of Analog Integrated Circuits
Ralf SOMMER
1
, Eckhard HENNIG
1
, Manfred THOLE
1
, Thomas HALFMANN
1
, Tim WICHMANN
1
ABSTRACT
In this paper an overview of the application of symbolic
analysis and computer algebra is given. After an intro-
duction requirements to symbolic analysis tools are for-
mulated, and a short abstract of a general symbolic
equation-based approximation algorithm is given. A
generic symbolic analysis flow is introduced and ap-
plied to derive a nonlinear behavioral model of an inte-
grated multiplier circuit using the symbolic analysis
toolbox Analog Insydes [1]. It will be shown how sym-
bolic analysis can assist behavioral model generation
and can help to better understand a circuit in order to
improve the quality of the design.
1. INTRODUCTION
Analog and mixed-signal design is of great importance
in microelectronics applications, like automotive and
telecommunication. The traditional design of analog in-
tegrated circuits relies largely on a mixture of expertise,
some manual calculations, and numerical circuit simu-
lation. Recent research and development in the field of
symbolic circuit analysis has produced results which
may have considerable impact on some parts of the tra-
ditional design flow, but few analog designers have
adopted symbolic analysis techniques as standard tools
in their CAD environments yet. To a large extent this
may be due to the lack of documented methodologies
which show what can be expected from symbolic anal-
ysis and how it can be efficiently employed to solve in-
dustrial circuit design problems. A tool assisting analog
expert designers in circuit sizing, optimization, and
characterization is now urgently needed to enhance de-
sign productivity in order to face shrinking time-to-
market schedules.
The application fields of symbolic analysis tech-
niques (in a close connection with numerical methods)
can be divided into the following four main categories,
which are essential tasks in the industrial design flow of
analog integrated circuits:
Circuit analysis
:
determine the influences of element parameters
on circuit behavior
extraction of dominant circuit behavior in a math-
ematical and interpretable form (also to be used
for circuit sizing)
error and tolerance analysis
Circuit modeling:
support of model generation for analog circuit
blocks (on different hierarchical levels)
allow for overall circuit simulation by use of
behavioral and macro-models
Circuit sizing:
support manual or computer-aided circuit synthe-
sis
derivation of symbolic (generic) sizing formulas
for circuit elements as functions of global circuit
specifications
Circuit optimization:
preprocessing of equations by e.g. elimination of
variables to allow for an efficient optimization
run
allow for application of optimization algorithms
already on system level
2. REQUIREMENTS FOR SYMBOLIC
ANALYSIS TOOLS
As a consequence of the large variety of application
fields summarized in the previous section it becomes
apparent that state-of-the-art symbolic analysis tools
have to be characterized by flexibility in their function-
ality as well as transparency in their data structures and
models. Moreover comfortable interfaces to the user on
the one hand and to numerical simulation environments
on the other hand must be provided because symbolic
analysis is no stand-alone application any more and has
to be embedded into the designers workflow. The fol-
lowing key requirements were identified in many tech-
nical discussions with circuit designers.
Equation formulation: To provide flexibility in
analysis modes as well as to assist a designer in model
development symbolic analysis tools should allow for
setting up circuit equations not only for linear circuits
in the frequency domain but also in the time domain for
both linear and nonlinear circuits and systems. For
modeling purposes and for better interpretability of ex-
pressions equation formulation should not be restricted
to special types of elements (conductances) or circuit
analysis representations (e.g. MNA).
1.
ITWM Institute of Industrial Mathematics,
Kaiserslautern, Germany Hierarchy: Since most analog circuits are designed
following a hierarchical approach a symbolic analysis
tool must allow for hierarchical circuit description in
terms of circuits, subcircuits and device models, and
must provide support for specification mapping and
propagation of parameters between hierarchy levels. In
addition circuit data representation must support the
parallel implementation of different abstraction levels
for a circuit block. Such partial abstractions and com-
putations with mixed hierarchy levels are just as impor-
tant for symbolic circuit analysis as for numerical
simulation. The underlying idea is to replace the sur-
rounding circuitry by a simpler behavioral description
of its input/output characteristics while only the block
under test is simulated at the device level.
Device modeling: Careful modeling of devices is
one of the main prerequisites for successful application
of symbolic circuit analysis. Failure to choose simple
models generally results in extremely large expressions
which cannot be interpreted or even computed at all.
Depending on its individual function, each device in a
circuit should be modeled in the simplest possible way
whose impact on overall simulation accuracy is still tol-
erable. This requires application-specific and even in-
stance-specific device modeling.
Determining the best compromise between model ac-
curacy and expression complexity is often an iterative
process in which various models must be tried for a de-
vice until a satisfactory analysis result is obtained. Se-
lecting and exchanging device models must therefore
be quick and easy, and should not involve tedious
netlist editing operations.
Data integration and interfaces: Circuit representa-
tion must fully integrate all symbolic and numerical
data which is necessary for model definition and expan-
sion, parameter translation and propagation, symbolic
approximation, etc. Since symbolic methods are hardly
ever applied independently of numerical circuit simula-
tion, simulation results, such as operating point data
and small-signal parameters, are always required as in-
put for symbolic approximation routines. Moreover,
symbolic analysis results should always be verified
against numerical simulation so that processing simula-
tor output data is an additional feature to reading in
netlists, model cards and operating-point information.
3. SYMBOLIC APPROXIMATION STRATEGIES
Practical application of symbolic analysis would
have been rather limited without application of symbol-
ic approximation techniques. Indeed these techniques
hold the key in modern symbolic circuit analysis. A lot
of research has been done and reported in this area re-
sulting in three different categories of approximation
strategies: Simplification after generation (SAG), Sim-
plification during generation (SDG), and Simplification
before generation (SBG).
One of the central prerequisites of the symbolic anal-
ysis flow presented in the next section was the develop-
ment and implementation of efficient symbolic
approximation algorithms which impose no restrictions
on the formulation of circuit equations, neither linear
nor nonlinear, or the set of circuit elements that may be
used.
Equation-based approximation procedures own all
these requested properties since they are already ap-
plied on the level of circuit equations before the solu-
tion is determined (SBG). The philosophy behind
equation-based approximation is to follow the method-
ology of a circuit designer who introduces his simplifi-
cations already when formulating equations. Thus the
complexity of the problem and the mathematical effort
to solve or process the system is reduced substantially.
Since this paper intends to give an overview of meth-
odologies and results, only the underlying principle of
equation-based approximation is presented. Figure 1
shows a general flow chart of the algorithm.
Equation-based approximation starts with the system
of symbolic linear or nonlinear equations and a list of
corresponding numerical reference values called design
point.
Based on these numerical reference values the sys-
tem of symbolic equations is evaluated and solved.
This information is subsequently used to generate a
term ranking. The term ranking mechanism plays a key
role in the algorithm. Its task is to compute an order of
all symbolic terms of the underlying equations such
that the terms are sorted with respect to their influence
on the solution. Ranking algorithms are an important
subject of research since a large variety of different cir-
cuit characteristics may be of interest which have to be
taken into account by the algorithm. For example in lin-
ear analysis magnitude, phase as well as pole and zero
locations are of interest while in nonlinear analysis DC
transfer, transient behavior, distortion, etc. are to be
captured by the approximated system.
In the next step the output of the ranking algorithm is
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