Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Volume 141　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1966

Low-Frequency Amplitude-Phase Analyzer for Studying the Relationships Between the Character of a Signal Spectrum and the Hardness of Ferromagnetic Materials

M. S. Roitman, E. I. Tsimbalist, N. P. Fefelov, V. Ya. Supyan

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

The use in industry of new structural materials and production technologies makes it necessary to develop new and improve existing methods for monitoring the quality of materials. At present, a whole series of methods and instruments has been developed that make it possible to monitor the parameters of specimens under test. With the aid of such instruments it is possible to solve separately the entire range of problems arising under production conditions, such as precise measurement of hardness, investigation of structure, measurement of the depth of cementation, surface hardening, etc. [1, 2, 3].

A number of the described methods for assessing materials by their qualitative characteristics are unsuitable for inspecting products in mass production. This applies above all to mechanical test methods, such as testing the hardness of materials by the Brinell, Rockwell, and Vickers methods [1].

In this connection, electromagnetic methods of inspection have found wide application. Instruments for hardness control are based on the use of a functional relationship between hardness and one of the basic electromagnetic characteristics—coercive force, residual induction, and magnetic permeability. However, the overwhelming majority of existing instruments do not meet practical requirements. Moreover, up to now there has been no scientifically grounded theory according to which the necessary apparatus could be created.

This is determined by a number of reasons, in particular by the absence in many cases of an unambiguous relationship between the hardness of a material and any electromagnetic characteristic.

The purpose of the present work was to create a low-frequency amplitude-phase harmonic analyzer that would make it possible experimentally to establish correlation functions between the spectrum of the signal at the output of the measuring circuit, the electromagnetic characteristics, and the structural-mechanical properties of ferromagnetic materials. Investigation of these dependences should clarify the question of which amplitude-phase changes in the spectrum of the output voltage depend most strongly on the hardness of the material for a given grade of steel.

This will make it possible to create accurate and convenient instruments for electrical monitoring of the hardness of ferromagnetic materials.

The setup (Fig. 1) is a combination of an amplitude-stable generator (Fig. 2), a measuring circuit, and indicators of modulus

and phase of the 1st (50 Hz), 3rd (150 Hz), and 5th (250 Hz) harmonics (Figs. 3, 4, 5). The generator voltage (Fig. 2) is stabilized by introducing into the feedback circuit a thermistor with indirect heating (TKP-300). The sensor (inductance coil \(L\)) operates in a mode close to that of a specified current, owing to the series-connected resistance \(R\).

Fig. 1. Block diagram of the harmonic analyzer.

Labels in the diagram: generator \(\sim 50\) Hz; \(U_0\) compensator; from sensor; amplifier; 100-Hz filter; cathode follower; phase shifter; 50-Hz filter; frequency multiplier 3; frequency multiplier 5; rectifier; power supply \(\sim 6.3\) V, \(+250\) V; SD; \(\varnothing_1\), \(\varnothing_2\).

Note:
1 — indicators of harmonic phase
2 — indicators of harmonic amplitude

In the absence of a specimen, the voltage drop across the coil is compensated by adjustments of the phase shifter (\(R_{25}\)) and the divider \(R_{25}\). Introducing a ferromagnetic body into the sensor will lead to the appearance of a voltage at the output of the measuring circuit. Since the sensor operates in the specified-current mode, the magnetic-field strength will practically be a sinusoidal function of time, while the induction will be nonsinusoidal.

The dependence of induction on time for a sinusoidal dependence of field strength (in the case of a symmetric dynamic hysteresis loop) has the form

\[ B(t)=B_1\sin(\omega t-\psi_1)+B_3\sin(3\omega t-\psi_3)+ \]
\[ +B_5\sin(5\omega t-\psi_5)+\ldots+B_n\sin(n\omega t-\psi_n). \]

It is clear that the amplitudes and phases of the odd harmonics of the field induction, as well as the emf of self-induction in the reactive coil, carry information about the electromagnetic properties of the core material under investigation.

The composition of, and the relationships between, the spectral components of the self-induction emf \(e\) will be determined by the course of the hysteresis loop and by its principal characteristics (the limiting magnetization \(B_m\), coer-

Block circuit

Notes:
○ — adjustment on the front panel
⊕ — adjustment on the front panel under the chassis
● — adjustment on the chassis under the chassis

Fig. 2. Schematic diagram of the low-frequency generator.

Translated inscriptions in the figure:

• Gain adjustment
• Gain adjustment of the 50-Hz channel
• 90° setting

Note:

○ — adjustment on the front panel
● — adjustment on the rear panel
PP-1–8 — PEV type

Fig. 3. Schematic diagram of the 50-Hz measuring channel.

Gain of the 150-Hz channel

Gain

90 degrees setting

Note

○ — adjustment on the front panel
● — adjustment on the chassis, by screwdriver slot
[[unclear: “PP1 = 1 … P13A”]]

Fig. 4. Basic circuit diagram of the 150-Hz measuring channel.

In the circuit diagram, the following annotations are visible: “Gain of the 250-Hz channel”; “Gain”; “50-Hz setting.”

Note.

○ — adjustment on the upper panel
\(P_1\)–\(P\)–\(R_{33}\)

● — adjustment on the chassis, by screwdriver slot

Fig. 5. Schematic diagram of the 250-Hz measuring channel.

coercive force \(F_c\), magnetic permeability \(\mu\)). These enumerated characteristics, in turn, indirectly characterize the hardness of the material.

Finding the amplitudes and phases of the harmonics in the voltage on the reactive coil reduces to finding the coefficients of the Fourier series. Indeed, let, as a rough approximation, the hysteresis loop be approximated by four broken straight lines (Fig. 6).

Then we obtain equations characterizing the idealized hysteresis loop:

1. On segment \(1 \div 2\), \(B(\omega t)=\mu(H-F_c)\).

2. On segment \(2 \div 3\), \(B(\omega t)=B_m=\mathrm{const}\).

3. On segment \(3 \div 4\), \(B(\omega t)=\mu(H+F_c)\).

4. On segment \(4 \div 1\), \(B(\omega t)=-B_m=\mathrm{const}\).

Owing to the flow of current through the reactive coil, an alternating magnetic-field intensity is produced:

\[ H=H_m\sin\omega t. \]

The e.m.f. of self-induction is

\[ e=-L\frac{d\Phi}{dt}=-\frac{L}{S}\frac{dB}{dt}= \begin{cases} -\dfrac{L\mu W}{S}\,I_m\cos\omega t & \text{(on seg. }1\div2\text{ and }3\div4),\\[6pt] 0 & \text{(on seg. }2\div3\text{ and }4\div1). \end{cases} \]

The function \(e(\omega t)\) will contain only odd harmonics

\[ \dot A_{2n+1}=\frac{2}{\pi}\int_{\omega t_1}^{\pi/2} -\frac{4\mu W}{S}\,I_m\cos\omega t\,e^{-j(2n+1)\omega t}\,d\omega t, \]

where \(n=0,1,2,3\ldots\)

\(\dot A_{2n+1}\) is the amplitude of the corresponding harmonic. For example, for the third harmonic the amplitude is equal to

\[ A_3=\frac{4\mu L F_c}{\pi S}\left(1-\frac{F_c}{H_m}\right), \tag{1} \]

and the phase

\[ \psi_3=2\arcsin\left(\frac{2F_c-H_m}{H_m}\right)-\frac{\pi}{2}. \tag{2} \]

From the formulas given it is clear that, in the prescribed-current mode, the amplitude of the third harmonic depends (1) both on the magnetic permeability of the core and on the coercive force. The phase of the 3rd harmonic, however, is unambiguously related to the change in the coercive force of the material (2).

The nonsinusoidal voltage from the output of the measuring circuit is fed to three channels, in each of which the amplitude and phase of the corresponding harmonic are determined.

To measure the amplitude and phase of the voltage, two synchronous detectors are used, whose control voltages are shifted by \(90^\circ\). By adjusting the phase shifter VTM-4, a zero reading of the phase indicator is obtained. In this case the amplitude is read from the scale of the electrical measuring instrument \((U_2)\), and the phase from the dial of the phase shifter.

The control voltage to the synchronous detectors in the channels for measuring the 3rd and 5th harmonics is supplied from the generator after frequency multiplication and corresponding filtering of the voltage of Figs. 4 and 5.

Fig. 6. Approximation of the hysteresis loop.

When a synchronous detector is switched by a sinusoidal voltage, for example 150 Hz, voltages with frequencies of 50 Hz and 250 Hz are not rectified, i.e., nearly ideal selectivity is achieved.

In practice, after frequency multiplication a certain magnitude of the voltage of the first harmonic remains in the control voltage. This leads to a reduction in selectivity. The analysis carried out by the authors shows that the rectification coefficient of the first harmonic by a synchronous detector when switched by higher harmonics is equal to

\[ \frac{\sqrt{2}}{\pi} q \cos \varphi, \tag{3} \]

where \(q\) is the ratio of the voltage of the first harmonic to the voltage at the switching frequency.

\(\varphi\) is the phase shift between the first harmonics in the signal and switching circuits.

The complete derivation of the formula presented is not given by us because of the very great bulkiness of the exposition. It follows directly from (3) that, in the channels for measuring higher harmonics, good filtering of the first harmonic is necessary, while the phase distortions of the useful signal must be minimal. Taking this into account, plug-in filters have been introduced in the higher-harmonic channels.

REFERENCES

1. USSR, State Standards. Methods of Mechanical and Technological Testing of Metals (official edition), Standartiz, Moscow, 1960.

2. Modern Methods of Nondestructive Testing of Materials. Edited by S. T. Nazarov, Mashgiz, Moscow, 1961.

3. I. I. Kifer. Testing of Ferromagnetic Materials. Gosenergoizdat, Moscow–Leningrad, 1962.