# Fault Modeling and Testing for Analog Circuits in Complex Space Based on Supply Current and Output Voltage

Authors: Hongzhi Hu; Shulin Tian; Qing Guo
Journal: Journal of Applied Mathematics
 (2015)

---

## Abstract

This paper deals with the modeling of fault for analog circuits. A two-dimensional (2D) fault model is first proposed based on collaborative analysis of supply current and output voltage. This model is a family of circle loci on the complex plane, and it simplifies greatly the algorithms for test point selection and potential fault simulations, which are primary difficulties in fault diagnosis of analog circuits. Furthermore, in order to reduce the difficulty of fault location, an improved fault model in three-dimensional (3D) complex space is proposed, which achieves a far better fault detection ratio (FDR) against measurement error and parametric tolerance. To address the problem of fault masking in both 2D and 3D fault models, this paper proposes an effective design for testability (DFT) method. By adding redundant bypassing-components in the circuit under test (CUT), this method achieves excellent fault isolation ratio (FIR) in ambiguity group isolation. The efficacy of the proposed model and testing method is validated through experimental results provided in this paper.


---

## Body

## 1. Introduction

Over the past two decades, fault detection and diagnosis of analog circuits become an important research area where a number of corresponding theories and techniques have been developed. Among the researches in this area, fault dictionary is one of the most important methods that have attracted great interests [1–5]. Various circuit variables (e.g., voltage, current, and frequency) have been used in fault dictionary to obtain fault signatures [6–8]. In addition to the commonly employed measurement of output voltage, supply current testing has been applied widely in analog and mixed signal integrated circuits (ICs) with excellent fault coverage, especially for the catastrophic fault of CMOS ICs [9–11].Despite the advantages of fault dictionary, there are still persistent challenges, such as test point selection and potential faults simulations, in applications of fault dictionary method for fault diagnosis of analog circuits. Wang and Yang proposed a slope fault mode, which not only reduces the required number of test points to two but also simplifies the potential fault simulation greatly [2]. However, the slope fault model is only limited to dynamic circuits. To achieve detection of parametric faults for analog circuits, Yang et al. proposes a complex-circle based fault model [3]. Although this proposed fault model is improved sequentially by optimal testing frequency selection [12] and fault location [13], the weakness of fault masking still remains. Moreover, a 3D model based on transfer function was proposed in [14]; it extends the distances between fault loci and improves the practicability for applications. Recently, Ma and Wang achieved detecting catastrophic faults and parametric faults, by analyzing the harmonic spectrum of output voltage and supply current [11]. Following the ideas of these researches, this paper proposes two improved fault models with satisfactory FDR based on collaborative analysis of output voltage and supply current. Moreover, to isolate ambiguity groups, a design for testability (DFT) method is proposed to achieve better fault isolation ratio (FIR) against the influence of analog tolerance and measurement errors.This paper is organized as follows. First, the theory of collaborative fault model on 2D complex plane is introduced in Section2. An example of a Sallen-Key filter is also provided to illustrate the theory in this section. In order to reduce the difficulty of fault location and isolation, an improved fault model in 3D complex space is proposed in Section 3. Furthermore, to deal with the ambiguity groups in the fault model, an innovative method of DFT is demonstrated in Section 4 to improve the FIR. Validation of the proposed method is provided by PSPICE. Finally, conclusions and future work are summarized in Section 5.


## 2. Collaborative Fault Model on 2D Complex Plane

### 2.1. Theorem of the 2D Collaborative Fault Model

A linear time-invariant passive CUTN is assumed to contain n components F
i
(
1
≤
i
≤
n
). N is assumed to be stimulated by a direct current (DC) power source U
d
d and a family of alternating current (AC) signals U
s
k
(
1
≤
k
≤
m
), where m is the number of AC signals. Then a (
m
+
1
)-dimension vector U
S is composed as follows:(1)
U
S
=
U
d
d
U
s
1
U
s
2
⋯
U
s
m
.Furthermore, a single componentP is assumed to be the potential failure component in N, where(2)
P
∈
F
i
∣
1
≤
i
≤
n
.The voltage across P is defined as u
F. According to the Substitution Theorem, a voltage source u
F can be used to replace the component P in the CUT. Therefore, both voltage u
F and vector U
S are regarded as the excitation of the CUT. Then the collaborative output F
c
· is defined as follows:(3)
F
c
s
,
P
=
U
o
I
d
d
c
0
c
1
,where U
o is the output voltage of CUT, I
d
d is the power supply current, and s is the Laplacian operator. c
0 and c
1 are predefined coefficients. Furthermore, the corresponding transfer function matrices H and H
F are defined as(4)
H
=
H
1
d
d
s
H
11
s
⋯
H
1
m
s
H
2
d
d
s
H
21
s
⋯
H
2
m
s
T
,
H
F
=
H
1
F
s
H
2
F
s
,where H
1
d
d
s, H
2
d
d
s, H
1
k
s
(
1
≤
k
≤
m
), H
2
k
s
(
1
≤
k
≤
m
), H
1
F
s, and H
2
F
s are transfer functions with respect to U
o and I
d
d. Then, F
c
· can be expressed as(5)
F
c
s
,
P
=
U
S
H
+
u
F
H
F
c
0
c
1
.In addition, according to the theories of circuit analysis,u
F can be further defined as(6)
u
F
=
u
oc
Z
F
Z
0
+
Z
F
,where u
oc is open circuit voltage for the failure component P; Z
0 is corresponding Thevenin equivalent impedance, while Z
F is the impedance of P. According to the Thevenin’s Theorem, u
oc and Z
0 are independent of Z
F. Therefore,(7)
F
c
s
,
P
=
U
S
H
c
0
c
1
+
u
oc
H
F
c
0
c
1
Z
F
Z
0
+
Z
F
.For any determinate frequencyω, the Laplacian operator s
=
j
ω is also an imaginary constant, where j is imaginary unit vector. In addition, without loss of generality, Z
F can be assumed to be a pure resistance R
F or reactance X
F. Therefore, taking R
F as an example,(8)
F
c
P
=
U
S
H
c
0
c
1
+
u
oc
H
F
c
0
c
1
R
F
Z
0
+
R
F
.With the following assumptions,(9)
U
S
H
c
0
c
1
=
a
1
+
j
b
1
,
u
oc
H
F
c
0
c
1
=
a
2
+
j
b
2
,
Z
0
=
R
0
+
j
X
0
,where a
1, b
1, a
2, b
2, R
0, and X
0 are all real numbers, then(10)
F
c
P
=
a
1
+
j
b
1
+
a
2
+
j
b
2
R
F
R
0
+
j
X
0
+
R
F
,
R
0
+
j
X
0
R
F
=
a
2
F
R
-
a
1
+
b
2
F
I
-
b
1
F
R
-
a
1
2
+
F
I
-
b
1
2
-
1
+
j
-
a
2
F
I
-
b
1
+
b
2
F
R
-
a
1
F
R
-
a
1
2
+
F
I
-
b
1
2
,where(11)
F
R
=
Re
F
c
P
,
F
I
=
Im
F
c
P
.IfR
0 and X
0 are all equal to zero, the CUT is equal to an ideal voltage source u
oc corresponding to the component P. This means that variation in Z
F will not affect u
F and F
c
P. Thus, we have (12)
R
0
R
F
=
a
2
F
R
-
a
1
+
b
2
F
I
-
b
1
F
R
-
a
1
2
+
F
I
-
b
1
2
-
1
,
X
0
R
F
=
-
a
2
F
I
-
b
1
+
b
2
F
R
-
a
1
F
R
-
a
1
2
+
F
I
-
b
1
2
.Then(13)
R
0
X
0
=
a
2
F
R
-
a
1
+
b
2
F
I
-
b
1
-
F
R
-
a
1
2
-
F
I
-
b
1
2
-
a
2
F
I
-
b
1
+
b
2
F
R
-
a
1
.Furthermore, (13) can be transformed into a formula of 2D circle:(14)
F
R
-
c
r
2
+
F
I
-
c
j
2
=
r
c
2
,
(15)
c
r
=
-
b
2
R
0
+
2
a
1
X
0
+
a
2
X
0
2
X
0
,
c
j
=
a
2
R
0
+
2
b
1
X
0
+
b
2
X
0
2
X
0
,
r
c
2
=
a
2
2
+
b
2
2
R
0
2
+
X
0
2
4
X
0
2
,where c
r and c
j are the real part and imaginary part of the circle center coordinates, respectively. r
c is the radius. It is confirmed that c
r, c
j, and r
c are independent of the parametric change of the failure component P. Therefore, fault loci fitting requires a small amount of simulation on potential faults, for both catastrophic faults and parametric faults. Alternatively, if Z
F is assumed to be a pure reactance X
F, similar conclusion can be reached.For a CUT withn components P
i
(
1
≤
i
≤
n
), fault modeling is to determine the model parameters c
r
i, c
j
i, and r
c
i for each P
i according to (15), where parameters c
r
i, c
j
i, and r
c
i are corresponding to c
r, c
j, and r
c, respectively. In most actual analog cases, (14) is hard to derive from transfer function directly. However, it is well known that three sets of distinct data are sufficient to determine a circle. In this regard, simulation is a simple method to establish the circle model in (14). Therefore, in the modeling process, three sets of distinct output data are obtained by parametric sweep simulations, corresponding to three distinct values of each component P
i. For the whole CUT, n parametric sweep simulations are needed. Three sets of distinct simulation data for each component are assumed as F
c
0, F
c
i
1, and F
c
i
2. Since F
c
0 is assumed to be the fault free output, it fits in with all components. The fault modeling process is illustrated in Figure 1. In addition, the following equation is used to calculate the parameters c
r
i, c
j
i, and r
c
i for component P
i
(
1
≤
i
≤
n
):(16)
Re
F
c
0
-
c
r
i
2
+
Im
F
c
0
-
c
j
i
2
=
r
c
i
2
,
Re
F
c
i
1
-
c
r
i
2
+
Im
F
c
i
1
-
c
j
i
2
=
r
c
i
2
,
Re
F
c
i
2
-
c
r
i
2
+
Im
F
c
i
2
-
c
j
i
2
=
r
c
i
2
.Figure 1
Flowchart of fault modeling.


### 2.2. Example of a Sallen-Key Filter

Compared with the difficulty of obtaining explicit mathematical expression, simulation is a relatively easy way for fault modeling. To demonstrate the method, a second-order Sallen-Key filter, as shown in Figure2, is adopted as an example of circuit under test (CUT). By parameter sweeping simulations with respect to the supply current I
d
d and the output response U
o in PSPICE, the 2D collaborative fault model can be achieved. The simulation results are listed in Table 1, in which parametric ratio means the ratio between failure value and nominal value for each component. Further results are also provided in Figure 3. The stimulation is a 3 kHz, 1 V sine signal, and the parameter sweeping range for each component is assumed to be from p
i
×
10
-
4 to p
i
×
10
4, where p
i
(
1
≤
i
≤
n
) is the fault-free value of the ith component F
i. In addition, the simulation range of R
3 and R
4 is reduced, because output power of the filter is limited. As shown in Figure 3, all potential fault statuses compose a family of loci on the complex plane, which is called F
i-loci. Every F
i-locus corresponds to all the potential fault statuses of a component, not only parametric faults but also catastrophic faults. Furthermore, F
i-loci converges in the fault free point A and zero point B.Table 1
Simulation results of 2D model for the Sallen-Key filter.
Parametric ratio
Collaborative simulation results
F
C
R
1
F
C
R
2
F
C
R
3
F
C
R
4
F
C
C
1
F
C
C
2
0.01
0.7977
-
j
1.0218
1.7383
-
j
0.3307
—
-
0.0364
-
j
0.4066
1.1249
+
j
0.6892
0.2792
-
j
0.8047
0.05
0.7959
-
j
1.0537
1.7801
-
j
0.4091
—
-
0.0369
-
j
0.4382
1.2076
+
j
0.6979
0.2816
-
j
0.8265
0.08
0.7925
-
j
1.0777
1.8094
-
j
0.4733
—
-
0.0373
-
j
0.4627
1.2755
+
j
0.7012
0.2830
-
j
0.8264
0.1
0.7906
-
j
1.0949
1.8271
-
j
0.5187
—
-
0.0375
-
j
0.4794
1.3282
+
j
0.7019
0.2839
-
j
0.8519
0.4
0.7063
-
j
1.3735
1.7917
-
j
1.4096
—
-
0.0348
-
j
0.7801
2.4864
+
j
0.2002
0.2905
-
j
1.0578
0.7
0.4829
-
j
1.6766
0.9513
-
j
2.0720
0.7186
-
j
3.9260
-
0.0105
-
j
1.2155
1.9996
-
j
2.3457
0.2565
-
j
1.3783
0.9
0.2312
-
j
1.8427
0.3137
-
j
2.0195
0.1476
-
j
2.2588
0.0354
-
j
1.6315
0.4333
-
j
2.2252
0.1647
-
j
1.7011
1.0
0.0767
-
j
1.8977
0.0767
-
j
1.8977
0.0767
-
j
1.8977
0.0767
-
j
1.8977
0.0767
-
j
1.8977
0.0767
-
j
.
8977
1.1
-
0.9320
-
j
1.9275
-
0.1008
-
j
1.7530
0.0385
-
j
1.6543
0.1390
-
j
2.2194
-
0.1113
-
j
1.6025
-
0.0746
-
j
2.1441
1.3
-
0.4407
-
j
1.8928
-
0.3146
-
j
1.4630
0.0013
-
j
1.3453
0.3832
-
j
3.0993
-
0.2476
-
j
1.1936
-
0.6829
-
j
2.6877
1.5
-
0.7474
-
j
1.7426
-
0.4093
-
j
1.2169
-
0.0151
-
j
1.1573
1.0301
-
j
4.5183
-
0.2751
-
j
0.9275
-
2.0075
-
j
2.9003
2
-
1.1278
-
j
1.1435
-
0.4420
-
j
0.8107
-
0.0304
-
j
0.9029
15.264
-
j
3.1208
-
0.2445
-
j
0.5844
-
2.8027
+
j
0.5055
3
-
0.9616
-
j
0.3874
-
0.3514
-
j
0.4570
-
0.0366
-
j
0.7040
—
-
0.1726
-
j
0.3330
-
0.7652
+
j
0.6990
5
-
0.5445
-
j
0.0583
-
0.2246
-
j
0.2339
-
0.0379
-
j
0.5686
—
-
0.1030
-
j
0.1748
-
0.2644
+
j
0.3346
10
-
0.2431
+
j
0.0129
-
0.1143
-
j
0.1029
-
0.0375
-
j
0.4793
—
-
0.0509
-
j
0.0800
-
0.0979
+
j
0.1423
30
-
0.0739
+
j
0.0104
-
0.0383
-
j
0.0314
-
0.0367
-
j
0.4248
—
-
0.0168
-
j
0.0252
-
0.0273
+
j
0.0425
100
-
0.0215
+
j
0.0036
-
0.0115
-
j
0.0091
-
0.0364
-
j
0.4067
—
-
0.0050
-
j
0.0074
-
0.0078
+
j
0.0124Figure 2
Second-order Sallen-key filter, whereR
1, R
2, R
3 and R
4 are 10 kΩ, C
1 and C
2 are 4.7 nF.Figure 3
2D collaborative fault model of the Sallen-Key filter. All potential fault statuses of the filter compose a family ofF
i-loci, and F
i-loci converge in the fault free point A and zero point B.Note that in some cases, distances between someF
i-loci are very small. In extreme cases, some F
i-loci may coincide with each other, which results in ambiguity group. Since the 2D collaborative fault model is derived from transfer function of CUT, ambiguity group may be an inevitable problem [15]. Table 2 lists the ambiguity groups of the Sallen-Key filter, and Figure 3 shows that while the F
i-loci of R
1, R
2, C
1, and C
2 are obviously different from others, fault status of R
3 and R
4 are located on the same locus. This means that potential faults of R
3 and R
4 cannot be isolated based on the fault model in Figure 3; therefore, R
3 and R
4 fall into the same ambiguity group {
R
3
,
R
4
}.Table 2
Ambiguity groups of the Sallen-Key filter.
Ambiguity group
Fault components
①
R
1
②
R
2
③
R
3, R
4
④
C
1
⑤
C
2


## 2.1. Theorem of the 2D Collaborative Fault Model

A linear time-invariant passive CUTN is assumed to contain n components F
i
(
1
≤
i
≤
n
). N is assumed to be stimulated by a direct current (DC) power source U
d
d and a family of alternating current (AC) signals U
s
k
(
1
≤
k
≤
m
), where m is the number of AC signals. Then a (
m
+
1
)-dimension vector U
S is composed as follows:(1)
U
S
=
U
d
d
U
s
1
U
s
2
⋯
U
s
m
.Furthermore, a single componentP is assumed to be the potential failure component in N, where(2)
P
∈
F
i
∣
1
≤
i
≤
n
.The voltage across P is defined as u
F. According to the Substitution Theorem, a voltage source u
F can be used to replace the component P in the CUT. Therefore, both voltage u
F and vector U
S are regarded as the excitation of the CUT. Then the collaborative output F
c
· is defined as follows:(3)
F
c
s
,
P
=
U
o
I
d
d
c
0
c
1
,where U
o is the output voltage of CUT, I
d
d is the power supply current, and s is the Laplacian operator. c
0 and c
1 are predefined coefficients. Furthermore, the corresponding transfer function matrices H and H
F are defined as(4)
H
=
H
1
d
d
s
H
11
s
⋯
H
1
m
s
H
2
d
d
s
H
21
s
⋯
H
2
m
s
T
,
H
F
=
H
1
F
s
H
2
F
s
,where H
1
d
d
s, H
2
d
d
s, H
1
k
s
(
1
≤
k
≤
m
), H
2
k
s
(
1
≤
k
≤
m
), H
1
F
s, and H
2
F
s are transfer functions with respect to U
o and I
d
d. Then, F
c
· can be expressed as(5)
F
c
s
,
P
=
U
S
H
+
u
F
H
F
c
0
c
1
.In addition, according to the theories of circuit analysis,u
F can be further defined as(6)
u
F
=
u
oc
Z
F
Z
0
+
Z
F
,where u
oc is open circuit voltage for the failure component P; Z
0 is corresponding Thevenin equivalent impedance, while Z
F is the impedance of P. According to the Thevenin’s Theorem, u
oc and Z
0 are independent of Z
F. Therefore,(7)
F
c
s
,
P
=
U
S
H
c
0
c
1
+
u
oc
H
F
c
0
c
1
Z
F
Z
0
+
Z
F
.For any determinate frequencyω, the Laplacian operator s
=
j
ω is also an imaginary constant, where j is imaginary unit vector. In addition, without loss of generality, Z
F can be assumed to be a pure resistance R
F or reactance X
F. Therefore, taking R
F as an example,(8)
F
c
P
=
U
S
H
c
0
c
1
+
u
oc
H
F
c
0
c
1
R
F
Z
0
+
R
F
.With the following assumptions,(9)
U
S
H
c
0
c
1
=
a
1
+
j
b
1
,
u
oc
H
F
c
0
c
1
=
a
2
+
j
b
2
,
Z
0
=
R
0
+
j
X
0
,where a
1, b
1, a
2, b
2, R
0, and X
0 are all real numbers, then(10)
F
c
P
=
a
1
+
j
b
1
+
a
2
+
j
b
2
R
F
R
0
+
j
X
0
+
R
F
,
R
0
+
j
X
0
R
F
=
a
2
F
R
-
a
1
+
b
2
F
I
-
b
1
F
R
-
a
1
2
+
F
I
-
b
1
2
-
1
+
j
-
a
2
F
I
-
b
1
+
b
2
F
R
-
a
1
F
R
-
a
1
2
+
F
I
-
b
1
2
,where(11)
F
R
=
Re
F
c
P
,
F
I
=
Im
F
c
P
.IfR
0 and X
0 are all equal to zero, the CUT is equal to an ideal voltage source u
oc corresponding to the component P. This means that variation in Z
F will not affect u
F and F
c
P. Thus, we have (12)
R
0
R
F
=
a
2
F
R
-
a
1
+
b
2
F
I
-
b
1
F
R
-
a
1
2
+
F
I
-
b
1
2
-
1
,
X
0
R
F
=
-
a
2
F
I
-
b
1
+
b
2
F
R
-
a
1
F
R
-
a
1
2
+
F
I
-
b
1
2
.Then(13)
R
0
X
0
=
a
2
F
R
-
a
1
+
b
2
F
I
-
b
1
-
F
R
-
a
1
2
-
F
I
-
b
1
2
-
a
2
F
I
-
b
1
+
b
2
F
R
-
a
1
.Furthermore, (13) can be transformed into a formula of 2D circle:(14)
F
R
-
c
r
2
+
F
I
-
c
j
2
=
r
c
2
,
(15)
c
r
=
-
b
2
R
0
+
2
a
1
X
0
+
a
2
X
0
2
X
0
,
c
j
=
a
2
R
0
+
2
b
1
X
0
+
b
2
X
0
2
X
0
,
r
c
2
=
a
2
2
+
b
2
2
R
0
2
+
X
0
2
4
X
0
2
,where c
r and c
j are the real part and imaginary part of the circle center coordinates, respectively. r
c is the radius. It is confirmed that c
r, c
j, and r
c are independent of the parametric change of the failure component P. Therefore, fault loci fitting requires a small amount of simulation on potential faults, for both catastrophic faults and parametric faults. Alternatively, if Z
F is assumed to be a pure reactance X
F, similar conclusion can be reached.For a CUT withn components P
i
(
1
≤
i
≤
n
), fault modeling is to determine the model parameters c
r
i, c
j
i, and r
c
i for each P
i according to (15), where parameters c
r
i, c
j
i, and r
c
i are corresponding to c
r, c
j, and r
c, respectively. In most actual analog cases, (14) is hard to derive from transfer function directly. However, it is well known that three sets of distinct data are sufficient to determine a circle. In this regard, simulation is a simple method to establish the circle model in (14). Therefore, in the modeling process, three sets of distinct output data are obtained by parametric sweep simulations, corresponding to three distinct values of each component P
i. For the whole CUT, n parametric sweep simulations are needed. Three sets of distinct simulation data for each component are assumed as F
c
0, F
c
i
1, and F
c
i
2. Since F
c
0 is assumed to be the fault free output, it fits in with all components. The fault modeling process is illustrated in Figure 1. In addition, the following equation is used to calculate the parameters c
r
i, c
j
i, and r
c
i for component P
i
(
1
≤
i
≤
n
):(16)
Re
F
c
0
-
c
r
i
2
+
Im
F
c
0
-
c
j
i
2
=
r
c
i
2
,
Re
F
c
i
1
-
c
r
i
2
+
Im
F
c
i
1
-
c
j
i
2
=
r
c
i
2
,
Re
F
c
i
2
-
c
r
i
2
+
Im
F
c
i
2
-
c
j
i
2
=
r
c
i
2
.Figure 1
Flowchart of fault modeling.


## 2.2. Example of a Sallen-Key Filter

Compared with the difficulty of obtaining explicit mathematical expression, simulation is a relatively easy way for fault modeling. To demonstrate the method, a second-order Sallen-Key filter, as shown in Figure2, is adopted as an example of circuit under test (CUT). By parameter sweeping simulations with respect to the supply current I
d
d and the output response U
o in PSPICE, the 2D collaborative fault model can be achieved. The simulation results are listed in Table 1, in which parametric ratio means the ratio between failure value and nominal value for each component. Further results are also provided in Figure 3. The stimulation is a 3 kHz, 1 V sine signal, and the parameter sweeping range for each component is assumed to be from p
i
×
10
-
4 to p
i
×
10
4, where p
i
(
1
≤
i
≤
n
) is the fault-free value of the ith component F
i. In addition, the simulation range of R
3 and R
4 is reduced, because output power of the filter is limited. As shown in Figure 3, all potential fault statuses compose a family of loci on the complex plane, which is called F
i-loci. Every F
i-locus corresponds to all the potential fault statuses of a component, not only parametric faults but also catastrophic faults. Furthermore, F
i-loci converges in the fault free point A and zero point B.Table 1
Simulation results of 2D model for the Sallen-Key filter.
Parametric ratio
Collaborative simulation results
F
C
R
1
F
C
R
2
F
C
R
3
F
C
R
4
F
C
C
1
F
C
C
2
0.01
0.7977
-
j
1.0218
1.7383
-
j
0.3307
—
-
0.0364
-
j
0.4066
1.1249
+
j
0.6892
0.2792
-
j
0.8047
0.05
0.7959
-
j
1.0537
1.7801
-
j
0.4091
—
-
0.0369
-
j
0.4382
1.2076
+
j
0.6979
0.2816
-
j
0.8265
0.08
0.7925
-
j
1.0777
1.8094
-
j
0.4733
—
-
0.0373
-
j
0.4627
1.2755
+
j
0.7012
0.2830
-
j
0.8264
0.1
0.7906
-
j
1.0949
1.8271
-
j
0.5187
—
-
0.0375
-
j
0.4794
1.3282
+
j
0.7019
0.2839
-
j
0.8519
0.4
0.7063
-
j
1.3735
1.7917
-
j
1.4096
—
-
0.0348
-
j
0.7801
2.4864
+
j
0.2002
0.2905
-
j
1.0578
0.7
0.4829
-
j
1.6766
0.9513
-
j
2.0720
0.7186
-
j
3.9260
-
0.0105
-
j
1.2155
1.9996
-
j
2.3457
0.2565
-
j
1.3783
0.9
0.2312
-
j
1.8427
0.3137
-
j
2.0195
0.1476
-
j
2.2588
0.0354
-
j
1.6315
0.4333
-
j
2.2252
0.1647
-
j
1.7011
1.0
0.0767
-
j
1.8977
0.0767
-
j
1.8977
0.0767
-
j
1.8977
0.0767
-
j
1.8977
0.0767
-
j
1.8977
0.0767
-
j
.
8977
1.1
-
0.9320
-
j
1.9275
-
0.1008
-
j
1.7530
0.0385
-
j
1.6543
0.1390
-
j
2.2194
-
0.1113
-
j
1.6025
-
0.0746
-
j
2.1441
1.3
-
0.4407
-
j
1.8928
-
0.3146
-
j
1.4630
0.0013
-
j
1.3453
0.3832
-
j
3.0993
-
0.2476
-
j
1.1936
-
0.6829
-
j
2.6877
1.5
-
0.7474
-
j
1.7426
-
0.4093
-
j
1.2169
-
0.0151
-
j
1.1573
1.0301
-
j
4.5183
-
0.2751
-
j
0.9275
-
2.0075
-
j
2.9003
2
-
1.1278
-
j
1.1435
-
0.4420
-
j
0.8107
-
0.0304
-
j
0.9029
15.264
-
j
3.1208
-
0.2445
-
j
0.5844
-
2.8027
+
j
0.5055
3
-
0.9616
-
j
0.3874
-
0.3514
-
j
0.4570
-
0.0366
-
j
0.7040
—
-
0.1726
-
j
0.3330
-
0.7652
+
j
0.6990
5
-
0.5445
-
j
0.0583
-
0.2246
-
j
0.2339
-
0.0379
-
j
0.5686
—
-
0.1030
-
j
0.1748
-
0.2644
+
j
0.3346
10
-
0.2431
+
j
0.0129
-
0.1143
-
j
0.1029
-
0.0375
-
j
0.4793
—
-
0.0509
-
j
0.0800
-
0.0979
+
j
0.1423
30
-
0.0739
+
j
0.0104
-
0.0383
-
j
0.0314
-
0.0367
-
j
0.4248
—
-
0.0168
-
j
0.0252
-
0.0273
+
j
0.0425
100
-
0.0215
+
j
0.0036
-
0.0115
-
j
0.0091
-
0.0364
-
j
0.4067
—
-
0.0050
-
j
0.0074
-
0.0078
+
j
0.0124Figure 2
Second-order Sallen-key filter, whereR
1, R
2, R
3 and R
4 are 10 kΩ, C
1 and C
2 are 4.7 nF.Figure 3
2D collaborative fault model of the Sallen-Key filter. All potential fault statuses of the filter compose a family ofF
i-loci, and F
i-loci converge in the fault free point A and zero point B.Note that in some cases, distances between someF
i-loci are very small. In extreme cases, some F
i-loci may coincide with each other, which results in ambiguity group. Since the 2D collaborative fault model is derived from transfer function of CUT, ambiguity group may be an inevitable problem [15]. Table 2 lists the ambiguity groups of the Sallen-Key filter, and Figure 3 shows that while the F
i-loci of R
1, R
2, C
1, and C
2 are obviously different from others, fault status of R
3 and R
4 are located on the same locus. This means that potential faults of R
3 and R
4 cannot be isolated based on the fault model in Figure 3; therefore, R
3 and R
4 fall into the same ambiguity group {
R
3
,
R
4
}.Table 2
Ambiguity groups of the Sallen-Key filter.
Ambiguity group
Fault components
①
R
1
②
R
2
③
R
3, R
4
④
C
1
⑤
C
2


## 3. Fault Modeling in 3D Complex Space

Based on the measurements of supply current and voltage response at any test point, the 2D collaborative fault model transforms all potential failure statuses into a family of loci. Therefore, both parametric faults and catastrophic faults can all be detected and located. However, due to the limits of component tolerance and measurement error, the 2D fault model is difficult to apply in actual analog circuits. As depicted in Figure3, between point A and point B, the R
2 and C
1-loci are too close to be distinguished. Therefore, to increase the practicability for actual applications, the 2D fault model is extended into 3D complex space.


### 3.1. Theorem of 3D Collaborative Fault Model

The 2D function in (3) is a linear combination of supply current I
d
d and voltage response U
o, based on the measurements of real part Re
· and imaginary part Im
· of each complex variable. I
d
d and U
o are easier to measure in-circuit from limited test points than other circuit variables, such as the input admittance. Furthermore, the complex modulus (absolute value) · is also employed to compose the 3D function. This helps to increase the distance between F
i-loci fault models and improve the FIR.Similarly, the componentP in (2) is still assumed to be the failure component. Therefore, by defining x, y, and z as the unit vectors in 3D complex space, a transformation function F
3
D
1
· can be composed as follow [14]:(17)
F
3
D
1
P
=
x
·
Re
F
U
P
+
y
·
Im
F
U
P
+
z
·
1
F
I
D
D
P
,where(18)
F
U
P
=
c
0
U
o
,
(19)
F
I
D
D
P
=
1
c
1
I
d
d
,where c
0 and c
1 are still adjustment coefficients as in (3). Taking the component pair R
2-C
1 as an example, the distance D
3
D
1
· between R
2 and C
1-loci is(20)
D
3
D
1
R
2
,
C
1
=
D
x
1
R
2
,
C
1
2
+
D
y
1
R
2
,
C
1
2
+
D
z
1
R
2
,
C
1
2
,where D
x
1
·, D
y
1
·, and D
z
1
· are distances corresponding to each axe in 3D complex space:(21)
D
x
1
R
2
,
C
1
=
Re
F
U
R
2
-
Re
F
U
C
1
,
D
y
1
R
2
,
C
1
=
Im
F
U
R
2
-
Im
F
U
C
1
,
D
z
1
R
2
,
C
1
=
1
F
I
D
D
R
2
-
1
F
I
D
D
C
1
.Although the distanceD
3
D
1
· in (20) has been proved to be better than that in 2D F
i-loci fault model [14], it is calculated based on voltage and input admittance, whose absolute values are usually far greater than supply current. Therefore, for an appropriate coefficient c
1, the following inequality can be guaranteed:(22)
1
F
I
D
D
P
=
c
1
I
d
d
<
1
.In addition, a new distanceD
z
2
· is defined as follows:(23)
D
z
2
R
2
,
C
1
=
F
I
D
D
R
2
-
F
I
D
D
C
1
,
D
z
2
R
2
,
C
1
=
D
z
1
R
2
,
C
1
1
/
F
I
D
D
R
2
·
1
/
F
I
D
D
C
1
>
D
z
1
R
2
,
C
1
.Therefore, a new transformation functionF
3
D
2
· for the 3D fault model can be defined as(24)
F
3
D
2
P
=
x
·
Re
F
U
P
+
y
·
Im
F
U
P
+
z
·
F
I
D
D
P
.Furthermore, the distanceD
3
D
2
· is defined for the new function F
3
D
2
·. Taking R
2 and C
1 as an example, it can be expressed as(25)
D
3
D
2
R
2
,
C
1
=
D
x
1
R
2
,
C
1
2
+
D
y
1
R
2
,
C
1
2
+
D
z
2
R
2
,
C
1
2
.


### 3.2. Simulation Example of Sallen-Key Filter for the 3D Model

For a given input of 1 V, 3 kHz sine signal, PSPICE results on the Sallen-Key filter are obtained. The key part of simulation results are listed in Table3. The original simulation results include U
o and I
d
d, corresponding to all potential failure status of each component, such as R
2 and C
1, and then all the data in Table 3 is obtained according to (18) and (19). Moreover, the simulation results are also illustrated integrally in Figure 4. It shows that all F
i-loci converge in the fault free point A, which is similar to the 2D F
i-loci. However, due to the definition in (19), some F
i-loci tend to be an infinity point B, instead of the determinate point B in 2D model. The associated ambiguity groups are the same as shown in Table 2.Table 3
Simulation results forR
2 and C
1 in 3D model.
Parametric ratio
Simulation results ofR
2
Simulation results ofC
1
F
U
R
2
F
I
D
D
R
2
c
1
I
d
d
F
U
C
1
F
I
D
D
C
1
c
1
I
d
d
0.01
2.052
-
j
0.040
2.3371
-
0.3137
-
j
0.2910
1.156
+
j
1.009
3.1270
-
0.0276
-
j
0.3186
0.05
2.115
-
j
0.121
2.2642
-
0.3349
-
j
0.2879
1.251
+
j
1.034
2.9564
-
0.0380
-
j
0.3361
0.08
2.161
-
j
0.189
2.116
-
0.3516
-
j
0.2843
1.328
+
j
1.051
2.8331
-
0.0472
-
j
0.3498
0.1
2.190
-
j
0.238
2.1782
-
0.3629
-
j
0.2812
1.382
+
j
1.061
2.7540
-
0.0538
-
j
0.3591
0.4
2.323
-
j
1.263
1.8144
-
0.5313
-
j
0.1466
2.804
+
j
0.713
1.6581
-
0.3176
-
j
0.5127
0.7
1.491
-
j
2.184
1.8142
-
0.5397
+
j
0.1120
2.743
-
j
2.293
1.3418
-
0.7434
-
j
0.0527
0.9
0.752
-
j
2.248
2.0242
-
0.4380
+
j
0.2285
0.929
-
j
2.460
1.8244
-
0.4953
+
j
0.2348
1.0
0.457
-
j
2.155
2.1775
-
0.3804
+
j
0.2573
0.457
-
j
2.155
2.1775
-
0.3804
+
j
0.2573
1.1
0.226
-
j
2.024
2.3555
-
0.3268
+
0.2710
0.185
-
j
1.854
2.5751
-
0.2959
+
j
0.2515
1.3
-
0.075
-
j
1.734
2.7645
-
0.2396
+
j
0.2710
-
0.051
-
j
1.413
3.3929
-
0.1970
+
j
0.2194
1.5
-
0.231
-
j
1.471
3.2227
-
0.1781
+
j
0.2541
-
0.134
-
j
1.114
4.2771
-
0.1410
+
j
0.1865
2
-
0.348
-
j
1.013
4.4787
-
0.0945
+
j
0.2023
-
0.166
-
j
0.716
6.5293
-
0.0787
+
j
0.1314
3
-
0.313
-
j
0.591
7.1763
-
0.0386
+
j
0.1339
-
0.133
-
j
0.414
11.0258
-
0.0401
+
j
0.0814
5
-
0.211
-
j
0.311
12.7565
-
0.0133
+
j
0.0773
-
0.084
-
j
0.220
20.3875
-
0.0193
+
j
0.0451
10
-
0.111
-
j
0.140
26.8834
-
0.0036
+
j
0.0370
-
0.043
-
j
0.101
43.6119
-
0.0083
+
j
0.0214
30
-
0.038
-
j
0.043
83.5634
-
0.0006
+
j
0.0120
-
0.014
-
j
0.032
136.5537
-
0.0025
+
j
0.0069
100
-
0.011
-
j
0.013
281.8780
-
0.0001
+
j
0.0035
-
0.004
-
j
0.009
461.8366
-
0.0007
+
j
0.0020Figure 4
Family of fault loci in 3D complex place for the Sallen-Key filter. AllF
i-loci converge in the fault free point A, while some F
i-loci have tendency to an infinity point B.To validate the improvement of fault modelF
3
D
2
· in (24), the 2D model proposed in Section 2 is used as a reference. In addition, both the 3D model F
3
D
1
· proposed in [14], and the complex-circle-based fault model proposed in [12, 13] are adopted for comparison. In the example of R
2 and C
1 in the Sallen-Key filter, distances between R
2 and C
1-loci are provided in Table 4, according to each fault model, respectively; the comparison range is from p
i
×
10
-
2 to p
i
×
10
2, where p
i is the fault-free value of R
2 and C
1. The results of comparison between different methods are also illustrated in Figure 5, in which the curve d
1 is the distance of R
2
-
C
1 in the 3D model F
3
D
2
·, the curve d
2 is the distance defined in the 2D model, the dash curve d
3 is defined by F
3
D
1
· [14], and dash curve d
4 is related to complex-circle-based fault model [12, 13]. As shown in Table 4, d
3 is better than d
4; however, because the biggest gap between d
3 and d
4 is less than 2.2%, d
4 is very close to d
3 in Figure 5. Moreover, d
1 is much better than all the others. Since d
3 is also based on the 3D model in [14], it is taken as an ideal reference. On average, d
1 is 9.6% bigger than d
3, in the value range from p
i
×
10
-
2 to p
i. Moreover, in the range from p
i to p
i
×
10
2, because I
d
d and U
o tend to be zero, d
1 increases continuously while d
3 reduces, and the advantage of d
1 is more significant. Thus, 3D fault model F
3
D
2
· results in better FIR against the measurement errors and parametric tolerance.Table 4
Distances betweenR2 and C1-loci with respect to different fault models.
Parametric ratio
d
1
d
2
d
3
d
4
0.01
1.5896
1.1902
1.3836
1.3805
0.05
1.6000
1.2463
1.4463
1.4453
0.08
1.6180
1.2902
1.4971
1.4963
0.1
1.6342
1.3184
1.5324
1.5294
0.4
2.0396
1.7533
2.0343
2.0336
0.7
1.3426
1.0835
1.2716
1.2567
0.9
0.3408
0.2379
0.2814
0.2761
1.0
0
0
0
0
1.1
0.2808
0.1509
0.1787
0.1750
1.3
0.7061
0.2776
0.3288
0.3219
1.5
1.1175
0.3190
0.3778
0.3700
2
2.0800
0.3004
0.3553
0.3483
3
3.8577
0.2176
0.2570
0.2523
5
7.6327
0.1353
0.1596
0.1568
10
16.7287
0.0675
0.0795
0.0782
30
52.9903
0.0223
0.0263
0.0259
100
179.9586
0.0067
0.0079
0.0077Figure 5
Distances betweenR
2 and C
1-loci with respect to different fault models. Curve d
1 is defined by the 3D model proposed in Section 3, d
2 is the 2D model proposed in Section 2, d
3 is corresponding to the 3D model proposed in [14], and d
4 is corresponding to the complex-circle-based fault model proposed in [12, 13].


## 3.1. Theorem of 3D Collaborative Fault Model

The 2D function in (3) is a linear combination of supply current I
d
d and voltage response U
o, based on the measurements of real part Re
· and imaginary part Im
· of each complex variable. I
d
d and U
o are easier to measure in-circuit from limited test points than other circuit variables, such as the input admittance. Furthermore, the complex modulus (absolute value) · is also employed to compose the 3D function. This helps to increase the distance between F
i-loci fault models and improve the FIR.Similarly, the componentP in (2) is still assumed to be the failure component. Therefore, by defining x, y, and z as the unit vectors in 3D complex space, a transformation function F
3
D
1
· can be composed as follow [14]:(17)
F
3
D
1
P
=
x
·
Re
F
U
P
+
y
·
Im
F
U
P
+
z
·
1
F
I
D
D
P
,where(18)
F
U
P
=
c
0
U
o
,
(19)
F
I
D
D
P
=
1
c
1
I
d
d
,where c
0 and c
1 are still adjustment coefficients as in (3). Taking the component pair R
2-C
1 as an example, the distance D
3
D
1
· between R
2 and C
1-loci is(20)
D
3
D
1
R
2
,
C
1
=
D
x
1
R
2
,
C
1
2
+
D
y
1
R
2
,
C
1
2
+
D
z
1
R
2
,
C
1
2
,where D
x
1
·, D
y
1
·, and D
z
1
· are distances corresponding to each axe in 3D complex space:(21)
D
x
1
R
2
,
C
1
=
Re
F
U
R
2
-
Re
F
U
C
1
,
D
y
1
R
2
,
C
1
=
Im
F
U
R
2
-
Im
F
U
C
1
,
D
z
1
R
2
,
C
1
=
1
F
I
D
D
R
2
-
1
F
I
D
D
C
1
.Although the distanceD
3
D
1
· in (20) has been proved to be better than that in 2D F
i-loci fault model [14], it is calculated based on voltage and input admittance, whose absolute values are usually far greater than supply current. Therefore, for an appropriate coefficient c
1, the following inequality can be guaranteed:(22)
1
F
I
D
D
P
=
c
1
I
d
d
<
1
.In addition, a new distanceD
z
2
· is defined as follows:(23)
D
z
2
R
2
,
C
1
=
F
I
D
D
R
2
-
F
I
D
D
C
1
,
D
z
2
R
2
,
C
1
=
D
z
1
R
2
,
C
1
1
/
F
I
D
D
R
2
·
1
/
F
I
D
D
C
1
>
D
z
1
R
2
,
C
1
.Therefore, a new transformation functionF
3
D
2
· for the 3D fault model can be defined as(24)
F
3
D
2
P
=
x
·
Re
F
U
P
+
y
·
Im
F
U
P
+
z
·
F
I
D
D
P
.Furthermore, the distanceD
3
D
2
· is defined for the new function F
3
D
2
·. Taking R
2 and C
1 as an example, it can be expressed as(25)
D
3
D
2
R
2
,
C
1
=
D
x
1
R
2
,
C
1
2
+
D
y
1
R
2
,
C
1
2
+
D
z
2
R
2
,
C
1
2
.


## 3.2. Simulation Example of Sallen-Key Filter for the 3D Model

For a given input of 1 V, 3 kHz sine signal, PSPICE results on the Sallen-Key filter are obtained. The key part of simulation results are listed in Table3. The original simulation results include U
o and I
d
d, corresponding to all potential failure status of each component, such as R
2 and C
1, and then all the data in Table 3 is obtained according to (18) and (19). Moreover, the simulation results are also illustrated integrally in Figure 4. It shows that all F
i-loci converge in the fault free point A, which is similar to the 2D F
i-loci. However, due to the definition in (19), some F
i-loci tend to be an infinity point B, instead of the determinate point B in 2D model. The associated ambiguity groups are the same as shown in Table 2.Table 3
Simulation results forR
2 and C
1 in 3D model.
Parametric ratio
Simulation results ofR
2
Simulation results ofC
1
F
U
R
2
F
I
D
D
R
2
c
1
I
d
d
F
U
C
1
F
I
D
D
C
1
c
1
I
d
d
0.01
2.052
-
j
0.040
2.3371
-
0.3137
-
j
0.2910
1.156
+
j
1.009
3.1270
-
0.0276
-
j
0.3186
0.05
2.115
-
j
0.121
2.2642
-
0.3349
-
j
0.2879
1.251
+
j
1.034
2.9564
-
0.0380
-
j
0.3361
0.08
2.161
-
j
0.189
2.116
-
0.3516
-
j
0.2843
1.328
+
j
1.051
2.8331
-
0.0472
-
j
0.3498
0.1
2.190
-
j
0.238
2.1782
-
0.3629
-
j
0.2812
1.382
+
j
1.061
2.7540
-
0.0538
-
j
0.3591
0.4
2.323
-
j
1.263
1.8144
-
0.5313
-
j
0.1466
2.804
+
j
0.713
1.6581
-
0.3176
-
j
0.5127
0.7
1.491
-
j
2.184
1.8142
-
0.5397
+
j
0.1120
2.743
-
j
2.293
1.3418
-
0.7434
-
j
0.0527
0.9
0.752
-
j
2.248
2.0242
-
0.4380
+
j
0.2285
0.929
-
j
2.460
1.8244
-
0.4953
+
j
0.2348
1.0
0.457
-
j
2.155
2.1775
-
0.3804
+
j
0.2573
0.457
-
j
2.155
2.1775
-
0.3804
+
j
0.2573
1.1
0.226
-
j
2.024
2.3555
-
0.3268
+
0.2710
0.185
-
j
1.854
2.5751
-
0.2959
+
j
0.2515
1.3
-
0.075
-
j
1.734
2.7645
-
0.2396
+
j
0.2710
-
0.051
-
j
1.413
3.3929
-
0.1970
+
j
0.2194
1.5
-
0.231
-
j
1.471
3.2227
-
0.1781
+
j
0.2541
-
0.134
-
j
1.114
4.2771
-
0.1410
+
j
0.1865
2
-
0.348
-
j
1.013
4.4787
-
0.0945
+
j
0.2023
-
0.166
-
j
0.716
6.5293
-
0.0787
+
j
0.1314
3
-
0.313
-
j
0.591
7.1763
-
0.0386
+
j
0.1339
-
0.133
-
j
0.414
11.0258
-
0.0401
+
j
0.0814
5
-
0.211
-
j
0.311
12.7565
-
0.0133
+
j
0.0773
-
0.084
-
j
0.220
20.3875
-
0.0193
+
j
0.0451
10
-
0.111
-
j
0.140
26.8834
-
0.0036
+
j
0.0370
-
0.043
-
j
0.101
43.6119
-
0.0083
+
j
0.0214
30
-
0.038
-
j
0.043
83.5634
-
0.0006
+
j
0.0120
-
0.014
-
j
0.032
136.5537
-
0.0025
+
j
0.0069
100
-
0.011
-
j
0.013
281.8780
-
0.0001
+
j
0.0035
-
0.004
-
j
0.009
461.8366
-
0.0007
+
j
0.0020Figure 4
Family of fault loci in 3D complex place for the Sallen-Key filter. AllF
i-loci converge in the fault free point A, while some F
i-loci have tendency to an infinity point B.To validate the improvement of fault modelF
3
D
2
· in (24), the 2D model proposed in Section 2 is used as a reference. In addition, both the 3D model F
3
D
1
· proposed in [14], and the complex-circle-based fault model proposed in [12, 13] are adopted for comparison. In the example of R
2 and C
1 in the Sallen-Key filter, distances between R
2 and C
1-loci are provided in Table 4, according to each fault model, respectively; the comparison range is from p
i
×
10
-
2 to p
i
×
10
2, where p
i is the fault-free value of R
2 and C
1. The results of comparison between different methods are also illustrated in Figure 5, in which the curve d
1 is the distance of R
2
-
C
1 in the 3D model F
3
D
2
·, the curve d
2 is the distance defined in the 2D model, the dash curve d
3 is defined by F
3
D
1
· [14], and dash curve d
4 is related to complex-circle-based fault model [12, 13]. As shown in Table 4, d
3 is better than d
4; however, because the biggest gap between d
3 and d
4 is less than 2.2%, d
4 is very close to d
3 in Figure 5. Moreover, d
1 is much better than all the others. Since d
3 is also based on the 3D model in [14], it is taken as an ideal reference. On average, d
1 is 9.6% bigger than d
3, in the value range from p
i
×
10
-
2 to p
i. Moreover, in the range from p
i to p
i
×
10
2, because I
d
d and U
o tend to be zero, d
1 increases continuously while d
3 reduces, and the advantage of d
1 is more significant. Thus, 3D fault model F
3
D
2
· results in better FIR against the measurement errors and parametric tolerance.Table 4
Distances betweenR2 and C1-loci with respect to different fault models.
Parametric ratio
d
1
d
2
d
3
d
4
0.01
1.5896
1.1902
1.3836
1.3805
0.05
1.6000
1.2463
1.4463
1.4453
0.08
1.6180
1.2902
1.4971
1.4963
0.1
1.6342
1.3184
1.5324
1.5294
0.4
2.0396
1.7533
2.0343
2.0336
0.7
1.3426
1.0835
1.2716
1.2567
0.9
0.3408
0.2379
0.2814
0.2761
1.0
0
0
0
0
1.1
0.2808
0.1509
0.1787
0.1750
1.3
0.7061
0.2776
0.3288
0.3219
1.5
1.1175
0.3190
0.3778
0.3700
2
2.0800
0.3004
0.3553
0.3483
3
3.8577
0.2176
0.2570
0.2523
5
7.6327
0.1353
0.1596
0.1568
10
16.7287
0.0675
0.0795
0.0782
30
52.9903
0.0223
0.0263
0.0259
100
179.9586
0.0067
0.0079
0.0077Figure 5
Distances betweenR
2 and C
1-loci with respect to different fault models. Curve d
1 is defined by the 3D model proposed in Section 3, d
2 is the 2D model proposed in Section 2, d
3 is corresponding to the 3D model proposed in [14], and d
4 is corresponding to the complex-circle-based fault model proposed in [12, 13].


## 4. DFT by Adding Switched Bypass-Components

### 4.1. Theory of Improved DFT Method

A common theory of ambiguity group in fault diagnosis has been proposed in previous research [15]. If the sensitivities of some analog components are linearly dependent on all frequencies, the change of these components cannot be distinguished, and thus these components fall into an ambiguity group. Based on this theory, components R
a and R
b in an analog CUT are assumed to fall into the ambiguity groups {
R
a
,
R
b
}; similar to R
3 and R
4 of the Sallen-Key filter in Figure 2, a linear ratio r is defined as(26)
r
=
R
b
R
a
.In addition,H
s is assumed to be the transfer function of CUT and it is a function of r as(27)
H
s
=
f
r
,where s is the Laplacian operator. Therefore, (28)
∂
H
s
∂
R
a
=
∂
f
∂
r
∂
r
∂
R
a
=
∂
f
∂
r
-
R
b
R
a
2
,
(29)
∂
H
s
∂
R
b
=
∂
f
∂
r
∂
r
∂
R
b
=
∂
f
∂
r
1
R
a
=
-
1
r
∂
H
s
∂
R
a
.Equation (29) shows the sensitivity dependence between R
a and R
b, which results in the ambiguity group {
R
a
,
R
b
}. Therefore, transforming the topological structure of CUT and breaking the linear sensitivity dependence r should be an effective DFT method. One of the most common DFT methods is bypassing and the traditional bypassing method is based on reducing the capacitive effects [16–18]. However, for the 3D model in complex space, keeping or increasing the capacitive effects is expected to be a far more effective method.For the purpose of increasing the capacitive effects,C
b is assumed to be added in CUT as a parallel branch of R
b, and then the impedance Z
b of the parallel branch is(30)
Z
b
=
R
b
×
1
/
s
C
b
R
b
+
1
/
s
C
b
=
R
b
1
+
s
C
b
R
b
.Thus the new linear ratior
′ is defined as(31)
r
′
=
Z
b
R
a
=
R
b
R
a
1
+
s
C
b
R
b
.So the corresponding transfer functionH
′
s is defined based on function f
·:(32)
H
′
s
=
f
r
′
,
∂
H
′
s
∂
R
a
=
∂
f
∂
r
′
∂
r
′
∂
R
a
=
∂
f
∂
r
′
R
b
1
+
s
C
b
R
b
-
1
R
a
2
,
∂
H
′
s
∂
R
b
=
∂
f
∂
r
′
∂
r
′
∂
R
b
=
∂
f
∂
r
′
1
R
a
1
1
+
s
C
b
R
b
2
,
∂
H
′
s
∂
C
b
=
∂
f
∂
r
′
∂
r
′
∂
C
b
=
∂
f
∂
r
′
R
b
R
a
-
s
R
b
1
+
s
C
b
R
b
2
;then(33)
∂
H
′
s
/
∂
R
a
∂
H
′
s
/
∂
R
b
=
-
R
b
1
+
s
C
b
R
b
R
a
,
∂
H
′
s
/
∂
R
a
∂
H
′
s
/
∂
C
b
=
s
R
b
1
+
s
C
b
R
b
R
a
,
∂
H
′
s
/
∂
R
b
∂
H
′
s
/
∂
C
b
=
-
1
s
R
b
2
.As shown in (33), the parameter sensitivities for R
a, R
b, and C
b are not linear obviously, which means that their faults can be isolated. In a similar way, if R
a and C
b fall into the same ambiguity group because their sensitivities are linearly dependent, some R
b can be paralleled with C
b for DFT purpose. Therefore, if the status of CUT is partitioned into working mode and testing mode, the DFT and testing flow proposed in previous can be illustrated in Figure 6.Figure 6
Flowchart of bypassing-based DFT and testing.


### 4.2. Example and Simulation on Sallen-Key Filter

In the Sallen-Key filter, not only the collaborative fault model on 2D Complex Plane but also theF
i-loci in 3D complex space show that R
3 and R
4 fall into an ambiguity group, which is also illustrated in Table 2. Therefore, for the purpose of DFT, the switched bypass-component C
3 is added by using a P-channel MOS switch Q
1 in Figure 7. When the control signal S
1
=
1, Q
1 is turned off, the CUT is in common working mode, and C
3 is bypassed by Q
1. Otherwise, when S
1
=
0, the CUT is switched into testing mode and C
3 is parallel to R
4 because Q
1 is turned on.Figure 7
The Sallen-Key filter after adding switched bypass-componentC
3. C
3 is added for the purpose of DFT, and MOS switch Q
1 is used to bypass C
3 in filter mode.With the input of 1 V, 3 kHz sine signal, PSPICE simulation results in Figure8 shows the F
i-loci of the Sallen-Key filter after DFT, where R
3 and R
4-loci separate obviously. All F
i-loci converge in the fault free point A, while points B and C are the other two intersections, and point B is located at infinity theoretically. Furthermore, the updated ambiguity groups are listed in Table 5 and R
3 and R
4 belong to different ambiguity groups. This means that adding switched bypass-components is an effective method to improve the FIR.Table 5
Updated ambiguity groups of the Sallen-Key filter by DFT.
Ambiguity group
Fault components
①
R
1
②
R
2
③
R
3
④
R
4
⑤
C
1
⑥
C
2
⑦
C
3Figure 8
3D fault model of the Sallen-key filter by DFT. Point A means the fault free status of allF
i-loci, and some F
i-loci are tendency to an infinity point B.In addition, compared with the DFT methods proposed in [16–18], the improved method given in this paper is also more effective to some extent. Both Sun [16] and Hong [17] reduce capacitive effects by bypassing the capacitors in original CUT, to achieve DFT in testing mode; however, this method cannot isolate the ambiguity group like {
R
3
,
R
4
} in the Sallen-Key filter. The DFT method given in this paper is based on the bypassing approach proposed in [16]. It improves DFT method by adding some redundant components in the CUT and bypassing these components only in working mode. It offers appealing potential to isolate some ambiguity groups compared with the methods in [16, 17]. Moreover, isolation of both catastrophic faults and parametric faults of analog circuits is possible with the improved DFT method.


## 4.1. Theory of Improved DFT Method

A common theory of ambiguity group in fault diagnosis has been proposed in previous research [15]. If the sensitivities of some analog components are linearly dependent on all frequencies, the change of these components cannot be distinguished, and thus these components fall into an ambiguity group. Based on this theory, components R
a and R
b in an analog CUT are assumed to fall into the ambiguity groups {
R
a
,
R
b
}; similar to R
3 and R
4 of the Sallen-Key filter in Figure 2, a linear ratio r is defined as(26)
r
=
R
b
R
a
.In addition,H
s is assumed to be the transfer function of CUT and it is a function of r as(27)
H
s
=
f
r
,where s is the Laplacian operator. Therefore, (28)
∂
H
s
∂
R
a
=
∂
f
∂
r
∂
r
∂
R
a
=
∂
f
∂
r
-
R
b
R
a
2
,
(29)
∂
H
s
∂
R
b
=
∂
f
∂
r
∂
r
∂
R
b
=
∂
f
∂
r
1
R
a
=
-
1
r
∂
H
s
∂
R
a
.Equation (29) shows the sensitivity dependence between R
a and R
b, which results in the ambiguity group {
R
a
,
R
b
}. Therefore, transforming the topological structure of CUT and breaking the linear sensitivity dependence r should be an effective DFT method. One of the most common DFT methods is bypassing and the traditional bypassing method is based on reducing the capacitive effects [16–18]. However, for the 3D model in complex space, keeping or increasing the capacitive effects is expected to be a far more effective method.For the purpose of increasing the capacitive effects,C
b is assumed to be added in CUT as a parallel branch of R
b, and then the impedance Z
b of the parallel branch is(30)
Z
b
=
R
b
×
1
/
s
C
b
R
b
+
1
/
s
C
b
=
R
b
1
+
s
C
b
R
b
.Thus the new linear ratior
′ is defined as(31)
r
′
=
Z
b
R
a
=
R
b
R
a
1
+
s
C
b
R
b
.So the corresponding transfer functionH
′
s is defined based on function f
·:(32)
H
′
s
=
f
r
′
,
∂
H
′
s
∂
R
a
=
∂
f
∂
r
′
∂
r
′
∂
R
a
=
∂
f
∂
r
′
R
b
1
+
s
C
b
R
b
-
1
R
a
2
,
∂
H
′
s
∂
R
b
=
∂
f
∂
r
′
∂
r
′
∂
R
b
=
∂
f
∂
r
′
1
R
a
1
1
+
s
C
b
R
b
2
,
∂
H
′
s
∂
C
b
=
∂
f
∂
r
′
∂
r
′
∂
C
b
=
∂
f
∂
r
′
R
b
R
a
-
s
R
b
1
+
s
C
b
R
b
2
;then(33)
∂
H
′
s
/
∂
R
a
∂
H
′
s
/
∂
R
b
=
-
R
b
1
+
s
C
b
R
b
R
a
,
∂
H
′
s
/
∂
R
a
∂
H
′
s
/
∂
C
b
=
s
R
b
1
+
s
C
b
R
b
R
a
,
∂
H
′
s
/
∂
R
b
∂
H
′
s
/
∂
C
b
=
-
1
s
R
b
2
.As shown in (33), the parameter sensitivities for R
a, R
b, and C
b are not linear obviously, which means that their faults can be isolated. In a similar way, if R
a and C
b fall into the same ambiguity group because their sensitivities are linearly dependent, some R
b can be paralleled with C
b for DFT purpose. Therefore, if the status of CUT is partitioned into working mode and testing mode, the DFT and testing flow proposed in previous can be illustrated in Figure 6.Figure 6
Flowchart of bypassing-based DFT and testing.


## 4.2. Example and Simulation on Sallen-Key Filter

In the Sallen-Key filter, not only the collaborative fault model on 2D Complex Plane but also theF
i-loci in 3D complex space show that R
3 and R
4 fall into an ambiguity group, which is also illustrated in Table 2. Therefore, for the purpose of DFT, the switched bypass-component C
3 is added by using a P-channel MOS switch Q
1 in Figure 7. When the control signal S
1
=
1, Q
1 is turned off, the CUT is in common working mode, and C
3 is bypassed by Q
1. Otherwise, when S
1
=
0, the CUT is switched into testing mode and C
3 is parallel to R
4 because Q
1 is turned on.Figure 7
The Sallen-Key filter after adding switched bypass-componentC
3. C
3 is added for the purpose of DFT, and MOS switch Q
1 is used to bypass C
3 in filter mode.With the input of 1 V, 3 kHz sine signal, PSPICE simulation results in Figure8 shows the F
i-loci of the Sallen-Key filter after DFT, where R
3 and R
4-loci separate obviously. All F
i-loci converge in the fault free point A, while points B and C are the other two intersections, and point B is located at infinity theoretically. Furthermore, the updated ambiguity groups are listed in Table 5 and R
3 and R
4 belong to different ambiguity groups. This means that adding switched bypass-components is an effective method to improve the FIR.Table 5
Updated ambiguity groups of the Sallen-Key filter by DFT.
Ambiguity group
Fault components
①
R
1
②
R
2
③
R
3
④
R
4
⑤
C
1
⑥
C
2
⑦
C
3Figure 8
3D fault model of the Sallen-key filter by DFT. Point A means the fault free status of allF
i-loci, and some F
i-loci are tendency to an infinity point B.In addition, compared with the DFT methods proposed in [16–18], the improved method given in this paper is also more effective to some extent. Both Sun [16] and Hong [17] reduce capacitive effects by bypassing the capacitors in original CUT, to achieve DFT in testing mode; however, this method cannot isolate the ambiguity group like {
R
3
,
R
4
} in the Sallen-Key filter. The DFT method given in this paper is based on the bypassing approach proposed in [16]. It improves DFT method by adding some redundant components in the CUT and bypassing these components only in working mode. It offers appealing potential to isolate some ambiguity groups compared with the methods in [16, 17]. Moreover, isolation of both catastrophic faults and parametric faults of analog circuits is possible with the improved DFT method.


## 5. Conclusion

In order to cope with the difficulties of fault diagnosis in analog circuits, a 2D fault model based on collaborative supply current and output voltage is first proposed in this paper. This 2D fault model is proved to be a family of circles on complex plane and helps to simplify the simulation of potential faults greatly. Then, the fault model is improved in the 3D complex space, and a family ofF
i-loci is obtained to illustrate the potential faults of each analog component. Since the distances between each F
i-locus are increased, the 3D model achieves a far better FIR against the measurement error and parametric tolerance. The proposed fault models are validated through examples on a Sallen-Key filter.However, it is worth noting that ambiguity group is still an inevitable problem. Therefore, to increase the FIR sequentially, an improved bypassing-based DFT method is introduced by using MOS switches and adding some redundant components in the CUT. The simulation results show that, except for few limited failure statuses, the parametric and catastrophic faults in the analog CUT can be isolated and the problem of ambiguity groups is well resolved with improved FIR. The fault models on 2D complex plane and 3D space are all based on the assumption of single fault; thus the improved bypassing methods proposed in this paper are also limited to single fault. In the presence of multifaults, this fault model becomes complicated and thus further research effort is needed in this area.



---

*Source: 851837-2015-03-09.xml*