Numerical modeling of magnetic induction tomography using the impedance method

Abstract

This article discusses the impedance method in the forward calculation in magnetic induction tomography (MIT). Magnetic field and eddy current distributions were obtained numerically for a sphere in the field of a coil and were compared with an analytical model. Additionally, numerical and experimental results for phase sensitivity in MIT were obtained and compared for a cylindrical object in a planar array of sensors. The results showed that the impedance method provides results that agree very well with reality in the frequency range from 100 kHz to 20 MHz and for low conductivity objects (10 S/m or less). This opens the possibility of using this numerical approach in image reconstruction in MIT.

Appendix

Using the quasi-static approximations for Maxwell’s equations, Pham and Peyton [12] obtained the distribution of electric field inside and outside a spherical object near a circular coil concentric with the object. This field has only an azimuthal component. From this result, we calculate the current density inside the sphere using the relationship j = σEand the magnetic induction outside the sphere using Faraday’s Law B = (i/ω)∇ × E, where σ is the conductivity of the sphere. The current density is given by Eq. A.1:

$$ j_{\phi } (r,\theta ) = {\frac{\sigma \gamma }{{2\alpha a\sqrt {ar} }}}\sum\limits_{n = 1}^{\infty } {{\frac{2n + 1}{n(n + 1)}}{\frac{{I_{n + 0.5} (\alpha r)}}{{I_{n - 0.5} (\alpha r)}}}\left( {{\frac{a}{{r_{o} }}}} \right)^{n + 1} } P_{n}^{1} \left( {\cos \theta_{o} } \right)P_{n}^{1} (\cos \theta ) $$

(A.1)

where I _n represents the modified Bessel functions of the first kind and P _n are the associated Legendre functions of the first kind. The constants that appear in this equation are: the sphere radius a, the polar coordinates of the coil (r _o , θ _o ) in relation to the sphere center, the propagation constant $\alpha = (1 + i)\sqrt {\omega \mu \sigma /2}$, and the constant $\gamma = i\omega \mu I_{\text{s}} r_{o} sin\theta_{o}$.

The primary field (B _pr and B _pθ ) and secondary field (B _sr and B _sθ ) in radial and polar directions outside the sphere are given by Eqs. A.2–A.5 below:

$$ B_{{{\text{p}}r}} = {\frac{ - i\gamma }{2\omega }}\sum\limits_{n = 1}^{\infty } {{\frac{{{{r^{n - 1} } \mathord{\left/ {\vphantom {{r^{n - 1} } {r_{o}^{n + 1} }}} \right. \kern-\nulldelimiterspace} {r_{o}^{n + 1} }}}}{n(n + 1)}}P_{n}^{1} \left( {\cos \theta_{o} } \right)} {\frac{d}{d(\cos \theta )}}\left[ {sin\theta P_{n}^{1} (\cos \theta )} \right] $$

(A.2)

$$ B_{{{\text{p}}\theta }} = {\frac{ - i\gamma }{2\omega }}\sum\limits_{n = 1}^{\infty } {\left[ {{\frac{{{{r^{n - 1} } \mathord{\left/ {\vphantom {{r^{n - 1} } {r_{o}^{n + 1} }}} \right. \kern-\nulldelimiterspace} {r_{o}^{n + 1} }}}}{n}}P_{n}^{1} \left( {\cos \theta_{o} } \right)P_{n}^{1} (\cos \theta )} \right]} $$

(A.3)

$$ \begin{aligned} B_{{{\text{s}}r}} & = {\frac{ - i\gamma }{2\omega }}\sum\limits_{n = 1}^{\infty } {{\frac{2n + 1}{n(n + 1)}}\left[ {{\frac{{I_{n + 0.5} (\alpha a)}}{{\alpha aI_{n - 0.5} (\alpha a)}}} - {\frac{1}{2n + 1}}} \right]} \\ {\frac{{\left( {{a \mathord{\left/ {\vphantom {a {r_{o} }}} \right. \kern-\nulldelimiterspace} {r_{o} }}} \right)^{n + 1} \left( {{a \mathord{\left/ {\vphantom {a r}} \right. \kern-\nulldelimiterspace} r}} \right)^{n} }}{{r^{2} }}}P_{n}^{1} \left( {\cos \theta_{o} } \right){\frac{d}{d(\cos \theta )}}\left[ {sin\theta P_{n}^{1} (\cos \theta )} \right] \\ \end{aligned} $$

(A.4)

$$ \begin{aligned} B_{{{\text{s}}\theta }} & = {\frac{ - i\gamma }{2\omega }}\sum\limits_{n = 1}^{\infty } {{\frac{2n + 1}{(n + 1)}}\left[ {{\frac{{I_{n + 0.5} (\alpha a)}}{{\alpha aI_{n - 0.5} (\alpha a)}}} - {\frac{1}{2n + 1}}} \right]} \\ {\frac{{\left( {{a \mathord{\left/ {\vphantom {a {r_{o} }}} \right. \kern-\nulldelimiterspace} {r_{o} }}} \right)^{n} \left( {{a \mathord{\left/ {\vphantom {a r}} \right. \kern-\nulldelimiterspace} r}} \right)^{n + 1} }}{{r \, r_{o} }}}P_{n}^{1} \left( {\cos \theta_{o} } \right)P_{n}^{1} (\cos \theta ) \\ \end{aligned} $$

(A.5)