2004 International Symposium on Nonlinear
Theory and its Applications (NOLTA2004)
Fukuoka, Japan, Nov. 29 - Dec. 3, 2004

Experimental-Based Approach for Macromodel Synthesis of Interconnects
Junji KAWATA† , Yuichi TANJI‡ , Yoshifumi NISHIO†† ,Tomoji YOSHIDA† , Tetsuo TADA† and Akio USHIDA††Faculty of Engineering, Tokushima Bunri University
1314-1 Shido, Sanuki-shi, Kagawa 769-2193 Japan‡Faculty of Engineering, Kagawa University††Faculty of Engineering, Tokushima University
Email: {kawata,yoshida,tada,ushida}@fe.bunri-u.ac.jp, tanji@eng.kagawa-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—In this paper, each element of the admittancematrix of RLCG interconnects is reduced by partial fraction consists of the exact poles of the admittance matrix
and the corresponding residues. The residues can be calculated by the least squares method so that the partial fraction
matches each element of the exact admittance matrix in thefrequency-domain. From the partial fraction, macromodels
which can be easily simulated with SPICE are synthesized.
Based on experimental studies, an approach for calculating
the residues that guarantee the passivity of the macromodels are presented.
1. Introduction
The analysis and design of high speed LSI chips are
becoming more and more important, because the subcircuits coupled with interconnects embedded in the substrate sometimes cause the fault switching operations dueto the signal delays, crosstalks, reflections and so on [1][5]. The Elmore resistance-capacitance (RC) delay metric is popular due to its simple closed-form expression,
computation speed and fidelity with respect to the simulation [4]. The closed-form combining with the delay andcrosstalk is firstly presented in the reference [5]. The modified algorithms are proposed later for the improvement of
the accuracy and the practical applications in the simulations [6]-[9].
AWE (asymptotic waveform evaluation method) [10] iswidely used as a reduction technique of the large scaleof linear networks, the algorithm is based on a momentmatching technique and Padé approximation. However,
the method sometimes become erroneous, if there exist
the poles far from the origin. To overcome the problem,
CFH (complex frequency hopping) [11] and multi-point
Padé approximation [12] methods are proposed. In these
reduction algorithms, the reduced circuits sometimes become unstable in the time domain even if all the poles arelocated in the left hand side of the complex plane. The
ill-condition can be overcome by PVL (Padé via Lanczos
process) [13], and PRIMA (passive reduced-order interconnect macromodeling algorithm) [2,14]. In order to apply
these algorithms to the interconnects, we need two steps
such that each interconnect is firstly modeled by a finite or-der system, and Arnoldi-based congruence transformation
is applied to the system to form its reduced order model.
In this paper, we consider LSIs such as ASIC or SoC(System on a Chip) are coupled with interconnects embedded in the substrate. In this case, the diffusion resistancecomponents of the interconnects are generally assumed tobe very large compared to those of PCBs [3] and the lengths
are very short. From the telegrapher’s equation of the interconnects, the admittance matrix is derived from the relations at the near and far ends [1]. Next, each of the elementsof the admittance matrix is approximated by partial fraction
using dominant poles around the origin. The poles can becalculated by a computational method proposed in our previous study [18]. The residues are subsequently evaluatedby the least squares method so that the frequency response
curves of the partial fractions match with those of the exact admittance matrix. Here the frequency response curvesof the partial fractions are compared with the exact ones.
Then, it will be investigated that how the passivity of partial fractions obtained from pole-residue pairs breaks andwhy inaccurate residues are obtained. Based on the considerations, we provide a procedure to enforce the passivityof the macromodels.Furthermore, SPICE simulations using the macromodel synthesized from the partial fractions
and built-in model of SPICE are carried out and compared.
2. Admittance Matrix of Interconnects and Partial
Fraction Approximation
The admittance matrix for a uniform N coupled RLCG
interconnect is given by [18]:−1Y11 (s) = Y22 (s) = Pc (s)diag[coth" γi (s)d]P#v (s)

1
(1)
−1 

Pv (s) 
Y12 (s) = Y21 (s) = −Pc (s)diagsinh γi (s)dwhere Pv (s) and Pc (s) are transfer matrices. The polesof the admittance matrix (1) must satisfy the relation
sinh γi (s)d = 0 and they can be calculated by applying
the Leverrier-Faddeeva algorithm [18]. We choose somedominant poles located around the origin of the complex
plane, because the poles have large impact on the transient
responses. Thus, the admittance matrix given by (1) are107

approximated by the partial fractions that are composed ofthe dominant poles and the corresponding residues, in thefollowing form:Ŷ11 (s) = Ŷ22 (s) =
N X
M
N
X
X
b11,i,n s + b10,i,n
k01,i
(2)
+
2+a
s
−
p
s
0,i
1,i,n s + a0,i,n
i=1 n=1i=1Ŷ12 (s) = Ŷ21 (s) =
N
X
k02,i
i=1

s − p0,i

+

N X
M
X
b21,i,n s + b20,i,n2
s + a1,i,n s + a0,i,n
i=1 n=1(3)

where N is the number of conductors and M shows thenumber of complex conjugate pairs. Also p0,i denotes areal pole and {a1,i,n , a0,i,n } are derived from complex conjugate poles for (1), respectively. Further the residues of (2)and (3) have the following relations [17, 18]:)
k02,i = −k01,i(4)
b21,i,n = (−1)n+1 b11,i,n , b20,i,n = (−1)n+1 b10,i,nthat is, we have to evaluate only the residues for (2) andthe values of (3) can be calculated using the relation (4).
Macromodels can be derived from the relations (2)–(4), but
omit in this paper due to space limitation (refer to [18] fordetail).
3. Approach for Calculating Residues
The residues satisfying (2) are calculated by the leastsquares method so that the frequency response curve of (2)fits to the exact one. In order to guarantee the passivityof the macromodels, the following trial-and-error approachfor calculating the residues is provided.

Case 1 Let the highest frequency and the number of poles
be fmax = 6 [GHz] and M = 10, respectively. In this casethe frequency corresponding to the pole p10 (the suffix i = 1
is omitted) with largest imaginary part is about 5 [GHz].
The frequency response curve obtained by substituting the
calculated poles and residues to the partial fraction (2) or(3) and the exact curve obtained from (1) are shown in Fig.
1. Here, the above-mentioned procedure of residues calculation is not used, and |Ŷ12 ( jω)| is calculated using therelation (4). The approximated curves by the partial fractions have relatively good agreement with the exact one up
to 11-th peak around 5 [GHz], because we took a real poleand 10 of complex conjugate poles (i.e., M = 10). From
the figure, it seems that the least squares fitting is successfully performed. However the calculated residues containnegative values as listed in the left-half of table 1, whichyields negative circuit parameters. Thus, it is clear that thepassivity is not satisfied in this case.
Case 2 For the case that the resistance component of interconnect becomes large, the least squares fitting fails toapproximate the frequency-domain response accurately, as
shown in Fig. 2. In this example, |Ŷ12 ( jω)| is also calculated using the relation (4). From these two examples, we
can conclude that the pole selection in this manner is notsuitable for our purpose.
Case 3 Next we tried to choose some extra poles beyond
the highest frequency fmax . After choosing 6 of extra poles
beyond fmax = 6 [GHz] (i.e., M = 18), the residues arecalculated by performing the least squares fitting, which
are listed in the right-half of table 1 and the frequency response curves are shown in Fig. 3. As expected, |Ŷ11 ( jω)|matches the exact one within the whole frequency range

Step 1) Determine the highest matched frequency fmax .
Step 2) Choose all poles with the imaginary part less thanor equal to 2π fmax and further several extra poles beyond the frequency. Let the number of the former andthe latter be M and α, respectively. Thus the number of poles required in the least squares method isreplaced with Mext = M + α. The appropriate initialvalue of α is about 5 ∼ 10.

Table 1: Residues obtained by the least squares method for Cases
1 and 3.
Case 3

Case 1

Step 3) Perform the least squares fitting and subsequently
truncate the extra α pole-residue pairs beyond 2π fmax .
Step 4) If all of the residues and the resulting macromodel
parameters are positive, finish the computation; otherwise increase the number of extra poles α and returnto Step 3.
It should be noted that the proposed approach enforces thepassivity of the interconnect macromodels. All the valuesof the passive elements included in the macromodels areconstrained to be positive. Hence, the macromodels areguaranteed to be passive.
We mention here the reason why the above residues calculation procedure is required, through the following numerical results.

108

n

b1,n

b0,n

b1,n

b0,n

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

3.815e−02
3.815e−02
3.816e−02
3.818e−023.820e−02
3.822e−02
3.825e−02
3.828e−023.829e−02
3.818e−02

3.979e−03
1.939e−03−1.568e−03−6.727e−03−1.385e−02−2.349e−02−3.664e−02−5.537e−02−8.508e−02−1.483e−01

4.000e−02
4.000e−02
4.000e−02
4.000e−02
4.000e−02
4.000e−02
4.000e−02
4.000e−02
4.000e−02
4.000e−02
4.000e−02
3.990e−02
4.149e−02
1.704e+00−1.902e+01
6.859e+01−9.761e+01
4.815e+01

5.002e−03
5.008e−03
5.017e−03
5.016e−03
5.004e−03
4.979e−03
4.940e−03
4.960e−03
5.049e−03
5.127e−03
4.623e−03
1.383e−02
3.551e+00−1.005e+02
6.750e+02−1.764e+03
1.972e+03−7.910e+02

0.5
0.4

exact
least square

0.3|Y12(jω)||Y11(jω)|

0.4

exact
least square

0.3
0.2

0.2
0.1

0.1
0

0
0

1

2

3
f [GHz]

4

5

6

0

1

(a) |Y11 ( jω)|2

3
f [GHz]456(b) |Y12 ( jω)|Figure 1: Frequency response curves for single conductor interconnect. (r = 0.5[Ω/mm], d = 5.0[mm], fmax = 6.0[GHz], M = 10)
0.04

0.04

exact
least square

exact
least square

0.03|Y12(jω)||Y11(jω)|

0.03
0.02
0.01

0.02
0.01

0

0
0

1

2

3
f [GHz]

4

5

6

0

1

(a) |Y11 ( jω)|2

3
f [GHz]456(b) |Y12 ( jω)|Figure 2: Frequency response curves for single conductor interconnect. (r = 25.0[Ω/mm], d = 5.0[mm], fmax = 6.0[GHz], M = 10)(0 ≤ f ≤ fmax ) of interest, shown in Fig. 3(a). Unfortunately, as shown in Fig. 3(b), |Ŷ12 ( jω)| obtained from (3)and (4) is entirely different from the exact one. Furthermore, the values of |Ŷ11 ( jω)| at certain frequency range beyond the highest matched frequency ( fmax = 6 [GHz]) become very large and differ completely from the exact one,as shown in Fig. 3(c). Note that the only difference of Figs.
3 (a) and (c) is the scales of axes.
From the right-half of table 1, the residues for n > 12,
which correspond to the extra poles, contain negative or
large values and consequently do not guarantee the passivity. Thus, it is concluded that disagreement of |Ŷ12 ( jω)| andhuge values of |Ŷ11 ( jω)| at frequency range beyond fmax are
due to the extra pole-residue pairs.
Case 4 In this example, to remove the pole-residue pairs
beyond the highest frequency, which are the factors disturbing our purpose, is considered. In the same conditions
as Case 3, 6 of extra pole-residue pairs for n > 12 are truncated and the frequency response curves are computed. The
results are illustrated in Fig. 4. The effect of truncation begins to appear at the last half of the frequency range; thatis, the accuracy of partial fractions (2) and (3) at higher
frequency range is degraded. However, at lower frequency
range both curves remain matched. Further the calculated
response |Ŷ12 ( jω)| using the relation (4) is also matched tothe exact one at lower frequency range, unlike the Case 3.
Note that the partial fractions (2) and (3) (or the resulting
asymptotic equivalent circuits) become more precise as thevalue of fmax increases, because matched frequency range
becomes wide accordingly. Further it has been confirmedthat positive residues and/or circuit elements of the macromodels can be obtained by increasing the number of extra
poles, even if negative residues and/or circuit elements appeared around fmax .

It is not easy to show whether the residues guaranteeing thepassivity can be always obtained by this approach, however
they could be obtained in our simulations conducted so far.
4. Transient Simulation Using Macromodels
The transient responses for a simple linear circuit with asingle-conductor interconnect are calculated. The SPICE
simulations using the macromodel are carried out withvarying the highest matched frequency fmax (accordingly
M is changed), and compared with the simulated resultsof built-in SPICE model. Figure 5 shows the transient
response waveforms for r = 0.5 [Ω/mm] and d = 5.0[mm]. All the waveforms have relatively good agreement,
although ringing occurs at rising and falling edges of waveforms. The ringing can be suppressed with increasing M,
but remarkable changes are not observed even for M = 100(in this case fmax = 50 [GHz]).
5. Conclusions
In this paper, we have provided an approach for calculating the residues that guarantee the passivity. Using theexact poles and the calculated residues, each element of
the admittance matrix of interconnects is approximated by
the partial fraction. Though the frequency response curves
calculated using the partial fractions become inaccurate
around the highest matched frequency fmax , the partial fractions and the resulting macromodels are always passive. In
this paper, some examples for single conductor interconnect have been presented, but our method can be effectively
applied to multi-conductors interconnects.
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109

0.04

0.04

exact
least square|Y11(jω)||Y11(jω)|

0.02

0.02
0.01

0.01

0

0
0

1

2

3
f [GHz]

4

5

0

6

2

6

8

10

(a) |Y11 ( jω)|(a) |Y11 ( jω)|
0.020
exact
least square

8

fmax

exact
least square

0.015|Y12(jω)||Y12(jω)|4f [GHz]

10

6
4

0.010
0.005

2
0
0

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SPICE
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v(0,t) [V]

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2

4

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yond fmax are truncated. (r = 25.0[Ω/mm], d = 5.0[mm], fmax =6.0[GHz], M = 12, α = 6)

25

0

8

(b) |Y12 ( jω)|

fmax

30

6

Figure 4: Frequency response curves for the case that 6 poles be-exact
least square

35

4f [GHz](b) |Y12 ( jω)|40

|Y11(jω)|exact
least square

fmax

0.03

0.03

8

10

f [GHz]32
10(c) |Y11 ( jω)|-1

Figure 3: Frequency response curves for the case that extra poles
beyond fmax are used. (r = 25.0[Ω/mm], d = 5.0[mm], fmax =6.0[GHz], M = 18)

0

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6

10

(a) near end.
2.0

SPICE
M=20
M=50

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