LIRA (Line Resonance Analysis System) is based on transmission line theory, an established and well documented theory that is at the base of two other existing cable fail detection techniques known as “Time Domain Reflectometry” (TDR) and “Frequency Domain Reflectometry” (FDR).
A transmission line is the part of an electrical circuit providing a link between a generator and a load. The behavior of a transmission line depends by its length in comparison with the wavelength λ of the electrical signal traveling into it. The wavelength is defined as:λ=v/f  (1)where v is the speed of the electric signal in the wire (also called the phase velocity) and f the frequency of the signal.
When the transmission line length is much lower than the wavelength, as it happens when the cable is short (i.e. few meters) and the signal frequency is low (i.e. few KHz), the line has no influence on the circuit behavior. Then the circuit impedance (Zin), as seen from the generator side, is equal to the load impedance at any time.
However, if the line length is higher than the signal wavelength, (L≥λ), the line characteristics take an important role and the circuit impedance seen from the generator does not match the load, except for some very particular cases.
The voltage V and the current I along the cable are governed by the following differential equations, known as the telephonists equations:
                                                        d              2                        ⁢            V                                dz            2                          =                              (                          R              +                              j                ⁢                                                                  ⁢                ω                ⁢                                                                  ⁢                L                                      )                    ⁢                      (                          G              +                              j                ⁢                                                                  ⁢                ω                ⁢                                                                  ⁢                C                                      )                    ⁢          V                                    (        2        )                                                                    d              2                        ⁢            I                                dz            2                          =                              (                          R              +                              j                ⁢                                                                  ⁢                ω                ⁢                                                                  ⁢                L                                      )                    ⁢                      (                          G              +                              j                ⁢                                                                  ⁢                ω                ⁢                                                                  ⁢                C                                      )                    ⁢          I                                    (        3        )            where ω is the signal radial frequency, R is the conductor resistance, L is the inductance, C the capacitance and G the insulation conductivity, all relative to a unit of cable length.
These four parameters completely characterize the behavior of a cable when a high frequency signal is passing through it. In transmission line theory, the line behavior is normally studied as a function of two complex parameters. The first is the propagation functionγ=√{square root over ((R+jωL)(G+jωC))}  (4)often written asγ=α+jβ  (5)where the real part α is the line attenuation constant and the imaginary part β is the propagation constant, which is also related to the phase velocity v, radial frequency ω and wavelength λ through:
                    β        =                                            2              ⁢                                                          ⁢              π              ⁢                                                          ⁢              f                        λ                    =                      ω            v                                              (        6        )            
The second parameter is the characteristic impedance
                              Z          0                =                                            R              +                              j                ⁢                                                                  ⁢                ω                ⁢                                                                  ⁢                L                                                    G              +                              j                ⁢                                                                  ⁢                ω                ⁢                                                                  ⁢                C                                                                        (        7        )            
Using (4) and (7) and solving the differential equations (2) and (3), the line impedance for a cable at distance d from the end is:
                              Z          d                =                                            V              ⁡                              (                d                )                                                    I              ⁡                              (                d                )                                              =                                    Z              0                        ⁢                                          1                +                                  Γ                  d                                                            1                -                                  Γ                  d                                                                                        (        8        )            
Where Γd is the Generalized Reflection CoefficientΓd=ΓLe−2γd  (9)and ΓL is the Load Reflection Coefficient
                              Γ          L                =                                            Z              L                        -                          Z              0                                                          Z              L                        +                          Z              0                                                          (        10        )            
In (10) ZL is the impedance of the load connected at the cable end.
From eqs. (8), (9) and (10), it is easy to see that when the load matches the characteristic impedance, ΓL=Γd=0 and then Zd=Z0=ZL for any length and frequency. In all the other cases, the line impedance is a complex variable governed by eq. (8), which has the shape of the curves in FIG. 1 (amplitude and phase as a function of frequency).
Existing methods based on transmission line theory try to localize local cable failures (no global degradation assessment is possible) by a measure of V (equation (2)) as a function of time and evaluating the time delay from the incident wave to the reflected wave. Examples of such methods are found in U.S. Pat. Nos. 4,307,267 and 4,630,228, and in US publications 2004/0039976 and 2005/0057259.
A method and a system for monitoring a condition of an electrical cable by analyzing a multifrequency signal applied to the cable is disclosed in U.S. Pat. No. 7,966,137B2. This disclosed method and system detect impedance changes along the cable. The monitoring system and method disclosed in U.S. Pat. No. 7,966,137B2 is referred to as a LIRA technique (Line Resonance Analysis System). The LIRA technique provides transformation into the domain of the line impedance (both amplitude and phase), also called the domain of t′, and applies frequency analysis in this domain. The steps involved in this process are:                1. Send an extended bandwidth signal through the cable and measure the reflected signal        2. Estimate the line impedance through the entire bandwidth on the basis of the sent and reflected signals        3. Analyse the line impedance to get information about cable properties, global cable condition, local degradation spots.        
LIRA (Line Resonance Analysis System) improves the detection sensitivity and accuracy by analyzing the cable input impedance (see equation (8) and FIG. 1). Local degradation detection and localization, as well as global degradation assessment is provided by:
Noiseless estimation of the line input impedance as a function of frequency (bandwidth 0-X MHz, where X depends on the cable length), and spectrum analysis of the line input impedance to detect and localize degradation spots (see detailed explanation). These steps are explained in detail later.
The LIRA method provides the possibility to detect degradations at an early stage, especially for cables longer than a few kilometers. In this case, LIRA can estimate the location of the challenged part with an estimation error within 0.3% of the cable length.
In addition, a global cable condition assessment is possible, which is important for cable residual life estimation in harsh environment applications (for example nuclear and aerospace applications).
The method described in U.S. Pat. No. 7,966,137B2 has however limitations as to the sensitivity at the cable termination and within areas of the cable where impedance changes have been identified and also with regard to the severity of the cable degradation.