EFFECT OF MAGNETIC-CORE CHARACTERISTICS ON PRODUCT QUALITY CONTROL BY COERCIVE FORCE AND RESIDUAL INDUCTION IN ALTERNATING CURRENT

V. P. Dolgopolov, I. G. Leshchenko

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

The literature [1] describes alternating-current coercimeters with magnetizing devices (sensors) in the form of a solenoid, into which the reference product and the product being inspected are placed. In this case the magnetic circuit is open, and obtaining large magnetizing fields in such sensors proves difficult. To obtain a large magnetic flux through the tested product, it is possible to use magnetizing devices with an additional magnetic core, made of magnetically soft material, closing the product.

The principle of operation of an alternating-current coercimeter consists in measuring the time interval between the instants at which the field intensity and the induction in the product being inspected pass through zero when it is remagnetized in a sinusoidal magnetic field. The magnetizing field (or current) will be sinusoidal in the case when the inductive reactance of the magnetizing device is much less than the active resistance of the magnetizing-current circuit, which is ensured by connecting a large active resistance in series with the magnetizing winding.

Fig. 1

Schematically, a sensor with a closing magnetic core is shown in Fig. 1a, where the following are designated: 1 — magnetic core, 2 — product being inspected, 3 — magnetizing winding, 4 — measuring winding. To simplify the analysis, we shall replace the magnetic core and the product by half-rings made of the corresponding materials, but with identical geometrical dimensions and with a winding uniformly distributed over both halves.

half-rings (Fig. 1b—the designations are the same as in Fig. 1a). Let us assume that the mean lengths of the half-rings are each equal to unity, i.e., \(l_1=l_2=l=1\). We note that the magnetic circuit consisting of half-rings is a series circuit; therefore the induction at every point of the circuit at any moment of time will be the same. In this case we shall not take into account the presence of leakage fluxes and air gaps, and we shall assume that the magnetic flux is uniformly distributed over the cross section of the half-rings.

Fig. 2

Let us now turn to Fig. 2. Suppose that the dynamic remagnetization loop 1 corresponds to magnetic core 1 in Fig. 1b, and loop 2—

to the product being inspected (the values of induction and field strength are taken in relative units). Since the induction in both half-rings is the same, then, taking the sum of the products of the amplitudes of the field strengths of the loops of each half-ring by their mean lengths, we find the amplitude of the total m.m.f. required to conduct the magnetic flux through both half-rings: $F_{mn}=H_{m1}\cdot l_1+H_{m2}\cdot l_2$. Knowing the amplitude value of the m.m.f., one can construct the entire sinusoidal curve of the m.m.f., expressing, on another scale, the current $i$ in the magnetizing winding. A part of it is shown in the lower part of Fig. 2 and is denoted by $F_n(i)$.

The subsequent constructions are performed as follows. We specify some value of the induction in the half-rings. At this induction value, on the loops we find the corresponding field strengths $H_1$ and $H_2$ for each half-ring. Multiplying these field strengths by the lengths of the half-rings and adding the obtained values, we find the value of the m.m.f. required to create the specified induction in the circuit. Having found this value on the sinusoid $F_n(i)$, we plot, on the corresponding scales, the specified induction value and the values of the m.m.f. in each of the half-rings. Carrying out the same operations for other induction values, we obtain the curves $F_1$, $F_2$, $B_{1,2}$. Let us note that the curves $F_1=f(\omega t)$ and $F_2=f(\omega t)$ vary non-sinusoidally with time, despite the fact that $F_n(i)$ is a sinusoidal function of time. The time interval between the instants at which the field-strength (current) sinusoid $F_n(i)$ and the induction $B_{1,2}$ pass through zero is expressed by the segment O—a on the $\omega t$ axis.

Let us now replace the inspected half-ring 2 by a half-ring having dynamic hysteresis loop 4 (Fig. 2). With such a replacement, at an unchanged total m.m.f., a redistribution of the m.m.f. falling on each of the half-rings occurs. In this case the maximum value of the induction in the half-rings decreased, and loop 1 changed into loop 3. We note that loop 4, in comparison with loop 2, has a considerably greater coercive force, but the same residual induction. Carrying out constructions analogous to the preceding case (with the same amplitude value $F_{mn}$), we obtain the m.m.f. curves $F_3$ and $F_4$, corresponding to loops 3 and 4, and the inductions $B_{3,4}$. From consideration of the resulting constructions it is seen that the time interval between the instants at which the m.m.f., or equivalently the field strength, and the induction pass through zero has increased considerably. Hence one may conclude that an increase in the coercive force of the material of the inspected half-ring 2 has led to an increase in the measured time interval between the instants at which the induction and the field strength pass through zero.

However, the question remains unresolved as to whether the change in the residual induction of the material of the half-rings affects the indicated interval (in the latter case a material with loop 4 was chosen, having the same residual induction as loop 3). To clarify this question, let us replace half-ring 1 by a half-ring having loop 5, with the same coercive force as loop 3 but with a greater residual induction. In Fig. 2 only the lower part of the loop is shown. Performing the same constructions again, we obtain the m.m.f. curve $F_5$, corresponding to loop 5, $F'_4$, corresponding to loop 4, and the induction curve $B_{5,4}$. It is seen from the figure that if, in the preceding case considered, the curve $F_4$ intersected the $\omega t$ axis to the left of zero, then now the curve $F'_4$, corresponding to the same loop 4, intersects the $\omega t$ axis to the right of zero. The curves $F_3$ and $F_5$ have also changed places; however, the time interval between the instants at which the induction and the field strength pass through zero has not changed and has remained equal, on the scale, to the segment O—б. Thus, one may conclude that a change in the residual induction of the material of any of the half-rings does not lead to a change in the time-

does not lead to a change in the interval; the time interval changes only when the coercive forces change.

If the coercive force of the monitored half-ring is measured as a quantity proportional to the instantaneous value of the magnetizing current or of the m.m.f. \(F_n\) at the moment when the induction passes through zero, then, for the case of half-rings with loops 3 and 4, we would find that it is expressed by the segment \(б—д\) (Fig. 2, lower part). The m.m.f. \(F_c\), expressed by this segment, may be represented as a certain “resultant” for both half-rings of the field strength \(H_{pc}\), multiplied by the total length of the half-rings \(l_1 + l_2 = 2l\):

\[ F_c = H_{pc} \cdot 2l . \]

But the total m.m.f. consists of two parts distributed over the half-rings:

\[ F_c = F_{c3} + F_{c4} = H_{c3} \cdot l_1 + H_{c4} \cdot l_2, \]

where \(H_{c3}, H_{c4}\) are the coercive forces corresponding to loops 3 and 4. But since \(l_1 = l_2 = l\), then

\[ F_c = H_{pc} \cdot 2l = (H_{c3} + H_{c4})l, \]

whence

\[ H_{pc} = \frac{H_{c3} + H_{c4}}{2}, \]

i.e., the measured “resultant” coercive force is equal to the arithmetic mean of the coercive forces of the half-rings making up the series magnetic circuit.

In an alternating-current coercimeter the time interval expressed in Fig. 2 by the segment \(0—б\) is measured. Let us denote \(0—б = \varphi\). It follows directly from the figure that \(F_c = F_{nm}\sin\varphi\), where \(F_{nm}\) is the amplitude value of the total m.m.f. For many steels subject to inspection the angle \(\varphi\) is rather small, so that it may be assumed that \(\sin\varphi \simeq \varphi\). Therefore one may write

\[ \varphi = \frac{F_c}{F_{nm}} = \frac{H_{pc} \cdot 2l}{F_{nm}} . \]

Substituting into the formula obtained the expression for \(H_{pc}\), written in terms of \(H_{c3}\) and \(H_{c4}\), and replacing \(F_{nm}\) by \(H_{nm} \cdot 2l\), where \(H_{nm}\) is the amplitude of the “resultant field strength,” we obtain

\[ \varphi = \frac{H_{c3} + H_{c4}}{2 \cdot H_m}. \]

Thus, a direct connection has been established between the measured interval and the coercive forces. It also follows from it that the smaller the coercive force of the magnetic core in comparison with the coercive force of the material being inspected, the more accurately the measurement will be made. Another conclusion following from the last formula is that the interval \(\varphi\) decreases as the amplitude of the magnetizing field \(H_m\) or current \(I_m\) increases. However, the question of selecting the magnitude of the magnetizing field is not part of the task of this work, and we shall not consider it here.

All that has been said above applies to the operating mode of the magnetic circuit which in electrical engineering is usually called the specified-current mode. In this mode the magnetizing current is sinusoidal and the induction is nonsinusoidal. With the inverse relationship between the shapes of the current and induction curves we have the specified-voltage mode: with a specified sinusoidal voltage applied to the magnetizing winding, the amplitude of the induction \(B_n\) in the half-rings is a constant quantity, and the change of induction with time occurs according to a sine law for any form of the remagnetization loop of each of the half-rings (Fig. 1б).

To obtain such a mode, it is necessary that the inductive resistance of the magnetizing winding be much greater than the active resistance of the winding and of the current source.

Let us consider the operation of the magnetizing device (Fig. 1b) under the same conditions as were assumed for the prescribed-current mode. Let loop 1 in Fig. 3 refer to half-ring 1 (the magnetic core), and loop 2 to half-ring 2 made of the material being inspected. Since the magnetic circuit is still a series circuit, the induction values at any instant of time will be identical in each of the half-rings. In the lower part, the letter B denotes the sinusoidal variation of induction with time (as a function of \(\omega t\)). The construction of the m.m.f. (or field-strength) curves is carried out approximately in the same way as in the case of the prescribed-current mode. Through the remagnetization loops, at some definite induction value, draw a line parallel to the \(H\) axis. From the points where it intersects the branches of the loops, the field strength at a definite instant of time in each of the half-rings can be determined. Multiplying it by the length \(l_1=l_2=l=1\), we obtain the m.m.f.’s \(F_1, F_2\) required to produce the selected value of induction in each of the half-rings. Adding these m.m.f.’s, we find the value of the total m.m.f. \(F_{2,1}\) required to carry the magnetic flux through both half-rings. On another scale, the total m.m.f. expresses the time variation of the current \(i\) in the magnetizing winding. Carrying out the same operations for other induction values, from the obtained values \(F_{n2,1}, F_2\) at the corresponding inductions \(B\) we construct the dependences \(F_{n2,1}=f(\omega t)\) and \(F_2=f(\omega t)\) on time—the lower part of Fig. 3. The time interval between the zero crossings of the induction and of the current \(i\) is then expressed by the segment \(A-a\).

Replace the right half-ring 2 (Fig. 1b), whose dependence of induction on field strength is described by loop 2, by a half-ring with loop 3. Let us note at once that the coercive force of loop 3 is considerably greater than the coercive force of loop 2 \((H_{c3}>H_{c2})\), while their residual inductions are identical. Carrying out constructions analogous to the preceding case, we obtain the curves \(F_{n1,3}=f(\omega t)\) and \(F_3=f(\omega t)\). We see that the curves \(F_{n1,3}=f(\omega t)\) and \(F_{n1,2}=f(\omega t)\) intersect the \(\omega t\) axis at the same point \(a\), i.e., the interval between the instants when the current and the induction pass through zero has remained the same, despite the fact that the coercive force of half-ring 3, substituted for half-ring 2, has increased considerably. Consequently, in the prescribed-voltage mode the time interval between the instants when the current and the induction pass through zero does not depend on changes in the coercive force of the inspected products.

The intersection of the curves \(F_{n1,2}\) and \(F_{n1,3}\) with the \(\omega t\) axis occurs at induction values for which the field strength in one half-ring is equal in absolute value and opposite in sign to the field strength in the other: \(H_1=-H_2\) (Fig. 3, top).

Let us suppose that the half-ring made of the inspected material has a loop intersecting the \(B\) axis at point б. In the upper part of Fig. 3 only part of this loop is shown, designated by the numeral 4. The line for which \(H_4=-H_1\) will move upward; the point \(d\) of intersection of the m.m.f. curve with the \(\omega t\) axis will also move toward smaller values of \(\omega t\), i.e., the time interval will increase. On this basis one may conclude that the time interval varies depending on the change in residual induction of the half-rings.

We have considered loop 4, the segment of whose descending branch near the \(B\) axis has the same inclination to the \(B\) axis as loop 3. However, such a case is possible when the loop intersects the induction axis \(B\) at point б, and its inclination to the \(B\) axis is such that it intersects loop 3 at point C, in

(Figure; no caption visible on this page.)

Visible labels and annotations in the graph:

• \(B\)
• \(H\)
• \(\omega t\) (electrical degrees)
• \(H_1 - H_2\)
• \(C\)
• \(H_{c3}\), \(H_{c2}\), \(H_{c1}\)
• \(F_3\), \(F_2\), \(F_{\mathrm{p}3,1(i)}\), \(F_{\mathrm{p}3,1}\)
• \(\alpha\), \(\beta\)
• \(n\)
• Curve and point labels visible: \(1\), \(2\), \(3\), \(4\), \(5\), \(0\), \(12\), \(14\)
• Axis/tick numerals visible include: \(30\), \(20\), \(10\), \(0\), \(2\), \(4\), \(6\), \(8\), \(12\), \(16\), \(40\), \(60\), \(90\), \(120\), \(180\)

for which the condition \(H_3=-H_1\) is satisfied. A part of such a loop is denoted by the number 5. When a half-ring with such a loop is placed in place of half-ring 3, we will observe no change in the time interval. Therefore, when determining the possibility of measuring residual induction in the specified-voltage mode, it is first necessary to establish how the branches of the loops near their intersection with the induction axis \(B\) vary from product to product.

Fig. 4

The conclusions obtained above for both operating modes of the magnetic circuit will be valid only if the saturation induction of the half-ring serving as the magnetic core is considerably greater than the saturation induction of the material being tested. This means that, when the tested half-ring is saturated, the magnetic core is far from saturation and the m.m.f. required to conduct the flux through it is small.

In Fig. 3 (in the lower part), the segment \(AC\), to scale, expresses the m.m.f. \(F_{n\,1,2}\) at the moment when the induction passes through zero. From the construction it follows that it is equal to the sum of the segments \(AB\) and \(BC\), which represent the m.m.f. expended on conducting the flux through each half-ring. Passing to field strengths, as in the case of the specified-current mode, we obtain

\[ H_{pc}=\frac{H_{c1}+H_{c2}}{2}. \]

Thus, in the specified-voltage mode it is possible to measure the coercive force of the dynamic hysteresis loop if the strength of the magnetizing field (or the current) is measured at the moment when the induction passes through zero.

A coercimeter operating on this principle can be built according to the block diagram shown in Fig. 4. A magnetizing current flows through coil 1. Block 2 generates a pulse at the moment when the induction passes through zero. The pulse is fed to input “In. 1” of block 3, which, during the action of the pulse, measures the instantaneous value of the voltage taken from resistance \(R\) and proportional to the magnetizing current. IP is an indicating instrument, which can be graduated in units of coercive force.

REFERENCES

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