Effect of Constant Bias Magnetization of the Control Winding of a Two-Phase Induction Motor on Its Dynamic Properties
Abstract Generated abstract
This paper analyzes how the constant component of an electronic amplifier output affects the mechanical and dynamic properties of a two-phase induction motor used in AC tracking systems. Using an equivalent motor and amplifier circuit, the authors derive current equations and an averaged electromagnetic torque expression for the RD-09 motor, showing that the starting torque is not affected by constant bias magnetization while the braking torque and damping coefficient increase. Calculated and experimental mechanical characteristics indicate that bias magnetization can partly offset the adverse effect of high active resistance in the control winding circuit, with the increase in damping most pronounced at small signal levels near equilibrium.
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PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV
Volume 145 1966
EFFECT OF CONSTANT BIAS MAGNETIZATION OF THE CONTROL WINDING OF A TWO-PHASE INDUCTION MOTOR ON ITS DYNAMIC PROPERTIES
V. A. VOROB’EV, G. P. TARASOV, B. A. KHOMYAKOV
(Presented by the scientific seminar of the Department of Electrical Machines.)
In instrument tracking systems using alternating current, two-phase induction motors are often employed, the excitation windings of which are supplied by voltage from the industrial power network, while the control windings are supplied from a low-power electronic amplifier with considerable internal resistance. The amplifier output voltage has a constant component, which causes bias magnetization of the motor. In a number of works [1, 3], the influence of a large active resistance in the circuit of the control winding on the dynamic properties of the motor has been investigated. It is noted there that the presence of the indicated resistance significantly worsens the dynamic characteristics of the motor as an element of a tracking system. Below, the influence of constant bias magnetization of the control winding on the mechanical characteristic and the dynamic properties of the motor is considered.
Fig. 1. Connection circuit of the RD-09 motor.
Figure 1 shows the connection circuit of the RD-09 motor. In the investigated mode, the equivalent circuit of the motor and amplifier can be represented as in Fig. 2. The electronic amplifier is replaced by two equivalent sources of e.m.f. with a common internal resistance
$R_i$; $E_{y0}$ is a dc emf source, $E_y$ is an ac emf source (because of the filtering properties of the motor, it is sufficient to take only the first harmonic). In the linear regime, for the equivalent circuit one may write the system of equations.
a) The equations of the motor circuits [1]:
Fig. 2. Equivalent circuit of the motor and of the output stage of the electronic amplifier.
\[ L_y \frac{di_y}{dt}+r_y i_y+M_y \frac{di_a}{dt}=u_y(t), \]
\[ L_{\mathrm{в}} \frac{di_{\mathrm{в}}}{dt}+r_{\mathrm{в}}i_{\mathrm{в}}+M_{\mathrm{в}}\frac{di_{\mathrm{б}}}{dt}=u_{\mathrm{в}}(t), \]
\[ L_p\frac{di_a}{dt}+r_p i_a+\left(L_p i_{\mathrm{б}}+M_{\mathrm{в}}i_{\mathrm{в}}\right)\Omega+M_y\frac{di_y}{dt}=0, \]
\[ L_p\frac{di_{\mathrm{б}}}{dt}+r_p i_{\mathrm{б}}-\left(L_p i_a+M_y i_y\right)\Omega+M_{\mathrm{в}}\frac{di_{\mathrm{в}}}{dt}=0. \tag{1} \]
Here
\[ i_a=i_3\cos\Theta-i_4\sin\Theta \]
is the current in the rotor windings, referred to the axis of the control winding,
\[ i_{\mathrm{б}}=i_3\sin\Theta+i_4\cos\Theta \]
is the current in the rotor windings, referred to the axis of the excitation winding.
b) The equations of the motor supply circuits:
\[ E_{y0}+E_y(t)=E_i(t), \]
\[ R_i(i_c+i_y)+\frac{1}{C_y}\int i_c(t)\,dt, \]
\[ u_{\mathrm{в}}(t)=u_2(t)-\frac{1}{C_{\mathrm{в}}}\int i_{\mathrm{в}}(t)\,dt, \tag{2} \]
\[ u_y(t)=u_c=\frac{1}{C_y}\int i_c(t)\,dt. \]
c) The expression for the motor electromagnetic torque averaged over a current period [1]:
\[ M_{\mathrm{av}}=\operatorname{Re}\left(M_{\mathrm{v}}\dot I_a^{*}\dot I_a-M_{\mathrm{y}}\dot I_y\dot I_b^{*}\right). \tag{3} \]
Since \(E_{\mathrm{r}}\) and \(u_2\) are known (Figs. 1, 2), the system of equations (1–2) can be solved for the currents. We shall seek the solution in complex form. If the excitation winding is supplied from an industrial-frequency network, then one may take
\[ u_2(t)=U_{vm}\cos\omega t \]
or, in complex form,
\[ \dot U_2=\dot U_{vm}. \tag{4} \]
Provided that only the first harmonic of the emf \(E_{\mathrm{r}}\) of the equivalent generator feeding the motor control winding is taken into account, we obtain
\[ E_{\mathrm{r}}(t)=E_{y0}+E_{ym}\cos(\omega t+\varphi) =E_{y0}\cos(\omega t-\omega t)+E_{ym}\cos(\omega t+\varphi) \]
or, in complex form:
\[ \dot E_{\mathrm{r}}=E_{y0}e^{-j\omega t}+E_{ym}e^{j\varphi}. \tag{5} \]
Here the constant component is represented by a vector rotating in the reverse direction with respect to \(e^{j\omega t}\). Simplifying the system of equations (1, 2), we write:
\[ (r_y+j\omega L_y)\dot I_y+j\omega M_y\dot I_a-\frac{1}{j\omega C_y}\dot I_c=0, \]
\[ \left(r_v+j\omega L_v+\frac{1}{j\omega C_v}\right)\dot I_v+j\omega M_v\dot I_b=\dot U_2, \tag{6} \]
\[ (r_r+j\omega L_r)\dot I_a+L_r\Omega \dot I_b+M_v\Omega \dot I_v+j\omega M_y\dot I_y=0, \]
\[ (r_r+j\omega L_r)\dot I_b-L_r\Omega \dot I_a-M_y\Omega \dot I_y+j\omega M_v\dot I_v=0, \]
\[ R_i\dot I_y+\left(R_i+\frac{1}{j\omega C_y}\right)\dot I_c=E_{\mathrm{r}}. \]
Solving this system with respect to the stator and rotor currents at rotor rotational speeds \(\Omega<\Omega_{\mathrm{synchr}}\), we may neglect the terms proportional to \(\Omega^2\); the solutions will have the form:
\[ \dot I_y=a_y\dot E_{\mathrm{r}}+b_y\Omega\dot U_2, \]
\[ \dot I_v=a_v\dot U_2+b_v\Omega\dot E_{\mathrm{r}}, \tag{7} \]
\[ \dot I_a=a_a\dot E_{\mathrm{r}}+b_a\Omega\dot U_2, \]
\[ \dot I_b=a_b\dot U_2+b_b\Omega\dot E_{\mathrm{r}}. \]
Here \(a_i\) and \(b_i\) are complex quantities; they are determined by the electrical parameters of the motor and the amplifier.
Calculating the motor electromagnetic torque by formula (3), we obtain
\[ M_{\mathrm{av}}=M_{\mathrm{p}}-D\Omega, \tag{8} \]
where
\[ M_{\mathrm{p}}=K_1U_vE_y, \tag{9} \]
\[ D=K_2\left(E_y^2+\frac{1}{2}E_{y0}^{2}\right)+K_3U_v^2. \tag{10} \]
The coefficients \(K_i\) are determined by the electrical parameters of the amplifier motor and by the phase shift between \(\dot U_2\) and \(\dot E_i\); \(E_y\) and \(U_v\) are rms values.
Analyzing expressions (8, 9, 10), the following conclusions may be drawn:
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The active component of the electromechanical torque \(M_p\) (starting torque) does not depend on magnetization by the constant component.
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The braking torque \(M_g = D\Omega\) increases as the constant magnetization increases. The increase in the damping coefficient \(D\) with increasing magnetization partially compensates the negative influence of the active resistance in the control circuit.
Example. Let the RD-09 motor be connected according to the circuit of Fig. 1 (standard amplifiers U-109, U-119). In this case the magnetization changes with the change in signal level. The emf of the equivalent sources is determined by:
Fig. 3. Families of mechanical characteristics of the RD-09 motor:
— — — — when supplied from a voltage generator \((R_i = 0)\);
—·—·— without magnetization \((R_i = 6.5\ \text{k}\Omega)\);
——— with magnetization \((R_i = 6.5\ \text{k}\Omega)\);
\(a\)—\(E_y = 0,\quad E_{y0} = 270\ \text{V}\);
\(b\)—\(E_y = 70\ \text{V},\quad E_{y0} = 205\ \text{V}\);
\(v\)—\(E_y = 130\ \text{V},\quad E_{y0} = 149\ \text{V}\);
\(g\)—\(E_y = 205\ \text{V},\quad E_{y0} = 83\ \text{V}\).
\[ E_y = \frac{\psi}{\sqrt{2}}\,U_c, \tag{11} \]
\[ E_{y0} = \frac{2\sqrt{2}}{\pi}\,(E_a - E_y), \tag{12} \]
where \(\mu\) is the gain of the tube;
\(U_g\) is the voltage on the grids of the tubes;
\(E_a\) is the anode voltage (we take \(E_a = 300\ \text{V}\) and the internal resistance of the amplifier \(R_i = 6.5\ \text{k}\)).
Figure 3 shows families of mechanical characteristics of the RD-09 motor. The dashed line shows the family of characteristics when the motor is supplied from voltage generators without bias magnetization; the dash-dotted line, when there is an active resistance \(R_i = 6.5\text{k}\) in the control circuit, with bias magnetization also absent. The solid line is for supply from an electronic amplifier, i.e., with bias magnetization and with resistance \(R_i = 6.5\ \text{k}\) in the control circuit. The first two families of characteristics were obtained experimentally.
The last family was obtained by calculation; the damping coefficient was calculated from formulas (10, 11, 12). Formula (10) may be transformed for ease of calculation. If \(D\), the damping coefficient in the absence of bias magnetization, is determined from the second family, then the damping coefficient \(D'\) in the presence of bias magnetization will be:
\[ D' = D + \frac{1}{2} K_2 E_{y0}^{2}, \tag{13} \]
where \(K_2 = 6 \cdot 10^{-6}\ \frac{\text{g cm sec}}{\text{V}^2}\) for the RD-09.
The calculated data for the damping coefficient, which determines the dynamic properties of the motor, are summarized in Table 1.
Table 1
| EMF of sources | EMF of sources | Damping coefficients | Damping coefficients |
|---|---|---|---|
| alternating current \(E_y\) [V] | direct current \(E_{y0}\) [V] | without bias magnetization \(\frac{\text{g cm sec}}{\text{V}^2}\) | with bias magnetization \(\frac{\text{g cm sec}}{\text{V}^2}\) |
| 0 | 270 | 0.27 | 0.49 |
| 70 | 205 | 0.28 | 0.41 |
| 130 | 149 | 0.34 | 0.42 |
| 205 | 83 | 0.44 | 0.46 |
Thus, in conclusion it may be emphasized that bias magnetization, varying with the signal level, increases the damping coefficient \(D'\), especially at small signals (when the tracking system approaches the equilibrium position).
REFERENCES
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N. P. Vlasov. Theory of Linear Systems Operating on Alternating Current. Energiya, Moscow, 1964.
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D. B. Polonnikov. Electronic Amplifiers of Automatic Compensators. FM, 1960.
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E. I. Baranchuk. Izvestiya VUZov. Instrument Engineering, No. 5, 1963.