Time-Domain SPICE Macromodel of High-Speed Modules from Y -parameter Data using Loewner Matrix Approach

Abstract

In macromodeling of high speed modules, recently introduced Loewner Matrix (LM) based algorithms were shown to be very accurate and efficient for generating time-domain macromodel of a structure with a large number of ports and poles, based on frequency-domain Y-parameter data. In this paper, we present a method to generate a SPICE netlist from the time-domain macromodel generated by LM method. This method generates minimum elements in the netlist, and it is shown to be efficient.

Full text

Time-Domain SPICE Macromodel of High-Speed Modules from
Y-parameter Data using Loewner Matrix Approach
Zhaoqing Liu (1), Muhammad Kabir (2) , and Roni Khazaka (3)
(1) (2) (3) Department of Electrical and Computer Engineering, McGill University, Montr´eal, Qu´ebec, Canada
Email: (1) zhaoqing.liu@mail.mcgill.ca, (2) muhammad.kabir@mail.mcgill.ca and (3) roni.khazaka@mcgill.ca
Abstract—In macromodeling of high speed modules, recently
introduced Loewner Matrix (LM) based algorithms were shown
to be very accurate and efficient for generating time-domain
macromodel of a structure with a large number of ports and
poles, based on frequency-domain Y-parameter data. In this
paper, we present a method to generate a SPICE netlist from the
time-domain macromodel generated by LM method. This method
generates minimum elements in the netlist, and it is shown to be
efficient.
Index Terms—Y-parameter, Macromodel, Loewner Matrix
method.
I. INTRODUCTION
In many microwave and high frequency applications, we
often encounter structures for which a physics based model
is very difficult to derive analytically. However, an accurate
frequency-domain representation (S/Y-parameter) is avail-
able through direct measurement or full-wave simulation.
SPICE-compatible lumped time-domain macromodel for those
structures can be obtained from the measured/simulated Y-
parameter data by applying an automatic algorithm. Physics
based model can then be obtained through SPICE. Vector
Fitting (VF) [1–5] is an effective algorithm to generate the
time-domain macromodel based on measured data. However
this method can have difficulties modeling systems with a
large number of poles and a large number of ports. More
recently, a new approach based on the Loewner Matrix (LM)
pencil has been proposed [6–9]. This method was shown to be
very efficient and accurate compared to Vector Fitting [7,9],
particularly for systems with a large number of ports. In this
paper, we propose a new method for generating SPICE netlist
based on the time-domain macromodel generated by LM
method automatically. This method uses some characteristics
of LM, and it allows efficiently obtaining a minimum number
of elements for the physics based SPICE model.
II. PROBLEM FORMULATION
Ap-port 3D high-speed module can be fully represented in
the frequency-domain by its Y-parameters. The Y-parameter
data can be obtained either by direct measurements or by
performing a full-wave simulation on the structure. The Y-
parameter data is expressed as:
{sk,Y(sk)}(1)
The authors would like to thank Natural Sciences and Engineering Re-
search Council of Canada (NSERC) and the Regroupement Strat´egique en
Microsyst´eme du Qu´ebec (ReSMiQ) for supporting this project.
where, skis the complex frequency, Y(sk)is the Y-parameters
at frequency skand k= 1,2,...,n where nis the number of
data points.
A. Time-Domain Macromodel
Our goal in this paper is to obtain a SPICE-compatible time-
domain macromodel that matches the frequency-domain data
in (1) and generate a SPICE netlist that can be used to perform
the transient analysis of the structure. The macromodel can be
expressed as:
E˙
x(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t) + Y∞˙
u(t)(2)
where, u(t)∈Rpand y(t)∈Rpcontain the vectors of
port voltages and currents respectively. The matrices E,A∈
Rm×m,B∈Rm×p,C∈Rp×mand D,Y∞∈Rp×pdefine
the LTI descriptor system and mis the order of the system.
III. LOEWNER MATRIX (LM) METHOD
In this section, we will present a brief review of the Loewner
Matrix (LM) method [6–9] which is used to obtain the time-
domain macromodel shown in (2) from the available Y-
parameter data shown in (1).
A. Splitting the Data
First the frequency-domain data is appended with the com-
plex conjugates at the negative frequencies, resulting in 2n
data points, and then split into two sets as follows.
{sk,Y(sk)} → {γi,Y(γi)},{µj,Y(µj)}(3)
There are a number of strategies for splitting the data and
these are outlined in [6–9]. The technique, used in this work,
is Matrix Format Tangential Interpolation (MFTI) [8,9].
B. Loewner Matrices
The next step is to construct the matrices L,σL,Fand W
block by block as follows [6–9]:
[Lj,i] = Y(µj)−Y(γi)
µj−γi
,[σLj,i] = µjY(µj)−γiY(γi)
µj−γi
.(4)
where j= 1,2,...,n,i= 1,2,...,n and [Lj,i],[σLj,i ]
represent the (j, i)th block entry of Land σLrespectively.
Similarly the matrices Fand Ware constructed as follows:
F=Y(µ1)T...Y(µn)TT,W= [Y(γ1)...Y(γn)] .(5)

2
Note that [Lj,i]and [σLj,i ]are block matrices of size p×p
and the size of Land σLis np ×np.
The Loewner Matrices as constructed in (4) and (5) are
complex. In order to obtain a real macromodel, the equivalent
real Loewner Matrices, Lr, σLr,Frand Wr, can be computed
using a similarity transformation [7,9].
C. Extract the Macromodel
The Loewner matrices, constructed in (4), (5) are directly
related to the system matrices shown in (2) [6,7]. In fact,
these matrices can be obtained by extracting the regular part
of the Loewner Matrices. In order to extract the regular part
a singular value decomposition (SVD) is performed on the
Loewner Matrix pencil (xLr−σLr)[6,7,9]:
xLr−σLr=ΛΣΨ∗(6)
where, x∈ {λi} ∪ {µj},Σis a diagonal matrix containing
the singular values, Λand Ψare the orthonormal matrices.
1) Order of the Regular Part: The first step of extracting the
regular part is to determine the order, mof the macromodel.
mis determined by examining the normalized singular values
of the Loewner Matrix pencil (xLr−σLr)[7,9].
2) Compute Orthonormal Bases: The next step is to com-
pute the orthonormal bases. This is done by preserving the first
mcolumns of the orthonormal matrices Λand Ψ, obtained
in (6):
ΛR= [λ1λ2... λi... λm],
ΨR= [ψ1ψ2... ψi... ψm].(7)
where, λiand ψirepresent the ith column of Λand Ψ
respectively.
3) Regular Part Extraction: The regular part of the LMs is
extracted by performing an oblique projection on the Loewner
Matrices using the orthonormal bases ΛRand ΨR, computed
in (7), as projectors:
ER=−Λ∗
RLrΨR,AR=−Λ∗
RσLrΨR,BR=Λ∗
RFr,
CR=WrΨ1,DR=0,Y∞
R=0.(8)
Note that the matrices DRand Y∞
Rare zero at this stage and
their contribution is embedded inside the other matrices.
4) Extraction of Dand Y∞:The regular part extracted in
(8) will give us the final macromodel if the Y-parameter data
in (1) contains no Dand Y∞.
E=ER,A=AR,B=BR,C=CR.(9)
However if the data contains nonzero Dand Y∞some unstable
poles are observed in the regular part extracted in (8) [7,9]. In
that case we have to extract Dand Y∞from the regular part
in order to make sure of the stability and the passivity of the
model [9]. The first step of the extraction is to decouple the
model in (8) into two systems such that
CR(sER−AR)−1BR=(10)
C(sE−A)−1B+CU(sEU−AU)−1BU
where, the system {A,E,B,C}is the desired system and the
system {AU,EU,BU,CU}contains the undesired poles that
are the artifact of embedding of Dand Y∞. The second step
is to compute Dand Y∞such that
D+sY∞≃CU(sEU−AU)−1BU(11)
this leads us to the final macromodel
E˙
x(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t) + Y∞˙
u(t)(12)
The matrices Dand Y∞are usually positive definite.
IV. PROPOSED APPROACH FOR GENERATING SPICE
NETLIST FROM THE MACROMODEL
The resulting matrices E,A,Band Care dense by construc-
tion. Usually dense matrices result in many internal nodes in
the SPICE netlist which makes the transient analysis ineffi-
cient. In this paper, we proposed an algorithm that generates
the equivalent sparse matrices from the dense macromodel
which are suitable for generating a SPICE netlist in order to
perform the transient analysis on the structure.
A. Generating the Sparse Macromodel
We can rewrite the time-domain macromodel shown in (12)
in frequency-domain as follows:
sEx=Ax+Bu,
y=Cx+Du+sY∞u.(13)
The matrix Eis usually singular or very close to singular
whereas the matrix Ais full-rank in the macromodel shown
in (13).
1) Sparsification of Matrix A:Since the matrix Ais full-
rank, the first Eq. in (13) is premultiplied by A−1in order to
obtain a sparse Amatrix:
sˆ
Ex=ˆ
Ax+ˆ
Bu,
y=Cx+Du+sY∞u.(14)
where, ˆ
E=A−1E,ˆ
B=A−1Band ˆ
A=I.
2) Sparsification of Matrix ˆ
E:In order to sparsify the
matrix ˆ
E, first an eigendecomposition is performed as follows:
ˆ
E=VΓV−1(15)
where, Γis the diagonal matrix containing all the eigenvalues
of ˆ
Eand Vis the corresponding eigenvector matrix. Then an
orthogonal projection is performed on the matrices in (14):
˜
E=V−1ˆ
EV =Γ,˜
A=V−1ˆ
AV =I,
˜
B=V−1ˆ
B,˜
C=CV.(16)
Since the orthogonal projection is performed by changing the
variable x, the matrices Dand Y∞will be unchanged. The
eigenvalues of ˆ
Ewill come in complex conjugate pairs. For a
real eigenvalue, ρappearing on (i, i)th entry:
Γi,i =ρ;ρ∈R(17)
On the other hand for a complex eigenvalue, α+jβ appearing
on (i, i)th entry:
Γi,i =α+jβ, Γi+1,i+1 =α−jβ;α, β ∈R(18)

3
The matrices ˜
E,˜
Band ˜
Care now complex. The equivalent
real matrices are obtained by performing another orthogonal
projection using transformation matrix P[10] as projector:
Es=P−1˜
EP =¯
Γ,As=P−1˜
AP =I,
Bs=P−1˜
B,Cs=˜
CP.(19)
where, P∈Cm×mis a block-diagonal matrix. Pi,i = 1 for
real eigenvalues shown in (17). For the complex eigenvalues
shown in (18), 

i i + 1
i1 1
i+ 1 +j−j


And ¯
Γis a block diagonal matrix containing the real and
imaginary part of the eigenvalues of ˆ
E. For real eigenvalues,
¯
Γi,i =ρ. For complex eigenvalues, the diagonal complex
conjugate entries transform into a block-diagonal real matrix:


i i + 1
i α β
i+ 1 −β α

(20)
The sparse form of the macromodel in (14) can be rewritten
in frequency-domain as follows:
sEs¯
x=As¯
x+Bsu,
y=Cs¯
x+Du+sY∞u.(21)
where, Es=¯
Γ,As=I,Bs=P−1V−1A−1Band Cs=
CVP. Comparing the original macromodel in (2) with the one
in (21), we can say that an excellent simplification has been
achieved. The matrix Esis sparse and real, Asis an identity
matrix. The matrices Bsand Csare still dense. However they
are the multiplier of uand ¯
xrespectively; so the sparsity of
these matrices are less concerned.
B. Generating the Netlist
The Modified Nodal Analysis (MNA) equation of a multi-
port network can be written as follows:
GX +sCX =b(22)
where, the matrix Gcontains all conductors and frequency-
independent stamps, the matrix Ccontains all the capacitor
and inductor values and other values that are associated with
dynamic elements, Xcontains all the unknown node volt-
ages and brunch currents and bcontains all the independent
sources.
In order make sure all the entries of Esgenerate a capacitor
stamp, the first Eq. in (21) is premultiplied by matrix Q:
sQEs¯
x=QAs¯
x+QBsu,
y=Cs¯
x+Du+sY∞u.(23)
where, Q∈Rm×mis a diagonal matrix with either +1 or −1
in the diagonal. For each block-diagonal entry shown in (20),
the entries of Q:
Qi,i =−1,Qi+1,i+1 = 1.
and Qi,i = 1 elsewhere. The two equations in the sparse
macromodel in (23) can then be combined to adopt in the
MNA equation in (22).
ˆ
Gˆ
X+sˆ
Cˆ
X=ˆ
b(24)
j
j′
+
−
vo
k
k′
gmvo
VCCS
Fig. 1. Voltage Controlled Current Source
where,
ˆ
G=D Cs
QBsQAs,ˆ
C=Y∞0
0−QEs
ˆ
X=u
¯
x,ˆ
b=y
0
Finally, SPICE netlist is generated by adding the MNA
stamps with suitable elements by evaluating the entries of ˆ
G
and ˆ
Crespectively [11].
1) Netlist Generation for ˆ
G:As As=I,QAswill be a
diagonal matrix containing +1 and −1. The generated netlist
stamps for QAswill all be resistors from corresponding node
to ground [11].
Matrices Bsand Csare full-column and full-row matrices
and each entry of these matrices are are different. Voltage
Controlled Current Source (VCCS) stamp [11] is used to
generate the netlist stamps for each entry of these matrices.
For the VCCS shown in Fig. 1 the stamps are as follows:


j j′
k gm−gm
k′−gmgm


Connecting nodes j′and k′to ground, jto control node and
kto output node the VCCS stamp becomes gm. As nodes
j′and k′are connected to ground, the rows and columns
corresponding to those nodes are eliminated.
As matrix Dis positive definite, it can be decomposed into
p(p−1)/2resistor stamps between the ports and presistor
stamps from ports to ground where pis the number of ports.
2) Netlist Generation for ˆ
C:Matrix −QEscontains diago-
nal entries which correspond to the real eigenvalues in (17) and
the block diagonal entries which correspond to the complex
eigenvalues in (20). A capacitor stamp from the corresponding
node to the ground can be generated for each of the nonzero
diagonal entries in ˆ
C. For the block diagonal entries in matrix
Esshown in (20) the entries in ˆ
C,−QEs, can be rewritten as
a summation of three block matrices:
α β
β−α=−β β
β−β+α+β+β−α
A capacitor stamp for each of the block matrices can then
be generated in order to account the block-diagonal entries
in ˆ
C. If −Eswas used instead of −QEs, we would end up
with some additional VCCS stamps. In this case our proposed
approach generates less number of VCCS stamps.
Similar to D, matrix Y∞can be decomposed into p(p−1)/2
capacitor stamps between ports and pcapacitor stamps from
ports to ground.

P2 P1
Fig. 2. Example 1: Microstrip line with coax to coax connection separation
P2 P1
Fig. 3. Example 2: Transmission line with via holes
V. SIMULATION RESULTS
Example 1 is a 2 port microstrip line structure with two
coaxial terminations shown in Fig. 2. At the termination there
is a 0.64 mm to 1 mm coaxial converter with separation in
the connection of two different sizes of the coaxial cable, t=
2mm. Example 2 is a 2 ports high-speed module. The module
is a combination of stripline and microstrip line with 4 via
holes in the line. Each port is terminated with a 1 mm coaxial
connector.
The Y-parameter data of bandwidth 0 to 10 GHz is gen-
erated from a commercial full-wave simulator for both of
the examples. The LM method is then applied to extract the
macromodel from the Y-parameter data. Finally the matrices
are sparsified and SPICE netlists are generated using the
proposed approach. The summary of the simulation results and
the generated netlists are shown in Table I and II respectively.
As we can see from the tables the proposed approach can
generate the netlist from the macromodel very efficiently.
Transient simulations on the example structures are performed
using an input pulse of 1 V peak, 20 ns width and 0.1
ns rise/fall time. The outputs are presented in Fig. 4 and 5
respectively.
VI. CONCLUSION
A method to generate the SPICE netlist from the time-
domain macromodel, extracted by LM method, was presented.
The method was shown to be efficient.
TABLE I
SUMMARY OF SIMULATION RESULTS
CPU time (S)
Ports Order Sparsification Netlist Generation
EX 1 2 25 0.011 0.020
EX 2 2 20 0.002 0.015
TABLE II
SUMMARY OF GENERATED NETLISTS
Resistor Capacitor VCCS
EX 1 25 37 100
EX 2 23 32 80
5 10 15 20 25
0
0.4
0.8
1.2
Time (n S)
Input
Output
Fig. 4. Example 1: Output at port 2 for an input pulse at port 1
5 10 15 20 25
−0.5
0
0.5
1
1.5
Time (n S)
Voltage (V)
Input
Output
Fig. 5. Example2: Output at port 3 for an input pulse at port 1
REFERENCES
[1] B. Gustavsen and A. Semlyen, “Rational approximation of frequency
domain responses by vector fitting,” IEEE Trans. Power Del., vol. 14,
no. 3, pp. 1052–1061, Jul. 1999.
[2] B. Gustavsen and A. Semlyen, “A robust approach for system identi-
fication in the frequency domain,” Power Delivery, IEEE Transactions
on, vol. 19, no. 3, pp. 1167–1173, July 2004.
[3] D. Deschrijver and T. Dhaene , “Passivity-based sample selection and
adaptive vector fitting algorithm for pole-residue modeling of sparse
frequency-domain data,” in Behavioral Modeling and Simulation Con-
ference, 2004. BMAS2004. Proceedings of the 2004 IEEE International,
Oct. 2004, pp. 68–73.
[4] B. Nouri, R. Achar, M. Nakhla, and D. Saraswat, “z-domain orthonormal
vector fitting for macromodeling high-speed modules characterized by
tabulated data,” in Signal Propagation on Interconnects, 2008. SPI 2008.
12th IEEE Workshop on, may 2008, pp. 1–4.
[5] A. Chinea and S. Grivet-Talocia, “A parallel vector fitting implemen-
tation for fast macromodeling of highly complex interconnects,” in
Electrical Performance of Electronic Packaging and Systems (EPEPS),
2010 IEEE 19th Conference on, oct. 2010, pp. 101–104.
[6] A. J. Mayo and A. C. Antoulas, “A framework for the solution of the
generalized realization problem,” Linear Algebra and its Application,
vol. 425, no. 2-3, pp. 634–662, Sep. 2007.
[7] S. Lefteriu and A. C. Antoulas, “A new approach to modeling multiport
systems from frequency-domain data,” IEEE Trans. Comput.-Aided
Design Integr. Circuits Syst., vol. 29, no. 1, pp. 14–27, Jan. 2010.
[8] Y. Wang, C. Lei, G. Pang, , and N. Wong, “MFTI: Matrix-format
tangential interpolation for modeling multi-port systems,” in Proceedings
IEEE/ACM Design Automation Conference (DAC’10), Anaheim, CA,
2010, pp. 683–686.
[9] M. Kabir and R. Khazaka, “Macromodeling of distributed networks from
frequency-domain data using the loewner matrix approach,” IEEE Trans.
Microw. Theory Tech., vol. 60, no. 12, pp. 3927–3938, 2012.
[10] G. Eason, B. Noble, and I. N. Sneddon, “On certain integrals of lipschitz-
hankel type involving products of bessel functions,” Phil. Trans. Roy.
Soc. London, vol. A247, pp. 529–551, April 1955.
[11] J. Vlach and K. Singhall, Computer Methods for Circuit Analysis and
Design, 2nd ed. Norwell, Massachusetts: Kluwer Academic Publishers,
2003.