Investigation of the Operation of a Device with an Electromagnetic Inverted Transducer
Abstract Generated abstract
This paper analyzes an automatic device for measuring the dynamic friction torque of instrument ball bearings, treating it as a low-power closed-loop automatic-control system with an electromagnetic inverse transducer in the feedback path. Using a structural-functional diagram, the authors derive transfer coefficients for the mechanical moving part, amplifier, electromagnet coil, and feedback transducer, then obtain the device transfer function and discuss its dependence on output current and measured torque. The analysis indicates that static sensitivity is determined primarily by the feedback transducer, while instability in the forward conversion chain has limited influence on measurement accuracy, and that appropriate positioning of the movable core can reduce the effect of additional feedback due to core displacement.
Full Text
INVESTIGATION OF THE OPERATION OF A DEVICE WITH AN ELECTROMAGNETIC INVERTED TRANSDUCER
E. F. Kolotii, I. G. Leshchenko
(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)
To investigate the properties of an automatic device measuring the dynamic friction torque of instrument ball bearings, the operating principle of which is described in [1], we propose to apply the mathematical apparatus of automatic-control theory, since the indicated device is a low-power automatic-control system. It is useful to begin the analysis by considering the structural-functional diagram of the device, which is presented in Fig. 1. As can be seen, the device under consideration is a closed automatic-control system. The input quantity of the system is the measured friction torque \(M_x\),
Fig. 1. Structural-functional diagram of an automatic device with an electromagnetic inverted transducer
the difference between the measured and compensating torques \(\Delta M\) causes, at the output of the direct-conversion chain \([W_1(p), W_2(p), W_3(p), W_4(p)]\), the appearance of a current, measured by the indicating instrument \(PP\), and entering the feedback circuit to create the compensating torque \(M_\beta\).
In the steady-state regime \((\Delta M = 0)\), from the readings of the instrument measuring the current \(I_{\text{out}}\), one can judge the magnitude of the friction torque \(M_x\).
In the feedback circuit there is only one electromagnetic inverted transducer, consisting of a coil and a ferromagnetic core drawn into it.
The purpose of the electromagnetic inverted transducer is to convert the output current of the instrument into a compensating moment \(M_\beta\) applied to the moving part. Rotation of the moving part through an angle \(\alpha\) leads both to a change in the current through the electromagnet coil and to a change in the position of the core relative to the coil. Both of these factors (the current \(I_{\text{out}}\) and the angle \(\alpha\)) cause a change in the magnitude \(M_\beta\); however, as is shown below, under certain conditions the additional feedback \((\alpha \to h \to M_\beta)\) present in the instrument can be almost eliminated, or, at least, its influence can be significantly reduced. Therefore such an instrument may, to some approximation, be regarded as a closed single-loop automatic-control system. The transfer function of the instrument will evidently be expressed as
\[ S(p)=\frac{I_{\text{out}}}{M_x}=\frac{\kappa(p)}{1+\kappa(p)\cdot \beta(p)}, \tag{1} \]
where
\[ \kappa(p)=\frac{I_{\text{out}}}{\Delta M} = W_1(p)\cdot W_2(p)\cdot W_3(p)\cdot W_4(p) \]
is the conversion coefficient of the forward-conversion chain;
\[ \beta(p)=\frac{M_\beta}{I_{\text{out}}}=W_\beta(p) \]
is the conversion coefficient of the feedback chain;
\(p\) is the operator.
Let us find the conversion coefficients of the individual elements of the instrument.
The equation of motion of the moving part of the inverted transducer under the action of the difference of moments \(\Delta M\), as is known, can be represented in the form
\[ J\frac{d^2\alpha}{dt^2}+P\frac{d\alpha}{dt}=\Delta M, \tag{2} \]
where \(J\) is the moment of inertia of the moving part;
\(P\) is the damping coefficient;
\(\alpha\) is the deflection angle of the moving part.
With zero initial conditions this equation in operator form will have the form
\[ (Jp^2+Pp)\alpha=\Delta M \]
and, consequently, the conversion coefficient of the first element of the forward-conversion chain is equal to
\[ W_1(p)=\frac{\alpha}{\Delta M}=\frac{1}{Jp^2+Pp}. \tag{3} \]
The conversion coefficient of the rotation angle \(\alpha\) into the input voltage of the amplifier \(U_{\text{in}}\), and the gain coefficient of the amplifier, will be considered constant and independent of frequency, i.e.,
\[ W_2(p)=\frac{U_{\text{in}}}{\alpha}=\kappa_2=\mathrm{const},\qquad W_3(p)=\frac{U_{\text{out}}}{U_{\text{in}}}=\kappa_3=\mathrm{const}. \tag{4} \]
The output voltage of the amplifier is expended in overcoming the active resistance \(R\) of the electromagnet coil, and also in overcoming its inductive reactance.
In accordance with this one may write
\[ U_{\text{out}}=L\frac{dI_{\text{out}}}{dt}+RI_{\text{out}}, \tag{5} \]
whence the transformation coefficient of the last link of the direct-conversion chain is equal to
\[ W_4(p)=\frac{I_{\text{out}}}{U_{\text{out}}}=\frac{1}{Lp+R}. \tag{6} \]
There is one electromagnetic inverse transducer in the feedback circuit, for which
\[ M_\beta=\frac{1}{2}I_{\text{out}}^2\frac{dL}{dx}, \tag{7} \]
whence
\[ W_\beta(p)=\frac{M_\beta}{I_{\text{out}}}=I_{\text{out}_0}\frac{dL}{dx}. \tag{8} \]
As L. F. Kulikovskii [2] showed, the quantity \(\frac{dL}{dh}\), which determines the tractive force acting on the electromagnet core, can in general form be expressed as
\[ \frac{dL}{dh}=-2L_{\text{cm}}mhe^{-mh^2}, \tag{9} \]
where \(h\) is the displacement of the electromagnet core;
\(L_{\text{cm}}\) is the maximum increment of the electromagnet-coil inductance when the core is fully inserted;
\(m\) is a coefficient depending on the ratios
\[ \lambda_{\text{n}}=\frac{l_k}{D_{\text{n}}} \quad \text{and} \quad \lambda_{\text{vn}}=\frac{l_k}{D_{\text{vn}}}. \]
On the other hand,
\[ \frac{dL}{dx}=\frac{dL}{dh}\cdot\frac{dh}{dx}=\frac{dL}{dh}\cdot\kappa, \tag{10} \]
where \(\kappa\) is the lever arm of action of the electromagnet’s tractive force.
From expressions (8), (9), and (10) we determine
\[ W_\beta(p)=2\kappa I_{\text{out}_0}L_{\text{cm}}mhe^{-mh^2}. \tag{11} \]
Fig. 2. Diagram of an electromagnet with a retracting core
After substituting expressions (3), (4), (6), and (11) into expression (1), we obtain
\[ S(p)= \frac{\kappa_2\kappa_3} {p(Jp+P)(Lp+R)+2\kappa\kappa_2\kappa_3 I_{\text{out}_0}L_{\text{cm}}mhe^{-mh^2}}. \tag{12} \]
Expression (12) makes it possible to assert that the transfer function (complex sensitivity) of the device under consideration does not remain constant when the output current \(I_{\text{out}}\) changes and, consequently, depends on the magnitude of the measured friction torque \(M_x\). Therefore, analysis of the dynamic properties of this device must be carried out both for small input values \(M_x\) and for large ones. For
To determine the frequency properties of the device, its speed of response, the quality of the transient process, etc., expression (12) must be analyzed by the usual methods of the theory of automatic control.
In the steady-state regime \((p=0)\), expression (12) assumes the simpler form:
\[ S(0)=\frac{\kappa_2\kappa_3}{2\kappa\kappa_2\kappa_3 I_{\text{out}}L_{\text{cm}}mhe^{-mh^2}}=\frac{1}{W_\beta(0)}, \tag{13} \]
i.e., the static sensitivity of the device under consideration is determined only by the conversion coefficient of the feedback circuit. This conclusion makes it possible to infer that the static error of the device will depend mainly only on the error of the inverse transducer. The instability of the transfer coefficients of the elements of the forward-conversion circuit cannot have a substantial effect on the measurement accuracy.
As for the additional feedback \((\alpha \to h \to M_\beta)\) present in the device, its influence on the magnitude of the moment \(M_\beta\) can be considerably reduced if the initial position of the movable core relative to the electromagnet coil is properly chosen. Figure 3 shows the dependence of the tractive force of the electromagnet on the position of the movable core at constant current through the electromagnet coil. Taking into account that the displacements of the movable core are sufficiently small, the working section \(h_p\) can be chosen near the maximum of the curve \(F=f(h)\). Then small displacements of the core will not cause any substantial change in the tractive force and, consequently, in the moment \(M_\beta\). The operation of an electromagnet with a pull-in core is discussed in greater detail in [2].
Fig. 3. Dependence of the tractive force of the electromagnet on the displacement of the core at \(I=\mathrm{const}\)
The simple analysis presented shows that investigation using the structural-functional diagram of the device makes it possible to take a deliberate approach to the selection of the device elements and to the determination of the requirements imposed on them.
References
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E. F. Kolotii, I. G. Leshchenko. On the question of measuring the friction moment of instrument ball bearings, Izvestiya TPI, vol. 141, 1955.
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L. F. Kulikovskii. Inductive Displacement Meters, GEI, 1961.