Commutation of an Asynchronous DC Motor
Abstract Generated abstract
The paper examines electromagnetic factors governing commutation in asynchronous DC motors, whose commutating behavior had not been sufficiently characterized. Using the motor vector diagram, the author derives expressions for the external-field commutating emf and for the angle between stator and resultant magnetomotive forces as functions of rotor resistance, rotor leakage reactance, magnetizing reactance, and slip. The analysis indicates that commutating emf is largest at zero slip and minimized at a critical slip set by rotor active resistance, while rotor inductive reactance generally worsens commutation, especially at high slip. The paper proposes commutation improvements, including reducing rotor leakage reactance, selecting rotor resistance to place critical slip near nominal operation, operating near that slip, adding short-circuited turns in stator slots, and using magnetic materials with wider hysteresis loops.
Full Text
COMMUTATION OF AN ASYNCHRONOUS DC MOTOR
N. A. Morshakov
(Recommended by the seminar of the Department of Electrical Machines and General Electrical Engineering)
At present, both in the USSR and abroad, a number of works have appeared devoted to asynchronous DC machines [1, 2].
The service life of these motors, as of ordinary DC machines, is determined by their commutating ability. However, the commutation of asynchronous DC motors has not yet been studied sufficiently, which does not make it possible to evaluate fully the operation of these motors. The purpose of the present article is to make up, in part, for the gap that exists in the field of commutation of asynchronous DC motors.
The basic circuit of an asynchronous DC motor is shown in Fig. 1 and requires no special explanation. An asynchronous DC motor differs from an ordinary asynchronous AC motor by the type of supply current, by the presence of commutator 1, and by a stator winding made in the manner of the usual armature windings of DC machines. The rotating field of an asynchronous DC motor, in contrast to asynchronous AC machines, is produced by the rotation of brushes 5 by drive motor 7.
Figure 2 presents the vector diagram of an asynchronous DC motor for a diametral pitch of the stator winding, for the fundamental harmonic of the magnetic field and without taking into account the angle of magnetic lag.
As can be seen from the figure, the vector diagram of an asynchronous DC motor differs from the vector diagram of an asynchronous AC motor by the absence in it of the reactive component of the voltage drop in the stator winding, which is explained by the unchanged magnitude of the current in the parallel branches of the stator winding during the commutation period [2].
On the basis of the vector diagram one may write the following equations:
\[ F_m = F_{1m}\cos\Theta - F_{2m}\sin\psi_2, \tag{1} \]
\[ F_{1m}\sin\Theta = F_{2m}\cos\psi_2, \tag{2} \]
where
\(F_m\) — the amplitude value of the magnetizing force (m.m.f.) of the motor;
\(F_{1m}\) — the amplitude value of the m.m.f. of the stator winding;
\(F_{m2}\) — the amplitude value of the m.m.f. of the rotor winding;
\(\psi_2\) — the angle between the rotor emf and current.
Fig. 1. Schematic diagram of an asynchronous DC motor.
1 — stator, 2 — rotor, 3 — external commutator, 4 — connecting wires, 5 — commutator brushes, 6 — rotating traverse, 7 — drive micromotor, 8 — contact rings, 9 — brushes of the contact rings, 10 — base, 11 — commutator
\[ \psi_2=\arccos \frac{r_2}{Z_2}, \tag{3} \]
\(r_2\) — active resistance of a rotor phase,
\(Z_2\) — magnitude of the total impedance of a rotor phase
\[ Z_2=\sqrt{r_2^2+x_2^2 s^2}, \tag{4} \]
\(x_2\) — inductive resistance of a rotor phase due to leakage fluxes,
\(s\) — slip
\[ s=\frac{n_1-n_2}{n_1}, \tag{5} \]
Fig. 2. Vector diagram of an asynchronous direct-current motor
\(n_1\) — speed of rotation of the field (commutator brushes);
\(n_2\) — speed of rotation of the rotor,
\(\Theta\) — spatial angle between the magnetomotive forces of the stator winding and the motor.
The quantities \(r_2\) and \(x_2\) can be determined in accordance with [3, 4, 5], and the magnitude of the angle \(\Theta\) can be determined after jointly solving equations (1) and (2) with the emf balance equation for the rotor phase \(E_{2s}\):
\[ E_{2s}=I_2 Z_2, \tag{6} \]
where \(I_2\) is the rms value of the rotor phase current. Solving (1, 2, and 6) jointly, we obtain:
\[ \cos \Theta = \frac{\sqrt{F_{1m}^{2}-F_{2m}^{2}\cos^{2}\psi_2}}{F_{1m}} = \sqrt{1-\frac{r_2^{2}x_m^{2}s^{2}}{z_2^{2}z_m^{2}}} = \sqrt{1-\frac{r_2^{2}x_m^{2}s^{2}}{(r_2^{2}+x_2^{2}s^{2})[r_2^{2}+(x_m+x_2)^{2}s^{2}]}}, \tag{7} \]
where \(x_m\) is the calculated inductive reactance of the stationary rotor phase to the magnetizing current.
\[ x_m=\frac{4W_2^2 k_{2c}^2 f_1 \lambda_1 Z_p}{\pi p}, \]
\(f_1\)—the frequency of reversal of the motor polarity
\[ f_1=pn_1, \tag{9} \]
\(p\)—the number of pairs of motor poles;
\(\lambda_1\)—the magnetic permeance of the motor flux for the fundamental harmonic of the magnetic field [3, 5]
\[ \lambda_1=\frac{\mu_0 S_\delta}{\delta k_\delta k_\mu}, \tag{10} \]
\(W_2\)—the number of turns of the rotor phase,
\(Z_p\)—the number of rotor slots,
\(k_{2c}\)—the slot-skew coefficient [3, 4, 5],
\(Z_m\)—a conditional calculated resistance of the rotor phase, making it possible to determine the ratio of the m.m.f.’s of the stator, rotor, and motor windings
\[ \frac{F_{1m}}{F_{2m}}=\frac{Z_m}{x_m s}, \tag{11} \]
\[ \frac{F_m}{F_{1m}}=\frac{Z_2}{Z_m}, \tag{12} \]
\[ Z_m=\sqrt{r_2^2+(x_m+x_2)^2s^2}. \tag{13} \]
The causes of sparking under the brushes of an asynchronous direct-current motor, by analogy with ordinary direct-current machines, may be divided into mechanical and electromagnetic [5, 6, 7, 8, 9].
In the present work only causes of an electromagnetic nature will be analyzed and, in particular, the magnitude of the commutating e.m.f. from the external field, which to a significant extent determines the operating capability of these motors. The magnitude of the e.m.f. of the commutating section from the external field may be determined on the basis of the vector diagram (Fig. 2).
\[ e_k=2w^1B_\delta lv\cos\Theta, \tag{14} \]
where \(w^1\)—the number of turns in a section of the stator winding,
\(B_\delta\)—the average value of the induction in the commutation zone
\[ B_\delta=\frac{2F_m\lambda_1}{\pi\tau l}, \tag{15} \]
\(l\)—the length of a half-turn of a section of the stator winding,
\(\tau\)—the value of the motor pole pitch,
\(v\)—the linear velocity of the field displacement relative to the stator
\[ v=2\tau f_1. \tag{16} \]
Solving (14–16) together, we obtain
\[ e_k=\frac{8w^1F_m\lambda_1 f_1}{\pi}\cos\Theta. \tag{17} \]
Expression (17) contains the quantities \(F_m\), \(\lambda_1\), and \(f_1\), which it is undesirable to reduce because of the need to preserve the motor’s operating parameters. Consequently, the principal quantities significantly affecting the value of \(e_k\) are the number of turns in a section \(w^1\) and the angle \(\Theta\).
In order to reduce the value of \(e_k\), as in ordinary direct-current machines—
current, one should strive for the smallest number of turns in the stator section of an asynchronous DC motor. The magnitude of the angle \(\Theta\) is the principal parameter of the motor, significantly reducing the value of the emf \(e_k\).
As is seen from (7), the quantity \(\cos \Theta\) depends on the design parameters of the motor and on the slip. To investigate the maximum and minimum of \(\cos \Theta\) as a function of slip, we take the first derivative \(\dfrac{d\cos\Theta}{ds}\)
\[ \frac{d\cos\Theta}{ds} = - \left( 1-\frac{r_2^{\,2}x_m^{\,2}s^2}{Z_m^{\,2}Z_2^{\,2}} \right)^{-1/2} \left\{ \frac{2r_2^{\,2}x_m^{\,2}sZ_m^{\,2}Z_2^{\,2}}{Z_m^{\,4}Z_2^{\,4}} - \frac{\left[2(x_m+x_2)^2sZ_2^{\,2}+2x_2^{\,2}sZ_m^{\,2}\right]r_2^{\,2}x_m^{\,2}s^2}{Z_m^{\,4}Z_2^{\,4}} \right\}. \tag{18} \]
Equating (18) to zero and solving it with respect to \(s\), we obtain
\[ s_k=\pm \frac{r_2}{\sqrt{x_2^{\,2}+x_2x_m}}, \tag{19} \]
where the plus sign is for the motor mode, and the minus sign for the generator mode of operation.
Substituting (19) into (7), we obtain
\[ \cos\Theta_k= \sqrt{ 1-\frac{x_m^{\,2}}{(2x_2+x_m)^2} }. \tag{20} \]
As is seen from (20), the magnitude of the angle \(\Theta\) does not depend on the value of the active resistance \(r_2\); its value determines only the point of critical slip at which the angle \(\Theta\) will be greatest.
Figure 3 presents the graphical dependence of the magnitude of the angle \(\Theta\) on the numerical values of \(r_2\) and \(x_2\) at various slips. The construction of this dependence was carried out according to expression (7) for an asynchronous DC motor with resistance \(x_m=365\cdot 10^{-5}\ \Omega\).
Fig. 3. Dependence of the magnitude of the angle \(\Theta\) on the rotor resistances at various slips
It is seen from the figure that the magnitude of the angle \(\Theta\) depends to a considerable degree on the resistances \(x_2\) and \(r_2\); the inductive resistance \(x_2\), in the general case, worsens the commutation conditions of the motor, having an especially
strongly in the zone of large slips. In order to improve the commutation of asynchronous DC motors, one should strive for the smallest value of the resistance \(x_2\); for this purpose the rotor slots should be made round.
On the basis of the analysis given, the following conclusion may be drawn:
-
The commutation of an asynchronous DC motor is determined mainly by the magnitude of the emf of the commutated section due to the external field.
-
The magnitude of the emf \(e_k\) depends on the operating mode of the motor (the magnitude of the slip): it has its greatest value at \(s=0\) and its smallest at \(s=s_k\) (see 19); in the remaining modes the emf \(e_k\) will have intermediate values.
-
The magnitude of the critical slip \(s_k\) may be varied as desired by changing the resistance \(r_2\).
In this connection the author considers unconvincing the statement in [2] concerning the commutation of an asynchronous DC motor: “During commutation of the machine, sparking was observed, which decreased with an increase in load. This is explained by the fact that, in manufacturing the machine, a commutator was used that operates normally at a voltage of 40 V, whereas during the tests the voltages were higher...”
Methods for Improving the Commutation of an Asynchronous DC Motor
An asynchronous DC motor represents a synthesis of DC and AC machines; therefore the means of improving the commutation of these motors may be divided into those already known for DC machines and those pertaining only to this type of machine.
The previously known methods for improving commutation (with the exception of additional poles, whose installation on the motor under study is impossible) of DC machines [10, 11] are fully applicable also for improving the commutation of an asynchronous DC motor.
Among the means that improve the commutation of an asynchronous DC motor and pertain only to this type of machine, the following should be included:
-
Reducing the inductive reactance of the rotor phase.
-
Selecting the active resistance of the rotor phase in order to obtain the largest angle \(\Theta\) in the zone of the nominal operating mode.
-
Operation of the motor in the zone of critical commutation slip.
-
Placing in the stator slots, together with the working winding, an additional short-circuited turn that makes it possible to increase the angle \(\Theta\) when the motor is operating at no load.
Experiments carried out by the author at FFVNIIZM show that, when the above-listed means of improving commutation for an asynchronous DC motor are applied, the angle \(\Theta\) approaches the commutation conditions of ordinary DC machines without additional poles.
Of great interest for this type of machine is obtaining an angle \(\Theta>90^\circ\), which makes it possible to bring their commutation conditions closer to those of ordinary DC machines with additional poles. This measure can be achieved by using a magnetic-core material with a widened hysteresis loop (this method is analogous to shifting the brushes from the geometric neutral in ordinary DC machines without additional poles).
References
-
Ya. I. Drobnin. Low-Frequency Converter Installations. “Energiya,” 1965.
-
S. M. Damyanov. On one variety of electrical machine (asynchronous direct-current machine). Izvestiya AN SSSR, “Energetika i transport,” No. 6, 1964.
-
G. N. Petrov. Electrical Machines, Part II, GEI, 1963.
-
N. V. Vinogradov, F. A. Goryainov, P. S. Sergeev. Design of Electrical Machines. GEI, 1956.
-
M. P. Kostenko, L. M. Piotrovskii. Electrical Machines, Part II. “Energiya,” 1965.
-
A. I. Skorospeshkin, B. I. Kostylev, Yu. P. Galishnikov. On certain questions of commutation in commutator electrical machines. Izvestiya GPI, Vol. 145, 1966.
-
A. I. Skorospeshkin, L. Ya. Zinner, A. I. Proshin. An instrument for studying mechanical factors in commutator electrical machines. Izvestiya GPI, Vol. 160, 1966.
-
M. F. Karasev. Commutation of DC Commutator Machines. Gosizdat, 1961.
-
O. G. Wegner. Theory and Practice of Commutation in DC Machines. GEI, 1961.
-
G. N. Petrov. Electrical Machines, Part III, 1968.
-
M. P. Kostenko, L. M. Piotrovskii. Electrical Machines, Part 1. “Energiya,” 1964.