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Transcript
Advanced Metal Detectors Dr. Serkan Aksoy 3.2.2. Natural (Mode) Resonance Signatures Finite lossy or lossless objects support natural electromagnetic resonances. In this sense, low frequency (EMI) source magnetic fields create magnetic dipoles in the finite conductive and/or magnetic objects. These dipoles fitting the target object’s dimension tend to resonate1 as similar to electric dipoles excited by radar wave impinges upon a metallic objects. The EMI (natural) resonance frequencies of low-loss dielectric materials are nearly real numbers, but they are purely imaginary for highly conducting materials at low frequencies2. For the latter case, the wavenumber is and converts the Helmholtz equation to the diffusion equation in which no √ wave phenomenon (no reflections) is present3. Therefore, the purely imaginary frequencies correspond to diffusive (signal) decay modes, physically. This needs existence of a “generalized sense of a wavelength” at resonance where the corresponding wavenumber is , , is a (characteristic) object dimension along the direction of induced magnetic currents [Shubitidze et al, 2002]. At low frequencies less than 1 MHz, the skin depth varies signifacantly. Therefore, the natural resonanance dependent decay-constant variability gives chance for discriminations. The natural resonances are used for object identification4 in the manner of Aspect Ratio Detection: Under axial and transverse excitation, the (natural) resonances in a cylinder of aspect ratio are expected when √ √ √ where and are the axial and transverse resonance frequencies for fundamental mode. It is clear that one can infer the aspect ratio ( ) noting and by quadrature peak frequency . Gnereally, the lowest mode dominates also for complex structures. Therefore, above-mentioned consideration is enough well for extraction of aspect ratio. In the case of high interior region comparing to low (generally ) exterior region, the interior depress (a kind of saturation) magnetic field inside the object. It means that the magnetic field will be confined within the object having low values at the ends. This resonance pattern is important to reveal object geometry. This phenomenon does not occur for weakly or non-magnetic objects. Increasing shifts the transverse reponse higher, the axial response lower. Objects with different degrees of sharpness generally have similar peak frequency patterns [Subitidze et al, 2002]. Target Transfer Function Extraction: In fact, a dielectric resonator model (modal expansion) of the highly conducting materials for the EMI resonant modes is appropriate as similar to the low-loss dielectric materials. The EMI resonance frequencies correspond to first-order poles in the complex frequency plane. Therefore, the target transfer function ( ) can be expressed as ( ) ∑ where ’s are the resonance frequencies with corresponding amplitudes of . They are relatively targetsensor position (aspect) independent. Therefore, they can be used for discrimination [Carin et al, 1999]. For , ( ) , for nonferrous materials and corresponds to low frequency region. It is principally useful for ferrous/nonferrous object discrimination. At high frequency limit, becomes dominant and minimal frequency variation is expected. When , ( ( )) ( ( )). The frequency domain 1 It means that resonant magnetic currents with current element lengths of where is the wavelength within the penetrable object should resonate [Shubitidze et al, 2002]. 2 The corresponding resonance frequencies are characteristic values of the wave number, [Shubitidze et al, 2002]. 3 The complex resonance frequencies ( ) in complex plane correspond to simple poles of scatterer transfer function. It is a set of relaxation curves in time domain ( ). 4 This is also known as “Magnetic Singularity Identification (MSI)”. Advanced Metal Detectors Dr. Serkan Aksoy EMI response is the most effective ( and responses are target dependent) around such frequencies. The EMI resonator supports an infinite number of modes [Carin et al, 1999]. In another form, the transfer function can be represented as ( ) ∑ ( ) ∑ ( ) where . This representation relates to the ( ) in section 2.1.1. By using the inverse Fourier transform, time domain transfer function is found to be ( ) ( ) ∑ ( ) where ( ) is Heaviside step function [Carin et al, 1999]. One can assume that a single (principal) mode is only excited, then ( ) where ( ( )) and ( ( )) are called corresponding late time impulse response is and components, respectively. Then, in time domain, ( ) Resonance Frequencies of Natural Modes: The EMI response of a conducting object can be characterized by (complex) resonance frequencies ( ) of the natural modes. They are determined by characteristic target lengths (spatial separation of scattering centers)5. They are aspect (observation point) and driving (excitation) function independent [Geng et al, 1999]. In principle, depending on the source frequency ( ), if - the field is characterized by nearly real frequencies, the field is evanescent (mode) and characterized by nearly imaginary frequencies where is the dominant mode frequency. In the latter case, the late-time fields decay as ( ) (damped exponetials), where and are the wire resistance and inductance, respectively6 and is known as damping coefficient (decay or time constants) which strongly dependent on target shape and material parameters (conductivity etc.). If the damping coefficient can be measured or calculated, it is possible to use it for target identification. This is also known as “Magnetic Singularity Identification (MSI)”. Opposing to Electromagnetic Magnetic Singularity Identification (EMSI) used for high frequencies (such as GPR), the MSI is less influenced by the soil properties due to low frequencies [Geng et al, 1999]. Geng et al extended concept of the natural resonance frequencies to highly conducting and permeable BOR shaped objects in free space. The decay coefficients and natural mode currents are calculated and validated by measured data. The circuits-based explanations for simple loop target and IE with MoM solution are considered for BOR and more complex object7 of only mode. In the case of two circular wire (multifilament) loops, the scattered magnetic field is formulated over impedance matrix with mutual inductances. Using the scattered magnetic field, the natural resonances of the loop target, corresponding to singularities are calculated for zero and non-zero couplings. The poles of the system are negative reel and correspond to pure exponential damping. For complex shaped objects, MoM solution of surface integral equation for complex natural resonance (natural complex frequencies) and resonant surface current is 5 For example, the resonance frequencies of a conducting wire of length is where and is an integer mode number. 6 The capacitance ( ) is taken zero due to the negligence of the displacement current. 7 There is no closed form solution available for cylinder natural resonances [Geng et al, 1999]. is the speed of light Advanced Metal Detectors Dr. Serkan Aksoy investigated by root (using search algorithm) calculations of MoM impedance matrix. The decay constants are extracted from measured noisy data. In this progress, Cramer-Rao Bound for estimation of the damping constant is investigated [Geng et al, 1999].