Method Of Symmetrical Components Research Articles

A-C servomotor operation from unbalanced voltages

MANY analytical studies directed at the predetermination of output characteristics of a-c servomotors have been predicated on the assumptions of constant source impedance and sinusoidal excitation. Under these assumptions, the method of symmetrical components serves admirably, particularly when steady-state characteristics are desired, and in cases where the duration of the mechanical transients makes the electrical transients insignificant.

Analysis of A-C servomotors operated from unbalanced nonsinusoidal voltage sources and nonlinear discontinuous source impedances

MANY ANALYTICAL studies directed at the predetermination of output characteristics of a-c servomotors have been based on the assumptions of constant-source impedance and sinusoidal excitation. Under these assumptions, the method of symmetrical components serves admirably, particularly when steady-state characteristics are desired, and in cases where the duration of the mechanical transients makes the electrical transients insignificant.

Induction motors with unbalanced rotor connections

THE USEFULNESS of the method of symmetrical components in analyzing the case of an induction motor operating with an unbalanced rotor circuit has been limited by a lack of conclusive evidence as to what assumptions can be made in applying this method. The nature of assumptions useful to this analysis has been established through comparison of numerous experimental and calculated speed-torque curves. An accounting for the variation of both rotor and stator winding resistances with frequency has been found necessary for accurate prediction of these characteristics. It has been found that a linear approximation of the resistance-frequency variation in these windings is sufficiently accurate for calculations on small wound-rotor induction motors. Blocked rotor tests over a range of frequencies have shown the resistance-frequency variation to be essentially linear for this type of motor. It is to be expected that certain cases involving a configuration of conductors in the motor windings such as deep bars or untransposed parallel windings may involve a departure from the linear resistance-frequency variation. In these cases either tests or design calculations can be used to establish a more suitable approximation.

Switching Transients in Single-Phase Induction Motors with Constant Speed

In the past, the problem of switching transients in polyphase induction motors has been studied in great detail by several investigators. There is, however, no published literature dealing with the analysis of the problem of transient conditions in single-phase induction motors of the split-phase variety with constant speed. In this paper, the method of symmetrical components of instantaneous potentials and currents has been applied to the analysis of transient conditions in a capacitor-start capacitor-run single-phase induction motor with constant speed, and with special reference to reclosing and plugging. Detailed computation has been carried out for the transient currents and torque produced when a capacitor motor running at full speed is suddenly taken off the lines, and then put back after an extremely brief interval. The problems of sudden short circuiting of stator terminals and plugging have also been analyzed. It has been found that if the motor is reclosed when the applied potential passes through its zero value there is a fundamental frequency slowly decaying torque superimposed on the average steady-state unidirectional torque. If, on the other hand, the motor is switched on to the supply when the applied potential passes through its maximum value, the developed electromagnetic torque settles down to its final value before the end of the third cycle of the applied a-c potential. In order to plug the motor satisfactorily it is absolutely necessary to first replace the running condenser by the starting condenser before the electric connections for one of the windings are interchanged.

The starting of single-phase induction motors having asymmetrical stator windings in quadrature

The operation of an asymmetrical 2-phase induction motor on single-phase supply system is analysed by the method of symmetrical components, and performance equations for the special problem of starting the machine are expressed in terms of three dimensionless parameters. The effects of variation of these parameters are studied in detail, and conditions for optimum starting performance are established. Starting characteristics are given for a wide range of the parameters, but it is shown that the best performance is obtained with a capacitive convertor selected to give minimum unbalance. Properly interpreted, the results are applicable to standard single-phase induction motors of any rating.

Dynamic braking of 3-phase motors by capacitors

When a series capacitor is introduced in one of the supply lines of a 3-phase induction motor working under normal load conditions, the torque developed by the motor is considerably reduced. If the speed falls below a certain value, the machine develops braking torque and decelerates to standstill and may even run in the opposite direction. This aspect was touched upon in a previous paper1 and is now investigated further, both theoretically and experimentally. Applying the method of symmetrical components, the current and torque equations are obtained for the machine during the braking period. It is suggested that the driving torque developed by the machine at high speeds, even with the introduction of the series capacitor, can be avoided either by working the machine as a self-excited induction generator or by introducing a suitable series reactor in another supply line. These methods of dynamic braking are compared with the conventional plugging method.

Application of a variable-reactor/capacitor combination for reversing and controlling the speed of polyphase induction motors

By introducing a variable reactor paralleled with a fixed capacitor in one of the supply lines of a 3-phase induction motor, the speed and direction of rotation can be controlled, by controlling the value of the the external effective reactance. Applying the method of symmetrical components, both the starting and running performances of such a motor are analysed. If the external reactance is made to vary automatically with speed, this gives a method of controlling, and reversing with dynamic braking, the speed of the motor.

A method of fault analysis for A-C aircraft electric power systems

THE increasing electric power requirements in today's aircraft make necessary the continual search for better methods of protection against faults which may cause the loss of electric power or otherwise endanger the mission. Power-system studies have been made in the past using a mock-up of the electric system composed of actual components, and have proved quite successful. However, it is desirable to obtain information faster, and before the development of complete apparatus. Parameters such as line voltages and currents, positive-, negative-, and zero-sequence voltages and currents must be known in order effectively to apply system protection, and they are of much value in regulator design. Faults occurring where they produce the same effect on all regulators, such as at the bus, can be calculated very simply by the method of symmetrical components. A means for detecting such faults by use of negative-sequence voltage is pointed out in another paper.1 However, in multigenerator systems, load-division circuits are employed in order that the load current may be properly divided between generators. Generator currents are usually sensed in a single phase through a current transformer; consequently, generators are forced to divide load current relative to the current in the current transformer associated with a particular generator feeder. Should a fault occur on the load-division phase at a point between a generator and its load-division current transformer, as shown in Fig. 1, that generator would become overexcited and supply practically all of the fault current.

Theory of the corner-driven square loop antenna

The general problem of determining the distribution of current and the driving point impedances of a square loop or frame antenna is formulated when arbitrary driving voltages are applied at each corner or when up to three of these voltages are replaced by impedances. The loop is unrestricted in size and account is taken of the finite cross-section of the conductors. Four simultaneous integral equations are obtained and then replaced by. four independent integral equations using the method of symmetrical components. These equations are solved individually by iteration and first-order formulas are obtained for the distributions of current and the driving-point admittances. By superposition the general solution for the abritrarily driven and loaded loop is obtained. Interesting special cases include a corner-reflector antenna and the square rhombic (terminated) antenna. An application of the principle of complementarity permits the generalization of the solution to the square slot antenna in a conducting plane when driven from a double-slot transmission line at one corner.

Sequence impedances of transformer connections

IN THE GENERAL APPLICATION of the method of symmetrical components to the solution of 3-phase problems, 3-phase circuits are resolved into three singlephase circuits, one for each of the sequences, positive, negative, and zero.

A symmetrical component synthesizer and analyzer

The basic formulas of the method of symmetrical components may be verified and the elliptical pattern produced by unbalanced currents can be analyzed to determine directly from the dimensions of the ellipse the magnitudes of the positive and negative sequence components by means of the polyphase oscilloscope developed at the State University of Iowa.

Determination of Inductive and Capacitive Unbalance for Untransposed Transmission Lines [includes discussion

The purpose of this paper is to present a method for the determination of inductive and capacitive unbalance for untransposed transmission lines. To accomplish this purpose, the conventional methods of symmetrical components are used. It is shown that the negative and zero-sequence currents in an untransposed line are functions of the positive-sequence load current, charging current, and line configuration. Calculated values of negative- and zero-sequence currents for typical configurations of high-voltage transmission lines are presented.

Vector power factor of 3-phase circuits

THE VECTOR power factor is a quantity which has been officially defined in the Standards of the American Standards Association1 and is the basis for many power rates which include lower power factor penalties. The effect of the unbalance in a polyphase load on this quantity can be analyzed by the method of symmetrical components.

Symmetrical Components as Appkied to the Single-Phase Induction Motor

THIS PAPER is intended to show in detail the application of the method of symmetrical components to the solution of single-phase-induction-motor performance characteristics. By the use of this method single-phase-motor theory follows very closely the simpler case of the balanced polyphase-motor theory. The author of this paper is of the opinion that a clearer concept of the physical theory can be obtained as contrasted to other methods — either from the revolving-field or the cross-field concepts.

Symmetrical components as applied to the single-phase induction motor

THIS PAPER is intended to show in detail the application of the method of symmetrical components to the solution of single-phase-induction-motor performance characteristics. By the use of this method single-phase-motor theory follows very closely the simpler case of the balanced polyphase-motor theory. The author of this paper is of the opinion that a clearer concept of the physical theory can be obtained as contrasted to other methods — either from the revolving-field or the cross-field concepts.

Asymmetrical Loading of Three-Phase Three-Winding Transformer Banks

It is accepted USSR practice to carry three-phase industrial load and single-phase traction load on a single three-phase three-winding transformer. For a given combination of loads it is necessary to determine the proper capacity relationships of the three windings. This paper presents an analysis of the problem by the methods of symmetrical components. The data obtained from this analysis were used in setting up the standards for selection of transformer size for combined three-phase industrial and single-phase traction loading.

An Extension of the Method of Symmetrical Components Using Ledder Networks

IN his now classical paper on symmetrical components Fortescue analyzed the behavior of synchronous and induction machinesoperating under unbalanced conditions. In this analysis Fortescue assumed that the stator and rotor windings themselves were symmetrical, that the rotor windings were either short-circuited or connected to balanced static circuits, and that the stator windings were connected to unbalanced circuits. Under these conditions there are unbalanced currents of a single frequency in the stator, and balanced currents of two different frequencies in the rotor. It has since been shown how the method of symmetrical components is applied when the rotor windings are connected to unbalanced circuits and the stator windings are connected to balanced circuits. Under these conditions there are unbalanced currents of a single frequency in the rotor, and balanced currents of two different frequencies in the stator.

An extension of the method of symmetrical components using ladder networks

IN his now classical paper on symmetrical components Fortescue analyzed the behavior of synchronous and induction machinesoperating under unbalanced conditions. In this analysis Fortescue assumed that the stator and rotor windings themselves were symmetrical, that the rotor windings were either short-circuited or connected to balanced static circuits, and that the stator windings were connected to unbalanced circuits. Under these conditions there are unbalanced currents of a single frequency in the stator, and balanced currents of two different frequencies in the rotor. It has since been shown how the method of symmetrical components is applied when the rotor windings are connected to unbalanced circuits and the stator windings are connected to balanced circuits. Under these conditions there are unbalanced currents of a single frequency in the rotor, and balanced currents of two different frequencies in the stator.

Transient analysis of symmetrical networks by the method of symmetrical components

Transient analysis of symmetrical networks by the method of symmetrical components

An extension of the method of symmetrical components using ladder networks

An extension of the method of symmetrical components using ladder networks