The document discusses Dr. Nasar's paper on developing a unified theory of electrical machines through applying electromagnetic theory and Maxwell's equations. It presents the advantages of this integrated treatment, including unifying different machine theories and solving problems not amenable to other methods. Various field-theory methods are outlined, such as energy-storage, energy-transfer, analogies to circuit theory, and special techniques like images and conformal transformations.
The document discusses Dr. Nasar's paper on developing a unified theory of electrical machines through applying electromagnetic theory and Maxwell's equations. It presents the advantages of this integrated treatment, including unifying different machine theories and solving problems not amenable to other methods. Various field-theory methods are outlined, such as energy-storage, energy-transfer, analogies to circuit theory, and special techniques like images and conformal transformations.
The document discusses Dr. Nasar's paper on developing a unified theory of electrical machines through applying electromagnetic theory and Maxwell's equations. It presents the advantages of this integrated treatment, including unifying different machine theories and solving problems not amenable to other methods. Various field-theory methods are outlined, such as energy-storage, energy-transfer, analogies to circuit theory, and special techniques like images and conformal transformations.
The document discusses Dr. Nasar's paper on developing a unified theory of electrical machines through applying electromagnetic theory and Maxwell's equations. It presents the advantages of this integrated treatment, including unifying different machine theories and solving problems not amenable to other methods. Various field-theory methods are outlined, such as energy-storage, energy-transfer, analogies to circuit theory, and special techniques like images and conformal transformations.
The author describes the advantages of an integrated treatment of the application of
electromagnetic theory, based on Maxwell's equations, to electrical machines. Dr.
Nasar is Assistant Professor of Electrical Engineering with the East Pakistan University of Engineering and Technology.
Unified theory of electrical machines
S. A. NASAR, B.SC, M.S., PH.D. Electrical machines have often been treated as lumpedparameter networks, and have been analysed by the techniques, such as matrixes and tensors, applicable to lumpedparameter linear systems. Based on this method, a generalised and unified theory for electrical machines has evolved. An alternative approach to the analysis of electrical machines is through the discipline of the continuum, such as the application of classical electromagnetic theory. The lumped-parameter circuit analysis and the continuousmedium field analysis both have their merits, and one cannot be substituted for the other. A survey of the literature on electrical machines indicates that the bulk of the work relates to network-theory methods. In my paper, an attempt is made to present an integrated treatment of the application of electromagnetic theory to electrical machines, thereby obtaining a unified machine theory. Such an approach to electrical machines has definite advantages, which are discussed later. The basic laws of electricity, which govern all electromagnetic phenomena, can be expressed as a set of equations, called Maxwell's equations. These equations are, naturally, also applicable to electrical machines. The object of analysing electrical machines, through the use of Maxwell's equations, may be one or more of the following: for purely academic and educational reasons to unify the apparently different theories for different machines to simplify, with rigour, the otherwise complicated and less rigorous methods to obtain the solution to problems that cannot be solved by other methods to determine the parameters of the machine, which may be treated subsequently by circuit theory. The advantages of such an approach lie in the objectives just enumerated. The field-theory methods of predetermining the performance of electrical machines can be roughly classified as: energy-storage methods energy-transfer methods methods derived from the analogies between field theory and circuit theory certain special techniques, such as the method of images and conformal-transformation and graphical methods. An outline of these methods is now presented. Energy-storage methods In order to make a machine problem amenable to field analysis, a model is chosen. The actual slotted structures are replaced by smooth structures, and the actual windings are replaced by infinitely thin current sheets. It is assumed that all the magnetic energy is stored in the air gap of the machine and the energy stored in the iron is negligible. The actual air-gap length is replaced by the effective air gap. The boundary-value problem is then solved, and the fields in the air gap are found by solving Laplace's equation for the magnetic vector or scalar potential. The fields may also be 194
obtained directly, without the use of the potential functions.
In any case, having determined the fields, the energy stored in the air gap is calculated. This yields the inductance coefficients and the equations of motion of the machine. The method is especially suitable for induction and synchronous machines. Energy-transfer methods An alternative method of obtaining the machine characteristics is by calculating the magnetic energy crossing the air gap of the machine. This is determined by obtaining the electric and magnetic fields, by solving Maxwell's equations, and then calculating the power density through the use of the Poynting vector. The losses can then be subtracted from the total energy crossing the air gap to obtain the output at the shaft. This method has been shown to be applicable to d.c. machines and synchronous and induction machines. Correlation between field theory and circuit theory An analogy has been made between field theory and circuit theory. The concept of wave impedance has been used in the analysis of transformers and induction motors. An appropriate model is chosen for which the characteristic impedance and the propagation constant are defined. The transmission-line theory is then used to determine the current and voltage distribution along an analogous transmission line. This yields the corresponding current density and flux density in the motor. The application of the Lorentz force equation gives the torque developed by the machine. Methods of images and conformal transformations The methods of images and conformal transformations have very important applications in electromagnetic theory. With regard to electrical machines, the method of images is useful in determining the end-connection leakages. A suitable model is chosen to replace the actual end connections. The equipotentials are then replaced by images, and the resultant fields are calculated. Several published papers are available on this topic. The method of conformal transformations is suitable to a machine problem where an irregular surface, such as salient poles, is to be reduced to a smooth surface to facilitate the solution of the boundary-value problem, so that energy methods may be applicable. Finally, the effects of space harmonics can be conveniently taken into account. It can be shown, from field analysis, that the different harmonics in the air-gap fields of a machine do not interact to produce any torque. The electromagnetic theory has a wide range of application to machine problems, and it leads to a unified machine theory. For instance, for a single-phase induction motor, the double-revolving-field theory and the crossfield theory can be blended into one via the field theory. Other applications include the analysis of d.c. machines; polyphase induction machines; salientand nonsalient-pole synchronous machines; sleeve-rotor, solid-rotor and eddy-current machines; hydromagnetic d.c. Paper 4479 convertors; and liquid-metal pumps. Electronics and Power June 1964