# Optimal Method for Catastrophic Faults Diagnosis in RC Ladder Network

Authors: Abhilasha Rani Goel; Mohd Wajid
Journal: Advances in Power Electronics
 (2016)

---

## Abstract

The RC ladder network has been analyzed for various catastrophic fault detection using minimal number of measurements. Generally, electronic circuit testing procedure is very exhaustive and includes higher cost; the presented approach will save fault diagnosis time. It is not possible to analyze the big RC ladder network to give the good fault coverage, so the ladder network has been broken into segments of different sizes. However, if segment size is small, it will cause more area overhead compared to bigger step size in terms of the interconnections and pins on the integrated circuit. A systematic and detailed analysis for one-step, two-step, three-step, and four-step RC ladder networks has been carried out for various faults and optimal step size is proposed. It has been investigated that three measurements are optimal to localize different catastrophic faults in a RC ladder network.


---

## Body

## 1. Introduction

The resistive ladder network has been analyzed for detecting various catastrophic faults associated with it, where only resistive components are considered [1]. However, there are many circuits which use capacitor also, like RC ladder network. The ladder networks with resistor and capacitors are widely used in filters, phase shifter circuits, oscillators, and so forth [2, 3]. As per the knowledge of authors, the related literature on the fault diagnosis in RC network is limited. In this and next paragraph, we have presented available existing methods/algorithms associated with fault identification in the circuits. Huang et al. have presented an approach based on an assemblage of learning machine that is trained to guide through diagnosis decision. It diagnoses the hard/soft fault by using defect filter. The hard faults have been diagnosed using the multiclass classifier and the soft fault is diagnosed using the inverse regression functions. The disadvantage of this method is in resolving the ambiguity; this method uses some auxiliary circuit specific fault diagnosis rules [4]. Kyzioł et al. [5] have given an algorithm to diagnose the catastrophic faults in analog circuits. This algorithm uses the Particle Swarm Optimization (PSO); it uses more than one dimension like load resistance and reactance, generator resistance, and reactance and generator frequency to diagnose the faults. They have shown that increasing the numbers of dimensions of search space influences the identification of states of circuit under test (CUT). The disadvantage of this method is that it can diagnose only single catastrophic fault of CUT.Starzyk et al. [6] have proposed an algorithm based on entropy index of available test points, where two-dimensional integer coded dictionary is created whose entries are measurements associated with faults and test points. Though this algorithm can be used for medium and large networks also, it is very costly. Huang et al. [7] have given a method to diagnose the local spot defects in analog circuit. This method is based on the combination of multiclass classifiers that are trained using data from fault simulation. The problem has been viewed as pattern recognition task. The method starts by inductive fault analysis (IFA) which results in a list of probable defects; then the defects are ranked based on their probability of occurrence. After that, multiclass classifiers are used to detect the fault. It is a probability based method, not so much accurate.In the present work, we have analyzed the RC ladder network (Figure1) for fault diagnosis. Two catastrophic faults, open and short, are considered in resistors and one catastrophic fault is considered in capacitors; that is, capacitor impedance is equal to zero. In integrated circuit, the separation between two wires cannot be very large, so infinite impedance of capacitor is not considered. Different size segments of RC ladder networks, that is, 1-step, 2-step, 3-step, and 4-step, are assessed. In this paper, the impedance X
c is varied as a function of n as given in (1), where 1
≤
n
≤
30 and other parameters R, ω, and C are kept constant (R
=
10 KΩ, C
=
100 pF, and ω
=
2
π
10
4 rad/sec):(1)
X
c
=
1
j
n
ω
c
.We have assumed that the network has only one fault at a time. Z
i
j is an equivalent impedance measured between ith and jth nodes (Figure 1). Notations gives the list of notations used.Figure 1
RC ladder network:N
i indicates node number.To calculate the fault coverage, two characteristics are considered, that is,distinguishability andambiguity. Distinguishability means that the impedance plot for any faulty case can be distinguished from the impedance plot of the faultless case, but it will not cover the overlapping of plots of any two distinct faults. Because distinguishability does not cover the overlapping of two plots for any two faulty cases, ambiguity is defined separately. Ambiguity means that the plots for any of the two faulty cases overlap each other and it does not cover the faultless case. A deviation d (see (2)) of 2% is considered as acceptable value in the given impedance plots as taken in [1]. If the plot for one fault lies within 2% of another plot for the given value of n, then both of those plots are considered as nondistinguishable or ambiguous:(2)
d
=
Z
i
j
f
o
r
f
a
u
l
t
l
e
s
s
c
a
s
e
-
Z
i
j
f
o
r
f
a
u
l
t
y
c
a
s
e
Z
i
j
f
o
r
f
a
u
l
t
l
e
s
s
c
a
s
e
×
100
.


## 2. Analysis

In this section, a large RC ladder network is broken into small size RC ladder networks, namely, one-step, two-step, three-step, and four-step RC network. These different step size networks are analyzed for fault detection for different values ofn, 1
≤
n
≤
30. In this paper, all graphs are shown only for n
=
5 to 30, because for n
=
1 to 4 some of the plots have very high values as compared to others, so that they cannot be plotted. Also, it has been assumed that a
=
ω
n
R
C. All the analysis is done mathematically and all the graphs are plotted in the MATLAB software.


### 2.1. Single-Step RC Ladder Network

The single-step RC ladder network is shown in Figure2. In this network, only three faults exist, that is, R
12 short circuit, R
12 open circuit, and C
20 which is short; that is, X
C
20
=
0:(3)
Z
01
2
=
1
+
ω
n
R
C
2
ω
2
n
2
C
2
.Figure 2
Single-step RC ladder network.It can be concluded from Table1 that all the three values of Z
01
2 for different fault types are distinct and faults can be detected with no ambiguity, so the fault coverage is 100%. But computational complexity and area overhead are very large, because number of measurements will be equal to number of components and many numbers of single-step RC ladder network have to be tested to identify the faults of a big network [8]. Also, large numbers of observable nodes are required, so this cannot be considered from integrated circuit point of view. When we are breaking the big RC network into small RC network, this will also use additional circuits for the switches which lead to increase in area overhead [1, 9].Table 1
Expressions for single-step RC ladder network.
Fault type
Z
01
2
R
12 short
1
ω
2
n
2
C
2
R
12 open
I
n
f
i
n
i
t
y
X
C
20 short
R
2


### 2.2. Two-Step RC Ladder Network

In this section, the detailed analysis of two-step RC ladder network as shown in Figure3 is conducted.Figure 3
Two-step RC ladder network.Table2 gives the Z
01
2, Z
02
2, and Z
13
2 expressions for considering each and every fault as well as for faultless case.Table 2
Expressions for two-step RC ladder network.
Fault type
Z
01
2
Z
02
2
Z
13
2
F
a
u
l
t
l
e
s
s
a
6
+
11
a
4
+
21
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
a
6
+
20
a
4
+
64
a
2
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
R
12 short
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
4
a
4
+
16
a
2
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
R
12 open
I
n
f
i
n
i
t
y
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
I
n
f
i
n
i
t
y
X
C
20 short
R
2
a
4
+
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
a
6
+
5
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
R
23 short
4
a
2
+
1
4
ω
2
n
2
C
2
1
4
ω
2
n
2
C
2
R
2
R
23 open
a
2
+
1
ω
2
n
2
C
2
1
ω
2
n
2
C
2
a
2
+
4
ω
2
n
2
C
2
X
C
30 short
a
6
+
5
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
0
a
6
+
5
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]Figures4, 5, and 6 show the values of magnitude of Z
01, Z
02, and Z
13, respectively, plotted against “n.”Figure 4
Plot ofZ
01 versus “n” for different faults of two-step network.Figure 5
Plot ofZ
02 versus “n” for different faults of two-step network.Figure 6
Plot ofZ
13 versus “n” for different faults of two-step network.Table3 shows the conclusion derived from Figures 4, 5, and 6 about faults distinguishability and required measuring parameters for the same.Table 3
Distinguishability for two-step RC ladder network.
Fault type
Distinguishability
By measuring
R
12 short
Distinguishable at alln
Z
13
R
12 open
Distinguishable at alln
Z
01 or Z
13
X
C
20 short
Distinguishable at alln
Z
01
R
23 short
Distinguishable at alln
Z
13
R
23 open
Distinguishable at alln
Z
02
X
C
30 short
Distinguishable at alln
Z
02Tables4, 5, and 6 show the ambiguity for each fault which is not distinguishable from the measurement of Z
01, Z
02, and Z
13, respectively.Table 4
Ambiguity fromZ
01 for two-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
X
C
20 short (n
=
8, 9), R
23 short (n
=
1), X
C
30 short (n
=
4)
R
12 open
No ambiguity
X
C
20 short
R
12 short (n
=
8, 9), R
23 short (n
≥
18)
R
23 short
R
12 short (n
=
1), X
C
20 short (n
≥
18), X
C
30 short (n
=
5)
R
23 open
X
C
30 short (n
=
11, 12, 13)
X
C
30 short
R
12 short (n
=
4), R
23 open (n
=
11, 12, 13), R
23 short (n
=
5)Table 5
Ambiguity from |Z
02| for two-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
12 open (all n), X
C
20 short (n
≥
10), R
23 short (n
≤
10)
R
12 open
R
12 short (all n), X
C
20 short (n
≥
10), R
23 short (n
≤
10)
X
C
20 short
R
12 short (n
≥
10), R
12 open (n
≥
10), R
23 short (n
=
9, 10), R
23 open (n
≥
26)
R
23 short
R
12 short (n
≤
10), R
12 open (n
≤
10), X
C
20 short (n
=
9, 10)
R
23 open
X
C
20 short (n
≥
26)
X
C
30 short
No ambiguityTable 6
Ambiguity from |Z
13| for two-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
23 short (n
≤
15), R
23 open (n
≥
21)
R
12 open
No ambiguity
X
C
20 short
X
C
30 short (all n)
R
23 short
R
12 short (n
≤
15)
R
23 open
R
12 short (n
≥
21)
X
C
30 short
X
C
20 short (all n)Table7 gives the conclusion derived from Tables 4, 5, and 6.Table 7
Ambiguity from |Z
01|, |Z
02|, and |Z
13| for two-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
23 short (n
=
1)
R
12 open
No ambiguity
X
C
20 short
No ambiguity
R
23 short
R
12 short (n
=
1)
R
23 open
No ambiguity
X
C
30 short
No ambiguityFigure7 shows the fault coverage for two-step RC ladder network. Fault coverage is 100% except for n
=
1. All three measurements, that is, Z
01, Z
02, and Z
13, are required to calculate the distinguishability and ambiguity.Figure 7
Fault coverage for two-step RC ladder network.


### 2.3. Three-Step RC Ladder Network

In this section, detailed analysis of three-step RC ladder network (Figure8) is done.Figure 8
Three-step RC ladder network.For faultless network, the expression forZ
01
2 is given by(4)
Z
01
2
=
a
10
+
23
a
8
+
165
a
6
+
378
a
4
+
244
a
2
+
9
ω
2
n
2
C
2
a
8
+
20
a
6
+
118
a
4
+
180
a
2
+
81
.Table8 gives the Z
01
2 expression for considering each and every fault. Similar analysis is done for Z
02
2 and Z
14
2 and the results are shown below.Table 8
Z
01
2 expression for three-step RC ladder network.
Fault location
Short faultZ
01
2
Open faultZ
01
2
R
12
a
8
+
17
a
6
+
80
a
4
+
73
a
2
+
9
ω
2
n
2
C
2
[
a
8
+
20
a
6
+
118
a
4
+
180
a
2
+
81
]
Infinity
X
C
20
R
2
None
R
23
16
a
6
+
84
a
4
+
112
a
2
+
9
ω
2
n
2
C
2
8
a
4
+
72
a
2
+
81
a
2
+
1
ω
2
n
2
C
2
X
C
30
a
6
+
5
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
None
R
34
16
a
6
+
120
a
4
+
193
a
2
+
9
ω
2
n
2
C
2
[
16
a
4
+
72
a
2
+
81
]
a
6
+
11
a
4
+
21
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
40
a
10
+
17
a
8
+
80
a
6
+
73
a
4
+
9
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
NoneFor a faultless network, the expression forZ
02
2 and Z
14
2 is given by(5)
Z
22
2
=
a
8
+
17
a
6
+
80
a
4
+
73
a
2
+
9
ω
2
n
2
C
2
a
8
+
20
a
6
+
118
a
4
+
180
a
2
+
81
,
Z
14
2
=
a
9
+
20
a
7
+
79
a
5
+
2124
a
3
2
+
2
a
8
+
34
a
6
+
224
a
4
+
720
a
2
2
ω
2
n
2
C
2
a
8
+
18
a
6
+
57
a
4
+
1336
a
2
+
144
.Table9 gives the Z
02
2 expressions for considering each and every fault and Table 10 gives the Z
14
2 expressions for each and every fault.Table 9
Z
02
2 expression for three-step RC ladder network.
Fault location
Short faultZ
02
2
Open faultZ
02
2
R
12
a
8
+
17
a
6
+
80
a
4
+
73
a
2
+
9
ω
2
n
2
C
2
[
a
8
+
20
a
6
+
118
a
4
+
180
a
2
+
81
]
a
8
+
17
a
6
+
80
a
4
+
73
a
2
+
9
ω
2
n
2
C
2
[
a
8
+
20
a
6
+
118
a
4
+
180
a
2
+
81
]
X
C
20
a
8
+
11
a
6
+
29
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
None
R
23
16
a
4
+
40
a
2
+
9
ω
2
n
2
C
2
[
16
a
4
+
72
a
2
+
81
]
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
30
a
4
+
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
None
R
34
4
a
4
+
13
a
2
+
9
ω
2
n
2
C
2
[
16
a
4
+
72
a
2
+
81
]
1
ω
2
n
2
C
2
X
C
40
0
NoneTable 10
Z
14
2 expression for three-step RC ladder network.
Fault location
Short faultZ
14
2
Open faultZ
14
2
R
12
2
a
7
+
34
a
5
+
512
a
3
-
288
a
2
+
2
a
8
+
34
a
6
+
128
a
4
+
1056
a
2
2
ω
2
n
2
C
2
a
8
+
18
a
6
+
57
a
4
+
1336
a
2
+
144
2
Infinity
X
C
20
a
10
+
17
a
8
+
80
a
6
+
73
a
4
+
9
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
None
R
23
4
a
7
+
50
a
5
+
208
a
3
+
288
a
2
+
6
a
6
+
48
a
4
+
96
a
2
2
ω
2
n
2
C
2
4
a
6
+
41
a
4
+
136
a
2
+
144
2
a
6
+
14
a
4
+
49
a
2
+
36
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
30
a
6
+
10
a
4
+
9
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
None
R
34
16
a
6
+
180
a
4
+
324
a
2
ω
2
n
2
C
2
[
16
a
4
+
72
a
2
+
81
]
a
6
+
14
a
4
+
49
a
2
+
36
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
40
a
10
+
17
a
8
+
80
a
6
+
73
a
4
+
9
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
NoneFigure9 shows the values of magnitude of Z
01 plotted against “n.” Fault o
p
n
R
12
-
Z
01 is not shown in Figure 9, as it has value of infinity. Therefore, eight curves are shown corresponding to other short and open faults.Figure 9
Plot ofZ
01 versus “n” for different faults of three-step network.Figure10 shows the values of magnitude of Z
02 plotted against “n.” A
c
t
u
a
l
-
Z
02, s
r
t
R
12
-
Z
02, and o
p
n
R
12
-
Z
02 overlap each other, so all these plots are represented by bold line in Figure 10; fault o
p
n
R
34
-
Z
02 is excluded from the graph because it has very high value as compared to other faults. Therefore, 6 lines are shown corresponding to other short and open faults.Figure 10
Plot ofZ
02 versus “n” for different faults of three-step network.Figure11 shows the values of magnitude of Z
14 versus “n.” Faults s
r
t
X
C
20
-
Z
14 and s
r
t
X
C
40
-
Z
14 are overlapping; o
p
n
R
23
-
Z
14 and o
p
n
R
34
-
Z
14 are overlapping; s
r
t
R
23
-
Z
14 and s
r
t
R
34
-
Z
14 are overlapping; one of the faults o
p
n
R
12
-
Z
14 is not shown in Figure 11, as it has value of infinity. Therefore only 5 curves are shown corresponding to other short and open faults.Figure 11
Plot ofZ
14 versus “n” for different faults of three-step network.Table11 shows the conclusion derived from Figures 9–11.Table 11
Distinguishability for three-step RC ladder network.
Fault type
Distinguishability
By measuring
R
12 short
Distinguishable at alln
Z
01
R
12 open
Distinguishable at alln
Z
01 or Z
14
X
C
20 short
Distinguishable at alln
Z
01
R
23 short
Distinguishable except in the rangen = 26–29
Z
01 (n
<
26), Z
14 (n
=
30)
R
23 open
Distinguishable except in the rangen = 22–29
Z
14
X
C
30 short
Distinguishable except atn = 30
Z
14
R
34 short
Distinguishable at alln
Z
02 (n
>
1), Z
14 (n
=
1)
R
34 open
Distinguishable except atn = 29
Z
02 (n
<
29), Z
14 (n
=
30)
X
C
40 short
Distinguishable at alln
Z
02Tables12, 13, and 14 show the ambiguity for each fault which is not distinguishable from |
Z
01
|, |
Z
02
|, and |
Z
14
|, respectively. Table 15 gives the conclusion derived from Tables 12, 13, and 14. Figure 12 shows the fault coverage for three-step RC ladder network. Fault coverage is 100% for some values of “n.” For n
=
2 to 10 and 13 to 20 the fault coverage is 100%, but for other values of n fault coverage is less than 100%. All the measurements, that is, Z
01, Z
02, and Z
14 are required to calculate the distinguishability and ambiguity.Table 12
Ambiguity fromZ
01 for three-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
X
C
20 short (n = 7–9), R
23 short (n = 1), X
C
30 short (n = 3)
R
12 open
No ambiguity
X
C
20 short
R
12 short (n = 7–9)
R
23 short
R
12 short (n = 1), R
23 open (n ≥ 21), R
34 short (n ≥ 27), R
34 open (n ≥ 11), X
C
40 short (n ≥ 26)
R
23 open
R
23 short (n ≥ 21), X
C
30 short (n = 11–13), R
34 short (n = 14–24), R
34 open (n ≥ 20), X
C
40 short (n ≥ 10)
X
C
30 short
R
12 short (n = 3), R
23 open (n = 11–13), R
34 short (n = 4, n
≥
23), R
34 open (n = 5), X
C
40 short (n = 10–17, 23–30)
R
34 short
R
23 short (n ≥ 27), R
23 open (n = 14–24), X
C
30 short (n = 4, n
≥
23), X
C
40 short (n ≥ 15), R
34 open (n = 6, 7, 30)
R
34 open
R
23 short (n ≥ 11), R
23 open (n ≥ 20), X
C
30 short (n = 5), R
34 short (n = 6, 7, 30), X
C
40 short (n ≥ 28)
X
C
40 short
R
23 short (n ≥ 26), R
23 open (n ≥ 10), X
C
30 short (n = 10–17, 23–30), R
34 short (n ≥ 15), R
34 open (n ≥ 28)Table 13
Ambiguity fromZ
02 for three-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
12 open (all n), X
C
20 short (n
≥
15), R
23 short (all n), R
23 open, X
C
30 short (n
≥
8), R
34 short (n
=
1), R
34 open (n
=
29  & 30)
R
12 open
R
12 short, R
23 short (all n), X
C
20 short (n
≥
15), R
23 open, X
C
30 short (n
≥
8), R
34 short (n
=
1), R
34 open (n
=
29  & 30)
X
C
20 short
R
12 short, R
12 open (n
≥
15), R
23 short (n
=
3  & ≥18), R
23 open (n
=
5
-
6  & ≥18), X
C
30 short (n
≥
13), R
34 short (n
=
3), R
34 open (n
=
29
-
30)
R
23 short
R
12 short, R
12 open (all n), X
C
20 short (n
=
3  & ≥18), R
23 open (n
≥
11), X
C
30 short (n
≥
7), R
34 short (n
=
1
–
3)
R
23 open
R
12 short, R
12 open (n
≥
8), X
C
20 short (n
=
5
-
6  & ≥18), R
23 short (n
≥
11), X
C
30 short (n
≥
10)
X
C
30 short
R
12 short, R
12 open (n
≥
8), X
C
20 short (n
≥
13), R
23 short (n
≥
7), R
23 open (n
≥
10), R
34 short (n
=
6), R
34 open (n
≥
25)
R
34 short
R
12 short, R
12 open (n
=
1), X
C
20 short (n
=
3), R
23 short (n
=
1
–
3), X
C
30 short (n
=
6)
R
34 open
R
12 short, R
12 open (n
=
29  & 30), X
C
20 short (n
=
29
-
30), X
C
30 short (n
≥
25)
X
C
40 short
No ambiguityTable 14
Ambiguity from|
Z
14
| for three-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
23 short, R
34 short (n
=
1)
R
12 open
No ambiguity
X
C
20 short
R
23 short, R
34 short (n
=
9, 10), X
C
30 short (n
=
1), X
C
40 short (all n)
R
23 short
R
12 short (n
=
1), R
23 open, R
34 open (n
≥
21), R
34 short (all n), X
C
20 short, X
C
40 short (n
=
9, 10)
R
23 open
R
23 short, R
34 short (n
≥
21), R
34 open (all n), X
C
30 short (n
=
11, 12)
X
C
30 short
X
C
20 short, X
C
40 short (n
=
1), R
23 open, R
34 open (n
=
11, 12)
R
34 short
R
12 short (n
=
1), R
23 open, R
34 open (n
≥
21), R
23 short (all n), X
C
20 short, X
C
40 short (n
=
9, 10)
R
34 open
R
23 short, R
34 short (n
≥
21), R
23 open (all n), X
C
30 short (n
=
11, 12)
X
C
40 short
R
23 short, R
34 short (n
=
9, 10), X
C
30 short (n
=
1), X
C
20 short (all n)Table 15
Ambiguity fromZ
01, Z
02, and Z
14 for three-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
23 short (n
=
1)
R
12 open
No ambiguity
X
C
20 short
No ambiguity
R
23 short
R
12 short (n
=
1), R
23 open (n
≥
21)
R
23 open
R
23 short (n
≥
21), X
C
30 short (n
=
11, 12)
X
C
30 short
R
23 open (n
=
11, 12)
R
34 short
No ambiguity
R
34 open
No ambiguity
X
C
40 short
No ambiguityFigure 12
Fault coverage for three-step RC ladder network.


### 2.4. Four-Step RC Ladder Network

In this section, detailed analysis of four-step RC ladder network (Figure13) has been performed. As in the previous case, this network is also analyzed for n
=
1 to 30.Figure 13
Four-step RC ladder network.For faultless network, the expression forZ
01
2 is(6)
Z
11
2
=
a
7
+
17
a
5
+
63
a
3
+
30
a
2
+
a
6
+
14
a
4
+
34
a
2
+
4
2
ω
2
n
2
C
2
a
6
+
16
a
4
+
52
a
2
+
16
2
.Table16 gives the expressions of Z
01
2 for all faults. In these expressions, (7)
a
=
ω
n
R
C
,where ω
=
2
π
f and f
=
10 KHz. The parameter “n” is varied from 1 to 30.Table 16
Z
01
2 expression for four-step RC ladder network.
Fault location
Short faultZ
01
2
Open faultZ
01
2
R
12
a
5
+
11
a
3
+
14
a
2
+
a
6
+
14
a
4
+
34
a
2
+
4
2
ω
2
n
2
C
2
a
6
+
16
a
4
+
148
a
2
+
16
2
Infinity
X
C
20
R
2
None
R
23
4
a
5
+
34
a
3
+
21
a
2
+
2
a
4
+
15
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
65
a
2
+
16
2
a
2
+
1
ω
2
n
2
C
2
X
C
30
a
6
+
5
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
None
R
34
4
a
5
+
24
a
3
+
26
a
2
+
4
a
4
+
14
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
20
a
2
+
16
2
a
6
+
11
a
4
+
29
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
40
a
10
+
17
a
8
+
80
a
6
+
73
a
4
+
9
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
None
R
45
4
a
5
+
37
a
3
+
29
a
2
+
4
a
4
+
25
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
33
a
2
+
16
2
a
5
+
11
a
3
+
14
a
2
+
a
4
+
8
a
2
+
3
2
ω
2
n
2
C
2
a
4
+
10
a
2
+
9
2
X
C
50
a
7
+
14
a
5
+
34
a
3
+
4
a
2
+
a
6
+
11
a
4
+
14
a
2
2
ω
2
n
2
C
2
a
6
+
13
a
4
+
26
a
2
+
1
2
NoneSimilar analysis is done forZ
02
2 and Z
15
2; for a faultless network the expression for Z
02
2 and Z
15
2 is given by(8)
Z
22
2
=
a
5
+
11
a
3
+
14
a
2
+
a
6
+
14
a
4
+
34
a
2
+
4
2
ω
2
n
2
C
2
a
6
+
16
a
4
+
52
a
2
+
16
2
,
Z
15
2
=
a
9
+
22
a
7
+
144
a
5
+
352
a
3
+
256
a
2
+
2
a
8
+
36
a
6
+
192
a
4
+
320
a
2
2
ω
2
n
2
C
2
a
8
+
20
a
6
+
116
a
4
+
224
a
2
+
64
2
.Table17 gives the Z
02
2 expressions for considering each and every fault and Table 18 gives the Z
15
2 expressions for each and every fault.Table 17
Z
02
2 expression for four-step RC ladder network.
Fault location
Short faultZ
02
2
Open faultZ
02
2
R
12
a
5
+
11
a
3
+
14
a
2
+
a
6
+
14
a
4
+
34
a
2
+
4
2
ω
2
n
2
C
2
a
6
+
16
a
4
+
52
a
2
+
16
2
a
5
+
11
a
3
+
14
a
2
+
a
6
+
14
a
4
+
34
a
2
+
4
2
ω
2
n
2
C
2
a
6
+
16
a
4
+
52
a
2
+
16
2
X
C
20
a
5
+
8
a
3
+
3
a
2
+
a
6
+
11
a
4
+
14
a
2
2
ω
2
n
2
C
2
a
6
+
13
a
4
+
26
a
2
+
1
2
None
R
23
4
a
3
+
13
a
2
+
4
a
4
+
25
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
33
a
2
+
16
2
a
3
+
5
a
2
+
a
4
+
8
a
2
+
3
2
ω
2
n
2
C
2
a
4
+
10
a
2
+
9
2
X
C
30
a
8
+
11
a
6
+
29
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
None
R
34
4
a
3
+
10
a
2
+
4
a
4
+
14
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
20
a
2
+
16
2
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
40
a
4
+
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
None
R
45
a
3
+
5
a
2
+
2
a
4
+
15
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
33
a
2
+
16
2
1
ω
2
n
2
C
2
X
C
50
0
NoneTable 18
Z
15
2 expression for four-step RC ladder network.
Fault location
Short faultZ
15
2
Open faultZ
15
2
R
12
2
a
7
+
28
a
5
+
128
a
3
+
192
a
2
+
2
a
8
+
36
a
6
+
192
a
4
+
320
a
2
2
ω
2
n
2
C
2
a
8
+
20
a
6
+
116
a
4
+
224
a
2
+
64
2
Infinity
X
C
20
a
11
+
22
a
9
+
162
a
7
+
500
a
5
+
576
a
3
+
64
a
2
+
a
10
+
19
a
8
+
118
a
6
+
288
a
4
+
224
a
2
2
ω
2
n
2
C
2
a
10
+
21
a
8
+
146
a
6
+
417
a
4
+
424
a
2
+
16
2
None
R
23
4
a
9
+
70
a
7
+
416
a
5
+
992
a
3
+
768
a
2
+
6
a
8
+
92
a
6
+
448
a
4
+
704
a
2
2
ω
2
n
2
C
2
4
a
8
+
65
a
6
+
344
a
4
+
656
a
2
+
256
2
a
9
+
19
a
7
+
118
a
5
+
288
a
3
+
224
a
2
+
2
a
8
+
34
a
6
+
188
a
4
+
384
a
2
+
192
2
ω
2
n
2
C
2
a
8
+
18
a
6
+
105
a
4
+
232
a
2
+
144
2
X
C
30
a
11
+
18
a
9
+
114
a
7
+
308
a
5
+
320
a
3
+
64
a
2
+
2
a
10
+
29
a
8
+
142
a
6
+
256
a
4
+
96
a
2
2
ω
2
n
2
C
2
a
10
+
16
a
8
+
88
a
6
+
193
a
4
+
4
a
2
+
16
2
None
R
34
4
a
5
+
28
a
3
+
48
a
2
+
8
a
4
+
32
a
2
2
ω
2
n
2
C
2
4
a
4
+
20
a
2
+
16
2
a
6
+
16
a
4
+
52
a
2
+
16
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
40
a
11
+
18
a
9
+
114
a
7
+
308
a
5
+
320
a
3
+
64
a
2
+
2
a
10
+
29
a
8
+
142
a
6
+
256
a
4
+
96
a
2
2
ω
2
n
2
C
2
a
10
+
16
a
8
+
88
a
6
+
193
a
4
+
4
a
2
+
16
2
None
R
45
4
a
7
+
54
a
5
+
200
a
3
+
192
a
2
+
6
a
6
+
68
a
4
+
176
a
2
2
ω
2
n
2
C
2
4
a
6
+
49
a
4
+
148
a
2
+
64
2
a
7
+
15
a
5
+
58
a
3
+
56
a
2
+
2
a
6
+
26
a
4
+
84
a
2
+
48
2
ω
2
n
2
C
2
a
6
+
14
a
4
+
49
a
2
+
36
2
X
C
50
a
11
+
22
a
9
+
162
a
7
+
500
a
5
+
576
a
3
+
64
a
2
+
a
10
+
19
a
8
+
118
a
6
+
288
a
4
+
224
a
2
2
ω
2
n
2
C
2
a
10
+
21
a
8
+
146
a
6
+
417
a
4
+
424
a
2
+
16
2
NoneFigure14 shows the values of magnitude of Z
01 on the y-axis, plotted against “n” on the x-axis. Fault o
p
n
R
12
-
Z
01 is not shown in Figure 14, as it has value of infinity. Therefore, 11 lines are shown corresponding to other short and open faults.Figure 14
Plot ofZ
01 versus “n” for different faults of four-step network.Figure15 shows the values of magnitude of Z
02 on the Y-axis, plotted against “n” on the X-axis. A
c
t
u
a
l
-
Z
02, s
r
t
R
12
-
Z
02, and o
p
n
R
12
-
Z
02 are overlapping, so all these plots are represented by bold line in Figure 15; fault o
p
n
R
45
-
Z
02 was excluded from the graph, since it has very high value compared to the curves. Therefore, 9 lines are shown corresponding to other short and open faults.Figure 15
Plot of|
Z
02
| versus “n” for different faults of four-step network.Figure16 shows the values of magnitude of Z
15 versus “n.” Faults s
r
t
X
C
20
-
Z
15 and s
r
t
X
C
50
-
Z
15 are overlapping; s
r
t
X
C
30
-
Z
15 and s
r
t
X
C
40
-
Z
15 are overlapping; s
r
t
R
23
-
Z
15 and s
r
t
R
45
-
Z
15 are overlapping; o
p
n
R
23
-
Z
15 and o
p
n
R
45
-
Z
15 are overlapping; one of the faults o
p
n
R
12
-
Z
15 is not shown in Figure 16, as it has value of infinity. Therefore 7 lines are shown corresponding to other short and open faults.Figure 16
Plot of|
Z
15
| versus “n” for different faults of four-step network.Table19 shows the conclusion derived from Figures 14, 15, and 16.Table 19
Distinguishability for four-step RC ladder network.
Fault type
Distinguishability
By measuring
R
12 short
Distinguishable at alln
Z
01 or Z
15
R
12 open
Distinguishable at alln
Z
01 or Z
15
X
C
20 short
Distinguishable at alln
Z
01 or Z
15
R
23 short
Distinguishable at alln
Z
01 or Z
15
R
23 open
Distinguishable except atn
=
29, 30
Z
01 (n
=
8, 9), Z
15 (all other n)
X
C
30 short
Distinguishable except atn
=
30
Z
15
R
34 short
Distinguishable except forn
≥
13
Z
15
R
34 open
Distinguishable except atn
=
30
Z
02 (n
=
5), Z
15 (all other n)
X
C
40 short
Distinguishable except atn
=
30
Z
15
R
45 short
Distinguishable at alln
Z
02 or Z
15
R
45 open
Distinguishable except atn
=
30
Z
02
X
C
50 short
Distinguishable at alln
Z
02 or Z
15Tables20, 21, and 22 show the ambiguity for each fault which is not distinguishable from the measurement of |
Z
01
|, |
Z
02
|, and |
Z
15
|, respectively.Table 20
Ambiguity from|
Z
01
| for four-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
X
C
20 short (n
=
5)
R
12 open
No ambiguity
X
C
20 short
R
12 short (n
=
5), R
23 short (n
=
8
–
10)
R
23 short
X
C
20 short (n
=
8
–
10)
R
23 open
X
C
30 short (n
=
11, 12), R
34 short (n
=
13
–
20), R
34 open (n
≥
15), X
C
40 short (n
≥
10), R
45 short, X
C
50 short (n
≥
12), R
45 open (n
≥
13)
X
C
30 short
R
23 open (n
=
11, 12), R
34 short (n
=
4, 5  & ≥20), R
34 open (n
=
6, 29, 30), X
C
40 short (n
=
10
–
17  & ≥22), R
45 short (n
=
5
–
10  & ≥23), R
45 open (n
=
5, 6  & ≥23), X
C
50 short (n
=
8
–
11  & ≥25)
R
34 short
R
23 open (n
=
13
–
20), X
C
30 short (n
=
4, 5  & ≥20), R
34 open (n
=
6
–
9  & ≥19), X
C
40 short (n
≥
13), R
45 short (n
≥
9), R
45 open (n
≥
5), X
C
50 short (n
≥
10)
R
34 open
R
23 open (n
≥
15), X
C
30 short (n
=
6, 29, 30), R
34 short (n
=
6
–
9  & ≥19), X
C
40 short (n
≥
19), R
45 short (n
=
5, 6  & ≥18), R
45 open (n
=
6
–
9,  & ≥15), X
C
50 short (n
=
3  & ≥16)
X
C
40 short
R
23 open (n
≥
10), X
C
30 short (n
=
10
–
17  & ≥22), R
34 short (n
≥
13), R
34 open (n
≥
19), R
45 short (n
≥
12), R
45 open (n
≥
14), X
C
50 short (n
=
5, 6  & ≥12)
R
45 short
R
23 open (n
≥
12), X
C
30 short (n
=
5
–
10  & ≥23), R
34 short (n
≥
9), R
34 open (n
=
5, 6  & ≥18), X
C
40 short (n
≥
12), R
45 open (n
≥
3), X
C
50 short (n
≥
9)
R
45 open
R
23 open (n
≥
13), X
C
30 short (n
=
5, 6  & ≥23), R
34 short (n
≥
5), R
34 open (n
=
6
–
9,  & ≥15), X
C
40 short (n
≥
14), R
45 short (n
≥
3), X
C
50 short (n
≥
11)
X
C
50 short
R
23 open (n
≥
12), X
C
30 short (n
=
8
–
11  & ≥25), R
34 short (n
≥
10), R
34 open (n
=
3  & ≥16), X
C
40 short (n
=
5, 6  & ≥12), R
45 short (n
≥
9), R
45 open (n
≥
11)Table 21
Ambiguity from|
Z
02
| for four-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
12 open (all n), X
C
20 short (n
≥
9), R
23 short (all n), R
23 open (n
≥
4), X
C
30 short (n
=
3  & ≥12), R
34 short (n
≥
12), R
34 open (n
≥
7), X
C
40 short (n
≥
10), R
45 open (n
=
30)
R
12 open
R
12 short (all n), X
C
20 short (n
≥
9), R
23 short (all n), R
23 open (n
≥
4), X
C
30 short (n
=
3  & ≥12), R
34 short (n
≥
12), R
34 open (n
≥
7), X
C
40 short (n
≥
10), R
45 open (n
=
30)
X
C
20 short
R
12 short, R
12 open (n
≥
9), R
23 short (n
≥
10), R
23 open, X
C
40 short (n
≥
12), X
C
30 short, R
34 open (n
≥
5), R
34 short (n
≥
13), R
45 open (n
=
30)
R
23 short
R
12 short, R
12 open (all n), X
C
20 short, X
C
40 short (n
≥
10), R
23 open (n
≥
5), X
C
30 short, R
34 short (n
≥
12), R
34 open (n
≥
8), R
45 open (n
=
30)
R
23 open
R
12 short, R
12 open (n
≥
4), X
C
20 short (n
≥
12), R
23 short (n
≥
5), X
C
30 short (n
≥
15), R
34 short, R
34 open, X
C
40 short (n
≥
8), R
45 open (n
=
30)
X
C
30 short
R
12 short, R
12 open (n
=
3  & ≥12), X
C
20 short (n
≥
5), R
23 short (n
≥
12), R
23 open (n
≥
15), R
34 short (n
≥
16), R
34 open (n
=
5, 6  & ≥18), X
C
40 short (n
≥
14), R
45 open (n
=
29, 30)
R
34 short
R
12 short, R
12 open, R
23 short (n
≥
12), X
C
20 short (n
≥
13), R
23 open (n
≥
8), X
C
30 short (n
≥
16), R
34 open (n
≥
10), X
C
40 short (n
≥
6), R
45 open (n
=
29, 30)
R
34 open
R
12 short, R
12 open (n
≥
7), X
C
20 short (n
≥
5), R
23 short, R
23 open (n
≥
8), X
C
30 short (n
=
5, 6  & ≥18), R
34 short, X
C
40 short (n
≥
10)
X
C
40 short
R
12 short, R
12 open, R
23 short, R
34 open (n
≥
10), X
C
20 short (n
≥
12), R
23 open (n
≥
8), X
C
30 short (n
≥
14), R
34 short (n
≥
6), R
45 short (n
=
5), R
45 open (n
≥
25)
R
45 short
X
C
40 short (n
=
5)
R
45 open
R
12 short, R
12 open, X
C
20 short, R
23 short, R
23 open (n
=
30), X
C
30 short, R
34 short (n
=
29, 30), X
C
40 short (n
≥
25)
X
C
50 short
No ambiguityTable 22
Ambiguity from|
Z
15
| for four-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
X
C
20 short, X
C
50 short (n
=
4, 6
–
21), R
23 short, R
45 short (n
=
1
–
4), R
34 short (n
=
1
–
3)
R
12 open
No ambiguity
X
C
20 short
R
12 short (n
=
4, 6
–
21), R
34 short (n
=
3), X
C
50 short (all n)
R
23 short
R
12 short (n
=
1
–
4), R
23 open, R
45 open (n
≥
18), R
34 short (n
=
1
–
5), R
34 open (n
=
8
–
10, & ≥16), R
45 short (all n)
R
23 open
R
23 short, R
45 short (n
≥
18), R
34 short (n
=
11
–
13), R
34 open (n
≥
21), R
45 open (all n)
X
C
30 short
R
34 short (n
≥
26), R
34 open (n
=
3), X
C
40 short (all n)
R
34 short
R
12 short (n
=
1
–
3), R
34 open (n
=
7), X
C
20 short, X
C
50 short (n
=
3), R
23 short, R
45 short (n
=
1
–
5), R
23 open, R
45 open (n
=
11
–
13), X
C
30 short, X
C
40 short (n
≥
26)
R
34 open
R
23 short, R
45 short (n
=
8
–
10 & ≥16), R
23 open, R
45 open (n
≥
21), X
C
30 short, X
C
40 short (n
=
3), R
34 short (n
=
7)
X
C
40 short
R
34 short (n
≥
26), R
34 open (n
=
3), X
C
30 short (all n)
R
45 short
R
12 short (n
=
1
–
4), R
23 open, R
45 open (n
≥
18), R
34 short (n
=
1
–
5), R
34 open (n
=
8
–
10 & ≥16), R
23 short (all n)
R
45 open
R
23 short, R
45 short (n
≥
18), R
34 short (n
=
11
–
13), R
34 open (n
≥
21), R
23 open (all n)
X
C
50 short
R
12 short (n
=
4, 6
–
21), R
34 short (n
=
3), X
C
20 short (all n)Table23 gives the conclusion derived from Tables 20, 21, and 22. Figure 17 shows the fault coverage for two-step RC ladder network. For n
=
1 to 12 and 18 to 20 the fault coverage is 100%, but for other values of “n” fault coverage is less than 100%. From Figure 17, it is also concluded that if we increase the value of “n” above 20, the fault coverage is decreasing.Table 23
Ambiguity from|
Z
01
|, |
Z
02
|, and |
Z
15
| for four-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
No ambiguity
R
12 open
No ambiguity
X
C
20 short
No ambiguity
R
23 short
No ambiguity
R
23 open
R
34 short (n
=
13), R
34 open (n
≥
21), R
45 open (n
=
30)
X
C
30 short
R
34 short (n
≥
26), X
C
40 short (n
=
14
–
17  & ≥22)
R
34 short
R
23 open (n
=
13), X
C
30 short, X
C
40 short (n
≥
26)
R
34 open
R
23 open (n
≥
21)
X
C
40 short
X
C
30 short (n
=
14
–
17  & ≥22), R
34 short (n
≥
26)
R
45 short
No ambiguity
R
45 open
R
23 open (n
=
30)
X
C
50 short
No ambiguityFigure 17
Fault coverage for four-step RC ladder network.


## 2.1. Single-Step RC Ladder Network

The single-step RC ladder network is shown in Figure2. In this network, only three faults exist, that is, R
12 short circuit, R
12 open circuit, and C
20 which is short; that is, X
C
20
=
0:(3)
Z
01
2
=
1
+
ω
n
R
C
2
ω
2
n
2
C
2
.Figure 2
Single-step RC ladder network.It can be concluded from Table1 that all the three values of Z
01
2 for different fault types are distinct and faults can be detected with no ambiguity, so the fault coverage is 100%. But computational complexity and area overhead are very large, because number of measurements will be equal to number of components and many numbers of single-step RC ladder network have to be tested to identify the faults of a big network [8]. Also, large numbers of observable nodes are required, so this cannot be considered from integrated circuit point of view. When we are breaking the big RC network into small RC network, this will also use additional circuits for the switches which lead to increase in area overhead [1, 9].Table 1
Expressions for single-step RC ladder network.
Fault type
Z
01
2
R
12 short
1
ω
2
n
2
C
2
R
12 open
I
n
f
i
n
i
t
y
X
C
20 short
R
2


## 2.2. Two-Step RC Ladder Network

In this section, the detailed analysis of two-step RC ladder network as shown in Figure3 is conducted.Figure 3
Two-step RC ladder network.Table2 gives the Z
01
2, Z
02
2, and Z
13
2 expressions for considering each and every fault as well as for faultless case.Table 2
Expressions for two-step RC ladder network.
Fault type
Z
01
2
Z
02
2
Z
13
2
F
a
u
l
t
l
e
s
s
a
6
+
11
a
4
+
21
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
a
6
+
20
a
4
+
64
a
2
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
R
12 short
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
4
a
4
+
16
a
2
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
R
12 open
I
n
f
i
n
i
t
y
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
I
n
f
i
n
i
t
y
X
C
20 short
R
2
a
4
+
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
a
6
+
5
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
R
23 short
4
a
2
+
1
4
ω
2
n
2
C
2
1
4
ω
2
n
2
C
2
R
2
R
23 open
a
2
+
1
ω
2
n
2
C
2
1
ω
2
n
2
C
2
a
2
+
4
ω
2
n
2
C
2
X
C
30 short
a
6
+
5
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
0
a
6
+
5
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]Figures4, 5, and 6 show the values of magnitude of Z
01, Z
02, and Z
13, respectively, plotted against “n.”Figure 4
Plot ofZ
01 versus “n” for different faults of two-step network.Figure 5
Plot ofZ
02 versus “n” for different faults of two-step network.Figure 6
Plot ofZ
13 versus “n” for different faults of two-step network.Table3 shows the conclusion derived from Figures 4, 5, and 6 about faults distinguishability and required measuring parameters for the same.Table 3
Distinguishability for two-step RC ladder network.
Fault type
Distinguishability
By measuring
R
12 short
Distinguishable at alln
Z
13
R
12 open
Distinguishable at alln
Z
01 or Z
13
X
C
20 short
Distinguishable at alln
Z
01
R
23 short
Distinguishable at alln
Z
13
R
23 open
Distinguishable at alln
Z
02
X
C
30 short
Distinguishable at alln
Z
02Tables4, 5, and 6 show the ambiguity for each fault which is not distinguishable from the measurement of Z
01, Z
02, and Z
13, respectively.Table 4
Ambiguity fromZ
01 for two-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
X
C
20 short (n
=
8, 9), R
23 short (n
=
1), X
C
30 short (n
=
4)
R
12 open
No ambiguity
X
C
20 short
R
12 short (n
=
8, 9), R
23 short (n
≥
18)
R
23 short
R
12 short (n
=
1), X
C
20 short (n
≥
18), X
C
30 short (n
=
5)
R
23 open
X
C
30 short (n
=
11, 12, 13)
X
C
30 short
R
12 short (n
=
4), R
23 open (n
=
11, 12, 13), R
23 short (n
=
5)Table 5
Ambiguity from |Z
02| for two-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
12 open (all n), X
C
20 short (n
≥
10), R
23 short (n
≤
10)
R
12 open
R
12 short (all n), X
C
20 short (n
≥
10), R
23 short (n
≤
10)
X
C
20 short
R
12 short (n
≥
10), R
12 open (n
≥
10), R
23 short (n
=
9, 10), R
23 open (n
≥
26)
R
23 short
R
12 short (n
≤
10), R
12 open (n
≤
10), X
C
20 short (n
=
9, 10)
R
23 open
X
C
20 short (n
≥
26)
X
C
30 short
No ambiguityTable 6
Ambiguity from |Z
13| for two-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
23 short (n
≤
15), R
23 open (n
≥
21)
R
12 open
No ambiguity
X
C
20 short
X
C
30 short (all n)
R
23 short
R
12 short (n
≤
15)
R
23 open
R
12 short (n
≥
21)
X
C
30 short
X
C
20 short (all n)Table7 gives the conclusion derived from Tables 4, 5, and 6.Table 7
Ambiguity from |Z
01|, |Z
02|, and |Z
13| for two-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
23 short (n
=
1)
R
12 open
No ambiguity
X
C
20 short
No ambiguity
R
23 short
R
12 short (n
=
1)
R
23 open
No ambiguity
X
C
30 short
No ambiguityFigure7 shows the fault coverage for two-step RC ladder network. Fault coverage is 100% except for n
=
1. All three measurements, that is, Z
01, Z
02, and Z
13, are required to calculate the distinguishability and ambiguity.Figure 7
Fault coverage for two-step RC ladder network.


## 2.3. Three-Step RC Ladder Network

In this section, detailed analysis of three-step RC ladder network (Figure8) is done.Figure 8
Three-step RC ladder network.For faultless network, the expression forZ
01
2 is given by(4)
Z
01
2
=
a
10
+
23
a
8
+
165
a
6
+
378
a
4
+
244
a
2
+
9
ω
2
n
2
C
2
a
8
+
20
a
6
+
118
a
4
+
180
a
2
+
81
.Table8 gives the Z
01
2 expression for considering each and every fault. Similar analysis is done for Z
02
2 and Z
14
2 and the results are shown below.Table 8
Z
01
2 expression for three-step RC ladder network.
Fault location
Short faultZ
01
2
Open faultZ
01
2
R
12
a
8
+
17
a
6
+
80
a
4
+
73
a
2
+
9
ω
2
n
2
C
2
[
a
8
+
20
a
6
+
118
a
4
+
180
a
2
+
81
]
Infinity
X
C
20
R
2
None
R
23
16
a
6
+
84
a
4
+
112
a
2
+
9
ω
2
n
2
C
2
8
a
4
+
72
a
2
+
81
a
2
+
1
ω
2
n
2
C
2
X
C
30
a
6
+
5
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
None
R
34
16
a
6
+
120
a
4
+
193
a
2
+
9
ω
2
n
2
C
2
[
16
a
4
+
72
a
2
+
81
]
a
6
+
11
a
4
+
21
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
40
a
10
+
17
a
8
+
80
a
6
+
73
a
4
+
9
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
NoneFor a faultless network, the expression forZ
02
2 and Z
14
2 is given by(5)
Z
22
2
=
a
8
+
17
a
6
+
80
a
4
+
73
a
2
+
9
ω
2
n
2
C
2
a
8
+
20
a
6
+
118
a
4
+
180
a
2
+
81
,
Z
14
2
=
a
9
+
20
a
7
+
79
a
5
+
2124
a
3
2
+
2
a
8
+
34
a
6
+
224
a
4
+
720
a
2
2
ω
2
n
2
C
2
a
8
+
18
a
6
+
57
a
4
+
1336
a
2
+
144
.Table9 gives the Z
02
2 expressions for considering each and every fault and Table 10 gives the Z
14
2 expressions for each and every fault.Table 9
Z
02
2 expression for three-step RC ladder network.
Fault location
Short faultZ
02
2
Open faultZ
02
2
R
12
a
8
+
17
a
6
+
80
a
4
+
73
a
2
+
9
ω
2
n
2
C
2
[
a
8
+
20
a
6
+
118
a
4
+
180
a
2
+
81
]
a
8
+
17
a
6
+
80
a
4
+
73
a
2
+
9
ω
2
n
2
C
2
[
a
8
+
20
a
6
+
118
a
4
+
180
a
2
+
81
]
X
C
20
a
8
+
11
a
6
+
29
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
None
R
23
16
a
4
+
40
a
2
+
9
ω
2
n
2
C
2
[
16
a
4
+
72
a
2
+
81
]
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
30
a
4
+
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
None
R
34
4
a
4
+
13
a
2
+
9
ω
2
n
2
C
2
[
16
a
4
+
72
a
2
+
81
]
1
ω
2
n
2
C
2
X
C
40
0
NoneTable 10
Z
14
2 expression for three-step RC ladder network.
Fault location
Short faultZ
14
2
Open faultZ
14
2
R
12
2
a
7
+
34
a
5
+
512
a
3
-
288
a
2
+
2
a
8
+
34
a
6
+
128
a
4
+
1056
a
2
2
ω
2
n
2
C
2
a
8
+
18
a
6
+
57
a
4
+
1336
a
2
+
144
2
Infinity
X
C
20
a
10
+
17
a
8
+
80
a
6
+
73
a
4
+
9
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
None
R
23
4
a
7
+
50
a
5
+
208
a
3
+
288
a
2
+
6
a
6
+
48
a
4
+
96
a
2
2
ω
2
n
2
C
2
4
a
6
+
41
a
4
+
136
a
2
+
144
2
a
6
+
14
a
4
+
49
a
2
+
36
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
30
a
6
+
10
a
4
+
9
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
None
R
34
16
a
6
+
180
a
4
+
324
a
2
ω
2
n
2
C
2
[
16
a
4
+
72
a
2
+
81
]
a
6
+
14
a
4
+
49
a
2
+
36
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
40
a
10
+
17
a
8
+
80
a
6
+
73
a
4
+
9
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
NoneFigure9 shows the values of magnitude of Z
01 plotted against “n.” Fault o
p
n
R
12
-
Z
01 is not shown in Figure 9, as it has value of infinity. Therefore, eight curves are shown corresponding to other short and open faults.Figure 9
Plot ofZ
01 versus “n” for different faults of three-step network.Figure10 shows the values of magnitude of Z
02 plotted against “n.” A
c
t
u
a
l
-
Z
02, s
r
t
R
12
-
Z
02, and o
p
n
R
12
-
Z
02 overlap each other, so all these plots are represented by bold line in Figure 10; fault o
p
n
R
34
-
Z
02 is excluded from the graph because it has very high value as compared to other faults. Therefore, 6 lines are shown corresponding to other short and open faults.Figure 10
Plot ofZ
02 versus “n” for different faults of three-step network.Figure11 shows the values of magnitude of Z
14 versus “n.” Faults s
r
t
X
C
20
-
Z
14 and s
r
t
X
C
40
-
Z
14 are overlapping; o
p
n
R
23
-
Z
14 and o
p
n
R
34
-
Z
14 are overlapping; s
r
t
R
23
-
Z
14 and s
r
t
R
34
-
Z
14 are overlapping; one of the faults o
p
n
R
12
-
Z
14 is not shown in Figure 11, as it has value of infinity. Therefore only 5 curves are shown corresponding to other short and open faults.Figure 11
Plot ofZ
14 versus “n” for different faults of three-step network.Table11 shows the conclusion derived from Figures 9–11.Table 11
Distinguishability for three-step RC ladder network.
Fault type
Distinguishability
By measuring
R
12 short
Distinguishable at alln
Z
01
R
12 open
Distinguishable at alln
Z
01 or Z
14
X
C
20 short
Distinguishable at alln
Z
01
R
23 short
Distinguishable except in the rangen = 26–29
Z
01 (n
<
26), Z
14 (n
=
30)
R
23 open
Distinguishable except in the rangen = 22–29
Z
14
X
C
30 short
Distinguishable except atn = 30
Z
14
R
34 short
Distinguishable at alln
Z
02 (n
>
1), Z
14 (n
=
1)
R
34 open
Distinguishable except atn = 29
Z
02 (n
<
29), Z
14 (n
=
30)
X
C
40 short
Distinguishable at alln
Z
02Tables12, 13, and 14 show the ambiguity for each fault which is not distinguishable from |
Z
01
|, |
Z
02
|, and |
Z
14
|, respectively. Table 15 gives the conclusion derived from Tables 12, 13, and 14. Figure 12 shows the fault coverage for three-step RC ladder network. Fault coverage is 100% for some values of “n.” For n
=
2 to 10 and 13 to 20 the fault coverage is 100%, but for other values of n fault coverage is less than 100%. All the measurements, that is, Z
01, Z
02, and Z
14 are required to calculate the distinguishability and ambiguity.Table 12
Ambiguity fromZ
01 for three-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
X
C
20 short (n = 7–9), R
23 short (n = 1), X
C
30 short (n = 3)
R
12 open
No ambiguity
X
C
20 short
R
12 short (n = 7–9)
R
23 short
R
12 short (n = 1), R
23 open (n ≥ 21), R
34 short (n ≥ 27), R
34 open (n ≥ 11), X
C
40 short (n ≥ 26)
R
23 open
R
23 short (n ≥ 21), X
C
30 short (n = 11–13), R
34 short (n = 14–24), R
34 open (n ≥ 20), X
C
40 short (n ≥ 10)
X
C
30 short
R
12 short (n = 3), R
23 open (n = 11–13), R
34 short (n = 4, n
≥
23), R
34 open (n = 5), X
C
40 short (n = 10–17, 23–30)
R
34 short
R
23 short (n ≥ 27), R
23 open (n = 14–24), X
C
30 short (n = 4, n
≥
23), X
C
40 short (n ≥ 15), R
34 open (n = 6, 7, 30)
R
34 open
R
23 short (n ≥ 11), R
23 open (n ≥ 20), X
C
30 short (n = 5), R
34 short (n = 6, 7, 30), X
C
40 short (n ≥ 28)
X
C
40 short
R
23 short (n ≥ 26), R
23 open (n ≥ 10), X
C
30 short (n = 10–17, 23–30), R
34 short (n ≥ 15), R
34 open (n ≥ 28)Table 13
Ambiguity fromZ
02 for three-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
12 open (all n), X
C
20 short (n
≥
15), R
23 short (all n), R
23 open, X
C
30 short (n
≥
8), R
34 short (n
=
1), R
34 open (n
=
29  & 30)
R
12 open
R
12 short, R
23 short (all n), X
C
20 short (n
≥
15), R
23 open, X
C
30 short (n
≥
8), R
34 short (n
=
1), R
34 open (n
=
29  & 30)
X
C
20 short
R
12 short, R
12 open (n
≥
15), R
23 short (n
=
3  & ≥18), R
23 open (n
=
5
-
6  & ≥18), X
C
30 short (n
≥
13), R
34 short (n
=
3), R
34 open (n
=
29
-
30)
R
23 short
R
12 short, R
12 open (all n), X
C
20 short (n
=
3  & ≥18), R
23 open (n
≥
11), X
C
30 short (n
≥
7), R
34 short (n
=
1
–
3)
R
23 open
R
12 short, R
12 open (n
≥
8), X
C
20 short (n
=
5
-
6  & ≥18), R
23 short (n
≥
11), X
C
30 short (n
≥
10)
X
C
30 short
R
12 short, R
12 open (n
≥
8), X
C
20 short (n
≥
13), R
23 short (n
≥
7), R
23 open (n
≥
10), R
34 short (n
=
6), R
34 open (n
≥
25)
R
34 short
R
12 short, R
12 open (n
=
1), X
C
20 short (n
=
3), R
23 short (n
=
1
–
3), X
C
30 short (n
=
6)
R
34 open
R
12 short, R
12 open (n
=
29  & 30), X
C
20 short (n
=
29
-
30), X
C
30 short (n
≥
25)
X
C
40 short
No ambiguityTable 14
Ambiguity from|
Z
14
| for three-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
23 short, R
34 short (n
=
1)
R
12 open
No ambiguity
X
C
20 short
R
23 short, R
34 short (n
=
9, 10), X
C
30 short (n
=
1), X
C
40 short (all n)
R
23 short
R
12 short (n
=
1), R
23 open, R
34 open (n
≥
21), R
34 short (all n), X
C
20 short, X
C
40 short (n
=
9, 10)
R
23 open
R
23 short, R
34 short (n
≥
21), R
34 open (all n), X
C
30 short (n
=
11, 12)
X
C
30 short
X
C
20 short, X
C
40 short (n
=
1), R
23 open, R
34 open (n
=
11, 12)
R
34 short
R
12 short (n
=
1), R
23 open, R
34 open (n
≥
21), R
23 short (all n), X
C
20 short, X
C
40 short (n
=
9, 10)
R
34 open
R
23 short, R
34 short (n
≥
21), R
23 open (all n), X
C
30 short (n
=
11, 12)
X
C
40 short
R
23 short, R
34 short (n
=
9, 10), X
C
30 short (n
=
1), X
C
20 short (all n)Table 15
Ambiguity fromZ
01, Z
02, and Z
14 for three-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
23 short (n
=
1)
R
12 open
No ambiguity
X
C
20 short
No ambiguity
R
23 short
R
12 short (n
=
1), R
23 open (n
≥
21)
R
23 open
R
23 short (n
≥
21), X
C
30 short (n
=
11, 12)
X
C
30 short
R
23 open (n
=
11, 12)
R
34 short
No ambiguity
R
34 open
No ambiguity
X
C
40 short
No ambiguityFigure 12
Fault coverage for three-step RC ladder network.


## 2.4. Four-Step RC Ladder Network

In this section, detailed analysis of four-step RC ladder network (Figure13) has been performed. As in the previous case, this network is also analyzed for n
=
1 to 30.Figure 13
Four-step RC ladder network.For faultless network, the expression forZ
01
2 is(6)
Z
11
2
=
a
7
+
17
a
5
+
63
a
3
+
30
a
2
+
a
6
+
14
a
4
+
34
a
2
+
4
2
ω
2
n
2
C
2
a
6
+
16
a
4
+
52
a
2
+
16
2
.Table16 gives the expressions of Z
01
2 for all faults. In these expressions, (7)
a
=
ω
n
R
C
,where ω
=
2
π
f and f
=
10 KHz. The parameter “n” is varied from 1 to 30.Table 16
Z
01
2 expression for four-step RC ladder network.
Fault location
Short faultZ
01
2
Open faultZ
01
2
R
12
a
5
+
11
a
3
+
14
a
2
+
a
6
+
14
a
4
+
34
a
2
+
4
2
ω
2
n
2
C
2
a
6
+
16
a
4
+
148
a
2
+
16
2
Infinity
X
C
20
R
2
None
R
23
4
a
5
+
34
a
3
+
21
a
2
+
2
a
4
+
15
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
65
a
2
+
16
2
a
2
+
1
ω
2
n
2
C
2
X
C
30
a
6
+
5
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
None
R
34
4
a
5
+
24
a
3
+
26
a
2
+
4
a
4
+
14
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
20
a
2
+
16
2
a
6
+
11
a
4
+
29
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
40
a
10
+
17
a
8
+
80
a
6
+
73
a
4
+
9
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
None
R
45
4
a
5
+
37
a
3
+
29
a
2
+
4
a
4
+
25
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
33
a
2
+
16
2
a
5
+
11
a
3
+
14
a
2
+
a
4
+
8
a
2
+
3
2
ω
2
n
2
C
2
a
4
+
10
a
2
+
9
2
X
C
50
a
7
+
14
a
5
+
34
a
3
+
4
a
2
+
a
6
+
11
a
4
+
14
a
2
2
ω
2
n
2
C
2
a
6
+
13
a
4
+
26
a
2
+
1
2
NoneSimilar analysis is done forZ
02
2 and Z
15
2; for a faultless network the expression for Z
02
2 and Z
15
2 is given by(8)
Z
22
2
=
a
5
+
11
a
3
+
14
a
2
+
a
6
+
14
a
4
+
34
a
2
+
4
2
ω
2
n
2
C
2
a
6
+
16
a
4
+
52
a
2
+
16
2
,
Z
15
2
=
a
9
+
22
a
7
+
144
a
5
+
352
a
3
+
256
a
2
+
2
a
8
+
36
a
6
+
192
a
4
+
320
a
2
2
ω
2
n
2
C
2
a
8
+
20
a
6
+
116
a
4
+
224
a
2
+
64
2
.Table17 gives the Z
02
2 expressions for considering each and every fault and Table 18 gives the Z
15
2 expressions for each and every fault.Table 17
Z
02
2 expression for four-step RC ladder network.
Fault location
Short faultZ
02
2
Open faultZ
02
2
R
12
a
5
+
11
a
3
+
14
a
2
+
a
6
+
14
a
4
+
34
a
2
+
4
2
ω
2
n
2
C
2
a
6
+
16
a
4
+
52
a
2
+
16
2
a
5
+
11
a
3
+
14
a
2
+
a
6
+
14
a
4
+
34
a
2
+
4
2
ω
2
n
2
C
2
a
6
+
16
a
4
+
52
a
2
+
16
2
X
C
20
a
5
+
8
a
3
+
3
a
2
+
a
6
+
11
a
4
+
14
a
2
2
ω
2
n
2
C
2
a
6
+
13
a
4
+
26
a
2
+
1
2
None
R
23
4
a
3
+
13
a
2
+
4
a
4
+
25
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
33
a
2
+
16
2
a
3
+
5
a
2
+
a
4
+
8
a
2
+
3
2
ω
2
n
2
C
2
a
4
+
10
a
2
+
9
2
X
C
30
a
8
+
11
a
6
+
29
a
4
+
4
a
2
ω
2
n
2
C
2
[
a
8
+
14
a
6
+
51
a
4
+
14
a
2
+
1
]
None
R
34
4
a
3
+
10
a
2
+
4
a
4
+
14
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
20
a
2
+
16
2
a
4
+
5
a
2
+
4
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
40
a
4
+
a
2
ω
2
n
2
C
2
[
a
4
+
2
a
2
+
1
]
None
R
45
a
3
+
5
a
2
+
2
a
4
+
15
a
2
+
4
2
ω
2
n
2
C
2
4
a
4
+
33
a
2
+
16
2
1
ω
2
n
2
C
2
X
C
50
0
NoneTable 18
Z
15
2 expression for four-step RC ladder network.
Fault location
Short faultZ
15
2
Open faultZ
15
2
R
12
2
a
7
+
28
a
5
+
128
a
3
+
192
a
2
+
2
a
8
+
36
a
6
+
192
a
4
+
320
a
2
2
ω
2
n
2
C
2
a
8
+
20
a
6
+
116
a
4
+
224
a
2
+
64
2
Infinity
X
C
20
a
11
+
22
a
9
+
162
a
7
+
500
a
5
+
576
a
3
+
64
a
2
+
a
10
+
19
a
8
+
118
a
6
+
288
a
4
+
224
a
2
2
ω
2
n
2
C
2
a
10
+
21
a
8
+
146
a
6
+
417
a
4
+
424
a
2
+
16
2
None
R
23
4
a
9
+
70
a
7
+
416
a
5
+
992
a
3
+
768
a
2
+
6
a
8
+
92
a
6
+
448
a
4
+
704
a
2
2
ω
2
n
2
C
2
4
a
8
+
65
a
6
+
344
a
4
+
656
a
2
+
256
2
a
9
+
19
a
7
+
118
a
5
+
288
a
3
+
224
a
2
+
2
a
8
+
34
a
6
+
188
a
4
+
384
a
2
+
192
2
ω
2
n
2
C
2
a
8
+
18
a
6
+
105
a
4
+
232
a
2
+
144
2
X
C
30
a
11
+
18
a
9
+
114
a
7
+
308
a
5
+
320
a
3
+
64
a
2
+
2
a
10
+
29
a
8
+
142
a
6
+
256
a
4
+
96
a
2
2
ω
2
n
2
C
2
a
10
+
16
a
8
+
88
a
6
+
193
a
4
+
4
a
2
+
16
2
None
R
34
4
a
5
+
28
a
3
+
48
a
2
+
8
a
4
+
32
a
2
2
ω
2
n
2
C
2
4
a
4
+
20
a
2
+
16
2
a
6
+
16
a
4
+
52
a
2
+
16
ω
2
n
2
C
2
[
a
4
+
8
a
2
+
16
]
X
C
40
a
11
+
18
a
9
+
114
a
7
+
308
a
5
+
320
a
3
+
64
a
2
+
2
a
10
+
29
a
8
+
142
a
6
+
256
a
4
+
96
a
2
2
ω
2
n
2
C
2
a
10
+
16
a
8
+
88
a
6
+
193
a
4
+
4
a
2
+
16
2
None
R
45
4
a
7
+
54
a
5
+
200
a
3
+
192
a
2
+
6
a
6
+
68
a
4
+
176
a
2
2
ω
2
n
2
C
2
4
a
6
+
49
a
4
+
148
a
2
+
64
2
a
7
+
15
a
5
+
58
a
3
+
56
a
2
+
2
a
6
+
26
a
4
+
84
a
2
+
48
2
ω
2
n
2
C
2
a
6
+
14
a
4
+
49
a
2
+
36
2
X
C
50
a
11
+
22
a
9
+
162
a
7
+
500
a
5
+
576
a
3
+
64
a
2
+
a
10
+
19
a
8
+
118
a
6
+
288
a
4
+
224
a
2
2
ω
2
n
2
C
2
a
10
+
21
a
8
+
146
a
6
+
417
a
4
+
424
a
2
+
16
2
NoneFigure14 shows the values of magnitude of Z
01 on the y-axis, plotted against “n” on the x-axis. Fault o
p
n
R
12
-
Z
01 is not shown in Figure 14, as it has value of infinity. Therefore, 11 lines are shown corresponding to other short and open faults.Figure 14
Plot ofZ
01 versus “n” for different faults of four-step network.Figure15 shows the values of magnitude of Z
02 on the Y-axis, plotted against “n” on the X-axis. A
c
t
u
a
l
-
Z
02, s
r
t
R
12
-
Z
02, and o
p
n
R
12
-
Z
02 are overlapping, so all these plots are represented by bold line in Figure 15; fault o
p
n
R
45
-
Z
02 was excluded from the graph, since it has very high value compared to the curves. Therefore, 9 lines are shown corresponding to other short and open faults.Figure 15
Plot of|
Z
02
| versus “n” for different faults of four-step network.Figure16 shows the values of magnitude of Z
15 versus “n.” Faults s
r
t
X
C
20
-
Z
15 and s
r
t
X
C
50
-
Z
15 are overlapping; s
r
t
X
C
30
-
Z
15 and s
r
t
X
C
40
-
Z
15 are overlapping; s
r
t
R
23
-
Z
15 and s
r
t
R
45
-
Z
15 are overlapping; o
p
n
R
23
-
Z
15 and o
p
n
R
45
-
Z
15 are overlapping; one of the faults o
p
n
R
12
-
Z
15 is not shown in Figure 16, as it has value of infinity. Therefore 7 lines are shown corresponding to other short and open faults.Figure 16
Plot of|
Z
15
| versus “n” for different faults of four-step network.Table19 shows the conclusion derived from Figures 14, 15, and 16.Table 19
Distinguishability for four-step RC ladder network.
Fault type
Distinguishability
By measuring
R
12 short
Distinguishable at alln
Z
01 or Z
15
R
12 open
Distinguishable at alln
Z
01 or Z
15
X
C
20 short
Distinguishable at alln
Z
01 or Z
15
R
23 short
Distinguishable at alln
Z
01 or Z
15
R
23 open
Distinguishable except atn
=
29, 30
Z
01 (n
=
8, 9), Z
15 (all other n)
X
C
30 short
Distinguishable except atn
=
30
Z
15
R
34 short
Distinguishable except forn
≥
13
Z
15
R
34 open
Distinguishable except atn
=
30
Z
02 (n
=
5), Z
15 (all other n)
X
C
40 short
Distinguishable except atn
=
30
Z
15
R
45 short
Distinguishable at alln
Z
02 or Z
15
R
45 open
Distinguishable except atn
=
30
Z
02
X
C
50 short
Distinguishable at alln
Z
02 or Z
15Tables20, 21, and 22 show the ambiguity for each fault which is not distinguishable from the measurement of |
Z
01
|, |
Z
02
|, and |
Z
15
|, respectively.Table 20
Ambiguity from|
Z
01
| for four-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
X
C
20 short (n
=
5)
R
12 open
No ambiguity
X
C
20 short
R
12 short (n
=
5), R
23 short (n
=
8
–
10)
R
23 short
X
C
20 short (n
=
8
–
10)
R
23 open
X
C
30 short (n
=
11, 12), R
34 short (n
=
13
–
20), R
34 open (n
≥
15), X
C
40 short (n
≥
10), R
45 short, X
C
50 short (n
≥
12), R
45 open (n
≥
13)
X
C
30 short
R
23 open (n
=
11, 12), R
34 short (n
=
4, 5  & ≥20), R
34 open (n
=
6, 29, 30), X
C
40 short (n
=
10
–
17  & ≥22), R
45 short (n
=
5
–
10  & ≥23), R
45 open (n
=
5, 6  & ≥23), X
C
50 short (n
=
8
–
11  & ≥25)
R
34 short
R
23 open (n
=
13
–
20), X
C
30 short (n
=
4, 5  & ≥20), R
34 open (n
=
6
–
9  & ≥19), X
C
40 short (n
≥
13), R
45 short (n
≥
9), R
45 open (n
≥
5), X
C
50 short (n
≥
10)
R
34 open
R
23 open (n
≥
15), X
C
30 short (n
=
6, 29, 30), R
34 short (n
=
6
–
9  & ≥19), X
C
40 short (n
≥
19), R
45 short (n
=
5, 6  & ≥18), R
45 open (n
=
6
–
9,  & ≥15), X
C
50 short (n
=
3  & ≥16)
X
C
40 short
R
23 open (n
≥
10), X
C
30 short (n
=
10
–
17  & ≥22), R
34 short (n
≥
13), R
34 open (n
≥
19), R
45 short (n
≥
12), R
45 open (n
≥
14), X
C
50 short (n
=
5, 6  & ≥12)
R
45 short
R
23 open (n
≥
12), X
C
30 short (n
=
5
–
10  & ≥23), R
34 short (n
≥
9), R
34 open (n
=
5, 6  & ≥18), X
C
40 short (n
≥
12), R
45 open (n
≥
3), X
C
50 short (n
≥
9)
R
45 open
R
23 open (n
≥
13), X
C
30 short (n
=
5, 6  & ≥23), R
34 short (n
≥
5), R
34 open (n
=
6
–
9,  & ≥15), X
C
40 short (n
≥
14), R
45 short (n
≥
3), X
C
50 short (n
≥
11)
X
C
50 short
R
23 open (n
≥
12), X
C
30 short (n
=
8
–
11  & ≥25), R
34 short (n
≥
10), R
34 open (n
=
3  & ≥16), X
C
40 short (n
=
5, 6  & ≥12), R
45 short (n
≥
9), R
45 open (n
≥
11)Table 21
Ambiguity from|
Z
02
| for four-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
R
12 open (all n), X
C
20 short (n
≥
9), R
23 short (all n), R
23 open (n
≥
4), X
C
30 short (n
=
3  & ≥12), R
34 short (n
≥
12), R
34 open (n
≥
7), X
C
40 short (n
≥
10), R
45 open (n
=
30)
R
12 open
R
12 short (all n), X
C
20 short (n
≥
9), R
23 short (all n), R
23 open (n
≥
4), X
C
30 short (n
=
3  & ≥12), R
34 short (n
≥
12), R
34 open (n
≥
7), X
C
40 short (n
≥
10), R
45 open (n
=
30)
X
C
20 short
R
12 short, R
12 open (n
≥
9), R
23 short (n
≥
10), R
23 open, X
C
40 short (n
≥
12), X
C
30 short, R
34 open (n
≥
5), R
34 short (n
≥
13), R
45 open (n
=
30)
R
23 short
R
12 short, R
12 open (all n), X
C
20 short, X
C
40 short (n
≥
10), R
23 open (n
≥
5), X
C
30 short, R
34 short (n
≥
12), R
34 open (n
≥
8), R
45 open (n
=
30)
R
23 open
R
12 short, R
12 open (n
≥
4), X
C
20 short (n
≥
12), R
23 short (n
≥
5), X
C
30 short (n
≥
15), R
34 short, R
34 open, X
C
40 short (n
≥
8), R
45 open (n
=
30)
X
C
30 short
R
12 short, R
12 open (n
=
3  & ≥12), X
C
20 short (n
≥
5), R
23 short (n
≥
12), R
23 open (n
≥
15), R
34 short (n
≥
16), R
34 open (n
=
5, 6  & ≥18), X
C
40 short (n
≥
14), R
45 open (n
=
29, 30)
R
34 short
R
12 short, R
12 open, R
23 short (n
≥
12), X
C
20 short (n
≥
13), R
23 open (n
≥
8), X
C
30 short (n
≥
16), R
34 open (n
≥
10), X
C
40 short (n
≥
6), R
45 open (n
=
29, 30)
R
34 open
R
12 short, R
12 open (n
≥
7), X
C
20 short (n
≥
5), R
23 short, R
23 open (n
≥
8), X
C
30 short (n
=
5, 6  & ≥18), R
34 short, X
C
40 short (n
≥
10)
X
C
40 short
R
12 short, R
12 open, R
23 short, R
34 open (n
≥
10), X
C
20 short (n
≥
12), R
23 open (n
≥
8), X
C
30 short (n
≥
14), R
34 short (n
≥
6), R
45 short (n
=
5), R
45 open (n
≥
25)
R
45 short
X
C
40 short (n
=
5)
R
45 open
R
12 short, R
12 open, X
C
20 short, R
23 short, R
23 open (n
=
30), X
C
30 short, R
34 short (n
=
29, 30), X
C
40 short (n
≥
25)
X
C
50 short
No ambiguityTable 22
Ambiguity from|
Z
15
| for four-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
X
C
20 short, X
C
50 short (n
=
4, 6
–
21), R
23 short, R
45 short (n
=
1
–
4), R
34 short (n
=
1
–
3)
R
12 open
No ambiguity
X
C
20 short
R
12 short (n
=
4, 6
–
21), R
34 short (n
=
3), X
C
50 short (all n)
R
23 short
R
12 short (n
=
1
–
4), R
23 open, R
45 open (n
≥
18), R
34 short (n
=
1
–
5), R
34 open (n
=
8
–
10, & ≥16), R
45 short (all n)
R
23 open
R
23 short, R
45 short (n
≥
18), R
34 short (n
=
11
–
13), R
34 open (n
≥
21), R
45 open (all n)
X
C
30 short
R
34 short (n
≥
26), R
34 open (n
=
3), X
C
40 short (all n)
R
34 short
R
12 short (n
=
1
–
3), R
34 open (n
=
7), X
C
20 short, X
C
50 short (n
=
3), R
23 short, R
45 short (n
=
1
–
5), R
23 open, R
45 open (n
=
11
–
13), X
C
30 short, X
C
40 short (n
≥
26)
R
34 open
R
23 short, R
45 short (n
=
8
–
10 & ≥16), R
23 open, R
45 open (n
≥
21), X
C
30 short, X
C
40 short (n
=
3), R
34 short (n
=
7)
X
C
40 short
R
34 short (n
≥
26), R
34 open (n
=
3), X
C
30 short (all n)
R
45 short
R
12 short (n
=
1
–
4), R
23 open, R
45 open (n
≥
18), R
34 short (n
=
1
–
5), R
34 open (n
=
8
–
10 & ≥16), R
23 short (all n)
R
45 open
R
23 short, R
45 short (n
≥
18), R
34 short (n
=
11
–
13), R
34 open (n
≥
21), R
23 open (all n)
X
C
50 short
R
12 short (n
=
4, 6
–
21), R
34 short (n
=
3), X
C
20 short (all n)Table23 gives the conclusion derived from Tables 20, 21, and 22. Figure 17 shows the fault coverage for two-step RC ladder network. For n
=
1 to 12 and 18 to 20 the fault coverage is 100%, but for other values of “n” fault coverage is less than 100%. From Figure 17, it is also concluded that if we increase the value of “n” above 20, the fault coverage is decreasing.Table 23
Ambiguity from|
Z
01
|, |
Z
02
|, and |
Z
15
| for four-step RC ladder network.
Fault
Ambiguity with the following
R
12 short
No ambiguity
R
12 open
No ambiguity
X
C
20 short
No ambiguity
R
23 short
No ambiguity
R
23 open
R
34 short (n
=
13), R
34 open (n
≥
21), R
45 open (n
=
30)
X
C
30 short
R
34 short (n
≥
26), X
C
40 short (n
=
14
–
17  & ≥22)
R
34 short
R
23 open (n
=
13), X
C
30 short, X
C
40 short (n
≥
26)
R
34 open
R
23 open (n
≥
21)
X
C
40 short
X
C
30 short (n
=
14
–
17  & ≥22), R
34 short (n
≥
26)
R
45 short
No ambiguity
R
45 open
R
23 open (n
=
30)
X
C
50 short
No ambiguityFigure 17
Fault coverage for four-step RC ladder network.


## 3. Conclusion

The fault coverage is 100% for single-step ladder network using only one measurement and fault coverage is 100% except forn
=
1 for two-step ladder network. But these have high area overhead which is a costly requirement for integrated circuit manufacturing. From Figures 12 and 17, it is shown that, for “n” ranging from 1 to 12, the four-step option is a better choice, and for “n” ranging from 13 to 30, the three-step option is a better choice. It has been concluded that only three measurements are required in different step size networks except single-step networks.



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*Source: 7176981-2016-07-11.xml*