Real Time Fault Location Using the Retardation Method

: A new method for short circuit fault location is proposed based on instantaneous signal measurement and its derivatives, and is based on the retardation phenomena. The difference between the times in which a signal is registered in two detectors is used to locate the fault. Although a description of faults in terms of a lumped circuit is useful for elucidating the methods for detecting the fault. This description will not sufﬁce to describe the fault signal propagation hence a distributed models is needed which is given in terms of the telegraph equations. Those equations are used to derive a transmission line transfer function, and an exact analytical description of the short circuit signal propagating in the transmission line is obtained. The analytical solution was veriﬁed both by numerical simulations and experimentally.


Introduction
One of the most important results of special relativity is the fact that no signal can 12 travel faster than the speed of light in vacuum c [1]. The same is true for a signal generated 13 from a fault occurring in a power transmission line such as short or a disconnection. As 14 the signal due to the fault reaches detectors along the line at different times (due to finite 15 propagation speed), one can use the differences in the time of arrival to locate the fault 16 along the line. This location technique is passive in the sense that one does not need to 17 inject any signal to the power line in order to locate the fault, rather the fault itself is the 18 source of the signal. Another advantage is that the detection and location are done in real 19 time. Still several issues are raised regarding the proposed technique: 20 • What is the signal velocity, and what is the needed sampling rate? 21 • What should be measured (voltage and/or current)? 22 • How many detectors are needed? 23 • What are the dispersion effects on the prorogation of the signal and how do they effect 24 accuracy?

25
• What are the best practices for signal processing in order to obtain an accurate time of 26 arrival. 27 • How does the current technique compare in terms of accuracy with previous art? 28 We will try to answer those questions in the current paper. The discussion will be 29 limited to low voltage transmission lines while the discussion of high voltage transmission 30 lines will be the subject of a future paper. 31 Power transmission lines have a broad range of faults. These fault classifications 32 appear in various previous articles [2 -9]. There are several approaches to fault location 33 algorithms, including various approaches regarding measurements and data processing 34 and their proposed applications. A bridge circuit method [10] employs an adjustable 35 impedance to calculate the location of the fault. M. N. Alam et al. [11] present a method 36 based on surface electromagnetic waves propagating along a transmission line. In M. 37 Aldeen and F. Crusca's study [12], the faults are modelled as unknown inputs, decoupled 38 from the state and output measurements through coordinate transformations, and then 39 estimated via the use of the observer theory. The article by Qais Alsafasfeh et al. [13] 40 presents a method that integrates the symmetrical components technique with principal 41 component analysis (PCA) for fault classification and detection. In another research [14], 42 Petri nets were used to obtain the modeling and location detection of faults in power 43 systems. Another widely used method is that of wavelet transform analysis [15][16][17][18]. We 44 will compare the accuracy of the current method to the accuracy of previous art in the 45 concluding section. 46 The plan of the paper is as follows: First we present the basic idea of the method. 47 Then we provide a description of the fault in terms of a lumped circuit which is useful 48 for elucidating the methods for detecting the fault, here we shall demonstrate the ability 49 to determine the signal arrival time using derivatives. This description will not suffice, 50 however, to describe the fault signal propagation hence a distributed models is needed 51 which will be given in terms of the telegraph equations [19][20][21][22]. After introducing the main 52 formalism and the telegraph equations of the distributed system we take a specific example 53 of a two-wire power line. We will present several models to describe the development 54 of a short and the signal generated in a possible detector due to that short. Those will 55 include exponential, Gaussian and step function forms. For the step function model an 56 inverse Laplace transform will allow us to determine the time dependent signal at the 57 sensor position analytically. At this stage we will compare the analytical and numerical 58 solutions. Next the experimental setup will be described. We will show a high level of 59 conformity of the theoretical and experimental measurements using the appropriate data 60 processing. Finally,we shall determine the system accuracy and compare it with previous 61 methods. 62 63 Consider a fault of unknown location which causes a signal propagating to both 64 sided of the transmission line (see figure 1). The signal is registered by a detector which 65 determines its time of arrival, t 1 for detector 1 and t 2 for detector 2.

Fault detection by retardation
In which d 1 and d 2 are the unknown distance from the fault to detectors 1 and 2 respectively. The total distance between the two detectors is known as: Defining the time difference t di ≡ t 2 − t 1 , the distance from detector 1 to the short may be written as follows: Hence if t di is a random variable of standard deviation σ t di , the corresponding standard deviation of d 1 is: provided we assume that the velocity v is known. Now σ t di can be evaluated as: In which the covariance C t 1 t 2 is given in terms of the following expectation value: If the uncertainty in t 1 is uncorrelated to the uncertainty in t 2 the covariance is null and we have a simplified expression: If σ t 1 = σ t 2 this will result in: Thus, we can rewrite σ d 1 in terms of σ t 1 as follows: Now suppose that the data is sampled at intervals of T, if the signal is detected initially at time t D it means that the signal arrived at any time between t D and t D − T. Since we do not have any information in which time the signal has really arrived, we assume that t 1 is a random variable distributed uniformly in the interval [t D − T, t D ]. The probability density function of t 1 is depicted in figure 2. Since the moments of a uniform distribution function are known, we can easily evaluate the expectation and standard deviation of t 1 as follows: Inserting the result of equation (10) into equation (9) will lead to: Thus the accuracy at which we need to know the location of the fault will determine the sampling rate f S according to the following formula: Thus, if the required distance precision is of about 1 meter, then the time measurement sampling rate should be: f S 6.12 10 7 Hz (14) This is much lower than the clock rate of current computer processors. We will not deal 67 here with additional sources of uncertainty such as the noise level of transmission lines 68 and will leave that for future work.

Signal detection
70 What kind of signal should we measure for fault location, should it be voltage or 71 current? And how should it processed in order to avoid accumulation of a large amount of 72 unnecessary data due to the high sampling rate dictated by equation (14)? We shall try to 73 answer those question using a lumped circuit model as follows [23][24][25]. The schematic description of the short circuit is presented in Figure 3. This is modelled in figure 4 such that the parts of the transmission line before and after the short circuit as a resistor and inductor connected in a series. In order to mathematically analyse this we use the Kirchhoff voltage law and Ohm's law: (15) in which V in is the source voltage, I 1 is the source current, V S is the short voltage, I 2 is the 76 load current, V out is the load voltage. R 1 and L 1 are resistance and inductance before the 77 short, R 2 and L 2 are resistance and inductance after the short and Z is the load impedance. 78

79
In the first step, we ignore the inductance for mathematical simplicity. We also notice that by Ohm's law: where R S (t) is the time dependent short resistance, and I S (t) is the current flowing through the short. According to Kirchhoff's current law:  Thus, the currents in this case can be calculated algebraically as follows: In the current model, the transmission line is a two-wire copper cable; each wire has a diameter d ,and the distance between them is D, the cable is depicted in figure 5. The total cable length is l. Values used for a demonstration are given in table 1. The surface resistance in ohm (Ω) may be written as follows [26]: in the above ω is the angular frequency, f is the frequency, µ c is the magnetic permeability of the material and σ c is the conductivity. Hence, the resistance per unit length [Ω/m] can be written as: 3.25 · 10 18 Ω Figure 6. Short circuit resistance The values used for the cable description appear in table 2. We notice that the value of 80 f = 50 Hz used above is typical for many power lines. However, as the typical duration 81 of the short is miniscule the signal generated by the short will include a broad band of 82 frequencies each suffering a different impedance. This will be dealt with in later part of 83 this paper. However, here we assume for simplicity that the resistance is constant.

84
The short circuit appears at distance l 1 from the input. The short-circuit R S (t) resistance is shown in figure 5. The short is described by a time dependent resistance which is assumed to behave exponentially: The initial resistance R S (0) = R S0 is assumed to be very large and represents the region's 85 air resistance. However, at time t = 0 a short is initiated causing an exponential decrease in 86 the short resistance at a typical duration of τ S = 10 − 100 [ns] which depends on conditions 87 and geometry of the short circuit region. The short circuit parameters are concentrated in 88 Table 3. Thus the current flows through the short instead of the load causing a decrease 89 in the load current. Moreover, since the overall impedance of the circuit is decreased and 90 is now dominated by the impedance of the short the current at the source becomes much 91 higher.

92
The transmission line resistances and the load impedance are described in table 4. The Table 4. Transmission line resistance and the load impedance 625 Ω where We shall now present the calculation 93 results. The source current just before the short is depicted in figure 7 and after the short is 96 depicted in figure 8.

97
Hence the current at the source is a highly sensitive indicator for the occurrence of a 98 short. To avoid storing unnecessary data and for precise timing of the short occurrence one 99 can look at the source current derivative (figure 9) this has a distinctive pulse shape. Hence 100 by taking the derivative of the signal and by fixing a high detection threshold one can avoid 101 recording unnecessary data. On the load side, the current vanishes after the short occurs 102 (Figure 10), hence the load current is also an excellent indicator of the short occurrence.

103
Again we see that on the load side, the current behavior allows for identification of 104 the short by taking the current derivative ( figure 11). This method allows precise timing 105 with the need to store a small amount of data as indicated above.  The short circuit pulse may also be detected by voltage measurement. For example, the voltage measured at half a distance between the source and the short will yield a voltage V 1 as follows: This voltage is depicted in figure 12.

108
Again the distinctive pulse shape of the voltage derivative (figure 13) is apparent with 109 the same advantage mentioned before.

110
The voltage at the short vanishes (figure 14), since the resistance approaches zero during the short creation, providing the same behavior that allows the pulse form in the voltage derivative ( figure 15). Similarly, the voltage V 2 at half a distance between the short and the load may be measured (figure 16) and the pulse may be detected using the voltage derivative (figure 17).
Summarizing the results of our first model, we saw that the current and voltage measure-111 ments enable the short pulse detection, indicating the short occurrence. Likewise, it was 112     Figure 18. The initial current shown that detection is possible both on the source and the load side. In order to detect 113 the pulse, a resolution of the τ S order is needed. In the current model, the inductance is ne-114 glected leading to a simplified description, in the next section we will look at the case were 115 induction is taken into account leading to somewhat more complex mathematical analysis. 116 Moreover, in a lumped model there is no pulse propagation along the transmission line 117 and for this purpose we will introduce a distributed model later in this paper. In this model, the transmission line inductance is no longer neglected as in previous section. Therefore, the equations given in (15) become coupled differential equations that can only be solved numerically. The two-wire cable induction can be calculated and is shown in Table 5. The current in the circuit before the short occurs can be calculated analytically and is given, as follows: where This current is depicted in figure 18. We will now study the numerical solutions 121 of equation (15) given the above initial form. The source current is demonstrated in figures 19 (right after the short) and 20 (a long 124 time after the short). In the previous model, the derivative had a pulse shape. In the current 125 model, the first derivative does not exhibit a pulse shape (figure 21), however, the second 126 derivative (figure 22) does exhibit a pulse shape with all the benefits mentioned previously. 127 Current measurements on the load side also provide the short pulse detection ability. The 128 load current is shown in figures 23 (right after the short) and 24 (a long time after the short). 129 Here the current decay is evident. The current's first and second derivatives are shown 130 in figures 25 and 26, respectively. We see that the short may be detected and accurately 131 timed by the current's second derivative, measured on either the source or the load side. 132 Next, we investigate the current at the short itself. It is obvious that it is zero before the 133     The short-circuit pulse may also be detected by voltage measurement between the source and the short. If the voltage is measured at half a distance, due to a high short current, the voltage will be: The voltage and its derivative are shown in figures 29 and 30 respectively. We deduce that 137 for the voltage case a first derivative will suffice for short location even when inductance is 138 not neglected. Figure 31 and 32 describe the voltage and its derivative at the short itself. 139 We see that the voltage difference on the short when its resistivity goes to zero is also zero, 140 and the pulse behavior of the voltage derivative is depicted. Analogous results may be obtained if the voltage is measured on the load side, even if the measurement is not on the load itself. For example, for half a distance voltage measurement, one obtains: Figures 33 and 34 show the voltage at half a distance between the short and the load. The 142 voltage in a brief duration after the short is formed is depicted in figure 33 and a longer 143 duration of the same is depicted in figure 34.

144
The voltage derivative displays pulse behavior (figure 35) and thus in this case a first derivative will suffice and a second derivative is not needed. Finally, the load voltage is described, which is proportional to the load current (see also equation (15)): The load voltage, after a brief duration since the short occurrence and later, are shown in 145 figures 36 and 37 respectively.

146
The first and the second load voltage derivatives are shown in figures 38 and 39. In 147 this case the desirable pulse shape is attained for the second derivative. The high sampling rate which is needed for accurate location as described in equation 150 (12) imposes a storage challenge as the amount of data accumulated may be prohibitive. 151 However, taking the derivative of the signal which may be voltage or current allows us to 152 over come this obstacle. By choosing a high detection threshold one avoids false positives 153 and allows us to store a relatively small amount of data which is sufficient for detection 154 and location of the short. In some cases a second derivative is required. 155 Generally, the first voltage derivative is enough (except for the load voltage), while in 156 the case of current measurement the second derivative is necessary. 157 We further notice an additional restriction on the required resolution T, in order to avoid the case that the pulse goes undetected, that is between sampling points we need to have a resolution smaller than the pulse duration which is of the same duration as the time it takes the short to form, hence: Now if τ S 10 −8 seconds this will mean that: This limitation is even more restrictive than the one appearing in equation (14) leading to: Obviously a lumped model neglects the effect of spatial distances and hence the effect 158 of signal propagation. To describe the effect of signal propagation properly a distributed 159 model is needed, in such a model we could study the short signal propagation and related 160 phenomena such as dispersion, this is discussed in the following section. Until now, we have ignored the signal propagation in the circuit and assumed that the 163 changes in the voltage and the current occur immediately and simultaneously everywhere. 164 This assumption is not compatible with the theory of special relativity, which states that any 165 signal must propagate with finite velocity, smaller than the speed of light in a vacuum [1]. 166 To describe this behavior, we use the transmission-line propagation model [26], a section of 167 which is depicted in figure 40. 168 This approach leads to the telegraph equations. We will describe this model in the 169 time and frequency domains and draw the relevant conclusion from each presentation. The equations that describe the voltage and the current dependence in the time domain are the telegraphs equations given below [26]: where R is the resistance, L is the inductance, G is the conductance and C is the capacitance, 172 per unit length each (see figure 40). This pattern is repeated indefinitely (see figure 41). The above equations can be solved for an infinite transmission line excited at x = 0 by 174 an excitation V in (t) representing the effect of the short on the voltage. The solution for the 175 voltage is: The above solution describes a voltage signal propagating at a velocity: and an exponential decay with a decay factor of: Assuming the transmission line to be a two-wire cable as described earlier, the following 177 parameters are obtained [26]: where each wire has a diameter d, and the distance between the wires is D. The material between the wires has a permittivity ε, permeability µ and (a very small) conductivity σ. The wire resistance is calculated, as in equation (20), using the surface resistance given in equation (19). The short propagation velocity (34) can be now calculated using the parameters of equation (36), to yield: where, n is the index of refraction around the transmission line. It should be noted that 179 the electromagnetic wave is propagating in the region between the conductors and not in 180 the conductors themselves, where the propagation is much slower and the decay is very 181 strong. We notice the if the two wires are surrounded by air n 1 we recover the velocity 182 of equation (13).

183
Notice that since according to equation (20) and equation (19) the resistance is fre-184 quency dependent as dictated by the skin effect, the resistance term in equation (32) is not a 185 simple multiplication and should be replaced by a convolution. Thus a frequency domain 186 formalism is more adequate to this type of problem as will be described next. In the frequency domain, the telegraph equation takes the form [26]: Combining these two equations we can separate the voltage and current variables, in terms of the following two equations: where we defined: These equations have a solution of the form: The functions V (±) (ω) are derived from the initial conditions. The impedance Z 0 is defined as follows: In the case where the resistivity and the leakage admittance are small enough, such that: we can approximate the impedance: and, the real and imaginary parts of γ take the form: As the frequency rises, the approximation becomes more accurate. Hence for the higher 189 frequency fourier components associated with the short formation this approximation is 190 more effective. Notice that while Re(γ) describes absorption and coincides with the same 191 expression for absorption obtained in equation (33), Im(γ) describes propagation. Steady state shorts are easily analyzed in transmission line theory, here we shall try to 194 elucidate the connection between the transient phenomena of the short appearance and 195 its asymptotic behaviour as a steady state phenomena. Our model is depicted in figure 42. 196 We assume that the short appears at some point (x=0) in the transmission line, in which 197 continuity of voltage and current dictate: We assume that the short current doesn't exist at t = 0; likewise, the short current after a long time can be calculated using the fact that the voltage on the short vanishes at t → ∞.
The second assumption can be formulated as: To obtain the asymptotic short current the following calculation is done. First we use the fact that after a considerable duration the voltage on the short vanishes to obtain the following: This leads to the result: Inserting this result into equation (41) and taking into account that V asymptotic (ω, −L 1 ) is 199 the source voltage leads to the following result: Hence: The asymptotic short current in the frequency domain can be calculated using equation (41) Taking into account V in (t) given in equation (22) (which is equivalent to a sum of delta functions in the frequency domain) the asymptotic short current in time domain is: The second assumption was that the short current vanishes at t = 0: In the current model, this requirement is fulfilled by multiplying the asymptotic expression with some reasonable function that vanishes at t → 0, and approached unity at t → ∞ for example: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 12 January 2022 doi:10.20944/preprints202201.0159.v1 In which u(t) is a step function. The calculation results for the short current on the load 203 side are as follows: Where v is a signal propagation velocity. The current in 205 the load side is affected by three terms each with unique retardation. The source voltage is 206 retarded by a time t d1 while the short current is retarded by a time t d for the direct signal and 207 by t d2 for the same signal reflected by the circuit source. Each retardation time is proportion 208 to the distance it needs to travel from its source and inversely proportional to velocity of 209 propagation. The model also show that the signals suffer attenuation proportional to the 210 distance they travel.

Signal dispersion 212
We now consider the problem of dispersion. This problem is interesting since we would like to know in what ways does the line distort signals propagating on it and in particular how the signal produced by the short is effected. We will start from the definition of γ given in equation (40) and we will assume that G = 0. Consequently, the propagation index is: Now let us investigate the condition: The resistance R(ω) is frequency dependent due to the skin effect and can be calculated from the surface resistance using equation (19) and equation (20) as follows: And thus we obtain the ratio: Thus the approximation given in equation (58) is even better for higher frequency components. Introducing the inductance impedance: we can cast equation (60) in the form: Hence taking into account that R ωL 1, γ(ω) of equation (57) has up to third order the form: This expression can be separated into an imaginary and real parts as follows. The imaginary part of γ takes the form: where we have taken in account equation (60). As equation (65) is linear in ω we conclude that there is no dispersion during the signal propagation which requires non linear phase terms. To appreciate the linearity of Imγ we depict it as function of the frequency in both figure 44 for low frequencies and in figure 45 for high frequencies without making an expansion approximation, the linearity and hence the lack of dispersion are apparent for a wide frequency range. For the real part of γ we obtain: The real part depends on the frequency, but this part is relatively small, compared to the imaginary part. To appreciate the frequency dependence of Reγ we have depicted its behaviour for low frequencies (figure 46), high frequencies (figure 47) and extremely high frequencies (figure 48). To appreciate how small is Reγ with respect to Imγ we plot the ratio of those quantities in figure 49. Another indication to the dominance of Imγ over Reγ is the frequency dependence of |γ| which follows quite closely the linear behaviour of Imγ as depicted in figure 50. Finally we remind ourselves that we assumed zero admittance G = 0 in our calculations. Practically this means that we have neglected the air admittance with respect to the capacitive admittance: To check if this assumption is justified we calculate the ratio of the air admittance to capacitive admittance: where we have used the parameters of table 3 for the air admittance, and equation (36) for 216 the capacitance. The ratio which is quite small become even smaller for higher frequencies 217    figure 51 and thus justifies our initial assumption. The conclusion of this 218 subsection is that dispersion is not significant in the media in which the pulse generate by 219 the short propagates. This will be further elaborated in the next subsection as we study the 220 prorogation of a signal along the line. We assume that at the entrance to a transmission line we are injecting a short signal of the form: Thus we assume that the short generates a pulse signal while the standard voltage is a periodic trigonometric function of frequency ω 0 , which is modulated by the pulse. The pulse is assumed to be Gaussian with a width σ 0 . The same signal in the frequency domain will take the form: this signal is depicted in figure 52. Now since that signal is injected at the entrance it can propagate in only one direction, hence equation (41) will take the form:  We now expand γ around ω 0 till the second order. This will result in: Taking into account equation (70), equation (71) and equation (72) and perform an inverse Fourier transform we arrive at the following expression for a propagating signal: Where the delay time t d is: hence this is approximately the distance divided by the velocity as expected. We notice that the expression: is the group velocity that is the velocity of a wave packet. The width of the signal is: Where: The term γ 0 x signifies the pulse broadening as it propagates along the line. Let us assume that σ 0 ≈ 10 −6 seconds. For a frequency of 1 MHz and distance of one kilometer: hence the dispersion is negligible . However, for a frequency of 1 kHz and a distance of 10 kilometers: γ 0 x ∼ = −6.39 × 10 −12 −2.85 × 10 −17 i ≈ σ 2 0 = 10 −12 (79) the widening is comparable to the initial width. Hence despite the fact that dispersion 223 seems small it accumulates over long distances. junction the signal is transmitted to the bifurcating channels and is also reflected to the original channel in which the short was originally formed. The amount of signal reflected or transmitted is quantified by the reflection R and transmission T coefficients. Those coefficients can in turn be calculated as follows: in which Z 1 is the impedance of the line before the bifurcation junction and Z 2 is the impedance of the line after the bifurcation junction. In case that the signal meets multiple bifurcation junctions along its path, multiple reflection occur as depicted schematically in figure 54. Multiple reflections will result in multiple signal arriving a the detector as depicted schematically in figure 55. We note that reflected signals arrive at the detector later and in reduced amplitude due to the longer path they need to travel and the additional attenuation the signal suffers during propagation (see equation (73)) and reflection. We note that if a line bifurcates into multiple identical lines as in figure 56, the total impedance  at the channel after the bifurcation will be equal to the original impedance Z 0 divided by the number of transmission line N, hence: Which implies according to equation (80) a transmission coefficient of: hence for a bifurcation to a large number of channels the reflection coefficient will tend to 226 one, while for a continuation into a single identical channel there will not be obviously a 227 reflection. 228 We will now examine the reflection effect and see if one can use the reflected signal instead of a second sensor thus reducing the amount of hardware needed in order to implement the method described. First let us look at figure 57, the short occurs between the sensor and the bifurcation point. Thus a signal is propagating from the short to the sensor and an additional signal propagates to the bifurcation point where it is reflected. Provided that the short will not introduce an impenetrable obstacle the signal will eventually reach the detector at a later time. The direct signal arrival time will satisfy according to equation (1): The reflected signal will arrive at a later time such that: Hence the time difference between the direct and reflected signal allows us to calculate the distance between the short and bifurcation point as:  now since the distance L between the sensor and bifurcation point is known in advance, we may calculate the distance between the short and the bifurcation point as follows: Thus in this case a single detector will suffice and we will not need two detectors as described in section 2. This will reduce the cost of the system and will make redundant issues like sensor synchronization. A last advantageous scenario is depicted in figure 58 The direct signal arrival time will satisfy according to equation (1): The reflected signal will arrive at a later time such that: The time difference in this case will yield: which does not reveal any information on the short location but rather some trivial information on the distance between the sensor and the bifurcation point which is already known. Of course if there exist additional bifurcation points on the signal path (for example left to the short) then we are at the previous case again and one sensor will suffice. We may deduce that putting sensors on bifurcation points will reduce the amount of sensors needed. Finally we look at the case in which the signal arrives to a sensor located after the branching point of the net as in figure 59. The sensor will receive a signal at:  We will now address the transmission line pulse propagation problem using the technique of Laplace analysis [27]. The simplified power system of our interest may be schematically presented in figure 60 . The fault produces two separate signals propagating towards the source and the load, where sensors are located. In each section, as a result of the fault, the voltage perturbation is perceived as an input signal, propagating towards a sensor, as shown in Figure 61. In the transmission line, we assume the validity of the telegraph equations (38) and replace iω → s. The Laplace form of the telegraph equations admits the solution: V(x, s) = V (−) (s)e −xγ(s) + V (+) (s)e xγ(s) (93) where, It is realistic to presume that the conductance G of the separating dielectric material between the wires, is insignificant. The series resistivity is calculated taking into account the skin effect: where d is a conductor wire diameter and: Consequently, the expression (96) can be written as follows: The sensor impedance Z L is designed to be infinite so as not to influence the measurement results. Substituting boundary conditions, and: which implies zero current at x = l: or: Combining the above equation with equation (101) yields: and The voltage signal at the sensor due to the fault is thus: Identifying the last expression as the summation of a geometric series, it can be re-written as the sum: Defining: and the expression for the voltage at the sensor equation (108) takes the form: This sum can be interpreted as the sum of multiple reflected waves each with its unique delay time. The system's transfer function may be now calculated as follows: this allows for obtaining the signal measured at the sensor, due to any fault waveform. For example, if the fault is a sudden short circuit at x = 0 and t = 0, the voltage at the fault location is: Using the superposition principle, V out (t) may be expressed as the sum of a DC input and a step function input response.
Since the impedance at the edge is infinite, the voltage along the line due to the DC input is simply V 0 . Moreover, the Laplace transform of a step function satisfies: hence: Hence using the transfer function definition equation (112), and performing an inverse transform back to the time domain, we obtain: Taking a known inverse Laplace transform [19]: The voltage at the sensor, as a result of fault, is: The above result will suffice if the rise time of the short is fast enough and could be ignored. 235  The scope channel data is exported and processed using the MATLAB software. Our 256 aim is of course to discover how long it takes the signal to arrive from the fault to the 257 measuring point. Because the waves arriving at the edge are reflected in the opposite 258 direction (towards the fault), where they are repeatedly reflected, one fault may supply 259 several data points that we can process to our advantage, depending on how fast the 260 transient signals decay. In this study, it was possible to measure five signals (peaks) each 261 time, and so choose the optimal data processing technique.

262
To obtain the best accuracy, we performed numerous Voltage smoothing and Voltage 263 derivative smoothing changing the time window and thus the number of samples. Obvi-264 ously the wider the window the less noise we have to distort our results. However, a wide 265 window may distort the reflection signal as well, compromising our accuracy. We used five 266 different averaging windows and thus obtained five different errors for each reflection. The 267 data is given in the appendix, in the table each line represent a different values of Voltage 268    Figure 67 shows that the voltage derivative peaks are much sharper; therefore, the 271 short circuit fault location is calculated by finding the time intervals between the local 272 extremum points in the voltage derivative at the transmission line edge. In addition, the 273 smoothing filter window may slightly change the results, and the optimum configuration 274 was 200 points in a window. In this situation, the accuracy in the short circuit fault location 275 detection is ±0.005% using the second peak.

276
This can be compared to the theoretical accuracy predicted by equation (11). In the 277 current case the sampling rate is 4 GHz and hence the time between samples is T = 2.5 10 −10 278 seconds and the velocity is v = c n = 1.62 10 8 meters/second. Thus the accuracy may be 279 as high as 0.008 meters which is ±0.008% quite close to the best experimental result. This 280 indicates that the theoretical limit is achievable provided the data is processed correctly.  figure 68. The unprocessed timings of various peaks are correlated with signal reflections and are given in table 6. The same results derived from a theoretical calculation are given in table 7. Obviously the theoretical distance evaluation is better than the experimental, however, processing dramatically improves the situation as we saw in the previous section. The distances can be derived from the time differences as follows:  In the above we denote ∆t i and ∆t i for one direction experimental and theoretical propaga-283 tion times, respectively. As we can see, the propagation times are sufficiently stable and 284 consistent with the theoretical model. Hence, the pulse location information can also be 285 obtained from the voltage signal (without signal derivative if precision is not required).

286
However, there is a difference between theoretical and the experimental graphs. In the 287 experimental data, there are additional peaks in the first reflected waves; after the third 288 (main) peak, the "oscillation" frequency is twice as great as in the theoretical model. This is 289 because of the reflected waves from the middle of the transmission line: in the experimental 290 setup, the line is not homogeneous as can be deduced from the additional peaks we receive. 291 Finally we present the comparison of theoretical and empirical derivatives given in 292 figure 65. The timing of peak derivatives seem to fit much better the theoretical curve and 293 the location errors are much smaller. As noted previously processing may improve the 294 accuracy drastically.

303
In the framework of the current research, a propagation of a short signal in a two-wire 304 cable was investigated. It was shown that the short can be detected either by voltage or 305 current measurements or by taking the voltage/current first or second derivatives. By 306 measuring the voltage or current derivatives, the short can be detected either on the source 307 or the load side. It was also shown that using bifurcations and reflection the amount 308 of detectors may be reduced. In order to achieve a high level of accuracy regarding the 309 short's location, the inter sampling duration should be of the nanosecond order (sampling 310 frequency of GHz). Of course, if there is no significant slope over a predetermined number 311 of samples (say 1,000 samples), there is no need to save them. Hence, the samples may 312 be written as a moving window to some buffer register and deleted if the gradient is not 313 significant. We have also studied the phenomena of dispersion of the short signal and 314 determined in what situations it is significant.

315
Our method is based on the idea that in any system information even in the form of 316 an electromagnetic wave will advance from one side of the system to the other in a fast but 317 finite velocity v stratifying v < c, c is the speed of light in vacuum.

318
When considering steady state we may neglect the propagation phenomena and 319 employ lumped models, however, this is not applicable for short durations as considered 320 here.

321
The accuracy is significantly better in the retardation approach than in various methods 322 mentioned in previous art (neglecting the propagation phenomena). In table 9 the accuracy 323 of the different methods is compared. 324 It is expected that in the future, better wave sampling techniques and devices will be 325 developed and better processing methodologies thus improving accuracy.

326
The above system description was derived for an electrical transmission line as the 327 wave behavior is well-known and there is considerable expertise in making the calculations; 328 however, the same holds true for optical fibers and optical reflections as well as for pipes, 329 such as water, gas or oil pipes and acoustic echoes. The calculations for the fault's location 330 are the same, although the propagation rates are different in the acoustic case. In the 331 optical case, the equation v = c n still holds true. In both cases, underground pipes and 332 underground fiber optic cables can be damaged at hard-to-reach locations, and an accurate 333 determination of the fault's location is helpful in knowing where to start digging.