### Table 6: Computed values of suspected components for R3 = 1k

1993

"... In PAGE 13: ...2 Example 2: R3 faulty For the second experiment the circuit is modi ed so that R3 is 1 k instead of correct 10 k . The measured values for this case are given in Table 5, computed values of the potentially faulty components in Table6 , and nal average values in Table 7, respectively. Due to the negative values computed for the normal mode, R2; R5; R6; C1 and C2 are ruled out.... ..."

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### Table 7: Ranking of the suspected components and their predicted values for R3 = 1k .

"... In PAGE 15: ... The measured values for this case are given in Table 5, and computed values of the potentially faulty components in Table 6. In Table7 we give the nal predicted mean values and ranking of the suspected components according to s= x. Due to the negative values computed for the normal mode, R2; R5; R6; C1 and C2 are ruled out.... ..."

### Table 4: Ranking of the suspected components and their predicted values for C1 = 1nF .

"... In PAGE 15: ...Table 4: Ranking of the suspected components and their predicted values for C1 = 1nF . Table4 gives the composite average values for the four circuit operation modes. Here again is the coe cient of variation s= x of C1 by an order of magnitude lower than the other values which con rms previously stated diagnosis.... ..."

### Table 3: Computed values of suspected components for C1 = 1nF

1993

"... In PAGE 12: ... Table 2 gives the measured values of gain and phase at selected frequencies of 100 Hz and 200 Hz for the four possible modes of circuit operation with C1 = 1 nF. Table3 presents the computed mean values x of suspected faulty components together with the corresponding coe cients of variation V for the given operation modes. Modes Frequency Gain Phase Gain di erence Phase di erence [Hz] [dB] [0] [dB] [0] normal 100 -13.... ..."

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### Table 1 summarizes the results of diagnosis. The example illustrates the basic idea of using CLP( lt;) to diagnoses soft faults. In our experiments with lters we do not take parameter tolerances into account. However, instead of one we take several measurements, and combine the computed values of suspected faulty components in order to rank them in decreasing likelihood of beeing faulty. Suspected Computed component value [ ] R1 250 500 R2 4000 8000

1993

"... In PAGE 6: ... However, instead of one we take several measurements, and combine the computed values of suspected faulty components in order to rank them in decreasing likelihood of beeing faulty. Suspected Computed component value [ ] R1 250 500 R2 4000 8000 Table1 : Computed actual values of potentially faulty resistors CLP( lt;) is resticted to systems of linear equations and inequalities. Non-linear con- straints are accepted but not resolved they are just delayed until (if) they eventually become linear.... In PAGE 15: ...3 Example 3: R5 faulty In the last experiment, we inserted a deviation fault of R5 (100 k instead of correct 10 k ) into the circuit. Like in previous examples, measurement results (Table 9), computed values of the assumed components for the four modes of operation ( Table1 0), and computed average values (Table 11) are given.... In PAGE 15: ...3 Example 3: R5 faulty In the last experiment, we inserted a deviation fault of R5 (100 k instead of correct 10 k ) into the circuit. Like in previous examples, measurement results (Table 9), computed values of the assumed components for the four modes of operation (Table 10), and computed average values ( Table1 1) are given.... In PAGE 16: ...423 x x x x 1005 0.002 Table1 0: Computed values of suspected components for R5 = 100k The di erence between the measurement results and the simulated values of gain and phase in the all-test mode indicate that the faulty element is one of the resistors. Computed values of V in the rst stage test mode con rm that R1 can be eliminated as well as R3.... In PAGE 17: ...0698 R3 997 3.371 Table1 1: Average values for R5 = 100k 5 Conclusion Achieved results show that the model-based diagnosis with CLP( lt;) can be used in auto- matic fault isolation of active analog lters designed in accordance with the proposed DFT methodology [9]. Although the results refer to a narrow problem domain, the applicability of the model-based approach to more general cases is by no means excluded.... ..."

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### Table 3: Computed values of suspected components for di erent test modes for C1 = 1nF . Inspection of coe cients of variation s= x for the normal mode immediately points to C1 as being faulty because it has minimum s= x and its value di ers by an order of magnitude from the others. Although the result is achieved in the rst step we proceed by the other three in order to illustrate the process of deriving the diagnosis as well as to con rm the initial result.

"... In PAGE 14: ... Table 2 gives the measured values of gain and phase at selected frequencies of 100 Hz and 200 Hz for the four possible modes of circuit operation with C1 = 1 nF. Table3 presents the computed mean values x of suspected faulty components together with the corresponding coe cients of variation s= x for the given operation modes. Modes Frequency Gain Phase Gain di erence Phase di erence [Hz] [dB] [0] [dB] [0] normal 100 -13.... ..."

### Table 10: Computed values of suspected components for R5 = 100k The di erence between the measurement results and the simulated values of gain and phase in the all-test mode indicate that the faulty element is one of the resistors. Computed values of V in the rst stage test mode con rm that R1 can be eliminated as well as R3. R2; R5, and R6 remain candidates also after inspecting the values in Table 11. The explanation can be found in the expression of the transfer function of the lter circuit, where the three resistors always appear in the same subexpression.

1993

"... In PAGE 15: ...3 Example 3: R5 faulty In the last experiment, we inserted a deviation fault of R5 (100 k instead of correct 10 k ) into the circuit. Like in previous examples, measurement results (Table 9), computed values of the assumed components for the four modes of operation ( Table10 ), and computed average values (Table 11) are given.... ..."

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### Table 1 summarizes the results of diagnosis. The example illustrates the basic idea of using CLP( lt;) to diagnose soft faults: each individual component is considered in turn and its actual value is computed from the measurements. If tolerances are taken into account then the computed value is a range, otherwise it is a single value. In our experiments with lters we do not take parameter tolerances into account. However, instead of one we take several measurements, and combine the computed values of suspected faulty components in order to rank them in decreasing likelihood of beeing faulty. CLP( lt;) is resticted to systems of linear equations and inequalities. Non-linear con- straints are accepted but not resolved | they are just delayed until (if) they eventually become linear. In order to make CLP( lt;) applicable to the diagnosis of analog circuits we have to make some assumptions. We have to restrict models to linear or piecewise linear circuits, and assume that there are no multiple faults (a single fault assumption). In the next section we give a formal argument which shows that under those assumptions, linear CLP( lt;) is indeed su cient for the diagnosis of single soft faults.

"... In PAGE 7: ...001, R1 = ab(500.0), R2 = ok Suspected Computed component value [ ] R1 250{500 R2 4000{8000 Table1 : Computed resistance ranges of potentially faulty resistors.... In PAGE 17: ...3 Example 3: R5 100k instead of 10k In the last experiment, we inserted a deviation fault of R5 (100 k instead of correct 10 k ) into the circuit. Like in previous examples, measurement results (Table 9), computed values of the suspected components for the four modes of operation ( Table1 0), and the computed mean values and ranking of the suspected components are given (Table 11). From the computed values of components for the normal mode, R3 can be eliminated due to its negative value.... In PAGE 18: ...423 x x x x 1005 0.002 Table1 0: Computed values of components for di erent test modes for R5 = 100k . values of gain and phase in the all-test mode indicate that the faulty element is one of the resistors.... In PAGE 19: ...0698 R3 997 3.371 Table1 1: Ranking of the suspected components and their predicted values for R5 = 100k . CLP( lt;) and model-based diagnosis techniques.... ..."

### Table 9: Measured gain and phase for the fault R5 = 100k .

"... In PAGE 17: ...3 Example 3: R5 100k instead of 10k In the last experiment, we inserted a deviation fault of R5 (100 k instead of correct 10 k ) into the circuit. Like in previous examples, measurement results ( Table9 ), computed values of the suspected components for the four modes of operation (Table 10), and the computed mean values and ranking of the suspected components are given (Table 11). From the computed values of components for the normal mode, R3 can be eliminated due to its negative value.... ..."