ABSTRACT 
 
 
 
 
Title of Document: PROGNOSTICS OF POLYMER POSITIVE 
TEMPERATURE COEFFICIENT 
RESETTABLE FUSES 
 
  
 Shunfeng Cheng, Doctor of Philosophy (Ph.D.), 
2011 
  
Directed By: Professor Michael G. Pecht 
Department of Mechanical Engineering 
 
Polymer positive-temperature-coefficient (PPTC) resettable fuse has been used 
to circuit-protection designs in computers, automotive circuits, telecommunication 
devices, and medical devices. PPTC resettable fuse can trip from low resistance to 
high resistance under over-current conditions. The increase in the resistance decreases 
the current and protects the circuit. After the abnormal current is removed, and/or 
power is switched off, the fuse resets to low resistance stage, and can be continuously 
operated in the circuit. The resettable fuse degrades with the operations resulting in 
loss or abnormal function of the protection of circuit. This thesis is focused on the 
prognostics methods for resettable fuses to provide an advance warning of failure and 
to predict the remaining useful life.  
The failure precursor parameters are determined first by systematic analysis 
using failure modes, mechanisms, and effects analysis (FMMEA) followed by a 
series of experiments to verify these parameters. Then the causes of the observed 
  
failures are determined by failure analyses, including the analyses of interconnections 
between different parts, the microstructures of the polymer composite, the properties 
(such as crystallinity) of the polymer composite, and the coefficient of thermal 
expansion (CTE) of different parts. The revealed failure causes include the cracks and 
gaps between different parts, the agglomerations of the carbon black particles, the 
change in crystallinity of the polymer composite, and the CTE-mismatches between 
different parts.  
Cross validation (CV) sequential probability ratio test (CVSPRT) is developed 
to detect anomalies. CV methods are introduced into SPRT to determine the model 
parameters without the need of experience and reduce the false and missed alarms. A 
moving window training updating based dynamic model parameter optimization 
(MW-DMPO) n-steps-ahead prognostics method is developed to predict the failure. 
MW methods update the training data for prediction models by a moving window to 
contain the latest degradation information/data and improve the prediction accuracy. 
For each updating of the training data, the model parameters for data-trending model 
are updated dynamically. Based on the developed MW-DMPO method, a MW cross 
validation support vector regression (MW-CVSVR) n-steps-ahead prediction is 
developed to predict failures of PPTC resettable fuses in this thesis. The cross 
validation method is used to determine the proper SVR model parameters. The 
CVSPRT anomaly detection method and MW-DMPO n-steps-ahead prognostics 
method developed in this thesis can be extended as general methods for anomaly 
detection and failure prediction.  
  
 
 
 
 
 
 
 
 
PROGNOSTICS OF POLYMER POSITIVE TEMPERATURE COEFFICIENT 
RESETTABLE FUSES  
 
 
 
By 
 
 
Shunfeng Cheng 
 
 
 
 
 
Dissertation submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy 
2011 
 
 
 
 
 
 
 
 
 
 
Advisory Committee: 
Professor Michael G. Pecht (Advisor), Chair 
Professor Donald Barker 
Professor Bilal Ayyub (Dean?s representative) 
Professor Peter Sandborn 
Associate Professor David Bigio 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Shunfeng Cheng 
2011 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ii 
 
Dedication 
This dissertation is dedicated to my lovely wife, Haiyan Zhang, my parents-in-law, 
Tiehua Zhang and Guangping Wei, and my father, An Cheng, who support me all the 
time. Especially, this dissertation is a memorial to my mother, Yaogui Zhang, who 
always gave me her endless love.  
 
 
iii 
 
Acknowledgements 
First, I would like to express my gratitude and respect to Prof. Michael Pecht for 
his guidance in this interesting dissertation topic and challenging me to set a higher 
standard. I appreciate him for his continuous support towards the progress of this 
work and the valuable help on my writing and presenting skills. 
I would also like to thank Prof. Donald Barker, Prof. Bilal Ayyub, Prof. Peter 
Sandborn, and Prof. David Bigio for serving on my dissertation committee and 
providing valuable suggestions.  
I would like to especially thank Dr. Michael Azarian and Dr. Diganta Das for 
their insightful advice to this work. I further thank the Prognostics and Health 
Management Consortium in Center for Advanced Life Cycle Engineering (CALCE) 
for its support. I would like to thank Dr. Michael Osterman, Dr. Nikhil Lakhkar, Dr. 
Alex Coppe, Mr. Bhanu Sood, Mr. Anshul Shrivastava, and Mr. Mark Zimmerman 
for their supports on my experiments and help on this thesis. 
I would like to thank my friends and colleagues at CALCE including Dr. Nikhil 
Vichare, Dr. Jie Gu, Dr. Lei Nie, Dr. Daeil Kwon, Dr. Sachin Kumar, Dr. Nishad 
Patil, Weiqiang Wang, Wei He, Xiaofei He, Jin Tian, Gilbert Haddad, Edwin Sutrisno, 
Vasilis Sotiris, Hyunseok Oh, Sandeep Menon, Mohammed Alam, and Sony Mathew 
for their help during my graduate experience. I would like to thank Dr. Larry Lai in 
Nano Center, Dr. Xin Zhang in Material Science, and Dr. Peter Zavalij in Department 
of Chemistry and Biochemistry for their help on the analyses of samples. 
Finally, I would like to extend very special thanks to my wife Haiyan Zhang and 
my family for their constant support and encouragement towards my work. 
 
iv 
 
Table of Contents 
 
Dedication .................................................................................................................ii 
Acknowledgements ................................................................................................. iii 
List of Publications ................................................................................................... vi 
List of Tables............................................................................................................ ix 
List of Figures ........................................................................................................... x 
Chapter 1: Introduction ........................................................................................ 1 
1.1 Positive Temperature Coefficient Effect ..................................................... 1 
1.2 Operation of PPTC Resettable Fuses .......................................................... 2 
1.3 Motivation ................................................................................................. 4 
1.4 Problem Statement and Research Work ...................................................... 5 
1.5 Overview of Thesis .................................................................................... 6 
Chapter 2: Literature Review ............................................................................... 7 
2.1 Operational Principle of PPTC Resettable Fuses......................................... 7 
2.2 Electrical Characteristics of PPTC Resettable Fuses ................................. 11 
2.3 Factors Affecting Characteristics of PPTC Resettable Fuses ..................... 13 
2.4 Reliability and Testing of PPTC Resettable Fuses .................................... 17 
2.5 Prognostics Methods ................................................................................ 19 
2.6 Summary .................................................................................................. 22 
Chapter 3: Failure Precursors of PPTC Resettable Fuses .................................... 23 
3.1 Potential Failure Precursor Parameters of PPTC Resettable Fuses ............ 23 
3.2 Experiments and Results .......................................................................... 29 
3.2.1 Trip cycle test ....................................................................................... 29 
3.2.2 Aging test ............................................................................................. 32 
3.2.3 Results ................................................................................................. 33 
3.3 Summary .................................................................................................. 41 
Chapter 4: Determination of Failure Causes ....................................................... 42 
4.1 Failure Analysis Methods ......................................................................... 42 
4.2 Analysis Results ....................................................................................... 45 
4.2.1 IR test results........................................................................................ 45 
4.2.2 Interconnection analysis results ............................................................ 48 
4.2.3 Microstructure analysis results ............................................................. 52 
4.2.4 DSC analysis results ............................................................................. 56 
4.2.5 CTE test results .................................................................................... 59 
4.3 Causes for Changes in Parameters ............................................................ 60 
4.3.1 Causes for changes in trip time and surface temperature ....................... 61 
4.3.2 Causes for changes in resistance after reset ........................................... 64 
4.4 Summary .................................................................................................. 65 
Chapter 5: Anomaly Detection Using Cross Validation Sequential Probability 
Ratio Test (CVSPRT) .............................................................................................. 66 
5. 1 Wald?s SPRT ........................................................................................... 67 
5. 2 Cross Validation SPRT ............................................................................ 71 
5. 3 Anomaly Detection Results of PPTC Resettable Fuses ............................. 76 
5. 4 Summary .................................................................................................. 80 
 
v 
 
Chapter 6: Moving Window based Dynamic Model Parameter Optimization N-
 steps-ahead Prediction ............................................................................................. 81 
6. 1 Moving Window based Dynamic Model Parameter Optimization (MW-
 DMPO) N-steps-ahead Prediction ........................................................................ 81 
6. 2 Development of MW-CVSVR.................................................................. 85 
6.2.1 SVR modeling for precursor parameters ............................................... 86 
6.2.2 Cross validation SVR (CVSVR) ........................................................... 93 
6.2.3 MW-CVSVR........................................................................................ 95 
6. 3 Prognostics of PPTC Resettable Fuses Using MW-CVSVR ..................... 96 
6. 4 Summary ................................................................................................ 100 
Chapter 7: Contributions and Future work ....................................................... 107 
Appendix A: Results of Failure Analyses ............................................................... 111 
Appendix B: Anomaly Detection Results by CVSPRT .......................................... 124 
Appendix C: 5-step-ahead and 10-step-ahead Predictions by MW-CVSVR ........... 128 
References ............................................................................................................. 136 
 
 
 
vi 
 
 
List of Publications 
 
Published Journal Papers: 
? S. Cheng, K. Tom, L. Thomas, and M. Pecht, ?A Wireless Sensor System for 
Prognostics?, IEEE Sensors Journal, Volume 10, Issue 4, pp. 856 ? 862, 2010. 
? S. Cheng, M. Azarian, and M. Pecht, ?Sensor System for Prognostics and Health 
Monitoring,? Sensors, No. 10, pp.5774-5797, 2010. 
? S. Cheng, K. Tom, and M. Pecht, ?Failure Precursors for Polymer Resettable 
Fuses,? IEEE Transactions on Devices and Materials Reliability, Vol.10, Issue.3, 
pp.374-380, 2010.  
? M. Pecht and S. Cheng, ?Prognostics and Health Management Method for Aging 
Systems?, U.S. Patent Application Publication, No. US 2010/0191681 A1, Jul. 
2010. 
Published Conference Papers  
? S. Cheng, K. Tom, and M. Pecht, ?Prognostics for Polymer Positive Temperature 
Coefficient Resettable Fuses,? Machinery Failure Prevention Technology 2011, 
Virginia Beach, VA, May. 2011. 
? S. Cheng and M. Pecht, ?A Fusion Prognostics Method for Remaining Useful Life 
Prediction of Electronic Products,? 5th Annual IEEE Conference on Automation 
Science and Engineering, Bangalore, India, pp. 102-107, August 22-25, 2009. 
? S. Cheng, L. Thomas, J. Cook, and M. Pecht, ?A Radio Frequency Sensor System 
for Prognostics?, ASME/CIE 2009, San Diego, CA, August 2009.  
? S. Cheng, M. Torres, L. Thomas, and M. Pecht, ?Autonomous Prognostic 
 
vii 
 
Monitoring Device,? Proceedings of the 62th Meeting of the Society for 
Machinery Failure Prevention Technology, pp.505-516, Virginia Beach, VA, 
May. 2008. 
? S. Cheng and M. Pecht, ?Multivariate State Estimation Technique for Remaining 
Useful Life Prediction of Electronic Products,? Symposium on Artificial 
Intelligence for Prognostics, pp.26-32, Arlington, VA, 2007. 
? B. Tuchband, S. Cheng, and M. Pecht, ?Technology Assessment of Sensor 
Systems for Prognostics and Health Monitoring,? IMAPS on Military, Aerospace, 
Space and Homeland Security: Packaging Issues and Applications (MASH), May. 
2007. 
Submitted Journal Papers 
? S. Cheng and M. Pecht, ?Using Cross validation for Model Parameter Selection of 
Sequential Probability Ratio Test,? Submitted to Journal Expert Systems with 
Applications on 06/17/11. 
? S. Cheng, D. Das, M. Pecht, ?Using FMMEA to Improve Medical Device Adverse 
Event Investigations and Reporting?, Submitted to journal: Expert Review of 
Medical Devices on 6/16/2011. 
? S. Cheng and M. Pecht, ?A Moving Window based Dynamic Model Parameter 
Optimization N-steps-ahead Prognostics Method,? Submitted to Journal Expert 
Systems with Applications on 06/11/2011. 
? S. Cheng, K. Tom, and M. Pecht, ?Anomaly Detection of Polymer Positive 
Temperature Coefficient Resettable Fuses,? Submitted to Journal IEEE 
Transactions on Power Electronics on 06/12/2011. 
 
viii 
 
? S. Cheng, K. Tom, and M. Pecht, ?Failure Analysis of Polymer Positive 
Temperature Coefficient Resettable Fuses?. Submitted to Journal IEEE 
Transactions on Electron Devices on 06/12/2011. 
? J. Sun, S. Cheng, and M. Pecht, ?Prognostics of Multilayer Ceramic Capacitors 
Via the Parameter Residuals,? submitted to IEEE Transactions on Devices and 
Materials Reliability in Dec. 2010.  
 
ix 
 
List of Tables 
Table 1: PPTC models ............................................................................................... 9 
Table 2: Some properties of LDPE and HDPE ......................................................... 16 
Table 3: Test results by White et al. [49] ................................................................. 18 
Table 4: FMMEA for PPTC resettable fuses ............................................................ 27 
Table 5: Potential failure precursor parameters ........................................................ 29 
Table 6: Trip cycle test matrix ................................................................................. 32 
Table 7: Aging test matrix ....................................................................................... 33 
Table 8: Correlation coefficients among parameters in trip cycle tests ..................... 37 
Table 9 Summary of observations ............................................................................ 38 
Table 10: Time needed to reach heat balance ........................................................... 48 
Table 11: Summary of interconnection analysis ....................................................... 48 
Table 12: Summary of microstructure analyses ........................................................ 53 
Table 13: Summary of DSC test .............................................................................. 58 
Table 14: Summary of CTE test ............................................................................... 60 
Table 15: Mapping between changes in parameters and failure analysis results ........ 62 
Table 16: Range and change interval of SPRT model parameters ............................. 74 
Table 17: Summary of anomaly detection results ..................................................... 80 
Table 18: Common Kernel functions ....................................................................... 90 
Table 19: Input and output pattern transformation for training data and process for n-
 steps-ahead prediction ............................................................................................. 92 
Table 20: Ranges of SVR model parameters ............................................................ 95 
Table 21: Summary of MW-CVSVR prediction results.......................................... 104 
Table 22: Comparison of 5-step-ahead predictions by MW-CVSVR and MW-SVR
  .............................................................................................................................. 106 
 
x 
 
List of Figures 
Figure 1: Resistance (Log (R)) vs. temperature curve of a PPTC resettable fuse ........ 2 
Figure 2: Operation process of PPTC resettable fuse .................................................. 3 
Figure 3: Picture of a radial through-hole PPTC resettable fuse.................................. 7 
Figure 4: Connection of a PPTC resettable fuse in a circuit ........................................ 7 
Figure 5: Conductive chain and thermal expansion model ........................................ 10 
Figure 6: Trip- and hold-current vs. temperature ...................................................... 12 
Figure 7: Recovery of PPTC fuse............................................................................. 12 
Figure 8: FMMEA process. ..................................................................................... 25 
Figure 9: Trip cycle test setup .................................................................................. 31 
Figure 10: Current profile in one trip cycle (current vs. time) ................................... 32 
Figure 11: Example of cracks and gaps .................................................................... 34 
Figure 12: Trip time at different conditions .............................................................. 34 
Figure 13: Minimum resistance after reset at different conditions ............................ 35 
Figure 14: Maximum surface temperature at different conditions ............................. 35 
Figure 15: Maximum resistance during trip endurance at different conditions .......... 36 
Figure 16: Actual hold current every 2000 cycles at different conditions.................. 36 
Figure 17: Resistance in aging tests at different conditions ...................................... 37 
Figure 18: Different areas for different analyses ...................................................... 43 
Figure 19: IR pictures at different moments (a new sample) ..................................... 46 
Figure 20: IR pictures at different moments (-10?C cycled sample) ......................... 47 
Figure 21: Normal interconnection (No cracks and gaps) ......................................... 49 
Figure 22: Example of internal gap .......................................................................... 50 
Figure 23: Example of cracks at solders and coatings; and gaps between coatings and 
electrodes ................................................................................................................ 50 
Figure 24: Example of cracks at solder .................................................................... 51 
Figure 25: Example of crack along the solder .......................................................... 51 
Figure 26: Example of voids and cracks at solder..................................................... 52 
Figure 27: Microstructure of CB/PE composite of a normal sample (Sample was 
freeze fractured in LN2) ........................................................................................... 54 
Figure 28: Typical CB aggregates of samples in aging test at 60?C (non-cycled, 
sample was freeze fractured in LN2) ........................................................................ 54 
Figure 29: Typical CB agglomerations of samples in trip cycle test at 60?C (Sample 
was freeze fractured in LN2) .................................................................................... 55 
Figure 30: Typical CB aggregates of samples in aging test at 60?C (Non-cycled, 
sample was etched by Argon ions) ........................................................................... 55 
Figure 31: Typical CB agglomerations of samples in trip cycle test at 60?C (Sample 
was etched by Argon ions) ....................................................................................... 56 
Figure 32: DSC curve of a sample under trip cycle test at RT .................................. 58 
Figure 33: CTE test direction ................................................................................... 59 
Figure 34: Example of CTE test (foil/polymer, Z direction) ..................................... 59 
Figure 35: Anomaly detection procedure using SPRT .............................................. 67 
Figure 36: SPRT hypotheses .................................................................................... 69 
Figure 37: SPRT procedure...................................................................................... 70 
Figure 38: Model parameter selection procedure by k-fold CV ................................ 74 
 
xi 
 
Figure 39: Adding an abnormal dataset into original validation dataset to calculate the 
missed alarm probability .......................................................................................... 76 
Figure 40: Anomaly detection on trip time under -10?C condition ........................... 78 
Figure 41: Anomaly detection on trip time under -10?C condition using model 
parameter (2, 2, 0.02, 0.01) (False alarms detected) ................................................. 79 
Figure 42: Anomaly detection on minimum resistance after reset under -10?C 
condition ................................................................................................................. 79 
Figure 43: Moving window training updating based prediction method ................... 83 
Figure 44: Non-linear SVR ...................................................................................... 87 
Figure 45: Schematic flowchart of SVR model ........................................................ 91 
Figure 46: k-fold CV for SVR parameter optimization ............................................. 93 
Figure 47: MW-CVSVR procedure.......................................................................... 95 
Figure 48: MSEs vs. number of prediction steps (trip time, -10?C). ......................... 97 
Figure 49: MSEs vs. number of prediction steps (min. resistance after reset, -10?C). 98 
Figure 50: Example of double-side MW 5-step-ahead prediction ........................... 101 
Figure 51: Example of extending window 5-step-ahead prediction......................... 101 
Figure 52: Example of double-side MW 5-step-ahead prediction ........................... 102 
Figure 53: Example of extending window 5-step-ahead prediction......................... 102 
Figure 54: All 5-step-ahead predictions (-10?C,trip time, double-side MW) ........... 103 
Figure 55: All 5-step-ahead predictions (-10?C, trip time, extending window) ....... 103 
Figure 56: All 10-step-ahead predictions (-10?C,trip time, double-side MW) ......... 103 
Figure 57: All 10-step-ahead predictions (-10?C,trip time, extending window) ...... 103 
Figure 58: Typical IR pictures of a new sample ..................................................... 111 
Figure 59: Typical IR pictures of tested sample (-10 ?C, trip cycle) ....................... 112 
Figure 60: Typical IR pictures of tested sample (RT, trip cycle) ............................. 114 
Figure 61: Typical IR pictures of tested sample (60 ?C, trip cycle) ......................... 114 
Figure 62: Typical IR pictures of tested sample (85?C, trip cycle) .......................... 115 
Figure 63: Typical microstructure of CB and PE of tested samples (-10?C, aging, 
freeze fractured) .................................................................................................... 116 
Figure 64: Typical microstructure of CB and PE of tested samples (-10 ? C, trip 
cycles, freeze fractured) ......................................................................................... 116 
Figure 65: Typical microstructure of CB and PE of tested samples (RT, aging, freeze 
fractured) ............................................................................................................... 117 
Figure 66: Typical microstructure of CB and PE of tested samples (RT, trip cycle, 
freeze fractured) .................................................................................................... 117 
Figure 67: Typical microstructure of CB and PE of tested samples (60? C, aging, 
freeze fractured) .................................................................................................... 118 
Figure 68: Typical microstructure of CB and PE of tested samples (60 ? C, trip cycles, 
freeze fractured) .................................................................................................... 118 
Figure 69: Typical microstructure of CB and PE of tested samples (85?C, aging, 
freeze fractured) .................................................................................................... 119 
Figure 70: Typical microstructure of CB and PE of tested samples (85 ? C, trip cycles, 
freeze fractured) .................................................................................................... 119 
Figure 71: Typical microstructure of CB and PE of tested samples (-10?C, aging, 
etched by Argon ions) ............................................................................................ 120 
 
xii 
 
Figure 72: Typical microstructure of CB and PE of tested samples (-10?C, trip cycle, 
etched by Argon ions) ............................................................................................ 120 
Figure 73: Typical microstructure of CB and PE of tested samples (RT, aging, etched 
by Argon ions) ....................................................................................................... 121 
Figure 74: Typical microstructure of CB and PE of tested samples (RT, trip cycle, 
etched by Argon ions) ............................................................................................ 121 
Figure 75: Typical microstructure of CB and PE of tested samples (60 ?C, aging, 
etched by Argon ions) ............................................................................................ 122 
Figure 76: Typical microstructure of CB and PE of tested samples (60 ?C, trip cycle, 
etched by Argon ions) ............................................................................................ 122 
Figure 77: Typical microstructure of CB and PE of tested samples (85 ?C, aging, 
etched by Argon ions) ............................................................................................ 123 
Figure 78: Typical microstructure of CB and PE of tested samples (85 ?C, trip cycle, 
etched by Argon ions) ............................................................................................ 123 
Figure 79: Anomaly detection results based on trip time (-10 ?C, trip cycle) .......... 124 
Figure 80: Anomaly detection results based on min. resistance after reset (-10 ?C, trip 
cycle)..................................................................................................................... 124 
Figure 81: Anomaly detection results based on trip time (RT, trip cycle) ............... 125 
Figure 82: Anomaly detection results based on min. resistance after reset (RT, trip 
cycle)..................................................................................................................... 125 
Figure 83: Anomaly detection results based on trip time (60 ?C, trip cycle) ........... 126 
Figure 84: Anomaly detection results based on min. resistance after reset (60 ?C, trip 
cycle)..................................................................................................................... 126 
Figure 85: Anomaly detection results based on trip time (85 ?C, trip cycle) ........... 127 
Figure 86: Anomaly detection results based on min. resistance after reset (85 ?C, trip 
cycle)..................................................................................................................... 127 
Figure 87: All 5-step-ahead predictions (-10?C,trip time, double-side MW) ........... 128 
Figure 88: All 5-step-ahead predictions (-10?C, trip time, extending window) ....... 128 
Figure 89: All 10-step-ahead predictions (-10?C,trip time, double-side MW) ......... 128 
Figure 90: All 10-step-ahead predictions (-10?C,trip time, extending window) ...... 128 
Figure 91: All 5-step-ahead predictions (-10?C, min. resistance after reset, double-
 side MW) .............................................................................................................. 129 
Figure 92: All 5-step-ahead predictions (-10?C, min. resistance after reset, extending 
window) ................................................................................................................ 129 
Figure 93: All 10-step-ahead predictions (-10?C, min. resistance after reset, double-
 side MW) .............................................................................................................. 129 
Figure 94: All 10-step-ahead predictions (-10?C, min. resistance after reset, extending 
window) ................................................................................................................ 129 
Figure 95: All 5-step-ahead predictions (RT, trip time, double-side MW) .............. 130 
Figure 96: All 5-step-ahead predictions (RT, trip time, extending window)............ 130 
Figure 97: All 10-step-ahead predictions (RT, trip time, double-side MW) ............ 130 
Figure 98: All 10-step-ahead predictions (RT, trip time, extending window) .......... 130 
Figure 99: All 5-step-ahead predictions (RT, min. resistance after reset, double-side 
MW) ...................................................................................................................... 131 
Figure 100: All 5-step-ahead predictions (RT, min. resistance after reset, extending 
window) ................................................................................................................ 131 
 
xiii 
 
Figure 101: All 10-step-ahead predictions (RT, min. resistance after reset, double-side 
MW) ...................................................................................................................... 131 
Figure 102: All 10-step-ahead predictions (RT, min. resistance after reset, extending 
window) ................................................................................................................ 131 
Figure 103: All 5-step-ahead predictions (60?C,trip time, double-side MW) .......... 132 
Figure 104: All 5-step-ahead predictions (60?C, trip time, extending window)....... 132 
Figure 105: All 10-step-ahead predictions (60?C,trip time, double-side MW) ........ 132 
Figure 106: All 10-step-ahead predictions (60?C,trip time, extending window) ...... 132 
Figure 107: All 5-step-ahead predictions (60?C, min. resistance after reset, double-
 side MW) .............................................................................................................. 133 
Figure 108: All 5-step-ahead predictions (60?C, min. resistance after reset, extending 
window) ................................................................................................................ 133 
Figure 109: All 10-step-ahead predictions (60?C, min. resistance after reset, double-
 side MW) .............................................................................................................. 133 
Figure 110: All 10-step-ahead predictions (60?C, min. resistance after reset, 
extending window) ................................................................................................ 133 
Figure 111: All 5-step-ahead predictions (85?C,trip time, double-side MW) .......... 134 
Figure 112: All 5-step-ahead predictions (85?C, trip time, extending window)....... 134 
Figure 113: All 10-step-ahead predictions (85?C,trip time, double-side MW) ........ 134 
Figure 114: All 10-step-ahead predictions (85?C,trip time, extending window) ...... 134 
Figure 115: All 5-step-ahead predictions (85?C, min. resistance after reset, double-
 side MW) .............................................................................................................. 135 
Figure 116: All 5-step-ahead predictions (85?C, min. resistance after reset, extending 
window) ................................................................................................................ 135 
Figure 117: All 10-step-ahead predictions (85?C, min. resistance after reset, double-
 side MW) .............................................................................................................. 135 
Figure 118: All 10-step-ahead predictions (85?C, min. resistance after reset, 
extending window) ................................................................................................ 135 
 
1 
 
Chapter 1: Introduction 
A polymer positive temperature coefficient (PPTC) resettable fuse can ?trip? 
from its normal operational state of low resistance to high resistance when overheated 
by ambient heat or the Joule heat generated by high current. It can reset to its normal 
operational state of low resistance when the heat is removed and/or the power is 
switched off.  
1.1 Positive Temperature Coefficient Effect 
The trip is caused by an effect of positive temperature coefficient (PTC) for 
some materials [1]-[6]. According to PTC effect, the resistance of a component 
increases (decreases) with the increases (decreases) in its temperature. A PPTC 
resettable fuse is made of polymer composites, which are mixed by conductive 
particle fillers, such as carbon black particles, and non-conductive semi-crystalline 
polymer matrix, such as polyethylene[4][5]. The mixing of the conductive particles 
into the polymer matrix changes the conductivity of the entire polymer composite and 
enables it to exhibit the PTC effect. Figure 1 is a resistance vs. temperature curve for 
a typical PPTC resettable fuse. A sharp increase in resistance is observed in a narrow 
temperature range (100?C -130?C in the example), lower than which the increase in 
resistance is not obvious with the increase in temperature. The PPTC resettable fuses 
utilize this sharp increase in resistance to reduce the current of a circuit quickly so as 
to protect the circuit.  
 
2 
 
 
-2
 0
 2
 4
 6
 8
 10
 -20 0 20 40 60 80 100 120 140
 L
 og
  (R
  )
 Temperature ?C 
 
Figure 1: Resistance (Log (R)) vs. temperature curve of a PPTC resettable fuse 
1.2 Operation of PPTC Resettable Fuses 
Figure 2 is a schematic picture to show the operational process of a typical 
PPTC resettable fuse [7]. Under normal ambient temperature, the fuse works in a low 
resistance state (like a wire) when the normal current (less than the hold current, Ihold, 
which is the maximum steady-state current the PPTC device can carry without 
tripping at the ambient temperature) passes through it. When a fault current (higher 
than trip current, Itrip, which is the minimum current that causes a PPTC device to trip 
at the ambient temperature) occurs, the resistance of the fuse increases sharply. 
Because of the sharp increase in resistance, the current is decreased to protect the 
circuit. The sharp increase in resistance is called trip. After the trip, a PPTC resettable 
fuse does not break as does a traditional fuse. Instead, it keeps the high resistance 
state and allows a small trickle current to pass through the circuit. The fuse will reset 
 
3 
 
to a low resistance state within a short time after the heat or fault current is removed 
and/or the power is switched off [1]-[7]. The trip time is defined as the time required 
for a PPTC fuse to decrease the current of the circuit to 50% of the trip current at the 
ambient temperature [6]. Typically, the hold current (Ihold ) is half of the trip current. 
When the current is higher than the trip current, the fuse will trip. When the current is 
lower than the hold current, the fuse will not trip. When the current is between the 
hold current and the trip current, the fuse may or may not trip [4][5][6]. After a 
number of trip-reset cycles, the PPTC resettable fuse will degrade and failures will 
occur [7].  
R
 es
 is
 ta
 nc
 e
 C
 ur
 re
 nt
 Time
 Normal 
operating 
current
 Fault 
current
 Power
 off
 Normal 
operating 
current
 Low
 resistance
 High
 resistance
 Low
 resistance
 Trip 
endurance
 Time
 Hold 
current
 Trip 
current
 Trip time
 Trickle
 current
 Cooling
 with 
power 
on
  
Figure 2: Operation process of PPTC resettable fuse 
 
4 
 
1.3 Motivation 
As a circuit protection device, polymer positive temperature coefficient (PPTC) 
resettable fuses are widely used in automotive circuits (e.g., the protection of micro-
 motors in window lifts, seats, and door locks), computers (e.g., the protection of the 
circuits in hard disk drives, interface ports, and cooling fan motors), 
telecommunication devices (e.g., cell phones), battery packs, power supplies, medical 
electronics, and so on [7]. The failure or abnormal behaviors of PPTC devices may 
cause damage to circuits, abnormal operation of circuits (e.g., inability to work at 
normal current), or unnecessary operations that force operators to switch off and on 
the power to reset the circuit. It is necessary to monitor the PPTC resettable fuses in-
 situ to provide advance warning of failures and predict the remaining useful life 
(RUL) to prevent damage to the circuits.  
Prognostics and health management (PHM) has been emerging as an effective 
method to achieve these requirements. PHM is an enabling discipline consisting of 
technologies and methods to assess the reliability of a product in its actual life cycle 
conditions to determine the advent of failure and mitigate system risk [8][9]. PHM 
generally combines sensing and interpretation of environmental, operational, and 
performance-related parameters to assess the health of a product and predict 
remaining useful life. Assessing the health of a product provides information that can 
be used to meet several critical goals: (1) providing advance warning of failures; (2) 
minimizing unscheduled maintenance, extending maintenance cycles, and 
maintaining effectiveness through timely repair actions; (3) reducing the life-cycle 
 
5 
 
cost of equipment by decreasing inspection costs, downtime, and inventory; and (4) 
improving qualification and assisting in the design and logistical support of fielded 
and future systems[8][9]. 
1.4 Problem Statement and Research Work 
As the literature review in chapter 2 shows, the research on prognostics of 
PPTC resettable fuses has been very limited. There are no established methods to 
provide the prognostics and health monitoring for PPTC resettable fuses. Firstly, the 
failure precursor parameters to be monitored have not been determined. This causes 
the difficulty or increases the complexity of in-situ monitoring. Furthermore, the 
causes of the observed failures of PPTC resettable fuses have not been determined. 
Without the determination of the root causes of the failures, it is difficult to identify 
the failure mechanisms and is hard to improve the design and the reliability of the 
products in the future. The last one, anomaly detection and failure prediction methods 
have not been developed for PPTC resettable fuses.  
This thesis is focusing on solving these problems by the following research 
work: 1) determining the failure precursors by failure modes, mechanisms, and 
effects analysis, followed by conducting a series of experiments to verify the 
parameters; 2) Determining the root causes of the failure by failure analysis methods; 
3) Developing data-driven anomaly detection and prediction methods to detect the 
anomalies and predict the failures of the resettable fuses in time.  
 
6 
 
1.5 Overview of Thesis 
Chapter 2 provides the literature review on principle of PPTC effect, reliability 
of PPTC resettable fuses, prognostics of PPTC resettable fuses, and describes the 
problems not covered in literature. Chapter 3 describes the processes and results of 
determining the failure precursor parameters. Chapter 4 reports the failure analysis 
methods and results to determine the root causes of the failures. Chapter 5 describes 
the development of anomaly detection method using cross validation sequential 
probability ratio test (CVSPRT). In Chapter 6, a moving window (MW) training 
updating based dynamic model parameter optimization n-steps-ahead prognostics 
method is developed. Based on this MW method, a MW cross validation support 
vector regression (MW-CVSVR) prognostics method is developed to conduct the 
failure prediction for PPTC resettable fuses. Chapter 7 discusses the contributions and 
future work. 
 
 
 
 
7 
 
Chapter 2:  Literature Review 
The literature review includes the following aspects: 1) the operational principle 
of PPTC resettable fuses; 2) electrical characteristics of PPTC resettable fuses; 3) 
factors affecting the characteristics of PPTC resettable fuses; 4) the reliability 
evaluation of the PPTC resettable fuses; and 5) prognostics methods. After the 
literature review, the unaddressed problems are described in the summary. 
2.1  Operational Principle of PPTC Resettable Fuses 
The understanding of the operational principle of the PPTC resettable fuse can 
help to identify its failure causes and mechanisms. In general, a PPTC resettable fuse 
includes electrodes, conductive polymer composite, and outside coating. The polymer 
composite is the functional part having PTC effect. It is a mixture of non-conductive 
polymer, such as polyethylene, and conductive fillers, such as carbon black (CB) 
particles.  
 
Figure 3: Picture of a radial through-hole 
PPTC resettable fuse 
Power M
 A BI
 PPTC 
fuse
  
Figure 4: Connection of a PPTC 
resettable fuse in a circuit 
 
8 
 
Figure 3 is a picture of a radial through-hole PPTC resettable fuse. There are 
other types of PPTC resettable fuses in the market, such as surface mount 
components. The fuse is connected with other components in series in a circuit, 
shown in Figure 4.  
Both the mechanisms of the conduction of polymer composite under normal 
conditions and the mechanisms of the increase in resistance under abnormal 
conditions should be understood. Although many models [1][2][10]-[19] have been 
proposed in an attempt to explain PTC effect in polymer composites, a model that can 
explain all the phenomenon of PPTC effects is not available yet. The main models 
include conductive chain and thermal expansion model by Kohler [11], tunneling 
conductance model by Ohe et al. [17], CB aggregation structure change and 
migration model by Klason et al.[18], and Ohm conductance and phase change model 
by Allak et al.[19]. The summary of these models is shown in Table 1. Currently, the 
conductive chain and thermal expansion model is used to explain the principle in 
most of literature, for example, Wei et al. [1], Luo et al. [2], Gorniak et al.[3], 
Doljack et al. [12], because of the ease of understanding. 
Figure 5, by Cheng et al. [7], shows the schematic picture of the structure of a 
radial through-hole resettable fuse shown in Figure 3. In general, radial through-hole 
PPTC fuses include conductive polymer composites, electrodes, and outside coatings 
[4][5]. Conductive polymer composite is generally manufactured as a thin sheet and 
consists of non-conductive polymer (e.g., polyethylene) and conductive particles 
(e.g., carbon black (CB)). An electrode is used to conduct and control the flow of 
electricity and is typically composed of foils and leads.  
 
9 
 
Table 1: PPTC models 
Models At low temperature 
At crystalline to amorphous phase transition 
temperature 
Defects or cannot 
explain 
Conductive chain 
and thermal 
expansion model by 
Kohler [11] 
Polymer is in semi-crystalline 
state; conductive fillers form 
conductive chains. 
Phase transition from crystalline to amorphous? 
volume is expanded ? conductive chains are 
broken 
Why the sharpness of 
resistance rise is not 
a function of volume 
change 
Tunneling 
conductance model 
by Ohe et al. [17]  
Distribution of conductive 
filler is uniform; gaps 
between the conductive 
fillers are small and electron 
tunnels are formed 
Distribution becomes random? gaps between 
conductive fillers increase to break the electron 
tunnels. 
Why the particle 
distribution changes 
from uniform to 
random distribution 
CB aggregation 
structure change and 
migration model by 
Klason et al.[18]  
CB aggregation structure is 
determined by crystalline 
phase of polymer; CB 
particles distribute in 
amorphous phase and form 
conductive chain. 
Phase transition from crystalline to amorphous 
breaks the CB aggregation structures ? a more 
homogenous particle distribution 
Lack of enough 
experiment results to 
support 
Ohm conductance 
and phase change 
model by Allak et 
al.[19] 
Linear I-V curve (Ohm 
behavior); CB resides in 
amorphous regions ? easily 
establish conductive chains 
Crystalline to amorphous with a sudden and large 
volume increase?gap between new formed 
amorphous and enlarged amorphous regions 
reduce?more resistive current pathways?break up 
the conductive chains 
Lacks of enough 
experiment results to 
support 
 
10 
 
Heated, phase transition, 
and expand
 Cooled down, recrystallize, 
and shrink
 Conductive chains of CB 
are broken ? increasing 
resistance
 Non-conductive 
polymer in 
amorphous state
 Conductive chains 
of carbon black 
(CB) particles 
Non-conductive 
polymer in semi-
 crystalline state
 Foil
 Lead 
Cold (normal)
 Low resistance
 Hot
 High resistance
 Solder
 I I I I
  
Figure 5: Conductive chain and thermal expansion model 
The foils are attached on both sides of the polymer sheet. One lead is connected to 
each foil by soldering. The dielectric coating material provides protection for the 
outside of the device. 
According to conductive chain and thermal expansion model, at normal 
temperatures, the conductive particles in the polymer form many conductive paths, 
which allow current to flow through the fuses without interruption. However, if the 
temperature rises around the crystalline to amorphous (C?A) phase transition 
temperature of the polymer, some crystalline state of the polymer will change to 
amorphous state. The expansion in volume of the polymer during the phase change 
separates the conductive particles and breaks the conductive paths. This results in a 
large nonlinear increase in the resistance of the PPTC device just in a narrow 
temperature range. The whole process of this model is shown in Figure 5 [7]. 
 
11 
 
2.2 Electrical Characteristics of PPTC Resettable Fuses 
Some electrical characteristics of PPTC composites should be concerned in the 
applications. For example, the relationship between resistance and temperature of the 
fuse is shown in Figure 1. The relationship between the volume (or thickness) of the 
fuse and the temperature were also introduced by Kohler [11], Doljack [12], and 
Founrnier et al.[14]. For example, Doljack [12] described that the electrical resistivity 
of conductive polymer composites was determined by the type of CBs and the 
volume ratio of total CB to polymer. The initial resistance of the devices is defined 
based on the maximum volume ratio of CB to polymer.  
Trip time is dependent upon the size of the current and the ambient temperature. 
The trip time with different current under a certain temperature should be given by 
manufacturer. This information is used to guide the users to select the proper devices 
for their applications. For example, Bourns [20] provide the curve of trip time vs. 
current under temperature 23 ?C in the datasheets. Trip time under 23 ?C is used as a 
reference for users to select the proper fuses for their applications. The trip time 
decreases with the increasing of the current through the fuse. Figure 6 illustrates 
schematically the hold- and trip-current behaviors of PPTC devices as a function of 
temperature [4][5]. Region A describes the combinations of current and temperature 
at which the PPTC device will trip and protect the circuit. Region B describes the 
combinations of current and temperature in which the device will allow for normal 
operation of the circuit. In region C, it is possible for the device to either trip or 
remains in the low resistance state, depending on the individual device resistance and 
 
12 
 
its environment. The specific hold current value under different temperature 
conditions are shown in the datasheet of the fuse. 
 
Figure 6: Trip- and hold-current vs. temperature 
 
Figure 7: Recovery of PPTC fuse 
The recovery of PPTC fuse occurs within the first several seconds. Figure 7 
show the process schematically. The resistance fully recovery is depended on both the 
design of the device and thermal environment. After tripped, the return to maximum 
Temperature (?C) 
-40 80 
C
 ur
 re
 nt
  (
 A)
  
Region A 
Region B 
Region C Trip current 
Hold current 
0 
Recovery time (s) 
0  
R
 es
 is
 ta
 nc
 e 
(OHM
 ) 
0 
1000 
 
13 
 
volume compaction of the polymer is a much slower process than simple cooling 
alone [11].  
2.3 Factors Affecting Characteristics of PPTC Resettable Fuses 
The factors that affect the characteristics of PPTC resettable fuses include the 
properties of the polymers, the particle fillers, and the processing conditions, which 
were mentioned by many researchers, such as Huang [21], Zhang et al. [22], Sau et 
al. [23], and Narkis et al. [24]. Different conductive particle fillers, such as metal 
powders or fibers [25], or CB [26], can be dispersed into a plastic matrix to enable the 
non-conductive plastic become conductive when the concentration of the filler 
reaches a threshold, which was discussed by many researchers, such as Zhang [22], 
Miyasakaet al. [27], and Lux [28]. Although metal particles are more conductive than 
CB, CB is applied as the common filler in the PPTC resettable fuses because of CB?s 
resistance on oxidization, which forms an insulating layer on the surface of the filler. 
CB is an amorphous form of carbon with a structure similar to disordered 
graphite. Huang [21] summarized five types of CBs in the CB industry: furnace 
black, thermal black, lamp black, channel black, and acetylene black. Over 90% of 
the CBs current produced are made by the furnace process. In this process, oil is 
thermally decomposed to form CB particles [29]. The general diameter range of the 
particles to fill in the polymer is less than 100 nm.  
Many researchers, such as Mironi-Harpaz et al. [30], Donnet [31], and Bourat 
[32], pointed out that CB particles have trends to aggregate together so that under 
electron microscope examination CB is composed of prime particles fused into 
 
14 
 
primary aggregates. The small aggregates also trend to aggregate together again to 
form larger agglomerates. Zhang et al. [22], Narkis et al. [24], and Xi et al. [33] 
stated that when the CBs particles or aggregates are filled into polymer matrix, they 
sit in the amorphous phase of the polymer and are separated by polymer matrix. But 
when the gap between aggregates is less than some critical distance, electrons can 
flow across the polymer barrier, and generate a high conductivity [33]. When the 
polymer is heated and the temperature increase to a certain level, the crystal in the 
polymer may change its phases from crystalline to amorphous. This phase transition 
increases the amorphous phases, which cause to the decrease in the density of CBs 
and increase in the distance between CBs, and cause to the increase in the resistance 
[33]. The CBs obtain more energy at high temperature and more of them are 
aggregating together.  
Researchers also investigated the effects of using different polymer matrix on 
the electrical and mechanical properties of PPTC polymer composites. For example 
Xi et al. [33] used polyethylene, Sumita et al. [34] and Huang et al. [35] used 
Ethylene-Vinyl Acetate Copolymer, and Tchoudakov et al. used the 
polypropylene/polyamide [36]. It is found that the threshold concentration is affected 
by the interaction between the polymer and CB, the crystallinity and melts viscosity 
of the polymer [21][34]-[37]. Huang [21] summarized that if the CB particle-to-
 particle aggregation is stronger than the adhesion between CB particles and polymer, 
the CB particles will aggregate and be hard to separate and disperse into the polymer 
matrix; but in the opposite, the particles will disperse into the polymer without 
contact, which affects the form of the conductive paths if the concentration of particle 
 
15 
 
is not too high. So with the increase of the surface tensions of the polymers, the 
threshold concentration increased. That means more particles are needed to fill into 
the polymer with high extension to obtain high conductivity.  
The other important effect of different polymer on the conductivity of 
composite is the crystallinity. Semi-crystalline polymer contains crystalline regions 
and amorphous regions. The crystallinity is used to measure the ratio of crystalline 
regions. Researches show the threshold concentration increases with the decrease in 
the crystallinity [38]-[46]. CB particles are dispersed uniformly in the amorphous 
regions and rejected from the crystalline regions, which was reported by He et al. 
[38], Park [40], Hunag et al.[41], Xu et al. [43], and Yui et al. [45].  
Experiments were conducted to evaluate the effect of different high-crystallinity 
resin and different content of CB on the electrical conductivity of PPTC device [38]-
 [40][45][46]. For example, Horibe et al. [46], investigated the relationship between 
the resistivity and the crystallinity, and the relationship between the resistivity and 
carbon content. During the test, the crystallinity of each polymer was determined by 
X-ray diffraction analysis, and the carbon particle size was directly measured by 
scanning electron microscopy (SEM). He found that high-density polyethylene 
(HDPE) has lowest resistivity at room temperature and the highest resistivity after 
polymer melting. The higher the crystallinity of HDPE is, the lower the resistivity of 
the composites. The resistivity of the composites decreased with increasing CB 
content.  
 
16 
 
The polymer composite tested in this thesis is made of carbon black particles 
and polyethylene (PE). PE is a semi-crystalline polymer. PE can be classified simply 
as low density PE (LDPE) and high density PE. Table 2 shows the different 
properties of them [48].  
Table 2: Some properties of LDPE and HDPE 
 
Property LDPE HDPE 
Formula ?(CH2-CH2)n? ?(CH2-CH2)n? 
Melting temperature 
(crystalline to amorphous 
transition temperature) 
Around 110?C Around 130?C 
Glass transition 
temperature 
Around -110?C Around -100?C 
Crystallinity < 50% >90% 
Density 0.91-0.94 g/cm3 0.95-0.97 g/cm3 
Researchers reported that CBs were dispersed into the amorphous region of the 
PE [38]-[42]. When the temperature is well below the C?A transition point of the 
PE, the crystalline and amorphous phase is in a solid state, and the resistivity of the 
composite does not rise significantly with the rise in temperature. Close to the C?A 
transition point of the polyethylene, the crystalline phase starts to form new 
amorphous regions. Such structural changes perturb the conducting pathways and 
consequently, the resistivity of the material increase. The C?A transition increases 
the volume of amorphous parts, which decrease the volume percent of particle fillers, 
causes to the increase in inter-distance between particles and the increase in the 
resistance. 
 
17 
 
In general, the aggregate of the particles increases the resistance compared with 
the resistance when the particles are distributed in uniformly, under which more 
random conductive networks are formed to decrease the resistance. Yui et al. [45] 
used SEM to investigate the aggregation of CB particles and found that CB particles 
in HDPE aggregated together but these aggregates did not contact together to form 
conductive networks, thus increased the resistance.  
2.4 Reliability and Testing of PPTC Resettable Fuses 
The research on the reliability of PPTC resettable fuses has been very limited. 
A surface-mount PPTC manufactured by Raychem Corporation was tested by White 
et al.[49]. The trip cycle life and the trip endurance life were evaluated from trip tests. 
The distribution was assumed as an exponential distribution, in which the constant 
failure rate was estimated through the 100 cycle test. The estimation was that there 
would be less than one failure out of 276816 trips. The failure of PPTC resettable fuse 
was defined as the observation of arcing or tripping below the hold current. The 
estimation for the trip endurance was that there would be less than one failure out of 
75,900 tripped hours at 95% confidence level [49].  
This work also tested the effect of electrical aging and environmental aging on 
the trip cycling and trip endurance. The data then were fitted in a statistical model to 
calculate the time to failure. No cause analyses of the failure were conducted. The 
device resistance increased with tripping cycles. One failure mode was that, after 
number of trip cycles or trip endurance, devices would become high enough in 
 
18 
 
resistance that they would not pass the normal operating current (hold current), and 
device would nuisance trip. 
The environmental aging was generated by the effects of temperature and 
humidity. Results showed that heat aging caused devices to decrease in resistance 
from their post-reflow resistance. Higher aging temperatures caused a faster decrease. 
The results were shown in the Table 3. 
Table 3: Test results by White et al. [49] 
  Resistance (Ohm) 
Trip Time 
(Sec) 
(15V 8A) 
Trip 
Current(A) 
(15V 8A) 
6000 trip cycles 
at 15V 40A, 10 
sec on, 50 sec 
off at 20 ?C 
Before 0.422?0.0219 0.013?0.0005 7.4?0.09 
After 4.25?2.24 0.004?0.0004 6.7?0.18 
125 days at 70 
?C 
Before 0.444?0.022 0.012?0.0008 7.4?0.08 
After 0.258?0.006 0.011?0.0006 7.4?0.07 
  
The failure of the fuse in field was defined as the fuse had not tripped properly. 
The possible failure modes include the device becoming high in resistance, arcing, or 
delaminating during trip. Some tests, such as trip cycle test and aging test, are 
conducted by manufacturing or certified institution based on the standard UL 1434 
[6]. Test results are listed in the datasheets of the products. However, these tests are 
not enough to evaluate the reliability of the PPTC resettable fuses. One of the reasons 
 
19 
 
is that the number of test cycles is not enough to observe the changes in the 
properties.  
2.5 Prognostics Methods  
Prognostics is an emerging technique in recent years. Anomaly detection and 
prognostics of PPTC have not been reported. This thesis will develop a prognostics 
method to provide advance warning of the failure of PPTC fuse and predict the 
remaining useful life of it. Cheng and Pecht [8][9] defined that PHM is an enabling 
discipline consisting of technologies and methods to assess the reliability of a product 
in its actual life cycle conditions to determine the advent of failure and mitigate 
system risk. PHM generally combines sensing and interpretation of environmental, 
operational, and performance-related parameters to assess the health of a product and 
predict remaining useful life. Traditionally, prognostics has been implemented by 
either a data-driven approach or a model-based approach. Gu et al. [50], Mathew et 
al.[51], Kumar et al. [52], and Pecht et al.[53] described these approaches and 
pointed out advantages and limitations of them.  
Data-driven methods use current and/or historical data to statistically derive 
decisions and predictions about the health and reliability of products [9]. Based on the 
type of the data available, a variety of data-driven methods can be used. These 
methods can be classified as three categories: supervised, semi-supervised, and 
unsupervised. Data-driven methods use a training process to define the healthy 
behavior or baseline of products. The training process reveals patterns or relationships 
between parameters, such as correlation, covariance, residual, and inference patterns, 
 
20 
 
from the historical healthy data or the initial healthy state of the detected electronic 
product [9]. The data-driven method then compares the current monitored data with 
the baseline to determine the health of the electronic product and predict its RUL. 
Data-driven methods do not require an understanding of complex failure mechanisms 
and damage assessment models. This advantage enables the data-driven methods to 
be used for the prognostics of complex products that have multiple complex failure 
mechanisms. There are many data-driven methods, such as autoregressive integrated 
moving average (ARIMA) developed by Box et al. [54] and demonstrated by Ho et 
al. [55][56], neural networks (NNs) shown by Xu et al. [57], support vector machines 
(SVMs) developed by Vapnik et al. [58], and demonstrated by Pai et al. [59], and 
Chen et al. [60], decision tree classifiers, used by Kuo et al. [61], and Freund et al. 
[62] on reliability prediction, principle component analysis (PCA), used by Choi et al. 
[63], and Cho et al. [64] on the fault detection, particle filtering (PF) used by Xu et al. 
[65], and Zhang et al. [66] on detection and prediction, and fuzzy logic used by 
Yadav et al. [67] on reliability improvement estimation, etc. However, the data-
 driven methods are dependent on the training data. If the training data does not 
contain the entire range of the healthy behavior of the product, or if the training data 
contains any degradation of the product, missed or false alarms may be generated. 
The data-driven method cannot distinguish between different failure modes or 
mechanisms.  
Pecht [9] stated that PoF methods utilize knowledge of a product's life cycle 
loading conditions, geometry, and material properties to identify potential failure 
mechanisms and estimate its remaining useful life. The advantages of the PoF method 
 
21 
 
include that providing the estimate of damage for given loading conditions and failure 
mechanisms; identifying the components which are critical to system reliability; 
estimating remaining life for different loading conditions, as long as the failure 
mechanisms and accelerated factors are identified; predicting the remaining life even 
in non-operating conditions. Several examples about PoF methods were shown by Gu 
et al. [68][69] and Mishra et al. [70]. However, it is difficult to estimate RUL using 
the PoF method if the failure mechanisms or models are unavailable. Sometimes, 
although the individual failure mechanisms are identified, dealing with the 
interactions between different failures mechanisms is still a challenge for the PoF 
method. Models with a combination of failure mechanisms are much more 
complicated than simply being the sum of the individual failure mechanisms. 
Selection of different data-driven methods depends on the features of data. For 
example, if the data is purely linear, ARIMA model may be suitable for the 
prediction; if the data is nonlinear, the NN and support vector regression (SVR) may 
be proper. These models can forecast the future value of the data based on the 
regression of the historical data. The support vector regression (SVR) has some 
advantages compared with other non-linear forecasting method mentioned by Pai et 
al. [59], and Chen et al. [60]. However, SVR has rarely been used in the prediction 
for the reliability of electronics. Most of research showed how to improve the fitting 
on the observed data [71]-[73], not the actual application on prediction of data in the 
future. 
 
22 
 
2.6 Summary 
Most of research on the PPTC polymer composites was focused on the physics 
of the conductivity and the electrical characteristics under different materials, 
contents, and structures of the fillers and polymers. Different models used to interpret 
the PPTC effects and behaviors were developed although none of them can explain 
all observed phenomenon.  
Prognostics and health monitoring of the PPTC devices have not been 
developed and reported. Failure precursors that can indicate the anomalies or failures 
of the PPTC resettable fuse have not been determined. Although White et al [49] 
reported that resistance after reset and the trip time changed before and after tests, he 
did not monitor these parameters continuously so that the comparison only before and 
after a test did not reveal the trends of the parameter changes and could not draw a 
conclusion about the precursors. The causes for the degradation and failures of PPTC 
resettable fuses as a component, which includes different parts, have not been 
analyzed. The reactions of different parts of the PPTC resettable fuses were not 
considered for the degradation or failures of PPTC resettable fuses. Anomaly 
detection and failure prediction of PPTC devices have not been reported. The failure 
mechanisms and models of the PPTC resettable fuses have not been determined and 
developed.  
 
 
23 
 
Chapter 3: Failure Precursors of PPTC Resettable Fuses 
The potential failure precursor parameters are determined by a systematic 
method, failure modes, mechanisms, and effects analysis (FMMEA). A series of 
experimental tests are then conducted to verify the precursors by identifying the 
correlations among the parameters and the mapping between the observed failures 
and the parameters. The monitoring of these precursor parameters enables the 
implementation of prognostics methodologies to detect anomalies and predict the 
failure of PPTC resettable fuses. 
3.1 Potential Failure Precursor Parameters of PPTC Resettable Fuses  
It is necessary to understand the environmental and usage conditions, 
structures, materials and geometry of the PPTC resettable fuses to identify the 
potential failure modes, mechanisms, and effects. A radial through-hole PPTC 
resettable fuse, as shown in Figure 3, was used as an example to show the 
determination of the failure precursor parameters. The methodologies used in this 
thesis can be extended to any kind of PTC devices.  
 Figure 5 shows schematic cross-section image of the device shown in Figure 3. 
In general, radial through-hole PPTC fuses include conductive polymer composites, 
electrodes, and outside coating. Conductive polymer composite is generally 
manufactured as a thin sheet by mixing the non-conductive polymer (e.g., 
polyethylene) and conductive particles (e.g., carbon black), being placed in a mold, 
 
24 
 
and then being punched out from the mold. An electrode is used to conduct and 
control the flow of electricity and is typically composed of foils and leads. The foils 
are attached on both sides of the polymer sheet. One lead is connected to each foil by 
soldering. The dielectric coating provides protection for the outside of the device.  
An example of the material sheet of different parts of the device, shown in 
Figure 3 and Figure 5 can be found in [74]. The operational temperature range of this 
PPTC resettable fuse is from -40?C to 85?C. The highest surface temperature is 
around 125?C. The fuse works in a trip-reset cycling way, or essentially a heating-
 cooling cycling way. Based on the conductive chain and thermal expansion model, 
device also works in a thermal expansion-shrink cycling way. 
The potential failure precursor parameters can be identified by failure modes, 
mechanisms, and effects analysis (FMMEA). FMMEA is a methodology used to 
identify failure mechanisms and models for all potential failure modes of a product 
under its operational and environmental conditions. Figure 8 is a schematic diagram 
of FMMEA. More detail information can be obtained from Ganesan et al.[75], Pecht 
et al.[76], and IEEE standard 1413 [77]. Identification of the potential failure modes, 
mechanisms, and models can help to determine the potential parameters to be 
monitored and the locations where the sensors should be placed. A series of 
experiment may be needed to verify the precursor parameters. 
 
25 
 
 
 
Figure 8: FMMEA process. 
Table 4 shows the FMMEA results of the radial through-hole PPTC resettable 
fuse shown in Figure 3. Potential failure modes include abnormal trip behaviors, 
shifts in parameters, and physical cracks and gaps. The failure criteria should be 
defined based on related standards, specifications, and customers? requirements. In 
this thesis, a failure was defined as any of the following:  
1) Fuse trips at less than normal current (?Ihold) at the specific ambient 
temperature. 
2) Fuse does not trip at fault current (?Itrip) at the specific ambient temperature. 
Identify life 
cycle profile 
Identify potential 
failure modes  
Identify potential 
failure mechanisms  
Identify failure models 
Define system and 
identify elements and 
functions to be 
analyzed 
Identify potential 
failure causes 
Document the process 
Prioritize failure 
mechanisms 
 
26 
 
3) Deviations in the trip time of the fuse impact the typical operations of the 
circuit. The criterion of failure in terms of trip time is application-
 dependent. A trip in a longer time increases the risk of damage to the circuit 
because of the longer exposure of the circuit to a high fault current. A 
shorter trip time makes the circuit more likely to be disturbed by noisy 
currents, which may result in unnecessary faults in circuit operation. For 
example, when a motor changes its rotating direction, a high peak current 
may be generated. If the trip time of the fuse becomes too short, the high 
peak current will trip the fuse, stop the motor, and the operator must switch 
off and on the power to reset the fuse. In some cases, the designer of a 
circuit does not consider the reliability issues of the PPTC resettable fuse, 
and therefore does not design sufficient margin for trip time changes. For 
example, the circuit may be designed to allow a 20A fault current to pass 
through the circuit for 3 to 5 seconds when the trip time of the selected 
PPTC fuse is 4 seconds. If the PPTC resettable fuse degrades, the trip time 
may become shorter than 3 seconds or longer than 5 seconds. If the trip time 
of the fuse is shorter than 3 seconds, it will stop the operation of the circuit 
even for short peak current noise. If the trip time becomes longer than 5 
seconds, it will damage the components in the circuit.  
 
 
 
 
 
 
 
 
 
 
 
27 
 
Table 4: FMMEA for PPTC resettable fuses 
(Structured as Figure 3) 
 
Potential 
Failure Sites 
Potential Failure 
Modes 
Potential Failure Causes 
Potential 
Failure 
Mechanisms 
Conductive 
polymer 
composite  
Abnormal trip 
behaviors 
(e.g., trip at 
normal current, 
no trip at fault 
current) 
Increase in the heat 
dissipation resistance, the 
cracks or gaps at the 
interconnections, the 
changes in the polymer 
properties, and the changes 
in the distribution of the CB 
particles. 
Degradation 
of the 
polymer; 
aggregation 
of CB 
particles 
 
Interconnection 
between foil 
and polymer 
composite 
Shift in resistance 
and surface 
temperature 
CTE-mismatch; gaps; 
manufacturing defects 
Fatigue 
Solder between 
lead and foil 
Shift in resistance Cracks Fatigue  
Outside 
coating 
Cracks, 
separation with 
foil, shift in 
surface 
temperature 
CTE-mismatch;  
deformation; manufacturing 
defects 
Fatigue 
 
4) A fuse becomes high in resistance after reset. One effect of the increase in 
resistance after reset is that it shortens trip time by generating more heat in 
the same amount of time. The other effect is that an increase in the 
resistance of the fuse decreases the voltage drops on other components, 
which may cause abnormal operation of the circuit.  
5) Physical cracks, breaks, separations, and/or degradation in the outside 
coating. Fuses with these types of degradation may still function; however, 
the internal parts of the fuse will lose protection due to the degradation of 
 
28 
 
the dielectric coating. For example, moisture will corrode the electrode 
more easily. Furthermore, degradation of dielectric materials also causes a 
safety issue for the operators. 
Potential causes of these failures include the degradation of materials include 
the increase in the heat dissipation resistance, the cracks or gaps at the 
interconnections, the changes in the polymer properties, such as the crystallinity, the 
changes in the distribution of the CB particles, and CTE-mismatch. Potential failure 
mechanisms include the degradation of the polymer, fatigue, and aggregation of CB 
particles. 
Table 5 shows the potential precursor parameters based on the FMMEA results. 
In actual applications, the current through the fuse, the voltage across the fuse, and 
the surface temperature of the fuse can be monitored in-situ. The resistance during the 
trip can be calculated by Ohm theory using the monitored current and voltage. The 
resistance after trip can be measured by a data-logger using a 4-wire connection if the 
circuit is not too complex. Trip time can be calculated by the difference between the 
time when the fault current occurs and the time when the current decreases to the hold 
current. This can be measured by a current meter or sensors. Actual hold current 
cannot be monitored in-situ based on the test method defined in standard [6] but it can 
be measured when maintenance is being conducted.  
 
 
 
29 
 
Table 5: Potential failure precursor parameters 
 
Potential Failure Precursors Can be monitored in-situ? 
Trip time Yes 
Resistance 
Resistance after reset Yes 
Resistance during trip 
Yes (by measure current and 
voltage of the device during trip) 
Surface temperature Yes 
Current 
Current through the devices at 
normal condition 
Yes 
Trickle current Yes 
Actual hold current No 
Voltage across the device Yes 
 
3.2 Experiments and Results 
In this thesis, the trip cycle tests and aging tests were designed to determine 
whether the trip time, current, resistance, and surface temperature are indicators of 
degradation in PPTC resettable fuses.  
3.2.1 Trip cycle test 
Referring to the manufacturer?s specifications for the fuse [20], standards [6], 
and the requirements of customers, trip cycle tests were conducted for at least 6000 
cycles at four different ambient temperature conditions, -10?C, room temperature 
(RT, 19-23?C), 60?C, and 85?C, respectively. Four samples were tested in each 
 
30 
 
condition, which was limited by our power supply. In each cycle, current through the 
fuse, voltage across the fuse, the surface temperature, trip time, and resistance in the 
resetting process (power switched off) of the fuse were monitored. The actual hold 
current was measured every 2000 cycles.  
Figure 9 shows the setup of the trip cycle test. The Labview program was used 
to control a data logger and the VEE program was used to control a 4-channel power 
supply. Each channel of the power supply provided power to one PPTC resettable 
fuse. The fuses were placed inside a temperature chamber or air conditioning room to 
perform the trip cycle test under different temperature conditions. The temperature in 
the chamber was constant. The VEE program recorded the time stamp when the 
initial high current occurred and the time stamp when the current reduced to the hold 
current at ambient conditions, and then calculated the trip time of each trip cycle. 
After trip, the VEE kept controlling the power supply on for additional time, and then 
switched off it. An Agilent Data Logger 34970A was used to monitor the resistance 
in the resetting process and the surface temperatures of each fuse. A 4-wire 
connection resistance measurement was used to remove the effects of the wire and the 
connection. The surface temperature was measured by thermocouples, each of which 
was attached on each side of the fuse; the maximum temperature of these two 
thermocouples was used to determine the surface temperature of the fuse.  
 
31 
 
Programmable 4-channel 
power supply: Agilent 
6705 (I, V, and Trip time)
 PPTC resettable 
fuse (in chamber for    
-10 ?C, 60 ?C and 
85 ?C), in air-
 conditioned room 
for RT.
 Thermocouple
 4-wire 
connection 
resistance 
measurement
 Agilent 
34970A data 
logger (surface 
temperature 
and resistance)
 VEE or Labview 
programs to 
communicate with 
power supply and 
data logger
  
Figure 9: Trip cycle test setup 
Figure 10 shows the current profile in one trip cycle. When the power was 
switched on, a high current was input to the fuse (20A for all conditions). The fuse 
was heated by Joule heat and tripped to a high resistance state in several seconds. 
When the current reduced to the specified hold current at the environmental 
temperature, power supply was kept on for 1 additional minute (called 1 minute trip 
endurance). After that, power was switched off for 7 minutes to cool the fuse and 
reset it to a low resistance state. Table 6 is the test matrix, which shows the input 
current the specified hold current. During each cycle, the fuses were tripped by the 
heating of fault current and reset by the natural cooling inside the chamber or under 
room condition. 
 
32 
 
Power off  
7 mins
 Trip time (Several seconds)
 20 A
 Hold 
current
 Current
 Time
 Trip endurance (1 min)
 Trickle current
  
Figure 10: Current profile in one trip cycle (current vs. time) 
 
 
Table 6: Trip cycle test matrix 
 
Temperature Input Current Specified Hold Current 
-10 ?C (4 samples) 20A 4.8A 
Room Temperature 
(RT, 4 samples) 
20A 3.75A 
60 ?C (4 samples) 20A 2.4A 
85 ?C (4 samples) 20A 1.5A 
 
3.2.2 Aging test 
When the trip cycle tests were in process, aging tests were being conducted 
under the same environmental condition. Only the resistances were measured using 
data-logger via 4-wire connection. Table 7 is the matrix of the aging test. The 
objectives of the aging tests included the followings: 1) to investigate if the resistance 
of the PPTC resettable fuse changes with time under a constant temperature 
condition; 2) to compare the experimental and failure analysis results of the cycled 
 
33 
 
samples (in trip cycle tests) and non-cycled samples (in aging tests) under the same 
environmental condition. 
Table 7: Aging test matrix 
 
Temperature Number of samples 
-10 ?C 8 
RT 8 
60 ?C 8 
85 ?C 8 
 
3.2.3 Results 
Failures were observed in trip cycle tests. No failures were observed in aging 
tests. Observed failures in trip cycle tests includes damage to the coating of the device 
(cracks and gaps) and shifting in some of the performance parameters, such as trip 
time and resistance after rest. The failures are not hard failures that stop the 
functioning of the fuse, but soft failures that affect the performance of the fuses.  
Figure 11 shows the examples of cracks and delimitation observed on the 
coating of the fuses after trip cycle tests under -10?C and RT conditions. These 
failures were not observed at any samples under 60?C and 85?C conditions. Figure 12 
to Figure 14 show the shifting in the monitored parameters, including trip time, 
resistance after reset, and surface temperature with increase in number of cycles. 
Figure 15 shows the maximum resistance that the device can reach during the trip 
endurance. Each point in these figures is the average of the corresponding parameter 
values of every 100 cycles. Figure 16 shows the changes in the offline measurements 
 
34 
 
of hold current, which was conducted every 2000 cycles. Figure 17 shows the 
changes in the resistance collected by every half hour in aging tests. Table 9 maps all 
the observations in trip cycle tests and aging tests and the parameter shifting under all 
test conditions. Table 8 shows the correlation coefficients between different 
parameters. 
Side view from leftObserved at all samples under -10 ?C
 and RT conditions
 mm
  
Figure 11: Example of cracks and gaps 
Number of Cycles
 T
 rip
  T
 im
 e 
(s
 )
 200180160140120100806040201
 11
 10
 9
 8
 7
 6
 5
 4
 3
 (X 100)
 85 ?C, 20A
 60 ?C, 20A
 RT, 20A
 -10 ?C, 20A
  
Figure 12: Trip time at different conditions 
(Each point is the average of every 100 cycles) 
 
35 
 
Number of Cycles
 M
 in
 im
 um
  R
 es
 is
 ta
 nc
 e 
(O
 hm
 )
 200180160140120100806040201
 0.12
 0.11
 0.10
 0.09
 0.08
 0.07
 0.06
 0.05
 0.04
 0.03
 (X 100)
 85 ?C, 20A
 60 ?C, 20A
 RT, 20A
 -10 ?C, 20A
  
Figure 13: Minimum resistance after reset at different conditions 
(Each point is the average of every 100 cycles) 
 
 
 
Number of Cycles
 M
 ax
 im
 um
  S
 ur
 fa
 ce
  T
 em
 pe
 ra
 tu
 re
  (
 ?C
 )
 200180160140120100806040201
 128
 126
 124
 122
 120
 118
 116
 114
 112
 110
 (X 100)
 85 ?C, 20A
 60 ?C, 20A
 RT, 20A
 -10 ?C, 20A
  
Figure 14: Maximum surface temperature at different conditions 
 (Each point is the average of every 100 cycles) 
 
 
36 
 
Number of Cycles
 M
 ax
 im
 um
  R
 es
 is
 ta
 nc
 e 
D
 ur
 in
 g 
T
 rip
  (
 O
 hm
 )
 1990017900159001390011900990079005900390019001
 10
 9
 8
 7
 6
 5
 4
 3
 85 ?C, 20A
 60 ?C, 20A
 RT, 20A
 -10 ?C, 20A
  
Figure 15: Maximum resistance during trip endurance at different conditions 
 (Each point is the average of every 100 cycles) 
 
 
 
Number of Cycles
 A
 ct
 ua
 l H
 ol
 d 
C
 ur
 re
 nt
  (
 A
 )
 200180160140120100806040201
 9
 8
 7
 6
 5
 4
 3
 2
 85 ?C, 20A
 60 ?C, 20A
 RT, 20A
 -10 ?C, 20A
 (X 100)
  
Figure 16: Actual hold current every 2000 cycles at different conditions 
 
 
 
37 
 
Aging Hours
 R
 es
 is
 ta
 nc
 e 
(O
 hm
 )
 16001400120010008006004002001
 0.07
 0.06
 0.05
 0.04
 0.03
 85 ?C, 20A
 60 ?C, 20A RT, 20A
 -10 ?C, 
20A  
Figure 17: Resistance in aging tests at different conditions 
 
Table 8: Correlation coefficients among parameters in trip cycle tests 
 
Conditions 
TT vs. 
MinR 
TT vs. 
MaxST 
MinR vs. 
MaxST 
TT vs. 
MaxR 
MinR vs. 
MaxR 
MaxST 
vs. MaxR 
-10 ?C -0.97 0.92 -0.90 -0.43 0.39 -0.43 
RT -0.98 0.92 -0.93 0.27 -0.32 0.36 
60 ?C -0.93 -0.46 0.68 -0.33 0.37 0.41 
85 ?C 0.77 0.85 0.72 -0.36 -0.42 -0.46 
 
Trip time: TT, Maximum surface temperature: MaxST, Minimum 
resistance after reset: MinR, Maximum resistance during trip: MaxR 
Under -10 ?C, RT, and 60?C conditions, decreases in trip times are correlated 
with the occurrences of cracks at coating or gaps between coating and foil (it was 
verified that internal gaps between coating and foil occurred in samples under 60?C 
condition). For 85 ?C condition, the changes in trip time are correlated with the 
absence of cracks and gaps. Most importantly, trip time shows the decreases much 
earlier than observable cracks or gaps. This indicated that trip time is precursor for 
cracks or gaps.  
 
38 
 
Table 9 Summary of observations 
 
Conditions 
Observable 
cracks or gaps 
Changes in 
trip time 
Changes in 
min. 
resistance 
after reset 
Changes in 
max. surface 
temperature 
Changes in 
actual hold 
current 
(every 2000 
cycle) 
Changes in 
(max. 
resistance or 
min. trickle 
current) 
during trip 
endurance 
Changes 
in 
resistance 
in aging 
test 
-10 ?C 
Yes (around 
4000 cycles) 
Increased by 
3% in the 
first 2000 
cycles, then 
decreased by 
35% of the 
initial value  
Decreased by 
7% in the first 
1000 cycles, 
then increases 
by 43% of the 
initial value  
Increased by 
4% in the 
first 3500 
cycles, then 
decreases by 
3% of the 
initial value  
 
 
Decreased by 
24% but still 
much higher 
than the 
value in 
mfr?s 
specification 
(6.5 vs.4.8A) 
 
No trends 
 
No trends 
 
RT 
Yes (around 
6000 cycles) 
 
Increased by 
7% in the 
first 1000 
cycles, then 
decreased 
from 5000 
cycles by 
24% of the 
initial value  
Decreased by 
4% in the first 
1000 cycles, 
then increases 
by 28% of the 
initial value  
Increased by 
3% in the 
first 4000 
cycles, then 
decreases by 
2% of the 
initial value  
Decreased by 
2% but still 
much higher 
than the 
value in 
mfr?s 
specification 
(5.8 
vs.3.75A) 
Decreased 
and then 
recover 
 
No trends 
 
 
39 
 
Conditions 
Observable 
cracks or gaps 
Changes in 
trip time 
Changes in 
min. 
resistance 
after reset 
Changes in 
max. surface 
temperature 
Changes in 
actual hold 
current 
(every 2000 
cycle) 
Changes in 
(max. 
resistance or 
min. trickle 
current) 
during trip 
endurance 
Changes 
in 
resistance 
in aging 
test 
60 ?C 
No (after 22000 
cycles, but 
internal gaps 
were identified 
by failure 
analysis) 
Increased by 
13% in the 
first 1000 
cycles, then 
decreased 
gradually 
Decreased by 
10% in the 
first 2000 
cycles, then 
increased by 
27% of the 
initial value 
Increased 
sharply by 
4% in the 
first 1000 
cycles, then 
gradually 
increased by 
3% of the 
initial value  
 
Decreased by 
6% but still 
much higher 
than the 
value in 
mfr?s 
specification 
(4.5 vs.2.4A) 
 
No trends 
 
No trends 
 
85 ?C 
No (after 21000 
cycles) 
 
Increased 
sharply by 
15% in the 
first 2000 
cycles, then 
no obvious 
changes  
Decreased by 
7% in the first 
1000 cycles, 
then increases 
by 38% of the 
start value  
Increased 
sharply by 
3% in the 
first 1000 
cycles, then 
gradually 
increased by 
3% of the 
initial value 
 
Decreased by 
10% but still 
much higher 
than the 
value in 
mfr?s 
specification 
(2.8 vs.1.5A) 
 
Gradually 
decreased 
and then 
recovered  
No trends 
 
 
40 
 
Under -10?C and RT conditions, surface temperature shows similar trends with 
trip time, shown in Figure 12, Figure 14, and Table 8, it changes in small range and 
later than the changes in trip time. For 60?C and 85?C conditions, surface temperature 
changes are not obvious (less than 5%). Although trip time can provide most of the 
information that surface temperature can provide, we suggest surface temperature 
should be monitored to be assistance for trip time. 
The increases in the trip time in the early period of experiment correlated with 
the decreases in the resistance after reset for all test conditions. This observation 
agrees with the principle of PPTC resettable fuses, in which the lower the resistance, 
the less heat is generated for the same input current, and the longer trip time is at the 
same ambient condition. After the early period, high positive correlation between the 
resistance after reset and the trip time was shown in the tests under -10?C and RT 
conditions. But weak correlation were observed for tests under the 60 ?C and 85?C 
conditions, under which the resistance increased more than 30% of the initial value 
but the trip time showed no obvious changes (less than 5% of the initial value). This 
observation inferred that trip time changes were caused by not only the resistance 
changes but other impact factors, and the resistance changes may be not the dominant 
one. The increases in resistance affect the performance of other components in the 
circuit and can indicate other failures of the entire fuses, such as the defects in 
interconnections between different parts and the degradation of CB/PE polymer. Thus 
it is a failure precursor parameter. Other parameters, such as the highest resistance (or 
the minimum trickle current) during the trip endurance and the actual hold current, 
did not present obvious changes in the tests. Although the actual hold currents under -
 
41 
 
10 ?C, 60?C and 85 ?C conditions have decreasing trends, it is much higher (more 
than twice) than the manufacturer specified hold currents. The correlation analysis 
among maximum resistance during the trip endurance and other parameters is shown 
in Table 8, which indicates that the maximum resistance during trip endurance has 
low correlation with others. Not enough evidences are observed to conclude that the 
highest resistance during trip endurance and the actual hold current are precursors at 
this stage. The resistance in the aging test does not change obviously, which showes 
that under normal aging conditions, the characteristics of the PPTC fuses are stable. 
3.3 Summary  
Trip time, resistance after reset, and surface temperature show high correlations 
with the observed failures, including the cracks at the coating, the gaps between 
coating and foil, and the performance shifting of the fuses. Changes in the trip time 
and the minimum resistance after reset affect the performance of the fuses directly 
and indicate the degradation of fuses. These parameters are failure precursors and 
must be monitored. High correlation between the trip time and surface temperature 
can simplify the post-analysis of the data. For example, it is enough to use trip time to 
predict the failure. But as a side-product of the performance of the fuse, surface 
temperature can be monitored to provide assistance information for trip time. Other 
parameters, including the maximum resistance during trip endurance and the actual 
hold current, did not exhibit obvious changes in all the conducted trip cycle tests. 
They are not considered as failure precursors at this stage. The resistances at aging 
tests under different conditions were stable thus were not failure precursors. 
 
42 
 
Chapter 4: Determination of Failure Causes 
Failures including cracks, gaps and performance parameter shifting were 
observed in trip cycle tests. Determination of the causes of the failures helps to 
identify the actual underlying failure mechanisms, develop proper data-driven and 
physics of failure models to predict the failure, and improve the design and reliability 
in the future.  
The PPTC resettable fuse includes polymer composite as a PTC functional 
component, foils, solders, electrodes, and outside coating materials. Degradation of 
any of them contributes to the performance degradation of the fuse. No literature has 
reported the cause analysis considering the entire resettable fuse. Potential failure 
causes of radial through-hole resettable fuses (shown in Figure 3) were identified by 
FMMEA and shown in Table 4. In this chapter, failure analyses considering all the 
parts of the fuse were conducted to determine the causes of the changes in trip time, 
resistance after reset, and surface temperature.  
4.1  Failure Analysis Methods  
Based on the potential failure cause analysis, some nondestructive and 
destructive analyses were conducted. First, heating process analysis (or surface 
temperature increase process) was conducted by infrared (IR) camera to investigate 
whether the heat dissipation changed after the trip cycle test. IR pictures were also 
used to identify the location of the internal gaps between different parts and to verify 
the effect of cracks and gaps on the changes in the surface temperature and the trip 
 
43 
 
time. The IR pictures were taken under RT condition. For each trip cycle test 
condition, one tested sample was picked for IR test without any cutting. 
The destructive analyses included: 1) interconnection analysis in order to 
determine the effects of the defects at the interconnection on failures of the fuse; 2) 
CB/PE microstructure analysis in order to investigate the effects of the distribution of 
the CB particles on failures of the fuse; 3) CB/PE crystallinity analysis in order to 
investigate the effects of the crystallinity changes in PE on failures of the fuse; 4) 
CTE test on a new sample to investigate the CTE-mismatch. The destructive analysis 
1), 2), and 3), were conducted on the cycled samples under each test condition. A fuse 
was cut to different small parts for different analysis, shown in Figure 18.  
2
 31
 4
  
Figure 18: Different areas for different analyses 
Area 1 was used for interconnection analysis to identify the cracks, voids, and 
gaps, which affect the mechanical, thermal, and electrical properties of the fuse. This 
area was cut from the fuse, cross-sectioned to different locations, and then 
investigated by environmental scanning electrons microscopy (ESEM). The 
concerned interconnections include those between coating and metal foil sheets, the 
 
44 
 
foil sheets and the leads, and the foil sheets and the polymer composite. Area 1 was 
also used to investigate the distributions of CBs in the polymer composite after some 
special process, such as ion etching by Argon ions, which can remove a thin layer of 
the PE and keep CB at its location. Aggregation of CB particles affects the 
conductivity of the polymer composite. 
Area 2 and area 3 were used to investigate the microstructure. The parts were 
freeze fractured. In freeze fracturing process, parts were immersed in liquid nitrogen 
for 1 hour to become fragile and easy to break, and then were broken in the liquid 
nitrogen. The surface at the broken site then was coated with a thin layer of gold by 
sputter coater. Comparing with ion-etched surface, freeze fractured surface kept all 
components of the polymer composite. The gold-coated surface of the section was 
investigated by ESEM to identify the microstructure of polymer composite including 
the CBs and PE networks. 
Area 4 was used to investigate the crystalline to amorphous (C?A) transition 
temperature and crystallinity of the polymer composite. These properties of the 
composite affect the heat required for the fuse trip. The polymer composite of this 
area was taken out and put into differential scanning calorimetry (DSC) to investigate 
the changes in C?A transition temperature and crystallinity. DSC is a technique for 
measuring heat capacity during a programmed change of temperature. Specific heat, 
which is the amount of energy needed to change the temperature of unit mass by unit 
degree at a constant pressure, was measured using a Pyris DSC. The specific heat 
curve measured over a wide temperature range shows how much energy is required to 
 
45 
 
the polymer phase transitions. For semi-crystalline polymers, the overall area of the 
transition peak increases with the degree of crystallinity, whilst its shape is 
determined by the crystal population present in the specimen. In DSC tests, sample 
was heated at 10 ?C /min from 30 ?C to 180 ?C and then cooled at 10 ?C /min to 30?C 
for 6 cycles.  
CTE tests, using Pyris TMA 7, of coating material and the combination of the 
foil and CB/PE composite were conducted to investigate the CTE mismatches, which 
were the possible causes for the gaps between different parts inside the fuses.  
4.2 Analysis Results 
4.2.1 IR test results 
Figure 19 and Figure 20 are the IR pictures of a new sample and a sample with 
cracks and gaps after cycle test under -10?C condition, shown in Figure 11. IR 
pictures reveal the presents and locations of the gaps between coating and foil, as well 
as the thermal distribution in the heating process. Table 10 summarizes the time 
needed to reach the heat balance for samples after cycle tests under different 
conditions. The tests results indicate that the samples, which have cracks and gaps, 
require longer time to dissipate the heat out. This means that the heat is kept inside 
for a longer time, and causes the faster heating of the internal CB/PE composite than 
that of the sample without cracks and gaps. This verifies that the gaps does cause the 
decrease in the trip time. Table 10 can also explain the reason of the a little increase 
in the trip time under 85?C, under which the heat dissipation is faster than the new 
 
46 
 
sample so longer time is needed to heat the sample up. All IR pictures are shown in 
Appendix A. 
 
0s 1s 2s 
 
3s 4s 5s 
 
6s 7s 8s 
 
 
Figure 19: IR pictures at different moments (a new sample) 
 
 
 
 
 
 
 
 
47 
 
0s 0.3s 0.5s 
 
2s 3s 4s 
 
5s 6s 7s 
 
8s 13s 24s 
 
 
Figure 20: IR pictures at different moments (-10?C cycled sample) 
 
 
 
 
           Gaps 
 
48 
 
Table 10: Time needed to reach heat balance 
 
Conditions 
Time needed to reach 
heat balance (s) 
Cracks or gaps 
New sample 8 No 
-10?C 24 Yes  
RT 15 Yes  
60?C 7 Internal gap only 
85?C 5 No 
 
4.2.2 Interconnection analysis results 
Figure 21 shows an example of new sample, in which no gaps and cracks were 
observed at all interconnections. Figure 11 shows an example of the observable 
cracks on the outside coating and gaps between coating and foil, which occurred at all 
the samples under -10 ?C and RT conditions. Figure 22 and Figure 23 show examples 
of the internal gap between coating and foil, which occurred at all the samples under -
 10 ?C, RT, and 60 ?C conditions. Figure 23 to Figure 26 show examples of different 
type of cracks and voids at the solders, which connect the electrode with the foil. The 
cracks were observed in all the samples under -10?C and RT conditions. Table 11 
summarizes the interconnection analysis results.  
Table 11: Summary of interconnection analysis 
 
Conditions 
Observable 
cracks and gap 
Internal gap 
Cracks at 
solder 
New No No No 
 
49 
 
Conditions 
Observable 
cracks and gap 
Internal gap 
Cracks at 
solder 
Trip cycle 
tests 
-10?C 
Yes (around 
4000 cycles, all 
samples) 
Yes (all samples) 
Yes (all 
samples) 
RT 
Yes (around 
6000 cycles, all 
samples) 
Yes (all samples) 
Yes 
(all samples) 
60?C 
No (after 22000 
cycles, all 
samples) 
Yes (all samples) 
No (after 
22000 cycles, 
all samples) 
85?C No (after 21000 cycles, all samples) 
Aging 
tests 
No (all samples under all conditions) 
 
 
Solder 
Cu 
CB/PE
 Coating 
Foil 
 
Figure 21: Normal interconnection (No cracks and gaps) 
 
 
50 
 
Gap
 Gap
 Foil 
Solder 
CB/PE 
Polymer 
Electrode
 (Cu) 
Coating 
 
Figure 22: Example of internal gap 
 
 
 
 
1 2
 1 2
 Solder 
Cracks 
Cu 
Solder 
Cracks 
Cu 
Gap Gap
 Crack
 Coating 
 
Figure 23: Example of cracks at solders and coatings; and gaps between coatings and 
electrodes 
 
51 
 
Solder 
Cracks 
 
Figure 24: Example of cracks at solder 
 
 
 
 
 
 
Solder 
Cracks 
 
Figure 25: Example of crack along the solder 
 
 
 
52 
 
Solder 
Cracks 
Cu 
Void CB/PE
 Coating 
Gap 
Foil 
Void
  
Figure 26: Example of voids and cracks at solder 
 
 
4.2.3 Microstructure analysis results 
Microstructure analysis revealed the distribution of PE and CBs. Figure 27 
shows how CBs distribute in PE matrix in a normal case using freeze fracturing 
method. The CBs have trends to aggregate together. Figure 28 shows a typical 
microstructure of a sample under 60?C aging test, in which the CBs aggregated in a 
relative low level. Figure 29 shows a typical microstructure of a sample under 60?C 
trip cycle tests, in which CBs aggregated more and formed large agglomerations. All 
microstructure analysis results using freeze fractured method are shown in appendix 
A.  
The analyses using ESEM after ion etching processing also observed the 
aggregates under different aging testing (non-cycled) conditions and agglomerations 
 
53 
 
of the CBs under different trip cycle test (cycled) conditions. Figure 30 shows a 
typical picture of a sample under 60?C aging condition using ESEM after etching by 
Argon ions. Figure 31 is atypical picture of a corresponding sample under 60 ? C trip 
cycle tests. All pictures of CBs aggregates and agglomerations under different 
conditions after ion etching are shown in appendix A.  
Agglomeration of CBs affects the form of CB conductive paths. Although more 
conductive paths are formed in the larger agglomeration, the gap between different 
agglomerations increases. These agglomerations actually reduce the global 
conductive paths and increase the resistance. Table 12 summarizes the microstructure 
analyses.  
 
Table 12: Summary of microstructure analyses 
 
Conditions CB Aggregates? 
CB Large 
Agglomerations? 
Resistance after 
reset 
Trip cycle 
tests 
-10?C Yes Yes  Increase 
RT Yes  Yes  Increase 
60?C Yes Yes  Increase 
85?C Yes Yes  Increase 
Aging 
tests 
CB aggregates were observed for all samples; No CB 
agglomerations were observed 
No obvious 
changes 
 
 
54 
 
 
Figure 27: Microstructure of CB/PE composite of a normal sample (Sample was 
freeze fractured in LN2) 
 
 
Figure 28: Typical CB aggregates of samples in aging test at 60?C (non-cycled, 
sample was freeze fractured in LN2) 
CB 
aggregates 
PE matrix 
CB 
aggregates 
PE matrix 
PE matrix 
 
55 
 
PE Matrix CB agglomerations
  
Figure 29: Typical CB agglomerations of samples in trip cycle test at 60?C (Sample 
was freeze fractured in LN2) 
 
 
Figure 30: Typical CB aggregates of samples in aging test at 60?C (Non-cycled, 
sample was etched by Argon ions) 
CB 
aggregates 
 
56 
 
 
Figure 31: Typical CB agglomerations of samples in trip cycle test at 60?C (Sample 
was etched by Argon ions) 
 
4.2.4 DSC analysis results 
DSC tests revealed the changes in the thermal properties of the CB/PE 
composite, including the C?A transition temperature and crystallinity of the semi-
 crystalline PE.. Sample was heated from 30 ?C to 180 ?C at 10 ?C /min and then 
cooled to 30?C at 10 ?C /min for 6 cycles. The results showed that the first heating 
curve was different with those of the other 5, which were vertically identical. All the 
cooling curves were identical.   
Figure 32 shows an example of DSC test results (only the first and the 6th cycle) 
of a sample after trip cycle test under RT condition. Table 13 summarizes the DSC 
test results and compares the C?A transition temperature and crystallinity between 
CB 
aggregates 
CB 
agglomerations 
 
57 
 
the cycled samples and non-cycled samples (aging samples) under each ambient 
condition. The DSC result of the first scan contained the sample?s thermal history 
including the heating/cooling cycles in the trip cycle tests and the temperature 
changes from experimental condition to room condition for sample preparation of 
DSC. We assumed that the thermal effects of sample preparations for all the samples 
were the same. The first scan then revealed the effects of trip cycle and aging tests on 
the thermal properties of polymer. Based on the results, all main C?A transition peak 
temperature of the cycled samples were higher than those of the aging samples. The 
percent crystallinity of the cycles samples were also a little higher than that of the 
aging samples. This indicated that the trip cycle increased the crystallinity of 
polymer. These results agreed with the results from some researchers. For example, 
Omastova et al. [78][79] conducted thermal cycle tests on polymer composite and 
found that the thermal cycles increased the crystallinity. The table also shows that the 
higher the ambient temperature, the higher the C?A transition temperature and 
crystallinity are. Since the highest temperature of the sample at different conditions is 
the same, under the natural cooling style, the higher the ambient temperature, the 
lower the cooling rate is. These results agreed with a results that the faster the 
cooling, the less crystallinity under RT is [80][81]. The increase of the crystallinity 
contributes to the increase of trip time because more heat is required for C?A 
transition. It also contributes to the decrease of resistance after reset because the less 
amorphous phase decreases the gaps between CB particles.  
 
 
58 
 
40             60              80             100           120            140            160            180
 Temperature (?C)
 H
 ea
 t F
 low
  E
 nd
 o 
D
 ow
 n 
(m
 W
 ) 1st heating: 
30?180 ?C
 6th heating: 
30?180 ?C
 1st cooling: 
180?30 ?C
 6th cooling: 
180?30 ?C
 Peak: 136?C
 ? H: -85 J/g Peak: 
120?C
 ? H: -75 J/g
 Peak: 117?C
 Peak: 117?C
 Peak: 
128?C
  
 
Figure 32: DSC curve of a sample under trip cycle test at RT 
 
 
 
 
Table 13: Summary of DSC test 
 
 
MainPeak_1st 
heating (?C) 
LowPeak_1st 
heating (?C) 
MainPeak_6th 
heating (?C) 
C%_1st heating 
(=?Hm/?Hm
 0) 
?100% 
? Hm
 0=285.8 J/g 
C%_6th heating 
(=?Hm/?Hm
 0) 
?100% 
? Hm
 0=285.8 J/g 
 Aging Cycle Aging Cycle Aging Cycle Aging Cycle Aging Cycle 
-10?C 130 135 120 119 128 128 27% 29% 24% 25% 
RT 130 135 119 120 128 128 28% 30% 26% 26% 
60?C 132 136 119 120 128 128 28% 29% 26% 25% 
85?C 133 137 120 119 128 128 29% 30% 25% 25% 
       
?Hm and ? Hm
 0 are C-A transition 
enthalpies of sample and 100% 
crystallization sample 
 
59 
 
4.2.5 CTE test results 
The CTE at X and Z directions of outside coating and the combination of the 
foil and polymer, shown in Figure 33, were measured using Pyris TMA 7. The 
temperature range is from -10 ?C to 140 ?C by 5 ?C/min.  
X Z
  
Figure 33: CTE test direction 
 
 
-10     0              20             40            60            80            100           120          140
 Temperature (?C)
 P
 ro
 be
  P
 os
 it
 io
 n (
 m
 m
 )
 Expansion Coefficient: 113.9 e-06/?C
 Expansion Coefficient: 1063.4 e-06/?C
 Expansion Coefficient: 269.2 e-06/?C
 Temperature: 110 ?C
 Position: 0.56mm
 Temperature: 130?C
 Position: 0.59mm
  
Figure 34: Example of CTE test (foil/polymer, Z direction) 
 
 
60 
 
CTE tests showed non-linear expansion at both X and Z directions. Figure 34 is 
an example of the test curve of the combination of foil and polymer (foil/polymer) at 
Z direction. Table 14 summarizes the CTE test results. CTE in temperature range 
from -10?C-110?C and range from 110?C -130?C were calculated individually, and 
then the overall CTE were calculated at range from -10?C -130?C. At Z direction, 
CTE-mismatch between coating and the foil/polymer combination were high at 
temperature range from 110?C -130?C, in which the C?A transition occurred and 
finished. At X direction, the CTE-mismatch was not that obvious. CTE-mismatch is 
one of the causes of the gaps and cracks shown in the interconnections.  
Table 14: Summary of CTE test 
 
 
CTE (E-06/?C) 
X direction Z direction 
Temperature range 
-10?C-
 110?C 
110?C 
-130?C 
-10?C -
 130?C 
-10?C-
 110?C 
110?C 
-130?C 
-10?C -
 130?C 
Coating 38.6 -105.4 20.6 105.4 -294.5 48.6 
Foil and CB/PE 20.2 -12.3 18.9 113.9 1063.4 269.2 
4.3 Causes for Changes in Parameters 
Table 15 shows the mapping of the changes in parameters (trip time and 
resistance after reset) and the failure analysis results. The causes for the changes in 
 
61 
 
the parameters can be determined based on this mapping and some theoretical 
analysis. 
4.3.1 Causes for changes in trip time and surface temperature 
Trip is caused by the PPTC effect when the temperature is increasing to a 
certain level. Trip time is affected by changes in the PTC effects and the rate of the 
temperature increase. PPTC effect is controlled by the properties of the CB/PE 
composite itself, such as the crystallinity of the polymer, which affects the heat 
required for phase transition, and the aggregation of the CBs, which affects the 
resistance changes. Increase in temperature is controlled by a heat transfer equation 
(1), and finally heating and dissipation reach to a balance shown in equation (2).  
 (1) 
(2) 
Where m = mass of the fuse, Cp = heat capacity of the fuse, ?T = change in fuse 
temperature, ?t = change in time, I = current flowing through the fuse, R = resistance 
of the fuse, U = overall heat-transfer coefficient, T = temperature of fuse, TA = 
ambient temperature. In equation (1), item 1 is the heat to raise the temperature of the 
fuse; item 2 is the generated Joule heat, which is the heat source; item 3 is the heat 
dissipated to environment. When the heat transfer reaches the balance, the all 
generated heat dissipates to environment, and temperature of the fuse does not 
increase any more. 
 
)(2)/( Ap TTURItTmC ?????
 )(2 ATTURI ??
       1                  2            3 
 
62 
 
Table 15: Mapping between changes in parameters and failure analysis results 
 
 
Trip time 
changes 
Resistance 
after reset 
changes 
Interconnection analysis 
results 
CBs 
agglomerations 
Time to heat 
balance (s) Crystallinity 
compared with 
non cycled 
samples Gap 
Cracks at 
solder 
New sample: 
8s 
-10?C Increased 
quickly in the 
early period, 
and then 
decreased 
Decreased in 
the early 
period, and 
then 
increased 
 
Observable 
cracks and 
gaps 
Yes Yes 24 Increased 
RT 
Observable 
cracks and 
gaps 
Yes Yes 15 Increased 
60?C 
Increased 
quickly in the 
early period, 
and then 
decreased 
gradually 
Internal gaps 
(observed by 
ESEM) 
No Yes 7 Increased 
85?C 
Increased 
quickly in the 
early period, 
and increased 
gradually 
No gap found No Yes 5 Increased 
 
 
63 
 
The external influence factors for temperature increasing include the current 
passing the fuse, the ambient temperature, and heat dissipation rate around the fuse. 
Temperature increasing is also affected by some internal factors including the start 
resistance of in each trip cycle (or the resistance after the previous cycle) and the heat 
dissipation ability of the device itself. In my experiment, the external effects were 
assumed to be constant. Only the internal effects were concerned in this thesis.  
Based on the analysis above and the mapping in Table 15, the causes for the 
trip time changes can be determined as the following:  
1) The increase in the trip time in the early period was caused by the decrease 
in resistance at that time period and the increase of the crystallinity, due to 
which more heat was required for C?A phase transition. 
2) The decreases in trip time after the earlier period were caused by the 
decrease in the heat dissipation ability due to the gaps and cracks at the 
interconnections between different parts. The increase of the resistance after 
reset contributed the decrease of the trip time but was not a dominant cause. 
The gaps between foil/polymer combination and coating were caused by the 
CTE-mismatch and the heating/cooling cycles. 
3) The gradually increase trend exhibited at 85 ?C was caused by the increase 
of the crystallinity and the increase of the heat dissipation ability.  
 
64 
 
Surface temperature is a side-product of the trip of the fuse. It was high 
correlated with trip time. The causes for the changes of these two parameters were the 
same. 
4.3.2 Causes for changes in resistance after reset 
Table 15 also shows the mapping between the changes in resistance after reset 
and the failure analysis results. Resistance after reset measured in the trip cycle test 
was the entire resistance of the fuses including the resistance of the CB/PE composite 
and the resistance of interconnections. Defects at the polymer and interconnection all 
contributed the changes in resistance after reset. The causes for changes in the 
resistance after rest included the following: 
1) The early decreases in resistance were caused by the increase in the 
crystallinity. Based on some research by He et al. [38], Park [40], Hunag et 
al.[41], Xu et al. [43], and Yui et al. [45], the CBs sit in the amorphous 
phase in PE. The increase in crystallinity reduces the amorphous phases 
resulting in the increase in the resistance. Furthermore, no cracks at 
interconnections were observed in this period. 
2) Cracks and gaps at the interconnections except for that between coating and 
foil contributed to the increases in the resistance after reset. 
3)  The formation of large CB agglomerations caused the increase in 
resistance. The local conductivity inside the agglomeration increases, but 
the conductivity between different agglomerations decreases due to the 
 
65 
 
increase in the gaps. The agglomerations increase the global resistance of 
the CB/PE composite.  
4) With the trip cycles, the effects of crystallinity changes on the resistance 
were ignorable comparing with the effects by the cracks at interconnections 
and CB agglomerations in CB/PE composite. 
4.4 Summary 
The causes for changes in parameters were determined in this chapter. Gaps 
between different parts were the main causes for decreases in the trip time. Without 
gaps, the properties change of the CB/PE, such as the increases in crystallinity, were 
the main causes for the increase of the trip time. The agglomerations of the CBs were 
the one of the main causes for the increase in the resistance after reset. With the 
increases in the cycle numbers, the cracks at the solder and the gaps between foil and 
CB/PE composite will control the increases in the resistance and finally cause an 
open current. The further analysis by CTE revealed that the CTE-mismatch caused to 
the gaps between coating and foil/polymer combination.  
 
 
 
 
 
 
66 
 
Chapter 5: Anomaly Detection Using Cross Validation 
Sequential Probability Ratio Test (CVSPRT) 
Trip time and resistance after reset were determined as precursors for gaps, 
cracks, and prosperity degradation of CB/PE composite, which have different failure 
mechanisms. Anomaly detection for PPTC resettable fuse should monitor and detect 
the abnormal behaviors exhibited in each of the precursor parameters. In this thesis, 
sequential probability ratio test (SPRT) is used to monitor the trip time and resistance 
after reset to detect the anomalies. The difficulty of model parameter selection for 
SPRT was solved by a cross validation method.  
Sequential probability ratio test (SPRT) has been used in in-situ monitoring, 
anomaly detection, and decision making for electronics, structures, and process 
controls. For example, Gross et al. used SPRT to monitor anomalies in computer 
servers [83][84], and equipment in nuclear plants [85]. Pecht et al. [53] used SPRT to 
detect anomalies in BGA solders by monitoring the resistance in the temperature 
cycles. SPRT was also used as an anomaly detection algorithm for an ePrognostics 
sensor tag [86]. Sohn [87] applied SPRT to conduct a statistical damage classification 
of a system, such as automotive system based on monitored vibration signals. 
Stoumbos et al. [88][89] utilized SPRT to generate a statistical process control model 
to monitor the process changes.  
 
67 
 
However, because model parameters for this method, such as the system 
disturbance magnitudes, and false and missed alarm probabilities, are selected by 
users primarily based on experience, the actual false and missed alarm probabilities 
are typically higher than the requirements of the users. This chapter presents a 
systematic method to select model parameters for the sequential probability ratio test 
without the need of experience by using a cross validation method. The presented 
method can improve the accuracy of sequential probability ratio test by reducing the 
false and missed alarm probabilities caused by improper model parameters. 
General anomaly detection using SPRT is shown in Figure 35. The training 
data, which are healthy data of the system, can be obtained from historic data or from 
the stable operational phases when the product is known to be healthy. Statistical 
features of the training data, such as the distribution, are extracted to create a 
detection baseline. In the detection procedure, in-situ monitored data is compared 
with the baseline sequentially using SPRT to detect the anomalies.  
 
 
 
 
5. 1 Wald?s SPRT  
SPRT is a sequential statistical binary hypothesis test introduced by Wald [90]. 
SPRT is different from traditional hypothesis testing where the data collection is often 
SPRT Anomaly 
detection 
In-situ 
monitored data 
 
Training data 
 
Figure 35: Anomaly detection procedure using SPRT 
 
68 
 
executed without analysis, and after all the data are collected analysis is conducted 
and conclusions are drawn [87]. In SPRT, data is monitored in-situ and is analyzed 
directly once it is collected. This ability enables SPRT for the detection of statistical 
changes against a null hypothesis at the earliest possible time or with the smallest 
sample size [90][91].  
The binary hypothesis of SPRT includes one null hypothesis and one or more 
alternative hypotheses, shown in Figure 36. For a normal distribution, the null 
hypothesis H0 represents the healthy state, with mean = 0 and standard deviation = ?; 
The alternative hypothesis includes four cases: 1) H1: the mean of the test data has 
shifted high to +M, with no change in standard deviation; 2) H2: the mean of the test 
data has shifted low to ?M, with no change in standard deviation; 3) H3: the variance 
of test data has increased to V?2, with no change in mean; 4) H4: the variance of test 
data has decreased to ?2/V, with no change in mean. M and V are the predetermined 
system disturbance magnitudes, which are decided by the user, and in general they 
are several times of the standard deviation of the training data.  
Figure 37 shows the procedure of SPRT. The SPRT calculates the SPRT index 
for each in-situ monitored data point and compares the SPRT index with the 
boundary to make a decision. The SPRT index is the natural logarithm of the ratio of 
the probability that accepts the null hypothesis to the probability that accepts the 
alternative hypothesis. Four SPRT index formulas for normal distribution are shown 
as Equations (3) to (6) [92]. 
 
 
69 
 
-M M
 H0: N(0, ?2)H2:N(-M, ?2) H1: N(M, ?2)
 H3:
 N(0, V?2)
 H4:
 N(0, ?2/V)H0: N(0, ?2)
 0
 0  
Figure 36: SPRT hypotheses 
 
(3) 
 
(4) 
 
 
(5) 
 
 (6) 
 
SPRT uses the probabilities of missed alarms and false alarms to create the 
thresholds of acceptance and rejection of the null hypothesis, as shown in equation 
(7).  
 
(7) 
Each SPRT index is compared to the decision boundaries (A, B). For each 
comparison, there are three possible outcomes: 1) the lower limit is reached, in which 
case the healthy hypothesis (H0) is accepted, the corresponding index is reset, and 
sampling continues; 2) the upper limit is reached, in which case the alarm is given, 
)
 2
 (
 1
 21 ?
 ?
 ??
 n
 i
 i
 M
 x
 M
 SPRT
 ?
 )
 2
 (
 1
 22 ?
 ?
 ???
 n
 i
 i
 M
 x
 M
 SPRT
 ?
 VnV
 x
 SPRT
 n
 i
 i
 ln)2/()1(
 2 2
 1
 2
 4 ???
 ?
 ?
 ?
 Vn
 V
 x
 SPRT
 n
 i
 i
 ln)2/()
 1
 1(
 2 2
 1
 2
 3 ???
 ?
 ?
 ?
 )
 1
 ln(  ),
 1
 ln(
 ?
 ?
 ?
 ? ?
 ?
 ?
 ? BA
 
70 
 
the corresponding index is reset, and sampling continues; or 3) neither limit is 
reached, in which case the information is not sufficient to make a conclusion, and the 
sampling continues.  
 
Figure 37: SPRT procedure 
Four model parameters should be determined for SPRT detection: the system 
disturbance magnitude, M, which is m times the standard deviation of the training 
data; the variation factor, V; the false alarm probability, ?; and the missed alarm 
probability, ?. In practice, these model parameters are selected by experience. Even 
though the system disturbance magnitudes selected are in this recommended range 
[92] [93], the actual false alarm probability and/or missed alarm probability may be 
higher than the user-allowed values. Developing a systematic model parameter 
selection method for SPRT is necessary to improve its performance and control the 
false alarms and missed alarms. In this thesis, a cross validation method was used as a 
systematic method to select a proper model parameter set (m, V, ?, ?) for SPRT 
without the need for experience.  
)
 )(
 )(
 ln(
 0
 4~1 ?
 ?
 ? ?
 k
 is s
 sj
 j xf
 xf
 SPRT
 Healthy 
Alarm 
Reset the 
corresponding 
SPRT index to 0 
Yes 
In (A, B)? 
<=A 
>=B 
Acquired data: 
xs 
k=k+1 
Initialization: 
i=k=1 
i,k=k+1 
 
71 
 
5. 2  Cross Validation SPRT 
Cross validation (CV) is a statistical method that can be used to evaluate the 
performance of machine-learning-based anomaly detection and prediction algorithms. 
In the process of CV, the labeled dataset, which is a known healthy dataset, is 
partitioned into two subsets: a training subset and a validation subset [94]. Machine-
 learning-based anomaly detection and prediction algorithms are trained by the 
training subset and tested by the validation subset. The performance of the anomaly 
detection algorithms can be estimated by the error rate of the detection results 
compared with the validation subset, in which data is labeled as healthy or abnormal. 
The performance of the prediction algorithms can be estimated by the mean squared 
error (MSE) calculated by the difference between the predicted data and the 
validation data. 
CV can also be applied to conduct the model parameter selection for anomaly 
detection algorithms [94] [95]. For example, it has been used to determine the number 
of neighbors of k-nearest-neighbor classification [96][97] and the number of hidden 
units for a one-layer neural network [98][99]. The performance of an anomaly 
detection algorithm with a certain model parameter can be calculated by error rate 
estimated by the CV process. The model parameter with which the algorithm has the 
lowest error rate is selected for the algorithm.  
Common CV methods include hold-out CV, k-fold CV, and leave-one-out CV 
(LOOCV) [100][101][102]. In hold-out CV, labeled data is split into two independent 
subsets randomly: a training subset and a validation subset. For example, 30% of 
 
72 
 
labeled data are randomly picked up as validation data and the remaining 70% of the 
labeled data is training data. The anomaly detection error rate is calculated by 
equation (8). 
(8) 
Hold-out CV avoids the overlap between the training subset and the validation 
subset. But it does not use all the labeled data to validate the model, and the detection 
error rate estimation is highly dependent on how to split the dataset. K-fold CV can 
overcome some of these problems [101][102]. In k-fold CV, a labeled dataset is 
partitioned into k equally or nearly equally sized subsets. An iterative process is then 
conducted. In each iteration, a different subset is selected as validation data to 
estimate the performance of the algorithm, and the remaining k-1 subsets are used as 
training data to train the algorithm. The detection error rate is calculated in each 
iteration using equation (9). The process is iterated k times until each of the subset has 
been selected as the validation data. After k iterations, k error rates are calculated. The 
mean of the k error rates, shown in equation (10), is considered to be the final 
detection error rate of the algorithm when the entire labeled dataset is used as the 
training data.  
 
(9) 
 
 
(10) 
 
datasetvalidationtheindataofNumber
 errorsofNumber
 E ?
 ?
 ?
 ?
 ?
 k
 i
 iv
 th
 th
 iv
 E
 K
 E
 iterationitheindatavalidationtheofNumber
 iterationitheinerrorsofNumber
 E
 1
 1
 
73 
 
All the labeled data is used as validation data in k-fold CV; therefore, the 
accuracy of the detection error rate estimation is improved. However, with the 
increase of k, the variance of the detection error rate estimation becomes large, and 
the entire CV process is more time-consuming.  
Leave-one-out CV (LOOCV) [102] is actually the extreme condition of k-fold 
CV, in which k equals to the number of data points in the labeled data, N. In each 
iteration, 1 different data point is selected as validation data and all the remaining 
data points are the training data. LOOCV runs N iterations, and the final detection 
error rate is estimated by equation (11). LOOCV can provide an unbiased estimation 
of the error rate, but the variance in the estimation is large.  
 
(11) 
The selection of the CV method can be based on the size of the labeled dataset. 
If the size is small, LOOCV can give unbiased error rate estimation. However, if the 
size of the labeled data is large, k-fold CV may reduce the computation time and 
maintain the accuracy of the estimation.  
The model parameter set (m, V, ?, ?) for SPRT can be selected by k-fold CV. 
We recommend the range for each parameter as shown in Table 16. The procedure of 
model parameter selection by k-fold CV is shown in Figure 38. The first step is to 
select the training dataset, which is all healthy data and is the training data for SPRT 
after the proper model parameter set is obtained. After the training data is selected, a 
model parameter set (m, V, ?, ?) is assigned to SPRT. For each assigned model 
?
 ?
 ?
 N
 i
 th iterationitheinerrorsofNumber
 N
 E
 1
 1
 
74 
 
parameter set, the actual false alarm probability (??) and missed alarm probability (??) 
are then calculated by k-fold CV.  
Table 16: Range and change interval of SPRT model parameters 
Parameter Range Start value Interval 
m 2 ~ 4 2 1 
V 2 ~ 4 2 1 
? 0.005 ~ 0.05 0.005 0.005 
? 0.005 ~ 0.05 0.005 0.005 
 
 
 
Training data
 Subset 1 Subset 2 Subset k? 
Subset iRemaining k-1 subsets
 SPRT
 False alarm probability 
?i? and missed alarm 
probability ?i? of ith
 iteration
 i=k?
 i=i+1
 Initial i=1
 False alarm probability ??
 and missed alarm 
probability ?? with
 (m,V,?,?) 
??<= ?,
 ??<= ?, and 
to Min. 
(??+??)?
 Next (m,V ,?,?) 
Initial
 (m,V,?,?) 
Best model parameters 
(m,V,?,?) 
No
 No
 Yes
 Yes
 Randomly 
split
 Abnormal 
dataset i
  
 
Figure 38: Model parameter selection procedure by k-fold CV 
 
 
The total training data are partitioned into k subsets of equal or nearly equal 
size. k iterations should be conducted to calculate the actual false alarm probability 
(??) and missed alarm probability (??). In each of the k iterations, one different subset 
is selected as the original validation data, and the remaining k-1 subsets are the 
 
75 
 
training data. If SPRT is only run on the original validation data, which is all healthy 
data, only false alarm probability can be calculated. In order to calculate the actual 
missed alarm probability of SPRT, an abnormal dataset should be added into the 
original validation dataset, as shown in Figure 38 and Figure 39. The abnormal 
dataset can be generated by randomly picking up data outside of the range of [?-m??, 
?+m??]. Here the ? and ? are the mean and standard deviation of the training data in 
each CV iteration (the remaining k-1 subsets of the entire training data).  
In each iteration, both false alarms and missed alarms can be identified based 
on the comparison of the detection results with the validation data and abnormal data. 
The false alarm probability and missed alarm probability can be calculated by 
equations (12) and (13). The process is repeated for k times until each of the subsets 
has been selected as the original validation dataset. The mean of the k false alarm 
probabilities and the mean of the k missed alarm probabilities are calculated by 
equations (14) and (15) as the final false alarm probability, (??), and final missed 
alarm probability, (??), when using the total entire training data (without the abnormal 
data). 
 
 
(12) 
 
 
(13) 
 
 
(14) 
 
 
(15) ?
 ?
 ?
 ?
 ?
 ?
 '
 1
 '
 '
 1
 '
 '
 '
 i
 i
 th
 th
 i
 th
 th
 i
 k
 k
 iterationiindataabnormalofNumber
 iterationiinalarmsmissedofNumber
 iterationiindatavalidationofNumber
 iterationiinalarmsfalseofNumber
 ??
 ??
 ?
 ?
 
76 
 
 
 
Iteration Subset 
1 
Subset 
2 
Subset 
3 ? 
Subset 
k 
1 V     
 
V A 
 
 
 
 
Figure 39: Adding an abnormal dataset into original validation dataset to calculate the 
missed alarm probability 
The actual false alarm probability and missed alarm probability are compared 
with the ones in the selected model parameter set. If both the final false alarm 
probability and the missed alarm probability are less than the user-specified false 
alarm probability and missed alarm probability, respectively, the model parameter set 
is considered as a proper model parameter set for SPRT, but may be not the optimal 
model set. The optimal model parameter set can be defined as the one with the 
minimum error probability, which is the minimum sum of the false and missed alarm 
probabilities. If the criteria are not satisfied, another model parameter set based on the 
interval in Table 16 should be re-assigned and the performance of the SPRT with the 
model parameter should be evaluated by k-fold CV again. A model parameter set 
should be updated when the training data is updated since the actual false alarm and 
missed alarm probabilities are calculated based on the specific training data. 
 
5. 3 Anomaly Detection Results of PPTC Resettable Fuses 
As demonstrations, trip time and resistance after reset under -10?C condition 
were used individually for the anomaly detection by CVSPRT. Trip time was plotted 
in Figure 12, and resistance after reset was plotted in Figure 13. Training data should 
Original validation dataset 
Abnormal dataset 
Updated validation dataset 
 
77 
 
be selected in the first step. It is proper to select the data in the early period as training 
data. The sequential changes in the data indicate the occurrences of failures. In this 
section, the first 20 TT data points in Figure 12, which represented the trip time 
values in the first 2000 cycles, were selected as training data for anomaly detection on 
trip time. The first 15 MinR data points in Figure 13, which represented the resistance 
after reset values in the first 1500 cycles, were selected as training data for the 
anomaly detection on resistance after reset. 
 The data may be normalized first using equation (16), where xi? is normalized 
data, xi is original data, and ? and ? are mean and standard deviation of the training 
data, respectively. The normality of the training data should also be checked using 
some tools. If the training data is not normally distributed, a transformation, such as 
the Box-Cox transformation [103] or the Johnson transformation [104], may be used 
to transform the data into normal distribution. Here the normalized training data was 
normally distributed. 
(16) 
Then the model parameters (m, V, ?, ?) were determined by the 5-fold CV 
process, in which an abnormal dataset with 20 abnormal data points, which were 
randomly generated out of the range of [?-m??, ?+m??] were added into each 
validation set. Here the ? and ? are the mean and standard deviation of the training 
data (the remaining k-1 subsets of the training data). The updated validation data set 
is used to calculate the actual false alarm and missed alarm probabilities in each 
iteration.  
?
 ??
 ?
 i
 i
 x
 x '
 
78 
 
After the 5-fold CV process, SPRT model parameter set (m=4, V=4, ? =0.005, 
and ?=0.01) was selected for trip time data under -10?C condition. SPRT then used 
these model parameters and the training data to detect the anomalies in-situ when a 
new trip time (TT) data was collected. Figure 40 shows the trip time anomaly 
detection results on trip time. Figure 41 is a plot of anomaly detection for the trip time 
when using a different model parameter set (2, 2, 0.02, 0.01), under which some false 
alarms were detected in the training data. Similarly, after 5-fold CV process, model 
parameter set (m=3, V=2, ? =0.005, and ?=0.015) was selected for SPRT for the in-
 situ anomaly detection of the resistance after reset data. Figure 42 shows the final 
minimum resistance after reset (MinR) anomaly detection results. All the detection 
results are shown in table and the figures are shown in Appendix B. 
Number of Cycles
 T
 ri
 p 
ti
 m
 e 
(s
 )
 The number of 
cycles @ Anomalies
 Number of Cycles
 Trip time data
 (X 100)
  
Figure 40: Anomaly detection on trip time under -10?C condition 
 
79 
 
Number of Cycles
 T
 ri
 p 
ti
 m
 e 
(s
 )
 The number of 
cycles @ Anomalies
 Number of Cycles
 Trip time data
 (X 100)
  
Figure 41: Anomaly detection on trip time under -10?C condition using model 
parameter (2, 2, 0.02, 0.01) (False alarms detected) 
 
Number of Cycles
 M
 in
 im
 um
  R
 es
 is
 ta
 nc
 e 
af
 te
 r 
R
 es
 t 
(O
 hm
 ) The number of 
cycles @ Anomalies
 Number of Cycles
 Minimum 
resistance after 
reset data
 (X 100)
  
Figure 42: Anomaly detection on minimum resistance after reset under -10?C 
condition 
 
80 
 
Table 17 summarizes the anomaly detection results based on trip time and 
resistance after reset for all the trip cycle test conditions. The anomaly detection for 
the component should provide the times (cycles) and the corresponding parameters, 
which triggered the alarm. The earliest anomaly alarm among all the parameters was 
used to trigger alarm for the anomaly of the component.  
Table 17: Summary of anomaly detection results 
 
 
Training 
data for 
TT 
SPRT 
model for 
TT 
First 
anomaly 
cycle for 
TT 
(x100) 
Training 
data for 
MinR 
SPRT 
model for 
MinR 
First 
anomaly 
cycle for 
MinR 
(x100) 
Earliest 
anomaly 
for the 
device 
(x100) 
-10?C 1-20 
(4,4,0.005,
 0.01) 
30 1-15 
(3,2,0.005,
 0.015) 
25 25 
RT 1-30 
(4,2,0.005,
 0.005) 
61 1-30 
(4,2,0.005,
 0.005) 
45 45 
60?C 10-34 
(4,4,0.035,
 0.005) 
47 7-24 
(4,2,0.005,
 0.005) 
37 37 
85?C 1-50 
(4,2,0.015,
 0.005) 
65 1-20 
(3,2,0.035,
 0.005) 
41 41 
 
5. 4 Summary 
CVSPRT is developed to select a proper model parameter set for SPRT without 
the need of experience, thus reduce the false alarm and missed alarm probability. 
Besides the anomaly detection for PPTC resettable fuses in this thesis, CVSPRT can 
be extended to conduct the anomaly detection for any conditions if the training data is 
available and the SPRT index can be calculated. 
 
 
 
81 
 
Chapter 6: Moving Window based Dynamic Model Parameter 
Optimization N-steps-ahead Prediction  
Prognostics of PPTC resettable fuses can be implemented using data trending or 
regression methods on the precursor parameters directly. When any anomalies are 
detected by the methods developed in chapter 5, failure prediction should be 
implemented to provide the information in the future. In this section, a moving 
window based dynamic model parameter optimization (MW-DMPO) n-steps-ahead 
prediction method is developed first as a general method for dynamic data trending 
and then moving window cross validation support vector regression (MW-CVSVR) 
n-steps-ahead prediction method is developed as a case study for the prediction of the 
PPTC resettable fuses. 
6. 1 Moving Window based Dynamic Model Parameter Optimization 
(MW-DMPO) N-steps-ahead Prediction  
The fixed training data can not contain the latest degradation information, thus 
cannot generate the prediction with high accuracy. Training data for prediction 
modeling should contain degradation features as much as or as latest as possible. 
Moving window method has been developed as the method to update the training 
data. For example, Djukanovic et al. [105] used MW on neural network and Ren et 
al. [106] applied MW on autoregressive quadratic model for short-term forecasting. 
Pecht and Cheng [107] used MW techniques on data driven methods for prognostics 
 
82 
 
of aging systems. However, the model parameter parameters, which determine the 
performance of the trending models, were not updated with each updating of the 
training data using MW. This caused the prediction inaccurate, even though the MW 
method was used to update the training data to catch the latest degradation 
information. A MW-DMPO n-steps-ahead prediction method is developed in this 
dissertation to solve this problem. 
The MW-DMPO n-steps-ahead prediction method is shown in Figure 43. Initial 
training data may be needed to transform to a certain pattern based on the requirement 
of the prediction model, and then used to train the prediction model, during the 
training process, best model parameters for the prediction model is identified by a 
model parameter optimization method. The n-steps-ahead predictions are calculated 
by the trained prediction model. N-steps-ahead predictions mean that only n values in 
the future are predicted in one prediction. If any of the n-steps predictions reaches the 
threshold, alarm is triggered and a failure is predicted. If all the n predictions do not 
reach the failure threshold, a conclusion should be drawn that the product will not fail 
in the future n steps (such as, times or cycles). And then the training data should be 
updated by a MW method to re-train the prediction model when the actual n 
observations are obtained to conduct the next n-steps-ahead predictions. Most 
importantly, for each training data updating, the model parameters should be 
optimized. That is called the dynamic model parameter optimization with each 
updating of training data.  
 
83 
 
f(x)
 x
 Prediction model
 Prediction 
error 
estimation
 ith-step-
 ahead 
prediction
 Failure 
criteria?
 Up to n
 steps?
 Update training data 
by moving window
 Alarm and RUL 
prediction
 Initial training 
data Extending window 
updating
 Double-side moving 
window updating
 f(x)
 x
 f(x)
 x
 No
 No
 Yes
 Yes
 Training 
window 
n-step-ahead
 prediction window
 Model parameter 
optimization 
 
Figure 43: Moving window training updating based prediction method 
Two MW methods were developed in this thesis to update the training data. 
The first one is called double-side MW updating, shown as the upper right in Figure 
43. In this method, actual observations in the n-steps-ahead prediction window are 
added into the training data, but at the same time, the same amount of the oldest 
training data is discarded. It looks like a fixed size of training window slide n-steps 
ahead. The second method, shown in the middle right of the Figure 43, called 
extending window updating, does not discard the oldest training data, but keep all the 
data when the actual observations in the n-steps-ahead window are added in. The 
selection of the MW updating methods depends of the features of the data itself, if the 
features do not change so much, the double-side moving window method maybe 
enough. If more historic data are needed to present the features, the extending 
 
84 
 
window method is suitable. After the training data is updated, the next n-steps-ahead 
prediction will be conducted. 
The most valuable information that n-steps-ahead prediction can provide is 
whether the product will fail in the n steps (such as, times or cycles) in the future. 
Most of researchers in prognostics try to calculate when the product will fail. But 
sometimes, it is more valuable to tell the heath information (health or failure) of a 
product in the near future. Then operators can have enough time but not too early to 
conduct maintenance or replacement. Besides this, n-steps-ahead may be more 
accurate than long-term prediction because of the accumulation of uncertainty of 
predictions. Furthermore, n-steps-ahead predictions can be conducted for individual 
products. It predicts the behavior in n steps in advance based on the historical 
information of the individual product itself. 
This MW-DMPO n-steps-ahead prediction method can be implemented by 
using different data trending and regression models, such as autoregressive integrated 
moving average (ARIMA), neural networks (NN), and support vector regression 
(SVR). As a case study, the MW-DMPO n-steps-ahead prediction method was 
applied on support vector regression (SVR) to predict the failure of PPTC resettable 
fuses in this chapter. This chapter also developed a cross validation method to select 
proper SVR model parameters, selections of some of which have not been considered 
in literature. The entire method was called moving window cross validation SVR 
(MW-CVSVR).  
 
 
85 
 
6. 2 Development of MW-CVSVR 
Many models have been used for time series forecasting in recent years, 
including ARIMA model [54][55] and neural network [56][57]. Selection of different 
modeling depends on the features of the data. For example, if the data is linear, 
ARIMA model is suitable, but if the data contains nonlinear components, neural 
network model has been thought as a universal approximator with an arbitrary 
accuracy [58][60]. 
 The data shown in Figure 12 and Figure 13 presented nonlinear features and 
ARIMA model may be not a good choice. As a novel neural network technique, 
support vector regression (SVR), developed by Vapnik [58], has some advantages 
compared with traditional neural network modeling. SVR can achieve an optimum 
network structure by balancing between the empirical error and the confidence 
interval. This balance eventually leads to better generalization performance than other 
neural-network models. Additionally, the SVR training process is equivalent to 
solving linearly constrained quadratic programming problems, and the SVR 
embedded solution meaning is unique, optimal and unlikely to generate local 
minima[58][60].  
However, using SVR for remaining useful life prediction of a product has been 
very limited in literature. Most of applications of SVR were focused on the 
forecasting of the time series data in financial area, such as stock price. Although 
Chen et al. [60], Tian et al. [71], Hong et al. [72], and Yang et al. [73] used SVR for 
 
86 
 
the reliability estimation, SVR was used only to predict the next reliability value 
based on the reliability values in the history. The research on using SVR for a real-
 time tool for the remaining useful life prediction based on actual in-situ monitored 
data directly has been very limited.  
In this chapter, a MW-CVSVR model was developed to capture the updating 
degradation features of the data and predict the future values. MW-CVSVR can solve 
the problems of traditional SVR on model parameter selection and training data 
updating.  
 
6.2.1 SVR modeling for precursor parameters  
The basic idea for SVR includes mapping the input data X into a high-
 dimensional feature space F by kernel functions, and then solve a linear regression 
problem in the feature space. This process is shown in Figure 44. The linear 
regression in the feature space is presented as equation (17): 
(17) 
Where, ?(x) is the feature in the feature space of data x in the input space. ? 
and b are coefficients, which are solved from the training data. Assuming a training 
data set including n data points, in which xi is the input vector and yi is target: 
Training = {(xi , yi ),i = 1,2,?, n} 
bxxf ?? )()( ??
 
87 
 
In ?-SVR (Vapnik [58]), the objective is to find a function f (x) to estimate the output 
of input x. The requirements for the function f(x) include two points: 1) for all the 
training data (xi,yi), the output estimation of input xi, f(xi), can only have at most ? 
deviation from the actually obtained targets yi; In other words, errors are not 
concerned as long as they are less than ?; 2) at the same time f(x) must be as flat as 
possible, which requires small ?. [108].  
f(x)
 x
 ?
 f(x)
 ?(x)
 +?
 -?
 0
 ?*
  
Figure 44: Non-linear SVR 
A regularized risk function can be defined by balancing these two requirements 
as equation (18), in which the first item is used as a measure of the flatness, and 
second item is used to measure empirical risk. Equation (19) defines a ?-insensitive 
loss function. The loss is zero if the estimate is within the ?-tube, shown in Figure 44. 
C is called cost function, which specifies the trade-off between the empirical risk and 
the model flatness. Both C and ? are user-determined parameters.  
(18) ??
 ??
 n
 i
 ii yfLn
 CR
 1
 2
 ),(
 1
 2
 1
 ??
 
88 
 
Where fi is the estimated output of input xi and yi is the actual output of xi. 
(19) 
The equation (19) has an assumption that function f actually exists and 
approximates all training data within ?-tube. However, sometimes, this may not be the 
case, or some small errors may be allowed. Slack variables, ?i and ?i
 *, are introduced 
into to form a soft margin, shown in Figure 44. The slack variables assume non-zero 
values outside ?-tube. The SVR fits f(x) to the data to satisfy that 1) the training error 
is minimized by minimizing ?i and ?i
 *, and 2) (???)/2 is minimized to raise the 
flatness of f(x). The entire task is to minimize the equation (20) 
(20) 
Subject to: 
(21) 
 
The minimization can be solved by introducing Lagrange multipliers and 
exploiting the optimality constraints [108].  
(22) 
 
?
 ?
 ? ???
 ?
 ?f-yyf
 otherwise
 yfL
          
                       0
 ),(
 ?
 ?
 ?
 ?
 ????
 n
 i
 CR
 1
 2
 )(
 2
 1
 ???
 , n., ibxy
 , n.,i
 , n.,iybx
 iii
 ii
 iii
 ?
 ?
 ?
 21    )(-
  21    0 ,
 21    )(
 *
 *
 ?????
 ??
 ??????
 ????
 ??
 ????
 ??
 ??
 ??
 ??
 ?
 ????????
 ????????
 n
 i
 iiii
 n
 i
 iiii
 n
 i
 iiii
 n
 i
 ybx
 bxyCL
 1
 **
 1
 1
 **
 1
 2*
 ).( ))((
  ))(-()(
 2
 1
 ),,(
 ?????????
 ???????????
 
89 
 
Finally, by applying Karush-Kuhn-Tucher (KKT) conditions for regression, the 
optimize formulation can be transformed into a Dual problem: 
Maxing:  
(23) 
Subject to  
(24) 
 Where, ?i and ?i
 * are Lagrange multipliers, and satisfy the equality, ?i? ?i
 *=0. 
By solving the Lagrange multipliers ?i and ?i
 *, the optimal weight vector of the 
regression is expressed as: 
(25) 
Then the regression is: 
(26) 
The nsv denotes the number of total support vectors. For ?-insensitive SVR, the 
vectors in the tube are not considered as support vectors. The training vectors, whose 
corresponding coefficients (?i-?i*) are non-zero values, have approximation error 
equal or larger than zero (on or outside the tube). Only these vectors are called 
support vectors. That means SVR do not need all training data as support vector, only 
a few vectors may generate a good regression. 
?
 ?
 ???
 nsv
 i
 iiiii bxxKxf
 1
 ** ),()(),,( ????
 ???
 ?
 ??
 ?
 ?
 ?
 ? ???????
 n
 ji
 jijjii
 n
 i
 ii
 n
 i
 iiiii xxKyQ
 1,11
 ),())((
 2
 1
 )()(),,( ????????????
 0)(
  21     ,0
 1
 *
 ??
 ???
 ?
 ?
 ?
 n
 i
 ii
 ii , n.,iC
 ??
 ?? ?
 )()(
 1
 i
 nsv
 i
 ii x???? ?
 ?
 ???
 
90 
 
  K(xi,xj) is the Kernel function, which is the inner product of two vectors xi and 
xj in the feature space, ?(xi ) and ?(xj), meaning that K(xi,xj)= ?(xi)? ?(xj). The kernel 
function is intended to handle any dimension feature space without the need to 
calculate ?(x) accurately. Any function can satisfy the Mercer?s conditions then can 
perform the non-linear mapping. Common kernel functions used in SVR are listed in 
Table 18.  
Table 18: Common Kernel functions 
 
Kernel Function Formula 
Polynomial  
Gaussian Radial Basis Function  
Exponential Radial Basis Function  
Multi-layer Perceptron  
Fourier Series  
B Splines  
Based on the understanding of the concepts of SVR, Figure 45 summarizes the 
procedures of SVR modeling [108]. In the training process, training data is used to 
obtain a model including the solutions of ? and b, support vectors, and all the 
corresponding mapped vectors in feature space. Support vectors are a subset of 
d
 ii bxxxxK ),(),( ??
 )
 2
 exp(),(
 2
 2
 ?
 i
 i
 xx
 xxK
 ??
 ?
 )
 2
 exp(),(
 2?
 i
 i
 xx
 xxK
 ??
 ?
 ),tanh(),( ?? ?? ii xxxxK
 ))(5.0sin(
 ))(2/1sin(
 ),(
 i
 i
 i xx
 xxN
 xxK
 ?
 ??
 ?
 ? ?5.0,5.0
 1
 12
 1 
),(
 ?
 ?
 ?
 ??
 ??
 k
 i
 i
 iNi
 Bwith
 xxBxxK
 
91 
 
training data. When a new vector is input to the model, it is mapped into the feature 
space by ?(X), and then the dot product of ?(X) with each existing mapped vectors in 
the feature space ?(Xi) obtained from training data are calculated. These two 
processes are conducted by kernel function, K(X,Xi), the output is the summation of 
the weighted dot products and the constant item b. The entire process is similar to 
regression in a neural network. The advantage over neural network is that SVR does 
not need all the training vectors but only a subset of it. 
 
? 
 
X1 
 
X2 
 
Xm 
 
X 
?(X1) ?(X2) ?(X) ?(Xm) ?  
?  
( ? )  ( ? ) ( ? ) ?  
Support vectors (subset 
of training data) 
A new input vector 
Mapped vectors in 
feature space 
Dot product of ?(X) 
and ?(Xi), =K(X, Xi) 
 
Output 
f(X)=?(?i-?i
 *)K(X, Xi)+b 
 
?1, ?1
 * 
 ?2, ?2
 * 
 
?m, ?m
 * 
 
b 
 
f(X) 
Figure 45: Schematic flowchart of SVR model 
 
92 
 
For the prediction model of univariate time series data, such as trip time or 
resistance after reset, the inputs data are past values, the lagged observations of the 
time series, while the outputs are the future values, shown in equation (27) 
(27) 
Table 19: Input and output pattern transformation for training data and process for n-
 steps-ahead prediction 
 
 Index of Pattern 
(i) 
Input Vector Xi Output (Target 
or Prediction) 
Training 
1 x1, x2, x3, ?, xp xp+1 
2 x2, x3, x4, ?, xp+1 xp+2 
3 x3, x4, x5, ?, xp+2 xp+3 
   
t-p xt-p, xt-p+1, xt-p+2, ?, xt-1 xt 
n-steps -
 ahead 
prediction 
t+1 xt-p+1, xt-p+2, xt-p+3, ?, xt x?t+1 
t+2 xt-p+2, xt-p+3, xt-p+4, ?, x?t+1 x?t+2 
t+3 xt-p+3, xt-p+4, ?, x?t+1, x?t+2 x?t+3 
   
t+nstep x(?)t-+nstep-p, ?, x?t+nstep-2, x?t+nstep-1 x?t+nstep 
Where data vector X: {xt-1, xt-2, ?, xt-p} is the input data, which is p lagged 
values, while xt is output or target. After training data is selected, it must be 
transformed into the certain X?Y pattern, in which X is the input vector and Y is the 
target, by a moving fixed-length window. The lag, which represents the number of 
),( ,,21 ptttt xxxfx ???? ?
 ?
  
?
  
?
  
?
  
?
  
?
  
 
93 
 
previous data or the length of the window, must be determined first. The 
transformation of the training data if the lag is p is shown in Table 19. Unfortunately, 
no systematic methods have been proposed to select the lag value.  
6.2.2 Cross validation SVR (CVSVR) 
In this thesis, Gaussian RBF is selected. It has only one variable to be 
determined, which can simplify the optimization of the model. In addition, SVR using 
RBF has excellent nonlinear regression ability. Four parameters, C, ?, ?, and lag, 
should be determined before the application of SVR model. Cross validation (CV) 
method has been used to select a proper C, ?, and ? for SVR. But the selection of lag 
value has not been considered in literature. In this thesis, all of four model parameters 
including the lag were selected by CV process. 
Transformed Training data
 Subset 1 Subset 2 Subset k? 
Subset iRemaining k-1 subsets
 Train SVR model
 Trained SVR model
 MSEiof ith iteration
 i=k?
 i=i+1
 Initial i=1
 CV MSE of SVR 
model with 
(C,?,?,lag) 
To Min. 
MSE?
 Next (C,? ,?,lag) 
Initial
 (C,? ,?,lag) 
No
 No
 Yes
 Yes
 Randomly 
split
 Training data Best 
(C,?,?,lag) 
 
 
 
Figure 46: k-fold CV for SVR parameter optimization 
 
94 
 
Figure 46 shows the procedure using k-fold CV to select parameters for SVR. 
For each model set (C, ?, ?, lag), k-fold CV is used to calculate the prediction 
accuracy. When the accuracy reaches the criteria, the SVR model with that parameter 
set is used to predict the output of new input data. Before the k-fold CV process, the 
training data should be transformed (shown in Table 19) based on the value of lag. In 
k-fold CV, the transformed training data set is split into k equal or nearly equal sized 
subsets, then an iterated process is conducted to calculate the prediction accuracy. In 
each iteration, one subset is selected as validation data, and the other k-1 subsets are 
used as training data to train the SVR model. The trained SVR model then is working 
on the validation data to predict the output for each input. The accuracy of prediction 
in the ith iteration is calculated by mean squared error (MSE), shown in equation (28): 
(28) 
Where nVal is the number of validation data.  
After k iterations, the prediction accuracy of the SVR model with the parameter 
set (C, ?, ?, lag) can be calculated by equation (29): 
(29) 
When the MSEfinal reaches the stop criteria, which can be defined as the minima 
of the MSEfinal, the selection process is stopped and the SVR model with the 
corresponding parameter set is the final trained model, which is used for the 
prediction of a new test data. If the criteria are not reached, a new updated model 
?
 ?
 ??
 nVal
 j
 jji xfynVal
 MSE
 1
 2))((
 1
 ?
 ?
 ?
 k
 i
 ifinal MSEk
 MSE
 1
 1
 
95 
 
parameter set is input to the CVSVR to repeat the process. Table 20suggests the range 
of each parameter and how the CV updates the model set.  
Table 20: Ranges of SVR model parameters 
 
Parameter Range Start value Interval 
Log2(C) -10 ~10 -10 0.5 
? 0.01 ~ 0.2 0.01 0.01 
Log2(?) -10 ~10 -10 0.5 
lag 5 ~15 5 1 
 
6.2.3 MW-CVSVR 
f(x)
 x
 K-fold CVSVR
 Trained SVR
 Prediction 
error 
estimation
 ith-step-
 ahead 
prediction
 Failure 
criteria?
 Up to n
 steps?
 Update training 
data by moving 
window
 Alarm and RUL 
prediction
 Initial training 
data
 Extending window 
updating
 Double-side moving 
window updating
 f(x)
 xNo
 No
 Yes
 Yes
 Training 
window 
n-step-ahead
 prediction window
 f(x)
 x
 Data-trans. and 
train SVR
  
 
Figure 47: MW-CVSVR procedure 
For failure prediction, training data for CVSVR modeling should contain the 
latest degradation features as much as possible. Using the developed MW training 
updating based prediction method in this thesis, a MW-CVSVR model was 
developed, shown in Figure 47.  
 
96 
 
Initial training data is transformed (shown in Table 19) and then used by k-fold 
CVSVR to select an optimal model parameter set for SVR. Then the training data is 
transformed again based on the optimal model parameters and used to train the SVR 
model, in which process the support vectors are identified and the coefficients and 
bias item are calculated. The n-steps-ahead predictions then can be calculated by this 
trained model. If any of the n-steps predictions reaches the threshold, alarm is 
triggered and a failure is predicted. If all the n predictions do not reach the failure 
threshold, a conclusion should be drawn to state that the fuse would not fail in the 
future n cycles. And then the training data for the prediction model should be updated 
by a MW method (double-side MW or extending window method) when the actual n 
observations are obtained. A new optimal model parameter set should be selected by 
cross validation process again. The model with the updated optimal model parameters 
should be trained by the updated training data to conduct the next n-steps-ahead 
predictions. This process is repeated until a failure is predicted. The prediction error 
is estimated by the residuals between the training data and its estimations. The actual 
application shows that the residuals are normally distributed. This MW-CVSPRT can 
always give the n-steps-ahead prediction to determine if failure will occur in near 
future, and the RUL prediction may be much closer to the actual failure.  
6. 3 Prognostics of PPTC Resettable Fuses Using MW-CVSVR 
The result of using MW-CVSVR on the failure prediction based on trip time 
data of PPTC fuses under -10?C condition is demonstrated in this section. Anomaly 
in terms of trip time under -10?C condition was detected at the 30th point (around 
 
97 
 
3000 cycles). So the prediction could be start at this point. The first 30 data points 
were used as the initial training data for MW-CVSVR. Gaussian radial basis function 
was chosen as the kernel function for MW-CVSVR and 3-fold cross validation was 
used to select the model parameters (C, ?, ?, lag).  
The accuracy of prediction is affected by the number of steps, n, and the MW 
methods (DMW and EW). The number of steps, n, also determines the number of 
predicted data in the future and the number of data used to update the training data (or 
steps in the window moving). The selection of n should be considered with a balance 
between the prediction error and the prediction distance. Figure 48 shows MSEs of 
predictions with different prediction steps, from 1 to 10, and different MW methods 
(DMW and EW) for trip time under -10?C conditions. Figure 49 shows the MSEs of 
predictions with different prediction steps and MW methods for minimum resistance 
after reset under -10?C conditions. Based on these two figures, 5-steps-ahead 
prediction is recommended as it balances accuracy and prediction distance. 
-0.02
 -0.01
 0
 0.01
 0.02
 0.03
 0.04
 0.05
 0 2 4 6 8 10 12
 MW style: EW
 MW style: DMW
 Prediction Steps
 M
 SE
  
Figure 48: MSEs vs. number of prediction steps (trip time, -10?C). 
 
98 
 
0.0E+00
 5.0E-07
 1.0E-06
 1.5E-06
 2.0E-06
 2.5E-06
 3.0E-06
 0 2 4 6 8 10 12
 MW style: EW
 MW style: DMW
 Prediction Steps
 M
 SE
  
Figure 49: MSEs vs. number of prediction steps (min. resistance after reset, -
 10?C). 
For comparison of accuracy for all conditions, 5-step-ahead prediction and 10-
 step-ahead prediction were conducted by MW-CVSVR. Both double-side MW 
updating and extending window updating methods were applied on this data. Figure 
50 to Figure 53 are examples of 5-step-ahead prediction using double-side MW 
updating and extending window updating methods at different moments, in which the 
Figure 50 and Figure 51 are the predictions at the start moment, and Figure 52 and 
Figure 53 show the predictions when failure occurred based on the failure criteria. 
The predicted failures were exact the same with the actual failure, but the failures 
were predicted in 3 points (300 cycles) in advance.  
The accuracy of prediction was evaluated by comparing the prediction results 
of all the n-steps-predictions at different moments with the original data, shown in 
Figure 54, Figure 55, Figure 56, and Figure 57. The figures indicate that the MW-
 CVSVR can catch the trends of the original data and the n-steps-ahead predictions 
 
99 
 
were close to the original data. Different training data updating methods and different 
prediction steps have different prediction accuracy. By comparing the MSE of 
different prediction methods, 5-step-ahead prediction with 5-step extending window 
training data updating had the best performance for the trip time data under -10?C 
condition.  
Table 21 summarizes the prediction results for the trip time and resistance after 
reset under all conditions. The figures of the applications using MW-CVSVR model 
on trip time and resistance after reset under all the conditions are shown in Appendix 
C. The failure criteria for trip time for each environmental condition are defined as 
the 25% increase or decrease of the mean of the trip time used as training data for 
anomaly detection, shown in Table 17. The failure criteria for minimum resistance 
after reset (MinR) is defined as the 25% increase of the mean of the MinR values 
used as the training data for anomaly detection, shown in Table 17. The prediction 
results show that n-steps-ahead predictions by MW-CVSVR can capture the trends of 
the data and the predicted failures were very close to the actual failure. The alarmed 
cycles that were the number of cycles when the prediction was conducted were not 
later than the actual failures. It means that the n-steps-ahead prediction using MW-
 CVSVR can predict the actual failure in advance but not too early. It is also shown 
that 5-step-ahead prediction was better than 10-step-ahead prediction based on the 
MSE values.  
Table 22 shows the comparison of 5-step-ahead prediction using MW-CVSVR 
and MW-SVR, in which the model parameters are not optimized for each updating of 
 
100 
 
training data. The MSEs of MW-CVSVR are smaller than those of MW-SVR. The 
alarmed cycles by MW-CVSVR are earlier than the actual failures while MW-SVR 
failed to predict some of the conditions. The comparisons indicate that the MW-
 CVSVR have much better performance than MW-SVR because of the contribution 
from dynamic model parameter optimization for each training data updating.  
6. 4 Summary  
Moving window based dynamic model parameter optimization (MW-DMPO) 
n-steps-ahead prediction method was developed in this chapter. This method keeps 
updating the training data by a moving window to contain the latest data so that the 
latest degradation information can be captured by the model. This method also 
optimizes the model parameters for the data trending models for each updating of the 
training data to ensure its high performance. As a case study of the developed MW-
 DMPO method, a moving window based cross validation support vector regression 
(MW-CVSVR) was developed for the prediction of PPTC resettable fuses. The cross 
validation technique solved the model parameter selection for SVR including the lag 
value. The application of MW-CVSVR on the PPTC resettable fuse shows that the 
developed method worked well on the prediction. The MW-DMPO method can be 
extended to apply on any data trending and regression methods, such as ARIMA and 
Neural networks.  
 
101 
 
 
Training data: 1-30
 5-step prediction: 31-35
 Model parameters: (C=16,?=0.125,?=0.01,lag=15)
 MSE:0.002
 Threshold
 Training data: 5-35
 5-step prediction: 36-40
 Model parameters: (C=11.3,?=0.71,?=0.02,lag=5)
 MSE:0.02
 Threshold
 Number of Cycles
 Number of Cycles (X 100)
 (X 100)
 T
 ri
 p 
T
 im
 e 
(s
 )
 T
 ri
 p 
T
 im
 e 
(s
 )
  
Figure 50: Example of double-side MW 5-step-ahead prediction 
(1-30?5-35, -10?C) 
Training data: 1-30
 5-step prediction: 31-35
 Model parameters: (C=16,?=0.125,?=0.01,lag=15)
 MSE:0.002
 Threshold
 Training data: 1-35
 5-step prediction: 36-40
 Model parameters: (C=2.83,?=2.83,?=0.02,lag=5)
 MSE:0.05
 Threshold
 Number of Cycles
 Number of Cycles (X 100)
 (X100)
 T
 ri
 p 
T
 im
 e 
(s
 )
 T
 ri
 p 
T
 im
 e 
(s
 )
  
Figure 51: Example of extending window 5-step-ahead prediction 
(1-30?1-35, -10?C) 
 
 
102 
 
 
 
 
Training data: 41-70
 5-step prediction: 71-75
 Model parameters: 
(C=128,?=0.125,?=0.02,lag=8)
 MSE:0.002
 Threshold
 Training data: 46-75
 5-step prediction: 76-80
 Model parameters: 
(C=45.3,?=0.125,?=0.03,lag=9)
 MSE:0.003
 Threshold
 Number of Cycles
 Number of Cycles (X 100)
 (X 100)
 T
 ri
 p 
T
 im
 e 
(s
 )
 T
 ri
 p 
T
 im
 e 
(s
 )
  
Figure 52: Example of double-side MW 5-step-ahead prediction 
(41-70?46-75, -10?C) 
Training data: 1-70
 5-step prediction: 71-75
 Model parameters: 
(C=256,?=0.007,?=0.01,lag=9)
 MSE:0.004
 Threshold
 Training data: 1-75
 5-step prediction: 76-80
 Model parameters: 
(C=724,?=0.02,?=0.01,lag=5)
 MSE:0.002Threshold
 Number of Cycles
 Number of Cycles (X 100)
 (X 100)
 T
 ri
 p 
T
 im
 e 
(s
 )
 T
 ri
 p 
T
 im
 e 
(s
 )
  
Figure 53: Example of extending window 5-step-ahead prediction 
(1-70?1-75, -10?C) 
 
103 
 
5
 6
 7
 8
 9
 10
 11
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 im
 e 
(s
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead double-side moving window
 MSE: 0.013
 Threshold
 Original data
  
 Figure 54: All 5-step-ahead predictions (-10?C,trip time, double-side MW) 
5
 6
 7
 8
 9
 10
 11
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead extending moving window
 MSE: 0.007
 Threshold
 Original data
  
Figure 55: All 5-step-ahead predictions (-10?C, trip time, extending window) 
5
 6
 7
 8
 9
 10
 11
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead double-side moving window
 MSE: 0.032
 Threshold
 Original data
  
Figure 56: All 10-step-ahead predictions (-10?C,trip time, double-side MW) 
5
 6
 7
 8
 9
 10
 11
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead extending moving window
 MSE: 0.027
 Threshold
 Original data
  
Figure 57: All 10-step-ahead predictions (-10?C,trip time, extending window) 
 
 
104 
 
Table 21: Summary of MW-CVSVR prediction results  
Conditions Parameters 
Moving 
window style 
(double-side 
or extending) 
Moving 
steps 
Prediction 
steps 
Alarmed cycle  
(The number of cycle 
when the prediction 
is conducted (?100) 
Predicted 
failure 
cycle 
(?100) 
Actual 
failure cycle 
(?100) 
MSE 
-10?C 
TT 
Double-side 5 5 75 78 
79 
0.013 
Extending 5 5 75 77 0.007 
Double-side 10 10 70 80 0.032 
Extending 10 10 80 81 0.027 
MinR 
Double-side 5 5 100 105 
108 
1.3E-06 
Extending 5 5 95 98 9.58E-07 
Double-side 10 10 105 107 2.34E-06 
Extending 10 10 95 98 2.05E-06 
RT 
TT 
Double-side 5 5 120 121 
121 
0.02 
Extending 5 5 120 122 0.015 
Double-side 10 10 110 116 0.076 
Extending 10 10 120 121 0.076 
MinR 
Double-side 5 5 115 117 
122 
1.96E-06 
Extending 5 5 115 119 1.86E-06 
Double-side 10 10 115 117 3.06E-06 
Extending 10 10 115 119 3.23E-06 
60?C 
TT 
Double-side 5 5 
No failure based on failure criteria 
3.8E-04 
Extending 5 5 3.6E-04 
Double-side 10 10 5.1E-04 
Extending 10 10 6.6E-04 
MinR 
Double-side 5 5 165 166 
167 
8.02E-07 
Extending 5 5 165 166 6.47E-07 
Double-side 10 10 165 166 1.61E-06 
Extending 10 10 165 166 1.47E-06 
 
105 
 
85?C 
TT 
Double-side 5 5 
No failure based on failure criteria 
4.8E-03 
Extending 5 5 1.1E-02 
Double-side 10 10 7.9E-03 
Extending 10 10 1.2E-02 
MinR 
Double-side 5 5 120 125 
126 
4.06E-06 
Extending 5 5 125 126 5.59E-06 
Double-side 10 10 115 125 6.86E-06 
Extending 10 10 125 126 8.76E-06 
 
 
 
 
 
 
 
106 
 
 
Table 22: Comparison of 5-step-ahead predictions by MW-CVSVR and MW-SVR 
 
  
5-step-ahead 
prediction and 5-
 step-ahead 
moving 
MSE Alarmed cycle (?100) 
Predicted failure cycle 
(?100) Actual failure 
cycle (?100) 
MW-
 CVSVR 
MW-SVR 
MW-
 CVSVR 
MW-SVR 
MW-
 CVSVR 
MW-SVR 
-10 ?C 
TT 
DMW 0.013 0.02 75 75  78  80  
79 
EW 0.007 0.01 75 75  77  78  
MinR 
DMW 1.3E-06 1.45E-06 100 115  105  116  
108 
EW 9.58E-07 2.4E-06 95 120  98  121  
RT 
TT 
DMW 0.02 0.03 120 125 121 126 
121 
EW 0.015 0.04 120 125 122 126 
MinR 
DMW 1.96E-06 2.0E-06 115 125 117 127 
122 
EW 1.86E-06 7.33E-06 115 150 119 152 
60 ?C  
TT 
DMW 3.8E-04 4.6E-04 
No failure based on failure criteria  
EW 3.6E-04 8.5E-04 
MinR 
DMW 8.02E-07 9.0E-07 165  165  166 170  
167 
EW 6.47E-07 8.12E-07 165  170  166 171  
85 ?C  
TT 
DMW 4.8E-03 5.0E-03 
No failure based on failure criteria  
EW 1.1E-02 1.15E-02 
MinR 
DMW 4.06E-06 5.39E-06 120  125 125 126 
126 
EW 5.59E-06 7.49E-06 125  130 126 132 
 
107 
 
Chapter 7: Contributions and Future work 
Contributions of this thesis include the following:  
1) Determined the failure precursor parameters for PPTC resettable fuses, shown 
in chapter 3. Potential precursor parameters were determined by failure modes, 
mechanisms, and effects analysis (FMMEA), and then a series of experiments 
were conducted to verify these parameters. Trip time, resistance after reset, and 
surface temperature have been determined as precursor parameters for observed 
failures, including the cracks at the coating and solders, gaps between different 
parts of the fuse, and the performance shifting of the fuses, such as tripping 
faster and faster and the resistance becoming higher and higher. These precursor 
parameters should be monitored when testing or in the actual application. Other 
parameters, including the maximum resistance during trip endurance and the 
resistance in aging test did not exhibit obvious changes (<5% of the initial 
values) under all test conditions. They are not considered as failure precursors. 
2) Determined the causes of the observed failures for PPTC resettable fuses 
considering the interconnections of different parts in the fuse, shown in chapter 
4. A series of failure analyses, including the heating process analyses by 
infrared camera, the interconnection analyses by cross-section and ESEM, the 
analyses of microstructures of carbon black(CB)/polyethylene (PE) composite, 
the thermal property analyses of CB/PE composite, and the CTE analyses of 
different parts of the component were conducted. The gaps between coating and 
the foil, caused by thermal fatigue and CTE-mismatches, are the main causes 
 
108 
 
for decreases in the trip time. Aside from the gaps, the increases in the 
crystallinity of CB/PE composite are the main causes for the increases in the trip 
time. The agglomeration of the CB particles is one of the main causes for the 
increase in the resistance after reset. With the increases in the cycle numbers, 
the cracks at the solder and the gaps between foil and CB/PE composite will 
control the increases in the resistance and finally cause an open of current.  
3) Developed an in-situ anomaly detection method using cross validation 
sequential probability ratio test (CVSPRT), shown in chapter 5. The cross 
validation was used to select optimal model parameters for SPRT to reduce the 
false alarm and missed alarm probabilities. The value of the approach is that 
CVSPRT can be extended to conduct in-situ anomaly detection for any 
conditions if the training data is available and the SPRT index can be calculated.  
4) Developed a moving window (MW) based dynamic model parameter 
optimization (MW-DMPO) n-steps-ahead prediction method, shown in chapter 
6. This method involves the latest monitored data by a moving window to 
update the training data for data trending and regression models to capture the 
latest degradation information, and for each updating of the training data, the 
model parameters are optimized by the dynamic model parameter optimization 
methods. Besides being used in the support vector regression (SVR) in this 
thesis, the developed MW-DMPO method can be extended as general training 
data updating methods for data trending and regression models, such as ARIMA 
and NN models, to enable more accurate failure prediction.  
 
109 
 
5) Developed a moving window cross validation SVR (MW-CVSVR) for n-steps-
 ahead prediction, shown in chapter 6. In MW-CVSVR n-steps-ahead prediction, 
the CV method is used to optimze model parameters for SVR including the lag 
value and then train the SVR model. The trained SVR model predicts n values 
of the monitored precursor parameter in the future. MW method is used to 
update the training data to ensure the SVR model capture the latest degradation 
features. MW-CVSVR was used on trip time and resistance after reset to predict 
the failures of PPTC resettable fuses in this thesis. The results of 5-step-ahead 
predictions and 10-step-ahead predictions using MW-CVSVR showed that the 
MW-CVSVR can capture the trends of the data and predict the failures 
accurately and not too early. MW-CVSVR can be used as a general method for 
nonlinear data regression. 
Future work may include the following: 
1) Determine the failure mechanisms of PPTC resettable fuses and develop PoF 
models to predict failures. The mechanisms of the polymer positive are complex 
and highly depend on the composition of the composite and the usage 
conditions. More experiments and analyses are required to reveal and model the 
formation of cracks and gaps, diffusion and aggregation of the CB fillers, and 
the changes in the properties of the polymer matrix.  
2) Develop a methodology to dynamically choose the double-side moving window 
training or extending window training for prediction.  
3) Develop a method to evaluate the uncertainty of the moving window based 
failure prediction methods. The impact factors may include the uncertainty from 
 
110 
 
the sensing, training, and modeling. For MW-CVSVR, the errors from the 
training data can indicate the prediction uncertainty in some degrees; however a 
better way is required to evaluate the prediction uncertainty.  
 
111 
 
Appendix A: Results of Failure Analyses 
0s 1s 2s 
 
3s 4s 5s 
 
6s 7s 8s 
 
Figure 58: Typical IR pictures of a new sample 
0s 0.3s 0.5s 
 
 
 
 
112 
 
0.8s 1s 1.5s 
 
2s 3s 4s 
 
5s 6s 7s 
 
8s 13s 24s 
 
Figure 59: Typical IR pictures of tested sample (-10 ?C, trip cycle) 
 
 
 
 
 
113 
 
 
0s 0.2s 0.4s 
 
0.6s 1s 2s 
 
3s 4s 5s 
 
6s 7s 8s 
 
 
 
 
 
 
 
114 
 
9s 10s 15s 
 
Figure 60: Typical IR pictures of tested sample (RT, trip cycle) 
 
0s 0.2s 1s 
 
2s 3s 4s 
 
5s 6s 7s 
 
Figure 61: Typical IR pictures of tested sample (60 ?C, trip cycle) 
  
 
115 
 
0s 0.2s 0.5s 
 
1s 2s 3s 
 
4s 5s 6s 
 
Figure 62: Typical IR pictures of tested sample (85?C, trip cycle) 
 
116 
 
 
Figure 63: Typical microstructure of CB and PE of tested samples (-10?C, aging, freeze fractured) 
 
Figure 64: Typical microstructure of CB and PE of tested samples (-10 ? C, trip cycles, freeze 
fractured) 
 
117 
 
 
Figure 65: Typical microstructure of CB and PE of tested samples (RT, aging, freeze fractured) 
 
Figure 66: Typical microstructure of CB and PE of tested samples (RT, trip cycle, freeze fractured) 
 
118 
 
 
Figure 67: Typical microstructure of CB and PE of tested samples (60? C, aging, freeze fractured) 
 
 
Figure 68: Typical microstructure of CB and PE of tested samples (60 ? C, trip cycles, freeze fractured) 
 
 
119 
 
 
 
Figure 69: Typical microstructure of CB and PE of tested samples (85?C, aging, freeze fractured) 
 
Figure 70: Typical microstructure of CB and PE of tested samples (85 ? C, trip cycles, freeze fractured) 
 
 
120 
 
 
Figure 71: Typical microstructure of CB and PE of tested samples (-10?C, aging, etched by Argon 
ions) 
 
Figure 72: Typical microstructure of CB and PE of tested samples (-10?C, trip cycle, etched by Argon 
ions) 
 
 
121 
 
 
Figure 73: Typical microstructure of CB and PE of tested samples (RT, aging, etched by Argon ions) 
 
Figure 74: Typical microstructure of CB and PE of tested samples (RT, trip cycle, etched by Argon 
ions) 
 
122 
 
 
Figure 75: Typical microstructure of CB and PE of tested samples (60 ?C, aging, etched by Argon 
ions) 
 
Figure 76: Typical microstructure of CB and PE of tested samples (60 ?C, trip cycle, etched by Argon 
ions) 
 
123 
 
 
Figure 77: Typical microstructure of CB and PE of tested samples (85 ?C, aging, etched by Argon 
ions) 
 
Figure 78: Typical microstructure of CB and PE of tested samples (85 ?C, trip cycle, etched by Argon 
ions) 
 
124 
 
Appendix B: Anomaly Detection Results by CVSPRT 
Number of Cycles
 T
 ri
 p 
ti
 m
 e 
(s
 )
 The number of 
cycles @ Anomalies
 Number of Cycles
 Trip time data
 (X 100)
  
Figure 79: Anomaly detection results based on trip time (-10 ?C, trip cycle) 
 
Number of Cycles
 M
 in
 im
 um
  R
 es
 is
 ta
 nc
 e 
af
 te
 r 
R
 es
 et
  (
 O
 hm
 ) The number of 
cycles @ Anomalies
 Number of Cycles
 Minimum 
resistance after 
reset data
 (X 100)
  
Figure 80: Anomaly detection results based on min. resistance after reset (-10 ?C, trip cycle) 
 
 
125 
 
Number of Cycles
 T
 ri
 p 
ti
 m
 e 
(s
 )
 The number of 
cycles @ Anomalies
 Number of Cycles
 Trip time data
 (X 100)
  
Figure 81: Anomaly detection results based on trip time (RT, trip cycle) 
 
Number of Cycles
 M
 in
 im
 um
  R
 es
 is
 ta
 nc
 e 
af
 te
 r 
R
 es
 et
  (
 O
 hm
 )
 The number of 
cycles @ Anomalies
 Number of Cycles
 Minimum 
resistance after 
reset data
 (X 100)
  
Figure 82: Anomaly detection results based on min. resistance after reset (RT, trip cycle) 
 
 
126 
 
 
Number of Cycles
 T
 ri
 p 
ti
 m
 e 
(s
 )
 The number of 
cycles @ Anomalies
 Number of Cycles
 Trip time data
 (X 100)
  
Figure 83: Anomaly detection results based on trip time (60 ?C, trip cycle) 
Number of Cycles
 M
 in
 im
 um
  R
 es
 is
 ta
 nc
 e 
af
 te
 r 
R
 es
 et
  (
 O
 hm
 )
 The number of 
cycles @ Anomalies
 Number of Cycles
 Minimum 
resistance after 
reset data
 (X 100)
  
Figure 84: Anomaly detection results based on min. resistance after reset (60 ?C, trip cycle) 
 
 
127 
 
Number of Cycles
 T
 ri
 p 
ti
 m
 e 
(s
 )
 The number of 
cycles @ Anomalies
 Number of Cycles
 Trip time data
 (X 100)
  
Figure 85: Anomaly detection results based on trip time (85 ?C, trip cycle) 
Number of Cycles
 M
 in
 im
 um
  R
 es
 is
 ta
 nc
 e 
af
 te
 r 
R
 es
 et
  (
 O
 hm
 )
 The number of 
cycles @ Anomalies
 Number of Cycles
 Minimum 
resistance after 
reset data
 (X 100)
  
Figure 86: Anomaly detection results based on min. resistance after reset (85 ?C, trip cycle) 
 
128 
 
Appendix C: 5-step-ahead and 10-step-ahead Predictions by MW-CVSVR 
5
 6
 7
 8
 9
 10
 11
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 im
 e 
(s
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead double-side moving window
 MSE: 0.013
 Threshold
 Original data
  
 Figure 87: All 5-step-ahead predictions (-10?C,trip time, double-side MW) 
5
 6
 7
 8
 9
 10
 11
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead extending moving window
 MSE: 0.007
 Threshold
 Original data
  
Figure 88: All 5-step-ahead predictions (-10?C, trip time, extending window) 
5
 6
 7
 8
 9
 10
 11
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead double-side moving window
 MSE: 0.032
 Threshold
 Original data
  
Figure 89: All 10-step-ahead predictions (-10?C,trip time, double-side MW) 
5
 6
 7
 8
 9
 10
 11
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead extending moving window
 MSE: 0.027
 Threshold
 Original data
  
Figure 90: All 10-step-ahead predictions (-10?C,trip time, extending window) 
 
129 
 
 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 M
 in
 . r
 es
 ist
 an
 ce
  a
 ft
 er
  re
 se
 t(
 O
 hm
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead double-side moving window
 MSE: 1.37E-06
 Original data
 Threshold
  
Figure 91: All 5-step-ahead predictions (-10?C, min. resistance after reset, 
double-side MW) 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 M
 in
 . r
 es
 ist
 an
 ce
  a
 ft
 er
  re
 se
 t(
 O
 hm
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead extending window
 MSE: 7.58E-07
 Original data
 Threshold
  
Figure 92: All 5-step-ahead predictions (-10?C, min. resistance after reset, 
extending window) 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 M
 in
 . r
 es
 ist
 an
 ce
  a
 ft
 er
  re
 se
 t(
 O
 hm
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead double-side moving window
 MSE: 2.34E-06
 Original data
 Threshold
  
Figure 93: All 10-step-ahead predictions (-10?C, min. resistance after reset, 
double-side MW) 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 M
 in
 . r
 es
 ist
 an
 ce
  a
 ft
 er
  re
 se
 t(
 O
 hm
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead extending window
 MSE: 2.05E-06
 Original data
 Threshold
  
Figure 94: All 10-step-ahead predictions (-10?C, min. resistance after reset, 
extending window) 
 
130 
 
3
 4
 5
 6
 7
 8
 9
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead double-side moving window
 MSE: 0.020
 Threshold
 Original data
  
Figure 95: All 5-step-ahead predictions (RT, trip time, double-side MW) 
3
 4
 5
 6
 7
 8
 9
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead extending moving window
 MSE: 0.015
 Threshold
 Original data
  
Figure 96: All 5-step-ahead predictions (RT, trip time, extending window) 
3
 4
 5
 6
 7
 8
 9
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead double-side moving window
 MSE: 0.076
 Threshold
 Original data
  
Figure 97: All 10-step-ahead predictions (RT, trip time, double-side MW) 
3
 4
 5
 6
 7
 8
 9
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead extending moving window
 MSE: 0.076
 Threshold
 Original data
  
Figure 98: All 10-step-ahead predictions (RT, trip time, extending window) 
 
131 
 
 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (Oh
 m
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead double-side moving window
 MSE: 1.96E-06
 Original data
 Threshold
  
Figure 99: All 5-step-ahead predictions (RT, min. resistance after reset, 
double-side MW) 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (Oh
 m
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead extending window
 MSE: 1.86E-06
 Original data
 Threshold
  
Figure 100: All 5-step-ahead predictions (RT, min. resistance after reset, 
extending window) 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (Oh
 m
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead double-side moving window
 MSE: 3.06E-06
 Original data
 Threshold
  
Figure 101: All 10-step-ahead predictions (RT, min. resistance after reset, 
double-side MW) 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0 20 40 60 80 100 120 140 160
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (Oh
 m
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead extending window
 MSE: 3.23E-06
 Original data
 Threshold
  
Figure 102: All 10-step-ahead predictions (RT, min. resistance after reset, 
extending window) 
 
132 
 
 
3
 3.5
 4
 4.5
 5
 0 50 100 150 200
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead double-side moving window
 MSE: 3.80E-04
 Threshold
 Original data
  
Figure 103: All 5-step-ahead predictions (60?C,trip time, double-side MW) 
3
 3.5
 4
 4.5
 5
 0 50 100 150 200
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead extending window
 MSE: 3.58E-04
 Threshold
 Original data
  
Figure 104: All 5-step-ahead predictions (60?C, trip time, extending window) 
3
 3.5
 4
 4.5
 5
 0 50 100 150 200
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead double-side moving window
 MSE: 5.12E-04
 Threshold
 Original data
  
Figure 105: All 10-step-ahead predictions (60?C,trip time, double-side MW) 
3
 3.5
 4
 4.5
 5
 0 50 100 150 200
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead extending window
 MSE: 6.63E-04
 Threshold
 Original data
  
Figure 106: All 10-step-ahead predictions (60?C,trip time, extending window) 
 
133 
 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0.09
 0 50 100 150 200
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (O
 hm
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead double-side moving window
 MSE: 8.02E-07
 Original data
 Threshold
  
Figure 107: All 5-step-ahead predictions (60?C, min. resistance after reset, 
double-side MW) 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0.09
 0 50 100 150 200
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (O
 hm
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead double-side moving window
 MSE: 1.61E-06
 Original data
 Threshold
  
Figure 108: All 5-step-ahead predictions (60?C, min. resistance after reset, 
extending window) 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0.09
 0 50 100 150 200
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (O
 hm
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead double-side moving window
 MSE: 1.61E-06
 Original data
 Threshold
  
Figure 109: All 10-step-ahead predictions (60?C, min. resistance after reset, 
double-side MW) 
0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0.09
 0 50 100 150 200
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (O
 hm
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead extending moving window
 MSE: 1.47E-06
 Original data
 Threshold
  
Figure 110: All 10-step-ahead predictions (60?C, min. resistance after reset, 
extending window) 
 
134 
 
  
3
 3.5
 4
 4.5
 5
 0 50 100 150 200
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead double-side moving window
 MSE: 0.005
 Threshold
 Original data
  
Figure 111: All 5-step-ahead predictions (85?C,trip time, double-side MW) 
3
 3.5
 4
 4.5
 5
 0 50 100 150 200
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead extending window
 MSE: 0.011
 Threshold
 Original data
  
Figure 112: All 5-step-ahead predictions (85?C, trip time, extending window) 
3
 3.5
 4
 4.5
 5
 0 50 100 150 200
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead double-side moving window
 MSE: 0.008
 Threshold
 Original data
  
Figure 113: All 10-step-ahead predictions (85?C,trip time, double-side MW) 
3
 3.5
 4
 4.5
 5
 0 50 100 150 200
 Number of Cycles (? 100)
 T
 ri
 p 
T
 ime
  (s
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead extending window
 MSE: 0.012
 Threshold
 Original data
  
Figure 114: All 10-step-ahead predictions (85?C,trip time, extending window) 
 
135 
 
0.05
 0.06
 0.07
 0.08
 0.09
 0.1
 0.11
 0.12
 0.13
 0 50 100 150 200
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (Oh
 m
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead double-side moving window
 MSE: 4.06E-06Original data
 Threshold
  
Figure 115: All 5-step-ahead predictions (85?C, min. resistance after reset, 
double-side MW) 
0.05
 0.06
 0.07
 0.08
 0.09
 0.1
 0.11
 0.12
 0.13
 0 50 100 150 200
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (Oh
 m
 )
 5-step-ahead predictions.
 Training updating style: 5-step-
 ahead extending window
 MSE: 5.59E-06Original data
 Threshold
  
Figure 116: All 5-step-ahead predictions (85?C, min. resistance after reset, 
extending window) 
0.05
 0.06
 0.07
 0.08
 0.09
 0.1
 0.11
 0.12
 0.13
 0 50 100 150 200
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (Oh
 m
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead double-side moving window
 MSE: 6.86E-06Original data
 Threshold
  
Figure 117: All 10-step-ahead predictions (85?C, min. resistance after reset, 
double-side MW) 
0.05
 0.06
 0.07
 0.08
 0.09
 0.1
 0.11
 0.12
 0.13
 0 50 100 150 200
 Number of Cycles (? 100)
 M
 in
 . r
 es
 is
 ta
 nc
 e 
af
 te
 r r
 es
 et
 (Oh
 m
 )
 10-step-ahead predictions.
 Training updating style: 10-step-
 ahead extending window
 MSE: 8.76E-06Original data
 Threshold
  
Figure 118: All 10-step-ahead predictions (85?C, min. resistance after reset, 
extending window) 
 
136 
 
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