TWI287746B - Projection-based model reductions for efficient interconnect modeling and simulations using the Arnoldi algorithm - Google Patents

Abstract

Projection-based model reductions become a necessity for efficient interconnect modeling and simulations. In order to choose the order of the reduced system that can really reflect the essential dynamics of the original interconnect, the residual error of the transfer function can be considered as a stopping criteria to terminate the Arnoldi iteration process. Analytical expressions of this residual error are derived in detail. Furthermore, it can be found that the approximate transfer function can also be expressed as the original interconnect model with some additive perturbations. The perturbation matrix only involves resultant vectors at the previous step of the Arnoldi algorithm. These error information will provide a guideline for the order selection scheme used in the Krylov subspace model-order algorithm.

Description

1287746 * IX. Description of the invention: [Technical field to which the invention pertains] - The invention belongs to the technical field of a simplified design method of a circuit model, in particular, a fast and accurate simplified design of an interconnected line model, which can be applied to a nano product. Model simplification of body circuit interconnection line signal analysis. [Prior Art] Press, as CMOS semiconductor process technology advances to nanotechnology, the parasitic effects of interconnect lines cannot be ignored. And because the complexity of the circuit is increased, the order of the model is also increased. Therefore, an efficient model simplification method has become an indispensable technology for the modeling and simulation of interconnect lines, as shown in US Pat. No. 6,023,573, US Pat. No. 6,041,170, US 6,687,658; Several model simplification methods have been proposed in recent years. The conventional techniques include: Asymptotic Waveform Evaluation (AWE); by L·Τ·Pillage :^R, A.Rohrer, “Asymptotic waveform evaluation for timing Analysis, ', IEEE Trans on Computer-Aided Design of Integrated Circuits and Systems, Vol. 9, No. 4, pp. 352-366, proposed in 990. PVL method (Pade via Lanczos); by P. Feldmann And R. W. freund, "Efficient linear circuit analysis by Pad" e approximation via the Lanczos process,'' IEEE Trans, on Computer-Aided Design of Integrated Circuits and Systems, Vol. 14 pp· 639-649, 1 9 9 5 proposed. 6 1287746 Symmetric Pade via Lanczos; by P. Feldmann and RW· Freund, “The SyMPVL algorithm and its applications to interconnect simulation,” Proc· 1997 Int· Conf· on Simulation of Semiconductor Processes and Devices, pp 113-116, 1997.

The Arnoldi method, as shown in U.S. Patent No. 681 0506, PRIMA method (Passive reduced-order interconnect macromodeling); by A· Odabasioglu, M. Celik and L. T. Pileggi, PRIMA· passive reduced-order interconnect macromodeling a 1 gor i Thin, ?? IEEE Trans· on

Computer-Aided Design of Integrated Circuits and

Systems, Vol. π pp. 645-653, proposed in 1998. The basic principle of all the above proposed methods is the rylov subspace projection method; the Quebec subspace projection method uses projection, and the operator system projects the state variables of the original circuit system (simplified state) = simplified m The state variable of the system; the projection operator can use the quebec: the order of the simplified circuit in the construction of the singular method is the subdivision of the iteration. The projection model simplifies the other and simplifies the important wei The circuit must find the appropriate order so that the simplified circuit accurately reflects the important dynamic behavior of the original circuit. The remainder error of the transfer function can be used as a basis for the conventional it t 1 勹 and the generation process is stopped; techniques such as Bai use the PVL algorithm to roll out the original circuit and simplify the transfer function error of the 1287746 circuit by five (4); Ζ· Bai, R· D. Slone, W. Τ.

Smith and Q· Ye “Error bound for reduced system model by

Pad 6 approximation via the Lanczos process, IEEE Trans, on Computer-Aided Design of Integrated Circuits and Systems, Vol. 18 pp. 133-141, 1 999; however, its error representation involves the decomposition of the original circuit The complex operation of the matrix billion-岣^, so the practical application will become very difficult. Therefore, the model simplification method used in the past does have the above-mentioned deficiencies, and the creator has studied and researched the problems faced by the above-mentioned existing model simplification method by conducting research and development experience in related technical fields for many years. Actively looking for a solution, after much improvement, finally created a model simplification method that can actually improve the above-mentioned defects. SUMMARY OF THE INVENTION The main object of the present invention is to provide a method for simplifying the interconnection model of a nano integrated circuit by using the Arnoldi algorithm error estimation, which can avoid the complicated operation of the #transition function error and achieve fast The accurate interconnection model simplifies the design. - In order to achieve the above object, the present invention provides a method for simplifying the interconnection of a nano-integrated circuit by a heterogeneous error estimation, the steps of which include: (a) input mesh circuit fe ^ , (b ) knowing a set of frequency expansion points, (c) establishing the state space of the circuit ^ Ί matrix, (d) estimating the residual error into the wedge type simplification; ^ the straw is derived in detail from the original circuit system and the simple remainder The error, using the Arnoldi algorithm to get the road between the introduction road system, and the correlation between the remaining 8 1287746 and the original circuit system. • In addition, verifying the transfer function of the original circuit adds some perturbations, ie, the table does not simplify the transfer function of the circuit. The perturbation matrix is only related to the resultant vector of the Arnoldi algorithm. In this way, the derived error formula can efficiently provide the basis for simplifying the circuit order of the Quirrov subspace model. In the following, a preferred embodiment will be described, and other purposes and effects of the present invention will be further explained with reference to the drawings and the drawings. This will enable the reviewer to have a more detailed understanding of the present invention and to familiarize himself with the The skilled artisan can use the actual target, and the following description is only for explaining the preferred embodiment, and is not intended to limit the scope of the present invention, and therefore, based on the inventive spirit of the present invention, Modifications or modifications of the form are intended to be within the scope of the invention. [Embodiment] When analyzing the linear circuit of the RLC interconnect line in the ultra-large integrated circuit, the large _ uses the positive node analysis (M〇difie (j Nodal Analysis, MNA) technique according to the Ketchhoff voltage and current law. (]^1^}^〇;^, "magic, can; circuit represented as the following state space ($tate $ pace) matrix form: λ / Γ dx (t) IT ~ Nx 妁 + b just, y (t) =cTχ[ή, (1) where, and 〆. where 'matrix M includes the capacitance value and the inductance value, matrix iV contains the conductance value of the spear resistance value, the shape of the matrix ice) contains the node voltage of the inductor and the branch current, "(4) is The round signal, γ(1) represents the output signal. (2) 1287746 and, let 』= ΛΠΜ and 'the equation (1) can be expressed as: ^ ^ =x(0-rw(0 5 y^)-cTx{t) 5 The purpose of model order simplification is to find a simplified circuit system with less order and effective reflection of the original circuit system, and equation (3) represents a state space matrix of the simplified circuit: (3) where f(i)EW, rVeW and g<</2. and let Z(4)=L[x(〇] and Z〇) = L[qiu]] be the impulse response of the original system and the simplified system in the Laplace Domain.忑(4) and (4) can be expressed as follows: Z(s) = (In - sAyW ^ X(s) = (In - sAYlr · · · · ( 4 ) where /" is the unit matrix of "X/2, 々 9X (? unit matrix.

The transfer function ugly of the original system (4) and the transfer function slice of the simplified system (4) can be expressed as /f(4)(4) and 泠(4)=(5rf〇), respectively. The projection method uses a projection operator to project the state variable of the original circuit system to obtain a state variable of the simplified circuit system. This positive parent projection can be generated by iterations via the Quebec subspace algorithm. Among them, the Quebec subspace K#, r) is produced by the combination of the matrix J and r, and the expression of σ Wang 5 is as follows:

Kq(A,r)^span(r,Ar,--^Aq']r) Kuilovian subspace K#,r) modified by Arnoldi algorithm Gram-Schmidt orthogonalization After the iteration, a set of units of Orthonormal Basis is produced: 々. From a mathematical point of view, the following formula can be obtained after the g-time iteration using the Arnoldi algorithm: ΑΚ =^Α+ν,Λ+ι< .... (6) where ' is the upper Hersenberg ( Upper Hessenberg) matrix: •V 厶22 ... V 厶21 厶22 * * # * * * ^2q 0 厶32 ^33... : 0 ... 0 ^ JL and the following orthogonal vector satisfies the following relationship: °曰罝+ V〕+· · ·+V, Vi, state κ: The state variable obtained by the simplified system is orthogonal to the unit orthogonal matrix P of the previous step by the inward v generated by the original system. To the $~ is the unit matrix 々 the first line vector. From this point of view, the state variable has a correlation x(〇-Vqx(t) where, ice) is the state variable of the original Lisi··· dimension. The state variable of the dimension is added, and the simplified system "brings equation (7) into equation (2) has the following transformation relationship: ', after the house is derived, the equation (1) can be obtained: Kr, bva7 ··· (8) The dichotomy method has several advantages, including maintaining the original 11 1287746 糸, first motion difference and stability and passiveness. In order to estimate the original system (formula (2)) and the simplified system (form (3)) The error. First, define the remainder error (8):

Er(s)^(In^sA)X(s)-r where z〇) is an approximate solution of z〇). If Ge (4) = z (5), then (4) = 〇. Although the Arno 1 d 1 algorithm is implemented, the approximate solution of equation (4) is not (4) must belong to the Quebec subspace, in this case, especially (8) =, meaning (7). The following is a brief description of some of the properties of the ideal approximation solution. After the Arnoldi algorithm is subjected to ^ iteration operations, the orthogonal matrix & and the corresponding Upper Hessenberg matrix 式 in equation (6) are obtained. Then let ζ(4) be the approximate solution of ζ(4) and 夕(4) be the approximate solution after the iteration of the 9th Arnoldi algorithm, that is, the remainder (2) is the remainder error. The following narrative will be correct. U) If you do # (4), then the Galerkin condition will be established:

VlEr (s) = VqT{(In - SA)X(s) - r} = 〇· .....( g ) (b) If the Ga 1 erk in condition is true, the remainder error is (4) can be expressed as (1 0 ): (10)

Er(s) = -shq+lqvq^eTq{Iq -sHqy]r 12 1287746 · The principle is as follows: 'After g-order Arno 1 di algorithm iterative operation, the order of equation (8) can be obtained - (simplified )system. (a) Because especially (1) (for 〇, z(二) can be obtained by linear combination of 行 row vectors, ie, ge (8) = 匕 and (8), where & (4) is a linear combination of coefficients. Here we hope? Red sj. The remainder error A 〇) is expressed as follows: EXs)^(In-sA)VqXq(s)-r is multiplied by a matrix V; [(In^sA)VqXq{s)-r] =KiWs) - + - Λ • =(Iq-sHq)Xq(s)—f Here, the unit orthogonal property of the Arno 1 di algorithm is used. When &(4)(8), the Galerkin condition is established. (b) When the remainder error can be expressed as follows:

Er(sXIn-SA)Vq(Iq-sHXf-r =[Vq(Iq -sHq)- shq+lqvq+leTq ]{Iq - sHqylVqTr - r 13 1287746 where K~. After algebraic operation, the following formula is obtained For the principle description, the range of error can be derived from UI & assume that all eigenvalues (Eigenvalue) are single, early U1 mP 1 e) and the eigenvalue of % is decomposed into equation (10) can be simplified as:

Er(s) = -shq^VqyqSg(Iq ^sAqrs-lr a h

々+V ^elSqZ(s)Sq -.(12) •βλ. Since ζ(4) is a high-pass matrix, llz(ML is done on the left and right sides of equation (12) by 4 norm (N〇rm). 13) where D is the condition number of the matrix (C〇nditi〇n Number). The above error estimation is only related to Lu and Lu. Compared with the error expression proposed by the prior art, it is rarely considered for the proposed formula. Cost (C〇St). Because it can reflect the perturbation behavior of the formula, but the transit is very time consuming. Therefore, the present invention only considers the process as the process of selecting the number of layers 14 1287746. The invention is not directly used; Use μ, ||^| as an indicator to stop the iterative process. If it is small enough, the simplified system will be very similar to the original system. As shown in Annex 1, the Arnoldi algorithm error is proposed for the present invention. Estimate the virtual program for road model simplification, first give the initial value, then carry out the iteration of Arno 1 d 1 derivation, gradually increase the order of the simplified model, and each iteration will generate a new unit orthogonal Vector, and calculate the 'proposed by the present invention' For Hfcll. When the Bu is small enough, the Arnoldi algorithm will stop the iteration, and the number of iterations at this time is the best order for simplifying the model. The advantages of the invention will be verified by way of example in the simple case. Presenting the simulation results. Adding the disturbance to the system below will deduce the simplified system that the original system is equivalent to after adding the disturbance: Μ - Δ) dt ^ y,it)-cTx,(t).^ ...·· 14) The transfer function of the original circuit adds some perturbations to represent the = function _ after the approach, as shown in the equation ((4). Under the condition of equation (15), the disturbance system, ", the transfer function 孖 △ (4) is equal to The simplified transfer function of the system is private.), lq+l, qVq+ • · (15) 15 1287746 The principle is as follows: Assume = (8), the following formula can be derived: (In - s(A - Δ))-V = Vq (Iq - sAylf... • · Multiply the left and right sides of equation (16) by (κ^Δ)): r=h(A - A)) Vq(Iq-sA)4 s(VqHq ^hg+hqvqyq ) + sAVq](iq then multiply F/ and rearrange (·7, ' = /, - s(Hq + hq+liqVq\+leTq ) + sVrAy and get: ' q if Α is set to ', Λ+ 1ν [, then ζ △ (4) (4). [The present invention [Embodiment] The invention of a bee line of the verification algorithm uses a simple embodiment as a test to verify the correctness. As shown in the first figure, a ten-circuit model is proposed. The line parameters are as follows: :UQ/cm; Capacitance: 5〇pF/^Inductance 1.5nH/cm; Drive resistance: 3Ω, and load capacitance: l〇pF. Each road length is 30_ and is divided into 50 small sections. Therefore, the dimension of the unitary matrix is n=1198. This embodiment uses the following frequency region {〇, 12 GHz}. The example is used to determine the voltage 4 of the frequency response of the RLC network circuit. When the Arno 1 d 1 algorithm starts to execute, the values of * and h are sequentially produced. As shown in the second figure, summarize the entire simulation results. Can be observed, 1 16 1287746

The Arnoldi algorithm is superimposed on the second dimension ^ Q 1, ~ relatively small. Therefore, it is recommended to set the order of the simplified system to 3彳. ^ ^ ^ 1 ugly (4), ugly (8), and foot (8) represent the original

The transfer function of the circuit, the transfer function of the system after the order reduction by the A 丄Arn〇ldl method, and the transfer function of the disturbance system of the original circuit. %n I _ 一加 / ^ 明 See the third picture of the brother, the comparison of these two systems under different 7. It can be equal to the film (4). The invention deduces the residual difference between the wide, # ρ τ Γ π, Α n ^ , and 13 connected lines and the simplified system, and also derives the original private loop function to add some disturbances to express the trend. The transfer function in the near future. 1 一 _ 管牛二Λ 八 Eight middle disturbance matrix only played with Arnoldi ‘the result of a different method of 异 叠 向 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 姜 ό ό ό ό ό ό ό ό ό ό This remainder, the difference knows; {, the simplification of the tactical karyotype, the uterus, and the important basis of the selection of the order. To sum up the above mentioned, only used, the gentleman’s political basis

The value is U 5 brothers, the work A of the present invention limits the scope of the present invention, and the average variation of the present invention is not the same as the scope of the present invention. Without departing from the spirit and scope of the present invention, it should not be disregarded, and the same technology is not seen in the same technical field: / the implementation of the step-by-step implementation or the approximation of the technology, should be paid The requirements of the invention patent are to apply for a patent in accordance with the law. And kicking the hair to the hair [Simplified illustration] Attachment 1 is the virtual program of Arnoldi of the present invention. h Difference Estimation Road Model Simplification The first diagram is an example of a simple mesh circuit of the present invention. = is a comparison of the present invention 'I am in different iterations.' - The system is compared with the three systems of the present invention. The comparison of the following 1 j ® generation times β 17 1287746 [Main component symbol description

128774-6 ·

Algorithm: Arno丨di (input: A, r ; outi3Ut: Vq,q) (1) :/*Initialize*/ 4=factory; μι == ι; (2) : /* Start Amoldi iteration*/ while (^ > tolerance ) (2.1) : /*Generates a new unit orthogonal vector */- U|,' - , (2.2) :/* Updates the remainder error ' to the next iteration*/ V: 4 V for t = \X...,q do

End for q two q + Y; end while ^: divination v2 ··· vj;

Claims (1)

1287746 X. Patent application scope: • 1 · A method for simplifying the interconnection model of nano-integrated circuits by Arno 1 di algorithm error estimation, the steps of which include: (a) input mesh circuit; (b) Enter a set of frequency expansion points; (c) Establish a state space matrix of the circuit; (d) Estimate the remainder error for model simplification. 2. As described in the first paragraph of the patent application scope, the Arno 1 di algorithm error estimation and measurement method is used to simplify the model of the nano-integrated circuit interconnection circuit. The estimated residual error is modeled and simplified. An Arno 1 di algorithm with the Quebec subspace method of iteration; (b) Adding a perturbation system; (c) Remainder error 仏(4) to (4)—“estimation. 3. As described in item 2 of the patent application scope In the method of Arnoldi algorithm error estimation, the nano-integrated circuit interconnection circuit model is simplified. In the |, the remainder error is modeled and simplified, and it has the Quellov subspace... The algorithm can continuously construct a near-simplified circuit in an iterative manner; as long as the order of the simplified circuit is increased by one order, the newly formed inter-banding circuit only needs to be superimposed once; therefore, the computational complexity can be effectively It is lower than the traditional method of adopting the non-inversion method. The method of estimating the error of Arno 1 di algorithm described in item 2 of the patent scope is the method of simplifying the interconnection circuit model of nano-integrated circuits. Modeling the remainder difference The addition of the disturbance system adds some disturbances to the original electric and private mountain numbers to indicate the transfer function after approaching (8): 19 1287746 If δ=Λ+1λ+1<established, the transfer function of the disturbance system is good △(4) is equal to 忾4 The transfer function slice of the post-system (4). The method of simplifying the nano-integrated circuit interconnection model by the Arn〇ldi algorithm error estimation as described in the second item of the patent application scope, wherein The remainder error is model simplified, and the remaining error estimates are calculated using the unit orthogonal property of the Arnoldi algorithm. The remainder error can be expressed as follows: = (L, sA)Vq(Iq - sH/f - r -K {Iq -sHq)^ shq+hMTq ]{Iq « sHXV;r . r After algebraic operation, get Er(s) - -shq+hqvq+1eTq(Iq-sHqylS-qlr schema as shown in next page 20