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A hybrid implicit-explicit FDTD (HIE-FDTD) algorithm is proposed in this paper. In the HIE-FDTD algorithm, semi-implicit- and explicit-difference schemes are imposed on two different field components, respectively, and the field component with

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3
10
7
S
/m, and the input modulated Gaussian pulse is
V
0
1.0 V,
16.68 ps, and
t
0
66.68 ps.As in Figure 4, the attenuation of the propagating sinusoidallymodulated Gaussian pulse is due mainly to the ohmic loss of boththe center strip and ground planes. Also, the attenuation is morerapid at the start of the pulse propagation distance.
4. CONCLUSIONS
Based on an improved empirical formula to calculate the ohmicloss of CPWs, the lossy effects in coplanar waveguides on thepulse propagation have been studied. These CPWs are assumed tobe fabricated on single-layer, low-dispersion and low-loss LTCC.For normal LTCC CPWs, it is shown that the total loss is contrib-uted mainly by the ohmic loss of the conductive planes in thefrequency range up to 40 GHz. The ohmic loss can lead to nearly50% pulse magnitude attenuation at the start of the propagationdistance of less than several millimeters.
REFERENCES
1. M. Riaziat, R. Majidi-Ahy, and I. Jaung Feng, Propagation modes anddispersion characteristics of coplanar waveguides, IEEE Trans Micro-wave Theory Tech MTT-41 (1993), 1499–1510.2. G. Gihione, A CAD-oriented analytical model for the losses of generalasymmetrical coplanar lines in hybrid and monolithic MICs, IEEETrans Microwave Theory Tech MTT-41 (1993), 1499–1510.3. H. Klingbeil and W. Heinrich, Calculation of CPW A.C. resistance andinductance using a quasi-static mode-matching approach, IEEE TransMicrowave Theory Tech MTT-42 (1994), 1004–1007.4. C.L. Hollowy and E.F. Kuester, A quasi-closed form expression forthe coplanar loss of CPW lines, with an investigation of edge shapeeffects, IEEE Trans Microwave Theory Tech MTT-43 (1995), 2695–2701.5. K.C. Gupta, R. Grag, I. Bahl, and P. Bhartia, Microstrip lines andslotlines, 2
nd
ed., Artech House, Boston, 1996.6. D. Lederer and J.P. Raskin, Substrate loss mechanisms for microstripand CPW transmission lines on lossy silicon wafers, 2002 IEEEMTT-S Dig, 2002, pp. 685–688.7. D.S. Phatak and A.P. Defonzo, Dispersion characteristics of opticallyexcited coplanar striplines: pulse propagation, IEEE Trans MicrowaveTheory Tech MTT-38 (1990), 654–661.8. C.L. Liao, Y.M. Tu, J.Y. Ke, and C.H. Chen, Transient propagation inlossy coplanar waveguides, IEEE Trans Microwave Theory TechMTT-44 (1996), 2605–2611.9. G.E. Ponchak, M. Magloubian, and L.P.B. Katehi, A measurement-based design equation for the attenuation of MMIC-compatible copla-nar waveguide, IEEE Trans Microwave Theory Tech MTT-47 (1999),241–243.10. D. Heo, A. Sutono, E. Chen, Y. Suh, and J. Laskar, A 1.9-GHz DECTCMOS power ampliﬁer with fully integrated multilayer LTCC pas-sives, IEEE Microwave Wireless Compon Lett 11 (2001), 249–251.11. K. Kageyama, K. Saito, H. Murase, H. Utaki, and T. Yamanoto,Tunable active ﬁlters having multilayer structure using LTCC, IEEETrans Microwave Theory Tech MTT-49 (2001), 2421–2424.© 2003 Wiley Periodicals, Inc.
A HYBRID IMPLICIT-EXPLICIT FDTDSCHEME WITH WEAKLY CONDITIONALSTABILITY
Binke Huang,
1
Gang Wang,
2
Yansheng Jiang,
1
and Wenbing Wang
1
1
Department of Telecommunication Engineering Xi’an Jiaotong University Xi’an 710049, P.R. China
2
Department of Telecommunication EngineeringJiangsu UniversityZhenjiang 212013, P.R. China
Received 13 March 2003
ABSTRACT:
A hybrid implicit-explicit FDTD (HIE-FDTD) algorithmis proposed in this paper. In the HIE-FDTD algorithm, semi-implicit-and explicit-difference schemes are imposed on two different ﬁeld com- ponents, respectively, and the ﬁeld component with implicit difference isupdated by solving simple tridiagonal matrix equations. The HIE-FDTDmethod, although conditionally stable, allows for larger time-step sizethan the conventional FDTD method because the stability condition of the HIE-FDTD method is weaker than that of the conventional Yee’sFDTD algorithm. The demonstrated numerical dispersion relation of the HIE-FDTD method is found to be identical to that of the ADI-FDTDmethod. The accuracy and efﬁciency of the HIE-FDTD are veriﬁed bynumerical simulation results.
© 2003 Wiley Periodicals, Inc. MicrowaveOpt Technol Lett 39: 97–101, 2003; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.11138
Key words:
implicit difference; explicit difference; weakly conditionalstability; FDTD; tridiagonal matrix equations
Figure 6
Propagation of a sinusoidally modulated Gaussian pulse inLTCC CPWs: (a) (
r
, tan
)
(4.2, 0.003); (b) (
r
, tan
)
(7.8, 0.0015)and (10.6, 0.001)
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 39, No. 2, October 20 2003
97
1. INTRODUCTION
In a conventional ﬁnite-difference time-domain (FDTD) scheme,the time-step size is strictly limited by the stability condition. Forthe analysis of structures with ﬁne-scale dimensions or that aresigniﬁcantly distinctive in different spatial dimensions, the maxi-mum time-step size is limited by the minimum grid size in thecomputational domain [1]. Smaller time-step size in simulationgenerally requires more computational resources. Therefore, someefforts have been made recently to develop more efﬁcient FDTDschemes that allow for larger time-step size. Along this line, analternative-direction implicit FDTD (ADI-FDTD) algorithm wasproposed by T. Namiki [2] to eliminate restraints on the time-stepsize. Indeed, the ADI-FDTD has proved to be very efﬁcient if theminimum cell size in the computational domain is much smallerthan the wavelength. However, the two sides of the updatingequations (Eqs. (1)–(4) in [3]) are not balanced with respect totime, which will result in larger numerical dispersion error [3].Moreover, three time steps are used for deﬁning the ﬁeld compo-nents and two sub-iterations are required for ﬁeld advancement,which will make the ADI-FDTD computationally inefﬁcient. Toovercome this drawback of the ADI-FDTD algorithm, an uncon-ditionally stable FDTD (US-FDTD) was proved by Zhao [3], inwhich a semi-implicit difference scheme for all ﬁeld componentsis applied simultaneously, and merely a single iteration is neededfor ﬁeld advancement. Unfortunately, the adjacent electric ﬁeldcomponents are coupling with each other at the same time step.Instead of the small tridiagonal equations as in the ADI-FDTDalgorithm, huge sparse matrix equations, neither banded nor diag-onally dominant, will be encountered for general two-dimensional(2D) or three-dimensional (3D) structures during the electric-ﬁeldupdating. Thus, the efﬁciency of US-FDTD is largely dependenton the solving of large-scale sparse matrix equations.In the present paper, a new FDTD algorithm, referred to ashybrid implicit-explicit difference FDTD (HIE-FDTD), is pro-posed by combining the advantages of ADI-FDTD and US-FDTD.In section 2, a numerical formulation of the HIE-FDTD for a 2DTE wave is presented. The HIE-FDTD uses implicit- and explicit-difference schemes for the two electric components, respectively.As will be shown, both sides of the difference equations arebalanced with respect to time and only one iteration is needed forﬁeld updating at each time step. Moreover, the ﬁeld componentwith implicit difference can be updated by solving simple tridi-agonal matrix equations. The numerical stability of the proposedHIE-FDTD algorithm is studied in section 3. It is shown that largertime-step size is allowed although the HIE-FDTD is conditionallystable. The accuracy and efﬁciency of HIE-FDTD will be illus-trated by numerical examples in section 4.
2. NUMERICAL FORMULATION FOR 2-D HIE-FDTDMETHOD
As an example, we consider a 2D TE wave. Maxwell’s equationsfor the 2D TE wave in an isotropic lossless medium are [1]:
E
x
t
H
z
y
, (1a)
E
y
t
H
z
x
, (1b)
H
z
t
E
y
x
E
x
y
, (1c)where
and
are the permittivity and permeability of the me-dium, respectively.As in the conventional FDTD scheme, a ﬁeld component
F
(
t
,
x
,
y
) can be denoted as
F
n
I
,
J
F
n
t
,
I
x
,
J
y
, (2)where
x
or
y
,
n
is time index,
I
and
J
are space indexes,
t
is the time-step size, and
x
and
y
are the spatial incrementsalong the
x
and
y
directions, respectively. We have assumed thatall cells in a computational domain have the same size. Theelectromagnetic ﬁeld components are arranged on the cells in thesame way as that used in the conventional Yee’s staggered FDTDalgorithm. The
E
y
ﬁeld is deﬁned at time steps
n
1/2 and
n
1/2, and the
E
x
and
H
z
ﬁelds at time steps
n
and
n
1. For the
E
y
component, we use the explicit-difference technique at integertime step
n
, whereas for the
E
x
and
H
z
components, we usesemi-implicit difference technique at time step
n
1/2. Note thatthe ﬁelds
E
x
and
H
z
at time step
n
1/2 are not available, and theﬁeld quantity at time step
n
1/2 will be calculated approxi-mately by averaging their values at time step
n
and
n
1 [3].Thus, the srcinal difference updating equations of Eq. (1) for theTE wave can be obtained as
t
E
x n
1
I
1/2,
J
E
x n
I
1/2,
J
12
y
H
zn
1
I
1/2,
J
1/2
H
zn
1
I
1/2,
J
1/2
H
zn
I
1/2,
J
1/2
H
zn
I
1/2,
J
1/2
, (3a)
t
E
yn
1/2
I
,
J
1/2
E
yn
1/2
I
,
J
1/2
1
x
H
zn
I
1/2,
J
1/2
H
zn
I
1/2,
J
1/2
, (3b)
t
H
zn
1
I
1/2,
J
1/2
H
zn
I
1/2,
J
1/2
1
x
E
yn
1/ 2
I
1,
J
1/2
E
yn
1/ 2
I
,
J
1/2
12
y
E
x n
1
I
1/2,
J
1
E
x n
1
I
1/2,
J
E
x n
I
1/2,
J
1
E
x n
I
1/2,
J
. (3c)Obviously, the left and right hands of Eq. (3a)–(3c) are balancedwith respect to time. Similar to the ADI-FDTD method, the up-dating of the
E
x
component, as shown in Eq. (3a), needs theunknown
H
z
component at the same time, thus the
E
x
componenthas to be updated implicitly. By substituting (3c) into (3a), theequation for
E
x
ﬁeld is given as
1
t
2
2
y
2
E
x n
1
I
1/2,
J
t
2
4
y
2
E
x n
1
I
1/2,
J
1)
E
x n
1
I
1/2,
J
1
]
t
y
H
zn
I
1/2,
J
1/2)
H
zn
I
1/2,
J
1/2
]
t
2
2
x
y
E
yn
1/2
I
1,
98
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 39, No. 2, October 20 2003
J
1/2)
E
yn
1/2
I
,
J
1/2
E
yn
1/2
I
1,
J
1/2
E
yn
1/2
I
,
J
1/2
]
t
2
4
y
2
E
x n
I
1/2,
J
1
2
E
x n
I
1/2,
J
E
x n
I
1/2,
J
1
]
E
x n
I
1/2,
J
. (4)Therefore, the updating of ﬁeld components in the HIE-FDTDalgorithm will be achieved by using Eqs. (3b), (3c), and (4). If theﬁeld component
E
y
is explicitly updated ﬁrst, the
E
x
componentwill be updated implicitly by solving the tridiagonal matrix equa-tion through Eq. (4). After the electric-ﬁeld components are de-rived at each time step, the
H
z
component can be explicitlyupdated straightforward by using (3c). If the
E
x
component isupdated explicitly ﬁrst, the
E
y
component will be updated implic-itly (note that the subscripts
x
and
y
should be exchanged in(3a)–(3c), and the spatial difference should also be changed ac-cordingly).
3. ANALYTICAL INVESTIGATION OF NUMERICAL STABILITY AND NUMERICAL DISPERSION
With no loss of generality, the ﬁeld components of a 2D TE wavecan be written as follows [2]:
E
x
A
n
exp
j
k
x
x
k
y
y
, (5a)
E
y
B
n
exp
j
k
x
x
k
y
y
, (5b)
H
z
C
n
exp
j
k
x
x
k
y
y
, (5c)where
j
1,
A
,
B
, and
C
are the amplitudes of the ﬁeldcomponents, respectively,
k
x
and
k
y
are wave numbers, and
indicates growth factor. In order to calculate
, we substitute(5a)–(5c) into (3a)–(3c) to obtain
t
A
1
j
C
y
1
sin
k
y
y
2
, (6a)
t
B
1
2
j
C
x
sin
k
x
x
2
1/ 2
, (6b)
t
C
1
2
j
B
x
sin
k
x
x
2
1/ 2
j
A
y
sin
k
y
y
2
1
.(6c)By eliminating
A
,
B
, and
C
of Eq. (6), we obtain
1
a
2
2
2
a
b
1
a
0, (7)where
a
t
2
y
2
sin
2
k
y
y
2
0, (8a)
b
4
t
2
x
2
sin
2
k
x
x
2
0. (8b)By solving Eq. (7), the growth factor
can be obtained
2
2
a
b
b
4
b
4
a
2
1
a
. (9)To satisfy the stability condition during ﬁeld advancement, themodule of growth factor
must be less than 1. From Eq. (9), for0
b
4 we have
1 so that HIE-FDTD is conditionallystable. The limitation for time-step size in the HIE-FDTD algo-rithm can be calculated from (8b):4
t
2
x
2
sin
2
k
x
x
2
4
f
t
x c
, (10)where
c
1/
is the speed of light in the medium.Note that the time-step size for conventional FDTD is
t
(
x
/
c
2) for square grids in the computational domain (that is,
x
y
) [1], it is obvious from Eq. (10) that, to meet the stabilitycondition of the HIE-FDTD algorithm, the time-step size needs tobe larger than that of the conventional FDTD method.Equation (10) also shows that the maximum time-step size forthe HIE-FDTD algorithm is only determined by one spatial incre-ment. Here the spatial increment in the
x
direction is due to theapplication of explicit difference for the
H
z
component in the
x
direction in (3b). This is especially useful when the simulatedstructure has a ﬁne-scale dimension in one direction: a smallspatial increment can be used in the direction with ﬁne scale anda larger spatial increment can be used in the direction with coarsescale. If we perform the implicit-difference scheme in the directionwith a larger spatial increment, the time-step size is thus deter-mined by the larger spatial increment. For example, the size of thestructure being analyzed in the
x
direction is larger than that of the
y
direction. By setting
x
4
y
and performing semi-implicitdifference for the
E
x
ﬁeld, the maximum time-step size meetingthe stability condition of the HIE-FDTD algorithm can be deter-mined by Eq. (10) as
t
x
/
c
4
y
/
c
, while the maximumtime-step size for the conventional FDTD method is
t
c
1/(
c
(1/
x
)
2
(1/
y
)
2
)
0.97
y
/
c
. As a result, computa-tional resources can be saved considerably.We now study the numerical dispersion in the proposed HIE-FDTD algorithm. The trial solutions of the ﬁelds for the TE wavecan be given as [1]:
E
x n
I
,
J
E
x
0
exp
j
k
x
I
x
k
y
J
y
n
t
, (11a)
E
yn
I
,
J
E
y
0
exp
j
k
x
I
x
k
y
J
y
n
t
, (11b)
H
zn
I
,
J
H
z
0
exp
j
k
x
I
x
k
y
J
y
n
t
, (11c)where
is the angular frequency. By substituting (11) separatelyinto (3a)–(3c) and eliminating
E
x
0
,
E
y
0
, and
H
z
0
, we obtain1
c
t
2
sin
2
t
2
1
x
2
sin
2
k
x
x
2
cos
2
t
2
1
y
2
sin
2
k
y
y
2
. (12)Equation (12) is the numerical dispersion relation of the HIE-FDTD algorithm for the TE wave.For comparison, we take a look at the numerical dispersionrelation of the conventional 2D FDTD method (Eq. (4.5) in [1]):1
c
t
2
sin
2
t
2
1
x
2
sin
2
k
x
x
2
1
y
2
sin
2
k
y
y
2
. (13)Compared to the dispersion Eq. (13) of the conventional 2D FDTDmethod, it can be obtained that there is a factor cos
2
(
t
/2) added
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 39, No. 2, October 20 2003
99
to the second term in the right-hand side of the numerical disper-sion relations of Eq. (12). If the limits
t
,
x
, and
y
all go tozero, the numerical dispersion relation of the HIE-FDTD method,Eq. (12), will reduce to the well-known dispersion relation for a2D plane wave, given by
k
2
2
c
2
k
x
2
k
y
2
, (14)where
k
/
c
is the wavenumber. In fact, the numerical disper-sion Eq. (12), compared to the dispersion equation of the 2DFDTD algorithm, is identical to the one of the
x
-directional 2DADI-FDTD method [4].To evaluate the performance of the different FDTD schemesdiscussed above, Figure 1 shows the normalized phase velocities(namely,
p
/
c
) for the conventional FDTD scheme and severalHIE-FDTD schemes, which are indicated by different Courant–Friedrich–Levy number (CFLN) values. The CFLN is deﬁned asthe ratio of the time-step size in the HIE-FDTD algorithm and themaximum time-step size in the conventional FDTD algorithm.Obviously, the CFLN value will depend on the spatial incrementsalong the
x
and
y
directions used in the FDTD algorithm. Figure1(a) shows the results for
x
y
and Figure 1(b) shows theresults for
x
5
y
. In our calculation,
x
is set to be
/20, with
the operating frequency. It can be found from Figure 1(a) and (b)that, for CFLN
1, the performance of the conventional FDTDalgorithm is superior to that of the HIE-FDTD algorithm. For
x
y
, increasing the CFLN value (within its limit
2) willlead to HIE-FDTD performance degradation. For
x
5
y
, thereis an optimal CFLN for the best HIE-FDTD performance and theoptimal value is around CFLN
2.89 [4]. Therefore, it is possibleto improve simulation efﬁciency through an increased time-stepsize in the HIE-FDTD method, if different spatial increments areset along different spatial directions.
4. NUMERICAL RESULTS
To demonstrate the accuracy and efﬁciency of the HIE-FDTDmethod, we will study a magnetic point radiation in the TE case infree space. The exciting pulse has a Gaussian envelope with acarrier frequency of 5 GHz, applied at the
H
z
component of thecentral cell of the computation domain. The computation domainis 85
85 cells, and the cell size is set as
x
y
1.5 mmso that 20 samples per wavelength at the center frequency of theexcitation are provided. The computation domain is truncated byan 8-cell split-ﬁeld PML layer with polynomial spatial scaling(
m
2) [5]. The theoretical reﬂection coefﬁcient is set as 10
6
todetermine
max
at the outer boundary of the PML medium. Theobservation point is set at 20 cells far from the source in the
y
direction.First, the time-step size is set as
t
2.5 ps to satisfy thelimitation of the 2D Courant–Friedrich–Levy (CFL) condition inthe conventional FDTD method. Figure 2 shows the results of the
H
z
component simulated by using the HIE-FDTD and conven-tional FDTD methods, respectively. It can be observed that the twonumerical results are in good agreement, except for a peak value
Figure 1
Normalized plane velocity of the HIE-FDTD method as functions of the CFLN: (a)
x
y
; (b)
x
5
y
Figure 2
Comparison of numerical results from conventional FDTD andHIE-FDTD methods under same time-step size
100
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 39, No. 2, October 20 2003
difference, which is due to the dispersion effect in solving thetridiagonal matrix equations, as in the HIE-FDTD method. Thetwo simulations are performed on a Pentium IV-M 1.4-GHz com-puter, and the 700 time step simulation takes 25.1 s for theconventional FDTD method and 24.4 s for the HIE-FDTD method,which indicates that they have almost the same efﬁciency underthe same time-step size (namely,
t
2.5 ps).For further comparison, we use different time-step sizes:
t
h
x
/
c
for the HIE-FDTD method and
t
c
x
/(
c
2) for theconventional FDTD method, both of which are the maximumtime-step sizes in order to ensure numerical stability. The simula-tion results of the
H
z
component are shown in Figure 3. The tworesults also agree very well. But the simulation of 1.8-ms propa-gation in free space takes 17.8 s for the conventional FDTDmethod and 11.5 s for the HIE-FDTD method. The time cost forthe conventional FDTD simulation is about
2 times that for theHIE-FDTD simulation, since the time-step size of the latter is
2times that of the former.As mentioned in section 3, unequal spatial increments can beused in different directions in the HIE-FDTD method. Therefore,larger time-step size can be introduced to further reduce the CPUtime. Assume that the computation domain is 0.3 m (in
x
)
0.15m (in
y
), which is discretized by a grid of 100
100 cells (that is,
x
2
y
3 mm). The maximum time increments are
t
c
0.8
y
/
c
4.472 ps for the conventional FDTD method and
t
h
2
y
/
c
10.0 ps for the HIE-FDTD method. Thus, wehave
t
h
/
t
c
2.24. The simulation results of the
H
z
componentare shown in Figure 4. The two results also agree very well.However, the simulation of 1.8-ms propagation in free space takes14.4 s for the conventional FDTD method and 6.0 s for theHIE-FDTD method. The time cost for conventional FDTD simu-lation is about 2.4 times that for the HIE-FDTD simulation, sincethe time-step size of the latter is 2.24 times that of the former.Therefore, the proposed HIE-FDTD algorithm is more efﬁcientthan the conventional FDTD algorithm.Note that the peak-value difference between the results simu-lated by using the conventional FDTD and HIE-FDTD methods, asshown in both Figures 3 and 4, may be due to the increase of numerical dispersion errors for larger time increments.
5. CONCLUSION
We have demonstrated a hybrid implicit-explicit FDTD algorithmbased on a consideration of the stability condition. The algorithmcomplexity of the HIE-FDTD method is the same as that of theconventional FDTD method, while the CFL condition restraint isweaker than that of the conventional FDTD method and, conse-quently, less CPU time is required. If the computation domain issigniﬁcantly distinctive in different spatial dimensions, the time-step size can be further increased to reduce CPU time by intro-ducing an unequal space increment in different dimensions in theHIE-FDTD method. Numerical simulation shows that the newalgorithm is efﬁcient and the results agree well with that of theconventional FDTD method. This paper has explained the HIE-FDTD method for a 2D TE wave; however, the method can also beapplied to a 2D TM wave and extended in a general sense to a fully3D wave.
REFERENCES
1. A. Taﬂove and S.C. Hagness, Computational electromagnetics: Theﬁnite-difference time-domain method, Artech House, Boston, 2000.2. T. Namiki, A new FDTD algorithm based on alternating directionimplicit method, IEEE Trans Microwave Theory Tech MTT-47 (1999),2003–2007.3. A. Zhao, More accurate and efﬁcient unconditionally stable FDTDmethod, Electron Lett 38 (2002), 862–864.4. A.P. Zhao, Analysis of the numerical dispersion of the 2-D alternating-direction implicit FDTD method, IEEE Trans Microwave Theory TechMTT-50 (2002), 1156–1164.5. J.P. Berenger, A perfectly matched layer for the absorption of electro-magnetic waves, J Comput Phys 114 (1994), 185–200.6. V. Van and S.K. Chaudhuri, A hybrid implicit-explicit FDTD schemefor nonlinear optical waveguide modeling, IEEE Trans MicrowaveTheory Tech MTT-47 (1999), 540–545.© 2003 Wiley Periodicals, Inc.
Figure 3
Comparison of numerical results for the two methods with themaximum time-step size
Figure 4
Comparison of numerical results for the two methods withdifferent spatial increments in different directions
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 39, No. 2, October 20 2003
101

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