Electromagnetic field solvers (or sometimes just field solvers) are specialized programs that solve (a subset of)
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
directly. They form a part of the field of
electronic design automation
Electronic design automation (EDA), also referred to as electronic computer-aided design (ECAD), is a category of software tools for designing Electronics, electronic systems such as integrated circuits and printed circuit boards. The tools wo ...
, or EDA, and are commonly used in the design of
integrated circuit
An integrated circuit (IC), also known as a microchip or simply chip, is a set of electronic circuits, consisting of various electronic components (such as transistors, resistors, and capacitors) and their interconnections. These components a ...
s and
printed circuit board
A printed circuit board (PCB), also called printed wiring board (PWB), is a Lamination, laminated sandwich structure of electrical conduction, conductive and Insulator (electricity), insulating layers, each with a pattern of traces, planes ...
s. They are used when a solution from first principles or the highest accuracy is required.
Introduction
The
extraction of parasitic circuit models is essential for various aspects of physical verification such as
timing
Timing is the tracking or planning of the spacing of events in time. It may refer to:
* Timekeeping, the process of measuring the passage of time
* Synchronization, controlling the timing of a process relative to another process
* Time metrolo ...
,
signal integrity
Signal integrity or SI is a set of measures of the quality of an electrical signal. In digital electronics, a stream of binary values is represented by a voltage (or current) waveform. However, digital signals are fundamentally analog signal, anal ...
,
substrate coupling, and power grid analysis. As circuit speeds and densities have increased, the need has grown to account accurately for
parasitic
Parasitism is a close relationship between species, where one organism, the parasite, lives (at least some of the time) on or inside another organism, the host, causing it some harm, and is adapted structurally to this way of life. The ent ...
effects for more extensive and more complicated interconnect structures. In addition, the electromagnetic complexity has grown as well, from
resistance and
capacitance
Capacitance is the ability of an object to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are two closely related ...
to
inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The electric current produces a magnetic field around the conductor. The magnetic field strength depends on the magnitude of the ...
, and now even full
electromagnetic wave
In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength, ...
propagation. This increase in complexity has also grown for the analysis of passive devices such as integrated inductors. Electromagnetic behavior is governed by
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
, and all
parasitic extraction In electronic design automation, parasitic extraction is the calculation of the parasitic effects in both the designed devices and the required wiring interconnects of an electronic circuit: parasitic capacitances, parasitic resistances and parasit ...
requires solving some form of Maxwell's equations. That form may be a simple analytic parallel plate capacitance equation or may involve a full numerical solution for a complex 3D
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
with wave propagation. In
layout extraction
The electric circuit extraction or simply circuit extraction, also netlist extraction, is the translation of an integrated circuit layout back into the electrical circuit (netlist) it is intended to represent. This extracted circuit is needed for ...
, analytic formulas for simple or simplified geometry can be used where accuracy is less important than speed. Still, when the geometric configuration is not simple, and accuracy demands do not allow simplification, a numerical solution of the appropriate form of
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
must be employed.
The appropriate form of Maxwell's equations is typically solved by one of two classes of methods. The first uses a differential form of the governing equations and requires the discretization (meshing) of the entire domain in which the electromagnetic fields reside. Two of the most common approaches in this first class are the
finite difference
A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.
The difference operator, commonly d ...
(FD) and
finite element
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat tran ...
(FEM) methods. The resultant linear algebraic system (matrix) that must be solved is large but
sparse (contains very few non-zero entries). Sparse linear solution methods, such as sparse factorization, conjugate-gradient, or
multigrid methods can be used to solve these systems, the best of which require CPU time and memory of O(N) time, where N is the number of elements in the discretization. However, most problems in
electronic design automation
Electronic design automation (EDA), also referred to as electronic computer-aided design (ECAD), is a category of software tools for designing Electronics, electronic systems such as integrated circuits and printed circuit boards. The tools wo ...
(EDA) are open problems, also called exterior problems, and since the fields decrease slowly towards infinity, these methods can require extremely large N.
The second class of methods are integral equation methods which instead require a
discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...
of only electromagnetic field sources. Those sources can be physical quantities, such as the surface charge density for the capacitance problem, or mathematical abstractions resulting from applying Green's theorem. When the sources exist only on two-dimensional surfaces for three-dimensional problems, the method is often called
method of moments (MoM) or
boundary element method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, ele ...
(BEM). For open problems, the sources of the field exist in a much smaller domain than the fields themselves, and thus the size of linear systems generated by integral equations methods are much smaller than FD or FEM. Integral equation methods, however, generate dense (all entries are nonzero) linear systems, making such methods preferable to FD or FEM only for small problems. Such systems require ''O(n
2)'' memory to store and ''O(n
3)'' to solve via direct Gaussian elimination or, at best, ''O(n
2)'' if solved iteratively. Increasing circuit speeds and densities require the solution of increasingly complicated interconnect, making dense integral equation approaches unsuitable due to these high growth rates of computational cost with increasing problem size.
In the past two decades, much work has gone into improving both the differential and integral equation approaches, as well as new approaches based on
random walk
In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space.
An elementary example of a rand ...
methods. Methods of truncating the discretization required by the FD and FEM approaches has greatly reduced the number of elements required. Integral equation approaches have become particularly popular for interconnect extraction due to sparsification techniques, also sometimes called matrix compression, acceleration, or matrix-free techniques, which have brought nearly ''O(n)'' growth in storage and solution time to integral equation methods.
Sparsified integral equation techniques are typically used in the IC industry to solve capacitance and inductance extraction problems. The random-walk methods have become quite mature for capacitance extraction. For problems requiring the solution of the full
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
(full-wave), both differential and integral equation approaches are common.
See also
*
Computational electromagnetics
Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment using computers.
It typically involve ...
*
Electronic design automation
Electronic design automation (EDA), also referred to as electronic computer-aided design (ECAD), is a category of software tools for designing Electronics, electronic systems such as integrated circuits and printed circuit boards. The tools wo ...
*
Integrated circuit design
Integrated circuit design, semiconductor design, chip design or IC design, is a sub-field of electronics engineering, encompassing the particular Boolean logic, logic and circuit design techniques required to design integrated circuits (ICs). A ...
*
Standard Parasitic Exchange Format
Standard Parasitic Exchange Format (SPEF) is an IEEE standard for representing parasitic data of wires in a chip in ASCII format. Non-ideal wires have parasitic resistance and capacitance that are captured by SPEF. These wires also have inductan ...
*
Teledeltos
Teledeltos paper is an electrically conductive paper. It is formed by a coating of carbon on one side of a sheet of paper, giving one black and one white side. Western Union developed Teledeltos paper in the late 1940s (several decades after it was ...
References
* ''Electronic Design Automation For Integrated Circuits Handbook'', by Lavagno, Martin, and Scheffer, {{ISBN, 0-8493-3096-3 A survey of the field of
electronic design automation
Electronic design automation (EDA), also referred to as electronic computer-aided design (ECAD), is a category of software tools for designing Electronics, electronic systems such as integrated circuits and printed circuit boards. The tools wo ...
. This summary was derived (with permission) from Vol II, Chapter 26, ''High Accuracy Parasitic Extraction'', by Mattan Kamon and Ralph Iverson.
Electronic design
Electronic design automation
Electronic engineering
Integrated circuits
Computational electromagnetics