CN107229762B - A Microwave Circuit Characteristic Analysis Method Containing Semiconductor Physical Model - Google Patents

Abstract

The invention discloses a method for analyzing the characteristics of a microwave circuit including a semiconductor physical model. This method mainly uses the field-circuit coupling algorithm of the time-domain volume integration method to analyze the structure of the microwave circuit with the semiconductor physical model. The field-circuit coupling algorithm requires the same current and potential at the junction of the electromagnetic structure and the circuit structure. The physical properties of the semiconductors are solved numerically by the time-domain spectral element method. Because the internal physical change process of the semiconductor is described by the drift-diffusion equation, the physical characteristics of the semiconductor are nonlinear. The calculation process of the electrical characteristics of the microwave circuit needs to use a discrete Newton iteration method, and an improved strictly synchronous coupling solution process. Improve the stability and efficiency of the algorithm.

Description

A Microwave Circuit Characteristic Analysis Method Containing Semiconductor Physical Model

technical field

The invention relates to the technical field of electromagnetic simulation, in particular to the analysis of the real physical characteristics of a microwave semiconductor circuit, and proposes a time-domain volume area integration method that can analyze the structure of an actual microwave circuit.

Background technique

Time-domain methods are very useful for the simulation of field-one-way mixing problems involving integrated circuit modules and substrate systems, because nonlinear components in circuits can be accurately modeled and broadband information can be obtained. The research on the coupling problem of the field in the time domain simulation mainly relied on the difference equation method, such as the finite difference method. addition of components. However, with the recent development of time-domain integral equation solving techniques, especially the improvement of its late-time stability, the reduction of computational complexity and the enhancement of computational performance with the adoption of fast algorithms, as well as new simulations for field-path coupling problems Continuing study of the program. All of these make the TDIE solution get extensive attention.

Initially, the TDIE method's efforts to solve the field-to-one hybrid problem simulation were based on the Partial Element Equivalent Circuit (PEEC) method. The key idea of PEEC is to convert the electromagnetic coupling between wires into an equivalent circuit and combine it with other Arbitrary lumped circuit models are linked together to form a circuit simulator, just like SPICE software. The recent improvements in stability and computational speed of TDIE have led to the idea of combining the integral equation method with the circuit analysis method, first in metallic structures, and then extended to metallic-dielectric hybrid structures. Generally, for the mixed metal/dielectric circuit problem, the TDIE method is combined with the modified nodal analysis (MNA) method of the circuit simulation method. The basis of the electromagnetic coupling scheme of solving circuit is the idea of coupling current, and the circuit part and the electromagnetic structure part in the area are connected by this coupling current. Since this method can add circuit excitation and field excitation at the same time, it can be used for signal integrity and EMI/EMC simulation.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a method for analyzing the characteristics of a microwave circuit including a semiconductor physical model, while considering the equation description of the real physical process in the microwave semiconductor circuit and coupling the field-circuit matrix equation using a field-circuit coupling algorithm, and finally through the improved Newton iteration The method is strictly synchronous to solve the physical parameters of the circuit.

The technical solution for realizing the purpose of the present invention is: a method for analyzing the characteristics of a microwave circuit containing a semiconductor physical model, and the steps are described as follows:

The first step is to establish the solution model of the microwave circuit structure. The metal part of the electromagnetic structure is divided into triangles, and the medium part is divided into tetrahedrons to obtain the structural information of the model, that is, the size of each triangle and tetrahedron. unit information;

The second step is to determine the time-domain electric field integral equation of the electromagnetic target from Maxwell's equations;

In the third step, it is necessary to discretize the surface current of the metal and the electric flux density of the dielectric body in the spatial and temporal domains, respectively. Here, the discretization method of the surface current on the metal surface is to use the RWG basis function to discretize in space, and the division method of the metal surface uses the triangular patch division method. The difference between the processing of the dielectric body and the metal is the spatial discretization of the electric flux density. In the spatial domain, the SWG basis function is used for discretization, and the tetrahedron is used for the division of the dielectric body. . In addition, in the time domain, the time domain volume integral equation is used to discretize the metal and dielectric parts using the 4th-order Lagrange interpolation time basis function.

The fourth step is to substitute the expanded expressions of the metal surface current and the electric flux density of the dielectric body in the third step into the time-domain electric field integral equation in the second step, and then use the time-domain electric field integral equation in the discrete form respectively. Point test, use Galerkin test in space, get the system impedance matrix equation;

The fifth step is to start from the semiconductor drift-diffusion equations, expand the required carrier concentration and potential at each node, use the Galerkin method to test the semiconductor drift-diffusion equations, and use the Newton iteration method to solve them. The carrier and potential distribution of the node, and finally the physical parameters inside the semiconductor;

In the sixth step, according to the system impedance matrix equation obtained in the fourth step and the semiconductor physical parameters obtained in the fifth step, a strictly synchronized field-circuit coupling matrix equation is established through the idea of field-circuit coupling, and then the microwave is obtained by the improved Newton iterative solution. The time-domain current distribution in the circuit structure, the physical parameters are obtained according to the time-domain current distribution, and the simulation process is completed.

Compared with the prior art, the present invention has significant advantages: (1) Compared with the time-domain surface integral equation method, the adopted time-domain volume integral equation method can handle more complex three-dimensional electromagnetic structures, such as metal-dielectric mixed microstrip transmission lines structure, etc. (2) The physical characteristic equations of microwave semiconductor circuits are based on the semiconductor drift-diffusion equations. The carrier concentration and potential to be solved are expanded at each node, and the equations are tested by the Galerkin method and the Newton iteration method. The carrier and potential distribution of each node are obtained by solving, which can more accurately analyze the physical process of the PIN tube compared with the equivalent circuit model, and can further analyze the electro-thermal integration of the microwave circuit. (3) Combine the physical model equation of the semiconductor into the MOT-based time domain volume integral equation to form a strictly synchronous field-circuit coupling solution method. The solution process adopts an improved Newton iteration method, which can ensure the calculation accuracy of the field-circuit coupling solution. while improving the computational efficiency to a certain extent.

Description of drawings

Figure 1 is the RWG basis function.

Figure 2 is the SWG basis function.

FIG. 3 is a cross-sectional view of a MOSFET.

Figure 4 is a three-dimensional electromagnetic structure model of a MOSFET amplifier circuit.

Fig. 5 is the simulation result of the MOSFET tube amplifier circuit under the action of sinusoidal continuous signal.

Detailed ways

The present invention combines the electromagnetic structure analysis process based on the time-step (MOT) time-domain volume integration method and the semiconductor circuit structure analysis process based on the time-domain spectral element method, that is, the nonlinear semiconductor physics equation is merged into the MOT-based time domain. A hybrid field-circuit solution is formed in the volume integral equation, which makes the analysis and calculation of the mutual coupling between the electromagnetic structure and the circuit structure more consistent and effective. Compared with the equivalent analytical model, the physical model is more in line with the real physical process when analyzing the characteristics of the actual structure, and the calculation analysis is more accurate. However, the analysis of the physical model also consumes more time and computing resources. For this problem, an improved Newton iterative solution is proposed, which can save some computing time. Due to the complexity of the simulation of the real physical process, there are few related reports on the field-circuit coupling analysis of the semiconductor physical process for microwave circuit problems, and no report has been reported in the time-domain volumetric integration method. This is also the innovation of the present invention.

The present invention will be described in further detail below with reference to the accompanying drawings.

1. The basic principle of the time-domain volumetric integration method

Assume that there is a metal-medium mixed target in free space, where the surface of the metal is denoted by S, the volume of the medium is denoted by V, and the medium is isotropic, non-magnetic, non-dispersive, and lossless, and its The dielectric constant is ε(r). The permittivity of free space is ε ₀ and the permeability is μ ₀ . When the time-domain incident wave E ^inc (r, t) irradiates the hybrid target, the induced surface current J ^s (r, t) and the polar body current J will be generated on the metal surface S and in the dielectric body V, respectively. ^v (r,t). Then, the scattered field in space includes the sum of the scattered fields jointly generated by the induced surface current and the polarizer current, namely:

and there is:

In addition, in the dielectric body, the relationship between the electric flux density D(r,t), the polar body current J ^v (r,t) and the total electric field E(r,t) satisfies the following expression:

D(r,t)=ε(r)E(r,t) (1.4)

Since there are two types of targets in space, metal and dielectric, the corresponding time domain integral equations must be constructed respectively. The time derivative of the total electric field satisfying the tangential direction on the metal surface S is zero, while the time derivative of the total electric field in the dielectric body V is zero. It is equal to the sum of the time derivative of the incident electric field and the time derivative of the scattering electric field, namely:

The above equations (1.6) and (1.7) together form a time-domain volumetric integral equation suitable for analyzing metal-dielectric mixed targets. Here, it is worth noting that the time domain volume integral equation in the above equation operates on the time derivation of both sides of the equation.

In order to solve the above-mentioned time-domain volume integral equation numerically, it is necessary to discretize the surface current J ^s (r,t) of the metal and the electric flux density D(r,t) of the dielectric body in the space and time domains, respectively. . Here, for the discretization method of the surface current J ^s (r, t) of the metal surface, the RWG basis function is used for spatial discretization, and the division method of the metal surface is a triangular patch. The difference between the processing of the dielectric body and the metal is the spatial discretization of the electric flux density D(r, t). In this paper, the SWG basis function is used to discretize in the spatial domain. The volume division of the tetrahedron. In addition, in the time domain, the time domain volume integral equation is used to discretize the metal and dielectric parts using the 4th-order Lagrange interpolation time basis function. Among them, the definition of the RWG basis function is shown in Figure 1 below, and the definition of the SWG basis function is shown in Figure 1.

和

共同组成了第n个基函数。l_n表示第n个基函数未知量边的边长，

分别表示上、下三角形的面积，

为空间矢量。第n个基函数为：Each complete RWG basis function consists of two different triangles, called the upper and lower triangles of the basis function.

and

Together they form the nth basis function. l _n represents the side length of the unknown side of the nth basis function,

are the areas of the upper and lower triangles, respectively.

is a space vector. The nth basis function is:

Using the RWG basis function given in Fig. 1 to expand the approximation to the surface current J(r) of an ideal conductor, it can be expressed as

Here, N represents the total number of basis functions obtained after triangular discretization of the entire target surface of the ideal conductor, In represents the current coefficient corresponding to the _nth RWG basis function, and J(r) is the current density (in ampere/meter or A /m).

和

公共面是第n个介质三角形，面积是a_n。此外，矢量

是上四面体的自由顶点指向源点r的矢量，反之可知

这样就可以写出与第n个三角形面片相关的SWG基函数的表达式为：The adjacent tetrahedra in Figure 2 are

and

The common face is the n-th dielectric triangle and the area is a _n . Also, the vector

is the vector of the free vertices of the upper tetrahedron pointing to the source point r, and vice versa

In this way, the expression of the SWG basis function related to the nth triangular patch can be written as:

分别对应于上、下四面体的体积。除了上述介绍的空间基函数外，本文在针对金属—介质混合问题时使用的4阶Lagrange插值时间基函数为：here

correspond to the volumes of the upper and lower tetrahedrons, respectively. In addition to the space basis functions introduced above, the fourth-order Lagrange interpolation time basis function used in this paper for the metal-dielectric mixing problem is:

In this way, J ^s (r, t) and D (r, t) can be discretely expanded into the following forms through the basis functions in space and time, namely:

各自是RWG基函数和SWG基函数，N_s、N_v各自是金属和介质体离散后的空间基函数的未知量数，N_t表示混合目标离散后的时间基函数的未知量数，

分别表示与金属和介质部分的基函数相关的未知系数，T_j(t)＝T(t-jΔt)，Δt为时间步的大小。in,

are the RWG basis function and the SWG basis function respectively, N _s and N _v are the unknown quantities of the space basis functions after the discretization of the metal and the dielectric body respectively, N _t represents the unknown quantities of the time base functions after the discretization of the mixed target,

represent the unknown coefficients related to the basis functions of the metal and dielectric parts, respectively, T _j (t)=T(t-jΔt), where Δt is the size of the time step.

和

分别对离散后的时域体面积分方程式(1.6)和式(1.7)进行空间上的伽辽金测试，并且在每个时刻t_j＝jΔt对离散后的时域体面积分方程进行时间上的点匹配，这样就可以得到一系列的方程组，可以写成矩阵方程的形式，即：After substituting equations (1.12) and (1.13) into equations (1.6) and (1.7), all the space basis functions are used

and

The Galerkin test in space is performed on the discrete time-domain volumetric integral equations (1.6) and (1.7), respectively, and the discrete time-domain volumetric integral equations are subjected to time points at each time t _j =jΔt Matching, so that a series of equations can be obtained, which can be written in the form of matrix equations, namely:

in,

Among them, j=1, . . . , N _t , and 〈·,·〉represents the inner product. By solving the matrix equation of equation (1.14) at each time t=jΔt, the unknown coefficients corresponding to all N _EM =N _s +N _v basis function unknowns at each time step can be obtained. —Analysis of the medium mixing target realizes the solution scheme in the time-stepping mode, and Equation (1.14) is also called the time-domain volumetric integral equation based on time-stepping.

2. Solving the physical model of semiconductor

为变量。Solve the transient drift-diffusion equation of the MOSFET with the coupled method, that is, solve the Poisson equation and the current continuity equation simultaneously, with the carrier concentration n, p and potential

for the variable.

The transient model equations for the MOSFET include:

Normalized Poisson equation:

In the Poisson equation of the above formula (2.1), Γ is the net doping concentration, ε ₁ , ε ₂ are the dielectric constants, which are expressed as:

Normalized electron current density equation:

Normalized hole current density equation:

Normalized electron current continuity equation:

Normalized hole current continuity equation:

Normalized compound rate model:

As shown in Figure 3, the boundary conditions of the MOSFET are:

For the Poisson equation, the solution region is the entire MOSFET and the boundary conditions are:

The gate, drain, source and base plates are fixed boundary conditions (metal boundary conditions):

Floating boundary conditions parallel to the x-axis

Si-SiO2 interface

For the current continuity equation, the solution region is semiconductor, excluding oxides, and the boundary conditions are:

Drain, source and base plates are fixed boundary conditions (metal boundary conditions):

N area: n=Г, p=1/ΓP area: n=-1/Γ, p=-Γ(2.10)

CD+EG+FH are floating boundary conditions

Note that the front and rear in the 3D model are set to floating boundary conditions.

Since both the current continuity equation and the Poisson equation are nonlinear, the Taylor expansion is used to linearize the equations.

The fully coupled method is used to solve the drift-diffusion equation, and the equation after Taylor expansion is written in the form of equation (2.12):

The final matrix form is obtained by appropriate derivation:

In formula (2.13), each matrix block is as follows:

For the drift-diffusion model, it is necessary to point out the treatment method of the avalanche generation term. Its expression is shown in (2.14):

In the above formula (2.14), the ionization coefficients of electrons and holes are:

Among them, T is the temperature at the current moment inside the device, T _ref is the initial ambient temperature, An , B _n , C _n , D _n and _Ap , B _p , C _p , D _{p are} constants.

后，便可以通过下面的公式求得相应极板的每一个节点的电流。Find the electron quasi-Fermi potential φ _n , the hole quasi-Fermi potential φ _p and the potential

After that, the current of each node of the corresponding plate can be obtained by the following formula.

Electron current at each node on the plate:

Hole current at each point on the plate:

Displacement current:

The sum of the currents at each node is the current generated by the corresponding carriers of the plate. In transient simulations, the displacement current cannot be ignored. The whole semiconductor solution process can be regarded as the current process of solving the semiconductor transistor output with known input voltage value.

3. Field-Road Coupling Algorithm

和

当作是等效的电压源的馈电电压，此外，流过半导体的电流等于垂直流过该加载边的电流，由于G位置处边界边的电流系数

表示的是垂直流过该边的电流密度，因此流过该In Figure 4, the RWG triangular boundary edges at positions G and D of the electromagnetic structure connected to the loaded semiconductor are regarded as equivalent voltage source edges, and the voltage on the source is

and

Treat the feed voltage as an equivalent voltage source, furthermore, the current flowing through the semiconductor is equal to the current flowing vertically through the loaded edge, due to the current coefficient of the boundary edge at the G position

represents the current density flowing vertically through the edge, so

得到混合场—路方程：The current of the side is equal to the current coefficient multiplied by the side length a, that is

Get the mixed field-road equation:

Since the gate current of the MOSFET is very small, it is treated as a zero value this time.

You can get:

The nonlinear field-circuit coupling system equation (3.3) in the time domain is simplified to:

只与电路未知量

有关，且电路未知量的个数N_CKT远小于场未知量的个数N_EM，那么就可以认为该系统方程组中非线性方程的维数远小于线性方程的维数。因此，传统的求解方案利用标准的牛顿迭代法对整个式(3.4)这样的如此大的矩阵系统进行求解，这并不是一个很有效的、优化的方案。此外，由于每个时间步的每个牛顿迭代步下，式(3.4)中矩阵维数为N_EM+N_CKT的雅可比矩阵都是变化的，这样不管使用直接解法还是使用迭代解法，对如此大维数的矩阵方程进行求解都是很耗时且繁琐的。因此，为了更有效地求解非线性的耦合系统方程式(3.4)，the nonlinear term of the equation

Only with circuit unknowns

and the number of circuit unknowns N _CKT is much smaller than the number of field unknowns N _EM , then it can be considered that the dimension of the nonlinear equation in the system of equations is much smaller than the dimension of the linear equation. Therefore, the traditional solution scheme uses the standard Newton iteration method to solve such a large matrix system as the whole formula (3.4), which is not a very efficient and optimal scheme. In addition, due to each Newton iteration step of each time step, the Jacobian matrix of the matrix dimension N _EM + N _CKT in Eq. (3.4) changes, so no matter whether the direct solution method or the iterative solution method is used, for this Solving matrix equations with large dimensions is time-consuming and tedious. Therefore, in order to solve the nonlinear coupled system equation (3.4) more efficiently,

Split into two equations, namely:

用电路未知量

来进行表示，可表示为：Convert the field unknown in Eq. (3.5) to

Using circuit unknowns

to express, it can be expressed as:

Next, substitute formula (3.7) into formula (3.6), and after sorting, we can get:

Express the nonlinear equation (3.8) as:

Then equation (3.8) can be abbreviated as:

The above nonlinear equation (3.11) is also solved according to the discrete Newton iteration method:

make

The final solution is x ⁿ⁺¹ = x ⁿ -[F'(x)] ^-1 ·F(x)(3.14)

之后，再将当前时刻求解出来的电路未知量代入到式(3.7)中去，就可以得到当前时刻的场未知量

其中，

是表示MOSFET求解的过程。

U_j，I_j为上一迭代步的电场，电流值。U^*＝U_j-ΔU，ΔU预先设为0.01V，幷由

求得电流I^*。然后，由(3.14)求得试探性的电压U_j+1和电流I_j+1。如此迭代，直至满足迭代精度停止。更新U_j＝U_j+1,I_j＝I_j+1。之后，再将当前时刻求解出来的电路未知量代入到式(3.7)中去，就可以得到当前时刻的场未知量

The solution vector of the equation can be obtained, that is, the circuit unknowns at each time step

After that, the circuit unknowns solved at the current moment are substituted into equation (3.7), and the field unknowns at the current moment can be obtained.

in,

is the process of representing the MOSFET solution.

U _j , I _j are the electric field and current value of the previous iteration step. U ^* =U _j -ΔU, ΔU is set to 0.01V in advance, and is set by

Find the current I ^* . Then, the tentative voltage U _j+1 and current I _j+1 are obtained from (3.14). Iterate in this way until the iteration accuracy is met. Update U _j =U _j+1 , I _j =I _j+1 . After that, the circuit unknowns solved at the current moment are substituted into equation (3.7), and the field unknowns at the current moment can be obtained.

It can be noticed that the dimension of the above nonlinear equation (3.11) is exactly the number of circuit unknowns N _CKT , so the dimension N _CKT of the above nonlinear equation is much smaller than the dimension N _EM +N of equation (3.4) _CKT 's. Since the above-mentioned improved solution of time-domain nonlinear coupled system equations only solves the nonlinear equation (3.11) by the Newton iteration method, the improved solution saves the solution time and greatly improves the computational efficiency compared with the traditional solution. When the Newton iteration meets the accuracy, the current and voltage distribution of the electromagnetic part of the microwave circuit structure and the voltage and current changes inside the semiconductor circuit can be obtained at each time step.

In the model shown in Figure 4, the length of the dielectric substrate is 17.526mm, the width is 16.256mm, the height of the dielectric substrate is 0.7874mm, the relative permittivity of the dielectric is 2.33, the length of the two metal microstrip lines is 7.763mm, and the width is 2.286 mm, the middle span is 2mm. The metal microstrip line of this model is divided into two sections, and 4 ports are defined, which correspond to 4 RWG basis functions of the metal surface. Among them, the input terminal is loaded with a small sinusoidal signal with a frequency of 1.34GHz, a voltage amplitude of 0.25V, a gate bias voltage of 0.5V, a drain bias voltage of 5V, and the bias and load resistances are both 50ohm. , and the other two ports respectively represent the gate-source port and the drain-source port of the FET. It is through these two ports G and D that the two metal microstrip lines in the figure are effectively connected. Use triangles to divide the metal surface of the model, and use tetrahedron to divide the model's dielectric body. After discretization, 160 triangular patches, 460 tetrahedra, 215 unknown metal inner edges, and 1102 dielectrics are obtained. The triangle is unknown, and the time step size is selected as Δt=0.002286lm. Through the analysis of the field-circuit coupling algorithm based on the time-domain integral equation, the time-domain waveforms of the voltage signals at the input and output ports of the FET are obtained.

Claims (3)

1. a microwave circuit characteristic analysis method containing semiconductor physical model, is characterized in that step is as follows: The first step is to establish the solution model of the microwave circuit structure. The metal part of the electromagnetic structure is divided into triangles, and the medium part is divided into tetrahedrons to obtain the structural information of the model, that is, the size of each triangle and tetrahedron. unit information; The second step is to determine the time-domain electric field integral equation of the electromagnetic target from Maxwell's equations; The third step is to discretize the surface current J ^s (r,t) of the metal and the electric flux density D(r,t) of the dielectric body respectively in the space and time domains; the discretization method of the surface current for the metal surface is: The RWG basis function is used to discretize the space, and the SWG basis function is used to discretize the electric flux density in the space domain; in addition, in the time domain, the time domain volume integral equation uses 4th-order Lagrange interpolation time for both the metal and dielectric parts. The basis function is discretized; The fourth step is to substitute the expanded expressions of the metal surface current and the electric flux density of the dielectric body in the third step into the time-domain electric field integral equation in the second step, and then use the time-domain electric field integral equation in the discrete form respectively. Point test, use Galerkin test in space, get the system impedance matrix equation; The fifth step is to start from the semiconductor drift-diffusion equations, expand the required carrier concentration and potential at each node, use the Galerkin method to test the semiconductor drift-diffusion equations, and use the Newton iteration method to solve them. The carrier and potential distribution of the node, and finally the physical parameters inside the semiconductor; In the sixth step, according to the system impedance matrix equation obtained in the fourth step and the semiconductor physical parameters obtained in the fifth step, a strictly synchronous field-circuit coupling matrix equation is established through the field-circuit coupling, and then the current-carrying current inside the semiconductor is solved through the improved Newton iteration. The distribution of electron and electric potential is finally obtained, and the transient current distribution in the microwave semiconductor circuit structure is finally obtained, and the simulation process is completed. 2. the microwave circuit characteristic analysis method containing semiconductor physical model according to claim 1 is characterized in that: the concrete processing of the 4th step is as follows: Consider the semiconductor-loaded RWG triangle side k ₁ as the equivalent voltage source side, and the voltage at the source V ₁ as the feed voltage of the equivalent voltage source, furthermore, the current flowing through the semiconductor is equal to the vertical flow through The current of the loaded edge _k1 , due to the current coefficient of the _k1th edge

Represents the current density that flows vertically through the edge, and the current flowing through the edge is equal to the current coefficient multiplied by the length of the edge

which is

Get the mixed field-road equation:

in,

represents the derivative with respect to time,

represents the field excitation source value and the circuit excitation source value,

represents the current coefficient vector of the RWG basis function at the jΔt time, and the matrices Z ₀ and Z _ji represent the mutual coupling between the current and past RWG basis functions of the electromagnetic structure, respectively. 3. the microwave circuit characteristic analysis method containing semiconductor physical model according to claim 1 is characterized in that: in the 6th step, establish the field-circuit coupling matrix equation of strict synchronization, then solve by improved Newton iteration, and concrete steps are as follows: 1) Give a coupling equation that can solve the time domain integral equation and the semiconductor nonlinear equation at the same time, the matrix equation form is as follows:

in,

is the vector composed of the node voltage and the unknowns of the current circuit on the voltage source branch,

contains the value representing the voltage source or current source in the circuit and the influence of the unknown quantity in the circuit at the historical moment on the current moment,

is the unknown nonlinear change in the circuit, Z ^CE and Z ^EC are sparse matrices, they are generated by the voltage-current relationship at the interface between the circuit and the electromagnetic structure, the matrix Z ^CE represents the influence of the circuit structure on the electromagnetic structure, including the coupling of the circuit ports voltage information; matrix Z ^EC represents the influence of electromagnetic structure on circuit structure, including the coupling current information of circuit ports; matrix Y represents linear time-invariant circuit elements, which is a sparse admittance matrix of size N _CKT ×N _CKT , Contains only N _CKT non-zero elements; the calculation of the voltage-to-time derivation operation adopts the third-order backward difference formula criterion, namely:

V _j , V _j-1 , V _j-2 , and V _j-3 in the above formula are the voltage values at the time of jΔt, (j-1)Δt, (j-2)Δt, and (j-3)Δt, respectively; The matrix Z ^CE contains the coefficients of the first term in the derivative expansion, and the rest of the terms are used to calculate

2) An improved matrix equation solution scheme is used to solve the above equations, and the specific implementation scheme is introduced as follows The nonlinear field-circuit coupling system equation (2) in the time domain is split into two equations, namely:

Convert the field unknown in Eq. (3) to

Using circuit unknowns

to represent it as:

Next, substitute formula (5) into formula (4) to get:

The nonlinear equation (6) is expressed as:

Then equation (6) is abbreviated as:

The above nonlinear equation (9) is also solved according to the discrete Newton iteration method: make

Finally, solve x ⁿ⁺¹ = x ⁿ -[F'(x)] ^-1 ·F(x), and find the solution vector of the equation, that is, the circuit unknowns at each time step

After that, the circuit unknowns solved at the current moment are substituted into equation (3), that is, the field unknowns at the current moment are obtained.

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