2. The Method MINIMAL TRANSIENT MODES FOR FAULTS DETECTION IN ANALOGUE VLSI CIRCUITS KADIM H.J. School of Engineering, Liverpool JM University, Byrom Street Liverpool L3 3AF, England, UK. E-mail: h.j.kadim@livjm.ac.uk Abstract. The method introduced here uses an investigation of the dominant natural mode to identify possible abnormalities in analogue circuits due to faults and parameter variations, and determining the reliability of circuits when operating in the presence of faults. 1. Introduction Testing and analysis of analogue circuits become extremely difficult due to the complexity inherited in their design. Techniques for testing analogue circuits were presented in [1-6], where it was shown and extensively applied that simulation, sensitivity, and other numerical and analytical methods can help in both analogue design and in analogue circuit testing. Due to the cost involved, most of the proposed methods of synthesis-for-testability and design-fortestability for identifying faults, parametric [5], [7, 8] and catastrophic [5, 6], in analogue circuits targeted small size circuits. Furthermore, these methods share a common ground, that is, the inclusion of all transient responses, which comprise the overall behaviour of the circuit under test. Transient response techniques involve applying simple waveforms into inputs and observing output responses. The approach proposed in [9] attempts to calculate or directly measure from the macro’s transient response certain characteristics of the circuit and then use multi-variable discriminant analysis to determine faults. Other researchers have concentrated on determining inputs which will improve diagnosability. If the fault spectrum for a circuit can be obtained, transient response analysis represents a possible method for testing complex circuits using pulse input stimuli [10]. Other work has taken this further [11]. Instead of testing individual macros, the complete circuit is tested as a unit. Particular test stimuli were applied to inputs, the characteristics of the transient test vectors being chosen to excite a transient response from the unit under test capable of propagation in both the analogue and the digital part (mixed testing). The transient response of a faulty device is subjected to correlation with the response of a fault free circuit to extract the correlation function. Analysis of these correlation functions is a means of indicating any disparity between faulty and fault free circuits. Other techniques use the frequency domain by comparing faulty and fault free behaviours of functional output voltage and power supply current measurements [12], [13]. Most of the existing techniques consider a circuit’s transient response, represented by all possible operational modes, which eventually increases the range of frequencies over which the circuit is to be tested. This paper focuses on reducing the number of modes when testing and analysing analogue circuits. 82 An analogue circuit’s behaviour can be represented by a number of operational modes (i.e. natural modes). To ensure the compliance of the circuit to a prescribed specification, it is important to test the circuit against the performance of its operational modes in the presence of soft and hard faults. One way to achieve this is to apply faults to the circuit and analyse the response of the modes accordingly. For higher order analogue circuits with a greater number of operational modes, it is costly in terms of time and effort to include all modes in the analysis. Some of the operational modes may be masked by other modes. The transient response of the latter may exhibit abnormalities in response to changes encountered in the behaviour of the masked modes, and therefore fewer numbers of modes can be considered when testing circuits. The method proposed in this paper considers the analysis of analogue integrated circuits in the time domain and the frequency domain based on investigations into the dominant pole characteristics. It also investigates the controllability and observability of an analogue circuit’s operational modes using state space concepts. An analogue circuit can be represented by its transfer function. This can reveal necessary information about the location of the circuit’s poles and zeros in the frequency domain and their contribution to the circuit’s behaviour in the time domain. From the transfer function a state space [14] representation can be obtained. Based on such a representation, further analysis can be carried out to determine the controllability and the observability of the natural modes. This information then determines the way the circuit is to be tested to identify possible faults. The circuit shown in Figure 1 is an example to investigate the characteristics of the dominant natural mode in the absence and presence of faults. Figure 1. Analogue circuit Using Hspice, the circuit has three zeros located in the complex plane [14] at -225.0812j, +225.0812j and -225.0812, and three poles located at -225.0812, -7.964-158.9515j and 7.964+158.9515j with the complex pair of poles representing the dominant transient. 2.1 The Dominant pole The influence of a particular pole (or pair of complex poles) on the response of a circuit is determined by the relative rate of decay of the transient term due to that pole, and the relative magnitude of the residue at the pole. The rate at which the transient term decays is determined by the real part of the pole. The step response of the poles of the circuit in Figure 1 is shown in Figure 2. R&I, 2003, ¹ 3 The controllability, Fc, and observability, Fo, matrices are given by 2 1.8 1.6 complex pair of poles s=-7.964r158.9515 Td Tr Ts 0.09 sec 0.288 sec 0.43 sec )c damping factor 0.05 1.4 1.2 1 )o 0.8 0.6 0.4 0.2 s=-225.0812 0 0 0.1 ª1 241 29171º «0 1 241 » «0 0 », 1 ¼ ¬ 7 º ª0 0 10 » « 7 7 9 u 10 «0 1 u 10 7 7». «¬0 65 u 10 1.4587 u 10 »¼ The ranks of the above matrices are as follows: rankFc = 3 0.2 0.3 Time (sec) 0.4 0.5 Figure 2 Exponential transient response terms Figure 2 shows that the response corresponding to e-225.08t dies away much faster than the response corresponding to e-(7.96±159)t. The graph also illustrates that the latter has the greater influence on the shape of the transient response. The term in the response due to the pole at s = -225.08 decays twenty eight times as fast as the term due to the poles at s=7.964±159. The residue at the pole at s=-225.08 is only 3.6% of that of the poles at s=-7.964±159 Because the contribution from the poles at s = -7.96±159 virtually masks the contribution from the pole at s=-225.08, they are described as providing the dominant transient behaviour of the circuit. The corresponding poles are called the dominant poles of the circuit. There are certain cases where the dominant pole could be cancelled by a zero. In this respect, the actual dominant pole of the circuit will be one of the remaining poles that has the smallest real part. rankFo = 2 The rank of the observability matrix proves that the mode of operation (natural mode), represented by the pole located at s=-225.0812 is unobservable. Therefore, the time that might be spent on testing this mode of operation can be saved since it is impossible to measure some behavioural characteristics of this particular pole to identify abnormalities caused by it. Saving time and effort will eventually result in reducing cost. 3. Faults Applied The faults were divided into two categories, hard faults, which are represented by changes in component values (these include short and open circuit faults), and soft faults, which are represented by slight variations in component values. Some of the results obtained are shown in Figure 3, Figure 4 and Figure 6. Hard faults: These are represented by short and open circuit faults as well as moderate changes in component values that are applied to the circuit of Figure 1. The graphs in Figure 3 illustrate that the dominant-response term is significantly affected by such changes. 2.2. Stability, Controllability and Observability For a circuit to be stable, the transient response must eventually die away. This is only achieved if all poles of the circuit’s transfer function lie in the left-hand side of the splane where the real part of ‘s’ is negative [14], [15]. Because the poles of the circuit in Figure 1 are located to the left of the imaginary axis of the s-plane, its dynamic behaviour will be stable. However, the introduction of a fault (e.g. short circuit at C2, slight variation in the value of C2; see section 3), may result in shifting one of the poles to the right (positive real part), and the behaviour of the circuit becomes unstable. The poles and zeros of a given circuit can also aid in determining the controllability and observability of analogue circuits. For example the transfer function above shows that the zero at s=225.0812 cancels the pole located at the same location. This pole is no longer controllable since it is always cancelled by a zero. The state space representation of the transfer function is given by ª x 1' º « '» «x 2 » «x '3 » ¬ ¼ ª 200 28900 5.701100º ª x 1 º ª1º « 0 » «x 2 » «0» VIN 0 0 « 0 » « x » «0 » 0 0 ¬ ¼¬ 3 ¼ ¬ ¼ VOUT R&I, 2003, ¹ 3 ªx1 º 0 21700 5701900 «x 2 » «x » ¬ 3¼ Figure 3. Faulty and fault-free dominant transient-response for the circuit of Figure 1 Soft faults: The value of C2 in Figure 1 is slightly changed to 0.715 Farads. The effect of this change on the dominant response is illustrated in Figure 4. The dominant pole is shifted to the right half side of the s-plane, and this indicates that the dynamic behaviour of the circuit of Figure 1 is unstable as a result of a change in the value of C2. 83 x 106 12 4 Figure 6 shows no changes in the dominant-transient response, but abnormalities can be identified in the nondominant pole characteristics. This can be due to: (i) relative pole coupling [14], and (ii) component value approximations. It is therefore necessary that a minimum number of nondominant modes have to be considered, as illustrated in section 4. 2 4. Optimal number of Poles 0 For an undriven analogue circuit or system [14]: 10 8 6 -2 d (x) dt -4 -6 0 0.1 0.2 0.3 0.4 0.5 Time (secs) Figure 4 Faulty dominate-response term (C2 = 0.715) The controllability and observability matrices, after introducing the fault, are: )c )o 0 0 12 2j ª 0 º « 0 » 1 9j 739 909j « » «¬6 4j 608 1416j 30553 95761j»¼ Since the poles are available separately at the diagonal matrix, the state vector c is related to the diagonal state vector d by: However, There is a possibility that soft faults may not exhibit visible abnormality in the dominant response. As an example, consider the circuit shown in Figure 5. The value of R4 is increased by 1% (i.e. its new value is 0.934). R2=0.383 + R in ohms C in farads C2=1 R4=0.924 C4=1 + C3=1 Vin 0 ...... 0 º ªO1 «0 O » ....... 2 « » « 0» « » ¬ 0 ....... 0 O n ¼ u106 Since the rank of each of the above matrices is 3, the circuit variables are still controllable, but this time they are observable. C1= 1 The diagonal matrix of A has the eignvalues of a circuit listed on its main diagonal. » ¼ 1 R1=2.613 R3=109 + RL (1) x(t): state vector (n elements); A: system-interconnection matrix (n.n). ª1 171 264j 8686 3.6607jº «0 1 171 264j » «0 ¬ Ax Vout Figure 5. An analogue circuit for soft and hard faults 1 ª χ 1º « . » « » « . » (2) « . » « » «¬ χ n »¼ where w is a matrix with the eignvectors as columns; the elements of this matrix give the distribution of a pole to various states. ª d 1º « . » « » « . » « » ¬d n ¼ ª w11 ..... ..... w1n º « . . »» « « . . » « » ¬wn1 .... .... wnn ¼ With reference to Equation (2) a nonzero value associated with an eignvector indicates a dependency between the corresponding c and d, otherwise c and d are decoupled. If one pole is selected from one group of dependent poles, the overall number of poles required for circuit verification can be reduced. Illustration: Consider the circuit of Figure 5. Using H-spice, the circuit has 4 poles located in the complex plane at s1 = – 60.9-147j, s2 = –60.9+147j, s3 = -147-60.938j and s4 = – 147+60.938j, with the complex pair of poles –60.9±147j representing the dominant transient. Using Matlab, and from Equation (1) damping factor 0.37 ª F1 º « » d «F 2 » dt «F3 » « » ¬F 4 ¼ ª 294 159.15 º ª F1 º «159.15 0.006 » «F » « »« 2 » « 121.8 159.17» «F3 » « »« » 159.17 0 ¼ ¬F 4 ¼ ¬ (3) with the diagonal matrix of the circuit is given by: damping factor 0.9 0 0 0 º ª 147 60.94 » « 0 147 60.94 0 0 » « » « 0 0 60.9 147 j 0 » « 0 0 0 60.9 147 j¼ ¬ Figure 6. Transient responses for the circuit of Figure 4 84 R&I, 2003, ¹ 3 and the distribution of the poles of the circuit to the states is given by: ª d1 º «d » « 2» «d 3 » « » ¬d 4 ¼ ªD E «D * E * « «0 0 « 0 ¬0 0 0 J J * 0 º ª F1 º 0 »» ««F 2 »» N » «F 3 » »« » N * ¼ ¬F 4 ¼ (4) where ‘*’: conjugate;a = -1.7-0.7j; b = -1.3-1.3j; g = -0.77-0.05j; k= 0.25-0.7j x3 and x4, but abnormalities can be identified in the characteristics of the transient corresponding to x1 and x2. The state variable x4, in Equation (3), is not dependent on either x1 or x2. Hence, parameter variations which trigger abnormalities in x1 or x2 would not affect x4. Catastrophic variations: These are represented by short and open circuits. The graphs in figure 4 show the changes in the transient responses associated with the state variables x1 – x4. By examining Equation (4), the state vectors d1 and d2 are dependent on c1 and c2, whereas d3 and d4 are dependent on c3 and c4. Therefore, changes in the transient response of the state variables x1 and x2 due to parameter variations may not be identified in the transient associated with x3 and x4. By examining equations (3) and (4), the optimal number of poles required are 2, e.g. the poles associated with d4 and d1. 4 2 2 2,3 4. Simulation 3 3,4 Parametric variations I: The value of R2 in Figure 5 is increased by 5%. The effect of this change on the pole corresponding to x3 and x4 is shown in Figure 7. 2 1 4 1.2 faulty response (s1=-19; R2=0.44) 1 Figure 9. Transient responses for the circuit in Figure 1 (hard faults) fault free response s1,2=-60.9r147j [ = 0.37 0.8 4. Discussion of Results It is seen that the characteristics of the dominant mode are significantly affected by the introduction of hard faults and moderately or slightly affected by soft faults. There is a possibility that soft faults may not exhibit visible abnormality in the dominant response (e.g. slight changes in R4 of Figure 5). The possible causes can be due to: (a) relative pole coupling, and (b) component value approximations. 0.6 faulty response (s2=-86.9; R2=0.4) 0.4 0.2 0 0 fault free response S3,4=-147r60.94j [ = 0.9 0.1 0.2 0.3 Time (sec) 0.4 0.5 Figure 7. Transient-responses for the circuit of Figure 5 1.2 faulty response s1,2=-60.9r147j 1 faulty response (s3=-86.9; R4=1) 0.8 fault free response s1,2=-60.9r147j 0.6 faulty response (s4=-207; R4=1) 0.4 0.2 0 0 fault free response for s3, s4 0.1 0.2 0.3 Time (sec) 0.4 0.5 Figure 8 Transient responses for the circuit in Figure 5 (parametric faults) Parametric variations II: The value of R4 is increased by 9%. Figure 3 shows no changes in the transient corresponding to R&I, 2003, ¹ 3 The transfer function in section 2 shows that one of the poles can be cancelled by one of the zeros and the consequence of this is that the natural mode associated with this pole is unobservable. This information can assist in identifying the possible conditions (e.g. frequency range, type of inputs, operation in steady state) under which circuits with such flaws could operate satisfactorily. In terms of testing, such information can help in avoiding unnecessary examinations such as determining the response of uncontrollable or unobservable modes. The dominant mode can aid in determining the possibility of whether an analogue circuit can operate in the presence of faults. Illustration: Figure 8 shows the transient responses when faults occur. For instance, a slight variation in the value of R4 affects the non-dominant response (e.g. the response of the pole at s=-147±60.94j). This pole, before the introduction of the fault, dies away at »0.028 sec and only slightly faster after the change in the value of R4. The fault does not affect the dominant response. This indicates that the circuit may exhibit malfunction within 0.028 sec of the start. If the circuit’s specifications permit such abnormalities within this time, then the circuit can operate in the presence of this fault. If, however, the effect of the fault introduced shifts the nondominant pole closer to the dominant pole, their characteristics may overlap. In this case, the range of frequencies over which the circuit is to be tested has overlapping frequency ranges 85 given by the dominant and non-dominant transient responses. Assuming that the dominant response is unaffected by the fault, it is possible to operate the circuit within the range of frequencies of the dominant transient response except for the overlapped frequencies (see Figure 8; faulty response represented by s=-86.9). If the dominant transient is directly affected by the fault introduced, it is important to determine the new characteristics of this response. If the variation in the characteristics is within a prescribed limit permitted by the specifications, then the circuit could be kept in operation, but with a lower performance. 5. Conclusion With reference to hard faults, the dominant-mode response can identify abnormalities in circuits’ behaviour. In the case of a circuit that experiences soft faults, the performance of the dominant mode is mainly influenced by the relative pole coupling. In such cases it is advantageous to include a relative pole in the analysis. This may increase the range of frequencies over which the circuit is to be tested, but it is limited to specific cases and the number of poles chosen is minimal. Finally, the dominant natural mode can provide a guide as to whether circuits can operate in the presence of faults. References: 1. Abdeerrahman A., Cerny E. Worst Tolerance Analysis and CLP-based Multifrequency Test Generation for Analogue Circuits”, IEEE Trans. Computer-Aided Design, Vol. 18, Mrach 1999. Р. 332-345. 2. Chao C., Lin H., Milor L. Optimal Testing of VLSI Analogue Circuits // IEEE Trans. Computer-Aided 86 Design, Vol.16, January 1997. Р. 58-77. 3. Devarayanadurg G., Soma M. Analytical Fault Modelling and Static Test Generation for Analogue Ics // Proc. IEEE ICCAD, 1994. Р. 44-47. 4. Wey C.-L. Built-in-self-test Structure for Analogue Circuit Fault Diagnosis / / IEEE Trans. Computer Aided Design, Vol. 39, no. 3, 1990. Р. 517521. 5. Lindermeir W., Graeb H., Antreich K. Analogue Testing by Characteristic Observation Inference // IEEE Trans. ComputerAided Design, Vol.18, 1999. Р. 1353-1368. 6. Povazanec J., Volek T., Taylor G.E. Analogue Test in Frequency and Time Domain”, International MixedSignal Testing Workshop, France 1995. P. 6671. 7. Nagi N., Chatterjee A., Balivada A., Abraham J. Fault-based Automatic Test Generator for Linear Analogue Circuits”, Proceedings of ICCAD, 1993, pp. 88-91. 8. Abderrahman A.,Cerny E., Kaminska B. Effective Test Generation for Analogue Circuits // International Mixed Signal Testing Workshop, France, 1995. Р. 86-91. 9. Chin K. Functional Testing of Circuits and SMD Board with Limited Nodal Acess // ITC, 1989. Paper 5.4. 10. Wang F.L., Schrieber H. A Pragmatic Approach to Automatic Test Generation and Failure Isolation of Analogue Systems // IEEE Transactions on Circuits and Systems. Vol.Cas. 26 no. 7, 1979. P.584. 11. Al-Qutayri M., Evans P., Sphepherd C. Testing Mixed Analogue/digital Circuitry Using Transient Response Techniques”, University of Bath, UK, 1990. 12. Povazanec J., Volek T., Taylor G.E. Analogue Test in Frequency and Time Domain // International Mixed Signal Testing Workshop, France 1995. P. 66-71. 13. Bushhnell M. L., Agrawal V.D. Essentials of Electronics Testing for Digital Memory & Mixed-signal VLSI Circuits”, Kluwer Academic Pub., usa, 2000, pp. 441-453. 14. Borgan W. Modern Control Theory // Printice Hall, NJ. USA, 1991. Р. 249, 486-488. 15. Franklin G.F., Powell J., Abbas E. Feedback Control of Dynamic Systems // Addison-Wesley Publishing Company, Inc., USA, 1994. R&I, 2003, ¹ 3

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