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Equation of the Synchronous Motor by the Method of Complex Variables

The power supplied by the generator to the motor will be:

In like manner, the electric power consumed by the motor will be

The power lost by ohmic resistance (Joule effect) is equal to the difference, or to

The diagram therefore determines, by the measurement of simple lines or areas, the current strength and the power of each of the two machines, for each value of the phase-angle.

Let us suppose that the vector OA (Fig. 13) is stationary, and that the phase-angle, Θ, varies. The point B' will describe a circle of radius OB'; the point D describes a circle constructed on OA as a diameter; finally, it is easily seen that the point C describes a third circle having for its center and for its radius the projections of O and of OG on a right line, AF, making the angle γ with OA. These three circles having been drawn, the conditions of operation can be followed on the diagram. Mr. Blakesley has thus determined the limits of stability and the maximum electrical efficiency.

But even for easy questions the use of this diagram is too complicated; the same thing is true of the diagrams devised by Kapp, after Blakesley, and based on the same principles. The author has also given a more convenient method of representation of power by circles which cut off segments directly from the lines OA and OB.

In all that follows, we will give preference to another graphical method, still more simple, which will be explained in Chapter II.

Equation of the Synchronous Motor by the Method of Complex Variables

Finally, to complete the review of the different methods of study proposed or used for the representation and study of the phenomena, attention should be given to the equations which translate the preceding diagram by the method of complex variables, already applied to alternating current problems, in the exponential form, by Oberbeck, Cornu, Chaperon, etc., and in the linear form by Kennelly Steinmetz, Guilbert, etc. It is merely an application of the ordinary geometrical representation of imaginary quantities.

We will employ the notation of Steinmetz modified by M. Guilbert (La Lumiere Electrique, Vol. L, p. 451, and l'Eclairage Electrique, Vol. XIV, p. 69), i.e., by expressing the impedance as (r+si) instead of (r - si), the latter expression being scarcely logical, since the reactance acts in the same direction as a resistance to reduce the current. It thus becomes possible to retain the ordinary axes.

Let OX, OY (Fig. 14) represent two rectangular co-ordinate axes. Let OA be a vector representing the sinusoidal function whose projections are x and y. Geometrically, the vector A is defined by the two projections, and analytically it is represented by a single imaginary value in which j is the imaginary symbol √-1.

All the harmonic functions may be also represented by imaginary quantities.
These quantities are added or subtracted the same as real quantities; the resultant quantities indicate immediately, by their real and imaginary portions, the magnitude and the phase of the resultant vector.
The multiplication of a complex quantity by a real quantity only changes the dimension of the vector, without changing its phase. On the contrary, the multiplication by an imaginary portion, such as jb modifies not only the magnitude of the vector, which thereby becomes multiplied by b, but it also modifies its phase, which is then made to advance by the amount ^π/₂ . In fact, we have:
j(x + jy) = - y + jx
i.e., the vector OA is then replaced by the vector OA'.
From this, it follows that the E.M.F. absorbed by an impedance, z, composed of a resistance, r, and of a reactance, s, in series, through which passes a current
i = x +jy
is obtained, in magnitude and phase, by forming the product
zi =(r + js) (x + jy)
The work done by an E.M.F. OB having the imaginary value
a + jb,
and a current OA having the imaginary value
x +jy
is easily obtained by decomposing it into the work of each of the components a and b of this E.M.F. The first component, a, does no work on the component y of the current with which it is "e;in quadrature"e; but it is "e;in phase"e; with the component x; likewise b is in quadrature with x but in phase with y. The power required is therefore the sum of the products of the analogous coefficients of E and of I. We will have
ax +by

Let us apply this notation to the generator and motor already considered. Let us direct the E.M.F. E₁ of the generator along the axis OX. E₂, lagging in phase by the angle Θ, measured from the opposite direction, will have the form

E₂(cos &Theta - j sin Θ)

The imaginary impedance is

R + ωLj

The resultant E.M.F. being

E_l - (E₂ cos Θ - jE₂ sin Θ)

the current will be obtained by taking the quotient

Separating the real and imaginary portions, we have

This equation, when transformed into finite values, and taking z to represent the impedance, gives

This equation defines the relation between I, E₁, E₂, and Θ in a synchronous motor.

The mean power, P₁, is obtained by simply multiplying the real portion by E₁ thus:

As for P₂, it can, according to the rule already stated, be expressed as follows:

This expression could have been written directly, by symmetry.

Such is the method of imaginaries for establishing the fundamental equations. It will be seen that the calculations thereby made are more simple than with the analytical method of Hopkinson, because they give, immediately, the general solution, and do not require any integration. In reality, however, complex variables only constitute an artifice for writing down the results of the graphical method. In place of detailed reasonings which are rendered more precise by means of diagrams, they substitute algebraical operations, which are effected mechanically, without benefit in helping the mind to understand the physical phenomena. In all that follows we will therefore adopt the graphical method.

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