A Dictionary Approach to Fault Diagnosis of Analog Circuits

Constantin Viorel Marin, Florin Constantinescu, Miruna Nitescu Departament of Electrical Engineering POLITEHNICA University Bucharest

Bucharest, Romania e-mail: viorel.marin@lce.pub.ro

Abstract—This paper deals with dictionary approach for single soft faults diagnosis in analog circuits. A method for selecting the optimum set of test frequencies is proposed. The first step uses circuit sensitivity analysis to choose the optimum frequency ranges. Then the optimum set of frequencies is selected using the entropy criterion.

Keywords-analog fault diagnosis , fault dictionary, optimum frequency selection

I. INTRODUCTION

The development of new circuits of high complexity raised the interest for diagnose techniques of faults of analog circuits that do not need direct access to individual elements of the circuit. There are two approaches: simulation after test (SAT) and simulation before test (SBT). In SAT, the results of measurements together with circuit topology and nominal values of components, considered known, are used in a computer program to determine the actual values of circuit elements. The components whose values overpass the tolerance limits imposed by design are considered faulty. So, the faulty components are then identified and located by comparison with their nominal values. Another approach in analog fault diagnosis is based on the concept of simulation of faults, called simulation before test (SBT), which leads to the generation of fault dictionaries. In this approach, the values of the various components are changed so as to simulate the failure of components and nodes voltages are calculated. Simulation results are then used to achieve fault dictionaries used to diagnose circuit faults. The main disadvantage of this approach is the large number of circuit simulations required to consider all possible failures of components. An important advantage of the method is that it can be used in any field (time, frequency or parametric) and for any circuit, linear or nonlinear.

Analog fault diagnosis is an important method for improving manufacturing technologies of integrated circuits. Recent works [1-8] deal with SBT techniques in diagnosing faults in analog circuits. The papers [1, 2] propose methods to select the frequencies of sinusoidal test signals based on sensitivity analysis for soft single fault diagnosis. The papers [3, 4, 5] propose methods to improve the selection of the set of test points for hard single faults diagnosis in fault dictionary approach. In [3] and [4] the measurements associated with faults and test points are used to build the fault dictionary.

Then, the test point is chosen as corresponding to the minimum entropy index in the coded fault table. In [6] and [7] some methods for test frequency selection in hard single fault diagnosis are proposed. An evolutionary method that uses entropy index to define a fitness function and employs genetic programming is proposed in [6], while a method based on simulated annealing with fuzzy objective function is developed in [7]. A method to improve an existing multi-frequency method where the gain signatures of faults and frequencies are used to realize the fault dictionary is presented in [8].

This paper proposes a new method for selecting the frequencies of sinusoidal test signals in fault dictionary techniques for single soft faults diagnosis in analog circuits. The method starts with the optimum frequency ranges determination based on circuit sensitivity analysis. Then the optimum set of frequencies is selected using the entropy criterion.

II. SELECTION OF OPTIMUM TEST FREQUENCIES

Soft or parametric faults are defined as a variation of parameter values of components that lead to abnormal functioning of the circuit. In [4] soft faults are considered as components whose parameter values are greater or less than their nominal values with an order of magnitude.

Identifying the most efficient test nodes is very important in order to build a fault dictionary, able to solve the problem of analog fault diagnosis in a reasonable time. In order to optimize the fault dictionary, the following criteria have to be considered:

- reducing the number of test nodes, because each of them requires measurements;

- selection of the characteristics to be extracted taking into account the impact on the volume of calculation determined by extraction algorithms.

The sensitivity analysis can be used to assess the ability of a stimulus to highlight a fault, by the spread of its effect to the test node. This capacity depends on type of stimulus, the location and typology of the fault, the circuit topology and selected test nodes. The sensitivity analysis could be used for the selection of the test frequency set of sinusoidal stimuli leading to a high computation efficiency.

The detectability is defined as the measure of capacity to

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identify and locate the fault. For the optimum test frequency range determination, two heuristic rules could be employed [1]: 1. A larger sensitivity of the signal in a test node with respect to a certain circuit parameter, leads to a higher detectability of that faulty element using measurements in that node. 2. A larger difference between two sensitivities of the signal in a test node leads to a greater detectability of the faulty element. These two rules are the intuitive concepts that allow the faults to be distinguished. The sensitivity analysis can determine the optimum frequency ranges, but can not give the optimum frequencies.

This paper proposes a new method for the selection of optimum test frequencies based on Shannon information theory [9]. The Shannon information content of an outcome x is inversely proportional with probability, measured in bits:

))(1(log)( 2 xPxh = (1)

For a source NxxxX ,...,, 21= that contains Nx symbols, the average information per symbol, named the entropy of the source, measured in bits, is:

))(1(log)()( 21

i

N

ii xPxPXH ∑

=

= (2)

Sometimes is convenient to consider )( pH instead of

H(X), where p is the probability vector Npp ,...,1 :

)1(log)( 21

i

N

ii pppH ∑

=

= (3)

Another name for the entropy H of a discrete random variable X is the uncertainty of X because the entropy is a measure of the amount of uncertainty associated with the value of X.

The minimization of the entropy is used as a criterion to select the measurements that contain the least amount of uncertainty. In the ideal case, the measurements at a single frequency are different for every fault, so all the faults are univocally identified, the probabilities for every fault are 1 and, as a consequence, the entropy is null. This event is rarely achieved. Usually, several frequencies have to be employed. If an inclusive algorithm is used, the frequency that corresponds to the minimum entropy is added. The stop test for the entropy minimization is null entropy value or two equal consecutive entropy values. If an exclusive algorithm is used, the frequency that corresponds to the maximum entropy is excluded.

The node selection technique differs from the frequency selection technique due to the fact that nodes are given by the circuit structure and the test frequencies are found using a search engine.

In this paper is presented an inclusive algorithm for building a fault dictionary. This algorithm has the following steps:

1. Finding an optimal range of the test frequencies using the sensitivity computation.

2. Completion of the fault table using circuit simulations for a set of test frequencies in the optimal range.

3. Estimation of the blind range due to deviations from nominal values of the circuit parameters within the tolerance limits.

4. Completion of the fault table including determination of the ambiguity sets for each test frequency.

5. Finding the test frequency order leading to an univocal identification of a maximum number of faults, based on the minimum entropy criterion.

6. Determination of the new ambiguity sets for each test frequency through elimination of the faults which were univocally identified at the previous test frequency.

The algorithm sweeps the test frequency set and stops if all faults are univocally identified (null entropy) or if no remaining fault can be univocally identified (minimum entropy remains the same).

III. EXAMPLE The circuit under test (CUT) is presented in figure 1. This

circuit represents a biquad filter which was described and analyzed with different methods in [1, 2, 7]. The faults are considered as components whose parameter values are greater or less than their nominal values with one order of magnitude. The problem is to determine the fault dictionary of this circuit.

Fig. 1. The circuit under test (CUT)

The nominal values of resistances and capacities are: Ω= kR 101 , Ω= kR 102 , Ω= kR 103 , Ω= kR 104 ,

Ω= kR 105 , Ω= kR 1006, Ω= kR 1007 , nFC 11 = , nFC 12 =

and the tolerances are 2% for resistances and 5% for capacities. Seven fault classes including three ambiguity groups are identified for this circuit [7]: class 1: 6R and/or 7R faulty;

class 2: 3R and/or 1C faulty; class 3: 4R and/or 2C faulty;

class 4: 5R faulty; class 5: 2R faulty; class 6 1R faulty; class 7 no faulty component. Each one of the first three classes defines an ambiguity group. The CUT was simulated with the

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program SAPWIN which computes the voltage gain )(sH . Its modulus is computed with MAPLE.

Fig. 2. The difference 1CMS - 2C

MS vs. frequency

The sensitivity vs. frequency is determined for each component from the fault list. The sensitivity functions used in this paper are the derivatives of network function module M=|H(s)| in symbolic form, with respect to each potential faulty parameter.

The optimum frequency range is chosen employing the heuristic rules presented in the third section. In order to use the

first rule we observe that for this circuit 1CMS and 2C

MS have the largest values. In order to use the second rule too, we take into

account the frequency characteristic of 1CMS - 2C

MS (Fig. 2) , appreciating that the optimum frequency range can be chosen as kHzfopt )355( ÷∈ .

IV. RESULTS The CUT is driven by the sinusoidal input stimulus of 4 V

[5] and the test frequencies were chosen heuristically as linearly spaced in the predetermined optimum range kHzkf k 5*= , 7,,2,1=k . The faults in the fault list are simulated with SPICE and the corresponding values of voltages were determined for the exit node N8.

The circuit parameter tolerances must be taken into account. For a resistor with nominal value R=1KΩ and a tolerance of 1% for its resistance it follows that RЄ[990,1010]Ω. Using the Monte-Carlo simulation [5, 6, 7] the worst case for the modulus (maximum deviation with respect to the nominal value) of the voltage in N8 can be computed taking into account the tolerances of all circuit parameters. The blind range DY, measured in Volts, is defined as the difference between the worst case modulus and the nominal modulus of the voltage in N8. Ymax is the worst case modulus of the voltage in N8 measured in percentage of the nominal value. The Monte Carlo simulations have been performed wit SPICE, the results DY and Ymax being written in TABLE 1.

The ambiguity sets are built for each frequency as follows:

• If the difference between two values is smaller than the maximum deviation DY, the corresponding faults are contained in the same ambiguity set.

• If the difference between several consecutive values is smaller than DY, the corresponding faults form a chain of faults and are contained in the same ambiguity set, despite the fact that the difference between nonconsecutive links of the chain could be larger than DY.

• If a medium fault of the chain is eliminated, the remaining ones could cease to belong to the same ambiguity set. This is why the ambiguity sets have to be refreshed after every step of the algorithm. As a consequence, the redundant measurements are avoided.

Each ambiguity set jiAS , is identified by two integer indices: the first one is the frequency number and the second one represents the number of the ambiguity set in the fault list for the given frequency.

Definition of the ambiguity sets for each frequency (unlike [6] and [7], in which the worst case is computed for all test frequencies) leads to ambiguity sets containing a smaller number of faults. In principle these faults could be identified using a smaller number of measurements.

Considering the hypothesis of equi-probability for the events contained in an ambiguity set, the probability ip is computed as the ratio between the number of the events contained in the ambiguity set and the number of faults. The single events that can univocally identify a fault have the value of certitude, the probability ip having the value 1 and the entropy being null.

In the first step the entropy is computed for all test frequencies. Consider f=10 KHz. For this frequency are identified: - a set containing four faults: (AS2,2)-F1, F7, F11, F14, - six sets conaining two faults: (AS2,1)-F0, F3, (AS2,5)-

F5, F9,(AS2,6)-F6, F10,(AS2,7)-F8, F12,(AS2,9)-F15, F18,(AS2,1)-F16, F17,

- three sets containig one fault: (AS2,3)-F2,( AS2,4)-F4 si (AS2,8)- F13 Using (3) it follows:

995.5)1(log)1(*3)2/19(log)19/2(*12)4/19(log)19/4(*4 222

=++=H

According to the entropy values in Table 1, this frequency has the lowest entropy, so it is chosen as the first frequency. Three faults F2 AS2,3, F4 AS2,4 and F13 AS2,8 are uniquely identified. The remained 16 faults are gathered in seven ambiguity sets: AS2,1=F0, F3, AS2,2=F1, F7, F11, F14, AS2,5=F5, F9, AS2,6=F6, F10, AS2,7=F8, F12, AS2,9=F15, F18, AS2,10=F16, F17.

The identified faults, F2, F4 and F13 are erased from the coded table, the new number of faults is 16 and the result is presented in Table 2. The second frequency is chosen 35 kHz. Three faults F0 (S2,1; S7,1), F3( S2,1; S7,4) and F14

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TABLE 1

Frequency [kHz] DY [V]

Ymax [%]

5 .2289

105.82

10 .3498 109.8

15 .3517

112.47

20 .3081 115.19

25 .2523

117.51

30 .1961 118.68

35 .1525

119.29 Faults/AS AS1,j AS2,j AS3,j AS4,j AS5,j AS6,j AS7,j

F0 (no fault) AS1,1 AS2,1 AS3,1 AS4,1 AS5,1 AS6,1 AS7,1 F1(R1*10) AS1,2 AS2,2 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F2(R1/10) AS1,3 AS2,3 AS3,3 AS4,3 AS5,3 AS6,3 AS7,3

F3(R2*10) AS1,1 AS2,1 AS3,1 AS4,4 AS5,4 AS6,4 AS7,4 F4(R2/10) AS1,4 AS2,4 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F5(R3*10) AS1,5 AS2,5 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F6(R3/10) AS1,1 AS2,6 AS3,4 AS4,1 AS5,1 AS6,5 AS7,5 F7(R4*10) AS1,6 AS2,2 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F8(R4/10) AS1,1/

AS1,10 AS2,7 AS3,5 AS4,5 AS5,5 AS6,6 AS7,6

F9(C1*10) AS1,5 AS2,5 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F10(C1/10) AS1,1 AS2,6 AS3,4 AS4,1 AS5,1 AS6,5 AS7,5 F11(C2*10) AS1,6 AS2,2 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F12(C2/10) AS1,1/

AS1,10 AS2,7 AS3,5 AS4,5 AS5,5 AS6,6 AS7,6

F13(R5*10) AS1,7 AS2,8 AS3,1 AS4,1 AS5,1 AS6,1 AS7,1

F14 (R5/10) AS1,2 AS2,2 AS3,2 AS4,2 AS5,2 AS6,2 AS7,7 F15(R6*10 ) AS1,8 AS2,9 AS3,6 AS4,6 AS5,6 AS6,5 AS7,8 F16(R6/10 ) AS1,9 AS2,10 AS3,7 AS4,7 AS5,7 AS6,7 AS7,9

F17(R7*10) AS1,9 AS2,10 AS3,7 AS4,7 AS5,7 AS6,7 AS7,9 F18(R7/10) AS1,8 AS2,9 AS3,6 AS4,6 AS5,6 AS6,5 AS7,8

H 6.569 5.995 7.711 7.659 7.659 7.134 6.569

TABLE 2

Frequency [kHz] DY [V]

Ymax [%]

5 .2289

105.82

15 .3517

112.47

20 .3081 115.19

25 .2523

117.51

30 .1961 118.68

35 .1525

119.29 Faults/AS AS1,j AS3,j AS4,j AS5,j AS6,j AS7,j

F0(nofault) AS1,1 AS3,1 AS4,1 AS5,1 AS6,1 AS7,1 F1(R1*10) AS1,2 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F3(R2*10) AS1,1 AS3,1 AS4,4 AS5,4 AS6,4 AS7,4 F5(R3*10) AS1,5 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F6(R3/10) AS1,1 AS3,4 AS4,1 AS5,1 AS6,5 AS7,5 F7(R4*10) AS1,6 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F8(R4/10) AS1,1/A

S1,10 AS3,5 AS4,5 AS5,5 AS6,6 AS7,6

F9(C1*10) AS1,5 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F10(C1/10) AS1,1 AS3,4 AS4,1 AS5,1 AS6,5 AS7,5 F11(C2*10) AS1,6 AS3,2 AS4,2 AS5,2 AS6,2 AS7,2 F12(C2/10) AS1,1/A

S1,10 AS3,5 AS4,5 AS5,5 AS6,6 AS7,6

F14 (R5/10) AS1,2 AS3,2 AS4,2 AS5,2 AS6,2 AS7,7 F15(R6*10 ) AS1,8 AS3,6 AS4,6 AS5,6 AS6,5 AS7,8 F16(R6/10 ) AS1,9 AS3,7 AS4,7 AS5,7 AS6,7 AS7,9

F17(R7*10) AS1,9 AS3,7 AS4,7 AS5,7 AS6,7 AS7,9 F18(R7/10) AS1,8 AS3,6 AS4,6 AS5,6 AS6,5 AS7,8

H 6.933- 6.933 6.792 6.792 6.183 5.621-----

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(S2,2; S7,7) are uniquely identified and eliminated. Similarly, in the next stage, corresponding to the frequency of 5 KHz, the fault F1 (S2, 2, S7, 2; S1, 2) is eliminated. In the last stage the entropy for all remaining frequencies (15, 20, 25, and 30 kHz) has the same value and the algorithm stops. Twelve faults, grouped in six ambiguity sets cannot be eliminated. Only the ambiguity groups they belong to can be identified. These ambiguity sets are: F7, F11, F5, F9, F6, F10, F8, F12, F15, F18, F16, F17.

In this example, faults are unequivocally identified through simulations at three frequencies that are determined in four iterations. In [1], that uses a global sensitivity approach and randomized algorithms to solve the same problem, seven test frequencies from the neighborhood of the cut-off frequency ( kHzf offcut 9,15=− ) have been selected.

In [7], that uses fuzzy fitness function to solve the same problem, three frequencies are employed too, but a number of 460 iterations are performed.

V. CONCLUSION This paper proposes an efficient method for the optimal

frequencies selection in fault diagnosis of linear analog circuits. This method is based on the sensitivity analysis and the entropy criterion. If the algorithm terminates with a non-zero entropy value, some faults cannot be unequivocally determined; in this case only the ambiguity sets these faults belong to can be determined in a simple manner.

An example is given, the computational effort being compared with that needed by other two known methods. Taking into account the number of AC analyses needed and the iterations number used to identify the faults, it follows that our algorithm is more efficient than those in [1] and [7].

ACKNOWLEDGMENT

The authors would like to acknowledge the financial support of the project ID_1698, contract nr. 683/2009, CNCSIS – UEFISCSU, Romania.

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