Engineer-Mechanic R. A. Voronov

Demagnetizing Effect of Alternating Current.

The simultaneous magnetization of an iron core by alternating and direct currents, or else the placement of iron in a magnetic field having direct and alternating components, gives rise to a number of phenomena which have, and may find in practice, various applications. The present article, describing these phenomena, presents part of the results of experiments carried out in the electrotechnical laboratory of the Siberian Technological Institute.

Let us suppose that on some core (Fig. 1) there are two coils, of which the first, having \(W_1\) turns, is supplied by a direct current of strength \(J_0\), while the second, having \(W_2\) turns, is supplied by an alternating current with amplitude \(J_m\) and angular velocity \(\omega = 2\pi c\), where \(c\) is the frequency. The magnetic flux passing through the core will have, when both coils are switched on simultaneously, both a direct and an alternating component. If it is assumed that the material of the core has a constant magnetic permeability independent of induction, then there must be a linear dependence between the instantaneous quantities

\[ \sum i W = a w = K \Phi \ldots\ldots\ldots (1). \]

where \(K\) is the coefficient of proportionality, and \(i\), \(aw\), and \(\Phi\) are the instantaneous values of the current strength, the total ampere-turns, and the magnetic flux. For a sinusoidal variation of the alternating current strength, the instantaneous value of the ampere-turns will be equal to

\[ aw = \sum i W = J_0 W_1 + J_m W_2 \cdot \operatorname{Sin}\omega t \ldots\ldots\ldots (2). \]

If the current is nonsinusoidal, then, expanding in a Fourier series, we obtain

\[ aw = \sum i W = J_0 W_1 + W_2 \cdot \sum J_{mn}\cdot \operatorname{Sin}(n\omega t + z_n)\ldots\ldots (2a). \]

Substituting the value of \(aw\) from equation (2) into equation (1), we find

\[ \Phi = \frac{J_0 W_1}{K} + \frac{J_m W_2}{K}\cdot \operatorname{Sin}\omega t = \Phi_0 + \Phi_m \operatorname{Sin}\omega t \ldots\ldots\ldots (3) \]

where \(\Phi_0\) is the direct component of the magnetic flux, and \(\Phi_m\) is the amplitude of its alternating component. Taking a nonsinusoidal form of current, in the same way we obtain from equation (2a)

\[ \Phi = \frac{J_0 W_1}{K} + W_2 \sum \frac{J_{mn}}{K}\cdot \operatorname{Sin}(n\omega t + z_n) = \Phi_0 + \sum \Phi_{mn}\cdot \operatorname{Sin}(n\omega t + z_n) \]

From the equations obtained we see that the direct component of the magnetic flux is completely independent of the alternating current, and the alternating component is independent of the direct current. The higher harmonics of the flux also depend only on the corresponding components of the current strength of the same order.

The voltage at the terminals of the alternating-current coil will be equal, if the ohmic resistance is considered very small, to the opposite of the counter-electromotive force,

\[ v = -e = -W_2 \frac{d\Phi}{dt} = -\Phi_m \cdot W_2 \cdot \omega \cdot \operatorname{Cos}\omega t \ldots\ldots (4) \]

From this equation we see that the alternating flux, and consequently also the current strength, at a definite frequency, depends only on the ...

from the applied voltage, and is entirely independent of the direct current.

In the direct-current coil, under the action of the alternating component of the flux, an electromotive force will likewise appear

\[ e=\Phi_m\cdot W_1\cdot \omega\cdot \cos\omega t. \]

As a result of this, an alternating current will begin to pass through the direct-current circuit, which will alter all the conclusions and distort the results obtained. In order to avoid such influence, it is necessary to use double circuits (see below in the description of the experiments), or else to include very large self-inductances in this circuit. In what follows we shall assume that there is no alternating current in the first coil.

In reality the magnetic permeability of the core cannot be constant, but varies depending on the induction, so that equation (1), and consequently all the remaining results, are correct only for coils without cores (air). It is impossible to express exactly such a dependence for variable permeability; its graphical representation (Fig. 2) is called the magnetization curve (initial magnetization) and the hysteresis loop (remagnetization). In the absence of hysteresis this dependence may be represented in the form of a series of odd powers

\[ aW=\sum iW=\kappa_1\Phi+\kappa_3\Phi^3+\kappa_5\Phi^5+\kappa_7\Phi^7+\ldots \tag{5} \]

where, for approximate conclusions, the first two or three terms are sufficient. As an example, Fig. 3 gives two experimental magnetization curves (dotted) and curves constructed from the equations:

\[ AW=13.2\left(\frac{B}{1000}\right)-0.39\left(\frac{B}{1000}\right)^3+0.00107\left(\frac{B}{1000}\right)^5 \]

for the upper one (A), and

\[ AW=3.5\left(\frac{B}{1000}\right)+0.365\left(\frac{B}{1000}\right)^3 \]

for the lower one (B). With the use of a larger number of terms, still better agreement of the curves is obtained.

The applied alternating voltage will produce in the core an alternating component of the magnetic flux. Since equation (4) is also retained for the present case, then, with sinusoidal voltage,

\[ \Phi=\frac{V_m}{w_2\omega}\cdot \cos\omega t+C=\Phi_0+\Phi_m\cdot \sin\left(\omega t+\frac{\pi}{2}\right) \]

Substituting this equation into equality (5), taking for simplicity of derivation only two terms, we obtain

\[ aW=\sum iW=\kappa_1\left[\Phi_0+\Phi_m\sin\left(\omega t+\frac{\pi}{2}\right)\right]+ \]

\[ +\kappa_3\left[\Phi_0+\Phi_m\cdot \sin\left(\omega t+\frac{\pi}{2}\right)\right]^3 \]

This expression, after a number of transformations and replacement of powers of the sine by functions of multiple angles, takes the form

\[ aW=\sum iW=\left[\kappa_1\Phi_0+\kappa_3\Phi_0^3+3\kappa_3\Phi_0\Phi_m^2\right]+\left[\kappa_1\Phi_m+3\kappa_3\Phi_m^3+\right. \]

\[ \left.+3\kappa_3\Phi_0^2\Phi_m\right]\sin\left(\omega t+\frac{\pi}{2}\right)+3\kappa_3\Phi_0\Phi_m^2\cdot \sin 2\omega t+ \]

\[ +K_3\Phi_m^3\cdot \sin\left(3\omega t+\frac{\pi}{2}\right)\ldots \tag{6}. \]

From this equation we see that, in order to obtain a constant component of the flux, it is necessary to have constant ampere-turns, since these cannot be obtained in an alternating-current coil (for this it is necessary that there be a constant electromotive force in it); they are equal to

\[ A W_0 = J_0 W_1 = \kappa_1 \Phi_0 + \kappa_3 \Phi_0^3 + 3 \kappa_3 \Phi_0 \Phi_m^2 \ldots \ldots (7). \]

If there were no alternating current, then the same current would correspond to the flux \(\Phi_0'\),

\[ A W_0 = J_0 W_1 = \kappa_1 \Phi_0' + \kappa_3(\Phi_0')^3 \ldots \ldots \ldots (8). \]

and to the same flux—ampere-turns \(A W_0'\), or the current \(J_0'\),

\[ A W_0' = J_0' W_1 = k_1 \Phi_0 + k_3 \Phi_0^3 \ldots \ldots \ldots (9) \]

Comparing equations 7, 8, and 9, we see that the greater the amplitude \(\Phi_m\), and consequently also the applied voltage, the greater the direct-current strength required to produce the same magnetic flux \(\Phi_0\). If, however, the strength of the direct current is unchanged, then the magnetic flux \(\Phi_0\) will be smaller than when the alternating current is switched off.

Thus, switching on the alternating current causes a decrease of the magnetic flux (of its constant component), i.e., demagnetizes the core. Switching on the direct current, without changing the amplitude of the flux (its alternating component), increases the strength of the alternating current and distorts it by harmonics of even orders.

If, in equation (5), not two but a larger number of terms is taken, the character of the conclusions obtained will not change; only new terms will be added. A similar result will be obtained if a nonsinusoidal form is adopted for the voltage curve.

When there is no alternating current, the point determined by the coordinates \(J_0\) and \(\Phi_0\) must lie on the magnetization curve. As soon as the alternating current is switched on, this point will move downward. Considering that it will nevertheless lie on the same curve, it must be admitted that the alternating current produces an effect analogous to constant ampere-turns. Indeed, under the influence of the alternating current, in equation (6) there appears the term

\[ 3 \kappa_3 \Phi_0 \Phi_m^2 = A W_0'' \]

which represents the constant component of the magnitude of the ampere-turns. This demagnetization must be compensated by a corresponding increase of the direct current. Equation (7) may be represented in the form

\[ A W_0' = A W_0 - A W_0'' = \kappa_1 \Phi_0 + k_3 \Phi_0^3 \]

where \(A W_0'\) is the apparent magnitude of the constant ampere-turns corresponding to the flux \(\Phi_0\).

In diagrams (4) and (5) the current curve has been constructed from the magnetization curve, with direct current switched on and without it. The segment \(oe\) represents the strength of the direct current, and \(eb\) the corresponding magnetic flux. When the alternating current is switched on, the constant component of the flux will decrease to a value equal to the segment \(ea = dc\), while the point \(a\) is found from the condition of equality of the areas of the current curve on both sides of the axis. The segment \(de\) will represent the demagnetizing influence of the alternating current, referred to the direct current.

If the constant component of the flux is caused not by the strength of a current but is residual magnetization, then in that case also there must be a demagnetizing action of the alternating current. This is fully confirmed by experiments and is used to destroy or reduce magnetization in steel and iron.

The presence of hysteresis only slightly changes the shape of the current curve, almost without affecting the demagnetizing action. In Figs. (6) and (7), curves obtained experimentally are given for comparison; here the curve of the magnetic flux was constructed by graphical integration of the electromotive-force curve. In Fig. (8) dynamic hysteresis loops are constructed for similar cases. The letter designations in these drawings are the same as in Figs. (4) and (5).

If the applied alternating voltage is gradually reduced, the demagnetizing action will become smaller and smaller, and when the voltage reaches zero the magnetic flux will attain its maximum value. This point, as it turns out, will not lie on the initial magnetization curve, but will always lie somewhat above it. For small values of the magnetic-field intensity, the percentage increase of induction may reach large values. This phenomenon is used to increase the magnetization of magnets in cases where it is impossible to obtain a large value of the constant ampere-turns. Such an increase can be explained by the fact that, in the presence of alternating current, the magnetic flux at certain moments assumes the value corresponding, according to the magnetization curve, to a direct current, and then no longer returns along the same curve, but always goes somewhat above it. This occurs because the magnetization curve, owing to hysteresis, always lies much closer to the lower part of the loop, while the alternating current destroys this influence. Fig. (9) shows such an increase of magnetization under the action of a gradual approach of the current strength to a certain point, with ever-decreasing deviations of the direct-current strength to one side and the other.

The greater the value of the initial amplitude of the alternating current, the stronger the magnetization will be when it is reduced to zero. Fig. (10) gives the character of the dependence of this increase for steel and iron, from which it is seen that it asymptotically approaches a certain limiting value. The induction obtained in this way is ideal, and the curve connecting these points will be the ideal magnetization curve. For the practical attainment of ideal values for steel, the alternating current must at first have a considerable amplitude (giving up to 100% and more of the direct ampere-turns), whereas for iron very small amplitudes are sufficient. The required amplitudes are connected with the ability of the material to retain residual magnetization.

In Fig. (11) are given experimentally obtained dependences between the induction \(B_0 = \dfrac{\Phi_0}{Q}\) and the constant ampere-turns \(AW_0\), for various values of the voltage and frequency of the alternating current, as well as the basic initial magnetization curve (A) and the ideal curve (B). The curves for 198 volts at 180 periods and for 55 volts at 50 periods almost coincide, which is a consequence of the equal amplitudes of the magnetic fluxes, since from equation (4) it follows that

\[ \Phi_m \cdot \left(W_2 \frac{\pi}{V\sqrt{2}}\right)=\frac{V}{c}=\frac{198}{180}=\frac{55}{50}. \]

In drawing (12) the magnetic permeabilities are given for the ideal and initial magnetization curves. In Fig. (13) are given the dependences of the percentage decreases of the magnetic flux \(\dfrac{\Phi_0' - \Phi_0}{\Phi_0}\) on the appli-

of the applied voltage, and in Fig. (14) similar dependences are given for the apparent demagnetizing ampere-turns \(\dfrac{AW_0''}{AW_0}\).

The form of the voltage curve has very little effect on the demagnetizing action. A peaked voltage curve, corresponding to a flat magnetic-flux curve, gives somewhat greater demagnetization, and conversely a trapezoidal voltage gives less. In Fig. (15) demagnetization curves are presented for the same magnitude of voltage, but of different form. Curve A refers to a sinusoidal form of voltage (form factor 1.11), and curve B to a trapezoidal one (form factor about 1.04). C is the ideal magnetization curve.

If the core is made up of different materials or sections, then the general character of all the phenomena will be the mean of those for the separate parts, and their influence will be the greater, the larger the share of ampere-turns falling to them out of the total number. The presence of air gaps strongly smooths out all the phenomena.

The practical application of the demagnetizing action of alternating current is possible in very varied cases. To clarify the peculiarities of some methods of application and to determine the degree of their suitability, experiments are at present being carried out, the results of which will be published subsequently.

In carrying out the experiments, an Ondograph Hospitalier and a mirror oscillograph were used for recording the current and voltage curves. For measuring the constant components of the magnetic flux, a ballistic galvanometer and a Grassot fluxmeter were used. In order that there should be no alternating current in the direct-current windings, there were two identical coils, connected according to the scheme of Fig. 16 in such a way that the electromotive forces in the magnetizing windings cancel one another. Instead of two coils it is also possible to use one, but with three cores (Fig. 18).

For measuring electromotive forces, the cores also had one coil each with terminals c and d. Ohmic resistances were inserted in series with the coils in the alternating-current circuit; to their terminals a and b there were connected, for recording the current curves, the ondograph or the loops of the oscillograph. To record the voltage curves, the same connections were made to terminals c and d.

For recording the magnetization curves, almost the same circuit was used (Fig. 17). A switch was inserted in the direct-current circuit, and in the alternating-current circuit a potentiometer for smoothly reducing the voltage to zero. The fluxmeter or ballistic galvanometer was connected into two additional coils, calculated so that no alternating current should penetrate into it. The ballistic galvanometer may likewise be connected into one coil, but then it must have a large inertia, since otherwise vibration of the frame and mirror results from the alternating current passing through it.

When measuring the ordinates of the ideal curve, the alternating current was gradually brought to zero, and only then was the direct current switched. After switching, the induction obtained was not the ideal one, but that lying on the initial curve. Consequently, instead of the half-difference of the readings on the instrument, it was necessary to take the difference between the total reading and the half-difference of the readings for the initial curve obtained at the same magnetizing current.

To the article by Eng. R. A. Voronov: “Demagnetizing Effect of Alternating Current”.

Fig. 1.

Labels: I, II, \(w_1\), \(w_2\).

Fig. 2.

Axis and curve labels: \(+B\), \(-B\), \(0\), \(-\alpha w(-H)\), \(+\alpha w(-H)\).

Fig. 3.

Axis and curve labels: \(B\), 15000, 10000, 5000, 100, 200, 300, 400, 500, \(\alpha w\), \(A\), \(B\).

Fig. 4.

Labels: \(+\varphi\), \(-\varphi\), \(-J\), \(0\), \(+J\), \(\varphi\), \(J\), \(v\).

Fig. 5.

Labels: \(+\varphi\), \(-\varphi\), \(-J\), \(0\), \(J_0\), \(+J\), \(B\), \(C\), \(d\), \(e\), \(a\), \(\varphi_{\mathrm{tot}}\), \(\varphi_{\sim}\), \(J_{\sim}\), \(\varphi_0\).

Typo-lithography of the Publishing House “Krasnoe Znamya”.

Labels in the figure: \(E\), \(I\), \(\Phi\), \(I_{\mathrm{const.}} = 0\).

Labels in the figure: \(E\), \(I\), \(\Phi\), \(I_{\mathrm{const.}} = 2A\).

Labels in the figure: \(+\Phi\), \(-\Phi\), \(-I\), \(+I\), \(I_0\), \(\Phi_0\).

Labels in the figure: \(\Delta \Phi\), steel, iron, \(\dfrac{I_m}{I_0}\).

Labels in the figure: \(\dfrac{A W'}{A W_0}\), \(I_0 = 5A\), \(I_0 = 1A\), \(V\).

To the article by Eng. R. N. [[unclear: author’s surname]]: “Demagnetizing Effect of Alternating Current.”

Fig. 9.

Labels visible in the figure: \(+B\), \(-B\), \(20000\), \(20000\), \(-H\), \(+H\), \(25\), \(25\), \(O\).

Fig. 12.

Labels visible in the figure: \(N\), \(3000\), \(2000\), \(A\), \(B\), \(aw\).

Fig. 11.

Labels visible in the figure: \(B\), \(15000\), \(10000\), \(5000\), \(A\), \(B\), \(198V\ 180^\circ\) [[unclear]], \(55V\ 500\) [[unclear]], \(110V\ 500\) [[unclear]], \(AW\), \(100\), \(200\), \(300\), \(400\), \(500\).

Fig. 13.

Labels visible in the figure: \(100\%\), \(\dfrac{\Phi}{\Phi_0}\cdot 100\), \(80\), \(60\), \(40\), \(20\), \(J_0=5A\), \(J_0=1A\), \(V\).

Fig. 15.

Labels visible in the figure: \(B\), \(C\), \(B\), \(A\), \(J\).

To the article by R. N. Voronov, “Demagnetizing Effect of Alternating Current.”

Fig. 16.

Fig. 17.

Fig. 18.