CN100452063C - Method for quick extraction of silicon integrated circuit substrate coupling parameters under multiple frequency points - Google Patents

Abstract

The present invention relates to a method for extracting liner multi frequency point integrated coupled parameters, which belongs to the IC-CAD technical field. The present invention is characterized in that the present invention provides a way for extracting the integrated coupled parameters at any frequency based on the similarity between liner coupled capacitor extraction and integrated parameter extraction. The way comprises two steps: in the first step, the coupled capacitance parameter is extracted; in the second step, capacitance is corrected accurately into the integrated coupled parameter under the frequency. The present invention provides simultaneously the efficient implementation way in the process of correction. When using the method to calculate the multi frequency point integrated coupled parameters, the capacitance parameter extraction only needs to be operated one time; the process of correction from the capacitance to the integrated parameters is only carried out one time at each frequency point, but, the correction process of the present invention can be completed effectively; a little calculation is needed at each time. Thus, the present invention has high efficiency to extract the integrated and coupled parameters at multi-frequency point, and simultaneously keeps the calculation accuracy based on strict mathematic derivation.

Description

The rapid extracting method of silicon integrated circuit substrate coupling parameters under multiple frequency points

Technical field

The comprehensive coupling parameter of three-dimensional substrate in the integrated circuit CAD (IC-CAD) extracts the software design field.

Background technology

(Integrated Circuit IC) is the foundation stone of current electronics industry and even information industry to SIC (semiconductor integrated circuit).Along with the development of circuit manufacturing technology, the integrated level of circuit constantly increases, and current a lot of chips contain several ten million and even device such as more than one hundred million CMOS.Meanwhile; RF (radio frequency) circuit, Digital Analog Hybrid Circuits and SoC (System on a Chip; System on Chip/SoC) widespread use (wherein an example is exactly the universal day by day of mobile phone terminal); make the very sensitive mimic channel of noise and the digital circuit of speed-sensitive switch operation (switch) are fabricated on the chip piece, share same substrate (substrate has people's translations substrate, substrate etc. again).This high integration has a lot of advantages, and is less such as energy consumption, cost is lower etc., but also brought some challenges.One of them is exactly, owing to share same substrate, substrate will be propagated some coupled noises, and Here it is so-called " substrate coupled noise ".On the one hand, the high-speed switch of digital circuit operation (switch) and other active connecting lines can inject noise current to substrate.(majority be silicon, Silicon) is not desirable insulator, a part of noise transfer can be given the mimic channel of sensitivity, the latter's normal function impacted, even its work is failed because backing material.Simultaneously, by the parasitic coupling capacitance between power lead, device etc. and the substrate, variation in voltage from power lead, device etc. will cause that the voltage of substrate floats, and this voltage floats can influence the size of the threshold voltage of substrate again, and then influences the circuit performance of other devices on the substrate.Under upper frequency, the equivalent resistance of substrate may bring energy loss etc., and this will produce material impact to circuit.Such as, the quality factor of inductance component in the RF circuit (qualify factor) is just influenced by the ohmic loss of substrate very much.In sum, the seriousness of substrate coupling makes that each stage all must this problem of concern in circuit design.Fig. 1 has shown a synoptic diagram, is the interface contact of substrate and other circuit in the frame of broken lines wherein, can be called contact, port again, hereinafter is abbreviated as port.In the process of follow-up extracting parameter, the coupling parameter that only needs to extract between these ports gets final product.

Current, the design cycle of integrated circuit as shown in Figure 2.At first to propose functional description, obtain describing the domain of semiconductor technology size, structure then through logical design, layout design.At this moment need to carry out " layout verification ", promptly wait and verify whether above-mentioned design can reach the requirement of setting originally by the computer software simulation.If meet the demands, just can carry out next step the manufacturing etc.; Otherwise return logical design etc. and carry out necessary correction.Repeat this iterative process, till layout verification shows that design can meet the demands really.In layout verification, an important link is called " parasitic parameter extraction ", wherein just comprises calculating substrate coupling parameter etc.Along with the continuous increase of circuit scale and constantly dwindling of characteristic dimension, simple estimation in the past even the way of directly ignoring these parameters have been difficult to the accuracy requirement that reaches enough.Have only accurate extraction (calculating) coupling parameter, just can carry out correct breadboardin and checking.

Along with the progress of technology and the raising of clock frequency, the extraction of substrate coupling parameter has obtained increasing concern from academia and industry member.Current, for extracting accurately, need to carry out numerical simulation to the three-dimensional model of circuit layout, its main method comprise domain type solution (finite difference method and Finite Element Method), based on the method for Green function and DIRECT BOUNDARY ELEMENT method (Boundary Element Method, BEM) etc.

Compare with the domain type solution, the advantage of Direct Boundary Element Method is the precision height, less discrete variable and than the ability of strength reason complex boundary shape.With compare based on the method for Green function, the DIRECT BOUNDARY ELEMENT method has advantages such as adapting to labyrinth.Though the DIRECT BOUNDARY ELEMENT method has above-mentioned advantage, because industrial actual demand is more and more, the three-dimensional substrate structure is also increasing, the counting yield that how to improve based on the three-dimensional extracting method of DIRECT BOUNDARY ELEMENT method has just become the task of top priority.

When frequency of operation was relatively low, the substrate coupling parameter showed as pure resistive, promptly had only dead resistance between the equivalent electrical circuit middle port (contact is promptly with the interface area of other circuit devcies etc., shown in Fig. 1 frame of broken lines).Its extracting method comprises a lot, wherein I document " Substrate Resistance Extraction with Direct Boundary Element Method; " Asiaand South Pacific Design Automation Conference (2005 Asia and South Pacific region Design Automation Conference), pp.208-211, among the Jan.2005 (below write a Chinese character in simplified form make ASP-DAC2005) the DIRECT BOUNDARY ELEMENT method is applied to during the substrate repeating resistance extracts, obtained good effect.

Under upper frequency, substrate coupling parameter (wherein comprising resistance, electric capacity) is along with frequency shift, and promptly existing resistance connects between the equivalent electrical circuit middle port, also have electric capacity to connect, and their numerical value is generally all with frequency change.This type of parameter hereinafter claims comprehensive coupling parameter.I document " A New Boundary Element Method for Accurate Modeling of LossySubstrates with Arbitrary Doping Profiles; " Asia and South Pacific Design Automation Conference, Yokohama, Japan, pp.683-688, Jan.2006 (below write a Chinese character in simplified form make ASP-DAC2006) is applied to the DIRECT BOUNDARY ELEMENT method in the extraction of comprehensive coupling parameter, has obtained effect preferably.Because this parameter has different values under different frequency, often need to calculate its concrete numerical value at a plurality of Frequency points, comes the comprehensive assessment coupling performance with this.Though above-mentioned second piece of document done some and improved to this, but still need to set up and find the solution the system of linear equations of a plural number at each Frequency point.This is when variable number is very big, and it is very low that efficient will become, because the system of equations solution procedure is often very slow, also needs more calculator memory simultaneously.What is worse, when needs calculated the following coupling parameter of a lot of Frequency points, overall computing time, (equal every frequency computing time add up and) meeting was very high, and efficient is very low.

Summary of the invention

The present invention provides the comprehensive coupling parameter of a kind of effective extraction method of (comprising resistance R, capacitor C two parts), and this method is higher to the computational problem efficient of coupling parameter under the multi-frequency point.

For ease of understanding, Fig. 3 has shown the synoptic diagram of a typical substrat structure.This method is called the comprehensive coupling parameter extraction apparatus of substrate, is abbreviated as SubRCExtract (Substrate coupling Resistance and Capacitance Extractor).Main innovate point is, is different from above that document ASP-DAC2006 uses the DIRECT BOUNDARY ELEMENT method to calculate comprehensive parameters (its process flow diagram is seen Fig. 4) successively under every Frequency point, but it is divided into two steps the extraction of comprehensive coupling parameter:

1.ExtractCap: extract coupling capacitance between the substrate port;

2.ReviseCap2RC: this electric capacity is modified to comprehensive parameters.

Wherein, the resistance extracting method very similar (seeing for details hereinafter) described in ExtractCap and the first piece of document may move slowly, but only need calculate once.ReviseCap2RC needs to carry out once at each Frequency point, but each seldom time-consuming.Generally speaking, SubRCExtract will have better efficient to multi-frequency point parameters calculation.Simultaneously,, and directly use DIRECT BOUNDARY ELEMENT extracting method (as above-mentioned document ASP-DAC2006) to compare, wherein do not have the loss of precision because be based on strict mathematical derivation.

The information that is input as geometries such as describing port and substrate dielectric of SubRCExtract method is output as the comprehensive coupling parameter matrix between the port.Method mainly comprises two modules:

[module 0] reads input file, obtains body information (containing substrate length, conductivity, specific inductive capacity, the position of port, size, frequency of operation etc.)

[module 1] carries out the extraction of substrate coupling capacitance, i.e. ExtractCap.

[module 2] carries out the mathematics correction with coupling capacitance, obtains comprehensive parameters, i.e. ReviseCap2RC.

Following merogenesis is introduced each module in detail.For the modification method in the better Understanding Module 2, after module 1, add and use DIRECT BOUNDARY ELEMENT to carry out the principle that comprehensive parameters extracts.

1 uses DIRECT BOUNDARY ELEMENT to extract coupling capacitance (ExtractCap)

It is very similar that boundary element among following principle and the document ASP-DAC2005 extracts repeating resistance, can provide both similarities and differences parts at last in this trifle.

Below discuss based on model tpo substrate 3 and launch, and make its interior all medium all be considered to insulator (being that the conductivity is 0), at this moment, the coupling between the port will be pure capacitive.Its coupling capacitance can be found the solution by following DIRECT BOUNDARY ELEMENT method and obtain.

Arbitrary medium M _iIn, voltage u satisfies the Lapalce equation:

${\▿}^{2}u=0,$ Medium M _iIn (1)

The mixed boundary condition of following formula is:

U=u (forces boundary condition) (2a) on the port surface

Q=0, (natural boundary conditions) (2b) on other surfaces

Wherein u is a voltage (1V or 0V) default on port.Q is a normal electric field intensity.In addition, on the interface of medium a and b, voltage and voltage shift are continuous:

u _a＝u _b， (3a)

ε _aq _a＝ε _bq _b， (3b)

ε wherein _aAnd ε _bBe respectively the specific inductive capacity of a and b, q is a normal electric field.

Solving equation formula (1) can directly obtain Electric Field Distribution.According to the electric charge on Electric Field Distribution numerical evaluation port surface, the numerical value of its numerical value and coupling capacitance is closely related then, shown in hereinafter formula (7).Be the method for specifically finding the solution Electric Field Distribution below.

Use the Green formula, and select for use free space Green function, formula (1) can be converted at medium M as weighting function _iBorder Γ _iThe boundary integral equation of last establishment:

${0.5u}_s+{\∫}_{{\mathrm{\Γ}}_i}{q}^{*}\mathrm{ud\Γ}={\∫}_{{\mathrm{\Γ}}_i}{u}^{*}\mathrm{qd\Γ},$ To medium M _i(4)

U wherein _sThe voltage (meaning of collocation point will explained after a while) of representing certain collocation point s.With border Γ _iBe divided into N _iBoundary element Γ _Ij, j=1,2 ..., N _iThe discrete form of formula (4) is:

${0.5u}_s+\underset{j=1}{\overset{N_i}{\mathrm{\Σ}}}u{\∫}_{{\mathrm{\Γ}}_{\mathrm{ij}}}{q}^{*}\mathrm{d\Γ}=\underset{j=1}{\overset{N_i}{\mathrm{\Σ}}}q{\∫}_{{\mathrm{\Γ}}_{\mathrm{ij}}}{u}^{*}\mathrm{d\Γ},$ To medium M _i(5)

Wherein, each boundary element Γ _IjOn voltage u and normal electric field q all be constant, so can be moved to outside the sign of integration.Wherein collocation point s (or be called sampled point, getting sampled point is the system of equations that contains unknown quantity in order to list, to solve unknown quantity) is taken as the central point of each boundary element.u ^*Be the elementary solution of Laplace equation, equal free space Green function, form is

Wherein (s v) represents the size of distance between collocation point s and variable point v to r, and variable point v also is taken as Γ _iGo up the central point of each boundary element, can overlap with a s.q ^*Be u ^*Normal direction outside the border

Derivative, expression formula is

Wherein

The expression distance vector.Each variable point is gone up (promptly on the Dui Ying boundary element) all two physical quantitys, i.e. voltage u and electric field q.According to boundary condition (2), if this boundary element is in port (contact) surface, u is known on it, i.e. Yu She voltage value, and electric field q the unknown; Otherwise promptly this boundary element is at non-ports zone, and electric field q is 0 on it, voltage u the unknown.So in fact each boundary element has only a variable, perhaps be u categorical variable or for the q categorical variable.By (5) as can be known, their coefficient form is respectively

With

Use interface equation (3a) and formula (3b), by change of variable (concrete mode sees below), the boundary integral equation formula (5) of All Media is coupled together, obtain unified system of equations, the writing matrix form is:

A _capx _cap＝b _cap，(6’)

Wherein, a collocation point s will produce A _CapA row element, i.e. u that when this collocation point, produces or the coefficient of q by formula (5), obviously, these coefficients can be expressed as

Or

Multiply by 1 or real number ε _b/ ε _a, ε wherein _aAnd ε _bBe respectively the medium a of boundary mutually and the specific inductive capacity of medium b.b _CapKnown voltage u (multiply by coefficient) is moved to the equation right-hand member and forms, concrete form is

Γ wherein _mIt is the surface of port m.x _CapComprise voltage u type and electric field q type variable in each medium.When needs are found the solution a lot to the coupling capacitance between port, system of equations (6 ') right-hand member will contain a plurality of vectors, write a Chinese character in simplified form and do:

A _CapX _Cap=B extracts (6) to electric capacity

Here the matrix formed by a plurality of vectors of B.Treat dematrix X _CapVoltage u and electric field q numerical value when comprising different bias voltages and being provided with.

Solving equation group (6) can directly obtain the numerical value of q.If preestablishing the port voltage that is numbered m is 1V, (k ≠ m) be made as 0V, the coupling capacitance between port m, the k is all the other port k

$C_{\mathrm{mk}}={\∫}_{{\mathrm{\Γ}}_k}\mathrm{\ϵqd\Γ},---\left(7\right)$

Γ wherein _kBe the surface of port k, q is Γ _kOn electric field, ε is a specific inductive capacity of being close to the medium of port k.This shows that being distributed by electric field q to be easy to calculate coupling capacitance ¹

Note that two couples of variable u are arranged in equation (3) _a, q _aAnd u _b, q _b, can only keep wherein a pair ofly, a pair ofly in addition obtain by substitution of variable.Such as, if keep u _b, q _b(belonging to medium b), and substitution formula (3) formula obtains u _aAnd q _a: u _a=u _b, q _a=ε _b/ ε _aq _bFor the collocation point among the medium b, variable q _bThe form of coefficient be

Γ wherein _BiIt is certain boundary element on the medium b; And for the collocation point among the medium a, variable q _bThe coefficient form will become Γ wherein _AiIt is certain boundary element on the medium a.

2 use DIRECT BOUNDARY ELEMENT to extract comprehensive parameters

Following principle is from document ASP-DAC2006.

Actual substrate all diminishes, and the medium conductivity that constitutes substrate is also non-vanishing.According to document S.Ramo, J.R.Whinnery, and T.van Duzer, Fields and Waves in Communication Electronics, 3 ^RdEd.New York:Wiley, pp.283-287, pp.572-676,1994, such " diminishing medium " can be regarded as " insulator " by approximate under certain frequency, have complex dielectric constant ε _c=ε-j σ/ω then should " diminishing medium " just can be used as such " insulator " and handle.For convenient, make the specific inductive capacity of " insulator " a and b be respectively ε here _{C, a}=ε _a-j σ _a/ ω, ε _{C, b}=ε _b-j σ _b/ ω, wherein σ _a, σ _bBe respectively the conductivity of medium a and b, ε _aAnd ε _bBe corresponding real number specific inductive capacity, ω is an angular frequency, equals 2 π f, and f is a frequency.

For calculating comprehensive parameters, we are provided with the voltage of 1V or 0V on port.Boundary condition formula (2a) and formula (2b) remain unchanged.Laplace equation (1) is still set up, and on medium a and b interface, voltage is still continuous, but formula (3b) becomes

ε _c，aq _a＝ε _c，bq _b.(3b’)

Formula (3b ') is done following distortion:

Here also has a details, according to document ASP-DAC2005, if respectively with the physical quantity ε in the formula (3b) _aAnd ε _bChange σ into _aAnd σ _b(being respectively the conductivity of medium a and b), other formula remain unchanged, and after generation and the solving equation group, can obtain repeating resistance R _Mk, wherein $\frac{1}{R_{\mathrm{mk}}}={\∫}_{{\mathrm{\Γ}}_A}\mathrm{\σ}{E}_n\mathrm{d\Γ},$ Find the solution formula (7) the spitting image of electric capacity in form.

(σ _a+jωε _a)q _a＝(σ _b+jωε _b)q _b. (3_freqRC)

Formula (3_freqRC) can be interpreted as total current conservation (comprising conduction current and displacement current).Use formula (3_freqRC) replacement formula (3b) keeps other constant.Adopt similar substitution of variable mode, can obtain a system of equations:

AX=B extracts (8) to comprehensive parameters

Wherein coefficient matrices A is a complex matrix.A above _CapSimilarly, the numerical value of element (being the coefficient of u or q) is among the A Or

Multiply by 1 or the plural number (σ _b+ j ω ε _b)/(σ _a+ j ω ε _a), σ wherein _a, σ _bAnd ε _a, ε _bBe respectively the medium a of boundary mutually and conductivity and the specific inductive capacity of medium b.As long as default bias voltage equates, it is consistent that the B in the formula (8) and B in the formula (6) form principle, and numerical value also is

Γ wherein _mIt is the surface of port m.

Find the solution formula (8), the electric current of certain port of flowing through will for

Like this, if predeterminated voltage is respectively 1V and 0V on the port of numbering m and k, the comprehensive coupling parameter between them is exactly:

$Z_{\mathrm{mk}}=\frac{1}{{\∫}_{{\mathrm{\Γ}}_k}(\mathrm{\σ}+\mathrm{j\ω\ϵ})\mathrm{qd\Γ}}---\left(9\right)$

Γ wherein _kBe the surface of port k, q is Γ _kOn electric field, σ and ε are respectively medium conductivity and the specific inductive capacity of being close to port k.

In document ASP-DAC2006, need under each Frequency point, set up and solving equation group (8), obtain the comprehensive coupling parameter under this frequency.But when the comprehensive coupling parameter under the many Frequency points of needs, the process that establishes an equation, solves an equation will repeat many times, and calculation cost is higher.Worse, system of equations (8) is often very big, and is plural system of equations, finds the solution than real number system of equations is more difficult.

In order to guarantee reasonable counting yield, can calculate coupling parameter under the multi-frequency point by following " correction " mode.3 modification method ReviseCap2RC from electric capacity to the comprehensive parameters

4.2 save the electric capacity extraction of described comprehensive parameters and 4.1 joints a lot of similaritys is arranged.This similarity can be fully utilized, and obtains a kind of new comprehensive coupling parameter method for solving.At first, can calculate coupling capacitance according to the principle of 4.1 joints.Next step is modified to comprehensive coupling parameter under certain frequency to these electric capacity.The first step need be set up and solving equation group (6).It is the real number system of equations, and it finds the solution information and the frequency-independent that obtains, and can be stored, and can read at any time when needed.Second step all need implement once at each Frequency point, and still, its needed calculating seldom.When need extract coupling parameter under a plurality of Frequency points, new method will have greater advantage, because the time-consuming just first step, second step only needed the seldom time.Simple and easy process flow diagram is seen Fig. 5, later, will provide more easy-operating flow process Figure 11.

Below, we will provide the detailed modification method from electric capacity to the comprehensive parameters.

Theorem 1: electric capacity extracts and comprehensive parameters extraction coefficient matrix similarity theorem

Same substrate geometry and same boundary element are divided matrix A in matrix A and the formula (6) in the formula (8) _CapFollowing relation: A=A is arranged _Cap+ UV ^T, matrix U and V ^TStructure see below.

Matrix A in matrix A in the system of equations (8) and the system of equations (6) _CapVery big similarity is arranged.At first how matrix A forms.Each substrate dielectric is all listed formula (5), and use the replacement of variable on the interface to be connected to become unified system of equations.Here with M ₁, M ₂Between interface Γ ₁₂On u type variable (unknown voltage) and q type variable (unknown normal electric field) be example.U, q variable in these interface both sides satisfies interface equation (3a) and (3_freqRC), does as the described substitution of variable of 4.1 joints.Here hypothesis keeps and belongs to medium M ₂Variable, respectively called after u and q ₁₂, then belong to medium M ₁Variable be 1 * u and f ₁₂* q ₁₂(according to interface equation (3a) and formula (3_freqRC)), wherein ${\mathrm{<figure-callout\; id="12"\; label="f"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">f</figure-callout>}}_{12}=\frac{{\mathrm{\&sigma;}}_2+{\mathrm{j\&omega;\&epsiv;}}_2}{{\mathrm{\&sigma;}}_1+{\mathrm{j\&omega;\&epsiv;}}_1}.$ Under this assumption, to M ₂Certain collocation point s ₂, the coefficient of variable u is an integrated value Variable q ₁₂Coefficient be

Γ wherein _2eBe medium M ₂Borderline certain boundary element (above-mentioned two integrated values all only and geological information relation is arranged); And to M ₁Certain collocation point s ₁, the coefficient of variable u is

Γ wherein _1eBe medium M ₁Borderline certain boundary element; Variable q ₁₂Coefficient then be

Wherein ${\mathrm{<figure-callout\; id="12"\; label="f"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">f</figure-callout>}}_{12}=\frac{{\mathrm{\&sigma;}}_2+{\mathrm{j\&omega;\&epsiv;}}_2}{{\mathrm{\&sigma;}}_1+{\mathrm{j\&omega;\&epsiv;}}_1}.$ Same, need do similar replacement to the variable on other dielectric interfaces.Obviously, have only that the coefficient of some variable is a more complicated on the interface

Other coefficient all can be expressed as

Or

Said process also can be referring to synoptic diagram 6, high spot reviews interface variable q ₁₂, q ₂₃Coefficient.S ₂₁As q ₁₂To M ₂In the coefficient of certain collocation point, equal and M ₂Relevant integration

By formula (3_freqRC), q ₁₂To M ₁In the coefficient of certain collocation point be and M ₁Relevant integration

Multiply by again ${\mathrm{<figure-callout\; id="12"\; label="f"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">f</figure-callout>}}_{12}=\frac{{\mathrm{\&sigma;}}_2+{\mathrm{j\&omega;\&epsiv;}}_2}{{\mathrm{\&sigma;}}_1+{\mathrm{j\&omega;\&epsiv;}}_1},$ F just ₁₂S ₁₂S ₃₂And f ₂₃S ₂₃, implication is similar: S ₃₂Be and M ₃Relevant integrated value; ${\mathrm{<figure-callout\; id="23"\; label="f"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">f</figure-callout>}}_{23}=\frac{{\mathrm{\&sigma;}}_3+{\mathrm{j\&omega;\&epsiv;}}_3}{{\mathrm{\&sigma;}}_2+{\mathrm{j\&omega;\&epsiv;}}_2},$ S ₂₃Be and M ₂Relevant integrated value.

For sake of convenience, f above ₁₂, f ₂₃Be called interface variable coefficient frequently, definition is: ${\mathrm{<figure-callout\; id="12"\; label="f"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">f</figure-callout>}}_{12}=\frac{{\mathrm{\&sigma;}}_2+{\mathrm{j\&omega;\&epsiv;}}_2}{{\mathrm{\&sigma;}}_1+{\mathrm{j\&omega;\&epsiv;}}_1},$ ${\mathrm{<figure-callout\; id="23"\; label="f"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">f</figure-callout>}}_{23}=\frac{{\mathrm{\&sigma;}}_3+{\mathrm{j\&omega;\&epsiv;}}_3}{{\mathrm{\&sigma;}}_2+{\mathrm{j\&omega;\&epsiv;}}_2}.$

In like manner, (see 4.1 joint principles for details) when using boundary element to carry out the electric capacity extraction, also need carry out substitution of variable.For simply, only with M ₁, M ₂Between interface Γ ₁₂On the electric field variable be example.Still hypothesis keeps and belongs to medium M ₂The electric field variable, called after q ₁₂, then, belong to medium M according to interface equation (3a) and formula (3_b) ₁The electric field variable then be c ₁₂* q ₁₂, wherein ${\mathrm{<figure-callout\; id="12"\; label="c"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">c</figure-callout>}}_{12}=\frac{{\mathrm{\&epsiv;}}_2}{{\mathrm{\&epsiv;}}_1}.$ To medium M ₂Certain collocation point s ₂, q ₁₂Coefficient be Γ wherein _2eBe medium M ₂Borderline certain boundary element.Then to M ₁Certain collocation point s ₁, q ₁₂Coefficient be

Γ wherein _1eBe medium M ₁Borderline certain boundary element, ${\mathrm{<figure-callout\; id="12"\; label="c"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">c</figure-callout>}}_{12}=\frac{{\mathrm{\&epsiv;}}_2}{{\mathrm{\&epsiv;}}_1}.$ Right.This process is referring to synoptic diagram 7, wherein ${\mathrm{<figure-callout\; id="23"\; label="c"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">c</figure-callout>}}_{23}=\frac{{\mathrm{\&epsiv;}}_3}{{\mathrm{\&epsiv;}}_2}.$

For sake of convenience, c above ₁₂, c ₂₃Be called the interface capacitance coefficient, definition is: $c_{12}=\frac{{\mathrm{\&epsiv;}}_2}{{\mathrm{\&epsiv;}}_1},$ $c_{23}=\frac{{\mathrm{\&epsiv;}}_3}{{\mathrm{\&epsiv;}}_2}.$

So, if we directly make in Fig. 6 ${\mathrm{<figure-callout\; id="12"\; label="f"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">f</figure-callout>}}_{12}=c_{12}=\frac{{\mathrm{\&epsiv;}}_2}{{\mathrm{\&epsiv;}}_1},$ ${\mathrm{<figure-callout\; id="23"\; label="f"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">f</figure-callout>}}_{23}=c_{23}=\frac{{\mathrm{\&epsiv;}}_3}{{\mathrm{\&epsiv;}}_2},$ This matrix just becomes the matrix of coefficients of system of equations (6), A as shown in Figure 7 _CapAs can be seen, the difference of two matrixes just and the interface variable (as q ₁₂, q ₂₃) relevant element.The difference of two matrixes is seen Fig. 8.This difference can be expressed as UV ^TForm wherein has only V ^TAnd frequency dependence.U and V can be highly sparse.Provide a kind of concrete make below.

Most of elements of U are 0, and only the coefficient of interface variable may non-zero, and nonzero value equals A _CapIn contain c ₁₂, c ₂₃Part, see Fig. 7 and Fig. 8 (saying that loaded down with trivial detailsly the element of U equals the coefficient to the interface electric field variable during collocation point in the little medium of interface sequence number) for details.U does not just need extra computation like this, only needs to extract A _CapRespective element get final product.

The shape of V is very simple, is diagonal matrix substantially, but except those element non-zeros of the field variable that powers on corresponding to interface, other elements all are 0; Nonzero element calculates also very simple, for interface frequency variable coefficient cuts 1 again divided by the interface capacitance coefficient, such as (f ₁₂-c ₁₂)/c ₁₂, (f ₂₃-c ₂₃)/c ₂₃, only depend on the physical property (conductivity, DIELECTRIC CONSTANT) and the angular frequency of interface both sides medium on the numerical value.As seen, V depends on frequency.Under the different frequency, U fixes, and V becomes.

In addition, a strict Woodbury formula of setting up is arranged on the mathematics:

(A+UV ^T) ^-1＝A ^-1-A ^-1U(I+V ^TA ^-1U) ^-1VTA ^-1，

A wherein, U, V are any matrixes, the condition of establishment is A and I+V ^TA ^-1U is reversible.Referring to document Weisstein EW.Concise Encyclopedia of Mathematics.Boca Raton, Fla.:CRC Press, 1999.

According to above theorem and Woodbury formula, can obtain another theorem:

Theorem 2: revise capacitance parameter and can get the comprehensive parameters theorem

Same substrate geometry and same boundary element are divided, system of equations formula (8) separate A ^-1But B through type (6) separate A _Cap ^-1B obtains:

$X=A^{-1}B={(A_{\mathrm{cap}}+{\mathrm{UV}}^T)}^{-1}B$

$=A_{\mathrm{cap}}^{-1}B-A_{\mathrm{cap}}^{-1}U{(I+V^T{A}_{\mathrm{cap}}^{-1}U)}^{-1}{V}^T{A}_{\mathrm{cap}}^{-1}B$

$=X_{\mathrm{cap}}-A_{\mathrm{cap}}^{-1}U{(I+V^T{A}_{\mathrm{cap}}^{-1}U)}^{-1}{V}^T{X}_{\mathrm{cap}}---\left(10\right)$

In other words, comprehensive parameters can obtain by revising capacitance parameter.Because X _CapAnd A _Cap ^-1B and frequency-independent, they can only be calculated once, store then.When calculating under arbitrary frequency comprehensive parameters, can call in X again _CapAnd A _Cap ^-1B, and use (the I+V that depends on frequency ^TA _Cap ^-1U) ^-1Do necessary " correction ", and then obtain the numerical value of comprehensive parameters under this frequency.

Seem that roughly formula (10) is though strictness is set up, and Practical significance is little, because calculate (I+V ^TA _Cap ^-1U) ^-1Directly find the solution A ^-1Equally complicated, the same difficulty, reason be they two wait size, all be the operation of inverting of difficulty.But the theorem below using can be calculated (I+V expeditiously ^TA _Cap ^-1U) ^-1:

Theorem 3:(I+V ^TA _Cap ^-1U) ^-1The efficient calculation theorem

Find the solution (I+V ^TA _Cap ^-1U) ^-1Be equivalent to and ask very minor matrix M contrary.M is by (I+V ^TA _Cap ^-1U) ^-1Middle nonzero block is formed, and these nonzero block do not comprise the subunit battle array on the principal diagonal.The size of M equals the sum of the boundary element on the interface between " difference " medium.Here, the meaning of " difference " is that interface both sides medium conductivity (σ) and specific inductive capacity (ε) do not equate entirely.

Specific practice is as follows.Because U and V ^TThe sparse property of height, I+V ^TA _Cap ^-1U, (I+V ^TA _Cap ^-1U) ^-1All very alike with unit matrix I, referring to Fig. 9.With I+V ^TA _Cap ^-1Nonzero block extracts (these nonzero block do not comprise the sub-piece of the non-zero that equals unit matrix on the principal diagonal) among the U, forms matrix M, writes down the dividing mode of each fritter of M.Calculate inverse matrix W=M ^-1, according to identical mode W is divided into a lot of fritters, then each piece according to counter the filling out and I+V of mode of " extraction " above ^TA _Cap ^-1In the unit matrix of size such as U, so just obtained (I+V ^TA _Cap ^-1U) ^-1But above process simple table is shown:

(11) equivalence is easy to obtain in theory proof: only need proof left figure of Fig. 9 and the right figure of Figure 10 inverse matrix each other really, just their product is a unit matrix; According to the matrix multiplication rule, be easy to verify that product is unit matrix really.

Significantly, M compares I+V ^TA _Cap ^-1U is little a lot.Its accurate size equals the total number of variable on the interface between " difference " medium, and the meaning of " difference " is that interface both sides medium conductivity (σ) and specific inductive capacity (ε) do not equate entirely.Reason illustrates as follows.Referring to matrix V among Fig. 8 ^TComposition.If medium M ₁And M ₂All identical σ and ε are arranged, then interface frequency variable coefficient $f_{}$ ${}_{12}=\frac{{\mathrm{\&sigma;}}_2+{\mathrm{j\&omega;\&epsiv;}}_2}{{\mathrm{\&sigma;}}_1+{\mathrm{j\&omega;\&epsiv;}}_1}=1,$ And capacitance coefficient $c_{}$ ${}_{12}=\frac{{\mathrm{\&epsiv;}}_2}{{\mathrm{\&epsiv;}}_1}=1.$ So, f ₁₂-c ₁₂Must equal 0, be embodied in the matrix V be exactly the element of relevant position be 0.If M ₁And M ₂σ and ε do not equate factor f so entirely ₁₂-c ₁₂And frequency dependence, just be not equal to 0, produce non-zero entry in the matrix V corresponding position.According to the matrix multiplication rule, has only V (perhaps V ^T) element of non-zero just can be at I+V ^TA _Cap ^-1The U generation (on the non-principal diagonal) non-zero entry.So the accurate size of M equals the total number of variable on the interface between " difference " medium.

Now, we can see, through type (10) is found the solution the efficient of comprehensive parameters will be very high, and this is that a demand is separated the inverse matrix of minor matrix M because under certain frequency.In addition, the operation of other matrix multiple is as A _Cap ^-1U, V ^TX _CapDeng also can being 0 to obtain quickening because of matrix U, the most of elements of V.Formula (10) is based on Woodbury formula strict on the mathematics, so be strict the establishment.

Formula (10) internal memory use amount is also not many, because have only A _Cap ^-1U, I+V ^TA _Cap ^-1U and (I+V ^TA _Cap ^-1U) ^-1In non-zero entry, also have X _CapNeed to use internal memory.Non-zero entry proportion wherein is very little, so the internal memory use amount can be not too many.Matrix A _Cap ^-1U, I+V ^TA _Cap ^-1U and (I+V ^TA _Cap ^-1U) ^-1Sparse property can be referring to example hereinafter.

4 method detailed process

Provide the method flow of practicability below.

1) input geometrical body information comprises: each medium M of substrate _iLength, conductivity, DIELECTRIC CONSTANT; The position of ports zone, size; Frequency of operation.

2) the coupling capacitance ExtractCap between the calculating substrate different port generates A _Cap, extract A _CapThe part element of middle part rows is formed U; Be specially:

Step 2.1. establishes 1V voltage on master port m, all the other ports (comprising k) are made as 0V;

Step 2.2. is medium M _iBoundary demarcation be N _iBoundary element Γ _Ij, j=1,2 ..., N _i

Step 2.3. is that collocation point is sampled point s with each boundary element central point respectively, lists system of equations A _CapX _Cap=B;

Wherein, if establish variable at boundary element Γ _IjOn, matrix A _CapElement value be:

The coefficient of voltage u type variable;

Or

The coefficient of non-interface electric field q type variable;

Or Interface Γ _AbThe coefficient of last q type variable, and collocation point s is at M _bOn;

Or

Interface Γ _AbThe coefficient of last q type variable, and collocation point s is at M _aOn;

Wherein, interface Γ _AbBe positioned at medium M _aAnd M _bBetween, a＜b; $u^{*}=\frac{1}{4\mathrm{\&pi;r}(s,v)},$ (s v) is the size of distance between collocation point s and variable point v to r, q ^*Be u ^*At Γ _IjOn the normal direction

Derivative, expression formula is $-\frac{1}{4\mathrm{\&pi;}{r}^2(s,v)}\frac{\&PartialD;\stackrel{\&RightArrow;}{r(s,v)}}{\&PartialD;\stackrel{\&RightArrow;}{n}},$ Wherein

The expression distance vector; $c_{\mathrm{ab}}=\frac{{\mathrm{\&epsiv;}}_b}{{\mathrm{\&epsiv;}}_a},$ ε _a, ε _bBe respectively medium M _aAnd M _bSpecific inductive capacity;

Each row of step 2.4. matrix B corresponding the setting of a master port m; To certain collocation point s, certain element value of B row is

Γ wherein _mBe the surface of master port m, q ^*Define the same;

Step 2.5. solving equation group A _CapX _Cap=B obtains $X_{\mathrm{cap}}=A_{\mathrm{cap}}^{-1}B;$ X _CapThe value that comprises electric field q type variable;

Step 2.6. can be by X _CapValue is calculated coupling capacitance:

$C_{\mathrm{mk}}={\&Integral;}_{{\mathrm{\&Gamma;}}_k}{\mathrm{\&epsiv;}}_{\left(k\right)}{q}_{\left(k\right)}\mathrm{d\&Gamma;}$

Γ wherein _kBe the surface of port k, q _(k)Be Γ _kOn electric field, ε _(k)It is the specific inductive capacity of being close to the medium of port k;

Step 2.7. generator matrix U: from A _CapMiddle replication form is

Element, all the other elements are made as 0;

Step 2.8. and step 2.4 are tried to achieve A similarly _Cap ^-1U;

3) to each Frequency point, carry out following operation, electric capacity is modified to comprehensive coupling parameter under this frequency:

Step 3.1. calculates the V matrix under this Frequency point: to those corresponding to medium M _aAnd M _bBetween the main diagonal element of electric field type variable on the interface, the numerical value of setting them is (f _Ab-c _Ab)/c _Ab, a＜b wherein, $f_{\mathrm{ab}}=\frac{{\mathrm{\&sigma;}}_b+{\mathrm{j\&omega;\&epsiv;}}_b}{{\mathrm{\&sigma;}}_a+{\mathrm{j\&omega;\&epsiv;}}_a},$ $c_{\mathrm{ab}}=\frac{{\mathrm{\&epsiv;}}_b}{{\mathrm{\&epsiv;}}_a},$ σ _a, σ _bAnd ε _a, ε _bBe respectively conductivity and the specific inductive capacity of medium a and b; All the other elements are made as 0;

Step 3.2. generates I+V ^TA _Cap ^-1U compresses it and to obtain less matrix M;

Step 3.3. inverts to M, obtains W=M ¹, and, obtain (I+V with the anti-unit matrix of filling out of W ^TA _Cap ^-1U) ^-1

Step 3.4. implements matrix and takes advantage of, and gets [A _Cap ^-1U] [(I+V ^TA _Cap ^-1U) ^-1] [V ^TX _Cap];

Step 3.5. matrix subtracts each other, and obtains X _Cap-{ [A _Cap ^-1U] [(I+V ^TA _Cap ^-1U) ^-1] [V ^TX _Cap];

Step 3.6. calculates comprehensive parameters by formula (9), and output;

Has coupling parameter all calculated and has finished under all Frequency points of step 3.7.? if not, then turn to step 3.1; Otherwise, turn to step 4;

4) finish.

Above process is seen Figure 11.

Description of drawings

Fig. 1 substrate coupling effect synoptic diagram.To influence some part of circuit by the coupled noise of substrate transfer, and and then influence overall performance.

The three-dimensional interconnect structure synoptic diagram of a limited area of Fig. 2 (need calculate the wherein coupling capacitance of leading body and other conductors).

The substrat structure synoptic diagram that Fig. 3 is made up of multilayered medium, a plurality of port (and ground plate).

The number in the figure meaning:

Mi: medium i, i=1～3

σ _i: conductivity, non-zero

ε _i: specific inductive capacity.

Γ ₁₂: the interface of medium M1 and medium M2

Γ ₂₃: the interface of medium M2 and medium M3

Fig. 4 directly uses Element BEM to extract the flow process of comprehensive parameters.

(referring to document " A New Boundary Element Method for Accurate Modeling of Lossy Substrates with ArbitraryDoping Profiles; " Asia and South Pacific Design Automation Conference, Yokohama, Japan, pp.683-688, Jan.2006)

The process flow diagram of higher-frequency coupling parameter computing method among Fig. 5 the present invention.

Annotate: the matrix element non-zero in the grey block.

Fig. 6 is to substrate shown in Figure 3, matrix A synoptic diagram in the formula (8).Matrix element non-zero in the grey block.

Annotate: the matrix element non-zero in the grey block.

Fig. 7 is to example shown in Figure 3, matrix A in the formula (6) _CapSynoptic diagram.

Fig. 8 matrix A and A _CapDifference.

Annotate:

The structure of U is consistent with A-Acap, and its non-zero entry numerical value equals the numerical value of Acap relevant position element.

The complete diagonal matrix of V right and wrong is made up of two diagonal matrixs, is respectively (f12-c12)/c12I1 and (f23-c23)/c23I2, and wherein I1 and I2 are unit matrix.

Fig. 9 left side figure is matrix I+V ^TA _Cap ^-1The structure of U, right figure is the equivalent matrice M after its compression.

Figure 10 obtains (I+V to counter the filling out in the unit matrix of the inverse matrix W=M-1 of matrix M shown in the left figure ^TA _Cap ^-1U) ^-1, as shown at right.

The detail flowchart of higher-frequency coupling parameter computing method among Figure 11 the present invention.

The simple substrate example that Figure 12 is made up of three layers of medium.

Symbol and relevant explanation among the figure:

σ: medium conductivity, unit is 1/ (Ω m));

ε _r: the relative dielectric constant of medium, unit are 1;

T: the height of medium, unit are um.

L: the length of medium, unit is um.

W: the width of medium, unit are um.

Two plane ports of top layer, size is 2um * 2um, and strict the symmetrical distribution is shown in right vertical view.

Embodiment

Extracting method of the present invention is realized with the C Plus Plus programming, can move on the (SuSE) Linux OS of UNIX operating system on the SUN SPARC series workstation and PC.The implementation that contains the SubRCExtract method below in conjunction with an instantiation explanation.

Figure 12 has shown a simple case.For asking simple, contact (port) is counted as the plane here, and is placed on the top.

Notice that here for the simple declaration implementation, specially the variable number with example falls very lowly.In following matrix, element is all only got one-bit digital behind the radix point.But in program is carried out, be that the floating number (float type or double type) with computer-internal is represented.For asking clear, the matrix null element is shown as sky.

1) the illustrated substrate information of input.

2) port one voltage being set is 1V, and port 2, ground plate are 0V.

Step 2.1. is divided into boundary element Γ to each dielectric boundaries _Ij

Step 2.2. is according to A _CapThe integral formula of element calculates its concrete numerical value and is: (matrix A _CapFor)

6.3	-0.8	-1.4	-1.4	13.3	-1.4	-13.3
																			-0.8	6.3	-1.4	-1.4	13.3	-1.4	-13.3
-1.4	-1.4	6.3	-0.8	13.3	-1.4	-13.3
																			-1.4	-1.4	-0.8	6.3	13.3	-1.4	-13.3
-1.4	-1.4	-1.4	-1.4	35.3	-0.8	-9.3
																			-1.4	-1.4	-1.4	-1.4	9.3	6.3	-35.3
					6.3	35.3	-1.4	-1.4	-1.4	-1.4	-0.8	-9.3
																								-1.4	13.3	6.3	-0.8	-1.4	-1.4	-1.4	-13.3
					-1.4	13.3	-0.8	6.3	-1.4	-1.4	-1.4	-13.3
																								-1.4	13.3	-1.4	-1.4	6.3	-0.8	-1.4	-13.3
					-1.4	13.3	-1.4	-1.4	-0.8	6.3	-1.4	-13.3
																								-0.8	9.3	-1.4	-1.4	-1.4	-1.4	6.3	-35.3
											6.3	109.8	-1.5	-1.5	-1.5	-1.5	0.5	0.5
																														-1.5	45.6	6.0	-1.0	-1.7	-1.7	0.9	0.7
											-1.5	45.6	-1.0	6.0	-1.7	-1.7	0.7	0.9
																														-1.5	45.6	-1.6	-1.6	5.9	-1.0	0.8	0.8
											-1.5	45.6	-1.6	-1.6	-1.0	5.9	0.8	0.8
																														-0.8	27.7	-1.9	-1.0	-1.3	-1.3	7.1	1.0
											-0.8	27.7	-1.0	-1.9	-1.3	-1.3	1.0	7.1

Generating the U most elements is 0, and all the other non-neck elements are A _CapA part, i.e. A _CapIn the part numeral that shows with black matrix.

						-13.3
																									-13.3

						-13.3
																									-13.3
						-9.3
																									-35.3
												-9.3
																															-13.3
												-13.3
																															-13.3
												-13.3
																															-35.3

3) solving equation group A _CapX _Cap=B, wherein B=

1.4

1.4

1.4

1.4

-6.3

0.8

0

0

0

0

0

0

0

0

0

0

0

0

0

$X_{\mathrm{cap}}=A_{\mathrm{cap}}^{-1}B=$

$1.0$

$1.0$

$1.0$

$1.0$

$-0.02$

$0.9$

$-0.02$

$0.8$

$0.8$

$0.8$

$0.8$

$0.8$

$-0.02$

$0.6$

$0.6$

$0.6$

$0.6$

$0.5$

$0.5$

Also can generate simultaneously $A_{\mathrm{cap}}^{-1}U=$

4) (be 10 with calculated rate here to next Frequency point ¹⁰Hz is that coupling parameter is an example under the 10GHz):

Step 4.1 generator matrix V ^T: two interfaces are calculated following coefficient:

${\mathrm{<figure-callout\; id="12"\; label="f"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">f</figure-callout>}}_{12}=\frac{{\mathrm{\&sigma;}}_2+{\mathrm{j\&omega;\&epsiv;}}_2}{{\mathrm{\&sigma;}}_1+{\mathrm{j\&omega;\&epsiv;}}_1}=\frac{2000+j6.28\&times;{10}^{10}\&times;11.8\&times;8.85\&times;{10}^{-12}}{{10+j6.28\&times;10}^{10}\&times;11.8\&times;8.85\&times;{10}^{-12}}={1.4\&times;10}^2-91.3i,$

${\mathrm{<figure-callout\; id="23"\; label="f"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">f</figure-callout>}}_{23}=\frac{{\mathrm{\&sigma;}}_3+{\mathrm{j\&omega;\&epsiv;}}_3}{{\mathrm{\&sigma;}}_2+{\mathrm{j\&omega;\&epsiv;}}_2}=\frac{1\&times;{10}^{-12}+j6.28\&times;{10}^{10}\&times;3.9\&times;8.85\&times;{10}^{-12}}{{2000+j6.28\&times;10}^{10}\&times;11.8\&times;8.85\&times;{10}^{-12}}={3.5\&times;10}^{-6}+1.1\&times;{10}^{-3}i,$

Annotate: j is identical with i, is complex unit, a square root just-1, down together.

${\mathrm{<figure-callout\; id="12"\; label="c"\; filenames="C20061001214000291.png,C20061001214000292.png"\; state="\{\{state\}\}">c</figure-callout>}}_{12}=\frac{{\mathrm{\&epsiv;}}_2}{{\mathrm{\&epsiv;}}_1}=\frac{11.8\&times;8.85\&times;{10}^{-12}}{11.8\&times;8.85\&times;{10}^{-12}}=1,$ $c_{23}=\frac{{\mathrm{\&epsiv;}}_3}{{\mathrm{\&epsiv;}}_2}=\frac{3.9\&times;8.85\&times;{10}^{-12}}{11.8\&times;8.85\&times;{10}^{-12}}=0.3.$

So, according to V ^TComputing formula, calculate V ^T:

						139.1-j91.

						3
												-1+j3.3e- 3

Step 4.2 generates I+V ^TA _Cap ^-1U compresses it and to obtain less matrix M;

$I+V^T{A}_{\mathrm{cap}}^{-1}U=$

Its compressed format matrix M is:

16.5-i10.2 -108.1+i71.0

-0.1+i3.6e-4 0.8+i7.3e-4

Step 4.3 couple M inverts, and obtains W=M ^-1, and, obtain (I+V with the anti-unit matrix of filling out of W ^TA _Cap ^-1U) ^-1

W＝

127.2-i

1.0-i5.9e-2

95.4

1.4e-1-i9.0e-3 19.4-i13.8

Fill out among the enough big unit matrix I W is counter, that obtain promptly is (I+V ^TA _Cap ^-1U) ^-1

																				1
		1
																						1
				1
																								1
						1.0-i5.9e-2						127.2-i 95.4
																										1
								1
																												1
										1
																														1
						1.4e-1-i9.0e- 3						19.4-i13.8
																																1
														1

															1
																																			1
																	1
																																					1

Step 4.4 is implemented matrix and is taken advantage of, $\left[A_{\mathrm{cap}}^{-1}U\right]\left[{(I+V^T{A}_{\mathrm{cap}}^{-1}U)}^{-1}\right]\left[V^T{X}_{\mathrm{cap}}\right]=$

$-3.3e-2+i3.0e-2$

$-3.3e-2+i3.0e-2$

$-3.3e-2+i3.0e-2$

$-3.3e-2+i3.0e-2$

$-8.8e-3+i8.1e-3$

$-6.5e-2+i5.9e-2$

$-1.5e-2+i6.1e-5$

$-1.2e-1+i5.9e-2$

$-1.2e-1+i5.9e-2$

$-1.2e-1+i5.9e-2$

$-1.2e-1+i5.9e-2$

$-1.7e-1+i6.0e-2$

$3.4e-3+i1.2e-3$

$-1.4e-1+i4.6e-2$

$-1.4e-1+i4.6e-2$

$-1.4e-1+i4.7e-2$

$-1.4e-1+i4.7e-2$

$-1.2e-1+i4.2e-2$

$-1.2e-1+i4.2e-2$

Step 4.5 matrix subtracts each other, and obtains $X_{\mathrm{cap}}-\left\{\right[A_{\mathrm{cap}}^{-1}U\left]\right[(I+V^T{A}_{\mathrm{cap}}^{-1}U){\mathrm{}}^{-1}\left]\right[V^T{X}_{\mathrm{cap}}\left]\right\}=$

$1.0+i3.0e-2$

$1.0+i3.0e-2$

$1.0+i3.0e-2$

$1.0+i3.0e-2$

$-6.3e-3-i8.1e-3$

$9.5e-1+i5.9e-2$

$-5.3e-2+i6.1e-5$

$9.5e-1+i5.9e-2$

$9.5e-1+i5.9e-2$

$9.5e-1+i5.9e-2$

$9.5e-1+i5.9e-2$

$9.5e-1+i6.0e-2$

$-1.8e-1+i1.2e-3$

$7.4e-1-4.6e-2$

$7.4e-1-i4.6e-2$

$7.5e-1-i4.7e-2$

$7.5e-1-i4.7e-2$

$6.7e-1-i4.2e-2$

$6.7e-1-i4.2e-2$

Step 4.6 is calculated comprehensive parameters by formula (9), and output:

Port one port 2 GroundPlane (ground connection

Plate)

Port one-3.37e+03+j1.35e+05-7.21e+04 1.02e+04

-j6.22e+05 -j1.63e+05

4.5) has coupling parameter all calculated and has finished under all Frequency points? if not, then turn to 4.1); If change 5).

5) finish.

Use identical example and identical parameter, the result of calculation of method is among the document ASP-DAC2006:

Port one port 2 GroundPlane (ground connection

Plate)

Port one-3.37e+03+j1.35e+05-7.21e+04 1.02e+04

-j6.22e+05 -j1.63e+05

Can find out that from last two forms the result of calculation of literature method is consistent with methods and results of the present invention, verify also that here theorem 2 (correction from electric capacity to the comprehensive parameters) do not lose computational accuracy.

Counting yield compares:

Above example has proved that method is correct.But, be not easy to relative efficiency because scale is too little, very little consuming time.

Another bigger calculated examples (variable number is 7252) is extracted comprehensive coupling parameter under arbitrary frequency, and the method among the present invention only needs 9.0 seconds, and method then needed more than 400 seconds among the document ASP-DAC2006, and speed-up ratio is more than 40 times.Simultaneously, method needs the 60MB internal memory among the present invention, and literature method needs the 45MB internal memory, and the method among the present invention needs more internal memory to be because need the extra matrix of storage, such as A _Cap ^-1B, M, W etc., but the use of these extra memorys can bring tens times speed-up ratio.

Claims (1)

1. The method for quickly extracting comprehensive coupling parameters at multiple frequency points on a silicon integrated circuit substrate has the following steps in turn: Step 1: Input geometric information to the computer, including a) the length, width and height of each medium _Mi of the substrate; b) the electrical conductivity σ and dielectric constant ε of each medium _{Mi of} the substrate; c) The location and size of the port area; d) Working frequency; Step 2: The computer calculates the coupling capacitance C _mk between the substrate main port m and port k according to the following steps: Step 2.1. Set a voltage of 1V on the main port m, and set a voltage of 0V on the remaining ports including k; Step 2.2. Divide the boundary of medium M _i into N _i boundary elements Γ _ij , j=1, 2, ..., N _i ; Step 2.3. Take the central point of each boundary element as the configuration point, that is, the sampling point s, and list the equation system A _cap X _cap = B; Among them, if the variable is set on the boundary element Γ _ij , the element value of the matrix A _cap is:

coefficient of the voltage u-variable; or

The coefficient of the q-type variable of the electric field at the non-interface; or

The coefficient of the q-type variable on the interface Γ _ab , and the configuration point s is on M _b ; or

The coefficient of the q-type variable on the interface Γ _ab , and the configuration point s is on _Ma ; _Among them, the interface Γ _ab is located between the media Ma and M _b , a<b; $u^{*}=\frac{1}{4\mathrm{\&pi;r}(\mathrm{the\; s},v)},$ r(s, v) is the distance between configuration point s and variable point v, q ^* is u ^* in the normal direction of Γ _ij

The derivative of , the expression is

in

represents the distance vector; $c_{\mathrm{ab}}=\frac{{\mathrm{\&epsiv;}}_b}{{\mathrm{\&epsiv;}}_a},$ ε _a , ε _b are the dielectric constants of media Ma and _{M b respectively} ; Step 2.4. Each column of the matrix B corresponds to the setting of a main port m; for a certain configuration point s, the value of a certain element in a certain column of B is

where _Γm is the surface of the main port m, and q ^* is defined as above; Step 2.5. Solve the system of equations A _cap X _cap = B, get $x_{\mathrm{cap}}=A_{\mathrm{cap}}^{-1}B;$ X _cap contains the value of the electric field q-type variable; Step 2.6. The coupling capacitance can be calculated from the X _cap value: ${\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; mk</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}}{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}$ where Γ _k is the surface of port k, q _(k) is the electric field on Γ _k , and ε _(k) is the dielectric constant of the medium next to port k; Step 2.7. Generate matrix U: Copy from A _cap as

The elements of , and the rest of the elements are set to 0; Step 2.8. Obtain A _cap ^-1 U similarly to step 2.4; Step 3 For each frequency point, perform the following correction operations: Step 3.1. Calculate the V matrix under this frequency point: for those main diagonal elements corresponding to the electric field type variable on the interface between the medium _Ma and M _b , set their values as (f _ab -c _ab ) /c _ab where a<b, $f_{\mathrm{ab}}=\frac{{\mathrm{\&sigma;}}_b+\mathrm{j\&omega;}{\mathrm{\&epsiv;}}_b}{{\mathrm{\&sigma;}}_a+\mathrm{j\&omega;}{\mathrm{\&epsiv;}}_a},$ $c_{\mathrm{ab}}=\frac{{\mathrm{\&epsiv;}}_b}{{\mathrm{\&epsiv;}}_a},$ σ _a , σ _b and ε _a , ε _b are the conductivity and permittivity of medium a and b respectively, ω is the angular frequency; the other elements are set to 0; Step 3.2. Generate a matrix I+V ^T A _cap ^-1 U, where I is a unit matrix of the same size as A _cap ; extract the non-zero blocks on the off-diagonal lines of the matrix to form a matrix M; Step 3.3. To the inverse matrix W of M, and backfill it to the identity matrix, get (I+V ^T A _cap ^-1 U) ^-1 ; Step 3.4. Implement matrix multiplication to get [A _cap ^-1 U][(I+V ^T A _cap ^-1 U) ^-1 ][V ^T X _cap ]; Step 3.5. Matrix subtraction, get $x=x_{\mathrm{cap}}-\left\{\right[A_{\mathrm{cap}}^{-1}u\left]\right[{(I+V^T{A}_{\mathrm{cap}}^{-1}u)}^{-1}\left]\right[V^T{x}_{\mathrm{cap}}\left]\right\};$ Step 3.6. From the value of the electric field q variable contained in X, calculate the comprehensive coupling parameter at this frequency ${\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; mk</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j\&omega;</span>}{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; q</span>}}_{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}}<span\; class="notranslate">\; ,</span>$ Where Γ _k is the surface of port k, q _(k) is the electric field value on Γ _k , σ _(k) and ε _(k) are the conductivity and permittivity of the medium close to port k, respectively; Step 3.7. Have the coupling parameters at all frequency points been calculated? If not, go to step 3.1; otherwise, go to step 4; Step 4 ends.

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