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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:

M¨q(t) + D ˙q(t) + Kq(t) = B1u(t)

y(t) = L1q(t) + L2˙q(t)(1)

where M,D,K∈Rn×n,B1∈Rn×p,L1,L2∈Rq×n,

q∈Rn,u∈Rp,y∈Rqand in which Mconsider to be

nonsingular. The objective of MOR is to produce a ROM:

˜

M¨qr(t) + ˜

D ˙qr(t) + ˜

Kqr(t) = ˜

B1u(t)

y(t) = ˜

L1qr(t) + ˜

L2˙qr(t)(2)

where ˜

M,˜

D,˜

K∈Rr×r,˜

B1∈Rr×p,˜

L1,˜

L2∈Rq×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 H∞norm

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≡ − 0−K

−K D ,E≡−K 0

0 M,

B≡0

B1,L≡L1L2

(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,U∈Rn×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:

T−1RUT =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 n−rstates corresponding to the smallest HSVs.

The BT method provides an error bound with respect to the

H∞norm 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×rΣ−1/2

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 M≡C,D≡G,K≡Γ,L1≡0, and L2≡BT

1, the

second-order formulation of an RLCK circuit is the following:

C¨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,Γ∈Rn×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(Z−ω1

−ω2

(iωE−A)−1dω+Zω2

ω1

(iωE−A)−1dω)(13)

The above matrix integral can be evaluated as:

F=1

πRe(Zω2

ω1

(iωE−A)−1dω)

=Re(i

πln((A+iω1E)−1(A+iω2E)))E−1

=E−1Re(i

πln((A+iω2E)(A+iω1E)−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 Bω≡EFB and Lω≡LFE), 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×rΣ−1/2

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= 10−2for 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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