Two-phase electrical conductivity meter for conductive liquids
Abstract Generated abstract
A contactless method is examined for measuring the electrical conductivity of conducting liquids by placing the medium in a traveling magnetic field produced by a two-phase inductor. The paper develops an equivalent circuit for a cylindrical inductor with an external magnetic core, deriving expressions from Maxwell’s equations for complex power, demagnetization and attenuation coefficients, introduced impedances, and the phase impedance as functions of channel geometry, wall conductivity, and liquid conductivity. It also gives conditions for selecting the phase-shifting resistance and capacitance to obtain the desired rotating magnetic field at a specified conductivity. The resulting analysis relates conductivity to measurable winding currents or voltage differences, indicating that the voltage difference can vary approximately linearly with liquid conductivity over a fairly wide range.
Full Text
UDC 538.4
TWO-PHASE ELECTRICAL CONDUCTIVITY METER FOR CONDUCTING LIQUIDS
Yu. B. Volynskii, V. A. Raskolenko
(Presented by the scientific seminar of the Faculty of Physics and Engineering)
A method is considered for contactless measurement of the electrical conductivity of conducting liquids, based on the creation of a traveling magnetic field in the monitored medium. An electrical circuit of a two-phase inductor of a traveling magnetic field is presented, and expressions are given that make it possible to determine the parameters of an equivalent substitution circuit and their relation to the electrical conductivity of the liquid.
Illustrations 1, bibliography 4.
If the monitored medium is placed in a traveling magnetic field produced by a two-phase inductor, then eddy currents are induced in the liquid, causing losses in the secondary system of the inductor. Since the magnitude of the losses depends substantially on the electrical conductivity of the liquid, it becomes possible to measure the monitored parameter by measuring the currents in the inductor windings or the phase shift between these currents [1].
Fig. 1. Electrical circuit of a two-phase electrical conductivity meter.
In the simplest case, a device for contactless measurement of electrical conductivity may consist of a two-phase inductor (Fig. 1), which creates a traveling magnetic field in the channel with the monitored medium. One phase of the inductor is connected directly to an alternating-voltage source, while the other is connected through a phase-shifting \(RC\) circuit. The electrical conductivity is determined from the difference of the voltages applied to the inductor windings.
For the case in which a cylindrical inductor with an external magnetic core creates a traveling magnetic field in a channel with outside diameter \(D_1\) and inside diameter \(D_2\), the solution of Maxwell’s equations [2] gives the following results, making it possible to determine the energy losses both in the entire secondary system and in the layer of liquid metal:
\[ \tilde{S} = jQ_0\, \frac{ \alpha\left[ I_0(\dot{\gamma}R_2)\dot{G}\cdot\dot{\gamma} + I_1(\dot{\gamma}R_2)\dot{H}\cdot\xi \right]\cdot I_0(\alpha R_1) }{ \xi\left[ I_0(\dot{\gamma}R_2)\dot{K}\cdot\dot{\gamma} + I_1(\dot{\gamma}R_2)\dot{L}\cdot\xi \right]\cdot I_1(\alpha R_1) }, \tag{1} \]
\[ Q_0 = \frac{ \pi R_1 l \mu_0 \dot{\delta}_m \omega I_1(\alpha R_1) }{ \alpha I_0(\alpha R_1) }, \tag{2} \]
\[ \widetilde{S}_{II}=jQ_{0II}\, \frac{I_1(\dot{\gamma}R_2)[a\dot{K}I_0(aR_2)+\dot{\xi}\dot{L}I_1(aR_2)]^{2*}I_0(\dot{\gamma}R_2)} {I_1(aR_2)[\dot{\gamma}K I_0(\dot{\gamma}R_2)+\dot{\xi}L I_1(\dot{\gamma}R_2)]^{2}aI_0(aR_2)}, \tag{3} \]
\[ Q_{0II}= \frac{\pi R_2 l \mu_0\delta_m^0\omega a I_0(aR_2)|\dot{M}|^2} {|a\dot{K}I_0(aR_2)+\dot{\xi}\dot{L}I_1(aR_2)|^2}, \tag{4} \]
where \(\widetilde{S}\) is the complex power of the secondary system,
\(S_{II}\) is the complex power of the layer of conducting liquid,
\[
a=\frac{\pi}{\tau},
\]
\(\tau\) is the pole pitch,
\[ \dot{\xi}^{2}=a^{2}+j\mu_0\omega\sigma_I, \tag{5} \]
\[ \dot{\gamma}^{2}=a^{2}+j\mu_0\omega\sigma_{II}, \tag{6} \]
\(\mu_0\) is the magnetic permeability;
\(\sigma_I\) is the electrical conductivity of the channel walls;
\(\sigma_{II}\) is the electrical conductivity of the liquid being monitored;
\(f\) is the frequency, \(\omega=2\pi f\);
\(l\) is the length of the inductor;
\(\delta_m\) is the linear current loading on the inner surface of the inductor;
\(I_0, I_1, K_0, K_1\) are modified Bessel functions of the zeroth and first orders of the first kind and of the zeroth and first orders of the second kind.
\[ \dot{G}=K_1(\dot{\xi}R_2)\cdot I_1(\dot{\xi}R_1)-I_1(\dot{\xi}R_2)\cdot K_1(\dot{\xi}R_1), \tag{7} \]
\[ \dot{H}=K_0(\dot{\xi}R_2)\cdot I_1(\dot{\xi}R_1)+I_0(\dot{\xi}R_2)\cdot K_1(\dot{\xi}R_1), \tag{8} \]
\[ \dot{K}=K_1(\dot{\xi}R_2)\cdot I_0(\dot{\xi}R_1)+I_1(\dot{\xi}R_2)\cdot K_0(\dot{\xi}R_1), \tag{9} \]
\[ \dot{L}=K_0(\dot{\xi}R_2)\cdot I_0(\dot{\xi}R_1)-I_0(\dot{\xi}R_2)\cdot K_0(\dot{\xi}R_1), \tag{10} \]
\[ \dot{M}=I_1(\dot{\xi}R_2)\cdot K_0(\dot{\xi}R_2)+K_1(\dot{\xi}R_2)\cdot I_0(\dot{\xi}R_2). \tag{11} \]
From expressions (2) and (4) it is easy to establish that
\[ X_{\Delta I}=\frac{\omega R_1\mu_0 m W^2}{p}\cdot\frac{I_1(aR_1)}{I_0(aR_1)}, \tag{12} \]
\[ X_{\Delta II}= \frac{\omega R_2\mu_0 m W^2 a^2 I_0(aR_2)I_1(aR_2)|\dot{M}|^2} {p|aI_0(aR_2)\dot{K}+\dot{\xi}I_1(aR_2)\dot{L}|^2} \tag{13} \]
where \(X_{\Delta}\) is the reactive resistance of the winding of the cylindrical inductor due to the magnetic field in the nonmagnetic gap under ideal no-load conditions,
\(X_{\Delta II}\) is the reactive resistance of the inductor winding due to the magnetic field in the zone with radius \(R_2\) for the case when the electrical conductivity of the liquid being monitored is zero;
\(m\) is the number of phases,
\(W\) is the number of turns per phase,
\(p\) is the number of poles.
The resistances introduced into the primary circuit of the inductor by losses in the secondary system can be taken into account as follows:
\[ \Delta Z=jX_{\Delta}(\dot{K}_r-1), \tag{14} \]
where \(\dot{K}_r\) is the demagnetization coefficient [3].
For our case, \(K_r\) is determined as the ratio of the normal components of the induction on the surface of the inductor in the presence of second—
of the system and, under ideal no-load conditions and an unchanged current load, is equal to
\[ \dot K_r = \frac{ a I_0(aR_1)\left[\dot\gamma I_0(\dot\gamma R_2)\dot G+\dot\xi I_1(\dot\gamma R_2)\dot H\right] }{ \xi I_1(aR_1)\left[\dot\gamma I_0(\dot\gamma R_2)\dot K+\dot\xi I_1(\dot\gamma R_2)\dot L\right] }. \tag{15} \]
The effect of losses caused by the presence of a conducting liquid in the channel may be taken into account as follows:
\[ \Delta Z_{\mathrm{II}} = jX_{\Delta \mathrm{II}}\left(\dot K_{r\mathrm{II}}\cdot \dot K_3^{*}-1\right), \tag{16} \]
where \(\Delta Z_{\mathrm{II}}\) is the resistance introduced into the primary circuit of the inductor;
\(\dot K_{r\mathrm{II}}\) is the demagnetization coefficient caused by the layer of conducting liquid;
\(\dot K_3^{*}\) is the attenuation coefficient caused by the layer of conducting liquid.
\(\dot K_{r\mathrm{II}}\) is determined by the ratio of the normal components of the induction on the inner surface of the channel in the presence and absence of the secondary system. The attenuation coefficient \(\dot K_3\) may be determined as the ratio of the tangential components of the induction on the inner surface of the channel in the presence and absence of the secondary system and with an unchanged current load.
In the case under consideration,
\[ \dot K_{r\mathrm{II}} = \frac{ I_1(\dot\gamma R_2)\left[a I_0(aR_2)\dot K+\dot\xi I_1(aR_2)\dot L\right] }{ I_1(aR_2)\left[\dot\gamma I_0(\dot\gamma R_2)\dot K+\dot\xi I_1(\dot\gamma R_2)\dot L\right] }, \tag{17} \]
\[ \dot K_3 = \frac{ \dot\gamma I_0(\dot\gamma R_2)\left[a I_0(aR_2)\dot K+\dot\xi I_1(aR_2)\dot L\right] }{ a I_0(aR_2)\left[\dot\gamma I_0(\dot\gamma R_2)\dot K+\dot\xi I_1(\dot\gamma R_2)\dot L\right] }. \tag{18} \]
The influence of the channel walls is taken into account by the introduced complex resistance, which is determined as follows:
\[ \Delta Z_{\mathrm{I}}=\Delta Z-\Delta Z_{\mathrm{II}}. \tag{19} \]
The total equivalent impedance of the inductor phase is
\[ Z_{\mathrm{eq}}=r+\Delta r+jX_{\Delta}+jX_s+j\Delta X = r_{\mathrm{eq}}+jX_{\mathrm{eq}}, \tag{20} \]
where \(r\) is the resistance of the inductor winding to direct current;
\(X_s\) is the leakage reactance;
\(\Delta r=-K_{rp}X_{\Delta}\) is the introduced active resistance;
\(\Delta X=X_{\Delta}(K_{ra}-1)\) is the introduced reactive resistance;
\(K_{ra}\) is the real part of the demagnetization coefficient;
\(K_{rp}\) is the imaginary part of the demagnetization coefficient.
To ensure a rotating magnetic field in the channel with a conducting medium, it is necessary that, for the specified value of the electrical conductivity \(\sigma_{\mathrm{II}0}\), the conditions [4] be satisfied:
\[ R=X_{\mathrm{eq}0}-r_{\mathrm{eq}0} = X_s+K_{rp0}X_{\Delta}+K_{ra0}X_{\Delta}-r, \tag{21} \]
\[ C=\frac{1}{\omega\left(X_{\mathrm{eq}0}+r_{\mathrm{eq}0}\right)} = \frac{1}{\omega\left(X_s+X_{\Delta}K_{ra0}+r-X_{\Delta}K_{rp0}\right)}. \tag{22} \]
In the general case, for currents of positive and negative sequence in phases A and B we have:
\[ \dot i_{A1}=U\, \frac{ Z_{\mathrm{eq}}+R-\dfrac{j}{\omega C}-jZ_{\mathrm{eq}} }{ 2Z_{\mathrm{eq}}\left(Z_{\mathrm{eq}}+R-\dfrac{j}{\omega C}\right) }, \tag{23} \]
\[ \dot{i}_{A2}=U\frac{Z_{\mathrm{eq}}+R-\dfrac{j}{\omega C}+jZ_{\mathrm{eq}}}{2Z_{\mathrm{eq}}\left(Z_{\mathrm{eq}}+R-\dfrac{j}{\omega C}\right)}, \tag{24} \]
\[ \dot{i}_{B1}=U\frac{Z_{\mathrm{eq}}-\dfrac{1}{\omega C}-j\left(Z_{\mathrm{eq}}+R\right)}{2Z_{\mathrm{eq}}\left(Z_{\mathrm{eq}}+R-\dfrac{j}{\omega C}\right)}, \tag{25} \]
\[ \dot{i}_{B2}=U\frac{Z_{\mathrm{eq}}+\dfrac{1}{\omega C}+j\left(Z_{\mathrm{eq}}+R\right)}{2Z_{\mathrm{eq}}\left(Z_{\mathrm{eq}}+R-\dfrac{j}{\omega C}\right)}. \tag{26} \]
The voltage drop across the winding of phase B is determined, for the positive- and negative-sequence currents, by
\[ \dot{U}_{B1}=U_A\frac{Z_{\mathrm{eq}}-\dfrac{1}{\omega C}-j\left(Z_{\mathrm{eq}}+R\right)}{2\left(Z_{\mathrm{eq}}+R-\dfrac{j}{\omega C}\right)}, \tag{27} \]
\[ \dot{U}_{B2}=U_A\frac{Z_{\mathrm{eq}}+\dfrac{1}{\omega C}+j\left(Z_{\mathrm{eq}}+R\right)}{2\left(Z_{\mathrm{eq}}+R-\dfrac{j}{\omega C}\right)}. \tag{28} \]
The difference between the voltages \(U_A\) and \(\dot{U}_B\) is equal to
\[ \Delta \dot{U}=U_A-\dot{U}_{B1}-\dot{U}_{B2}. \tag{29} \]
If the negative-sequence currents are absent, which corresponds to the specified value of the electrical conductivity of the monitored medium, then \(\dot{U}_B\) is equal to \(U_A\). Analysis of expression (29) shows that a linear dependence between \(\Delta U\) and \(\sigma\) can occur when the electrical conductivity of the liquid varies within fairly wide limits.
REFERENCES
- Yu. B. Volynskii. Method for contactless determination of the electrical conductivity of a liquid. Inventor’s certificate No. 211645, Bulletin No. 8, 1968.
- E. T. Külm, H. I. Yanes. Proceedings of TPI. Tallinn, Series A, No. 231, 1965, 3.
- N. M. Okhremenko. Elektrichestvo, No. 8, 1964, 18.
- Yu. S. Chechet. Electrical Micromachines of Automatic Devices. Moscow, Energiya, 1964.