Padmanabhan Rajaraman, Qinghai Shi, Olfa Kanoun
IMAGE LICENSED BY INGRAM PUBLISHING
Fault analysis is of vital importance in power systems, and the techniques employed must be reliable, accurate, and fast. With the advent of time- and frequency-domain analyses in power systems, reflectometry techniques have been introduced.
In time-domain reflectometry (TDR), a test signal is used that is transmitted along the cable length; when it encounters a discontinuity in the cable, which is seen as an impedance, a portion of the signal is reflected. Based on signal processing techniques, the fault location and nature of the fault can be determined. The frequency-domain reflectometry (FDR) method is faster and has better accuracy than the TDR approach because power signals are used as incident signals, and the Fourier transform, which has good resolution in the frequency domain, is applied to locate wire faults. However, it is difficult to identify the types of the individual wire faults using the FDR method.
Besides these techniques, a novel approach to detect and locate faults in a single wire system using impedance spectroscopy (IS) is explicated. IS is applied to measure the input impedance of the wiring system based on a frequency sweep. The impedance of the single wire fault can be obtained from the input impedance to identify the nature of the fault. The input impedance of the cable is a periodic function of the frequency of the input test signal, and the period of this function is proportional to the distance to the fault. The fast Fourier transform (FFT) of the input impedance function gives a single sharp pulse that can be used to locate the wire fault. Therefore, using IS, both the location and characterization of the fault are possible using a frequency-domain analysis of the input impedance of the system.
Coaxial cables are growing in use in applications, and with their increased usage in devices and networks, the need arises for efficient and accurate techniques to detect and characterize faults along their length. Faults may be of two types: hard faults, which comprise open and short circuit faults, and soft faults, which include small faults, such as chafes, frays, and joints.
TDR was one of the first techniques to be proposed in this regard. The method involves a fast-rising step signal that is transmitted along the length of a cable. Whenever the signal encounters a discontinuity or a load impedance, a portion of the signal is reflected, which is based on the reflection coefficient. The reflected signal can be processed to get information regarding the fault location and characterization of the fault.
Contrary to this method is the FDR procedure, where digital signal processing in the frequency domain gives this approach better location accuracy than TDR. The main disadvantage of this technique is that it cannot identify the type of fault. IS is a new method that uses the principle of abstraction of the load impedance from a measured set of input impedance data from the cable under test. Using appropriate transmission line modeling and the FFT technique, the location and nature of the fault can be determined in an accurate and fast manner.
Reflectometry is a technique that provides for the study of the properties of a given medium. This technique employs the analysis of the reflection of waves at a given interface. The input signal propagates into a medium, and, when it encounters a discontinuity (which is seen as impedance), a portion of its energy is transmitted, and the rest is reflected back to the injection point. A coaxial cable can be modeled as an ideal, lossless, unbalanced transmission line (Fig. 1).
Fig 1 (a) A cross-section of a coaxial cable and (b) a transmission line model of a coaxial cable.
The principle behind reflectometry arises from the behavior of signals in systems. An arbitrary signal that is transmitted along a system continues to propagate [according to the laws of propagation in the medium, discussed by Magnusson et al. (1979)] with its propagation velocity, defined as \[{\upsilon} = \frac{c}{\sqrt{{\varepsilon}_{r}}}, \tag{1} \] where c is the speed of light in a vacuum (3 × 108 m/s), and ${\varepsilon}_{r}$ is the relative permittivity of the cable material.
The characteristic impedance of a transmission line or cable is defined as the ratio of the amplitudes of the voltage and current of a single wave propagating along the line. This excludes any reflected waveforms along the length of the line. The characteristic impedance as shown by Schelkunoff (1934) is given by \[{Z}_{0} = \sqrt{\frac{{R} + {j}\cdot\mathit{\omega}\cdot{L}}{{G} + {j}\cdot\mathit{\omega}\cdot{C}}}{,} \tag{2} \] where R, L, G, and C, respectively, represent the per-unit length resistance, inductance, conductance, and capacitance of the line or cable network.
Due to the location of the cable fault or load impedance, the signal encounters a discontinuity in the medium of propagation, and this is seen as an impedance at the location of the fault. This causes a portion of the signal to be absorbed at the location, and the remaining portion is reflected back to the source end. This portion of the signal interferes destructively or constructively with the original signal, and its effect is determined by the nature of the fault.
The amount of signal that is reflected back is calculated based on the reflection coefficient, which is given by Schelkunoff (1934) as \[{\rho} = \frac{{Z}_{L}{-}{Z}_{0}}{{Z}_{L} + {Z}_{0}}, \tag{3} \] where ${Z}_{L}$ is the fault impedance, and ${Z}_{0}$ is the characteristic impedance of the system. This is in accordance with the work of Rachidi and Tkachenko (2008) in the fields of transmission line modeling and electromagnetic field coupling to cables.
If ${Z}_{L} = {\infty}$ (i.e., ${\rho} = + {1}{)}$, then the reflected waveform is equal to the incident waveform, and they superimpose constructively at the point of discontinuity. If ${Z}_{L}{>}{Z}_{0}$ (i.e., ${0}{<}{\rho}{<}{1}{),}$ then a portion of the incident waveform is reflected, and this superimposes constructively over the test signal at the input end. Therefore, the amplitude of the net signal is the sum of the amplitudes of the incident and reflected waveforms.
If ${Z}_{L} = {Z}_{0}$ (i.e., ${\rho} = {0}{),}$ then the incident waveform is unaffected, as no portion of the waveform is reflected. The incident signal is completely absorbed at the point of discontinuity. If ${Z}_{L}{<}{Z}_{0}$ (i.e., ${\rho}{<}{-}{1}{),}$ then a portion of the incident waveform is reflected, and this superimposes destructively over the test signal at the input end. Therefore, the amplitude of the net signal is the difference between the amplitudes of the incident and reflected waveforms.
If ${Z}_{L} = {0}$ (i.e., ${\rho} = {-}{1}{),}$ then the reflected waveform equals the inverse of the incident signal, and they superimpose destructively at the point of discontinuity. Therefore, the amplitude of the net signal is zero.
IS is a method of calculating the impedance of a network as seen from two test electrodes, which are connected across the input of the system under test. The electrodes are kept in contact with the surface of the domain being tested, and a voltage/current is applied. The induced voltages/currents provide a map of the internal conductivity and permittivity in the interface of interest.
The impedance values of electrical transmission lines vary with the frequency. The IS meter used (Agilent 4294 A) provides a frequency sweep of 40 Hz to 110 MHz, which can be used to study the electrical properties of a device under test, particularly the input impedance. This technique measures the impedance of a system over a range of frequencies, and, therefore, the frequency response of the system, including the energy storage and dissipation properties, is revealed.
Often, data obtained by the IS method are expressed graphically in a Bode or Nyquist plot. Using an appropriate transmission line model, the fault impedance can be abstracted from the knowledge of the input impedance. Fault location requires signal processing in the frequency domain.
TDR is a pulse sampling technique that gives information regarding the distributed electrical properties of transmission lines. A TDR system (shown in Fig. 2) implements low-amplitude, high-frequency, fast-rising pulses, which are transmitted along the line, cable, or waveguide under test. The reflectometry equipment also samples the reflected signal amplitudes, which are displayed on a device with a calibrated timescale. In this way, cable impedance changes and discontinuities can be spatially located and assessed.
Fig 2 The TDR technique.
The characteristic impedance, propagation velocity, and reflection coefficient determine the voltage–current relationship of a time-varying signal along the length of the coaxial cable, as reported by Magnusson (1979). The nature of the fault can be calculated from the amplitude and energy of the reflected signal, from which the reflection coefficient can be ascertained, and, in turn, the fault impedance can be calculated.
The distance to the fault can also be determined by the simple expression ${d} = {v}\cdot{t}{/}{2},$ where v is the propagation velocity of the signal in the medium of the cable, as defined earlier, and t is the time of flight, which is the difference between the time instances of the signal injection and its reception at the source end. The propagation velocity of the signal is dependent on the frequency of the incident signal and properties of the medium of the cable/line.
The major drawback with TDR, as reported by Furse et al. (2006), is that it fails in the case of long cable lengths, where the attenuation and dispersion of the signal in the medium become major problems. The attenuation of the signal is proportional to the cable length, and the dispersion of the signal is due to dielectric losses between the two conductors in a cable. Another issue is that cable faults with minute impedance variations cannot be detected using TDR. It is also expensive, as a rapid step-pulse generator and a fast analog-to-digital converter (ADC) is required for processing the signals.
FDR implements a waveform generator, which creates a sequence of sinusoidal waves that is swept over a given bandwidth (from the initial frequency ${f}_{i}$ to the end frequency ${f}_{e})$ with a frequency step size $\Delta{f}{.}$ A coupler is used to separate the received signal from the incident signal. The mixer performs the operation of convolution, which is basically the multiplication of the two signals in the time domain. This results in a signal with two frequency components, which are the sum and difference of the original two frequencies.
The phase shift between the incident and reflected signal can determine the length of the cable and help to locate the cable fault. A low-pass filter is applied to remove the high-frequency components, as only the difference of the frequencies is of interest. An ADC is used to read the mixed output signal. Then, the signals are digitized and sent for processing. The FFT is used to locate the cable fault. This can be performed by locating the frequency of the single spike in the waveform signal which points to the location of the fault.
With digital signal processing, the FDR method is far more accurate than the TDR approach. FDR has the advantage of adjusting the bandwidth of the frequency sweep to cater to the limited bandwidth systems. The main drawback of the method is that it cannot be used to identify the type of cable fault.
With the superimposition of the incident and reflected signals at the point of the input, we subtract the superimposed signal from the incident one to get the reflected waveform alone, which gives us the phase-shift relationship between the two signals. The phase shift is a direct indicator of the fault location. An FFT operation is performed that extracts the culmination of the standing waves of the reflected response (which may include multiple impedance mismatches). The setup is as shown in Fig. 3.
Fig 3 The FDR technique.
The main advantage of FDR over TDR is that the user can program the acquisition system so as to correct the cable loss (determined by the type of cable used) due to attenuation of the signal. Therefore, the length of the cable does not affect the accuracy of this technique. The various advantages of FDR over the conventional TDR and its efficacy to detect both soft and hard faults render it more useful than TDR. As reported by Mukhopadhyay (2011), both of these reflectometry methods are useful in various applications, such as structural testing and diagnosis in transmission cables. For branched networks, he also describes the use of the reflectometry techniques in collaboration with efficient network topology, extracting algorithms to make the technique more practical.
The technique proposed using IS is that the measurement of input impedance of a cable or line network can be used to abstract the fault impedance value. The knowledge of the line modeling is a necessity for this method, as the characteristic impedance (which is frequency dependent) is calculated before determining the fault impedance. As defined by Schelkunoff (1934), the propagation constant of the transmission line with an amplitude attenuation constant ${\alpha}$ and phase lag constant ${\beta}$ is given by ${\alpha} + {j}\cdot{\beta}{:}$ \[{\gamma} = \sqrt{\left({{R'} + {j}\cdot{\omega}\cdot{L'}}\right)\cdot{(}{G'} + {j}\cdot{\omega}\cdot{C'}{)}}{.} \tag{4} \]
The input impedance of a given coaxial cable is first obtained for a sweep range in frequency. Knowing the characteristic impedance of the cable using (2), the propagation constant ${\gamma},$ and the length of the cable l, the fault impedance is then abstracted using the relation \[{Z}_{\text{in}}{(}{l}{)} = {Z}_{0}\times\frac{{[}{Z}_{L} + {Z}_{0}\cdot\tanh{(}{\gamma}\cdot{l}{)]}}{{[}{Z}_{0} + {Z}_{L}\cdot\tanh{(}{\gamma}\cdot{l}{)]}}, \tag{5} \] from which we obtain the value of ${Z}_{L},$ the fault impedance at the point of discontinuity. The input impedance is a periodic function of the frequency, and the period of this function is proportional to the distance to the fault, which is caused by the phase difference between the incident and reflected signals (as in FDR). The longer the distance to the fault, the greater the number of time periods of the signal in the defined frequency range.
To locate the fault in the cable, the pseudofrequency domain is implemented. Bogert et al. (1963) defined the power cepstrum of a signal as the spectrum of the logarithm of the frequency spectrum of the signal. A comparable methodology for fault location in coaxial cables was achieved using the pseudofrequency-domain analysis. Using the FFT, the input impedance is converted to the pseudofrequency domain, and, at the location of a possible fault, a sharp rise is observed (the Dirac delta function of the frequency). The pseudofrequency domain is obtained by the cepstrum of the input impedance signal, which is defined as the inverse Fourier transform of the logarithm of the squared magnitude of the estimated spectrum of a time-varying signal.
The pseudofrequency of the input impedance ${f'}_{0}$ is proportional to the position of the wire fault: \[{f}^{'}_{0} = \frac{l}{{\upsilon}{/}{2}}{.} \tag{6} \]
The value of the velocity of the propagation of a signal in the medium depends on the material in question. The material taken for our study is an RG58 coaxial cable, which has a relative dielectric constant of 2.25. Therefore, the speed of propagation in the cable is 3 × 108/(2.25)1/2, which is 2 × 108 m/s.
The peaks of the FFT occur periodically as shown, and the value of the frequency at which the local maximum is reached is taken into account to determine the fault distance. It is necessary to calculate the range of the low-frequency band before proceeding for this technique. The low-frequency range is fixed based on the distance to the wire or cable fault. If the distance to the fault is large, the low-frequency range will be correspondingly small, as attenuation and dispersion of the signal will be an issue in that case. According to the studies of Shi and Kanoun (2013), the distance to the fault was fixed to a maximum of a tenth of the minimum wavelength $(\text{v}/{f}_{\max})$ of the input signal. Also, the starting boundary of this low-frequency range was fixed at a quarter of the range itself.
Therefore, for a 50-m fault, the maximum frequency was designed as 0.4 MHz, and the upper frequency boundary was calculated as 0.1 MHz. Accordingly, the local peak (maximum) was determined in the region greater than the low-frequency range. For a 20-m fault, the maximum frequency would be designed as 1 MHz, and the upper frequency boundary for this region would be calculated as 0.25 MHz.
Therefore, by calculating the frequency at which the peak value occurs and knowing the velocity of the propagation of the signal in the medium of the cable, the fault can be located very easily using the (6). As observed in Fig. 4, for a fault at 50 m on the cable length, the frequency of the signal at which the first (highest) peak is obtained in the defined frequency range is around 2.001 MHz. Therefore, the distance to the fault can be verified by (6), which gives a calculated fault distance of 49.975 m.
Fig 4 The amplitude of the FFT of the impedance versus frequency.
Open circuits and broken cables are modeled as shunt capacitors in the line, and, in the low-frequency range, the real part (resistance) is neglected. Therefore, the capacitance of the line at the point of the fault is given by the imaginary part of the admittance at each frequency. The value of the capacitance is linearly proportional to the distance of the fault: \[{C} = \frac{1}{{\omega}\times{\text{imaginary}}\left({{\text{Z}}^{\text{open}}}\right)}{.} \tag{7} \]
Similarly, short circuits are modeled as inductors in series with the line, and, in the low-frequency range, the real part (resistance) is neglected. Therefore,, the inductance of the line at the point of the fault is given by the imaginary part of the impedance at each frequency. The value of the inductance is linearly proportional to the distance of the fault: \[{L} = \frac{{\text{imaginary}}\left({{\text{Z}}^{\text{short}}}\right)}{\omega}{.} \tag{8} \]
The input impedance is a direct periodic function of its frequency, and the number of periods of the function depends directly upon the distance of the fault point. Therefore, the larger the distance to the fault, the greater the number of periods of the input impedance. This observation is evident from the given figures and can be easily used to deduce the location of the fault by simple linear extrapolation. Figures 5 and 6 show the effect of the frequency and value of the fault impedance (considered as purely resistive in this case) on the real and imaginary parts of the impedance. The uniform and periodic variation is particularly useful in ascertaining the nature of fault and distance to the fault.
Fig 5 The real part of the impedance measured versus the frequency of the system for (a) a 10-m and (b) a 20-m fault point.
Fig 6 The imaginary part of the impedance measured versus the frequency of the system for (a) a 10-m and (b) a 20-m fault point.
Now, the fault distance can be directly estimated using (6), and the first highest peak in the defined frequency range is chosen as the point of the maximum amplitude of the FFT, which, in turn, corresponds to the location of the cable fault. The results of cable faults located at 20 and 50 m on a given coaxial cable are given in Fig. 7.
Fig 7 The fault location procedure in the pseudofrequency domain: the amplitude of the FFT versus the length to fault for (a) a 20-m and (b) a 50-m fault point.
A program with a graphical user interface was designed using LabVIEW to control and interface with the Agilent 4294A impedance spectrometer analyzer. The frequency and sweep data were programmed onto the instrument. The impedance data were acquired and stored in the PC, where a MATLAB program was implemented for processing the impedance function in the frequency domain and performing the fault analysis procedure. The unknown parameter—namely, the location of the fault (l) can be determined. Also, the value of the fault impedance $({Z}_{L})$ can be determined from the value of the real/imaginary part of the impedance at the starting frequency. The experiment was repeated for a set of seven cable lengths ranging from 2 to 100 m and with various fault impedances. The maximum error obtained was around 4%, and the time of processing was found to be a maximum of 5 s.
Using IS and frequency-domain analysis, an efficient and fast algorithm to deduce the location of a single fault point in a cable was performed. This method can be extended to multiple faults on a single line, as the amplitudes of the FFT of the impedances would be local maxima at the points of fault. The chosen value of the step size in the frequency (df) was 200 kHz after an analysis of the relative accuracies of all possible values. Also, using this value of df, faults that are closer can be more sharply detected by this technique, as the values of the function maxima at which the peaks are detected would be at closer frequencies, that is, a smaller bandwidth in the frequency domain. Therefore, this algorithm has proven to be successful and satisfyingly accurate with a reasonably low processing time.
The first author would like to thank Qinghai Shi, Dipl.-Ing., and Dr. Olfa Kanoun for their support and guidance in the various phases of the project and research work carried out.
Padmanabhan Rajaraman (ashwath.raja@gmail.com) was currently pursuing the final year of his B.Tech. degree in electrical and electronics engineering at the National Institute of Technology, Tiruchirappalli, Tiruchirappalli 620015, India, at the time of the writing of this article.
Qinghai Shi (qinghai.shi@etit.tu-chemnitz.de) earned his M.S. degree in electrical engineering from the Technical University of Dresden, Germany, in 2008. He was a scientist and Ph.D. student with the chair of measurement and sensor technology at the Chemnitz University of Technology, D-09107 Chemnitz, Germany, at the time of the writing of this article.
Olfa Kanoun (olfa.kanoun@etit.tu-chemnitz.de) studied electrical engineering and information technology at the Technical University in Munich from 1989 to 1996, where she specialized in the field of electronics. She was a university professor and the chair of measurement and sensor technology at the Chemnitz University of Technology, D-09107 Chemnitz, Germany, at the time of the writing of this article. She has authored several reputed publications and is recognized for her work in impedance spectroscopy. She is a Senior Member of IEEE.
Digital Object Identifier 10.1109/MPOT.2014.2377272