The effects of eddy currents in transformer windings are considered, and a method is derived for calculating the variation of winding resistance and leakage inductance with frequency for transformers with single-layer, multilayer and sectionalised windings. The method consists in dividing the winding into portions, calculating the d.c. resistances and d.c. leakage inductances of each of these portions, and then multiplying the d.c. values by appropriate factors to obtain the corresponding a.c. values. These a.c. values are then referred to, say, the primary winding and summed to give the total winding resistance and leakage inductance of the transformer. Formulas are derived and quoted for calculating the d.c. resistances and leakage inductances of the winding portions. Theoretical expressions are derived for the variation with frequency etc. of the factors by which the d.c. values must be multiplied to obtain the corresponding a.c. values. These expressions are presented in the form of graphs, permitting the factors to be read as required.

Effects of eddy currents in transformer windings

The classical one dimensional magnetic field and eddy current distribution ("proximity effect") within a series connected multiple layer coil is reexamined with regard to power losses withinthe windings. When the lengthand number of layers ina coilare fixed, the power dissipation within each layer can be minimized by choosing a specific radial thickness for each layer. Above or below this thickness, the losses within the winding increase. The conductor thickness which results in minimum dissipation depends on the relative position of the layer. When compared to a design having a constant thickness for each layer (chosen for minimum total dissipation), it is found that substantial savings in power consumption can be realized by employing a variable thickness of conductor. The one dimensional solution in cylindrical coordinates for the eddy current and skin effect in amultiple layer series connected coil is alsopresented. By solving the problem n cylindrical coordinates, the effect of curvature on the power consumption within each layer is apparent. This analysis should have application to the design of power transformers, armature windings, and inductors for power transmission lines.

Multiple Layer Series Connected Winding Design for Minimum Losses

IN any multilayer winding carrying an alternating current, such as the windings illustrated in figures 1, 2, and 3, each layer of copper lies in the alternating magnetic field set up by the current in all the other layers. Eddy currents are set up in each layer in a direction to partly neutralize the magnetic intensities in the interior of the copper wire in each layer. As a result of the eddy-current losses in the copper, the effective resistance of the winding to the alternating current it carries may be many times its resistance to continuous currents.

Effective resistance to alternating currents of multilayer windings

Operation of 3-Phase Induction Motors on Unbalanced Voltages [includes discussion]

For problems involving induction motors other than the usual balanced winding type operating under steady-state conditions, a general method of analysis is necessary. In this paper, a general analysis is presented, and in particular, its use in short-circuit problems is described. The comparison between tests and calculated results indicates that the method does properly describe induction motor phenomena.

http://ieeexplore.ieee.org/iel7/6413714/6540033/06540046.pdf

An Analysis of the Induction Motor

By making use of suitably chosen components of current and voltage, the method of symmetrical components, which is so useful in the analysis of symmetrically wound polyphase a-c machines, can be extended readily to the analysis of unsymmetrical 2 phase induction machines, such as the capacitor or split phase motor in which the windings are not in space quadrature. Such extensions to this method, illustrated by a typical numerical example, are presented in this paper.

Analysis of Unsymmetrical Machines

The principal object of this paper is to show how hyperbolic functions of complex angles may be applied to the solution of the problem of heat losses in rectangular conductors that are embedded in open slots. A certain knowledge of the functions themselves is presupposed. Inasmuch, however, as they are handled like trigometric functions of real angles?except in regard to the plus and minus signs?it is a simple matter to acquire the requisite technical skill to use them. The hyperbolic function of a complex angle, consisting as it does of a real and an imaginary part, may represent a vector?the real part being the component of the vector along the horizontal, and the imaginary part, component along the vertical. Thus, for example, A sinh (x + j x) represents a vector just as A e j ? A/?, A (cos ? + j sin ?) represent vectors. Considerable experience has shown that the vector method for handling a-c. problems is much superior to the original method in which simple trigonometric functions were used. With this lesson before us, it should require but little contact with the problem at hand to demonstrate the superiority of the vector method, even though it employs the possibly unfamiliar hyperbolic quantities. These hyperbolic vectors have been used for a number of years in the analysis of problems involving a-c. circuits, which have distributed inductance and capacitance, and have proved their usefulness.

Heat Losses in the Conductors of Alternating-Current Machines

The vector method is just as useful in solving problems involving transient conditions in electric circuits as it has proved to be when the currents and potentials are steady sinusoids. As far as the writer is aware, the vector method for determining transients in rotating electric machines was first used by L. Dreyfus. Previously the method had been applied to fixed combinations of resistances, inductances and capacitances by Kennelly and others. By making certain assumptions that are, however, quite reasonable in many cases, the transient currents in nearly all of the common types of electric machinery are damped sinusoids. Fortunately the damping is exponential and is thus readily accounted for. It is interesting to trace the development of the method. In the solution of all problems in direct currents the potentials, currents, and circuit constants are real numbers. In the corresponding problem in which the applied potentials are steady sinusoids, these quantities are all represented by complex numbers. In all other respects the working out of the solution is identical with that followed in the direct-current case. When the currents are damped sinusoids, they and the potentials and the circuit constants can still be represented by complex numbers. There is this difference, however; the vectors which represent the currents and potentials shrink exponentially as they rotate and the values of the circuit constants depend not only upon the frequency of the current, but also upon its rate of shrinking. Again the solution of any problem follows the same procedure that would the corresponding one in which the currents are steady sinusoids. In both the steady and damped sinusoidal cases the circuit constants depend upon the angular velocity of the vectors which represent the currents. In the former, the angular velocity is purely imaginary while in the latter it is complex, the real part being the rate at which the current vector shrinks and the imaginary portion, its angular velocity. In electric machinery in which rotating magnetic fields are produced, these fields shrink exponontially as they rotate when the currents are damped sinusoids. If these rotating magnetic fields are represented by vectors, the vectors will have a complex angular velocity just as do the currents. The e. m. f. which is produced by a steady sinusoidal variation of flux lags the flux by 90 degrees, whereas if the flux variation is a damped sinusoid, the angle of lag is less than 90 degrees, depending upon the damping. The mathematical relation, however, is the same, vis., the e. m. f. is proportional to the negative of the product of the flux and its angular velocity. It is then readily appreciated that the form of the solution for the transient state is the same as that which is used for the steady state. Before the method can be expected to give as accurate results as are obtained when predicting the steady operation, considerable experimental data must be obtained in order to determine the best methods of measuring the necessary constants, for these may be somewhat different during the transient period than during steady operation.

Transient Conditions in Electric Machinery

Mechanical oscillations of synchronous motors following the application of abrupt shaft loads have not been hereto-fore easily calculated for cases of large angular swings, taking into account the damping currents in the rotor, except by tedious point-by-point methods. The chief difficulty has been due to the form of the basic differential equation upon which the solution of hunting problems depends. Within the lastfew years Dr. V. Bush and others at the Massachusetts Institute of Technology have devised the integraph which is capable of solving the equations of the synchronous motor. Into the integraph are put curves representing the non-linear differential equations and the boundary conditions. The desired results come from the integraph as curves. In this paper the problem of sudden load on the non-salient pole alternator is solved by the integraph for enough different conditions so that the performance of practically any machine of this type may be easily predicted from the compiled results. Knowing the moments of inertia of the machine and its load, the characteristics of the machine running as a synchronous motor and as an induction motor, the maximum amount of sudden load for which the machine will remain in synchronism may be determined for different values of initial load. Other curves give the maximum angle of the first swing of the rotor and the time interval for the rotor to change from its initial position to this maximum angle. The simplest type of equation representing an ideal synchronous machine is solved in this paper.

Transient Torque-Angle Characteristics of Synchronous Machines

In the present paper, which is a continuation of one presented at the last annual convention, the author extends the method of complex hyperbolic functions to the solution of the problem of heat losses in stranded conductors embedded in rectangular slots. In the preceding paper the discussion was confined to solid conductors and to those having an infinite number of strands. In the latter case, the insulation between the strands was assumed to have no appreciable thickness. In the present paper, conductors are considered which have a finite number of strands separated by insulation of appreciable thickness. In the mathematical development which is to follow, free use is made of the results obtained in the first paper.

Waldo V. Lyon

Papers

Analysis of Unsymmetrical Machines

Heat Losses in the Conductors of Alternating-Current Machines

Transient Conditions in Electric Machinery

Transient Torque-Angle Characteristics of Synchronous Machines

Heat Losses in Stranded Armature Conductors