1. Introduction

At present, the commercial PCB simulation software mainly includes: Sigrity of cadence company, siwave / HFSS of ANSYS company, CST of CST company, HyperLynx of mentor company, si9000 of polar company, etc. The electromagnetic field solvers used by different simulation software are different, but they can be roughly divided into several categories

According to the simulation dimension: 2D, 2.5D, 3D

According to the approximation type: static, quasi-static, TEM wave, full wave

The following table lists the characteristics of various electromagnetic field solvers and their applicable structures and occasions.

2. Classify field solvers by dimension

2D solver

2D solver is the simplest and most efficient, which is only suitable for simple applications. For example, 2D static solver can extract capacitance parameters of on-chip interconnect cross section. The 2D quasi-static solver can extract the low frequency RLGC parameters per unit length on the cross section of uniform multiconductor transmission lines. 2D full wave solver can extract full frequency RLGC parameters on the cross section of uniform multiconductor transmission lines. Typical 2D full wave calculation methods include 2D boundary element method, 2D finite difference method and 2D finite element method.

2.5D solver

The concept of 2.5D was put forward by rautio in the 1980s when he was studying for a doctor’s degree in Syracuse University. At that time, he did research on planar mom algorithm with the support of Ge electronics laboratory under Professor Roger. At that time, people only had the concept of 2D current (XY direction) and 3D electromagnetic field. People in Ge Electronics Lab pay more attention to current, which is called 2D, while Professor Roger focuses on electromagnetic field, which is called 3D. Rautio cooperated with the two teams. At that time, he was reading a book on fractal theory, in which the concept of fractal dimension was clearly defined. Therefore, rautio was inspired and proposed the concept of 2.5D, which was the first time that fractal dimension theory was used in electromagnetic field.

“2.5D solver” means that the solver uses the full wave formula, which contains six electromagnetic field components (XYZ electric field E and XYZ magnetic field H) and two conductive current components (such as X and Y directions) in multilayer media. Using the full wave Green’s function of multi-layer media and the method of moments, a 3D problem is reduced to a metal surface problem. In this way, there is no need to mesh the whole three-dimensional space, just mesh the metal surface. In addition, 2.5D means that the metal thickness of the transmission line is ignored, which is a good approximation for planar circuit structures (PCB Applications) where the linewidth is greater than the metal thickness. It can even be said that the accuracy of the half solution Green’s function is higher than that of the general 3D solver in the calculation of multilayer dielectric structures.

The 2.5D solver with metal thickness and Z-direction conduction current is called 3D plane algorithm. 3D means that this solver can be used as a company of multilayer media to solve some 3D structures, such as transmission lines or vias. However, bondwire can not be done in this way. Full wave means that radiation is considered in the formula, or the displacement current component is considered in Maxwell equations.

The 2.5D TEM solver is suitable for the case that the structure is dominated by TEM mode, that is, there is no electric field and magnetic field component in the direction of electromagnetic field propagation, and the power plane with low working frequency is suitable for the structure. However, 3D effect, coplanar setting or lack of reference plane design will reduce the accuracy of this method.

The 2.5D BEM / mom solver is a full wave solver, which is based on the boundary element method (BEM) or the method of moment (MOM) formula and uses the Green’s function of layered medium to solve the problem. It is usually assumed that the plane with infinite layers of medium is the plane. However, for the 3D edge effect in package and package PCB junction, the accuracy of 3D geometry and finite dielectric layer is not high. Representative software: ANSYS designer, microwave office, IE3D, FEKO, sonnet.

3D solver

3D quasi-static solver is suitable for most 3D structures in chip package circuit board system, but it is effective for low frequency, and the error of high frequency results is large. If the structure is large, the calculation time will be long, and the memory consumption will be large.

3D full wave solver is the most accurate solver to model the actual situation. It can simulate all the effects of RF, Si, PI, EMI and so on. Typical 3D full wave solvers include: boundary element method (si9000), finite difference method (CST, keysight empro / FDTD) and finite element method (ANSYS HFSS, keysight empro / FEM).

3. Field solvers are classified by approximation type

Quasi static electromagnetic algorithm

It needs a three-dimensional structural model. The so-called “quasi-static” means that the system must support the existence of electrostatic field and steady current, which is manifested as the field type of electrostatic field and static magnetic field. More precisely, the rate of change of magnetic flux or displacement current is very small, so the partial conductance terms of B and D to time can be ignored in Maxwell equations, and the corresponding Maxwell equations are called quasi-static and quasi-static respectively. The algorithm derived from this is called quasi electrostatic algorithm and quasi magnetostatic algorithm. This kind of algorithm is mainly used in EMC Simulation of power frequency or low frequency power system or motor equipment. For example, the quasi-static electromagnetic algorithm can be used to extract the distribution parameters between the converter bus and the cabinet. It is obvious that quasi-static approximation can be used for high voltage insulation devices, while quasi-static algorithm is preferable for high current devices, such as converters, motors, transformers, etc.

Full wave electromagnetic algorithm

In short, it is an algorithm to solve the complete form of Maxwell’s equation. The full wave algorithm is divided into time domain algorithm and frequency domain algorithm.

Finite difference method (FD), finite integral method (FI), transmission line matrix method (TLM), finite element method (FEM), boundary element method (BEM), method of moment (MOM) and multi-layer fast multipole method (mlfmm) are all full wave algorithms. All full wave algorithms need volume mesh or surface mesh segmentation in the simulation area. The first three methods (FD, FI and TLM) are mainly explicit in time domain with sparse matrix, and the simulation time and memory are proportional to the first power of the number of grids; the last four methods (FEM, BEM, mom and mlfmm) are all implicit in frequency domain. FEM is also a sparse matrix, and the simulation time and memory are proportional to the square of the number of grids; while BEM and mom are dense matrices, so the time and memory are proportional to the third power of the number of grids. FD, fi, TLM and FEM are suitable for arbitrary structure and arbitrary medium, BEM and mom are suitable for arbitrary structure but uniform non rotating medium distribution, while mlfmm is mainly suitable for metal convex structure, although mlfmm has superlinear grid convergence, which is well known as nlogn computation.

Full wave algorithm, also known as low frequency or accurate algorithm, is an accurate method to solve electromagnetic compatibility problems. For a given computer hardware resource, there is an upper limit of the electrical size that can be simulated by this method. Generally speaking, without any restriction, that is, any structure and any material, TLM and FI can simulate the largest electrical size, followed by FD, FEM, mom and BEM. For metal convex structures, mlfmm is a full wave algorithm that can simulate the largest electrical size.

The inherent advantage of time domain algorithm is that it is very suitable for UWB simulation. EMC itself is an ultra wideband problem. For example, the national military standard GJB151A re102 involves extremely wide frequency bands ranging from 10kHz to 40GHz. In addition, the time-domain algorithm is natural, efficient and accurate for the simulation of transient electromagnetic effects, such as the simulation of transient impulse voltage induced by intense electromagnetic pulse irradiating wire bundle.

4. Electromagnetic field solver algorithm

There are many theories about electromagnetic model extraction, but no one has absolute advantage in accuracy and efficiency. Different algorithms have different advantages and are suitable for different applications.

Method of moment (MOM)

Mom is an algorithm in frequency domain. The characteristics of the algorithm make it suitable for the analysis of multilayer planar structures, such as PCB routing analysis, system level package (SIP) and integrated circuit package analysis.

In many electromagnetic simulation theories, mom is one of the algorithms which is not easy to implement by program. Because this algorithm must solve the integral equation of Green’s functions and electromagnetic coupling skillfully. Maxwell’s equation can be transformed into integral equation. The characteristic of this transformation is that the main unknown term of mom is the current distribution on the metal surface, while the main unknown terms of other electromagnetic simulation algorithms are the electric field and magnetic field in the structure.

Because only the electric six parts on the metal surface must be considered in the grid, the number of grids can be greatly reduced. This technique allows mom to calculate complex structures more efficiently, but it is also limited to the problem of only analyzing the multi-layer plane (3D planer), which is not suitable for 3D solid structures.

With the increase of the complexity of electronic products, the computation time of electromagnetic simulation is too long to solve the problem of high complexity. A lot of matrix operations are needed to complete electromagnetic simulation. For mom, the main bottleneck is how to calculate and store a large number of coupling matrices. A network with n Unknown items needs to spend N2 proportion of space in memory, and the computing time will be increased by N3 (if direct solver is used) or N2 (if iterative solver is used).

The figure below shows the reference value of ADS software to improve the software algorithm to optimize the simulation speed and memory utilization with the version update.

Finite element method (FEM)

Compared with mom, the application scope of FEM algorithm is much wider, because FEM is a full 3D algorithm, which can be used to analyze the structure of any shape, such as bond wire, solver balls or other structures with any shape in z-axis direction. The FEM simulator can also simulate media blocks or limited size substrates. Many applications, such as resonator design, require this feature. FEM is also a frequency domain technology. But the simulation time of FEM is usually longer than that of mom, especially in the part of multilayer planar structure.

The package structure with bond wire shown in the figure below is suitable for FEM analysis, but not for mom analysis.

FEM algorithm will divide a large structure into many small areas, and use three-dimensional network to calculate the field strength of each small area. The geometric model can be automatically divided into a large number of tetrahedrons, each tetrahedron is composed of four triangles. These tetrahedrons are called finite element meshes. The field values of the top tangent of the triangle cone, the three sides and the center point of each side are stored. The field pattern inside each triangle cone can be calculated by interpolation. In this way, the large structure can be transformed into a small structure, Maxwell’s equation can be transformed into a matrix problem, and the S parameters of any shape can be extracted by mathematical calculation. The following figure shows the grid diagram of the three-dimensional structure.

The convergence of FEM is usually judged by comparing the results of two previous operations. If the error range is less than a certain specification, it can be judged that it is close to convergence. If the error is still too large, the mesh will be redefined and the mesh density will be increased to enhance the convergence. However, in three-dimensional structure, some areas such as surface, corner and material interface have poor convergence, which leads to a large amount of memory and computing time consumption. Therefore, in recent years, with the help of multi-core operation, it is also very important to improve the convergence of the structure and the efficiency of matrix solving.

Finite difference time domain method (FDTD)

FDTD is also a full wave electromagnetic simulation algorithm, which can be used to analyze any 3D structure and directly solve the Maxwell’s program in time domain. The unknown terms in the matrix, like FEM, are the electric and magnetic fields in the three-dimensional structure space. However, the network of FEM is triangular cone shape, and the mesh of FDTD is usually expressed as a positive cube (Yee). The FDTD method can update the electric field and magnetic field in three-dimensional space in real time when the electromagnetic wave passes through the three-dimensional structure by using the program of real-time operation in time domain. Therefore, unlike FEM, it is necessary to complete all the convergence and post-processing operations to get the S parameter. FDTD can update the current S-parameter value at any time. FDTD simulation can provide data in a very wide frequency range.

Due to its simple and reliable characteristics and the ability to handle linear and nonlinear materials and devices, FDTD can be used in many applications: antenna design, microwave circuits, biological / electromagnetic effects, EMC / EMI problems, optoelectronics and so on. FDTD is an inherent parallel algorithm, which can make full use of the latest CPU (general purpose processor) and GPU (graphics processor) hardware resources. Now the speed of complex engineering simulation problems can be 20 ~ 40 times faster than the traditional CPU.

Boundary element method (BEM)

The boundary element method is a more accurate and effective method developed after the finite element method. It is also called boundary integral equation boundary element method. It takes the boundary integral equation defined on the boundary as the governing equation, and discretizes it into algebraic equations by interpolation. Compared with the domain method based on partial differential equation, it reduces the dimension of the problem and the number of degrees of freedom significantly. The discretization of the boundary is much more convenient than that of the domain. It can accurately simulate the shape of the boundary with simpler elements, and finally get the lower order linear algebraic equations. Because it uses the analytic fundamental solution of differential operator as the kernel function of boundary integral equation, it has the characteristics of combining analytical and numerical, and usually has high accuracy. Especially for the problems with large gradient of boundary variables, such as stress concentration problems or crack problems with singularity of boundary variables, BEM is considered to be more accurate and efficient than FEM. Because the basic solution of differential operator used by BEM can automatically satisfy the condition of infinite distance, BEM is especially convenient to deal with infinite and semi infinite problems. The main disadvantage of BEM is that its application scope is based on the existence of the basic solution of the corresponding differential operator, and it is difficult to apply to the problems of non-uniform medium, so its application scope is far less than that of finite element method. Moreover, the coefficient matrix established by BEM for solving algebraic equations is asymmetric full matrix, which limits the scale of solving problems. For the general nonlinear problem, the integral term in the domain will appear in the equation, which partly counteracts the advantage of BEM as long as the boundary is discrete.