Cable network time domain coupling characteristic prediction method in external random electromagnetic environment

Abstract

The invention discloses a cable network time-domain coupling characteristic prediction method in an external random electromagnetic environment, and the method comprises the steps: firstly carrying out the normalization modeling of an external source multi-dimensional random variable, and selecting a corresponding chaotic polynomial according to the probability distribution of the random variable; based on an admission basis concept, constructing a multi-dimensional admission sparse basis function and a sample point set isomorphic with the multi-dimensional admission sparse basis function by using a one-dimensional chaotic polynomial; then, deterministic solution is carried out on the coupling response of the cable network and the external electromagnetic environment on the sample points, and corresponding deterministic response is obtained; solving coefficients of the primary function based on the deterministic response; and finally, forming an explicit function by using the constructed multi-dimensional primary function and the coefficient corresponding to each primary function, and carrying out random sampling by using the explicit function to obtain an uncertainty analysis result. According to the method, the statistical characteristics responded by the complex cable network system in practice can be well calculated, the calculation amount of high-dimensional randomness analysis is greatly reduced through generalized sparse grids, and the problem of curse of dimensionality is solved to a certain extent.

Description

Technical Field

[0001] The invention relates to a method for predicting time domain coupling characteristics of a cable network under an external random electromagnetic environment, and belongs to the field of electromagnetic compatibility prediction. Background Art

[0002] With the development of high-power pulse source technology, modern electronic and electrical systems are constantly facing the threat of strong electromagnetic pulses. In actual engineering applications, cables, as the main body of electromagnetic energy transmission and information transmission, have a large diameter / transverse size ratio and a structure similar to a wire antenna. They are the main channel for introducing external harmful signals into the radio frequency system.

[0003] Cable networks are often in an external random electromagnetic environment. At the same time, factors such as cable bundle bundling, relative cable arrangement position, routing trajectory, dielectric material and other electromagnetic parameters as well as the direction of strong electromagnetic pulse waves are uncertain, resulting in strong randomness of coupled surge signals on the cable network. In view of the above-mentioned transient coupling characteristics of cable networks, specific protection measures can be taken, in addition to reasonable layout of cable networks, such as installing protection circuits and adding shielding structures. However, the premise for all protection measures to work effectively is to accurately predict the characteristic parameters of hazardous signals along the transmission path of the cable network and at the input port of the electrical and electronic system to avoid "over-design" or "under-design" of protection.

[0004] In order to solve this problem, some scholars have proposed random analysis methods including Monte Carlo method, random Galerkin method, random point configuration method, etc., and have made important progress in the random analysis of external source and cable coupling characteristics. Monte Carlo method is an exhaustive solution method for each case, that is, the statistical characteristics of the cable network system are obtained by continuous repeated experiments. Although this method is simple in theory, it has a huge amount of calculation. The random Galerkin method is a random theoretical analysis method based on generalized polynomials. It is necessary to expand the generalized polynomials of all quantities related to random variables. There is a problem of programming difficulty. It is an intrusive random analysis method. It is necessary to change the algorithm of the original transmission line network solver and it is difficult to integrate with the existing solver. The random point configuration method is also a method based on generalized polynomials. Unlike the random Galerkin method, this method is a non-intrusive method that can be applied to the random analysis of complex systems. It is a method that is extremely suitable for complex cable network systems. However, all current random analysis methods face the difficulty that the amount of calculation increases exponentially with the increase of the dimension of random variables-dimensionality disaster. There is a lack of an analysis method that can reduce the amount of calculation of high-dimensional random analysis. Therefore, it is becoming increasingly important to accurately predict the coupling response of complex cable networks with high-dimensional uncertain parameters and the external electromagnetic environment. This will strongly support the EMC (electromagnetic compatibility) design and cable network optimization solutions for the new generation of electronic and electrical systems and their equipment platforms, and improve their EMC performance. Summary of the invention

[0005] In view of the above-mentioned deficiencies in the prior art, the purpose of the present invention is to provide a method for predicting the time domain coupling characteristics of a cable network under an external random electromagnetic environment. The present invention can be directly used to calculate the statistical characteristics of the response of an actual complex cable network system, which helps to better solve the randomness problems of complex cables with high-dimensional random factors, and the prediction is accurate, with greatly improved prediction efficiency.

[0006] The technical solution of the present invention is achieved in this way:

[0007] A method for predicting the time domain coupling characteristics of a cable network under an external random electromagnetic environment is carried out in the following steps:

[0008] 1) Determine the random variables of the external electromagnetic environment and cable coupling analysis, and perform normalized modeling;

[0009] 2) According to the probability distribution of random variables, select the corresponding chaotic polynomial;

[0010] 3) Construct a multidimensional access sparse basis using one-dimensional chaotic polynomials of all random variables;

[0011] 4) Construct a set of sample points isomorphic to the multi-dimensional access sparse basis, each sample point corresponds to the specific value of all random variables in the external electromagnetic environment, and the sample point set is used as the external electromagnetic environment sample input of the field line coupling solver; perform deterministic calculations on the field line coupling solver on the sample point set to obtain the state quantity to be calculated corresponding to each sample point;

[0012] 5) Based on the sample point set obtained in step 4) and the state quantity to be determined corresponding to each sample point, the coefficients of the basis function in the multidimensional access sparse basis constructed in step 3) are calculated using a numerical integration method;

[0013] 6) The basis functions in the constructed multidimensional access sparse basis and the coefficients corresponding to each basis function are composed into an explicit function, which is used to replace the field line coupling solver and randomly sampled to obtain the statistical value of the state quantity to be determined, thereby realizing the prediction of the time domain coupling characteristics of the cable network under an external random electromagnetic environment.

[0014] Furthermore, the specific implementation process of step 1) is as follows:

[0015] Let the random variable θ of the external electromagnetic environment and cable coupling analysis be i Satisfied in [a i -b i ,a i +b i ] is uniformly distributed, as shown in formula (1);

[0016] θ i =a i ±b i ,i=1,…,d (1)

[0017] Where θ i represents a random variable, a i -b i represents the lower limit of the distribution, a i +b i represents the upper limit of the distribution, and d represents the number of random variables;

[0018] By normalizing the random variable, the random variable is transformed into the normalized random variable p in formula (2): i ,θ i The randomness of the random variable is transferred to the normalized random variable p i superior;

[0019] θ i (p i )=a i +b i p i ,i=1,…,d (2)

[0020] pi , i=1,…,d all satisfy the normalized uniform distribution on the interval [-1,1];

[0021] Let p be a random vector space consisting of all normalized random variables p i The vector formed is shown in formula (3);

[0022] p={p 1 ,p 2 ,…p d} (3)

[0023] The vector space in equation (3) represents all the random variables in the input of the field line coupling solver.

[0024] Furthermore, step 2) selects the corresponding chaotic polynomial according to the probability distribution of the random variable. The specific implementation process is as follows:

[0025] The relationship between the state quantity V to be determined and the random variable p is expressed in the form of a polynomial: that is, the chaotic polynomial containing the random variable p is used as a basis function to expand the relationship between the state quantity V to be determined and p, as shown in formula (4);

[0026]

[0027] In the formula, φ n (p) is a chaotic polynomial. This polynomial φ n (p) is called the basis function and is presented in the form of a random variable polynomial; c n are the coefficients of the basis functions; the set of basis functions {φ 0 ,…φ l} is called a basis; n is the order of the basis function, and l is the order of the expanded basis.

[0028] Furthermore, step 3) uses the one-dimensional chaotic polynomials of all random variables to construct a multi-dimensional access sparse basis. The specific implementation process is as follows:

[0029] Multidimensional Basis Function Φ under Multiple Random Variables n (p) As shown in formula (9),

[0030]

[0031] Where n=(n 1 ,n 2 ,…n d ) is the multidimensional basis function Φ n The subscript of (p), n i ≥0,n i is the i-th unidimensional random variable p i The order of the chaotic polynomial is is the 1-norm of vector n and also represents the multidimensional basis function Φ n The order of (p);

[0032] The relationship between the state quantity to be determined and the random variable is expressed in the form of a set of multidimensional basis functions of order 0-1, as shown in formula (10):

[0033]

[0034] Where c n is the coefficient of the basis function, Φ n (p) is a multidimensional basis function, V(p) is the state quantity to be determined; the set of this multidimensional basis function is defined as the multidimensional classical basis l is a multidimensional classical basis The order of

[0035] If a multidimensional basis function Φ n (p) belongs to the base Then all lower-order multidimensional basis functions Φ m (p) must also belong to the base base The expression of is shown in formula (12):

[0036]

[0037] Where m=(m 1 ,m 2 ,…m d ) is the multidimensional basis function Φ m The subscript of (p), Φ m (p) is the ratio Φ n (p) is a multidimensional basis function with a lower order, namely The basis that meets this condition That is the constructed multi-dimensional access sparse basis;

[0038] Multidimensional access to sparse basis All basis functions Φ in n The subscript set consisting of the subscript n corresponding to (p) is defined as This results in the following explicit function:

[0039]

[0040] Furthermore, the specific implementation process of step 4) is as follows:

[0041] 4.1) Construct a set of sample points isomorphic to the multi-dimensional access sparse basis;

[0042] Multidimensional access sparse basis Each basis function has a corresponding subset of sample points Θ n, let Φ n (p) corresponds to the sample point subset Θ n is the tensor product form of the single-dimensional Gaussian integral point in equation (14);

[0043]

[0044] In the formula represents the tensor product operator, n i is a multidimensional basis function Φ n The order of the i-th random variable in (p) is the integration point of the generalized sparse grid integral.

[0045]

[0046] That is, a set of sample points isomorphic to the multi-dimensional access sparse basis is obtained;

[0047] 4.2) Perform deterministic calculations on the field line coupling solver;

[0048] The transmission line equations of the field line coupling solver are described by equations (16) and (17):

[0049]

[0050]

[0051] In the formula, C represents the capacitance matrix per unit length, G represents the conductance matrix per unit length, L represents the inductance matrix per unit length, and R represents the resistance matrix per unit length;

[0052] When the cable spacing is large, the inductance and capacitance of the cable are calculated by equations (18)-(21):

[0053]

[0054]

[0055] C=μεL -1 (20)

[0056]

[0057] V(z,t) and I(z,t) represent the voltage and current column vectors on the transmission line with position and time; V F and I F Represents the column vector of the equivalent inductive distributed voltage and current sources on the transmission line, which is determined by the external electric field;

[0058]

[0059]

[0060] E T 、E L They are the electric fields perpendicular to the cable. Integrate and sum the electric field parallel to the cable The external field is calculated by the three-dimensional finite difference time domain method; the transmission line equations (16) and (17) of the field line coupling solver are discretized, and combined with the Thevenin equivalent terminal load condition, the following FDTD iterative calculation formula is obtained:

[0061]

[0062]

[0063]

[0064]

[0065] The sample point set is used as the input of iterative calculation formulas (24)-(27); the desired state quantity corresponding to each sample point can be obtained.

[0066] Further, in step 5), the coefficients of the basis functions in the constructed multidimensional access sparse basis are calculated, and the specific implementation process is as follows:

[0067] Φ n The inner product of (p) and V(p) is calculated as follows:

[0068]

[0069] According to the orthogonality of chaotic polynomials, the inner product is not 0 when and only when n=m; the basis function coefficients can be calculated by the following formula:

[0070] <Φ n (p),V(p)>=c n <Φ n (p),Φ n (p)> (29)

[0071]

[0072] The solution of the basis function coefficients is transformed into the solution of the inner product. The generalized Smolyak sparse grid integral is used to calculate the inner product. The calculation formula is:

[0073]

[0074] χ n =<Φ n (p),Φ n (p)> (32)

[0075] The integral is represented by the tensor product operator Q of order greater than or equal to n k calculate;

[0076] The tensor product operator Q k The set of index k is recorded as

[0077] Tensor product operator Q k It is composed of the one-dimensional integral operator of each dimension i=1,…,d is constructed by tensor product, and the integrand r(p)=Φ n The integral expression of (p)V(p) is:

[0078]

[0079] In the formula represents the tensor product operator, k = (k 1 ,k 2 ,…k d ) is the index of the tensor product operator, i=1,…,d is w i (p i ) is the weight function with respect to the single variable p i K i +1 point Gaussian integral operator, w i (p i ) is the random variable p i The probability distribution of is the corresponding k i +1 single-dimensional integration point and integration coefficient;

[0080] when When , that is, when the integral grid is isomorphic to the sparse basis, multiply the tensor product operator by the corresponding coefficient α k , which is the generalized Smolyak sparse grid integral operator Q sparse :

[0081]

[0082] In formula (35), Q k is the tensor product operator, α k represents the tensor product operator Q k The combination coefficient of is the index set of the generalized Smolyak sparse grid. The coefficients of the generalized sparse grid are a set of The only constant that determines this is calculated as

[0083]

[0084] Where "∧" means "and", and "1" means all-1 vector; a property of Smolyak sparse grid is

[0085] Compared with the prior art, the present invention has the following beneficial effects:

[0086] 1. The present invention uses transmission line equations, generalized sparse grids and random point configuration methods to establish an input-output model of random input parameters and port coupling output responses of an external random electromagnetic environment based on chaotic polynomials, and realizes the randomness analysis of the cable port response under the irradiation of external strong electromagnetic pulses. It can accurately predict the statistical characteristics of the coupling response of external strong electromagnetic pulses and cable networks when there are random parameters, and physically reveals the coupling mechanism of cables and cable networks with external strong electromagnetic pulses.

[0087] 2. The present invention is based on generalized polynomial theory to form a non-invasive randomness analysis method. It does not require changing the original deterministic solver and can be easily combined with the cable network field line coupling solver. It can be applied to any (including time-domain finite difference method and moment method, etc.) cable network response solver, and can well calculate the statistical characteristics of the response of complex cable network systems in practice. It has good versatility and can be used even when the solver is complex or the solver is not open source.

[0088] 3. The random point configuration method based on generalized sparse grid proposed in the present invention can handle electromagnetic coupling problems involving multiple random variables, and by constructing a multi-dimensional access sparse basis, the number of integral points is reduced, further reducing the huge amount of computation caused by the dimensionality disaster in high-dimensional randomness, and solving the dimensionality disaster to a certain extent. When the dimension of the random variable reaches more than six dimensions, the computational efficiency of this method is improved by more than 9 times compared with the traditional random point configuration method, and this efficiency continues to increase with the increase of the dimension of the random variable and the order of the basis function, providing feasibility for the high-dimensional randomness analysis of the field line coupling of the cable network. BRIEF DESCRIPTION OF THE DRAWINGS

[0089] Figure 1 - Example diagram of the third-order admission basis, non-admission basis, classical basis and q-pseudo-norm basis in the two-parameter case. Each square represents a basis function, and the number (0,3) on the square indicates that this square represents the basis function Φ (0,3) (p).

[0090] Figure 2 - Illustration of the construction process of the third-order two-variable generalized Smolyak sparse grid integral in the embodiment (taking two parameters, w(p)=w1(p1)w2(p2)=0.5·0.5 as an example).

[0091] Figure 3-Schematic diagram of the routing and cross-section of multi-conductor transmission lines under different conditions. (a) is the electric field loop integral diagram of (n+1) multi-conductor cables placed in parallel; (b) is the current closed surface integral diagram of (n+1) multi-conductor cables placed in parallel; (c) is the cross-sectional structure diagram of (n+1) multi-conductor cables placed in parallel.

[0092] Figure 4 -Schematic diagram of the double-conductor transmission line structure under strong electromagnetic pulse irradiation.

[0093] Figure 5 - Comparison diagram of mean values ​​of remote load voltage in the embodiment. MC represents the calculation result of the Monte Carlo method, and SCM represents the calculation result of the random point configuration method based on the generalized sparse grid of the present invention.

[0094] Figure 6 - Comparison diagram of mean square error of voltage of remote load in the embodiment. Wherein, MC represents the calculation result of Monte Carlo method, and SCM represents the calculation result of random point configuration method based on generalized sparse grid of the present invention. DETAILED DESCRIPTION

[0095] The present invention proposes an analysis method for the coupling characteristics of a cable network and an external random electromagnetic environment based on a random point configuration method. The method first normalizes and models the multidimensional random variables such as the amplitude and incident direction of the external source, and selects the corresponding chaotic polynomial according to the probability distribution of the random variables; based on the concept of the admission basis, a multidimensional admission sparse basis and a set of sample points isomorphic to the sparse basis are constructed using a one-dimensional chaotic polynomial; then, the coupling response of the cable network and the external electromagnetic environment is deterministically solved at the sample points using a finite difference time domain algorithm to obtain the corresponding deterministic response at each sample point; then, based on the deterministic response results, the coefficients of the basis function are solved using numerical integration technology; finally, the constructed multidimensional basis function and the coefficients corresponding to each basis function are combined into an explicit function, and the explicit function is used to replace the field line coupling solver and perform random sampling to obtain its uncertainty analysis results, forming a new method for high-dimensional randomness analysis of transmission line networks. Due to its non-invasive characteristics, this method can well calculate the statistical characteristics of the response of complex cable network systems in practice, and greatly reduces the computational complexity of high-dimensional randomness analysis by generalizing sparse grids, thus solving the curse of dimensionality to a certain extent.

[0096] The specific embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0097] The present invention provides a method for predicting the time domain coupling characteristics of a cable network under an external random electromagnetic environment, which is performed according to the following steps:

[0098] (1) Determine the random variables of the external electromagnetic environment and cable coupling analysis and perform normalized modeling;

[0099] Random variable modeling is to use random variables to express the random events of the external random electromagnetic environment in mathematical language. It is the preprocessing part before uncertainty analysis. Only after completing the random variable modeling can the corresponding uncertainty analysis method be used for solving. In the actual cable network and the external random electromagnetic environment coupling process, the general random variables mainly include the amplitude, pulse width, incident angle, etc. of the external strong electromagnetic pulse. Considering the characteristics of the external strong electromagnetic pulse environment, the random factor (random variable) θ of the external electromagnetic environment can be made i Satisfied in [a i -b i ,a i +b i ] is uniformly distributed, as shown in formula (1).

[0100] θ i =a i ±b i ,i=1,…,d (1)

[0101] Where θ i represents a random variable, a i -b i represents the lower limit of the distribution, a i +b i represents the upper limit of the distribution, and d represents the number of random variables. After further normalization, the random variable is transformed into the normalized random variable p in formula (2): i ,θ i The randomness of the random variable is transferred to the normalized random variable p i superior.

[0102] θ i (p i )=a i +b i p i ,i=1,…,d (2)

[0103] p i , i=1,…,d all satisfy the normalized uniform distribution on the interval [-1,1]. Let p be the random vector space, which is composed of all normalized random variables p i The vector composed is shown in the following formula.

[0104] p={p 1 ,p 2 ,…p d} (3)

[0105] The vector space in equation (3) can represent all the uncertainties in the input of the coupling model between the cable network and the external random electromagnetic environment.

[0106] It is worth noting that whether using the Monte Carlo method or the generalized polynomial approximation theory, it is necessary to ensure that all random variables in the random variable space are independent of each other before uncertainty analysis can be performed. If the random variables are not independent of each other, the Karhunen-Loeve decomposition technique can be used to decouple the random variable space before uncertainty analysis is performed.

[0107] (2) Select the corresponding chaotic polynomial according to the probability distribution of the random variable;

[0108] The earliest generalized polynomial approximation theory expressed the Gaussian random process in the form of Hermite orthogonal polynomials. Later, some scholars used the Askey orthogonal polynomial system to extend it to the case of non-Gaussian random variables. The generalized polynomial approximation theory can be divided into two categories: the random Galerkin method and the random point collocation method. However, no matter which method is used, the first element of the generalized polynomial approximation theory is which basis function terms are selected to form the approximation function. This is because different probability distributions of generalized polynomials have different orthogonal polynomial bases. Choosing the correct orthogonal basis can ensure the accuracy and efficiency of the approximation. According to the Askey rule in the generalized polynomial approximation theory, for uniform distribution, Legendre polynomials are commonly used orthogonal bases; for normal distribution, Hermite polynomials are more suitable.

[0109] Taking the case of one-dimensional variables as an example, a random variable p is used as the independent variable of the approximation function, and a state quantity V to be determined is used as the dependent variable of the approximation function. The chaotic polynomial containing the random variable p is used as the basis function to expand the relationship between V and p. The explicit function is shown in formula (4). Then the coefficient c of each term of the function is calculated using the numerical integration method. n , that is, we get an explicit expression that can be used to express the functional relationship between V and p. We use the polynomial form to express the relationship between the state quantity to be determined and the random variable:

[0110]

[0111] In formula (4), φ n (p) is a chaotic polynomial. In the present invention, this polynomial φ n (p) is called the basis function, which is presented in the form of a random variable polynomial; c n is the coefficient of the basis function, which is the constant to be solved; and the set of basis functions {φ 0 ,…φ l} is called a basis. n is the order of the basis function, and l is the order of the expanded basis.

[0112] In the present invention, since the random variables of the external electromagnetic environment are uniformly distributed, Legendre polynomials are used to expand the cable coupling response V. Generally, l=3rd order expansion is sufficient to meet the accuracy requirements of most electromagnetic compatibility problems. The third order chaotic polynomial under the condition of a single random variable is shown below.

[0113] φ 0 (p)=1 (5)

[0114]

[0115]

[0116]

[0117] (3) Construct a multidimensional access sparse basis using one-dimensional chaotic polynomials of all random variables;

[0118] We have already completed the selection and expansion of one-dimensional chaotic polynomials. However, in actual situations, there are often multiple random variables at the same time. In this case, we need to construct a multidimensional random variable p = (p 1 ,p 2 ,p 3 …p d ) and the desired state quantity V. Based on the access conditions, this section uses one-dimensional chaotic polynomials to construct a multidimensional arbitrary sparse access basis. The basis functions in the basis are multiplied by the corresponding coefficients to form an explicit function with multidimensional random variables as input and the desired state quantity (cable terminal coupling voltage / current) as output in the form of sum.

[0119] Considering that each random variable is independent or can be decoupled by decomposition technology, the construction of n-dimensional classical basis function can be directly obtained by multiplying chaotic polynomials under single random variable conditions in the form of tensor product. The multidimensional basis function Φ under multiple random variables is n (p) is as shown in formula (9).

[0120]

[0121] Where n=(n 1 ,n 2 ,…n d ) is the multidimensional basis function Φ n Subscript of (p), n i ≥0,n i is the i-th unidimensional random variable p i The order of the chaotic polynomial is is the 1-norm of vector n and also represents the order of the multidimensional basis function.

[0122] The relationship between the state quantity to be determined and the random variable is also expressed in the form of a set of multidimensional basis functions of order 0-1:

[0123]

[0124] Where c n is the coefficient of the basis function, Φ n (p) is a multidimensional basis function, and V(p) is the state quantity to be determined. We define this set of multidimensional basis functions as a multidimensional classical basis l is a multidimensional classical basis The order (base The order of the highest-order basis function contained in ).

[0125] Among them: The relationship between the number of classical bases H and the dimension d and order l is shown in formula (11).

[0126]

[0127] It is obvious that the number of classical bases increases rapidly with the increase of dimension d, resulting in the dimensionality curse of high-dimensional parameterization problems. Some theories believe that the root cause of the dimensionality curse is the existence of too many mixed terms in the basis (i.e., terms with two or more different parameters, representing the interaction between the parameters, such as p 1 、p 2 ), if the number of basis functions in the basis can be reduced, the dimensionality curse can be delayed. Therefore, the present invention adopts a class of extremely flexible sparse basis, which is called multi-dimensional access sparse basis. It only needs to meet the access condition: if a basis function Φ n (p) belongs to the base Then all lower-order basis functions Φ m (p) must also belong to the base Its expression is shown in formula (12):

[0128]

[0129] Where m=(m 1 ,m 2 ,…m d ) is the multidimensional basis function Φ m The subscript of (p), Φ m (p) is the ratio Φ n (p) is a multidimensional basis function of lower order (i.e. ), the basis that satisfies this condition This is the constructed multidimensional access sparse basis. From formula (12), we can see that the access basis is defined in a recursive form rather than a closed form. It is not a basis, but a general basis. It is not difficult to verify that the classical basis and the q-pseudo-norm basis are its special forms. Figure 1Some examples of the admission basis in the two-dimensional parameter case are given. In the figure, (1,2) represents the basis function Φ in the two-dimensional parameter case. (1,2) (p), (1,1) represents the basis function Φ in the two-dimensional parameter case (1,1) (p). It can be seen that in geometry, the admittance basis can be figuratively interpreted as a convex set without "depressions", which is equivalent to stacking square wooden blocks one by one without collapse. In algebra, the admittance basis will not skip any low-order terms. Figure 1 (b) is not an admittance basis because it contains the term Φ (1,2) (p) but skips the low-order term Φ (1,1) (p) (i.e. the depression in the figure). Figure 1 (c) and Figure 1 (d) An example diagram of the third-order classical basis and q-pseudo-norm basis in the two-parameter case is given.

[0130] Multidimensional access to sparse basis All basis functions Φ in n The subscript set consisting of the subscript n corresponding to (p) is defined as This results in the following explicit function:

[0131]

[0132] (4) Construct a set of sample points isomorphic to any access sparse basis as the external electromagnetic environment sample input of the field line coupling solver, and use the FDTD (finite difference time domain) method to perform deterministic calculations on the field line coupling solver based on the FDTD on the sample point set to obtain the state quantity V (such as port induced voltage, current, etc.) corresponding to each sample point;

[0133] 4.1) Selection of sample point set

[0134] As mentioned above, third-order accuracy is sufficient to meet the requirements of most electromagnetic compatibility problems, so the selection of the sample point set only requires the first three-order Gaussian integral points of the one-dimensional chaotic polynomial as shown below.

[0135]

[0136] Considering that the condition for the generalized Smolyak sparse grid integral to match any sparse basis (12) is that the Smolyak sparse grid is isomorphic to the basis from a topological point of view, the following constructs a set of sample points that is isomorphic to any sparse basis (12). That is, any sparse basis Each basis function has a corresponding subset of sample points Θ n , for example Φ n (p) corresponds to the sample point subset Θ n is the tensor product form of the single-dimensional Gaussian integral point in equation (14).

[0137]

[0138] In the formula represents the tensor product operator, n i is a multidimensional basis function Φ n The order of the i-th random variable in (p) is the integration point of the generalized sparse grid integral.

[0139]

[0140] Any sparse basis All basis functions Φ in n (p) The corresponding subscript n consists of the subscript set definition Formula (15) is the set of sample points isomorphic to the multi-dimensional access sparse basis.

[0141] 4.2) Deterministic calculation of the field line coupling solver. The present invention considers a uniform transmission line system consisting of (n+1) parallel conductors excited by an external electromagnetic field, such as Figure 3 shown.

[0142] According to Faraday's theorem, along Figure 3 The loop C around the shaded area in (a) i By loop integration, the first equation in the transmission line equation can be derived; according to the charge continuity theorem, Figure 3 (b) The shaded area S i and S e The current density is integrated on the closed surface composed of the two electrodes, and then the second equation in the transmission line equation can be derived using Gauss's theorem. Therefore, the transmission line equation of field line coupling can be described by the following formula:

[0143]

[0144]

[0145] In the expression, C represents the capacitance matrix per unit length, G represents the conductance matrix per unit length, L represents the inductance matrix per unit length, and R represents the resistance matrix per unit length. w ), according to mathematical derivation, the inductance and capacitance of the cable can be approximately calculated by the following analytical formula:

[0146]

[0147]

[0148] C=μεL -1 (20)

[0149]

[0150] d i0 , d j0 , d ij , r w0 , r wi , r ij The collective parameters of cables such as Figure 3 As shown in (c), μ, ε, and σ are the magnetic permeability, dielectric constant, and conductivity of the medium between cables.

[0151] V(z,t) and I(z,t) represent the voltage and current column vectors on the transmission line with position and time. F and I F Represents the column vector of the equivalent inductive distributed voltage and current sources on the transmission line, which is determined by the external electric field.

[0152]

[0153]

[0154] E T 、E L They are the electric fields perpendicular to the cable. Integrate and sum the electric field parallel to the cable The external field is calculated by the three-dimensional finite difference time domain method. The transmission line equations (16) and (17) of field line coupling are discretized and combined with the Thevenin equivalent terminal load condition to obtain the following FDTD iterative calculation formula:

[0155]

[0156]

[0157]

[0158]

[0159] The influence of dielectric loss G is ignored in the above iterative formula.

[0160] The sample point set is used as the input of iterative calculation formulas (24)-(27); the desired state quantity corresponding to each sample point can be obtained.

[0161] (5) Using numerical integration methods, calculate any sparse basis constructed The basis function Φ n (p) Individual coefficients.

[0162] Φ n The inner product of (p) and V(p) is calculated as follows:

[0163]

[0164] According to the orthogonality of chaotic polynomials, the inner product is not 0 when and only when n=m. Therefore, the basis function coefficients can be calculated as follows:

[0165] <Φ n (p),V(p)>=c n <Φ n (p),Φ n (p)> (29)

[0166]

[0167] The solution of the basis function coefficients is transformed into the solution of the inner product. Next, the generalized Smolyak sparse grid integral is used to calculate the inner product. This method converts the Φ corresponding to each sample point p into n (p)V(p) is weighted summed to get the inner product Φ n (p),V(p)>value(Φ n (p) is the constructed basis function, which is known and can be directly substituted into p for evaluation). The calculation formula is as follows:

[0168]

[0169] χ n =<Φ n (p),Φ n (p)> (32)

[0170] The integral is represented by the tensor product operator Q of order greater than or equal to n k Calculation (low-order integrals are not included in the calculation).

[0171] To explain the sparse grid coefficient α k and the tensor product operator Q k The present invention has previously defined any access to the sparse basis The set composed of the subscript n of each basis function in is recorded as Now let the tensor product operator Q k The set of index k is recorded as

[0172] First, we introduce the tensor product operator Q k , which is composed of the one-dimensional integral operators of each dimension i=1,…,d is constructed by tensor product, and the integrand r(p)=Φ n The integral expression of (p)V(p) is:

[0173]

[0174] In the formula represents the tensor product operator, k = (k 1 ,k 2 ,…k d ) is the index of the tensor product operator, i=1,…,d is w i (p i )(that is, the random variable p i The probability distribution of the weight function is the probability distribution of the single variable p i K i +1 point Gaussian integral operator, is the corresponding k i +1 single-dimensional integration point and integration coefficient. It is worth noting that and a single-parameter orthogonal polynomial All are for the weight function w i (p i ), and k i +1 Gaussian integration point is exactly The zero point.

[0175] About the single-dimensional parameter p i Under the condition of uniform distribution on [-1, 1] (i.e. w i (p i )=0.5), k i +1 point single-dimensional integral coefficient as follows:

[0176]

[0177] when When , that is, when the integral grid is isomorphic to the sparse basis, multiply the tensor product operator by the corresponding coefficient α k , which is the generalized Smolyak sparse grid integral operator Q sparse :

[0178]

[0179] In formula (35), Q k is the tensor product operator, α k represents the tensor product operator Q k The combination coefficient of (called grid coefficient). is the index set of the generalized Smolyak sparse grid (also the index basis of the sparse access basis, because the two are isomorphic). The coefficients of the generalized sparse grid are a set of constants uniquely determined by K, and the calculation formula is

[0180]

[0181] Where "∧" means "and", and "1" is a vector of all 1s. Formula (36) means to calculate the grid coefficient α corresponding to the index k k , need to Find the index that is not less than it but not greater than it plus 1, and then sum it. One property of the Smolyak sparse grid is

[0182] (6) The basis function coefficients obtained by solving equation (31) are multiplied by the corresponding basis functions and the sum is obtained to obtain an explicit function that replaces the field line coupling solver. The explicit function is quickly randomly sampled to obtain the statistical value of the state quantity V (cable terminal coupling voltage) to be determined.

[0183] The explicit function that replaces the field line coupling solver is

[0184]

[0185] Repeated random experiments are performed on the obtained polynomial function input random parameter p to obtain the desired state quantity V(p), and the expectation q and mean square error p of the desired state quantity V(p) are obtained according to formulas (38) and (39).

[0186]

[0187]

[0188] Where q is the expectation, p is the mean square error, N is the number of random experiments, V(p) i is the state quantity to be determined in each random experiment.

[0189] The present invention is further described below with reference to specific examples and accompanying drawings.

[0190] Consider Figure 4 The two-conductor transmission line system shown in Figure 1 is excited by an external plane wave. The transmission line configuration is as follows: the length is L = 50m, the spacing between the cables is h = 5cm, and the wire radius is r = 2mm. The distributed parameters of the cable are calculated to be L = 1.2876uH / m and C = 8.6414pF / m. The plane wave incident direction and the electric field polarization direction are shown in Figure 4 As shown, the pitch angle is a random variable θ = 135° + 22.5°p 1 , Azimuth The polarization direction is α = 180°. The near end is connected to a 50Ω linear load, and the far end load is 200Ω. The incident plane wave is a Gaussian pulse modulated by the medium wave, and its amplitude is a random variable. The analytical function form can be described as:

[0191] E(t)=-(1+0.1p 2 )cos(ωt)exp(-4π(tt0 ) 2 / τ 2 )V / m(40)

[0192] Where τ=2us, t 0 =1.6us,

[0193] p 1 and p 2 are random variables, all of which satisfy the uniform distribution on the interval [-1,1]. And p is a random vector space, which is composed of p 1 and p 2 The vector is as follows

[0194] p={p 1 ,p 2} (41)

[0195] The vector space in equation (41) can represent all uncertainties in the model input.

[0196] Below Figure 2 The construction process of the generalized Smolyak sparse grid shown in equation (12) is introduced in detail. The example admission basis used is a third-order two-variable basis, which contains 8 basis functions, namely like Figure 2 (a) as shown (with Figure 1 (a) is the same as the basis). Figure 2 (b) for Figure 2 (a) is a generalized Smolyak sparse grid for sparse admission basis matching. Then the basis function Φ (0，0) The corresponding sparse grid integration point of (p) is Basis function Φ (0,1) The corresponding sparse grid integration point of (p) is Access sparse basis The sparse grid integration points corresponding to each basis function in are as follows Figure 2 As shown in (c), these point sets are merged to obtain a complete set of sample points as follows Figure 2 (d) as shown.

[0197] Deterministic field line coupling is solved at these points to obtain the corresponding responses, which are used to calculate the basis function coefficients.

[0198] According to the above formula (31), the basis function Φ (1,1) The coefficient c of (p) (1,1) Should only be Q (1,1) Calculate; and the basis function Φ (1,0) The coefficient c of (p) (1,0) Should be composed of 4 tensor product rules Q (1,0) , Q(2,0) , Q (3,0) , Q (1,1) Calculate together (see Figure 2 (b) dark blue part), and the calculation formula is (-1)·Q (1,0) +0·Q (2,0) +1 Q (3,0) +1 Q (1,1) , where -1,0,1,1 are the four Q k The grid factor α k . Excluding Q with a grid factor of 0 (2,0) , the other three integral rules have a total of 10 points, so c (1，0) It can be written as a linear combination of the function values ​​at these 10 points, that is

[0199]

[0200] In formula (42), 1 is The first real vector is the integral coefficient, "·" represents the inner product of two real vectors, p (m) , m=1,…,10 are the above integration points, and the second real vector is Φ (1,0) The values ​​of (p)r(p) at these points. The coefficients of other basis functions are calculated using similar steps.

[0201] The explicit function is used to replace the field line coupling model (solver) and repeated random experiments are performed. The statistical characteristics (mean and mean square error) of the induced voltage at the remote load of the cable at each moment are calculated using formulas (38) and (39), and the results are compared with those of 500 Monte Carlo random experiments. Figure 5 , 6 shown.

[0202] It can be found that the statistical feature prediction results of the method proposed in the present invention at each time are consistent with the Monte Carlo method results and have high accuracy. This shows that the random point configuration method based on generalized sparse grid can effectively analyze the random response of cable networks under strong electromagnetic pulse irradiation.

[0203] The randomness analysis method proposed in the present invention can be applied to any cable network response solution method (including time-domain finite difference method and moment method, etc.), and has universal applicability. It can also be directly used to calculate the statistical characteristics of the response of complex cable network systems in practice, which helps to better solve the randomness problem of complex cables with high-dimensional randomness factors, and provides prediction methods and data support for complex cable network analysis and prediction and targeted electromagnetic interference suppression.

[0204] Finally, it should be noted that the above examples of the present invention are merely examples for illustrating the present invention, and are not intended to limit the embodiments of the present invention. Although the applicant has described the present invention in detail with reference to the preferred embodiments, for those of ordinary skill in the art, other different forms of changes and modifications can be made based on the above description. It is impossible to list all the embodiments here. Any obvious changes or modifications derived from the technical solution of the present invention are still within the scope of protection of the present invention.

Claims

1. A method for predicting the time domain coupling characteristics of cable networks under external random electromagnetic environments. Features: Follow the steps below, 1) Determine the random variables of the external electromagnetic environment and cable coupling analysis, and perform normalized modeling; 2) According to the probability distribution of random variables, select the corresponding chaotic polynomial; 3) Construct a multidimensional access sparse basis using one-dimensional chaotic polynomials of all random variables; 4) Construct a set of sample points isomorphic to the multi-dimensional access sparse basis, each sample point corresponds to the specific value of all random variables in the external electromagnetic environment, and the sample point set is used as the external electromagnetic environment sample input of the field line coupling solver; perform deterministic calculations on the field line coupling solver on the sample point set to obtain the state quantity to be calculated corresponding to each sample point; 5) Based on the sample point set obtained in step 4) and the state quantity to be determined corresponding to each sample point, the coefficients of the basis function in the multidimensional access sparse basis constructed in step 3) are calculated using a numerical integration method; 6) The basis functions in the constructed multidimensional access sparse basis and the coefficients corresponding to each basis function are composed into an explicit function, which is used to replace the field line coupling solver and randomly sampled to obtain the statistical value of the state quantity to be determined, thereby realizing the prediction of the time domain coupling characteristics of the cable network under an external random electromagnetic environment.

2. According to the method for predicting the time domain coupling characteristics of a cable network under an external random electromagnetic environment according to claim 1, Features: The specific implementation process of step 1) is as follows: Let the random variable θ of the external electromagnetic environment and cable coupling analysis be i Satisfied in [a i -b i ,a i +b i ] is uniformly distributed, as shown in formula (1); θ i =a i ±b i ,i=1,…,d (1) Where θ i represents a random variable, a i -b i represents the lower limit of the distribution, a i +b i represents the upper limit of the distribution, and d represents the number of random variables; By normalizing the random variable, the random variable is transformed into the normalized random variable p in formula (2): i ,θ i The randomness of the random variable is transferred to the normalized random variable p i superior; θ i (p i )=a i +b i p i ,i=1,…,d (2) p i , i=1,…,d all satisfy the normalized uniform distribution on the interval [-1,1]; Let p be a random vector space consisting of all normalized random variables p i The vector formed is shown in formula (3); p={p 1 ,p 2 ,…p d } (3) The vector space in equation (3) represents all the random variables in the input of the field line coupling solver.

3. According to the method for predicting the time domain coupling characteristics of a cable network under an external random electromagnetic environment as described in claim 1, Features: Step 2) Select the corresponding chaotic polynomial according to the probability distribution of the random variable. The specific implementation process is as follows: The relationship between the state quantity V to be determined and the random variable p is expressed in the form of a polynomial: that is, the chaotic polynomial containing the random variable p is used as a basis function to expand the relationship between the state quantity V to be determined and p, as shown in formula (4); In the formula, φ n (p) is a chaotic polynomial. This polynomial φ n (p) is called the basis function and is presented in the form of a random variable polynomial; c n are the coefficients of the basis functions; the set of basis functions {φ 0 ,…φ l } is called a basis; n is the order of the basis function, and l is the order of the expanded basis.

4. According to the method for predicting the time domain coupling characteristics of a cable network under an external random electromagnetic environment as described in claim 3, Features: Step 3) Use the one-dimensional chaotic polynomials of all random variables to construct a multi-dimensional access sparse basis. The specific implementation process is as follows: Multidimensional Basis Function Φ under Multiple Random Variables n (p) As shown in formula (9), Where n=(n 1 ,n 2 ,…n d ) is the subscript of the multidimensional basis function Φn(p), n i ≥0,n i is the i-th unidimensional random variable p i The order of the chaotic polynomial is is the 1-norm of vector n and also represents the multidimensional basis function Φ n The order of (p); The relationship between the state quantity to be determined and the random variable is expressed in the form of a set of multidimensional basis functions of order 0-1, as shown in formula (10): Where c n is the coefficient of the basis function, Φn(p) is the multidimensional basis function, V(p) is the state quantity to be determined; the set of this multidimensional basis function is defined as the multidimensional classical basis l is a multidimensional classical basis The order of If a multidimensional basis function Φ n (p) belongs to the base Then all lower-order multidimensional basis functions Φ m (p) must also belong to the base base The expression of is shown in formula (12): Where m=(m 1 ,m 2 ,…m d ) is the subscript of the multidimensional basis function Φm(p), which is a multidimensional basis function of lower order than Φn(p), that is, The basis that meets this condition That is the constructed multi-dimensional access sparse basis; Multidimensional access to sparse basis All basis functions Φ in n The subscript set consisting of the subscript n corresponding to (p) is defined as This results in the following explicit function:

5. According to the method for predicting the time domain coupling characteristics of a cable network under an external random electromagnetic environment according to claim 4, Features: Step 4) The specific implementation process is as follows: 4.1) Construct a set of sample points isomorphic to the multi-dimensional access sparse basis; Multidimensional access sparse basis Each basis function has a corresponding subset of sample points Θ n , let Φn(p) correspond to the sample point subset Θ n is the tensor product form of the single-dimensional Gaussian integral point in equation (14); In the formula represents the tensor product operator, ni is the multidimensional basis function Φ n The order of the i-th random variable in (p) is the integration point of the generalized sparse grid integral. That is, a set of sample points isomorphic to the multi-dimensional access sparse basis is obtained; 4.2) Perform deterministic calculations on the field line coupling solver; The transmission line equations of the field line coupling solver are described by equations (16) and (17): In the formula, C represents the capacitance matrix per unit length, G represents the conductance matrix per unit length, L represents the inductance matrix per unit length, and R represents the resistance matrix per unit length; When the cable spacing is large, the inductance and capacitance of the cable are calculated by equations (18)-(21): C=withL -1 (20) V(z, t) and I(z, t) represent the voltage and current column vectors on the transmission line with position and time; V F and I F Represents the column vector of the equivalent inductive distributed voltage and current sources on the transmission line, which is determined by the external electric field; E T 、E L They are the electric fields perpendicular to the cable. Integrate and sum the electric field parallel to the cable The external field is calculated by the three-dimensional finite difference time domain method; the transmission line equations (16) and (17) of the field line coupling solver are discretized, and combined with the Thevenin equivalent terminal load condition, the following FDTD iterative calculation formula is obtained: The sample point set is used as the input of iterative calculation formulas (24)-(27); The desired state quantity corresponding to each sample point can be obtained.

6. According to the method for predicting the time domain coupling characteristics of a cable network under an external random electromagnetic environment as described in claim 4, Features: In step 5), the coefficients of the basis functions in the constructed multidimensional access sparse basis are calculated. The specific implementation process is as follows: Φ n The inner product of (p) and V(p) is calculated as follows: According to the orthogonality of chaotic polynomials, the inner product is not 0 when and only when n=m; the basis function coefficients can be calculated by the following formula: <Φ n (p),V(p)>=c n <Φ n (p),Φ n (p)> (29) The solution of the basis function coefficients is transformed into the solution of the inner product. The generalized Smolyak sparse grid integral is used to calculate the inner product. The calculation formula is: x n =<Φ n (p),Φ n (p)> (32) The integral is represented by the tensor product operator Q of order greater than or equal to n k calculate; The tensor product operator Q k The set of index k is recorded as Tensor product operator Q k It is composed of the one-dimensional integral operator of each dimension It is constructed by tensor product. At this time, the integrand r(p) = Φ n The integral expression of (p)V(p) is: In the formula represents the tensor product operator, k = (k 1 ,k 2 ,…k d ) is the index of the tensor product operator, So w i (p i ) is the weight function with respect to the single variable p i K i +1 point Gaussian integral operator, w i (p i ) is the random variable p i The probability distribution of is the corresponding k i +1 single-dimensional integration point and integration coefficient; when When , that is, when the integral grid is isomorphic to the sparse basis, multiply the tensor product operator by the corresponding coefficient α k , which is the generalized Smolyak sparse grid integral operator Q sparse : In formula (35), Q k is the tensor product operator, α k represents the tensor product operator Q k The combination coefficient of is the index set of the generalized Smolyak sparse grid. The coefficients of the generalized sparse grid are a set of The only constant that determines this is calculated as Where "∧" means "and", "1" means all 1 vector; a property of Smolyak sparse grid is

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