interconnect model reductions by using the aora ... - IEEE Xplore
Interconnect Model Reductions by Using the AORA Algorithm With Considering the Adjoint Network Chia-Chi Chu
Herng-Jer Lee
Wu-Shiung Feng and Ming-Hong Lai
Department of Electrical Engineering EverCAD Corp. Chang Gung University Hsin-Chu 300, Taiwan, R.O.C. Kwei-San, Tao-Yuan 333, Taiwan, R.O.C. Email: [email protected] Email: [email protected]
Abstract— This work proposes a hybrid method for interconnect model-order reductions. First, the Adaptive-Order Rational Arnoldi (AORA) method will be investigated. An extension of the classical multi-point Pad´e approximation by using the rational Arnoldi iteration approach will be studied. The adaptiveorder can be achieved by choosing the expansion frequency corresponding to the maximum output moment error. Secondly, the adjoint network technique will be studied. By exploring symmetric properties of the MNA formulation, the computational cost of constructing the congruence transformation matrix can be reduced by 50% compared with the conventional methods.
I. I NTRODUCTION Interconnect plays a significant role in the recent development of high-speed VLSI design. Due to the continuous increasing in component densities and clock rates, the signal integrity problems naturally arise in the interconnect structure. For efficient simulations, it is necessary to construct a low-order macro-model whose terminal behaviors essentially capture the complicated interactions. Several methods that are based on Pad´e synthesis have been applied to improve the model-order reduction techniques recently. Among all existing methods, the class of Krylovspace methods seems to be more accurate because it can avoid the ill-conditional problems. However, these conventional approximation methods tend to converge in a local fashion around a single frequency because Pad´e approximation is exact at the point while accuracy is lost away from it. To overcome this difficulty, multiple point Pad´e moment matching techniques have been proposed recently [1], [2], [6]. The straightforward way for the multi-point moment matching is to apply the Krylov subspace algorithm at various expansion frequencies. This is the so-called rational Krylov algorithm [3], [5]. This paper will further investigate the adaptive-order rational Arnoldi (AORA) method without determining the order of moments at each expansion frequency in advance. We will apply this technique to interconnect model order reductions [4]. In addition, the adjoint network method will be considered. By exploring symmetric properties of the interconnect MNA formulation, system moments of the adjoint network can be directly calculated from those of the original RLC network. Therefore, computational cost can be reduced significantly.
0-7803-8834-8/05/$20.00 ©2005 IEEE.
Department of Electronic Engineering Chang Gung University Email: [email protected] [email protected]
II. BACKGROUND In analyzing an RLC interconnect, the following modified nodal analysis (MNA) will be utilized: [6].
where
and
(1)
Matrices , , , and contain capacitances, inductances, conductances, and resistances, which are all symmetric positive definite. presents the incident matrix that satisfies Kirchhoff’s current law. The state variable includes node voltages and branch currents of inductors . In practical
simulations of large-scale circuits, all the above matrices are large and sparse. Since the computational cost for simulating a large circuit is indeed tremendously huge, model-order reduction techniques have been proposed recently to reduce the computational complexity [6]. The transfer functions of state variables and those of the and outputs are defined as , respectively. Given an expansion frequency , let matrix and matrix , where is assumed to be nonsingular. By at , we have taking the Taylor expansion of ½ , where is called the th-order system moment at . Similarly, the th-order output moment at is calculated as . The model-order reduction problem is to seek a -order system, where , such that
(2) where , ¢ , ¢ , and ¢ . The aim of moment matching is to establish a reduced-order system such that for , where is the order of moment matching.
One efficient way of obtaining a reduced-order system is to use the multi-point Pad´e approximation [1]. The multipoint Pad´e approximation requires that the output moment of the original system equals that of the reduced system, .
1278
However, this method usually yields ill-conditioned problem. Recent works have proposed Krylov subspace projection methods to avoid such numerical difficulties [6]. The reducedorder system is constructed by projecting an original largedimensional problem into a low-dimensional Krylov subspace. Given a square matrix ¢ and a vector , the -th Krylov sequence, defined as follows: is a sequence of column vector and the corresponding column space is called the th Krylov space. The Arnoldi algorithm can be applied to iteratively generate an orthonormal basis ¢ from the successive Krylov subspace span span . If we set and , it has been shown that the Krylov subspace is indeed spanned by for . The reduced-order can be constructed by using the orthogonal projection . In this case, the reduced system in Eq. (2) can be defined by the congruence transformation [6],
and (3)
The number of moments in the reduced system is exactly the number of moments in the original system at an expansion frequency , up to the order of . The multiple point Pad´e approximation can be achieved by the rational Arnoldi method. Let represent the set of predetermined expansion frequencies. be the set of the number of the Let matched moments at each corresponding frequency. The rational Arnoldi method will generatea reduced-order system , which matches -order ( ) moments of the , at the expansion points , . original system Implementing the rational Arnoldi method is equivalent to implementing the Arnoldi method times at expansion frequencies. That is, the first iterations correspond to the expansion frequency ; the next iterations are associated with , and so on. Each Arnoldi iteration generates orthonormal vectors. Then,
is the desired orthonormal matrix generated from a union Krylov space at various expansion points, as stated by
of expansion points for and the number of matched moments about each is by no means trivial, for simplicity, it is assumed that expansion points for are determined in advance. Suppose that the AORA algorithm have been performed iterations, where , the output moments of the original system and that of the reduced-order system are matched in the following sense: , for . After simple manipulations, the transfer function error can be represented as
(4) where is the th-order moment of at . The basic concept of the AORA method is to select an expansion point among all expansion points in in the )st iteration such that . The new ( expansion point and the new orthonormal vector
can be generated to achieve this additional order of the moment matching. Under this situation, the transfer function error in the st iteration corresponding to the expansion point can be expressed as
!
This means that the order of moment matching at the other remains the same as that in the th in the iteration for , iteration. That is, £ , and . The following theorem presents an exact formula for the output moment errors. for Theorem 2: [4] Suppose that and . The system matrices of reduced-order system are generated by the congruence transformation with the orthonormal matrix using the AORA algorithm, where . At each expansion point , the magnitude error between the th-order moments and Once the orthonormal matrix has been formed by ap- can be expressed: " # . plying the rational Arnoldi method, the reduced-order system where the normalized coefficient " # can be obtained using the congruence transformation. st-order reduced system with the In order to get the Theorem 1 : [4] Let be the orthonormal matrix generated greatest moment improvement, it is intuitive to choose the by the rational Arnoldi algorithm with iterations. Since expansion point such that the th-order moment of colspan for and , we have , and , , is maximal. As shown in Theorem 2, can be related to the residue vector # from . the th iteration. if has been selected in the th iteration, III. AORA A LGORITHM almost no additional computations of are needed. has been chosen, the residue Generally speaking, Pad´e based methods can not guaran- Once the expansion point tee to yield the reduced-order models with the best overall vector # can be normalized to be the new basis vector performances in the entire frequency domain. Only the local . The details of the AORA method includes the following approximation around the expansion point can be achieved. main steps: of This is also true for our AORA method. Table I outlines the Step (1): Initialize the first vector $ adaptive-order rational Arnoldi method. Since selecting a set the Krylov sequence for each expansion point , where
On the other hand, for the other expansion point not chosen as the new expansion point in the st iteration, the corresponding transfer function error is
1279
. Since the reduced-order model and the orthonormal
A DAPTIVE -O RDER R ATIONAL A RNOLDI (AORA) A LGORITHM
"
Algorithm 1: AORA (input: : /* Initialize */ for each do
#
TABLE I
matrix are not yet determined, the residue for each is set to . The normalization coefficient about each , , is initialized to be one. Step (2.1): Choose an expansion frequency such that gives the greatest difference between the th-order output moment of the original system and that of the reducedorder system . As presented in Theorem 2, max ¾ max ¾
$
The corresponding reduced-order model is yielded by using the congruence transformation matrix . can indeed match -order output moments of at . The scalar is obtained in Step (2.2) of the last iteration. The chosen expansion frequency in the th iteration is called . Step (2.2): After the chosen expansion point in the th iteration has been determined, the single-point Arnoldi method is applied at the expansion point . The new orthnormal vector is incorporated into the orthnormal matrix . The normalization coefficient if has been selected in the th iteration. Step (2.3): Determine the new residual at each expansion point . The calculation involves a projection with the new orthonormal matrix . The next vector at the frequency must be updated to enable further matching st iteration. Since no of the output moment in the improvement is obtained at the other unselected frequency , the vector at frequency in the current iteration remains , which was obtained in the preceding iteration. Step (3): Generate the real orthogonal matrix by using the reduced QR factorization if there exists any complex expansion points.
"
#
"
$
$
# #
end for : /* Begin AORA Iterations */ for do : /* Select the Expansion Frequency with the Maximum Output Moment Error*/ Choose as the giving max Set be the expansion frequency in the th iteration : /* Generate the Orthonormal Vector at */
: /* Update for the Next Iteration */ for each do if then else
end if
for do
$
end for end for end for
(3): /* Yield Real for Complex Expansion Points */ if there exists any such that is not real then real imag
qr end if
)
; output:
If the port driving-point impedance is concerned, we have , and . Substituting (6) into
, the state variables of the adjoint network and those of the original system have the following relationship:
(7) The adjoint network (or the dual system [6]) of the system Thus can also be calculated directly from .
(1) is represented as The following theorem further ensures that the moments of (5) at can be matched up to st-order by applying another congruence transformation matrix % . The system transfer function and its th-order system moment Theorem 3: Suppose that colspan for and , , and . is the orthonormal about are defined as respectively. matrix generated by the proposed AORA algorithm. Let The congruence transformation can be applied to construct % be the congruence transformation matrix , for a reduced system. If the matrix % is chosen as the congru- for model-order reductions, then ence transformation matrix such that and . colspan % for , $ $ , V. H YBRID A LGORITHM Now we are in the position to develop our proposed hybrid and , thenthe moment matching can also be for $ , algorithm. It is assumed the order of the reduced system is . preserved: The algorithm consists of two phases: and [6]. The computational cost of constructing the congruence ¯ In the first phase, the -th -order AORA algorithm will transformation matrix can be further reduced by exploring the be employed. Let the output be . MNA formulation. Let the signature matrix be defined as ¯ In the second phase, the adjoint network will be utilized. &' ( ( . The MNA formulation of the interconnect The congruence transformation matrix is % . has symmetric characteristics [6]: Since linear independence of columns in the Krylov sequence and (6) is generally lost gradually, numerical instability may occur IV. A DJOINT N ETWORKS F OR M ODEL R EDUCTIONS
1280
TABLE III
T1
T1
Vout
C OMPARISON STUDIES AMONG VARIOUS REDUCTION METHODS 6.75pF T2
T2
6.75pF T2
T1
6.75pF T2
T1 6.75pF
Projector Time (s) Avg Rel. Err. (%)
T1
6.75pF T2
T2
6.75pF
-
6.75pF 40ohm Vin
Fig. 1.
The mesh circuit.
S ELECTION SCHEME AT EACH EXPANSION FREQUENCY. Frequency ( Order ( ) Order ( ) 2 8 2 8 16
4 10 4 10 15 7 7 11 19 5 5 12 18 6 6 14 3 9 3 9 17 1 1 13 20
A mesh twelve-line circuit, presented in Fig. 1, is studied to demonstrate the efficiency of the proposed method. The line parameters of the horizontal lines are , , and those of the vertical lines are . Each line is 30mm long and divided into 20 sections. Therefore, , ,
) ,*++ -.*++ ) *++ -.*++ + (a)
0.025
0.02
Mag.
0.015
0.01
The Original System U=Vq U=V2q U=[Vq SVq] 0.5
1
1.5
2
2.5
3
3.5
4
5 9
x 10
(b) 0
Rel. Err. (%)
10
−5
10
−10
10
−15
10
0
0.5
1
1.5
2
2.5
Freq. (Hz)
3
3.5
4
9.23 4.04
,/
can be found in [4]. The frequency responses of the original model and the reduced-order models that are generated by , and (3) the following projectors: (1) ; (2)
, are compared in Fig. 2(a). Both the orthnormal matrices and are yielded by the AORA algorithm with 20 and 40 iteration numbers, respectively. Table II summarizes the order of moments to be matched at each expansion points of the AORA algorithm. The waveform of the original model and those of the reduced-order models generated by and are indistinguishable. The corresponding relative error is displayed in Fig. 2(b). The computational time to generate each reduced-order model and the average 1-norm relative error are summarized in Table III. It can be found that only 60 work is needed by using the adjoint network reduction method. VII. C ONCLUSION This paper presents a hybrid method for interconnect reductions. In the first phase, the adaptive-order rational Arnoldi method is utilized. The corresponding reduced-order model will yield the greatest improvement in output moments among all reduced-order models of the same order. In the second phase, the adjoint network technique is employed to further reduce the computational cost of constructing the congruence transformation matrix. Experimental results have demonstrated the accuracy and the efficiency of the proposed hybrid method. ACKNOWLEDGEMENT The authors would also like to thank the National Science Council, R.O.C., for financially supporting this research under Contract No. NSC92-2213-E-182-001.
%
%
%
R EFERENCES
4.5
Freq. (Hz)
%
VI. S IMULATION R ESULTS
0 0
%
TABLE II
0.005
15.94 1.22
0 Other expansion points has also be applied. Technical details
during the orthonormalization process. If only half of the orthonormalization iterations are performed, the procedure seems to be more numerically stable.
*++ ,*++
8.71 20.34
and . The frequency response between 0 and at the voltage is investigated and a total of 1001 frequency points distributed uniformly for simulations. Consider the frequency response of the current that leaves from the voltage source. Thus the circuit forms a one-port and the same system. Set the iteration number expansion points
6.75pF
T1
6.75pF
4.5
5 9
x 10
Fig. 2. (a) Frequency responses and (b) relative errors of the current leaving from the voltage source in the mesh circuit.
[1] M. Celik, O. Ocali, M. A. Tan, A. Atalar, Pole-zero computation in microwave circuits using multipoint Pad´e approximation, IEEE Trans. Circuits Syst. I-Fundam. Theor. Appl., 42 (1):6–13, 1995. [2] I. M. Elfadel, D. D. Ling, A block rational Arnoldi algorithm for multipoint passive model-order reduction of mutiport RLC networks, in: Proc. ICCAD, pp. 66–71, 1997. [3] K. Gallivan, E. J. Grimme, P. V. Dooren, A rational Lanczos algorithm for model reduction, Numer. Algorithms, 12:33–63, 1996. [4] H. J. Lee, C. C. Chu, and W. S. Feng. An adaptive-order rational arnoldi method for model-order reductions of linear time-invariant systems. Accepted for publication in Linear Algebrs and Its Aplications, Special Issue on Order Reduction of Large-Scale Systems, 2005. [5] A. Ruhe, Rational Krylov sequence methods for eigenvalue computation, Linear Alg. Appl., 58:391–405, 1984. [6] J. M. Wang, C. C. Chu, Q. Yu, E. S. Kuh, On projection-based algorithms for model-order reduction of interconnects, IEEE Trans. Circuits Syst. I-Fundam. Theor. Appl. 49 (11):1563–1585, 2002.
1281
Herng-Jer Lee
Wu-Shiung Feng and Ming-Hong Lai
Department of Electrical Engineering EverCAD Corp. Chang Gung University Hsin-Chu 300, Taiwan, R.O.C. Kwei-San, Tao-Yuan 333, Taiwan, R.O.C. Email: [email protected] Email: [email protected]
Abstract— This work proposes a hybrid method for interconnect model-order reductions. First, the Adaptive-Order Rational Arnoldi (AORA) method will be investigated. An extension of the classical multi-point Pad´e approximation by using the rational Arnoldi iteration approach will be studied. The adaptiveorder can be achieved by choosing the expansion frequency corresponding to the maximum output moment error. Secondly, the adjoint network technique will be studied. By exploring symmetric properties of the MNA formulation, the computational cost of constructing the congruence transformation matrix can be reduced by 50% compared with the conventional methods.
I. I NTRODUCTION Interconnect plays a significant role in the recent development of high-speed VLSI design. Due to the continuous increasing in component densities and clock rates, the signal integrity problems naturally arise in the interconnect structure. For efficient simulations, it is necessary to construct a low-order macro-model whose terminal behaviors essentially capture the complicated interactions. Several methods that are based on Pad´e synthesis have been applied to improve the model-order reduction techniques recently. Among all existing methods, the class of Krylovspace methods seems to be more accurate because it can avoid the ill-conditional problems. However, these conventional approximation methods tend to converge in a local fashion around a single frequency because Pad´e approximation is exact at the point while accuracy is lost away from it. To overcome this difficulty, multiple point Pad´e moment matching techniques have been proposed recently [1], [2], [6]. The straightforward way for the multi-point moment matching is to apply the Krylov subspace algorithm at various expansion frequencies. This is the so-called rational Krylov algorithm [3], [5]. This paper will further investigate the adaptive-order rational Arnoldi (AORA) method without determining the order of moments at each expansion frequency in advance. We will apply this technique to interconnect model order reductions [4]. In addition, the adjoint network method will be considered. By exploring symmetric properties of the interconnect MNA formulation, system moments of the adjoint network can be directly calculated from those of the original RLC network. Therefore, computational cost can be reduced significantly.
0-7803-8834-8/05/$20.00 ©2005 IEEE.
Department of Electronic Engineering Chang Gung University Email: [email protected] [email protected]
II. BACKGROUND In analyzing an RLC interconnect, the following modified nodal analysis (MNA) will be utilized: [6].
where
and
(1)
Matrices , , , and contain capacitances, inductances, conductances, and resistances, which are all symmetric positive definite. presents the incident matrix that satisfies Kirchhoff’s current law. The state variable includes node voltages and branch currents of inductors . In practical
simulations of large-scale circuits, all the above matrices are large and sparse. Since the computational cost for simulating a large circuit is indeed tremendously huge, model-order reduction techniques have been proposed recently to reduce the computational complexity [6]. The transfer functions of state variables and those of the and outputs are defined as , respectively. Given an expansion frequency , let matrix and matrix , where is assumed to be nonsingular. By at , we have taking the Taylor expansion of ½ , where is called the th-order system moment at . Similarly, the th-order output moment at is calculated as . The model-order reduction problem is to seek a -order system, where , such that
(2) where , ¢ , ¢ , and ¢ . The aim of moment matching is to establish a reduced-order system such that for , where is the order of moment matching.
One efficient way of obtaining a reduced-order system is to use the multi-point Pad´e approximation [1]. The multipoint Pad´e approximation requires that the output moment of the original system equals that of the reduced system, .
1278
However, this method usually yields ill-conditioned problem. Recent works have proposed Krylov subspace projection methods to avoid such numerical difficulties [6]. The reducedorder system is constructed by projecting an original largedimensional problem into a low-dimensional Krylov subspace. Given a square matrix ¢ and a vector , the -th Krylov sequence, defined as follows: is a sequence of column vector and the corresponding column space is called the th Krylov space. The Arnoldi algorithm can be applied to iteratively generate an orthonormal basis ¢ from the successive Krylov subspace span span . If we set and , it has been shown that the Krylov subspace is indeed spanned by for . The reduced-order can be constructed by using the orthogonal projection . In this case, the reduced system in Eq. (2) can be defined by the congruence transformation [6],
and (3)
The number of moments in the reduced system is exactly the number of moments in the original system at an expansion frequency , up to the order of . The multiple point Pad´e approximation can be achieved by the rational Arnoldi method. Let represent the set of predetermined expansion frequencies. be the set of the number of the Let matched moments at each corresponding frequency. The rational Arnoldi method will generatea reduced-order system , which matches -order ( ) moments of the , at the expansion points , . original system Implementing the rational Arnoldi method is equivalent to implementing the Arnoldi method times at expansion frequencies. That is, the first iterations correspond to the expansion frequency ; the next iterations are associated with , and so on. Each Arnoldi iteration generates orthonormal vectors. Then,
is the desired orthonormal matrix generated from a union Krylov space at various expansion points, as stated by
of expansion points for and the number of matched moments about each is by no means trivial, for simplicity, it is assumed that expansion points for are determined in advance. Suppose that the AORA algorithm have been performed iterations, where , the output moments of the original system and that of the reduced-order system are matched in the following sense: , for . After simple manipulations, the transfer function error can be represented as
(4) where is the th-order moment of at . The basic concept of the AORA method is to select an expansion point among all expansion points in in the )st iteration such that . The new ( expansion point and the new orthonormal vector
can be generated to achieve this additional order of the moment matching. Under this situation, the transfer function error in the st iteration corresponding to the expansion point can be expressed as
!
This means that the order of moment matching at the other remains the same as that in the th in the iteration for , iteration. That is, £ , and . The following theorem presents an exact formula for the output moment errors. for Theorem 2: [4] Suppose that and . The system matrices of reduced-order system are generated by the congruence transformation with the orthonormal matrix using the AORA algorithm, where . At each expansion point , the magnitude error between the th-order moments and Once the orthonormal matrix has been formed by ap- can be expressed: " # . plying the rational Arnoldi method, the reduced-order system where the normalized coefficient " # can be obtained using the congruence transformation. st-order reduced system with the In order to get the Theorem 1 : [4] Let be the orthonormal matrix generated greatest moment improvement, it is intuitive to choose the by the rational Arnoldi algorithm with iterations. Since expansion point such that the th-order moment of colspan for and , we have , and , , is maximal. As shown in Theorem 2, can be related to the residue vector # from . the th iteration. if has been selected in the th iteration, III. AORA A LGORITHM almost no additional computations of are needed. has been chosen, the residue Generally speaking, Pad´e based methods can not guaran- Once the expansion point tee to yield the reduced-order models with the best overall vector # can be normalized to be the new basis vector performances in the entire frequency domain. Only the local . The details of the AORA method includes the following approximation around the expansion point can be achieved. main steps: of This is also true for our AORA method. Table I outlines the Step (1): Initialize the first vector $ adaptive-order rational Arnoldi method. Since selecting a set the Krylov sequence for each expansion point , where
On the other hand, for the other expansion point not chosen as the new expansion point in the st iteration, the corresponding transfer function error is
1279
. Since the reduced-order model and the orthonormal
A DAPTIVE -O RDER R ATIONAL A RNOLDI (AORA) A LGORITHM
"
Algorithm 1: AORA (input: : /* Initialize */ for each do
#
TABLE I
matrix are not yet determined, the residue for each is set to . The normalization coefficient about each , , is initialized to be one. Step (2.1): Choose an expansion frequency such that gives the greatest difference between the th-order output moment of the original system and that of the reducedorder system . As presented in Theorem 2, max ¾ max ¾
$
The corresponding reduced-order model is yielded by using the congruence transformation matrix . can indeed match -order output moments of at . The scalar is obtained in Step (2.2) of the last iteration. The chosen expansion frequency in the th iteration is called . Step (2.2): After the chosen expansion point in the th iteration has been determined, the single-point Arnoldi method is applied at the expansion point . The new orthnormal vector is incorporated into the orthnormal matrix . The normalization coefficient if has been selected in the th iteration. Step (2.3): Determine the new residual at each expansion point . The calculation involves a projection with the new orthonormal matrix . The next vector at the frequency must be updated to enable further matching st iteration. Since no of the output moment in the improvement is obtained at the other unselected frequency , the vector at frequency in the current iteration remains , which was obtained in the preceding iteration. Step (3): Generate the real orthogonal matrix by using the reduced QR factorization if there exists any complex expansion points.
"
#
"
$
$
# #
end for : /* Begin AORA Iterations */ for do : /* Select the Expansion Frequency with the Maximum Output Moment Error*/ Choose as the giving max Set be the expansion frequency in the th iteration : /* Generate the Orthonormal Vector at */
: /* Update for the Next Iteration */ for each do if then else
end if
for do
$
end for end for end for
(3): /* Yield Real for Complex Expansion Points */ if there exists any such that is not real then real imag
qr end if
)
; output:
If the port driving-point impedance is concerned, we have , and . Substituting (6) into
, the state variables of the adjoint network and those of the original system have the following relationship:
(7) The adjoint network (or the dual system [6]) of the system Thus can also be calculated directly from .
(1) is represented as The following theorem further ensures that the moments of (5) at can be matched up to st-order by applying another congruence transformation matrix % . The system transfer function and its th-order system moment Theorem 3: Suppose that colspan for and , , and . is the orthonormal about are defined as respectively. matrix generated by the proposed AORA algorithm. Let The congruence transformation can be applied to construct % be the congruence transformation matrix , for a reduced system. If the matrix % is chosen as the congru- for model-order reductions, then ence transformation matrix such that and . colspan % for , $ $ , V. H YBRID A LGORITHM Now we are in the position to develop our proposed hybrid and , thenthe moment matching can also be for $ , algorithm. It is assumed the order of the reduced system is . preserved: The algorithm consists of two phases: and [6]. The computational cost of constructing the congruence ¯ In the first phase, the -th -order AORA algorithm will transformation matrix can be further reduced by exploring the be employed. Let the output be . MNA formulation. Let the signature matrix be defined as ¯ In the second phase, the adjoint network will be utilized. &' ( ( . The MNA formulation of the interconnect The congruence transformation matrix is % . has symmetric characteristics [6]: Since linear independence of columns in the Krylov sequence and (6) is generally lost gradually, numerical instability may occur IV. A DJOINT N ETWORKS F OR M ODEL R EDUCTIONS
1280
TABLE III
T1
T1
Vout
C OMPARISON STUDIES AMONG VARIOUS REDUCTION METHODS 6.75pF T2
T2
6.75pF T2
T1
6.75pF T2
T1 6.75pF
Projector Time (s) Avg Rel. Err. (%)
T1
6.75pF T2
T2
6.75pF
-
6.75pF 40ohm Vin
Fig. 1.
The mesh circuit.
S ELECTION SCHEME AT EACH EXPANSION FREQUENCY. Frequency ( Order ( ) Order ( ) 2 8 2 8 16
4 10 4 10 15 7 7 11 19 5 5 12 18 6 6 14 3 9 3 9 17 1 1 13 20
A mesh twelve-line circuit, presented in Fig. 1, is studied to demonstrate the efficiency of the proposed method. The line parameters of the horizontal lines are , , and those of the vertical lines are . Each line is 30mm long and divided into 20 sections. Therefore, , ,
) ,*++ -.*++ ) *++ -.*++ + (a)
0.025
0.02
Mag.
0.015
0.01
The Original System U=Vq U=V2q U=[Vq SVq] 0.5
1
1.5
2
2.5
3
3.5
4
5 9
x 10
(b) 0
Rel. Err. (%)
10
−5
10
−10
10
−15
10
0
0.5
1
1.5
2
2.5
Freq. (Hz)
3
3.5
4
9.23 4.04
,/
can be found in [4]. The frequency responses of the original model and the reduced-order models that are generated by , and (3) the following projectors: (1) ; (2)
, are compared in Fig. 2(a). Both the orthnormal matrices and are yielded by the AORA algorithm with 20 and 40 iteration numbers, respectively. Table II summarizes the order of moments to be matched at each expansion points of the AORA algorithm. The waveform of the original model and those of the reduced-order models generated by and are indistinguishable. The corresponding relative error is displayed in Fig. 2(b). The computational time to generate each reduced-order model and the average 1-norm relative error are summarized in Table III. It can be found that only 60 work is needed by using the adjoint network reduction method. VII. C ONCLUSION This paper presents a hybrid method for interconnect reductions. In the first phase, the adaptive-order rational Arnoldi method is utilized. The corresponding reduced-order model will yield the greatest improvement in output moments among all reduced-order models of the same order. In the second phase, the adjoint network technique is employed to further reduce the computational cost of constructing the congruence transformation matrix. Experimental results have demonstrated the accuracy and the efficiency of the proposed hybrid method. ACKNOWLEDGEMENT The authors would also like to thank the National Science Council, R.O.C., for financially supporting this research under Contract No. NSC92-2213-E-182-001.
%
%
%
R EFERENCES
4.5
Freq. (Hz)
%
VI. S IMULATION R ESULTS
0 0
%
TABLE II
0.005
15.94 1.22
0 Other expansion points has also be applied. Technical details
during the orthonormalization process. If only half of the orthonormalization iterations are performed, the procedure seems to be more numerically stable.
*++ ,*++
8.71 20.34
and . The frequency response between 0 and at the voltage is investigated and a total of 1001 frequency points distributed uniformly for simulations. Consider the frequency response of the current that leaves from the voltage source. Thus the circuit forms a one-port and the same system. Set the iteration number expansion points
6.75pF
T1
6.75pF
4.5
5 9
x 10
Fig. 2. (a) Frequency responses and (b) relative errors of the current leaving from the voltage source in the mesh circuit.
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