CN100517339C - Transfer Function Recursion Method and Model Simplification of RLC Interconnection Line and Transmission Line Model - Google Patents

Abstract

The invention provides a group of strict, accurate and effective recursive methods for establishing transfer function models of RLC distributed interconnection lines and transmission lines in 2n order frequency domain. Wherein the RLC components may be uniformly distributed or of various values. Interconnects and transmission lines can be by themselves or with their source and load sections. The main features include its two recursive polynomials and its two recursive algorithms that interact in a loop, and the efficiency and accuracy of said recursive method. It further facilitates the practice of simulation methods and various model simplifications and their optimization. The present invention also further provides a simplified and optimized model method for uniformly distributed interconnection and transmission lines of uniform length order. The method has high precision in frequency domain, low computational complexity and low computational time consumption, and the model is stable and synthesizable.

Description

The transport function recurrence method and the model simplification thereof of RLC interconnection line and transmission line model

One. TECHNICAL FIELD OF THE INVENTION

[0001] the present invention relates to RLC interconnection line and transmission line, generate the recurrence method of the transfer function model of their frequency field fast and accurately, and the emulation of feature and evolution, and to the implementation method of its various model simplifications.

In order to narrate simplification, below " interconnection line and (perhaps) transmission line " abbreviated as " interconnection line ".

Two. the background technology of invention

[0002] current large scale integrated circuit has become bigger, has more, littler transistor.Along with the rapid raising of integrated level and speed, the interconnection line of integrated circuit has become a main limiting factor of large scale integrated circuit design performance today.The time delay of interconnection line has become the major part of current deep-submicron large scale integrated circuit time delay.Constantly meticulous along with technology, particularly improving constantly of chip speed, it is more serious that the influence of interconnection line time delay is just becoming.High speed and deep-submicron large scale integrated circuit Progress in technique require chip interconnecting line and packaging line distributed circuit modeling [" Applied Introductory Circuit Analysis for Electrical and Computer Engineering ", M.Reed and R.Rohrer, Prentice Hall, Upper Saddle River, NJ, USA, 1999].Finally cause the analysis of large-scale RLC and RC linear circuit.In the transmission line field, well-known transmission line should be used the distributed circuit modeling, also causes large-scale RLC and RC linear circuit on the other hand.And when chip speed and signaling rate improved fast, the inductance characteristic of interconnection line must be considered.

[0003] in circuit design, the modeling fast and accurately of interconnection line be necessary also be difficult.The emulation fast and accurately of circuit performance is important, particularly to VLSI (very large scale integrated circuit), up to a million circuit components is arranged on one of them chip.

[0004] increase of integrated system scale has caused the surge of interconnection line modeling complicacy.According to the demand of actual design, reasonably in the time circuit performance and feature are being assessed the exponent number that just must make great efforts to simplify the interconnection line circuit.

[0005] for the circuit of design complexity rightly, just needs the performance of accurate characterization interconnection line and the transition of signal.And in large scale integrated circuit interconnecting construction single line normally, tree or network.But one bar single line is the fundamental element of a tree and a network.Therefore be basic to the interconnection line characterisation process of a single line with important.

[0006] current model simplification has the whole bag of tricks, as Elmore time delay model, the time series analysis of progressive waveform appraisal (AWE), PVL (the Pad é with the Lanczos method is approximate), the decomposition in Klyrov space, based on the reduced-order model of Klyrov-Amoldi, BTM (balance method for cutting) and even length [cutting apart] exponent number (ELO) model.

[0007] still in order to obtain a good simplified model, the Model Simplification Method of nearly all state space all needs from an accurate state space high-order model, Klyrov space-wise for example, BTM, ELO, PVL and need interconnection line state space system matrix A and input matrix B based on the Arnoldi method.On the other hand, the Model Simplification Method by transport function also needs from above-mentioned accurate state-space model or accurate transfer function model in frequency field.Elmore method for example, AWE and ELO method.

[0008] the original accurate model of describing with state space equation and transport function is important, and this is not only the basis of the accurate starting point of various Model Simplification Method, and is the basis of the comparison of the approximate performance of the various Model Simplification Method of check.

[0009] notices that to simplify starting point be need very high computation complexity to current the whole bag of tricks in order to obtain an accurate state-space model, even disregard the computation complexity of model simplification technology itself.The RLC interconnection line can be described with the following matrix differential equation based on KCL (kirchhoff current law) or KVL (kirchhoff voltage law):

$\mathrm{Gx}\left(t\right)+C_{\mathrm{LC}}\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=\mathrm{bu}\left(t\right)---\left(1\right)$

Wherein G and C _LCBe parameter matrix, relevant for the resistance of interconnection line, electric capacity and inductance parameters, and line, the structure of tree and network, u (t) is an input source, x (t) is a node voltage, the vector that inductive current or node voltage derivative are formed.The state-space model of RLC interconnection line A, and B, C, D} is

$\stackrel{\·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right),$ y(t)＝Cx(t)+Du(t)， (2)

State variable x (t) ∈ R wherein ²ⁿ, input variable u (t) ∈ R, output variable y (t) ∈ R, exponent number 2n are the numbers of circuit (single line, tree or network) middle electric capacity and inductance.Clearly, obtain matrix A and matrix B the state-space model, essential compute matrix C from equation (1) _LCContrary and inverse matrix C _LC ^-1With the product of matrix G and vectorial b, perhaps corresponding matrix decomposition and multiplication.As everyone knows, only be that the computation complexity of matrix inversion is O (n ²) ~ O (n ³), depend on the structure and the inversion algorithms of matrix, and the computation complexity of n * n matrix product also is O (n usually ³).To the unusual matrix of high-order, because the ill-condition number of matrix, matrix inversion operation causes singularity problem, just produces another dyscalculia problem.To a distributed model, 2n should be big as much as possible, and exponent number can be up to thousands of in a typical macroreticular on the other hand.

[0010], gets a suitable little or medium sized exponent number and the even length of utilization usually and cut apart and have the original basis that method that parameter is proportional to its length is asked distribution RLC interconnection line for fear of this difficulty.But this has obviously brought suitable initial error.

[0011] conventional limited exponent number or limit number is the transient response that can not assess the node of underdamped RLC interconnection line rightly, and its need one very the model of high-order transient response is accurately described.But high-precision signal transient need to be estimated, is not only critical performance mode and network analysis for large scale integrated circuit, and is to accurately giving the potentially dangerous in the newspaper switch.The performance requirement that improves constantly forces the safety allowance that is reduced in the worst case design, also needs a more accurate time delay forecast.

Therefore [0012] definite original high-order model is very important, is not only the starting point as all Model Simplification Method, and is evaluation criteria to the model of having simplified as all.Wherein, the definite master pattern of interconnection line is important at all, because it is a basic structure of interconnection line, and is a tree and network structure that a fundamental element is used to constitute interconnection line.But because the huge exponent number of original interconnection line model, the aspect of an important difficulty is how to find a method reasonably and in computing time cheaply to try to achieve its master pattern at one.

[0013] when so that consider when uncertain, also need thorough careful interconnection line knowledge, just its precise analytic model during the robustness of the large scale integrated circuit performance of research interconnection line.

[0014] method of the linear model that seek to distribute is normally from the method for s-territory utilization Kirchhoff law and algebraic equation or from the method for the time domain utilization Kirchhoff law and the differential equation.But in various classic methods, this will run into certainly and calculate the contrary of unusual higher-dimension matrix number.For example 10 ⁶* 10 ⁶Matrix, thus wish to have the solution of a new state-space model and effectively the transport function recursive algorithm to RLC distribution interconnect line, in the hope of reducing computation complexity widely.And then develop based on the emulation on these models on strictness or quite high precision.

[0015] notices that some simple algorithm of seeking the interconnection line transport function exists.But these only are a kind of very coarse being similar to definite transport function, the coupling effect between for example not having to consider mutually.The example of simple two joint RC interconnection lines has illustrated this approximate error.Simple rough method is got the product of each single-unit transport function, causes a transport function to be

$T_{12}\left(s\right)=\frac{1}{1+R_1{C}_{1}s}\&CenterDot;\frac{1}{1+R_2{C}_{2}s}=\frac{1}{R_1{C}_1{R}_2{C}_2{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="C20051007826600191.png,C20051007826600201.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+(R_1{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826600161.png,C20051007826600162.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+R_2{C}_2)s+1}$

$=\frac{1/\left(R_1{C}_1{R}_2{C}_2\right)}{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="C20051007826600191.png,C20051007826600201.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+(\frac{1}{R_1{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826600161.png,C20051007826600162.png"\; state="\{\{state\}\}">C</figure-callout>}}_1}+\frac{1}{R_2{C}_2})s+\frac{1}{R_1{C}_1{R}_2{C}_2}}---\left(3\right)$

But correct transport function should be

$T_{12}\left(s\right)=\frac{1}{R_1{C}_1{R}_2{C}_2{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="C20051007826600191.png,C20051007826600201.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+(R_1{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826600161.png,C20051007826600162.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+R_2{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826600161.png,C20051007826600162.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+R_2{C}_2)s+1}$

$=\frac{1/\left(R_1{C}_1{R}_2{C}_2\right)}{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="C20051007826600191.png,C20051007826600201.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+(\frac{1}{R_1{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826600161.png,C20051007826600162.png"\; state="\{\{state\}\}">C</figure-callout>}}_1}+\frac{1}{R_2{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="C20051007826600191.png,C20051007826600201.png"\; state="\{\{state\}\}">C</figure-callout>}}_2}+\frac{1}{R_1{C}_2})s+\frac{1}{R_1{C}_1{R}_2{C}_2}}---\left(4\right)$

Difference between visible significantly equation (4) and the equation (3).

[0016] interconnection line recursive algorithm of the present invention comprises the recursion of reporting to the leadship after accomplishing a task of two variablees (s-polynomial expression).This method and algorithm have disclosed a very effective method definitely and have tried to achieve a definite RLC interconnection line transport function, both systematizations, low-down again computation complexity.Therefore, this fact means all one step of traditional single argument recursive algorithm or out of true, or is not efficient on computation complexity.

[0017] in a word, present various classic methods lack definite original high-order state-space model and the transport function that a kind of effective method is fast tried to achieve RLC distribution interconnect line.

Three. summary of the invention

[0018] by last finding, energy of demand accurately reflects RLC interconnection line circuit transfer function model and analytical approach and system with effective account form significantly.

[0019] fundamental purpose design of the present invention is for the RLC interconnection line provides a kind of systems approach, sets up the strict accurate transfer function model of frequency field with effective recurrence method.

[0020] the present invention and then the response of described accurate model as the frequency field of the RLC interconnection line of various model simplifications existing and that develop thus of assessment utilization or approximation method is provided.

[0021] the present invention and then a kind of method is provided, system and basis, the model simplification that a kind of naive model that uses above-mentioned strict precise analytic model and various Model Simplification Method to seek the RLC interconnection line is simplified or optimized.

[0022] the present invention and then provide above-mentioned systems approach with effective calculation.

What [0023] said system method provided by the invention had numerical evaluation stability and limit stability and physics can be comprehensive.

[0024] in brief, fundamental purpose of the present invention is that recurrence method by described frequency domain provides the strict accurate 2n rank model of RLC interconnection line and the simple algorithm of model simplification and optimization thereof.

[0025] in order to reach above-mentioned and other purpose, the present invention provides the calculating effective method, and computation complexity is low, is simple multiplication.The present invention guarantees the stability of lower-order model, and for classic method, this is a useful feature.

[0026] it is as follows to set up the system of original 2n rank model: the exponent number of distributed circuit as supposition be taken as 2n.So the RLC interconnection line has the n section as shown in Figure 1, i=1 ..., n, every section has a distributed resistance R _iWith distributed inductance L _iThe node and the distributed capacitance C that connect two adjacency _iFrom the node to ground, input end connects a source voltage v _In(t), so output terminal has a voltage v _o(t).Subscript is according to the order of sequence from the terminal to the input end, is different from general from the input end to the output terminal.Node i and node voltage v _i(t) also so number, i=1 ..., n.When the development recursive algorithm, this strong point of handling mode will show.The general interconnection line has a source resistance R _s, a pull-up resistor R ₀With a load capacitance C ₀, this moment, its source voltage was designated as v _In(t)=v _s(t).Claim that this is a circuit model 1, as shown in Figure 1.

[0027] consider circuit model 1, get state variable vector x (t), input variable u (t) and output variable y (t) are respectively

$x\left(t\right)={\left[\begin{array}{cccc}{v}^T& \left(t\right)& {\stackrel{\&CenterDot;}{v}}^T& \left(t\right)\end{array}\right]}^T,$ v(t)＝[v _n(t)，v _n-1(t)，…，v ₁(t)] ^T，u(t)＝v _in(t)，y(t)＝v _o(t)＝v ₁(t)，(5)

State variable x (t) ∈ R wherein ²ⁿ, input variable u (t) ∈ R, output variable y (t) ∈ R.The distribution interconnect line circuit of being considered, the state-space model of its distribution rlc circuit shown in Figure 1 A, and B, C, D} is

$\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right),$ y(t)＝Cx(t)+Du(t) (6)

A ∈ R wherein ^{2n * 2n}, B ∈ R ^{1 * 2n}, C ∈ R ^{2n * 1}And D ∈ R.

This is the state-space model of strictness of the 2n rank distribution interconnect line of Fig. 1 model 1, wherein n＞＞1 usually.

[0028] the another kind of common model of describing Circuits System is its transport function.It has set up the relation from input signal vector to the output signal vector in the S territory (frequency field).The transport function of the distribution RLC interconnection line circuit model 1 among Fig. 1 is to be derived by a following effective recurrence method of the present invention.

[0029] the distribution rlc circuit of consideration Fig. 1.So Fig. 1 circuit is from input v _In(t) to output v _o(t) transport function and recursive algorithm thereof are as follows:

$T_n\left(s\right)=\frac{N_n\left(s\right)}{D_n\left(s\right)}---\left(7\right)$

N _n(s)＝1 (8)

${\mathrm{\&Delta;}}_1\left(s\right)=(C_1+C_0)s+\frac{1}{R_0},$ $D_1\left(s\right)=s(L_{1}s+R_1)(C_1+C_0)+1+\frac{L_{1}s+R_1}{R_0}---\left(9\right)$

Δ _j(s)＝sC _jD _j-1(s)+Δ _j-1(s)，D _j(s)＝(L _js+R _j)Δ _j(s)+D _j-1(s)，j＝2，…，n-1(10)

Δ _n(s)＝sC _nD _n-1(s)+Δ _n-1(s)，D _n(s)＝(L _ns+R _n+R _s)Δ _n(s)+D _n-1(s)，n＞2(11)

Wherein 2n is the exponent number of circuit, and the branch submultinomial of transport function is N _n(s), its proper polynomial (denominator polynomial expression) is D _n(s).

[0030] so far, the transport function recursive algorithm of 1 voltage shines following steps from the source voltage signal to terminal node:

[0031] method TF1 (transfer function model 1)

(i) put the molecule of transport function as (8)

N _n(s)＝1； (12)

(ii) put the recursion initial value $D_1\left(s\right)=s(L_{1}s+R_1)({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826600161.png,C20051007826600162.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_0)+1+\frac{L_{1}s+R_1}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="C20051007826600172.png,C20051007826600191.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}$ With ${\mathrm{\&Delta;}}_1\left(s\right)=({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826600161.png,C20051007826600162.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_0)s+\frac{1}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="C20051007826600172.png,C20051007826600191.png"\; state="\{\{state\}\}">R</figure-callout>}}_0};---\left(13\right)$

(iii)For?j＝2；…，n-1

Δ _j(s)＝sC _jD _j-1(s)+Δ _j-1(s) (14)

D _j(s)＝(L _js+R _j)Δ _j(s)+D _j-1(s) (15)

end

(iv) put Δ _n(s)=sC _nD _N-1(s)+Δ _N-1(s) and D _n(s)=(L _nS+R _n+ R _s) Δ _n(s)+D _N-1(s) to n＞2; (16)

(v) put transport function T _n(s)=N _n(s)/D _n(s). (17)

[0032] model is the distribution interconnect line itself that does not have source and load effect among Fig. 2, can regard the special circumstances of model 1 among Fig. 1 as, by put source electricity value and load capacitance value be 0 and the pull-up resistor value be infinity,

R _s＝0，C ₀＝0，1/R ₀＝0. (18)

[0033] the another one special circumstances among Fig. 1 are among uniform interconnection line such as Fig. 3

R _i＝R，C _i＝C，L _i＝L，i＝1，…，n. (19)

The pass of the parasitic parameter of itself and interconnection line is

R＝R _t/n，C＝C _t/n，L＝L _t/n (20)

Dead resistance R wherein _t, stray capacitance C _tAnd stray inductance L _tBe " always " resistance of interconnection line, " always " electric capacity and " always " inductance." always " with the band quotation marks is because in fact this distributes here, is not total.

[0034] the another one special circumstances are that even interconnection line among Fig. 3 does not have source and load elements as shown in Figure 4.Therefore, this also is that Fig. 2 is that equally distributed special circumstances are as (19) and (20).

[0035] corresponding frequency domain and the time-domain analysis transport function that can easily pass through to lead is carried out bode and the step order of MATLAB.

[0036] considers the model simplification of interconnection line, propose the model of an above-mentioned derivation gained in easy comprehensive and attainable 2m rank here, have the performance index minimum of optimization model simplification that optimized parameter makes definition.This also often tends to the simplicity of equally distributed 2m rank model for simplified model.Above-mentioned transport function recursive algorithm is used to seek the parameter of optimum simplified model.Because simplified model is just like Fig. 1-4 same structure, so that the corresponding low order 2m rank RLC interconnection line among utilization Fig. 1-4 comes is comprehensive, have obvious can be comprehensive.

[0037] transport function of the present invention can be further used for the parasitic RLC interconnection line in 2m rank is done the proximate analysis of the simplified model of even length section.We claim that the parasitic model in these 2m rank is even length rank (ELO) simplified models.So method of the present invention and algorithm can be used for assessing the ELO simplified model.

[0038] a preferred mode among the present invention is any abnormal Qu Bianhua that the model simplification of RLC interconnection line itself does not comprise its source and loading section.Then this simplified model is connected to its actual source and loading section.So this simplified model is not subjected to the influence of each provenance and loading section and can be connected with it.

[0039] various forms of the present invention can comprise that the invention of arbitrary portion among the present invention and present arbitrarily relevant RLC interconnection line and the modeling of transmission line and the known method of analysis combine, perhaps and the combination of arbitrarily present known method combine.

Four. description of drawings

[0040] all accompanying drawing is as follows:

[0041] Fig. 1 shows any RLC interconnection line of the broad sense that has source resistance and pull-up resistor and electric capacity or transmission line (model 1)

[0042] Fig. 2 shows any broad sense RLC interconnection line or transmission line itself (model 2)

[0043] Fig. 3 shows an equally distributed RLC interconnection line or a transmission line (model 3) that has source resistance and pull-up resistor and electric capacity

[0044] Fig. 4 shows equally distributed RLC interconnection line or transmission line itself (model 4)

[0045] Fig. 5 shows the Bode figure of 200 rank master patterns of the RLC interconnection line example of Fig. 4 model 4.

[0046] Fig. 6 shows the step transient response of simplified model of 2 rank (n=1) ELO of Fig. 4 RLC interconnection line example, is tried to achieve by transport function.

[0047] Fig. 7 shows the Bode figure of simplified model of 2 rank (n=1) ELO of Fig. 4 RLC interconnection line example.

[0048] Fig. 8 shows the step transient response of simplified model of 4 rank (n=2) ELO of Fig. 4 RLC interconnection line example, is tried to achieve by transport function.

[0049] Fig. 9 shows the Bode figure of simplified model of 4 rank (n=2) ELO of Fig. 4 RLC interconnection line example.

[0050] Figure 10 shows the step response of ELO model on 20 rank (n=10) of Fig. 4 RLC interconnection line example, is tried to achieve by transport function.

[0051] Figure 11 shows the Bode figure of ELO model on 20 rank (n=10) of Fig. 4 RLC interconnection line example.

[0052] Figure 12 shows the Bode figure of Fig. 3 RLC interconnection line example.

Five. embodiment

[0053] of the present inventionly preferred embodiment will be described in detail at this.

[0054] A joint narration is calculated accurately in the method for RLC interconnection line from the transport function that is input to its outlet terminal.The B joint is discussed the method for trying to achieve state-space model from transfer function model.The C joint is discussed the model simplification of those interconnection lines.The D joint is set forth and is utilized the approximate of Performance Evaluation decision simplified model.The E joint is discussed the stability and the complexity characteristics of described method.Last F joint provides experimental result.

A. recursion calculation of transfer function

[0055] this section is to set forth how to use the top notion that is developed to come recursion calculating RLC interconnection line and transmission line from being input to the transport function of its outlet terminal.Consider general distribution RLC interconnection line and transmission line among Fig. 1-4.

A.1. model 1-band source and load

[0056] this section is showed the transport function T of the RLC interconnection line model 1 of using above-mentioned recursive algorithm (7)-(11) generation distribution _n(s) method, wherein 2n is the circuit exponent number.

[0057] model TF1-1:

(i) input data: n; R _i, i=1 ... n; C _i, i=1 ... n; L _i, i=1 ... n; R _s1/R ₀C ₀(21)

(ii) put new

R _n＝R _n+R _s，C ₁＝C ₁+C ₀； (22)

(iii) put the molecule of transport function

N _n(s)＝1； (23)

(iv) put the recursion initial value

$D_1\left(s\right)=s(L_{1}s+R_1){\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826600161.png,C20051007826600162.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+1+\frac{L_{1}s+R_1}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="C20051007826600172.png,C20051007826600191.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}$ With ${\mathrm{\&Delta;}}_1\left(s\right)=C_{1}s+\frac{1}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="C20051007826600172.png,C20051007826600191.png"\; state="\{\{state\}\}">R</figure-callout>}}_0};---\left(24\right)$

If (v) n=1, yet

T ₁(s)＝N ₁(s)/D ₁(s) (25)

Stop.

If next step is changeed in n＞1.

If (vi) n＞1, yet to j=2 ..., n

Δ _j(s)＝sC _jD _j-1(s)+Δ _j-1(s) (26)

D _j(s)＝(L _js+R _j)Δ _j(s)+D _j-1(s) (27)

End.

(vii) transfer function model is T _n(s)=N _n(s)/D _n(s). (28)

A.2. model 2-is not with source and load:

[0058] this section provides the transport function recursive algorithm of the RLC interconnection line of Fig. 2 model 2.A method is by above-mentioned recursive algorithm TF1-1.

[0059] model 2 of Fig. 2 can be regarded a special case of model 1 among Fig. 1 as, and putting source resistance and load capacitance is 0, and pull-up resistor is infinitely great, as going on foot at (i)

R _s＝0，C ₀＝0，1/R ₀＝0 (29)

Use recursive algorithm TF1-1 (21)-(28) then.Here it is recurrence method TF2-1.

[0060] another kind of method is with the step (i) in new simple steps (i) the replacement said method and (ii) as follows.

[0061] method TF2-2:

(i) input data: n; R _i, i=1 ... n; C _i, i=1 ... n; L _i, i=1 ... n; (30)

(ii) put the transport function molecule

N _n(s)＝1； (31)

(iii) put the recursion initial value

D ₁(s)=s (L ₁S+R ₁) C ₁+ 1 and Δ ₁(s)=C ₁S; (32)

If (iv) n=1, yet

T ₁(s)＝N ₁(s)/D ₁(s) (33)

Stop.

If next step is changeed in n＞1.

If (v) n＞1, yet to j=2 ..., n

Δ _j(s)＝sC _jD _j-1(s)+Δ _j-1(s) (34)

D _j(s)＝(L _js+R _j)Δ _j(s)+D _j-1(s) (35)

End.

(vi) produce transfer function model-T _n(s)=N _n(s)/D _n(s). (36)

A.3. model 3-evenly distributes, band source and load:

[0062] this section provides the even distribution RC interconnection line of model 3 among Fig. 3 and the transport function recurrence method of transmission tape source and load.

[0063] method TF3-1.Be TF1-1, but R wherein _i=R, C _i=C, and L _i=L, i=1 ..., n.

[0064] method TF3-2:

(i) input data: n, R, R _s, 1/R ₀, C ₀, C and L; (37)

(ii) put the transport function molecule

N _n(s)＝1； (38)

If (iii) n=1, yet

$T_1\left(s\right)=\frac{N_1\left(s\right)}{D_1\left(s\right)}=\frac{1}{D_1\left(s\right)},$ $D_1\left(s\right)=s(\mathrm{Ls}+R+R_s)(C+C_0)+1+\frac{\mathrm{Ls}+R+R_s}{R_0}---\left(39\right)$

Stop.

If next step is changeed in n＞1.

(iv) put the recursion initial value

$D_1\left(s\right)=s(\mathrm{Ls}+R)(C+C_0)+1+\frac{\mathrm{Ls}+\mathrm{<figure-callout\; id="0"\; label="R\; R"\; filenames="C20051007826600172.png,C20051007826600191.png"\; state="\{\{state\}\}">R</figure-callout>}}{R_{}}$ and ${\mathrm{\&Delta;}}_1\left(s\right)=(C+C_0)s+\frac{1}{R_0}---\left(40\right)$

If (v) n=2, change (if vii). n＞2, change next step.

If (vi) n＞2, yet to j=2 ..., n-1

Δ _j(s)＝sCD _j-1(s)+Δ _j-1(s) (41)

D _j(s)＝(Ls+R)Δ _j(s)+D _j-1(s) (42)

End

(vii) put Δ _n(s)=sCD _N-1(s)+Δ _N-1(s) (43)

(viii) put D _n(s)=(Ls+R+R _s) Δ _n(s)+D _N-1(s) (44)

(ix) produce transport function T _n(s)=N _n(s)/D _n(s). (45)

A.4. method 4-evenly distributes, and is not with source and load:

[0065] this section provides the production method of the transport function of the equally distributed RLC interconnection line of model 4 among Fig. 4 itself.This model is not subjected to the influence and the distortion of any source and load.The recurrence method of the transport function of this model of output is

T _n(s)＝N _n(s)/D _n(s)，N _n(s)＝1，Δ _j(s)＝sCD _j-1(s)+Δ _j-1(s)，D _j(s)＝(Ls+R)Δ _j(s)+D _j-1(s)，j＝2，…，n(46)

Have initial value

D ₁(s)=s (Ls+R) C+1 and Δ ₁(s)=Cs. (47)

[0066] some specific algorithms are as follows.

[0067] method TF4-1: application method TF1-1 puts R _s=0, C ₀=0,1/R ₀=0, R _i=R, L _i=L, C _i=C, i=1 ..., n.

[0068] method TF4-2: application method TF3-2 puts R _s=0, C ₀=0,1/R ₀=0.

[0069] method TF4-3:

(i) input data: n, R, L and C;

(ii) put the transport function molecule

N _n(s)＝1； (48)

(iii) put the recursion initial value

D ₁(s)=s (Ls+R) C+1 and Δ ₁(s)=Cs; (49)

If (iv) n＞1, yet to j=2 ..., n

Δ _j(s)＝sCD _j-1(s)+Δ _j-1(s) (50)

D _j(s)＝(Ls+R)Δ _j(s)+D _j-1(s) (51)

End

(v) produce transport function T _n(s)=N _n(s)/D _n(s). (52)

B. ask state-space model by transfer function model

[0070] this section is to set forth how to use the top notion that is developed to be asked the state-space model of RLC interconnection line by transfer function model.Consider general distribution RLC interconnection line and transmission line among Fig. 1-4.

[0071] tries to achieve by recurrence method owing to transfer function model, so can easily try to achieve interconnection line by the state-space model that is input to its output node { A, B, C, D} by MATLAB instruction tf2ss.Because transfer function model is accurate, so state-space model also is quite accurate.But the state-space model of trying to achieve thus is to provide a canonical form, and it does not have virgin state spatial model { A, B, C, the careful structure of the supplemental characteristic of the interconnection line that closed type reflected of D} of another invention of the inventor.

C. model simplification and approximate exponent number

[0072] illustrated the method for how trying to achieve strict transport function by above-mentioned recurrence method.But calculating these strict models is up to thousands of to the typical exponent number of distribution interconnect line greatly.In practice, there is no need to calculate the RLC interconnection line of high-order like this, because transient performance can enough lower-order models accurately characterize, for example, with minority leading pole (common tens limits).Basis and starting point that present above-mentioned accurate transfer function model that produces fast and state-space model provide model simplification or model to block and further compared.For example, balance intercept method (BTM) can apply to above-mentioned state-space model and do model simplification.The transport function of gained can be used for the Model Simplification Method by frequency field in addition, as AWE, and approximate and other methods of Pade.By the comparison to the master pattern performance, according to required approximate performance, for example precision and frequency range can determine the approximate exponent number of simplified model.

[0073] above-mentioned model is very effective for the relation that discloses between ELO simplified model and the original high-order model.Its method is as follows.The circuit of considering an equally distributed RLC interconnection line in 2n rank as shown in Figure 4, its total length resistance R _t, the total length inductance L _iAnd total length capacitor C _t, shown in (20).The ratio that the exponent number of its 2m rank ELO model is simplified is

r＝n/m. (53)

[0074] the 2m rank transport function of ELO model is

T _em(s)＝N _em(s)/D _em(s)，N _em(s)＝1， (54)

Try to achieve by recurrence method

Δ _e，j(s)＝srCD _e，j-1(s)+Δ _e，j-1(s)，D _e，j(s)＝r(Ls+R)Δ _e，j(s)+D _e，j-1(s)，j＝2，…，m，(55)

Its initial value is

D _e1(s)＝r ²s(Ls+R)C+1，Δ _e1(s)＝rCs. (56)

When m=1, r=n, its algorithm deteriorates to

N _e1(s)＝1，D _e1(s)＝n ²s(Ls+R)C+1，T _e1(s)＝1/[n ²LCs ²+n ²RCs+1].(57)

[0075] method of the present invention can apply to the circuit of the equally distributed RLC interconnection line of Fig. 3 one band source and load, and is as described below.

[0076] its transport function is T _Em(s)=N _Em(s)/D _Em(s), N _Em(s)=1, D _Em(s) try to achieve by following recurrence method and initial value

[0077] if initial value is put in m＞1

D _e，1(s)＝r ²Cs(Ls+R)(1+C ₀/rC)+1+r·(Ls+R)/R ₀，Δ _e，1(s)＝rC(1+C ₀/rC)s+1/R ₀(58)

To j=2 ..., m-1 carries out recursive algorithm

Δ _e，j(s)＝srCD _e，j-1(s)+Δ _e，j-1(s)，D _e，j(s)＝r(Ls+R)Δ _e，j(s)+D _e，j-1(s)(59)

The most rearmounted

Δ _e，m(s)＝srCD _e，m-1(s)+Δ _e，m-1(s)，D _em(s)＝r(Ls+R+R _s/r)Δ _em(s)+D _e，m-1(s)(60)

So get R _Em(s)=1/D _Em(s).

[0078] when m=1, r=n, transport function is

T _e1(s)＝1/D _e1(s)，D _e1(s)＝n ²Cs(Ls+R+R _s/n)(1+C ₀/nC)+1+n·(Ls+R+R _s/n)/R ₀(61)

[0079] the said method model that disclosed ELO band source and load depends on that the parameter of its distribution parameter and external parameter compares R/R _s, R/R ₀, C/C ₀, (R _t/ R _s, R _t/ R ₀, C _t/ C ₀) and exponent number simplify and to compare r.The extreme case that two kinds of external parameters are arranged: a kind of is not have external parameter promptly to have only interconnection line itself, does not contain any abnormal song, and another kind is to contain active big external parameter.Normal conditions be between between this two extreme case.But the simplified model to interconnection line itself can be used for connecting various external sources and load parameter.

D. determine transient response and Bode figure

[0080] and then, above-mentioned master pattern and simplified model can be used for determining and research Bode figure (frequency response) and transient response, i.e. their frequency domain performance and time domain performance.For example (n d) makes time domain step response, and (n d) makes frequency domain Bode figure to bode with some simple MATLAB instruction step.These performance map and data also can compare master pattern and its simplified model easily.

E. computation complexity and stability features

[0081] computation complexity of the RLC interconnection line transfer function model of the new method of foregoing invention is O ((n-1) ²) ≈ O (n ²), much smaller than the complexity O (n of classic method ^k), k＞2 or k＞3, n is the RLC joint number, exponent number is 2n.What need emphasize here is the number of times that said here computation complexity is based on multiplication and division.And classic method is sometimes only based on the number of times by the element pathway.That yes is linear to this.So the computation complexity here is stricter, and is more accurate.The O of transport function ((n-1) here ²) computation complexity is owing to adopted new recursive algorithm, it comprises simple multiplication.

[0082] still, to equally distributed RLC interconnection line, said recursive algorithm is simpler.And interconnection line and transmission line, tree and network usually are to be made of with time transmission line equally distributed interconnection line.So this neoteric method computation complexity is used to set or network will be a structurally associated with these trees and network.

[0083] these methods have caused the strict precise analytic model of 2n rank distribution RLC interconnection line and transmission line system.So, the stability of the model that these methods assurances are derived.And its numerical evaluation also is stable, and this is the model to any rank.These methods also can combine with the engineer's scale method and the other technologies of data.

[0084] the inventive method is the modeling that is effective in interconnection line distribution of impedance characteristic especially, because it so is easy to the recursive algorithm of transfer function model, adds its high precision.

F. experimental result

[0085] described transport function is very useful to frequency domain emulation and assessment, particularly makes frequency domain analysis and assessment Bode figure commonly used.If a system is described with state-space model, it will at first convert the transport function of frequency domain to and scheme in the hope of Bode.

[0086] described method will be used to calculate the transient response of step input of two even distribution RLC interconnection lines and the Bode figure of frequency response now.Example 1 is interconnection line and transmission line itself, and example 2 is interconnection tape source and load.Then, the step response of the master pattern of gained and Bode figure will compare respectively with the step response and the Bode figure of its ELO simplified model.

[0087] example 1: consider an equally distributed RLC interconnection line model 4,0.01cm is long, distribution characteristics data: resistance 5.5k Ω/m and electric capacity 94.2pF/m.It has R=5.510 to one 200 rank model as master pattern ^-3Ω and C=9.4210 ^-5PF, its inductance value is tried to achieve by the light velocity in the material and capacitance and is L=2.83110 ^-13H.

[0088] example 2: consider to be same as the even distribution RLC interconnection line in the example 1, but have a source resistance R _s=500 Ω, pull-up resistor R ₀=1M Ω and load capacitance C ₀=1pF as shown in Figure 3.Here, these external parameters are compared with distribution parameter R, and L and C dominate.

[0089] routine 1A: application process SS4 is in the model 4 of example 1, and the original transport function on its 200 rank can be tried to achieve by recurrence method.But, can use in conjunction with the engineer's scale method because distribution parameter is very little.

[0090] Fig. 5 has shown 200 rank master pattern Bode figure, tries to achieve from the transport function that described recurrence method obtains.

[0091] Bode figure both can try to achieve from above-mentioned state-space model, and perhaps the transport function that obtains from recurrence method is tried to achieve.But it is the most accurate from the Bode figure that the transport function of recursive algorithm is done.Master pattern has shown that when frequency is higher than certain limit when frequency increased, Bode figure had the decay of increase.But its Bode figure curve can not be followed this characteristic when simplified model was higher than a certain frequency when frequency, and we claim that this frequency is distortion (or separation) frequency of model approximation.But when asking time domain step response, state-space model most convenient and the most accurate.

[0092] routine 1B: experimental data comprises master pattern and 2,4 and 20 rank ELO simplified models in the example 1.The ELO simplified model promptly is that 2m rank model 4 has R, and L and C be its fragment length in proportion to.The ELO model is that the method with foregoing invention obtains.

[0093] Fig. 6 has shown its 2 rank (n=1) ELO model step response, is tried to achieve by described transfer function model.

[0094] Fig. 7 has shown the Bode figure of its 2 rank (n=1) ELO model, is tried to achieve by the transport function that described recurrence method obtains.

[0095] Fig. 8 has shown its 4 rank (n=2) ELO model step response, is tried to achieve by described transfer function model.

[0096] Fig. 9 has shown the Bode figure of its 4 rank (n=2) ELO model.

[0097] Figure 10 has shown its 20 rank (n=10) ELO model step response, is tried to achieve by described transfer function model.

[0098] Figure 11 has shown the Bode figure of its 20 rank ELO model.

[0099] these emulation have illustrated that the ELO simplified model of low order can not represent original equally distributed RLC interconnection line well.

[0100] clearly also very natural, the exponent number of ELO model is high more, and it is approximate good more.

[0101] routine 2A: use the model 3 of said method in example 2, its master pattern is 200 rank.

[0102] Figure 12 has shown this master pattern but C ₀=0 Bode figure, the transport function of trying to achieve from described recurrence method is so that show the special characteristic of some RLC interconnection lines.

This shows that [0103] in sum, method of the present invention is useful, and is stable, accurate, they are again easy on the other hand, and are simple, effectively, and consumption when having low computation complexity and low calculating.

Claims (11)

1. A recursive method of setting up the transfer function model of RLC interconnection line or transmission line, the method has the following steps: (a) Set the order of the transfer function model to an even number 2n; (b) Set the interconnection line or transmission line model as n nodes in series, with a source end and n nodes, wherein n nodes are a terminal and n-1 intermediate nodes, and each node has a resistance and an inductance are connected in series, and a capacitor is connected to the ground from the lower node of the section, the source voltage is taken as the input variable, and the terminal voltage is used as the output variable; (c) Assign the parameters of n sections: the values of resistors, inductors and capacitors; Its characteristics are: (d) Each node and its lower node are sequentially numbered i from the terminal to the source, i=1,...,n, and the corresponding resistance, inductance and capacitance parameters of each node are R _i , L _i and C _i ; (e) set transfer function numerator to 1; (f) Based on the first section parameters, set the initial value of the first recursive s-polynomial (coefficient) and the initial value of the second recursive s-polynomial (coefficient); (g) Execute n-1 recursive loops as follows: i) If n=1, stop the recursive loop, ii) If n>1, execute n-1 loops for the loop index j=2,...,n, in each loop, first update the first recursive s-polynomial, which is based on the j section parameters and The first recursive formula consisting of the current values of the second recursive s-polynomial is updated; the second recursive s-polynomial is then updated according to the parameters of section j and the current values of the first and second recursive polynomials The second recursive formula update of where the first recursive s-polynomial has just been updated; (h) setting the second recursive s-polynomial obtained through n-1 cycles as the denominator polynomial of the transfer function; Thus, the recursive method establishes the transfer function model of the RLC interconnection or transmission line, which can be used as the basis for simulation, model simplification, verification, performance analysis or circuit design. 2. The method according to claim 1, further comprising: a) mark the first recursive s-polynomial as Δ _j (s), and the second recursive s-polynomial as D _j (s); b) The initial value of the first recursive s-polynomial is Δ ₁ (s)=C ₁ s, and the initial value of the second recursive s-polynomial is D ₁ (s)=s(L ₁ s+R ₁ ) C ₁ +1; c) said first recursive formula is Δ _j (s)=sC _j D _j-1 (s)+Δ _j-1 (s); d) The second recursive formula is D _j (s)=(L _j s+R _j )Δ _j (s)+D _j-1 (s). 3. The method according to claim 1, further comprising: a) The said interconnection line or transmission line is uniform, that is, the parameters of each section are the same, which are a resistance R, an inductance L and a capacitance C, so R _i =R, L _i =L and C _i = C,i=1,...,n; b) mark the first recursive s-polynomial as Δ _j (s), and the second recursive s-polynomial as D _j (s); c) The said initial value is Δ ₁ (s)=Cs and D ₁ (s)=s(Ls+R)C+1; d) said first recursive formula is Δ _j (s)=sCD _j-1 (s)+Δ _j-1 (s); e) said second recursive formula is D _j (s)=(Ls+R)Δ _j (s)+D _j-1 (s); Thus, the recursive method establishes a transfer function model of a uniform RLC interconnection line or transmission line, providing the basis for simulation, model simplification, performance analysis, optimization, or circuit design. 4. The method according to claim 3, further comprising the steps of: (i) use the method for claim 3, replace n with m wherein, m＜n, set up the transfer function model of the 2m order of a low order; (ii) Set a simplified performance index of the optimal model to represent the deviation of the 2m-order model from the original 2n-order model; (iii) obtain the parameter in the 2m order model, be called optimal parameter, make the performance index that said optimal model simplifies is minimum; (iv) Establish a new 2m order transfer function model with the obtained optimal parameters; This method provides an optimized 2m order transfer function simplified model for establishing uniformly distributed RLC interconnection lines or transmission lines. 5. The method according to claim 1, further comprising the steps of: a) said interconnection or transmission line has a source resistance R _s , a load resistance R ₀ and a load capacitance C ₀ , wherein the source resistance is connected in series at the source terminal, and the load resistance and capacitance are connected in parallel between the terminal and ground; b) mark the first recursive s-polynomial as Δ _j (s), and the second recursive s-polynomial as D _j (s); c) The initial value of the first recursive s-polynomial is ${\mathrm{\&Delta;}}_1\left(\mathrm{the\; s}\right)=(C_1+C_0)\mathrm{the\; s}+\frac{1}{R_0},$ The initial value of the second recursive s-polynomial is ${\mathrm{D.}}_1\left(\mathrm{the\; s}\right)=\mathrm{the\; s}(L_1\mathrm{the\; s}+R_1)(C_1+C_0)+1+\frac{L_1\mathrm{the\; s}+R_1}{R_0};$ d) In the last cycle, i.e. j=n, update said second recursive formula to include source resistance R _s ; The transfer function model established by this method is suitable for an interconnection or transmission line with source resistance and load resistance and capacitance. 6. The method according to claim 5, further comprising: (i) said interconnection line or transmission line is uniform, that is, the parameters of each section are the same, which are a resistance R, an inductance L and a capacitance C, so R _i =R, L _i =L and C _i =C,i=1,...,n; (ii) The initial value of the first recursive s-polynomial is ${\mathrm{\&Delta;}}_1\left(\mathrm{the\; s}\right)=(C+C_0)\mathrm{the\; s}+\frac{1}{R_0},$ The initial value of the second recursive s-polynomial is ${\mathrm{D.}}_1\left(\mathrm{the\; s}\right)=\mathrm{the\; s}(\mathrm{ls}+R)(C+C_0)+1+\frac{\mathrm{ls}+R}{R_0};$ (iii) said first recursive formula degenerates into Δ _j (s)=sCD _j-1 (s)+Δ _j-1 (s), j=2,...,n; (iv) said second recursive formula degenerates into D _j (s)=(Ls+R)Δ _j (s)+D _j-1 (s), j=2,..., n-1, and D _n (s)=(Ls+R+ _Rs ) _Δn (s)+Dn _-1 (s); Thus, the recursive method builds a transfer function model for a uniform RLC interconnect or transmission line with source and load sections. 7. The method according to claim 5, further comprising the steps of: (i) use the method of claim 5 to replace n with m, m<n, and set up a low-order 2m-order transfer function model; (ii) Set a simplified performance index of the optimal model to represent the deviation of the 2m-order model from the original 2n-order model; (iii) obtain the parameter in the 2m order model, be called optimal parameter, make the performance index that said optimal model simplifies is minimum; (iv) Establish a new 2m order transfer function model with the obtained optimal parameters; Based on this, an optimized 2m-order reduced-order model is established for the RLC interconnection line or transmission line with source and load. 8. The method according to claim 1, further comprising the steps of: a) use the method of claim 1 to replace n with m, m<n, and set up a low-order 2m-order transfer function model; b) Set a simplified performance index of the optimal model to represent the deviation of the 2m-order model from the original 2n-order model; C) obtain the parameter in the 2m order model, be called optimal parameter, make the performance index that said optimal model simplifies is minimum; d) Establish a new 2m order transfer function model with the obtained optimal parameters; Based on this method, an optimized 2m-order reduced-order model is established for RLC interconnection lines or transmission lines. 9. The method of claim 1, further comprising: The assigned inductance and capacitance parameter values are scaled to facilitate simulation, analysis or design. 10. The method of claim 1, used in simulation, performance analysis, model reduction, or technical processing of circuit design. 11. The method according to claim 1, further comprising: being used in the manufacturing process of VLSI chip or transmission line, wherein the interconnection line or transmission line of VLSI chip is to use the method described in claim 1 to carry out performance analysis, model simplification, or designed.

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