Gabriel Kron

A Set of Principles to Interconnect the Solutions of Physical Systems

A set of principles and a systematic procedure are presented to establish the exact solutions of very large and complicated physical systems, without solving a large number of simultaneous equations and without finding the inverse of large matrices. The procedure consists of tearing the system apart into several smaller component systems. After establishing and solving the equations of the component systems, the component solutions themselves are interconnected to obtain outright, by a set of transformations, the exact solution of the original system. The only work remaining is the elimination or solution of the comparatively few superfluous constraints appearing at the points of interconnection.The component and resultant solutions may be either exact or approximate and may represent either linear or, with certain precautions, nonlinear physical systems. The component solutions may be expressed in numerical form or in terms of matrices having as their elements real or complex numbers, functions of time, ...

Tensorial Analysis of Integrated Transmission Systems Part I. The Six Basic Reference Frames

A method of analysis is outlined to solve steady-state problems arising in the operation of large interconnected transmission systems exchanging power between their own operating divisions and outside companies. Special attention is paid to the calculation of total and incremental losses for economic loading studies and to the allocation of incremental losses between the various subsidiary and outside companies. This first part restricts itself to lay the necessary foundations for the numerical study made in a companion paper1 on the system of the American Gas and Electric Company.

The Doubly Fed Machine

SYNCHRONOUS machines, operating with a-c excitation on both stator and rotor are used in many applications, for example, as induction frequency converters, as power and instrument Selsyn drives, and as variable speed power drives. Reference 1 has mentioned particularly the variable speed fan drive, and presented equations for the small oscillations of one such doubly fed machine. Reference 2 has also previously given the equations of hunting of the doubly fed machine (part XIV, section IV) in connection with the general study of oscillations of rotating machines. However, since the present authors have been using in their own work equations which seem to them to be more convenient and simpler in form for calculations, and since it now seems desirable to present not only general equations but also some of the more fundamental and significant performance characteristics of these machines, it is thought that this paper may now be appropriate. The form of the equations developed possesses the additional novelty of facilitating the setting up of equivalent circuits for hunting on the a-c network analyzer, and allowing the quick determination of the damping and synchronizing torques directly by wattmeter readings.

Electric circuit models of partial differential equations

Electrical models of linear partial differential equations may serve several practical purposes: 1. If the networks are physically constructed, they actually may solve the equations within an accuracy of, say, one to five per cent, which is acceptable in many engineering applications. 2. If the networks are constructed only on paper, they supply a visualizable schedule of operations for the numerical solution of the equations or for the improvement of the results found by the network analyzer or by other methods. 3. The networks may serve to check the accuracy and self-consistency of results arrived at by other methods, approximate or exact. 4. In many problems where the fields have boundaries of unusual shape, or where both fields and circuits are present and mutually are influencing each other, it is next to impossible to formulate the problem mathematically. In such cases the electrical model representation offers a practical means for formulating and solving the problem.

Self-Excited Oscillations of Capacitor-Compensated Long-Distance Transmission Systems

THE transmission of electric power over long distances has been the subject of considerable study,1 and it has been concluded that an effective way of increasing the permissible straightaway transmission distance is by means of series-capacitor compensation of part of the transmission-line inductive reactance. The introduction of longer lines together with capacitor compensation brings up several technical problems which must be considered if proper operation is to be assured. Studies of transient stability and switching times following faults and of abnormal overvoltages which may result from interaction of line capacitance and transformer exciting impedance at no load are presented in references 2 and 3 respectively. This paper presents the results of a study of hunting and self-excitation during normal operation as affected by line and machine characteristics.

Tensorial Analysis of Integrated Transmission Systems; Part IV. The Interconnection of Transmission Systems [includes discussion]

A radically new method is developed for solving integrated transmission systems (as well as other types of extensive physical systems) without calculating the inverse of large matrices, or without using a large a-c analyzer. The physical system is torn up into several parts, each part is solved separately (either on an a-c analyzer or by a digital computer), and afterward the component solutions are interconnected into the solution of the original system by a series of transformations. The method is illustrated by establishing the individual and total loss equations of a power pool, whose component companies form several closed loops. A numerical example is worked out in a companion paper.1

Damping and Synchronizing Torques of Power Selsyns

SELSYN systems are used very widely as power drives and as indicators when it is necessary to have the position of an element correspond synchronously with that of a controlling element which is remotely located. The receiver Selsyn is used either directly as the output driving motor or as an input to a following power-amplifying stage of a control system.∗ In the study of such Selsyn systems and of the control systems of which Selsyn systems form one of the links, it becomes necessary to determine the hunting-torque characteristics of these machines (that is, the damping and synchronizing torques for small oscillations) as affected by speed, load, oscillation frequency, and, of course, the various design factors. The direct calculation of these hunting-torque characteristics is difficult, but within the past few years methods have been developed whereby the machine performance may be represented by equivalent static electric circuits. The purposes of this paper are: 1. To show how these hunting-torque characteristics can be simply determined by measurements on equivalent electric circuits set up on an a-c network analyzer. 2. To present some of the results of studies made of specific Selsyn systems.

Tensorial Analysis of Integrated Transmission Systems; Part II. Off-Nominal Turn Ratios

When the turn-ratios of transformers differ from the ratios of nominal values assigned to the different voltage levels of transmission, the a-c network analyzer representation of a transmission system requires autotransformers. The currents flowing in the latter form quite a large percentage of the total current entering the network and consequently the auto-transformer currents cannot be ignored. Their resultant is subdivided into a no-load current and a turn-ratio current. The no-load current is neglected and the turn-ratio current is eliminated from view by a transformation. Afterward the steps given in the first part of this serial for setting up the loss equations L = MP are retraced here. A change occurs only in the calculation of C23. A numerical example appears in reference 1.

A Method of Solving Very Large Physical Systems in Easy Stages

Physical systems with a very large number of variables (say with tens of thousands) may be solved with available digital computers by tearing the system apart into a large number of small subdivisions. After solving each subdivision separately, the partial solutions are interconnected by a set of transformations so as to obtain the exact solution of the original system. Among the many advantages of the tearing method is the reduction of the amount of original calculations to a small fraction of about 2/n2, where n is the number of subdivisions. Another advantage is the reduction of the number of nonzero elements in inverse matrices to a fraction smaller than 1/√n. The same labor saving appears also in smaller systems using slide-rule calculations. This paper illustrates the solution of Maxwell two-dimensional field-equations by tearing their electriccircuit models apart into a convenient number of subdivisions.

Stationary Networks and Transmission Lines Along Uniformly Rotating Reference Frames

This paper represents the first of a series of studies on the physical and mathematical analysis of the steady-state stability of conventional and long-distance transmission systems as influenced by the presence of voltage regulators and other control devices. As a first step attention is restricted to the transmission line as a component part of the over-all system and practical equivalent circuits are established for its representation, when the connected synchronous machines oscillate. For that purpose the basic equations of stationary networks are set up when viewed from uniformly rotating reference frames. An isolated inductor and capacitor, then a transformer, and finally a transmission line are used as examples. Their equations viewed from a simultaneously rotating and oscillating reference frame are given in reference 1.