CN100517338C - State space direct methods of RLC interconnect and transmission line model and model predigestion - Google Patents

Abstract

The invention provides a strictly accurate and efficient closed- form method used for establishing 2n order time domain state space model of RLC distributive interconnecting lines and transmission lines, wherein, the RLC component can be evenly distributed or different values; the interconnecting lines and the transmission lines can be themselves only, or can have a source and a load with the lines. The closed- form method is characterized in that the closed- form state space model (A, B, C, D) is of simplicity and accuracy, without matrix inversion or matrix decomposition and matrix multiplication. The computation complexity of the multiplication and division of the state space model is greatly reduced to O (n <2>). To the evenly distributed interconnecting lines and transmission lines, the computation complexity of the closed form of the state space model is only a fixed constant, that is, O (1), so as to facilitate the simplification and optimization of simulation methods and various models. The invention also further gives the method of the model simplification and optimization of the even length order of the evenly distributed interconnecting lines and the transmission lines. The method is of high accuracy, low computation complexity and low computation time consumption. In addition, the model is stable and integrative.

Description

The state space direct 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 state-space model of their time domain 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-Arnoldi, 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

$\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right),$ 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 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 chases after and makes 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 direct closed 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 exploitation is described transient response on strictness or quite high precision based on the emulation on these models.

[0015] notices the closed type that does not also have simple and accurate RLC interconnection line state-space model so far, so that try to achieve state-space model with this direct method.

[0016] 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

[0017] by last finding, energy of demand accurately reflects RLC interconnection line circuit model and the analytical approach and the system of the transient response of each node of master pattern with effective account form significantly.

[0018] fundamental purpose design of the present invention is for the RLC interconnection line provides a kind of systems approach, sets up the state-space model of strict accurate time domain with effective closed type.

[0019] the present invention and then the transient response of described accurate model as the RLC interconnection line of various model simplifications existing and that develop thus of assessment utilization or approximation method is provided.

[0020] of the present invention and then a kind of method is provided, a kind of naive model that the above-mentioned strict precise analytic model of system and basis utilization and various Model Simplification Method are sought the RLC interconnection line is simplified or the model simplification of optimization.

[0021] the present invention and then provide above-mentioned systems approach with effective calculation.

What [0022] said system method provided by the invention had numerical evaluation stability and limit stability and physics can be comprehensive.

[0023] in brief, fundamental purpose of the present invention is that state space closed type by described time domain provides the strict accurate 2n rank model of RLC interconnection line and the simple algorithm of model simplification and optimization thereof.

[0024] in order to reach above-mentioned and other purpose, the present invention provides the calculating effective method, and computation complexity is O (n ²) multiplication.For equally distributed RLC interconnection line, the computation complexity of the closed type of described state-space model only is O (l), only is a fixed constant.The present invention guarantees the stability of lower-order model, relative AWE or the like additive method, and this is a useful feature.

[0025] 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 developing the recursive algorithm of transport function afterwards, 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.

[0026] 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)，(3)

State variable x (t) ∈ R wherein ²ⁿ, input variable u (t) ∈ R, output variable y (t) ∈ R or be many outputs, 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) (4)

$A=\left[\begin{array}{cc}0& I\\ A_{21}& A_{22}\end{array}\right],---\left(5\right)$

$A_{21}=$

$B=\left[\begin{array}{c}0\\ B_1\end{array}\right],$ $B_1=\left[\begin{array}{c}\frac{1}{C_n{\mathrm{<figure-callout\; id="0"\; label="L\; n"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">L</figure-callout>}}_n}\\ 0\\ .\\ .\\ .\\ 0\end{array}\right],$ C＝[J?0]，J＝[0…?0?1]，D＝0 (8)

A ∈ R wherein ^{2n * 2n}, A ₂₁∈ R ^{N * n}, A ₂₂∈ R ^{N * n}, B ∈ R ^{1 * 2n}, B ₁∈ R ^{1 * n}, C ∈ R ^{2n * 1}, J ∈ R ^{N * 1}And D ∈ R.

This is the closed type of state-space model of strictness of the 2n rank distribution interconnect line of Fig. 1 model 1, wherein n＞＞1 usually.But this closed type is set up any n＞1.

[0027] to special circumstances n=1, above-mentioned model is degenerated, and will narrate at the aftermentioned chapters and sections.It is normal relevant with model simplification, and distribution interconnect line feature is characterized by very high 2n rank.

[0028] it is as follows to set up the method for state-space model:

[0029] method SS1 (state-space model 1):

(i) exponent number 2n. is set

State matrix A (ii) is set as (5)

A ₁₁=0, n * n null matrix; (9)

A ₁₂=I, n * n unit matrix; (10)

A ₂₁As (6), (11)

A ₂₂As (7), put (12) then

$A=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]---\left(13\right)$

(iii) put input matrix B, 2n * 1 column matrix is as (8)

(a) b _i=0, i=1 ... .2n, (b) b then _N+1=1/C _nL _n(14)

(iv) putting output matrix C is that 1 * 2n row matrix is as (8)

(a) c _i=0, i=1 ..., 2n, (b) c then _j=1, j ∈ I[1, n]; Common j=n, (15)

Wherein n-j+1 node is selected output node.

(v) put direct output scalar matrix D=0.

So (vi) the state-space model of model 1 A, B, C, D} sets up, stops.

[0030] 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 tried to achieve by the instruction ss2tf of state-space model of the present invention by MATLAB.

[0031] 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 be 0 and pull-up resistor be infinity,

R _s=0, C ₀=0, and 1/R ₀=0. (16)

[0032] the another one special circumstances among Fig. 1 are among uniform interconnection line such as Fig. 3

R _i=R, C _i=C and L _i=L, i=1 ..., n. (17)

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

R=R _t/ n, C=C _t/ n and L=L _t/ n (18)

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.

[0033] 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 (17) and (18).

[0034] the corresponding time and frequency domain analysis state space closed type that can easily pass through to lead is carried out step and the bode order of MATLAB.

[0035] 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 state space closed type 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.

[0036] closed type 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.

[0037] 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.

[0038] 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

[0039] all accompanying drawing is as follows:

[0040] 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)

[0041] Fig. 2 shows any broad sense RLC interconnection line or transmission line itself (model 2)

[0042] 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

[0043] Fig. 4 shows equally distributed RLC interconnection line or transmission line itself (model 4)

[0044] Fig. 5 shows the step transient response of 200 rank master patterns of the RLC interconnection line example A of Fig. 4 model 4.

[0045] Fig. 6 shows the Bode figure of 200 rank master patterns of Fig. 4 RLC interconnection line example, is tried to achieve by state-space model.

[0046] Fig. 7 shows the step transient response of simplified model of 2 rank (n=1) ELO (evenly length rank) of Fig. 4 RLC interconnection line example.

[0047] Fig. 8 shows the Bode figure of simplified model of 2 rank (n=1) ELO of Fig. 4 RLC interconnection line example, is tried to achieve by state-space model.

[0048] Fig. 9 shows the step transient response of ELO model on 4 rank (n=2) of Fig. 4 RLC interconnection line example.

[0049] Figure 10 shows the Bode figure of ELO model on 4 rank (n=2) of Fig. 4 RLC interconnection line example, is tried to achieve by state-space model.

[0050] Figure 11 shows the step transient response of ELO model on 20 rank (n=10) of Fig. 4 RLC interconnection line example.

[0051] Figure 12 shows the Bode figure of ELO model on 20 rank (n=10) of Fig. 4 RLC interconnection line example, is tried to achieve by state-space model.

[0052] Figure 13 shows the step transient response of Fig. 3 RLC interconnection line example.

[0053] Figure 14 shows the Bode figure of Fig. 3 RLC interconnection line example, is tried to achieve by state-space model.

[0054] Figure 15 shows the step transient response of the BTM model on Fig. 3 RLC interconnection line example master pattern and 1 rank and 2 rank.

[0055] Figure 16 shows the Bode figure of the BTM model on Fig. 3 RLC interconnection line example master pattern and 1 rank and 10 rank.

Five. embodiment

[0056] of the present inventionly preferred embodiment will be described in detail at this.

The method closed type as described of the state-space model parameter of RLC interconnection line is sought in the narration of [0057] A joint.The B joint is discussed and is tried to achieve transfer function model from state-space 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. the direct calculating of state-space model

A.1. model 1-band source and load:

[0058] preferred mode is that model 1 can have following manner to produce by closed type as (5)-(8) with the state-space model (4) that sketch the front as shown in Figure 1:

$A=\left[\begin{array}{cc}0& I\\ A_{21}& A_{22}\end{array}\right],---\left(19\right)$

$B=\left[\begin{array}{c}0\\ {\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">B</figure-callout>}}_1\end{array}\right],$ ${\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">B</figure-callout>}}_1={\left[\begin{array}{cccc}\frac{1}{C_{\mathrm{<figure-callout\; id="0"\; label="n\; L\; n"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">n</figure-callout>}}{L}_n{}_{}}& 0& ...& 0\end{array}\right]}^T,$ C＝[J?0]，J＝[0…0?1]，D＝0 (22)

[0059] subsystem matrix A here ₂₁3 diagonal entries are arranged: go up diagonal line, diagonal line, following diagonal line; And last column vector.Its i is capable, i=2 ..., n-2 has Elements C _N-i+1, L _N-i+1And L _N-i+1, and the going and equal 0 of its 3 diagonal entries, i=2 ..., n-2.Its n-1 is capable and equal

Its last column vector element all has pull-up resistor R ₀The 1st row has Elements C _n, L _n, L _N-1, R _n, R _N-1And source resistance R _sIts last column has Elements C ₁, L ₁, R ₁With pull-up resistor R ₀And capacitor C ₀, it is gone and is

Its diagonal entry is born, for

$a_{21}^{\mathrm{ii}}=-\frac{1}{C_{n-i+1}}(\frac{1}{L_{n-\mathrm{<figure-callout\; id="1"\; label="i\; +"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">i</figure-callout>}+1}}+\frac{1}{L_{n-1}}),$ i＝1，…，n-1， $a_{21}^{\mathrm{nn}}=-\frac{1}{({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_0)L_1}(\frac{R_1}{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}+1),$ n≥2 (23)

Diagonal entry is positive on it, for

$a_{21}^{i,i+1}=\frac{1}{C_{n-i+1}{L}_{n-i}},$ i＝1，…，n-2，n＞2， $a_{21}^{n-1,n}=\frac{1}{C_2{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">L</figure-callout>}}_1}+\frac{1}{R_0{C}_2}(\frac{R_1}{L_1}-\frac{R_2}{L_2})---\left(24\right)$

Diagonal entry is positive under it, for

$a_{21}^{i,i-1}=\frac{1}{C_{n-i+1}{L}_{n-i+1}},$ I=2 ..., n-1 and $a_{21}^{n,n-1}=\frac{1}{({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_0){\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">L</figure-callout>}}_1},$ n≥2. (25)

Matrix A ₂₁Every other element all be 0.

[0060] subsystem matrix A ₂₂It is upper triangular matrix.Its element relates to resistance R _iSame with it section inductance L _iRatio value, a capacitor C _iWith a capacitor C _jRatio value.Its last column has element L ₁, R ₁, C ₁Pull-up resistor R ₀And capacitor C ₀The 1st row has element R _n+ R _s, L _nAnd L _N-1Its i is capable element R _N-i+1, R _N-i, L _N-i+1, L _N-iAnd C _N-i+1For i=1 ..., n-1.Its i shows Elements C _N-i+1, i=2 ..., n.These features have reflected the structure of distribution interconnect line and the relation of its element subscript sequence.

[0061] input matrix B only has a non-element in addition at (n+1) OK:

b _n+1＝1/C _nL _n， (26)

The row of output matrix C has only a non-element 1 in addition:

c _n＝1， (27)

Directly output matrix D is 0.So the computation complexity of this state-space model only is O (n ²) multiplication.

[0062] voltage of i node if desired just makes output matrix C that its i bit element is 1, and all the other elements are 0, C=[0 ... 1 ... 0], and its system matrix A, input matrix B and direct output matrix D are all constant separately in (19)-(22).So the voltage of any node can be represented and detect to this state-space model, only need to adjust its corresponding output matrix C.

[0063] equation (19)-(22) are closed types of the state-space model of the strictness of 2n rank distribution interconnect line model 1 among Fig. 1, common here n＞＞1.But this closed type all is effective to any n＞1.Its usually corresponding method and algorithm are shown in the method SS1 of an as above chapter.

[0064] work as n=1, above-mentioned model deteriorates to:

$A_{21}=-\frac{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">R</figure-callout>}}_0+{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">R</figure-callout>}}_1+R_s}{({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_0)L_1{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">R</figure-callout>}}_0}=-\frac{1}{C_1{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">L</figure-callout>}}_1}\&CenterDot;\frac{1+({\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">R</figure-callout>}}_1/R_0)+(R_s/R_0)}{1+{\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">C</figure-callout>}}_0/{\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">C</figure-callout>}}_1},$ $A_{22}=-\frac{{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">R</figure-callout>}}_1+R_s}{L_1}-\frac{1}{({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_0)R_0}---\left(28\right)$

$B={\left[\begin{array}{cc}0& \frac{1}{({\mathrm{<figure-callout\; id="1"\; label="C"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">C</figure-callout>}}_1+C_0)L_1}\end{array}\right]}^T,$ C＝[1?0] (29)

This usually relates to the simplified model of exponent number 2, and the feature of RLC distribution interconnect line is to characterize on very high 2n rank.

A.2. model 2-is not with source and loading section:

[0065] this section provides the closed type method of the state-space model that produces model 2.Model 2 shown in Figure 2 is an interconnection line itself, does not have influence and perturbation (distortion) from various not homologies and load.This situation is very important, does not have the distortion of loaded impedance and source impedance because it has described the propagation delay feature of a distribution RLC interconnection line.The closed type of its state-space model is as follows:

$A=\left[\begin{array}{cc}0& I\\ A_{21}& A_{22}\end{array}\right]---\left(30\right)$

$B={\left[\begin{array}{cccccccc}& & & .& & & & \\ 0& ...& 0& .& 1/\left(C_n{L}_n\right)& 0& ...& 0\\ & & & .& & & & \end{array}\right]}^T,$ $C=\left[\begin{array}{cccccccc}& & & & .& & & \\ 0& ...& 0& 1& .& 0& ...& 0\\ & & & & .& & & \end{array}\right],$ D＝0，n≥1. (33)

[0066] when n=1, its state-space model is

$A_{21}=-\frac{1}{C_1{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">L</figure-callout>}}_1},$ $A_{22}=-\frac{R_1}{{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">L</figure-callout>}}_1},$ $B={\left[\begin{array}{cc}0& \frac{1}{C_1{L}_1}\end{array}\right]}^T,$ ${\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">B</figure-callout>}}_1=\frac{1}{C_1{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="C20051007826400231.png,C20051007826400232.png"\; state="\{\{state\}\}">L</figure-callout>}}_1},$ C＝[1?0]，D＝0. (34)

This shows that [0067] closed type formula (30)-(33) also are effectively to n=1, if get matrix A ₂₁And A ₂₂Lower right corner unit, B matrix neutron matrix B ₁Top unit, first right side unit is as the depression of order form in the C matrix.

[0068] similar in appearance to aforesaid method, a method and a computational algorithm of trying to achieve model is as follows:

[0069] method SS2:

I) put exponent number 2n.

Ii) put matrix A as (30),

A ₁₁=0, n * n null matrix; (35)

A ₁₂=I, n * n unit matrix; (36)

A ₂₁As (31), (37)

A ₂₂As (32), yet put (38)

$A=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]---\left(39\right)$

Iii) put input matrix B, 2n * 1 column matrix is as (33)

(a) b _i=0, i=1 ..., 2n, yet (b) $b_{n+1}=\frac{1}{C_n{L}_n}---\left(40\right)$

Iv) put output matrix C, 1 * 2n row matrix as

(a) c _i=0, i=1 ..., 2n, yet (b) c _j=1, j ∈ I[1, n]; (j=n usually), (41)

In order to select node n-j+1 as output node;

V) put direct output matrix D=0;

Vi) stop

[0070] just state-space model (A, B, C, D} have been set up by closed type (30)-(33) for model 2.

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

[0071] this section provides the method for the strict state-space model that the uniform RLC interconnection line of the model of setting up among Fig. 33 has source and load as follows:

$A=\left[\begin{array}{cc}0& I\\ A_{21}& A_{22}\end{array}\right]---\left(42\right)$

$B={\left[\begin{array}{cc}0& B_1\end{array}\right]}^T={\left[\begin{array}{cccccccc}0& 0& ...& 0& \frac{1}{\mathrm{<figure-callout\; id="0"\; label="CL"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">CL</figure-callout>}}& 0& ...& 0\end{array}\right]}^T,$ $C=\left[\begin{array}{cccccccc}& & & & .& & & \\ 0& ...& 0& 1& .& 0& ...& 0\\ & & & & .& & & \end{array}\right],$ D＝0，n＞1. (45)

[0072] to special circumstances n=1, above-mentioned model deteriorates to

$A_{21}=-\frac{{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">R</figure-callout>}}_0+R+R_s}{(C+C_0){\mathrm{<figure-callout\; id="0"\; label="LR"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">LR</figure-callout>}}_0}=-\frac{1}{\mathrm{CL}}\&CenterDot;\frac{1+(R/R_0)+(R_s/R_0)}{1+{\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">C</figure-callout>}}_0/C},$ $A_{22}=-\frac{R(1+R_s/R)}{L}-\frac{1}{C(1+{\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">C</figure-callout>}}_0/C)R_0}---\left(46\right)$

$B={\left[\begin{array}{cc}0& \frac{1}{\mathrm{CL}(1+{\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">C</figure-callout>}}_0/C)}\end{array}\right]}^T,$ C＝[1?0]，D＝0. (47)

[0073] closed type has shown that source and the influence that loads in the model 3 are the passing ratio factor R _s/ R, R/R ₀And C ₀/ C.When these scale factors very hour, they can be ignored respectively.An extreme special case of model 3 is models 4, and it leaves out all these factors in the closed type.

[0074] should be understood that with ben be multiplication and the division that above-mentioned closed type only comprises fixing very limited number of times, to any high exponent number n (n＞＞1).Its computation complexity be fix less than O (n), only be O (1)!

[0075] obtaining the method for model and the algorithm of calculating is similar in appearance to aforesaid as follows.

[0076] method SS3:

I) put exponent number 2n.

Ii) put system matrix A as (42)

As n=1, put

A ₂₁As (46), (48)

A ₂₂As (46), yet put (49)

$A=\left[\begin{array}{cc}0& 1\\ A_{21}& A_{22}\end{array}\right];$ Put (50)

$B={\left[\begin{array}{cc}0& \frac{1}{\mathrm{CL}(1+{\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">C</figure-callout>}}_0/C)}\end{array}\right]}^T,$ C=[1 0] and D=0 (51)

Stop.

If the step is down changeed in n＞1.

Iii) put

A ₁₁=0, n * n null matrix, (52)

A ₁₂=I, n * n unit matrix, (53)

A ₂₁As (43), (54)

A ₂₂As (44), yet put (55)

$A=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]---\left(56\right)$

Iv) put input matrix B, 2n * 1 column vector is as (45)

(a) b _i=0, i=1 ..., 2n, yet (b) b _N+1=1/CL (57)

V) put output matrix C, the capable vector of 1 * 2n as

(a) c _i=0, i=1 ..., 2n, yet (b) c _j=1, j ∈ I[1, n]; (j=n usually), (58)

In order to select node n-j+1 as output node.

Vi) put direct output matrix D=0.

Vii) stop.

[0077] so model 3 has been set up the model 3 of state space by { A, B, C, D } above-mentioned closed type (42)-(45).

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

[0078] this section provides the method for the state-space model that produces equally distributed interconnection line itself.Fig. 4 has provided the circuit of model 4.It does not have any distortion of each provenance and load.The closed type of its state-space model is as follows.

$A=\left[\begin{array}{cc}0& I\\ A_{21}& A_{22}\end{array}\right]---\left(59\right)$

$A_{22}=-\frac{R}{L}\&CenterDot;I_{n\&times;n}---\left(61\right)$

$B={\left[\begin{array}{cc}0& B_1\end{array}\right]}^T={\left[\begin{array}{cccccccc}0& 0& ...& 0& \frac{1}{\mathrm{<figure-callout\; id="0"\; label="CL"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">CL</figure-callout>}}& 0& ...& 0\end{array}\right]}^T,$ $C=\left[\begin{array}{cccccccc}& & & & .& & & \\ 0& ...& 0& 1& .& 0& ...& 0\\ & & & & .& & & \end{array}\right],$ D＝0，n≥1. (62)

[0079] when the special circumstances of n=1, above-mentioned model deteriorates to

$A_{21}=-\frac{1}{\mathrm{CL}},$ $A_{22}=-\frac{R}{L},$ $B={\left[\begin{array}{cc}0& \frac{1}{\mathrm{CL}}\end{array}\right]}^T,$ C＝[1?0]，D＝0. (63)

This shows that closed type formula (59)-(62) also are effective to n=1.

[0080] should point out emphatically above-mentioned closed type only need 2 multiplication and 2 divisions to no matter arbitrarily big n (n＞＞1).This shows that computation complexity is a constant 4, just O (l)!

[0081] method that produces even interconnection line and transmission line model and algorithm are to above-mentioned similar as follows.

[0082] method SS4:

I) put exponent number 2n.

Ii) put system matrix A as (59)

If n=1 puts

A ₂₁=-1/CL is as (63), (64)

A ₂₂=-R/L is as (63), yet puts (65)

$A=\left[\begin{array}{cc}0& 1\\ A_{21}& A_{22}\end{array}\right],$ Put (66)

B=[0 1/CL] ^T, C=[1 0], D=0 is as (63) (67)

Stop.

If next step is changeed in n＞1.

Iii) put

A ₁₁=0, n * n null matrix; (68)

A ₁₂=1, n * n unit matrix; (69)

A ₂₁As (60), (70)

A ₂₂As (61), yet put (71)

$A=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]---\left(72\right)$

Iv) put output matrix B, 2n * 1 column vector is as (62)

(a) b _i=0, i=1 ..., 2n, yet (b) b _N+1=1/CL (73)

V) put output matrix C, the capable vector of 1 * 2n as

(a) c _i=0, i=1 ..., 2n, yet (b) c _j=1, j ∈ I[1, n]; (j=n usually), (74)

In order to select node n-j+1 as output node.

Vi) put direct output matrix D=0.

Vii) stop.

[0083] so set up its state-space model { A, B, C, D} by closed type (59)-(62) of described model 4.

B. ask transfer function model by state-space model

[0084] this section is to set forth how to use the top notion that is developed to ask the transport function of RLC interconnection line to its arbitrary output node by state-space model.Consider general distribution RLC interconnection line and transmission line among Fig. 1-4.

[0085] tries to achieve by closed type owing to state-space model, so can easily try to achieve interconnection line by the transport function T that is input to its arbitrary output node by MATLAB instruction ss2tf _n(s).Because state-space model is accurate, so transport function also is quite accurate.

C. model simplification and approximate exponent number

[0086] illustrated and how to try to achieve the method for strict state-space model and further to try to achieve transfer function method quite accurately by above-mentioned accurate closed type.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).Present above-mentioned accurate state-space model and the transfer function model that produces fast provides model simplification or model to block, and further relatively basis and starting point.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.

[0087] 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.

[0088] 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 _tAnd total length capacitor C _t, shown in (18).Its even distributed model in original 2n rank is shown in (59)-(63).Its 2m rank ELO model { A _Em, B _Em, C _Em, D} is

$A_{\mathrm{em}}=\left[\begin{array}{cc}0& I\\ A_{\mathrm{em}21}& A_{\mathrm{em}22}\end{array}\right]---\left(75\right)$

$A_{\mathrm{em}22}=-\frac{R}{L}\&CenterDot;I_{m\&times;m}---\left(76\right)$

$B_{\mathrm{em}}={\left[\begin{array}{cc}0& B_{\mathrm{em}1}\end{array}\right]}^T=\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="C20051007826400241.png,C20051007826400261.png"\; state="\{\{state\}\}">r</figure-callout>}}^2\mathrm{<figure-callout\; id="0"\; label="CL"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">CL</figure-callout>}}{\left[\begin{array}{cccccccc}0& 0& ...& 0& 1& 0& ...& 0\end{array}\right]}^T,$ $C_{\mathrm{em}}=\left[\begin{array}{cccccccc}& & & & .& & & \\ 0& ...& 0& 1& .& 0& ...& 0\\ & & & & .& & & \end{array}\right],$ D＝0，n≥1 (77)

A wherein _Em∈ R ^{2m * 2m}, A _Em21∈ R ^{M * m}, A _Em22∈ R ^{M * m}, B _Em∈ R ^{2m * 1}, C _Em∈ R ^{1 * 2m}, and R, L and C are the parameters of original 2n rank model, the ratio that its exponent number is simplified is

r＝n/m. (78)

[0089] 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.Its 2n rank master pattern is (42)-(45).Then, its 2m rank ELO state-space model { A _Em, B _Em, C _Em, D} is

$A_{\mathrm{em}}=\left[\begin{array}{cc}0& I\\ A_{\mathrm{em}21}& A_{\mathrm{em}22}\end{array}\right]---\left(79\right)$

$B_{\mathrm{em}}={\left[\begin{array}{cc}0& B_{\mathrm{em}1}\end{array}\right]}^T=\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="C20051007826400241.png,C20051007826400261.png"\; state="\{\{state\}\}">r</figure-callout>}}^2\mathrm{<figure-callout\; id="0"\; label="CL"\; filenames="C20051007826400242.png,C20051007826400252.png"\; state="\{\{state\}\}">CL</figure-callout>}}{\left[\begin{array}{cccccccc}0& 0& ...& 0& 1& 0& ...& 0\end{array}\right]}^T,$ $C_{\mathrm{em}}=\left[\begin{array}{cccccccc}& & & & .& & & \\ 0& ...& 0& 1& .& 0& ...& 0\\ & & & & .& & & \end{array}\right],$ D＝0，n＞1 (82)

A wherein _Em∈ R ^{2m * 2m}, A _Em21∈ R ^{M * m}, A _Em22∈ R ^{M * m}, B _Em∈ R ^{2m * 1}, C _Em∈ R ^{1 * 2m}, and R, C and L are the parameters of original 2n rank model.The model simplification ratio is r=n/m.

[0090] 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

[0091] and then, above-mentioned master pattern and simplified model can be used for determining and research transient response and Bode figure (frequency response), i.e. their time domain performance and frequency domain performance.For example with some simple MATLAB instruction step (A, B, C, D) or step (n d) makes time domain step response, bode (n, d) and bode (C D) makes frequency domain Bode and schemes for A, B.These performance map and data also can compare master pattern and its simplified model easily.

E. computation complexity and stability features

[0092] computation complexity of the RLC interconnection line state-space model of the new method of foregoing invention is O (n ²), n is the RLC joint number, exponent number is 2n, much smaller than the complexity O (n of classic method ^k), k＞2 or k＞3.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 complexity O (n here ²) be because the closed type method of new state-space model has been avoided matrix inversion or matrix decomposition and matrix multiplication.

[0093] still, to equally distributed RLC interconnection line, the closed type computation complexity of said state-space model only is a fixing constant, i.e. O (1).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 state-space model method computation complexity is used to set or network will be that these are set and the index of network multiply by O (l), this will be much smaller than O (n), and wherein n is the exponent number of total tree or network.

[0094] 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.

[0095] the inventive method is the modeling that is effective in interconnection line distribution of impedance characteristic especially, because its so simple state-space model calculates and so easily tries to achieve interconnection line by the transfer function model that is input to its arbitrary output node, adds its high precision.

F. experimental result

[0096] closed type of described state-space model is very useful for the emulation of time domain.Particularly for the commonly used test of time domain and the step response of assessment.If a system is described with transport function, it often will at first convert the state space of time domain in the hope of step response.But 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.

[0097] 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 corresponding step response and the Bode figure of its BTM and ELO simplified model.

[0098] 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.

[0099] 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.

[0100] routine 1A: application process SS4 is in the model 4 of example 1, the virgin state spatial model S={A on its 200 rank, and B, C, D} is

A ₂₂＝-1.9435·10 ¹⁰·I _100＞100， $A=\left[\begin{array}{cc}0& I\\ A_{21}& A_{22}\end{array}\right],$

B＝[0?B ₁] ^T＝[0?0…0.37511·10 ²⁸?0…0] ^T， $C=\left[\begin{array}{cccccccc}& & & & .& & & \\ 0& ...& 0& 1& .& 0& ...& 0\\ & & & & .& & & \end{array}\right]$ and?D＝0.

Its transport function 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.

[0101] Fig. 5 has shown 200 rank master pattern step responses, is tried to achieve by the closed type of described state-space model.

[0102] Fig. 6 has shown the original Bode figure in 200 rank, is tried to achieve by described state-space model method.

[0103] Bode figure both can try to achieve from above-mentioned state-space model, and also available MATLAB instruction ss2tf gets its transport function from above-mentioned state-space model and tries to achieve, and perhaps the transport function that obtains from its another invention recurrence method is tried to achieve.But it is the most accurate from the Bode that the transport function of its another invention recursive algorithm is done.Master pattern has shown that when frequency is higher than certain limit Bode figure had the decay of increase when frequency increased.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.

[0104] 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.

[0105] Fig. 7 has shown its 2 rank (n=1) ELO model step response, is tried to achieve by the closed type of described state-space model.

[0106] Fig. 8 has shown the Bode figure of its 2 rank (n=1) ELO model, is tried to achieve by described state-space model method.

[0107] Fig. 9 has shown its 4 rank (n=2) ELO model step response, is tried to achieve by the closed type of described state-space model.

[0108] Figure 10 has shown the Bode figure of its 4 rank (n=2) ELO model, is tried to achieve by described state-space model method.

[0109] Figure 11 has shown its 20 rank (n=10) ELO model step response, is tried to achieve by the closed type of described state-space model.

[0110] Figure 12 has shown the Bode figure of its 20 rank (n=10) ELO model, is tried to achieve by described state-space model method.

[0111] these emulation have illustrated that the ELO simplified model of low order can not represent original equally distributed RLC interconnection line well.

[0112] thus need from the ELO model of time domain response one high-order, to the enough good 200 rank master patterns that are similar to.Clearly also very natural, the exponent number of ELO model is high more, and it is approximate good more.

[0113] routine 2A: use the model 3 of said method in example 2, its 200 rank master pattern S is

$A=\left[\begin{array}{cc}0& I\\ A_{21}& A_{22}\end{array}\right],$

B＝[0?B ₁] ^T＝[0?0…0?3.7511·10 ²⁸?0…0] ^T， $C=\left[\begin{array}{cccccccc}& & & & .& & & \\ 0& ...& 0& 1& .& 0& ...& 0\\ & & & & .& & & \end{array}\right]$

[0114] Figure 13 has shown an above-mentioned master pattern but C ₀=0 step response is tried to achieve by the closed type of described state-space model, so that show the special characteristic of some RLC interconnection lines.

[0115] Figure 14 has shown this master pattern but C ₀=0 Bode figure is tried to achieve by described state-space model method.

[0116] routine 2B: consider even distribution RLC interconnection line among the routine 2A but C ₀The master pattern on=0,200 rank, and its BTM simplified model of the method generation of consideration A3 and B3 joint.

[0117] Figure 15 has shown the step response of this master pattern and 1 rank and 2 rank BTM models.Wherein the curve of steepest is the curve of master pattern, and beneath curve is the curve of 2 rank BTM models, and the highest curve is the curve of 1 rank BTM model.

[0118] Figure 16 has shown the Bode figure of this master pattern and 1 rank and 10 rank BTM models.Wherein the curve of multimodal is the curve of master pattern, and the curve of inferior multimodal is the curve of 10 rank BTM models, and straight curve is the curve of 1 rank BTM model.

[0119] error of the BTM model simplification of these low orders is to show see big.This illustrates that also described new method and technology relatively are very useful and effective to modeling and model simplification and model.

This shows that [0120] 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 (13)

1. a method that generates the state-space model of RLC interconnection line or transmission line is used for emulation, performance evaluation, and model simplification, or circuit design, this method has following step:

(a) configuration state spatial model exponent number is an even number 2n;

(b) putting said interconnection line or transmission line model is the series connection of n joint, has a source end and n node, and wherein n node is a terminal and n-1 intermediate node, and every joint has a resistance and an inductance to be in series, and has an electric capacity to be connected to ground from the following leaf of this joint;

(c) resistance R of distributing n to save _i, inductance L _iAnd capacitor C _iParameter value, i=1 ..., n;

(d) get the state variable vector that n node voltage and derivative thereof are said state-space model, source voltage is input variable, and getting a certain node is output terminal, and its voltage is output variable; It is characterized in that:

(e) structure 2n * 2n maintains the system matrix A, and 4 n * n dimension submatrix A is arranged ₁₁, A ₁₂, A ₂₁, A ₂₂Constitute, for $A=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right],$ Submatrix A wherein ₁₁Be null matrix, submatrix A ₁₂Be unit matrix I, submatrix A ₂₁Diagonal line, last diagonal line and following diagonal entry are made of submatrix A said inductance and capacitance ₂₂Be a triangular matrix, the element of its non-zero is by said resistance, and inductance and capacitance are formed;

(f) structure 2n * 1 dimension input matrix B, it has a nonzero element that is positioned at (n+1) row to be made up of the inductance value and the capacitance of a joint of direct connection source end;

(g) structure output matrix C has one 1 * 2n dimension row vector only to have a nonzero element to be positioned in [1, n] row, with the output node voltage that will select corresponding to selection;

(h) structure one is 0 direct output matrix D;

(i) form said state-space model A, B, C, D} be by said matrix A, B, C, D forms;

Said thus state variable vector is followed a differential equation and is called system equation $\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right),$ Output variable is followed an algebraically equation

Be called output equation y (t)=Cx (t)+Du (t), wherein x (t) is the state variable vector, u (t) is an input source voltage, and y (t) is for output vector contains output end voltage, and said method provides a method of setting up the state-space model of RLC interconnection line or transmission line.

2. according to the said method of claim 1, its further feature is:

(i) i joint and node i thereof, resistance R _i, inductance L _iAnd capacitor C _iNumbering rise to source extreme direction serial number by terminal;

(ii) the state variable vector is designated as $x\left(t\right)={\left[\begin{array}{cccccc}{v}_n\left(t\right)& \&CenterDot;\&CenterDot;\&CenterDot;& v_1\left(t\right)& {\stackrel{\&CenterDot;}{v}}_n\left(t\right)& \&CenterDot;\&CenterDot;\&CenterDot;& {\stackrel{\&CenterDot;}{v}}_1\left(t\right)\end{array}\right]}^T,$ V wherein _i(t) be the voltage of node i, i=1 ..., n;

(iii) said submatrix A ₂₁For

Said submatrix A ₂₂For

(iv) said input matrix B is $B={\left[\begin{array}{cccccccc}0& \&CenterDot;\&CenterDot;\&CenterDot;& 0& \begin{array}{c}\&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\end{array}& 1/\left(C_n{L}_n\right)& 0& \&CenterDot;\&CenterDot;\&CenterDot;& 0\end{array}\right]}^T$ ；

(it is 1 that the row vector of v) said output matrix C has non-zero entry in the n+1-k position, corresponding to the voltage of selecting the k node, 1≤k≤n.

3. according to the said method of claim 1, it is characterized in that further comprising the steps:

(i) said interconnection line or transmission line have the source resistance R _s, pull-up resistor R ₀With load capacitance C ₀, wherein the source resistance string is associated in the source end, pull-up resistor and electric capacity be connected in parallel terminal and ground;

(ii) put said submatrix A ₂₁One column element is arranged by pull-up resistor R ₀, capacitor C _i, inductance L _i, resistance R _i, i=1 ..., n forms, and wherein has an element further to have the source resistance R _s, and another element further has load capacitance C ₀, have a non-principal diagonal unit further to have C ₀

(iii) put said submatrix A ₂₂There is a non-zero row element to have R _s, have a non-zero column element to have C ₀, wherein have an element to have R ₀

The state-space model of being set up is applicable to and has source resistance thus, the interconnection line of pull-up resistor and electric capacity or the synthetic model of transmission line.

4. the described method of claim 1, its feature further comprises the steps:

(i) (m＜n), use the method for claim 1 sets up the state-space model on 2m rank by replace n with m;

The performance index that the simplification of one optimization model (ii) is set are represented 2m rank model departing from respect to original 2n rank model;

(iii) try to achieve 2m rank Model parameter, be called optimized parameter, the performance index minimum that makes said optimization model simplification;

(iv) set up a new 2m scalariform state space model that has the optimized parameter of trying to achieve;

Method provides the state space simplified model on the 2m rank of an optimization thus.

5. the described method of claim 1, its feature further comprises:

The inductance and the capacitance parameter value of the n joint that (i) is distributed adopt engineer's scale, make things convenient for modeling, and emulation is analyzed or design; Perhaps

(ii) similarity transformation by a nonsingular matrix T act on model A, B, C, on the D}, with a scale model.

6. the described method of claim 1 is used for emulation, performance evaluation, and model simplification, or in the technical finesse of circuit design.

7. the described method of claim 1, its feature further comprises: apply in the manufacture process of VLSI chip or transmission line, wherein the interconnection line of VLSI chip or transmission line are that the described method of utilization claim 1 is carried out performance evaluation, model simplification, or design.

8. a method of setting up the state-space model of uniform RLC interconnection line or transmission line is used for emulation, performance evaluation, and model simplification or circuit design, this method has following step:

(a) configuration state spatial model exponent number is an even number 2n;

(b) putting described interconnection line or transmission line model is the series connection of n joint, one source end and n node are arranged, wherein node comprises a terminal and a middle n-1 node, and every joint evenly equates to have a resistance and an inductance to be in series, and has an electric capacity to be connected to ground from the following leaf of this joint;

(c) distribute one group of identical uniform resistance R, the parameter value of inductance L and capacitor C saves to each;

(d) get the voltage of n node and the state variable vector that derivative is said state-space model thereof, source voltage is input variable, and getting a certain node is output terminal, and its voltage is output variable; It is characterized in that:

(e) structure 2n * 2n maintains the system matrix A 4 n * n dimension submatrix A ₁₁, A ₁₂, A ₂₁, and A ₂₂Constitute, for $A=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right],$ Submatrix A wherein ₁₁Be 0 matrix, submatrix A ₁₂Be unit matrix I, submatrix A ₂₁Diagonal line, last diagonal line and following diagonal entry are made of submatrix A said inductance value L and capacitance C ₂₂For the non-zero diagonal entry is made up of resistance value R and inductance value L;

(f) structure 2n * 1 dimension input matrix B has a nonzero element that is positioned at (n+1) row to be made up of inductance value L and capacitance C;

(g) structure output matrix C has one 1 * 2n dimension row vector only to have a nonzero element to be positioned in [1, n] row, with corresponding to selecting selected output node voltage;

(h) structure one is 0 direct output matrix D;

(i) form state-space model A, B, C, D} be by said matrix A, B, C, D forms;

Said thus state variable vector is followed a differential equation and is called system equation $\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right),$ Output variable is followed an algebraically equation and is called output equation y (t)=Cx (t)+Du (t), wherein x (t) is said state variable vector, u (t) is a source voltage, y (t) contains output end voltage for output vector, said method provides a method of setting up the state-space model of RLC interconnection line or transmission line, has reduced computation complexity.

9. method according to claim 8, its further feature is:

(i) i joint, and the numbering of node i is risen to source extreme direction serial number by terminal;

(ii) the state variable vector is designated as $x\left(t\right)={\left[\begin{array}{cccccc}{v}_n\left(t\right)& \&CenterDot;\&CenterDot;\&CenterDot;& v_1\left(t\right)& {\stackrel{\&CenterDot;}{v}}_n\left(t\right)& \&CenterDot;\&CenterDot;\&CenterDot;& {\stackrel{\&CenterDot;}{v}}_1\left(t\right)\end{array}\right]}^T,$ V wherein _i(t) be the voltage of node i, i=1 ..., n;

(iii) said submatrix A ₂₁For

(iv) said submatrix A ₂₂It is a diagonal matrix $A_{22}=-\frac{R}{L}\&CenterDot;I_{n\&times;n};$

(v) said input matrix B is $B={\left[\begin{array}{cccccccc}0& \&CenterDot;\&CenterDot;\&CenterDot;& 0& \begin{array}{c}\&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\end{array}& 1/\left(CL\right)& 0& \&CenterDot;\&CenterDot;\&CenterDot;& 0\end{array}\right]}^T$ ；

(it is 1 that the row vector of vi) said output matrix C has non-zero entry in the n+1-k position, with the voltage corresponding to selection k node, 1≤k≤n; The computation complexity of said method is O (1) thus, and said method provides further emulation, and model simplification is just being examined, performance evaluation, the basis of optimization or circuit design.

10. according to right 8 described methods, it is characterized in that further comprising the steps:

(i) settle the numbering i of every joint and its node to rise to source extreme direction serial number by terminal, i=1 ..., n;

(ii) the state variable vector is designated as $x\left(t\right)={\left[\begin{array}{cccccc}{v}_n\left(t\right)& \&CenterDot;\&CenterDot;\&CenterDot;& v_1\left(t\right)& {\stackrel{\&CenterDot;}{v}}_n\left(t\right)& \&CenterDot;\&CenterDot;\&CenterDot;& {\stackrel{\&CenterDot;}{v}}_1\left(t\right)\end{array}\right]}^T,$ V wherein _i(t) be the voltage of node i, i=1 ..., n;

(iii) said interconnection line or transmission line have the source resistance R _s, pull-up resistor R ₀With load capacitance C ₀, wherein the source resistance string is connected on the source end, and pull-up resistor and electric capacity are connected in terminal and ground in parallel;

(iv) said submatrix A ₂₁For

Or $A_{21}=\frac{1}{\mathrm{CL}}\left[\begin{array}{cc}-2& 1-\frac{R_s}{R_0}\\ \frac{1}{1+C_0/C}& -\frac{{R+R}_0}{R_0(1+C_0/C)}\end{array}\right],n=2;$

(v) said submatrix A ₂₂For

Or $A_{22}=\left[\begin{array}{cc}-\frac{R+R_s}{L}& -\frac{R_s(1+C_0/C)}{L}\\ 0& -\frac{R}{L}-\frac{1}{R_0(C_1+C_0)}\end{array}\right],n=2;$

(vi) said input matrix B is $B={\left[\begin{array}{cccccccc}0& \&CenterDot;\&CenterDot;\&CenterDot;& 0& \begin{array}{c}\&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\end{array}& 1/\left(CL\right)& 0& \&CenterDot;\&CenterDot;\&CenterDot;& 0\end{array}\right]}^T$ ；

(the vectorial non-zero entry of the row of vii) said output matrix C is 1 in the n+1-k position, corresponding to the voltage of k node, and 1≤k≤n;

This method computation complexity is low to moderate O (1) thus, and further emulation is provided, and model simplification is just being examined, performance evaluation, the basis of optimization or circuit design.

11. method according to claim 8, its feature further comprises the steps:

(i) (m＜n), use the method for claim 8 sets up the state-space model on the 2m rank of a low order by replace n with m;

The performance index that the simplification of one optimization model (ii) is set are represented 2m rank model departing from respect to original 2n rank model;

The 2m rank model parameter of (iii) trying to achieve one group of optimum makes the performance index minimum that said optimization model is simplified;

(iv) set up the interconnection line on 2m rank of a 2m rank model parameter that has an optimum of trying to achieve or the state-space model of transmission line;

Method provides the 2m scalariform state space simplified model of an optimization thus.

12. the described method of claim 8 is used for emulation, performance evaluation, and model simplification, or in the technical finesse of circuit design.

13. the described method of claim 8, its feature further comprises: apply in the manufacture process of VLSI chip or transmission line, wherein the interconnection line of VLSI chip or transmission line are that the described method of utilization claim 1 is carried out performance evaluation, model simplification, or design.

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