CN114139372A - Broadband electromagnetic simulation method using efficient adaptive frequency scanning - Google Patents

Broadband electromagnetic simulation method using efficient adaptive frequency scanning Download PDF

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CN114139372A
CN114139372A CN202111434445.7A CN202111434445A CN114139372A CN 114139372 A CN114139372 A CN 114139372A CN 202111434445 A CN202111434445 A CN 202111434445A CN 114139372 A CN114139372 A CN 114139372A
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赵智英
赵鹏
陈世昌
袁博
王高峰
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Hangzhou Dianzi University
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Abstract

The invention discloses a broadband electromagnetic simulation method using efficient adaptive frequency scanning. The invention provides a method for judging frequency response convergence by using an error formula, so that a proper frequency sampling point can be obtained, and compared with other methods, the required sampling point is reduced, and the simulation speed is accelerated. The invention provides a method for converting a nonlinear problem into two linear problems, calculates the accurate values of poles and residues through iteration, solves a linear equation, obtains an accurate frequency response fitting value, accelerates the calculation rate and improves the accuracy.

Description

Broadband electromagnetic simulation method using efficient adaptive frequency scanning
Technical Field
The invention belongs to the field of electromagnetic field numerical calculation, and particularly relates to a broadband electromagnetic simulation method using efficient adaptive frequency scanning.
Background
Full-wave simulation of electromagnetic problems is one of the most challenging problems in modern computing technology. The moment method based on the integral equation is used as an effective solving method and is widely applied. For many practical applications, the electromagnetic problem needs to be solved in a wide frequency band. However, for the frequency domain solver, many frequency points need to be calculated to describe the frequency response, especially for the problem that the frequency response changes rapidly over a wide frequency band.
To overcome this problem, many scanning techniques have been developed, such as Adaptive Multipoint (AMP) method, Asymptotic Waveform Evaluation (AWE). One of the most popular methods is an adaptive frequency sweep method, which can greatly reduce the number of frequency points in direct calculation by using frequency sweep based on interpolation in electromagnetic simulation. However, for a fast changing frequency response, the error threshold should be set to a very small value, and the number of frequency points calculated by the original adaptive frequency sweep may still be correspondingly very large, and the time consumed may be longer.
The invention provides a broadband electromagnetic simulation method using efficient adaptive frequency scanning. The method adopts a novel self-adaptive frequency scanning technology, combines a frequency fitting technology and a dichotomy, and can quickly and accurately obtain a response result in a frequency band, so that high-efficiency self-adaptive frequency scanning is realized.
Disclosure of Invention
The invention aims to provide a broadband electromagnetic simulation method for efficient adaptive frequency scanning, aiming at overcoming the defects of the prior art.
The method comprises the following steps:
step (1) regarding the antenna as being in the incident field (E)inc,Hinc) Ideal conductor under irradiation, the conductor being located at an electromagnetic parameter of (epsilon)(e),μ(e)) In the medium space of (a); the electric field and the magnetic field in the outer region of the conductor are respectively Ee、HeThe electric field and the magnetic field in the conductor are both 0; epsilon is dielectric constant, mu is magnetic conductivity;
step (2), according to the frequency range [ f ] set by the antenna1,f2]Initializing the end points and the intermediate value f of the selected frequency rangemin，fmid，fmaxAs a sample of the frequency, the frequency of the sample,wherein f ismin＝f1，fmid＝(f1+f2)/2，fmax＝f2(ii) a Then solving the electromagnetic field in the conductor by adopting a moment method, and calculating to obtain a corresponding frequency response theoretical value S (f)min)，S(fmid)，S(fmax)；
Step (3), obtaining frequency response fitting values corresponding to all frequency points in a frequency range;
and (4) judging whether all frequency response fitting values under the current iteration reach a convergence standard, if so, outputting all frequency response fitting values, otherwise, performing frequency sampling point addition by using a dichotomy, and returning to the step (2).
It is another object of the present invention to provide an electronic device comprising a processor and a memory, said memory storing machine executable instructions capable of being executed by said processor, said processor executing said machine executable instructions to implement a method of broadband electromagnetic simulation using efficient adaptive frequency scanning as described above.
It is a further object of the present invention to provide a machine-readable storage medium having stored thereon machine-executable instructions that, when invoked and executed by a processor, cause the processor to implement a method of broadband electromagnetic simulation using efficient adaptive frequency scanning as described above.
The invention has the beneficial effects that:
compared with the traditional method, the method can obtain the frequency response of one frequency band more quickly and accurately.
The invention provides a method for judging frequency response convergence by using an error formula, so that a proper frequency sampling point can be obtained, and compared with other methods, the required sampling point is reduced, and the simulation speed is accelerated.
The invention provides a method for converting a nonlinear problem into two linear problems, calculates the accurate values of poles and residues through iteration, solves a linear equation, obtains an accurate frequency response fitting value, accelerates the calculation rate and improves the accuracy.
Drawings
FIG. 1 is a flow chart of a broadband electromagnetic simulation method of the present invention implementing efficient adaptive frequency scanning;
FIG. 2 is a butterfly patch antenna model for testing of the present invention;
figure 3 is the result of the inventive test.
Detailed Description
The present invention will be described in detail with reference to the accompanying drawings.
For example, fig. 1 shows a method for wideband electromagnetic simulation with efficient adaptive frequency scanning, which uses a rational function to approximate the measured or calculated frequency domain response by replacing a set of starting poles with a set of modified poles by a scaling procedure.
The method comprises the following steps:
step (1) regarding the antenna as being in the incident field (E)inc,Hinc) Ideal conductor under irradiation, the conductor being located at an electromagnetic parameter of (epsilon)(e),μ(e)) In the medium space of (a); the electric field and the magnetic field in the outer region of the conductor are respectively Ee、HeThe electric field and the magnetic field in the conductor are both 0; epsilon is dielectric constant, mu is magnetic conductivity;
step (2), according to the frequency range [ f ] set by the antenna1,f2]Initializing the end points and the intermediate value f of the selected frequency rangemin，fmid，fmaxAs frequency sampling points, where fmin＝f1，fmid＝(f1+f2)/2，fmax＝f2(ii) a Then solving the electromagnetic field in the conductor by adopting a moment method, and calculating to obtain a corresponding frequency response actual value S (f)min)，S(fmid)，S(fmax)；
The method for solving the electromagnetic field inside the conductor by adopting the moment method is concretely as follows:
2-1, solving the electric field in the conductor by adopting a moment method:
integral equation of electric field inside conductor:
Figure BDA0003381327640000031
in the formula
Figure BDA0003381327640000032
Representing the conductor normal unit vector, eta the wave impedance of the medium space, L the linear operator, J the current density, EincRepresenting the incident electric field;
RWG basis functions are defined as two adjacent triangles with common edges, and can simulate the surface electric and magnetic current distribution of an object in any shape;
the RWG basis function is defined as:
Figure BDA0003381327640000033
in the formula InRepresenting the common edge length of the nth pair of adjacent triangles,
Figure BDA0003381327640000034
and
Figure BDA0003381327640000035
are respectively adjacent triangles
Figure BDA0003381327640000036
And
Figure BDA0003381327640000037
the area of (a) is,
Figure BDA0003381327640000038
is a triangle
Figure BDA0003381327640000039
Points to the vector of points r,
Figure BDA00033813276400000310
is a triangle
Figure BDA00033813276400000311
Is directed to the vector of point rAn amount; f. ofn(r) represents the nth RWG basis function in the field region;
the unknown current on the conductor surface can be expanded into the following form with the RWG basis function:
Figure BDA00033813276400000312
in the formula InRepresents an unknown current expansion coefficient (unknown quantity), N represents the number of unknowns;
substituting equation (3) into equation (1) yields:
Figure BDA0003381327640000041
wherein L (f)n(r)) represents the RWG basis function fn(r) a linear equation;
with RWG basis functions as the test functions, the test electric field integral equation can be given as:
wherein f ism(r) represents the mth RWG basis function in the field region;
because f ism(r) is always tangential to the surface of the conductor, so the following is rewritten:
Figure BDA0003381327640000042
further equation (6) can be written in matrix form:
ZI ═ V formula (7)
Wherein Z represents an NxN moment method impedance matrix, I and V are both Nx1 column vectors, and the two represent a current vector and a voltage vector of the moment method respectively;
the elements of the m-th row and n-th column of the matrix Z are represented as follows:
Figure BDA0003381327640000043
Figure BDA0003381327640000044
in the formula
Figure BDA0003381327640000045
ω ═ 2 pi f, f denotes frequency; r denotes a field point, r denotes a source point, G (R) denotes a Green's function,
Figure BDA0003381327640000046
(uniform unbounded space), R ═ R-R', meaning the distance of the field point to the source point,
Figure BDA0003381327640000047
the gradient operator is represented by a gradient operator,
Figure BDA0003381327640000048
expressing a divergence operator, ds expressing a field area vector infinitesimal, and ds' expressing a source area vector infinitesimal; f. ofn(r') represents the nth RWG basis function in the source region;
substituting the equations (8) - (9) into the matrix equation (7), and solving the matrix equation to obtain a current I; the current I at a fixed voltage is expressed as the Y admittance parameter; converting the Y admittance parameter into an S parameter, namely a frequency response theoretical value;
2-2, solving the internal magnetic field of the conductor by adopting a moment method:
the integral equation of the magnetic field inside the conductor is as follows:
Figure BDA0003381327640000049
where K denotes a linear operator, HincRepresenting an incident magnetic field;
taking a smooth surface model as an example, using RWG basis function as an expansion function, and using Galois method to test, the following formula is obtained:
Figure BDA0003381327640000051
writing the matrix form as ZI ═ V;
wherein, the matrix elements of Z and V are respectively expressed as:
Figure BDA0003381327640000052
Figure BDA0003381327640000053
wherein the content of the first and second substances,
Figure BDA0003381327640000054
is the nth basis function fn(r') the triangle pair (source triangle pair),
Figure BDA0003381327640000055
is the m-th basis function fm(r) the triangle pair (source triangle pair) in which it is located;
substituting equations (12) - (13) into a matrix equation ZI ═ V, and solving the matrix equation to obtain a current I; the current I at a fixed voltage is expressed as the Y admittance parameter; converting the Y admittance parameter into an S parameter, namely a frequency response theoretical value;
step (3), obtaining frequency response fitting values corresponding to all frequency points in a frequency range;
3-1 an approximation of a given value is found by fitting the ratio of two polynomials:
Figure BDA0003381327640000056
wherein d is1-dNAll represent coefficients of a molecular polynomial, b1-bNEach representing a coefficient of a denominator polynomial,
Figure BDA0003381327640000057
a frequency response fit value representing the frequency f.
Due to the rational function approximation, equation (14) can be rewritten as:
Figure BDA0003381327640000058
in the formula anRepresents the pole, cnResidues are indicated, d and h both represent linear constants.
Known residue cnAnd pole anEither a real quantity or a complex conjugate pair, and d and h are real quantities. Due to unknown number anAppearing in the denominator, which will be a non-linear problem, so during the vector fitting process, a is takennSolving as a known pole, and then converting the nonlinear problem into two linear problems;
3-2, simplifying the formula (15) and solving poles and residues; the method comprises the following specific steps:
1) initializing a randomly defined pole estimation value and a residue estimation value (1 in the embodiment), wherein the iteration number i is equal to 1;
Figure BDA0003381327640000061
formula (17)
The formula (16) is simplified to obtain:
Figure BDA0003381327640000062
equation (18) can be expressed in the form Ax ═ b:
Figure BDA0003381327640000063
wherein x represents the number of frequency sampling points;
solving equation (19) to obtain c1…cN、d、h、
Figure BDA0003381327640000066
3) Constructing a matrix A-BC according to the known pole estimated value and the residue estimated value obtained by solving in the step 2); solving the characteristic value of the matrix A-BC to obtain a pole value;
Figure BDA0003381327640000065
4) updating the pole estimated value to the pole value obtained in the step 2), and returning to the step 2), wherein the iteration number i is updated to i +1 until the iteration number i reaches the maximum iteration number, so as to obtain a pole accurate value;
5) updating iteration residues to obtain more accurate residues;
taking the pole accurate value obtained in the step 4) as a pole estimated value, and rewriting the formula (15) into a matrix form again:
Figure BDA0003381327640000071
solving to obtain residue accurate value c1…cN、d、h。
6) And (3) substituting the pole accurate value and the residue accurate value obtained by the solution in the steps 4) and 5) into a formula (15) according to the solution in the formula (21) to obtain a frequency response fitting value.
And (4) judging whether all frequency response fitting values under the current iteration reach a convergence standard, if so, outputting all frequency response fitting values, otherwise, performing frequency sampling point addition by using a dichotomy, and returning to the step (2).
The new frequency sampling point by using the dichotomy is to divide the frequency range into fmin,fmid]And [ fmid,fmax]Taking an end point and a middle value of each newly divided frequency range as frequency sampling points, and repeating the step (2) to obtain all frequency sampling points corresponding to the frequency response theoretical value;
the convergence standard adopts an error formula
Figure BDA0003381327640000074
Wherein S (f) represents a theoretical value of frequency response,
Figure BDA0003381327640000073
representing a frequency response fit value; if err < threshold, convergence is reached, otherwise convergence is not reached.
The frequency response of the butterfly patch antenna of fig. 2 was analyzed in a simulation of this example. The frequency band of 8-12GHz needs to be analyzed actually, and the traditional analysis method is to calculate the average 0.1GHz interval. The method provided by the invention can fit the frequency response of the whole frequency band by using fewer frequency sampling points.
Firstly, selecting three sampling points in the first iteration process: f. ofmin＝8GHz,fmid＝10GHz,fmaxPassing the frequency response S (f) at the frequency point in step (2) at 12GHzmin)，S(fmid)，S(fmax)。
By step 3-2, initially let the pole estimate be 1, a set of solutions is solved by equation (15): c. C1…cN、d、h、
Figure BDA0003381327640000072
By then using the above results, the eigenvalues of (A-BC), which are the poles, are solved in combination with equation (16)
Figure BDA0003381327640000081
And replacing the initial pole estimated value with the calculated more pole accurate value, and repeatedly calculating for 4-5 times to obtain a more accurate pole accurate value.
Substituting the pole accurate value calculated above into formula (12);
calculate more accurate c1…cN、d、h。
In summary, all unknowns are solved in the first iteration, so that any frequency response fitting value can be calculated through formula (6);
for the calculated frequency band response fit value, it is determined whether the error is within an error range, and the result of the first iteration for this example is not converged.
Thus dividing the original frequency band into f by bisectionmin,fmid]And [ fmid,fmax]；
And (3) a second iteration process, namely increasing frequency points: f. of1＝(fmid-fmin)/2，f2＝(fmax-fmid)/2；
Calculating the response S (f) of the frequency point1) And S (f)2)；
The above steps are repeated, and the result converges through three iterations in this example.
Fig. 3 is a test result of the present embodiment.
The above embodiments are not intended to limit the present invention, and the present invention is not limited to the above embodiments, and all embodiments are within the scope of the present invention as long as the requirements of the present invention are met.

Claims (6)

1. A method for broadband electromagnetic simulation using efficient adaptive frequency scanning, comprising the steps of:
step (1) regarding the antenna as being in the incident field (E)inc,Hinc) Ideal conductor under irradiation, the conductor being located at an electromagnetic parameter of (epsilon)(e),μ(e)) In the medium space of (a); the electric field and the magnetic field in the outer region of the conductor are respectively Ee、HeThe electric field and the magnetic field in the conductor are both 0; epsilon is dielectric constant, mu is magnetic conductivity;
step (2), according to the frequency range [ f ] set by the antenna1,f2]Initializing the end points and the intermediate value f of the selected frequency rangemin，fmid，fmaxAs frequency sampling points, where fmin＝f1，fmid＝(f1+f2)/2，fmax＝f2(ii) a Then solving the electromagnetic field in the conductor by adopting a moment method, and calculating to obtain a corresponding frequency response theoretical value S (f)min)，S(fmid)，S(fmax)；
Step (3), obtaining frequency response fitting values corresponding to all frequency points in a frequency range;
3-1 obtaining a frequency response fitting value of the frequency f by fitting the ratio of the two polynomials
Figure FDA0003381327630000011
Figure FDA0003381327630000012
Wherein d is1…dNAll represent coefficients of a molecular polynomial, b1…bNCoefficients that each represent a denominator polynomial;
due to the rational function approximation, equation (14) is rewritten as:
Figure FDA0003381327630000013
in the formula anRepresents the pole, cnRepresents a residue, d and h both represent linear constants;
3-2, simplifying the formula (15) and solving poles and residues; the method comprises the following specific steps:
1) initializing a random pole-defining estimated value and a residue estimated value, wherein the iteration number i is 1;
2) introducing a function σ (f) to equation (15):
Figure FDA0003381327630000014
Figure FDA0003381327630000021
in the formula
Figure FDA0003381327630000022
The pole estimate is represented as a function of,
Figure FDA0003381327630000023
representing residue estimates;
the formula (16) is simplified to obtain:
Figure FDA0003381327630000024
equation (18) is further expressed in the form Ax ═ b:
Figure FDA0003381327630000025
wherein x represents the number of frequency sampling points;
solving equation (19) to obtain c1…cN、d、h、
Figure FDA0003381327630000026
3) Constructing a matrix A-BC according to the known pole estimated value and the residue estimated value obtained by solving in the step 2); solving the characteristic value of the matrix A-BC to obtain a pole value;
Figure FDA0003381327630000027
4) updating the pole estimated value to the pole value obtained in the step 2), and returning to the step 2), wherein the iteration number i is updated to i +1 until the iteration number i reaches the maximum iteration number, so as to obtain a pole accurate value;
5) updating iteration residues to obtain residue accurate values; the method comprises the following specific steps:
taking the pole accurate value obtained in the step 4) as a pole estimated value, and rewriting the formula (15) into a matrix form again:
Figure FDA0003381327630000031
solving to obtain residue accurate value c1～cN、d、h；
6) Substituting the pole accurate value and the residue accurate value obtained by the solving in the steps 4) and 5) and d and h obtained by the solving in the formula (21) into the formula (15) to obtain a frequency response fitting value;
and (4) judging whether all frequency response fitting values under the current iteration reach a convergence standard, if so, outputting all frequency response fitting values, otherwise, performing frequency sampling point addition by using a dichotomy, and returning to the step (2).
2. The method according to claim 1, wherein the step (2) of solving the electromagnetic field inside the conductor by using a moment method is as follows:
2-1, solving the electric field in the conductor by adopting a moment method:
integral equation of electric field inside conductor:
Figure FDA0003381327630000032
in the formula
Figure FDA0003381327630000033
Representing the conductor normal unit vector, eta the wave impedance of the medium space, L the linear operator, J the current density, EincRepresenting the incident electric field;
RWG basis functions are defined as two adjacent triangles with common edges, and can simulate the surface electric and magnetic current distribution of an object in any shape;
the RWG basis function is defined as:
Figure FDA0003381327630000034
in the formula InRepresenting the common of the nth pair of adjacent trianglesThe length of the side is longer than that of the side,
Figure FDA0003381327630000035
and
Figure FDA0003381327630000036
are respectively adjacent triangles
Figure FDA0003381327630000037
And
Figure FDA0003381327630000038
the area of (a) is,
Figure FDA0003381327630000039
is a triangle
Figure FDA00033813276300000310
Points to the vector of points r,
Figure FDA00033813276300000311
is a triangle
Figure FDA00033813276300000312
Points to the vector of point r; f. ofn(r) represents the nth RWG basis function in the field region;
the unknown current on the conductor surface can be expanded into the following form with the RWG basis function:
Figure FDA0003381327630000041
in the formula InRepresenting an unknown current expansion coefficient, N representing the number of unknowns;
substituting equation (3) into equation (1) yields:
Figure FDA0003381327630000042
wherein L (f)n(r)) represents the RWG basis function fn(r) a linear equation;
with RWG basis functions as the test functions, the test electric field integral equation can be given as:
Figure FDA0003381327630000043
wherein f ism(r) represents the mth RWG basis function in the field region;
because f ism(r) is always tangential to the surface of the conductor, so the following is rewritten:
Figure FDA0003381327630000044
further equation (6) is written in matrix form:
ZI ═ V formula (7)
Wherein Z represents an NxN moment method impedance matrix, I and V are both Nx1 column vectors, and the two represent a current vector and a voltage vector of the moment method respectively;
the elements of the m-th row and n-th column of the matrix Z are represented as follows:
Figure FDA0003381327630000045
Figure FDA0003381327630000046
in the formula
Figure FDA0003381327630000047
ω ═ 2 pi f, f denotes frequency; r denotes a field point, r 'denotes a source point, G (R) denotes a Green's function,
Figure FDA0003381327630000048
(uniform unbounded space), R ═ R-R', meaning the distance of the field point to the source point,
Figure FDA0003381327630000049
the gradient operator is represented by a gradient operator,
Figure FDA00033813276300000410
expressing a divergence operator, ds expressing a field area vector infinitesimal, and ds' expressing a source area vector infinitesimal; f. ofn(r') represents the nth RWG basis function in the source region;
substituting the equations (8) - (9) into the matrix equation (7), and solving the matrix equation to obtain a current I; the current I at a fixed voltage is expressed as the Y admittance parameter; converting the Y admittance parameter into an S parameter, namely a frequency response theoretical value;
2-2, solving the internal magnetic field of the conductor by adopting a moment method:
the integral equation of the magnetic field inside the conductor is as follows:
Figure FDA0003381327630000051
where K denotes a linear operator, HincRepresenting an incident magnetic field;
the RWG basis function is used as an expansion function, and the Galileo method is adopted for testing, so that the following results are obtained:
Figure FDA0003381327630000052
writing the matrix form as ZI ═ V;
wherein, the matrix elements of Z and V are respectively expressed as:
Figure FDA0003381327630000053
Figure FDA0003381327630000054
wherein the content of the first and second substances,
Figure FDA0003381327630000055
is the nth basis function fn(r') is located in the triangular pair,
Figure FDA0003381327630000056
is the m-th basis function fm(r) the triangle pair in which it is located;
substituting equations (12) - (13) into a matrix equation ZI ═ V, and solving the matrix equation to obtain a current I; the current I at a fixed voltage is expressed as the Y admittance parameter; and converting the Y admittance parameters into S parameters, namely frequency response theoretical values.
3. The method according to claim 1, wherein the step (4) of adding new frequency samples by dichotomy is dividing the frequency range into [ f [ [ f ]min,fmid]And [ fmid,fmax]And (3) taking the end point and the middle value of each newly divided frequency range as frequency sampling points, and repeating the step (2) to obtain all frequency sampling points corresponding to the frequency response theoretical value.
4. The method for wideband electromagnetic simulation using efficient adaptive frequency scanning as defined in claim 1, wherein the convergence criterion in step (4) is an error formula
Figure FDA0003381327630000057
Wherein S (f) represents a theoretical value of frequency response,
Figure FDA0003381327630000058
representing a frequency response fit value; if err < threshold, convergence is reached, otherwise convergence is not reached.
5. An electronic device comprising a processor and a memory, the memory storing machine executable instructions executable by the processor, the processor executing the machine executable instructions to implement a method of broadband electromagnetic simulation using efficient adaptive frequency scanning as claimed in any one of claims 1 to 4.
6. A machine-readable storage medium having stored thereon machine-executable instructions which, when invoked and executed by a processor, cause the processor to implement a method of broadband electromagnetic simulation using efficient adaptive frequency scanning as claimed in any of claims 1-4.
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WO2024099061A1 (en) * 2022-11-09 2024-05-16 深圳飞骧科技股份有限公司 Frequency sweeping method and system for adaptive frequency point sampling, and related device

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Publication number Priority date Publication date Assignee Title
WO2024099061A1 (en) * 2022-11-09 2024-05-16 深圳飞骧科技股份有限公司 Frequency sweeping method and system for adaptive frequency point sampling, and related device
CN116341454A (en) * 2023-03-31 2023-06-27 北京百度网讯科技有限公司 Method, device and medium for generating coupling-off point information of superconducting quantum chip
CN116341454B (en) * 2023-03-31 2024-05-28 北京百度网讯科技有限公司 Method, device and medium for generating coupling-off point information of superconducting quantum chip

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