A Neural Network Diagnosis Approach for Analog Circuits.
This paper presents a neural network system for the diagnosis of analog circuits and shows how the performance of such a system can be affected by the choice of different techniques used by its submodules. In particular we discuss the influence of feature extraction techniques such as Fourier Transforms, Wavelets and Principal Component Analysis. The system uses several different power supplies and as many neural networks in parallel. Two different algorithms that can be used to combine the candidate sets produced by each network are also presented. The system is capable of diagnosing multiple faults even if trained on single ones.
Introduction
During the past years, the authors have been involved
in several projects on analog circuit diagnosis
and quality control of electrical components.
The aim of this paper is to present the diagnostic
system developed by them and to show how
the performance of such a system can be affected
by the choice of different techniques used by its
submodules.
The system is based on neural networks , and
is used for off-line diagnosis of analog circuits affected
by catastrophic multiple faults . It may handle
linear and nonlinear circuits in transient or
steady state behavior.
1.1. Analog circuit faults
Fault diagnosis of analog circuits is a complex
problem. Classical solutions require either a huge
amount of calculation if parameter identification
methods are used, or a great number of simulations
of faulty conditions if fault dictionary methods
are used [23, 24].
The faults in analog circuits may be catastrophic
faults, that cause a large and sudden variation of
the circuit parameter values, and deviation faults,
associated to slight variations of the circuit parameter
values from their nominal values [2]. Since
statistics have shown that 80-90% of analog circuit
faults are catastrophic [19], we chose to study
faults of this kind, such as short circuits and open
circuits between two terminals of a component.
In some applications (regulation systems, nuclear
plants, etc.) a prompt fault detection is
necessary to avoid damaging the controlled process
any further. A diagnostic system capable of
detecting a fault during its occurrence performs
what is called on-line diagnosis.
In many other applications (quality control
of circuits, post-mortem diagnosis of electronic
boards, etc.) the diagnostic procedure may be
applied in an off-line fashion, in the sense that
the diagnosed device need not be operative. In
these cases there is no strict time constraint and
even computationally intensive diagnostic sys-
tems, such as those based on parameter identification
or fault dictionary methods [23, 24], qualitative
reasoning [11, 13] model-based and rule-based
expert systems [7, 28] etc., may be used. In the
case of electric circuits, off-line diagnosis offers an
additional advantage: suitable voltage supply configurations
may be chosen in order to maximize
the observability of the faults [10].
1.2. Diagnosis as pattern recognition
Classically, a pattern recognition system is composed
of three modules [12]. A transducer acquires
data on a physical device and passes them
to a feature extractor whose purpose is to reduce
the data by computing certain features (or prop-
erties). These features will be used by a classifier
to make a final decision on the state of the device.
A circuit diagnostic system is a particular pattern
recognition system, in which the physical device
is an analog circuit, and the state that must
be recognized is the set of faulty components. In
particular, in the diagnostic system we have de-
veloped, the classifier is a neural network.
This type of diagnostic system offers some advantage
over other classical diagnostic methods.
Rule-based systems. These diagnostic systems
use compiled sets of rules to associate a symptom
to its cause. On the contrary, a neural
network automatically derives the symptom-
cause correspondence during the training, and
does not require an explicit formalization. It
is well known that this formalization is the
bottleneck of rule-based system technology.
Note that there is a small price to pay for this.
A rule-based system has a symbolic-heuristic
approach to diagnosis and is generally able to
justify its deduction from the rules used to
compute the diagnosis. A neural network, on
the contrary, has a numerical-algorithmic approach
and the knowledge is implicitly memorized
in the weights of its synapses. Thus, to
justify its deduction a neural network requires
additional rule extraction techniques [9].
Model-based systems. These diagnostic systems
usually require the complete knowledge
of the circuit scheme and a model of its be-
havior. Using neural networks it is possible
to avoid the problems connected with the calculation
of circuit parameters and in general
to the modeling.
Fault dictionary method. This method can be
used to identify only those faults whose signature
has been previously computed and added
to the dictionary. Neural networks on the
contrary - as reported in several works -
may be able to recognize fault configurations
not explicitly included in the training set. In
[14, 21, 31] neural networks trained to recognize
single faults are successfully used to diagnose
multiple faults. In [36] neural networks
accurately classify previously unseen fault signatures
belonging to a deviation fault class
known by a few samples.
There have been several works where neural
networks have been compared with other pattern
classifiers in diagnosis applications. In the domain
of single fault diagnosis of circuits, a comparison
with Gaussian maximum likelihood and K-nearest
neighbors is presented in [26] where neural net-
works, once trained, are shown to significantly
reduce the time of the diagnosis, although they
do not offer improvements in the diagnostic accu-
racy. The same result was independently reported
in [36], comparing neural networks and K-nearest
neighbors classifiers in the diagnosis of deviation
faults.
In the diagnostic system we present, the transducer
is an acquisition board that measures the
voltage values in a given set of test points . Other
choices are possible as we will discuss in Section 2.
As an example, Spence et al. [34, 35] have used
nonintrusive circuit measurements (such as infrared
images or magnetic field images); however,
nonintrusive measurements have been proved to
be very ineffective, in the sense that they can only
be used to recognize a limited number of faults.
Kirkland and Dean [22] obtained good results using
current measurements; however, current mea-
Neural Diagnosis for Analog Circuits 3
surements are often impractical, since they would
require the opening of the circuit, and this is
clearly not possible on printed circuits.
We have investigated several feature extraction
techniques and have studied their influence on
the performance of the diagnostic system. In
this paper we compare Fourier Transforms [14],
Wavelets [8], Principal Components Analysis [15],
and Sampling. In [16] Mean and Root-Mean-
Square Values of the test point voltages were used
as features, but due to the large amount of lost
information they could only recognize a limited
number of faults.
We also observed that the performance of the
diagnostic system heavily depends on the choice
of power supplies. In particular, it is often the
case that a given supply can only lead to the detection
of a particular subset of all possible faults.
A suitable set of different supplies may be used to
build a diagnostic system that combines different
diagnoses (one for each supply) dramatically improving
the performance of the diagnostic system.
In the paper we also present two algorithms that
can be used to combine these different diagnoses.
The paper is structured as follows. In Section 2
we recall relevant work on the use of neural networks
for circuit diagnosis. In Section 3 we describe
the architecture of the proposed diagnostic
system and discuss the important issue of simulation
versus acquisition. In Section 4 we discuss
the choice of power supplies and how this affect
the diagnosis. In Section 5 we describe different
techniques that can be used by the features extraction
module to compactly represent the behavior
of the circuit. In Section 6 we present the structure
of neural network classifier and show how
it is trained. In Section 7 we present two algorithms
that can be used to combine the diagnosis
computed by different networks. In Section 8 we
present statistics of the system performance when
diagnosing two different circuits: a board part of
a DC motor drive, and an oscillator.
2. Relevant work
Other approaches to the use of neural networks
for circuit diagnosis have recently been published.
Keagle et al. [21] discuss how networks trained
to recognize single faults may be used to detect
multiple faults. Tests are performed on a digital
circuit consisting of nine logical gates affected by
stuck-at 1 or stuck-at 0. The paper also presents
results on the performance of the diagnostic system
as a function of the network architecture.
Meador et al. [26] compare feedforward neural
network performance with other classifiers: gaussian
maximum likelihood and K-nearest neigh-
bors. In each experiment a single parameter deviation
fault on an operational amplifier circuit is
considered. The classifiers must separate the input
patterns corresponding to the correct behavior
and to the faulty one.
Parten et al. [29] propose using neural networks
as part of a model-based expert system for diagnosing
lumped parameter devices. The purpose of
the net would be that of solving the equations ruling
the behavior of the diagnosed device, modeled
as a set of interconnected components.
Thompson et al. [38] consider the problem of
diagnosing an IC board with approximately
components, both analog and digital. They use
a backpropagation neural network with a modular
structure, i.e., each part of the net recognizes
a particular fault.
Totton and Limb [39] use neural networks to diagnose
a circuit board part of a digital telephone
exchange. They observed from historical data that
failures on four types of components account for
more than 85% of all faults. This led them to
construct a network whose four outputs signal the
presence of a faulty component of a given type,
i.e., the network does not pinpoint the faulty component
but simply detects what type of component
is faulty.
Spence et al. [34] use a different approach to
the single fault diagnosis of printed circuit boards
(PCB). The difference between the malfunctioning
infrared image and the image of a correctly
functioning PCB is interpreted by an artificial
neural network to diagnose some types of
faulty components. In a subsequent work Spence
[35] presents a different test method based on the
interpretation of the magnetic field close to the
PCB. Although these methods can only recognize
a limited number of faults, they have the advantage
of requiring nonintrusive measurements.
Rutkowski [31] was the first to suggest the use of
neural networks for the diagnosis of multiple faults
on analog DC circuits. In this introductory work,
4 Fanni, Giua, Marchesi and Montisci
the main focus is on testing the capability of the
network to generalize from single to double fault
diagnosis. In the application example presented
in the paper, only a limited number of faults are
considered.
Bernieri et al. [3] use a neural net-work
for on-line analysis of dynamic discrete-time
systems whose input/output behavior is
ruled by equations of the form: y
f(y . The network
at the k\Gammath instant receives as inputs the value
of y and is trained to
estimate the value of given parameters that rule
the behavior of the system. Parameter deviations
over a given threshold are symptoms of faults.
Kirkland and Dean [22] have reported using input
current measurements as circuit images.
Gu et al. [17] combine neural networks and expert
systems into a single diagnostic system. To
each component is associated a neural network
trained to recognize the component's fault. The
expert system acts as a coordinator between the
different neural networks, supplying suitable inputs
to the networks and deriving a diagnosis from
the analysis of the networks' output.
Spina and Upadhyaya [36] have considered the
problem of diagnosing deviation faults in linear
circuits. A white noise source is used to automatically
generate test patterns. Fault signatures
are generated associating to a single component a
value equal to the nominal value plus 50%. The
network can correctly classify previously unseen
patterns corresponding to deviation faults of different
magnitude.
All these works highlight the prominence in a
neural diagnostic system of the aspects related to
feature extraction and circuit supplies, thus leading
us to a systematic exploration of these issues.
The present paper summarizes the results that its
authors have obtained throughout a long period of
time and that have only partially been presented
in the papers referenced in the rest of this section.
In [16] is discussed how networks trained to recognize
single faults on analog circuits in dynamic
behavior may be used to detect multiple faults.
The neural network identifies the faulty components
from the mean values of the voltage measurements
in a given set of test points. In general
it was observed that the network is able to diagnose
multiple faults on two and three components,
although less sharply than in the single fault case,
due to the presence of false alarms. The set of multiple
faults was chosen among those single faults
well recognized by the network.
In [14] Fourier transforms are used as features
of the circuit image, and multiple neural networks
were used in parallel by the diagnostic system.
This improved the performance of the diagnostic
system with respect to the previous one.
In [8] Wavelet transforms are used as features.
Wavelets proved to be a good data compression
technique when the circuit is studied during a
transient. In fact, one can increase the number
of wavelets only in particular time intervals depending
on the degree of approximation required.
In [15] Principal Component Analysis is used in
the feature extraction phase. The main advantage
of such a technique lies in the fact that it gives a
simple automatic procedure to compress the data.
3. Architecture of the diagnostic system
The architecture of the proposed diagnostic system
is shown in Figure 1.
Testing procedure (horizontal path)
Given a circuit to diagnose, we apply a suitable
power supply and acquire the voltage signals
at a given set of test points, constructing the circuit
image. We extract significant features, as discussed
in Section 5, from the image and use them
as inputs to a neural network that has been previously
trained to recognize single faults on that
circuit. The neural network will generate the candidate
set , i.e., the set of components recognized
as faulty.
In Section 4, we will show that to increase the
number of detectable faults it is necessary to use
different supplies. Consequently, we will have several
neural networks, one for each supply consid-
ered. Repeating the procedure described above
for all supplies, we obtain several candidate sets.
These sets will be combined to derive a single diagnosis
using suitable algorithms, as described in
Section 7.
Training procedure (vertical path)
The diagnostic system is built training the neural
networks that will be used in the testing procedure

Neural Diagnosis for Analog Circuits 5
Each neural network is trained using a set
of patterns corresponding to all possible single-
faults, as detailed in Section 5 and 6. The training
patterns are constructed from the faulty circuit
images using the same feature extraction technique
that will be used in the testing.
It may be possible to obtain each faulty circuit
image using an acquisition board. One has to pro-
duce, one by one, all single faults on the circuit
and then has to acquire the corresponding faulty
circuit image. This procedure is not practical in
many cases. Thus we resorted to PSpice simulation
of the circuit behaviour in faulty conditions.
On the contrary, when constructing the circuit
image in fault-free condition, both real acquisitions
and PSpice simulations are possible. As we
will later discuss, several real acquisitions will be
used to estimate the magnitude of the measurement
noise.
Our results showed that if the circuit PSpice
model is accurate enough, there is no difference
between a network trained with "simulated" patterns
and a network trained with "acquired" pat-
terns. In fact, the distance between a simulated
and an acquired pattern has the same order of
magnitude of the distance (due to measurement
noise and component parameter tolerance) between
two patterns acquired during the same fault
condition.
4. Power supplies
One of the main problems in the diagnosis of circuits
is the presence of undistinguishable and undetectable
faults.
Consider two (or more) components, say k and
k 0 in parallel as in Figure 2.(a). Clearly the behavior
of the circuit is the same whenever component
k or component k 0 is short circuited. The
same problem appears when we consider open circuit
faults of series components as in Figure 2.(b).
Faults of this kind are called undistinguishable, in
the sense that they produce the same voltage configuration
at the available test points.
A similar problem may arise when a fault is un-
detectable. In this case, the measured behavior of
the fault-free circuit is the same as the measured
behavior of the faulty circuit.
The presence of undistinguishable and undetectable
faults may have different causes.
ffl Topology of the circuit , as in the examples discussed
above.
ffl Limited number of test points , that may not
allow detection of an abnormal behavior of the
circuit.
ffl Components whose measured behavior is the
same when faulty or correctly functioning. We
recall some of the possible causes.
Operating point of the component. Consider
the diode in Figure 2.(c). It is reverse
biased and thus for all practical purposes
its behavior is the same when the diode is
functioning well or when it is affected by
an open circuit fault.
Frequency content of the supplies. Some
frequency components may not be suitable
for highlighting a given fault. In DC
steady state, for instance, capacitors behave
as open circuits and inductors behave
as short circuits, as shown in Figure 2.(d)
and

Figure

2.(e), respectively.
Protection subcircuits. The behavior of
the protection components is not supposed
to affect the overall behavior unless other
faults are present.
There is little we can do to resolve the ambiguity
due to the topology of the circuit or due to the
choice of test points. However, we may try to
resolve the ambiguity due to the behavior of the
circuit by an appropriate choice of power supplies.
As an example, a different choice of supply, such
a high frequency square voltage, force the diode in

Figure

2.(c) to alternatively switch from reverse to
forward bias, and the capacitor and inductance in

Figure

2.(d),(e) to work in AC.
This problem has also been discussed by Dague
et al. [10]. These authors add an external stimulation
in suitable points so as to disturb the circuit
operating conditions.
We will train different networks to process the
data collected for each different supply configura-
tion. Thus, our diagnostic system is composed of
several neural networks, each one specialized in
detecting a given set of faults. When the system
is used to diagnose a circuit, each network will
produce a set of candidates, i.e., of possibly faulty
6 Fanni, Giua, Marchesi and Montisci
combination of
candidate sets
power
supplies
circuit
under test
feature
extractors
neural nets
PSpice model
of circuit
under test
acquisition
board diagnosis
feature
extractor
single faults
simulations
training
Fig. 1. The proposed diagnostic system architecture.
components. The overall diagnosis can be computed
by means of different algorithms, given in
Section 7.
5. Feature extraction techniques
We assume that the information on the circuit be-
havior, i.e., the circuit image, is given by the voltage
measurements in a set of available test points.
These points are usually given by the circuit board
manufacturer and cannot be arbitrarily chosen.
(a) (b)
(c) (d) (e)
Fig. 2. Examples of undistinguishable and undetectable
faults.
Since the voltage signal at each test point is a
function of time, we need to extract significant features
to compactly represent the circuit behavior.
Extensive experimental studies showed the influence
of the particular feature chosen. The feature
used in [16] was the mean value (MV). The diagnostic
system performances improved when root-
mean-square values (RMSV) or a combination of
MV and RMSV were used. When MV or RMSV
are used, all the information on the dynamic behavior
is lost. Thus other feature extraction techniques
are required. We discuss here four different
techniques: Fourier Transforms , Wavelets , Principal
Components Analysis , and Sampling.
During the training , the goal of the feature extraction
procedure is to construct an (s\Thetar) matrix
X . Each row of this matrix represents the circuit
behavior during one of the s acquisitions and each
column represents the value of a particular fea-
ture. Each row of X is use as a training pattern
input for the neural network, hence there will be
r nodes in the network input layer, and s training
patterns, as discussed in Section 6.
During the testing of a circuit, the same feature
extraction procedure is used to derive the inputs
that will be given to the neural network.
In this section, we mainly discuss the feature
extraction module as used during the training.
Consider a circuit with n components and a
given set of m test points.
Neural Diagnosis for Analog Circuits 7
The voltage of all test points is measured on a
real circuit by an acquisition board during p acquisitions
in the absence of faults. These measurements
will be used to estimate the magnitude of
measurement noise.
On the contrary, the faulty circuit images, i.e.,
the voltage of all test points in presence of a fault,
are constructed via PSpice simulation. We consider
two single faults for each bipolar component:
open circuit and short circuit. We also considered
faults on components with more than two
terminals. As an example, in the circuit shown
in

Figure

4, there are trimmers and operational
amplifiers. We considered two possible faults on a
trimmer (cursor stuck up and cursor stuck down)
and just one single fault on an operational amplifier
(it was made inoperative by feeding with
exceedingly high voltage).
In general, let s 0 be the number of the single
faults taken into account; then one needs to collect
images.
5.1. Fourier transforms
A simple technique for compacting the information
given by the circuit image without losing the
dynamics of the system is given by the Fourier
analysis that converts the signals into frequency
components [37].
We compute the Fast Fourier Transform
of the sampled voltage signal measured at each
test point. If we have t voltage samples, we obtain
- for each test point -
components and we take the amplitude of each
component.
We are now ready to construct the input pattern
matrix . The matrix has initially s rows, one
for each acquisition, and m \Delta q columns, one for
each feature, i.e., frequency component computed
at each test point. Thus the input pattern matrix
takes the form X
qg. The first p rows of X 0 are associated to the
fault-free acquisitions.
Matrix X 0 is still unusable because of its high di-
mensionality. Domain dependent knowledge may
be used to further reduce its number of columns
[37].
The data reduction algorithm we propose, requires
two phases.
1. Remove features that give no information. We
compute for each column j the difference \Delta j
between its maximum and minimum element.
We also compute the difference ffi j between the
maximum and minimum element in the first p
rows of the column: this is an index of the numerical
uncertainty associated to the value of
feature j during the p different fault-free ac-
quisitions. Fix a threshold ' ? 1. If \Delta j  'ffi j
then the variation of the feature j has the
same order of magnitude of the numerical uncertainty
and column j will be removed. We
used a value of
2. Scale the inputs. To improve separability between
patterns we scale the columns of the
input pattern matrix in the interval [\Gamma1; 1].
3. Select a subset of significant features. The
idea is to keep only those columns that are
necessary to distinguish between different pat-
terns. Fix a threshold oe ! 1. If j x 0
oe then the variation of feature j is not large
enough to distinguish pattern i from pattern
We used a value of
We proceed as follows.
begin
let the initial set of significant features be S :=
for i := 2; s (* compare each row i of X 0
with all previous ones *)
begin
ffl let S i;i 0
oeg be the set
of those features, i.e., columns, that may
be used to distinguish between patterns i
and
is not empty then
add to S the most significant feature, i.e.,
feature j such that j x 0
We thus obtain a new matrix X of order (s \Theta r)
with r  m \Delta q.
The data reduction algorithm we use with FFT
falls into the category of unsupervised feature extraction
methods [4], i.e., methods that do not use
information on the target data. Note, however,
that the data reduction is performed opportunis-
tically, by projecting the features onto a subspace
8 Fanni, Giua, Marchesi and Montisci
that still contains all information required to separate
the input patterns.
5.2. Wavelets
The origins of Wavelets date back to 1909, when
Haar proposed them as a viable solution to function
decomposition problems. In fact Fourier se-
ries, as stated in its original formulation, show a
non-uniform convergence even for particular continuous
functions. Wavelets approach is more
suitable than Fourier one, especially when signals
are non-stationary. Both "time-frequency" and
"time-scale" wavelets are suited to signal analysis
ranging from "quasi-stationary" to fractal structure
type. Mathematicians speak of "atomic de-
composition" of signals, where wavelets are the
elementary constituents. The various wavelets
are obtained from a single wavelet by scaling and
shifting operations.
There are several definition of wavelets. One
possible is the following [27]: a wavelet is a function
y(x) in L 2 (IR) such that 2 j=2 y(2 is an
orthonormal basis for L 2 (IR). The most frequently
used wavelets are the Grossmann-Morlet wavelets,
that are also similar to Daubechies wavelets and
to Gabor-Malvar wavelets. The last algorithm is
of time-frequency type, while the former is a time-scale
algorithm.
In the wavelet theory [30, 25] any signal of finite
energy can be represented as a linear combination
of wavelets whose coefficients represent the features
we want to extract, and indicate how close
the signal is to a particular basis function.
Discrete wavelet transform (DWT) is a relatively
recent method whose biggest potential has
been found to be signal compression. The two major
advantages of the wavelet transformation are
that it can zoom in time discontinuity and that
it is possible to construct an orthonormal basis,
localized in time and frequency.
An important issue of wavelet analysis is the
choice of the proper type of wavelet and of
the methodology to use, i.e., time-scale, time-frequency
or a combination of the two.
In our diagnostic system, Haar wavelets are chosen
to realize data compression of circuit-image
information. Decomposition proposed by Haar results
as follows:
R 1f(t) h i (t) dt , and s n (t) is the
n\Gammath order summation which uniformly converges
to the signal f(t), and Haar wavelets are defined
as:
Here, the scaling factor is a power of 2, and
k defines the time shift with respect to the basic
wavelet H, that is the unit square window func-
tion. The various wavelets (n ? are obtained
starting from the basic wavelet
ing, scaling and shifting operations. It is important
to note that the time range has to be limited
in the [0; 1] interval. This is not limiting because
real signals always have a finite time length and
this will become the new time unit. It is also possible
to realize a suitable time windowing of the
signal.
Thus, it is possible to project the time signal
onto a set of mutually orthonormal wavelets. The
number of the wavelets may be arbitrary, depending
on the required approximation in reconstruction
or, as in the present case, on the amount of
information to extract from the signal.
Because a circuit image results from a set of
digital acquisition, signals are not continuous in
time, but discrete due to sampling. Hence, a discrete
transform has to be used and particular care
is required to compute the inner products.
The construction of the input matrix X using
wavelets follows the same procedure presented in
Section 5.1 for Fourier transforms and will not be
repeated here.
5.3. Principal components analysis
Principal Component Analysis (PCA) is another
unsupervised feature extraction method. Compression
by means of PCA is accomplished by projecting
each data vector along the directions of the
individual orthonormal eigenvectors of the covariance
matrix of data. As the first few eigenvalues
of the covariance matrix contain most of the signal
energy, the dimensionality of the data can be
Neural Diagnosis for Analog Circuits 9
greatly reduced without losing much information
on the input data.
It may happen that the information associated
to the discarded PC subspace is important for the
subsequent classification phase [4] and in this case
PCA is not suitable. However, PCA is a potentially
useful method because it works in many ap-
plications. In [1] PCA is used for terrain classifi-
cation, and it is shown that it can lead to a significant
improvement in the classifier performance.
In [6] there is a comparison between Gabor filters
and PCA as feature extraction methodologies
applied to SAR images segmentation with neural
networks.
Let s be the number of the circuit behaviors
taken into account, and t be the number of samples
for each test point voltage. Each circuit image
is represented by m \Delta t values. We have a (s \Theta m \Delta t)
data matrix X 0 which could be used as input for
the neural network.
As previously mentioned, preprocessing is necessary
to extract from these data the salient fea-
tures. We would like to reduce the number of
columns of this matrix from m \Delta t to r  m \Delta t,
with acceptable loss of information. Using PCA
[20] this compression is accomplished projecting
the s circuit images along the directions of the
principal eigenvectors of the covariance matrix of
Given the data matrix (X th column
represents a circuit im-
age, the covariance matrix of these data is the
s
The eigenvectors of this matrix form an orthonormal
basis, and any vector ~x i can be represented
with respect to this basis by means of a coefficient
vector with m \Delta t elements.
To reduce the data dimension, it is possible to
consider only those eigenvectors associated to the
dominant eigenvalues of C. Fix a threshold c 2
be the ordered set
of eigenvalues of C, i.e.,  j   j+1 . We say that
there are r dominant eigenvalues if
are the eigenvectors associated to the
dominant eigenvalues, we may use as compressed
representation of a vector ~x i the coefficient vector:
We used a value of
Thus, the data matrix X 0 is reduced to a (s \Theta
r) matrix X . The same compression technique
will be used on subsequent circuit images acquired
during the test phase.
5.4. Sampling
Given the circuit image (i.e., the sampled voltage
signals at all test points) one may compact the
data retaining just a limited number q of the t
samples.
Experimental results [16] showed that this is not
a viable technique if the circuit is in AC steady-state
or if there are many test points. In fact, this
leads to a neural network with too many nodes
in the input layer, i.e., too many features. This
may reduce the performance of the classification
system and leads to a higher computational cost
of the training.
However, this technique was effective when
studying short transients on circuits with a limited
number of test points. The choice of the samples
to retain must be opportunistic, and depends on
the signal variation pattern.
6. Neural model
As proposed in most of the literature discussed in
Section 2, we use a three level neural network with
sigmoid activation functions and backpropagation
learning with generalized delta rule.
6.1. Fault coding
The network has r input nodes, i.e., as many as
there are columns in the input pattern matrix X
derived with any of the different feature extraction
procedures previously described. The output
nodes of the network are as many as the number
of circuit components n.
We construct the s input-output patterns that
will be used to train the neural network for the
diagnosis of the circuit as follows. Each pattern is
given by a pair (~x i ; ~y i ). The vector ~x i is the th
row of matrix X while the associated vector ~y i is
defined as follows:
component k is not faulty
during the th acquisition;
component k is faulty
during the th acquisition.
This general scheme must be altered to take into
account undistinguishable faults.
Topologically undistinguishable faults are easy
to deal with. From an inspection of the circuit
a list of all sets of parallel components is made.
Then, a single short circuit fault acquisition for
each set C i of parallel components is considered.
There will be a single training pattern (~x
such a fault. The vector ~y i is such that y i
for all k 2 C i , while all other components have a 0
value. A dual procedure takes care of sets of series
components.
Two faults i and i 0 are behaviorally undistinguishable
if
where oe is the threshold introduced in section 5.1.
A fault i is behaviorally undetectable if the condition
k1 oe is satisfied for all input
vectors ~x 0 obtained in the faulty free condition.
We combine the patterns of behaviorally undistinguishable
faults (as we did for topologically undistinguishable
faults) and remove from the training
set the patterns associated to undetectable faults.
The fault coding here described is different from
the one presented in [16], that defined the vector
~y i as follows:
component k is short circuited
during the th acquisition;
0:5 if component k is not faulty
during the th acquisition;
component k is open circuited
during the th acquisition.
The new coding gives sharper identification of the
faulty component and is more robust when diagnosing
multiple faults because the values of interest
(0 and 1) are obtained by "pushing" the sigmoid
function toward saturation. Note also that
there is a difference with the coding in [31] where
each output node is associated to a catastrophic
fault and not to a component.
Once the net has been trained, it may be used to
perform the diagnosis of the circuit. The net must
be given the features extracted from the measured
test point voltages as input vector ~x. The net
will produce an output vector ~y; a value of y(k)
close to 1 will pinpoint a fault of component k; a
value close to 0 will denote that the component is
correctly functioning.
Although the net has been trained with the results
of single fault acquisitions, it is potentially
able to diagnose multiple faults. In this case, two
or more elements of ~y will be close to 1.
6.2. Network structure and training
The basic architecture we used consists of a three
layers backpropagation network. Since the input
patterns have been preprocessed to eliminate
undistinguishable faults, and thus they are sepa-
rable, we are sure that eventually there will be a
network capable of correctly learning all patterns.
We use early-stopping [4] to avoid overfitting.
This consists in measuring, during the training,
the error with respect to an independent set of
patterns, called validation set , and in stopping the
training when this error reaches a minimum.
Caruana [5] has shown that if early-stopping
is used the number of nodes in the hidden layer
may vary without appreciably affecting the performance
of a neural network, provided it is sufficiently
large. The results of our simulations, not
reported in this paper, seem to confirm this general
rule.
The validation set used for the stopping is independent
from the training set. We construct it by
performing a new set of PSpice simulations (one
for each fault) randomly changing the parameter
values of the components within their tolerance
range and by adding to the voltage signals of the
test points a noise whose magnitude is equivalent
to the measurement noise observed during the p
fault-free acquisitions.
7. Combining different diagnosis
In the diagnosis of circuits, we have underlined the
importance of using more than one power supply.
In fact, it is often the case that a given supply can
only lead to the detection of a particular subset
Neural Diagnosis for Analog Circuits 11
of all possible faults. The use of different supplies
leads to the use of several neural networks N i , each
of which produces its own candidate set A i . The
final diagnosis must be computed combining these
sets of candidates.
The combination of neural networks is a problem
that has been discussed in the literature and
is reviewed in [32]. In particular, since we use neural
networks that are all trained on the same task,
our approach falls into the ensemble (or commit-
It is clear that the "union" of two sets of candidates
magnifies the influence of false alarms, while
the "intersection" can be used to filter false alarms
at the risk of removing some faulty components
from the diagnosis. Keeping this in mind, we propose
two different ensemble algorithms.
Let us first give the following definitions. For
each candidate k let v(k) be the number of votes
it receives, i.e., the number of nets that consider k
malfunctioning, and let
v(k). We consider
all non-empty intersections of  v candidate
sets; assume there are ff of such intersections and
denote them R u , with We also define
u the index of the intersection R u with the
smallest cardinality (should there be more than
one such intersection we randomly pick one).
Algorithm 1
The first algorithm considers as faulty all those
candidates that have received the highest number
of votes. The corresponding diagnosis is:
ff
R u
Algorithm 2
The second algorithm considers as faulty all those
candidates that have received the highest number
of votes and that belong to the intersection with
the smallest cardinality. By considering only the
smallest intersection we hope to filter out some
false alarms. The corresponding diagnosis is:
An example is shown in Figure 3. Here D
Note that these algorithms do not give different
weight to the candidate sets of each network,
but simply perform boolean operations on these
sets. We are currently investigating the possibility
of associating to each candidate set a different
weight, depending on how the network has learned
to recognize the single fault on each component
that belongs to the candidate set.
8. Experimental results
We discuss the results obtained by the different
diagnostic strategies presented in this paper. Two
circuits are studied: a DC motor drive board, and
an astable multivibrator.
Training As discussed above, we use early stop-
ping, hence we need both a training and a validation
set of patterns.
The training patterns corresponding to each single
fault condition are constructed using a PSpice
model of the circuit. This choice gives patterns
corresponding to a circuit where the component
parameters have nominal values and the voltage
signals in each test point are noise free.
The validation set is constructed by performing
a new set of PSpice simulations where parameter
tolerance and measurement noise are introduced.
Testing During the test phase, we consider a real
circuit and the different faults are implemented by
manually shortcircuiting or opening each component
terminals. The circuit measurements are col-
A 3
Fig. 3. Example of diagnosis combination.
lected through a National Instrument Corporation
acquisition board.
Thus the test patterns are determined independently
of the training patterns. Furthermore, the
test patterns are affected by measurement noise
and by the error due to the parameter tolerance
of the circuit components.
When diagnosing a circuit, we observe the net-work
output corresponding to the input pattern
derived from the measurements. Let us recall that
the network output layer has as many nodes as
there are components. During the training phase
we have coded a fault on component k assigning
a value 1 to the corresponding output node, while
a value 0 was assigned to the output node of a
fault-free component.
In general, during the test phase the value of
each output node may take any value between 0
and 1. A value close to 0 (1) of an output node
will be interpreted as the absence (presence) of a
fault on the corresponding component. Threshold
values need to be set to discriminate between these
two cases.
Let vmax be the maximum value of all output
nodes. If vmax ! 0:2 we consider the circuit
as fault-free and the candidate set will be
empty. If vmax  0:2 we consider the circuit as
faulty, and the candidate set will contains all components
whose corresponding output node has a
value greater than 0:5v max .
8.1. DC motor drive board
We present the results obtained diagnosing the
circuit in Figure 4, part of a DC motor drive.
The same circuit has also been diagnosed in
[8, 13, 14, 15, 16]. In the figure, the
test points are marked by numbers within circles,
while the are labeled by numbers
in square brackets.
Training There are 70 single faults to consider
on this circuit. In fact, the circuit is composed of
36 components but only one fault is considered for
each of the two operational amplifiers. Thus, the
overall training set should consist of 76 training
patterns - the additional six being obtained by
acquisitions of the circuit behavior in absence of
fault.
The following sets contain topologically undistinguishable
faults:
28s, 29sg,
17og, 30/31o =f30o, 31og. Here 10s represents
a short circuit fault on component 10, 16o represents
an open circuit fault on component 16, etc.
Thus, the training set is reduced to 62+6 patterns
by combining the conflicting patterns as discussed
in Section 6.1.
We have used three different voltage supplies
and thus three different networks.
1. The first network N 1 is trained with patterns
acquired when the circuit has close to nominal
voltage supplies: V 1
\Gamma12 (V).
Fourier The number of significant frequency
components is This gives rise to
columns in the input matrix X 0 ,
that are reduced to r = 14 in the matrix
X .
The set of behaviorally undetectable faults
for this net is: f3o, 6o, 11o,12s, 15o, 23o,
24o, 25s, 27o, 29o, 31s, 32s, 33o, 34og.
The sets of behaviorally undistinguishable
faults are: f4o, 6sg, f21o, 27/28/29s, 30sg,
f34/35s, 36og, f 35o, 36sg.
Wavelets The number of significant wavelets
is 8. This gives rise to m \Delta
columns in the input matrix X 0 , that are
reduced to r = 15 in the matrix X .
The set of behaviorally undetectable faults
for this net is: f3o, 6o, 11o, 12s, 15o, 23o,
24o, 25s, 27o, 29o, 31s, 32s, 33o, 34og.
The sets of behaviorally undistinguishable
faults are: f21o, 27/28/29s, 30sg, f34/35s,
36og, f35o, 36sg.
PCA Assuming a threshold 0:999, the
number of dominant eigenvalues (i.e., the
number of columns of the X matrix) is
25.
The set of behaviorally undetectable faults
for this net is: f3o, 6o, 11o, 12s, 15o, 23o,
24o, 25s, 27o, 29o, 31s, 32s, 33o, 34og.
Neural Diagnosis for Analog Circuits 13
Fig. 4. DC motor drive board.
The sets of behaviorally undistinguishable
faults are: f4o, 6sg, f21o, 27/28/29s, 30sg,
f34/35s, 36og, f35o, 36sg.
2. The second network N 2 is trained with
patterns acquired when the circuit has
far from nominal periodic voltage supplies:
are zero-mean square
waves with 160 Hz frequency, 4 V peak-to-
Fourier The number of significant frequency
components is 8. This gives rise to
columns in the input matrix X 0 ,
that are reduced to r = 15 in the matrix
X .
The set of behaviorally undetectable faults
for this net is: f10o, 11o, 12s, 21o, 22o,
23o, 24o, 25s, 27/28/29s, 28o, 29o, 30s,
31s, 32sg.
The sets of behaviorally undistinguishable
faults are: f1o, 3sg, f4o, 6sg, f34/35s,
36og.
Wavelets The number of significant wavelets
is 9. This gives rise to m \Delta
columns in the input matrix X 0 , that are
reduced to in the matrix X .
The set of behaviorally undetectable faults
for this net is: f8 up , 10o, 11o, 12s, 14o, 17s,
21o, 22o, 23o, 24o, 25s, 27/28/29s, 28o,
29o, 30s, 31s, 32sg.
The sets of behaviorally undistinguishable
faults are: f4o, 6sg, f14/15s, 16sg, f18o,
19og, f34/35s, 36og.
PCA Assuming a threshold 0:999, the
number of dominant eigenvalues is
The set of behaviorally undetectable faults
for this net is: f10o, 11o, 12s, 21o, 22o,
23o, 24o, 25s, 27/28/29s, 28o, 29o, 30s,
31s, 32s, 34/35s, 36og.
The set of behaviorally undistinguishable
faults is: f1o, 3sg.
3. The third network N 3 is trained with patterns
acquired when the circuit has step voltage
supplies:
(V).
Fourier The number of significant frequency
components is 8. This gives rise to
columns in the input matrix X 0 ,
14 Fanni, Giua, Marchesi and Montisci
that are reduced to r = 15 in the matrix
X .
The set of behaviorally undetectable faults
for this net is: f10o, 12s, 14o, 21o, 23o, 25s,
27o, 28o, 29o, 31s, 32s, 34og.
The sets of behaviorally undistinguishable
faults are: f1o, 3sg, f4o, 6sg, f22o,
27/28/29s, 30sg.
Wavelets The number of significant wavelets
is This gives rise to m \Delta
columns in the input matrix X 0 , that are
reduced to r = 17 in the matrix X .
The set of behaviorally undetectable faults
for this net is:f10o, 12s, 21o, 23o, 25s, 27o,
28o, 29o, 31s, 32s, 34og.
The sets of behaviorally undistinguishable
faults are: f1o, 3sg, f4o, 6sg, f22o,
27/28/29s, 30sg.
PCA Assuming a threshold 0:999, the
number of dominant eigenvalues is
The set of behaviorally undetectable faults
for this net is: f10o, 12s, 21o, 23o, 25s, 27o,
28o, 29o, 31s, 32sg.
The sets of behaviorally undistinguishable
faults are: f1o, 3sg, f4o, 6sg, f22o,
27/28/29s, 30sg.
Testing We are now ready to study the performance
of the neural diagnostic systems previously
constructed.
In the initial phase, we test the systems on a
fault-free circuit. We observed that when diagnosing
a real circuit in absence of faults, all networks
correctly identify this behavior, in the sense
that all output nodes have a value less than the
assigned threshold of 0:2 and thus the candidate
set is always empty.
In a second phase, we consider faulty circuits.

Table

1 compares the performance (in percent) of
the different systems. The first columns of the table
shows the diagnosis of N 1 , N 2 , and N 3 and
the diagnosis obtained combining the candidate
sets of the three nets with Algorithm 1 and Algorithm
2, using Fourier, Wavelets, and PCA, re-
spectively. The last two columns show the results
obtained combining the candidate sets of the nine
nets (three for each feature extraction technique)
with Algorithm 1 and Algorithm 2.
We consider a fault correctly diagnosed if the
candidate set of the net contains a subset of
the components associated to this fault, taking
into account topologically undistinguishable fault
classes. Let us consider some examples in the circuit
of Figure 4. The fault 16o belongs to the
topologically indistinguishable fault class 16/17o;
we say that it is correctly identified if the candidate
set is either f16g or f17g or f16, 17g. The
fault 16s is correctly identified if the candidate set
is f16g.
Single faults
The first row block of Table 1 shows the diagnosis
of the 62 possible single faults. There are
three different classes of diagnosis.
Class A 1 Faults correctly diagnosed.
Undistinguishable faults: these are the
faults that we have classified as behaviorally
undistinguishable during the training. As an
example, in net N 1 with Fourier, we have identified
9 undistinguishable faults, i.e., 14% of
the total 62 faults.
Class C 1 Undetected faults: these are the faults
that we have classified as behaviorally undetectable
during the training.
Double faults
The second row block of Table 1 shows the performance
of the different systems when diagnosing
double faults. Each double fault consists in the
simultaneous presence of two faulty components.
Note that not all possible pairs of single faults
constitute a double fault: e.g., a bipolar component
cannot be simultaneously open- and short-
circuited. We have considered a sample of 168
different double faults randomly chosen from the
total population. This sample was large enough
to satisfy the  2 test for the six different classes
of diagnosis.
These are the classes considered.
Class A 2 Both faults correctly diagnosed.
Only one fault correctly diagnosed.
Class C 2 At least one fault correctly diagnosed
with one or two false alarms.
Class D 2 At least one fault correctly diagnosed
with more than two false alarms.
Only false alarms.
Neural Diagnosis for Analog Circuits 15
Triple faults
The third row block of Table 1 shows the performance
of the different systems when diagnosing
triple faults. We have considered a sample of 181
different faults randomly chosen out of the total
population.
These are the classes considered.
Class A 3 All three faults correctly diagnosed.
Only one or two faults correctly diagnosed

Class C 3 At least one fault correctly diagnosed
with one or two false alarms.
Class D 3 At least one fault correctly diagnosed
with more than two false alarms.
Class F 3 Only false alarms.
Discussion
In the case of multiple faults, we consider correct
all diagnoses in class A and in class B. In fact,
starting from class B we may use an incremental
repair procedure, substituting the faulty components
one by one. Diagnosis in class C may also
be useful.
From the table it can be seen that the use of
several networks improves the system performance
provided that a good procedure is used to combine
the results of the networks. In particular, Algorithm
1 and Algorithm 2 give the same results
when diagnosing: (a) single faults; (b) multiple
faults using a system composed of many nets in
parallel. When diagnosis multiple faults, if the
system is composed by a small number of neural
nets Algorithm 2 performs better because it exalts
the filtering effect of the intersection operator, reducing
the number of diagnoses in class A but
increasing the total number of diagnoses in class
A+B.
All three feature extraction techniques give
comparable results. PCA performs better than
the other two when diagnosing single and double
faults, but seems to be less robust when diagnosing
three simultaneous faults. Unlike Fourier and
Wavelets, PCA requires less data preprocessing in
the feature extraction phase, as discussed in Section
5.
8.2. Astable multivibrator
Dague et al. in [10] remarked that oscillators are
difficult to diagnose because most faults cause
the same type of symptoms. This is exactly the
case in which a proper choice of the power supplies
can improve the diagnosability of the circuit.
They proposed using an external "stimulation"
and showed the results obtained using their diagnostic
expert system on the astable multivibrator
shown in Figure 5. In this section we present results
obtained using our diagnostic system on the
same circuit.
Training We chose test-points in the
nodes labeled 1 and 2 in the figure. The number of
components is single faults have been
considered. In fact, we consider 6 faults for each
transistor: short circuit between base and emitter
(QBEs), base and collector (QBCs),collector and
emitter (QCEs); open circuit on the base (QBo),
collector (QCo), and emitter (QEo). The circuit
does not contain topologically undistinguishable
faults.
To be able to compare the results of our diagnostic
system with the system developed by
Dague, we used the same voltage supply proposed
in [10]. It consists of the superposition of the
nominal supply PS (a continuous voltage signal of
+5V) and of an external stimulation EP (a voltage
pulse of 10V amplitude, applied in
and lasting 1s). Since we consider a unique sup-
ply, we use a single neural network for each feature
extraction.
We used a PSpice model of the oscillator to collect
the training and validation patterns for all
faulty conditions, as previously described.
The circuit has been studied in transient behaviour
and the voltage signals in each test point
have been collected in the interval 0:9 \Xi 40s with
a sampling interval of t 0:1s. This gives rise
to a circuit image before feature extraction composed
of for each test point.
We have used all different feature extraction
techniques described in Section 5.
Fourier The number of significant frequency
components is This gives rise
to m \Delta columns in the input matrix

Table

1. Diagnosis of the circuit in Figure 4 (in percent).
Fourier Wavelets PCA F+W+P
Single faults (62 fault cases)
Double faults (168 fault cases)
Triple faults (181 fault cases)
A3
28 19 21 62 74 23 21 23
are reduced to in the matrix
X .
There are no behaviorally undetectable faults.
The sets of behaviorally undistinguishable
faults are: fRC2s, Q2CEsg, fRB2s, Q2BEsg.
Wavelets The number of significant wavelets is
This gives rise to m \Delta columns
in the input matrix X 0 , that are reduced to
in the matrix X .
There are no behaviorally undetectable faults.
The sets of behaviorally undistinguishable
PS
RI
Fig. 5. Astable multivibrator. PS is the nominal power
generates the stimulation voltage pulse.
faults are: fC1o, RC2s, RB2s, Q2CEs,
Q2BEsg.
PCA Assuming a threshold 0:999, the number
of dominant eigenvalues (i.e., the number
of columns of the X matrix) is
There are no behaviorally undetectable faults.
The sets of behaviorally undistinguishable
faults are: fRC2s, Q2CEsg, fRB2s, Q2BEsg.
Sampling The Sampling feature extraction retains
samples (out of a total of
for each test point spaced with an hyperbolic
law so as to have more samples during the
initial phase of the transient and only a few
as the circuit reaches the steady state. For

Table

2. Diagnosis of the circuit in Figure 5 (in percent).
Fourier Wavelets PCA Sampling
Single faults (24 fault cases)
Double faults (140 fault cases)
28 8 23 41
Neural Diagnosis for Analog Circuits 17

Table

3. A comparison between the diagnosis of the circuit in Figure 5 done with Dague's Expert System and with Neural
Network with Sampling.
Defect Expert System Neural Network
(Dague et al.) with Sampling
RC1s RC1, Q1, RC1
fC2g \Theta fPS, CX, C1, EP, Q2, RB1, RB2, RC2, RIg
RB1s (*) Q1 RB1
double candidates C1
Q1CEs Q1, C2, Q1
fRC1g \Theta fCX, C1, EP, Q2, RB2, RC2, RIg
Q1BEs Q1, C1, RB1 Q1
fC2g \Theta fPS, EP, Q2, RB2, RC2, RIg
RC1o RC1, Q1, C2 RC1
RB1o RB1, Q1, C1 RB1
C1o C1, Q2, RC2 fC1, RC2g
Q1Eo Q1 Q1
Q1Bo Q1 Q1
Q1Co Q1, C1, RB1, C2, RC1 Q1
(*) Note that a short-circuit on RB1 induces destruction of Q1.
Then the ae-th retained sample is the  ae -th
sample, as shown in Figure 6.
There are no behaviorally undetectable faults.
The sets of behaviorally undistinguishable
faults are: fC1o, RC2og, fRC2s, Q2CEsg,
fRB2s, Q2BEsg.
Testing In a first phase, we test the systems on a
fault-free circuit. We observed that when diagnos-
tr
r
Fig. 6. Sampling of the total measurements.
ing a real circuit in absence of faults, all networks
correctly identify this behavior.
In a second phase, we consider faulty circuits.

Table

2 compares the performances (in percent)
of the different neural diagnostic systems. The
classes of diagnosis are the same defined in the
previous example.
The first row block of Table 2 shows the diagnosis
of the 24 single faults considered.
The second row block of Table 2 shows the diagnosis
of the 140 double faults considered. Note
that in this case the total number of possible double
faults is 240.
We can see that in this particular case Sampling
appears to be the most effective feature extraction
technique (and it is also the easiest to implement).
As remarked before, however, it is a viable solution
only because we have a small number of test
points (two in this example) and the circuit has a
short transient.
Comparison with Dague's Expert System
In

Table

3 we compare the results obtained using
our diagnostic system with sampling feature
extraction (Neural Network with Sampling) with
the results obtained by Dague's expert system.
Note that although only 12 faults are considered
by Dague, our diagnostic system has been trained
to recognize all 24 single faults.
We observe that the neural network has been
able to correctly classify 11 faults in class A and
only 1 fault (C1o) in class B. The expert system,
on the contrary, can very rarely correctly identify
the faulty component (only 3 diagnosis in class A
including the diagnosis of RB1s).
The results show that the neural network performs
much better than the expert system. This
is a consequence of its ability to exploit the information
on each single-fault behavior of that particular
circuit and to generalize. This information
is not taken into account by the expert system,
that reasons on more abstract principles.
9. Conclusions
We have shown how a neural network, trained to
recognize catastrophic single faults, may be used
to diagnose multiple faults on analog circuits.
In general we observe that the network is almost
always able to learn and recall the single fault
patterns presented during the training. Multiple
faults on two and three components may also be
diagnosed, although less sharply than in the single
fault case, due to the presence of false alarms. In
most cases, however, the network is able to detect
at least one of the malfunctioning components.
Thus one may use an incremental repair proce-
dure, substituting the faulty components one by
one.
We consider several different power supplies in
order to detect those faults that do not modify
the circuit behavior under nominal supplies. We
use several neural networks "in parallel", one for
each different supply configuration. Each network
is specialized in detecting a given set of faults.
Thus, it is not necessary to force a network to
recognize a fault that is more easily detected by
another one.
The use of different networks, leads to the problem
of composing different sets of candidates into
a single diagnosis. We showed that a suitable
choice of the composition algorithm may dramatically
improve the system performance, especially
when diagnosing multiple faults.
We compared the results obtained by our system
when using different feature extraction tech-
niques. In fact, the performance of the diagnostic
system is noticeably affected by the choice of
features that we consider as representative of the
device behavior.
Although we have only presented two simple examples
of diagnosis, extensive experiments convinced
us that this approach is fairly general and
that it gives better results than other diagnostic
systems, such as expert systems, whenever it can
be applied.



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multiple fault diagnosis;analog circuits;neural networks