Higher-Order Non-Standard FDTD Modeling of Complicated EMC ...

Higher-Order Non-Standard FDTD Modeling of Complicated
EMC Structures in 3-D Curvilinear Lattices

Nikolaos V. Kantartzis Theodoros D. Tsiboukis

Department of Electrical and Computer Engineering,

Aristotle University of Thessaloniki, Thessaloniki, GR-54124, GREECE

e-mail: [email protected]; [email protected]

Abstract: A systematic higher-order FDTD-PML methodology for Dual lattice D D Dual
the accurate simulation of complex curvilinear 3-D EMC problems B H H components
is presented in this paper. Developing an efficient covariant-
contravariant vector classification, the novel technique introduces a Secondary
parametric family of higher-order non-standard schemes for the source ρ
suppression of the critical dispersion errors. The wider spatial
stencils near boundary walls are treated by self-adaptive compact E
operators, while for the temporal variable a multi-stage leapfrog
integration is utilized. Moreover, this optimal formulation leads to Primary E E
enhanced curvilinear PMLs that significantly annihilate outgoing components B B
waves. Numerical validation indicates the benefits of the proposed
algorithm via several demanding and practical EMC applications. E Primary lattice

Keywords: Advanced FDTD-PML method, Higher-order schemes, Figure 1. A dual generalized curvilinear FDTD lattice
Curvilinear coordinates, EMC numerical modeling and predictions
The Curvilinear HO Non-Standard FDTD Method
Introduction
An alternative to cell division comes from the HO FDTD schemes
The progressively increasing needs for the correct realization of which offer increased accuracy for a given node density [16]-[17].
modern scientific concepts in the EMC field have lately established Despite the evident improvement, these concepts are still prone to
challenging design or fabrication trends. The excessively complex dispersion errors, especially in curvilinear domains. Actually, this
profile of such devices along with their large electrical size turns remark has comprised the primary motive for our algorithm.
ordinary treatments virtually insufficient to follow the underlying
physical phenomena. So, the development of robust schemes that Theoretical Formulation
merge optimal performance with affordable cost and concurrently
provide insightful deductions is required. Among existing cases, The superiority of the 3-D HO non-standard concepts stems from
EMC modeling via the Finite-Difference Time-Domain (FDTD) the low wavelength-to-stencil ratio and the reduced number of
method [1] has gained a significant thrust [2]-[12]. However, when iterations. We start by introducing three spatial operators which
numerically solving a curved problem via a staircase regime, correspond to u,v,w axes of a general coordinate system (u,v,w) –
dispersion errors place serious limits. Unfortunately, finer mesh defined as Su[.], Sv[.], Sw[.] – and one temporal operator T[.]. Thus,
resolutions are translated to inhibiting computational overheads.
On the other hand, trustworthy simulations are also attained by the Su  f t = b1Pδuu  f t  + b2P3uδ u  f t
correct truncation of infinite curvilinear regions. Therefore, soon  u ,v, w  u,v,w   u ,v,w
after the initial advent of the remarkable Berenger perfectly (1)
matched layer (PML), an important research has been devoted to ( ) δ+b3
its unsplit-field extension in non-Cartesian meshes [13]-[15]. f − f /t
u−δ u / 2,v,w
It is the objective of this paper to introduce a novel fully non- t u,
orthogonal higher-order (HO) FDTD algorithm for the precise u − 3δ u / 2,v , w
solution of complex 3-D EMC structures in generalized curvilinear
coordinates and the accomplishment of reliable electromagnetic ( )T
interference (EMI) predictions. The basic premise of the method  f t = f t+δt / 2 − f t −δ t / 2 / cT (kδ t)
lies on the representation of electromagnetic fields by means of  u,v,w  u,v,w u,v,w
accurate non-standard concepts and self-adaptive compact central
differences forming thus, a topologically consistent HO covariant Ts 1  cT (kδ t ) τ −1 ∂τ f n (2)
and contravariant framework. Analyzing the inherent mechanisms (odd τ!  2  ∂tτ
of discrete space, the proposed procedure launches supplementary ∑−2b3 ,
degrees of freedom, whereas to avoid erroneous instabilities, a dual τ = 3 )
grid that preserves global convergence near material interfaces is u,v,w
developed (see Fig. 1). Hence, the new FDTD-PML framework
achieves a critical attenuation of outgoing waves and subdues with Sv[.], Sw[.] designed for derivatives along v and w. By setting
dispersion or anisotropy deficiencies. Additionally, its profile is b1 = 9/8, b2 = –1/24, b3 = 0, we get the basic class member, while
further enhanced by a general leapfrog integrator. The proposed for b1 = 11/12, b2 = 1/24, b3 = 1/24 the second counterpart is
methodology receives an extensive numerical verification in terms extracted, with δu, δv, δw its spatial and δt its temporal increments.
of various practical 2- and 3-D EMC arrangements with curvilinear Parameter τ, in (2), runs from 3 to Ts, where Ts is an odd prefixed
parts that can not be adequately modeled by usual techniques. limit that controls the order of time derivative approximation. In
(1), Ph[.] for h = δu, 3δu, is the 3-D non-standard form, defined as

1
cS (kh)
{Phu
 f t ≡ pA D(A)  f t
 u , v , w u,h  u , v , w

}+ (3)

pB D(B)  f t  + pC D(C)  f t .
u,h  u,v,w  u,h  u , v , w

D(A) D(B) D(C) z
u,h u,h u,h
+1 h +1 h +1 h

r1

-1 -1 -1 θr Q

PEC r0
wall
φ y
x
Figure 2. Graphical depiction of D operators Scatterer

Functions cS(kh), cT(kδt) are selected to minimize the lattice errors ΩCS
and guarantee the smooth transition from the continuous to the Ω PML
discrete space. To pick an acceptable kh for several wavenumbers
k, the Fourier transform of the already computed transient electric Figure 3. The geometry of space Ω in spherical coordinates
(magnetic) components, at predefined grid points, is utilized. After
their frequency range is obtained, we find the spectrum’s vector of HO operators, JE the electric current density column
maximum value. Obviously, the same holds for cT(kδt). Operators vector, Yt the constitutive matrix featuring all materials, GH the
D(i) for i = A,B,C in (3), lead to a set of formulae which offer a
viable interconnection. For example, postulating the general metric tensor and TE the matrix of HO temporal derivatives.
geometry of Fig. 2 (the +1 and -1 integers at the faces indicate the
sign of summation for the values of function f), D(C) is given by To overcome the difficulty of the widened stencils near perfectly
electric conducting walls (PEC), a modified set of self-adaptive
 f t + f t + f t  curvilinear compact operators is developed and expressed as
 + h/ h / 2,−δ v,0 h / 2,0,δ w 
 − 2,δ v,0 ∑1 n n )n
 s i+s i
Du(C,h)  f t = 1  f t − f t − f t , (4) = −1 (aA fu + aB fu + aC fu i−s =0, (9)
 u,v,w  4 h / 2,0,−δ w −h / 2,δ v,0 −h / 2,−δ v,0 

f t − f t  where aA, aB, aC are pre-selected real numbers. For a stable update
−h / 2,0,δ w −h / 2,0,−δ w process, we use a generalized leapfrog formulation which divides
each time-step to a number of stages equal to the order of
where only the respective increments along the u,v,w are indicated approximation and hardly influences the total burden. Finally, the
(i.e. h/2,–δv,0 means u+h/2,v–δv,w). Of particular importance, also, considerably improved dispersion relation can be written as (with
are the pi parameters. A typical and instructive choice is, c΄ the numerical velocity, β = 2π/(kδw) and θ the incident angle)

pA = q +η(1− q) / 3 , pB = η (1 − q) / 3 , (5)  c nst 5π 7 1 + 12cos(4θ )
 c′  670β 7 19
pC = 1 − q − 2η(1 − q) / 3 , (6) ≅1− . (10)

while η, q are computed through the components of wavevector k HO Reflectionless Curvilinear PMLs
properly defined according to the coordinate system under study.
The prior topological perspective has an additional merit; it reveals The construction of curvilinear unsplit-field PMLs requires the
the pitfalls of the second-order outlook, which lead to inaccuracies, division of a 3-D space Ω into two regions: ΩCS (computational
more frequently encountered in curvilinear implementations. domain) and ΩPML (PML area). Focusing on spherical coordinates
(r,θ,φ), we examine an isotropic, dielectric medium, described by

HO Modeling of a Non-Orthogonal Domain jωε E′ = ∇′ × H′ , ∇′ ⋅ (ε E′) = 0 , (11)
− jωµH′ = ∇′ × E′ , ∇′ ⋅ (µH′) = 0 . (12)
Consider a 3-D domain discretized into uniform cells which form
two dual grids. Covariant hm and contravariant hl components (l,m Here, ΩCS is a sphere of radius r0, and ΩPML must be truncated up
= u,v,w) of magnetic vectors H are located at dual edges remaining to a radius r1 by a PEC wall, as in Fig. 3. The optimized PML, is
interleaved with em and el quantities of electric vectors E at primary built through the proper scaling between the independent (primed)
edge centers. The new scheme hosts the idea of fluxes for the fields and dependent variables in (11), (12) which retain the original field
across the faces defined by f (l) = g1/2f l (f = e,h), with glm the variation in ΩCS and launch a coordinate stretching in ΩPML. So,
coordinate metrics. fm are represented with f l via the rules fm = gmlf l
and f l = glmfm. Next, we introduce the linear operator Q(l) such that F′ = F, for r′ ∈ ΩCS for , (13)
f (l) = Q(l)[fm], which uses the local fm and the neighboring ones fm+1, diag{ζ r ,ξr−1,ξr−1}F, r′ ∈ ΩPML
fm+2 (m+1, m+2 denote a cyclic permutation of u,v,w), multiplied
by the discrete metric terms. In this manner, Q(u)[hu] becomes with r΄ = diag{ξr,1,1}, r = (r,θ,φ)T and F = E, H. Application of

1(u) n+1/ 2 (13) to (11) gives (Ξ = diag{ξr2ζ , ζ −1 , ζ r−1} is the material tensor)
Q h  = s h + 4 s (h + h )u i−1/2, j,k r
uu n+1/ 2 uv n+1/ 2 n+1/ 2 r
i−1/ 2, j,k u i−1/ 2, j,k i, j,k v i, j−1/ 2,k v i, j+1/ 2,k
jωεΞ ⋅ E = ∇ × H , − jωµΞ ⋅ H = ∇ × E , (14)
n+1/ 2 n+1/ 2
v i−1, j+1/ 2,k w i, j,k+1/ 2
uv n+1/ 2

+ s (h + h ) + s (h + h ) (7)i−1, j,k v i−1, j−1/ 2,k
uw n+1/ 2 ζ r (r,ω) = [1 + σ r (r) / jω]−1 , ξr (r,ω) = 1 + σ r′ (r) / jω , (15)
i, j,k w i, j,k−1/ 2

uw n+1/ 2

+ s (h + h ),i−1, j,k w i−1, j,k−1/ 2
n+1/ 2 max ]md σ r′ (r) = r −1 r ( s )ds ,
w i−1, j,k+1/ 2 ∫for r r0
σ (r) = σ [(r − r0 ) / δ ; md ≥ 0 , σ
r r
where slm = g1/2glm. Through (1), (2) Ampere’s law now yields
with ΩPML restricted to a sphere of radius r0 + δ and δ the layer’s
∇×H = ∂tD + JE ⇒ (I + 1 Yt )Ecnv+1 (8) depth. Considering Ampere’s law in (14) and using (15), we obtain
2

= (I − 1 Yt )Ecnv − cT (kδ t )ε −1J n +1 / 2 + G HS[H n +1 / 2 ] + TEn +1/2 . T[Br ] + σ r′ (r)Br = gϕSϕ [Eθ ] − gθSθ [Eϕ ] , (16)
2 E , cv cv with T[Br ] + σ r (r )Br = µT[H r ] + µσ r′ (r)H r , (17)

in which cv are the covariant components, S = [Su, Sv, Sw] is the

port 3 Magnitude of Sij conducting a
plane
0
region C

-10 region A b

port 2 a3 x΄ -20
y΄ dB
θ L h1 θ s1
s2
-30 L
w
a1 -40 Modified modal analysis
Proposed HO FDTD for S11
a2 -50 Proposed HO FDTD for S12 11 12 Ey h2
b1 8 Proposed HO FDTD for S13
Hx region B
b2 b3 port 1 9 10
Frequency (GHz) (c)
(a)
(b)

Electric shielding effectiveness Magnitude of Sij
50
-10
θ = 30 deg

θ = 45 deg
40 θ = 60 deg

30 f = 300 MHz l l1 -15 Modified modal analysis for S11
dB dB Proposed HO FDTD for S11

1 2nd-order FDTD for S11

20 -20 Modified modal analysis for S12
ab Proposed HO FDTD for S12
l2
2nd-order FDTD for S12

10 2
f = 600 MHz
-25

0 waveguide or w 10 11 12 13 14
012 34 567 8 coaxial cable Frequency (GHz)
Number of slots
(f)
(d) (e)

Figure 4. (a) A sidewall coupled elliptical cavity and (b) the magnitude of its S-parameters. (c) An elliptical cavity with multiple
inclined slots and (d) its shielding effectiveness. (e) An aperture with two elliptical slots and (f) the magnitude of its S-parameters

in which Br = µξrζrHr and gm (m = r,θ,φ) the spherical system Next, we will explore wave penetration in an elliptical cavity with
metrics. Similar expressions are extracted for Bθ and Bφ. It is multiple rectangular inclined slots placed in a conducting plane
stressed that the proposed strategy enables HO PMLs to introduce (Fig. 4c). It is stressed that this problem with horizontal or vertical
supplementary attenuation terms along more directions in the layer apertures has been analytically [3] and numerically [5] studied.
which entail a substantial suppression of anisotropy discrepancies. However, the simulation here becomes more hard because of the
cavity’s elliptical cross-section. A typical set of dimensions is: a =
Numerical Results 10 cm, b = 5 cm, L = 20 cm, s1 = 1 cm, s2 = 3 cm, h1= 1.7 cm, h2 =
4.5 cm and w = 0.2 cm, while Fig. 4d demonstrates the shielding
The new method combined with the improved PMLs is validated effectiveness of the structure. All HO computations are performed
via various practical 3-D curvilinear EMC applications. Towards using an 8-cell PML and three cases are examined, i.e. θ = 30o, 45o,
this aim, we compare our HO FDTD outcomes with analytical 60o. Results depicted in Fig. 4d confirm the very sufficient EMI
solutions or measured data using very coarse grid resolutions. prediction profile of the HO schemes, which do not producing any
Furthermore, special attention has been paid to the choice of the late time instabilities, even for the frequency of 300 MHz.
correct PML conductivity profiles and depths in an effort to
achieve a faster dampening of all outgoing waves. The preceding In a similar way, Fig. 4f shows the behavior of S-parameters for the
prescriptions become critical in the case of non-Cartesian lattices aperture of Fig. 4e fed by two waveguides or coaxial cables. The
which entail a very cautious field-to-grid correspondence owing to dimensions of the device are a = 8.62 mm, b = 20.71 mm, l1 = 4.51
the gradual cell growth. Hence, a fairly smaller amount of time- mm and l2 = 1.93 mm. Observe the very satisfactory agreement
steps is needed without any degrading of the convergence rate. between the theoretical [11] and the HO plots as well as the
inability of the usual FDTD technique to give acceptable results.
Let us, first, study an elliptical cavity with a narrow inclined slot
coupled to a rectangular waveguide (Fig. 4a), basically utilized in Finally, the proposed algorithm is applied to modern broadband
the design of circular waveguide dual-mode filters. The numerical dual-polarized patch antennas. Fig. 5a describes the structure fed
analysis of this T-junction is difficult due to its curvilinear shape by a capacitatively- and an H slot-coupled feed. Specifically port 1
and the arbitrary inclination of the slot. Herein, we have selected a1 is cut in the ground and centered below the radiating patch. Results
= 22.86 mm, a2 = 12.63 mm, a3 = 35.91 mm and b1 = 7.39 mm, b2 of the return loss are given in Figs 6a, b and compared to the
= 12.16 mm, b3 = 20.44 mm. The domain is discretized into measurements of [12]. Evidently, our scheme is proven to be
20×35×110 cells, with δx = 1.734 mm, δy = 1.845 mm, δz = 0.931 remarkably accurate. Conversely, Fig. 5b presents a more complex
mm and δt = 1.794 ps. Also, the slot inclination angle is θ = 60ο, device: a two-element elliptical patch antenna with high isolation
while the open waveguide ends are terminated by a 6-cell PML. rates. Now, port 1 consists of two capacitaitvely-coupled feeds
Fig. 4b gives the magnitude of S-parameters (excitation is launched with 0o and 180o phase difference provided by a Wilkinson power
at port 2) compared with those obtained by [11]. As can be divider, while port 2 is connected to two elliptical slots. The design
observed, the HO non-standard FDTD algorithm is proven to be frequency for Fig. 6c is 1.8 GHz and the rest of the parameters are:
extremely accurate and the most important; it requires significantly a1 = 25 mm, b1 = 10 mm, d1 = 16 mm, a2 = 35 mm, b2 = 14 mm, d2
lower computer resources (almost 80%) than the Yee’s scheme. = 22 mm, g = 3.2 mm, w = 1.2 mm, h = 13.6 mm and l = 12 mm.

patch Top view ground plane port 2 a2 conducting strip
antenna
via hole b2 elliptical patch d2
antenna g
w

strip a1 180o conducting post
b1 l d1 l4
a
port 2 dl 100 Ω
h
H slot 0o w l2
l1 dw port 1
feed line ground
plane l3
power divider
port 1

feed substrate (εr = 4.4) 50 Ω microstrip

(a) (b)
Figure 5. Geometry of (a) a dual-polarized circular patch antenna, (b) a two-element dual-polarized elliptical patch antenna

Return loss Return loss Return loss
0 0 -10

1.841 2.036

-10 -10 -20

dB 1.632 dB 1.601 1.795 2.088
-20
1.793 dB 1.879

-20 1.936 -30
-30
1.791 2nd-order FDTD 2nd-order FDTD Slot-coupled feed (port 1)
1.5 Measured data [12] Measured data [12] Slot-coupled feed (port 2)
Proposed HO FDTD Proposed HO FDTD 1.852

1.921

1.6 1.7 1.8 1.9 2.0 -30 1.6 1.7 1.8 1.9 2.0 -40 1.8 1.9 2.0 2.1 2.2
Frequency (GHz) 1.5 Frequency (GHz) 1.7 Frequency (GHz)

(a) (b) (c)
Figure 6. Return loss for (a) port 1 of Fig. 5a, (b) port 2 of Fig. 5a and (c) the elliptical slot-coupled feeds of Fig. 5b

Conclusions FDTD implementation,” IEEE Trans. Electromagn. Compat., vol. 43,
pp. 504-514, Nov. 2001.
The efficient development of a HO non-standard FDTD method for [9] Dawson, J., T. Konefal, A. Marvin, S. Porter, M. Robinson, C.
the meticulous analysis of complicated 3-D EMC problems and Christopoulos, D. Thomas, A. Denton, T. Benson, “Intermediate level
EMI predictions in curvilinear lattices has been presented in this tools for emissions and immunity: Enclosure contents to aperture
paper. Taking avail of a covariant/contravariant vector regime coupling,” IEEE EMC Symp., Montreal, 2001.
along with advanced time integrators and powerful dispersionless [10] Xiao, F., W. Liu, and Y. Kami, “Analysis of crosstalk between finite-
PML absorbers, the novel scheme conducts notably accurate length microstrip lines: FDTD approach and circuit-concept
simulations and achieves serious computational savings. modelling,” IEEE Trans. Electromagn. Compat., vol. 43, pp. 573-578,
Nov. 2001.
References [11] Wu, K.-L., M. Yu, and A. Sivadas, “A novel modal analysis of a
circular-to-rectangular waveguide T-junction and its application to
[1] Taflove, A. (ed), Advances in Computational Electrodynamics: The design of circular waveguide dual-mode filters,” IEEE Trans.
Finite-Difference Time-Domain Method. Boston, Artech House, 1998. Microwave Theory Tech., vol. 50, pp. 465-473, Feb. 2002.
[12] Wong, K.-L. and T.-W. Chiou, “Broad-band dual-polarized patch
[2] D’Amore, M. and M. S. Sarto, “Theoretical and experimental antennas fed by capacitatively coupled feed and slot-coupled feed,”
characterization of the EMP-interaction with composite-metallic IEEE Trans. Antennas Propagat., vol. 50, pp. 346-351, Mar. 2002.
enclosures,” IEEE Trans. Electromagn. Compat., vol. 42, pp. 152- [13] Teixeira, F. L. and W. C. Chew, “PML-FDTD in cylindrical and
163, Jan. 2000. spherical coordinates,” IEEE Microwave Guided Wave Lett., vol. 7,
pp. 285-287, Sept. 1997.
[3] Park, H. and H. Eom, “Electromagnetic penetration into a rectangular [14] Roden, J. A. and S. D. Gedney, “Efficient implementation of the
cavity with multiple rectangular apertures in a conducting plane,” uniaxial-based PML media in three-dimensional nonorthogonal
IEEE Trans. Electromagn. Compat., vol. 43, pp. 573-578, Aug. 2000. coordinates with the use of the FDTD technique,” Microwave Opt.
Tech. Lett., vol. 14, pp. 71-75, 1997.
[4] Orlandi, A. and C. R. Paul, “An efficient characterization of [15] Petropoulos, P. G., L. Zhao, and A. C. Cangellaris, “A reflectionless
interconnected multiconductor-transmission-line networks,” IEEE sponge layer absorbing boundary condition for the solution of
Trans. Microwave Theory Tech., vol. 48, pp. 466-470, Mar. 2000. Maxwell’s equations with high-order staggered finite differences,” J.
Comp. Physics, vol. 139, pp. 184-208, 1998.
[5] Li, M., J. Nuebel, J. L. Drewniak, R. E. DuBroff, T. H. Hubing, and T. [16] Grivet-Talocia, S. and F. Canavero, “Wavelet-based high-order,
P. Van Doren, “EMI from airflow aperture arrays in shielding adaptive modeling of lossy interconnects,” IEEE Trans. Electromagn.
enclosures – experiments, FDTD, and MoM modeling,” IEEE Trans. Compat., vol. 43, pp. 471-484, Nov. 2001.
Electromagn. Compat., vol. 42, pp. 265-275, Aug. 2000. [17] Georgakopoulos, S. V., C. R. Birtcher, C. A. Balanis, and R. A.
Renaut, “Higher-order finite-difference schemes for electromagnetic
[6] Feliziani, M. and F. Maradei, “EMI prediction inside conductive radiation, scattering, and penetration, part I: Theory,” IEEE Antennas
enclosures with attached cables,” IEEE EMC Symp., Montreal, 2001. Propagat. Mag., vol. 44, pp. 134-142, Feb. 2002.

[7] Kwon, D.-H., R. Burkholder, and P. H. Pathak, “Efficient method of
moments formulation for large PEC scattering problems using
asymptotic phasefront extraction (APE),” IEEE Trans. Antennas
Propagat., vol. 49, pp. 583-591, Apr. 2001.

[8] Scarlatti, A. and C. L. Holloway, “An equivalent transmission-line
model containing dispersion for high-speed digital lines – with an