OSCILLATIONS OF AN ELECTROMAGNETIC VIBRATOR OPERATING FROM A SINUSOIDAL VOLTAGE SOURCE WITHOUT A RECTIFIER

Yu. Ya. Kovylin

(Presented by the scientific seminar of the departments of the Faculty of Automatic Systems)

Figure 1 shows the simplest electromagnetic vibrator, consisting of a ferromagnetic armature 1, mounted on a spring 2, and an electromagnet 3. The winding of the electromagnet is supplied from a sinusoidal voltage source.

The initial system of differential equations may be written in the following form:

\[ U_M \cos(\omega t + \psi) = Ri + \frac{d}{dt}(Li), \tag{1} \]

\[ i(U_M \cos(\omega t + \psi))\,dt = Ri^2\,dt + d\frac{Li^2}{2} + P\,dY, \tag{2} \]

\[ M\frac{d^2Y}{dt^2} + H\frac{dY}{dt} + CY = P. \tag{3} \]

Fig. 1.

Here \(U_M\)—amplitude of the supply voltage;
\(\omega\)—angular frequency of the supply,
\(t\)—time;
\(\psi\)—initial phase of the voltage;
\(i\)—instantaneous value of the current in the winding;
\(L\)—inductance of the magnet winding;
\(R\)—active resistance of the electrical circuit:
\(P\)—electromagnetic force applied to the armature;

\(Y\)—the displacement of the armature, measured from the position of its static equilibrium at \(i=0\), with a positive displacement of the armature corresponding to a decrease in the gap in the magnetic circuit;

\(M\)—the reduced mass of the moving parts of the vibrator;

\(H\)—a coefficient taking into account energy absorption in the mechanical circuit;

\(C\)—the spring stiffness coefficient reduced to the armature.

From the simultaneous solution of (2) and (3), assuming linearity of the magnetization characteristic of the magnet, we obtain the formula for the tractive force of the magnet

\[ P=\frac{i^{2}}{2}\frac{dL}{dY}. \tag{4} \]

We shall write the general solution of equation (1) in the form

\[ i=\frac{U_M}{\omega L}e^{-\int \frac{R}{\omega L}\,d\omega t} \left[ \int e^{\int \frac{R}{\omega L}\,d\omega t}\cos(\omega t+\psi)\,d\omega t+K_0 \right]. \tag{5} \]

Here \(K_0\) is a constant of integration determined from the initial conditions.

The inductance of the magnet winding may approximately be expressed as

\[ L=\frac{L_0}{1-\dfrac{Y(t)}{S}}, \tag{6} \]

where \(L_0\) is the winding inductance at \(Y=0\);

\(S\)—the reduced air gap between the armature and the stator.

Let us assume that the law of motion of the armature can be represented by the Fourier series

\[ Y=A_0+A_2\sin 2\omega t-A_4\sin(4\omega t-\varphi_4)+\ldots \tag{7} \]

To simplify the exposition, we introduce the notation:

\[ \left. \begin{aligned} &\frac{Y}{S}=y;\quad \frac{A_0}{S}=a_0;\quad \frac{A_2}{S}=a_2;\quad \frac{A_4}{S}=a_4;\ \ldots\\[4pt] &\frac{H}{M}=h;\quad \frac{C}{M}=\omega_0^2;\quad \omega t=x;\\[4pt] &\frac{R}{\omega L_0}=q;\quad \frac{a_2q}{2}=\Delta. \end{aligned} \right\} \tag{8} \]

If the frequency of the system’s natural undamped oscillations \(\omega_0\) does not differ greatly from the frequency \(2\omega\) of the forced oscillations, then

\[ \int \frac{R}{\omega L}\,d\omega t \approx qx+\Delta\cos 2x. \]

Further, since usually \(\Delta<0.2\div 0.3\), even in the case when the oscillation amplitude is regulated by means of a rheostat, from (5), with sufficient accuracy for practical calculations, we have

\[ i=\frac{U_M}{\omega L}e^{-\Delta\cos 2x} \left\{ \frac{I_0(\Delta)}{1+q^2}\,[q\cos(x+\psi)+\sin(x+\psi)] +K_0e^{-qx} \right\}. \]

Here \(I_0(\Delta)\) is the modified Bessel function of the first kind of zero order.

For the steady-state regime \((x \to \infty)\), independently of the initial conditions, taking (6) into account we obtain a simple formula for the current:

\[ i=(1-y)\frac{U_M}{\omega L_0} I_0(\Delta)e^{-\Delta\cos 2x}\xi\sin(x+\psi+\delta), \tag{9} \]

where

\[ \xi=\frac{1}{\sqrt{1+q^2}};\quad \delta=\operatorname{arc\,tg} q. \]

Substituting (9) into (4), we determine the tractive force of the electromagnet:

\[ P=\frac{L_0}{2S}\left(\frac{U_M}{\omega L_0}\right)^2 I_0^2(\Delta)e^{-2\Delta\cos 2x}\cdot \frac{\xi^2}{2}\,[1-\cos 2(x+\psi+\delta)]. \tag{10} \]

We expand the factor \(e^{-2\Delta\cos 2x}\) on the right-hand side of (10) in a Fourier series,

\[ e^{-2\Delta\cos 2x}\sim I_0(2\Delta)-2I_1(2\Delta)\cos 2x+\cdots, \tag{11} \]

in which, to simplify the subsequent calculations, we shall retain only the first two terms, since for the above-specified values of \(\Delta\) this series converges very rapidly. Now, substituting (11) into (10), taking (8) into account, the differential equation (3) can be represented in the form

\[ \frac{d^2y}{dt^2}+h\frac{dy}{dt}+\omega_0^2y= \omega_0^2y_C I_0^2(\Delta)\frac{\xi^2}{2} (Q+B_2\sin 2x+C_2\cos 2x+ \]

\[ +B_4\sin 4x+C_4\cos 4x), \tag{12} \]

where

\[ y_C=\frac{L_0}{CS^2}\left(\frac{U_\Э}{\omega L_0}\right)^2; \tag{13} \]

\(U_\Э\) is the rms line voltage, equal to \(\dfrac{U_M}{\sqrt{2}}\);

\[ Q=I_0(2\Delta)+I_1(2\Delta)\cos 2(\psi+\delta); \tag{14} \]

\[ B_2=I_0(2\Delta)\sin 2(\psi+\delta); \tag{15} \]

\[ C_2=-2I_1(2\Delta)-I_0(2\Delta)\cos 2(\psi+\delta); \tag{16} \]

\[ B_4=-I_1(2\Delta)\sin 2(\psi+\delta); \tag{17} \]

\[ C_4=I_1(2\Delta)\cos 2(\psi+\delta). \tag{18} \]

As we see, the spectrum of the disturbing force agrees with the form of the solution (7) assumed earlier. Substituting (7) into (12) and equating the coefficients of identical trigonometric functions on the left and on the right, we obtain a system of equations for determining the unknown amplitudes and phases:

\[ a_0=y_C I_0^2(\Delta)\frac{\xi^2}{2}Q; \tag{19} \]

\[ a_2\left[1-\left(\frac{2\omega}{\omega_0}\right)^2\right] =y_C I_0^2(\Delta)\frac{\xi^2}{2}B_2; \tag{20} \]

\[ a_2h\frac{2\omega}{\omega_0^2} =y_C I_0^2(\Delta)\frac{\xi^2}{2}C_2; \tag{21} \]

\[ a_4\left[1-\left(\frac{4\omega}{\omega_0}\right)^2\right]\cos\varphi_4 +a_4 h\frac{4\omega}{\omega_0^2}\sin\varphi_4 =y_c I_0^2(\Delta)\frac{\xi^2}{2}B_4; \tag{22} \]

\[ -a_4\left[1-\left(\frac{4\omega}{\omega_0}\right)^2\right]\sin\varphi_4 +a_4 h\frac{4\omega}{\omega_0^2}\cos\varphi_4 =y_c I_0^2(\Delta)\frac{\xi^2}{2}C_4. \tag{23} \]

From (20) and (21), taking (15) and (16) into account, we find the formula for determining the frequency \(\omega_0\) for a given amplitude of the first harmonic of oscillation and the initial phase of the supply voltage:

\[ \frac{2\omega}{\omega_0} = \sqrt{ \left[ 1 \mp \sqrt{ \frac{y_c}{2a_2}I_0^2(\Delta)I_0(2\Delta)\xi^2 } \right]^2 - \left[ \frac{y_c}{a_2}I_0^2(\Delta)I_1(2\Delta)\xi^2 +\frac{h_0}{\omega_0} \right]^2 }; \tag{24} \]

\[ \operatorname{tg}2(\psi+\delta) = \frac{ \left(\dfrac{2\omega}{\omega_0}\right)^2-1 }{ \dfrac{y_c}{a_2}I_0^2(\Delta)I_1(2\Delta)\xi^2+\dfrac{h_0}{\omega_0} }. \tag{25} \]

The last two formulas were derived using Bock’s well-known hypothesis on the independence of the energy losses in the mechanical circuit from the oscillation frequency. Formula (24) makes it possible to construct the amplitude–frequency curve of the vibrator in the region of the principal resonance, under the assumption \(\omega=\mathrm{const}\) and \(\omega_0=var\). All values of the sum of angles determined from (25) lie within the limits

\[ 45^\circ<\delta+\psi<135^\circ. \]

The constant component of the displacement \(a_0\) is readily determined directly by means of (19), (14), or through the initial quantities:

\[ a_0 = y_c I_0^2(\Delta)\frac{\xi^2}{2} \left[ I_0(2\Delta)-2\frac{I_1^2(2\Delta)}{I_0(2\Delta)} \right] - a_2\frac{h_0}{\omega_0}\frac{I_1(2\Delta)}{I_0(2\Delta)}. \tag{26} \]

From the simultaneous solution of (22) and (23), taking (17) and (18) into account, the amplitude of the second harmonic of the armature oscillation and its initial phase are determined:

\[ a_4 = \frac{ y_c I_0^2(\Delta)I_1(2\Delta)\dfrac{\xi^2}{2} }{ \sqrt{ \left[1-\left(\dfrac{4\omega}{\omega_0}\right)^2\right]^2 + \left(\dfrac{h_0}{\omega_0}\right)^2 } }; \tag{27} \]

\[ \varphi_4=\gamma-90^\circ-2(\psi+\delta). \tag{28} \]

Here

\[ \gamma = \operatorname{arctg} \frac{ \dfrac{h_0}{\omega_0} }{ 1-\left(\dfrac{4\omega}{\omega_0}\right)^2 }; \quad 0\leq\gamma\leq180^\circ. \tag{29} \]

From (24) it is not difficult to determine the value of the force coefficient \(y_c\) required to maintain a given oscillation amplitude \(a_2\):

\[ \frac{y_c}{a_2} = \frac{ \dfrac{h_0}{\omega_0}I_1(2\Delta) }{ I_0^2(\Delta)\xi^2 \left( \dfrac{I_0^2(2\Delta)}{4}-I_1^2(2\Delta) \right) } + \]

\[ +\frac{ \sqrt{ \left(\frac{h_0}{\omega_0} I_1(2\Delta)\right)^2 + \left(\frac{I_0^2(2\Delta)}{4}-I_1^2(2\Delta)\right) \left\{\left[1-\left(\frac{2\omega}{\omega_0}\right)^2\right]^2+\left(\frac{h_0}{\omega_0}\right)^2\right\} } }{ I_0^2(\Delta)\xi^2\left(\frac{I_0^2(2\Delta)}{4}-I_1^2(2\Delta)\right) }. \tag{30} \]

The effective current drawn from the network is determined by the well-known formula

\[ i_{\mathrm{e}}=\sqrt{\frac{1}{2\pi}\int_0^{2\pi} i^2(x)\,dx}. \tag{31} \]

Substituting here the value of \(i\) from (9), and noting that in the region of the principal resonance \(a_4 \ll a_2\), we obtain, in the first approximation, without taking into account the influence of the second harmonic of the armature oscillation,

\[ i_{\mathrm{e}}= \frac{U_{\mathrm{e}}}{\omega L_0} I_0(\Delta)\xi \sqrt{ \left[(1-a_0)^2+\frac{a_2^2}{4}\right] \left[I_0(2\Delta)+I_1(2\Delta)\cos 2(\psi+\delta)\right] - (1-a_0)a_2 I_0(2\Delta)\sin 2(\psi+\delta) +\frac{a_2^2}{4}I_0(2\Delta) }. \tag{32} \]

The active power in the electric circuit is determined as

\[ w_a= \frac{U_{\mathrm{e}}^2}{\omega L_0} I_0(\Delta)\xi \left\{ (1-a_0)\left[I_0(\Delta)\sin\delta-I_1(\Delta)\sin(2\psi+\delta)\right] -\frac{1}{2}a_2 I_0(\Delta)\cos(2\psi+\delta) \right\}, \tag{33} \]

where, as in the preceding case, small terms proportional to \(a_4\) have been omitted.

Finally, the power factor, \(\cos\varphi\), of the electromagnetic vibrator when operating in the region of the principal resonance is equal to

\[ \cos\varphi= \frac{ (1-a_0)\left[I_0(\Delta)\sin\delta-I_1(\Delta)\sin(2\psi+\delta)\right] -\frac{1}{2}a_2 I_0(\Delta)\cos(2\psi+\delta) }{ \sqrt{ \left[(1-a_0)^2+\frac{a_2^2}{4}\right] \left[I_0(2\Delta)+I_1(2\Delta)\cos 2(\psi+\delta)\right] - (1-a_0)a_2 I_0(2\Delta)\sin 2(\psi+\delta) +\frac{a_2^2}{4}I_0(2\Delta) } }. \tag{34} \]

When a half-wave rectifier is present in the magnet winding circuit, the problem is solved, in general, by an analogous method. However, in view of the special features that arise, this problem is considered separately.