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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
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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)
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842 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 17, NO. 3, JULY
2002
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 .
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NODA AND YOKOYAMA: THIN WIRE REPRESENTATION IN FDTD SURGE
SIMULATION 843
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
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844 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 17, NO. 3, JULY
2002
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
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NODA AND YOKOYAMA: THIN WIRE REPRESENTATION IN FDTD SURGE
SIMULATION 845
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
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846 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 17, NO. 3, JULY
2002
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.
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NODA AND YOKOYAMA: THIN WIRE REPRESENTATION IN FDTD SURGE
SIMULATION 847
REFERENCES[1] H. W. Dommel, Digital computer solution of
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[2] K. S. Yee, Numerical solution of initial boundary value
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[3] K. S. Kunz and R. J. Luebbers, The Finite Difference Time
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[4] R. F. Harrington, Field Computation by Moment Methods. New
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[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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