On the Dynamics of Direct Current Electromagnets
Abstract Generated abstract
The paper addresses the calculation of dynamic transient processes in direct current electromagnets, focusing on time dependent current and armature motion, or velocity for a U shaped electromagnet with a transverse armature. It derives finite difference forms of the electrical and mechanical equations and combines them with flux linkage versus displacement curves to produce a graphical calculation procedure for successive motion intervals. The method is intended to require relatively simple computations and to limit error accumulation through repeated joint graphical solution of the governing equations. Applied to a sample electromagnet, the calculated dynamic characteristics reportedly agreed with oscillograms within about 10 to 15 percent, with discrepancies attributed mainly to inaccuracies in the flux linkage characteristics.
Full Text
On the Dynamics of Direct-Current Electromagnets
V. N. Gurnitskii
(Recommended by the scientific seminar of the Electromechanical Faculty)
A large number of works [Refs. 1–8, etc.] have been devoted to the calculation of the dynamic modes of direct-current electromagnets. The dynamic characteristics of an electromagnet provide a real picture of all the complex electromagnetic, electromechanical, and, ultimately, thermal processes occurring in it.
However, up to the present time there has been no sufficiently simple and intuitive method that would make it possible, with the aid of a small number of computational and graphical operations and with a simple mathematical apparatus, to calculate the dynamic transient processes of an electromagnet. The aim of the present work is, to some extent, to fill this gap.
The principal dynamic characteristics of an electromagnet are considered to be the dependences of the armature travel and of the current in the winding on time. For a U-shaped electromagnet with a transversely moving armature, which is an element of a linear electromagnetic motor (Fig. 1), the principal dynamic characteristics should be regarded as the curves of armature velocity and current in the winding.
The transient processes in an electromagnet, as is known, are completely described by the following equations
\[ U = ir + \frac{d\psi}{dt}, \tag{1} \]
\[ m\frac{d^2 x}{dt^2} = m\frac{dV}{dt} = F_{\mathrm{t}} - \sum F_{\mathrm{pr}}, \tag{2} \]
where \(U\) is the voltage applied to the winding;
\(i\) is the instantaneous value of the current;
\(r\) is the active resistance of the winding circuit;
\[
\frac{d\psi}{dt}
\]
is the rate of change of the flux linkage of the winding with time;
Fig. 1. U-shaped electromagnet with a transversely moving toothed armature.
\(m\) — the mass of the moving parts reduced to the armature;
\(x\) — armature travel;
\(V\) — instantaneous armature velocity;
\(F_{\mathrm{t}}\) — electromagnetic tractive force developed by the armature;
\(\sum F_{\mathrm{pr}}\) — total opposing force.
Let us express from equations (1) and (2) \(V=\dfrac{\Delta x}{\Delta t}\) in finite differences. Omitting the intermediate transformations, from equation (1) we obtain
\[ V'=\frac{\left[U-\left(i_{\mathrm{p}}+\frac{\Delta i}{2}\right)r\right]\Delta x}{\Delta\psi}, \tag{3} \]
and from equation (2)
\[ V''=\frac{V_{\mathrm{p}}}{2}+\sqrt{\frac{V_{\mathrm{p}}^{2}}{4}+\frac{\Delta x}{m}\left(F_{\mathrm{t}}-\sum F_{\mathrm{pr}}\right)}. \tag{4} \]
where \(\dfrac{\Delta i}{2}\) and \(\dfrac{\Delta\psi}{2}\) are the mean values of the increments of current and flux linkage over the path segment \(\Delta x\);
\(i_{\mathrm{p}}\) — the value of the current at the end of the interval preceding the one under consideration;
\(V_{\mathrm{p}}\) — the armature velocity at the end of each preceding interval.
The electromagnetic tractive force \(F_{\mathrm{t}}\) on each interval is approximately determined as
\[ F_{\mathrm{t}}=\frac{i_{\mathrm{p}}+\dfrac{\Delta i}{2}}{2}\cdot\frac{\Delta\psi}{\Delta x}. \tag{5} \]
To construct the dynamic characteristics of an electromagnet it is necessary to know the dependences of the variation of the flux linkage of its winding along the armature displacement path. Let us construct these characteristics (Fig. 2) for different currents differing by \(\Delta i=\mathrm{const}\).
Before the beginning of the armature’s equation of motion, the magnetic circuit of most electromagnets is not saturated; therefore the current and the pickup time are determined by the usual formulas
\[ i_{\mathrm{t}}=\sqrt{\frac{2\left(\sum F_{\mathrm{pr}}\right)_{\mathrm{n}}}{\left(\dfrac{dL}{dx}\right)_{\mathrm{n}}}}, \tag{6} \]
\[ t_{\mathrm{c}}=\frac{L_{\mathrm{n}}}{r}\cdot\ln\frac{1}{1-\dfrac{r}{U}\cdot i_{\mathrm{t}}}, \tag{7} \]
where \(\left(\sum F_{\mathrm{pr}}\right)_{\mathrm{n}}\) is the total opposing force at the moment of the beginning of armature motion;
Fig. 2. Construction of the transient curve of flux linkage.
\(\left(\dfrac{dL}{dx}\right)_{\mathrm{n}}\) — derivative of the winding inductance at the initial point of the path (determined by the tangent method);
\(L_{\mathrm{n}}\) — initial inductance.
The mean value of the current \(i\) over the first segment of motion \(\Delta x_1\) is equal to
\[ i_1 = i_{\mathrm{r}} + \frac{\Delta i}{2}. \]
Suppose that the armature has moved from the initial position \(x_{\mathrm{n}}\) by a small distance \(\Delta x_1'\); from the curves \(\psi(x)\) (Fig. 2) we find the corresponding increment \(\Delta \psi_1'\). Using formulas (3), (4), (5), we determine \((V')_1'\) and \((V'')_1'\).
For another increment \(\Delta x_1''\), in an analogous manner we find \((V')_1''\) and \((V'')_1''\).
Let us connect the points \((V')_1'\) and \((V')_1''\), and also \((V'')_1'\) and \((V'')_1''\), by straight-line segments (Fig. 3). Their intersection gives the graphical solution of equations (3) and (4) on the first interval: the velocity \(V_1\) and the path \(x_1\).
Fig. 3. Construction of the velocity characteristic of an electromagnet armature as a function of path.
The change of all the other quantities on the segment \(\Delta x_1\) is then readily determined. On all subsequent segments the constructions are carried out in the same way. Some quantities in the course of the dynamic process (current, flux linkage) may decrease. Then their increments must be taken with the “minus” sign. In this case the quantities themselves pass through an extremum, which is apparent in the course of the construction.
The considered method for constructing the dynamic characteristics of direct-current electromagnets was used in calculating the dynamic characteristics of the electromagnet shown in Fig. 1. The errors of the computations in comparison with the corresponding oscillograms did not exceed 10–15%, which is explained primarily by inaccuracies in representing the dependences \(\psi(x)\) for \(i = \mathrm{var}\).
Conclusions
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Having no restrictions except for the neglect of eddy currents, the proposed method can evidently also be applied to the calculation of dynamic transient processes in other direct-current electromagnetic mechanisms.
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No accumulation of error occurs in the course of the construction, since at each interval it is corrected by the joint graphical solution of the working equations (3) and (4).
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In the case of a nonzero initial velocity of the armature, the dynamic characteristics of the electromagnet can be constructed using the same formulas.
References
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