A New Symbolic Method For Analog Circuit Testability ... - IEEE Xplore
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A New Symbolic Method for Analog Circuit Testability Evaluation Giulio Fedi, Member, IEEE, Antonio Luchetta, Member, IEEE, Stefano Manetti, Member, IEEE, and Maria Christina Piccirilli
Abstract— Testability is a very useful concept in the field of circuit testing and fault diagnosis and can be defined as a measure of the effectiveness of a selected test point set. A very efficient approach for automated testability evaluation of analog circuits is based on the use of symbolic techniques. Different algorithms relying on the symbolic approach have been presented in the past by the authors and in this work noteworthy improvements on these algorithms are proposed. The new theoretical approach and the description of the subsequent algorithm that optimizes the testability evaluation from a computational point of view are presented. As a result, in the computer implementation the roundoff errors are completely eliminated and the computing speed is increased. The program which implements this new algorithm is also presented. Index Terms— Analog circuits, analog system fault diagnosis, analog system testing, circuit testing, fault diagnosis, fault location.
I. INTRODUCTION
I
N EVERY field of engineering, testing is a fundamental step for the validation of design, being the most direct way to verify that a product meets its specifications. If the desired performance is not achieved, testing should identify all the causes of malfunctioning and indicate suitable corrective actions. While testing methodologies are fully automated for digital devices, there is a lack of efficient and systematic methods of testing for analog and mixed signal devices. Furthermore this problem is even going to be more critical with technology improvements and increasing number of mixed signal applications, especially in the communication field. In the case of analog electrical circuits, deviations from the desired circuit behavior are generally caused by variations of the components from their nominal value (parametric fault). For this reason it is very important that an analog circuit testing procedure allows us to determine an estimate of component values simply and quickly. Moreover, an essential task in analog circuit testing is the selection of an optimum set of measurements, that is an optimum set of test points. To do this, it is necessary to have a quantitative index to compare different solutions. For analog circuits, the testability concept provides this index. Manuscript received April 11, 1997; revised December 4, 1998. G. Fedi, S. Manetti, and M. C. Piccirilli are with the Department of Electronic Engineering, University of Florence, 3-50139 Firenze, Italy. A. Luchetta is with the Department of Engineering and Environmental Physics, University of Basilicata, 85100 Potenza, Italy. Publisher Item Identifier S 0018-9456(98)09856-8.
The most used definition of testability evaluation was given by Saeks [1]–[4], who defined it as a measure of solvability of the nonlinear fault diagnosis equations, indicating the ambiguity which results from an attempt to solve such equations in a neighborhood of almost any failure. Then testability is strictly tied to the concept of network-elementvalue-solvability [5]–[8], that is, it allows us to establish a priori if it is possible to uniquely determine component values, given a circuit structure and some input–output (I/O) relations to be measured. If a unique solution is not achievable, other I/O measurements, that is other test points, must be added or a reduced number of unknown parameters must be used. So testability information is essential, either to designers who must know which test points to make accessible for testing or to test engineers who must plan tests and know how many and what parameters can be uniquely isolated by these tests. Algorithms for evaluating the testability as defined by Saeks have been developed by the authors in the past years, at first using a numerical approach [9], [10]. Subsequently a symbolic approach has been used in order to eliminate the unavoidable roundoff errors introduced by numerical algorithms, which render the obtained testability only an estimate. The symbolic analysis is a technique that permits one to obtain, as result of a computer program, circuit network functions in closed form, where some or all the circuit elements and the complex frequency are represented by symbolic parameters. Hence symbolic analysis is particularly suitable for the testability evaluation problem, in which the component values are the unknown quantities. It allows realization of very simple testability measurement algorithms which are based on the determination of the nonlinear fault diagnosis equations in symbolic form with respect to both complex frequency s and component values [11], [12] and on the subsequent sensitivity analysis of these equations [13]. Also in the symbolic approach, as in the numerical one, testability evaluation is made by selecting integer numerical values for the entries of the matrix whose rank determines the testability [14], because the circuit testability is independent with respect to component values [1]–[4]. Unfortunately the testability evaluation algorithms based on symbolic approach presented in the past reduce but do not eliminate roundoff errors. Nevertheless, by exploiting the previously obtained results, it has been possible to develop a new symbolic procedure that completely eliminates the roundoff error problem.
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In this paper, a new symbolic procedure is presented. Starting from the theoretical background developed in previous works [15]–[17] and summarized in Section II, new theorems and the subsequent new algorithm that optimizes the testability evaluation from a computational point of view have been developed. They are presented in Section III, while in Section IV explanatory examples are proposed. Finally, a program which implements this new algorithm has been realized and its description is presented in Section V. II. TESTABILITY EVALUATION An analog, linear, time-invariant circuit can be described in the following way, by using the modified nodal analysis (MNA): (1) is the vector of the parameters where which are potentially faulty, assuming that all the faults can be expressed as parameter variations, without influencing the circuit topology (i.e., faults as short and open are not is considered); the vector of the output test points (voltages and/or currents); is the vector of the inaccessible node voltages and/or currents of all the elements that have not an admittance representation; is the characteristic matrix, conformable to the vectors; is the input vector. It is worth pointing out that the parameters assumed not faulty have their own nominal value. The fault diagnosis equations of the circuit under test are constituted by the network functions relevant to each test point output and to each input. They can be obtained from (1) by applying the superposition principle and have the following form:
(2)
minor of the matrix and the with th output due to the contribution of input only. As it can be easily noted the total number of the fault diagnosis equations is equal to the product of the number of outputs and inputs. The testability measure , evaluated in a suitable neighborhood of the nominal value of the potentially faulty parameter vector , is given by the maximum number of linearly associated independent columns of the Jacobean matrix with the fault diagnosis equations in (2) [1]–[4] rankcol
(3)
where the entries of the Jacobean matrix are polynomial of the functions in , evaluated in the nominal value can be written parameter vector [8]. In fact the matrix
in the following way:
(4) polynomial matrix. So the testability value coincides with . The with the number of linearly independent columns of evaluation, starting from (4), is very difficult to handle and subject to roundoff errors, especially for large circuits, if a numerical approach is used [10]. It has been proven [15] that the number of linearly indepenis equal to the rank of a matrix dent columns of the matrix constituted by the coefficients of the polynomial functions . Then the entries of the matrix are independent of with respect to the complex frequency . In other words, by in the following way: expressing (5)
and deg deg , we have rank , where is a matrix of order rankcol ( number of potentially faulty parameters, ) of the following form: with
deg
(6)
. whose entries are the coefficients of the polynomials are in comThis result is very useful if the coefficients pletely symbolic form. In fact, in this case, they are functions of circuit parameters to which we can assign arbitrary values because the testability is independent of component values. is essentially a sensitivity Furthermore, since the matrix matrix of the circuit under test, starting from a fully symbolic generation of the network functions corresponding to the selected fault diagnosis equations, it is very easy to obtain symbolic sensitivity functions [11]–[13]. As a consequence, the use of a symbolic approach simplifies the testability measure procedure and reduces roundoff errors, because the components of are not affected by any computational error. An important simplification of this procedure, reported in [16], results from the fact that the testability measure can be constituted by the derivatives evaluated as rank of a matrix of the coefficients of the fault diagnosis equations with respect to the potentially faulty circuit parameters. For the sake of simplicity, let us consider a circuit with only one test point and
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only one input. In this case there is only one fault diagnosis equation that must be expressed in the following form:
(7)
vector of potentially faulty pawith and degrees of numerator and denominator, rameters, of order , constituted respectively. The matrix in (7) with by the derivatives of the coefficients of respect to the unknown parameters, is the following:
A. Theoretical Basis: Case of Single Fault Diagnosis Equation For the sake of simplicity, let us consider a circuit with only one test point and only one input. Then there is only one fault diagnosis equation to consider. The theoretical basis of the new approach regards essentially the following theorem. Hypothesis: Let be the fault diagnosis equation, the vector of the potentially faulty with and coefficients formed by sum of parameters and products of at least one element of the vector (this means that the derivatives of and with respect to the potentially faulty parameters are not equal to zero). Hence, if we define a , with rows and columns, as follows: matrix
(8) (9)
has the same rank of the As shown in [16] the matrix previously defined matrix , because the rows of are linear . Then the testability value combination of the rows of by assigning can be computed as the rank of the matrix arbitrary values to parameters . If the circuit under test is a multiple-input–multiple-output system, that is, if there is more than one fault diagnosis equation, the same result can be easily obtained. This is a noteworthy simplification from a computational point of view, because derivatives of coefficients of fault diagnosis equations are simpler to compute with respect to derivatives of fault diagnosis equations. A limitation of this method is the need to get rational functions whose denominator has a unitary coefficient in the highest order term, while, in most cases, this coefficient is different from one. This limitation has a not negligible impact on the computation task, because, in order to obtain such a kind of normalization, it is necessary to divide all the coefficients of the rational functions by the coefficient of the highest term of the denominator, with a consequent complication in the evaluation of the derivatives (derivative of a rational function instead of a polynomial function) and the introduction of roundoff errors (in any case lower than those relevant to completely numerical approaches). The symbolic approach for testability evaluation presented in this paper removes this limitation, providing a complete elimination of roundoff errors and an increase of computing speed, as will be shown in the following section. III. NEW APPROACH FOR TESTABILITY EVALUATION The symbolic approach for testability evaluation proposed in this paper is based on a new theorem and subsequent new corollaries discussed in the following.
is different from the null one. each row of Let us suppose also that the vector is . linearly independent with respect to the rows of Let be the function obtainable from by dividing numerator and denominator for (maximum degree coefficient of denominator) and the rows and columns whose following matrix of rank is equal to the circuit testability value
(10)
FEDI et al.: NEW SYMBOLIC METHOD FOR ANALOG CIRCUIT
Fig. 1. General survey in the case of each scalar term h independent with respect to the vector of potentially faulty parameters
Thesis: The rank of is equal to the rank of and indicates the testability value of the circuit (case B of Fig. 1). and Proof: The generic elements of are derivatives of a ratio, hence they have the following form:
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p.
(11.a)
(11.b)
(12)
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So can be written as shown in (12) at the bottom of the can be split in the following previous page. Furthermore, way:
where the right side of the identity is linearly independent by hypothesis, hence rank is at least equal to . , for obtaining rank On the other hand, if rank at least equal to , it must be (16) , then . In the right term of , that are supposed linearly the identity there are rows of is linearly independent with independent. Also the vector . In fact, if it were not so, we could respect to the rows of affirm, without loss of generality, that
But
(17) (13)
with have:
and
scalar terms different from zero. So we should (18)
is obtained by adding to the Then the first row of , multiplied for , the vector first row of multiplied for . The second row of is obtained by adding to , multiplied for ( , the vector the second row of multiplied for and so on for every row of . , for obtaining in at least Hence, if rank linearly independent rows, it must be (14)
and are but that is an impossible relation, because different from the null row and, with respect to the vector , is they are linearly independent by hypothesis. Hence rank at least and so the thesis is proved. without null rows The request to consider the matrix is justified by the following corollary. ) in the matrix Corollary 1: If there are null rows ( and the vector is linearly independent with respect to rank (case A of Fig. 1). the rest of them, rank involves Proof: The hypothesis of null rows in and/or in which there are that there are coefficients not potentially faulty parameters. The presence of null rows involves also that in the matrix there are rows in of the following kind: (19)
But row of for
, with and
generic row of scalar value equal to and equal to . Then
generic for
with as previously defined. These rows belong to a vectorial hyperspace of dimension equal to one ( is the generating rank . In fact, if vector), hence, if rank , we have there are linearly independent rows in
(20)
(15)
where the right term of the identity generates a vectorial , because the rows of hyperspace of dimension and the vector are linearly independent by hypothesis.
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More in general the rank of cannot be greater or . In fact the lower than one with respect to the rank of and the row two matrices obtained by adding to have the same rank always, because they are obtainable each by the other with elementary operations on the rows. Then, if is linearly , rank dependent with respect to the rows of rank rank . Now, if is linearly dependent rank rank , also with respect to the rows of otherwise, if is linearly independent with respect to the rows rank rank rank , that is of rank . rank From the previous observation the following corollaries are developed. Corollary 2: Let be linearly dependent with respect to the rows, which are all different from the null row. Then: rank if is a linear combination 1) rank and of the fault diagnosis equaof coefficients plus a constant term without elements tion belonging to the vector . rank if is a linear combination 2) rank and of the fault diagnosis equation of coefficients (cases D and E of Fig. 1). be a linear combination of the rows of Proof: Let . Then can be expressed in the following way:
and being by hypothesis, there are linearly dependent at least one and, in particular, is independent with respect rows in . Hence rank rank (case to the rows of D of Fig. 1). is linearly dependent with respect to the Corollary 3: If and has null rows, rank rank rows of and is a linear combination of coefficients and of the with or without the adding fault diagnosis equation of a constant term different from zero and with no elements belonging to (case C of Fig. 1). has null rows, is linearly dependent Proof: If (see the proof of the Corollary with respect to the rows of is linearly dependent with respect to the 1). Then, since by hypothesis, rank rank and rows of ( can be also equal to zero). Now we know all the possible relationships which can exist and , if is a scalar independent with between respect to the vector of potentially faulty parameters . Being the testability measure equal to rank , we can summarize all the obtained results as shown in Fig. 1, in which the permits us to generalize the argumentation constant term to the case that not all the circuit parameters can be faulty. However, all the possible kinds of fault diagnosis equations are not yet been considered, as it will be shown in the following. B. Use of an Auxiliary Matrix
(21) are the linearly independent rows of , where indicates in a general way the coefficients and of the are scalar terms indefault diagnosis equation pendent with respect to the vector . This equation involves is a linear combination of coefficients and that then it can be expressed in the following way: (22) with generic constant term without elements belonging to . can be As previously shown, a generic row of , then, we have expressed as (23) that
is .
If
,
and is a linear combination of rows of . So, by , we have indicating in a general way can be that the condition . This means that, written as , with constant term if is a linear combination without elements of and then rank rank of rows of (case E of Fig. 1). If , this , that is, being means that
In the previous paragraph it has been shown that the linear with respect to the rows of the independence of the vector is given by the following relation among the matrix : coefficients of the fault diagnosis equation (24) and without elements belongwith the scalar quantities ing to . In fact, by the derivatives of (24) (considered with the sign “ ”) with respect to the circuit parameters, we obtain . In order to verify a linear combination of the rows of of order if expression (24) is satisfied, an auxiliary matrix can be advantageously used, where is the number of coefficients of the fault diagnosis equation and is the number of the different addends which is are present in the coefficients. The generic element of equal to the numeric value of the addend if the addend is present in the coefficient, zero otherwise. The last row of is , the last column to the constant relevant to the coefficient addends. As an example, referring to the case that not all the circuit parameters are considered potentially faulty, if with
(25) the matrix
is shown in (26) at the bottom of the next page.
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It is easy to understand that a linear combination of rows of is a linear combination of the fault equation coefficients and . If in the triangularization of the last row becomes is a linear combination of the other coefficients. It null, is important to notice that the last column, relevant to the constant addends, is useful to select between the cases D and triangularization the last row E of Fig. 1. In fact, if in the becomes null except for the entry relevant to the last column, (case E of Fig. 1). this means that If also the last column entry is null, we are in the case of (case D of Fig. 1). Moreover the rank of , evaluated without considering the last row and the last column, helps in the determination of circuit testability, as it will be more clearly shown in the following paragraph. C. Functional Relations Among Coefficients Until now we have considered linear combinations of fault indiagnosis equation coefficients with the scalar terms dependent with respect to the potentially faulty parameters . Now let us suppose that the scalar terms of the linear . combinations are not constant with respect to For example, if (we suppose that not all the circuit parameters are potentially is the following: faulty), the matrix
(27) could be considered linearly independent with Hence and because in the triangularization of respect to the last row is different from zero, but it is not true from a , functional point of view. In fact then the scalar terms change their values with the variations of and . In this case there is a functional1 the parameters dependence, that we will call dynamic. Otherwise we will call static the scalar dependence. For the considered example and are the following:
(28.a) (28.b) with of
dynamically dependent with respect to the rows . From a general point of view, if we obtain the
1 In particular the functional dependence is of polynomial type because there are not rational coefficients. This is very important from a computational point of view.
last row of different from null row and dynamically , there is a dependent with respect to the rows of and the other fault equation dynamic combination among rank coefficients. In the example we have also rank (as previously specified, we remember that rank is evaluated : this by excluding the last row and the last column from information tells us that there is an absence of functional and without dependence among the other coefficients . Instead, if rank rank , there is considering and (except functional dependence among coefficients ), independently on functional dependence of with respect to the other coefficients that is highlighted by the . functional dependence of with respect to the rows of For example, if (we suppose again that not all the circuit parameters are is the following: potentially faulty), the matrix
(29)
is functionally dependent with respect to In this case and , furthermore is both functionally and statically independent with respect to the other coefficients. This means relevant to , and are linearly that the rows of independent and the last row of does not become null during and are the following: its triangularization.
(30.a) (30.b) (in fact with independent with respect to the rows of is functionally independent with respect to the other coefficients). The functional dependence of with respect to and involves rank rank (rank rank . The determination of functional dependencies is very important for the testability evaluation. In Fig. 2 a general survey, considering both dynamic and static cases, is shown (the first example falls in the case 5, the second example falls in the case 2): the cases 3, 4, 7–9 are of nonfunctional type (static), as described in the previous paragraphs. For the remaining cases (dynamic) demonstrations similar to those of static type can be applied:
(26)
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Fig. 2. Testability evaluation: a general scheme.
Case 1: In this case there are dynamic combinations among coefficients that determine the presence of null rows during triangularization and is dynamically, but not the . Then statically, dependent with respect to the rows of rank (same demonstration of the static case C). rank Case 2: In this case there are dynamic combinations among coefficients that determine the presence of null rows during triangularization and is both dynamically and the . Then statically independent with respect to the rows of rank (same demonstration of the static rank case A). Case 5: In this case there are not dynamic combinations among coefficients that determine the presence of null rows triangularization and is dynamically, but during the . not statically, dependent with respect to the rows of could exist for the presence of static linear Null rows in rank (same demonstration combinations, then rank of the static case C). Case 6: In this case there are dynamic combinations among coefficients that determine the presence of null rows during triangularization and is statically dependent with the . Then rank rank (same respect to the rows of demonstration of the static case C). In brief, if only the matrix is considered, only static cases can be located. To reveal also dynamic (functional) cases it , compare it with is necessary always to determine rank is dependent or not with respect to the rank and verify if , as shown in Fig. 2. rows of
D. Testability in the Case of Multiple Fault Diagnosis Equations The previous results can be extended to the multiple fault diagnosis equation case and then also to the multiple test point case. The theoretical basis is the following theorem. Hypothesis: L.et us suppose that is the generic th test points (we fault diagnosis equation of a circuit with suppose that there is only one input, but the same reasoning is the product between number of test can be applied if points and number of inputs) and is the vector of the potentially faulty circuit parameters. If we fall in one of the cases 1, 4–7, and 9 of Fig. 2, the is the following, as defined for corresponding matrix the case of single test point:
(31)
and it has
rows and
columns.
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If we fall in the cases 2 or 3 of Fig. 2, we define as the matrix of rows and the matrix columns obtained from the union of the vector with as defined in (31)
obtain
(34)
The matrix
associated to
is the following:
(32)
(35)
Finally, if we fall in the case 8 of Fig. 2, that is if for the , the generic th fault diagnosis equation is defined as follows: matrix
(33)
has in this case rows and columns due to the elimination of one of the rows related to the coefficients . with rows and Thesis: The matrix columns, obtained by the union of matrices , has rank selected test equal to the testability value of a circuit with fault diagnosis equations. points or, more in general, with Proof: The proof is an immediate consequence of the relations found in the previous paragraphs. In fact, if we as the function obtained from by consider dividing numerator and denominator coefficients with respect (maximum degree coefficient of the denominator), we to
has rows and columns and its rank is equal to the testability value with respect to generic th test point [16]. If for the generic th fault diagnosis equation we fall in one of the cases 1, 4–7, and 9 of Fig. 2, rank rank , then the matrix to consider is as defined in the case of single test point. If we fall in the cases 2 or 3 of is independent with respect to the rows Fig. 2, the vector and rank rank . Then we consider of as . Finally, if we fall in the the matrix is dependent with respect to the rows case 8, the vector and rank rank . Then we consider as of the matrix defined in the hypothesis for this case (the elimination of an independent row decreases the matrix rank has the same rank of of one). In this way each matrix . Then the matrix with rows and columns, union of the matrices , has the same rank of . Being the matrix obtained by the union of the matrices the rank of this last matrix equal to the circuit testability in represents the case of multiple test points [15], [16], rank the testability value in the multiple fault equation case. IV. EXAMPLES In this section some examples of circuits relevant to different cases of testability evaluation reported in Fig. 2 are presented. For the sake of clarity, we refer to the situation of single
(36)
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Also has one null row and rank rankA = 2. is dynamically dependent with respect to the rows of , rank . then we fall in the dynamic case 5 and In order to verify this result, we determine the matrix obtained starting from the previous fault diagnosis equation . is the with all the coefficients divided for following:
Fig. 3. Circuit example.
(40) test point and, for the sake of brevity, we analyze only some possible cases. Let us consider the circuit in Fig. 3. The output voltage of the last operational amplifier is the selected test point. The fault diagnosis equation is shown in (36) at the bottom of the previous page. , and Case 5: If we suppose potentially faulty only , the parameters , and do not have to be considered in symbolic form. By choosing, for example, , the fault diagnosis equation is the following:
As it can be easily verified, the linearly independent rows of are two, then rank . , and Case 9: If we suppose potentially faulty only , while , the fault diagnosis equation is the following: (41) The matrix following:
, the matrix
, and the vector
are the
(37) The matrix
(42)
is the following:
(38) (43.a) without the last column and last row Only the part of is considered for the evaluation of rank : the last column is useful only for the discrimination of the cases 8 and 9, the last row, relevant to the maximum degree coefficient of the fault equation denominator, is useful only to establish if this coefficient is statically dependent with respect to the other coefficients of the fault equation. In this case the last row of is independent with respect to the other rows and has one null row (except the last column relevant to constant addends). and the vector are the following: The matrix
(39.a) (39.b)
(43.b) is dependent with respect to the other The last row of rows if the last column of , that is different from zero, is excluded during the triangularization procedure. This means that, in the triangularization, the last column entry, relevant , is different from zero. Furthermore, does not have to rank . is obviously dependent null rows, rank , then we fall in the static with respect to the rows of rank . case 9 and As it can be noted, in the considered examples the number of potentially faulty elements is lower than the total number of circuit parameters. This involves a partially symbolic form of the fault diagnosis equation and the possibility that combinations (static and/or dynamic) of fault equation coefficients exist. In fact, if all the circuit parameters are considered
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Fig. 4. Image taken from the software program TEST.
potentially faulty, the fault equation coefficients are dimensionally different and only a dynamic, but not static, relation of coefficients can exist. Furthermore, in the considered examples only one test point corresponding to an adimensional fault diagnosis equation is considered. This means that there is not maximum testability if all the circuit parameters are considered potentially faulty, because adimensional functions are invariant with respect to normalization of the circuit parameters. It is then often convenient to suppose that only some components are potentially faulty. Finally, we note that, in the circuit shown in Fig. 3, the testability is not equal in the two considered cases. But this should not be surprising because a different fault diagnosis equation is associated to each case.
V. THE PROGRAM TEST A new program, called TEST, that realizes the proposed testability evaluation algorithm has been developed and inserted in the project SAPEC (symbolic analysis program for electric circuits) [11], [12], [18], [19]. The starting point is the drawing of the circuit topology and the choice of the test points. Then the symbolic form of the corresponding fault diagnosis equations can be obtained: they constitute the basic . information for the construction of the matrices and In the triangularization method used to determine rank and rank the divisions are completely avoided and the testability is evaluated by using integer operations: in fact we recall that, since the testability value is independent of the component values [1]–[4], the rank is easily evaluated by assigning integer is values to the components [14]. Moreover, the matrix sparse, so its rank calculation is very quickly evaluated. All this allows us to completely eliminate the inevitable roundoff errors introduced by the numerical algorithms and by the not
optimized symbolic algorithms which were realized in the past years by the authors [15]–[17]. Once the testability value has been determined, the circuit parameters that contribute to the rank evaluation can also be obtained: this information is particularly helpful for the case of multiple test points, where we have a global matrix formed by the union of the matrices relevant to the single test points. An example of TEST program application is shown in Fig. 4, in which all the components of the circuit under test are considered potentially faulty. In Fig. 4 the outputs of TEST are shown: as it can be noted, besides the testability value, also the list of parameters that contribute to the obtained testability value is reported. However, it is worth pointing out that this information does not have an absolute validity because of the arbitrary selection of the columns in the triangularization procedure.
VI. CONCLUSIONS A new symbolic approach for testability evaluation of analog linear circuits has been presented. It allows us to realize a very simple testability measurement algorithm that is based on the derivatives of polynomial functions instead of rational functions. In this way the testability evaluation is optimized from a computational point of view. Furthermore in the computer implementation of this algorithm, the inevitable roundoff errors introduced by numerical algorithms and by nonoptimized symbolic algorithms realized in the past years are completely avoided and the computing speed is increased. The new proposed approach is particularly suitable for the testability evaluation in the case of medium and large circuit size and in presence of multiple test points.
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REFERENCES [1] R. Saeks, “A measure of testability and its application to test point selection theory,” in Proc. 20th Midwest Symp. Circuits Syst., Texas Tech. Univ., Lubbock, Aug. 1977. [2] N. Sen and R. Saeks, “Fault diagnosis for linear systems via multifrequency measurement,” IEEE Trans. Circuits Syst., vol. CAS-26, pp. 457–465, July 1979. [3] H. M. S. Chen and R. Saeks, “A search algorithm for the solution of multifrequency fault diagnosis equations,” IEEE Trans. Circuits Syst., vol. CAS-26, pp. 589–594, July 1979. [4] R. Saeks, N. Sen, H. M. S. Chen, K. S. Lu, S. Sangani, and R. A. De Carlo, “Fault analysis in electronic circuits and systems,” Tech. Rep., Texas Tech. Univ., Lubbock, Jan. 1978. [5] R. S. Berkowitz, “Conditions for network-element-value solvability,” IEEE Trans. Circuit Theory, vol. CT-9, pp. 24–29, 1962. [6] J. W. Bandler and A. E. Salama, “Fault diagnosis of analog circuits,” Proc. IEEE, vol. 73, pp. 1279–1321, Aug. 1985. [7] G. N. Stenbakken and T. M. Souders, “Test point selection and testability measures via QR factorization of linear models,” IEEE Trans. Instrum. Meas., vol. IM-36, pp. 406–410, June 1987. [8] G. N. Stenbakken, T. M. Souders, and G. W. Stewart, “Ambiguity groups and testability,” IEEE Trans. Instrum. Meas., vol. 38, pp. 941–947, Oct. 1989. [9] G. Iuculano, A. Liberatore, S. Manetti, and M. Marini, “Multifrequency measurement of testability with application to large linear analog systems,” IEEE Trans. Circuits Syst., vol. CAS-23, pp. 644–648, June 1986. [10] M. Catelani, G. Iuculano, A. Liberatore, S. Manetti, and M. Marini, “Improvements to numerical testability evaluation,” IEEE Trans. Instrum. Meas., vol. IM-36, pp. 902–907, Dec. 1987. [11] A. Liberatore and S. Manetti, “SAPEC—A personal computer program for the symbolic analysis of electric circuits,” in Proc. 1988 IEEE Int. Symp. Circuits Syst., Helsinki, Finland, June 1988, pp. 897–900. [12] S. Manetti, “A new approach to automatic symbolic analysis of electric circuits,” Proc. Inst. Elect. Eng. Pt. G: Electron. Circuits Syst., vol. 138, pp. 22–28, Feb. 1991. [13] A. Liberatore and S. Manetti, “Network sensitivity analysis via symbolic formulation,” in Proc. 1989 Int. Symp. Circuits Syst., Portland, OR, May 1989, pp. 705–708. [14] R. W. Priester and J. B. Clary, “New measures of testability and test conplexity for linear analog failure analysis,” IEEE Trans. Circuits Syst., vol. CAS-28, pp. 1088–1092, Nov. 1981. [15] R. Carmassi, M. Catelani, G. Iuculano, A. Liberatore, S. Manetti, and M. Marini, “Analog network testability measurement: A symbolic formulation approach,” IEEE Trans. Instrum. Meas., vol. 40, pp. 930–935, Dec. 1991. [16] A. Liberatore, S. Manetti, and M. C. Piccirilli, “A new efficient method for analog circuit testability measurement,” in Proc. IEEE Instrum. Meas. Tech. Conf., Hamamatsu, Japan, May 1994, pp. 193–196. [17] A. Liberatore, A. Luchetta, S. Manetti, and M. C. Piccirilli, “Analog network testability measurement using symbolic techniques,” in Proc. SMACD, Sevilla, Spain, Oct. 1994, pp. 167–176. , “A new symbolic program package for the interactive design [18] of analog circuits,” in Proc. Int. Symp. Circuits Syst., Seattle, WA, May 1995, pp. 2209–2212. [19] A. Luchetta, S. Manetti, and M. C. Piccirilli, “A windows package for symbolic and numerical simulation of analog circuits,” in Proc. Electrosoft, San Miniato, Italy, May 1996, pp. 115–123.
Giulio Fedi (S’94–M’95) was born in Milano, Italy, in 1970. He received the Laurea degree in electronic engineering (summa cum laude) from the University of Florence, Italy, in 1995. Since November 1995, he has been a Ph.D. student in electrical sciences at the University of Pisa. His research activities regard analog circuit fault diagnosis using symbolic techniques and neural network applications to circuit modeling. Mr. Fedi is a member of AEI.
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Antonio Luchetta (M’96) received the Laurea degree in eletronic engineering from the University of Florence, Italy, in 1993. He is a Researcher in the Department of Engineering and Environmental Physics at the University of Basilicata, Italy. His research interests include software and algorithms for electric network simulation, neural network application to forecasting and control problems, and elaboration of meteorological data. Mr. Luchetta is a member of AEI.
Stefano Manetti (M’96) received the laurea degree in electronic engineering from the University of Florence, Italy, in 1977. From 1977 to 1979, he had a research fellowship at the Engineering Faculty, University of Florence. He was an Assistant Professor of applied electronics at the Accademia Navale of Livorno from 1980 to 1983 and a Researcher at the Electronic Engineering Department , University of Florence, from 1983 to 1987. Since May 1987, he has been an Associate Professor of network theory at the same university. In 1994, he joined the University of Basilicata, Potenza, Italy, as a Full Professor of electrical sciences. Since November 1996, he has been a Full Professor of electrical sciences, at the University of Florence. His research interests are in the fields of circuit theory, neural networks, and fault diagnosis of electronic circuits. Mr. Manetti is a member of ECS and AEI.
Maria Cristina Piccirilli received the Laurea degree in electronic engineering from the University of Florence, Italy, in 1987. From 1988 to 1990, she had a research fellowship from the University of Pisa, Italy. From March 1990 to October 1998, she was a Researcher at the Department of Electronic Engineering, University of Florence. Since November 1998, she has been an Associate Professor of network theory in the same department, where she works in the area of circuit theory, fault diagnosis of electronic circuits, and symbolic analysis.
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 47, NO. 2, APRIL 1998
A New Symbolic Method for Analog Circuit Testability Evaluation Giulio Fedi, Member, IEEE, Antonio Luchetta, Member, IEEE, Stefano Manetti, Member, IEEE, and Maria Christina Piccirilli
Abstract— Testability is a very useful concept in the field of circuit testing and fault diagnosis and can be defined as a measure of the effectiveness of a selected test point set. A very efficient approach for automated testability evaluation of analog circuits is based on the use of symbolic techniques. Different algorithms relying on the symbolic approach have been presented in the past by the authors and in this work noteworthy improvements on these algorithms are proposed. The new theoretical approach and the description of the subsequent algorithm that optimizes the testability evaluation from a computational point of view are presented. As a result, in the computer implementation the roundoff errors are completely eliminated and the computing speed is increased. The program which implements this new algorithm is also presented. Index Terms— Analog circuits, analog system fault diagnosis, analog system testing, circuit testing, fault diagnosis, fault location.
I. INTRODUCTION
I
N EVERY field of engineering, testing is a fundamental step for the validation of design, being the most direct way to verify that a product meets its specifications. If the desired performance is not achieved, testing should identify all the causes of malfunctioning and indicate suitable corrective actions. While testing methodologies are fully automated for digital devices, there is a lack of efficient and systematic methods of testing for analog and mixed signal devices. Furthermore this problem is even going to be more critical with technology improvements and increasing number of mixed signal applications, especially in the communication field. In the case of analog electrical circuits, deviations from the desired circuit behavior are generally caused by variations of the components from their nominal value (parametric fault). For this reason it is very important that an analog circuit testing procedure allows us to determine an estimate of component values simply and quickly. Moreover, an essential task in analog circuit testing is the selection of an optimum set of measurements, that is an optimum set of test points. To do this, it is necessary to have a quantitative index to compare different solutions. For analog circuits, the testability concept provides this index. Manuscript received April 11, 1997; revised December 4, 1998. G. Fedi, S. Manetti, and M. C. Piccirilli are with the Department of Electronic Engineering, University of Florence, 3-50139 Firenze, Italy. A. Luchetta is with the Department of Engineering and Environmental Physics, University of Basilicata, 85100 Potenza, Italy. Publisher Item Identifier S 0018-9456(98)09856-8.
The most used definition of testability evaluation was given by Saeks [1]–[4], who defined it as a measure of solvability of the nonlinear fault diagnosis equations, indicating the ambiguity which results from an attempt to solve such equations in a neighborhood of almost any failure. Then testability is strictly tied to the concept of network-elementvalue-solvability [5]–[8], that is, it allows us to establish a priori if it is possible to uniquely determine component values, given a circuit structure and some input–output (I/O) relations to be measured. If a unique solution is not achievable, other I/O measurements, that is other test points, must be added or a reduced number of unknown parameters must be used. So testability information is essential, either to designers who must know which test points to make accessible for testing or to test engineers who must plan tests and know how many and what parameters can be uniquely isolated by these tests. Algorithms for evaluating the testability as defined by Saeks have been developed by the authors in the past years, at first using a numerical approach [9], [10]. Subsequently a symbolic approach has been used in order to eliminate the unavoidable roundoff errors introduced by numerical algorithms, which render the obtained testability only an estimate. The symbolic analysis is a technique that permits one to obtain, as result of a computer program, circuit network functions in closed form, where some or all the circuit elements and the complex frequency are represented by symbolic parameters. Hence symbolic analysis is particularly suitable for the testability evaluation problem, in which the component values are the unknown quantities. It allows realization of very simple testability measurement algorithms which are based on the determination of the nonlinear fault diagnosis equations in symbolic form with respect to both complex frequency s and component values [11], [12] and on the subsequent sensitivity analysis of these equations [13]. Also in the symbolic approach, as in the numerical one, testability evaluation is made by selecting integer numerical values for the entries of the matrix whose rank determines the testability [14], because the circuit testability is independent with respect to component values [1]–[4]. Unfortunately the testability evaluation algorithms based on symbolic approach presented in the past reduce but do not eliminate roundoff errors. Nevertheless, by exploiting the previously obtained results, it has been possible to develop a new symbolic procedure that completely eliminates the roundoff error problem.
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In this paper, a new symbolic procedure is presented. Starting from the theoretical background developed in previous works [15]–[17] and summarized in Section II, new theorems and the subsequent new algorithm that optimizes the testability evaluation from a computational point of view have been developed. They are presented in Section III, while in Section IV explanatory examples are proposed. Finally, a program which implements this new algorithm has been realized and its description is presented in Section V. II. TESTABILITY EVALUATION An analog, linear, time-invariant circuit can be described in the following way, by using the modified nodal analysis (MNA): (1) is the vector of the parameters where which are potentially faulty, assuming that all the faults can be expressed as parameter variations, without influencing the circuit topology (i.e., faults as short and open are not is considered); the vector of the output test points (voltages and/or currents); is the vector of the inaccessible node voltages and/or currents of all the elements that have not an admittance representation; is the characteristic matrix, conformable to the vectors; is the input vector. It is worth pointing out that the parameters assumed not faulty have their own nominal value. The fault diagnosis equations of the circuit under test are constituted by the network functions relevant to each test point output and to each input. They can be obtained from (1) by applying the superposition principle and have the following form:
(2)
minor of the matrix and the with th output due to the contribution of input only. As it can be easily noted the total number of the fault diagnosis equations is equal to the product of the number of outputs and inputs. The testability measure , evaluated in a suitable neighborhood of the nominal value of the potentially faulty parameter vector , is given by the maximum number of linearly associated independent columns of the Jacobean matrix with the fault diagnosis equations in (2) [1]–[4] rankcol
(3)
where the entries of the Jacobean matrix are polynomial of the functions in , evaluated in the nominal value can be written parameter vector [8]. In fact the matrix
in the following way:
(4) polynomial matrix. So the testability value coincides with . The with the number of linearly independent columns of evaluation, starting from (4), is very difficult to handle and subject to roundoff errors, especially for large circuits, if a numerical approach is used [10]. It has been proven [15] that the number of linearly indepenis equal to the rank of a matrix dent columns of the matrix constituted by the coefficients of the polynomial functions . Then the entries of the matrix are independent of with respect to the complex frequency . In other words, by in the following way: expressing (5)
and deg deg , we have rank , where is a matrix of order rankcol ( number of potentially faulty parameters, ) of the following form: with
deg
(6)
. whose entries are the coefficients of the polynomials are in comThis result is very useful if the coefficients pletely symbolic form. In fact, in this case, they are functions of circuit parameters to which we can assign arbitrary values because the testability is independent of component values. is essentially a sensitivity Furthermore, since the matrix matrix of the circuit under test, starting from a fully symbolic generation of the network functions corresponding to the selected fault diagnosis equations, it is very easy to obtain symbolic sensitivity functions [11]–[13]. As a consequence, the use of a symbolic approach simplifies the testability measure procedure and reduces roundoff errors, because the components of are not affected by any computational error. An important simplification of this procedure, reported in [16], results from the fact that the testability measure can be constituted by the derivatives evaluated as rank of a matrix of the coefficients of the fault diagnosis equations with respect to the potentially faulty circuit parameters. For the sake of simplicity, let us consider a circuit with only one test point and
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only one input. In this case there is only one fault diagnosis equation that must be expressed in the following form:
(7)
vector of potentially faulty pawith and degrees of numerator and denominator, rameters, of order , constituted respectively. The matrix in (7) with by the derivatives of the coefficients of respect to the unknown parameters, is the following:
A. Theoretical Basis: Case of Single Fault Diagnosis Equation For the sake of simplicity, let us consider a circuit with only one test point and only one input. Then there is only one fault diagnosis equation to consider. The theoretical basis of the new approach regards essentially the following theorem. Hypothesis: Let be the fault diagnosis equation, the vector of the potentially faulty with and coefficients formed by sum of parameters and products of at least one element of the vector (this means that the derivatives of and with respect to the potentially faulty parameters are not equal to zero). Hence, if we define a , with rows and columns, as follows: matrix
(8) (9)
has the same rank of the As shown in [16] the matrix previously defined matrix , because the rows of are linear . Then the testability value combination of the rows of by assigning can be computed as the rank of the matrix arbitrary values to parameters . If the circuit under test is a multiple-input–multiple-output system, that is, if there is more than one fault diagnosis equation, the same result can be easily obtained. This is a noteworthy simplification from a computational point of view, because derivatives of coefficients of fault diagnosis equations are simpler to compute with respect to derivatives of fault diagnosis equations. A limitation of this method is the need to get rational functions whose denominator has a unitary coefficient in the highest order term, while, in most cases, this coefficient is different from one. This limitation has a not negligible impact on the computation task, because, in order to obtain such a kind of normalization, it is necessary to divide all the coefficients of the rational functions by the coefficient of the highest term of the denominator, with a consequent complication in the evaluation of the derivatives (derivative of a rational function instead of a polynomial function) and the introduction of roundoff errors (in any case lower than those relevant to completely numerical approaches). The symbolic approach for testability evaluation presented in this paper removes this limitation, providing a complete elimination of roundoff errors and an increase of computing speed, as will be shown in the following section. III. NEW APPROACH FOR TESTABILITY EVALUATION The symbolic approach for testability evaluation proposed in this paper is based on a new theorem and subsequent new corollaries discussed in the following.
is different from the null one. each row of Let us suppose also that the vector is . linearly independent with respect to the rows of Let be the function obtainable from by dividing numerator and denominator for (maximum degree coefficient of denominator) and the rows and columns whose following matrix of rank is equal to the circuit testability value
(10)
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Fig. 1. General survey in the case of each scalar term h independent with respect to the vector of potentially faulty parameters
Thesis: The rank of is equal to the rank of and indicates the testability value of the circuit (case B of Fig. 1). and Proof: The generic elements of are derivatives of a ratio, hence they have the following form:
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p.
(11.a)
(11.b)
(12)
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So can be written as shown in (12) at the bottom of the can be split in the following previous page. Furthermore, way:
where the right side of the identity is linearly independent by hypothesis, hence rank is at least equal to . , for obtaining rank On the other hand, if rank at least equal to , it must be (16) , then . In the right term of , that are supposed linearly the identity there are rows of is linearly independent with independent. Also the vector . In fact, if it were not so, we could respect to the rows of affirm, without loss of generality, that
But
(17) (13)
with have:
and
scalar terms different from zero. So we should (18)
is obtained by adding to the Then the first row of , multiplied for , the vector first row of multiplied for . The second row of is obtained by adding to , multiplied for ( , the vector the second row of multiplied for and so on for every row of . , for obtaining in at least Hence, if rank linearly independent rows, it must be (14)
and are but that is an impossible relation, because different from the null row and, with respect to the vector , is they are linearly independent by hypothesis. Hence rank at least and so the thesis is proved. without null rows The request to consider the matrix is justified by the following corollary. ) in the matrix Corollary 1: If there are null rows ( and the vector is linearly independent with respect to rank (case A of Fig. 1). the rest of them, rank involves Proof: The hypothesis of null rows in and/or in which there are that there are coefficients not potentially faulty parameters. The presence of null rows involves also that in the matrix there are rows in of the following kind: (19)
But row of for
, with and
generic row of scalar value equal to and equal to . Then
generic for
with as previously defined. These rows belong to a vectorial hyperspace of dimension equal to one ( is the generating rank . In fact, if vector), hence, if rank , we have there are linearly independent rows in
(20)
(15)
where the right term of the identity generates a vectorial , because the rows of hyperspace of dimension and the vector are linearly independent by hypothesis.
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More in general the rank of cannot be greater or . In fact the lower than one with respect to the rank of and the row two matrices obtained by adding to have the same rank always, because they are obtainable each by the other with elementary operations on the rows. Then, if is linearly , rank dependent with respect to the rows of rank rank . Now, if is linearly dependent rank rank , also with respect to the rows of otherwise, if is linearly independent with respect to the rows rank rank rank , that is of rank . rank From the previous observation the following corollaries are developed. Corollary 2: Let be linearly dependent with respect to the rows, which are all different from the null row. Then: rank if is a linear combination 1) rank and of the fault diagnosis equaof coefficients plus a constant term without elements tion belonging to the vector . rank if is a linear combination 2) rank and of the fault diagnosis equation of coefficients (cases D and E of Fig. 1). be a linear combination of the rows of Proof: Let . Then can be expressed in the following way:
and being by hypothesis, there are linearly dependent at least one and, in particular, is independent with respect rows in . Hence rank rank (case to the rows of D of Fig. 1). is linearly dependent with respect to the Corollary 3: If and has null rows, rank rank rows of and is a linear combination of coefficients and of the with or without the adding fault diagnosis equation of a constant term different from zero and with no elements belonging to (case C of Fig. 1). has null rows, is linearly dependent Proof: If (see the proof of the Corollary with respect to the rows of is linearly dependent with respect to the 1). Then, since by hypothesis, rank rank and rows of ( can be also equal to zero). Now we know all the possible relationships which can exist and , if is a scalar independent with between respect to the vector of potentially faulty parameters . Being the testability measure equal to rank , we can summarize all the obtained results as shown in Fig. 1, in which the permits us to generalize the argumentation constant term to the case that not all the circuit parameters can be faulty. However, all the possible kinds of fault diagnosis equations are not yet been considered, as it will be shown in the following. B. Use of an Auxiliary Matrix
(21) are the linearly independent rows of , where indicates in a general way the coefficients and of the are scalar terms indefault diagnosis equation pendent with respect to the vector . This equation involves is a linear combination of coefficients and that then it can be expressed in the following way: (22) with generic constant term without elements belonging to . can be As previously shown, a generic row of , then, we have expressed as (23) that
is .
If
,
and is a linear combination of rows of . So, by , we have indicating in a general way can be that the condition . This means that, written as , with constant term if is a linear combination without elements of and then rank rank of rows of (case E of Fig. 1). If , this , that is, being means that
In the previous paragraph it has been shown that the linear with respect to the rows of the independence of the vector is given by the following relation among the matrix : coefficients of the fault diagnosis equation (24) and without elements belongwith the scalar quantities ing to . In fact, by the derivatives of (24) (considered with the sign “ ”) with respect to the circuit parameters, we obtain . In order to verify a linear combination of the rows of of order if expression (24) is satisfied, an auxiliary matrix can be advantageously used, where is the number of coefficients of the fault diagnosis equation and is the number of the different addends which is are present in the coefficients. The generic element of equal to the numeric value of the addend if the addend is present in the coefficient, zero otherwise. The last row of is , the last column to the constant relevant to the coefficient addends. As an example, referring to the case that not all the circuit parameters are considered potentially faulty, if with
(25) the matrix
is shown in (26) at the bottom of the next page.
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It is easy to understand that a linear combination of rows of is a linear combination of the fault equation coefficients and . If in the triangularization of the last row becomes is a linear combination of the other coefficients. It null, is important to notice that the last column, relevant to the constant addends, is useful to select between the cases D and triangularization the last row E of Fig. 1. In fact, if in the becomes null except for the entry relevant to the last column, (case E of Fig. 1). this means that If also the last column entry is null, we are in the case of (case D of Fig. 1). Moreover the rank of , evaluated without considering the last row and the last column, helps in the determination of circuit testability, as it will be more clearly shown in the following paragraph. C. Functional Relations Among Coefficients Until now we have considered linear combinations of fault indiagnosis equation coefficients with the scalar terms dependent with respect to the potentially faulty parameters . Now let us suppose that the scalar terms of the linear . combinations are not constant with respect to For example, if (we suppose that not all the circuit parameters are potentially is the following: faulty), the matrix
(27) could be considered linearly independent with Hence and because in the triangularization of respect to the last row is different from zero, but it is not true from a , functional point of view. In fact then the scalar terms change their values with the variations of and . In this case there is a functional1 the parameters dependence, that we will call dynamic. Otherwise we will call static the scalar dependence. For the considered example and are the following:
(28.a) (28.b) with of
dynamically dependent with respect to the rows . From a general point of view, if we obtain the
1 In particular the functional dependence is of polynomial type because there are not rational coefficients. This is very important from a computational point of view.
last row of different from null row and dynamically , there is a dependent with respect to the rows of and the other fault equation dynamic combination among rank coefficients. In the example we have also rank (as previously specified, we remember that rank is evaluated : this by excluding the last row and the last column from information tells us that there is an absence of functional and without dependence among the other coefficients . Instead, if rank rank , there is considering and (except functional dependence among coefficients ), independently on functional dependence of with respect to the other coefficients that is highlighted by the . functional dependence of with respect to the rows of For example, if (we suppose again that not all the circuit parameters are is the following: potentially faulty), the matrix
(29)
is functionally dependent with respect to In this case and , furthermore is both functionally and statically independent with respect to the other coefficients. This means relevant to , and are linearly that the rows of independent and the last row of does not become null during and are the following: its triangularization.
(30.a) (30.b) (in fact with independent with respect to the rows of is functionally independent with respect to the other coefficients). The functional dependence of with respect to and involves rank rank (rank rank . The determination of functional dependencies is very important for the testability evaluation. In Fig. 2 a general survey, considering both dynamic and static cases, is shown (the first example falls in the case 5, the second example falls in the case 2): the cases 3, 4, 7–9 are of nonfunctional type (static), as described in the previous paragraphs. For the remaining cases (dynamic) demonstrations similar to those of static type can be applied:
(26)
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Fig. 2. Testability evaluation: a general scheme.
Case 1: In this case there are dynamic combinations among coefficients that determine the presence of null rows during triangularization and is dynamically, but not the . Then statically, dependent with respect to the rows of rank (same demonstration of the static case C). rank Case 2: In this case there are dynamic combinations among coefficients that determine the presence of null rows during triangularization and is both dynamically and the . Then statically independent with respect to the rows of rank (same demonstration of the static rank case A). Case 5: In this case there are not dynamic combinations among coefficients that determine the presence of null rows triangularization and is dynamically, but during the . not statically, dependent with respect to the rows of could exist for the presence of static linear Null rows in rank (same demonstration combinations, then rank of the static case C). Case 6: In this case there are dynamic combinations among coefficients that determine the presence of null rows during triangularization and is statically dependent with the . Then rank rank (same respect to the rows of demonstration of the static case C). In brief, if only the matrix is considered, only static cases can be located. To reveal also dynamic (functional) cases it , compare it with is necessary always to determine rank is dependent or not with respect to the rank and verify if , as shown in Fig. 2. rows of
D. Testability in the Case of Multiple Fault Diagnosis Equations The previous results can be extended to the multiple fault diagnosis equation case and then also to the multiple test point case. The theoretical basis is the following theorem. Hypothesis: L.et us suppose that is the generic th test points (we fault diagnosis equation of a circuit with suppose that there is only one input, but the same reasoning is the product between number of test can be applied if points and number of inputs) and is the vector of the potentially faulty circuit parameters. If we fall in one of the cases 1, 4–7, and 9 of Fig. 2, the is the following, as defined for corresponding matrix the case of single test point:
(31)
and it has
rows and
columns.
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If we fall in the cases 2 or 3 of Fig. 2, we define as the matrix of rows and the matrix columns obtained from the union of the vector with as defined in (31)
obtain
(34)
The matrix
associated to
is the following:
(32)
(35)
Finally, if we fall in the case 8 of Fig. 2, that is if for the , the generic th fault diagnosis equation is defined as follows: matrix
(33)
has in this case rows and columns due to the elimination of one of the rows related to the coefficients . with rows and Thesis: The matrix columns, obtained by the union of matrices , has rank selected test equal to the testability value of a circuit with fault diagnosis equations. points or, more in general, with Proof: The proof is an immediate consequence of the relations found in the previous paragraphs. In fact, if we as the function obtained from by consider dividing numerator and denominator coefficients with respect (maximum degree coefficient of the denominator), we to
has rows and columns and its rank is equal to the testability value with respect to generic th test point [16]. If for the generic th fault diagnosis equation we fall in one of the cases 1, 4–7, and 9 of Fig. 2, rank rank , then the matrix to consider is as defined in the case of single test point. If we fall in the cases 2 or 3 of is independent with respect to the rows Fig. 2, the vector and rank rank . Then we consider of as . Finally, if we fall in the the matrix is dependent with respect to the rows case 8, the vector and rank rank . Then we consider as of the matrix defined in the hypothesis for this case (the elimination of an independent row decreases the matrix rank has the same rank of of one). In this way each matrix . Then the matrix with rows and columns, union of the matrices , has the same rank of . Being the matrix obtained by the union of the matrices the rank of this last matrix equal to the circuit testability in represents the case of multiple test points [15], [16], rank the testability value in the multiple fault equation case. IV. EXAMPLES In this section some examples of circuits relevant to different cases of testability evaluation reported in Fig. 2 are presented. For the sake of clarity, we refer to the situation of single
(36)
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Also has one null row and rank rankA = 2. is dynamically dependent with respect to the rows of , rank . then we fall in the dynamic case 5 and In order to verify this result, we determine the matrix obtained starting from the previous fault diagnosis equation . is the with all the coefficients divided for following:
Fig. 3. Circuit example.
(40) test point and, for the sake of brevity, we analyze only some possible cases. Let us consider the circuit in Fig. 3. The output voltage of the last operational amplifier is the selected test point. The fault diagnosis equation is shown in (36) at the bottom of the previous page. , and Case 5: If we suppose potentially faulty only , the parameters , and do not have to be considered in symbolic form. By choosing, for example, , the fault diagnosis equation is the following:
As it can be easily verified, the linearly independent rows of are two, then rank . , and Case 9: If we suppose potentially faulty only , while , the fault diagnosis equation is the following: (41) The matrix following:
, the matrix
, and the vector
are the
(37) The matrix
(42)
is the following:
(38) (43.a) without the last column and last row Only the part of is considered for the evaluation of rank : the last column is useful only for the discrimination of the cases 8 and 9, the last row, relevant to the maximum degree coefficient of the fault equation denominator, is useful only to establish if this coefficient is statically dependent with respect to the other coefficients of the fault equation. In this case the last row of is independent with respect to the other rows and has one null row (except the last column relevant to constant addends). and the vector are the following: The matrix
(39.a) (39.b)
(43.b) is dependent with respect to the other The last row of rows if the last column of , that is different from zero, is excluded during the triangularization procedure. This means that, in the triangularization, the last column entry, relevant , is different from zero. Furthermore, does not have to rank . is obviously dependent null rows, rank , then we fall in the static with respect to the rows of rank . case 9 and As it can be noted, in the considered examples the number of potentially faulty elements is lower than the total number of circuit parameters. This involves a partially symbolic form of the fault diagnosis equation and the possibility that combinations (static and/or dynamic) of fault equation coefficients exist. In fact, if all the circuit parameters are considered
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Fig. 4. Image taken from the software program TEST.
potentially faulty, the fault equation coefficients are dimensionally different and only a dynamic, but not static, relation of coefficients can exist. Furthermore, in the considered examples only one test point corresponding to an adimensional fault diagnosis equation is considered. This means that there is not maximum testability if all the circuit parameters are considered potentially faulty, because adimensional functions are invariant with respect to normalization of the circuit parameters. It is then often convenient to suppose that only some components are potentially faulty. Finally, we note that, in the circuit shown in Fig. 3, the testability is not equal in the two considered cases. But this should not be surprising because a different fault diagnosis equation is associated to each case.
V. THE PROGRAM TEST A new program, called TEST, that realizes the proposed testability evaluation algorithm has been developed and inserted in the project SAPEC (symbolic analysis program for electric circuits) [11], [12], [18], [19]. The starting point is the drawing of the circuit topology and the choice of the test points. Then the symbolic form of the corresponding fault diagnosis equations can be obtained: they constitute the basic . information for the construction of the matrices and In the triangularization method used to determine rank and rank the divisions are completely avoided and the testability is evaluated by using integer operations: in fact we recall that, since the testability value is independent of the component values [1]–[4], the rank is easily evaluated by assigning integer is values to the components [14]. Moreover, the matrix sparse, so its rank calculation is very quickly evaluated. All this allows us to completely eliminate the inevitable roundoff errors introduced by the numerical algorithms and by the not
optimized symbolic algorithms which were realized in the past years by the authors [15]–[17]. Once the testability value has been determined, the circuit parameters that contribute to the rank evaluation can also be obtained: this information is particularly helpful for the case of multiple test points, where we have a global matrix formed by the union of the matrices relevant to the single test points. An example of TEST program application is shown in Fig. 4, in which all the components of the circuit under test are considered potentially faulty. In Fig. 4 the outputs of TEST are shown: as it can be noted, besides the testability value, also the list of parameters that contribute to the obtained testability value is reported. However, it is worth pointing out that this information does not have an absolute validity because of the arbitrary selection of the columns in the triangularization procedure.
VI. CONCLUSIONS A new symbolic approach for testability evaluation of analog linear circuits has been presented. It allows us to realize a very simple testability measurement algorithm that is based on the derivatives of polynomial functions instead of rational functions. In this way the testability evaluation is optimized from a computational point of view. Furthermore in the computer implementation of this algorithm, the inevitable roundoff errors introduced by numerical algorithms and by nonoptimized symbolic algorithms realized in the past years are completely avoided and the computing speed is increased. The new proposed approach is particularly suitable for the testability evaluation in the case of medium and large circuit size and in presence of multiple test points.
FEDI et al.: NEW SYMBOLIC METHOD FOR ANALOG CIRCUIT
REFERENCES [1] R. Saeks, “A measure of testability and its application to test point selection theory,” in Proc. 20th Midwest Symp. Circuits Syst., Texas Tech. Univ., Lubbock, Aug. 1977. [2] N. Sen and R. Saeks, “Fault diagnosis for linear systems via multifrequency measurement,” IEEE Trans. Circuits Syst., vol. CAS-26, pp. 457–465, July 1979. [3] H. M. S. Chen and R. Saeks, “A search algorithm for the solution of multifrequency fault diagnosis equations,” IEEE Trans. Circuits Syst., vol. CAS-26, pp. 589–594, July 1979. [4] R. Saeks, N. Sen, H. M. S. Chen, K. S. Lu, S. Sangani, and R. A. De Carlo, “Fault analysis in electronic circuits and systems,” Tech. Rep., Texas Tech. Univ., Lubbock, Jan. 1978. [5] R. S. Berkowitz, “Conditions for network-element-value solvability,” IEEE Trans. Circuit Theory, vol. CT-9, pp. 24–29, 1962. [6] J. W. Bandler and A. E. Salama, “Fault diagnosis of analog circuits,” Proc. IEEE, vol. 73, pp. 1279–1321, Aug. 1985. [7] G. N. Stenbakken and T. M. Souders, “Test point selection and testability measures via QR factorization of linear models,” IEEE Trans. Instrum. Meas., vol. IM-36, pp. 406–410, June 1987. [8] G. N. Stenbakken, T. M. Souders, and G. W. Stewart, “Ambiguity groups and testability,” IEEE Trans. Instrum. Meas., vol. 38, pp. 941–947, Oct. 1989. [9] G. Iuculano, A. Liberatore, S. Manetti, and M. Marini, “Multifrequency measurement of testability with application to large linear analog systems,” IEEE Trans. Circuits Syst., vol. CAS-23, pp. 644–648, June 1986. [10] M. Catelani, G. Iuculano, A. Liberatore, S. Manetti, and M. Marini, “Improvements to numerical testability evaluation,” IEEE Trans. Instrum. Meas., vol. IM-36, pp. 902–907, Dec. 1987. [11] A. Liberatore and S. Manetti, “SAPEC—A personal computer program for the symbolic analysis of electric circuits,” in Proc. 1988 IEEE Int. Symp. Circuits Syst., Helsinki, Finland, June 1988, pp. 897–900. [12] S. Manetti, “A new approach to automatic symbolic analysis of electric circuits,” Proc. Inst. Elect. Eng. Pt. G: Electron. Circuits Syst., vol. 138, pp. 22–28, Feb. 1991. [13] A. Liberatore and S. Manetti, “Network sensitivity analysis via symbolic formulation,” in Proc. 1989 Int. Symp. Circuits Syst., Portland, OR, May 1989, pp. 705–708. [14] R. W. Priester and J. B. Clary, “New measures of testability and test conplexity for linear analog failure analysis,” IEEE Trans. Circuits Syst., vol. CAS-28, pp. 1088–1092, Nov. 1981. [15] R. Carmassi, M. Catelani, G. Iuculano, A. Liberatore, S. Manetti, and M. Marini, “Analog network testability measurement: A symbolic formulation approach,” IEEE Trans. Instrum. Meas., vol. 40, pp. 930–935, Dec. 1991. [16] A. Liberatore, S. Manetti, and M. C. Piccirilli, “A new efficient method for analog circuit testability measurement,” in Proc. IEEE Instrum. Meas. Tech. Conf., Hamamatsu, Japan, May 1994, pp. 193–196. [17] A. Liberatore, A. Luchetta, S. Manetti, and M. C. Piccirilli, “Analog network testability measurement using symbolic techniques,” in Proc. SMACD, Sevilla, Spain, Oct. 1994, pp. 167–176. , “A new symbolic program package for the interactive design [18] of analog circuits,” in Proc. Int. Symp. Circuits Syst., Seattle, WA, May 1995, pp. 2209–2212. [19] A. Luchetta, S. Manetti, and M. C. Piccirilli, “A windows package for symbolic and numerical simulation of analog circuits,” in Proc. Electrosoft, San Miniato, Italy, May 1996, pp. 115–123.
Giulio Fedi (S’94–M’95) was born in Milano, Italy, in 1970. He received the Laurea degree in electronic engineering (summa cum laude) from the University of Florence, Italy, in 1995. Since November 1995, he has been a Ph.D. student in electrical sciences at the University of Pisa. His research activities regard analog circuit fault diagnosis using symbolic techniques and neural network applications to circuit modeling. Mr. Fedi is a member of AEI.
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Antonio Luchetta (M’96) received the Laurea degree in eletronic engineering from the University of Florence, Italy, in 1993. He is a Researcher in the Department of Engineering and Environmental Physics at the University of Basilicata, Italy. His research interests include software and algorithms for electric network simulation, neural network application to forecasting and control problems, and elaboration of meteorological data. Mr. Luchetta is a member of AEI.
Stefano Manetti (M’96) received the laurea degree in electronic engineering from the University of Florence, Italy, in 1977. From 1977 to 1979, he had a research fellowship at the Engineering Faculty, University of Florence. He was an Assistant Professor of applied electronics at the Accademia Navale of Livorno from 1980 to 1983 and a Researcher at the Electronic Engineering Department , University of Florence, from 1983 to 1987. Since May 1987, he has been an Associate Professor of network theory at the same university. In 1994, he joined the University of Basilicata, Potenza, Italy, as a Full Professor of electrical sciences. Since November 1996, he has been a Full Professor of electrical sciences, at the University of Florence. His research interests are in the fields of circuit theory, neural networks, and fault diagnosis of electronic circuits. Mr. Manetti is a member of ECS and AEI.
Maria Cristina Piccirilli received the Laurea degree in electronic engineering from the University of Florence, Italy, in 1987. From 1988 to 1990, she had a research fellowship from the University of Pisa, Italy. From March 1990 to October 1998, she was a Researcher at the Department of Electronic Engineering, University of Florence. Since November 1998, she has been an Associate Professor of network theory in the same department, where she works in the area of circuit theory, fault diagnosis of electronic circuits, and symbolic analysis.
