Accounting for Electromagnetic Energy Losses in a Slot-Type Eddy-Current Sensor

D. V. MILYAEV, V. K. ZHUKOV

(Presented by the Scientific and Technical Seminar of the Faculty of Automation and Computer Engineering)

Noncontact monitoring of the geometrical dimensions and electrical conductivity of products is successfully carried out by the eddy-current method. The principle of operation of instruments working by this method is described in the literature in considerable detail. We are interested in monitoring the dimensions of cylindrical nonmagnetic products. In such cases, along with feed-through sensors, eddy-current sensors with an annular magnetic core are currently used [1].

The existing theory of a slot sensor with an annular magnetic core does not take into account energy losses in the magnetic core, nor the occurrence of eddy currents in the sensor winding and the phenomena associated with these energy losses. At high frequencies, the losses mentioned may be significant, which undoubtedly leads to discrepancies between experimental and theoretical data.

Let us determine the energy losses of the electromagnetic field in the sensor winding. To do this, we shall represent the sensor winding as a turn of infinite length, to the ends of which a sinusoidal voltage is applied. The alternating magnetic field produced by the current flowing through the turn induces eddy currents in the winding. The magnetic-field intensity and the current density in the turn are determined by solving the system of electromagnetic-field equations:

\[ \operatorname{rot} \overline{H} = \gamma \overline{E}, \tag{1} \]

\[ \operatorname{rot} \overline{E} = -\mu \mu_0 \frac{d \overline{H}}{dt}. \tag{2} \]

\[ \operatorname{div} \overline{D} = 0, \tag{3} \]

\[ \operatorname{div} \overline{B} = 0. \tag{4} \]

The simultaneous solution of (1), (2), (3), (4) leads to the wave equation

\[ \nabla^2 \overline{H} - i \omega \gamma \mu \mu_0 \overline{H} = 0, \tag{5} \]

in which \(\gamma, \mu\) are the specific electrical conductivity and magnetic permeability of the material of the sensor winding (for copper \(\mu = 1\)). To solve (5), it is necessary to use a cylindrical coordinate system—

In a cylindrical coordinate system the wave equation will have the form

\[ \frac{d^{2}\dot H}{dr^{2}}+\frac{1}{r}\frac{d\dot H}{dr}-i\omega\gamma\mu_{0}\dot H=0. \tag{6} \]

The integral of equation (6) is a Bessel function of the first and second kind of zero order

\[ \dot H=\dot C_{1}J_{0}(x\sqrt{-i})+\dot C_{2}N_{0}(x\sqrt{-i}), \tag{7} \]

where \(\dot C_{1}\) and \(\dot C_{2}\) are constants of integration, \(x\sqrt{-i}\) is the argument of the functions, \(x=\kappa r\), \(\kappa=\sqrt{\omega\gamma\mu_{0}}\).

The current density \(\dot\delta=-\dfrac{d\dot H}{dr}\), in accordance with (7), is equal to

\[ \dot\delta=\kappa\sqrt{-i}\,[\dot C_{1}J_{1}(x\sqrt{-i})+\dot C_{2}N_{1}(x\sqrt{-i})]. \tag{8} \]

The constants of integration are found from the boundary conditions: on the outer surface of the turn the magnetic-field intensity is equal to zero, i.e.

\[ \dot C_{1}J_{0}(x\sqrt{-i})+\dot C_{2}N_{0}(x\sqrt{-i})=0, \tag{9} \]

and on the inner surface the intensity is determined by the current in the turn

\[ \dot H=\dot H_{s}=\dot C_{1}J_{0}(x\sqrt{-i})+\dot C_{2}N_{0}(x\sqrt{-i}). \tag{10} \]

In the last expressions

\[ x_{1}=\kappa r_{1},\quad r_{1}\text{ is the inner radius of the turn,} \]

\[ x_{2}=\kappa r_{2},\quad r_{2}\text{ is the outer radius of the turn,}\quad H_{s}=Iw,\quad I\text{ is the current in the turn.} \]

From (9) and (10), by solving them jointly, \(\dot C_{1}\) and \(\dot C_{2}\) are found. After determining the constants of integration, for the magnetic-field intensity and the current density we find

\[ \dot H=\sqrt{2}I\, \frac{ N_{0}(x_{2}\sqrt{-i})J_{0}(x\sqrt{-i})-J_{0}(x_{2}\sqrt{-i})N_{0}(x\sqrt{-i}) }{ J_{0}(x_{1}\sqrt{-i})N_{0}(x_{2}\sqrt{-i})-N_{0}(x_{1}\sqrt{-i})J_{0}(x_{2}\sqrt{-i}) }. \tag{11} \]

\[ \dot\delta=\kappa\sqrt{-i}\sqrt{2}I\, \frac{ N_{0}(x_{2}\sqrt{-i})J_{0}(x\sqrt{-i})-J_{0}(x_{2}\sqrt{-i})N_{0}(x\sqrt{-i}) }{ J_{0}(x_{1}\sqrt{-i})N_{0}(x_{2}\sqrt{-i})-N_{0}(x_{1}\sqrt{-i})J_{0}(x_{2}\sqrt{-i}) }. \tag{12} \]

According to the equation of electrodynamics

\[ \dot U_{(t)}=\int_{s}\frac{\dot\delta}{\gamma}\,ds+\frac{d}{dt}\int_{\sigma}\dot B\,d\sigma, \tag{13} \]

where \(s\) and \(\sigma\) are the area and the cross section of the turn, respectively. Substituting expressions (11) and (12) into (13) and taking the integral, we obtain

\[ \dot U_{(t)}=\sqrt{2}\,i\omega\mu_{0}\pi r_{1}^{2}I+ \frac{2\sqrt{2}\pi\sqrt{-i}\,x_{1}I}{\gamma}\, \frac{ N_{0}(x_{2}\sqrt{-i})J_{1}(x_{1}\sqrt{-i})- }{ J_{0}(x_{1}\sqrt{-i})N_{0}(x_{2}\sqrt{-i})- } \to \]

On the other hand,

\[ \to \frac{ -J_{0}(x_{2}\sqrt{-i})N_{1}(x_{1}\sqrt{-i}) }{ -N_{0}(x_{1}\sqrt{-i})J_{0}(x_{2}\sqrt{-i}) }. \tag{14} \]

\[ \dot U_{(t)}=I Z_{\mathrm{ob}} \tag{15} \]

Comparing (14) and (15), we find an expression for the total resistance of the sensor winding.

\[ Z_{\mathrm{ob}}=i\omega\mu_0\pi r_1^2\sqrt{2}+ \frac{2\sqrt{2}\pi\sqrt{-i}\,x_1}{\gamma}\times \]

\[ \times \frac{ N_0(x_2\sqrt{-i})J_1(x_1\sqrt{-i})- J_0(x_2\sqrt{-i})N_1(x_1\sqrt{-i}) }{ J_0(x_1\sqrt{-i})N_0(x_2\sqrt{-i})- N_0(x_1\sqrt{-i})J_0(x_2\sqrt{-i}) }. \tag{15} \]

Let us next determine the energy losses in the ferrite magnetic core. In the general case, the energy losses consist of hysteresis losses and eddy-current losses. The magnetization-reversal losses, i.e., hysteresis losses, can be taken into account by introducing the complex absolute magnetic permeability:

\[ \dot{\mu}_c=\frac{\dot{B}_m}{\dot{H}_m}=\mu_c e^{-i\delta}. \tag{17} \]

The wave equation for a ferromagnetic medium in the case of a cylindrical wave, taking (17) into account, takes the form

\[ \frac{d^2\dot{H}}{dr^2}+\frac{1}{r}\frac{d\dot{H}}{dr} -i\omega\gamma_c\dot{\mu}_c\dot{H}=0, \tag{18} \]

where \(\gamma_c\) is the electrical conductivity of the ferromagnetic material. Introduce the notation

\[ -i\omega\gamma_c\dot{\mu}_c=\kappa^2, \]

\[ |\kappa|=\sqrt{\omega\gamma_c|\mu_c|}=M, \tag{19} \]

\[ \Theta=\frac{1}{2}\delta+\frac{\pi}{4}, \]

with which the wave equation will have the form

\[ \frac{d^2\dot{H}}{dr^2}+\frac{1}{r}\frac{dr\dot{H}}{dr} +(Me^{-i\Theta})^2\dot{H}=0. \tag{20} \]

The general solution of this equation is given by Bessel functions of the first and second kinds of zero order,

\[ \dot{H}=\dot{C}_1J_0(\kappa r)+B N_0(\kappa r). \tag{21} \]

It follows from the boundary conditions that \(H=0\) at \(r=0\) and \(H=H_0\) at \(r=a\); therefore \(B=0\),

\[ \dot{C}=\frac{\dot{H}_0}{J_0(\kappa a)}. \tag{22} \]

The distribution of the magnetic-field intensity over the radius is

\[ \bar{\dot{H}}=\bar{I}_z\dot{H}_0 \frac{J_0(Me^{-i\Theta}\cdot r)} {J_0(Me^{-i\Theta}a)}. \tag{23} \]

According to Maxwell’s first equation,

\[ \dot{\bar{E}}=\frac{1}{\gamma}\operatorname{rot}\bar{\dot{H}} \tag{24} \]

or, taking (24) into account,

\[ \dot{\bar{E}}=-\bar{I}_{\varphi}\frac{1}{\gamma_c}\dot{H}_0Me^{-i\Theta} \frac{J_1(Me^{-i\Theta}r)} {J_0(Me^{-i\Theta}a)}. \tag{25} \]

The losses in the core are determined by the flux of the electromagnetic wave that penetrates through the lateral surface. The flux of electromagnetic energy, or the Umov–Poynting vector, averaged over a period

in the case of a sinusoidal variation in time of the components of the electromagnetic field, is determined by the vector product

\[ \bar{\Pi}=\frac{1}{2}\,[\bar{E}\,\bar{H}^{*}], \tag{26} \]

in which \(\bar{H}^{*}\) is the conjugate complex magnetic-field intensity.

The power absorbed by the ferrite core is determined by the total flux of electromagnetic energy through the lateral surface, i.e.,

\[ \dot{P}_{\mathrm{f.c}}=\int_s \bar{\Pi}\,d\bar{s}. \tag{27} \]

For engineering calculations it is convenient to introduce an expression for the power released in the core per unit of its length,

\[ \bar{\Pi} = -\mathbf{1}_r\,\frac{1}{2}\, \frac{\dot{H}_0^{\,2}}{\gamma_c}\, M e^{-i\theta} \frac{J_1\!\left(M e^{-i\theta} r\right)J_0\!\left(M e^{i\theta} r\right)} {J_0\!\left(M e^{+i\theta} a\right)J_0\!\left(M e^{-i\theta} a\right)}. \tag{28} \]

The minus sign in the general expression for the flux indicates the motion of the wave from the larger radius to the smaller one. Since the sign is of no interest to us, it may be omitted:

\[ \dot{P}_{\mathrm{f.c}} = \frac{\dot{H}_0^{\,2}}{\gamma_c}\, \pi a\,M e^{-i\theta} \frac{J_1\!\left(M e^{-i\theta} a\right)} {J_0\!\left(M e^{-i\theta} a\right)}. \tag{29} \]

Part of the energy supplied to the sensor is expended in the ferrite core, i.e.,

\[ \dot{P}_{\mathrm{f.c}}=I^2 \dot{Z}_{\mathrm{f.c}}. \tag{30} \]

From equality of the right-hand sides of expressions (29) and (30) it follows that

\[ \dot{Z}_{\mathrm{f.c}} = \frac{2w_{10}^{\,2}\pi a}{\gamma_c}\, M e^{-i\theta} \frac{J_1\!\left(M e^{-i\theta} a\right)} {J_0\!\left(M e^{-i\theta} a\right)}. \tag{31} \]

where \(w_{10}\) is the number of turns per unit length of the sensor, and \(Z_{\mathrm{f.c}}\) is the resistance of the core per unit length.

The expression obtained takes into account all losses occurring in the core; therefore it can be used to carry out a detailed analysis of the processes taking place in a slot-type eddy-current sensor.

REFERENCES

1. V. K. Zhukov, V. E. Dreizin, I. G. Leshchenko. Induction–impedance sensor with an annular magnetic core, Defektoskopiya, No. 2, 1966.