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Frequency-Limited Reduction of RLCK Circuits via Second-Order Balanced Truncation

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Second-order formulation using susceptance elements has become very effective in modeling on-chip inductive couplings. Several prior works have proposed model order reduction techniques for RLCK circuits, mostly based on balanced truncation (BT) and moment matching, providing reduced-order models (ROMs) that can be simulated over the whole frequency range. However, in most applications, the ROMs are simulated only at specific frequency windows, which means that established methods usually provide models that may become unnecessarily large to achieve approximation over all frequencies. In this paper, we present a second-order frequency-limited approach for RLCK circuits, which may be combined with efficient low-rank Lyapunov solvers, leading to ROMs which are either smaller or exhibit better accuracy compared to an established second-order BT method. Experimental results on interconnect bus structures demonstrate the advantages of the proposed method.
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Frequency-Limited Reduction of RLCK Circuits
via Second-Order Balanced Truncation
Olympia Axelou, Dimitrios Garyfallou, George Floros
Department of Electrical and Computer Engineering, University of Thessaly, Volos, Greece
{oaxelou, digaryfa, gefloros}@e-ce.uth.gr
Abstract—Second-order formulation using susceptance ele-
ments has become very effective in modeling on-chip inductive
couplings. Several prior works have proposed model order re-
duction techniques for RLCK circuits, mostly based on balanced
truncation (BT) and moment matching, providing reduced-order
models (ROMs) that can be simulated over the whole frequency
range. However, in most applications, the ROMs are simulated
only at specific frequency windows, which means that established
methods usually provide models that may become unnecessarily
large to achieve approximation over all frequencies. In this
paper, we present a second-order frequency-limited approach for
RLCK circuits, which may be combined with efficient low-rank
Lyapunov solvers, leading to ROMs which are either smaller or
exhibit better accuracy compared to an established second-order
BT method. Experimental results on interconnect bus structures
demonstrate the advantages of the proposed method.
Index Terms—model order reduction (MOR), balanced trun-
cation (BT), circuit simulation, second-order systems.
I. INTRODUCTION
As integrated circuits move to advanced technology nodes,
2.5D and 3D chiplet-based architectures have become key
enablers for overcoming the related manufacturing chal-
lenges [1]. In this type of integration, the challenge of ef-
ficient parasitic modeling is completely different compared
to monolithic 2D design, since inductive effects have to be
taken into account as well. These effects are more pronounced
in power delivery and clock networks, as well as at long
and wide bus structures, and can affect the power and signal
integrity of the chiplet. The electrical models of the above are
usually formulated as RLCK circuits in second-order form,
since many desired properties of the susceptance matrix can
be preserved in this form [2]. However, the large and dense
inductance matrix, mostly due to mutual inductance, hinders
the application of numerical simulation methods. Model order
reduction (MOR) provides efficient techniques to reduce the
model complexity by replacing the original model with a much
smaller one, while achieving accurate approximation of the
input–output port behavior.
MOR methods that have been applied in second-order sys-
tems are divided into two main categories. Moment matching
(MM) techniques [3], [4] are well established due to their
computational efficiency in producing reduced-order models
(ROMs). Their drawback is that the ROM depends only
on the number of matching moments and the quality of
the produced Krylov subspace. On the other hand, system
theoretic techniques, such as balanced truncation (BT) [5], [6],
provide very satisfactory and reliable bounds for the approxi-
mation error. However, BT techniques require the solution of
Lyapunov matrix equations which are very computationally
expensive, and also involve storage of dense matrices, even
if the system matrices are sparse. In order to make such
techniques amenable to large circuit models, low-rank solution
methods, such as the extended Krylov subspace (EKS), have
been developed [7].
The majority of the aforementioned methods focus on ap-
proximating the original model over the whole frequency range
(from DC to infinity). In most practical applications, however,
we are only interested in a specific finite frequency range.
Frequency-limited BT methods have been proposed in the past
[8], [9], where a user-specified frequency range is given in
order to obtain solutions of Lyapunov matrix equations that
improve the accuracy of the ROM in this particular range. The
problem is that the existing frequency-limited BT techniques
are only applied in first-order systems and cannot handle
RLCK circuits, where the susceptance matrix is utilized.
In this paper, we introduce a frequency-limited second-
order BT method for RLCK circuits, which extends [9] in
order to handle second-order Gramians, as defined in [10]. In
contrast to the first-order frequency-limited BT, the proposed
methodology produces ROMs which preserve the structure
information that is inherent to the RLCK circuits. Finally, we
evaluate our methodology on actual multi-line bus examples,
and we demonstrate that frequency-limited second-order BT
may produce ROMs with either smaller size or superior accu-
racy compared to standard BT in a specific frequency range.
The rest of the paper is organized as follows. Section II
presents the theoretical background of second-order BT meth-
ods for the reduction of RLCK circuit models. Section III
presents our main contributions in the application of the
frequency-limited framework to second-order BT methods.
Section IV presents our experimental results, while conclu-
sions are drawn in Section V.
II. SECOND-OR DE R BT F OR RLCK CIRCUITS
Consider the second-order LTI system:
q(t) + D ˙q(t) + Kq(t) = B1u(t)
y(t) = L1q(t) + L2˙q(t)(1)
where M,D,KRn×n,B1Rn×p,L1,L2Rq×n,
qRn,uRp,yRqand in which Mconsider to be
nonsingular. The objective of MOR is to produce a ROM:
˜
qr(t) + ˜
D ˙qr(t) + ˜
Kqr(t) = ˜
B1u(t)
y(t) = ˜
L1qr(t) + ˜
L2˙qr(t)(2)
where ˜
M,˜
D,˜
KRr×r,˜
B1Rr×p,˜
L1,˜
L2Rq×r,qr
Rr, and in which the order r << N and the output error
is bounded as ||˜y(t)y(t)||2< ε||u(t)||2for given input
u(t)and given small ε. The bound in the output error can
be equivalently written in the frequency domain as ||˜y(s)
y(s)||2< ε||u(s)||2via Plancherel’s theorem [11]. If
H(s)=(L1+sL2)(s2M+sD+K)1B1
˜
H(s)=(˜
L1+s˜
L2)(s2˜
M+s˜
D+˜
K)1˜
B1
are the transfer functions of the original model and the ROM,
then the output error in the frequency domain is:
||˜y(s)y(s)||2=|| ˜
H(s)u(s)H(s)u(s)||2
|| ˜
H(s)H(s)||||u(s)||2
(3)
where ||.||is the induced L2matrix norm, or Hnorm
of a rational transfer function. Therefore, in order to bound
the output error, we need to bound the distance between the
transfer functions as || ˜
H(s)H(s)||< ε.
In order to directly apply BT and reduce the second-order
system of (1), the basic idea is to first transform the second-
order system into an equivalent first-order form as
Edx(t)
dt =Ax(t) + Bu(t)
y(t) = Lx(t)
(4)
and then obtain the balancing matrices by a standard BT
procedure. To this end, the second-order Gramians, which
are introduced in [10], are formed based on the first-order
realization of the state-space formulation of (1) with 2n
dimensions, xT= [q,˙q], and
A 0K
K D ,EK 0
0 M,
B0
B1,LL1L2
(5)
The first-order realization of (4) has the same input–output
behavior as the second-order system of (1). In order to
reduce the second-order system, first we need to compute the
Gramians for the first-order realization, which are derived by
the solution of the Lyapunov matrix equations [12]:
EPAT+APET=BBT
ETQA +ATQE =LTL(6)
If we conformally partition the computed first-order Gramians
of (6) as defined in [10], we obtain
P=R S
STO,Q=U X
XTH(7)
In the above, R,URn×nsubmatrices are the second-order
controllability and observability matrices, respectively. Finally,
there exists a similarity transformation matrix T, which makes
the eigenvalues of the Gramians product invariant such as:
T1RUT =diag(σ2
1, σ2
2, ...., σ2
n)(8)
where σ1, σ2...,σnare the Hankel singular values (HSVs)
of the model. The most controllable and observable states
correspond to the largest HSVs, and rof them are kept by
truncating the nrstates corresponding to the smallest HSVs.
The BT method provides an error bound with respect to the
Hnorm such as:
|| ˜
H(s)H(s)||2(σr+1 +σr+2 +... +σn)(9)
Finally, in order to compute the projection matrix, the
singular value decomposition (SVD) of UΣV =ZRKZUhas
to be computed, where ZRand ZUare the Cholesky factors
of Rand U, respectively, i.e., R=ZRZR,U=ZUZU.
The projection matrices onto a lower dimensional subspace are
defined as Wl=Wr=ZRUn×rΣ1/2
r×r. Then the reduced-
order matrices are:
˜
M=WT
lMWr,˜
D=WT
lDWr,˜
K=WT
lKWr
˜
B1=WT
lB1,˜
L1=L1Wr,˜
L2=L2Wr
(10)
Regarding the RLCK circuits, they can be formulated as a
second-order system of (1) using nodal analysis (NA). By
setting MC,DG,KΓ,L10, and L2BT
1, the
second-order formulation of an RLCK circuit is the following:
q(t) + G ˙q(t) + Γq(t) = B1u(t)
y(t) = BT
1˙q(t)(11)
where u(t),y(t)Rpare the input currents and output volt-
ages, ˙q(t)Rnare the nodal voltages, and C,G,ΓRn×n
are the capacitance, conductance, and susceptance matrices,
respectively. The capacitance matrix is considered nonsingular.
III. SECOND-ORD ER BT FO R RLCK CIRCUITS
IN LIMITED FR EQUENCY INTERVALS
The solution of the Gramians that appear in (6) consider
the entire frequency interval, i.e., (−∞,). If we restrict the
interval to a certain range, i.e., [ω2,ω1][ω1, ω2], we can
obtain the frequency-limited Gramians, Pωand Qω, which
may be derived by the solution of the following modified
Lyapunov equations [13]:
APωET+EPωAT=(EFBBT+ (EFBBT)T)
ATQ0
ωE+ETQ0
ωA=((LTLFE)T+LTLFE)(12)
where
F=1
2π(Zω1
ω2
(EA)1+Zω2
ω1
(EA)1)(13)
The above matrix integral can be evaluated as:
F=1
πRe(Zω2
ω1
(EA)1)
=Re(i
πln((A+1E)1(A+2E)))E1
=E1Re(i
πln((A+2E)(A+1E)1))
(14)
where ln() is the function of the matrix logarithm, which
represents the inverse of the matrix exponential, i.e., a matrix
Xsuch that exp(X) = Y.
The frequency-limited Gramians characterize the controlla-
bility and observability of the model in the selected frequency
range, and the process of balancing and truncation eliminates
states that are difficult to control and observe inside this
frequency range. This means that more states can be eliminated
for a given tolerance in (9), which would not have been
eliminated otherwise (e.g., states which are easily controllable
and observable in other frequencies), leading to lower ROM
order r(or alternatively, lower error in the frequency range
for a given order r). In order to compute Pω,Qωby solving
(12), we have to deal with the two different right-hand sides
(RHS) of frequency-limited Lyapunov equations, which are
in the forms of (BωBT+BBT
ω)and (LT
ωL+LTLω)
(where BωEFB and LωLFE), instead of the standard
forms BBTand LTLof (6). However, efficient Lyapunov
solvers, which can efficiently deal with the matrix logarithm
and the modified RHS, have been already presented in the
literature [9]. The main steps of the proposed frequency-
limited BT for second-order RLCK models are summarized
in Algorithm 1.
Algorithm 1: Second-order frequency-limited reduction
of RLCK circuits by BT
Input: System matrices C,G,Γ,B1, and frequency
range [ω1, ω2]
Output: ROM matrices ˜
C,˜
G,˜
Γ,˜
B1
Function so_freq_lim_BT(G,C,Γ,B1,[ω1, ω2]):
1) Solve the frequency-limited Lyapunov equations of
(12) to obtain the Gramian matrices Pω,Qωwith
respect to the first-order realization matrices of (5).
2) Partition the Cholesky factors, R=ZRZR,
U=ZUZU, according to (7).
3) Compute the singular value decomposition (SVD)
of the product of the Cholesky factors, i.e.,
UΣV =ZRΓZU.
4) Compute the truncated part of the balancing
transformations using Wl=Wr=ZRUn×rΣ1/2
r×r.
5) Compute the corresponding ROM matrices as:
˜
C=WT
lCWr,˜
G=WT
lGWr,
˜
Γ=WT
lΓWr,˜
B1=WT
lB1.
return ˜
C,˜
G,˜
Γ,˜
B1
End Function
Finally, it should be noted that BT-type MOR methods
do not generally preserve the passivity of the original model
(due to the similarity transformation involved in the balanc-
ing before truncation). However, instead of provably passive
models, MOR techniques have been focused on passivity
enforcement after efficient reduction. A wealth of passivity
enforcement techniques, such as [14], have been developed to
assure passivity of the ROMs obtained by frequency-limited
second-order BT.
IV. EXP ER IM EN TAL RESULTS
For the experimental evaluation of the proposed methodol-
ogy, we created 5 benchmarks of 3D geometry bus structures
using FastHenry [15], which consist of layers of parallel wires.
More specifically, different layers are planes across the z-
axis, different wires are line segments across the y-axis, and
each wire extends on the x-axis. The characteristics of these
benchmarks are shown in Table I, where #layers is the number
of layers of each benchmark, #wires per layer represents the
number of wires that each layer has, and #filaments per wire
denotes the number of filaments that each wire is broken into.
The wires composed of multiple filaments are split along their
width. In all benchmarks, the wires have length 1mm and
cross section 1µm2. The distance between successive layers is
2µm and the distance between wires of the same layer is 1µm.
TABLE I
CHARACTERISTICS OF THE EVALUATION BENCHMARKS
Benchmark #layers #wires per layer #filaments per wire
interc1 9 91 1
interc2 3 273 2
interc3 3 410 1
interc4 3 410 2
interc5 3 546 1
The second-order frequency-limited BT was implemented
with the procedure of Algorithm 1 for the frequency range of
[ω1, ω2] = [105,109], and was compared against SBPOR [5].
In both methods, the Lyapunov equations were solved with the
method presented in [9]. The ROMs of SBPOR and second-
order frequency-limited BT were compared with respect to
both their order for given tolerance εand their accuracy for
given ROM order. In the first case, the error tolerance was
chosen as ε= 102for both methods, while in the second
case, the order that resulted from the execution of SBPOR was
reused for the truncation of the HSVs of the frequency-limited
Gramians. All experiments were executed with MATLAB
R2021a on a Linux workstation, having a 3.6GHz Intel Core
i7 processor with 32GB memory.
Our experimental results are reported in Table II, where
Max error refers to the maximum error between the infinity
norms of the transfer functions of the original model and the
ROM in the selected frequency range, i.e., ||H(s)˜
H(s)||,
Time refers to the computational time (in seconds) needed
to generate the ROMs, while Reduction percentage refers to
the percentage of ROM reduction for the same error bound
between SBPOR and second-order frequency-limited BT. As
can be seen, the proposed second-order frequency-limited BT
can produce ROMs which, in the selected frequency range,
exhibit either smaller size for given error, or smaller error for
given order in comparison to SBPOR. The proposed method
provides ROMs which have up to 50% less states with respect
to SBPOR in interc1 benchmark, while achieving up to 2615×
less error in interc5 benchmark. Note that the execution time
of second-order frequency-limited BT was slightly larger than
SBPOR due to the matrix logarithm calculation.
TABLE II
REDUCTION RESULTS OF FREQUENCY-LI MI TED B T VE RSU S SBPOR IN THE [105
,109]FR EQU ENC Y IN TERVAL
Benchmark #nodes #ports
Second-order BT (SBPOR) Second-order frequency-limited BT (Proposed)
ROM Max error Time (s) ROM order Reduction Max error Time (s)
order for same error percentage for same order
interc1 4095 32 64 1.9e-03 35.48 32 50% 6.9e-05 40.74
interc2 4095 64 98 4.6e-02 41.23 70 28.57% 7.0e-05 47.85
interc3 6150 96 197 3.3e-02 144.56 152 22.84% 2.2e-04 186.21
interc4 6150 192 480 3.4e-02 312.09 312 35% 1.3e-05 354.52
interc5 8190 256 704 6.4e-02 653.90 430 38.92% 9.2e-05 733.88
To demonstrate the accuracy of our method, we compare
the transfer functions of the original model and the ROMs
generated by the second-order frequency-limited BT and SB-
POR. Fig. 1 presents the transfer functions of ROMs produced
by second-order frequency-limited BT and SBPOR for the
interc2 benchmark, in the frequency interval [105,109], along
with the absolute errors induced over the original model for
the selected benchmark in the same frequency range. As can
be seen, the response of second-order frequency-limited BT
ROM is performing very close to the original model, while
the response of SBPOR exhibits a clear deviation.
Fig. 1. Comparison of ROM transfer functions and absolute error magnitudes
obtained by the proposed second-order frequency-limited BT and SBPOR in
the range [105
,109]at port (3,3) of interc2 benchmark.
V. CONCLUSION
In this paper, we presented a frequency-limited methodology
for reducing second-order systems arising in the modelling
of RLCK circuits, which requires only the specification of
the end frequencies. Experimental results indicate that our
approach provides clear improvements in model accuracy or
size with respect to SBPOR, while retaining its benefits of
specified error bounds, introducing only a small overhead in
the reduction process.
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