17-09-2012, 10:43 AM
Topological Analysis and Diagnosis of Analog Circuits
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Introduction
Did you ever wonder what graphs, Kirchhoff laws, the Internet, rough sets, neural
networks, and brain organization have in common? The answer may be very simple - the
system topology. Whether it is a flow -graph that describes the flow of signals between
the nodes of a graph, Kirchhoff laws that describe relation ships between currents or
voltages in an electronic network, or the Internet that uses a web of interconnected
computers to move packets of data between the end users, they all rely on specific
topological information about the system structure. A similar argument can be used for
rough sets that describe features of the information system; neural networks that
implement the connectionist concept of massively parallel interconnec t structures of
processing elements; or the human brain - the most complex, and still only sketchily
described, system of interconnected neurons. In all of these systems topology determines
how the system operates. Topology is a silent system of constrai nts imposed on an
electronic network, governing the signal flow between its components. Thus it is used in
all aspects of system design from system analysis and synthesis through diagnosis.
Over many years computer analysis of large analog circuits was an important
research topic presented in many monographs and research papers [23], [55], [56], [104],
[138], [174], [193], [203], [281]. The main objective of these works was to improve
computational efficiency of the computer analysis methods (like accuracy of the results,
analysis time, memory requirements, numerical stability and convergence, etc.) and to
obtain full, accurate, and illustrative information about the analyzed circuits. These were
also the objectives of the symbolic or semi symbolic network analyses [5], [83], [145],
[205], [206], [215], [216]. Since it was difficult to develop effective programs of
topological analysis for large networks , the development in these years was focused on
numerical methods for sparse matrices [27], [55], [99], [105] or eigenvalues methods
[41], [121], [139], [155], [177], [202], [218]. Since then computer aided analysis and
computer aided design of electronic circuits developed into a leading industry behind the
microelectronic revolution with many professional conferences, design tools, software
vendors, design houses, and fabrication facilities. In this development the symbolic
analysis methods played an important role.
Graphs and Network Topology
Topological methods of circuit analysis and diagnosis relate to the study of
electronic circuits. Study of these methods was initiated by Kirchhoff [123] at the end of
the 19th century and intensified in the sixties and seventies [71], [119], [48], [173], [24],
[25], [63], [254], to a large degree due to the development of computer technology a nd
related devices requiring advanced methods of electronic circuit analysis and design.
Concurrently, algebraic methods that represent network topology and can be used
for its analysis were developed, most notably by Wang [272], [258], [261], [68] and
Bellert [14], [15], [16], [17], [87]. The most attractive feature of topological methods at
this early stage of their development was their ability to obtain transfer functions directly
from the circuit netlist or from its graph description.
Graph based methods used signal flow graphs (Coates [57], [58] and Mason’s
[157], [158] graphs), linear graphs (current -voltage [163], [212] and nullator-norator [62],
[63] graphs), and directed graphs (unistor [159], [45] and distor [47], [45] graphs) to
describe network topology. Specialized analysis methods were developed for each of
these graph representations and there was no unifying method that would handle these
various representations or reuse results from one form of graph representation to another.
The presented work illustrates a unifying approach to circuit analysis using network
topology and its various graph representations. First, basic notations from graph th eory
are presented to provide a tool for network analysis and diagnosis.
Graph Decomposition
Graph decomposition is used in many applications dealing with large systems like
linear programming [59], [112] or the shortest path problem [116], [94], [85], information
encoding [59], synthesis of VLSI circuits [54], partition of sparse matrices [190], job
shop scheduling [282], gene assembly [72], software synthesis [126], etc. Its aim is to
improve the algorithmic performance of problems represented by a system graph.
Network decomposition is used in analysis of computer and communication networks
[74], [81], [122], [151], [172], [180]. Decomposition plays an important role in stability
analysis of large systems [30], [97], [66] or layout compaction in very large scale
integrated (VLSI) circuits [243]. In circuit analysis we distinguish diakoptics [138],
[175], [251], generalized hybrid analysis [55], and topological analysis with nodal
decomposition [134], [135].