close

Вход

Забыли?

вход по аккаунту

?

Minimal transient modes for faults detection in analogue VLSI circuits.

код для вставкиСкачать
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
Документ
Категория
Без категории
Просмотров
4
Размер файла
1 760 Кб
Теги
faults, mode, transiente, detection, circuits, vlsi, minimax, analogues
1/--страниц
Пожаловаться на содержимое документа