USE OF TWO-FREQUENCY MAGNETIZATION TO OBTAIN THE CURVE OF DIFFERENTIAL DYNAMIC PERMEABILITY

B. B. VINOKUROV, I. G. LESHCHENKO

(Presented by the scientific and technical seminar of the Department of Information and Measuring Technology)

The use of two alternating magnetic fields for magnetizing a ferromagnet has the special feature that the remagnetization process is accompanied by the appearance of partial loops, whose shape and position on the low-frequency hysteresis loop are determined by the ratio of the frequencies, amplitudes, and, in a number of cases, also the phases of the magnetizing fields [1]. The appearance of partial loops is associated with the presence, in the curve of the resultant field, of periodically alternating maxima and minima.

An analysis of the process of remagnetization of a ferromagnet in a complex field [2] made it possible, by expansion in a Fourier series, to obtain the complete expression for the emf of the measuring winding of the transducer. It follows from it that at the output of a two-frequency transformer transducer there is a complex signal, which includes low-frequency and high-frequency components. Restricting consideration to only the high-frequency components, one may note that the amplitudes of the individual harmonics do not remain constant, but are modulated according to a quite definite law with twice the frequency of the low-frequency field. The character of the envelopes is determined both by the properties of the ferromagnet being remagnetized and by the absolute values of the parameters of the magnetizing fields and their ratio.

Let us single out from the entire spectrum the component varying with the frequency of the exciting high-frequency field. Its envelope, according to [2], has the form

\[ U_{m\Omega}^{1\omega}(t) = W_2 S \left[ \frac{\mu_{\mathrm{e}}(t)+\mu_r(t)}{2}\,\omega H_{m\omega} + \frac{\mu_{\mathrm{e}}(t)-\mu_r(t)}{\pi}\,\Omega H_{m\Omega}\cos\Omega t \right], \tag{1} \]

where \(\omega\), \(\Omega\), \(H_{m\omega}\), \(H_{m\Omega}\) are, respectively, the frequencies and amplitudes of the high-frequency and low-frequency exciting fields; \(\mu_{\mathrm{e}}(t)\) is the “equivalent permeability of increase,” the physical meaning of which is considered in [1]; \(\mu_r(t)\) is the time dependence of the reverse permeability. For the time dependence \(\mu_{\mathrm{e}}(t)\), the expression is valid

\[ \mu_{\mathrm{e}}(t) = \mu_d(t) - \frac{\Delta H_{-}(t)}{\Delta H_{+}(t)} \left[\mu_d(t)-\mu_r(t)\right]. \tag{2} \]

In this expression

\[ \mu_d(t) \]

is the differential dynamic permeability, and \(\Delta H_-(t)\) and \(\Delta H_+(t)\) are, respectively, the negative and positive increments of the total field over the period of the low-frequency component.

Analysis of expression (2) shows that the “equivalent permeability of increase” can take values of the differential or reversible permeability, or values lying between them (Fig. 1).

Of interest is the case of magnetization of a ferromagnet by fields whose parameters satisfy the inequality

\[ \pi \Omega H_{m\Omega} > 2H_{m\omega}\omega \tag{3} \]

For the region in which this inequality is fulfilled, there are regions on the low-frequency hysteresis loop that are free of partial cycles. Therefore, in expression (1) for the envelope of the first harmonic, it is necessary to substitute \(\mu_d(t)\) instead of \(\mu_e(t)\) and \(\mu_r(t)\). For this reason the second term in (1) becomes zero, and

\[ U_{m\Omega}^{(1\omega)}(t)=W_2S\omega H_{m\omega}\cdot\mu_d(t). \tag{4} \]

Fig. 1. Dependence of the permeability \(\mu_e(t)\) on the frequency of the high-frequency magnetizing field for \(H_{m\Omega}=10\) Oe

The more strongly the inequality indicated above is satisfied, the smaller the role played by the second term in the formation of the envelope of the first harmonic, and the more its shape resembles the time dependence of the differential dynamic permeability. At the same time, however, it should not be forgotten that an excessive increase of inequality (3) leads to a decrease in the overall level of the output signal.

With the coefficients preceding \(\mu_d(t)\) in expression (4) constant, in order to obtain the time dependence of the differential permeability it is sufficient to extract the envelope of the first harmonic. This method can be implemented by an arrangement whose block diagram is shown in Fig. 2.

Fig. 2. Block diagram of the experimental setup

The ferromagnetic specimen is remagnetized in two alternating magnetic fields, the sources of which are the generators \(G_\omega\) and \(G_\Omega\). The measuring winding \(W_2\) is connected to a selective amplifier tuned to the fundamental harmonic of the high-frequency exciting field. The amplitude-modulated signal from the amplifier output is detected by a linear detector and fed to the vertical input of an electronic oscilloscope. To compensate the initial component of the first harmonic caused by nonideality of the coil filling factor, a compensating device with separate adjustment of the phase and magnitude of the compensating voltage is used. When the oscilloscope operates in sweep mode, on the screen

there is a time dependence of the differential dynamic permeability. In the case of feeding the horizontal plates from the reference signal, this is the dependence \(\mu_d(t)=F(H)\).

Fig. 3

The studies were carried out at frequencies of the exciting fields \(\Omega/2\pi = 50\) Hz and \(\omega/2\pi = 10\ \text{kHz} \div 200\ \text{kHz}\). As an example, one may cite the curve of the differential dynamic permeability shown in Fig. 3.

REFERENCES

1. B. B. Vinokurov, I. G. Leshenko. Investigation of the regularities of two-frequency magnetization of ferromagnets. In: Multiparameter inductive quality control of machine parts by the higher-harmonics method. Kiev, RDNTP, 1972.

2. B. B. Vinokurov, I. G. Leshenko. Theoretical study of the method of two-frequency magnetization of ferromagnets. In: Proceedings of the First All-Union Interuniversity Conference on Electromagnetic Methods of Quality Control of Materials and Products, Part II, Moscow, Rotaprint MEI, 1972.