A Solenoid in an Imperfect Dielectric

V. K. Zhukov, V. Yu. Sosnin

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

The eddy-current method can be successfully applied to monitoring the concentration of two-component electrolyte solutions. Theoretically, this problem has been considered in a number of works [1, 2]. However, the authors of these works restrict themselves only to the case in which the solution being monitored is outside the solenoid; in addition, in considering this question they neglect displacement currents in the electrolyte.

In the general case, eddy currents in an electrolyte solution are determined both by displacement currents and by conduction currents. Consequently, it appears possible, by monitoring changes in the parameters of a solenoid caused by immersing it in a solution, thereby to monitor the electrical conductivity and dielectric permittivity of weak solutions.

If there is a one-to-one dependence between the concentration and the conductivity of two-component solutions, then the conductivity value can be used to judge the concentration of the solution.

If the solution flows around the solenoid both inside and outside, then the sensitivity of the solenoid to changes in the solution parameters will be higher than in the case where the solution is located only outside the solenoid.

Let us find the dependence of the total resistance of the solenoid on the parameters of a solution that surrounds the solenoid on all sides and that may be regarded as an imperfect dielectric with dielectric permittivity \(\varepsilon\) and specific electrical conductivity \(\gamma\). To this end, we find the solution of the wave equation

\[ \nabla^2 \vec{H} = -\omega^2 \mu_0 \dot{\varepsilon}\,\vec{H} \tag{1} \]

for the internal and external regions of the solenoid (Fig. 1).

The wave equation contains the complex dielectric permittivity of the solution

\[ \dot{\varepsilon} = \varepsilon - j\frac{\gamma}{\omega}. \tag{2} \]

If the length of the solenoid is many times greater than its diameter, then without substantial error one may assume

\[ \vec{H} = \vec{I}_z H', \]

\[ \vec{E} = \vec{I}_{\varphi} E'. \tag{3} \]

Owing to the symmetry of the system,

\[ \frac{\partial \dot H}{\partial \varphi}=0 \quad \text{and} \quad \frac{\partial \dot H}{\partial z}=0 . \tag{4} \]

Taking into account the simplifying expressions (3) and (4), the wave equation (1) takes the form:

\[ \frac{\partial^2 \dot H}{\partial r^2}+\frac{1}{r}\frac{\partial \dot H}{\partial r}+\kappa^2\dot H=0, \tag{5} \]

\[ \kappa^2=\omega^2\cdot\mu_0\cdot\varepsilon . \tag{6} \]

The general solution of the wave equation in the internal medium will be

\[ \dot H_1=\dot A_1\cdot J_0(\kappa r)+\dot B_1\cdot N_0(\kappa r). \]

and in the external medium—

\[ \dot H_2=\dot A_2\cdot J_0(\kappa r)+\dot B_2\cdot N_0(\kappa r). \]

From the condition that \(\dot H_1\) be finite at \(r=0\), it follows that \(\dot B_1=0\). Then

Fig. 1.

\[ \dot H_1=\dot A_1\cdot J_0(\kappa r). \tag{7} \]

In order that, as \(r\to\infty\), in the conducting external medium \(\dot H_2\) tend to zero, it is necessary to take \(\dot A_2=0\), since \(J_0(\kappa r)\to\infty\) for complex \(\kappa\) and \(r\to\infty\). Thus, we obtain

\[ \dot H_2=\dot B_2\cdot N_0(\kappa r). \tag{8} \]

We shall assume that the diameter of the solenoid is much greater than the diameter of the wire from which it is wound; consequently, the gap between the boundaries of the media, in comparison with the radius of the solenoid \(r_0\), is negligibly small, so that

\[ r_2-r_1\simeq 0. \]

Under this assumption, the electric-field intensity at the inner boundary of the external medium will be equal to:

\[ \dot{E}_2=\frac{\dot{e}_{2.0}}{2\pi\cdot r_0}. \tag{9} \]

where \(\dot{e}_{2.0}\) is the e.m.f. at the boundary layer. The electromotive force induced in the boundary layer is determined by the magnetic flux \(\dot{\Phi}_{\mathrm{in}}\) penetrating the contour formed by this layer:

\[ \dot{e}_{2.0}=-j\omega\cdot \dot{\Phi}_{\mathrm{in}}. \tag{10} \]

We find the magnetic flux \(\dot{\Phi}_{\mathrm{in}}\) by integration

\[ \dot{\Phi}_{\mathrm{in}} = \mu_0\int_s \dot{H}_1\,ds = \mu_0\cdot \dot{A}_1 \int_0^{2\pi} d\varphi \int_0^{r_0} J_0(kr)\cdot r\cdot dr = \]

\[ = 2\pi\cdot\mu_0\cdot \dot{A}_1\cdot \frac{r_0}{k}J_0(kr). \tag{11} \]

Using (10) and (11), we determine the electric-field intensity

\[ \dot{E}_{2.0} = -j\omega\mu_0\cdot \dot{A}_1\cdot \frac{1}{k}J_0(kr_0). \tag{12} \]

From the equation \(\operatorname{rot}\dot{\bar{H}}=j\omega\cdot\varepsilon\cdot\dot{\bar{E}}\) it follows that

\[ \dot{E}_2=-\frac{1}{j\omega\varepsilon}\frac{\partial \dot{H}_2}{\partial r}, \]

whence, after substituting \(\dot{H}_2\) from (8), we obtain

\[ \dot{E}_2=\frac{k}{j\omega\cdot\varepsilon}\dot{B}_2\cdot N_1(kr). \tag{13} \]

At the interface, for \(r=r_2\simeq r_0\), the electric-field intensity will be equal to

\[ \dot{E}_{2.0} = \frac{k}{j\omega\varepsilon}\cdot \dot{B}_2\cdot N_1(kr_0). \tag{14} \]

Equating the right-hand sides of expressions (12) and (14), we obtain the first equation necessary for determining the integration constants \(\dot{A}_1\) and \(\dot{B}_2\):

\[ \dot{B}_2\cdot N_1(kr_0)=\dot{A}_1\cdot J_1(kr_0). \tag{15} \]

From the condition of equality of the magnetic-field intensities at the boundary of the inner medium,
\(\dot{H}_{1.0}=\dot{H}_0-\dot{H}_{2.0}\), we find another equation necessary for determining \(\dot{A}_1\) and \(\dot{B}_2\):

\[ \dot{A}_1\cdot J_0(kr_0)=\dot{H}_0-\dot{B}_2\cdot N_0(kr_0). \tag{16} \]

As a result of the simultaneous solution of equations (15) and (16), we determine the integration constant \(\dot{A}_1\):

\[ \dot{A}_1= \frac{\dot{H}_0}{ J_0(kr_0)+J_1(kr_0)\dfrac{N_0(kr_0)}{N_1(kr_0)} }. \tag{17} \]

Using the found value of the integration constant, we-

expression (11) for the magnetic flux penetrating the solenoid can be represented in the form

\[ \dot{\Phi}_{\mathrm{in}}=\dot{\Phi}_{0}\cdot F(\kappa r_{0}), \tag{18} \]

where

\[ F(\kappa r_{0})= \frac{\dfrac{2}{\kappa r_{0}}} {\dfrac{J_{0}(\kappa r_{0})}{J_{1}(\kappa r_{0})} + \dfrac{N_{0}(\kappa r_{0})}{N_{1}(\kappa r_{0})}}, \tag{19} \]

\[ \dot{\Phi}_{0}=\pi r_{0}^{2}\mu_{0}\dot{H}_{0}. \tag{20} \]

The voltage drop produced on the solenoid when a current \(I_{1}\) flows through it is determined by the total impedance \(z\) of the solenoid and is equal to

\[ U_{c}=I\cdot z=I\cdot R_{0}+j\omega\cdot w\cdot \dot{\Phi}_{\mathrm{in}} \tag{21} \]

where \(R_{0}\) is the dc resistance of the solenoid, and \(w\) is its number of turns.

On the basis of the total-current law one may write

\[ \dot{H}_{0}=\frac{I\cdot w}{l}, \tag{22} \]

after which, from expression (21), taking into account (18) and (22), one can determine the total impedance of a solenoid of length \(l\), immersed in the solution,

\[ Z=R_{0}+jX=R_{0}+j\omega\cdot w^{2}\pi r_{0}^{2}\mu\cdot \frac{1}{l}F(\kappa r_{0}). \tag{23} \]

The impedance of this same solenoid in air is equal to

\[ Z_{0}=R_{0}+jX_{0}=R_{0}+j\omega\cdot w^{2}\pi r_{0}^{2}\mu_{0}\cdot \frac{1}{l}. \tag{24} \]

The change in total impedance caused by immersing the solenoid in the solution,

\[ \Delta Z=j\omega\cdot w^{2}\cdot \pi r_{0}^{2}\frac{\mu_{0}}{l}\,[F(\kappa r_{0})-1], \]

in normalized form will have the form:

\[ \frac{\Delta Z}{X_{0}}=F(\kappa r_{0})-1. \tag{25} \]

Separating the real and imaginary parts in expression (25), we obtain

\[ \frac{\Delta R}{X_{0}}=\operatorname{Re}[F(\kappa r_{0})-1]. \tag{26} \]

\[ \frac{\Delta X}{X_{0}}=\operatorname{Im}[F(\kappa r_{0})-1]. \]

With the aid of formulas (26) one can find the change in the active and reactive components of the total impedance of the solenoid when it is immersed in an electrolyte solution. Analysis of these formulas will make it possible to identify the conditions under which the optimum sensitivity of the solenoid to the parameters of the monitored solution is ensured.

REFERENCES

1. N. N. Shumilovskii, G. G. Yarmolchuk, V. P. Grabovetskii, M. A. Prusov. The Eddy-Current Method for Monitoring Production Parameters, ILIM, 1964.

2. A. A. Desova. Measurement of the concentration of two-component electrolyte solutions by the eddy-current method, Encyclopedia of Measurement, Control, Automation, vol. 3, 1964.