CN108595770B - Mathematical model for accurately fitting plate parameters and fitting method thereof - Google Patents
Mathematical model for accurately fitting plate parameters and fitting method thereof Download PDFInfo
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Abstract
The invention discloses a mathematical model for accurately fitting plate parameters and a fitting method thereof in the technical field of circuit boards, wherein in the mathematical model, the resonance of each molecule or atom in a dielectric medium is established into a respective independent harmonic oscillator model, then the harmonic oscillator models are superposed, and a multipole point model with dielectric constant is formed by performing segmented fitting on a plurality of different frequencies; the fitting method comprises the steps of determining the line width of the PCB and the thickness from the PCB to a reference plane, simulating by software, listing the variables of the fitting model, setting the optimization target of simulation software, automatically optimizing, obtaining the optimal result and the like. The invention can ensure that each small frequency band interval has good fitting precision, then the fitting precision of the full frequency band can be high after superposition, and meanwhile, the automatic optimization of software replaces manual scanning adjustment, thereby facilitating the adjustment of performance optimization.
Description
Technical Field
The invention relates to the technical field of circuit boards, in particular to a mathematical model for accurately fitting plate parameters and a fitting method thereof.
Background
Printed Circuit Boards (PCB), also called PCB and PCB, are important components for physical support and signal transmission of electronic products, and the most important component is transmission lines. The PCB board is a carrier for carrying signals, and the dielectric constant (DK) and the loss factor (DF) of different PCB boards are different in terms of electric signals. The skilled in the art knows that DK can influence the impedance and the time delay of transmission line, and DF directly determines the loss grade of panel, and the material that DF is low can have less loss under the condition of the same PCB routing length and line width, can transmit longer routing under the same agreement loss standard, is favorable to the performance of system, so how can more accurate data fitting through real test go out accurate panel DK and DF value, plan device layout and walk the length and have vital effect when the predesigned.
Under the condition of a specific plate, the line width and length of the PCB wiring, the thickness of the reference plane and the roughness of the copper foil of the wiring can influence the loss of the PCB wiring, so that the industry uses a digital model of infinite points to model the plate parameters.
However, the infinite point model cannot have good fitting precision for all plates; moreover, the model is self-contained in software, so that the software cannot be automatically optimized and can only be manually adjusted. Therefore, how to find a better fitting model and an optimization algorithm to ensure higher-frequency fitting accuracy becomes an urgent problem to be solved.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a mathematical model for accurately fitting plate parameters and a fitting method thereof.
The technical scheme of the invention is as follows:
on the one hand, the mathematical model for accurately fitting the parameters of the plate is characterized in that in a dielectric medium, the resonance of each molecule or atom is respectively established into independent harmonic oscillator models, then the harmonic oscillator models are superposed, and the multipole point model of the dielectric constant is formed by performing segmented fitting on a plurality of different frequencies.
The present invention according to the above aspect is characterized in that, in establishing the harmonic oscillator model for each molecule or atom:
(1) firstly, a mechanical model of a spring is utilized to establish a damping harmonic oscillator differential equation
Wherein m is mass point, n is damping coefficient, k is spring coefficient, FejωtIs a driving force;
(2) secondly, substituting an electric field force formula F (q × E), an electric dipole moment p (q × x), an electric polarizability x (p/epsilon 0 × E) and a relative dielectric constant epsilon r (1 + x) into a damping harmonic oscillator differential equation;
obtaining a mathematical model of the dielectric constant
Wherein m1 and m2 are constants, omega 1 and omega 2 are constant angular velocities, and omega is an angular velocity variable;
normalizing the constants of the mathematical model of the dielectric constant to obtain a simplified model of the dielectric constant
Wherein a and b are constants, f1 and f2 are upper and lower cut-off frequencies, and freq is a frequency variable.
The invention according to the above aspect is characterized in that, in the process of superimposing the harmonic oscillator models:
(1) the relative dielectric constant is expressed by the sum of the responses of n independent harmonic oscillator models
Wherein j is the complex number in a complex function;
(2) setting Δ ∈' indicates that the lower frequency limit ω 1 is 10m1And the upper frequency limit ω 2 is 10m2And it is uniformly distributed on the scale which is the frequency logarithm axis, and if infinite terms are taken, the above formula can be simplified as follows:
(3) by using inverse derivation processes, prototypes of multi-pole models are obtained, i.e.
On the other hand, the fitting method of the mathematical model for accurately fitting the parameters of the plate is characterized by comprising the following steps of:
step 3, listing variables of the fitting model, including a constant a of a real part, an imaginary part constant bn of each frequency band interval, a value range fn of each frequency band interval and a frequency variable freq;
step 5, automatically optimizing the software;
and 6, stopping the simulation process after the optimal result is obtained in the simulation process, and displaying an optimized result graph.
The present invention according to the above aspect is characterized in that, in step 4, the optimization target of the simulation software includes: the phase of the test data is consistent with that of the fitting data, and the loss of the test data is consistent with that of the fitting data.
The present invention according to the above aspect is characterized in that, in step 5, the values of the variables are substituted and then error calculation is performed, the result of the minimum error is sought, and the values of the variables are updated once for each substitution until the optimum error is found.
The method is characterized in that curves of the dielectric constant and the loss factor of the plate respectively changing along with the frequency are extracted according to a model after final optimization; and then, performing early evaluation on the loss according to the corresponding numerical values in the curve.
The invention according to the scheme has the advantages that: the multipole point model can relatively and independently decompose the full frequency band to be fitted into a plurality of frequency band intervals for fitting respectively, can ensure that each small frequency band interval has good fitting precision, and can achieve high fitting precision of the full frequency band after superposition. According to the method, after the multi-pole model is fitted, the DK and DF parameters of the plate can be more accurately extracted, so that the method has great guiding significance for the layout of the system device at the early stage and the evaluation of the wiring length. The invention also lists the simulation optimized variables one by one for optimization through a method of editing a formula in simulation software, and the automatic optimization of the software replaces the manual scanning adjustment, thereby facilitating the adjustment of performance optimization.
Drawings
Fig. 1 is a graph of loss curves of transmission lines actually measured in an embodiment of the present invention.
Fig. 2 is a graph of the phase of a transmission line actually measured in an embodiment of the present invention.
FIG. 3 is a schematic diagram of a lamination pattern confirmed by a board manufacturer in an embodiment of the invention.
FIG. 4 is a schematic view of another overlay pattern identified by the plate mill in the embodiment of the present invention.
FIG. 5 is a microscopic section of an embodiment of the present invention.
FIG. 6 is a diagram illustrating editing a multi-pole model using simulation software according to an embodiment of the present invention.
Fig. 7 is a schematic diagram of editing values of constants and frequency ranges according to an embodiment of the present invention.
FIG. 8 is a diagram illustrating setting of software optimization objectives according to an embodiment of the present invention.
FIG. 9 is a diagram illustrating an optimized constant value and a frequency variable according to an embodiment of the present invention.
FIG. 10 is a diagram illustrating optimization of the c variable in an embodiment of the present invention.
FIG. 11 is a graph of loss versus frequency for the optimized results obtained by the present invention.
Fig. 12 is a graph of the phase versus frequency curve obtained from the optimization of the present invention.
Fig. 13 is a graph of loss factor versus frequency extracted by the present invention.
FIG. 14 is a graph showing the variation of dielectric constant with frequency extracted by the present invention.
Detailed Description
The invention is further described with reference to the following figures and embodiments:
a mathematical model accurately fits the parameters of a sheet in which the polarization of the actual dielectric contains different numbers of ionic, molecular and electronic polarizations, and which causes multiple resonances over a wide frequency band due to the complex frequency dependence of the real and imaginary parts of the dielectric constant. Therefore, independent harmonic oscillator models are established for the resonance of each molecule or atom, and then the harmonic oscillator models are superimposed and segmented into a multipole dielectric constant model by several different frequencies.
1. And establishing a harmonic oscillator model of each molecule or atom independently.
(1) Firstly, a mechanical model of a spring is utilized to establish a damping harmonic oscillator differential equation
Wherein m is mass point, n is damping coefficient, k is spring coefficient, FejωtIs the driving force.
(2) Substituting the calculation formulas of electric field force, electric dipole moment, electric polarizability and relative dielectric constant into a differential equation of the damping harmonic oscillator to obtain a mathematical model of the dielectric constant:
electric field force formula: f ═ qxE
Electric dipole moment: p is q x
Electric susceptibility: χ ═ p/ε 0 × E
Relative dielectric constant: ε r ═ 1+ χ
Obtaining a mathematical model of the dielectric constant by solving operations
Wherein m1 and m2 are constants, omega 1 and omega 2 are constant angular velocities, and omega is an angular velocity variable.
(3) Normalizing the constants of the mathematical model of the dielectric constant to obtain a simplified model of the dielectric constant
Wherein a and b are constants, f1 and f2 are upper and lower cut-off frequencies, and freq is a frequency variable.
2. And superposing the harmonic oscillator models.
(1) For a dielectric with n natural frequencies, the relative permittivity can be expressed as the sum of the responses of n independent harmonic oscillator models:
simplified model of dielectric constant through infinite poles
Stacking a plurality of independent dielectric constants such as ion polarization, molecular polarization and electron polarization expressed in different frequency intervals to obtain
Where j is the complex number in a complex function, j2＝-1。
(2) Assuming a linear decrease on the logarithmic scale axis, taking a certain number of terms, the following simplification can be made:
setting Δ c' represents the lower frequency limitω1＝10m1And the upper frequency limit ω 2 is 10m2The total deviation between the two is uniformly distributed on the scale which is the frequency logarithm axis, thus, the deviation of the ten-frequency multiplication logarithm is the linear attenuation on the logarithm scale axis, and when an infinite item is taken, the above formula can be simplified as follows:
(3) by using inverse derivation processes, prototypes of multi-pole models are obtained, i.e.
The dielectric constant epsilon is subjected to piecewise fitting by using a plurality of different frequencies f, so that the respective fitting precision of each frequency band interval can be ensured, and the precision of a required full frequency band is ensured.
A fitting method of a mathematical model for accurately fitting plate parameters is characterized in that a single-pole model is evolved into a multi-pole model with 4 poles aiming at fitting of a 20GHz frequency band which is customary in the industry. In the aspect of the fitting method, the optimization function of the software is utilized, the fitted formula is edited on the software, then each variable is listed, and the optimal result can be automatically found by the software by setting the optimization target. The specific fitting process specifically comprises the following steps:
1. as shown in fig. 1-2, a loss curve and a phase curve of a transmission line obtained by real test are obtained first.
2. The linewidth and thickness to the reference plane of the PCB that was finally processed is determined by the manufacturer's final laminate (as shown in fig. 3-4) or actual profile image (as shown in fig. 5), using the loose M6G board as an example.
After obtaining the exact routing line width and the exact lamination thickness, the method is built in ADS software, and the following steps are shown: the values we have determined include thicknesses H1 and H2 above and below, line width W of the traces, line spacing S, Length, parameters needed to obtain DK and DF, and roughness rough (c).
3. As shown in FIG. 6, the formula of the multi-pole model as dielectric constant and the formula of the loss factor are edited in the simulation software
In this embodiment, the multipole model for 4 poles is:
the loss fitting formula in the embodiment is divided into 4 frequency band intervals of f1 to f4 to be fitted respectively, so that the fitting is more accurate than the fitting of a universal single pole.
The function of the dielectric constant is DK real (m), real (m) is the real part of the calculation m,
the function of the loss factor is DF ═ abs (imag (m))/(real (m)), and abs (m) is the absolute value of the calculated m. The DF is derived by dividing the imaginary part by the real part of m and then their absolute value.
4. And listing variables of the fitting model, including a constant a of a real part, an imaginary constant bn of each frequency band interval, a value range fn of each frequency band interval and a frequency variable freq.
As shown in fig. 7, in the present embodiment, the variables of the fitting model include a, b1, b2, b3, b4, c and the medium frequencies f1, f2, f3, f4 of each frequency band interval. Wherein, a is a constant value of a real part, b1, b2, b3 and b4 are respectively imaginary constant values of 4 frequency band intervals, and f1, f2, f3 and f4 are range values of the 4 frequency band intervals.
5. And setting an optimization target of the simulation software.
As shown in fig. 8, the optimization objectives of the simulation software include: the phase of the test data is consistent with that of the fitting data, and the loss of the test data is consistent with that of the fitting data.
Wherein the meaning of phase (S (1,2)) -phase (S (3,4)) is the phase of the test S-parameter minus the phase of the simulated S-parameter;
the meaning of dB (S (1,2)) -dB (S (3,4)) is the loss of the test S parameter minus the loss of the simulated S parameter.
6. And (3) automatically optimizing software, respectively substituting the value of each variable, then carrying out error calculation, seeking a minimum error result, and updating the value of each variable once every substitution until an optimal error is found. FIGS. 9-10 are screenshots of the optimization process, respectively.
Fig. 9 is a diagram of automatically optimizing variables a, b1, b2, b3, b4, f1, f2, f3 and f4 to achieve the goal of phase agreement between test data and simulation data, with the boxes representing the phase difference plots of the real-time fit.
FIG. 10 is a graph of loss difference curves fitted in real time, with blocks representing the automatic optimization of the c-variable to achieve the goal of consistent loss for test data and simulation data.
7. As shown in fig. 11-12, after the optimal result is obtained by the simulation, the simulation process is stopped and the optimized result graph is displayed.
As shown in fig. 13-14, according to the finally optimized model, the curves of the dielectric constant and the loss factor of the plate respectively changing with the frequency are extracted; and then, performing early evaluation on the loss according to the corresponding numerical values in the curve.
After the model is fitted, the DK and DF values fitted by the model are directly used for accurately estimating the early stage of loss of other transmission structures (such as line width, line distance and lamination) of the same plate. The fitting of different plates proves that the fitting model and the algorithm have high fitting precision and obvious universality for various universal plates in the industry.
It will be understood that modifications and variations can be made by persons skilled in the art in light of the above teachings and all such modifications and variations are intended to be included within the scope of the invention as defined in the appended claims.
The invention is described above with reference to the accompanying drawings, which are illustrative, and it is obvious that the implementation of the invention is not limited in the above manner, and it is within the scope of the invention to adopt various modifications of the inventive method concept and technical solution, or to apply the inventive concept and technical solution to other fields without modification.
Claims (3)
1. A fitting method of a mathematical model for accurately fitting plate parameters is characterized by comprising the following steps:
step 1, obtaining a loss curve and a phase curve of a transmission line obtained by a real test;
step 2, determining the line width of the PCB to be processed and the thickness of the PCB to a reference plane through the final laminated layer or the actual section image given by a manufacturer;
step 3, editing a formula of the multi-pole model, a function of the dielectric constant and a function of the loss factor in simulation software, wherein the formula of the multi-pole model is
m is the particle, the dielectric constant is function DK ═ real (m), the loss factor is function DF ═ abs (imag (m))/(real (m)), real (m) is the real part of calculated m, and abs (m) is the absolute value of calculated m;
step 4, listing variables of the fitting model, including a constant a of a real part, an imaginary part constant bn of each frequency band interval, a value range fn of each frequency band interval and a frequency variable freq;
and 5, setting an optimization target of the simulation software, wherein the optimization target of the simulation software comprises the following steps: the phase of the test data is consistent with that of the fitting data, and the loss of the test data is consistent with that of the fitting data;
step 6, automatically optimizing the software;
and 7, stopping the simulation process after the optimal result is obtained in the simulation process, and displaying an optimized result graph.
2. The method of claim 1, wherein in step 5, the values of each variable are substituted and then subjected to error calculation, the result of the minimum error is sought, and the values of the variables are updated once each substitution until the optimal error is found.
3. The method for fitting a mathematical model for accurately fitting parameters of a sheet material according to claim 1, wherein curves of the dielectric constant and the loss factor of the sheet material varying with frequency respectively are extracted according to the model after final optimization; and then, performing early evaluation on the loss according to the corresponding numerical values in the curve.
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| CN110290637B (en) * | 2019-07-30 | 2021-02-02 | 广东浪潮大数据研究有限公司 | Method, system and related assembly for reducing signal crosstalk of PCB (printed circuit board) |
| CN111814358A (en) * | 2020-08-06 | 2020-10-23 | 中国电子科技集团公司第四十六研究所 | Multi-factor co-optimization design method for comprehensive performance of microwave composite dielectric substrate |
| TWI783719B (en) * | 2021-10-07 | 2022-11-11 | 南亞塑膠工業股份有限公司 | Method for calculating the dielectric constant and dielectric loss of a polymer material |
| CN115954071B (en) * | 2023-03-10 | 2023-05-16 | 西南交通大学 | Method for determining dielectric constant of wave-absorbing material |
| CN116562053B (en) * | 2023-07-05 | 2023-09-26 | 广汽埃安新能源汽车股份有限公司 | Method and device for determining broadband material parameters |
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