Download 3.2.2. Natural (Mode) Resonance Signatures

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
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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].