Qualitative Modelling of Linear Networks in EGAD Applications

Abstract

There is a real need for qualitative circuit analysistools in the ECAD application area . Recent simu-lators based on finite many valued resistance andcurrent models (switch-level models) have beenshown to have serious limitations. Through inde-pendent parallel research the QR community hasproduced qualitative electrical models and simu-lators that have been successfully used in softwaretools for specific engineering tasks. Many of thesesystems are based on steady-state models usingthree valued resistance networks.This paper examines the relationship betweenqualitative and more conventional electrical cir-cuit modelling and explores the characteristics ofcertain qualitative formulations with a view tosatisfying the needs of future intermediate levelmodels . An existing three valued resistance modelis analysed and then extended to deal with ad-ditional cases. The problems associated withbridge circuits may pose a major barrier to fu-ture progress and this is examined in detail andsome solutions are described.

The Importance of Qualitative Conceptsin ECAD

There has been a long standing interest in the ECADcommunity to find an intermediate modelling level be-tween the gate-level simulators and other analysis toolsused in digital logic and the electrical analogue simula-tors that work at the transistor level. Circuit analysisoften involves predictions of both logical/state informa-tion and details of electrical parameters and ideally thisshould be performed in an integrated and coherent en-vironment. While it is possible to model a complete cir-cuit on an analogue simulator this does not capture thedistinct qualitative states that characterise importantabstractions valued by engineers. Even more importantis the problem of maintaining close affinities betweenthe properties of the model and the preferred notationsand concepts used by human engineers. This is verydifficult to achieve with the voluminous numeric outputgenerated by conventional simulators . This problem isconfirmed by reports from ECAD experts who have ac-cess to considerable data on engineers' experience, e.g .

QR99 Loch Awe, Scotland

Mark H. LeeCentre for Intelligent Systems

Department of Computer ScienceUniversity of Wales, Aberystwyth, UK

mhlUaber.ac.uk

"Graphical interfaces can present simulation datain more convenient forms, but they do not interprettheir meaning nor do they reduce the number ofsimulations required in order to gain a feel aboutsome circuit performance. It has thus become clearthat numerical descriptions of circuit behaviour donot convey understanding about the operation ofa circuit . In fact, the use of such numerical-onlysystems can impose a barrier to the developmentof insight." (Makris and Toumaaou 1992)

An early attack on this problem produced the"Switch-Level Models" developed by Hayes and oth-ers in the last decade . This work aimed to build cir-cuit models intermediate between gate-level simulatorsand analogue electrical models (Hayes 1986, 1987). Thevariables, known as "magnitude classes", had valuepairs consisting of a four-valued voltage level and ann-valued signal strength (or current) value.

After a great deal of development in the 1980s, Hayesfinal paper in 1992 described the severe accuracy limita-tions that prevented any further development of switch-level models (Cerny et al . 1992) . Thorough analysisshowed that the accuracy of the higher order modelswas no better than those with only two or three levels :

"A major conclusion from our work is that switch-level simulators which use many signal strengthclasses may have no better accuracy than simula-tors in which only two or three strengths are usedto distinguish very large differences in resistancevalues." (Cerny et al . 1992)

Hayes has recently stated that there is now a renewedand pressing need to overcome these problems and pro-vide an intermediate-valued model for use at the impor-tant interface between dense numerical detail and thesparseness of extreme qualitative representations .

The Significance of QR Circuit ModelsWithout any apparent knowledge of the switch-levelmodel experiments, the QR community has indepen-dently developed qualitative electrical models and in-vestigated their application to significant problems indesign and analysis . There are various approaches to

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modelling electrical circuits using qualitative abstrac-tions of electrical properties, a classic study being (deKleer 1984), but we are concerned with steady-staterather than dynamic models. Quiescent current resis-tance models have been examined recently by severalauthors. (Struss et al . 1996) used a qualitative resis-tance model in applications in diagnosis and FMEA and(Mauss and Neumann 1996) have investigated the use ofthe series/parallel/star/delta replacement rules to con-vert any resistive mesh into a single equivalent resistor .Flores and Farley have applied similar series/paralleldecompositions to analyse alternating current circuits(Flores and Farley 1996) . Most of these methods havein common the idea of a three valued resistive mesh ab-straction which was first explored by (Lee and Ormsby1992).The motivation for the QR work has been to address

various significant tasks like diagnosis, design and anal-vsis . Software tools are now being developed for suchapplications and embody qualitative models in contrastto the numerical equation solvers used in most electri-cal simulation packages . In this paper we mainly drawon our own work on qualitative circuit theory, known asCIRQ (Lee and Ormsby 1993), which addresses the re-quirements of the analysis task of FMEA (Failure ModeEffects Analysis). This theory underpins one of themost advanced commercial systems in regular use forelectrical FMEA in the automobile industry (Price etal . 1992, 1995) .The switch-level modelling work was built from

many-valued logic and hence had a symbolic ratherthan a numeric grounding. It was different in its as-sumptions, approach and target circuits from QR workbut in many ways had very similar aims . The volt-age variable was 4-valued and therefore was essentiallyqualitative . Voltage values were taken from [+, -, 0,U]which represented the two supply potentials, an inter-mediate potential, and an open circuit, respectively. InCIRQ the symbol set for voltage is [0, +, ^,, 0] where 0and + are the supply terminals, 0 is a special symbolimplying a "floating" or disconnected node and - in-dicates a voltage between the supply potentials' . It isclear that the semantics of these are practically identicaland the equivalence between these independent devel-opments is most striking .In most of the recent steady state resistance models

qualitative resistance values are three valued and of theform [0, +, oe] representing a conductor, an energy ab-sorbing load and an insulator respectively. Switch-levelmodels had a finite set of resistance values which alsoincluded 0 and oo but gave scope for finer resolution byincluding n - 1 intermediate levels, where Ri+i > Ri .With a fixed voltage, Ohm's law gives an equivalentrange of current levels corresponding in number to theresistances. The combination of the voltage and n + 1current values gives 3n + 1 symbols that could be as-

'Previous work has also used

0] as an alternateSYmbol set.

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signed to circuit elements . In the simulators described,current usually had 3 or 4 levels and so the number ofdiscrete labels was 10 or 13 .

The CIRQ AlgorithmWe now briefly describe the main features of the CIRQapproach . Circuits are constructed from library com-ponents which are specified as resistive meshes betweendefined terminals. This allows the whole circuit tobe modelled as a graph E(T, R) containing T nodesand R weighted edges corresponding to resistive ele-ments . The edges are assigned qualitative resistancevalues from the set [0, n, oo] where any positive inte-ger, n E N, signifies a load resistance . Normally werestrict all resistive loads to unity. As described above,qualitative voltage values are [0, +, -, 0] and qualitativecurrent is two-valued [0, +] corresponding to absence orpresence of electrical activity in an edge .The physics of series and parallel circuit reduction

require the numeric summation of their resistances ortheir conductances respectively. The standard qualita-tive version of this uses Max and Min as the series andparallel reduction rules, given the ordinal relationship0 < n < oo . For reasons that will become clear, wemodify this slightly and use Sum, rather than Max forthe series reduction rule. As we are using positive inte-gers this has no effect whatsoever on the results as anypositive integer produced from Sum remains distinctfrom 0 and oo . The Sum function adds the values ofload resistances on serial paths and we define the resultof this as the path resistance, PR(s, t), between anytwo nodes s and t . When parallel branches are com-bined the Min function ensures that shortest paths arereturned and so the single equivalent resistance valuefound by applying series/parallel reduction transformsto a network between terminals s and t is the same asthe value of the shortest path between s and t using theresistive edge weights.The labelling algorithm is a shortest distance algo-

rithm that begins from one power terminal and assignsthe minimum path resistance value to each node's for-ward label, then the reverse labels are assigned by arepeat process starting from the other terminal . Thesevariables are stored in each node as forward and re-verse path resistances, known as f/r . All f/r valuesare initialised to oo/oo because any sub-graphs discon-nected from the supply terminals are not traversed andso return appropriate values . Figure 1 illustrates thisprocess. As a new edge from v to w is explored itsresistance, R, is added to the parent node value usingthe series rule and this result is then combined with anyexisting value representing a parallel path using the par-allel rule . In figure 1(a) all edges have resistance valueunity and the node u has produced labels for its succes-sors w and v. Node v is currently being processed andso 4 + 1 is to be combined with the existing value of 4at node w . Thus a final result of 4 is produced .

Clearly, any quantitative form of this traversal pro-cess would fail because combining the branches of a

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0

1

t3

'l 4

3

Figure 1: The CIRQ labelling process

parallel circuit would affect the previously calculatedvalues on any series sections . It might be expectedthat this problem would also occur in qualitative caseswhere one series branch has a much higher value thananother. However, this does not happen because the la-belling process always explores the lowest valued nodesfirst . Figure 1(b) shows how a long series branch is la-belled . This topology causes the algorithm to enter theright-hand branch from both ends and thus each nodeis assigned its correct minimum path length from thestart node.When all the nodes are labelled their qualitative volt-

ages are assigned according to their f/r values as shownin table 1.

Table 1 : Mapping f/r values into voltage

During the traversal all electrically identical nodes(i .e . connected by apath of zero resistance) are groupedinto supernodes . The supernode concept facilitates ablock detection algorithm that assigns zero current toany dead or shorted branches . All 2-connected blocksare identified then all edges in the block containingthe power terminals are labelled active and all edgesin other blocks are labelled inactive . Figure 2 showsthe supernode structure of an example circuit with theactive parts marked .

Figure 2: Supernodes in a circuit

Why does CIRQ work?Circuits that can be reduced to a single equivalent re-sistance by repeated application of only series/parallelreduction rules are called SP reducible. However, manycircuits have a topology that is not SP reducible, see ex-amples in figure 3.

co/oo

Figure 3: Examples of non SP reducible circuits

For numerical solution, any non SP reducible circuitsrequire the application of additional transform rulessuch as the star/delta conversion equations. Other au-thors (Mauss and Neumann 1996) have stated that al-gorithms like CIRQ can not cope with non SP reduciblenetworks but this is incorrect. It is perhaps surprisingthat using only series/parallel reduction rules CIRQ cananalyse any topology of circuit, including non SP re-

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Node value Interpretationf r Condition Voltage0 0 Short across supply0 + Active path through node +0 00 Dead node ++ 0 Active path through node 0+ + Active path through node+ 00 Dead node +00 0 Dead node 000 + Dead node 0oc o0 11 Dead sub-graph 0

ducible networks . This raises some important issueswhich we now examine .We defined the path resistance PR(s, t) between two

nodes, s and t, as the value of a single equivalent re-sistance found by applying reduction transforms to thesub-network that comprises all paths from s to t . Ourchoice of Sum and Min for the reduction functions com-bines the requirements of3-valued resistance transformswith shortest path criteria . The reduction operationson 3-valued series and parallel circuits are correctly per-formed for qualitative evaluation of SP reducible cir-cuits while also being directly equivalent to the shortestpath in terms of the resistive edge weights . This equiva-lence allows us to employ a traversal algorithm to labelthe nodes . From the node resistance labels the voltagesare determined and then loops and dead branches areremoved by graph-theoretic methods . For SP reduciblenetworks it is clear that CIRQ is complete and in ad-dition is able to produce flow directional labels . Edgecurrent flow direction are related to the node resistancevalues and so flow direction can be determined as fol-lows . Let a simple series path exist between end nodesu and v and define

then

F(u) =

f(u)f (u) -i- r(u)

from u to v

if F(u) < F(v)direction of flow is : _

from v to u

if F(u) > F(v)ambiguous

if F(u) = F(v)This can be used to assign labels for SP reducible net-works, see (Lee and Ormsby 1993) for examples of pre-vious path following algorithms .We now need to consider the non SP reducible case .

The difference between SP reducible and irreduciblecircuits is the existence of bridging edges . Non SPreducible circuits have two types of edges : main flowedges that can be labelled for current magnitude anddirection just as for SP reducible circuits, and "bridges"that connect different flow paths and in which both flowdirection and magnitude are ambiguous . The circuitsin figure 3 are all non SP reducible examples where thecurrent magnitude and direction can be labelled in thenon-bridge (unmarked) edges but are ambiguous in thebridge (marked) edges . If bridges have identical voltagepotential at each end then they have zero current flowand are said to be balanced . To illustrate the subtletiesof bridge configurations, figure 4 shows the three possi-ble assignments for the central bridge in the circuit offigure 3(c) . It is interesting that a configuration existswhere reverse flow occurs in the central bridge .

In all cases except for bridges CIRQ correctly la-bels both current directions and current magnitude (2-valued) . For any bridges CIRQ always labels currentas active and direction as either ambiguous (correct) orin the direction of majority flow paths (strictly incor-rect) . Figure 5 shows a case where the labelling schemehas given plausible directions to the central bridges but

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Figure 4: The 3 flow cases for the central edge in figure3(c)

these are essentially ambiguous as reverse flow is possi-ble in some unusual configurations .

Figure 5 : Bridging edges erroneously labelled for direc-tion

These assignments by CIRQ are acceptable for theFMEA application where bridges are treated separatelyand flow direction is of secondary importance . In gen-eral circuit analysis, however, the bridge topology is arecurring difficulty. This is because the direction of cur-rent flow depends upon the exact quantitative values ofresistance in the associated bridge components . This isa problem for all qualitative resistance representationsas it is impossible to deduce the state of a bridge el-ement without quantitative values . Consequently thebest any quantitative analysis of bridge circuits can re-turn is an "ambiguous" direction label . This is what isdone in the methods that use series/parallel/star reduc-tion rules (Mauss and Neumann 1996) and comparisonswith CIRQ show that this produces results with iden-tical qualitative meaning .

Limitations of Qualitative ResistanceModels

For FMEA of the kind required for automotive circuitanalysis, CIRQ type models are effective and efficient .Flow directions are not essential and balanced bridgesare rare and can be isolated . CIRQ satisfies the spec-ification of the FMEA task as open-circuit and powershort-circuit faults have topological interpretations thatare correctly reflected in the model. See (Lee 1999) forfull details of this application.

However, for general use we must be aware of theeffects of any limitations of qualitative circuit models .There are two questions that arise: how often do bridgecircuits actually occur, and how might they be han-dled . For the first question we notice that in the generalcase of a resistive mesh with a single source and sinkterminal, all edges are candidate bridge edges exceptthose incident on the supply terminals . In the case of aplanar mesh, the extreme edges bounding the networkare also exempt . This means that in large highly con-nected meshes the majority of edges could be bridges .However, in practical applications such circuit topolo-gies rarely occur. This is because most flow paths areconcerned with a particular causal sequence through achain of components running from one supply termi-nal to the other. Hence the topology of many realisticcircuits has a branching structure with diverging andconverging patterns. In addition, any bridging pathsare usually clearly designed to provide flow in one di-rection only . Experience with many automotive circuitsconfirms that ambiguous or balanced bridges rarely oc-cur in practice. In applications of CIRQ in automotiveFMEA, (Snooke and Price 1997) report that 85% of thecircuits encountered could be handled immediately withthe remaining 15% requiring special treatment to dealwith complex behaviour. These were handled either bybuilding complex component models out of the primi-tive components, by constructing dependency links be-tween active states and other components, or by apply-ing approximating simplifications, which either reducedetail or even ignore some behaviour . There were nocases of balanced bridges and no other directional flowproblems that could not be resolved .The second question refers to the treatment of bridges

in qualitative simulators . Given that inference of cur-rent from solely qualitative values is incomplete for anynon SP system then at least we can detect whethera circuit is SP reducible or not . We have examinedseveral methods for preprocessing circuits with bridgedetection algorithms . We first notice that during theabove directional labelling process, if the circuit is SPreducible then the label `ambiguous' can not occur.Conversely, appearance of this label indicates a nonSP reducible circuit . Unfortunately this criteria doesnot provide a complete test as some bridges may erro-neously receive consistent directional assignments, anexample of this was shown in figure 5.

VVe have designed a bridge detector using a graph-

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theoretic approach . An obvious algorithm for bridge de-tection is to enumerate all paths from source to sink andlabel each edge with the direction of traversal . Thenany edges that have both directional labels are can-didate bridges. However, such algorithms are closelyrelated to Hamiltonian paths and have complexities inthe region of O(n!) . A more efficient algorithm is givenby a breadth-first scan of the circuit, starting at thesource terminal, and assigning "dewey decimal" typelabels at each branch point . A label consists of a vec-tor of indices, xl, x2, Xi . . .xm where each index recordsthe successor branch number at level i in the structure .When a new node is encountered the vector increasesto add x�1+i which takes the value 1,2, . ..n for eachsuccessor branch . As the circuit diverges the labels ex-pand, and then on convergence they either recombine(without conflict) or they clash, indicating a bridge .procedure bridge-test (node, node-label, level)put node at start of Queuewhile not(empty(Queue)) do

remove node w from Queueif w = null then return "no bridges"

else for each node v E adjacent(w) doif node-label(v) = null then new-label(v)

else update-label(v), if label- clash(v)return "bridge-found"

add v to end of Queueend-while

end bridge-testRecombination involves the removal of the last vectorelement when all branches have been recovered andclashes occur when level j completes recombination be-fore level i, where j < i. The algorithm is efficient intime but the node vector space demands considerablestorage.

This bridge detector function gives a binary outputfor presence or otherwise of bridges, that is, it stopswhen the first bridge has been detected . Further devel-opment of the algorithm is required if the location ofall bridges are to be found.

Solving circuits with orders-of-magnituderesistanceAnother approach to the problem is to increase thenumber of resistance values to gain finer resolution . Wefound the orders-of-magnitude concept (Raiman 1991)offers a very promising solution . There is some evi-dence that five valued variables are preferred by engi-neers (Brna and Caiger 1992) and we introduce a qual-itative resistance set [0, lo, med, hi, oc] which is orderedby << . The semantics of lo, med and hi can be re-lated to various ranges of numeric resistance values suchas 10,100, lk ohms or 10,1k,100k ohms with orders ofmagnitude separation . Resistance is one of the physi-cal variables with a vast range spanning many ordersof magnitude and a promising possibility for applica-tions is to separate by two orders of magnitude, e.g .

15 0

100, 10k, 100M ohms .

However, for simplicity and il-lustration we will use the scheme [0,1,10,100, oo] inthe CIRQ algorithm. The ordering relation maintainsthe shortest path criterion and so the three-valued se-mantics of [0, n, oo] ensure that the nodes will be la-belled with voltage exactly as before . However the nodepath resistance values now produce much more infor-mation about flow directions . Previously the shortestpath found was the route through the least number ofresistive edges, now the result is the lowest valued re-sistance path . Figure 6 shows the result of CIRQ onthe bridge circuit of figure 3(c) . The high valued edges(marked hi) cause the main flow to follow the low re-sistance branches and the unusual reverse flow in thecentre branch has been detected (q.v . figure 4(c)) .

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Figure 6: O-o-M resistance finds correct flow paths innon SP circuit

This is clearly seen in the node path resistance valueswhich trace the longer main route which now involves9 edges rather than the 5 of the shortest path .Other circuits that can have bridge directions re-

solved by orders-of-magnitude resistance are shown infigure 7. Various constraints act between bridge con-figurations, for example in figure 7(a) the branches aand b can not both be zero and of the 9 possible circuitlabellings (3 for each of a and b) only 5 are physicallyrealisable . For figure 7(b) at least 2 of the 4 bridgesmust be non-zero and in figure 7(c) the central branche has many configurations for downward flow but only3 for zero and 2 for upward flow .

The Contribution from QRGiven that, in general, bridges cannot be resolved nei-ther for current magnitude nor direction without precisequantitative values, we have explored the possibilities

Figure 7: Non SP reducible circuits

of coping with this limitation in possible quantitativemodels for EGAD . There are three levels of difficulty :SP reducible circuits with no bridges, SP irreducible cir-cuits with non-zero bridges, and SP irreducible circuitswith balanced bridges.We have designed methods for detecting bridges so

that any SP reducible circuit can be immediately pro-cessed for complete results . We noticed that detect-ing bridges by ambiguity labels is rather like a semi-decidable problem and not reliable for non SP reduciblecases.

For SP irreducible circuits with non-zero (unbal-anced) bridges the orders-of-magnitude formulation isparticularly promising. By using a larger set of resis-tance values separated by an orders-of-magnitude rela-tion, directional flows can be decided that would oth-erwise remain ambiguous. The only remaining problemis the case of balanced bridges which will of necessityinvolve local resistance values from the same orders-of-magnitude band . As mentioned before, balance, can notbe determined by any qualitative method under suchconditions . However, by increasing the number of re-sistance values we can resolve more and more cases ofambiguity. It will be interesting to consider the limitsand trade-offs that occur from increasingly finer resolu-tion .The CIRQ process with m valued resistance, Rl . .�,

is correct for directional labels provided the number ofresistances in a series path is always less than n whereRi+i > 2nRi . Thus for a separation of 2 orders ofmagnitude we may allow up to 50 resistors in a serieschain. This offers exciting possibilities for new modelsof circuits that give full directional routing of currents .

It is important to realise that we do not encounterthe accuracy problems of the switch-level methods be-cause we do not generate finer current values from theincreased resistance set. Applying our fixed voltage andm resistance values to Ohm's law would produce m val-ued current but this is avoided andcurrent remains two-

valued. Our approach can be seen as an overloading ofthe resistance value set for two different purposes : firstas a 3-valued qualitative analysis to determine voltageand the existence (or not) of edge current, and secondly,as a tool for directional labelling in the active branches .As we do not calculate current magnitudes we do notneed star/delta transform rules - series/parallel reduc-tions in a shortest path traversal process is both suffi-cient and efficient for all topologies .

Qualitative models have strong intuitive appeal withengineers and often have more affinity with the realworld problem than mathematical formulations. Wehope the methods described here are a step towardsan intermediate level that can be used in more conven-tional electrical circuit modelling environments .

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