing grounding transientphenomena, and its accuracy systematically checked throughcomparisons with corresponding measured and/or MoM- orFDTD-computed results for different grounding electrodeconfigurations.

To handle soil–air interface for calculating resistance, capac-itance, and inductance parameters of a buried conductor on thebasis of circuit and transmission line approaches, the conven-tional image method including and neglecting images of con-duction currents along conductors was employed for groundingperformance analysis [4], [10]. The method yields reasonablyaccurate results as far as the transverse electromagnetic mode isdominant [11].

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YUTTHAGOWITH et al.: APPLICATION OF THE PEEC METHOD TO ANALYSIS OF TRANSIENT POTENTIAL RISES IN GROUNDING SYSTEMS 727

The modified image method introduced by Takashima etal. [28] was formulated by considering only the charge densityor transverse currents along a conductor. Physically, the trans-verse current produced by charge consists of two components,a displacement current and a conduction current dissipated inthe media. When the media is free space or air, the conductioncurrent dissipated in the media can be neglected. Although it isknown that the use of the modified image theory is somewhatless rigorous, it is useful if it gives reasonably accurate results inanalyzing grounding transient phenomena. For a high-frequencyrange, employing the modified image method [28] gives under-estimation of magnetic vector potential and overestimation ofscalar potential in comparison with employing the numericalintegration of Sommerfeld’s integrals as the results reported byDawalibi and Selby [13]. However, the modified image methodincluding images of the conduction currents along conductorshas been adapted with the MoM computations carried out byGrcev [14]. The results in [14] still show good agreement withthe corresponding results computed with a more accurate modelin [12].

In the experiment, the voltage difference between a consid-ered point and a distant reference point is selected as represen-tative of the ground potential rise and measured via an insulatedvoltage reference wire, but it is quite difficult to measure aground potential rise in practice. The voltage calculation re-quires the integration of electric fields. As it is well known, thevoltage difference is path dependent or not uniquely defined indynamic electromagnetic field. On the integration path perpen-dicular to the ground structure, the potential difference equalsvoltage difference since there is no contribution of magneticvector potential in electric fields and the scalar potential alsobecomes the voltage at low-frequency consideration. Therefore,the scalar potential of the considered point should be moreappropriate to use for the analysis of grounding performancebecause it is uniquely defined. This is in accordance with therecommendation of Grcev and Rachidi [29].

The aim of this paper is to calculate transient potential risesin grounding systems using the PEEC method. Based on thePEEC method described in Appendix A, the shunt admittance(C and G) and the series impedance (R and L) representing ef-fects of the charge density and the conduction current alongthe conductor, respectively, are given. The admittance and theimpedance are calculated on the basis of the structure of theconductors and their images. In this paper, the PEEC methodin the frequency domain is employed. In this method, self andparasitic impedances and admittances are calculated. The meth-ods allow separating the calculations of the admittance and theimpedance. The neglecting conduction currents along conduc-tors adopting in the modified image method was proposed tocompensate the overestimation of scalar potential and the un-derestimation of magnetic vector potential for a high-frequencyrange. The simulation results are compared with experimentaldata collected from the literature and those simulated by theMoM and the FDTD methods in some cases. It is shown that forcalculations of grounding transient potential rises in Section II,the proposed methods are as effective as the MoM or the FDTDmethod. The effect of retardation in the PEEC method is also

Fig. 1. Configuration of electrodes in the experiments (from [14] and [30]).

investigated in Section III. The ground potential rise calculatedin this paper gives roughly and reasonably accurate results incomparison with MoM-computed results [14], [15].

All computations with the PEEC method were carried outin this paper using MATLAB in a personal computer with a2.0 GHz core 2 Duo CPU and a 2-GB memory.

II. VALIDITY OF THE PEEC METHOD IN GROUNDING

TRANSIENT ANALYSIS

In this section, transient potential rises in grounding systemscalculated by the PEEC method considering retardation effectand adopting a modified image method are compared with thosemeasured by Rochereau [30] and Ametani et al. [1]. Note thatsome of those calculated and measured results are comparedwith MoM-calculated results by Grcev [14], [15]. In addition,the results calculated by the proposed method are comparedwith the corresponding results calculated by the FDTD method.

The proposed method throughout this section has two optionseither including or neglecting images of conduction currentsalong conductors for calculating series impedances.

Note that the PEEC method interprets the Maxwell’s equa-tions to a circuit domain so a current lead wire is not necessaryin the method. By the circuit domain, it allows to insert any cur-rent and/or voltage sources at any point of the conductor systemwithout a current lead wire. The configuration in the simulationdoes not completely agree with the actual experiment. However,neglecting a current lead wire will reduce number of elements inthe calculation and the calculated results still show satisfactoryagreement with experimental results.

A. Comparison With Measured Results by Rochereau andMoM-Computed Results by Grcev for Simple Structure

The configuration of two electrodes and soil parameters inthe experiments performed by the Electricite de France (EDF)[30] are illustrated in Fig. 1. In the experiment, the voltageswere measured by a 60-m-long resistive voltage divider withmeasuring bandwidth of 3 MHz, and currents were measured bycoaxial shunt resistor, as shown in Fig. 2. In the first experiment,an impulse current was injected into the top of a 6-m-longvertical electrode of 8 mm radius. The voltages were measuredat the top of the electrode. In the second experiment, an impulsecurrent was injected to the sending end of a 15-m-long horizontalelectrode with a radius of 12 mm. The voltages were measuredat the sending end of the electrode, 3.5 and 7 m away from thesending end of the electrode, respectively.


Fig. 2. Simplified schematic diagram of the experiments in Refs. [14] and [30].

Fig. 3. Measured transient voltages, calculated ground potential rises, and aninjected current for a case of the vertical electrode.

The same conditions were simulated by the PEEC method ina frequency range up to 5 MHz with a frequency step of 19.53kHz and frequency range up to 10 MHz with a frequency step of39.06 kHz for the first and the second cases, respectively. Thetime interval is from 0 to 3 μs and from 0 to 700 ns for compar-ison with the first and second experiments, respectively. Eachcase calculated by the PEEC method required a computationtime of around 5 s.

Figures 3 and 4 show PEEC-computed waveforms and corre-sponding measured and MoM-computed waveforms [14]. A cir-cle mark was used for the starting point of the results which doesnot start from zero and collected from [14]. Good agreement isobserved between PEEC-computed and MoM-computed wave-forms and satisfactory agreement is observed when the com-puted waveforms are compared with the measured waveforms.

B. Comparison With Measured Results by Ametani et al. for aCounterpoise Electrode

The configuration and parameters of an experiment carriedout by Ametani et al. [1], [8] are illustrated in Fig. 5. In theexperiment, an impulse current was injected into the sendingend of a counterpoise electrode of 10 mm radius and 1 m indepth. The voltages were measured at points A, B, C, D, and E.The currents were measured at points A, B, C, and D. A voltagereference insulated wire of about 100 m was laid on the earthsurface to measure the voltages. A Peason CT of model 110A was employed to measure currents. The CT was put under-ground and linked with the grounding electrode. The voltage

Fig. 4. Measured transient voltages and calculated ground potential rises alongthe horizontal electrode and an injected current. (a) Measured waveforms [30].(b) Simulation results

Fig. 5. Configuration of a counterpoise electrode in the experiment [1].

and current were transmitted to an oscilloscope by an opticalfiber via electrical to optical (E/O) conversion.

The same conditions were simulated by the PEEC methodin the frequency range from up to 80 MHz with a frequencystep of 312.5 kHz. The soil relative permittivity was set to 25.Computation time is around 28 s.

Figure 6 shows PEEC-computed waveforms and corre-sponding measured waveforms [1]. Satisfactory agreement isobserved.

C. Comparison With FDTD-Calculated Results byTheethayi et al.

The configuration and parameter of the FDTD simulationsperformed by Theethayi et al. [11] are illustrated in Fig. 7. Abare horizontal wire having 0.23 m radius and l m length is


Fig. 6. Measured transient voltages, calculated ground potential rises, andinjected currents along the counterpoise electrode.

Fig. 7. Configuration of an electrode in the simulation.

buried with a depth 2 m. The soil conductivity was set to be 0.4and 5 mS/m, and relative permittivity is set to be 10. An impulsecurrent with a peak of 1 A was injected in the middle point ofthe grounding electrode. The voltages at points 0 m, l/4, l/2, and3 l/4 were calculated.

The same conditions as the FDTD computation [11] weresimulated by the PEEC method in the frequency range up to25 MHz with a frequency step of 48.83 kHz. The time intervalwas from 0 to 5 μs for comparison with the simulated results bythe FDTD method in the cases of Figs. 8 and 9. The frequencyrange was up to 1.6 MHz with a frequency step of 3.125 kHz andthe time interval was from 0 to 30 μs for comparison with thesimulated results by the FDTD method for a case of Fig. 10. Twotypes of injected current waveforms, which comprise doubleexponentials, were used in those cases. The formulation of theapplied current was selected as:

I(t) = k(e−αt − e−βt), (1)

where I(t) is an applied current and k is a constant in unit ofampere corresponding to 1 A peak current. The constant α is1 × 104 s−1 for both the fast and slow impulses. The constantβ are 5 × 107 and 4 × 105 s−1 for a fast impulse and a slowimpulse, respectively. The fast impulse has a rise time of about0.1 μs and the slow impulse has a rise time of about 10 μs.The two waveforms are used to validate the proposed model

Fig. 8. PEEC-computed and FDTD-computed waveforms of current and volt-age for a 40-m horizontal electrode buried in depth 2 m for a fast impulse currentinjection. The soil conductivity is σs = 0.4 mS/m.

neglecting and including images of conduction currents alongconductors. In each case, computation time by employing thePEEC method is around 11 min.

In the PEEC method, neglecting images of conduction cur-rents along conductors increases total series impedances of el-ements. It reduces transverse currents and also increases con-duction currents along conductors. Therefore, in Figs. 8–10, thePEEC-computed results neglecting images of the conductioncurrents along the conductors show a better agreement with theFDTD-computed results for a high-frequency range or fast risetime of the injected current.

D. Comparison With MoM-Calculated Results by Grcev forComplex Structures

Here, two kinds of complex structures of grounding systemswere selected as test cases. The results calculated by the pro-posed method are compared with the MoM-calculated resultsby Grcev [14], [15].

The first structure is a grounding grid system composed ofhorizontal electrodes. The sizes of 1 × 1, 2 × 2, and 6 × 6grids are considered and the configurations are illustrated inFig. 11. The grounding grids are comprised of copper wires of7 mm radius, buried with depth 0.5 m. The soil conductivity isset to be 1 mS/m, and the relative permittivity is set to be 9. A1-A impulse current, which was found in [31], was injected topoint A of the ground grid system. The voltages at point A werecalculated.

The same conditions were simulated by the PEEC methodin the frequency range up to 10 MHz with a frequency stepof 19.53 kHz. The time interval was from 0 to 20 μs for


Fig. 9. PEEC-computed and FDTD-computed waveforms of current and volt-age for a 40-m horizontal electrode buried in depth 2 m for a fast impulse currentinjection. The soil conductivity is σs = 5 mS/m.

comparison with the simulated results [15]. An injected cur-rent waveform is taken from Refs. [15] and [31] by piecewiselinear approximation, as shown in Fig. 12 for comparison withthe MoM-calculated results by Gcrev [15]. In the case of the 6×6 grid size, the computation time by employing PEEC methodis around 11 min. Figure 13 shows the calculated results bythe proposed method. The PEEC-calculated results agree wellwith the MoM-calculated results, which are taken from [15] byvirtual inspection.

The second structure is a grounding grid system composedof horizontal electrodes and vertical electrodes at corners. Theconfiguration and parameters in the simulation carried out byGrcev [14] are illustrated in Fig. 14. The grounding systemwas buried with 0.6 m in depth constructed by horizontal andvertical electrodes which have a 6.54 mm radius (2.0 AWG).The ground grid size is 60 × 60 m2 with a 6 by 6 m2 grid. Thesoil is assumed to be homogeneous with 0.01 S/m conductivityand with 36 relative permittivity. An impulse current with 1 kApeak was applied to point A of the ground grid system. Thevoltages were calculated at the points A, B, and C.

The same conditions were simulated by the PEEC methodin the frequency range up to 5 MHz with a frequency step of19.53 kHz. The time interval is from 0 to 20 μs for the compari-son with the experimental results. An injected current waveformwhich is a double exponential function, as given in (1) and takenfrom [14], was employed in this case. The k corresponding toa 1 kA peak current is 1.1152 × 103 A. The constants α andβ corresponding to a 1/20 μs waveform are 4 × 104 and 1.8× 106 s−1 , respectively. In this case, the computation time for

Fig. 10. PEEC-computed and FDTD-computed waveforms of current andvoltage for a 600-m horizontal electrode buried in depth 2 m for a slow impulsecurrent injection. The soil conductivity is σs = 0.4 mS/m.

Fig. 11. The structures of the grids 1 × 1, 2 × 2, and 6 × 6 in the simulation[15].

Fig. 12. Applied current in the simulation (from [15] and [31]).


Fig. 13. Comparison of PEEC-calculated waveforms and MoM-calculatedwaveforms [15] at point A of the grid grounding systems. (a) Including imagesof conduction currents along conductors. (b) Neglecting images of conductioncurrents along conductors.

Fig. 14. Configuration of grid electrodes in the simulation [14].

calculating 301 node voltages on a ground grid by employingthe PEEC method is around 7 min.

Figure 15 shows comparison results calculated by the MoM[14] and by the proposed method. A circle mark was used forthe starting point of the results which does not start from zeroand collected from [14]. Good agreements are observed. Figure16 shows the distribution of ground potential rise on the groundgrid at different time points by neglecting images of conductioncurrents along conductors.

Note that good agreements are observed by comparison of thePEEC-computed results including images of conduction cur-rents along conductors and MoM-computed results by Grcev[14], [15], since the same modified image method including im-

Fig. 15. Comparison of results for calculated ground potential rises alongthe grid grounding system. (a) Including images of conduction currents alongconductors. (b) Neglecting images of conduction currents along conductors.

ages of conduction currents along conductors was employed inthe simulations.

III. EFFECT OF RETARDATION EFFECT IN THE PEEC METHODS

In the case of the PEEC method neglecting retardation, themethod is still represented by R, L, C, and, G elements andcoupling effects among elements and the term of retardation inthe Green function are set to be 1.

The formulation matrices of the method neglecting retarda-tion in the frequency domain and the time domain are passive.The models of reduction (MOR) [32]–[34] are applicable andthe instability problem is avoided. The solution adopted with theMOR is computationally efficient. However, neglecting retarda-tion can be applicable only when the minimum wavelength ofapplied sources is much larger than the dimensions of the con-sidered computational domain.

In this section, effects of retardation in the PEEC methodare investigated by comparisons between results calculated byconsidering retardation and results calculated by neglecting re-tardation.

To investigate the effect of retardation, the transient behav-ior of horizontal grounding conductors with different lengths,buried in the soil with permittivity of εr = 4 and conductivityσ = 1 mS/m is investigated. The 0.6 m buried depth of theelectrodes with a radius of 5 mm and 20, 40, 80, and 160 min length are considered. Two injected current waveforms, as in(1), are considered with different front time characteristics. The


Fig. 16. Distribution of ground potential rise on the 60 m × 60 m groundinggrid at different time points.

fast front current with α = 2.7 × 104 s−1 and β = 5.6 × 106

s−1 has a rise time of 0.36 μs. The slow front current with α =1 × 104 s−1 and β = 4 × 106 s−1 has a rise time of 10 μs. Theinjected current was applied at the sending end of the electrode.Figures 17 and 18 show calculated results for potentials andcurrents by the PEEC method either neglecting retardation orincluding retardation in the case of a 160-m-long electrode. The

Fig. 17. Currents and potential rises along 160 m grounding electrode with5 mm radius and 0.6 m buried depth for the fast impulse current.

Fig. 18. Currents and potential rises along 160 m grounding electrode with 5mm radius and 0.6 m buried depth for the slow impulse current.

potentials at the points 20, 40, 80, and 120 m from the currentinjected point are calculated. Figures 19 and 20 show calculatedresults for potentials by the PEEC method either neglecting re-tardation or including retardation in the cases of 20-, 40-, 80-,and 160-m-long electrodes. The same input current waveformsemployed in Figs. 17 and 18 were injected at the sending end


Fig. 19. Transient potential rises at the sending end of horizontal groundingelectrodes with 20, 40, 80, and 160 m in length for the fast impulse current.

Fig. 20. Transient potential rises at the sending end of horizontal groundingelectrodes with 20, 40, 80, and 160 m in length for the slow impulse current.

of the electrode in this simulation. The potentials at the currentinjected points were calculated.

Comparing the simulation results in Figs. 17 and 18 basedon the PEEC methods, it can be observed that the accuracy ofthe ground potential rise calculated by the PEEC method ne-glecting retardation effect is not satisfactory; even though com-parisons of the calculated current waveforms and peaks showgood agreement. The peaks of ground potential rises calculatedby the PEEC method neglecting retardation effect are greaterthan those calculated by the PEEC method considering retar-dation effect. Differences between potential rises calculated bythe PEEC neglecting retardation and those calculated consider-ing retardation effect are about 10%–30%. The PEEC methodneglecting retardation gives over estimated results.

Comparing the simulation results, potential rises, and currentsin Figs. 19 and 20, it is observed that the PEEC method neglect-ing retardation effect is valid in the case of the slow rise timecurrent when the length of the grounding conductor is muchsmaller than the effective length.

IV. CONCLUSION

The PEEC methods adopting a modified image method havebeen applied to analyzing transient potential rise calculations ingrounding systems, and the validity of these methods has beenexamined in comparison with simulation results calculated bythe MoM and the FDTD method and with experimental results.The PEEC method adopting a modified image method shows

satisfactory accuracy in comparison with experimental resultsand with simulation results calculated by the MoM and theFDTD method.

Neglecting the image of conduction currents along conduc-tors influences the calculated results for a high-frequency rangeor a fast rise time of the injected current. The results calculatedby the PEEC method considering retardation effect and neglect-ing the image of the conduction currents along conductors andcalculated by the FDTD method agree very well at the begin-ning time and agree quite well in a long time period. However,the peak differences of results between neglecting images ofconduction currents along conductors and including images ofconduction currents along conductors are around 5%–10%.

The PEEC method neglecting the retardation effects is noteffective for transient calculations of a grounding system be-cause the method is time consuming when it is compared withthe circuit theory and transmission-line-based approaches.

The PEEC method has been shown to be as effective as theMoM and the FDTD method for the grounding transient poten-tial rise calculations. The main advantage of the PEEC methodover the MoM and the FDTD method is that it does not re-quire any post processing to evaluation potential rises along thegrounding system.

APPENDIX A

PEEC METHOD FOR THE THIN WIRE STRUCTURE

The theoretical derivation of the PEEC method for the thinwire structure starts from a total electric field on a wire surface.From the thin wire approximation, currents and charge densitiesare assumed to be distributed along the contour of wire axis(C(�r)). The boundary condition on the surface of the thin wire,as illustrated in Fig. 21, is that the total tangential electric fieldshall be as (A1):

�s · �Et(�r) = �s · ( �Ei(�r) + �Es(�r)) = �s ·�J(�r)σ

, (A1)

where �s is a unit tangential vector along C(�r), �s ′ is a unit tan-gential vector on the wire surface, the �Ei is an incident electricfield, and �Es is the scattered electric field which represents thereaction of the wire to the incident field corresponding to (A1).

The scattered field is calculated by volume current densitiesand volume charge densities which can be expressed in termsof a magnetic potential vector ( �A(�r)) and an electric scalarpotential (φ(�r)) at point �r. Assuming thin wires, �A(�r) and φ(�r)can be written in forms of a current along a conductor (I(�r ′))and line charge density (ρl(�r ′)), as given in (A2) and illustratedin Fig. 21:

�Es = −jω �A −∇φ (A2.1)

�A(�r) =μ

4π

∫

c

�s ′I(�r ′)e−ΓR

Rds′ (A2.2)

φ(�r) =1

4πε

∫

c

ρl(�r ′)e−ΓR

Rds′ (A2.3)


Fig. 21. Problem geometry.

Fig. 22. Configuration of elements in the medium domain. (a) Elements ofthe first group. (b) Elements of the second group.

where ds is a small distance along C(�r), the ε = ε + σ/(jω),R = |�r − �r ′|, and Γ in (A3) is the propagation constant of theconsidered medium

Γ =√

jωμ(σ + jωε). (A3)

The time dependence of the variables in the frequency do-main is exp(jωt). σ, ε, μ, and ω are conductivity, permittivity,and permeability of the medium, and the angular frequency,respectively.

Substituting (A2) into (A1), (A4) is obtained

�s · �Ei(�r) − �s · I(�r)σl

− jωμ

4π

∫

c

�s · �s′I(�r′)e−ΓR

Rds′ − dφ

ds= 0,

(A4)where σl is length conductivity of a conductor.

As illustrated in Fig. 22, the conductors in the media are seg-mented into two groups. The first group in Fig. 22(a) has Nelements and the second group in Fig. 22(b) has M elements.The elements in two groups are interleaved each other. The con-figurations of elements in the first group and the second groupare used to calculate series impedances and shunt admittances,respectively.

From (A4), an element m with length Lm for nodes k and l inthe first group is considered.

Rearranging and integrating (A4) along the segment m froma point k to a point l, yields

l∫

k

dφ

dsds = −jω

μ

4π

s ′= l∫

s ′=k

∫

c

�s · �s′I(�r′)e−ΓR

Rds′ds (A5.1)

−s ′= l∫

s ′=k

I(�r)σl

ds +

l∫

k

�s · �Ei(�r)ds,

φl − φk = φlk = −jωμ

4π

s ′= l∫

s ′=k

∫

c

�s · �s′I(�r′)e−ΓR

Rds′ds

−s ′= l∫

s ′=k

I(�r)σl

ds +

l∫

k

�s · �Ei(�r)ds. (A5.2)

The first term on the right hand side in (A5) can be expressedas follows:

jωμ

4π

s ′= l∫

s ′=k

∫

c

�s · �s′I(�r′)e−ΓR

Rds′ds′ =

N∑n=1

(ZLmnILn )

(A6.1)

ZLmn = −jωμ

4π

∫

Lm

∫

Ln

e−ΓR cos φ

Rdlndlm . (A6.2)

The current along each small element is assumed to be constant.Also, the propagation times between any points on the element nto any other point on the element m are assumed to be identical.(A6.2) can be derived from (A6.1).

The second and the third terms represent resistance and anadditional voltage source of the element.

The potential difference on both ends of each element iscalculated by (A5) and (A6).

Consider the element of which center is at node k in thesecond group, as shown in Fig. 22(b). Assuming a constantcharge density along the element and conservation of charge,the charge density can be calculated by the following equation(A7):

ρl = (ILm + ILo + ILp)/(jωLk ) = IT k/(jωLk ) (A7)

The average potential along the element at node k can becalculated by the following (A8):

φk =1

Lk

∫

Lk

φ(�r)dlk

=1

4π (σ + jωε) Lk

∫

Lk

∫

c

IT i

Li× e−ΓR

Rds′dlk (A8.1)

φk =M∑i=1

ZT kiIT i (A8.2)


Fig. 23. Equivalent circuit model of a PEEC cell.

ZT ki =1

4π (σ + jωε) LkLi

∫

Lk

∫

Li

e−ΓR

Rdlidlk (A8.3)

where φlk is the potential difference between both ends (k andl) of the element m, and φk is an average voltage of the node k.lm is a local distance along element m employed for integrationof (A6) and (A7). Subscripts m and n indicate elements m andn in the first group, respectively, and subscripts k and i corre-spond elements k and i in the second group, respectively. R isthe distances between the element m and element n and cosφis a cosine function of an angle between the element m and ncorresponding to Fig. 22. Note that the same kind of the formulafor calculating the precisely equivalent impedance including re-tardation by using the familiar Neumann formula type integralswas previously introduced by Visacro and Soares [20].

From those formulations, an element or a cell can be rep-resented in the form of an equivalent circuit, as illustrated inFig. 23. The series impedance (Rmm and Lmm ) and the voltagesource (Vkl) are derived from (A5) and (A6). The shunt admit-tance (Gk , Gl , Ck , and Cl) and the current sources (Ik and Il)are derived from (A8).

Equations (A6) and (A8) are applied to all elements. Thecoupling effects among elements and the propagation effectsare included in this method. The potential difference and thepotential average of the elements are written in forms of matriceswhich are composed by node potentials (potentials at the endsof an element), the conduction currents along conductors, thetransverse currents, and incident electric fields. The potential ofeach node can be calculated from the equations employing nodalanalysis approaches in the frequency domain at each frequencythrough the considered frequency range. The frequency stepin the PEEC method depends on the maximum consideringtime, and the frequency range depends on the circuit condition.The length of a small element shall be much smaller than theminimum wavelength which is inversely proportional to themaximum frequency.

The ground effect can be taken into account by employingthe image methods.

From applying the modified inverse fast Fourier transform(MIFFT) [35], all quantities in the time domain are found. Theaccuracy of this method is dependent on element length and thenumber of frequencies to be used.

ACKNOWLEDGMENT

The authors would like to thank Prof. Rachidi and threeanonymous reviewers for their valuable comments on the paper.

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Peerawut Yutthagowith (S’09) received the B. Eng.and M.Eng. degrees from Chulalongkorn University,Bangkok, Thailand, in 1998 and 2002, respectively,and the Ph.D. degree from Doshisha University, Ky-oto, Japan, in 2010.

In 2007, he joined King Mongkut’s Institute ofTechnology, Ladkrabang, as a Lecturer. He is cur-rently in the Department of Electrical Engineering,Doshisha University, Kyoto. His research interestsare in area of a high voltage equipment modeling andelectromagnetic transients in power systems.

Dr. Peerawut is also a member of International Council on Large ElectricSystems WGC4.501.

Akihiro Ametani (M’71–SM’83–F’92–LF’10) wasborn on February 14, 1944. He received the Ph.D.degree from the University of Manchester (UMIST),Manchester, U.K., in 1973.

From 1971 to 1974, he was at UMIST. From1976 to 1981, he was in Bonneville Power Admin-istration to develop electromagnetic transient pro-gram (EMTP) for summer. He was a Professor atthe Catholic University of Leaven, Belgium, in 1988.Since 1985, he has been a Professor at Doshisha Uni-versity, Kyoto, Japan. He was the Director of the

Institute of Science and Engineering from 1996 to 1998 and Dean of Libraryand Computer/Information Center from 1998 to 2001. He is also a CharteredEngineer in U.K.

Dr. Ametani is a Distinguished Member of the International Council on LargeElectric Systems and a Fellow of IET. He was awarded D.Sc. degree (higherdegree in UK) from the University of Manchester in 2010.

Naoto Nagaoka (M’87) was born in Nagoya, Japanon October 21, 1957. He received the B.Sc., M.Sc.,and Dr. Eng. degrees from Doshisha University, Ky-oto, Japan, in 1980, 1982, and 1993, respectively.

In 1985, he joined the Faculty of Engineering,Doshisha University, where he has been a Professorsince 1999 and the Dean of the Student AdmissionCenter since 2008.

Dr. Nagaoka is a member of IET.

Yoshihiro Baba (S’95–M’99) received the B.S.,M.S., and Ph.D. degrees in electrical engineeringfrom the University of Tokyo, Tokyo, Japan, in1994,1996, and 1999, respectively.

From April 2003 to August 2004, he was a Visit-ing Scholar at the University of Florida, Gainesville,on sabbatical leave from Doshisha University. He iscurrently an Associate Professor in the Department ofElectrical Engineering, Doshisha University, Kyoto,Japan.

Dr. Baba is a member of the American Geophys-ical Union (AGU) and the Institution of Engineering and Technology (IET).From 2009, he has served as an Editor of the IEEE TRANSACTION ON POWER

DELIVERY.