Fast universal passive macro-modeling method based on Hamiltonian matrix pure virtual feature values

Abstract

The invention discloses a fast universal passivity macro-modeling method based on Hamiltonian matrix pure virtual feature values. The method comprises the steps that n-port passive network parameters obtained through simulation or measurement serve as an original data set; rational fitting is carried out on the passive data set, and a macro model is established; calculating a pure virtual characteristic value of a Hamiltonian matrix M of the macro model, and judging the passivity of the macro model; carrying out passivity repair on the initial model according to a passivity judgment result; and inputting the passive macro model into a circuit simulator for simulation. According to the method, only the pure virtual feature value of the Hamiltonian matrix is calculated to achieve rapid passivity detection of the macro model, the method is suitable for macro modeling of symmetric and asymmetric models, the method has the characteristic of universality, and when the method is applied to integrated circuit design, the convergence of circuit simulation can be improved, the macro modeling time is shortened, the circuit simulation is accelerated, and the circuit simulation efficiency is improved. And the design and market period of electronic products is shortened.

Description

Technical Field

[0001] The invention relates to the technical field of circuit simulation, and in particular to a fast universal passive macro modeling method based on pure imaginary eigenvalues ​​of a Hamiltonian matrix. Background Art

[0002] In high-speed signal systems, interconnection and packaging will cause signal and power integrity problems, resulting in ringing, reflection, distortion, crosstalk, and delay. If effective measures are not taken in circuit design, serious performance problems or even failures will occur in electronic systems. Time is very valuable in the design cycle of electronic products. In order to bring the designed products to market as quickly as possible, iterative modifications to the system should be avoided as much as possible. Macro models are widely used in signal and power integrity assessments to estimate transient waveforms of signals and power supplies, and to verify whether they meet design constraints and avoid design risks.

[0003] The macro model is usually derived from the parameters of the n-port passive network, and the macro model is constructed using the vector fitting algorithm. However, the vector fitting algorithm cannot guarantee that the established macro model is passive. If the passivity constraint is not met, it may cause the transient simulation to not converge, that is, the system will amplify the input energy, which is inconsistent with the inherent properties of the physical device. Therefore, generating a macro model that meets the passivity is very important for signal and power integrity analysis.

[0004] The passivity detection process of the model usually takes up most of the time during the macro modeling process. The passivity evaluation scheme of the macro model generally adopts the method of calculating the eigenvalues ​​of the Hamiltonian matrix. A typical evaluation method is the scattering parameter equivalent circuit passivity correction system and method disclosed in patent document CN102486810B. This is an algebraic method that uses the call of the eigenvalue function to obtain all the eigenvalues ​​of the Hamiltonian matrix (the eigenvalue is a complex number), and then detects the interval of passivity violation. However, in some complex packaging cases, the macro model generated by vector fitting may have a very high order N, such as thousands or even tens of thousands. In addition, the algorithm complexity of solving all the eigenvalues ​​of the Hamiltonian matrix is ​​O(N 3 ), where N is the order. For a model with tens of thousands of orders, it often takes several hours to solve all the eigenvalues ​​of the Hamiltonian matrix. The process of solving all the eigenvalues ​​of the Hamiltonian matrix seriously restricts the speed of passive macro modeling, which in turn determines the circuit design cycle. Therefore, the need to reduce the time for solving eigenvalues ​​is very urgent.

[0005] In addition, there is a half-size detection matrix, which is only half the size of the Hamiltonian matrix and can calculate eigenvalues ​​8 times faster. Although the half-size detection matrix has the advantage of faster performance, it is only applicable to symmetric models, that is, all ports have common poles, and cannot be applied to asymmetric models, that is, matrix fitting. However, in large-port passive network parameters, matrix fitting has performance and accuracy issues, so the use scenarios of the half-size detection matrix are limited.

[0006] Therefore, developing a general fast passivity macromodeling algorithm that is applicable not only to symmetric models but also to asymmetric models is of great significance for accelerating circuit simulation and promoting circuit design and development. Summary of the invention

[0007] In order to overcome the deficiencies in the prior art, the present invention provides a fast and universal passivity macro modeling method based on the pure imaginary eigenvalues ​​of the Hamiltonian matrix. The method realizes fast passivity detection of the macro model by only calculating the pure imaginary eigenvalues ​​of the Hamiltonian matrix. The method is suitable for macro modeling of symmetric and asymmetric models and has universal characteristics. By applying the method to integrated circuit design, the convergence of circuit simulation can be improved, the macro modeling time can be reduced, and the circuit simulation can be accelerated, thereby shortening the design and market launch cycle of electronic products.

[0008] The embodiment of the present invention provides the following solution:

[0009] An embodiment of the present invention provides a fast universal passivity macromodeling method based on pure imaginary eigenvalues ​​of a Hamiltonian matrix, the method comprising:

[0010] Step 1: The n-port passive network parameters obtained by simulation or measurement are used as the original data set and pre-processed as the passive data set;

[0011] Step 2: Perform rational fitting on the passive data set to establish a macro model;

[0012] Step 3: Calculate the pure imaginary eigenvalues ​​of the Hamiltonian matrix M of the macro model to determine the passivity of the macro model;

[0013] Step 4: According to the result of passivity judgment, the initial model is repaired for passivity;

[0014] Step 5: Input the passive macro model into the circuit simulator for simulation.

[0015] In an optional embodiment, the macro model described in step 2 is:

[0016] H(s)=C(sI-A) -1 B+D

[0017] Among them, A represents the pole matrix, B represents the state matrix, C represents the residue matrix, D represents the constant matrix, s represents the complex frequency, and I represents the identity matrix.

[0018] In an optional embodiment, step three includes the following process:

[0019] S3.1. For the Hamiltonian matrix M of the S parameter,

[0020]

[0021] Adaptive sampling strategy is adopted to divide the frequency domain interval into passive area and suspicious area, where the suspicious area is the area where the passive area is removed;

[0022] S3.2. Perform complex shifts in suspicious areas using binary search to construct a matrix Where v is the complex shift iteratively generated during the binary division process, and the calculation matrix Low-order Galerkin projections on d-dimensional Krylov subspaces The characteristic value ξ j , based on the eigenvalue ξ j Calculate the eigenvalue λ of the Hamiltonian matrix M j The approximate eigenvalues ​​of

[0023] S3.3. Based on the approximate eigenvalue Pure imaginary eigenvalues ​​are screened out to obtain the passivity violation interval.

[0024] In an optional embodiment, in step S3.3, the Heisenberg matrix is ​​constructed using the Arnoldi algorithm

[0025] In an optional embodiment, the Heisenberg matrix The construction includes the following steps: randomly initialize vector v1, construct The d-dimensional Krylov subspace formed with v1 is: Using the Arnoldi algorithm to calculate the d-dimensional orthogonal basis V of the Krylov subspace d =[v1,v2,…,v d ], then the Hessenberg matrix is

[0026] In an optional embodiment, step four includes the following process: if the detection result is passive and the initial model meets the passivity condition, the initial model is output as a passive model; if the detection result is active and the initial model does not meet the passivity condition, passivity repair is performed.

[0027] In an optional embodiment, step four includes the following process: the passive repair adopts a residue perturbation strategy, using a lateral squeezing method and / or a longitudinal squeezing method to make the initial model free of pure imaginary eigenvalues.

[0028] In an optional embodiment, step four includes the following process: if the lateral squeezing method is adopted, the eigenvalues ​​of the Hamiltonian matrix are disturbed so that the pure imaginary eigenvalues ​​leave the imaginary axis.

[0029] In an optional embodiment, step four includes the following process: if the longitudinal squeezing method is adopted, the singular value at the maximum violation in the passivity violation interval is disturbed so that the maximum singular value is less than or equal to 1.

[0030] The beneficial effects of the present invention based on its technical solution are:

[0031] The invention provides a fast universal passivity macro modeling method based on the pure imaginary eigenvalue of the Hamiltonian matrix. By using adaptive sampling and Arnoldi algorithm, only the eigenvalue of the pure imaginary part of the Hamiltonian matrix is ​​calculated to realize fast passivity detection of the macro model. Theoretically, the algorithm complexity can be reduced to O(N 2 ), its accuracy meets the needs of passive macro modeling, has good feasibility, and has no restrictions on model symmetry and asymmetry. It has universal characteristics. Applying it to integrated circuit design can improve the convergence of circuit simulation, reduce macro modeling time and accelerate circuit simulation, thereby shortening the design of electronic products and the time to market. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] In order to more clearly illustrate the embodiments of this specification or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this specification. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0033] Figure 1 A schematic flow chart of a fast universal passivity macromodeling method based on pure imaginary eigenvalues ​​of a Hamiltonian matrix provided by the present invention.

[0034] Figure 2 This is a singular value curve diagram before and after the sample points are refined in this embodiment.

[0035] Figure 3 It is a schematic diagram of the result of estimating the pure imaginary eigenvalues ​​of the Hamiltonian matrix in this embodiment.

[0036] Figure 4 FIG. 1 is a schematic diagram of an output waveform obtained by the passive macro model of this embodiment in circuit simulation. DETAILED DESCRIPTION

[0037] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention belong to the scope of protection of the embodiments of the present invention.

[0038] Reference Figure 1 The embodiment of the present invention provides a fast universal passivity macro modeling method based on pure imaginary eigenvalues ​​of Hamiltonian matrix, the method comprising:

[0039] Step 1: The n-port passive network parameters obtained by simulation or measurement are used as the original data set and preprocessed as the passive data set.

[0040] In this embodiment, the n-port passive network parameters are taken as S parameters. The processing of Y parameters and Z parameters is similar to that of S parameters.

[0041] The form of the n-port S-parameter matrix is:

[0042]

[0043] Its singular value decomposition is U k represents the kth left singular matrix, V k represents the kth right singular matrix, Σ k is the singular value matrix, K is the number of sample points of the S parameter, and

[0044]

[0045] If the frequency w k The S parameter at is passive, that is, σ1≤1, where σ1 is the maximum singular value; if the frequency point w k The S parameter at is active, that is, σ1>1, The data of the frequency point is corrected and saved.

[0046] Finally, the passivity-corrected S parameters, namely the passive data set, are formed.

[0047] Step 2: Perform rational fitting on the passive data set. You can use matrix fitting, column fitting and element fitting. Matrix fitting requires all ports to have common poles, column fitting only requires the same port to have common poles, and element fitting can fit the S parameter curve with different poles. According to vector fitting, establish its initial transfer function model H(s) = C(sI-A) -1 B+D.

[0048] Among them, A represents the pole matrix, B represents the state matrix, C represents the residue matrix, D represents the constant matrix, s represents the complex frequency, and I represents the identity matrix, all of which are determined by rational fitting. For any S parameter curve S ij ,have:

[0049]

[0050] In the formula, p n =α n +jβ n For curve S ij The nth pole of n For p n The corresponding residue, d ij For curve S ij The constant term of .

[0051] Step 3: Calculate the pure imaginary eigenvalue of the Hamiltonian matrix M of the macro model to determine the passivity of the macro model. This includes the following steps:

[0052] S3.1, using adaptive sampling strategy, divide the frequency domain interval into passive area and suspicious area, where the suspicious area is the area without passive area. It includes the following process:

[0053] (1) According to the Arnoldi algorithm, for the Hamiltonian matrix M of the S parameter,

[0054]

[0055] Estimate the largest eigenvalue λ of the Hamiltonian matrix M max , then the pure imaginary eigenvalue λ∈[0,w max ], where w max =|λ max |.

[0056] (2) Using poles to adaptively generate sampling frequencies:

[0057]

[0058] Among them, w n,r represents the rth sampling point of the nth pole, β n represents the extreme point p n The imaginary part, α n represents the extreme point p n The real part of R represents the number of initial sampling points used to control the generation of each pole. Sort the sampling frequency points generated by each pole and limit w n,r ∈[0,w max ]. Large macro models may result in adjacent sampling frequencies being very close, so it is necessary to limit the minimum frequency spacing Δw between sampling points. And ρ>>1, ρ represents the frequency interval control factor. Finally, the sampling frequency points are further processed according to Δw, and the sampling points whose adjacent intervals are less than Δw are removed to form the initial sample points.

[0059] Reference Figure 2 , which is a schematic diagram of a singular value curve of the sampling points and the S parameters of the corresponding frequency points generated according to the pole distribution after vector fitting in this embodiment. Figure 2 As shown in (1), the initial sampling points provide a rough description of passivity. Especially at low frequencies, there are fewer sample points, and the initial samples cannot accurately describe the changes in singular values. Therefore, it is necessary to refine the low frequencies and add more frequency points.

[0060] (3) Refine the sample points based on singular value decomposition and correlation measurement. The initial sample points only roughly depict the passivity of the transfer function. For the boundary area between passive and active and the area where the singular value changes rapidly, these sample points cannot effectively reflect the change of passivity. Therefore, an adaptive strategy is needed to refine the initial sampling points. Assume that two adjacent sample points are S(w k ) and S(w k+1 ):

[0061]

[0062] Let σ k is the set of diagonal elements, that is, σ k =diag(Σ k ).

[0063] According to the singular value vector σ k and σ k+1 Calculate the cosine similarity, that is:

[0064]

[0065] If r k <γ, then at the adjacent frequency point w k and w k+1 Insert midpoint And repeat the above refinement process. Usually set γ = 0.9, and the degree of refinement can be controlled by changing the value of γ. The above refinement strategy is simple and easy to implement, and only needs to calculate the singular value vector, which saves more memory and computing resources. Figure 2 The right figure in (2) is the result of refinement based on the initial sample points according to the above process. The refined sample points add sample points at low frequencies and where the singular values ​​change rapidly, so more details can be obtained. This is conducive to the subsequent division of passive areas and avoids omissions and misjudgments.

[0066] (4) According to the refined sample points, [0, w max] is divided into passive area Ω passive and suspicious areaΩ suspect , which are defined as follows:

[0067]

[0068] Ω suspect ∪Ω passive =[0,w max ]

[0069] Usually, setting ε = 0.01 can avoid misjudgment. Figure 3 , the area below the horizontal dotted line represents the passive area, and the area above the horizontal dotted line represents the suspicious area. The diamond represents the eigenvalue of the Hamiltonian matrix, the horizontal axis represents the real part of the eigenvalue, and the vertical axis represents the imaginary part of the eigenvalue. For clarity, only the eigenvalues ​​with positive imaginary parts are displayed. Pure imaginary eigenvalues ​​are distributed on the imaginary axis.

[0070] S3.2, in the suspicious area Ω suspect The binary method is used to perform complex shifts until the convergence circle covers the entire suspicious interval, thereby quickly estimating all pure imaginary eigenvalues ​​of the Hamiltonian matrix. The following process is included:

[0071] (1) Assume the suspicious interval Ω q ∈Ω suspect , and Ω q =[w q-1 , w q ].

[0072] For a complex shift θ, substitute Computing low-order Galerkin projections onto d-dimensional Krylov subspaces The characteristic value ξ j , the construction process is:

[0073]

[0074] Among them, A -θ =(A+θI) -1 , A θ =A +θ =(A-θI) -1 , H -θ =D-CA -θ B, H θ =D-CA θ B.

[0075] (2) Constructing the Heisenberg matrix The construction includes the following steps: randomly initialize vector v1, construct The d-dimensional Krylov subspace formed with v1 is: Using the Arnoldi algorithm to calculate the d-dimensional orthogonal basis V of the Krylov subspace d =[v1,v2,…,v d ], then the Hessenberg matrix is

[0076] (3) Calculate the Heisenberg matrix The characteristic value ξ j , then That is, the Hamiltonian matrix is ​​closest to λ j Some approximate eigenvalues ​​of Correspondingly, the convergence circle corresponding to the complex shift θ is

[0077] Φ(s,θ)={s:|s-θ|≤τ}

[0078]

[0079] Where τ is the convergence radius. Then, we only need to q =[w q-1 , w q ], we can ensure that the convergence circle Φ(s,θ) covers Ω by using the bisection method to determine the complex shift θ according to the convergence radius τ. q . In Ω suspect By repeating the above process, all the pure imaginary eigenvalues ​​of the Hamiltonian matrix can be found, thereby achieving the purpose of rapid passivity detection.

[0080] Reference Figure 3 , the five-pointed star represents the complex shift θ selected by the dichotomy method, and the circle represents the convergence circle corresponding to the complex shift θ. The eigenvalues ​​in the convergence circle are all eigenvalues ​​estimated by the Arnoldi algorithm. The four convergence circles cover the entire suspicious area, indicating that all pure imaginary eigenvalues ​​of the Hamiltonian matrix have been found.

[0081] (4) Assume that the estimated m pure imaginary eigenvalues ​​are These purely imaginary eigenvalues ​​correspond to the boundaries of the passivity violation. Therefore, in order to obtain the passivity violation interval, we also need to calculate midpoint according to Judgment interval Is it active or passive? Finally, the complete passivity violation interval Ω is obtained.

[0082] Step 4: According to the result of passivity judgment, the initial model is repaired for passivity: if the detection result is passive and the initial model meets the passivity condition, the initial model is output as a passive model; if the detection result is active and the initial model does not meet the passivity condition, the passivity repair is performed.

[0083] Passivity repair adopts the residue perturbation strategy. If the lateral squeezing algorithm is used, that is, the eigenvalues ​​of the Hamiltonian matrix are perturbed so that the two pure imaginary eigenvalues ​​move toward the midpoint and collide, and gradually leave the pure imaginary axis to achieve the purpose of passivity repair. According to the above principle, the controllability Gramian factor W is calculated. c The corresponding Cholesky factorization Q c , build an optimization model:

[0084] stZξ=η

[0085]

[0086] Solve the above optimization model to get ξ, and then recover ΔC, then the perturbed model is

[0087] If the longitudinal squeezing algorithm is used, that is, the singular value at the maximum violation in the passivity violation interval is disturbed so that the maximum singular value is less than or equal to 1. In order to facilitate the subsequent description, the function Θ(jω) = H(jω) is introduced. The passivity detection divides the frequency axis into passive and active regions. The active region For each passivity violation interval [w k , w k+1 ], find the maximum singular value of Θ(jω) (the processing process of eigenvalue is similar, Θ(jω)=IH(jω) H H(jω)), that is

[0088]

[0089] Make the maximum singular value of each passivity violation interval less than or equal to 1, then the interval can be made passive, that is, after the disturbance, it satisfies

[0090]

[0091] According to the above principles, an optimization model is established:

[0092]

[0093] Repeat steps 3 and 4 until the perturbed model satisfies passivity.

[0094] Step 5: Input the passive macro model into the circuit simulator for simulation. The output waveform is as follows: Figure 4 The waveform diagram shows that the circuit has successfully converged to a steady state, and the resonant frequency meets the circuit design expectations.

[0095] Those skilled in the art will appreciate that embodiments of the present invention may be provided as methods, systems, or computer program products. Therefore, the present invention may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present invention may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0096] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (modules, systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded computer, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0097] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0098] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0099] Although the preferred embodiments of the present invention have been described, those skilled in the art may make other changes and modifications to these embodiments once they have learned the basic creative concept. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present invention.

[0100] Obviously, those skilled in the art can make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include these modifications and variations.

Claims

1. A fast general passivity macromodeling method based on pure imaginary eigenvalues ​​of Hamiltonian matrix, characterized in that: The method comprises: Step 1, the n-port passive network parameters obtained by simulation or measurement are used as the original data set, and pre-processed as the passive data set; Step 2: Perform rational fitting on the passive data set to establish a macro model; Step 3: Calculate the pure imaginary eigenvalues ​​of the Hamiltonian matrix M of the macro model to determine the passivity of the macro model; Step 4: According to the result of passivity judgment, the initial model is repaired for passivity; Step 5: Input the passive macro model into the circuit simulator for simulation.

2. The fast universal passivity macromodeling method based on pure imaginary eigenvalues ​​of Hamiltonian matrix according to claim 1, characterized in that: The macro model described in step 2 is: H(s)=C(sI-A) -1 B+D Among them, A represents the pole matrix, B represents the state matrix, C represents the residue matrix, D represents the constant matrix, s represents the complex frequency, and I represents the identity matrix.

3. The fast universal passivity macromodeling method based on pure imaginary eigenvalues ​​of Hamiltonian matrix according to claim 2, characterized in that: Step 3 The process includes: S3.

1. For the Hamiltonian matrix M of the S parameter, Adaptive sampling strategy is adopted to divide the frequency domain interval into passive area and suspicious area, where the suspicious area is the area where the passive area is removed; S3.

2. Use binary division to perform complex shift in suspicious areas and construct a matrix Where v is the complex shift iteratively generated during the binary division process, and the calculation matrix Low-order Galerkin projections on d-dimensional Krylov subspaces The characteristic value ξ j , based on the eigenvalue ξ j Calculate the eigenvalue λ of the Hamiltonian matrix M j The approximate eigenvalues ​​of S3.

3. Based on the approximate eigenvalue Pure imaginary eigenvalues ​​are screened out to obtain the passivity violation interval.

4. The fast universal passivity macromodeling method based on pure imaginary eigenvalues ​​of Hamiltonian matrix according to claim 3, characterized in that: In step S3.3, the Heisenberg matrix is ​​constructed using the Arnoldi algorithm 5. The fast universal passivity macromodeling method based on pure imaginary eigenvalues ​​of Hamiltonian matrix according to claim 3, characterized in that: The Heisenberg matrix The construction includes the following steps: randomly initialize vector v1, construct The d-dimensional Krylov subspace formed with v1 is: Using the Arnoldi algorithm to calculate the d-dimensional orthogonal basis V of the Krylov subspace d =[v1,v2,…,v d ], then the Hessenberg matrix is 6. The fast universal passivity macromodeling method based on pure imaginary eigenvalues ​​of Hamiltonian matrix according to claim 1, characterized in that: Step 4 includes the following process: if the detection result is passive and the initial model meets the passivity condition, the initial model is output as a passive model; if the detection result is active and the initial model does not meet the passivity condition, the passivity repair is performed.

7. The fast universal passivity macromodeling method based on pure imaginary eigenvalues ​​of Hamiltonian matrix according to claim 6, characterized in that: Step 4 includes the following process: Passivity repair adopts a residue perturbation strategy, using a lateral squeezing method and / or a longitudinal squeezing method to make the initial model have no pure imaginary eigenvalues.

8. The fast universal passivity macromodeling method based on pure imaginary eigenvalues ​​of Hamiltonian matrix according to claim 7, characterized in that: Step 4 includes the following process: if the lateral squeezing method is used, the eigenvalues ​​of the Hamiltonian matrix are perturbed so that the pure imaginary eigenvalues ​​leave the imaginary axis.

9. The fast universal passivity macromodeling method based on pure imaginary eigenvalues ​​of Hamiltonian matrix according to claim 7, characterized in that: Step 4 includes the following process: if the longitudinal squeezing method is used, the singular value at the maximum violation in the passivity violation interval is disturbed so that the maximum singular value is less than or equal to 1.

Images