BRNO UNIVERSITY OF TECHNOLOGY

Faculty of Electrical Engineering and Communication

Department of Radio Electronics

Ing. Martin Štumpf, Ph.D.

EM FIELD CHARACTERIZATION OF PLANAR CIRCUITS: AN APPROACH BASED ON THE TIME-DOMAIN CONTOUR

INTEGRAL METHOD

CHARAKTERIZACE EM POLE PLANÁRNÍCH OBVODŮ: PŘÍSTUP ZALOŽENÝ NA METODĚ HRANIČNÍHO INTEGRÁLU

V ČASOVÉ OBLASTI

SHORT VERSION OF HABILITATION THESIS

BRNO 2017

i

KEYWORDS

Contour integral method; integral equations; electromagnetic compatibility;electromagnetic interference; reciprocity theorem; numerical analysis.

KLICOVA SLOVA

Metoda hranicnıho integralu; integralnı rovnice; elektromagneticka kompati-bilita; elektromagneticka interference; teorem reciprocity; numericka analyza.

THE THESIS IS AVAILABLE AT:

Brno University of Technology, Dept. of Radio Electronics, Technicka 3082/12,616 00 Brno, The Czech Republic.

ISSN 1213-418X

ii Contents

Contents

1 Introduction 1

2 Problem description 2

3 Problem formulation 4

4 Real-FD numerical solution 5

4.1 Point-matching solution . . . . . . . . . . . . . . . . . . . . . 64.2 Pulse-matching solution . . . . . . . . . . . . . . . . . . . . . 74.3 A numerical example . . . . . . . . . . . . . . . . . . . . . . . 8

5 TD numerical solution 10

5.1 Pulsed EM emissions and susceptibility . . . . . . . . . . . . . 125.2 A numerical example . . . . . . . . . . . . . . . . . . . . . . . 13

6 Conclusions 16

Contents iii

About the author

Martin Stumpf was born in Caslav, The CzechRepublic. He received his PhD-degree (2011) inElectrical Engineering from the Brno University ofTechnology (BUT), Brno, The Czech Republic. Af-ter his PhD defense he spent a year and a half asa Postdoctoral Fellow at the ESAT-TELEMIC divi-sion, Katholieke Universiteit Leuven, Leuven, Bel-gium (2011–2012) and he currently is an AssistantProfessor at the BUT, Department of Radio Elec-tronics. He also spent several months as a visiting

scientist at the Institute for Fundamental Electrical Engineering and Elec-tromagnetic Compatibility, Otto-von-Guericke University, Magdeburg, Ger-many (2008) and at the Laboratory of EM Research, Delft University ofTechnology, Delft, The Netherlands (2009–2010, 2014).

Dr. Stumpf received a Young Scientist Best Paper Award (3rd prize) anda Young Scientist Award at the URSI International Symposium on Electro-magnetic Theory in Hiroshima, Japan (2013), a Young Scientist Award atthe URSI International Symposium on Electromagnetic Theory in Berlin,Germany (2010), a Radioengineering Young Scientist Award at the Ra-dioelektronika conference (2011) and a Best Diploma Thesis Award fromthe MTT/AP/ED/EMC joint Chapter of the Czechoslovak Section of IEEE(2008).

His main research interests include electromagnetic, acoustic and elasto-dynamic wave and diffusive phenomena, with an emphasis on antenna engi-neering, wave propagation and electromagnetic compatibility.

1

1 Introduction

The analysis of high-speed multilayered printed circuit boards (PCBs) canbe in principle carried out with the aid of standard full-wave numerical tech-niques (e.g. [6]). The multi-scale nature of such complex systems along withthe ever increasing edge rates of transmitted signals, however, lead to compu-tational efforts that are still beyond the capability of the available hardwarewithin an acceptable computational time. A way out of this limitation issplitting the problem configuration to elementary components that admit the(physics-based) equivalent network representation [8, 18]. An integral part ofsuch equivalent networks is the parallel-plane impedance matrix that rep-resents the signal transmission characteristics of a (parallel-plane) cavity ina multilayered PCB. Introducing a unified, reciprocity-based computationalmethodology for electromagnetic (EM) analysis of such a general parallel-plane structure is hence exactly the main objective of (the short version of)this thesis1.

Previous works on the subject are largely focused on applications of theclassic frequency-domain (FD) contour integral method (CIM) [5, Chapter3] to analyzing transmission parameters and radiated EM emissions of gen-eral power-ground structures [17] and its further developments addressingnon-uniform field distributions along the port periphery [3], the inclusion ofsubstrate inhomogeneities [7] and the efficient speed-up techniques [19], forinstance. More recently, the contour-integral approach has also been formu-lated directly in the time domain (TD) [12, 14], thereby introducing a novelTD integral-equation numerical method. This computational technique isformulated using the EM reciprocity theorem of the time-convolution type[2, Sec. 28.2] and is known as the time-domain contour integral method (TD-CIM). It has been recently shown that the TD-CIM is an expedient tool forstudying TD EM transmission/reception reciprocity properties of parallel-plane structures [10] and provides an efficient means for evaluating theirspace-time mutual EM coupling [13] and pulsed EM plane-wave responses[9]. In particular, it has been demonstrated that these calculations are ex-tremely fast for rectangular parallel-plane structures (e.g. [9, Sec. 4.2]) forwhich the ray-type TD analytical description is available in closed form [11].

Firstly, the problem configuration that is analyzed is described in thefollowing Sec. 2. A generic, complex-FD reciprocity-based relation fromwhich the CIM can be derived both in real-FD and TD is then introducedin Sec. 3. This general formulation represents the starting point for all nu-

1This short version of the thesis is largely based on a review paper [16]. The permissionto reuse the material originally published in the Radioengineering journal is gratefullyacknowledged.

2 2 Problem description

merical schemes presented throughout this work. Subsequently, the relationbetween the general formulation and the classic FD-CIM is shown in Sec. 4.In this section, the outcomes of two FD numerical schemes are examined withrespect to a FD closed-form analytical description concerning a rectangularparallel-plane structure. Subsequently, Sec. 5 introduces a numerical solutionof the TD reciprocity relation. The computational scheme is validated usingan alternative (three-dimensional) computational method. This section isalso supplemented with the Appendix, where some general features of thetypical (Boltzmann-type) relaxation models are discussed. Furthermore, ap-plications of the TD-CIM to the evaluation of pulsed EM radiated emissionsand susceptibility of a parallel-plane structure are briefly described. Finally,the introduced methodology is validated by a numerical experiment employ-ing the fundamental property of self-reciprocity. This experiment combinesthe TD-CIM with an independent numerical tool and is conducted directlyin TD.

2 Problem description

To localize a spatial point in the problem configuration we employ the orthog-onal right-handed reference frame defined via its origin O and the standardbasis i1, i2, i3. The position vector is then x1i1 + x2i2 + x3i3, which onx3 = 0 reduces to x = x1i1 + x2i2. The time coordinate is t. The par-tial differentiation with respect to x1, x2 and t is denoted by ∂1, ∂2 and ∂t,respectively. The continuous time-convolution is denoted by ∗ and the time-integration operator is ∂−1

t . The Heaviside unit step function is H(t). TheDirac delta distribution is denoted by δ(t).

We shall analyze a parallel-plane structure shown in Fig. 1. The struc-ture consists of a dielectric slab placed in between two perfectly electri-cally conducting (PEC) planes. The thickness of the slab d is assumedto be so small with respect to the spatial support of the excitation pulse,which implies that only x3-independent EM field components E3, H1, H2 =E3, H1, H2(x, t) are dominant. The EM field components are excited t = 0.Prior to this instant these fields are zero throughout the problem configura-tion. The vanishing thickness of the slab allows to define a pulsed voltageV (x, t) = −dE3(x, t). The spatial solution domain is then bounded by a(time-invariant) domain Ω extending over the PEC plate. The correspond-ing bounding contour is denoted by ∂Ω whose outward unit normal vector isν = ν(x).

The EM properties of the analyzed structure are described by the dielec-

3

×Oi3

i2i1

D0 ǫ0, µ0

ν

Ω

∂Ω

Figure 1: Parallel-plane structure.

tric relaxation function κ(t) whose generic form is written as

κ(t) = ǫ[δ(t) + χ(t)] (1)

where χ = χ(t) is the (causal) relaxation function (see the Appendix), i.e.χ(t) = 0 for all t < 0, and ǫ is the (real-valued and positive) electric permittiv-ity. The structure is assumed to be instantaneously-reacting in its magneticbehavior, which implies the (impulsive) form of its magnetic-permeability re-laxation function µ0δ(t). The corresponding EM wave speed is c = (ǫµ0)

1/2

with (.)1/2 > 0. The structure is placed in the linear, homogeneous andisotropic embedding D0 whose EM properties are described by real-valuedand positive constants ǫ0, µ0. The EM wave speed in the embedding isc0 = (ǫ0µ0)

1/2 > 0.

The one-sided Laplace transformation of a causal and bounded space-timequantity f(x, t) is defined as

f(x, s) =

∫

∞

t=0

exp(−st)f(x, t)dt (2)

with s ∈ C; Re(s) > 0 such that f does exist. The corresponding inversionis denoted by L−1. A limiting case is arrived at via s = iω + δ as δ ↓ 0 withω ∈ R;ω > 0 being the angular frequency. This special case is of interestwithin the scope of the (real-)FD analysis. Examples of the latter will begiven in Sec. 4.

4 3 Problem formulation

3 Problem formulation

The application of the complex-FD reciprocity theorem of the time-convolutiontype [2, Sec. 28.4] to the actual field state and the testing field state (denotedby superscript ‘B’) and to domain Ω leads to the following interaction quan-tity

χΩ(xS)

∫

x∈Ω

V (x, s)J B3 (x|xS, s)dA(x)

−∫

x∈∂Ω

V (x, s)ν(x) · ∂JB(x|xS, s)dl(x)

=

∫

x∈Ω

V B(x|xS, s)J3(x, s)dA(x)

−∫

x∈∂Ω

V B(x|xS, s)ν(x) · ∂J(x, s)dl(x) (3)

where χΩ(x) is the characteristic function χΩ(x) = 1, 1/2, 0 for x ∈Ω, ∂Ω,Ω′ (Ω′ denotes the complement of Ω in R2), J3 is the electric-currentvolume density and is the electric-current surface density ∂J = H2i1− H1i2.In Eq. (3), the test wave fields are linearly related to their source via

V B(x|xS, s) = sµ0d

∫

xT∈R2

G∞[r(x|xT ), s]J B3 (xT |xS, s)dA(xT ) (4)

∂J Bκ (x|xS, s) = −

∫

xT∈R2

∂κG∞[r(x|xT ), s]J B3 (xT |xS, s)dA(xT ) (5)

for κ = 1, 2, where G∞(r, s) is the (bounded) fundamental solution ofthe two-dimensional modified Helmholtz equation in R2 and r(x|xT ) is theEucledian distance between the points specified by position vectors x andxT . Note that when specifying a testing-field quantity, x|xS stands for thefield and source position vectors, respectively. For a general form of the TDcounterpart of G∞(r, s) we refer the reader to the Appendix.

To get a boundary-contour relation, the testing surface electric currentdensity is next applied to the periphery of the circuit, i.e. we take J B

3 (x|xS, s) =∂J B

3 (x|xS, s)δ(x− xS), where δ(x− xS) being the Dirac delta distribution

5

operative along xS ∈ ∂Ω. In this way we get the following reciprocity relation

12

∫

x∈∂Ω

E3(x, s)∂JB3 (x|xS, s)dl(x)

−∫

x∈∂Ω

E3(x, s)ν(x) · ∂JB(x|xS, s)dl(x)

=

∫

x∈Ω

EB3 (x|xS, s)J3(x, s)dA(x)

−∫

x∈∂Ω

EB3 (x|xS, s)ν(x) · ∂J(x, s)dl(x) (6)

with

V B(x|xS, s) = sµ0d

∫

xT∈∂Ω

G∞[r(x|xT ), s]∂J B3 (xT |xS, s)dl(xT ) (7)

∂J Bκ (x|xS, s) = −

∫

xT∈∂Ω

∂κG∞[r(x|xT ), s]∂J B3 (xT |xS, s)dl(xT ) (8)

for κ = 1, 2. Reciprocity relation (6) with Eqs. (7) and (8) serve as thebasis for the numerical methods whose description follows.

4 Real-FD numerical solution

The classic FD-CIM is a well-established numerical technique for an efficientanalysis of parallel-plane structures of arbitrary shape [5]. The followingsubsections illustrate that the standard FD-CIM formulation is a specialcase of the starting complex-FD reciprocity relation (6).

The FD-CIM formulation [5, Eq. (3.1)] is arrived at upon combiningEqs. (6)–(8) with the (bounded) solution of the modified Helmholtz equa-tion, G∞(r, s) = K0[Γ(s)r/c]/2π, and the Dirac-delta testing source density∂J B

3 (x|xS, s) = δ(x− xS), viz

V (xS, iω) = (k/2i)

∫

x∈∂Ω

V (x, iω)H(2)1

[

kr(x|xS)]

cos[θ(x|xS)]dl(x)

− (ωµ0d/2)

∫

x∈∂S

ν(x) · ∂J(x, iω)H(2)0

[

kr(x|xS)]

dl(x) (9)

where cos(θ) = ν1∂1r + ν2∂2r and we have taken the limit s = iω + δ asδ ↓ 0 and used [1, (9.6.4)] to express the modified Bessel functions with thecomplex argument using the Hankel functions of the second kind. In Eq. (9),k = −iΓ(iω)/c is the (complex-valued) wave number (see also Eq. (42)).

6 4 Real-FD numerical solution

In order to solve the resulting Fredholm integral equation of the sec-ond kind numerically, the circuit’s rim is first approximated by a set of linesegments ∂Ω ≃ ∪N

m=1Ω[m]. Subsequently, upon employing the piecewise-constant expansion, the equation is cast into its matrix form [5, Eq. (3.2)]

U · V = H · I (10)

in which V is the voltage 1D-array of [N × 1] (unknown) coefficients, I isthe electric-current 1D-array of [N × 1] (prescribed) excitation coefficientsand U and H are [N × N ] 2D-arrays. Clearly, the corresponding [N × N ]impedance matrix directly follows from Eq. (10) as Z = U−1 · H , whereU−1 is the matrix inverse for U .

4.1 Point-matching solution

In the first step, the unknown voltage distribution along the approximatedcircuit’s periphery is expanded in terms of the rectangular functions Π, i.e.

V (x, iω) =

N∑

m=1

v[m]Π[m](x) (11)

where v[m] are (yet unknown) the voltage expansion coefficients of vectorV . Upon enforcing the equality in Eq. (9) at isolated points located at thecenters of the dividing segments denoted by x[m;c], for m = 1, · · · , N, weend up with (cf. [5, Eq. (3.26)])

(U)S,m = − (kΩ[m]/2i)H(2)1 [kr(x[m;c]|x[S;c])] cos[θ(x[m;c]|x[S;c])] (12)

(H)S,m = (ωµ0d/2)H(2)0 [kr(x[m;c]|x[S;c])] (13)

for all S 6= m and

(U)S,m = 1 (14)

(H)S,m =ωµ0d

2

1− 2i

π

[

ln

(

kΩ[m]

4

)

− 1 + γ

]

(15)

for all S = m with γ = 0.57721... being Euler’s constant. The latter ex-pression has been found using the small-argument expansion of the Hankelfunction (see [1, Eq. (9.1.8)]).

4.2 Pulse-matching solution 7

0

0.05

0.1

0

0.05

0.1

0.15

x1 (m)x2 (m)

Figure 2: The analyzed rectangular parallel-plane structure.

4.2 Pulse-matching solution

Instead of applying the point-matching procedure we can start over withthe real-FD version of the reciprocity-based relation (6) and associate thecorresponding testing-source density with the rectangular function, i.e. welet ∂J B

3 (x|xS, s) = Π[S](x). This choice in combination with the piecewise-constant expansion (11) leads to the system of algebraic equations (10) with

(U)S,m = −(kΩ[m]/2i)

∫ 1

λ=0

dλ

∫ 1

λT=0

H(2)1

kr[x(λ)|xT (λT )]

cos

θ[x(λ)|xT (λT )]

dλT (16)

(H)S,m = (ωµ0d/2)

∫ 1

λ=0

dλ

∫ 1

λT=0

H(2)0

kr[x(λ)|xT (λT )]

dλT (17)

in which

x(λ) = x[m] + λ(

x[m+1] − x[m])

∈ Ω[m] (18)

xT (λT ) = x[S] + λT(

x[S+1] − x[S])

∈ Ω[S] (19)

for all S 6= m, where x[n] localizes the position of the n-th nodal point.Similarly to the previous section, the diagonal terms are handled analytically.After a few steps of algebra we get

(U)S,m = 1 (20)

(H)S,m =ωµ0d

2

1− 2i

π

[

ln

(

kΩ[m]

2

)

− 3

2+ γ

]

(21)

8 4 Real-FD numerical solution

for all S = m. Finally it is noted that Eqs. (12)–(13) can be understood asa special case of (16)–(17) to which the 1-point Gaussian quadrature (see [1,Eq. (25.4.30)]) is applied.

4.3 A numerical example

In this section we shall analyze a rectangular parallel-plane structure of di-mensions L = 0.10 (m) and W = 0.15 (m) (see Fig. 2). The thickness ofthe structure is d = 1.50 (mm). The EM properties of the dielectric slab aredescribed by its electric permittivity ǫ = 4.50 ǫ0 and magnetic permeabilityµ = µ0. The circuit is assumed to show low losses such that the correspond-ing (complex-valued) wavenumber k can be approximated according to [5,Sec. 2.2.1]

k ≃ (ω/c) 1 + [tan(δ) + δs/d] /2i (22)

where the dielectric loss is accounted for via tan(δ) = 0.0045, the skindepth of the conductor is found from δs =

√

2/ωµ0σ with conductivityσ = 5.80 · 107 (S/m). The structure is activated using the excitation verticalport that has its center at xS = (75.0, 112.5) (mm) and radius 1.50 (mm). Allthe calculations that follow are performed in the frequency range 50 ≤ f =ω/2π ≤ 2000 (MHz) at 200 uniformly-spaced frequency points. The circuit’sboundary is discretized such that maxn(|Ω[n]|) < 0.12 c/max(f). The in-tegrations in Eqs. (16) and (17) are carried out using the Gauss-Legendrequadrature, symbolically written as

∫ 1

λ=0

f(λ)dλ ≃K∑

k=1

wkf(λk) (23)

where the corresponding abscissas λk and weights wk for 0 ≤ λ ≤ 1,K = 1, . . . , 8 of the quadrature can be found in [1, p. 921], for example.Recall that for K = 1 for which w1 = 1, λ1 = 1/2, Eqs. (16)–(17) becomefully equivalent to Eqs. (12)–(13).

For validation purposes, the input impedance is also evaluated using theeigenfunction-expansion approach via the double-summation formula (see [5,Sec. 2.4.1]), namely

Z(iω) =iωµ0d

LWlim

M,N→∞

M∑

m=0

N∑

n=0

e2me2n

k2m + k2n − k2F SmnF

Pmn (24)

with em = 1 for m = 0 and em =√2 for m 6= 0 and km = mπ/L, kn = nπ/W

with F Smn = F P

mn, where

F Smn = cos(kmx

S1 ) cos(knx

S2 )sinc(kmW

S1 /2)sinc(knW

S2 /2) (25)

4.3 A numerical example 9

0 500 1000 1500 2000

10−1

100

101

102

f (MHz)

|Z(i

ω)|

(Ω)

EIG-E

FD-CIM: point-match.

(a)

0 500 1000 1500 2000

10−1

100

101

102

f (MHz)

|Z(i

ω)|

(Ω)

EIG-E

FD-CIM: K = 6

(b)

0 500 1000 1500 200010

−3

10−2

10−1

100

101

102

f (MHz)

|δZ

(iω)|

(Ω)

point-match.

K = 6

K = 12

(c)

Figure 3: Input impedance of the rectangular circuit evaluated using FD-CIM and the eigenfunction expansion method (EIG-E). (a) Point-matchingsolution with EIG-E. (b) Pulse-matching solution with EIG-E. (c) Absoluteerror of the FD-CIM solutions with respect to the referential EIG-E formula.

10 5 TD numerical solution

where we take W S1 = W S

2 = 1.0 (mm). In the actual calculations, the sum-mations in Eq. (25) are truncated to N =M = 103.

The obtained results are summarized in Fig. 3. In Figs. 3a and 3b wehave shown the point-matching and pulse-matching solutions, respectively,together with the FD response calculated according to the analytical solu-tion (24). As can be observed, the both CIM-based solutions agree withthe reference well. In order to clearly assess the accuracy of the numericalsolutions, the absolute error of the calculated input impedance with respectto the referential solution (24) has been plotted in Fig. 3c. Apparently, thecalculated error curves attain their peak values at the cavity resonance fre-quencies. Comparing the point-matching and pulse-matching approaches,the latter solution leads, except for the very high-frequency part of the fre-quency range, to more accurate results. The difference is most evident atlow frequencies. Moreover, it has been observed that doubling the number ofthe integration from K = 6 to K = 12 does not imply a significant precisionincrease. Finally, as the pulse-matching solution requires computation of in-tegrals, one should carefully consider whether the improvement with respectto the point-matching solution is worth the effort for practical purposes.

5 TD numerical solution

The first step in solving the TD counterpart of Eq. (6) numerically consists ofthe discretization of the bounding contour ∂Ω ≃ ∪N

m=1Ω[m] and of the timeaxis T = tk ∈ R; tk = kt,t > 0, k = 1, ..., NT. The piecewise-linearspace-time expansion of the voltage distribution in terms of the triangularfunctions T , i.e.

V (x, t) =

N∑

m=1

NT∑

k=1

v[m][k] T

[m](x)T[k](t) (26)

where v[m][k] are (yet unknown) the voltage expansion coefficients, along with

the piecewise linear spatial expansion of the testing current density, i.e.

J B3 (x|xS, t) = T [S](x)δ(t) (27)

allow to cast (the TD counterpart of) Eq. (6) into the following system ofalgebraic equations (cf. [12, Eq. (14)])

(

I −Q[0]

) · V [p] =

p−1∑

k=1

Q[p−k] · V [k] + F [p] (28)

11

that can be solved in a step-by-step manner for all p = 1, · · · , NT. Here,V [p] is a 2D-array of [N ×NT ] unknown voltage coefficients at tp = pt, Iis a three-diagonal [N ×N ] 2D-array with elements

(I)S,m =(

Ω[S−1]/6)

δS−1,m +(

Ω[S−1]/3

+Ω[S]/3)

δS,m +(

Ω[S]/6)

δS+1,m (29)

and Q[p−k] is a time-dependent [N ×N ×NT ] 3D-array whose elements aregiven as [12, Eq. (16)]

(Q[p−k])S,m =1

πct

∫

xT∈∂Ω

T [S](xT )

∫

x∈∂Ω

T [m](x)

Ψ[r(x|xT ), (p− k)t] cos[θ(x|xT )]dl(x)dl(xT ) (30)

for all S = 1, · · · , N, m = 1, · · · , N and t ∈ T . The excitation is ac-counted for via an [N×NT ] 2D-array F [p] whose elements read [12, Eq. (17)]

(F [p])S =1

π|∂S|

∫

xT∈∂Ω

T [S](xT )

∫

x∈∂S

Φ[r(x|xT ), pt]dl(x)dl(xT ) (31)

for all S = 1, · · · , N and all t ∈ T , where |∂S| denotes the arc length of∂S ⊂ ∂Ω. The time-dependent functions in Eqs. (30) and (31) are

Ψ(r, t) = ψ(r, t+t)− 2ψ(r, t) + ψ(r, t−t) (32)

ψ(r, t) = L−1

Γ(s)K1[Γ(s)r/c]

/s2 (33)

Φ(r, t) = µ0d∂tI(t) ∗ L−1

K0[Γ(s)r/c]

(34)

where I(t) is the source signature of the electric current injected into the cir-cuit periphery and I(t) = 0 for t < 0. For a special case of an instantaneously-reacting dielectric slab (see Eq. (42)) we have Γ(s) = s, which simply yields

ψ(r, t) = (c2t2/r2 − 1)1/2H(t− r/c) (35)

Φ(r, t) = µ0d∂tI(t) ∗ (t2 − r2/c2)−1/2H(t− r/c) (36)

Combining Eqs. (28) with (29)–(34), the space-time voltage distributionalong the discretized circuit periphery and time axis is readily evaluated.In contrast to the real-FD solution that requires the matrix inverse at alarge number of frequency points in combination with the inverse FFT, the

12 5 TD numerical solution

0

0.05

0.1

0

0.05

0.1

0.15

x1 (m)x2 (m)

PROBE 1

PROBE 2

PORT

Figure 4: The analyzed irregularly-shaped parallel-plane structure.

marching on-in-time scheme (28) furnishes the stable TD voltage response ina given time window T via a single matrix inversion of the (time-independent)matrix I −Q[0].

For more details on the typical relaxation models and their behavior werefer the reader to the Appendix. Finally note that a dedicated numericaltechnique for handling L−1 has been proposed and successfully validated in[14].

5.1 Pulsed EM emissions and susceptibility

With the space-time voltage distribution at our disposal, the pulsed EM emis-sions radiated from a thin parallel-plane structure follow from [10, Eq. (9)]

∂−1t ET;∞(ξ, t) = −c−1

0 ξ ×∫

x∈∂Ω

V (x, t+ ξ · x/c0)τ (x)dl(x) (37)

where ET;∞ is the amplitude radiation characteristic of the electric-type [2,Sec. 26.12], ξ is the unit vector of observation and τ = i3×ν is the unit vectortangential to ∂Ω. Remarkably, the fundamental property of self-reciprocityallows to go yet another step further and carry out the radiated susceptibilityanalysis of the structure. Namely, if the emitting circuit is activated by theelectric-current pulse I(t) applied at xS we may write [10, Eq. (22)]

V (xS, t) = α · ET;∞(−β, t) (38)

provided that

ei(t) = µ0∂tI(t) (39)

5.2 A numerical example 13

where V (xS, t) is the pulsed voltage response of the structure to a pulsedincident plane wave defined by its pulse shape ei(t) and by its direction ofpropagation and polarization specified by unit vectors β and α, respectively.

In a related work [9], the combination of Eqs. (37)–(39) with the TDray-type field expansion (see [11]) has proved to be computationally veryefficient. Moreover, it has been recently shown that the voltage responsegiven by Eq. (38) can be interpreted as the Thevenin voltage in the equivalentKirchhoff-network representation [15].

5.2 A numerical example

In this section we shall analyze the irregularly-shaped parallel-plane structureshown in Fig. 4). The thickness of the structure is d = 1.50 (mm). TheEM properties of the dielectric slab are described by its dielectric relaxationfunction (43) with ǫ = 4.50ǫ0 and σ = 0.02 (S/m). The structure is atxS = (25, 37.5) (mm) excited by the electric-current pulse described as

I(t) = 2Im

[

(t/tw)2H(t)− 2(t/tw − 1/2)2H(t− tw/2)

+ 2(t/tw − 3/2)2H(t− 3tw/2)

− (t/tw − 2)2H(t− 2tw)]

(40)

where we take Im = 1.0 (A) and ctw = 0.10 (m) (see Fig. 5a). The radius ofthe excitation port is 1.0 (mm). The resulting pulsed voltage response hasbeen evaluated at two field points placed at x = (50, 0) (mm) (PROBE 1)

and x = (100, 112.5) (mm) (PROBE 2) within the finite time window ofobservation T = 0 ≤ ct ≤ 2.0 (m) at 401 uniformly spaced time points.The circuits boundary is discretized such that maxn(|Ω[n]|) < 0.063 ctw.

In order to validate the computational scheme described in Sec. 5, the TDvoltage responses have also been calculated using the finite integration tech-nique (FIT) as implemented in CST Microwave Studior. The resulting pulseshapes are shown in Fig. 6. As can be seen, the pulse shapes agree well. Theobservable discrepancies can be largely attributed to different boundary con-ditions along ∂Ω. While the TD-CIM model prescribes the perfect magneticwall along the circuit’s rim, the referential (three-dimensional) FIT modelhas the ‘open’ boundary condition on the faces of its square bounding box.Thanks to the spatial dimensionality reduction from 3 to 1 that is implicit inany CIM formulation, the reduction of computational costs with respect tostandard direct-discretization computational techniques is typically (at least)3 orders of magnitude.

14 5 TD numerical solution

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

ct (m)

I(t

)(A

)

(a)

0 0.5 1 1.5 2−4

−2

0

2

4

ct (m)

ei(t

)=

µ0∂

tI(t

)(k

V/m

)

(b)

Figure 5: Excitation pulse shapes. (a) Electric-current pulse (transmittingstate); (b) Plane-wave pulse (receiving state).

5.2 A numerical example 15

0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

ct (m)

V(x

,t)

(V)

TD-CIM

CST-FIT

(a)

0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

ct (m)

V(x

,t)

(V)

TD-CIM

CST-FIT

(b)

Figure 6: The pulsed-voltage response at (a) PROBE 1 and (b) PROBE 2 asevaluated using TD-CIM and FIT.

16 6 Conclusions

0 0.5 1 1.5

−3

−2

−1

0

1

2

ct (m)

voltage

resp

onse

(V)

V (xS, t)

α · ET;∞(−β, t)

Figure 7: Illustration of the property of TD self-reciprocity.

In the final example, the evaluated space-time voltage distribution along∂Ω×T is used in Eq. 37 to calculate the TD radiated amplitude at a chosendirection specified by ξ = (0,

√2/2,

√2/2). For validation purposes, the FIT

model has been now irradiated by a uniform EM plane wave whose pulseshape ei(t) is related to the excitation electric current according to Eq. (39)(see Fig. 5) and whose direction of propagation and polarization are describedby unit vectors β = −ξ = (0,−

√2/2,−

√2/2) and α = (0,−

√2/2,

√2/2),

respectively. Figure 7 shows the FIT-based plane-wave voltage responseV (xS, t) together with the TD-CIM-based, co-polarized radiation amplitudein the backward direction α · ET;∞(−β, t). A good agreement between thecalculated signals thus validates the relations discussed in Sec. 5.1 as well asthe numerical solution given in Sec. 5 once more.

6 Conclusions

A unified reciprocity-based contour-integral approach for analyzing trans-fer and EM emission/reception characteristics of a dispersive parallel-planestructure has been presented. It has been shown that the complex-FDreciprocity-based relation of the time-convolution type is a convenient start-ing point for proposing new CIM computational schemes in both real-FD andTD. Examples of such numerical schemes have been given and validated us-ing a real-FD closed-form solution and TD numerical full-wave simulations.Subsequently, a few applications of the TD-CIM have been discussed. Inparticular, it has been shown that the TD radiated emissions can be readily

17

calculated either from the voltage space-time distribution along the circuit’speriphery or from the pulsed plane-wave response in the relevant receivingsituation. An evidence of the property of self-reciprocity has been furnishedvia a numerical experiment employing both the described TD-CIM and the(three-dimensional) FIT.

Appendix

The Green’s function satisfies the scalar 2D wave equation

(∂21 + ∂22)G∞ − c−2∂2t (G∞ + χ ∗G∞) = −δ(x)δ(t) (41)

with the zero-value initial conditions G∞(x, 0) = 0 and ∂tG∞(x, 0) = 0 forall x ∈ R2. Here, c is the wave speed (a positive and real-valued constant)and χ = χ(t) is the relaxation function. Typically, the relaxation functionhas the following forms, e.g.

χ(t) = 0 (42)

for an instantaneously-reacting dielectric slab or

χ(t) = (σ/ǫ)H(t) (43)

for a dielectric slab with conductive losses described by conductivity σ or

χ(t) = (ǫr/ǫ∞ − 1)τ−1r exp(−t/τr)H(t) (44)

for the Debye-type relaxation with the characteristic relative permittivitiessatisfying 0 < ǫ∞ < ǫr and the relaxation time τr > 0.

The bounded complex-FD solution of Eq. (41) can be expressed as

G∞(r, s) = (1/2π)

∫

∞

τ=r/c

exp[

−Γ(s)τ] dτ

(τ 2 − r2/c2)1/2(45)

where Γ(s) = s(1+χ)1/2 with Re[(.)1/2] > 0 and r = (x21+x22)

1/2 > 0. Since, ingeneral, the exponential kernel does not have a closed form Laplace-transforminversion, we start with the relevant Bromwich inversion integral, viz

E(t, τ) = (1/2πi)

∫

s∈B

exp(st) exp[

−Γ(s)τ]

ds (46)

where B is the Bromwich contour. For standard relaxation models Γ(s) showsalgebraic branch points due to square roots for which we choose Re[(.)1/2] > 0

18 6 Conclusions

for all s ∈ C. This implies (possibly overlapping) branch cuts along thenegative real axis in the complex s-plane. Moreover, the following asymptoticexpansion applies

Γ(s) = Γ∞(s) +O(s−1) (47)

as |s| → ∞, where Γ∞(s) = s + Λ, Λ ∈ R; Λ ≥ 0. In virtue of Jordan’slemma we therefore rewrite (46) as

E(t, τ) = (1/2πi)

∫

s∈B

exp(st)

exp[

−Γ(s)τ]

− exp[

−Γ∞(s)τ]

ds

+ (1/2πi)

∫

s∈B

exp(st) exp[

−Γ∞(s)τ]

ds (48)

Obviously, the second term in Eq. (48) represents the (attenuated) Diracdelta distribution

(1/2πi)

∫

s∈B

exp(st) exp[

−Γ∞(s)τ]

ds

= δ(t− τ) exp(−Λτ) (49)

while the first term can be handled numerically using the method introducedin [11]. In this process, the Bromwich contour is for t > τ closed to the rightby supplementing it with a semi-circle of radius ∆ → ∞ and the resultingcontour is in view of Cauchy’s theorem contracted into the hyperbolic con-tour Γ ∪ Γ∗ provided that all singularities are enclosed. Once the numericalintegration is carried out, we arrive at

E(t, τ) = F (t, τ)H(t− τ) + δ(t− τ) exp(−Λτ) (50)

Upon substituting Eq. (50) into the TD counterpart of (45) we end up with

G∞(r, t) = (1/2π)(t2 − r2/c2)−1/2H(t− r/c) exp(−Λt)

+ (1/2π)

∫ t

τ=r/c

(τ 2 − r2/c2)−1/2F (t, τ)dτ (51)

where we have used the sifting property of the Dirac distribution. Thefirst part in Eq. (51) is the (attenuated) fundamental solution of the two-dimensional wave equation, while the second one represents the effect of(Boltzmann-type) relaxation. The latter is negligible close to the wavefrontas t ↓ r/c, which is a general feature of dispersive phenomena [4].

References 19

References

[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions.New York, NY: Dover Publications, 1972.

[2] A. T. de Hoop. Handbook of Radiation and Scattering of Waves. London,UK: Academic Press, 1995.

[3] X. Duan, R. Rimolo-Donadio, H.-D. Bruns, and C. Schuster. Exten-sion of the contour integral method to anisotropic modes on circularports. IEEE Transactions on Components, Packaging and Manufactur-

ing Technology, 2(2):321–331, February 2012.

[4] L. B. Felsen. Propagation and diffraction of transient fields in non-dispersive and dispersive media. In Felsen L. B., editor, Transient Elec-tromagnetic Fields, chapter 1, pages 1–72. Berlin, Germany: Springer -Verlag, 1976.

[5] T. Okoshi. Planar Circuits for Microwaves and Lightwaves. SpringerSeries in Electrophysics. Berlin: Springer-Verlag, 1985.

[6] M. Piket-May, A. Taflove, and J. Baron. FD-TD modeling of digitalsignal propagation in 3-D circuits with passive and active loads. IEEE

Transactions on Microwave Theory and Techniques, 42(8):1514–1523,August 1994.

[7] J. B. Preibisch, X. Duan, and C. Schuster. An efficient analysis ofpower/ground planes with inhomogeneous substrates using the contourintegral method. IEEE Transactions on Electromagnetic Compatibility,56(4):980–989, August 2014.

[8] R. Rimolo-Donadio, X. Gu, Y. H. Kwark, M. B. Ritter, B. Archam-beault, F. de Paulis, Y. Zhang, J. Fan, H.-D. Bruns, and C. Schuster.Physics-based via and trace models for efficient link simulation on multi-layer structures up to 40 GHz. IEEE Transactions on Microwave Theory

and Techniques, 57(8):2072–2083, August 2009.

[9] M. Stumpf. The pulsed EM plane-wave response of a thin planar an-tenna. Journal of Electromagnetic Waves and Applications, 30(9):1133–1146, 2016.

[10] M. Stumpf. Pulsed EM field radiation, mutual coupling, and reciprocityof thin planar antennas. IEEE Transactions on Antennas and Propaga-

tion, 62(8):3943–3950, August 2014.

20 References

[11] M. Stumpf. Time-domain analysis of rectangular power-ground struc-tures with relaxation. IEEE Transactions on Electromagnetic Compat-

ibility, 56(5):1095–1102, October 2014.

[12] M. Stumpf. The time-domain contour integral method – an approachto the analysis of double-plane circuits. IEEE Transactions on Electro-

magnetic Compatibility, 56(2):367–374, April 2014.

[13] M. Stumpf. Time-domain mutual coupling between power-ground struc-tures. In Proc. IEEE Int. Symp. EMC, pages 240–243, Raleigh, NorthCarolina, August 2014.

[14] M. Stumpf. Analysis of dispersive power-ground structures using thetime-domain contour integral method. IEEE Transactions on Electro-

magnetic Compatibility, 57(2):224–231, April 2015.

[15] M. Stumpf. The equivalent Thevenin-network representation of a pulse-excited power-ground structure. IEEE Transactions on Electromagnetic

Compatibility, 59(1):249–255, February 2017.

[16] M. Stumpf. Modeling of electromagnetic fields in parallel-plane struc-tures: a unified contour-integral approach. Radioengineering, April 2017.[invited].

[17] M. Stumpf and M. Leone. Efficient 2-D integral equation approachfor the analysis of power bus structures with arbitrary shape. IEEE

Transactions on Electromagnetic Compatibility, 51(1):38–45, February2009.

[18] X. C. Wei, E.-P. Li, E. X. Lu, and R. Vahldieck. Efficient simulationof power distribution network by using integral-equation and modal-decoupling technology. IEEE Transactions on Microwave Theory and

Techniques, 56(10):2277–2285, October 2008.

[19] H. Zhao, E. X. Liu, J. Hu, and E.-P. Li. Fast contour integral equa-tion method for wideband power integrity analysis. IEEE Transactions

on Components, Packaging and Manufacturing Technology, 4(8):1317–1324, August 2014.

References 21

Acknowledgments

In the first place I wish to express my thanks to Professor Adrianus T. de Hoopfor many stimulating discussions and critical remarks, which, as I stronglybelieve, improved the quality of the thesis.

Further I desire to thank Professor Guy A. E. Vandenbosch for his con-tinuous encouragement and valuable advices that help me to carry on withmy research.

Finally I wish to thank Professor Zbynek Raida for giving me the opportu-nity to develop my master thesis at the Institute for Fundamental ElectricalEngineering and ElectroMagnetic Compatibility (IGET), Otto-von-GuerickeUniversity Magdeburg, where I got acquainted with EMC applications of thecontour integral method. The latter would not be possible without the pro-fessional scientific supervision of Professor Marco Leone who has drawn myattention to this numerical method and to the field of EMC as a whole.

Much of the research work included in this thesis has received fundingfrom the Czech Ministry of Education, Youth and Sports under projectLD15005 of the COST CZ (LD) program and under grant LO1401 of theNational Sustainability Program. The author takes this opportunity to grate-fully acknowledge this financial support.

22 References

Abstract

In this short version of the habilitation thesis, a unified reciprocity-basedmodeling approach for analyzing electromagnetic fields in dispersive pla-nar structures of arbitrary shape is introduced. It is shown that the ap-plication of the reciprocity theorem of the time-convolution type leads toa global contour-integral interaction quantity from which novel both time-and frequency-domain numerical schemes can be arrived at. Applicationsof the numerical method concerning the time-domain radiated interferenceand susceptibility of parallel-plane structures are discussed and illustratedon numerical examples.

Abstrakt

V teto zkracene verzi habilitacnı prace je predstaven sjednoceny prıstupmodelovanı elektromagnetickeho pole v planarnıch strukturach libovolnehotvaru. Je ukazano, ze aplikace casove-konvolucnıho teoremu reciprocity vedeke globalnı interakcnı velicine definovane krivkovym integralem, ze ktere lzevyvodit nova numericka schemata jak v casove, tak i ve frekvencnı oblasti.Uzitı numericke metody na resenı problemu elektromagneticke interferencea susceptibility planarnıch struktur je diskutovano a nazorne objasneno nanumerickych prıkladech.