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A two-dimensional numerical analysis of the cross section of a homopolar inductor alternator supplying a rectifier load is developed. Besides the effects of the nonlinearities of the magnetic materials, the eddy currents induced in the rotor by the nonlinear load are considered. Results for a 95-kVA homopolar inductor alternator are presented and include magnetic induction and flux distribution at two instants of time, output voltage, eddy current densities, and loss. The theoretical results have been found to be in good agreement with available test data on the alternator.

In the subtransient analysis of salient pole alternators the reactances X11d and X11q are generally assumed to be constants. In saturated alternators these reactances are in reality functions of the pre-transient load and saturationix This gaper is an extension of former papers by Fuchs, et it dealing with steady state, and transient reactances. The methods used are the same as in [1] and [2] with the addition of damper windings effects. Notations not explained here are those of Concordia4 and of Fuchs3. The analysis is applied to a G. E. alternator with the following name-plate data: 82.5 MVA, 16.5 kV, wye-connected, 40 poles and 60 Hz, described in [1].

In former papers 1,2,3 the authors have analyzed the behavior of homopolar inductor alternators, considering all non-linearities and load conditions. As a logical sequence, a means to find the main dimensions of a new design for a specific aerospace power system is presented. The limitations imposed by the high-speed operation are pointed out. Dimensional analysis is used as a method for optimizing alternators for minimum armature leakage for frequency-changing power system. The relation of slot width to slot depth of the armature is optimized for minimum excitation.

A computer program for establishing the equipotential plots in the cross section of a dc machine is described. The necessary steps are as follows: 1) scanning of the gridlines in the cross section to find the points of definite magnetic potentials from the discrete vector potential; 2) finding the location of the point of the mesh section containing the given vector potential by interpolation; 3) expressing the positions of the points in rectangular coordinates and finding and printing out all points of the same vector potentials by a CalComp computer. This procedure is carried out for several values of the vector potentials to obtain a comprehensive plot of flux lines.

A description of different types of eddy-current clutches is given. The difficulties encountered in designing these are explained. A method to predict the torque-speed characteristic of eddy-current devices based on dimensional analysis and model theory is developed. Experimental data show the calculations made with the method are sufficiently accurate for engineering purposes.

Based on methods described in Part I1 of the paper, the airgap radial inductions are calculated for four different rotor positions at full load and at one rotor position at twice full load. Line over-relaxation and additive acceleration of convergence techniques are used. Convergence criteria are described. Results of experiments to find suitable over-relaxation factors and expedient line integral paths for convergence accelerations are given. A computer program is detailed and applied to a 500 hp, 750 V, 4-pole, 300/900 r/min, dc motor.

The method in Ref. [1] is used to determine synchronous reactances for the turbogenerator at operating conditions in Part I [2]. The influence of saturation on the direct and quadrature reactances is studied and the difficulties caused by linear parameters in nonlinear theory are discussed. A modified phasor-diagram taking the small angles ?d and ?q into account is developed. The results are given for a Westinghouse Electric Corp. 200 MVA, 15 kV, wye-connected, four-pole, 50 Hz generator. Available test data are compared with computations.

The magnetic induction in homopolar alternators is found by a method developed in a companion paper. The mathematical procedure and the selection of the mesh system and the identification system for the different mesh points is explained. A computer program for the numerical solution by overrelaxation of the boundary value problem is developed and applied to a 95 KVA, 115/220V, 3,300 cps. 40,800 rpm. alternator. The results show the decrease of the airgap induction fundamentals with increased excitation.

In the steady-state analysis of salient pole alternators the reactances Xd and Xq are generally assumed to be constant. This is not the case in saturated machines. The reactances are functions of the operational regime of the alternators, and for an exact analysis of the alternator's behavior the appropriate values of Xd and Xq have to be used. The magnetic induction is found in the cross section of the alternator by difference methods established at the Universityl and then saturated Xd and Xq are determined from flux plots. The results confirm that Xd and Xq are load dependent. The analysis is applied to a General Electric alternator with nameplate data 82.5 MVA, 16.5 kV, wye-connected, 40 poles and 60 Hz.

The usual two-reaction theory of synchronous machines yields results that neglect to take saturation into account. Based on a method developed in a companion paper [1], the radial airgap induction is found in synchronous machines when supplying balanced loads. The armature currents are considered to be distributed uniformly in the upper and lower layer of the armature windings. The partial differential equations, the boundary conditions, and the method of solution are reviewed. Numerical results are obtained for a 2750 kVA, 2300 V, 60 c/s, three-phase alternator at seven different internal power-factor angles. Calculated results compare well with tests.

A former paper of the authors is extended to find the radial component of the magnetic induction in the airgap of homopolar alternators when supplying balanced loads. The armature currents are represented by sinusoidal current sheets at the stator bore. The partial differential equations, the boundary conditions, and the method of solution are reviewed. Numerical results were obtained for a 95 KVA, 115/200 v., 3,400 cps developmental homopolar alternator. Equations for the calculations of the armature impedances are given. A phasor diagram for the finding of the terminal voltage of the alternator at balanced loads is explained. Calculated results compare well with tests.

The flux densities under load are established in the cross section of salient pole alternators of low pole numbers. The two-dimensional partial differential equations describing the magnetic vector potentials are first transformed in partial difference equations and a new algorithm is given. A combination of rectangular and polar grids is used to follow the contours of the alternator. The reluctivity of the material is assumed to be the function of the magnetic induction. An iteration procedure consisting of relaxation of the vector potentials using a new form of blockrelaxation and underrelaxation of reluctivities is described. A computer program in FORTRAN language is given and applied to find the airgap radial inductions for a 2750-kVA, 2300-volt, 60-Hz, three-phase, eight-pole alternator.

The two-reaction theory of synchronous generators yields only qualitative results when applied to homopolar machines under load. Saturation has to be considered for more exact results. The method developed in two companion papers is used to find the airgap radial induction when the armature carries purely reactive balanced currents. The additional boundary conditions are explained and a computer program established. Numerical results were obtained for a 95 kvs. 115/200 volts, 3,300 cps, 40,800 rpm alternator. The results are shown in curves.

The commutating impedance of homopolar alternators of medium frequency, as described by Trutt and Erdelyi [1], has to be kept at a low value in order to enable the various arrangements made up from solid-state elements to convert frequencies of 3500 Hz to the usual 400 or 60 Hz used in aerospace supplies. This can be achieved by fitting damping devices into homopolar inductor alternators. To study theoretically the effect of these devices, the inductances of the various windings must be known. This problem is treated here. For this purpose, the two-dimensional model of the alternator and the vector potential analysis of the cross section (as shown by Schenk et al. [2]) is used. Because of the varying position of the stator winding with respect to the rotor teeth, these reactances are time-dependent, and the coefficients of the differential equations describing the commutating (subtransient) regime are time-varying. The calculations were carried out for a 95-kVA, 115/200 V, 5 rotor teeth, 3400-Hz homopolar inductor alternator. The air gap of the alternator was 0.030 inch. Detailed data of the homopolar inductor alternator are in [1] and [6].

Based on the concept of the magnetic vector potential the partial differential equations and the boundary conditions of the electromagnetic field in the different regions in the cross section of heteropolar machines are developed. These are transformed into finite difference equations, that do not neglect iron non-linearities and the effect of current carrying regions. A method of finding a numerical solution for the vector potential and for the magnetic induction is explained. A simplified cross-section of a synchronous generator is used to illustrate the application of the method.

The two dimensional, non-linear potential equation for saturated homopolar generators is established in circular cylindrical coordinates. The boundary conditions are discussed. The partial differential equation and the boundary conditions are transformed into difference form as preparation for a computer program. A non-uniform polar system of meshes is described and the difference formulae for these is given. The flow chart of a computer program applicable to an IBM 7090 computer is diagrammed. Over-relaxation and alternating direction relaxation techniques are explained. The flux distribution at no-load in a 95 KVA, 115/200 V, 3,400 cps three phase alternator is found with this program. Computed results compare very closely with test results.

It is shown how ever-increasing enrollments, the explosion of knowledge, advances in technology, and social changes have affected undergraduate electrical engineering curricula to a point where they are seemingly unsatisfactory for a career in the power field. A program is outlined to provide undergraduate and graduate training to a limited number of engineers serving the public utilities. Concrete suggestions are made for the organization and financing of the proposed scheme. The importance of research at universities is emphasized.

The calculation of flux densities in dc machines using digital computers requires a representation of the magnetization curve of the materials either as H = f1(B) or B = f2(H). A figure of merit of the approximation is defined. The representation can be achieved by the use of simple algebraic or transcendental functions. For satisfactory fit, however, the B-H characteristic has to be subdivided into several parts. As a better alternative a numerical method is proposed in which the magnetization curve is subdivided into a large number of sections. A method of linear interpolation is then employed. A computer program to accomplish this task is described.

Improvements in the method of determining the magnetic fields of nonlinear dc machines are introduced in part I of the paper. A modified, successive line overrelaxation technique to satisfy boundary conditions is developed. A novel block relaxation technique, the additive acceleration of convergence, is explained. The improved method also takes care of relative position changes between field and armature. In part II [8] the improved method is applied to a large dc machine. A new computer flux plotting procedure is developed in part III [9].

The effect of slotting not considered in previous papers, is analyzed by taking into account the actual slots of the stator. The equations and methods of Ref.l and 2 are used with some modifications. The boundary conditions at no-load on the lines of symmetry have been replaced by the value of the potentials and thus, improved convergence has been achieved. The flow chart of the main program for an IBM 7090 and the flow chart of an IBM 1620 program for calculating the potentials on the lines of symmetry are given. In a simplified solution, the actual slotted region is replaced by a smooth airgap and an equivalent " synthetic" region of homogeneous iron. The methods are used to calculate the no-load flux distribution and no-load voltage of a 95 KVA, three-phase 115/200 v, 3,400 cps., high-speed inductor alternator. The fundamentals and higher harmonics of the different solutions are compared. Calculated no-load voltages compare with test results very closely.

The partial differential equations of the potential and the boundary conditions of fields in highly saturated rotating machines at no-load are developed. These are expressed in finite difference form, and in the numerical solution the potentials and the permeabilities are over-relaxed alternately. The choice of a non-uniform system of meshes, and a method of identifying the numerical operations at the mesh points are given. Companion papers illustrate the application of the method.

The eddy current losses in rotor tooth-heads of homopolar alternators are generally high and a limiting factor of design. A method is established to calculate eddy current losses in rotor designs with titanium wedges in the intertooth space. A computer program solves numerically the boundary value problem that yields the electric field intensity in the conducting materials. Inverter loads on an experimental 95 KVA, 3,400 cps, three-phase homopolar alternator are calculated to be approximately five percent.

The general differential equations for the circuits of a homopolar inductor alternator are established. The system of equations is modified for the case of nonsymmetrical subtransient operation. The Runge-Kutta and Adams-Moulton methods are used to solve the system of differential equations with variable coefficients. The solution is obtained for different initial conditions. Several damper winding designs are analyzed. A step-by-step correction method in the time domain is used to improve the approximate initial value of the subtransient reactance. This method takes care of saturation, which changes greatly during a subtransient. The analysis of the most advantageous design of the axial damper winding located in the armature slots is given in this paper. All calculations were carried out for the same alternator as used in the companion paper [1].

The partial differential equation for the vector potential of salient-pole heteropolar machines is established by considering nonlinearities of the iron and current density fields. The nonlinear equation, similar to Poisson's equation, is transformed into a partial difference equation to find the magnetic induction at discrete points in the cross section of the heteropolar machine. Alternating relaxation methods are used to obtain numerical solutions. The relaxation procedure is accelerated considerably by a new method of adjusting the vector potential between successive iterations. The method is used to find the no-load characteristic and the no-load air-gap flux distributions of a medium-sized 8-pole salient-pole alternator. Calculated voltages and test results show good agreement.

The partial differential equation for the vector-potential of heteropolar machines is established taking into account the non-linearities of the materials and the current distributions. The non-linear equation, similar to Poisson's equation, is transformed into a partial difference equation. The magnetic induction is found at discrete points in the cross section of the machine. A new relaxation procedure, accelerated greatly by the use of a new physical method of adjusting the vector potential between two successive iterations, is used. The method is shown for an aerospace synchronous generator. Calculated and test results of the no-load characteristic agree quite well. Pole tip saturation is fully considered.

In the transient analysis of salient pole alternators, the reactances X'd and X'q are assumed to be constants. In saturated machines, however, these reactances are functions of the pre-transient load of the alternator. For more exact analyses saturated reactances should be used. The expression of rotor quantities in stator terms, or vice versa, at nonlinear condition is explained. The operational impedances Xd(p), Xq(p) and the various time constants of alternators without amortisseur windings are developed from leakage reactance concepts. After a first approximation of the transient reactances is obtained from the operational impedances, a step-by-step correction procedure is used for finding the operationally dependent direct transient reactance. This is obtained from a simulated dynamical test. Only a few step-by-step corrections are necessary to yield good values for the transient reactances. Since there is a slight difference between Xq and X'q, the analysis will concentrate only to X'd. Notations not explained here are the same as in Concordia [1]. The analysis is applied to a General Electric alternator with nameplate data 82.5 MVA, 16.5 kV, wye-connected, 40 poles and 60 Hz, described in [2].

In the first part of the optimization process, the width and depth of the rotor tooth are incrementally varied for two airgap radial lengths and two field excitations. In the second part the tooth shape that would yield a sinusoidal voltage is investigated. The numerical results are represented by curves for the no-load radial flux density in a 95 KVA, 120/208 v 3,400 cps alternator. The results show that the fundamental airgap flux density depends on the length of the airgap and on the rotor tooth width. It depends only slightly on the rotor tooth depth and excitation. A more sinusoidal output voltage can be obtained by flaring the tips of the rotor tooth and widening the rotor tooth.

In this paper the effects of the time harmonics on the output voltage are investigated. Instead of a complete analysis, the waveshape of the line current is considered as recorded on an oscilloscope when the alternator supplies an inverter. The minimum airgap induction occurs when the magnetomotive forces of the fifth and seventh time harmonic oppose the field excitation. The reduction in the amplitude of the fundamental of the airgap induction caused by the time harmonics was found to be of the order of fifteen to twenty percent for a typical cyclo-inverter load.

The theoretical analysis based on electromagnetic field theory is given for high speed aerospace induction motors whose solid rotor is provided with layers of highly permeable and highly conductive materials. Numerical solutions indicate that the efficiency of the rotor as compared with a solid iron rotor is increased considerably. The torque-speed characteristic of the composite rotor motor rises steeply and is nearly flat between 20 and 100 percent slip. The design approaches the characteristic of a multiple cage motor. Operating temperatures are considerably reduced.

The method of finding flux plots in the cross section of salient pole rotating machines is extended to large, internally cooled turbogenerators. The relaxation procedure used by the authors in former papers [1,2,3] is assumed to be known to the reader and is not reviewed here. The grid systems for the partial differential equations are described in detail for a Westinghouse Electric Corp. 200 MVA, 15 kV, wye-connected, four-pole, 1,500 r/min, 50 Hz generator. Flux plots in the cross section of the alternator and the distribution of the radial magnetic induction on the stator bore are determined at no-load, three-phase short circuit, and at balanced lagging and leading loads. Available test data compare well with calculated results.

The consideration of slotting in the armatures of dc machines increases the number of grid points to be considered beyond the capacity of normally available computers. To reduce the memory storage, a combination coordinate system characteristic to the parts of the lamination is proposed. To satisfy a large number of boundary conditions, a scheme is described to separate the rotor and stator lattices used for the calculation of the vector potentials. A modular scheme to take care of the profile repetitions in the armature is described. The flow charts of subprograms to be used with existing computer programs are given.

In this paper, a method is provided for finding the magnetic fields in dc machines on-load and overloads. Mathematically, the problem is that of solving a quasi-Poissonian equation through use of a modified relaxation method. The differential equations are first replaced by difference equations; this constitutes a set of algebraic equations which are solved by a procedure consisting of relaxation and block relaxation of the potentials, and of underrelaxing the reluctivities. Currents and nonlinearities caused by the saturation of the iron are fully considered. A flow chart of a computer program is given, and the procedure is applied for finding the fields in a 500-hp, 4-pole dc motor at rated load, at twice rated load, and at six-times rated load with a high-speed computer. Flux plots in the region between the commutating pole and the armature show the spread of the main pole flux into the commutating region at overloads.

A method based on beam theory is developed to describe the bending vibrations of nonthin, complete circular rings of rectangular cross section. Bending, shear, and extensional strain energies are considered together with translational and rotatory kinetic energies. The notion of the average shear angle is used. It is also shown that a slightly simplified analysis, neglecting extensional energies, avoids certain computational difficulties. The solution for resonant frequencies is given in a nondimensional form and the theoretical results compared with the results of a thin-ring theory and with experiments.

Based on a theory presented by Ahamed and Erdelyi [1], a method has been developed to find the magnetic flux densities in the air gaps, armatures, poles, and frames of dc machines, taking full account of nonlinearity caused by saturation. A digital computer program to obtain the no-load characteristics of large compensated dc machines is given and applied in analyzing 500-hp, 4-pole dc motors. Computer results agree well with tests.