Equivalent circuits have been found valuable in the solution of certain problems in that they simplify the analysis. Furthermore, equivalent circuits provide a means of replacing magnetic coupling with simple impedance links thereby permitting the use of the calculating board for solution. In Part I of this paper, the general equivalent mesh for the n-winding transformer is derived, and other less rigorous transformer circuits are discussed and compared. In Part II the mesh equivalent for the general network having (m + l) points of entry is derived and its application is explained. The generalizations of Parts I and II provide a background for the particular problem treated in Part III in which the study of system networks involving groups of parallel transmission lines is considered. The usually difficult problem of accurately determining ground fault currents in such a system is readily solved by applying the special equivalent circuits developed in Part III to a calculating board set-up of the system. These circuits are particularly valuable in the solution of any problem involving an analysis of the zero phase impedance diagram of a system. The methods and circuits developed in this paper are perfectly general and have many evident applications other than those treated.

Equivalent Circuits-I

We suggest that Whites’ declining share of the U.S. population threatens their status as the most prototypical ethnic group in America. This prototypicality threat should lead to growing resistance toward diversity, motivated by the desire to reassert Whites’ standing as prototypical Americans. In Study 1, how dramatically Whites perceived their share of the population to decline predicted support for cultural assimilation, mediated by prototypicality threat (controlling for realistic and symbolic threat). This relationship held only among Whites who felt that ethnic groups differ in their prototypicality, not among those who saw all groups representing America equally. Study 2 experimentally manipulated exposure to demographic projections such that Whites who saw their group shrinking showed weaker diversity endorsement relative to those who believed their share to be stable, again mediated by prototypicality threat. These findings reveal Whites’ threatened prototypicality as a novel, emerging source of ...

No Longer “All-American”? Whites’ Defensive Reactions to Their Numerical Decline

Tracing a route through the recent ‘ugly history’ of British citizenship, this article advances two central claims. Firstly, British citizenship has been designed to fail specific groups and populations. Failure, it argues, is a design principle of British citizenship, in the most active and violent sense of the verb to design: to mark out, to indicate, to designate. Secondly, British citizenship is a biopolitics – a field of techniques and practices (legal, social, moral) through which populations are controlled and fashioned. This article begins with the 1981 Nationality Act and the violent conflicts between the police and black communities in Brixton that accompanied the passage of the Act through the British parliament. Employing Michel Foucault's concept of state racism, it argues that the 1981 Nationality Act marked a pivotal moment in the design of British citizenship and has operated as the template for a glut of subsequent nationality legislation that has shaped who can achieve citizenship. The c...

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Designed to fail: A biopolitics of British citizenship

The objective of this two-part paper is to provide clarity to physical concepts used in the field of transformer modeling, to dispel common misconceptions regarding numerical instabilities, and to present unified modeling techniques for low-frequency transients. This paper focuses on proper modeling of nonlinearities (magnetizing branches) since these components are critical to determine the low-frequency behavior. A good low-frequency model should properly represent: normal operation, inrush currents, open and short circuit, out-of-phase synchronization transient of step-up transformers, geomagnetic-induced currents, ferroresonance, and harmonics. This paper discusses the derivation of electrical dual models from the equivalent (magnetic) reluctance networks and the direct application of the principle of duality. It is shown that different dual models need to be derived for different transformer geometries and the advantages and disadvantages of each method are discussed. This paper also compares double-sided versus single-sided dual models, and shows that the double-sided model is a more general approach. The mathematical equivalency of several leakage models (negative inductance, mutual coupling, and BCTRAN) is demonstrated for three-winding transformers. It is also shown that contrary to common belief, a negative inductance is not the source of numerical oscillations, but they occur due to the use of noncorrect topological models for representing the core.

Duality Derived Transformer Models for Low-Frequency Electromagnetic Transients—Part I: Topological Models

A modular transformer arrangement providing power to a multi-level power converter includes a plurality of three phase transformers having delta or extended delta connected primary windings and delta or extended delta or wye or zig zag wye connected secondary windings and the multi-level power converter has full bridge power modules inputs connected to isolated secondaries of the transformer modules and outputs connected in series and/or in parallel to provide increased voltage and/or current or multiple phases which can be phase shifted with respect to the input to the transformer.

Modular transformer arrangement for use with multi-level power converter

Symmetrical-components concepts are developed for the currents, voltages, and impedances of a four-phase system, and formulas are given for various types of short circuits. The idea of symmetry among the members of each set is based on the angular displacement between the members of the set, rather than that of phase rotation which fits the three-phase case very nicely but is inadequate for the four-phase case. Aside from its possible value for four-phase problems, which are rather rare, the paper is presented here in the expectation of an additional benefit, namely, that the parallelisms and contrasts between the four-phase and the three-phase formulas may throw further light on the general theory and applications of symmetrical components.

Symmetrical-Components Analysis of the Four-Phase System

The rotor moment of inertia of a separately excited d-c motor constitutes a true capacitance, capable of resonating with the armature inductance, so the equivalent armature circuit consists of resistance, inductance, and capacitance in series. However, data presented indicate that the armature moment of inertia would have to be reduced radically to render the motors suitable for 60 cycles.

The D-C motor capacitor

THE PHYSICAL THEORY of the characteristic oscillations of a high-voltage winding as a primary, that is, when the voltage disturbances are impressed directly across its terminals, is now well known.1 However its behavior as a secondary with respect to the disturbances is not as well known. Although papers2–4 have been published on this latter subject, this paper presents a physical theory believed to be novel and helpful.

Oscillations of a High-Voltage Secondary Winding

A child's idea of an angle is a corner or edge; the sharper the corner or edge, the smaller is the angle, and hence expressions like “acute (sharp) angle,” “obtu e (blunt) angle,” etc. With a little schooling his idea of an angle is broadened into that of turning or rotation and finds a geometrical expression in a circle. The amount of turning is represented by the area of the sector of the circle like a piece of pie gone over by the turning radius. This broadened dea is still in agreement with the earlier idea because the sector is a surface having a corner at the center of the circle, and the sharpness or bluntness of this corner varies with the area. With further progress, he also learns to associate the angle with the arc of the circle, for, evidently, the length of the arc is a measure of the amount of turning and is proportional to the area of the sector. The angle when represented by the arc of the circle loses all resemblance to a corner, which latter idea however is discarded as unnecessary, emphasis now being laid on “turning” which is well represented by the arc of the circle. Mathematical analysis is then applied which still further broadens the scope of angles and gives rise to the so called “imaginary” angles which are called “hyperbolic” in contrast with ordinary “real” angles which are called “circular,” combinations of these two kinds of angles being called “general” or “complex.” At this stage, any physical interpretation or geometrical representation of an angle practically vanishes, for, a hyperbolic angle can not be conceived of as a rotation, not even as a rotation around an “imaginary” axis, for, how can one conceive of an “imaginary” axis? If the “imaginary” axis be defined as a new axis at right angles to the old axis, rotation around it can easily be conceived of, but it does not correspond to facts, because a hyperbolic angle implies no change in direction, and hence no rotation, but merely affects the magnitude of a quantity. Inasmuch as problems can be solved mathematically without any visualization, the mathematician discards visual interpretations and bases his conceptions on formulas. However, the engineering type of mind finds it both difficult and distasteful to be dependent on symbolic definitions, and craves for visualization. He feels that if physical problems lead to “imaginary” or “complex” angles, these angles must stand for some concrete physical facts and must therefore be capable of a physical interpretation. A broad and yet simple physical interpretation applicable to both circular and hyperbolic angles is developed below, which also applies to their trigonometric functions, as follows: A quantity may be expressed or specified either by its own dimensions or as a percentage of another quantity. The first kind of specification is its “absolute” measure, the second kind is its “angular” measure. Thus “angle” and “percentage” are synonymous terms. If a quantity changes by a certain percentage (compound percentage as in compound interest) that “percentage” is the “angular” change in the quantity in radians. According to this interpretation, neither corner, edge, turning nor rotation are essential characteristics of an angle; they are the characteristics of a limited class only. If the percentage change in a quantity is like itself, geometrically parallel to itself, the per cent change, that is, the angle, is called a hyperbolic angle, and there is no rotation. If the quantity varies without any change in magnitude but only with a change in direction, geometrically perpendicular to itself, the per cent change, that is, the angle, is called a circular angle, and there is rotation. Thus, hyperbolic and circular angles are at right angles to each other, the first being a per cent change parallel to the quantity, and the second perpendicular to it. “Imaginary” thus means a quadrature. However, it is not rotation around a quadrature axis but per cent change in a quadrature direction. This new interpretation is in agreement with the representation by arcs in a more extended vectorial sense in which one is not limited to the arc of a circle but arcs of any shape have an angular interpretation. Thus, consider a portion of an arc of any arbitrary shape, and resol e it into two components, one circular (at right angles to the radius drawn to the point) and the other straight and radial (parallel to the radius at that point). These two arcs expressed as a percentage of the radius represent circular and hyperbolic angles, respectively, and being in quadrature with each other their vector sum is the “complex” angle of the arc. In this interpretation a hyperbola has nothing to do with a hyperbolic angle, and the name is unfortunate. It would have been more intelligible to speak of the ordinary circular angle as a “quadrature” angle, and of the so called hyperbolic angle as a “linear” angle or a “radial” angle. This new interpretation of an angle is then utilized to interpret the physical significance of sines and cosines of angles, whether circular or hyperbolic. The common idea of a cosine as the projection of the radius vector on the X axis, and of sine as the projection on the Y axis, is found inadequate, this idea being limited to circular angles and not being of much value in explaining their occurrence in physical problems. A cosine is then interpreted as the mean vector value of a unit quantity for a given per cent increase and an equal per cent decrease, or a given value of rotation positive and negative. Examples are given illustrating the application of this vector idea in physical problems, as in the resolution of a harmonic wave into two oppositely rotating vectors, solution of differential equations, etc. A geometrical construction is given for the sines and cosines of complex angles, which is not merely a representation but photographically true, so to speak.

A. Boyajian

Papers

Symmetrical-Components Analysis of the Four-Phase System

The D-C motor capacitor

Oscillations of a High-Voltage Secondary Winding

Physical Interpretation of Complex Angles and Their Functions

Electrical frequency changer