Design of Optimized Reference Signal for Joint Time-Frequency ...
Design of Optimized Reference Signal for Joint Time-Frequency Domain Reflectometry-based Wiring Diagnostics Philip Crapse, Jingjiang Wang, Yong-June Shin, and Roger Dougal
Trang Mai, Joseph Molnar, and Lan Tran
University of South Carolina Columbia, SC 29208 Telephone: (803) 777–9569 Email: [email protected]
Naval Research Laboratory Washington, DC Telephone: (202)767–3161
Abstract— In order to maintain the integrity and safe operation of a power system, a state-of-the-art wiring diagnostic technique is imperative. Joint Time-Frequency Domain Reflectometry (JTFDR) is proposed as an ideal solution due to it’s customizable reference signal and unique time-frequency cross-correlation function. The reference signal depends on three parameters: center frequency, bandwidth, and time duration. Previously, these parameters were chosen based on the frequency characteristics of the cable under test. This paper will fully analyze the changing effects of a wide-range of parameter combinations on two different types of defects. It is determined that there exist optimal reference signal parameters for particular defect types. With this knowledge, JTFDR is able to more sensitively detect various defect types over a longer distance than previously possible. Keywords—Joint Time-Frequency Domain Reflectometry (JTFDR), diagnostics, optimization, reference signal, reflectometry
I. I NTRODUCTION The integrity of the wiring in an electric power system is essential to the safe and effective operation of the system [1],[2]. The wires and cables that transmit power and information are the arteries to the systematic power network body. If even just one of the wires or cables in a system is damaged or severed, the whole system suffers. The common hazards that pose a threat to these vital conductors of information and electricity are well-documented and vary depending on the type and location of the system [3],[4]. Thus far, most available wiring diagnostic techniques are destructive, need to be performed in a laboratory setting, or cannot provide information about the remaining useful life of the cables [4]. However, reflectometry - in which a reference signal is propagated through a cable under test and any reflections are analyzed to determine the status of the defects on the cable - has been proven to be an extremely effective class of wiring diagnostics. Nevertheless, detection and location of defects by the classical techniques such as time-domain reflectometry (TDR) and frequency-domain re-
flectometry (FDR) are limited by the fact that they analyze reflections only in the time domain or only in the frequency domain, respectively. Also, ideally a defect would be detected before it ever occurs to prevent any chance of harmful effects, which neither of these techniques can do. An innovative solution to collectively resolve each of these problems has been proposed: Joint Time-Frequency Domain Reflectometry (JTFDR) [5],[6]. Combining the classical timebased and frequency-based reflectometry techniques in such a manner as to capitalize on each technique’s advantages and alleviate their disadvantages, the state-of-the-art JTFDR has been verified as a successful diagnostic tool [5],[6]. Like other reflectometry techniques, the capability of JTFDR to detect, locate, and prognose a hard or incipient defect depends greatly upon the reference signal. By design, one of the greatest advantages of JTFDR is the reconfigurability of its reference signal. Thus, the JTFDR diagnostic/prognostic results in the past have been accomplished using a reference signal designed according to the frequency characteristics of the cable under test. However, it would be beneficial to search for the “optimal” reference signal designed specifically for the defect being detected. If this goal is achieved, one can take advantage of the versatility of the Gaussian-chirp reference signal by analyzing different defect types and environments with a wide range of reference signal parameters. In Sec. II, the basics of the theory behind JTFDR will be reviewed. In Sec. III, the process of optimizing JTFDR’s customizable reference signal will be explained. The experimental setup is described in Sec. IV and the results of the comprehensive experiments are analyzed in Sec. V. The multiple reference signals will have varying center frequencies, frequency bandwidths, and time durations, and will allow for a more thorough analysis of the various defect types. Conclusions are drawn based on the analyzed results in Sec. VI, and the necessary future work is suggested.
when it is passed into a cable of length 10 m with a damage at 5 m. It is very difficult to see any semblance of the defect at 5 m in the time domain waveform, but it is very apparent in the time-frequency cross-correlation provided in Fig. 2-(b). The reflection due to the end of the cable is apparent in the time domain and the peak of the cross-correlation corresponding to the end of the cable is 1, the maximum value, at 10 m. The peak corresponding to the damage is around 0.5 at 5 m. The quality of reference signal optimization and the severity of the defect are the main contributing factors to the peak at 5 m. These peaks are used to detect and locate any changes in the
1
0
−0.5
Fig. 1.
1
2
3
4 Time (ns)
5
6
7
8
Reference signal of the JTFDR in time domain
II. J OINT T IME -F REQUENCY D OMAIN R EFLECTOMETRY JTFDR captures the advantages of time-domain and frequency-domain techniques and alleviates their disadvantages by employing a strategically designed reference signal and using advanced digital signal processing techniques to analyze any reflections from cable defects. The reference signal applied by JTFDR is a linear chirp signal with a Gaussian envelope, allowing it to be well-localized in both the time and frequency domains simultaneously: s(t) = (α/π)1/4 e−α(t−t0 )
2
/2+jβ(t−t0 )2 /2+jω0 (t−t0 )
(1)
where coefficient α determines the time duration of the reference signal; coefficients α and β together determine the bandwidth of the reference signal; and ω0 is the center frequency. An example of this reference signal created in MATLAB can been seen in Fig. 1. The user may configure the three main parameters of the reference signal: the center frequency (ω0 ), the bandwidth (B) around that center frequency, and the time duration (T ) of the signal. The reference signal is created based on the chosen values of those parameters and propagated into a cable under test. After obtaining the reflections due to defects in the cable, JTFDR uses a predetermined kernel (the Wigner distribution is used in this paper) to find the time-frequency distributions of the reference signal and all reflections. The Wigner distribution of the time signal, s(t), is obtained by the following transformation: 1 1 1 s∗ (t − τ ) · s(t + τ )e−jτ ω dτ (2) W (t, ω) = 2π 2 2 JTFDR then computes the time-frequency cross-correlation between the reference signal and the reflected signal(s) with the following equation: t =t+Ts 1 Wr (t , ω)Ws (t − t, ω)dωdt Csr (t) = Es Er (t) t =t−Ts (3) where Wr (t, ω) is the distribution of the reflected signal; Ws (t, ω) is the distribution of reference signal; and Es and Er are normalization factors. The normalization factors bound the cross-correlation between 0 and 1. Fig. 2-(a) shows an example of JTFDR’s reference signal
Voltage (V)
−1
0.5 0 −0.5 0
2
4 6 Distance (m)
8
10
(a) Incident and reflected waveforms of JTFDR in the time domain.
Cross-Correlation
Voltage (V)
0.5
1 0.5 0
0
2
8 6 4 Distance (m)
10
(b) Corresponding time-frequency cross-correlation. Fig. 2. JTFDR Reference and reflected signals along with the corresponding time-frequency cross-correlation.
impedance of the cable (incipient or hard defects). A higher time-frequency cross-correlation peak means that there is a greater chance of the JTFDR algorithm detecting the defect. Therefore, these peaks are also the metrics that will be used to quantify the effectiveness of a particular combination of reference signal parameters in Sec. V. III. O PTIMIZATION OF JTFDR R EFERENCE S IGNAL In previous papers, JTFDR was demonstrated to have the ability to accurately detect and locate defects [5],[6]. However, previously the parameters of the reference signal were determined based mainly upon the frequency characteristics of the cable under test. The purpose of this paper is to investigate the effect of various combinations of these parameters on multiple defect types to determine if there is an optimal set of parameters based on the expected type of defect. How those sets of optimal parameters relate to each other and to the typical set of parameters implemented based solely upon the cable under test is also of interest in this study. The reference signal of JTFDR is characterized by three variable parameters (center frequency (ω0 ), bandwidth (B), time duration (T)). The ranges over which each parameter is tested are as follows:
ω0 B T
: 150 - 750 MHz, with 50 MHz increments : 0, 25, 100, 300 MHz : 3 - 8 ns, with 1 ns increments
The center frequency bounds are chosen based on the capabilities of the waveform generator being used and the typical frequency characteristics of the common cables in power systems. Most coaxial cables suffer severe attenuation after 700 MHz. The lower bound of the time duration was chosen large enough that a sufficient number of periods still existed in the reference signal for the lower frequencies. The upper bound determines the dead zone - the distance from the beginning of the cable to the first position on the cable at which a defect can be detected. A time duration of 8ns corresponds to a dead zone of almost 5 meters, which would only be acceptable on very long cables. The chosen approach was to set one parameter as a
2) The second step is portrayed by the boxes with vertical lines and darker shading. The first step is basically repeated, except with the center frequency held constant at the optimal value determined in step 1. For every bandwidth, each time duration was implemented incrementally, and the waveforms were acquired, analyzed 3 times, and averaged (and vice versa). With this information the behavior of the cross-correlation at each bandwidth at each time duration (and vice versa) is known. This information was used to determine if there is an optimal bandwidth and, along with the information from step 1, if there is an optimal time duration. By monitoring the particular peak of the cross-correlation function corresponding to the defect at a known location in these ways, ideal parameters for particular defect types may be obtained. Optimal parameters will allow JTFDR to more sensitively detect defects of a lower level of severity at a greater distance, thereby ensuring the integrity of the vital wiring in a power system. IV. E XPERIMENTAL S ETUP
Fig. 3.
Mapping paths of reference signal on time-frequency plane.
constant, vary the other two parameters, and analyze how the peaks of the time-frequency cross-correlation change. After every adaptation of the reference signal, the new reference and reflected signals were obtained and passed through JTFDR’s algorithm 3 times, and then those 3 times were averaged together to attenuate the effects of any out-liars. This procedure, conveyed in Fig. 3, had two major steps: 1) The first step is portrayed by the boxes with horizontal lines and lighter (light gray) shading. The bandwidth was chosen to be the constant parameter. For every center frequency, each time duration was implemented incrementally, and the waveforms were acquired, analyzed 3 times, and averaged. For every time duration, the same was done at each center frequency. With this information the behavior of the time-frequency cross-correlation at each center frequency at each time duration (and vice versa) and the behavior of the cross-correlation of each center frequency averaged over all time durations (and vice versa) is known. Judging from this information, an optimal center frequency is determined for step 2.
The JTFDR experimental setup consists of four (4) main components: (1) a Tektronix Arbitrary Waveform Generator (AWG) 610, (2) an Agilent Infiniium 54754A Oscilloscope, (3) a Hittite GaAs MMIC SPDT Switch, and (4) a desktop computer. The computer utilizes MATLAB to implement a graphical user interface (GUI) to control the entire process. The GUI first takes as input the parameters for the reference signal and tells the AWG to create the particular Gaussianchirp signal. The GUI also tells the AWG to create a PWM signal to send to the switch. The PWM signal is high for the duration of the reference signal and low otherwise. The switch functions as a large-bandwidth circulator, taking as input the reference signal and PWM signal, allowing the reference signal to propagate into the cable, and passing any reflections on to the oscilloscope. The reference signal is also passed directly from the AWG to the scope. The GUI then acquires the reference signal and any reflections from the scope and imports them into MATLAB. Once the data for the signals is in MATLAB, an algorithm based on advanced digital signal processing performs diagnostic and/or prognostic analyses. The cable used for these experiments is RG-58 coaxial cable. The RG-58 is a 50 Ω coaxial cable consisting of a solid bare copper inner conductor, a solid polyethylene dielectric, a braided tinned copper outer conductor, and a PVC outer jacket. Two different cables, both of RG-58, with two different types of defects will be investigated. The two defect types are: 1) Outer Conductor / Insulation Removal 2) Water Damage The outer conductor / insulation removal defect is characterized by the removal of the outer black PVC jacket and the tinned copper outer conductor around half of the circumference of the cable over a 3cm segment, but leaving the dielectric and inner conductor unscathed. This type of defect corresponds
to constant, concentrated vibration, chemical exposure, or any other degrading external influence. The water damage defect was created by puncturing the cable with a needle nine times over a 1 cm segment of the cable and placing that segment in water over night. The nine needle punctures penetrated to the copper inner conductor, but did not harm the inner conductor, and were evenly distributed over the circumference of that 1 cm segment. Both of these defects are located at 10m on a 20m cable. V. R ESULTS AND A NALYSIS As described in Sec. III, the behavior of the time-frequency cross-correlation function versus each parameter will be investigated for each type of defect to determine if there exist optimal parameter settings unique to each defect and, if so, if those settings are similar or different. The RG-58 typically begins to suffer a higher level of attenuation (>11 dB) at 450 MHz. Based on this knowledge, the typical reference signal design would consist of a center frequency of 450 MHz or less, a time duration low enough to obtain a dead zone of 3m or less (∼5 ns), and a bandwidth of ∼25 MHz. The optimal parameters found for each defect should be compared to these parameters that would be chosen based solely upon the characteristics of the cable under test. A. Center Frequency The first step was to analyze the behavior of the timefrequency cross-correlation peaks versus the center frequency of the reference signal. This step (outlined in Sec. III) was performed for each defect type, averaging the peaks of each center frequency over all time durations with the bandwidth held constant at 25 MHz. The results can be seen in Fig. 4. It is a notable fact that the peak - center frequency relationship provided in Fig. 4 for the 3cm defect is inverseparabolic. Most of the highest center frequency peaks reside in the 300-450 MHz range. Again, the higher peaks are 1 3cm Water Damage
Peak
0.8
0.6
0.4
0.2
0 150 200 250 300 350 400 450 500 550 600 650 700 750 CF (MHz) Fig. 4. Peak value of time-frequency cross correlation with different center Frequency for 3 cm and water damage defect.
optimal because the higher the level of the time-frequency cross-correlation peak, the greater the chance of the defect being detected. All of the peaks in this case would have been detected, but a less severe (shorter segment) defect of the same type (PVC jacket and outer conductor removal) would simply shift this curve down. The center frequency corresponding to the highest point on the peak curve, 350 MHz, was chosen as the optimal center frequency for the 3cm defect. The peak - center frequency relationship for the water damage defect is at first inverse-parabolic, but levels out at a very low value for the higher frequencies. The peak of the initial inverse parabola is clearly 200 MHz with a more severe drop-off in peak value beginning at 400 MHz. It is therefore determined that 200 MHz is the optimal center frequency for the water damage defect. The different shapes and the different climaxes of the peak center frequency curves for the 3cm and water damage defects indicate that the two defects have very different characterization functions. The entire curve for the 3cm defect is greater than the highest point of the water damage curve, therefore the impedance change due to the 3cm of jacket and outer conductor removal is greater than that of the water penetrating to the inner conductor, albeit in pin-hole size punctures. Whereas the 3cm defect would have been detected no matter which center frequency was chosen, the water damage defect may have been missed if a center frequency greater than 400 MHz had been selected. B. Frequency Bandwidth The second step was to determine the behavior of the peaks of the time-frequency cross-correlation versus the frequency bandwidth (B) of the reference signal. The experimental results are provided in Fig. 5. The data points for each bandwidth represent the average peak over all time durations with the center frequency held constant at the optimal center frequency: 350 MHz for the 3cm defect and 200 MHz for the water damage defect. It is clear that the effect of bandwidth at the optimal frequency has a relatively small effect on the peak of the cross-correlation compared to the effect of varying the center frequency in Sec. V-A. However minimal, the trend is obviously downward. As the bandwidth is increased, the peak tends to slightly decrease at the optimal center frequency for both defects. It is noted that the peaks at 25 MHz correspond to the peaks at the optimal frequencies in Fig. 4 as well ( 0.7 at 350 MHz for the 3cm defect and 0.4 at 200 MHz for the water damage defect) in which the bandwidth was held constant at 25 MHz. The purpose of having a bandwidth rather than a single frequency in the JTFDR algorithm is to reduce the width of the time frequency cross-correlation peaks corresponding to defects in the cable. The greater the bandwidth, the slimmer the Gaussian-shaped peaks and the greater the resolution (cite). Therefore, it is determined that there must be a balance between having a bandwidth low enough not to seriously attenuate the peaks of the cross-correlation and hamper the
1 0.9
C. Time Duration
0.8
Having performed both steps of the procedure outlined in Sec. III, the peak - time duration curve may be obtained in one of two ways: (1) average the peaks corresponding to each time duration for all center frequencies while the bandwidth is held constant, from part 1, or (2) average the peaks corresponding to each time duration for all bandwidths while the center frequency is held constant, from part 2. Theoretically, the results would be identical. But, in actuality, they were indeed very similar, therefore only the first option will be presented. The peak - time duration curves can be seen in Fig. 6. The thinner, patterned (also colored) curves with circular markers refer to the peaks of the cross-correlation function at particular frequencies (100 MHz intervals between 150-750 MHz) at each time duration. The thicker, solid (black) curve with square markers was obtained by averaging the peaks corresponding to each time duration over all center frequencies while the bandwidth was maintained at 25 MHz. The plots show that the overall trend for both defects is that as the time duration increases, the peak of the cross-correlation decreases. But as was the case with bandwidth, the decrease is minimal. Also in common with the bandwidth curves, the overall level of the peaks is higher for the 3cm defect than for the water damage defect. The range of peak values is also greater for the 3cm defect. The peak of the cross-correlation for the 3cm defect dips as low as 0.22 at 750 MHz - 7 ns and rises as high as 0.75 at 350 MHz - 4 ns, for a difference of 0.53. The average value for all center frequencies (thickest line) ranges from 0.6 - 0.45. On the other hand, the peak of the cross-correlation for the water damage defect is as low as 0.02 at 750 MHz - 7 ns and as high as 0.4 at 250 MHz - 3 ns, for a difference of 0.38. The water damage average
0.7 Peak
ability of the algorithm to detect defects and high enough to provide the desired resolution for defect location.
150 MHz 250 MHz 350 MHz 450MHz 550 MHz 650 MHz 750 MHz Average
0.6 0.5 0.4 0.3 0.2 0.1 0 3
4
5
6
7
8
TD (ns) (a) Peak vs. Time Duration for 3cm defect.
1
150 MHz 250 MHz 350 MHz 450MHz 550 MHz 650 MHz 750 MHz Average
Peak
0.8
0.6
0.4
0.2
0 3
4
5
6
7
8
TD (ns) (b) Peak vs. Time Duration for water damage defect. Fig. 6.
Peak vs. Time Duration curves for the two different defects.
1 3cm Water Damage
Peak
0.8
0.6
0.4
0.2
0 0
25
100
400
BW (MHz) Fig. 5. Peak value of time-frequency cross correlation with different frequency bandwidth for 3 cm and water damage defect.
peak values for all center frequencies ranges from 0.2 - 0.15, not much of an overall change at all. Plots are only shown for increments of 100 MHz because including plots for each frequency would have made the data indistinguishable. Summarizing the experimental data, it is clear that an increased time duration of the JTFDR reference signal results in the decreased peak value of the the time-frequency crosscorrelation function. However, as with frequency bandwidth, one should note to the fact that the time duration serves other significant purposes. For example, the longer the time duration of the reference signal the more energy the signal contains, and therefore the further it can propagate down a cable and return for acquisition and analysis. But the higher time duration will also sacrifice resolution of the defect location. Therefore, a desirable alternative may be to amplify the reference signal to increase the energy level instead of increasing the time duration of the reference signal.
VI. C ONCLUSION One of JTFDR’s main advantages is the ability to customize the design of it’s Gaussian-chirp reference signal through three parameters: center frequency (ω0 ), bandwidth (B), and time duration (T ). Previously, these parameters have been chosen based upon the frequency characteristics of the cable under test. The goal of this paper was to determine if there exist an optimal set of parameters for different defect types, and how those parameters relate to each other and the typical parameters that would be chosen based solely upon the frequency characteristics of the cable under test. Through a two-step process it was determined that the optimal reference signal center frequency is 350 MHz for a 3 cm jacket / outer conductor defect and 200 MHz for a water damage defect. It was observed that as the bandwidth of the reference signal increases, the peak of the cross-correlation decreases, but only minimally. It is necessary to compromise between the increased attenuation of the peak of the cross-correlation and the increase in defect location resolution of the algorithm that both come with increasing the bandwidth. It was also determined that as the time duration of the signal increases, the peak of the cross-correlation decreases minimally. Comparatively, the determined optimal center frequencies were very different. The experiments also revealed different characteristic functions for each defect type through the different reactions of the time-frequency cross-correlation peak to various reference signal center frequencies. The trends for the peak versus the bandwidth and time duration were negative for both defects. However, whereas the range of decrease of cross-correlation peak versus bandwidth was similar for the two defects, the peak value decrease from the minimum time duration to the maximum time duration was greater for the 3cm defect. Therefore, the compromise between the time duration and peak attenuation for the 3cm defect will be more sensitive. The experiments presented in this paper were successful in determining the relationships between the peak of the timefrequency cross-correlation to each of the three parameters used in the design of the JTFDR reference signal. It is clear from these experiments that the different defect types have unique, particular optimal parameters to make each of them more susceptible to detection. These optimal parameters may not be the same as the parameters that could be chosen from the general range of possible parameters based solely upon the cable under test. With the same knowledge for a wide variety of defect types, the JTFDR reference signal may be customized to best fit whichever type of defect is expected on a cable under test by only changing three parameters. This novel ability will allow for the maintenance of the integrity and performance of any power system diagnosed and an increase in reliability of that system. The results of these experiments revealed the optimal parameters for two defect types, but future work is needed to analyze other defect types and to take into account important aspects other than the time-frequency cross-correlation peak values such as defect
location resolution and the ability to accurately quantify the severity of a defect.
ACKNOWLEDGMENT The work reported in this paper was supported by the U.S. ONR Electric Ship Research and Development Consortium under Grants N00014-08-1-0080 and National Science Foundation Grants #0747681, “CAREER: Diagnostics and Prognostics of Electric Cables in Aging Power Infrastructure.” R EFERENCES [1] L. A. Griffiths, R. Parakh, C. Furse, and B. Baker, ”The Invisible Fray: A Critical Analysis of the Use of Reflectometry for Fray Location,” IEEE Sensors Journal, Vol. 6, No. 3, pp. 697-706, June 2006. [2] P. Smith, C. Furse, and J. Gunther, ”Analysis of Spread Spectrum Time Domain Reflectometry for Wire Fault Location,” IEEE Sensors Journal, Vol. 5, No. 6, pp. 1469-1478, December 2005. [3] S. J. Grayson, et al., ”Fire Performance of Electric Cables,” Interscience Communications, Ltd., United Kingdom, 2000. [4] C. Furse, Y. C. Chung, R. Dangol, M. Nielsen, G. Mabey, and R. Woodward, ”Frequency-Domain Reflectometry for on-Board Testing of Aging Aircraft Wiring,” IEEE Transactions on Electromagnetic Compatibility, Vol. 45, No.2, pp. 306-315, May 2003. [5] Y. J. Shin, E. J. Powers, T. S. Choe, C. Y. Hong, E. S. Song, J. G. Yook, J. B. Park, “Application of Time-Frequency Domain Reflectometry for Detection and Localization of a Fault on a Coaxial Cable,” IEEE Transactions on Instrumentation and Measurement, Vol. 54, No. 6, Dec. 2005. [6] Jingjiang Wang, Philip Crapse, John Abrams, Yong-June Shin, Roger Dougal, Trang Mai, Lan Tran and Joseph Molnar, “Diagnostics and Prognostics of Wiring Integrity via Joint Time-Frequency Domain Reflectometry,” 10th Joint FAA/DoD/NASA Conference on Aging Aircraft, Palm Springs, CA, April 2007.
Trang Mai, Joseph Molnar, and Lan Tran
University of South Carolina Columbia, SC 29208 Telephone: (803) 777–9569 Email: [email protected]
Naval Research Laboratory Washington, DC Telephone: (202)767–3161
Abstract— In order to maintain the integrity and safe operation of a power system, a state-of-the-art wiring diagnostic technique is imperative. Joint Time-Frequency Domain Reflectometry (JTFDR) is proposed as an ideal solution due to it’s customizable reference signal and unique time-frequency cross-correlation function. The reference signal depends on three parameters: center frequency, bandwidth, and time duration. Previously, these parameters were chosen based on the frequency characteristics of the cable under test. This paper will fully analyze the changing effects of a wide-range of parameter combinations on two different types of defects. It is determined that there exist optimal reference signal parameters for particular defect types. With this knowledge, JTFDR is able to more sensitively detect various defect types over a longer distance than previously possible. Keywords—Joint Time-Frequency Domain Reflectometry (JTFDR), diagnostics, optimization, reference signal, reflectometry
I. I NTRODUCTION The integrity of the wiring in an electric power system is essential to the safe and effective operation of the system [1],[2]. The wires and cables that transmit power and information are the arteries to the systematic power network body. If even just one of the wires or cables in a system is damaged or severed, the whole system suffers. The common hazards that pose a threat to these vital conductors of information and electricity are well-documented and vary depending on the type and location of the system [3],[4]. Thus far, most available wiring diagnostic techniques are destructive, need to be performed in a laboratory setting, or cannot provide information about the remaining useful life of the cables [4]. However, reflectometry - in which a reference signal is propagated through a cable under test and any reflections are analyzed to determine the status of the defects on the cable - has been proven to be an extremely effective class of wiring diagnostics. Nevertheless, detection and location of defects by the classical techniques such as time-domain reflectometry (TDR) and frequency-domain re-
flectometry (FDR) are limited by the fact that they analyze reflections only in the time domain or only in the frequency domain, respectively. Also, ideally a defect would be detected before it ever occurs to prevent any chance of harmful effects, which neither of these techniques can do. An innovative solution to collectively resolve each of these problems has been proposed: Joint Time-Frequency Domain Reflectometry (JTFDR) [5],[6]. Combining the classical timebased and frequency-based reflectometry techniques in such a manner as to capitalize on each technique’s advantages and alleviate their disadvantages, the state-of-the-art JTFDR has been verified as a successful diagnostic tool [5],[6]. Like other reflectometry techniques, the capability of JTFDR to detect, locate, and prognose a hard or incipient defect depends greatly upon the reference signal. By design, one of the greatest advantages of JTFDR is the reconfigurability of its reference signal. Thus, the JTFDR diagnostic/prognostic results in the past have been accomplished using a reference signal designed according to the frequency characteristics of the cable under test. However, it would be beneficial to search for the “optimal” reference signal designed specifically for the defect being detected. If this goal is achieved, one can take advantage of the versatility of the Gaussian-chirp reference signal by analyzing different defect types and environments with a wide range of reference signal parameters. In Sec. II, the basics of the theory behind JTFDR will be reviewed. In Sec. III, the process of optimizing JTFDR’s customizable reference signal will be explained. The experimental setup is described in Sec. IV and the results of the comprehensive experiments are analyzed in Sec. V. The multiple reference signals will have varying center frequencies, frequency bandwidths, and time durations, and will allow for a more thorough analysis of the various defect types. Conclusions are drawn based on the analyzed results in Sec. VI, and the necessary future work is suggested.
when it is passed into a cable of length 10 m with a damage at 5 m. It is very difficult to see any semblance of the defect at 5 m in the time domain waveform, but it is very apparent in the time-frequency cross-correlation provided in Fig. 2-(b). The reflection due to the end of the cable is apparent in the time domain and the peak of the cross-correlation corresponding to the end of the cable is 1, the maximum value, at 10 m. The peak corresponding to the damage is around 0.5 at 5 m. The quality of reference signal optimization and the severity of the defect are the main contributing factors to the peak at 5 m. These peaks are used to detect and locate any changes in the
1
0
−0.5
Fig. 1.
1
2
3
4 Time (ns)
5
6
7
8
Reference signal of the JTFDR in time domain
II. J OINT T IME -F REQUENCY D OMAIN R EFLECTOMETRY JTFDR captures the advantages of time-domain and frequency-domain techniques and alleviates their disadvantages by employing a strategically designed reference signal and using advanced digital signal processing techniques to analyze any reflections from cable defects. The reference signal applied by JTFDR is a linear chirp signal with a Gaussian envelope, allowing it to be well-localized in both the time and frequency domains simultaneously: s(t) = (α/π)1/4 e−α(t−t0 )
2
/2+jβ(t−t0 )2 /2+jω0 (t−t0 )
(1)
where coefficient α determines the time duration of the reference signal; coefficients α and β together determine the bandwidth of the reference signal; and ω0 is the center frequency. An example of this reference signal created in MATLAB can been seen in Fig. 1. The user may configure the three main parameters of the reference signal: the center frequency (ω0 ), the bandwidth (B) around that center frequency, and the time duration (T ) of the signal. The reference signal is created based on the chosen values of those parameters and propagated into a cable under test. After obtaining the reflections due to defects in the cable, JTFDR uses a predetermined kernel (the Wigner distribution is used in this paper) to find the time-frequency distributions of the reference signal and all reflections. The Wigner distribution of the time signal, s(t), is obtained by the following transformation: 1 1 1 s∗ (t − τ ) · s(t + τ )e−jτ ω dτ (2) W (t, ω) = 2π 2 2 JTFDR then computes the time-frequency cross-correlation between the reference signal and the reflected signal(s) with the following equation: t =t+Ts 1 Wr (t , ω)Ws (t − t, ω)dωdt Csr (t) = Es Er (t) t =t−Ts (3) where Wr (t, ω) is the distribution of the reflected signal; Ws (t, ω) is the distribution of reference signal; and Es and Er are normalization factors. The normalization factors bound the cross-correlation between 0 and 1. Fig. 2-(a) shows an example of JTFDR’s reference signal
Voltage (V)
−1
0.5 0 −0.5 0
2
4 6 Distance (m)
8
10
(a) Incident and reflected waveforms of JTFDR in the time domain.
Cross-Correlation
Voltage (V)
0.5
1 0.5 0
0
2
8 6 4 Distance (m)
10
(b) Corresponding time-frequency cross-correlation. Fig. 2. JTFDR Reference and reflected signals along with the corresponding time-frequency cross-correlation.
impedance of the cable (incipient or hard defects). A higher time-frequency cross-correlation peak means that there is a greater chance of the JTFDR algorithm detecting the defect. Therefore, these peaks are also the metrics that will be used to quantify the effectiveness of a particular combination of reference signal parameters in Sec. V. III. O PTIMIZATION OF JTFDR R EFERENCE S IGNAL In previous papers, JTFDR was demonstrated to have the ability to accurately detect and locate defects [5],[6]. However, previously the parameters of the reference signal were determined based mainly upon the frequency characteristics of the cable under test. The purpose of this paper is to investigate the effect of various combinations of these parameters on multiple defect types to determine if there is an optimal set of parameters based on the expected type of defect. How those sets of optimal parameters relate to each other and to the typical set of parameters implemented based solely upon the cable under test is also of interest in this study. The reference signal of JTFDR is characterized by three variable parameters (center frequency (ω0 ), bandwidth (B), time duration (T)). The ranges over which each parameter is tested are as follows:
ω0 B T
: 150 - 750 MHz, with 50 MHz increments : 0, 25, 100, 300 MHz : 3 - 8 ns, with 1 ns increments
The center frequency bounds are chosen based on the capabilities of the waveform generator being used and the typical frequency characteristics of the common cables in power systems. Most coaxial cables suffer severe attenuation after 700 MHz. The lower bound of the time duration was chosen large enough that a sufficient number of periods still existed in the reference signal for the lower frequencies. The upper bound determines the dead zone - the distance from the beginning of the cable to the first position on the cable at which a defect can be detected. A time duration of 8ns corresponds to a dead zone of almost 5 meters, which would only be acceptable on very long cables. The chosen approach was to set one parameter as a
2) The second step is portrayed by the boxes with vertical lines and darker shading. The first step is basically repeated, except with the center frequency held constant at the optimal value determined in step 1. For every bandwidth, each time duration was implemented incrementally, and the waveforms were acquired, analyzed 3 times, and averaged (and vice versa). With this information the behavior of the cross-correlation at each bandwidth at each time duration (and vice versa) is known. This information was used to determine if there is an optimal bandwidth and, along with the information from step 1, if there is an optimal time duration. By monitoring the particular peak of the cross-correlation function corresponding to the defect at a known location in these ways, ideal parameters for particular defect types may be obtained. Optimal parameters will allow JTFDR to more sensitively detect defects of a lower level of severity at a greater distance, thereby ensuring the integrity of the vital wiring in a power system. IV. E XPERIMENTAL S ETUP
Fig. 3.
Mapping paths of reference signal on time-frequency plane.
constant, vary the other two parameters, and analyze how the peaks of the time-frequency cross-correlation change. After every adaptation of the reference signal, the new reference and reflected signals were obtained and passed through JTFDR’s algorithm 3 times, and then those 3 times were averaged together to attenuate the effects of any out-liars. This procedure, conveyed in Fig. 3, had two major steps: 1) The first step is portrayed by the boxes with horizontal lines and lighter (light gray) shading. The bandwidth was chosen to be the constant parameter. For every center frequency, each time duration was implemented incrementally, and the waveforms were acquired, analyzed 3 times, and averaged. For every time duration, the same was done at each center frequency. With this information the behavior of the time-frequency cross-correlation at each center frequency at each time duration (and vice versa) and the behavior of the cross-correlation of each center frequency averaged over all time durations (and vice versa) is known. Judging from this information, an optimal center frequency is determined for step 2.
The JTFDR experimental setup consists of four (4) main components: (1) a Tektronix Arbitrary Waveform Generator (AWG) 610, (2) an Agilent Infiniium 54754A Oscilloscope, (3) a Hittite GaAs MMIC SPDT Switch, and (4) a desktop computer. The computer utilizes MATLAB to implement a graphical user interface (GUI) to control the entire process. The GUI first takes as input the parameters for the reference signal and tells the AWG to create the particular Gaussianchirp signal. The GUI also tells the AWG to create a PWM signal to send to the switch. The PWM signal is high for the duration of the reference signal and low otherwise. The switch functions as a large-bandwidth circulator, taking as input the reference signal and PWM signal, allowing the reference signal to propagate into the cable, and passing any reflections on to the oscilloscope. The reference signal is also passed directly from the AWG to the scope. The GUI then acquires the reference signal and any reflections from the scope and imports them into MATLAB. Once the data for the signals is in MATLAB, an algorithm based on advanced digital signal processing performs diagnostic and/or prognostic analyses. The cable used for these experiments is RG-58 coaxial cable. The RG-58 is a 50 Ω coaxial cable consisting of a solid bare copper inner conductor, a solid polyethylene dielectric, a braided tinned copper outer conductor, and a PVC outer jacket. Two different cables, both of RG-58, with two different types of defects will be investigated. The two defect types are: 1) Outer Conductor / Insulation Removal 2) Water Damage The outer conductor / insulation removal defect is characterized by the removal of the outer black PVC jacket and the tinned copper outer conductor around half of the circumference of the cable over a 3cm segment, but leaving the dielectric and inner conductor unscathed. This type of defect corresponds
to constant, concentrated vibration, chemical exposure, or any other degrading external influence. The water damage defect was created by puncturing the cable with a needle nine times over a 1 cm segment of the cable and placing that segment in water over night. The nine needle punctures penetrated to the copper inner conductor, but did not harm the inner conductor, and were evenly distributed over the circumference of that 1 cm segment. Both of these defects are located at 10m on a 20m cable. V. R ESULTS AND A NALYSIS As described in Sec. III, the behavior of the time-frequency cross-correlation function versus each parameter will be investigated for each type of defect to determine if there exist optimal parameter settings unique to each defect and, if so, if those settings are similar or different. The RG-58 typically begins to suffer a higher level of attenuation (>11 dB) at 450 MHz. Based on this knowledge, the typical reference signal design would consist of a center frequency of 450 MHz or less, a time duration low enough to obtain a dead zone of 3m or less (∼5 ns), and a bandwidth of ∼25 MHz. The optimal parameters found for each defect should be compared to these parameters that would be chosen based solely upon the characteristics of the cable under test. A. Center Frequency The first step was to analyze the behavior of the timefrequency cross-correlation peaks versus the center frequency of the reference signal. This step (outlined in Sec. III) was performed for each defect type, averaging the peaks of each center frequency over all time durations with the bandwidth held constant at 25 MHz. The results can be seen in Fig. 4. It is a notable fact that the peak - center frequency relationship provided in Fig. 4 for the 3cm defect is inverseparabolic. Most of the highest center frequency peaks reside in the 300-450 MHz range. Again, the higher peaks are 1 3cm Water Damage
Peak
0.8
0.6
0.4
0.2
0 150 200 250 300 350 400 450 500 550 600 650 700 750 CF (MHz) Fig. 4. Peak value of time-frequency cross correlation with different center Frequency for 3 cm and water damage defect.
optimal because the higher the level of the time-frequency cross-correlation peak, the greater the chance of the defect being detected. All of the peaks in this case would have been detected, but a less severe (shorter segment) defect of the same type (PVC jacket and outer conductor removal) would simply shift this curve down. The center frequency corresponding to the highest point on the peak curve, 350 MHz, was chosen as the optimal center frequency for the 3cm defect. The peak - center frequency relationship for the water damage defect is at first inverse-parabolic, but levels out at a very low value for the higher frequencies. The peak of the initial inverse parabola is clearly 200 MHz with a more severe drop-off in peak value beginning at 400 MHz. It is therefore determined that 200 MHz is the optimal center frequency for the water damage defect. The different shapes and the different climaxes of the peak center frequency curves for the 3cm and water damage defects indicate that the two defects have very different characterization functions. The entire curve for the 3cm defect is greater than the highest point of the water damage curve, therefore the impedance change due to the 3cm of jacket and outer conductor removal is greater than that of the water penetrating to the inner conductor, albeit in pin-hole size punctures. Whereas the 3cm defect would have been detected no matter which center frequency was chosen, the water damage defect may have been missed if a center frequency greater than 400 MHz had been selected. B. Frequency Bandwidth The second step was to determine the behavior of the peaks of the time-frequency cross-correlation versus the frequency bandwidth (B) of the reference signal. The experimental results are provided in Fig. 5. The data points for each bandwidth represent the average peak over all time durations with the center frequency held constant at the optimal center frequency: 350 MHz for the 3cm defect and 200 MHz for the water damage defect. It is clear that the effect of bandwidth at the optimal frequency has a relatively small effect on the peak of the cross-correlation compared to the effect of varying the center frequency in Sec. V-A. However minimal, the trend is obviously downward. As the bandwidth is increased, the peak tends to slightly decrease at the optimal center frequency for both defects. It is noted that the peaks at 25 MHz correspond to the peaks at the optimal frequencies in Fig. 4 as well ( 0.7 at 350 MHz for the 3cm defect and 0.4 at 200 MHz for the water damage defect) in which the bandwidth was held constant at 25 MHz. The purpose of having a bandwidth rather than a single frequency in the JTFDR algorithm is to reduce the width of the time frequency cross-correlation peaks corresponding to defects in the cable. The greater the bandwidth, the slimmer the Gaussian-shaped peaks and the greater the resolution (cite). Therefore, it is determined that there must be a balance between having a bandwidth low enough not to seriously attenuate the peaks of the cross-correlation and hamper the
1 0.9
C. Time Duration
0.8
Having performed both steps of the procedure outlined in Sec. III, the peak - time duration curve may be obtained in one of two ways: (1) average the peaks corresponding to each time duration for all center frequencies while the bandwidth is held constant, from part 1, or (2) average the peaks corresponding to each time duration for all bandwidths while the center frequency is held constant, from part 2. Theoretically, the results would be identical. But, in actuality, they were indeed very similar, therefore only the first option will be presented. The peak - time duration curves can be seen in Fig. 6. The thinner, patterned (also colored) curves with circular markers refer to the peaks of the cross-correlation function at particular frequencies (100 MHz intervals between 150-750 MHz) at each time duration. The thicker, solid (black) curve with square markers was obtained by averaging the peaks corresponding to each time duration over all center frequencies while the bandwidth was maintained at 25 MHz. The plots show that the overall trend for both defects is that as the time duration increases, the peak of the cross-correlation decreases. But as was the case with bandwidth, the decrease is minimal. Also in common with the bandwidth curves, the overall level of the peaks is higher for the 3cm defect than for the water damage defect. The range of peak values is also greater for the 3cm defect. The peak of the cross-correlation for the 3cm defect dips as low as 0.22 at 750 MHz - 7 ns and rises as high as 0.75 at 350 MHz - 4 ns, for a difference of 0.53. The average value for all center frequencies (thickest line) ranges from 0.6 - 0.45. On the other hand, the peak of the cross-correlation for the water damage defect is as low as 0.02 at 750 MHz - 7 ns and as high as 0.4 at 250 MHz - 3 ns, for a difference of 0.38. The water damage average
0.7 Peak
ability of the algorithm to detect defects and high enough to provide the desired resolution for defect location.
150 MHz 250 MHz 350 MHz 450MHz 550 MHz 650 MHz 750 MHz Average
0.6 0.5 0.4 0.3 0.2 0.1 0 3
4
5
6
7
8
TD (ns) (a) Peak vs. Time Duration for 3cm defect.
1
150 MHz 250 MHz 350 MHz 450MHz 550 MHz 650 MHz 750 MHz Average
Peak
0.8
0.6
0.4
0.2
0 3
4
5
6
7
8
TD (ns) (b) Peak vs. Time Duration for water damage defect. Fig. 6.
Peak vs. Time Duration curves for the two different defects.
1 3cm Water Damage
Peak
0.8
0.6
0.4
0.2
0 0
25
100
400
BW (MHz) Fig. 5. Peak value of time-frequency cross correlation with different frequency bandwidth for 3 cm and water damage defect.
peak values for all center frequencies ranges from 0.2 - 0.15, not much of an overall change at all. Plots are only shown for increments of 100 MHz because including plots for each frequency would have made the data indistinguishable. Summarizing the experimental data, it is clear that an increased time duration of the JTFDR reference signal results in the decreased peak value of the the time-frequency crosscorrelation function. However, as with frequency bandwidth, one should note to the fact that the time duration serves other significant purposes. For example, the longer the time duration of the reference signal the more energy the signal contains, and therefore the further it can propagate down a cable and return for acquisition and analysis. But the higher time duration will also sacrifice resolution of the defect location. Therefore, a desirable alternative may be to amplify the reference signal to increase the energy level instead of increasing the time duration of the reference signal.
VI. C ONCLUSION One of JTFDR’s main advantages is the ability to customize the design of it’s Gaussian-chirp reference signal through three parameters: center frequency (ω0 ), bandwidth (B), and time duration (T ). Previously, these parameters have been chosen based upon the frequency characteristics of the cable under test. The goal of this paper was to determine if there exist an optimal set of parameters for different defect types, and how those parameters relate to each other and the typical parameters that would be chosen based solely upon the frequency characteristics of the cable under test. Through a two-step process it was determined that the optimal reference signal center frequency is 350 MHz for a 3 cm jacket / outer conductor defect and 200 MHz for a water damage defect. It was observed that as the bandwidth of the reference signal increases, the peak of the cross-correlation decreases, but only minimally. It is necessary to compromise between the increased attenuation of the peak of the cross-correlation and the increase in defect location resolution of the algorithm that both come with increasing the bandwidth. It was also determined that as the time duration of the signal increases, the peak of the cross-correlation decreases minimally. Comparatively, the determined optimal center frequencies were very different. The experiments also revealed different characteristic functions for each defect type through the different reactions of the time-frequency cross-correlation peak to various reference signal center frequencies. The trends for the peak versus the bandwidth and time duration were negative for both defects. However, whereas the range of decrease of cross-correlation peak versus bandwidth was similar for the two defects, the peak value decrease from the minimum time duration to the maximum time duration was greater for the 3cm defect. Therefore, the compromise between the time duration and peak attenuation for the 3cm defect will be more sensitive. The experiments presented in this paper were successful in determining the relationships between the peak of the timefrequency cross-correlation to each of the three parameters used in the design of the JTFDR reference signal. It is clear from these experiments that the different defect types have unique, particular optimal parameters to make each of them more susceptible to detection. These optimal parameters may not be the same as the parameters that could be chosen from the general range of possible parameters based solely upon the cable under test. With the same knowledge for a wide variety of defect types, the JTFDR reference signal may be customized to best fit whichever type of defect is expected on a cable under test by only changing three parameters. This novel ability will allow for the maintenance of the integrity and performance of any power system diagnosed and an increase in reliability of that system. The results of these experiments revealed the optimal parameters for two defect types, but future work is needed to analyze other defect types and to take into account important aspects other than the time-frequency cross-correlation peak values such as defect
location resolution and the ability to accurately quantify the severity of a defect.
ACKNOWLEDGMENT The work reported in this paper was supported by the U.S. ONR Electric Ship Research and Development Consortium under Grants N00014-08-1-0080 and National Science Foundation Grants #0747681, “CAREER: Diagnostics and Prognostics of Electric Cables in Aging Power Infrastructure.” R EFERENCES [1] L. A. Griffiths, R. Parakh, C. Furse, and B. Baker, ”The Invisible Fray: A Critical Analysis of the Use of Reflectometry for Fray Location,” IEEE Sensors Journal, Vol. 6, No. 3, pp. 697-706, June 2006. [2] P. Smith, C. Furse, and J. Gunther, ”Analysis of Spread Spectrum Time Domain Reflectometry for Wire Fault Location,” IEEE Sensors Journal, Vol. 5, No. 6, pp. 1469-1478, December 2005. [3] S. J. Grayson, et al., ”Fire Performance of Electric Cables,” Interscience Communications, Ltd., United Kingdom, 2000. [4] C. Furse, Y. C. Chung, R. Dangol, M. Nielsen, G. Mabey, and R. Woodward, ”Frequency-Domain Reflectometry for on-Board Testing of Aging Aircraft Wiring,” IEEE Transactions on Electromagnetic Compatibility, Vol. 45, No.2, pp. 306-315, May 2003. [5] Y. J. Shin, E. J. Powers, T. S. Choe, C. Y. Hong, E. S. Song, J. G. Yook, J. B. Park, “Application of Time-Frequency Domain Reflectometry for Detection and Localization of a Fault on a Coaxial Cable,” IEEE Transactions on Instrumentation and Measurement, Vol. 54, No. 6, Dec. 2005. [6] Jingjiang Wang, Philip Crapse, John Abrams, Yong-June Shin, Roger Dougal, Trang Mai, Lan Tran and Joseph Molnar, “Diagnostics and Prognostics of Wiring Integrity via Joint Time-Frequency Domain Reflectometry,” 10th Joint FAA/DoD/NASA Conference on Aging Aircraft, Palm Springs, CA, April 2007.
