CN108108557A - Nport problems auto-adapted fitting and emulation mode based on Vector fitting method - Google Patents
Nport problems auto-adapted fitting and emulation mode based on Vector fitting method Download PDFInfo
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Abstract
The present invention provides a kind of nport problems auto-adapted fittings and emulation mode based on Vector fitting method, comprise the following steps：(1) auto-adapted fitting is carried out to S parameter with Vector fitting method, obtains the optimal fitting result S of fitting exponent number K and time delay dij(s), wherein 1≤i, j≤n；(2) by Laplace inverse transformations, the transmission function S on frequency domainij(s) the excitation function s being converted into time domainij(t)；(3) convolution of S parameter is calculated with recursion method, so as to obtain the algebraic equation of nport models.The nport problems auto-adapted fitting and emulation mode based on Vector fitting method of the present invention, auto-adapted fitting is carried out to S parameter using Vector fitting method and recursive convolution method quickly calculates the convolution of S parameter, higher fitting precision, better fitting effect are reached, and the computation scheme that a kind of form is simple, precision is higher are provided by Taylor approximations.
Description
Technical Field
The invention relates to the technical field of circuit simulation, in particular to an nport problem adaptive fitting and simulation method based on a vector matching method.
Background
Vector matching is a stable and efficient fitting method proposed by Gustavsen in the literature "Rational adaptation of frequency domain responses by vector matching" (published in "IEEE Transactions on Power Delivery" volume 14, 3 rd). The vector matching method is particularly suitable for modeling related S parameters (Y parameters or Z parameters) in circuit simulation, and has the following advantages compared with other fitting methods:
(1) The vector matching method is to give an initial pole in principle, solve two linear least square equations to obtain a modified pole, generally only need iteration of limited steps to obtain the pole meeting the requirements, and has high convergence speed;
(2) The S parameters obtained by actual measurement usually only contain limited range frequency response, and the result fitted by the vector matching method can theoretically process the frequency response in any range;
(3) Other fitting methods suffer from numerical problems when fitting a measured frequency response over a wide frequency range using high-order rational functions, especially in the case of noisy frequency responses, while the vector matching method is not affected.
The method adopts a rational function to approximately fit a network function H(s), and the partial sum form is as follows:
wherein r is a constant term, being a real number, p k And q is k Respectively, the pole and the residue. The approximate fitting method is to replace the initial poles with a group of corrected poles, the corrected poles are obtained by a pole repositioning method based on linear least square, and the fitting order is equal to the number of the initial poles. The selection of each pair of conjugate poles is as follows:
to ensure stability of the fit, the selected poles should all lie in the left half plane of the complex plane.
The recursive convolution method is a method for calculating convolution proposed in the literature, "transition Simulation of loss interconnection based on the recursive convolution equation" (published in "EEE Transactions on Circuits and Systems" journal No. 39, vol. 11), by Shen Lin and Ernest S.Kuh. When the convolution is calculated by the traditional method, in order to obtain the convolution of the current time, the convolution needs to be integrated from zero time to the current time, and the convolution calculation needs to be subjected to inverse folding, moving, multiplying and adding, so that the convolution operation is very complicated, and the simulation time is difficult to bear. When the convolution of the current moment is calculated by the recursive convolution method, only the convolution result of the previous moment is needed to be utilized, so that the defect is avoided, and the calculation efficiency is greatly improved.
In rf circuit analysis, it is difficult to measure the current and voltage on each port directly, so the n-port network problem is usually described by the S-parameter. The scattering parameter is a network parameter based on the relationship between incident and reflected waves, and describes the network in terms of the reflected signal at a port of the device and the signal transmitted from that port to another port. The scattering matrix is a matrix reflecting the relationship between the incident and reflected waves at the port. FIG. 1 shows a two-port network model, where A 1 ,A 2 Is an incident wave, B 1 ,B 2 Is a reflected wave, V 1 ,V 2 Is the voltage, I 1 ,I 2 Is an electric current. FIG. 2 shows an n-port network model, let A i ,B i Incident wave and reflected wave, V, of the ith port i ,I i Respectively, voltage and current, S = (S) ij ) I is more than or equal to 1 and less than or equal to n, j is more than or equal to 1 and less than or equal to n, and the incident wave and the reflected wave have the following relationship:
wherein the content of the first and second substances,R i is a reference resistance.
Thus, there is a formula in the frequency domain
Conversion to time domain has
Wherein the convolutions ij (t)＝L -1 (S ij ) Is an excitation function, L -1 Representing the inverse Laplace transform. Substitution intoTo the above formula, get the formula
Simplifying to obtain a model equation of the ith port:
note that the key to solving the above model equation is how to compute the excitation function s ij Is performed. The traditional method transforms S parameters (scaling parameters) in the frequency domain into excitation functions in the time domain by Fourier inverse transform, and then calculates convolution by numerical discrete method. When convolution is performed by using the method, in order to obtain convolution of the current time, integration from zero time to the current time is requiredThe longer the simulation time, and thus the more costly it is to compute the convolution at the current time, the computation time is often intolerable.
In order to reduce the time of convolution calculation, a recursive method is often considered to calculate the convolution. The recursion method has the advantages that in order to obtain the convolution at the current moment during convolution, only the convolution result at the previous moment is needed, so that complex calculation is avoided, and the calculation efficiency is greatly improved. Moreover, compared with the conventional method, the recursive method does not need to perform causal correction (correction) but only needs to perform passive correction (passive correction) when calculating the convolution for the S parameter. Reference is made to the document "Passive Based Sample Selection and Adaptive Vector Fitting Algorithm for policy-response Modeling of spark Frequency Domain Data" (behavial Modeling and mapping reference, 2004.Proceedings of the 2004IEEE International).
The general procedure for solving the nport problem by the recursion method is that a rational formula is fitted to the S parameters in a given discrete frequency domain by a vector matching method, then an expression of the S parameters in a time domain is solved, and finally the convolution of the S parameters in the time domain is calculated by the recursion method, so that an algebraic equation of the nport model (1) is obtained.
nport models also typically take into account the delay effect, i.e., the time delay (time delay) that a signal experiences as it passes through an nport device, which is manifested in the frequency domain as a shift in the amplitude and phase of the signal. The delay effect is related to the properties of the nport device and the frequency of the input signal. In practical applications, we usually assume that the delay time is constant. Such an assumption is reasonable over a range of frequencies.
When solving the nport problem by using a recursion method, how to effectively extract the delay time of the S parameter and solve the rational-factorial fitting of the optimal order and how to combine the circuit simulation characteristics and quickly calculate the convolution by using the recursion method have very important significance for improving the simulation precision and speed of the nport problem.
Disclosure of Invention
In order to solve the defects in the prior art, the invention provides an nport problem self-adaptive fitting and simulation method based on a vector matching method. When the S parameter is fitted, a rational fraction fitting type with higher precision is obtained by automatically selecting a fitting order and extracting the optimal delay time; meanwhile, when the convolution of the S parameter is calculated, a recursive calculation format with simple form and high precision is obtained through Taylor approximation according to the characteristics of circuit simulation.
In order to achieve the purpose, the nport problem self-adaptive fitting and simulation method based on the vector matching method comprises the following steps:
(1) Carrying out self-adaptive fitting on the S parameter by using a vector matching method to obtain a fitting result S with optimal fitting order K and delay time d ij (s), wherein i is greater than or equal to 1 and j is less than or equal to n;
(2) Transfer function S in frequency domain by inverse Laplace transform ij (s) conversion to an excitation function s in the time domain ij (t)；
(3) The convolution of the S-parameters is computed using a recursive method to obtain the algebraic equation for the nport model.
Further, the step (1) further comprises:
for S parameter S ij Carrying out passive correction, wherein i is more than or equal to 1, and j is more than or equal to n;
assume delay time d =0, for K =2,4 max Fitting S by K order vector matching method ij Calculating the fitting error of the K-order vector matching method, and selecting K with the smallest fitting error or meeting the precision requirement firstly;
if the fitting error does not meet the accuracy requirement, assume delay time d>, 0, orderFor M =1,2,. Lamda., M, d m = m Δ d, calculating the extraction delay time d m Last S parameterAnd toCarrying out passive correction, and fitting by using K-order vector matching methodCalculating the fitting errorThe delay time with the smallest fitting error is selected.
Further, the pair K =2,4 max Fitting S by K order vector matching method ij The calculation formula of (2) is as follows:
in the formula, r ij,K Is a constant term, being a real number, p ij,k And q is ij,k Respectively, the pole and the residue.
Further, a calculation formula of the fitting error of the K-order vector matching method is as follows:
fitting error of K order vector matching method:
further, the S parameter after the delay time is calculated and extractedThe calculating method comprises the following steps:at port i, j and frequency s l Value of (A)
Further, in step (2), the transfer function S in the frequency domain is transformed by inverse Laplace transform ij (s) conversion to an excitation function s in the time domain ij (t)。
Let the transfer function S ij (s) has the fitting formula
Then conversion is made to the time domain with an excitation function
Where δ (·) denotes a dirac function, and d is a delay time.
Further, the step (3) of calculating the convolution of the S parameter by using a recursive method, Y ij (t)＝s ij (t)*X j Recursive calculation format of (t):
Y ij (t)＝r ij X j (t-d)+Z ij (t)
wherein, the first and the second end of the pipe are connected with each other,
X j (t)＝V j (t)+R j I j (t)
d is the delay time for extraction.
The invention mainly considers two aspects involved in calculating nport problem by using a recursion method: fitting by a vector matching method and recursive computation of convolution. The nport problem self-adaptive fitting and simulation method based on the vector matching method has the following advantages: (1) Compared with the common vector matching method for fixing the fitting order, the self-adaptive fitting method can automatically select the order, so that the fitting accuracy is higher; meanwhile, the delay effect of the S parameter is considered, and the fitting effect is better by extracting the delay time. (2) When the convolution of the S parameter is calculated by using a recursion method, a calculation format with simple form and high precision is provided by Taylor approximation in consideration of the fact that the time step of general simulation is small.
Additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.
Drawings
The accompanying drawings are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention. In the drawings:
FIG. 1 is a schematic diagram of a two-port network model according to the prior art;
FIG. 2 is a schematic diagram of a prior art n-port network model according to the present invention;
FIG. 3 is a flow chart of an nport problem adaptive fitting and simulation method based on a vector matching method according to the present invention.
Detailed Description
The preferred embodiments of the present invention will be described in conjunction with the accompanying drawings, and it should be understood that they are presented herein only to illustrate and explain the present invention and not to limit the present invention.
Fig. 3 is a flowchart of the nport problem adaptive fitting and simulation method based on the vector matching method according to the present invention, and the nport problem adaptive fitting and simulation method based on the vector matching method according to the present invention will be described in detail with reference to fig. 3.
First, in step 101, the S parameter is adaptively fitted by using a vector matching method. The actually measured S parameters have a rational formula sum form theoretically after being fitted by a vector matching method. In practical applications, we generally have a fitting form considering the delay effect in the time domain
Wherein i is more than or equal to 1 and less than or equal to n, j is more than or equal to 1 and less than or equal to n, r ij Is a constant term (real number), p k And q is k Respectively the pole and the residue, K ij Order of rational fraction, d ij Is a delay time, and r ij Is a real number, p k And q is k Either real or respectively complex conjugates occurring in pairs. K ij Is a positive integer (K is always set in the vector matching method) ij Even number), d ij Are non-negative real numbers. In practical applications, we can assume K ij And d ij The value at each port being the same, i.e. K ij ＝K,d ij D (for all i, j), where K is a positive even number and d is a non-negative real number.
Let L denote the total number of samples in the S parameter, S l Indicates the frequency(s) of the l-th sample point l ＝2πf l ，f l Denotes the frequency, 1. Ltoreq. L.ltoreq.L), S ij,l Represents the value of S parameter, S, corresponding to the ith sampling point ij (s l ) Expressing rational components fitted according to the vector matching method at s l The calculated value is processed.
S parameter S of i, j port by using hypothesis vector matching method ij Fitting results having the form:
definition of S ij Relative error of fit:
relative error e ij To measure the accuracy of the fit. Theoretically, the larger the order K of fitting is, the higher the fitting precision is; however, the larger K, the more computation is performed in the recursive method to calculate convolution, and the slower the computation is. Therefore, the temperature of the molten metal is controlled,a balance needs to be maintained when selecting the fitting order K. We use the tolerance tol to control the order of the fit.
Constants in the fit are taken as follows. Tolerance tol =1.0e-3, maximum fitting order K max =64, maximum delay time d max =10ns (1ns =1e-9 s). For maximum delay time d max M times of aliquoting (no M = 100) and noting Δ d = d max and/M. Note K opt ,d opt Respectively, are the best values of the fit.
The basic method for carrying out self-adaptive fitting on the S parameters by using a vector matching method comprises the following steps: (1) Without considering the time delay (i.e. d) m = 0), fitting the S parameter by a vector matching method directly, and gradually increasing the fitting order K from 2 to K max If the fitting precision meets the requirement in the process, the fitting is successful; if K = K max If the fitting accuracy still does not meet the requirement, selecting the K with the maximum fitting accuracy (not setting) And (2) turning to fitting by a vector matching method with time delay; (2) Let delay time d increase gradually from 0 to d max For each d, useFitting by an order vector matching method, wherein if the fitting precision in the process meets the requirement, the fitting is successful; if d = d max If the fitting accuracy still does not meet the requirement, selecting the d with the maximum fitting accuracy (not setting) Outputting fitting results
The following describes a specific implementation process of the nport problem adaptive fitting method based on the vector matching method in detail with reference to specific embodiments.
Step 1 for S parameter S ij (i is more than or equal to 1, j is less than or equal to n) to carry out passivityCorrection, setting initial value m =0, assuming delay time d m ＝0。
Step 2 for K =2,4 max ，
(2.1) fitting S by using K order vector matching method ij (1. Ltoreq. I, j. Ltoreq. N) to obtain
(2.2) calculation of S ij Error of fit
Then the fitting error of the K order vector matching method is calculated as
(2.3) if e K Total less than or equal to tol, successful fitting, K opt K, exit.
Otherwise: if K is<K max K = K +2, step (2.1); if K = K max Go to step 3.
Step 3 orderFor M =1,2,. Multidot.m, d m ＝mΔd,
(3.1) calculating S parameter after extracting delay timeAt port i, j and frequency s l Has a value of
(3.2) pairsPerforming passive correction, and usingFitting by order vector matching method, calculating fitting error
(3.3) ifThe fitting is successful and the fitting is successful,d opt ＝d m and exiting.
Otherwise: if m is&M, then M = M +2, d m = m Δ d, go to step (3.1);
if M = M, takeAnd (6) exiting.
Then the fitting error of the K order vector matching method is calculated as
The above implementation steps are explained as follows:
(1) In the above step, when the S parameter is fitted in step 2, the S parameters S of all ports ij (1. Ltoreq. I, j. Ltoreq. N) with exactly the same fitting order. If S-parameters S of different ports are assumed ij (i is more than or equal to 1, j is less than or equal to n) has different orders, and only the step 2 needs to be changed as follows: fix i, j, to parameter S ij Respectively using K ij (2≤K ij ≤K max ) Fitting by order vector matching method to find out relative error e ij,K Minimum (or first satisfied with e) ij,K Not more than tol) ij As S ij The fitting result of (2).
(2) In the above step, when the delay time of the S parameter is extracted in step (3.1), it is assumed that S parameters S of all ports ij (1. Ltoreq. I, j. Ltoreq. N) with identical delay times. This has the advantage of being convenientAnd (4) performing passive correction (because S parameters of all ports are required to be used simultaneously when performing passive correction before fitting by using a vector matching method). If S-parameters S of different ports are assumed ij Extracted delay time d ij Different from each other, the passive correction is performed before the delay time of each port i, j is extracted, and thus the fitting efficiency is reduced (especially when the number of ports is large).
(3) In the above process, if the given S parameter has no time delay (i.e. d = 0), a more accurate fitting result can be obtained quickly through steps 1 and 2.
(4) In the fitting process, the constant K max ,d max M, etc. can be determined autonomously based on the particular problem.
(5) The fitting process described above mainly considers how to autonomously select the fitting order and the delay time. In practical application, the fitting order and the delay time are key factors for restricting the fitting accuracy, and other factors, such as the properties of devices, instruments for measuring S parameters and the like, also influence the fitting result, so that the thought method can be used together with other technical means.
Next, in step 102, the transfer function S in the frequency domain is transformed by inverse Laplace transform ij (s) conversion to an excitation function s in the time domain ij (t)。
Setting the vector matching method to the S parameter S ij The result of the adaptive fitting is
Conversion to the time domain has
Where δ (·) denotes a dirac function.
Finally, the convolution of the S-parameters is computed in step 103 using a recursive method, resulting in the algebraic equation for the nport model (1). The recursive calculation format of the S-parameter convolution is given below.
In formula (1), let X j (t)＝V j (t)+R j I j (t), then the following convolution needs to be calculated:
s ij (t)*(V j (t)+R j I j (t))＝s ij (t)*X j (t)＝:Y ij (t) (5)
Y ij the recursive computation format of (t) is derived as follows:
here, the first and second liquid crystal display panels are,
let Δ t be the time step of the current time, then
The last step of the above equation uses Taylor approximation
e x ＝1+x+O(x 2 )(x→0)
Since the time step Δ t is small in the actual circuit simulation, the above approximation is reasonable; and e is approximated by 1+ x in the computer operation process x It would be calculated faster.
From (3) - (7), recursive calculation format
Y ij (t)＝r ij X j (t-d)+Z ij (t) (9)
Wherein the content of the first and second substances,
and finally, substituting (9) - (10) into (1) to obtain an algebraic equation of the nport model.
The recursive calculation format for calculating the convolution of the S parameter is explained as follows:
(1) In deriving the above-mentioned recursion relationship, we assume that the delay times of the S-parameters at each port are the same, i.e., d ij D (for all i, j). If d is ij The recursive format is also applicable to different values of i and j, and only the formula (9) is changed into
Y ij (t)＝r ij X j (t-d ij )+Z ij (t)
Elsewhere, it is unchanged.
(2) In the above recursive relationship, we assume that the delay time d is constant regardless of the frequency of the incident wave. If the delay time d is related to the frequency of the incident wave, the above-mentioned recursive relation is no longer true, and the convolution cannot be accurately calculated by the recursive method.
(3) From the Taylor equation, we can also estimate the local stage error of equation (8). In fact, a Taylor approximation is made to the second term after the 3 rd equal sign in equation (8), having
Thereby having an estimationTherefore, the local truncation error of equation (8) is O ((Δ t) 2 )。
(4) The mode of calculating convolution by using a recursion method is mature, and the recursion format is different from other formats in that the delay time is considered, approximation is carried out when delta t is small, the recursion relation is simple, and meanwhile, the precision is high.
Those of ordinary skill in the art will understand that: although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that changes may be made in the embodiments and/or equivalents thereof without departing from the spirit and scope of the invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (7)
1. An nport problem self-adaptive fitting and simulation method based on a vector matching method comprises the following steps:
(1) Carrying out self-adaptive fitting on the S parameter by using a vector matching method to obtain a fitting result S with optimal fitting order K and delay time d ij (s), wherein i is greater than or equal to 1, and n is greater than or equal to j;
(2) Transfer function S in frequency domain is transformed by Laplace inverse transformation ij (s) conversion to an excitation function s in the time domain ij (t)；
(3) The convolution of the S parameters is computed by a recursive method, thus obtaining the algebraic equation of the nport model.
2. The nport problem adaptive fitting and simulation method based on the vector matching method as claimed in claim 1, wherein the step (1) further comprises:
for S parameter S ij Carrying out passive correction, wherein i is more than or equal to 1, and j is more than or equal to n;
let delay time d =0, for K =2,4 max Fitting S by K order vector matching method ij Calculating the fitting error of the K-order vector matching method, and selecting K with the minimum fitting error or meeting the precision requirement firstly;
if the fitting error does not meet the accuracy requirement, assume delay time d>, 0, orderFor M =1,2,. Lamda., M, d m = m Δ d, calculating the extraction delay time d m Last S parameterAnd are aligned withMaking passive corrections and then usingFitting by order vector matching methodCalculating the fitting errorSelecting d with the smallest fitting error m 。
3. The nport problem adaptive fitting and simulation method based on the vector matching method as claimed in claim 2, wherein the pairs K =2,4 max Fitting S by K order vector matching method ij The calculation formula of (2) is as follows:
in the formula, r ij,K Is a constant term, being a real number, p ij,k And q is ij,k Respectively pole and residue.
4. The nport problem adaptive fitting and simulation method based on the vector matching method as claimed in claim 2, wherein the calculation formula for calculating the fitting error of the K-th order vector matching method is:
fitting error of K order vector matching method:
5. the nport problem adaptive fitting and simulation method based on the vector matching method as claimed in claim 2, wherein the S parameter after the delay time is calculated and extractedThe calculation method comprises the following steps:at port i, j and frequency s l Value of (A)
6. The nport problem adaptive fitting and simulation method based on the vector matching method as claimed in claim 1, wherein the step (2) is to transform the transfer function S in the frequency domain through Laplace inverse transformation ij (s) conversion to an excitation function s in the time domain ij (t): let the transfer function S ij (s) has the fitting formula
The excitation function s in the time domain ij (t) is in the form of
Where δ (·) denotes a dirac function, and d is the delay time of the extraction.
7. The adaptive fitting and simulation method for nport problem based on vector matching method as claimed in claim 1, wherein the step (3) of calculating convolution of S parameters by using recursive method: y is ij (t)＝s ij (t)*X j (t) a recursive convolution calculation format of
Y ij (t)＝r ij X j (t-d)+Z ij (t)
Wherein, the first and the second end of the pipe are connected with each other,
X j (t)＝V j (t)+R j I j (t)
d is the delay time for extraction.
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| CN111144461B (en) * | 2019-12-16 | 2023-11-24 | 华南理工大学 | Dynamic clustering vector fitting method for port loading effect |
| CN112859086A (en) * | 2021-01-25 | 2021-05-28 | 聚融医疗科技(杭州)有限公司 | Self-adaptive rapid arc tangent system and method and ultrasonic imaging device |
| CN112859086B (en) * | 2021-01-25 | 2024-02-27 | 聚融医疗科技(杭州)有限公司 | Self-adaptive rapid arctangent system, method and ultrasonic imaging device |
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