EXPERIMENTAL DETERMINATION OF THE TIME CONSTANT OF AN EDDY-CURRENT CIRCUIT

A. P. ZAITSEV, A. D. MITAENKO

(Presented by the Scientific and Technical Seminar of the Scientific Research Institute of AEM)

In the process of pulsed regulation of the excitation of electrical machines, accounting for the influence of eddy currents in the massive parts of the magnetic core, which exert a damping effect on the course of electromagnetic transients, becomes of substantial importance [1]. Analytical accounting for this influence involves well-known mathematical difficulties. Therefore one often turns to indirect methods of accounting for the influence of eddy currents, of which the simplest appears to be the method using the equivalent replacement of a set of eddy-current contours by a short-circuited winding with corresponding parameters. Assuming that the main magnetizing circuit is magnetically coupled without losses with the equivalent circuit, one obtains the reduced equivalent circuit of the excitation-winding circuit (Fig. 1) [2], which takes into account the presence of leakage fluxes (the quantity \(L_\sigma\)) and eddy currents (the quantity \(r\)). On the basis of the equivalent circuit of the excitation winding of a dc electrical machine, one can obtain an expression for determining the time constant of the eddy-current circuit \(T_\mathrm{e}\), using experimental data. If a harmonic action \(U \cdot \sin \omega t\) is applied to the input of the excitation-winding circuit represented by the equivalent circuit in Fig. 1, then the expression for the complex resistance of the circuit under consideration is written as follows:

Fig. 1

\[ Z=z\cdot e^{j\varphi}=r_\mathrm{B}+j\omega L_\sigma+\frac{j\omega L_\mu\cdot r_\mathrm{e}}{j\omega L_\mu+r_\mathrm{e}}. \tag{1} \]

In this expression \(r_\mathrm{B}\) is the resistance of the excitation winding;
\(L_\sigma\) is the leakage inductance;
\(L_\mu=L_\mathrm{e}\) is the inductance of the magnetizing circuit;
\(r_\mathrm{e}\) is the resistance of the equivalent eddy-current circuit.

Separating in equality (1) the real and imaginary components, we obtain

\[ Z=r_\mathrm{B}+\frac{\omega^2 T_\mathrm{e} L_\mu}{1+\omega^2 T_\mathrm{e}^{\,2}} +j\omega\left(L_\sigma+\frac{T_\mathrm{e}r_\mathrm{e}}{1+\omega^2 T_\mathrm{e}^{\,2}}\right), \tag{2} \]

where \(T_{\mathrm{e}}=\dfrac{L_{\mu}}{r_{\mathrm{e}}}\) is the time constant of the equivalent eddy-current circuit.

Using the expression for the square of the modulus \(z^2=|Z|^2\), obtained on the basis of equality (2), we determine the value of the time constant \(T_{\mathrm{e}}\):

\[ T_{\mathrm{e}1,2}=\frac{r_{\mathrm{v}}}{h}\pm \sqrt{\frac{r_{\mathrm{v}}^{2}}{h^{2}}+\frac{2L_{\sigma}+L_{\mu}}{h}-\frac{1}{\omega^{2}}}, \tag{3} \]

where

\[ h=\frac{z^{2}-r_{\mathrm{v}}^{2}-\omega^{2}L_{\sigma}^{2}}{L_{\mu}}. \tag{4} \]

On the basis of expression (3), a method may be proposed for determining the value of the time constant of the eddy-current circuit \(T_{\mathrm{e}}\), using experimental data. In this case the excitation winding EW is supplied from series-connected sources of direct and alternating voltages (Fig. 2).

The magnitude of the direct voltage is determined by the specified value of the mean current in the excitation winding, which, in turn, determines the magnitude of the inductance of the excitation winding. The alternating-voltage source is necessary in order to make it possible to measure the modulus of the complex resistance of the excitation winding

\[ z=\frac{U_{m\sim}}{I_{m\sim}}, \]

where \(U_{m\sim}\) is the voltage amplitude of the source \(U\sim\), and \(I_{m\sim}\) is the amplitude of the alternating component of the current in the excitation winding, measured with the aid of the additional resistance \(r_{\mathrm{d}}\).

Fig. 2

In this case the value \(r_{\mathrm{v}}\) in expressions (3) and (4) is taken with allowance for the value \(r_{\mathrm{d}}\), and the values of the inductances \(L_{\mu}\) and \(L_{\sigma}\) are considered known or are calculated by a known method.

Determination of the time constant of the eddy-current circuit \(T_{\mathrm{e}}\) by expression (3) and by methods known from the literature gives satisfactory agreement of the results.

However, the proposed method for determining \(T_{\mathrm{e}}\) is distinguished by its simplicity and by the possibility of taking into account such factors as hysteresis of the magnetic material and the frequency of the alternating component of the excitation current, which are difficult to account for analytically. At the same time, there is no need to determine the geometrical dimensions, the configuration of the magnetic system, or the winding data of the electrical machine. Attention need only be paid to the reliability of determining the value of the inductance \(L_{\sigma}\), and the experiment should be carried out at the appropriate frequency.

REFERENCES

1. M. D. Nakhodkin, G. V. Vasilenko, M. A. Kozorezov, D. M. Lupkin. Design of Traction Electrical Machines. Moscow, “Transport,” 1967.

2. O. V. Slezhanovskii. On taking into account and compensating for the influence of eddy currents in control systems for the excitation flux of electrical machines. “Elektrichestvo,” 1962, No. 9.