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- Synchronous motors: An introduction
- Chapter I: Synchronous motors General Principles
- Efficiency Synchronous Motors and Experimental Properties
- Stalling of a Synchronous Motor
- Over-excited synchronous motor
- Necessity of synchronism and stability of synchronous operation
- Explanation of Single-Phase Synchronous Motors
- Equations of Synchronous Motors; Analytical Theory
- Symmetrical Polyphase Motors
- Equation of the Synchronous Motor by the Method of Complex Variables
- Excitation of Synchronous Motors
- Shunt-excitation
- Chapter II: Operation of synchronous motor
- Installation of Synchronous Motors
- Current controller
- Starting of Single-Phase Machine
- Starting of Machine with Laminated Field Poles
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Symmetrical Polyphase Motors
Case of Symmetrical Polyphase Motors
Case of Symmetrical Polyphase Motors
In the case of polyphase motors the same considerations and equations remain applicable to each of the symmetrical circuits, if only care be taken to include, in the self-induction of these circuits, their mutual induction effects. By reason of symmetry itself, the result is obtained by a simple increase of the coefficient, L, in a ratio which depends on the number of phases in the machine.
The currents are, in general, approximately equal in effective values in the different circuits, if there are no defects in construction. It is sufficient, therefore, to apply the reasoning to one of the circuits only, and to multiply the power by the number of circuits.
It should be noted, moreover, that, in these motors, the pulsations of the power derived from the different circuits, occurring at different intervals, compensate each other, from the standpoint of the total power, which becomes constant. This is easily shown by taking the sum of the powers. For example, in a three-phase motor, the pulsations at the instant 2ωt will have the form
It is known that the sum of the sines of three angles differing from each other by 120° is identically equal to zero. Therefore the "e;resultant"e; pulsation is equal to zero. The same thing would be true for any number whatever of equidistant and symmetrical phases.
The result is that the torque is constant (to a sufficient degree of approximation for this theory, which neglects the higher harmonics of the field-distortions), whereas, in a single-phase motor, it undergoes heavy periodical variations. In the latter case, the inertia of the armature plays the role of a flywheel storing and restoring energy twice during each period; but if the inertia is insufficient, the velocity of the armature will experience slight variations which will greatly interfere with the stability of operation. Polyphase motors are, in this respect, superior to single-phase motors; they are also superior to them in being lighter for a given output (less weight per kilo-volt-ampere) and also in having higher efficiency.
Graphical Representation of Operative Conditions. Blakesley's Method
Mr. Blakesley was the first to apply Fresnel's method of vectors to the study of alternating currents.
The following principles constitute the basis of Fresnel's method:
Any sinusoidal function can be represented in magnitude and in phase by a vector, or a segment of a right line, whose length is proportional to the amplitude of the function and whose phase is measured by an angle reckoned from some other vector serving as a point of origin.
The addition or subtraction of sinusoidal functions may be made on the graph by a geometrical addition or subtraction of the vectors of these functions.
The mean product of two functions is equal to the work done by one of the vectors on the other, i.e. it is equal to the area of the triangle constructed on vectors which are equal to the effective values of the variables, or to half this area when the triangle is constructed with the amplitudes of the sinusoidal functions.
This method, with which the reader is supposed to be familiar, has been applied by Blakesley to the problem of synchronous motors in the following manner:
Let us represent the amplitudes of the E.M.F.'s. of the generator and of the motor, E1√2, and E2√2,, respectively, by means of two vectors, OA and OB, (Fig. 12), having between them the angle π+Θ. These E.M.F.'s are approximately opposed to each other, as we have seen in what precedes.
Let OB' be equal but opposed to OB. The E.M.F. ε√2 , which is the resultant of E1√2 and E2√2, will be represented by the vector B'A which is equal to the resultant of OA and OB.
Let us again designate by γ the phase-angle between the current and the resultant E.M.F. defined by the relation
and let this angle be drawn with respect to the point A. The direction CA will represent the current which is in phase. Let CA be the projection of B'A . We will have
The length of CA is therefore proportional to I.