PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
ORDER OF THE OCTOBER REVOLUTION AND ORDER OF THE RED BANNER OF LABOR
named after S. M. KIROV

Vol. 194　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1972

ON THE INFLUENCE OF HIGHER HARMONICS OF THE MAGNETIZING CURRENT ON THE ERROR IN MEASURING DYNAMIC COERCIVE FORCE

I. G. LESHCHENKO, V. P. DOLGOPOLOV

(Presented by the scientific seminar of the Department of Information and Measurement Engineering)

In measuring the dynamic coercive force by the time interval between the instants at which the magnetic-field strength and flux pass through zero values [1, 2], it was assumed that magnetization is produced by a sinusoidal current. In this case the value of the dynamic coercive force is

\[ H_{\mathrm{cd}} = H_{\mathrm{m}}\cdot \sin \omega t_{\phi}. \]

For small values of the ratio \(H_{\mathrm{cd}}/H_{\mathrm{m}}\) (usually small \(H_{\mathrm{cd}}\)),

\[ H_{\mathrm{cd}} = H_{\mathrm{m}}\cdot \omega t_{\phi}, \]

where \(\omega t_{\phi}\) is the phase shift between \(H\) and \(\Phi\) at the instants when they pass through zero.

The magnitude of the driving current, and consequently also of the maximum magnetic-field strength, is set or monitored by an instrument responding to the rms value of the current. In the presence of higher harmonics in the magnetizing current, as a rule odd ones, an error arises in the measurement of the dynamic coercive force, depending on the amplitude and phase of the higher harmonics.

We shall estimate the possible errors for an amplitude of the third harmonic equal to 1% of the fundamental harmonic of the magnetizing current, neglecting higher harmonics. As is seen from Fig. 1,a—for a sinusoidal current—and Fig. 1,b—for a nonsinusoidal current, the time intervals \(t_{\phi}\) and \(\tau_{\phi}\), corresponding to the specified value of \(H_{\mathrm{cd}}\), are not equal and, consequently, an error will appear in the measurement of \(H_{\mathrm{cd}}\) and an error in rejecting products. This error in the general case cannot be compensated, since the amplitude and phase not only of the third harmonic but also of the higher harmonics are not known.

We shall analyze the influence of higher harmonics for the purpose of estimating the error first for the third harmonic; in doing so we shall assume that the value of \(H_{\mathrm{cd}}\) and \(H_{\mathrm{m}}\) is not changed by the higher harmonics, owing to their smallness. In this case the instantaneous values of the currents corresponding to the specified \(H_{\mathrm{cd}}\) and \(H_{\mathrm{m}}\) for sinusoidal and nonsinusoidal current waveforms must be equal to one another, i.e.,

\[ I_{\mathrm{m}}\cdot \sin \omega t_{\phi} = I_{\mathrm{m1}}\cdot \sin(\omega \tau_{\phi}+\varphi_{1}) + I_{\mathrm{m3}}\sin(3\omega \tau_{\phi}+\varphi_{3}), \tag{1} \]

where

\(\varphi_1\) is the phase shift between the current of the fundamental harmonic and the nonsinusoidal current curve at the instant when they pass through zero values.

\(\varphi_3\) is the phase shift between the first and third harmonics of the magnetizing current,

\(\omega t_\phi\) is the angle corresponding to \(H_{\text{cd}}\) (the phase shift between \(H\) and \(\Phi\) for a sinusoidal current).

Since the rms value of the current for the nonsinusoidal form is equal to that for the sinusoidal current form \((I_{\text{dc}} = I_{\text{dn}})\), then, for the adopted value \(I_{\text{m3}} = 0.01 I_{\text{m1}}\),

\[ I_{\text{dc}} = I_{\text{d1}} \sqrt{1 + \left(\frac{I_{\text{d3}}}{I_{\text{d1}}}\right)^2} = 1.00005 I_{\text{d1}}. \]

and equation (1) takes the form

\[ 1.00005 \cdot \sin \omega t = \sin(\omega \tau_\phi + \varphi_1) + 0.01 \sin(3\omega \tau_\phi + \varphi_3). \tag{2} \]

This equation cannot be solved with respect to \(\tau_\phi - t_\phi\). To find this difference, and hence the relative error, we first determine \(\varphi_1\), which depends on the ratio of the amplitudes of the third and first harmonics and on \(\varphi_3\). Indeed, under the initial conditions \((t_\phi = 0\) and \(\tau_\phi = 0)\), from expression (2),

\[ \sin\varphi_1 = -0.01 \sin\varphi_3. \]

Fig. 1

Assigning various values to \(\varphi_3\), we find \(\varphi_1\), the values of which are given in the table.

Table

Next, for the adopted \(K_3 = 0.01\), by assigning \(\varphi_3\) from \(0^\circ\) to \(360^\circ\) and the calculated \(\varphi_1\), we compute the dependence of \(\omega t_\phi\) on \(\omega \tau_\phi\), the difference \(\Delta = \omega \tau_\phi - \omega t_\phi\), and the relative error \(\gamma = \Delta / \omega t_\phi\).

The results of the calculations for values of \(\varphi_3\) at every \(30^\circ\) are presented in Fig. 2. As is evident from the error curves constructed for all possible ratios \(H_{\text{cd}}/H_{\text{m}}\), they vary from zero to approximately 10%. The envelope of the errors, which represents the maxi-

small relative errors, is 3%; for values of \(H_{\mathrm{cd}}/H_m\) greater than zero but less than 0.9, it then increases markedly. The region of increased errors corresponds to materials with a rectangular hysteresis loop and appeared as a result of the fact that the maximum values of the magnetic-field strength for sinusoidal and nonsinusoidal magnetizing currents are not equal to one another.

Fig. 2

In the case where control of the magnetizing current is carried out by the current amplitude, i.e., for sinusoidal and nonsinusoidal currents the condition \(I_{\mathrm{mc}} = I_{\mathrm{mn}}\) is satisfied, the calculation formula has the form

\[ I_{\mathrm{mc}}\sin \omega t_{\phi} = I_{m1}\sin(\omega t_{\phi 1}+\varphi_1) + I_{m3}\sin(3\omega t_{\phi}+\varphi_3). \]

To determine the errors from this expression, one first finds the conditions for different \(\varphi_3\) under which the equality \(I_{\mathrm{mc}} = I_{\mathrm{mn}}\) is fulfilled. It is not difficult to verify that for \(K = 0.01\) the relative error will vary very slightly for small ratios \(H_{\mathrm{cd}}/H_m\), and up to “normal” values (3%) for ratios \(H_{\mathrm{cd}}/H_m\) close to unity. In Fig. 2 this maximum relative error is indicated by a dashed line. Carrying out analogous reasoning and calculations for the fifth and other harmonics of the magnetizing current, one can estimate the maximum relative errors by specifying the coefficients \(K_5\), etc. Thus, for the fifth harmonic with \(K_5 = 0.01\) relative to the fundamental harmonic, the maximum relative errors for various \(\varphi_5\) have the values indicated in Fig. 2 by the dash-dotted line. If it is taken into account that, as the harmonic number increases, their amplitude decreases, then the share of the introduced error will also be smaller than that from the third harmonic. The total value of the errors from the higher harmonics can be estimated from Fig. 2.

Conclusions and Recommendations

1. The nonsinusoidal character of the magnetizing current leads to additional errors that depend on the amplitudes and phases of the higher harmonics and on the ratio of the measured dynamic coercive force to the maximum magnetic-field strength. To reduce the errors to permissible values, the higher harmonics of the magnetizing current must be shunted.

2. It is advisable to use methods for measuring the dynamic coercive force that are independent of, or only weakly dependent on, the current waveform—for example, oscillographic and differential methods (admission control), the method of measuring the instantaneous value of the current \((H_{\mathrm{d}})\), etc.

3. Current monitoring should be carried out by its maximum value.

References

1. I. G. Leshchenko, V. A. Malin. An apparatus for sorting parts by coercive force. Advanced Scientific-Technical and Production Experience, Moscow, GOSINTI, 1964, No. 2-64-282/8.

2. A. M. Gruznov, V. P. Dolgopolov, I. G. Leshchenko, V. D. Sartakov. Measurement of the hardness of steel products using an alternating-current coercimeter. Defektoskopiya, No. 4, 1967.