Mathematical Problems in Engineering

Volume 2018, Article ID 9714206, 11 pages

https://doi.org/10.1155/2018/9714206

## Test Point Selection Method for Analog Circuit Fault Diagnosis Based on Similarity Coefficient

^{1}School of Instrumentation Science and Opto-Electronics Engineering, Beihang University, 37 Xueyuan Road, Beijing 100191, China^{2}School of Physics and Electronic Information, Huaibei Normal University, 100 Dongshan Road, Huaibei 235000, China

Correspondence should be addressed to Qingfeng Ma; moc.qq@efiqam

Received 17 September 2017; Revised 20 December 2017; Accepted 8 January 2018; Published 4 February 2018

Academic Editor: Konstantinos Karamanos

Copyright © 2018 Qingfeng Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The demand for testability analysis has increased with the integration densities and complexity of circuits. As an important part of testability analysis, the test point selection method needs to be researched in depth. A new similarity coefficient criterion is proposed to determine the fault isolation degree because output responses of a circuit with component tolerance are approximately subject to the normal distribution. Then, a new test point selection method is proposed based on the fault-pair similarity coefficient criterion information table. Simulation experiments are used to validate the accuracy of the proposed method in terms of the optimum test point set and fault isolation degree. The results show that the proposed method improves the performance of test point selection by comparing with the other reported methods.

#### 1. Introduction

Testability analysis is an important research topic for fault diagnosis in analog circuits. It performs test generation and test point selection in order to improve observability of a circuit under test (CUT). Test generation technique is used to gain the optimal test excitation signals for a CUT and test point selection technique is used to search the optimal test point set for a CUT. This paper mainly studies the test point selection method.

Generally, a CUT often includes many test points, but not every test point in the CUT is necessary or measurable. The optimal test point selection technique can reduce the fault dictionary dimension and save the computational cost by eliminating redundant and immeasurable test points. The total test cost of the CUT can be reduced greatly. Although the exhaustive search method can select a global minimum test point set, the papers [1–3] pointed out that the method is NP-hard and only suitable for small scale analog systems because of its high computational cost. Now, test point selection for analog circuit has become very difficult due to the high dense packages of chips and complex electronic system. The compromise approach is to find a local minimum test point set [1, 3].

There have been many researches about test point selection methods in the past years. A heuristic method for test points selection based on the concept of confidence levels was proposed in paper [4]. The concept of ambiguity sets and developed logical rules to select test points was proposed by Hochwald and Bastian [5]. Lin and Elcherif [6] proposed two heuristic methods based on the criteria proposed by Hochwald and Bastian. Van Spaandonk and Kevenaar [7] combined the decomposition method of system sensitivity matrix and an iterative algorithm to search a set of test points for analog circuits. Prasad and Babu [8] proposed four algorithms based on inclusive approaches and exclusive approach. An entropy index method was proposed to search for a local minimum test point set [1]. A genetic algorithm was proposed to determine the optimal test point set, which effectively enabled the results to avoid being trapped into local minimums [9]. In paper [10], the test point selection procedure was transformed into a graph node expanding procedure and utilized entropy of information to guide graph search, and the method is subsequently improved by Gao et al. [11]. A greedy randomized adaptive search algorithm was proposed to find the global optimal test point set [12]. A multiobjective fruit fly optimization algorithm was proposed to enhance the global test point selection ability [13]. All test point selection methods reported above are based on an integer-coded dictionary technique.

Generally, analog CUT output values change in intervals because an analog signal has continuity and parameters of analog component have tolerances. Hence, the responses of some analog circuit faults usually overlap each other. In order to discriminate these ambiguous faults and construct an integer-coded dictionary for analog circuit, the ambiguity set and the ambiguity gap (i.e., a diode drop 0.7 V) between ambiguity sets were introduced by Hochwald and Bastian firstly [5]. Subsequently, most of test point selection algorithms employ 0.7 V as the ambiguity gap, but it is proved by many practical test results that 0.7 V is not always effective and accurate [2, 3]. In paper [14], 0.2 V was chosen as the ambiguity gap and an accurate fault-pair Boolean table technique for the test point selection was proposed, which overcame the shortcoming of the traditional integer-coded table that only a part of the faults can be isolated. In paper [2], ambiguity gaps were determined by means and variances of the circuit responses because the tolerances of component parameters obey the normal distribution. According to the normal distribution characteristics, an overlapped area method was proposed to improve the accuracy of selecting the optimal test point set. Subsequently, the fault-pair isolation table technique was proposed by Zhao and He to guide the test point selection algorithm. The table consisted of the isolation probability of fault-pairs, which was gained by calculating ambiguity gap [3].

This paper proposes a similarity coefficient criterion to compute each fault-pair isolation capability. The larger the coefficient is, the higher the isolation probability of a fault-pair is. According to the trait of a similarity coefficient criterion, the fault-pair similarity coefficient criterion information table is constructed and a new test point selection method is proposed. And then the proposed method is validated by two filter benchmark circuits in terms of the optimal test point set and fault isolation degree. The results show that the new method is effective and accurate.

The remainder of this paper is arranged in the following order. Section 2 introduces the new method to construct the fault-pair similarity coefficient criterion information table. Then a new test point selection algorithm is proposed based on the table. The simulation details are described and the simulation results are discussed in Section 3. In the end, brief conclusions are summarized in Section 4.

Nomenclature of the paper is listed in Nomenclature.

#### 2. New Test Point Selection Method

##### 2.1. Similarity Coefficient Criterion

The core idea of the similarity coefficient criterion is to estimate the overlapping degree of a fault-pair. Assume that there are two faults and , and their samples follow the normal distribution. The probability density function curves are and , respectively. Three pairs of fault curves with different overlapping degree are shown in Figure 1. Because and are one-dimensional nonnegative real functions, they satisfy the condition of Cauchy-Schwarz inequality. The following inequality can be deduced according to Cauchy-Schwarz inequality [15]:The similarity coefficient criterion is defined as the following mathematical formula: so its interval is from 0 to 1. If the curves and almost coincide with each other as shown in Figure 1(a), and cannot be isolated and . If the curves and are partially overlapping as shown in Figure 1(b), faults and cannot be isolated partially and . If the curves and are not overlapping absolutely as shown in Figure 1(c), faults and can be isolated completely and . For example, assume that the mean and variance of fault are 3.32 and 0.24; the probability density function of fault is expressed asassume that the mean and variance of fault are 4.33 and 0.13:According to formula (2), the calculation of is 0.999. If is rounded to 2 decimal places, and the fault-pair can be isolated.