电磁兼容论文1

Estimating Radio-Frequency Interference to an

Antenna Due to Near-Field Coupling Using

Decomposition Method Based on Reciprocity Hanfeng Wang,Student Member,IEEE,Victor Khilkevich,Member,IEEE,Yao-Jiang Zhang,Senior Member,IEEE,

and Jun Fan,Senior Member,IEEE

Abstract—In mixed radio-frequency(RF)and digital designs, noise from high-speed digital circuits can interfere with RF re-ceivers,resulting in RF interference issues such as receiver desen-sitization.In this paper,an effective methodology is proposed to estimate the RF interference received by an antenna due to near-?eld coupling,which is one of the common noise-coupling mecha-nisms,using decomposition method based on reciprocity.In other words,the noise-coupling problem is divided into two steps.In the ?rst step,the coupling from the noise source to a Huygens surface that encloses the antenna is studied,with the actual antenna struc-ture removed,and the induced tangential electromagnetic?elds due to the noise source on this surface are obtained.In the second step,the antenna itself with the same Huygens surface is studied. The antenna is treated as a transmitting one and the induced tan-gential electromagnetic?elds on the surface are obtained.Then, the reciprocity theory is used and the noise power coupled to the antenna port in the original problem is estimated based on the results obtained in the two steps.The proposed methodology is val-idated through comparisons with full-wave simulations.It?ts well with engineering practice,and is particularly suitable for prelayout wireless system design and planning.

Index Terms—Decomposition,mixed radio-frequency(RF)/ digital design,near-?eld coupling to antenna,prelayout system design and planning,reciprocity,RF interference.

I.I NTRODUCTION

A S a special type of intrasystem electromagnetic compati-

bility(EMC)problem,radio-frequency(RF)interference from digital circuits to RF receivers is becoming increasingly critical in the design of mixed RF/digital systems[1].This is due to the increasing speed in digital integrated circuits(ICs)and the decreasing form factor of the systems.There are many pos-sible mechanisms for digital noise to couple to RF receivers.RF antennas/receivers can pick up digital noise through direct?eld coupling inside a metal chassis,such as from high-speed printed circuit board(PCB)traces,connectors,?ex cables,and IC heat sinks.Digital noise can also couple to RF antennas/receivers

Manuscript received August29,2012;revised November29,2012;accepted January10,2013.Date of publication March8,2013;date of current version December10,2013.This work was supported in part by the National Science Foundation(NSF)under Grant0855878.

The authors are with the EMC Laboratory,Missouri University of Science and Technology,Rolla,MO65409USA(e-mail:jfan@https://www.360docs.net/doc/392342119.html,).

Color versions of one or more of the?gures in this paper are available online at https://www.360docs.net/doc/392342119.html,.

Digital Object Identi?er

10.1109/TEMC.2013.2248090

Fig.1.RF interference from digital units to analog units.

through power distribution networks and other conducted paths

such as chassis and ground planes.

In this study,the focus is on the direct?eld coupling from

a digital noise source to an RF antenna,as shown in Fig.1.

Similar to a conventional noise-coupling problem[2],there are

three major components:digital IC as the noise source,electro-

magnetic?eld coupling as the noise-coupling mechanism,and

RF antenna as the victim.Due to the small dimensions in many

mixed RF/digital systems,such noise coupling could occur in

the near-?eld region.The main contribution of this paper is to

propose an ef?cient method to estimate the near-?eld coupling

for RF interference evaluations.

When the noise coupling occurs in the far-?eld region,the

Friis transmission equation can be used to estimate the induced

noise power at the RF antenna port.Treat the digital noise source

as a transmitting“antenna,”and calculate the received power

using[3]

P r

P t

=e cdt e cdr

1?|Γt|2

1?|Γr|2

×

λ

4πR

2

D t(θt,φt)D r(θr,φr)|?ρt·?ρr|2(1)

where P r,P t are the receiving power of the RF antenna and the

transmitting power of the noise-source“antenna,”respectively;

e cdr and e cdt are their radiation ef?ciency and the subscript“cd”

indicates that both are combinations of conduction ef?ciency

and dielectric ef?ciency;Γr andΓt are the re?ection coef?-

cients looking into the load and looking into the noise source

“antenna”at the antenna ports,respectively;λis the wavelength; 0018-9375?2013IEEE

R is the distance between the two antennas;D r and D t are the

directivity of the two antennas;?ρr and ?ρ

t are the polarization unit vectors of the two antennas.The Friis transmission equation is a straightforward way to evaluate the receiving power (noise power in this case)at the RF antenna port.However,it only works for far-?eld coupling.It assumes 1/R distance depen-dence for the ?eld strength;however,in the near ?eld,there are 1/R 2and 1/R 3terms.Further,the Friis transmission equation is based on the plane-wave receiving properties of each antenna,which are built into the directivities.In the near-?eld case,the ?eld strength can vary strongly over the dimensions of the an-tenna and the plane wave behavior is not expected.In compact mobile devices,the coupling between the IC noise source and the RF antenna is mostly in the near-?eld region.Thus,the Friis transmission equation is no more suitable for this kind of prob-lems,which will be demonstrated using numerical simulations in Section IV .

Then,to estimate the noise power coupled to the RF antenna port from the digital circuit in near-?eld region,full-wave sim-ulation of the entire structure including the noise source,the RF receiving antenna,and all the relevant scatterers/materials is the straightforward,brute-force solution.However,this approach is time consuming for practical applications such as the prelayout system design and optimization in terms of the minimized RF interference,since full-wave model needs to be built and run for each con?guration.A more practical strategy is to generate “libraries”for digital noise source (IC)and antenna models,where interference estimation between the noise source and an-tenna models in a speci?c mechanical design can be ef?ciently obtained without the need to run lengthy full-wave simulations whenever a model,its location,or a mechanical feature changes.In this paper,a decomposition method based on reciprocity is introduced for estimating the received digital noise power at the RF antenna port,which ?ts very well with the engineering practice described earlier.First,a Huygens box is introduced to enclose the RF antenna so that it can be removed from the original structure.Then,the tangential electromagnetic ?elds on the Huygens box for the original structure with the RF antenna removed are obtained through a full-wave simulation (the noise source model obtained in [4]or similar ones can be used as the excitation).Next,the RF antenna alone is simulated as a transmitting antenna and the tangential electromagnetic ?elds on the same Huygens box are obtained.In other words,an antenna model is https://www.360docs.net/doc/392342119.html,stly,the induced noise power at the RF antenna port is obtained by using the two sets of the tangential electromagnetic ?elds on the Huygens box based on the reciprocity theorem.The bene?t of the proposed method is that the noise source and the antenna models remain the same as long as the same digital ICs and the same RF antenna are used.When component position changes or other mechanical design changes,only the electromagnetic ?elds on the Huygens box due to the noise source need to be recalculated,which is more ef?cient and faster than the full-wave simulation of the entire structure.Similarly,if several antenna designs are to be evaluated and compared,only individual antenna models need to be constructed (all other steps only need to be run once).This makes the prelayout design and optimization more ?exible and

ef?cient.

Fig.2.Simpli?ed model to illustrate the decomposition method.

II.D ECOMPOSITION M ETHOD

A simpli?ed model in Fig.2is used to illustrate the decom-position method.Two small probes are created above a large ground plane.The dimensions of the ground plane are 4in ×2in,and the distance between the two probes is 2000mils.The ground plane has a thickness of 1mil and the probes have a height of 100mils.Two ports are de?ned between the probes and the ground plane for either excitation or termination.One probe is used to represent a simpli?ed digital noise source and the other probe represents an RF antenna.

The ultimate goal is to estimate the induced noise power on the “RF antenna”port generated from the “digital noise source.”One straightforward way to estimate this induced noise on the antenna port is to run the simulation of the entire model in a full-wave simulation tool such as HFSS [5].For the reasons discussed earlier,an alternative method is proposed herein.In this method,the entire geometry is divided into two parts by creating an observation box that surrounds the “RF antenna,”as shown in Fig.3.The observation box is a cubic with a side length of 200mils in this example.Then,the original struc-ture with the “RF antenna”removed,as shown in Fig.3(a),is studied,and simulated using HFSS in this study,to obtain the tangential electromagnetic ?elds on the observation box.This set of the tangential ?elds will be used as the excitation to the “RF antenna”when it is in the receiving mode.The step to cal-culate the antenna output using the ?eld excitation is referred to as the “forward”problem herein.Next,as shown in Fig.3(b),only the “RF antenna”is studied as a transmitting antenna,again using HFSS in this study,to obtain the tangential electromag-netic ?elds on the same observation box.This step is referred to as the “reverse”problem herein.The induced noise output at the “RF antenna”can then be calculated from these two sets of tangential electromagnetic ?elds on the observation box.The detailed algorithm will be introduced in the following section.III.R ECIPROCITY T HEOREM AND I NTERFERENCE E STIMATION An ef?cient and fast method for calculating transfer functions between equivalent current sources and antenna was introduced in [6]based on the reciprocity theorem.In this study,similar derivations are extended to calculate the induced antenna output from the two sets of the tangential electromagnetic ?elds on the observation box obtained using the decomposition method.

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Fig.3.Divide the whole model into two parts.(a)Part1including“digital noise source”and coupling path.(b)Part2including“RF antenna”only. The reciprocity theorem can be expressed using the?eld and source quantities in the forward and reserve problems as[7]:

?

S

ˉ

E rev×ˉH fwd?ˉE fwd×ˉH rev

·ds =

V

ˉ

E rev×ˉJ fwd+ˉH fwd×ˉM rev

dv

?

V

ˉ

E fwd×ˉJ rev+ˉH rev×ˉM fwd

dv(2)

whereˉJ is the electric current source andˉM is the magnetic current source.The superscript denotes the problem type,“fwd”for the forward problem and“rev”for the reverse one.If the integral is evaluated over the entire space(which means the boundary surface S is at in?nity),the left side of(2)is zero. Thus,(2)can be simpli?ed as

V ˉ

E rev

c

·ˉJ fwd

c

?ˉH rev c·ˉM fwd

c

dv

=

V

ˉ

E fwd

a

·ˉJ rev a?ˉH fwd a·ˉM rev a

dv(3)

where the subscript denotes the location of the source and?eld quantities:“c”for cells on observation box and“a”for the RF antenna port.Therefore,in the reverse problem,sources at the

antenna port are de?ned asˉJ rev

a andˉM rev

a

,and the resulting

?elds on the observation box are de?ned asˉE rev

c andˉH rev

c

.In

the forward problem,equivalent sources on the observation box

are de?ned asˉJ fwd

c andˉM fwd

c

,and the resulting?elds on the

antenna port are de?ned asˉE fwd

a andˉH fwd

a

.

The electric term at the left-hand side of(3)can be derived as

V

ˉJ fwd

c

·ˉE rev c dv=

S c

ˉJ fwd

c

·ˉE rev c ds=

cells

ˉJ fwd

c

·ˉE rev c S cell

=

cells

?n×ˉH fwd

c

·ˉE rev c S cell(4)

where S c is the surface of the observation box.The volume

integral becomes a surface integral since the equivalent electric

current densities in(4)only reside on the surface of the obser-

vation box.If the surface of the box is meshed into uniform

square cells,and the meshed cells are small enough so that the

?elds are approximately uniform inside each cell,the integral

can be transformed into summations where S cell is the area of

each mesh cell.Finally,the equivalent electric current densities

on the surface of the observation box can be calculated from the

H-?eld,where n is the inward unit vector normal to the surface

of the observation box.

Similarly,the magnetic term at the left side of(3)can be

derived as

V

ˉM fwd

c

·ˉH rev c dv=

S c

ˉM fwd

c

·ˉH rev c ds

=

cells

ˉM fwd

c

·ˉH rev c S cell=

cells

ˉE fwd

c

×?n·ˉH rev c S cell.(5)

The electric term at the right-hand side of(3)can be derived

similarly as in[8]as

V

ˉE fwd

a

·ˉJ rev a dv=?

S a

E fwd

a

J rev

a

ds=?I rev

a

U fwd

a

(6)

where the volume integral is transformed into a surface integral

at the antenna port surface.The negative sign is becauseˉE fwd

a

andˉJ rev

a

have opposite directions.S a is the cross-sectional

surface of the antenna port.U fwd

a

and I rev

a

are the voltage and

current at the antenna port in the forward and reverse problems,

respectively.

The magnetic term at the right-hand side of(3)is?rst con-

verted to an electric term as

V

ˉH fwd

a

·ˉM rev a dv=

V

ˉ

E rev

a

×?n

·ˉH fwd a dv

=

V

ˉE rev

a

·

?n×ˉH fwd

a

dv=

V

ˉE rev

a

·ˉJ fwd

a

dv.(7)

Then,(7)can be similarly derived as in(6)as

V

ˉE rev

a

·ˉJ fwd

a

dv=

S a

J fwd

a

E rev

a

ds=I fwd

a

U rev

a

.(8)

Substituting(4)–(8)back into(3)results in

cells

ˉn×ˉH fwd

c

·ˉE rev c S cell+

cells

ˉn×ˉE fwd

c

·ˉH rev c S cell

=?I rev

a

U fwd

a

?I fwd a U rev a=?

1

Z in

+

1

Z L

U fwd

a

U rev

a

(9)

where Z in is the input impedance of the RF antenna in the trans-

mitting mode in the reverse problem,Z L the load impedance

at the antenna port in the receiving mode(50Ωin our case)in

the forward problem,and U rev

a

is the exciting voltage of the RF

1128IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY ,VOL.55,NO.6,DECEMBER 2013

antenna in the transmitting mode (1V in our case).From (9),

the voltage induced at the RF antenna port U fwd

a

can be solved from the tangential E and H ?elds on the observation box in both the reverse and the forward problems as follows:

U fw d a =?

Z in Z L

U rev a (Z in +Z L

)×

cells

ˉn ×ˉH fw d c ·ˉE rev c S cell + cells

ˉn ×ˉE fw d c ·ˉH rev c S cell .(10)

It needs to be pointed out that the simple decomposition

method proposed in this study assumes that the multiple scat-tering effects between the noise source and the RF antenna are negligible.

IV .R ESULTS AND V ALIDATIONS

A.Simple Model

The simple probe model in Fig.2is ?rst used to prove the necessity of the study of near-?eld coupling.The Friis trans-mission equation [3]is used to estimate the induced power at the port of the receiving “RF antenna,”and it will be shown that this equation only works when the noise-source antenna and the “RF antenna”are located suf?ciently far away to each other.In this example,since both the noise-source and the “RF”antennas have the same probe structure.Their parameters,such as the directivity,the gain and the input impedance,can be obtained by simulating one probe antenna alone in HFSS.The peak gain of the probe antenna is 2.97and the input impedance is Z in =(3.719–161.5j)Ω.The peak gain is the multiplication of the peak directivity and the radiation ef?ciency.The re?ection coef?cient can be calculated as:

|Γ|=

Z in ?Z L Z in +Z L

(11)where Z in is the input impedance of the probe antenna and

Z L =50Ω,assuming both probes are either terminated with a 50Ωload or connected with a source with a 50Ωsource resistance.Since the two probe antennas are aligned in their maximum radiation direction and are also polarization-matched,the receiving power at the “RF antenna”can be easily calculated from (1).A sinusoidal wave of magnitude of 1V is assumed as the source voltage for the transmitting antenna in the results shown below.

Meanwhile,the power induced at the port of the receiving antenna can also be obtained from the full-wave simulation of the entire structure with both probes (denoted direct method)and from the proposed decomposition method using (10).The mesh size on the observation box,when using the proposed method,is 5mils,which was found to be suf?ciently small to obtain the converged results in this example.The results of the three methods are compared in Fig.4,with the distance between the two probes changing from 500to 2000mils.

It can be clearly seen that the difference between the Friis transmission equation and the full-wave simulation becomes larger when the distance between the two probes becomes smaller.This means that the Friis transmission equation

can

https://www.360docs.net/doc/392342119.html,parison of the Friis transmission equation,the direct full-wave simulation,and the proposed decomposition method at 10GHz.

only be used to estimate the receiving power for far-?eld cou-pling.However,for the RF interference problems in compact mobile devices,near-?eld coupling could be dominant.There-fore,the Friis transmission equation is no longer suitable in these cases.The results of the direct full-wave simulation and the proposed decomposition method show a strong agreement for all the cases.The frequency under study in this example is 10GHz.

Further,the same model in Fig.2,when the distance between the two probes is 500mils,is used to validate the proposed method at different frequencies.The induced power at the re-ceiving antenna port at each frequency point was calculated using (10)?rst,and then obtained from the full-wave simula-tion of the entire structure.The two sets of results are compared at ten different frequency points from 1to 10GHz in Fig.5.The difference between the proposed method and the direct full-wave simulation of the entire structure is within 0.06dB.With the decrease of frequency,the electrical distance between the two probes is reduced in terms of the wavelength and the er-ror using the Friis transmission equation increases signi?cantly,as clearly shown in Fig.5.At the frequencies above 8GHz,the difference between the Friis transmission equation and the other two methods increases again.This is possibly due to insuf?cient mesh size at these frequencies.Since the probe is very small,the simulated frequency range is out of the resonant frequency of the antenna.Thus,the coupled power to the antenna port is very small.Two more practical examples are used to validate the proposed method in the following sections.B.Trace +PIFA

A more practical case,with a section of microstrip trace and a planar inverted-F antenna (PIFA)[9],is shown in Fig.6.The ground plane size is 4in ×2in and the thickness of the plane is 1mil.The patch of the PIFA has dimensions of 500mils ×500mils and is 10mils above the ground plane.The radius of

W ANG et al.:ESTIMATING RADIO-FREQUENCY INTERFERENCE TO AN ANTENNA DUE TO NEAR-FIELD COUPLING

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Fig.5.Output power comparison among the Friis transmission equation,the proposed method,and the direct full-wave simulation for the simpli?ed model shown in Fig.

2.

Fig.6.Near-?eld coupling from a microstrip trace to a PIFA antenna.

the short via and feeding probe of the PIFA is 10mils,and the antipad radius of the feeding port is 30mils.The trace (digital noise source)is excited with a 1V voltage source with a 50Ωsource resistance at one end and terminated with a 50Ωat the other end.The detailed dimensions for the trace can be found in [4].The distance between the trace and the PIFA is 2000mils.This example represents a typical near-?eld coupling problem from a digital trace to an RF antenna in modern mobile devices.The proposed decomposition method is used to calculate the induced power at the PIFA port.An observation box with the dimensions of 1000mils ×1000mils ×200mils is created to enclose the PIFA antenna.The mesh size on the observation box is 10mils.The calculated power is validated through the com-parisons with the full-wave simulation of the entire structure,as shown in Fig.7.

The agreement between the two methods is excellent in gen-eral in the entire frequency range from 3.1to 5.1GHz,with some small discrepancies close to 4.1GHz,the resonant frequency of the PIFA

antenna.

Fig.7.Output power comparisons between the proposed method and the direct method for the trace +PIFA structure shown in Fig.6.

There are several factors that may in?uence the accuracy of the proposed decomposition method.Based on the derivation of the reciprocity theorem,the observation box only needs to be big enough to enclose the RF antenna.Numerical experiments have also validated this conclusion.The distance between the noise source and the antenna is varied,and results have shown excellent agreement of the proposed method compared with the full-wave simulation of the entire structure,even for the very closely coupled cases as long as the noise source does not in?uence the radiation pattern of the antenna and the multiple scattering effects between the noise source and the antenna can be neglected.The mesh size on the observation box has been found very critical for the proposed method.Since the integral is approximated using the summation,the tangential components of the ?elds need to be approximately constant in each mesh cell.Thus,the size of the mesh cell is determined by both the wavelength and the variation of the ?elds on the observation box.The mesh cell size needs to be smaller than 1/10th of the wavelength.Furthermore,if the box is at the very near ?eld of the antenna (meaning the ?eld is changing dramatically on the surface of the box),the mesh cell size needs to be further smaller.C.DECT +IC

The last example to validate the proposed method is extracted from a real product [10].The near-?eld coupling between a Dig-ital Enhanced Cordless Telecommunications (DECT)antenna and an IC is studied as shown in Fig.8.The DECT antenna is on the edge of a printed circuit board,and one I/O port of the IC is excited as the noise source.The detailed dimensions of the DECT antenna are shown in Fig.9,and the thickness of the antenna is 0.709mils.The distance between the centers of the IC to the DECT antenna is 1949mils.The size of the die of the IC is 141.7mils ×433.1mils,and it has 27×2pins.The detailed modeling information of the IC is described in [10].

1130IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY ,VOL.55,NO.6,DECEMBER

2013

Fig.8.Near-?eld coupling from an IC to a DECT antenna on a real printed circuit

board.

Fig.9.Detailed dimensions of the DECT antenna (the values outside the parentheses are in millimeters and the values inside the parentheses are in

mil).

Fig.10.Output power comparisons between the proposed method and the

direct method for the DECT +IC structure shown in Fig.8.

The induced power on the DECT antenna port is evaluated us-ing both the proposed decomposition method and the full-wave simulation method of the entire structure.The dimensions of the observation box enclosing the DECT antenna are 551.1mils ×1732mils ×551.1mils,and the mesh size on the observation box is 39.37mils.The results are compared in Fig.10.

The resonant frequency of the DECT antenna is 1840MHz.The results of the output power at the antenna port from the two methods from 1340to 2340MHz with a 100MHz

step

Fig.11.

PIFA antenna rotated by 90?from Fig.

6.

Fig.12.Output power comparisons between the proposed method and the direct method for the trace +PIFA structure shown in Fig.11.

are compared in Fig.10.The difference between the proposed method and the direct full-wave simulation is within 0.8dB,which demonstrates that the proposed method is valid even when the antenna is at the edge of the board.D.Application in Prelayout Design

The proposed decomposition method breaks the modeling of the entire interference problem into two separate simulations.The main advantage of the proposed method is its application in prelayout planning and design.At this early design stage,both the antenna type and the component placement need to be determined.To achieve optimal performance in terms of the minimized RF interference,different antennas and/or different IC/antenna placements may need to be evaluated.If the direct full-wave method is used,multiple time-consuming simulations are needed whenever the antenna con?guration or the compo-nent placement changes.In other words,if K antenna con?gura-tions and L placement con?gurations are to be evaluated,totally K ×L full-wave simulations may have to be performed.How-ever,using the proposed decomposition method,only K +L full-wave simulations may be necessary.When K and L val-ues are larger,K +L <

W ANG et al.:ESTIMATING RADIO-FREQUENCY INTERFERENCE TO AN ANTENNA DUE TO NEAR-FIELD COUPLING1131

method is much more ef?cient than the direct method in this kind of engineering applications.

To demonstrate the idea,the PIFA and trace example in Fig.6 are revisited.The PIFA antenna is simply rotated by90?and the same Huygens box is used,as shown in Fig.11.

In this example,no new simulations are necessary.The an-tenna model obtained in Section IV-B is simply rotated by90?through coordinate transformation,and then the noise power received at the antenna is estimated using the decomposition method.The result is compared with the direct full-wave simula-tion in Fig.12.Again,the difference is small,with the maximum value at the resonant frequency of the antenna less than3dB.

V.C ONCLUSION

A decomposition method is introduced for estimating the near-?eld noise coupling from a digital noise source to an RF antenna.The reciprocity theorem is used to obtain the antenna noise output based on the modeling of the forward and reserve problems.Three examples,from a simpli?ed model to a practi-cal model extracted from a real product,are used to validate the proposed method.The calculated results using the decomposi-tion method based on the reciprocity theorem show excellent agreement of the induced noise at the RF antenna port with the direct full-wave simulations of the original geometry.The proposed method?ts well with typical engineering practice and can be used for ef?cient prelayout design and optimization.The multiple scattering effects between the antenna and other scat-ters/sources in the system are neglected in this study.

A CKNOWLEDGMENT

The authors would like to thank H.Li with the Electromag-netic Compatibility Laboratory,Missouri University of Science and Technology,Rolla,MO,USA,for providing the DECT+IC HFSS model for the noise-coupling study in this paper.

R EFERENCES

[1]K.Slattery and H.Skinner,Platform Interference in Wireless Systems:

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[2] C.R.Paul,Introduction to Electromagnetic Compatibility.Hoboken,

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[3] C.A.Balanis,Antenna Theory:Analysis and Design,3rd ed.Hoboken,

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[4]Z.Yu,J.A.Mix,S.Sajuyigbe,K.P.Slattery,and J.Fan,“An improved

dipole-moment model based on near-?eld scanning for characterizing near-?eld coupling and far-?eld radiation from an IC,”IEEE https://www.360docs.net/doc/392342119.html,pat.,vol.55,no.1,pp.97–108,2013.

[5]HFSS.(2011).[Online].Available:https://www.360docs.net/doc/392342119.html,/staticassets/

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tions between equivalent current sources and antennas,”to be published.

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USA:Wiley,1989,pp.323–325.

[8]J.H.Richmond,“A reaction theorem and its applications to antenna

impedance calculations,”IRE Trans.Antennas Propag.,vol.9,no.6, pp.515–520,Nov.1961.

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[10]H.Li,V.Khilkevich,D.Pommerenke,“Identi?cation and visualization

of electromagnetic coupling path—Part II:Practical application,”to be

published.

Hanfeng Wang(S’09)received the B.S.and M.S.

degrees in electronic engineering from Tsinghua Uni-

versity,Beijing,China,in2005and2008,respec-

tively.He is currently working toward the Ph.D.

degree from the Electromagnetic Compatibility Lab-

oratory,Missouri University of Science and Technol-

ogy,Rolla,USA.

In2008,he joined the Electromagnetic Compati-

bility Laboratory,Missouri University of Science and

Technology.His current research interests include

signal integrity,power integrity,and advanced nu-merical

modeling.

Victor Khilkevich(M’08)received the Ph.D.de-

gree in electrical engineering from Moscow Power

Engineering Institute,Technical University,Moscow,

Russia,in2001.

He is currently a Research Associate Professor

at the Missouri University of Science and Tech-

nology,Rolla,USA.His current research interests

include computational electrodynamics,automotive

electromagnetic compatibility modeling,and high-

frequency measurement

techniques.

Yao-Jiang Zhang(S’97–M’01–SM’11)received

B.E.and M.E.degrees in electrical engineering from

the University of Science and Technology of China,

Hefei,China,in1991and1994,respectively,and

the Ph.D.degree in physical electronics from Peking

University,Peking,China,in1999.

From1999to2001,he was a Postdoctoral

Research Fellow at Tsinghua University,Beijing,

China.From August2001to August2006,he was a

Senior Research Engineer at the Institute of High

Performance Computing(IHPC),Agency for Sci-ence,Technology and Research(A?STAR),Singapore.From September2006 to September2008,he was a Postdoctoral Research Fellow in the EMC Labora-tory,Missouri University of Science and Technology(M S&T,formerly known as the University of Missouri-Rolla).From September2008to April2010,he was a Research Scientist at IHPC.Since April2010,he has been a Research Associate Professor in the EMC Laboratory,M S&T.His current research in-terests include analytical and numerical methods in electromagnetics,and their applications in signal/power integrity and electromagnetic interference issues for high-speed electronic packages or printed circuit

boards.

Jun Fan(S’97–M’00–SM’06)received the B.S.and

M.S.degrees in electrical engineering from Tsinghua

University,Beijing,China,in1994and1997,respec-

tively,and the Ph.D.degree in electrical engineering

from the Missouri University of Science and Technol-

ogy(formerly University of Missouri-Rolla),Rolla,

USA,in2000.

From2000to2007,he was a Consultant En-

gineer with the NCR Corporation,San Diego,CA,

USA.In July2007,he joined the Missouri Univer-

sity of Science and Technology(formerly University of Missouri-Rolla),where he is currently an Associate Professor with the EMC Laboratory.His current research interests include signal integrity and electro-magnetic interference(EMI)designs in high-speed digital systems,dc power-bus modeling,intrasystem EMI and radio-frequency interference,printed circuit board noise reduction,differential signaling,and cable/connector designs.

Dr.Fan was the Chair of the IEEE EMC Society TC-9Computational Elec-tromagnetics Committee from2006to2008and was a Distinguished Lecturer of the IEEE EMC Society in2007and2008.He is currently the Vice Chair of the Technical Advisory Committee of the IEEE EMC Society.He received the IEEE EMC Society Technical Achievement Award in August2009.He is cur-rently an Associate Editor of the IEEE T RANSACTIONS ON E LECTROMAGNETIC C OMPATIBILITY and EMC Magazine.