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.
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 (Figure 1) 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 Xc 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=2π104 rad/sec):(1)Xc=1jnωc.We have assumed that the network has only one fault at a time. Zij is an equivalent impedance measured between ith and jth nodes (Figure 1). Notations gives the list of notations used.
RC ladder network: Ni indicates node number.
To calculate the fault coverage, two characteristics are considered, that is, distinguishability and ambiguity. 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=Zijforfaultlesscase-ZijforfaultycaseZijforfaultlesscase×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 of n, 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=ωnRC. 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 Figure 2. In this network, only three faults exist, that is, R12 short circuit, R12 open circuit, and C20 which is short; that is, XC20=0:(3)Z012=1+ωnRC2ω2n2C2.
Single-step RC ladder network.
It can be concluded from Table 1 that all the three values of Z012 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].
Expressions for single-step RC ladder network.
Fault type
Z012
R12 short
1ω2n2C2
R12 open
Infinity
XC20 short
R2
2.2. Two-Step RC Ladder Network
In this section, the detailed analysis of two-step RC ladder network as shown in Figure 3 is conducted.
Two-step RC ladder network.
Table 2 gives the Z012, Z022, and Z132 expressions for considering each and every fault as well as for faultless case.
Expressions for two-step RC ladder network.
Fault type
Z012
Z022
Z132
Faultless
a6+11a4+21a2+4ω2n2C2[a4+8a2+16]
a4+5a2+4ω2n2C2[a4+8a2+16]
a6+20a4+64a2ω2n2C2[a4+8a2+16]
R12 short
a4+5a2+4ω2n2C2[a4+8a2+16]
a4+5a2+4ω2n2C2[a4+8a2+16]
4a4+16a2ω2n2C2[a4+8a2+16]
R12 open
Infinity
a4+5a2+4ω2n2C2[a4+8a2+16]
Infinity
XC20 short
R2
a4+a2ω2n2C2[a4+2a2+1]
a6+5a4+4a2ω2n2C2[a4+2a2+1]
R23 short
4a2+14ω2n2C2
14ω2n2C2
R2
R23 open
a2+1ω2n2C2
1ω2n2C2
a2+4ω2n2C2
XC30 short
a6+5a4+4a2ω2n2C2[a4+2a2+1]
0
a6+5a4+4a2ω2n2C2[a4+2a2+1]
Figures 4, 5, and 6 show the values of magnitude of Z01, Z02, and Z13, respectively, plotted against “n.”
Plot of Z01 versus “n” for different faults of two-step network.
Plot of Z02 versus “n” for different faults of two-step network.
Plot of Z13 versus “n” for different faults of two-step network.
Table 3 shows the conclusion derived from Figures 4, 5, and 6 about faults distinguishability and required measuring parameters for the same.
Distinguishability for two-step RC ladder network.
Fault type
Distinguishability
By measuring
R12 short
Distinguishable at all n
Z13
R12 open
Distinguishable at all n
Z01 or Z13
XC20 short
Distinguishable at all n
Z01
R23 short
Distinguishable at all n
Z13
R23 open
Distinguishable at all n
Z02
XC30 short
Distinguishable at all n
Z02
Tables 4, 5, and 6 show the ambiguity for each fault which is not distinguishable from the measurement of Z01, Z02, and Z13, respectively.
Ambiguity from Z01 for two-step RC ladder network.
Fault
Ambiguity with the following
R12 short
XC20 short (n=8, 9), R23 short (n=1), XC30 short (n=4)
R12 open
No ambiguity
XC20 short
R12 short (n=8, 9), R23 short (n≥18)
R23 short
R12 short (n=1), XC20 short (n≥18), XC30 short (n=5)
R23 open
XC30 short (n=11, 12, 13)
XC30 short
R12 short (n=4), R23 open (n=11, 12, 13), R23 short (n=5)
Ambiguity from |Z02| for two-step RC ladder network.
Fault
Ambiguity with the following
R12 short
R12 open (all n), XC20 short (n≥10), R23 short (n≤10)
R12 open
R12 short (all n), XC20 short (n≥10), R23 short (n≤10)
XC20 short
R12 short (n≥10), R12 open (n≥10), R23 short (n=9, 10), R23 open (n≥26)
R23 short
R12 short (n≤10), R12 open (n≤10), XC20 short (n=9, 10)
R23 open
XC20 short (n≥26)
XC30 short
No ambiguity
Ambiguity from |Z13| for two-step RC ladder network.
Fault
Ambiguity with the following
R12 short
R23 short (n≤15), R23 open (n≥21)
R12 open
No ambiguity
XC20 short
XC30 short (all n)
R23 short
R12 short (n≤15)
R23 open
R12 short (n≥21)
XC30 short
XC20 short (all n)
Table 7 gives the conclusion derived from Tables 4, 5, and 6.
Ambiguity from |Z01|, |Z02|, and |Z13| for two-step RC ladder network.
Fault
Ambiguity with the following
R12 short
R23 short (n=1)
R12 open
No ambiguity
XC20 short
No ambiguity
R23 short
R12 short (n=1)
R23 open
No ambiguity
XC30 short
No ambiguity
Figure 7 shows the fault coverage for two-step RC ladder network. Fault coverage is 100% except for n=1. All three measurements, that is, Z01, Z02, and Z13, are required to calculate the distinguishability and ambiguity.
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 (Figure 8) is done.
Three-step RC ladder network.
For faultless network, the expression for Z012 is given by(4)Z012=a10+23a8+165a6+378a4+244a2+9ω2n2C2a8+20a6+118a4+180a2+81.
Table 8 gives the Z012 expression for considering each and every fault. Similar analysis is done for Z022 and Z142 and the results are shown below.
Z012 expression for three-step RC ladder network.
Fault location
Short fault Z012
Open fault Z012
R12
a8+17a6+80a4+73a2+9ω2n2C2[a8+20a6+118a4+180a2+81]
Infinity
XC20
R2
None
R23
16a6+84a4+112a2+9ω2n2C28a4+72a2+81
a2+1ω2n2C2
XC30
a6+5a4+4a2ω2n2C2[a4+2a2+1]
None
R34
16a6+120a4+193a2+9ω2n2C2[16a4+72a2+81]
a6+11a4+21a2+4ω2n2C2[a4+8a2+16]
XC40
a10+17a8+80a6+73a4+9a2ω2n2C2[a8+14a6+51a4+14a2+1]
None
For a faultless network, the expression for Z022 and Z142 is given by(5)Z222=a8+17a6+80a4+73a2+9ω2n2C2a8+20a6+118a4+180a2+81,Z142=a9+20a7+79a5+2124a32+2a8+34a6+224a4+720a22ω2n2C2a8+18a6+57a4+1336a2+144.
Table 9 gives the Z022 expressions for considering each and every fault and Table 10 gives the Z142 expressions for each and every fault.
Figure 9 shows the values of magnitude of Z01 plotted against “n.” Fault opnR12-Z01 is not shown in Figure 9, as it has value of infinity. Therefore, eight curves are shown corresponding to other short and open faults.
Plot of Z01 versus “n” for different faults of three-step network.
Figure 10 shows the values of magnitude of Z02 plotted against “n.” Actual-Z02, srtR12-Z02, and opnR12-Z02 overlap each other, so all these plots are represented by bold line in Figure 10; fault opnR34-Z02 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.
Plot of Z02 versus “n” for different faults of three-step network.
Figure 11 shows the values of magnitude of Z14 versus “n.” Faults srtXC20-Z14 and srtXC40-Z14 are overlapping; opnR23-Z14 and opnR34-Z14 are overlapping; srtR23-Z14 and srtR34-Z14 are overlapping; one of the faults opnR12-Z14 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.
Plot of Z14 versus “n” for different faults of three-step network.
Table 11 shows the conclusion derived from Figures 9–11.
Distinguishability for three-step RC ladder network.
Fault type
Distinguishability
By measuring
R12 short
Distinguishable at all n
Z01
R12 open
Distinguishable at all n
Z01 or Z14
XC20 short
Distinguishable at all n
Z01
R23 short
Distinguishable except in the range n = 26–29
Z01 (n<26), Z14 (n=30)
R23 open
Distinguishable except in the range n = 22–29
Z14
XC30 short
Distinguishable except at n = 30
Z14
R34 short
Distinguishable at all n
Z02 (n>1), Z14 (n=1)
R34 open
Distinguishable except at n = 29
Z02 (n<29), Z14 (n=30)
XC40 short
Distinguishable at all n
Z02
Tables 12, 13, and 14 show the ambiguity for each fault which is not distinguishable from |Z01|, |Z02|, and |Z14|, 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, Z01, Z02, and Z14 are required to calculate the distinguishability and ambiguity.
Ambiguity from Z01 for three-step RC ladder network.
Fault
Ambiguity with the following
R12 short
XC20 short (n = 7–9), R23 short (n = 1), XC30 short (n = 3)
R12 open
No ambiguity
XC20 short
R12 short (n = 7–9)
R23 short
R12 short (n = 1), R23 open (n ≥ 21), R34 short (n ≥ 27), R34 open (n ≥ 11), XC40 short (n ≥ 26)
R23 open
R23 short (n ≥ 21), XC30 short (n = 11–13), R34 short (n = 14–24), R34 open (n ≥ 20), XC40 short (n ≥ 10)
XC30 short
R12 short (n = 3), R23 open (n = 11–13), R34 short (n = 4, n≥23), R34 open (n = 5), XC40 short (n = 10–17, 23–30)
R34 short
R23 short (n ≥ 27), R23 open (n = 14–24), XC30 short (n = 4, n≥23), XC40 short (n ≥ 15), R34 open (n = 6, 7, 30)
R34 open
R23 short (n ≥ 11), R23 open (n ≥ 20), XC30 short (n = 5), R34 short (n = 6, 7, 30), XC40 short (n ≥ 28)
XC40 short
R23 short (n ≥ 26), R23 open (n ≥ 10), XC30 short (n = 10–17, 23–30), R34 short (n ≥ 15), R34 open (n ≥ 28)
Ambiguity from Z02 for three-step RC ladder network.
Fault
Ambiguity with the following
R12 short
R12 open (all n), XC20 short (n≥15), R23 short (all n), R23 open, XC30 short (n≥8), R34 short (n=1), R34 open (n=29 & 30)
R12 open
R12 short, R23 short (all n), XC20 short (n≥15), R23 open, XC30 short (n≥8), R34 short (n=1), R34 open (n=29 & 30)
XC20 short
R12 short, R12 open (n≥15), R23 short (n=3 & ≥18), R23 open (n=5-6 & ≥18), XC30 short (n≥13), R34 short (n=3), R34 open (n=29-30)
R23 short
R12 short, R12 open (all n), XC20 short (n=3 & ≥18), R23 open (n≥11), XC30 short (n≥7), R34 short (n=1–3)
R23 open
R12 short, R12 open (n≥8), XC20 short (n=5-6 & ≥18), R23 short (n≥11), XC30 short (n≥10)
XC30 short
R12 short, R12 open (n≥8), XC20 short (n≥13), R23 short (n≥7), R23 open (n≥10), R34 short (n=6), R34 open (n≥25)
R34 short
R12 short, R12 open (n=1), XC20 short (n=3), R23 short (n=1–3), XC30 short (n=6)
R34 open
R12 short, R12 open (n=29 & 30), XC20 short (n=29-30), XC30 short (n≥25)
XC40 short
No ambiguity
Ambiguity from |Z14| for three-step RC ladder network.
Fault
Ambiguity with the following
R12 short
R23 short, R34 short (n=1)
R12 open
No ambiguity
XC20 short
R23 short, R34 short (n=9, 10), XC30 short (n=1), XC40 short (all n)
R23 short
R12 short (n=1), R23 open, R34 open (n≥21), R34 short (all n), XC20 short, XC40 short (n=9, 10)
R23 open
R23 short, R34 short (n≥21), R34 open (all n), XC30 short (n=11, 12)
XC30 short
XC20 short, XC40 short (n=1), R23 open, R34 open (n=11, 12)
R34 short
R12 short (n=1), R23 open, R34 open (n≥21), R23 short (all n), XC20 short, XC40 short (n=9, 10)
R34 open
R23 short, R34 short (n≥21), R23 open (all n), XC30 short (n=11, 12)
XC40 short
R23 short, R34 short (n=9, 10), XC30 short (n=1), XC20 short (all n)
Ambiguity from Z01, Z02, and Z14 for three-step RC ladder network.
Fault
Ambiguity with the following
R12 short
R23 short (n=1)
R12 open
No ambiguity
XC20 short
No ambiguity
R23 short
R12 short (n=1), R23 open (n≥21)
R23 open
R23 short (n≥21), XC30 short (n=11, 12)
XC30 short
R23 open (n=11, 12)
R34 short
No ambiguity
R34 open
No ambiguity
XC40 short
No ambiguity
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 (Figure 13) has been performed. As in the previous case, this network is also analyzed for n=1 to 30.
Four-step RC ladder network.
For faultless network, the expression for Z012 is(6)Z112=a7+17a5+63a3+30a2+a6+14a4+34a2+42ω2n2C2a6+16a4+52a2+162.
Table 16 gives the expressions of Z012 for all faults. In these expressions, (7)a=ωnRC,where ω=2πf and f=10 KHz. The parameter “n” is varied from 1 to 30.
Similar analysis is done for Z022 and Z152; for a faultless network the expression for Z022 and Z152 is given by(8)Z222=a5+11a3+14a2+a6+14a4+34a2+42ω2n2C2a6+16a4+52a2+162,Z152=a9+22a7+144a5+352a3+256a2+2a8+36a6+192a4+320a22ω2n2C2a8+20a6+116a4+224a2+642.
Table 17 gives the Z022 expressions for considering each and every fault and Table 18 gives the Z152 expressions for each and every fault.
Figure 14 shows the values of magnitude of Z01 on the y-axis, plotted against “n” on the x-axis. Fault opnR12-Z01 is not shown in Figure 14, as it has value of infinity. Therefore, 11 lines are shown corresponding to other short and open faults.
Plot of Z01 versus “n” for different faults of four-step network.
Figure 15 shows the values of magnitude of Z02 on the Y-axis, plotted against “n” on the X-axis. Actual-Z02, srtR12-Z02, and opnR12-Z02 are overlapping, so all these plots are represented by bold line in Figure 15; fault opnR45-Z02 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.
Plot of |Z02| versus “n” for different faults of four-step network.
Figure 16 shows the values of magnitude of Z15 versus “n.” Faults srtXC20-Z15 and srtXC50-Z15 are overlapping; srtXC30-Z15 and srtXC40-Z15 are overlapping; srtR23-Z15 and srtR45-Z15 are overlapping; opnR23-Z15 and opnR45-Z15 are overlapping; one of the faults opnR12-Z15 is not shown in Figure 16, as it has value of infinity. Therefore 7 lines are shown corresponding to other short and open faults.
Plot of |Z15| versus “n” for different faults of four-step network.
Table 19 shows the conclusion derived from Figures 14, 15, and 16.
Distinguishability for four-step RC ladder network.
Fault type
Distinguishability
By measuring
R12 short
Distinguishable at all n
Z01 or Z15
R12 open
Distinguishable at all n
Z01 or Z15
XC20 short
Distinguishable at all n
Z01 or Z15
R23 short
Distinguishable at all n
Z01 or Z15
R23 open
Distinguishable except at n=29, 30
Z01 (n=8, 9), Z15 (all other n)
XC30 short
Distinguishable except at n=30
Z15
R34 short
Distinguishable except for n≥13
Z15
R34 open
Distinguishable except at n=30
Z02 (n=5), Z15 (all other n)
XC40 short
Distinguishable except at n=30
Z15
R45 short
Distinguishable at all n
Z02 or Z15
R45 open
Distinguishable except at n=30
Z02
XC50 short
Distinguishable at all n
Z02 or Z15
Tables 20, 21, and 22 show the ambiguity for each fault which is not distinguishable from the measurement of |Z01|, |Z02|, and |Z15|, respectively.
Ambiguity from |Z01| for four-step RC ladder network.
Fault
Ambiguity with the following
R12 short
XC20 short (n=5)
R12 open
No ambiguity
XC20 short
R12 short (n=5), R23 short (n=8–10)
R23 short
XC20 short (n=8–10)
R23 open
XC30 short (n=11, 12), R34 short (n=13–20), R34 open (n≥15), XC40 short (n≥10), R45 short, XC50 short (n≥12), R45 open (n≥13)
XC30 short
R23 open (n=11, 12), R34 short (n=4, 5 & ≥20), R34 open (n=6, 29, 30), XC40 short (n=10–17 & ≥22), R45 short (n=5–10 & ≥23), R45 open (n=5, 6 & ≥23), XC50 short (n=8–11 & ≥25)
R34 short
R23 open (n=13–20), XC30 short (n=4, 5 & ≥20), R34 open (n=6–9 & ≥19), XC40 short (n≥13), R45 short (n≥9), R45 open (n≥5), XC50 short (n≥10)
R34 open
R23 open (n≥15), XC30 short (n=6, 29, 30), R34 short (n=6–9 & ≥19), XC40 short (n≥19), R45 short (n=5, 6 & ≥18), R45 open (n=6–9, & ≥15), XC50 short (n=3 & ≥16)
XC40 short
R23 open (n≥10), XC30 short (n=10–17 & ≥22), R34 short (n≥13), R34 open (n≥19), R45 short (n≥12), R45 open (n≥14), XC50 short (n=5, 6 & ≥12)
R45 short
R23 open (n≥12), XC30 short (n=5–10 & ≥23), R34 short (n≥9), R34 open (n=5, 6 & ≥18), XC40 short (n≥12), R45 open (n≥3), XC50 short (n≥9)
R45 open
R23 open (n≥13), XC30 short (n=5, 6 & ≥23), R34 short (n≥5), R34 open (n=6–9, & ≥15), XC40 short (n≥14), R45 short (n≥3), XC50 short (n≥11)
XC50 short
R23 open (n≥12), XC30 short (n=8–11 & ≥25), R34 short (n≥10), R34 open (n=3 & ≥16), XC40 short (n=5, 6 & ≥12), R45 short (n≥9), R45 open (n≥11)
Ambiguity from |Z02| for four-step RC ladder network.
Fault
Ambiguity with the following
R12 short
R12 open (all n), XC20 short (n≥9), R23 short (all n), R23 open (n≥4), XC30 short (n=3 & ≥12), R34 short (n≥12), R34 open (n≥7), XC40 short (n≥10), R45 open (n=30)
R12 open
R12 short (all n), XC20 short (n≥9), R23 short (all n), R23 open (n≥4), XC30 short (n=3 & ≥12), R34 short (n≥12), R34 open (n≥7), XC40 short (n≥10), R45 open (n=30)
XC20 short
R12 short, R12 open (n≥9), R23 short (n≥10), R23 open, XC40 short (n≥12), XC30 short, R34 open (n≥5), R34 short (n≥13), R45 open (n=30)
R23 short
R12 short, R12 open (all n), XC20 short, XC40 short (n≥10), R23 open (n≥5), XC30 short, R34 short (n≥12), R34 open (n≥8), R45 open (n=30)
R23 open
R12 short, R12 open (n≥4), XC20 short (n≥12), R23 short (n≥5), XC30 short (n≥15), R34 short, R34 open, XC40 short (n≥8), R45 open (n=30)
XC30 short
R12 short, R12 open (n=3 & ≥12), XC20 short (n≥5), R23 short (n≥12), R23 open (n≥15), R34 short (n≥16), R34 open (n=5, 6 & ≥18), XC40 short (n≥14), R45 open (n=29, 30)
R34 short
R12 short, R12 open, R23 short (n≥12), XC20 short (n≥13), R23 open (n≥8), XC30 short (n≥16), R34 open (n≥10), XC40 short (n≥6), R45 open (n=29, 30)
R34 open
R12 short, R12 open (n≥7), XC20 short (n≥5), R23 short, R23 open (n≥8), XC30 short (n=5, 6 & ≥18), R34 short, XC40 short (n≥10)
XC40 short
R12 short, R12 open, R23 short, R34 open (n≥10), XC20 short (n≥12), R23 open (n≥8), XC30 short (n≥14), R34 short (n≥6), R45 short (n=5), R45 open (n≥25)
R45 short
XC40 short (n=5)
R45 open
R12 short, R12 open, XC20 short, R23 short, R23 open (n=30), XC30 short, R34 short (n=29, 30), XC40 short (n≥25)
XC50 short
No ambiguity
Ambiguity from |Z15| for four-step RC ladder network.
Fault
Ambiguity with the following
R12 short
XC20 short, XC50 short (n=4, 6–21), R23 short, R45 short (n=1–4), R34 short (n=1–3)
R12 open
No ambiguity
XC20 short
R12 short (n=4, 6–21), R34 short (n=3), XC50 short (all n)
R23 short
R12 short (n=1–4), R23 open, R45 open (n≥18), R34 short (n=1–5), R34 open (n=8–10, & ≥16), R45 short (all n)
R23 open
R23 short, R45 short (n≥18), R34 short (n=11–13), R34 open (n≥21), R45 open (all n)
XC30 short
R34 short (n≥26), R34 open (n=3), XC40 short (all n)
R34 short
R12 short (n=1–3), R34 open (n=7), XC20 short, XC50 short (n=3), R23 short, R45 short (n=1–5), R23 open, R45 open (n=11–13), XC30 short, XC40 short (n≥26)
R34 open
R23 short, R45 short (n=8–10 & ≥16), R23 open, R45 open (n≥21), XC30 short, XC40 short (n=3), R34 short (n=7)
XC40 short
R34 short (n≥26), R34 open (n=3), XC30 short (all n)
R45 short
R12 short (n=1–4), R23 open, R45 open (n≥18), R34 short (n=1–5), R34 open (n=8–10 & ≥16), R23 short (all n)
R45 open
R23 short, R45 short (n≥18), R34 short (n=11–13), R34 open (n≥21), R23 open (all n)
XC50 short
R12 short (n=4, 6–21), R34 short (n=3), XC20 short (all n)
Table 23 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.
Ambiguity from |Z01|, |Z02|, and |Z15| for four-step RC ladder network.
Fault
Ambiguity with the following
R12 short
No ambiguity
R12 open
No ambiguity
XC20 short
No ambiguity
R23 short
No ambiguity
R23 open
R34 short (n=13), R34 open (n≥21), R45 open (n=30)
XC30 short
R34 short (n≥26), XC40 short (n=14–17 & ≥22)
R34 short
R23 open (n=13), XC30 short, XC40 short (n≥26)
R34 open
R23 open (n≥21)
XC40 short
XC30 short (n=14–17 & ≥22), R34 short (n≥26)
R45 short
No ambiguity
R45 open
R23 open (n=30)
XC50 short
No ambiguity
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 for n=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.
NotationsActual-Z01:
Z012 when there is no fault in the network
srtRij-Z01:
Z012 when Rij is short circuit
opnRij-Z01:
Z012 when Rij is open circuit
srtXCij-Z01:
Z012 when XCij is short circuit
Actual-ZLL:
ZLL2 when there is no fault in the network
srtRij-ZLL:
ZLL2 when Rij is short circuit
opnRij-ZLL:
ZLL2 when Rij is open circuit
srtXCij-ZLL:
ZLL2 when XCij is short circuit
Actual-Z1L:
Z1L2 when there is no fault in the network
srtRij-Z1L:
Z1L2 when Rij is short circuit
opnRij-Z1L:
Z1L2 when Rij is open circuit
srtXCij-Z1L:
Z1L2 when XCij is short circuit.
SymbolsRij:
The resistance between nodes i and j
XCij:
The impedance due to capacitor between nodes i and j.
Subscriptsi and j:
The node numbers
L:
The last node number in the RC circuit.
Competing Interests
The authors declare that they have no competing interests.
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