STUDY OF THE PROPERTIES OF A FERROMAGNETIC MATERIAL UNDER MAGNETIZATION BY A PERIODIC MAGNETIZING FORCE

R. A. Vaynshteyn, L. I. Voronova, A. V. Shmoilov

(Presented by the scientific seminar of the Department of Electrical Stations)

As shown in [1], in order to determine the constant component and the modulation depth of the inductance from the known parameters of the coil of a parametric amplifier, or to solve the inverse problem, it is necessary to know two values of the magnetic permeability between which the latter varies periodically.

At the same time it must be taken into account that, for example, the maximum value of the differential magnetic permeability and other properties of a ferromagnetic material depend very strongly on the magnetization conditions. Practically every partial cycle corresponds to its own value of the maximum and initial permeability.

In this connection, in measurements it is desirable to approximate the actual magnetization conditions as closely as possible. The method used here for measuring dynamic permeability satisfies this requirement to a considerable degree. The measurement of differential permeability is based on using the voltage across a periodically varying inductance when a direct current flows through it. If a direct current flows through the periodically varying inductance \(l(t)\), then the voltage across the inductance is

\[ U_L = I \frac{dl(t)}{dt}. \]

By integrating the voltage \(U_L\), it is easy to obtain a voltage proportional to \(l(t)\):

\[ l(t) = \frac{1}{I} \int U_L\,dt. \]

Thus, in order to obtain an oscillogram of the curve of dynamic inductance as a function of time, it is necessary to integrate the voltage \(U_L\), which can be done, for example, by using an \(R_u—C_u\) circuit with appropriately selected parameters.

If, instead of a time sweep, a voltage proportional to the strength of the magnetizing field is applied to the plates of the cathode-ray tube of the oscilloscope, then we obtain the curve \(l(H)\), which, apart from scale factors, reproduces the curve \(\mu_d(H)\).

The scale of the differential-permeability oscillogram will be equal to

\[ m_l = m_0 \frac{R_u C_u l}{2 I q W^2}, \]

where

\(m_0\) — the scale of the oscilloscope in V/mm;
\(R_u, C_u\) — parameters of the integrating circuit;
\(I\) — auxiliary direct current;
\(q\) — cross-sectional area of the specimen;
\(l\) — length of the mean line of force of the specimen;
\(W\) — number of turns of the measuring winding.

The measurement circuit is shown in Fig. 1.

In the diagram: to the horizontal plates of the oscilloscope; test specimen; to the vertical plates of the oscilloscope; \(R_{\text{sh}}\), \(mA\), \(V\), \(I_{\text{add}}\), \(R_{\text{add}}\), \(W_n\), \(W_u\), \(R_u\), \(C_u\), \(mA\), “l.v.”

Fig. 1. Circuit for studying the properties of a ferromagnetic material under magnetization by a periodic magnetizing force

Although the measurements are carried out on a specimen made of ferromagnetic material, which is intended to be used for a parametric amplifier (PA), we shall, here as well, using the terminology adopted in the literature, call the magnetizing current the pumping current. The measurement results discussed below were obtained on a specimen made of transformer steel USh-12, grade E-41, used by us for manufacturing the PA.

Having a voltage proportional to the differential permeability under periodic magnetization of the specimen, we can study changes in permeability as a function of the magnitude and waveform of the pumping current. In this connection we shall be interested in the following questions:

1) the stability of the modulation depth of the permeability when the voltage of the pumping-current source is varied;

2) the conditions under which the depth of modulation of the permeability has the greatest value;

3) the conditions under which the form of the curve of differential permeability as a function of time, \(\mu(t)\), approaches the optimal one [2].

We shall investigate the following cases:

1. Feeding the pump circuit through a valve in which the reverse current may be neglected in comparison with the forward current (Fig. 2, a).

Fig. 2. Circuit diagrams for producing various magnetization conditions

In Fig. 3 the dependence is shown of the depth of modulation of the permeability as a function of the voltage of the pump-current source, reduced to one turn of the pump winding and to a unit cross section of the specimen. From these curves it follows that, beginning with a specific voltage of approximately \(15 \cdot 10^2\ \mathrm{V}/\mathrm{m}^2\cdot\text{turn}\) and above, the depth of modulation of the permeability \(\Delta\mu\) changes very little.

Fig. 3. Dependence of the depth of modulation of permeability on the specific voltage when the pump circuit is fed through a valve with negligibly small reverse current

The form of the curve \(\mu_d(t)\) also practically does not change, beginning with the same value of the specific pump voltage, as is seen from the oscillograms of the curve \(\mu_d(t)\), taken at

\[ U_* = 15 \cdot 10^{-2}\ \mathrm{V}/\mathrm{m}^2\cdot\text{turn} \]

and at

\[ U_* = 27 \cdot 10^{+2}\ \mathrm{V}/\mathrm{m}^2\cdot\text{turn} \]

(Fig. 4). Such a relatively high stability of the depth of modulation of the permeability and of the form of the curve \(\mu_d(t)\) is explained by the small deformation of the curve \(\mu_d(H)\) in its initial part when the pump voltage is varied. This is illustrated by the oscillograms of the curve \(\mu_r(H)\) in Fig. 4.

2. Feeding the pump winding through a valve with a finite value of reverse current.

The circuit is shown in Fig. 2, b. The reverse current of the required magnitude is produced by the resistance \(R_{\mathrm{rev}}\). In the presence of a reverse current we pass from an asymmetric hysteresis cycle \(B(H)\), located entirely in the first quadrant of the \(B\)—\(H\) plane, to another asymmetric cycle, on which the maximum value of the dynamic permeability is considerably larger. At the same time, the form of the curve \(\mu_d(t)\) remains favorable, since the reverse current is so small that it does not cause the appearance of a noticeable second harmonic in the curve \(\mu_d(t)\) owing to a partial transition of the curve \(B(H)\) into the third quadrant of the \(B\)—\(H\) plane.

Fig. 4. Oscillograms of the curves \(Md(t)\) and \(Md(H)\)
a) for \(U_* = 15 \cdot 10^2\ \mathrm{V}/\mathrm{m}^2 \cdot \text{turn}\),
b) for \(U_* = 27 \cdot 10^2\ \mathrm{V}/\mathrm{m}^2 \cdot \text{turn}\)

However, in this case, to each value of the direct current there corresponds one and only one value of the reverse resistance at which the optimal parameters of the curve \(\mu_d(t)\) are obtained. Moreover, the value of the reverse current to which the maximum value \(\Delta\mu\) corresponds practically coincides with the current at which the best form of the curve \(\mu_d(t)\) is attained. From what has been said above it follows that both the depth of modulation of the permeability and the form of the curve \(\mu_d(t)\) will change considerably when the pumping voltage is changed. This circumstance limits the use of the favorable influence of the reverse current to those cases in which the voltage of the pumping-current source changes little. This is illustrated by oscillograms. Figure 5 gives oscillograms of \(\mu d(t)\) at different field intensities produced by the reverse current, and \(U_* = 20.0 \cdot 10^2\ \mathrm{V}/\mathrm{m}^2 \cdot \text{turn}\). It is already evident from these oscillograms that there exists an optimal value of the reverse current. The distortion of the curve \(\mu d(t)\) at \(H_{\mathrm{rev}} > H_{\mathrm{opt}}\) is explained by the appearance of a second harmonic owing to a considerable transition of the curve \(B(H)\) into the third quadrant of the \(B\)—\(H\) plane. This is seen from the oscillograms of the curve \(\mu d(H)\) shown in Fig. 5.

The presence of a reverse current has practically no effect on the operating mode of the pumping-current source, since the required reverse current is considerably smaller than the direct current. For example, for the case shown in Fig. 5

case when, at \(U_* = 20.7 \cdot 10^2\ \mathrm{V/m^2 \cdot turn}\), the amplitude of the dc-field intensity is \(9350\ \mathrm{A/m}\), and the optimum amplitude of the reverse-field intensity is \(71\ \mathrm{A/m}\).

Figure 6 gives the dependences of the depth of modulation of the permeability on the magnitude of the reverse-field intensity for two values of the pump voltage. These dependences are characterized by the fact that the reverse-field intensity at which the maximum \(\Delta\mu\) is reached is almost independent of the pump voltage. This circumstance may be used to determine the optimum value of the reverse resistance and indicates that, at \(R_{\mathrm{rev}}=\mathrm{const}\), the optimum supply conditions of the pump circuit will be disturbed when the pump voltage changes. This circumstance is illustrated once again in Fig. 7, where the dependence of the depth of modulation of the permeability on the pump voltage at \(R_{\mathrm{rev}}=\mathrm{const}\) is presented. As was to be expected, the stability of \(\Delta\mu\) is low here. These curves are characterized by the presence of a maximum with respect to \(U_*\), the position of which depends on the magnitude of the reverse resistance.

It should be noted that the conditions for parametric amplification on the descending branch of the curve \(\Delta\mu(U_*)\) are worse than on the ascending branch, because its shape is distorted by the second harmonic.

3. Supply of the pump circuit by a symmetric periodic current (Fig. 2, c).

In this case the remagnetization of the cores takes place along a symmetric cycle; therefore the frequency of variation of the inductance is twice the frequency of the pump current. With this supply mode of the pump circuit, parametric amplification and the generation of oscillations with a frequency equal to the frequency of the pump current are possible. Figure 8 gives the dependence \(\Delta\mu(U_*)\). The degree of stability of the depth of modulation of the permeability here is lower than in case 1. The absolute value of \(\Delta\mu\) is also smaller here. In addition, the shape of the curve \(\mu_d(t)\) here deviates considerably from a rectangular one. Figure 9 gives oscillograms of the curves \(\mu_d(t)\) and the corresponding

Fig. 5. Oscillograms of the curves \(\mu_d(t)\) and \(\mu_d(H)\) at \(U_* = 20 \cdot 10^2\ \mathrm{V/m^2 \cdot turn}\).
a) at \(H_{\mathrm{rev}} = 0\),
b) at \(H_{\mathrm{rev}} = H_{\mathrm{opt}} = 71\ \mathrm{A/m}\),
c) at \(H_{\mathrm{rev}} = 120\ \mathrm{A/m}\).

Fig. 6. Dependence of the depth of modulation of permeability on the magnitude of the reverse-field intensity.
1 — \(U_* = 9 \cdot 10^2\ \mathrm{V/m^2 \cdot turn}\),
2 — \(U_* = 20 \cdot 10^2\ \mathrm{V/m^2 \cdot turn}\).

Fig. 7. Dependence of the depth of permeability modulation on the specific magnetizing voltage.
1 — at \(R_{\mathrm{ob}} = 15\ \text{k}\Omega\),
2 — at \(R_{\mathrm{ob}} = 50\ \text{k}\Omega\),
3 — at \(R_{\mathrm{ob}} = 100\ \text{k}\Omega\)

Fig. 8. Dependence of the depth of permeability modulation on the specific magnetizing voltage under supply by a symmetrical periodic current

Fig. 9. Oscillograms of the curves \(\mu_d(t)\) and \(\mu_d(H)\) under supply of the magnetizing circuit by a symmetrical periodic current
\(a\) — at \(U_* = 3\text{–}5 \cdot 10^2\ \text{V}/\text{m}^2\cdot\text{turn}\),
\(b\) — at \(U_* = 20 \cdot 10^2\ \text{V}/\text{m}^2\cdot\text{turn}\)

oscillograms of \(\mu_d(H)\) at a pumping voltage corresponding to the maximum \(\Delta \mu\) and at \(U_* = 20.0 \cdot 10^2\ \text{V}/\text{m}^2 \cdot \text{turn}\).

Conclusions

1. The greatest stability of the depth of modulation of the permeability is observed when the pumping circuit is fed through a valve with a very large reverse resistance.

2. A considerable increase in the depth of modulation of the permeability is achieved without changing the power consumed by the pumping circuit, by setting the corresponding reverse current. The stability of \(\Delta \mu\) in this mode is considerably lower.

3. In the supply modes according to items 1 and 2, the shape of the inductance variation curve is close to rectangular.

4. When magnetization is carried out by a symmetrical periodic current, the stability of \(\Delta \mu\) is low. The shape of the curve differs substantially from rectangular.

LITERATURE

1. R. A. Vainshtein. Parametric amplifier. Author’s certificate No. 178862 with priority of 15.12.1964.

2. N. V. Zernov, V. G. Karpov. Theory of Radio-Engineering Circuits. “Energiya,” 1965.