This section focuses on the parametrization of the dipole model, i.e., the MPT and in particular its eigenvalues. The literature offers a variety of options for representing them. Finding these unknown parameter values based on measured EMI data is an inverse problem, which can be solved by mathematical optimization. However, such
2As described above, the size of the object approaches zero asymptotically as the distance from the coils approaches infinity. This also to say that uniformity of the field values is not explicitly required; yet the accuracy of the model increases as the size of the object gets smaller.
2.3. Parametric models for EMI response presentation 13 inverse optimization algorithms are not covered here, but information about the possible algorithms is available in the literature for each parametric model.
The MPT eigenvalues essentially define thetransfer function of the system formed by the input signal, the target object, and the output signal. For example, in a pulsed EMI system the eigenvalues can be thought to define the impulse response of the particular system. The transfer function can be defined using the Laplace transform. The so-called poles and zerosof the Laplace-space transfer function then essentially define the behaviour of the system. The Laplace transform space is also called the complex frequency plane as the poles are defined to be complex valuess=σ+jω. See, e.g., [66] for details.
Using this idea, Baum [67] introduced a methodology called the singularity expansion method (SEM)to represent the EMI response of conductive metallic targets, independent of the exciting signal waveform, in terms of singularities in the Laplace transform plane [68]. In particular, according to Baum, the Laplace-plane poles represent complexnatural frequencies of the target, and reveal its intrinsic properties. According to the established theory, a low-frequency EMI response of highly conducting, permeable objects can be characterized by natural (complex Laplace-plane) frequencies that are real and negative [51, 69]. Geng et al. [70] provide a thorough explanation of the theory and show that each eigenvalue λi can be modeled as a sum ofN Laplace-plane poles, given by
λ(s) =
whereAn is thenth expansion coefficient, andζn is thenth pole. Note that the notation here is altered from the original version given by Geng et al. [70]. According to the authors, one or two of these poles are usually necessary to represent the measured response, whereas Riggs et al. [69] state that most EMI responses of (conductive) objects can be characterized by only two or three poles. For example, Tarokh et al. [71] have used this approach to represent the MPT eigenvalues of CW EMI data. Similarly, Carin et al.
[59] have modeled the pulsed EMI response of finite length metallic cylinders and rings by using two or three poles, and state that the approach can also be applied to general rotationally symmetric targets.
Additionally, real and negative poles, according to the theory [66], correspond to damped exponentials that define how the signal decays as a function of time. According to Baum, the response can be represented as a sum of damped exponentials or exponentially decreasing sinusoids [67]. Therefore, the two representations are equivalent. The time-dependent decay of the receive coil signal is of special interest for pulsed EMI systems.
Hence, a common approach in the literature (see, e.g., Baum [51] and Collins et al. [72]) has been to model the time-domain EMI response of a permeable, conducting target as a sum of damped exponentials, given by
λi(t) =
N
X
n=1
Ane−αnt, (2.7)
whereAn is an amplitude factor that depends on the size of the target and on its distance from the sensor, andαn is a decay parameter [51, 72]. Similarly, Pasion and Oldenburg [73] argue that the time decay behaviour of dipoles along each axis, i.e., each eigenvalue, depends linearly on
λi(t) =κi(t+αi)−ψie−t/γi, (2.8)
where κi, αi, ψi, and γi are the decay parameters, and their values depend on the size, shape, conductivity and permeability of the target object. The authors propose a nonlinear inversion process to be used for finding the parameter values [73]. As an example of this general approach, Geng et al. [70] have shown that a pulsed EMI response of conducting and permeable bodies of revolution (BoR) can be modeled as a sum of damped exponentials, and that the damping constants are strongly dependent on the shape, conductivity, and permeability of the target.
Modeling the wideband frequency response (spectrum) of an EMI signal has also been studied. For example, Gao et al. [43] have used a so-calledMethod of Moments analysis to model the EMI spectra of objects, assuming that they are BoR. The benefit of knowing the EMI spectrum is that different frequencies reveal distinct characteristics of the objects.
For example, Chilaka et al. [74] state that discrimination of thick-walled and thin-walled ferrous cylinders necessitates the use of low frequencies (< 30 Hz). Above these frequencies, wall thickness does not affect the response and distinct cylinders look almost identical [74].
Furthermore, Miller et al. [48] have proposed three parametric models to estimate the EMI spectra of different types of objects. The models are based on analytical solutions found in the literature, namely for a sphere, a cylinder, and multiple conducting loops.
The proposed three-parameter model is for permeable spheres and cylinders, the four-parameter version for wire loops, and the five-four-parameter version for complex targets.
Their results show that the EMI response of most targets can be modeled accurately by using only a few parameters. Furthermore, Bell et al. [62] state that the four-parameter model can be used to successfully present the frequency domain EMI response of a variety of compact objects. The model is given by
f(ω) =R(ω) +jQ(ω) =A{s+(jωυ)ς−2
(jωυ)ς+ 1}, (2.9) whereω is the frequency, A is an amplitude,υ is a response time constant, and whereς determines the width of the response spectrum, andsis a factor controlling the relative magnitudes of response asymptotes at low and high frequencies [62]. Recently, to enable faster inversion, this model has been reduced to a two-parameter version by Ramachandran et al. [75] by using a gradient angle model.
A somewhat similar approach, the discrete spectrum of relaxation frequencies (DSRF) (see, e.g., studies by Wei et al. [76, 77]) is a model that describes the EMI spectrum of an object as a discrete set of pairs{ζK,cK}, whereζk = 1/τkis a relaxation frequency, andτk
is the corresponding relaxation time, andck is the amplitude related to the corresponding frequency. These pairs define the frequency bins of the spectrum. The spectrum can be solved analytically for basic shapes such as spheres and cylinders. The relaxation frequencies are position and rotation invariant, but the amplitudes are not. The DSRF contains information about the shape, size, orientation, permeability, and conductivity of the object, and using it, the frequency spectrum of an EMI signal can be presented by
Ψ(ω) =c0+
where c0 is a shift term, N is the number of relaxations, i.e., the model order, cn are the real spectral amplitudes, andζn the relaxation frequencies. Wei et al. [76, 77] have provided methods for estimating the DSRF parameters. Furthermore, Tantum et al. [78]
2.4. Extensions of the dipole model and representing heterogeneous objects 15