Electromagnetic field solver
Encyclopedia
Electromagnetic field solvers (or sometimes just field solvers) are specialized programs that solve (a subset of) Maxwell's equations
directly. They form a part of the field of electronic design automation
, or EDA, and are commonly used in the design of integrated circuit
s and printed circuit board
s. They are used when a solution from first principles is needed, or the highest accuracy is required.
is important for various aspects of physical
verification such as timing
, signal integrity
, substrate coupling
, and power grid analysis. As circuit
speeds and densities have increased, the need has grown to account accurately for
parasitic effects for larger and more complicated interconnect structures. In addition, the
electromagnetic complexity has grown as well, from resistance
and capacitance
, to inductance
,
and now even full electromagnetic wave
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
, and all parasitic extraction
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
complicated 3D geometry
with wave propagation. In layout extraction
,
analytic formulas for simple or simplified geometry can be used where
accuracy is less important than speed, but when the geometric configuration is not simple
and accuracy demands do not allow simplification, numerical solution of the appropriate
form of Maxwell's equations
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
(FD) and
finite element (FEM) method. 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 method
s 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
(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 of only the sources of electromagnetic field. Those sources can be physical
quantities, such as the surface charge density for the capacitance problem, or mathematical
abstractions resulting from the application of Green's theorem. When the sources exist
only on two-dimensional surfaces for three-dimensional problems, the method is often
called a boundary element method
(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, as
illustrated for a small portion of two signal lines in Figure 1.
Integral equation methods, however, generate dense (all entries are nonzero) linear systems
which makes 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 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
.
In the IC industry, sparsified integral equation techniques are typically used 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
(full-wave), both differential and integral equation approaches
are common.
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
directly. They form a part of the field of electronic design automation
Electronic design automation
Electronic design automation is a category of software tools for designing electronic systems such as printed circuit boards and integrated circuits...
, or EDA, and are commonly used in the design of integrated circuit
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...
s and printed circuit board
Printed circuit board
A printed circuit board, or PCB, is used to mechanically support and electrically connect electronic components using conductive pathways, tracks or signal traces etched from copper sheets laminated onto a non-conductive substrate. It is also referred to as printed wiring board or etched wiring...
s. They are used when a solution from first principles is needed, or the highest accuracy is required.
Introduction
The extraction of parasitic circuit modelsParasitic extraction
In electronic design automation, parasitic extraction is calculation of the parasitic effects in both the designed devices and the required wiring interconnects of an electronic circuit: detailed device parameters, parasitic capacitances, parasitic resistances and parasitic inductances, commonly...
is important for various aspects of physical
verification such as timing
Static timing analysis
Static Timing Analysis is a method of computing the expected timing of a digital circuit without requiring simulation.High-performance integrated circuits have traditionally been characterized by the clock frequency at which they operate...
, signal integrity
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 waveform. However, digital signals are fundamentally analog in nature, and all signals are subject to effects such as noise,...
, substrate coupling
Substrate coupling
In an integrated circuit, a signal can couple from one node to another via the substrate. This phenomenon is referred to as substrate coupling or substrate noise coupling....
, and power grid analysis. As circuit
speeds and densities have increased, the need has grown to account accurately for
parasitic effects for larger and more complicated interconnect structures. In addition, the
electromagnetic complexity has grown as well, from resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
and capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
, to inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
,
and now even full electromagnetic wave
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
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
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
, and all parasitic extraction
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 it is intendedto represent...
requires solving some form of Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
. That form may be a simple
analytic parallel plate capacitance equation, or may involve a full numerical solution for a
complicated 3D geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
with wave propagation. In layout extraction
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 it is intendedto represent...
,
analytic formulas for simple or simplified geometry can be used where
accuracy is less important than speed, but when the geometric configuration is not simple
and accuracy demands do not allow simplification, numerical solution of the appropriate
form of Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
must be employed.
The appropriate form of Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.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
Finite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
(FD) and
finite element (FEM) method. The resultant linear algebraic system (matrix) that must
be solved is large but sparse
Sparse matrix
In the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros . The term itself was coined by Harry M. Markowitz....
(contains very few non-zero entries). Sparse linear solution
methods, such as sparse factorization, conjugate-gradient, or multigrid method
Multigrid method
Multigrid methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior...
s 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
Electronic design automation is a category of software tools for designing electronic systems such as printed circuit boards and integrated circuits...
(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 of only the sources of electromagnetic field. Those sources can be physical
quantities, such as the surface charge density for the capacitance problem, or mathematical
abstractions resulting from the application of Green's theorem. When the sources exist
only on two-dimensional surfaces for three-dimensional problems, the method is often
called a boundary element method
Boundary element method
The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations . It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture...
(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, as
illustrated for a small portion of two signal lines in Figure 1.
Integral equation methods, however, generate dense (all entries are nonzero) linear systems
which makes 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 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
.
In the IC industry, sparsified integral equation techniques are typically used 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
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
(full-wave), both differential and integral equation approaches
are common.
See also
- Electronic design automationElectronic design automationElectronic design automation is a category of software tools for designing electronic systems such as printed circuit boards and integrated circuits...
- Integrated circuit designIntegrated circuit designIntegrated circuit design, or IC design, is a subset of electrical engineering and computer engineering, encompassing the particular logic and circuit design techniques required to design integrated circuits, or ICs...
- Standard Parasitic Exchange FormatStandard Parasitic Exchange FormatStandard Parasitic Exchange Format is an IEEE standard for representing parasitic data of wires in a chip in ASCII format. Resistance, capacitance and inductance of wires in a chip are known as parasitic data...