On the Theory of a Single-Phase Motor with an Auxiliary Winding
Abstract Generated abstract
This paper develops a simplified analytical treatment of a single-phase induction motor with an auxiliary winding, using symmetrical components for an equivalent two-phase system. Starting from relations between the direct and quadrature magnetomotive-force components and the transformation coefficients of the main and auxiliary windings, it derives reduced expressions for positive- and negative-sequence equivalent currents, line current, auxiliary-winding current, electromagnetic power, efficiency, and power factor. Under the assumption of similarly shaped distributed windings with small space-harmonic content and conductor cross sections inversely proportional to turns, the auxiliary-winding parameters are eliminated through transformation-coefficient relations. The resulting formulas reduce the number of equivalent impedances and complex-number operations needed for calculation, and are presented as a basis for further analysis of motor currents and power.
Full Text
ON THE THEORY OF A SINGLE-PHASE MOTOR WITH AN AUXILIARY WINDING
Yu. S. Dyachenko
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
Suur [1], using the method of symmetrical components for two-phase systems and applying the concepts of longitudinal and transverse transformation coefficients, obtained the currents (the transverse and the resultant longitudinal currents) along the mutually perpendicular axes of a single-phase motor with split poles and distributed windings.
In addition, he expressed the actual currents of the main and auxiliary windings through the symmetrical components of the positive and negative sequences of the resultant current along the longitudinal axis and the above-mentioned transformation coefficients.
In turn, he determined the symmetrical components of the resultant current along the longitudinal axis by extremely complicated formulas which contain, in addition to the supply voltage, various relations among six values of the equivalent resistances (of the positive and negative sequences of the circuits of the main and auxiliary windings, as well as the resistance of mutual induction between them).
The difficulty in the calculations also consists in the fact that, in the general case, all the resistances listed are calculated independently of one another.
Taking all the foregoing into account, the author of the present work set himself the following tasks: 1) to facilitate the calculation of currents and electromagnetic powers by simplifying the corresponding expressions; 2) thereby to make possible an analysis of the expressions for calculating the electromagnetic power and currents of the motor under consideration.
With the same form of the main and auxiliary windings, when the cross sections of the conductors are inversely proportional to the numbers of their turns, all the corresponding equivalent active and inductive resistances of the main and auxiliary windings, as well as the resistances of mutual induction between these windings, are related through the transformation coefficients (see “Notation”).
In this case it proved possible to carry out the simplification of the above-mentioned expressions.
The immobile wave of induction in the air gap of the machine, pulsating at the supply frequency and produced by the magnetomotive force of the winding, can have a strictly sinusoidal spatial distribution only if winding, tooth, and saturation-induced harmonics are absent. However, there is no necessity to manufac-
replace the windings by sinusoidal ones—the latter can be made simpler and are characterized by a very small content of higher space harmonics.
In this case, expansion in a Fourier series makes it possible to take into account only the fundamental space harmonic in the conductor linear-density distribution curves.
The fundamental space harmonic of the magnetomotive force along the pole axis (Fig. 1).
Fig. 1. Schematic diagram of a single-phase motor with split poles: a) basic, b) equivalent, c) equivalent two-phase.
\[ \dot{F}_d=\dot{i}w_1-\dot{i}_{\mathrm{B}}w_{1\mathrm{B}}\cos\Theta \tag{1} \]
(For the notation, see the end of the paper.)
The fundamental space harmonic of the magnetomotive force across the pole axis is
\[ \dot{F}_q=\dot{i}_{\mathrm{B}}w_{1\mathrm{B}}\sin\Theta . \tag{2} \]
The longitudinal magnetomotive force can be represented as follows:
\[ \dot{F}_d=\dot{i}'w_1=\dot{i}w_1-k_d\dot{i}_{\mathrm{B}}w_1 . \tag{3} \]
The transverse magnetomotive force is written as
\[ \dot{F}_q=\dot{i}'_{\mathrm{B}}w_1=k_q\dot{i}_{\mathrm{B}}w_1 . \tag{4} \]
The resultant current \(\dot{i}'\) along the longitudinal axis can be decomposed into its symmetrical components, i.e., into currents \(\dot{i}'_1\) and \(\dot{i}'_2\) of the positive and negative sequences,
\[ \dot{i}'=\dot{i}'_1+\dot{i}'_2 \tag{5} \]
just as the current \(\dot{i}\) of the main winding,
\[ \dot{i}=\dot{i}_1+\dot{i}_2 . \tag{6} \]
Similarly, we decompose the resultant current along the transverse axis into symmetrical components,
\[ \dot{i}'_{\mathrm{B}}=\dot{i}'_{\mathrm{B}1}+\dot{i}'_{\mathrm{B}2} \tag{7} \]
just as the currents
\[ \dot{i}_{\mathrm{B}}=\dot{i}_{\mathrm{B}1}+\dot{i}_{\mathrm{B}2}. \tag{8} \]
Between the symmetrical components for two-phase systems there exist the relations
\[ \dot I'_{B1}=j\dot I'_1, \tag{9} \]
\[ \dot I'_{B2}=-j\dot I'_2. \tag{10} \]
From equations (4), (7), and (8) we obtain the relation
\[ \dot I'_{B1}+\dot I'_{B2}=\kappa_q \dot I_{B1}+\kappa_q \dot I_{B2}, \tag{11} \]
which makes it possible to write that
\[ \dot I_{B1}=\frac{1}{\kappa_q}\dot I'_{B1}, \tag{12} \]
\[ \dot I_{B2}=\frac{1}{\kappa_q}\dot I'_{B2}. \tag{13} \]
Using equations (9), (10), (12), and (13), one can obtain
\[ \dot I_{B1}=\frac{j}{\kappa_q}\dot I'_1, \tag{14} \]
\[ \dot I_{B2}=\frac{-j}{\kappa_q}\dot I'_2. \tag{15} \]
With the aid of equations (3), (6), and (8), the current of the main winding can be represented as
\[ \dot I=\dot I'_1+\dot I'_2+\kappa_d(\dot I_{B1}+\dot I_{B2}). \tag{16} \]
Substitution of equations (14) and (15) into equation (16) gives
\[ \dot I=\dot I'_1+\dot I'_2+j\frac{\kappa_d}{\kappa_q}(\dot I'_1-\dot I'_2) =\dot I'_1\left(1+j\frac{\kappa_d}{\kappa_q}\right) +\dot I'_2\left(1-j\frac{\kappa_d}{\kappa_q}\right). \tag{17} \]
This makes it possible to write that
\[ \dot I_1=\dot I'_1\left(1+j\frac{\kappa_d}{\kappa_q}\right), \tag{18} \]
\[ \dot I_2=\dot I'_2\left(1-j\frac{\kappa_d}{\kappa_q}\right). \tag{19} \]
Relations between the voltage, current, and total impedance of the motor
The fundamental spatial harmonic of the voltage applied to the motor terminals is balanced by the voltage drop caused by the flow of the current \(\dot I\) of the main winding (Fig. 2),
\[ \dot I_1 Z_1+\dot I_2 Z_2 \tag{20a} \]
and by the electromotive force of mutual induction induced in the main winding by the current \(\dot I_B\) flowing in the longitudinal \(\kappa_d w_1\) turns of the auxiliary winding,
\[ -(\dot I_{B1}Z_{AB1}+\dot I_{B2}Z_{AB2}), \tag{20б} \]
i.e.
\[ \dot U_1=\dot I_1Z_1-\dot I_{B1}Z_{AB1}+\dot I_2Z_2-\dot I_{B2}Z_{AB2}. \tag{20} \]
Using the relations expressed by equations (14), (15), (18) and (19), equation (20) can be represented in the form
\[ \dot U=\dot I'_1\left[\left(1+j\frac{k_d}{k_q}\right)Z_1-\frac{j}{k_q}Z_{AB1}\right]+ \]
\[ +\dot I'_2\left[\left(1-j\frac{k_d}{k_q}\right)Z_2+\frac{j}{k_q}Z_{AB2}\right]. \tag{21} \]
Fig. 2. Equivalent circuits of a single-phase motor: a) for positive-sequence currents; b) for negative-sequence currents.
Similarly, the fundamental spatial harmonic of the voltage in the auxiliary-winding circuit is balanced by the voltage drop due to the flow of the current \(\dot I_B\)
\[ \dot I_{B1}Z_{B1}+\dot I_{B2}Z_{B2} \tag{22a} \]
and by the electromotive force of mutual induction induced in the \(k_d\cdot w_1\) turns when the current \(\dot I\) flows in the \(w_1\) turns of the main winding,
\[ -(\dot I_1Z_{AB1}+\dot I_2Z_{AB2}), \tag{22b} \]
i.e.
\[ \dot U_{B1}=\dot I_{B1}Z_{B1}-\dot I_1Z_{AB1}+\dot I_{B2}Z_{B2}-\dot I_2Z_{AB2}. \tag{22} \]
Substitution of equations (14), (15), (18) and (19) into equation (22) gives
\[ \dot U_{B1}=\dot I'_1\left[\frac{j}{k_q}Z_{B1}-\left(1+j\frac{k_d}{k_q}\right)Z_{AB1}\right]- \]
\[ -\dot I'_2\left[\frac{j}{k_q}Z_{B2}+\left(1-j\frac{k_d}{k_q}\right)Z_{AB2}\right]. \tag{23} \]
In the case of the use of distributed windings with an insignificant content of spatial harmonics (both in the main and in the auxiliary windings), the system of equations (21) and (23) can be solved for the currents \(\dot I'_1\) and \(\dot I'_2\). Under these conditions (when the auxiliary winding is short-circuited)
\[ \dot U_1=\dot U, \]
\[ \dot U_{B1}=0. \]
The equivalent current along the longitudinal axis of the positive sequence is
\[ \dot I_1'=\frac{\dot U}{2}\, \frac{\kappa^2 Z_2-\kappa_d(\kappa_d+j\kappa_q)Z'_{R2}} {\kappa^2 Z_1Z_2-Z_{AB1}Z_{AB2}} . \tag{24} \]
The equivalent current along the longitudinal axis of the negative sequence is
\[ \dot I_2'=\frac{\dot U}{2}\, \frac{\kappa^2 Z_1-\kappa_d(\kappa_d-j\kappa_q)Z'_{R1}} {\kappa^2 Z_1Z_2-Z_{AB1}Z_{AB2}} . \tag{25} \]
Substitution of equations (24) and (25) into equation (17) makes it possible to find the actual value of the current drawn by the motor from the network,
\[ \dot I=\frac{\dot U}{2}\, \frac{\kappa^2(Z_1+Z_2)} {\kappa^2 Z_1Z_2-Z_{AB1}Z_{AB2}} . \tag{26} \]
By substituting equations (14), (15), (24), and (25) into equation (8), the current of the auxiliary winding is found:
\[ \dot I_B=\frac{\dot U}{2}\, \frac{Z_{AB1}+Z_{AB2}-j\kappa_q(Z_1-Z_2)} {\kappa^2 Z_1Z_2-Z_{AB1}Z_{AB2}} . \tag{27} \]
The total mechanical power of the motor is
\[ P'_R=2\bigl(|\dot I_1'|^2 r'_{R1}-|\dot I_2'|^2 r'_{R2}\bigr)\nu . \tag{28} \]
The power on the motor shaft is
\[ P_R=P'_R-(p_{\text{mech}}+p_{\text{add}}). \tag{29} \]
The power consumed by the motor is
\[ P_s=U\cdot I_a . \tag{30} \]
The efficiency of the motor is
\[ \eta=\frac{P_R}{P_s}. \tag{31} \]
The power factor of the motor is
\[ \cos\varphi=\frac{I_a}{|\dot I|}. \tag{32} \]
Conclusions
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By eliminating the parameters of the auxiliary winding, the number of equivalent impedances entering expressions (24–28) for calculating the electromagnetic power and the currents of the main and auxiliary windings of the motor under consideration has been reduced.
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As a result, the number of computational operations with complex numbers has been reduced in determining the electromagnetic power and the currents of the main and auxiliary windings.
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These simplifications create conditions for carrying out investigations of the indicated expressions.
Notation
\[ \dot I'=\dot I'_1+\dot I'_2=\dot I-\kappa_d \dot I_{\mathrm{B}} \]
— equivalent current along the direct axis in \(w_1\) turns of the main winding.
\[ \dot I'_1 \text{ and } \dot I'_2 \]
— the positive- and negative-sequence components of the current \(\dot I'\).
\[ \dot I \]
— the line current of the main winding.
\[ \kappa_d \dot I_{\mathrm{B}} \]
— the direct component of the current of the auxiliary winding.
\[ \dot I'_{\mathrm{B}}=\dot I'_{\mathrm{B}1}+\dot I'_{\mathrm{B}2}=\kappa_q \dot I_{\mathrm{B}} \]
— equivalent current along the quadrature axis in \(w_1\) turns of the main winding.
\[ \dot I_{\mathrm{B}} \]
— the actual current of the auxiliary winding.
\[ \kappa=\frac{w_{1\mathrm{B}}}{w_1}; \qquad \kappa_d=\kappa\cdot\cos\Theta; \]
\[ \kappa_q=\kappa\cdot\sin\Theta \]
— transformation coefficients.
\[ w_1 \text{ and } w_{1\mathrm{B}} \]
— the fundamental space harmonics for the distribution curves of the linear density of the conductors of the main and auxiliary windings.
\[ \Theta \]
— the angle between the axes of the windings \(w_1\) and \(w_{1\mathrm{B}}\).
\[ r'_{R1};\ x'_{R1} \text{ and } r'_{R2};\ x'_{R2} \]
— equivalent active and inductive resistances of the parallel branches of the rotor—magnetizing branch of the equivalent circuit of the main winding for positive- and negative-sequence currents.
\[ x'_{R1}= \frac{[r_R x_m+(1-\nu)x_R r_m][r_R+(1-\nu)r_m]-(1-\nu)[r_R r_m-} {[r_R+(1-\nu)r_m]^2+(1-\nu)^2(x_R+x_m)^2} \]
\[ \frac{-(1-\nu)x_R x_m](x_R+x_m)} {[r_R+(1-\nu)r_m]^2+(1-\nu)^2(x_R+x_m)^2}; \]
\[ r'_{R1}= \frac{[r_R r_m-(1-\nu)x_R x_m][r_R+(1-\nu)r_m]+} {[r_R+(1-\nu)r_m]^2+(1-\nu)^2(x_R+x_m)^2} \]
\[ \frac{+(1-\nu)[r_R x_m+(1-\nu)x_R r_m](x_R+x_m)} {[r_R+(1-\nu)r_m]^2+(1-\nu)^2(x_R+x_m)^2}. \]
Replacing the multiplier \((1-\nu)\) by \((1+\nu)\) gives \(x'_{R2}\) and \(r'_{R2}\), respectively.
\[ r_R \text{ and } x_R \]
— the active and inductive resistances of the rotor winding, referred to the number of turns of the main winding.
\[ r_m \text{ and } x_m \]
— the active and inductive resistances of the magnetizing branch in the equivalent circuit of the main winding.
\[ r_s \text{ and } x_s \]
— the active and inductive resistances of the main winding on the stator.
\[ \nu=\frac{n}{n_s} \]
— the relative rotational speed of the rotor, in fractions of the synchronous rotational speed of the magnetic field.
\[ s=1-\nu \]
— rotor slip.
\[ \dot U \]
— voltage at the terminals of the main winding.
\[ \begin{aligned} Z_{\mathrm{AB}1} &= k_d Z'_{R1} \\ Z_{\mathrm{AB}2} &= k_d Z'_{R2} \end{aligned} \]
are the total mutual-inductance impedances between the main and auxiliary windings for the positive and negative sequences.
\[ Z_1 = Z_s + Z'_{R1} = r_s + r'_{R1} + j(x_s + x'_{R1}) \]
is the equivalent total impedance of the main winding for the positive sequence, and
\[ Z_2 = Z_s + Z'_{R2} = r_s + r'_{R2} + j(x_s + x'_{R2}) \]
is the equivalent total impedance of the main winding for the negative sequence, respectively.
\[ \begin{aligned} Z_{B1} &= k^2 Z_1 \\ Z_{B2} &= k^2 Z_2 \end{aligned} \]
are the equivalent total impedances of the auxiliary winding for the positive and negative sequences, respectively, if the configurations of the main and auxiliary windings are identical and the wire cross sections are inversely proportional to the corresponding numbers of turns.
REFERENCES
F. W. Suhr. A Theory for Shaded—Pole Induction Motors, Power and Apparatus Systems, August, n. 37, 1958.