Paul C. Krause

Induction Machine Analysis for Arbitrary Displacement Between Multiple Winding Sets

The steps in t he applied voltages of a three-phase, inverter-induction motor drive system cause undesirable pulsations in motortor By using multiple inverters connected to a multiphase machine with appropriate winding displacements, significant improvement in system performance is possible. The winding displacements required, however, are not necessarily the symmetrical displacements used in standard multiphase machines. This paper sets forth an improved method for analyzing many types of unsymmetrical, as well as symmetrical, multiphase induction machines. The simulation of a 7.5-hp induction machine with two three-phase sets of stator windings fed from two six-step inverters illustrates the application of the method of analysis and shows how the machine torque characteristic may be improved by proper spacing of the winding sets.

The Theory of Neglecting Stator Transients

For more than four decades it has been thought that neglecting the so-called ph terms in Parks equations is equivalent to neglecting the electric transients due to the stator windings of a synchronous machine. Over the years, trial and error results have indicated that a more accurate prediction of synchronous machine performance could be obtained if the Aw(p6) terms in the stator voltage equations were also neglected. This paper provides a theoretical explanation of this experimental observation. In particular, it is established theoretically that in order to neglect the stator transients both the p* and Aw terms must be set equal to zero. This is accomplished by proving that the equivalence of neglecting the pq terms in the synchronous reference frame is to neglect the p* and Aw terms in all other reference frames. R-L transmission systems, synchronous and induction machines are considered.

Accuracy of a Reduced Order Model of Induction Machines in Dynamic Stability Studies

Previously published information regarding the accuracy of the eigenvalues calculated from the linearized reduced order model of an induction machine, wherein stator transients are neglected, is shown to be incorrect, nearly opposite to the actual situation. In this paper the error in the previous work is pointed out and the parameters of the induction machine which influence the accuracy of the linearized reduced order model are established analytically. A ratio of parameters is suggested which could be used as an indication of the accuracy of the reduced order model. A salient conclusion is that the accuracy of the linearized reduced order model increases as the horsepower rating of the machine increases. This conclusion is completely opposite to that previously reported.

Theory and Comparison of Reduced Order Models of Induction Machines

Three methods of deriving linearized, reduced order models of induction machines are presented and compared. The first is the familiar method of neglecting stator transients (p¿terms). Although the development of the second method is similar to the first, it yields a model with substantially different dynamic characteristics. In the third method, only the fast components of the stator flux linkages are neglected in the solution of the slower rotor variables rather than neglecting the rate of change of stator flux linkages (p¿terms). This refinement markedly improves the accuracy of the resulting reduced order model. The improvement is demonstrated by comparison of the response characteristics predicted by the three reduced models with that predicted by the complete or detailed model.

Analysis of a Permanent Magnet Synchronous Machine Supplied from a 180° Inverter with Phase Control

Reference frame theory is used to establish the equations which describe the steady-state and dynamic behavior of an electric drive system consisting of a permanent magnet synchronous machine supplied from an inverter operating in the 180° conduction mode and with provisions to shift the phase of the stator voltages relative to the rotor position. An expression for the phase-shift angle which yields maximum torque is derived. It is shown that a comparison of the stator time constant and the no-load rotor speed without phase shift, can be used to anticipate the increase in average torque achievable by phase shifting. It is also shown that advancing the phase of the stator voltages advances the phase of the fundamental component of the stator phase currents relative to the phase voltages. A method of calculating the steady-state harmonic currents and torque is also given. The information given in this paper should serve as a guide for the operation of brushless dc motors and the design of speed or position controls that employ phase shifting techniques along with or instead of pulse width modulation.

Shaft torques during out-of-phase synchronization

As a consequence of shaft failures in two units in a Consumers Power gas turbine installation, a computer study was Performed to determine the torques In the turbine-generator shaft system during out-of-phase synchronization. Prior to this study, the possible occurrence of excessive torques in the turbine-generator shaft system without damage to the stator windings due to large currents was questioned. If the gas turbine-generator unit is designed to sustain the stator currents and the instantaneous shaft torques resulting from a three phase short circuit at the machine terminals, then the results of this investigation revealed that during out-of-phase synchronization the design torques may be exceeded by 2 to 3 times without exceeding the amplitude of the short circuit stator currents. From this it seems reasonable to infer that shaft failure could occur during out-of-phase synchronization without damage to stator windings due to excessive currents.

Solution of State Equations in Terms of Separated Modes with Applications to Synchronous Machines

Using a special linear transformation, a linear time invariant state model can be changed into a new form defined as the separated mode model. As this transformation is repeated, the state model tends to be separated into two decoupled or independent parts. Each of them is a lower dimensional state model. The solutions of the two are the fast and slow components of the original state model respectively. Applying this approach to induction and synchronous machines, new linearized reduced order models are obtained. Computation results have shown that the new models have lower dimension and higher accuracy than the common one which neglects stator transients (p¿ terms).

The Method of Symmetrical Components Derived by Reference Frame Theory

In 1965 it was shown that all known real transformations commonly used in the analysis of induction machines were contained in one general change of variables. It is now shown that complex transformations, in particular the method of symmetrical components, are also contained in this same general transformation. Hence all change of variables (transformations), real or complex, commonly used in ac machine analysis stem from one general transformation. Moreover, even without prior knowledge of the existence of symmetrical components, this method naturally evolves from the general transformation as the logical technique to solve unbalanced operation of induction machines. This rigorous approach to the sOblution of unbalanced stator or rotor conditions enables one to appreciate the power and flexibility of the method of symmetrical components in induction machine analysis. The general transformation (arbitrary reference frame) is used to establish the positively and negatively rotating balanced sets for multi-frequency phase variables which need only be periodic. The method of symmetrical components is then derived from the expressions in the arbitrary reference frame. -An open circuited stator phase and unbalanced rotor resistors are used as two specific examples to illustrate the facility of this approach in establishing the symmetrical component voltage equations necessary to analyze these modes of operation.

Modeling of Transient Electrical Torques in Solid Iron Rotor Turbogenerators

It has previously been shown that it is necessary to employ a synchronous machine representation with unequal mutual reactances in the direct axis in order to calculate accurately the transient field current. This paper shows that if the stator current is calculated accurately by a synchronous machine model then the instantaneous electromagnetic torque is also accurately predicted. It is further shown that this may be accomplished by a model with two rotor windings in each axis and without representation of unequal mutual reactances. Results are presented to demonstrate the degree of accuracy possible with this model.

Analysis and Hybrid Computer Simulation of Multiconductor Transmission Systems

A method of analysis convenient for the simulation of transposed and untransposed transmission systems is set forth, and the facility of the hybrid computer in studying the traveling waves on these systems is illustrated. In particular, computer results are given showing the electric transients during line charging of a 3-phase, transposed system as well as the traveling waves on transposed and untransposed 3- phase systems during 3-phase and line-to-ground faults. Also presented are computer results showing the line-charging transients of two mutual- ly coupled, untransposed, 3-phase transmission systems. The material presented in this paper should permit a broader base from which to evaluate the various methods of analysis and simulation of traveling waves on transmission systems.