Method and system for monitoring a condition of electrical cables

Abstract

A method and a system are described for monitoring a condition of an electrical cable. The method comprises applying a broadband signal wave to a first end of the electrical cable, wherein the broadband signal wave is phase and amplitude modulated and acquiring the phase and amplitude modulated broadband signal wave transmitted and reflected by the electrical cable. Signal analyses is performed which results in establishing a relationship between a real and an imaginary part of a phase impedance spectrum Fourier transform of the power spectrum in an interval around at least one of the locations along the cable, and identifying impedance changes in these locations. Analyzing a second order reflection of the broadband signal in said identified fault locations may establish a local degradation severity of an identified fault in the cable insulation in at least one of the locations along the cable.

Claims

1. A method for monitoring a condition of an electrical cable, said method comprising: applying, by a signal generator, a broadband signal wave to a first end of the electrical cable, wherein the broadband signal wave is phase and amplitude modulated by at least a cable impedance of the electrical cable, acquiring, by an acquisition module, at the first end of the cable the phase and amplitude modulated broadband signal wave transmitted and reflected by the electrical cable, calculating, by an analyzer, a complex cable impedance Z.sub.DUT as a function of frequency f specified by an amplitude and a phase, for the acquired reflected broadband signal wave, translating, by the analyzer, the calculated complex cable impedance into a time domain t; calculating, by the analyzer, a frequency f in the time domain t, wherein the frequency f is the fundamental frequency of a pseudo-periodic function of radial frequency w and amplitude A in the time domain t due to wave reflection of the broadband signal wave at a distance d from an end of the cable, and wherein the frequency f is calculated by applying: $f^{}=\frac{{}^{}}{2}=\frac{2d}{v_r{v}_0}$ in which v.sub.0 is the speed of light in a vacuum and v.sub.r, is an estimated relative phase velocity of an electrical signal in the cable; performing, by the analyzer, a power spectrum analysis of both amplitude and phase of the complex cable impedance in the time domain t to find and localize any local degradation to the insulation of the cable; identifying, by the analyzer, frequency components f.sub.1, f.sub.2, f.sub.n in the power spectrum of the time domain t due to wave reflections of the broadband signal wave at locations x.sub.1, x.sub.2, . . . , x.sub.n along the cable, the wave reflections being due to discontinuities in electrical parameters of the cable; calculating, by the analyzer, each of the locations x.sub.i by applying: $x_i=\frac{v_r{v}_0{f}_i^{}}{2};$ establishing, by the analyzer, a relationship between a real and an imaginary part of a phase impedance spectrum Fourier transform of the power spectrum in an interval around at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n along the cable; identifying, by the analyzer, impedance changes in at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n along the cable from said real and imaginary part; displaying, by a display, the identified impedance changes in at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n as a function of cable length on a display for visual representation of the condition of the cable; and analyzing, by the analyzer, the visual representation of the identified impedance changes to identify a cause of the impedance change of the cable in at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n on the cable.

2. The method of claim 1, wherein identifying impedance changes includes identifying at least one of a step up impedance change, a step down impedance change, a high impedance spot or a low impedance spot.

3. The method of claim 1, wherein a high impedance spot is identified when the imaginary part is zero and the real part is positive.

4. The method of claim 1, wherein a low impedance spot is identified when the Imaginary part is zero and the real part is negative.

5. The method of claim 1, wherein a step up impedance change is identified when the real part is zero and the imaginary part is positive.

6. The method of claim 1, wherein a step down impedance change is identified when the real part is zero and the imaginary part is negative.

7. The method of claim 1, further comprising identifying a cable segment with a lower impedance as a step-down impedance change in the beginning of the cable segment followed by a step-up impedance change at the end of the cable segment.

8. The method of claim 1, further comprising identifying a cable segment with a higher impedance as a step-up impedance change in the beginning of the cable segment followed by a step-down impedance change at the end of the cable segment.

9. A method for monitoring a condition of an electrical cable, said method comprising: applying, by a signal generator, a broadband signal wave to a first end of the electrical cable, wherein the broadband signal wave is phase and amplitude modulated by at least a cable impedance of the electrical cable, acquiring, by an acquisition module, at the first end of the cable the phase and amplitude modulated broadband signal wave transmitted and reflected by the electrical cable, estimating/calculating, by an analyzer, a complex cable impedance Z.sub.DUT as a function of frequency f specified by an amplitude and a phase, for the acquired reflected broadband signal wave, translating, by the analyzer, the calculated complex cable impedance into a time domain t; calculating, by the analyzer, a frequency f in the time domain t, wherein the frequency f is the fundamental frequency of a pseudo-periodic function of radial frequency w and amplitude A in the time domain t due to wave reflection of the broadband signal wave at a distance d from an end of the cable, and wherein the frequency f is calculated by applying: $f^{}=\frac{{}^{}}{2}=\frac{2d}{v_r{v}_0}$ in which v.sub.0 is the speed of light in a vacuum and v.sub.r, is an estimated relative phase velocity of an electrical signal in the cable; performing, by the analyzer, a power spectrum analysis of both amplitude and phase of the complex cable impedance in the time domain t to find and localize any local degradation to the insulation of the cable; identifying, by the analyzer, frequency components f.sub.1, f.sub.2, . . . f.sub.n in the power spectrum of the time domain t due to wave reflections of the broadband signal wave at locations x.sub.1, x.sub.2, . . . , x.sub.n along the cable, the wave reflections being due to discontinuities in electrical parameters of the cable; calculating, by the analyzer, each of the locations x.sub.1 by applying: $x_i=\frac{v_r{v}_0{f}_i^{}}{2};$ displaying, by a display, a visual representation of the identified frequency components of the power spectrum as a function of a length of the cable; establishing, by the analyzer, a local degradation severity of an identified fault in the cable insulation in at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n along the cable by analyzing a second order reflection of the broadband signal in said identified fault location; and displaying, by the display, a visual representation of the local degradation severity of the insulation in said identified fault location.

10. The method according to claim 9, further comprising establishing a difference between a height of a first order reflection peak and the height of the second order reflection peak in the power spectrum and to evaluate the cable attenuation in order to normalize the height of any peak at any distance from the termination.

11. The method according to claim 1, further comprising establishing a measure of the condition of the insulation of cable termination by analyzing a termination peak in the power spectrum, comprising establishing a relationship between the difference dy between the two valleys on each side of the termination peak and a height dz of the termination peak and displaying a visual representation of the condition of the cable termination.

12. The method of claim 11, further comprising: estimating, using the analyzer, the estimated relative phase velocity v.sub.r, wherein the estimating the estimated relative phase velocity v.sub.r includes: evaluating at least two resonance frequencies of the complex cable impedance Z.sub.DUT; identifying two consecutive resonance frequency values f.sub.k and f.sub.k+ respectively, of the complex cable impedance Z.sub.DUT; calculating a first value of a relative phase velocity v.sub.r of the cable by applying
v.sub.r=2L(f.sub.k+1f.sub.k)/v.sub.0, where L is the length of the cable; calculating the fundamental frequency f of the cable, in the time domain t using the first relative phase velocity v.sub.r and applying $f^{}=\frac{2L}{v_r{v}_0};$ calculating a second value f of the fundamental frequency f by finding a maximum peak value in the time domain t in a selectable interval around f; and calculating the estimate of the relative phase velocity v.sup.final.sub.r by applying $v_r^{\mathrm{final}}=\frac{2L}{v_0{f}^{}}.$

13. A system for monitoring a condition of an electrical cable, said system comprising: an analyzer for calculating a complex cable impedance Z.sub.DUT as a function of a frequency f specified by an amplitude and phase; a generating circuit for generating a broadband signal wave that is to be phase and amplitude modulated by a cable impedance, the broadband signal wave being applied to a first end of the cable; an acquisition module for acquiring at the first cable end the broadband signal wave phase and amplitude modulated by the cable impedance; a translating circuit for translating the complex cable impedance Z.sub.DUT into a time domain t, an analyzer for calculating a frequency f in the time domain t, wherein the frequency f is the fundamental frequency of a pseudo-periodic function of radial frequency and amplitude A in the time domain t due to wave reflection of the broadband signal wave at a distance d from an end of the cable, and wherein the frequency f is calculated by applying: $f^{}=\frac{{}^{}}{2}=\frac{2d}{v_r{v}_0}$ in which v.sub.0 is the speed of light in a vacuum and v.sub.r is an estimated relative phase velocity of an electrical signal in the cable, wherein said analyzer performs a power spectrum analysis of both amplitude and phase of the complex cable impedance in the time domain t to find and localize any local degradation to insulation of the cable, wherein said analyzer identifies frequency components f.sub.1, f.sub.2, . . . f.sub.n in the power spectrum of the time domain t due to wave reflections at locations x.sub.1, x.sub.2, . . . , x.sub.n along the cable, the wave reflections being due to discontinuities in electrical parameters of the cable, wherein said analyzer calculates each of the locations x.sub.i by applying: $x_i=\frac{v_r{v}_0{f}_i^{}}{2},$ wherein said analyzer establishes a relationship between a real and an imaginary part of a phase impedance spectrum Fourier transform of the power spectrum in an interval around at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n along the cable, wherein said analyzer identifies impedance changes in at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n along the cable from said real and imaginary part, wherein a display displays the identified impedance changes in at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n as a function of cable length on a display for visual representation of the condition of the cable, and wherein said analyzer analyzes the visual representation of the identified impedance changes to identify a cause of the impedance change of the cable in at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n on the cable.

14. The system of claim 13, wherein identifying impedance changes includes identifying at least one of a step up impedance change, a step down impedance change, a high impedance spot or a low impedance spot.

15. The system of claim 13, wherein a high impedance spot is identified when the Imaginary part is zero and the real part is positive.

16. The system of claim 13, wherein a low impedance spot is identified when the Imaginary part is zero and the real part is negative.

17. The system of claim 13, wherein a step up impedance change is identified when the real part is zero and the imaginary part is positive.

18. The system of claim 13, wherein a step down impedance change is identified when the real part is zero and the imaginary part is negative.

19. The system of claim 13, further comprising identifying a cable segment with a lower impedance as a step-down impedance change in the beginning of the cable segment followed by a step-up impedance change at the end of the cable segment.

20. The system of claim 13, further comprising identifying a cable segment with a higher impedance as a step-up impedance change in the beginning of the cable segment followed by a step-down impedance change at the end of the cable segment.

21. The system of claim 13, further comprising establishing a local degradation severity of an identified fault in the cable insulation in at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n along the cable by analyzing a second order reflection of the broadband signal in said identified fault location.

22. The system according to claim 21, further comprising establishing a difference between a height of a first order reflection peak and the height of the second order reflection peak in the power spectrum and normalizing the height difference to a height of a termination peak in the power spectrum.

23. The system according to claim 13, further comprising establishing a measure of the condition of the cable end by analyzing a termination peak in the power spectrum, comprising establishing a relationship between the difference dy between the two valleys on each side of the termination peak and a height dz of the termination peak.

24. The system of claim 13, wherein said analyzer is operable to: evaluate at least two resonance frequencies of the complex cable impedance Z.sub.DUT; identify two consecutive resonance frequency values f.sub.k and f.sub.k+1 respectively, of the complex cable impedance Z.sub.DUT; calculate a first value of a relative phase velocity v.sub.r of the cable by applying
v.sub.r=2L(f.sub.k+1f.sub.k)/v.sub.0, where L is the length of the cable; calculate the fundamental frequency f of the cable using the first relative phase velocity v.sub.r and applying $f^{}=\frac{2L}{v_r{v}_0},$ calculate a second value f of the fundamental frequency f by finding a maximum peak value in the time domain t in a selectable interval around f; and calculate the estimate of the relative phase velocity v.sup.final.sub.r by applying $v_r^{\mathrm{final}}=\frac{2L}{v_0{f}^{}}.$

25. The method according to claim 9, further comprising establishing a measure of the condition of the cable end by analyzing a termination peak in the power spectrum, comprising establishing a relationship between the difference dy between the two valleys on each side of the termination peak and a height dz of the termination peak.

26. The method of claim 25, further comprising: estimating, using an analyzer, the estimated relative phase velocity v.sub.r, wherein the estimating the estimated relative phase velocity v.sub.r includes: evaluating at least two resonance frequencies of the complex cable impedance Z.sub.DUT; identifying two consecutive resonance frequency values f.sub.k and f.sub.k+ respectively, of the complex cable impedance Z.sub.DUT; calculating a first value of a relative phase velocity v.sub.r of the cable by applying
v.sub.r=2L(f.sub.k+1f.sub.k)/v.sub.0, where L is the length of the cable; calculating the fundamental frequency f of the cable, in the time domain t using the first relative phase velocity v.sub.r and applying $f^{}=\frac{2L}{v_r{v}_0};$ calculating a second value f of the fundamental frequency f by finding a maximum peak value in the time domain t in a selectable interval around f; and calculating the estimate of the relative phase velocity v.sup.final.sub.r by applying $v_r^{\mathrm{final}}=\frac{2L}{v_0{f}^{}}.$

27. The method according to claim 1, wherein the cause of the impedance change of the cable in at least one of the locations x.sub.1, x.sub.2, . . . x.sub.n on the cable is due to at least one of the following parameters: temperature increase or temperature decrease; water/moisture intrusion into the cable; decrease in insulation resistance; mechanical impact to the cable; and gamma irradiation.

28. The system according to claim 13, wherein the cause of the impedance change of the cable in at least one of the locations x.sub.1, x.sub.2, . . . , x.sub.n on the cable is due to at least one of the following parameters: temperature increase or temperature decrease; water/moisture intrusion into the cable; decrease in insulation resistance; mechanical impact to the cable; and gamma irradiation.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) Embodiments of the invention will now be described with reference to the following drawings, where:

(2) FIG. 1 shows a graphical representation of a complex line impedance amplitude and phase as a function of frequency according to equation (8);

(3) FIG. 2 shows a functional diagram of a spot detection algorithm;

(4) FIG. 3 is a power spectrum of a phase impedance in a domain of t, where the x-axis has been scaled to distance from cable start (d=300 m);

(5) FIG. 4 is a power spectrum of phase impedance in the domain of t, where the frequency component at x=200 is visible, due to a capacity change of 20 pF in a 30 cm cable segment;

(6) FIG. 5 is a signal resulting from the real part of equation (18), which is a dumped sine wave of radial frequency and amplitude A in the domain of t;

(7) FIG. 6 is a plot of the phase of Z in the domain of t for A=1 in equation (19);

(8) FIG. 7 is a plot of a semi periodic function of the phase function (18);

(9) FIG. 8 is a plot of the components of the function f(x) in equation (28);

(10) FIG. 9 shows an example of a LIRA display of the first three harmonics of equation (29) according to an embodiment of the present invention;

(11) FIG. 10 is a schematic illustration of the effects of C and L on the LIRA aging indicators according to an embodiment of the present invention;

(12) FIG. 11 illustrates the LIRA Signature from a segment of a cable with a higher impedance, and where this segment is longer than the resolution of the LIRA according to an embodiment of the present invention;

(13) FIG. 12 shows a display image of a LIRA Signature for an impedance step-up (5 pF/m, resolution=1.5 m) as illustrated in FIG. 11 from 50 m to 60 m from cable start according to an embodiment of the present invention;

(14) FIG. 13 illustrates the LIRA Signature from a segment of a cable with lower impedance, where the segment is longer than the LIRA resolution according to an embodiment of the present invention;

(15) FIG. 14 shows a display image of a LIRA Signature for an impedance step-down (+5 pF/m, resolution=1.5 m) as illustrated in FIG. 13 from 50 m to 60 m from cable start according to an embodiment of the present invention;

(16) FIG. 15 is a display image of the amplitude in Volt of a cable impedance (real and imaginary parts) spectrum at the step position as a function of dot distance for a single impedance step, negative (SP), and where the spectrum at the impedance step position is purely imaginary and negative according to an embodiment of the present invention;

(17) FIG. 16 shows the complex value of the phase impedance spectrum at the step location shown in FIG. 15, where the complex value is pure imaginary and negative for SP according to an embodiment of the present invention;

(18) FIG. 17 is a display image of the amplitude in Volt of a cable impedance (real and imaginary parts) spectrum at the step position as a function of dot distance for a single impedance step, positive (SP+) and where the spectrum at the impedance step position is purely imaginary and positive according to an embodiment of the present invention;

(19) FIG. 18 shows the complex value of the phase impedance spectrum at the impedance step location shown in FIG. 17, where the complex value is pure imaginary and positive for SP+ according to an embodiment of the present invention;

(20) FIG. 19 illustrates a segment of a cable with a higher impedance and the corresponding LIRA Signature shown below, and where this cable segment is shorter than the LIRA resolution according to an embodiment of the present invention;

(21) FIG. 20 illustrates a segment of a cable with a lower impedance and the corresponding LIRA Signature shown below, and where this cable segment is shorter than LIRA resolution according to an embodiment of the present invention;

(22) FIG. 21 is a display image of the amplitude in Volt of a cable impedance (real and imaginary parts) spectrum at the step position as a function of dot distance (m) for a low impedance spot, as shown in FIG. 20, where the impedance spot is shorter than the resolution of the LIRA (DP); the spectrum at the spot is purely real and negative according to an embodiment of the present invention;

(23) FIG. 22 shows the complex value of the phase impedance spectrum at the impedance step location from FIGS. 20 and 21: where the complex value is pure real and negative for DP according to an embodiment of the present invention;

(24) FIG. 23 is a display image of a LIRA Signature for a DP spot (+5 pF/m for 1 m, resolution=1.5 m) as shown in FIG. 20-22 at a position of 50 m from the start of the cable according to an embodiment of the present invention;

(25) FIG. 24 is a spectrum (real and imaginary parts) of a high impedance spot of the cable as shown in FIG. 25, where the length of the impedance spot is shorter than the resolution of the LIRA (DP+), and where the spectrum at the spot is purely real and positive according to an embodiment of the present invention;

(26) FIG. 25 shows the complex value of the phase impedance spectrum from FIG. 24 at the impedance step location; where the complex value is pure real and positive for DP+ according to an embodiment of the present invention;

(27) FIG. 26 shows the LIRA Signature for a DP+ spot (5 pF/m for 1 m, resolution=1.5 m) as shown in FIGS. 19, 24, 25 at a distance of 50 m from start of the cable according to an embodiment of the present invention;

(28) FIG. 27 shows a display image in which it is shown how a local degradation severity is correlated to the spike heights of the LIRA signature according to an embodiment of the present invention;

(29) FIG. 28 shows a display image of a Balanced termination Signature (BTS) at a cable end for a cable in balancing conditions (good cable) BTS=0, according to an embodiment of the present invention;

(30) FIG. 29 shows a display image of a Balanced termination Signature (BTS) at a cable end for a cable with a high capacitance (low impedance) termination, where (BTS>0), where BTS=Y/Z*100, according to an embodiment of the present invention; and

(31) FIG. 30 shows a display image of a Balanced termination Signature (BTS) at a cable end for a cable with a low capacitance (high impedance) termination, where BTS<0, according to an embodiment of the present invention.

DETAILED DESCRIPTION

(32) Throughout the description and in the claims the term monitor is to be widely interpreted and includes e.g. global/local condition monitoring, real-time diagnosis, and fault detection.

(33) The monitoring and analyzing system according to the invention may be implemented as software modules. The monitoring system may be implemented in a portable hardware system. The hardware comprises an input module for connection to the cable/wire to be monitored. The monitoring and analyzing software may perform the method and provide the analyzing result for the cable/wire automatically. Manual intervention by an operator of the monitoring and analyzer system is however also possible. The LIRA simulator module. LIRA simulator works in frequency domain applying the transmission line equations (Eqs. 1 to 10). In addition to that, it employs a stochastic model to evaluate the uncertainties in the cable electrical parameters due to manufacturing tolerances and environment changes. Cable parameters and load parameters for the actual cable connection are input to the LIRA simulator module. The stochastic model evaluates and applies statistical variations (using a normal distribution with user selected standard deviation) in the electrical parameters (L, C and R) along the cable, due to manufacturing tolerances and environment noise. The LIRA Analyzer module. It can be operated in real or simulation mode. In the first case it takes the input from the acquisition board signal acquiring module, in the second case the input comes from the LIRA simulator module. The LIRA Analyzer is the core of the wire/cable monitoring system. The LIRA Analyzer works both in frequency and time domain, performing the following tasks: Estimate and display the frequency spectrum of the line input impedance. Calculate the resonance frequencies. Resonance frequencies are calculated from the impedance spectrum and correspond to frequency values where the phase is zero. Estimate the cable characteristic impedance Z.sub.DUT. It is also calculated from the impedance spectrum. The characteristic impedance is the value of the impedance amplitude at any local maximum (or minimum) of the impedance phase. Estimate the cable length, if not known. Detect local degradation areas and localize it. Detect load changes. Analyzing and displaying the frequency spectrum in order to identify impedance changes and impedance spots, calculating real and imaginary components of the impedance phase spectrum and analyzing spike heights.
Diagnosis and Localisation of Local Degradation

(34) The line impedance is calculated as the averaged windowed transfer function from the reference signal CH0 to the impedance modulated signal CH1, which result in the calculation of the amplitude and phase of the line impedance Z.sub.DUT as a function of frequency (See Eq. (11)). Once the line impedance is calculated, the cable state is analyzed by the examination of the frequency content of the amplitude and phase components of the complex impedance. This will be explained in the following. Zdut may be calculated in hardware, and in this case we do not see CH0 and CH1.

(35) Eq. (8) is the mathematical expression of the function in FIG. 1. Actually the line impedance Z.sub.d (for a cable at distance d from the end of the cable), is a complex parameter and FIG. 1 shows both the amplitude and phase of it. The pseudo-periodic shape of the phase is due to the periodicity of .sub.d, Eq. (9), that can be rewritten as:
.sub.d=.sub.Le.sup.2de.sup.2jd(12)
where the amplitude is decreasing with d (the cable length) because of the attenuation (the phase is periodic if =0). The period of d (and then of the line impedance phase) is , considering d as the independent variable, or d, considering the independent variable (as in FIG. 1).

(36) Using the expression for the propagation constant from Eq. (6), Eq. (12) can be rewritten as:

(37) $\begin{array}{cc}{}_d={}_L{e}^{-2d}{e}^{\frac{-2jd}{v}}& \left(13\right)\\ {}_d={}_L{e}^{-2d}{e}^{\frac{-4j\mathrm{fd}}{v}}& \left(14\right)\end{array}$
where f is the frequency of the applied signal that travels along a cable of length d and v is the phase velocity of the electrical signal in the cable.

(38) Assuming f as the independent variable and writing the following translations:

(39) $\begin{array}{cc}f.fwdarw.t^{}& \left(15\right)\\ \frac{4d}{v_r{v}_0}.fwdarw.{}^{}& \left(16\right)\end{array}$
where v.sub.r=v/v.sub.0, v.sub.r being the relative phase velocity of the electrical signal in the cable, and v.sub.0 the speed of light in vacuum.
.sub.Le.sup.2d=A(17)

(40) Eq. (14) becomes:
.sub.d=Ae.sup.jt(18)

(41) Equation (18) is the mathematical expression (in complex notation) of a pseudo-periodic function of radial frequency and amplitude A. In the lossless case (=0) A=1, in real life lossy cables is an increasing function of signal frequency, so that amplitude A is a decreasing function of t, resulting in the damped oscillation of FIG. 1. The frequency of this function (in the domain of t) is:

(42) $\begin{array}{cc}{f}^{}=\frac{{}^{}}{2}=\frac{2d}{v_r{v}_0}& \left(19\right)\end{array}$
where f is the fundamental frequency of the phase function in the domain of t due to the wave reflection at distance d (the cable termination). Note that the expression of f has the dimension of time and it is the time elapsed for a wave to reach the termination at the distance d and be reflected back. The Fourier transform (power spectrum) of the impedance phase, in the domain of t, looks e.g. like FIG. 3, where the x-axis has been scaled to d at the fundamental frequency given in Eq. (19). In FIG. 3, the x-axis has been scaled to distance from cable start (d=300 m).

(43) When at a distance x the wave finds a discontinuity in the electrical parameters of the cable (for example a small change in the insulation dielectric), another reflection would be visible from distance x, which would add a new frequency component in the power spectrum of the impedance phase, where the frequency (from Eq. (19)) would be:

(44) $\begin{array}{cc}{f}^{}=\frac{2x}{v_r{v}_0}& \left(20\right)\end{array}$
And so:

(45) 0$\begin{array}{cc}\frac{f^{}}{f^{}}=\frac{d}{x}& \left(21\right)\end{array}$

(46) If the cable length is known, the knowledge of f and f from the power spectrum of the impedance phase (in the domain of t) is sufficient for the calculation of the x location:

(47) $\begin{array}{cc}x=d\frac{f^{}}{f^{}}& \left(22\right)\end{array}$

(48) If d is not known, the knowledge of the relative phase velocity v.sub.r (from the cable datasheet or by measuring it on a cable sample of the same type) can be used to calculate the x location based on Eq. (19):

(49) $\begin{array}{cc}x=\frac{v_r{v}_0{f}^{}}{2}& \left(23\right)\end{array}$

(50) The final result is then a spike at any position where a change (even a very small change) of electrical parameters (mainly a dielectric value change) produces a reflected wave of the applied reference signal. This reflection appears as a frequency component in the phase/amplitude spectrum of the line impedance. The frequency of the reflected wave is a linear function of the distance from the cable end to the deviation. FIG. 4 shows a Fourier transform (power spectrum) of phase impedance in the domain of t, where a reflection due to a change of electrical parameters of the cable at location x=200 m from cable start is visible as a frequency component at x=200. This spike is in the case of FIG. 4, due to a capacity change of 20 pF in a 30 cm segment of the cable under test.

(51) When condition monitoring a cable, a number of discontinuities (n) in the electrical parameters of the cable may be present. Each of these discontinuities will appear in the power spectrum as distinct frequency components spikes f.sup.n, and their positions x.sub.n identified as explained above.

(52) In order to have good sensitivity and digital resolution, it is important to operate with the highest possible bandwidth, which is however limited by the cable attenuation. Typical examples of bandwidth are 100 MHz bandwidth for 30 m cables ( ) and 20 KHz bandwidth for cables up to 120 km ( ) Long cables require narrow bandwidths to overcome the increasing cable attenuation, which is a function of frequency.

(53) Global Degradation Monitoring

(54) Monitoring of global changes in the cable condition is provided by: Method 1: Relative Phase Velocity estimation and monitoring. Method 2: CBAC method (Central Band Attenuation Correlation)

(55) As for the local degradation and diagnosis, the method for global degradation is also based on applying a reference signal CH0 to the cable, the reference signal having a frequency bandwidth from w.sub.1 to w.sub.2, which is then phase and amplitude modulated by the cable impedance Z.sub.DUT of the cable under test, providing the signal CH1.

(56) Method 1: The Relative Phase Velocity is calculated by LIRA through a two step process: 1. A first approximate value is estimated using 2 consecutive resonance frequency values in the line impedance, applying the following equations: At any resonance, the cable length L is exactly equal to half wavelength or any multiple of that (this is true when the cable is open ended, but different load reactances can be easily accounted for), or, using Eq. (1):

(57) $\begin{array}{cc}L=\frac{v_r{v}_0}{2f_k}k& \left(24\right)\end{array}$ where L is the cable length, v.sub.0 is the light speed in vacuum, v.sub.r is the relative phase velocity and f.sub.k is the k.sup.th resonance peak frequency. Applying Eq. (24) to two consecutive resonance peaks, we get:
v.sub.r=2L(f.sub.k+1f.sub.k)/v.sub.0.(25) Eq. (25) is used by LIRA to evaluate a first value of v.sub.r, after the estimation of the cable input impedance and the calculation of the resonance frequencies. Note that any load reactance shift would be eliminated by the difference term in the equation. The reason why this value is approximate is that v.sub.r is a slow function of f, but it has been assumed constant in Eq. (25). 2. The value of v.sub.r found in step 1 is used to calculate the approximate value of the fundamental frequency f (domain of t, see description above and FIG. 3), as:

(58) $\begin{array}{cc}{f}^{}=\frac{2L}{v_r{v}_0}& \left(26\right)\end{array}$ LIRA searches the maximum peak f in the Fourier transform (power spectrum) in the domain of t, in a user selectable interval around f. When an accurate value of f (called f) is found from the spectrum, Eq. (19) is solved for v.sub.r as:

(59) $\begin{array}{cc}{v}_r^{\mathrm{final}}=\frac{2L}{v_0{f}^{}}& \left(27\right)\end{array}$ which is the final and accurate value of the phase velocity. The phase velocity decreases with the degradation of the cable insulation. Method 2: CBAC method (Central Band Attenuation Correlation)
.sub.d=Ae.sup.jt(18)

(60) Equation (18) is the equation (in complex notation) of a sine wave of radial frequency and amplitude A (considering only the real part of the complex function in eq. 18), in the domain of t. If the attenuation is not 0, A is a decaying constant and the resulting signal is a dumped sine wave as shown in FIG. 5. FIG. 5 shows the GRC (real part) as a function of t.

(61) Writing eq. (8) for the input line impedance Z using eq. (18) for .sub.d, we get:

(62) $\begin{array}{cc}Z=Z_0\frac{1+A{e}^{-j{}^{}{t}^{}}}{1-A{e}^{-j{}^{}{t}^{}}}& \left(27\right)\end{array}$

(63) Equation (27) is a complex function representing the amplitude and phase of the line impedance Z at distance d from the cable end. Plotting the phase of the line impedance Z in the domain of t when A=1 (zero attenuation and termination open or shorted), provides the graph as shown in FIG. 6.

(64) The plot in FIG. 6 is a square wave of radial frequency (eq. 16) and amplitude /2. If the attenuation is not zero (the real case), so that A is less than 1 and decaying to zero as t moves toward infinity, the shape of the phase of function (16) moves toward a semi periodic function with the same frequency , as shown in FIG. 7.

(65) The reason for this change can be explained if we write the equation of a square wave in Taylor's series:

(66) $\begin{array}{cc}f\left(x\right)=\frac{4}{}\underset{n=1,3,5,\mathrm{.Math.}}{.Math.}\frac{1}{n}\sin \left(\frac{nx}{d}\right)& \left(28\right)\end{array}$

(67) Eq. (28) is the infinite sum of the odd harmonics starting with the fundamental sine wave of period 2d. FIG. 8 plots these components and the final result. Physically speaking, in a transmission line the fundamental sine wave of period 2d correspond to the first reflection from the cable termination. The harmonic components are due to higher order reflections and their amplitude (in the domain of t) decrease with the term 1/n, where n is the harmonic order. For this reason when the attenuation term is not considered (as in eq. (28)), the infinite number of reflections result in a square impedance phase, as shown in FIG. 6 or FIG. 8.

(68) The addition of the attenuation term results in a progressive attenuation of the amplitude of the harmonics, having as a consequence that the fundamental wave increases its importance in respect to the higher harmonics (FIG. 7). Including the attenuation term (and the reflection coefficient, if the termination is not open or shorted), eq. (28) becomes:

(69) $\begin{array}{cc}f\left(x\right)=\frac{4}{}\underset{n=1,3,5,\mathrm{.Math.}}{.Math.}\frac{1}{n}\sin \left(\frac{nx}{d}\right){}_R^n{e}^{-2\mathrm{dn}}& \left(29\right)\end{array}$

(70) Integrating along d and converting in dB units, we arrive at:

(71) $\begin{array}{cc}{}_{\mathrm{dB}\text{/}\mathrm{km}}=\frac{\left({\left({\mathrm{TP}}_1-{\mathrm{TP}}_3\right)}_{\mathrm{dB}}-9.54+20\log \left({}_R^2\right)\right)}{d}250& \left(30\right)\end{array}$

(72) Where TP1 and TP2 are the amplitudes of the fundamental and third harmonics of the impedance phase and a the cable attenuation per unit of length at the middle of the bandwidth. When the cable termination is shorted or open, the reflection coefficient is 1 (or 1), so that the log term in eq. (30) disappears.

(73) Eq. (30) shows that the mid-bandwidth cable attenuation is proportional to the difference of TP1 and TP2, which LIRA can estimate with accuracy. FIG. 9 shows an example of LIRA display of the first 3 harmonics. The terminal peak vs third harmonic is thus a stable indicator of global thermal aging of the cable.

(74) Extensive tests at EPRI and TECNATOM demonstrate that a good correlation exists between the thermal aging condition of a cable (especially for EPR insulated cables) and its mid-band attenuation.

(75) These tests show also that the low frequency attenuation is practically unchanged, the characteristic impedance is almost unchanged (with large standard deviation) and the phase velocity tends to decrease (with large standard deviation). The mid-band attenuation was found to be larger for unaged cables, in all cases.

(76) These results seem to confirm that thermal ageing produce a significant increase in the cable inductance, together with some change in the dielectric capacitance.

(77) In LIRA is normalized to a reference cable length (by default 20 m), to take care of the differences between long and short cables.

(78) Attenuation is given by:

(79) 0$\begin{array}{cc}=K\frac{R}{2}\sqrt{\frac{C}{L}}& \left(31\right)\end{array}$

(80) Where R is the wire resistance (invariant with ageing), C the insulation capacitance and L the cable inductance.

(81) Equation (31) shows that attenuation is a function of the ratio between the insulation capacitance C and the cable inductance L, which exhibit a change with aging. The resultant correlation between aging and a might be complex and not monotonic, while monotonic behavior is an important requirement for an aging indicator.

(82) To solve this problem, the attenuation value is corrected using the characteristic impedance and the phase velocity ratio, both calculated by LIRA.

(83) When is divided by the characteristic impedance Z0 and for the square of the relative phase velocity VR, we have:

(84) $\begin{array}{cc}=K\frac{R}{2}\sqrt{\frac{C}{L}}\sqrt{\frac{L}{C}}{\left(\frac{K^{}}{\sqrt{\mathrm{LC}}}\right)}^2=K^{}{C}^2=\mathrm{CBAC}& \left(32\right)\end{array}$

(85) CBAC is then an indicator sensitive only to changes in the insulation capacitance.

(86) If we multiply, instead for the square of VR:

(87) $\begin{array}{cc}=K\frac{R}{2}\sqrt{\frac{C}{L}}\sqrt{\frac{L}{C}}{\left(\frac{K^{}}{\sqrt{\mathrm{LC}}}\right)}^2=K^{}{L}^{-2}& \left(33\right)\end{array}$

(88) The inverse of it is CBAL, an indicator sensitive only to changes in the cable inductance.

(89) FIG. 10 illustrates the concept of the isolation effects of the insulation capacitance C and the cable inductance L on the LIRA aging indicators.

(90) Spectral Decomposition of Cable Impedance Changes: Real-Imaginary Relationship in Phase Impedance Spectrum to Calculate Spot Direction and Size (the BTS Approach)

(91) As explained in detail above, the Line Resonance Analysis System disclosed in U.S. Pat. No. 7,966,137B2 provides a LIRA Signature. An example of a LIRA signature is shown in FIG. 11. The LIRA Signature tracks changes in cable impedance along the cable. The LIRA Signature is a power spectrum transformed to a function of distance from the beginning of the cable to which the CH0 signal is applied.

(92) As shown in FIG. 11 the LIRA Signature has identified a change in the cable impedance in positions a and b in FIG. 11. The LIRA resolution is shown by the vertical lines in FIG. 11. However, as the LIRA only detects changes in the cable impedance, the LIRA Signature as shown in FIG. 11 is silent about the cable condition between the two cable impedance changes as identified in a and b. As also the impedance changes are within the resolution of the LIRA system, further information of the nature of the impedance change may not be provided by the prior art LIRA system in U.S. Pat. No. 7,966,137B2.

(93) Cable impedance increases (step-up) as a result of a change in one or more of the following parameters: 1. Dielectric capacitance decrease 2. Cable self-inductance increase 3. Conductor or/and shield resistivity increase 4. Insulation conductivity decrease (rare)

(94) The effects 1 and 2 are more sensitive at high frequency, whereas the effects from 3 and 4 are more sensitive at low frequency.

(95) Cable impedance changes in the opposite direction cause an impedance decrease (step-down).

(96) Frequent factors for impedance changes are: Temperature increase (decrease) (dielectric capacitance up (down), impedance down (up)) Water/moisture intrusion (dielectric capacitance up, impedance down) Insulation resistance down (conductivity up, impedance down) Mechanical impact (any direction) Gamma irradiation (normally impedance up)

(97) A problem with the LIRA Signature, as described in U.S. Pat. No. 7,966,137B2, is that a spike in the signature may be caused by any of the following factors listed below. The different factors are identified by an event name. This event name will be used throughout the rest of the description.

(98) The resolution of LIRA is a function of the applied bandwidth and it is about 1.5 m at 100 MHz. The resolution is decreasing as the bandwidth is reduced.

(99) 1. SP+ (single peak, positive): A step-up change in impedance. This situation occurs when there is a stable increase of the cable impedance for a distance longer than the actual resolution. This is illustrated in FIGS. 11 and 12 Feil! Fant ikke referansekilden. FIG. 11 illustrates the LIRA Signature from a segment of a cable with higher impedance, and where this segment is longer than the resolution of the LIRA. At the beginning of the segment there is an impedance change (impedance steps up (SP+)) within the resolution of the LIRA Signature. The impedance step up results in a spike in the LIRA Signature, as the LIRA Signature detects only changes in the impedance along the cable. At the end of the segment, there is another impedance change, where the impedance steps down (SP) and again results in a spike in the LIRA Signature. FIG. 12 shows a display image of a LIRA Signature for an impedance step-up (5 pF/m, resolution=1.5 m) as illustrated in FIG. 11. The impedance step up is represented by the first peak at 50 m and the impedance step down appears at 60 m from cable start.

(100) 2. SP (single peak, negative): A step-down change in impedance. This situation occurs when there is a stable decrease of the cable impedance for a distance longer than the actual resolution of LIRA. FIGS. 13 and 14 show an example of a long segment (relatively to the bandwidth) of lower impedance. The two spikes in FIG. 13 represent the beginning and end of the low impedance segment. At the beginning of the segment there is an impedance change (impedance steps down (SP)) within the resolution of the LIRA Signature. The impedance step-down results in a spike in the LIRA Signature, as the LIRA Signature detects only changes in the impedance along the cable. At the end of the segment, there is another impedance change, where the impedance steps up (SP+) and again results in a spike in the LIRA Signature. As can be seen from the Figures, the LIRA signature in FIG. 14 is identical to the LIRA signature in FIG. 12. Both a cable impedance increase and a cable impedance decrease over a distance longer than the resolution for the actual applied bandwidth of the CH0 signal applied to the cable, result in the same LIRA signature; i.e. that is a SP spike followed by a SP+ spike.

(101) 3. DP+: A short cable segment (shorter than the actual resolution) with a higher impedance. This is what is normally called a spot. In this event the cable impedance is the same before and after the spot, but it is higher inside the short cable segment. See FIG. 19 for an example of high impedance spot and FIG. 26 for the corresponding LIRA signature. As can be seen from FIG. 19 a high impedance spot results in one spike only in the LIRA signature.

(102) 4. DP: A short cable segment (shorter than the actual resolution) with a lower impedance. In this event the cable impedance is the same before and after the spot, but it is lower inside the short cable segment. See FIG. 20 for an example of high impedance spot and FIG. 23 for the corresponding LIRA signature. As can be seen from FIG. 20 a high impedance spot results in one spike only in the LIRA signature.

(103) As can be seen from FIGS. 11, 13, 19 and 20 the different impedance changes described above results in the same spike in the LIRA signature. In summary, there are four event types that the normal LIRA signature thus cannot discriminate: SP+: step-up impedance change. This also includes the final termination, when the cable is open ended (or the attached load has an impedance higher than the cable characteristic impedance). SP: step-down impedance change. This also includes the final termination, when the cable is short ended (or the attached load has an impedance lower than the cable characteristic impedance). DP+: High impedance spike DP: Low impedance spike
The Real-Imaginary Relationship (BTS, Balanced Termination Signature) Approach to Detect the Four Possible Events

(104) A Real-Imaginary Relationship (BTS, Balanced Termination Signature) approach may be used to detect the four possible events listed above (SP+, SP, DP+, DP). As it follows from eq. (8) the line impedance Z.sub.DUT is a complex variable. The shape of the curve that follows from eq. (8) is shown in FIG. 1 for amplitude and phase as a function of frequency.

(105) The present invention provides further analyses of the spikes identified at particular positions in the Fourier transform of the impedance phase, the position of which are calculated by eq. (23) and e.g. shown in FIGS. 3 and 4.

(106) The BTS approach is based on a calculation of a real and an imaginary component of the impedance phase spectrum Fourier transform, at the position of the event.

(107) The step-up and step-down (SP+ and SP) events are characterized by a pure imaginary value of the transform at the event position, as described in FIGS. 15 to 18. While the real component is 0 (Re=0), the imaginary component (Im=+90) is positive for a SP+ event and negative (Im=90) for a SP event.

(108) The high and low impedance spikes are characterized by a close combination of two SP events of opposite sign; SP+ followed by SP, or SP followed by SP+. This results in the imaginary components (+90+90, 90++90) being cancelled out. The result of this is a pure real value of the spectrum transform at the event position with a positive real component for DP+(FIG. 25) and a negative real component for DP(FIG. 22).

(109) The impedance phase spectrum Fourier transform for a DP event is shown in FIG. 21. The spectrum at the spot is purely real and negative. The impedance phase spectrum Fourier transform for a DP+ event is shown in FIG. 24. The spectrum at the spot is purely real and positive.

(110) These calculations may be performed automatically by the LIRA system according to the invention. As is shown in the display images in FIGS. 12, 14, 23 and 26, the impedance spikes in the LIRA signatures are presented in a table on the display image identifying each spike as one of SP+, SP, DP+, DP.

(111) Location Accuracy for SP and DP Events

(112) There are however some differences in the localization process between the two SP steps events (SP+, SP) and the two DP spike events (DP+, DP).

(113) The detected location of a SP+(step-up) event is the cable position where one or more electrical parameters change resulting in an increase in the cable impedance. Similarly, the location of a SP event is the position on the cable where the local impedance decreases.

(114) A DP+(DP) event is the superimposition of a SP+(SP) event followed by a SP(SP+) event at a distance closer than the current resolution resulting from the applied bandwidth of the signal CH1 applied to the cable. For this reason, the detected event location is in the middle of the short segment where the change occurs.

(115) Local Degradation Severity (DNORM Algorithm)

(116) The severity of the local degradation in the cable insulation is correlated to the spike heights in the LIRA signature. However, the spike height for the same damage to the insulation, also depends on the position of this cable degradation (insulation damage) along the cable. The height of the spike in the LIRA signature decreases with the distance of the cable degradation from the beginning of the cable to which the signal CH0 is applied. This means that as the fault moves towards the end of the cable, the height of the spikes decreases. This effect was also automatically compensated for in the analyses performed by the LIRA system as described in U.S. Pat. No. 7,966,137B2 through plot normalization.

(117) Such normalization is a good feature for detecting easily significant spikes, but it is not reliable as a distance compensation as the fault (cable insulation degradation) moves towards the end of the cable. The present invention provides a DNORM algorithm which may be expressed as follows:

(118) Any local spot (change in cable impedance) produces a specular image on the other side of the terminal peak (see FIG. 27). This is due to a second order reflection of the damaged section. The second order spike in the power spectrum is like a new spot of the same severity at a distance 2LSL, where L is the cable length and SL is the spot location on the cable.

(119) Since the distance between the two spots (a local spot and its specular image) is known (after the spot localization), it is possible to draw and calculate a trendline (the straight line between the spots in FIG. 27) that represents how the spike size changes with the distance, due to the cable attenuation. In DNORM the difference between the value of the trend line and the termination spike (spike delta) is normalized to the termination position (TP), regardless of where the real spot is located. This difference (DNORM) represents a measure of the severity of the local degradation for the particular spot.

(120) Termination End

(121) The broadband signal CH0 applied to the cable is reflected at the termination end of the cable. The signal from the termination end provides a spike in the spot signature at a distance from start of the cable being the cable length. Any impedance changes occurring at the end of the cable is thus within the shadow of this termination spike/peak. The present invention using the BTS and DNORM approach as explained above provides the possibility to establish whether the termination end of the cable segment is of good or bad quality. The termination end of the cable provides a spot (DP+ or DP). At the cable termination, the relation between the delta difference dy between the two sides of the termination peak and the peak height dz; BTS=dy/dz*100 is a measure of the condition of the insulation of the cable termination.

(122) FIG. 28 shows a display image of a Balanced termination Signature (BTS) at a cable end for a cable in balancing conditions (good cable) where BTS=0. FIG. 29 shows a display image of a Balanced termination Signature (BTS) at a cable end for a cable with a high capacitance (low impedance) termination, where (BTS>0). FIG. 30 shows a display image of a Balanced termination Signature (BTS) at a cable end for a cable with a low capacitance (high impedance) termination, where BTS<0.

(123) Establishing the condition of the cable end is particularly important in environments as e.g. oil and gas industry, downhole in oil wells, subsea, nuclear power plants, power transmission distribution and other hard to reach locations.

(124) Having described preferred embodiments of the invention it will be apparent to those skilled in the art that other embodiments incorporating the concepts may be used. These and other examples of the invention illustrated above are intended by way of example only and the actual scope of the invention is to be determined from the following claims.