A. Potebnya.

On the Theory

OF PARALLEL OPERATION

OF ALTERNATORS.

Tomsk.

Steam Typo-Lithography of P. I. Makushin, Blagoveshch. Lane, own house.

1903.

Over the last two decades a whole series of articles in the electrotechnical literature has been devoted to the theory of the parallel operation of alternators. The beginning of this series may be considered to be an article by J. Hopkinson, published in 1884 in Jour. Soc. telegr. Engin., in which the author for the first time indicates the influence of self-induction on the success of the parallel connection of alternating-current machines and arrives at the most advantageous relation between the ohmic resistance and the reaction of self-induction of the armature winding of an alternator:

\[ r : \omega L = 1, \]

where \(r\) is the ohmic resistance, \(L\) is the coefficient of self-induction, \(\omega = 2\pi\infty\), and \(\infty\) is the number of periods per second.

Hopkinson’s ideas gave rise among electricians to lively disputes concerning the influence of self-induction. There appeared advocates both of the greatest possible increase of self-induction in machines and of its reduction to the limit indicated by Hopkinson.

Subsequent practice showed, however, that the relative magnitude of self-induction plays by no means the first role in the phenomena occurring in the parallel connection of alternators, and that the success of their operation is influenced by a whole series of other causes, chiefly the excitation of both dynamos, the degree of uniformity of motion of the prime movers, and the ratio of the moments of inertia of the rotating masses to the axis of rotation.

The newest works treating the question of interest to us concern chiefly the investigation of the relative oscillations performed by the rotating parts of machines connected in parallel. As early as 1892 Boucherot¹ gave a formula

¹ “La. Lum. El.” Vol. 45, p. 265.

for determining the duration of these oscillations. In the same year, a similar investigation was given by Blondel¹. He also gives a formula for calculating the period of relative oscillations, expanding these oscillations in a Fourier series. Both authors point out the danger of interference between the oscillations of the driving machine and the natural oscillations of the rotating parts of the alternator. In this connection Blondel pays special attention to the magnitude of the moment of inertia, whereas Boucherot assigns to it a comparatively insignificant role. At the same time, Hutin and Leblanc² proposed their device for damping oscillations. None of the named authors gives a mathematical theory of damping.

In 1899 G. Kapp³ gave a very simple method for calculating the period of oscillations of alternating-current machines operating in parallel. He sees the chief difficulty of the parallel connection of alternators in the interference of oscillations. In 1900 Görges⁴, in the article “Ueber das Verhalt. parall. gesch. Wechselstrommaschinen,” gives a simple graphical method of representing the phenomena occurring when alternators are connected in parallel, and, assuming the voltage at the terminals of all alternators connected in parallel to be constant, derives the differential equation of the oscillation of each group (the prime mover with the corresponding alternator) separately. Integrating it, he determines the period of oscillation and the conditions for stability of the motion. Finally, in 1899 Dettmar⁵ points out certain difficulties in the parallel connection of alternators driven by gas engines and the means of avoiding them, and Meyer⁶ in 1901 attempts, from experimental data, to establish the limiting values of the synchronizing current.

A special case of the problem under consideration is the operation of a synchronous motor in an alternating-current circuit. Studying the phenomena of parallel operation of alternators, it is not difficult to become convinced, both theoretically and experimentally, of the reversibility of alternating-current machines, i.e. of the possibility of using one—

by it and at the same time by the alternator as a generator or as a motor, as desired, provided that the motor has first been brought up to a speed corresponding to the same number of periods as that of the generator supplying it.

Among the investigations that have appeared recently on the theory of synchronous motors, one may cite: G. Ossana, E. T. Z. 1896, and P. Steinmetz, “Theory of the Synchronous Motor” (Trans. of the Amer. Inst. E. E., Oct. 17, 1894, and June 18, 1902). In addition, A. Blondel’s book, Moteurs Synchrones à courants alternatifs, 1901, gives a very complete theory of the synchronous motor.

Although, as is evident from the foregoing list, the theory of the parallel operation of alternators has already been developed in many respects, I take the liberty of presenting the proposed article, first, because in Russian technical literature this question has been touched upon comparatively little, and second, because in deriving certain formulas I have tried to avoid the assumption of constancy of the voltage at the terminals, which is laid at the foundation of the arguments of Görges and others, as a result of which the formulas proposed here have a more general character.

I. Derivation of the Basic Formulas.

1. Recently the problem of the parallel connection of alternators has acquired especially great importance: first, because of the ever greater spread of alternating current in comparison with direct current; second, because of the tendency, at central stations, for reasons of economy, to use units of the largest possible size; third, because of the increasingly common direct coupling of the dynamo and the steam engine, whereby, obviously, irregularities in the running of the driving engine have a far stronger effect on the oscillations of the electromotive force than with belt or rope transmission; and finally, fourth, because of the spread of indirect distribution of energy by means of rotary transformers (especially for purposes of electric traction), which convert alternating current into direct current and represent

on the alternating-current side are nothing other than synchronous motors.

The basic condition for the possibility of parallel operation of alternating-current machines is equality of the numbers of periods.

This condition imposes a requirement on the machines driving the alternators: the transmission of power from the engines to the dynamos must take place in such a way that, when the numbers of periods are equal, there is no slipping of the belts, and so that neither the belts nor the corresponding transmission organs are subjected to excessive stress.

A rigid mechanical coupling of alternating-current machines with one another cannot be considered a good means of synchronization, since it does not permit mutual regulation of the phases between the machines. Such a coupling would, as it were, turn the whole series of interconnected machines into a single dynamo. Since a change of the phase angle with such a coupling is impossible, inequality of excitation and, consequently, a difference between the electromotive forces of the machines will produce excessive parasitic currents. Such an arrangement can be encountered only in very old installations, for example at the municipal station in Vevey—Montreux in Switzerland, built in 1887.

The second important condition for satisfactory operation of alternators connected in parallel is uniform speed of rotation, i.e., constancy of the number of periods.

Let us imagine two alternators rotated by two single-cylinder steam engines independent of one another. If their crankpins happen to be at an angle of \(90^\circ\), the turning moment of one machine will attain its greatest value at the time when the piston of the other is in the dead position, and conversely. As a result, one alternator begins to rotate faster, while the rotation of the other slows down. If the flywheels are not sufficiently heavy, a noticeable oscillatory motion results and, because of this, flickering of the lamps connected into the circuit, corresponding to the period of rotation of the steam engine, owing to the transfer of part of the load from one alternator to the other. This phenomenon may reach such proportions that further joint operation of the machines becomes impossible.

Obviously, among the various types of modern prime movers, turbines, water and steam, and water wheels best satisfy the second condition, while heat engines working by explosions satisfy it worst of all.

In considering the phenomena that occur during the parallel operation of alternating-current machines, one must distinguish two periods: the period of unsteady motion, when a new machine is connected to a number of machines already in operation, or when accidentally one of the machines operating in parallel receives a disturbance that brings it out of synchronism with the others; and the working period, when all the machines are in synchronism.

As is known, before two alternators are connected in parallel, they must be brought to approximately equal periodicity and equal electromotive forces, and then switched in at the moment of approximate coincidence of phases, which is judged by phase lamps or by the indications of other analogous instruments.

With the modern construction of machines, equality of electromotive forces is far less important than equal periodicity. Owing to the reversibility of alternators, complete coincidence of phases at the moment of switching in is also not of very great importance. If there is a difference of phase between the two machines, besides the working current there is also established a current that does not pass into the external circuit. This current accelerates the motion of one machine and retards the other until complete equality of phase and periodicity is established.

In modern alternators with iron in the armature, this exchange of energy almost never, even with a large departure from synchronism, reaches dangerous proportions. The synchronizing current is weakened by the reaction of self-induction.

In machines with very small armature reaction, before switching in it is necessary with much greater care to make sure of equality of periodicity, excitation, and phase; otherwise the exchange of current reaches such a magnitude that the machine is endangered by damage as a result of the shock. Machines without iron in the armature sometimes prove so sensitive in this respect that it is necessary to abandon their parallel connection.

Once brought into synchronism, almost all types operate satisfactorily. Nevertheless, in modern machines

they try to keep the magnitude of the armature reaction within certain limits*).

If the armature reaction is too weak, then in the event of some accident (a short circuit, a sudden opening of the external circuit, etc.) the machine may suffer serious damage. If the armature reaction is very large, then the driving force must be very constant, since the synchronizing force of the dynamo machines will be insufficient to compensate for the irregularity of running of the prime movers.

Having briefly set forth the essence of the phenomena occurring in the parallel operation of alternators, we shall proceed to their analytical investigation, which is the purpose of this article.

2. In the following considerations we shall assume both the currents and the electromotive forces to be sinusoidal.

The vectors representing these quantities will be denoted symbolically by complex quantities, with the axis of real values taken as positive to the right, and the axis of imaginary values upward. To the factor \(j=\sqrt{-1}\) we shall ascribe the property of rotating the vector multiplied by it through the angle \(\frac{\pi}{2}\) counterclockwise, and in the same direction we shall count positive angles of phase difference and the arguments of vectors.

Then the electromotive force, or voltage, will be represented in the form

\[ E=e+je', \]

and the current

\[ J=i+ji', \]

where \(e, i\) are the projections of the corresponding vectors on the axis of abscissas, and \(i', e'\)—on the axis of ordinates.

The apparent resistance will also be represented by a vector

\[ Z=r+js, \]

where \(r\) is the ohmic resistance, and \(s\) the reactance of self-induction and capacitance:

*) In the newest machines \(\omega L=(2—10)\,r\).

\[ s = 2\pi \infty L - \frac{1}{2\pi \infty C} \]

where \(\infty\) is the number of periods, \(L\) and \(C\) the coefficients of self-induction and capacitance.

Remark. The expression of the apparent resistance in the form of the vector \(r + js\) was proposed by Steinmetz and modified by Guilbert (Lum. El. Vol. 50, p. 451); Steinmetz proposed the form \(r - js\). However, there is no reason to take the second term with the sign \((-)\), since, when the current vector is multiplied by \(s\), we obtain the vector of the electromotive force of self-induction, which is ahead of the current vector by an angle of \(90^\circ\); and if the current vector coincides with the positive direction of the real axis, then the product of this vector by \(s\) will be a vector coinciding in direction with the positive axis of imaginary values.

Representing the sinusoidal functions that we introduce into the formulas in the form \(x + jy\), where \(\sqrt{x^2 + y^2}\) is the amplitude of the wave, we assume the periodicity of all functions to be the same, and all \(x, y\) to be constant. But the same representation may also be used for different numbers of periods entering into the functions under consideration, if we stipulate in a certain way the application of complex quantities, and if we regard \(x\) and \(y\) as functions of time.

Let us have two sinusoidal functions

\[ \alpha = A \sin(\omega_1 t + \gamma_1) \]

\[ \beta = B \sin(\omega_2 t + \gamma_2) \]

where

\[ \omega_1 = 2\pi \infty_1,\qquad \omega_2 = 2\pi \infty_2 \]

and \(\gamma_1, \gamma_2\) are positive or negative lead angles of these functions relative to a certain sinusoid, the direction of whose vector we have taken as the origin for measuring angles.

The instantaneous value of the resultant function is

\[ c = A \operatorname{Sin}(\omega_1 t + \gamma_1) + B \operatorname{Sin}(\omega_2 t + \gamma_2). \]

Let us write the second term on the right-hand side as follows:

\[ B \operatorname{Sin}(\omega_2 t + \gamma_2) = B \operatorname{Sin}\left[\omega_1 t + (\omega_2 - \omega_1)t + \gamma_2\right], \]

i.e. let us represent the wave \(\beta\) as having the same periodicity, and consequently the same angular velocity \(\omega_1\), as the wave \(\alpha\), but with a variable angle of lead

\[ \gamma' = \gamma_2 + (\omega_2 - \omega_1)t. \]

Then

\[ c = [A \operatorname{Cos}\gamma_1 + B \operatorname{Cos}\gamma'] \operatorname{Sin}\omega_1 t + [A \operatorname{Sin}\gamma_1 + B \operatorname{Sin}\gamma'] \operatorname{Cos}\omega_1 t, \]

i.e., from the formal point of view, the resultant wave is represented in the same way as in the case \(\omega_1 = \omega_2 = \omega\); only, of course, it will no longer be a sinusoid, since the expressions in brackets depend on time.

What has been said has the following geometrical meaning: if the sinusoids \(\alpha\) and \(\beta\), having different periodicity, are represented in the form of vectors, referring them to one and the same system of axes, then the vector \(\beta\) will rotate in the direction of increasing positive angles with velocity \(\omega_2 - \omega_1\), and the instantaneous value of the resultant function will be equal to the value of the sinusoid with period

\[ T = \frac{1}{\infty_1}, \]

represented at the given moment by the geometrical sum of the vectors \(\alpha\) and \(\beta\).

Let us explain this by an example. In Fig. 1 is shown the case of addition of vectors under the assumption that the direction of the vector \(\alpha\) is taken as the axis, and that both vectors are constant in magnitude. Then the value of the resultant function for the moment \(t\) will be equal to the value, for the same moment, of the sinusoid represented by the vector \(ON\), where the point \(N\) is the intersection of the circumference described with radius \(\beta\) from the end \(O_1\) of the vector \(\alpha\), with the straight line \(O_1N\) drawn

at an angle \(NO_1X=\gamma_2+(\omega_2-\omega_1)t\) to the axis \(OX\), i.e., for each instant we shall have

\[ e=ON\sin(\omega_1 t+\angle NO_1X). \]

Taking into account all that has been said, we may make use of the symbolic representation of electromotive forces and currents also in the case of unequal periodicity of the quantities introduced into the calculation, remembering that the formulae obtained represent not constant, but instantaneous values.

The case shown in Fig. 1 assumes constancy of the amplitudes of the sinusoids \(\alpha\) and \(\beta\); if these amplitudes were functions of time, then the locus of the point \(N\) would not be a circle, but some curve not known a priori, which, however, can be constructed for each given case.

3. Let us have \(n\) alternating-current machines operating in parallel on one common external circuit (Fig. 2). We shall find the relations between the electromotive forces of the alternators, the voltage on the busbars, the currents delivered by each of the alternators, and the current passing along the busbars into the external circuit, as functions of the ohmic resistances and self-inductions of the armatures of the machines and of the external circuit.

Let \(E_1, E_2 \ldots E_n\) be the vectors representing the electromotive forces of the alternators; \(J_1, J_2 \ldots J_n\) the currents flowing through their windings; \(Z_1, Z_2 \ldots Z_n\) the apparent resistances of their armature windings; \(E_0\) the vector representing the difference of potentials on the busbars; \(J_0, Z_0\) the vectors of the current and the apparent resistance of the external circuit.

Then, as is known,

\[ \begin{aligned} E_1 &= E_0 + J_1 Z_1\\ E_2 &= E_0 + J_2 Z_2\\ &\ldots\ldots\ldots\\ E_n &= E_0 + J_n Z_n\\ E_0 &= J_0 Z_0 \end{aligned} \qquad\} \tag{1} \]

i.e., the vectors of the electromotive forces of the alternators are equal to the geometric sums of the vector of the voltage on the busbars and the vector of the electromotive force generated as a result of the apparent—

of the resistance of the armature winding, and the vector of the voltage on the busbars is equal to the electromotive force lost as a result of the apparent resistance of the external circuit.

Between the quantities \(J_0, J_1 \ldots J_n\) there exists the relation

\[ J_0 = J_1 + J_2 + \ldots + J_n \tag{1'} \]

i.e. the vector of the current flowing in the external circuit is equal to the geometrical sum of the vectors of the currents flowing through the windings of the alternators.

Representing all the enumerated vectors symbolically in the form of complex quantities, we shall have

\[ \begin{aligned} E_0 &= e_0 + je'_0, \quad E_1 = e_1 + je'_1, \ldots \ldots E_n = e_n + je'_n \\ J_0 &= i_0 + ji'_0, \quad J_1 = i_1 + ji'_1 \ldots \ldots J_n = i_n + ji'_n \\ Z_0 &= r_0 + js_0, \quad Z_1 = r_1 + js_1 \ldots \ldots Z_n = r_n + js_n \end{aligned} \tag{2} \]

Concerning the values of all these quantities, the following must be said: the expressions \(\sqrt{e_k^2 + e_k'^2}\) and \(\sqrt{i_k^2 + i_k'^2}\) we shall, so as not to introduce a correction factor, consider equal to the effective voltages and currents. In addition, we shall assume the quantities \(\sqrt{e_k^2 + e_k'^2}\) to depend only on the number of revolutions of the machine and on the current strength in the exciting winding of the magnets, and not to depend on the demagnetizing action of the armature.

In reality, \(s_k\) is the sum of the self-induction reactance and of a term depending on the demagnetizing action of the armature, so that

\[ s_k = 2\pi \omega L_k + K_k \sin(E_k J_k) \]

where \(K_k\) is a coefficient depending on the design features of the machine*). In making the second assumption, we neglect the second term in the expression for \(s_k\).

All these expressions and all the subsequent considerations refer to single-phase machines or to one phase of polyphase machines.

) See G. Kapp, Dyn. f. Gl. und Wechs*, 2nd ed., § 103.

Substituting expressions (2) into equations (1) and \((1')\), and equating separately the real and imaginary parts, we obtain a system of \(2n+4\) equations:

\[ \left. \begin{aligned} e_0+i_1 r_1-i'_1 s_1&=e_1\\ e_0+i_1 s_1+i'_1 r_1&=e'_1\\ &\cdots\\ &\cdots\\ e_0+i_n r_n-i'_n s_n&=e_n\\ e_0+i_n s_n+i'_n r_n&=e'_n\\ e_0-i_0 r_0+i'_0 s_0&=0\\ e'_0-i_0 s_0-i'_0 r_0&=0\\ i_0-\sum_1^n i_k&=0\\ i'_0-\sum_1^n i'_k&=0 \end{aligned} \right\} \tag{3} \]

The case of greatest practical interest is that in which, alongside \(n-1\) machines already in established synchronous motion, a new, \(n\)-th dynamo-machine is connected.

We may represent the aggregate of the previously operating \(n-1\) machines by a single fictitious machine, whose resistance \(R\) and armature reaction \(S\) are determined by the equations

\[ \frac{1}{R}=\sum_1^{n-1}\frac{1}{r_k},\quad \frac{1}{S}=\sum_1^{n-1}\frac{1}{s_k}. \]

This fictitious machine must obviously supply a current \(A=a+ja'\) equal to the sum of the currents supplied by the \(n-1\) actual machines, and therefore

\[ a=\sum_1^{n-1} i_k,\quad a'=\sum_1^{n-1} i'_k. \tag{4} \]

The electromotive force of the fictitious machine \(E=\varepsilon+j\varepsilon'\) must be equal to the geometric sum of the actual voltage at

at the busbars \(e_0 + j e'_0\) and of the electromotive force lost as a result of the passage of the fictitious current \(a + ja'\) through the armature with fictitious apparent resistance \(R + jS\), i.e.

\[ \begin{aligned} \varepsilon &= e_0 + aR - a'S,\\ \varepsilon' &= e'_0 + aS + a'R \end{aligned} \tag{5} \]

The last 6 equations of system (3) and equations (7) are sufficient for determining \(e_0, e'_0, a, a'_1, i'_n, i'_n\) as functions of the quantities \(\varepsilon, \varepsilon', e_0, e'_0\).

We shall find the distribution of currents among all \(n - 1\) alternators from the first \(2(n-1)\) equations of system (3) and equations (4), assigning values to the electromotive forces of each alternator.

4. On the basis of what has been said, and also in view of the complexity of the formulas obtained in solving the complete system of equations (3), we shall confine ourselves to investigating the system of equations for the case of two alternators connected in parallel.

This system, consisting of eight equations, besides the quantities \(r_0, r_1, r_2, s_0, s_1, s_2\), which we may regard as constant, contains twelve variables

\[ e_1, e'_1, e_2, e'_2, i_1, i'_1, i'_2, i'_2, e_0, e'_0, i_0, i'_0 \]

and has the form

\[ \left. \begin{aligned} e_0 + i_1 r_1 - i'_1 s_1 &= e_1\\ e'_0 + i_1 s_1 + i'_1 r_1 &= e'_1\\ e_0 + i_2 r_2 - i'_2 s_2 &= e_2\\ e'_0 + i_2 s_2 + i'_2 r_2 &= e'_2\\ e_0 - i_0 r_0 + i'_0 s_0 &= 0\\ e'_0 - i_0 s_0 - i'_0 r_0 &= 0\\ i_0 - i_1 - i_2 &= 0\\ i'_0 - i'_1 - i'_2 &= 0 \end{aligned} \right\} \tag{6} \]

Of the twelve variables entering into these equations, the quantities \(e_1, e'_1, e_2, e'_2\) are in fact arbitrary: only them can we change at will by varying the exciting current ...

or the rotational speed. As a function of these four quantities we shall express the remaining 8 unknowns.

The determinant of the system of equations (6) is

\[ \Delta = \begin{vmatrix} 1 & 0 & r_1 & -s_1 & 0 & 0 & 0 & 0\\ 0 & 1 & s_1 & r_1 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & r_2 & -s_2 & 0 & 0\\ 0 & 1 & 0 & 0 & s_2 & r_2 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 & r_0 & -s_0\\ 0 & 1 & 0 & 0 & 0 & 0 & s_0 & r_0\\ 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 \end{vmatrix} \]

\[ = S(r_1^2+s_0^2)(r_2^2+s_2^2)+2S(s_1s_2+r_1r_2)(r_0^2+s_0^2) \tag{7} \]

(where \(S\) denotes the sum of three terms obtained from one another by cyclic permutation of the indices \(0, 1, 2\), for example,

\[ S\,s_1s_2=s_1s_2+s_2s_0+s_0s_1). \]

For the magnitudes of the projections of the current vectors \(J_1\) on the coordinate axes we shall have the expressions

\[ \left. \begin{aligned} i_1 &= e_1\,\frac{\Delta_{31}+e_1'\Delta_{32}+e_2\Delta_{33}+e_2'\Delta_{34}}{\Delta},\\[4pt] i_1' &= e_1\,\frac{\Delta_{41}+e_1'\Delta_{42}+e_3\Delta_{43}+e_2'\Delta_{44}}{\Delta}, \end{aligned} \right\} \]

and similar expressions for the projections \(i_2, i_2'\) of the current \(J_2\), where \(\Delta_{\alpha\beta}\) is the minor determinant corresponding to the \(\alpha\)-th column and the \(\beta\)-th row.

Considering the determinant \(\Delta\), it is not difficult to find that

\[ \begin{aligned} \Delta_{31} &= \Delta_{42}=A_1,\\ \Delta_{32} &= -\Delta_{41}=B_1,\\ \Delta_{33} &= \Delta_{44}=\Delta_{51}=\Delta_{62}=-C,\\ \Delta_{34} &= -\Delta_{43}=\Delta_{52}=-\Delta_{61}=-D,\\ \Delta_{53} &= \Delta_{64}=A_2,\\ \Delta_{54} &= -\Delta_{63}=B_2. \end{aligned} \]

where

\[ \left. \begin{aligned} A_1&=(r_1+r_2)(r_0^2+s_0^2)+(r_0+r_1)(r_2^2+s_2^2)+2r_1(r_0r_2+s_0s_2),\\ B_1&=(s_1+s_2)(r_0^2+s_0^2)+(s_0+s_1)(r_2^2+s_2^2)+2s_1(r_0r_2+s_0s_2),\\ A_2&=(r_1+r_2)(r_0^2+s_0^2)+(r_0+r_2)(r_1^2+s_1^2)+2r_2(r_0r_1+s_0s_1),\\ B_2&=(s_1+s_2)(r_0^2+s_0^2)+(s_0+s_2)(r_1^2+s_1^2)+2s_2(r_0r_1+s_0s_1),\\ C&=(r_1+r_2)(r_0^2+s_0^2)+r_0(r_1r_2-s_1s_2)+s_0(r_1s_2+s_1r_2),\\ D&=(s_1+s_2)(r_0^2+s_0^2)+s_0(s_1s_2-r_1r_2)+r_0(r_1s_2+s_1r_2). \end{aligned} \right\} \tag{8} \]

and the expressions for the quantities \(i_1,\ i_1',\ i_2,\ i_2'\) will finally be written as follows:

\[ \left. \begin{aligned} i_1&=\frac{e_1A_1+e_1'B_1-e_2C-e_2'D}{\Delta},\\[1.2em] i_1'&=\frac{-e_1B_1+e_1'A_1+e_2D-e_2'C}{\Delta},\\[1.2em] i_2&=\frac{-e_1C-e_1'D+e_2A_2+e_2'B_2}{\Delta},\\[1.2em] i_2'&=\frac{e_1D-e_1'C-e_2B_2+e_2'A_2}{\Delta}. \end{aligned} \right\} \tag{9} \]

From the last two equations of group (6) we shall have

\[ \left. \begin{aligned} i_0&=\frac{e_1M_1+e_1'N_1+e_2M_2+e_2'N_2}{\Delta},\\[1.2em] i_0'&=\frac{-e_1N_1+e_1'M_1-e_2N_2+e_2M_2}{\Delta}, \end{aligned} \right\} \tag{10} \]

where, for brevity,

\[ \left. \begin{aligned} M_1&=A_1-C=(r_0+r_1)(r_2^2+s_2^2)+r_0(r_1r_2+s_1s_2)+s_0(r_1s_2-s_1r_2),\\ N_1&=B_1-D=(s_0+s_1)(r_2^2+s_2^2)-r_0(r_1s_2-s_1r_2)+s_0(r_1r_2+s_1s_2),\\ M_2&=A_2-C=(r_0+r_2)(r_1^2+s_1^2)+r_0(r_1r_2+s_1s_2)-s_0(r_1s_2-s_1r_2),\\ N_2&=B_2-D=(s_0+s_2)(r_1^2+s_1^2)+r_0(r_1s_2-s_1r_2)+s_0(r_1r_2+s_1s_2). \end{aligned} \right\} \tag{11} \]

Substituting into the fifth and sixth equations of group (6) the values \(i_0\) and \(i_0'\) from (10), and putting, for brevity,

\[ \begin{aligned} K_1&=M_1 r_0+N_1 s_0=(r_0 r_1+s_0 s_1)(r_2^2+s_2^2)+(r_0^2+s_0^2)(r_2^2+s_2^2)+{}\\ &\qquad +(r_0^2+s_0^2)(r_1 r_2+s_1 s_2),\\ L_1&=M_1 s_0-N_1 r_0=(s_0 r_1-r_0 s_1)(r_2^2+s_2^2)+(r_1 s_2-s_1 r_2)(r_0^2+s_0^2),\\ K_2&=M_2 r_0+N_2 s_0=(r_0 r_2+s_0 s_2)(r_1^2+s_1^2)+(r_0^2+s_0^2)(r_1^2+s_1^2)+{}\\ &\qquad +(r_0^2+s_0^2)(r_1 r_2+s_1 s_2),\\ L_2&=M_2 s_0-N_2 r_0=(s_0 r_2-r_0 s_2)(r_1^2+s_1^2)-(r_1 s_2-s_1 r_2)(r_0^2+s_0^2). \end{aligned} \tag{12} \]

we obtain the projection of the voltage vector on the busbars

\[ \left. \begin{aligned} e_0&=\frac{e_1 K_1-e_1' L_1+e_2 K_2-e_2' L_2}{\Delta}\\[6pt] e_0'&=\frac{e_1 L_1+e_1' K_1+e_2 L_2+e_2' K_2}{\Delta} \end{aligned} \right\} \tag{13} \]

By means of equations (9) we can express the powers of both alternators as functions of the projections of the vectors of their electromotive forces on two mutually perpendicular axes; if \(W_1\) and \(W_2\) are these powers, then

\[ \left. \begin{aligned} W_1&=e_1 i_1+e_1' i_1' =\frac{(e_1^2+e_1'^2)A_1-C(e_1 e_2+e_1' e_2')-D(e_1 e_2'-e_2 e_1')}{\Delta}\\[6pt] W_2&=e_2 i_2+e_2' i_2' =\frac{(e_2^2+e_2'^2)A_2-C(e_1 e_2+e_1' e_2')+D(e_1 e_2'-e_2 e_1')}{\Delta} \end{aligned} \right\} \tag{14} \]

Equations (9), (10), (13), and (14) represent quantities characterizing the connection of the alternators under consideration; moreover, these formulas have a constant value if the periodicity of the electromotive forces of both alternators is the same (\(e_1, e_1', e_2, e_2'\) are constant), or a time-dependent value, only for the given instant, if \(E_1, E_2\) have different periodicities (all or some of the quantities \(e_1, e_1', e_2, e_2'\) are variable). See § 2.

1. In “Electrical Revue. London 1893” Whitwell published an article* containing the derivation of formulas for the currents and powers of two alternators connected in parallel, under the supposition

* in extenso in Handb. d. El. v. Heinke B. IV. S. 186 (Niethammer).

identity of the machines and equality of the amplitudes of the electromotive forces \(E_1\) and \(E_2\).

The form of Whitwell’s formulas is different from that obtained by us, but after certain transformations they are not difficult to obtain as a special case of formulas (9) and (14).

Let us take as the direction of the axis of real values the direction of one of the vectors of electromotive forces, for example \(E_1\); then, if \(\gamma\) is the angle of phase difference of the electromotive forces, it is evident that in formulas (9) and (14) one must put

\[ e_1=E_1,\quad e_1'=0 \]

\[ e_2=E_2\operatorname{Cos}\gamma,\quad e_2'=E_2\operatorname{Sin}\gamma \]

and we shall have

\[ \left. \begin{aligned} i_1&=\frac{E_1 A_1-E_2(C\operatorname{Cos}\gamma+D\operatorname{Sin}\gamma)}{\Delta}\\[6pt] i_1'&=\frac{-E_1 B+E_2(D\operatorname{Cos}\gamma-C\operatorname{Sin}\gamma)}{\Delta}\\[6pt] i_2&=\frac{E_2(A_2\operatorname{Cos}\gamma+B_2\operatorname{Sin}\gamma)-E_1 C}{\Delta}\\[6pt] i_2'&=\frac{E_2(A_2\operatorname{Sin}\gamma-B_2\operatorname{Cos}\gamma)+E_1 D}{\Delta} \end{aligned} \right\} \tag{15} \]

Further, from equations (7) and (8) it is not difficult to see that

\[ C^2+D^2=(r_0^2+s_0^2)\Delta, \]

\[ A_1^2+B_1^2=\{(r_0+r_2)^2+(s_0+s_2)^2\}\Delta, \]

\[ A_2^2+B_2^2=\{(r_0+r_1)^2+(s_0+s_1)^2\}\Delta, \]

Therefore, if we introduce new quantities \(\psi_1,\ \psi_2,\ \psi_3\), putting

\[ \operatorname{tg}\psi_1=\frac{C}{D},\quad \operatorname{tg}\psi_2=\frac{A_1}{B_1},\quad \operatorname{tg}\psi_3=\frac{A_2}{B_2}, \]

we shall have from (15)

\[ \left. \begin{aligned} i_1 &= E_1\frac{\sqrt{(r_0+r_2)^2+(s_0+s_2)^2}}{\sqrt{\Delta}}\sin\psi_2 - E_2\frac{\sqrt{r_0^2+s_0^2}}{\sqrt{\Delta}}\sin(\gamma+\psi_1), \\[6pt] i'_1 &= -\,E_1\frac{\sqrt{(r_0+r_2)^2+(s_0+s_0)^2}}{\sqrt{\Delta}}\cos\psi_2 + E_2\frac{\sqrt{r_0^2+s_0^2}}{\sqrt{\Delta}}\cos(\gamma+\psi_1), \\[6pt] i_2 &= E_2\frac{\sqrt{(r_0+r_1)^2+(s_0+s_1)^2}}{\sqrt{\Delta}}\sin(\gamma+\psi_3) - E_1\frac{\sqrt{r_0^2+s_0^2}}{\sqrt{\Delta}}\sin\psi_1, \\[6pt] i'_2 &= -\,E_2\frac{\sqrt{(r_0+r_1)^2+(s_0+s_1)^2}}{\sqrt{\Delta}}\cos(\gamma+\psi_3) + E_1\frac{\sqrt{r_0^2+s_0^2}}{\sqrt{\Delta}}\cos\psi_1 \end{aligned} \right\} \tag{16} \]

The power of the first machine is

\[ W_1 = E_1^2\frac{\sqrt{(r_0+r_2)^2+(s_0+s_2)^2}}{\sqrt{\Delta}}\sin\psi_2 - E_1E_2\frac{\sqrt{r_0^2+s_0^2}}{\sqrt{\Delta}}\sin(\gamma+\psi_1) \tag{17} \]

The power of the second:

\[ W_2 = E_2^2\frac{\sqrt{(r_0+r_2)^2+(s_0+s_2)^2}}{\sqrt{\Delta}}\sin\psi_3 + E_1E_2\frac{\sqrt{r_0^2+s_0^2}}{\sqrt{\Delta}}\sin(\gamma-\psi_1), \tag{17 bis} \]

The angle of lag of the current behind the electromotive force in the first alternator will be
\(\alpha=\operatorname{arctg}\frac{i'_1}{i_1}\), and in the second \(\gamma+\beta\), where

\[ \beta=\operatorname{arctg}\frac{i'_2}{i_2} \]

For identical and equally excited machines we shall have

\[ E_1 = E_2 = E,\quad \psi_2 = \psi_3 = \psi \]

\[ r_1 = r_2 = r,\quad s_1 = s_2 = s \]

\[ \Delta = (r^2+s^2)\{r^2+s^2+4(r_0^2+s_0^2)+4(rr_0+ss_0)\} \]

Assuming that the electromotive force of the first alternator leads in phase the electromotive force of the second, i.e. taking in formulas (16) the angle \(\gamma\) to be negative, we obtain for the instantaneous values of the current strength of the leading machine the expression

\[ J_1=\sqrt{2}\,E\sqrt{X_1^2+Y_1^2}\,\sin(\omega t-\varepsilon), \]

of the lagging one

\[ J_2=\sqrt{2}\,E\sqrt{X_2^2+Y_2^2}\,\sin(\omega t-\xi \]

where

\[ X_1= \frac{\sqrt{(r_0+r)^2+(s_0+s)^2}}{\sqrt{\Delta}}\sin\psi - \frac{\sqrt{r_0^2+s_0^2}}{\sqrt{\Delta}}\sin(\psi_1-\gamma) \]

\[ Y_1= \frac{\sqrt{(r_0+r)^2+(s_0+s)^2}}{\sqrt{\Delta}}\cos\psi - \frac{\sqrt{r_0^2+s_0^2}}{\sqrt{\Delta}}\cos(\psi_1-\gamma) \]

\[ X_2= \frac{\sqrt{(r_0+r)^2+(s_0+s)^2}}{\sqrt{\Delta}}\sin(\psi-\gamma) - \frac{\sqrt{r_0^2+s_0^2}}{\sqrt{\Delta}}\sin\psi_1 \]

\[ Y_2= \frac{\sqrt{(r_0+r)^2+(s_0+s)^2}}{\sqrt{\Delta}}\cos(\psi-\gamma) - \frac{\sqrt{r_0^2+s_0^2}}{\sqrt{\Delta}}\cos\psi_1 \]

and

\[ \operatorname{tg}\varepsilon=\frac{Y_1}{X_1},\quad \operatorname{tg}\xi=\frac{Y_2}{X_2} \]

Expressions (17) and \((17^{\mathrm{bis}})\) will then take the form: for the leading machine,

\[ W_1=E^2X_1, \]

whereby the current lags behind the electromotive force by the angle $\varepsilon$, and for the lagging one

\[ W_2 = E^2 (X_2 \operatorname{Cos}\gamma - Y_2 \operatorname{Sin}\gamma) \]

at the angle of lag of the current $\gamma + \xi$.

In this form, the expressions for $J_1$, $J_2$, $W_1$, and $W_2$ were obtained in the cited article by Whitewell. These forms have no particular advantage over those obtained by us, and their derivation has been given here only because Whitewell’s theory is presented as an example of the mathematical theory of the parallel operation of alternators in the above-mentioned edition by V. Heinke, which is widely circulated among electrical engineers.

II. Graphical Investigation.

6. Equations (14) represent the dependence of the two quantities $W_1$ and $W_2$ on the vectors of the electromotive forces, or on their projections on mutually perpendicular axes. It is more convenient to investigate this dependence by representing it geometrically.

Let us take, as in the preceding paragraph, the direction of the vector $E_1$ as the axis of real values, and assume that in magnitude this vector remains constant. Then equations (14) become

\[ \left. \begin{aligned} \Delta W_1 &= A_1 E_1^2 - C E_1 e_2 - D E_1 e'_2 \\ \Delta W_2 &= A_2 (e_2^2 + e_2'^2) - C E_1 e_2 + D E_1 e'_2 \end{aligned} \right\} \tag{18} \]

These equations are obviously the equations of the geometric loci on which the end of the vector $E_2 = e_2 + j e'_2$ must remain for given magnitudes $W_1$ and $W_2$. We shall call these geometric loci the lines of power of the first and second alternator.

The first equation represents, with the variable parameter $W_1$, a bundle of parallel straight lines inclined to the $X$ axis (Fig. 3) at the angle $KOX$, determined from the equation

\[ \operatorname{tg} KOX = -\frac{C}{D} \tag{19} \]

The second equation (18) is the equation of a circle, the coordinates of whose center \(N\) are

\[ \xi=\frac{E_1}{2}\frac{C}{A_2},\quad \eta=-\frac{E_1}{2}\frac{D}{A_2}. \tag{20} \]

and the radius

\[ \rho=\frac{1}{2A_2}\sqrt{(C^2+D^2)E_1^2+4A_2\Delta W_2}. \tag{21} \]

The coordinates of the center do not depend on \(W_2\), and consequently all circles corresponding to different \(W_2\) are concentric.

The common center lies, as is seen from equations (20), at the intersection of the perpendicular \(NS\) to the axis \(OX\), erected from the point \(S\) lying at the distance

\[ OS=\frac{C}{A_2}\frac{E_1}{2} \]

from the origin of coordinates, to the right if \(\dfrac{C}{A_2}>0\), and to the left when \(\dfrac{C}{A_2}<0\), and of the inclined line to \(OX\) drawn from the point \(O\) at the angle \(G_2OX\), which we find from the condition

\[ \tg G_2OX=-\frac{D}{C}. \tag{22} \]

The formulae we have derived apply, as was mentioned above, to the most general case of the parallel operation of two alternators, which includes both the case of two generators working in parallel on an external circuit, and the case of the transmission of energy by means of two alternators, of which one serves as a motor and the other as a generator.

In the first case both alternators deliver electrical energy to the external circuit; in both, the angle between the vectors of electromotive force and current will be less than \(\dfrac{\pi}{2}\), and consequently, in this case

\[ W_1>0,\quad W_2>0. \]

In the second case, one alternator (let it be the first) generates electrical energy by absorbing mechanical energy, and feeds the external circuit and the other alternator, which absorbs electrical energy and develops mechanical energy. In the second alternator, as is known, the angle between the vectors of electromotive force and current will be greater than \(\frac{\pi}{2}\), and consequently in this case we shall have

\[ W_1 > 0,\quad W_2 < 0. \]

On the diagram the signs of \(W_1\) and \(W_2\) are determined in the following manner.

The circle corresponding to the power \(W_2 = 0\) will have radius

\[ \rho_0 = \frac{\sqrt{C^2 + D^2}}{A_2}\,\frac{E_1}{2} \]

and passes through the origin of coordinates. The radii of the circles of negative powers \(W_2\), i.e. of the circles corresponding to the case when the second alternator operates as a motor, absorbing the energy developed by the first alternator, will evidently be smaller than the value \(\rho_0\), since \(A_2\) is essentially positive; and consequently the circles of powers \(W_2 < 0\) will lie inside the circle \(W_2 = 0\). The circles of positive powers \(W_2\), corresponding to the cases when either both alternators operate as generators, or the second alternator feeds, in addition to the external circuit, the first—operating as a motor—will lie outside the circle \(\rho_0\).

The greatest absolute value, for the case when the second alternator operates as a motor, is attained by the power \(W_2\) at \(\rho = 0\). Then

\[ W_2 = -\,\frac{C^2 + D^2}{4 A_2 \Delta}\,E_1^{\,2} = -\,\frac{r_0^{\,2} + s_0^{\,2}}{4 A^2}\,E_1^{\,2}; \]

this is the greatest power that can be developed by a motor connected in parallel with an external circuit having apparent resistance \(r_0 + j s_0\), when the electromotive force of the feeding alternator is \(E_1\).

Let us turn to the straight lines of power of the first alternator. The straight line passing through the origin corresponds to

\[ W_1 = \frac{A_1}{\Delta} E_1^2; \]

this quantity is always positive. The straight line of power \(W_1 = 0\) intersects the \(X\)-axis at a distance from the origin

\[ \xi = \frac{A_1}{C} E_1. \]

The sign of \(\xi\) depends on the sign of \(C\); if \(C < 0\), the straight lines of power go, in the order of increasing power, from left to right; when \(C > 0\), from right to left. The sign of the coefficient \(C\) depends chiefly on the magnitude \(r_0\); usually this quantity is comparatively insignificant in relation to \(s_1, s_2\), and the term \((-r_0 s_1 s_1)\) in the expression for \(C\) in equations (8) acquires the predominant significance, i.e. \(C < 0\). For the case \(r_0 = \infty\), i.e. for the case of transmission of energy by two alternators without branching of current into the external circuit, \(\xi = E_1\).

1. Having constructed, for given \(s_0, s_1, s_2, r_0, r_1, r_2\) and a given voltage, i.e. assuming the speed of rotation to be constant, for a given excitation of one of the alternators, the network of straight lines and circles of power, we shall easily find the power of each of the alternators, knowing the excitation of the second alternator and the angle of phase difference of their electromotive forces.

In Fig. 3, for an example (*), the lines of power of two alternators connected in parallel are constructed for the case

\[ E_1 = 2500 \text{ volts} \]

\[ r_1 = r_2 = 1,\quad s_1 = s_2 = 10,\quad r_0 = 0.6,\quad s_0 = 0.8. \]

Then

\[ \sqrt{r_0^2 + s_0^2} = 1 \]

\[ A_1 = A_2 = 180.8 \]

\[ C = -41.4 \]

\[ D = 111.2 \]

\[ \text{*) The example is taken from the book by C. P. Steinmetz, T. u. B. d. W., p. 291.} \]

We shall call the normal power the power that the alternators will have when their electromotive-force vectors coincide in magnitude and direction. By formula (14) we find \(W_1 = W_2\) norm. \(= 99\ \mathrm{KW}\). On the diagram circles have been drawn for the normal power \(W_2\), \(3/4\), \(1/2\), and \(1/4\) of it. Choosing the voltage scale \(25\) volts \(= 1^{\mathrm{m/m}}\), by formula 21 we find:

\[ \begin{aligned} \text{for}\quad & W_2 \text{ norm.} = 99\ \mathrm{KW} && \rho = 116^{\mathrm{mm}}.\\ \text{for}\quad & 0.75\,W_2 \text{ norm.} && \rho = 102^{\mathrm{mm}}.\\ \text{for}\quad & 0.5\,W_2 \text{ norm.} && \rho = 87^{\mathrm{mm}}.\\ \text{for}\quad & 0.25\,W_2 \text{ norm.} && \rho = 65.8^{\mathrm{mm}}. \end{aligned} \]

For the power \(W_1\), the straight lines of normal power, \(0.75\) and \(0.50\) of it, have been drawn. These straight lines are constructed by knowing

\[ \operatorname{tg} KOX = -\frac{C}{D} = 0.372 \]

and by calculating, from the equation of the straight line of power, the distances of the points of intersection of this straight line with the ordinate axis from the origin of coordinates:

\[ OF_1 = -38^{\mathrm{mm}},\quad OF_2 = 12.2^{\mathrm{mm}},\quad OF_3 = 62.3^{\mathrm{mm}}. \]

The position of the vector \(E_2\) on the drawing corresponds to certain values of \(W_1\) and \(W_2\), determined by the straight line and the circle of powers passing through the point \(\varepsilon_2\).

Let us imagine that we decrease the inflow of energy to the driving machine of the second alternator, leaving the electromotive forces \(E_1\) and \(E_2\) constant in magnitude, i.e., leaving the excitation of both alternators constant. Then, obviously, the power of the second alternator will decrease, and the point \(\varepsilon_2\) will be forced to move to a circle of smaller radius corresponding to a certain value of power \(W_2 - \delta W_2\), where \(\delta W_2\) is the change in power, and to occupy the position \(\varepsilon'_2\). The power of the first alternator will in this case increase to the value \(W_1 + \delta W_1\), determined by the straight line of power passing through the point \(\varepsilon'_2\). The angle of phase difference of the electromotive forces in this case, as is seen from the diagram, will increase. Continuing to decrease the inflow of energy to the driving machine

of the second alternator, we shall see that the point \(\varepsilon_2\) of the vector \(O\varepsilon_2\) will move along the circle \(G_1 \varepsilon_2 G_2\) in the direction of clockwise motion, while the power of the first alternator will increase, and consequently the supply of energy to the prime mover of the first alternator must increase. The power \(W_1\) will increase as the angle of phase difference of the electromotive forces is increased, until the point \(\varepsilon_2\) comes to the point \(G_1\), at which the circle \(G_2 \varepsilon_2 G_1\) is tangent to the straight line of power passing through this point. If we continue to increase the angle of phase difference of the electromotive forces, the power \(W_2\) will obviously begin to decrease, and the equilibrium between the driving force and the electrical energy will be disturbed. The straight line \(OG_1\) in the diagram will be the boundary of stability of the motion of the alternators connected in parallel.

If, starting from the position of the vector \(O\varepsilon_2\) in the diagram, we assume the supply of energy to the prime mover of the second alternator to be increasing, we shall in a similar manner see that the power \(W_1\) will decrease, while the power \(W_2\) will increase, until the point \(\varepsilon_2\), moving along the circle \(G_2 \varepsilon_2 G_1\) against the direction of clockwise motion, comes to the point \(G_2\), at which this circle is tangent to the circle of power passing through this point. The straight line \(OG_2\) will be another limit of stability of the motion. From conditions (19) and (22) we have

\[ \operatorname{tg} G_2 OX \cdot \operatorname{tg} KOX = 1; \]

the straight line \(G_1 O\) is perpendicular to the straight line \(KO\), consequently

\[ \angle G_2 OX = \angle G_1 OX, \]

i.e. the limiting angle of phase difference of the electromotive forces of alternators operating in parallel is determined by the condition

\[ \operatorname{tg}\gamma_{\max} = \frac{D}{C} \quad \text{(absolute value.)} \tag{23} \]

This quantity obviously depends only on the apparent resistances of the armature windings of both alternators and of the external circuit, and plays a very important role in the question of stability of mot—

of the alternators for the connection under consideration, since the larger the angle

\[ \gamma_{\max}=\angle G_1OX=\angle G_2OX, \]

the greater the angle by which the vectors of the electromotive forces may diverge without disturbing the stability of the motion of the system; \(\gamma\) is the angle in the diagram; in reality a considerably smaller angle corresponds to it. If, for example, both alternators have the same number of pairs of poles \(p\), then to the angle \(2\pi\) in the diagram there corresponds an angle of rotation of the armature \(\dfrac{2\pi}{p}\), and to the angle of phase difference of the electromotive forces \(\gamma\) there corresponds the angle of divergence of the axes of like-named poles \(\dfrac{\gamma}{p}\).

Hence it is clear that the greater the number of poles, the less, under otherwise identical conditions, alternators are suited to parallel operation.

As was said, almost all authors of articles on the question under consideration, beginning with Hopkinson, have paid attention to the influence of the ratio of the armature self-induction reactance to its ohmic resistance on the ability of an alternator to be connected in parallel. Let us examine formula (23) in this respect.

Let

\[ \sigma_1=\frac{s_1}{r_1},\quad \sigma_2=\frac{s_2}{r_2}. \]

Then equation (23) may be written thus:

\[ \operatorname{tg}\gamma_{\max}= \frac{ (r_0^2+s_0^2)\left(\dfrac{\sigma_1}{r_2}+\dfrac{\sigma_2}{r_1}\right) +s_0(\sigma_1\sigma_2-1)+r_0(\sigma_1+\sigma_2) }{ (r_0^2+s_0^2)\left(\dfrac{1}{r_1}+\dfrac{1}{r_2}\right) +r_0(1-\sigma_1\sigma_2)+s_0(\sigma_1+\sigma_2) } \qquad \text{(abs. val.) }(23^{\mathrm{bis}}) \]

If we form the partial derivatives of this expression with respect to \(\sigma_1\) and \(\sigma_2\), we shall see that these derivatives are always positive and do not vanish for real values of \(\sigma_1\) and \(\sigma_2\). This means that, as \(\sigma_1\) and \(\sigma_2\) increase, \(\operatorname{tg}\gamma_{\max}\) increases, whereby its absolute value increases if \(\operatorname{tg}\gamma_{\max}\) is positive, and decreases in the opposite case. The most advantageous, in the sense of the stability of alternators connected in parallel in a circuit—

of the alternators will be the case when \(\gamma_{\max.} = \pm \dfrac{\pi}{2}\), i.e. when \(\operatorname{tg}\gamma_{\max.} = \pm \infty\), and this will occur in the case of such a combination of the ratios of the self-induction reactances of the armatures to their ohmic resistances \(\sigma_1\) and \(\sigma_2\), for which the denominator of expression \(23^{\mathrm{bis}}\) becomes zero, i.e. when

\[ r_0\sigma_1\sigma_2 - s_0(\sigma_1+\sigma_2) - (r_0^2+s_0^2)\left(\frac{1}{r_1}+\frac{1}{r_2}\right)-r_0=0. \]

In the case where we have two identical alternators, i.e.

\[ \sigma_1=\sigma_2=\sigma,\quad r_1=r_2=r, \]

then \(\gamma_{\max.}=\pm\dfrac{\pi}{2}\) when

\[ \sigma= \frac{ s_0+\sqrt{s_0^2+4r_0\left[\dfrac{2}{r}(r_0^2+s_0^2)+r_0\right]} }{ 2r_0 }. \]

The absolute value of expression \(23_{\mathrm{bis}}\) when \(\sigma_1=\sigma_2=\infty\) is \(\dfrac{s_0}{r_0}\), and when \(\sigma_1=\sigma_2=0\), this expression becomes

\[ \frac{s_0}{(r_0^2+s_0^2)\left(\dfrac{1}{r_1}+\dfrac{1}{r_2}\right)+r_0}. \]

These expressions show that both with very large self-induction in comparison with the ohmic resistance, and with very small self-induction, stable motion of alternators connected in parallel is possible only with considerable self-induction in the external circuit.

In practice it turns out that the presence of iron in the armatures of alternators of modern designs guarantees a magnitude of self-induction for which the angle \(\gamma_{\max.}\) remains sufficiently large.

§ 8. The formulas and diagrams derived above are not difficult to apply to the case of transmission of power by means of two alternators, of which one operates as a generator and the other as a synchronous motor, and from the diagram of Fig. 3 to obtain the well-known Blondel diagram*). Putting \(r=\infty\) in formulas (9), (10), (13), and (14), we obtain

) A. Blondel, Moteurs Synchrones, p. 82. See also C. Thompson, Dynamo-Electric Machinery*. Russian translation by Golov, pp. 841 and 847.

\[ \left. \begin{aligned} i_1=-i_2&=\frac{(r_1+r_2)(e_1-e_2)+(s_1+s_2)(e_1'-e_2')} {(r_1+r_2)^2+(s_1+s_2)^2},\\[6pt] i_1'=-i_2'&=\frac{(s_1+s_2)(e_1-e_2)-(r_1+r_2)(e_1'-e_2')} {(r_1+r_2)^2+(s_1+s_2)^2},\\[6pt] i_0&=0,\quad i_0'=0 \end{aligned} \right\} \tag{24} \]

\[ \left. \begin{aligned} e_0&=\frac{ e_1\left[r_2(r_1+r_2)+s_2(s_1+s_2)\right] +e_2\left[r_1(r_1+r_2)+s_1(s_1+s_2)\right] +(r_1s_2-s_1r_2)(e_2'-e_1') } {(r_1+r_2)^2+(s_1+s_2)^2},\\[8pt] e_0'&=\frac{ e_1'\left[r_2(r_1+r_2)+s_2(s_1+s_2)\right] +e_2'\left[r_1(r_1+r_2)+s_1(s_1+s_2)\right] +(r_1s_2-s_1r_2)(e_2-e_1) } {(r_1+r_2)^2+(s_1+s_2)^2} \end{aligned} \right\} \tag{25} \]

\[ \left. \begin{aligned} W_1&=\frac{ (r_1+r_2)(e_1^2+e_1'^2-e_1e_2-e_1'e_2')-(s_1+s_2)(e_1e_2'-e_2e_1') } {(r_1+r_2)^2+(s_1+s_2)^2},\\[8pt] W_2&=\frac{ (r_1+r_2)(e_2^2+e_2'^2-e_1e_2-e_1'e_2')+(s_1+s_2)(e_1e_2'-e_2e_1') } {(r_1+r_2)^2+(s_1+s_2)^2} \end{aligned} \right\} \tag{26} \]

The equations of the power lines will, under the assumption \(r=\infty\), take the following form:

\[ \left. \begin{aligned} \left\{(r_1+r_2)^2+(s_1+s_2)^2\right\}W_1 &=(r_1+r_2)E_1^2-(r_1+r_2)E_1e_2\\ &\quad -(s_1+s_2)E_1e_2',\\[6pt] \left\{(r_1+r_2)^2+(s_1+s_2)^2\right\}W_2 &=(r_1+r_2)(e_2^2+e_2'^2)-(r_1+r_2)E_1e_2\\ &\quad +(s_1+s_2)E_1e_2' \end{aligned} \right\} \tag{27} \]

in which, as before, \(e_2\) and \(e_2'\) are taken as the current coordinates. The diagram will have the form of Fig. 4. In this diagram the designations remain the same as in Fig. 3.

The center of the power circles lies at the intersection of the perpendicular erected from the midpoint of the segment \(OE_1\) and the inclined line drawn from the point \(O\) at the angle \(NOE_1=\gamma_{\max}\) to \(OE_1\), where

\[ \tg \gamma_{\max}=\frac{s_1+s_2}{r_1+r_2}. \]

The power circle \(W_2 = 0\) passes through the points \(O\) and \(E_1\), while the power straight line \(W_1 = 0\) passes through the point \(E_1\) and is tangent at this point to the circle \(W_2 = 0\). It is obvious that if the point \(E_2\) lies in the part of the plane (shaded in the drawing) enclosed between the circle \(W_2 = 0\) and the straight line \(W_1 = 0\), then both alternators operate as generators, i.e., they absorb mechanical energy and develop a current, the entire energy of which goes into heating the windings; in this case there will be no transmission of energy.

The vector \(E_1 E_2\) in the diagram evidently represents the loss of electromotive force due to the apparent resistance in the armature and in the connecting wires of the alternators \((r_1, r_2, s_1\) and \(s_2\) we assume to include also the ohmic resistances and the self-induction reactances of the connecting wires). In magnitude the vector of the current flowing through both armatures will be equal to

\[ J = \frac{E_1 E_2}{\sqrt{(r_1+r_2)^2+(s_1+s_2)^2}}; \]

since the denominator of this expression is, for the given alternators, a constant quantity, the segment \(E_1 E_2\) may be regarded as equal to the current \(J\), measuring it on a scale \(\sqrt{(r_1+r_2)^2+(s_1+s_2)^2}\) times larger than the scale of the electromotive forces.

Remark. In equation (24) the current vectors \(J_1, J_2\) were obtained with opposite signs, although evidently these vectors represent one and the same current, flowing successively through the windings of both alternators. This apparent inconsistency arises because we consider the direction of the current with respect to the electromotive forces \(E_1, E_2\), whose vectors, for convenience, we assume to be arranged at an angle \(\gamma\) smaller than \(\dfrac{\pi}{2}\). If we had arranged the vectors \(E_1\) and \(E_2\) at an angle \(\pi+\gamma\), the signs of \(J_1\) and \(J_2\) would be the same.

The limit of negative work will be, as in the general case, that value of the quantity \(W_2\) at which the radius of the power circle \(W_2\) becomes zero. We obtain this quantity from the condition

\[ \rho=\frac{\sqrt{(r_1+r_2)^2+(s_1+s_2)^2}}{r_1+r_2} \sqrt{\left(\frac{E_1}{2}\right)^2+(r_1+r_2)W_z}=0 \]

whence

\[ W_z=-\frac{E_1^2}{4(r_1+r^2)}. \tag{28} \]

This is a theoretical limit which the magnitude of the power never reaches, since such power always corresponds to a current considerably greater than that which the armature windings of the alternators can withstand without dangerous heating. If this latter current is known, then, describing from the point \(E_1\) as from a center a circle with radius equal to this current, in the corresponding scale, we shall have a boundary within which the point \(E_2\) must remain.

In the same way as we constructed the lines of equal powers, one can construct lines of equal angles of phase difference between the current and the electromotive forces \(E_1\) and \(E_2\), i.e. lines having the property that, so long as the end of the vector \(OE_2\) remains on them, the angle between the vectors \(E_1\) and \(J\), or \(E_2\) and \(J\), remains constant.

The direction of the current always makes with the direction of the vector \(E_2E_1\) (taking the point \(E_2\) as the pole) an angle \(\gamma_m\), for which
\[ \tg \gamma_m=\frac{s_1+s_2}{r_1+r_2}. \]
It is not difficult to see that the angle between the current and the electromotive force \(E_1\) is equal to the angle \(E_2E_1N\), and is negative (the current lags) when the point \(E_2\) lies to the left of the straight line \(NE_1\), and positive (the current leads the electromotive force) when the point \(E_2\) is to the right of this straight line, for the case when the first alternator works as a generator and the second as a motor, and conversely in the opposite case. It is evident that the line of angle of phase difference \((E_1,J)=0\) will be the straight line \(NE_1\), while the lines of the other angles \((E_1,J)\) are straight lines passing through the point \(E_1\) and inclined to the \(X\)-axis at these angles. From what has been said it is clear that, with constant excitation of the first alternator, the angle of phase difference between its electromotive force and current depends only on the excitation of the second alternator. The excitation of the second alternator for which the angle under consideration is equal to zero is called normal. It is evident that overexcitation of the alternator

operating in the alternating-current network as a synchronous motor, will cause the current to lead the primary voltage and thus compensate the effect of the self-induction of the circuit and of the transformers, asynchronous motors, and arc lamps connected into it. The question of the profitability, in various cases, of compensating the self-induction of the circuit by means of synchronous motors has been analyzed in detail by A. Blondel in the booklet Moteurs synchrones, pp. 112–150; we shall not concern ourselves with it here.

In order to construct the lines of equal angles of phase difference between the current and the electromotive force of the second alternator \(E_2\), we note that if \(\angle(E_2,J)=\mathrm{Const}\), then it must also be that \(\angle OE_1E_2=\mathrm{Const}\). Consequently, the required loci will be circles passing through the points \(O,E_1,E_2\). The circle of angle \((E_2,J)=0\) will evidently correspond to \(\angle OE_2E_1=\pi-\gamma_m\), and consequently will be tangent to the straight lines \(NO\) and \(NE_1\). The circle of angle \((E_1,J)=\dfrac{\pi}{2}\) will coincide with the circle of powers \(W_2=0\).

Further analysis of the case of energy transmission by means of two alternators, i.e. the case \(r_0=\infty\), does not fall within the scope of the present article*). Here we shall point out only one simplification of Blondel’s diagrams. We shall take the magnitude \(E_1\) to be the voltage at the terminals of the second alternator, assuming, as before, that \(E_1=\mathrm{Const}\). This assumption is equivalent to assuming that the ohmic resistance and the self-induction reactance of the first alternator are so small that the change in terminal voltage due to a change in load may be neglected; in other words, that \(r_1=s_1=0\). Then the entire diagram of Fig. 4 may be applied without change to the case of a single alternator operating in a constant-voltage network, whether as a motor or as a generator.

Remark. Formulas (24–26) apply to this case if the quantities in parentheses in them are replaced by the quantity \(r_2\), and formula (28) will take the form

\[ W_{2\min}=-\frac{E_1^2}{4r_2} \]

) In detail on this, see A. Blondel, loco cit.*

In this limiting case the point \(E_2\) coincides with \(N\), and the length of the vector of the lost voltage will be

\[ \frac{E_1}{2\operatorname{Cos}\gamma_m} = E_1\sqrt{\frac{r_2^2+s_2^2}{2r_2}}, \]

the corresponding current intensity

\[ J=\frac{E_1}{2r_2}, \]

and the efficiency at the greatest power of the alternator operating as a motor is

\[ \eta = \frac{\dfrac{E_1^2}{4r_2}} {\dfrac{E_1^2}{4r_2} + \dfrac{E_1}{2r_2}\cdot \dfrac{E_1\sqrt{r_1^2+s_1^2}}{2r_2} \operatorname{Cos}\gamma_m} =0.5 \]

The equations obtained show that the law of greatest power (Jacobi’s law for direct-current motors) is applicable to synchronous motors in the following form: the power of a motor, at a given voltage at the terminals, will be greatest when, in synchronism, its field is excited in such a way that the current passing through the armature winding reaches half the value that it would have with the excitation entirely absent, or with the armature stationary. The efficiency in this case is \(\eta=50\%\).

In the case under consideration \((r_1=s_1=0)\), \(W_1\) will be the power measured at the terminals of the alternator, and \(W_2\), neglecting losses due to friction and hysteresis, the electrical power equal to the mechanical power supplied to the alternator when it operates as a generator, or developed by it when it operates as a synchronous motor.

Görges*) in constructing the diagram of the parallel operation of alternating-current machines makes use of the indicated simplification of the construction, taking as the power lines of the alternators ...

not the line \(W_2\) but the line \(W_1\), which is equivalent to discarding as comparatively small the quantity of the ohmic loss of energy.

If, in addition, we assume that the ohmic voltage losses in the armatures of the machines are so small that they may be neglected in comparison with the magnitudes of the voltages lost as a result of self-induction reactions—in other words, if we assume

\[ \frac{r_2}{s_2}=0, \]

then the circles and the straight lines of power (eq. 26) become straight lines parallel to the \(X\)-axis, i.e. to the vector of voltage at the terminals \(E_1\). The diagram of the parallel operation of alternators under these assumptions is greatly simplified and takes the form used by S. Hefter in his memoir). Such an assumption is also laid at the basis of Kapp’s theory of the parallel operation of alternators*).

1. The authors named—Görges, Kapp, Hefter, and some others—in their reasoning assume constancy of the voltage at the terminals. Diagrams based on this assumption have the advantage that with their aid one may represent the operation of any number of alternators connected in parallel, whereas without this assumption the formulas representing the operation of more than two alternators cannot be used because of their complexity. However, constancy of the voltage at the terminals when the load changes can be attained only by a corresponding change in the excitation of the dynamos; and if we consider so short an interval of time that we cannot change the excitation, and consequently the vectors of the electromotive forces remain constant in magnitude and only the angle between them changes (which will occur in the theory of oscillations of alternators connected in parallel), then it is more correct, as we have done, to regard the vectors \(E_1\) and \(E_2\) as independent variables, and the vector \(E_0\) as a function of them.

Analytically, if the projections of the vectors \(E_1\) and \(E_2\) on two mutually perpendicular axes are given, the projections of the vector \(E_0\) on the same axes are found from equations (13). If we wish to solve the problem gra—

* ) Görges, E. T. Z. 1900. 188 and 1902. 1054.
* ) S. Hefter. Bulletin 5—6—7 de 1901 de l’Ass des Ing. électr. sortis de l’Inst. électr. Montefiore. There is a Russian translation by the author himself.
** ) Kapp. Dynamomasch. f. G. u. W. 3rd ed., p. 373 ff.

then, in order to construct the vector \(E_0\) on a diagram like Fig. 3, we proceed as follows.

Let in Fig. 5 the vectors \(O E_1 = E_1\) and \(O E_2 = E_2\) of the electromotive forces of the alternators be given. Having constructed, as was indicated, the triangle \(O N O_1\), we describe from the point \(N\), as from a center, a circle passing through the point \(E_2\) (the circle of power \(W_2\)), and through the same point \(E_2\) draw the straight line \(E_2 F\), perpendicular to the straight line \(N O_1\) (the straight line of power \(W_1\)). Then, on the basis of equations (18) and (21), we shall have:

\[ \left. \begin{aligned} W_1 &= \frac{E_1}{\triangle}\left(A_1 \cdot E_1 - D \cdot O F\right) \\ W_2 &= \frac{A_2}{\triangle} N E_2 - \frac{(r_0^2 + s_0^2)}{4 A_2} E_1^2 \end{aligned} \right\} \qquad 29 \]

where the segments \(O F\) and \(N E_2\) are measured on the voltage scale.

Further, we may regard each of the alternators connected to a network with voltage \(E_0\) as a system for transmitting energy by means of two alternators, with a total resistance and self-induction reactance respectively \(r_1\) and \(s_1\), or \(r_2\) and \(s_2\). The electromotive force of one alternator will be, respectively, \(E_1\) or \(E_2\), and that of the other (fictitious) one, \(E_0\).

If, as the \(X\)-axis, we take respectively the directions of the vectors \(E_1\) and \(E_2\), then the equations of the straight lines of power of the acting alternators will be

\[ W_1 = \frac{r_1}{r_1^2 + s_1^2} E_1^2 - \frac{r_1 E_1}{r_1^2 + s_1^2} e_0 - \frac{s_1 E_1}{r_1^2 + s_1^2} e_0' \tag{a} \]

\[ W_2 = \frac{r_2}{r_2^2 + s_2^2} E_2^2 - \frac{r_2 E_2}{r_2^2 + s_2^2} e_0 - \frac{s_2 E_2}{r_2^2 + s_2^2} e_0' \tag{b} \]

We take the quantities \(W_1\) and \(W_2\) from equations (29). The intersection of the straight lines expressed by the latter equations gives the point \(E_0\), determining, in magnitude and direction, the vector \(E_0 = O E_0\).

On the diagram we construct the straight line (a), noting that it is perpendicular to the straight line intersecting the straight line \(O E_1\) at the point \(E_1\) at the angle \(\alpha_1\), for which \(\operatorname{tg}\alpha_1 = \dfrac{s_1}{r_1}\), and passes through the point \(Q_1\) le-

lying in the direction of the vector \(OE_2\) and distant from the pole \(O\) by

\[ OQ_1 = E_1 - \frac{r_1^2+s_1^2}{r_1 E_1}\,W_1. \]

If the point \(Q_1\) were to fall outside the limits of the drawing, one would have to make use of the point \(P_1\), lying on the perpendicular to the vector \(OE_1\) at a distance from the pole \(O\)

\[ OP_1=\frac{r_1}{s_1}E_1-\frac{r_1^2+s_1^2}{s_1E_1}\,W_1. \]

In the same manner construct the straight line expressed by equation (b), from the angle \(OP_2E_2=\alpha_2\), where \(\tan\alpha_2=\dfrac{s_2}{r_2}\), and from the point \(P_2\), lying on the perpendicular \(OP_2\) to the vector \(OE_2\) at a distance from the point \(O\)

\[ OP_2=\frac{r_2}{s_2}E_2-\frac{r_2^2+s_2^2}{s_2E_2}\,W_2. \]

The equation of the geometrical locus of the end \(E_0\) of the vector \(E_0\), on which it will remain if, with constant excitation of the alternators, the angle of phase difference of their electromotive forces is varied, will be found from equations (13). Putting in them

\[ e_1=E_1,\quad e_1'=0,\quad e_2=E_2\cos\gamma,\quad e_2'=E_2\sin\gamma \]

and eliminating the variable \(\gamma\), we shall find, taking equations (12) into account, the required equation

\[ \left. \begin{aligned} e_0^2+e_0'^2 &=\frac{2K_1E_1}{\Delta}\,e_0-\frac{2L_1E_1}{\Delta}\,e_0' \\ &\quad+\frac{(r_1^2+s_1^2)}{\Delta}\left\{(r_2^2+s_2^2)E_1^2-(r_1^2+s_1^2)E_2^2\right\}=0 \end{aligned} \right\} \tag{30} \]

This is the equation of a circle, the coordinates of whose center \(N_1\) are

\[ \xi=\frac{K_1}{\Delta}E_1,\quad \eta=\frac{L_1}{\Delta}E_1. \]

and the radius

\[ R = E_2 \sqrt{\frac{(r_0^2+s_0^2)(r_1^2+s_2^1)}{\Delta}} \]

If the circle (30) is first constructed for the given \(E_1, E_2, r_1, r_2, s_1, s_2, r_0, s_0\), then, in order to determine the point \(E_0\), it will suffice to construct one line of power, for example \(P_1 Q_2\), corresponding to the power \(W_2\).

Figure 5 shows the construction of the required geometric locus for the case

\[ E_1 = E_2 = 2500 \text{ volts} \]

\[ r_1 = r_2 = 1,\quad s_1 = s_2 = 10,\quad r_0 = 16,\quad s_0 = 12. \]

When the angle \(\gamma\) changes, the point \(E_0\) slides along the circle \(O E_0 E_0'\). The point \(E_0\) corresponds to the angle \(\gamma\) in the drawing; the point \(E_0'\) corresponds to the angle \(\gamma=0\). The points \(E_0''\) and \(E_0'''\) correspond to the two limits of stability. The segment \(e E_0'\) is the difference between the greatest and the least value of the voltage at the terminals.

In the diagram the circle \(O E_0 E_0'\) passes through the origin of coordinates. This occurs as a consequence of the assumption that the two alternators are identical. Under this assumption, the constant term of equation (30) vanishes.

Let us now find at what angle of phase difference of the electromotive forces of the two alternators, for a given excitation of the field magnets, the voltage on the busbars—and consequently also the current strength in the external circuit—attains its greatest value.

If the angle of phase difference is \(\gamma\), then the required condition will be

\[ \frac{d(e_0^2+e_0'^2)}{d\gamma}=0 \tag{31} \]

or

\[ \frac{d(i_0^2+i_0'^2)}{d\gamma}=0 \tag{31 bis} \]

Equations (31) and (31 bis) give identical results. We shall consider the second, since the coefficients entering into it are simpler. Carrying out the differentiation, we shall have

\[ i\frac{di}{d\gamma}+i'\frac{di'}{d\gamma}=0 . \tag{32} \]

Taking, as everywhere, the axis of real values to coincide in direction with the vector \(\mathbf{E}_1\), we shall have

\[ \frac{de_1}{d\gamma}=\frac{de_1'}{d\gamma}=0,\quad \frac{de_2}{d\gamma}=-e_2,\quad \frac{de_2'}{d\gamma}=e_2, \]

and equation (32), on the basis of equations (10), we write thus:

\[ (e_1M_1+e_2M_2+e_2'N_1)(e_1N_2-e_2'M_2) -(e_1N_1+e_2N_2-e_2'M_2)(e_1N_2+e_2M_2)=0 \]

whence

\[ \operatorname{tg}\gamma=\frac{e_2'}{e_2} =\frac{M_1N_2-N_1M_2}{M_1M_2+N_1N_2}. \]

Substituting from equations (11) the values \(M_1, N_1, M_2, N_2\), we obtain

\[ \begin{aligned} M_1N_2-N_1M_2&=(r_1s_2-s_1r_2)\Delta,\\ M_1M_2+N_1N_2&=(r_1r_2+s_1s_2)\Delta, \end{aligned} \]

where \(\Delta\) is determinant (7), and consequently the desired magnitude of the angle \(\gamma\) is determined by the equation

\[ \operatorname{tg}\gamma=\frac{r_1s_2-s_1r_2}{r_1r_2+s_1s_2}. \]

The condition under which expression (33) becomes zero, i.e. under which the greatest voltage at the terminals will be obtained when the phases of the electromotive forces of both alternators coincide, is:

\[ \frac{s_1}{s_2}=\frac{r_1}{r_2}. \]

1. The magnitude of the voltage on the busbars is fully determined by the current in the external circuit; but obviously, for one and the same voltage on the busbars, i.e. for one and the same current delivered by the alternators to the external circuit, the distribution of this current between the two alternators may vary depending on the relation of the vectors \(\mathbf{E}_1\) and \(\mathbf{E}_2\) of the electromotive forces of the alternators.

There always exists the equality

\[ \mathrm{E}_1-\mathrm{E}_2=\mathrm{J}_1 Z_1-\mathrm{J}_2 Z_2 \tag{34} \]

where \(\mathrm{J}_1, \mathrm{J}_2, Z_1, Z_2\) have their previous meanings.

If the electromotive forces of both dynamo machines are equal in magnitude and direction, then, denoting the currents passing through the windings of their armatures under this condition respectively by \(\mathrm{J}_{01}\) and \(\mathrm{J}_{02}\), we shall have:

\[ \mathrm{J}_{01} Z_1=\mathrm{J}_{02} Z_2 \tag{35} \]

If the voltage on the busbars is constant, then evidently

\[ \mathrm{J}_1+\mathrm{J}_2=\mathrm{J}_{01}+\mathrm{J}_{02}=\mathrm{J}_0 \]

and this is possible only when

\[ \mathrm{J}_1=\mathrm{J}_{01}+\mathrm{J}_s,\quad \mathrm{J}_2=\mathrm{J}_{02}-\mathrm{J}_s \tag{36} \]

where \(\mathrm{J}_s\) is a certain as yet unknown vector. Substituting the values of \(\mathrm{J}_1\) and \(\mathrm{J}_2\) from (36) into (34), we shall have

\[ \mathrm{E}_1-\mathrm{E}_2=(\mathrm{J}_{01}+\mathrm{J}_s)Z_1-(\mathrm{J}_{02}-\mathrm{J}_s)Z_2 \]

whence, on the basis of equation (35),

\[ \mathrm{J}_s=\frac{\mathrm{E}_1-\mathrm{E}_2}{Z_1+Z_2} \tag{37} \]

i.e. the vector \(\mathrm{J}_s\) represents the current excited in the circuit with apparent resistance \(Z_1+Z_2\) by the electromotive force \(\varepsilon=\mathrm{E}_1-\mathrm{E}_2\). This current, as a vector, may be represented in the form of a complex quantity \(\mathrm{J}_s=i_s+j i'_s\), where \(i_s\) and \(i'_s\) are found by separately equating the imaginary and real quantities in equation (37).

\[ \left. \begin{aligned} i_s&=\frac{(e_1-e_2)(r_1+r_2')+(e'_1-e'_1{}^{2})(s_1+s_2)} {(r_1+r_2)^2+(s_1+s_2)^2} \\[6pt] i'_s&=\frac{(e_1-e_2)(s_1+s_2)+(e'_1-e'_2)(r_1+r_2)} {(r_1+r_2)^2+(s_1+s_2)^2} \end{aligned} \right\} \tag{37 bis} \]

From the equations found it is seen that the phenomenon takes place as though, in the windings of the alternators, there flowed currents \(J_{01}\) and \(J_{02}\), corresponding to the equality, in magnitude and direction, of the vectors of the electromotive forces of the parallel-connected dynamo machines at the same voltage on the busbars, and, in addition, in the circuit \(a I b d II c a\) (Fig. 6) there circulates a current \(J_s = i_s + j i'_s\), determined by equations (37).

It must be noted that the component of the current \(J_s\) is not in fact accessible to direct measurement. Directly we can measure only \(J_1\) and \(J_2\). The vector \(J_s\) is, strictly speaking, a fictitious quantity, introduced by us only for convenience of representation. The vectors \(J_{01}\) and \(J_{02}\) are determined from the system of equations (6) under the assumption \(e_1=e_2\), \(e'_1=e'_2\).

The current \(J_s\) must perform a certain amount of work. There will correspond to it a certain power of each alternator. It is obvious that if the vector of the current \(J_s\) forms an acute angle with the vector of the electromotive force of one alternator, then with the vector of the electromotive force of the other alternator it will form an obtuse angle. This means: of the two alternators, if only the current \(J_s\) is considered, one acts as a generator (the current \(J_s\) brakes it), the other as a synchronous motor (the current \(J_s\) develops a torque in the same direction as the torque of the mechanical motor rotating the armature of this alternator).

The powers of the alternators corresponding to the current \(J_s\) are evidently represented by equations (26). In distinction from the total powers \(W_1\) and \(W_2\), we shall denote them by the symbols

\[ W_{s1}\ \text{and}\ W_{s2}. \]

One of the quantities \(W_{s1}\) and \(W_{s2}\) (always, of course, the smaller in absolute value) will be negative, the other positive, and their sum, i.e. the difference of the absolute values, must represent the energy going to heat the windings of both alternators as a consequence of the circulation of the current \(J_s\). Indeed, from equations (26),

\[ W_{s1}+W_{s2}=W_{sr}= \frac{(r_1+r_2)\{(e_1+e_2)^2+(e'_1-e'_2)^2\}} {(r_1+r_2)^2+(s_1+s_2)^2} \tag{38} \]

The current \(J_s\) is called the equalizing current (Ausgleichstrom), since by means of it the voltage at the terminals of both alternators is equalized. Another name, synchronizing current, derives from the above-mentioned property of this current: to act in a braking manner on the leading alternator and in an accelerating manner on the lagging one, and thus to reduce the angle of phase difference of the electromotive forces. Irrespective of the action which this current has, the most correct name will be exchange current, since it circulates only in the circuit of the two alternators, without branching into the external circuit.

In the article “Ueber den Parallelbetrieb mit Wechselstrommaschinen,” E. T. Z. 1899, Benitschke, using for the determination of the synchronizing current the formula

\[ J_s=\frac{\varepsilon}{\sqrt{R^2+L^2\omega^2}} \]

where \(\varepsilon\) is the difference of the electromotive forces of the machines operating in parallel, \(L\) and \(R\) are the sums of the coefficients of self-induction and the ohmic resistances of the armature windings of both alternators together with the connecting leads, and \(\omega=2\pi\infty\), by a formula equivalent to equation (37), expresses the proposition that this current is wattless (Wattlos) when \(E_1\) and \(E_2\) are in phase and differ only in amplitude, i.e., when the positions of the poles coincide, but the machines are excited differently.

This view is repeated by many other authors (e.g. Niethammer)*. As may be seen from formulas (26), this view is valid only when the ohmic resistance of the machines may be regarded as infinitely small in comparison with the reactance of self-induction of their windings. Indeed, let us assume, for clarity, that \(W_{s1}>0,\ W_{s2}<0\). The power \(W_{s1}\) corresponding to the current \(J_s\), supplied by the first alternator in the case of a wattless current, must go entirely into heating the windings, while the mechanical effect of this current must be zero, i.e., the power \(W_{s2}\), increasing the torque of the second alternator, must be zero:

\[ W_{s2}=0 \]

* Handbuch d. Elektrtät. v. Heinke. B. IV. 181.

or

\[ (r_1+r_2)\{(e_2-e_1)+e'_2(e'_2-e'_1)\}-(s_1+s_2)(e'_1e_2-e'_2e_1)=0 \tag{39} \]

The condition for equality of the phases of the electromotive forces of both alternators is expressed thus:

\[ \frac{e_1}{e'_1}=\frac{e_2}{e'_2}, \]

and consequently condition (39) will take the form

\[ e_2(e_2-e_1)+e'_2(e'_2-e'_1)=0, \]

which is obviously possible only when not only the phases but also the amplitudes of the electromotive forces coincide. In other words, however the electromotive forces of two alternators may differ, there is always, in addition to losses due to heating, a transfer of energy from one alternator to the other, and the case of a wattless current does not exist. The difference between the cases of inequality of phase angles and inequality of excitation consists in the fact that in the first case the synchronizing current, as already stated, accelerates the motion of one alternator and retards the other, reducing the angle of phase difference of the electromotive forces, whereas in the second, without changing the speed of rotation, it only unloads the driving machine of the less excited alternator at the expense of the work of the driving machine of the alternator with the more excited field.

1. With constant excitation of the magnetic fields of the alternators, i.e., with constant \(E_1\) and \(E_2\), the magnitude of the mutually synchronizing action of two parallel-connected alternating-current machines depends exclusively on the angle \(\gamma\) between the vectors of the electromotive forces.

Let us find the value of the angle \(\gamma\) at which the synchronizing power reaches its greatest value, confining ourselves here to the most important case \(E_1=E_2\). In this case the synchronizing power \(W_{s2}\) is represented by the second equation (26), which, putting in it \(E_1=E_2\) and taking as the \(X\)-axis the direction of the vector \(E_1\) of the leading machine, we may write in the form

\[ W_{s2}=\frac{E_1^2}{(r_1+r_2)^2+(s_1+s_2)^2}\left\{(r_1+r_2)(1-\Cos\gamma)-(s_1+s_2)\Sin\gamma\right\} \]

Whence, putting

\[ \frac{s_1+s_2}{r_1+r_2}=\tg\psi \]

we obtain for \(W_{s2}\) the expression

\[ W_{s2}=\frac{E_1^2}{r_1+r_2}\,\frac{1-\Cos\gamma+\tg\psi\Sin\gamma}{1+\tg^2\psi} \]

Equating to zero the derivative of \(W_{s2}\) with respect to \(\gamma\), we obtain:

\[ \Sin\gamma+\tg\psi\Cos\gamma=0, \]

or

\[ \tg\gamma=-\tg\psi, \]

i.e. the greatest synchronizing action will be at an angle of phase difference equal to \(\arctg\left(\frac{s_1+s_2}{r_1+r_2}\right)\).

In order to find at what ratio of the sums of the self-induction reactances and ohmic resistances of the alternators the greatest synchronizing action is obtained, i.e. at what ratio of these quantities the system will be most stable, we equate to zero the derivative of \(W_{s2}\) with respect to \(\psi\). Performing the differentiation and equating the resulting expression to zero, after reductions we shall have

\[ \frac{\Sin\gamma}{1-\Cos\gamma}=\frac{2\tg\psi}{1-\tg^2\psi} \]

or

\[ \tg 2\psi\,\tg\frac{\gamma}{2}=1. \]

As is evident from the equation obtained, the sought quantity \(\psi\) depends on the angle of phase difference of the electromotive forces \(\gamma\). If we assume their directions to coincide, i.e. \(\gamma=0\), then

\[ \tg 2\psi=\infty, \]

whence

\[ \psi = 45^\circ \]

and

\[ \frac{s_1+s_2}{r_1+r_2}=\operatorname{tg}45^\circ=1, \]

which is nothing other than the condition for the most advantageous ratio of the self-induction reactance and the ohmic resistance, as was obtained by Hopkinson.

Steinmetz *) gives another derivation of the formulas representing the synchronizing action of two alternators in parallel operation, and obtains somewhat different results.

For comparison, let us present his reasoning.

The power of the first machine, measured at the terminals, is

\[ W_{e1}=e_0 i_1+e'_0 i'_1 \]

of the second

\[ W_{e2}=e_0 i_2+e'_0 i'_2. \]

The difference of the powers of the two machines is

\[ W_{e1}-W_{e2}=e_0(i_1-i_2)+e'_0(i'_1-i'_2) \]

Steinmetz calls the synchronizing action of the machines one half of the derivative of the difference \(W_{e1}-W_{e2}\) with respect to the angle \(\gamma\), i.e. the quantity

\[ q=\frac{1}{2}\frac{d(W_{e1}-W_{e2})}{d\gamma}; \]

this is evidently the power transferred from one machine to the other when the phase angle between the electromotive forces of the two alternators is changed.

Let us now see whether the quantity \(q\) can serve as a measure of the synchronizing action of the machines to the same extent as the quantity \(W_{s2}\) obtained by us earlier.

) C. P. Steinmetz, Theorie und Ber. der Wechselstr.* Chap. XVII. The form of Steinmetz’s formulas is somewhat different from the formulas obtained by us below, which depends on a different choice of the direction of the coordinate axes.

\(W_{e1}\) and \(W_{e2}\) may be represented in the form

\[ \begin{aligned} W_{e1}&=W_{t1}+W_s,\\ W_{e2}&=W_{t2}-W_s \end{aligned} \]

where \(W_{t1}, W_{t2}\) are the powers delivered by each machine to the external circuit, and \(W_s\) is the power transmitted from one machine to the other. Subtracting one expression from the other, we obtain

\[ W_{e1}-W_{e2}=W_{t1}-W_{t2}+2W_s. \]

The quantity \(q\) can serve as a measure of the synchronizing action only when

\[ q=\frac{1}{2}\frac{d(W_{e1}-W_{e2})}{d\gamma}=\frac{dW_s}{d\gamma}, \]

and this will be only under the assumption \(W_{t1}-W_{t2}=\mathrm{Const}\).

But the assumption \(W_{t1}-W_{t2}=\mathrm{Const}\) is equivalent to the assumption of constancy of the voltage at the terminals. Moreover, it is obvious that \(W_s\), thus defined, in addition to the power converted into mechanical power, i.e. the synchronizing power in the proper sense of the word, also includes the power lost in heating the armature winding of the lagging alternator owing to the circulation of the balancing current \(J\).

Steinmetz examines in detail the case of two identical generators when

\[ \sqrt{e_1^2+e_1'^2}=\sqrt{e_2^2+e_2'^2} \]

and in this case obtains the maximum value for \(W_s\) at \(\gamma=\dfrac{\pi}{2}\), and for the magnitude of the synchronizing action at \(\gamma=0\).

On the basis of what has been said, it seems more correct, in determining the synchronizing interaction of alternators, to proceed from equation (37).

III. Oscillations of Alternators Connected in Parallel

11. Up to this point we have assumed that both alternators are in steady motion, i.e., that both rotate with constant angular velocity, develop constant powers, and that the field excitations and the angles of phase difference of their electromotive forces are constant. Let us now suppose that the power of at least one of the engines driving our alternators—or, in the case of a synchronous motor, the power absorbed by external mechanical resistances—changes. In doing so we shall assume that the excitation of both alternators remains constant and shall suppose that changes in the electromotive forces, owing to oscillations of the speed, may be neglected, i.e., that

\[ e_1^2 + e_1'^2 = \mathrm{Const.},\quad e_2^2 + e_2'^2 = \mathrm{Const.} \]

Let the change of power occur on the side of the second alternator; the power \(W_2\) will receive some increment \(\pm \delta W_2\). Referring to the diagram in Fig. 3, we see that this change of power will correspond to a displacement of the point \(E_2\) along the arc of the circle \(E_2' E_2 E_2''\) to the intersection of this circle with the circle of power \(W_2 + \delta W_2\) at the point \(E_2''\), or respectively with the circle of power \(W_2 - \delta W_2\) at the point \(E_2'\). Through the points \(E_2'\) and \(E_2''\) will pass the straight lines of powers \(W_1 + \delta W_1\) and \(W_1 - \delta W_1\). When the power of one of the alternators changes, the power of the other changes likewise. In passing from one line of power to another, the angle of phase difference of the electromotive forces \(\gamma\) will evidently change, receiving an increment \(\pm \xi\). After this the motion will again become steady.

So long as both alternators rotate with constant speed, the electrical power may be taken as equal (neglecting losses due to friction and hysteresis) to the mechanical power on the shaft of the alternator. During the period in which the angle \(\gamma\) changes to the value \(\gamma \pm \xi\), this equality does not exist, since this change can be attained only by a temporary change in the rotational speeds of the alternators, whereby part of the mechanical ener-

will go toward changing the speed. In this case four possibilities may occur: 1) the alternator operates as a generator, and the speed increases (the acceleration is positive); in this case the mechanical energy supplied by the engine is greater than the electrical energy of the dynamo; 2) the alternator operates as a generator with decreasing speed (the acceleration is negative); the mechanical energy supplied is less than the electrical energy developed by the generator; 3) the alternator operates as a motor with increasing speed; the electrical energy is greater than the mechanical energy developed at the pulley; 4) the alternator operates as a motor with decreasing speed; the electrical energy is less than the mechanical energy.

In all the cases enumerated, the difference between the electrical and mechanical energies is equal to the energy going into increasing the kinetic energy of the rotating parts, or given up by these rotating parts to the shaft of the dynamo when their kinetic energy decreases as a result of a decrease in speed.

If, at a certain angle of phase difference \(\gamma\), corresponding to the mechanical powers \(W_1, W_2\), the kinetic energies of the rotating parts, by virtue of inertia, differ from the values corresponding to the established motion, they will tend to change the angle \(\gamma\). When this angle changes, synchronizing currents must evidently arise between the armatures of the two alternators, whose energy will be directed toward changing the kinetic energies of the rotating masses, bringing them closer to the values corresponding to the established motion. Having attained a certain greatest value, the angle \(\gamma + \xi\) will begin to decrease, will come to the value \(\gamma\), pass through it by virtue of inertia, and the phenomenon will repeat in the previous order. An oscillation will result about a certain mean position, characterized by the angle of phase difference of the electromotive forces \(\gamma\).

If the decrement of the amplitude of this oscillation is positive, i.e. if this amplitude decreases with time, then in the end the vector \(E_2\) assumes, relative to the vector \(E_1\), the position characterized by the angle of phase difference \(\gamma\), and the motion becomes uniform. If the decrement is negative, the amplitude of the oscillation increases without bound, and the synchronizing current circulating between the armature windings of both alternators

as a result of these oscillations, reaches such a magnitude that further operation of the machines becomes impossible (the fuses melt). There is also possible the case when the decrement is zero. Then the magnitude of the angle of difference of phase of the electromotive forces oscillates about the mean value \(\gamma\) with constant amplitude.

12. Let \(\xi\) be the increment of the angle \(\gamma\) at some instant \(t\). Generally speaking, \(\gamma\) is a periodic function of time, with a number of periods per second considerably smaller than the normal number of periods of the electromotive force of the alternators. Let, further, \(\Theta_1,\ \Theta_2\) be the moments of inertia, with respect to the axis of rotation, of the rotating masses of both alternators. If the transmission of mechanical energy is effected by a belt or a rope, then only the masses of the alternators enter into \(\Theta_1\) and \(\Theta_2\). If the armature of the machines is seated directly on the shaft of the engine machine, \(\Theta_1,\Theta_2\) will include also the masses of the flywheels; in a similar way, in \(\Theta_1\) and \(\Theta_2\) there will enter the masses set into rotational motion by the alternator when it works as a motor, if these masses are seated directly on its shaft, as is the case in rotating transformers.

Let, then, \(\Omega_1,\ \Omega_2\) be the angular velocities of the rotating parts of the alternators at the given instant \(t\); \(\Omega_{01},\ \Omega_{02}\)—these same angular velocities corresponding to the established motion and to the normal number of periods of the current per second.

Denote by \(x_1,x_2\) fractions representing the change of the angular velocities \(\Omega_1,\Omega_2\) in comparison with the velocities \(\Omega_{01},\Omega_{02}\) in the established motion, i.e. put

\[ \Omega_1=\Omega_{01}(1+x_1),\quad \Omega_2=\Omega_{02}(1+x_2). \]

\(x_1\) and \(x_2\), like \(\xi\), are periodic functions of time. The angle of difference of phase of the electromotive forces of the alternators at each instant, so long as the oscillations continue, is \(\gamma+\xi\).

If \(\omega_1,\omega_2\) are the angular velocities of the alternators, referred to a two-pole field, at each instant, \(\omega_{10},\omega_{20}\)—the same quantities in the established motion, \(n_1,n_2,n_{01},n_{02}\)—the corresponding numbers of revolutions of the armature per minute, \(p_1,p_2\) the numbers of pairs of field poles of the alternator, then

\[ \omega_1=\frac{2\pi n_1 p_1}{60}=\Omega_1 p_1,\quad \omega_2=\frac{2\pi n_2 p_2}{60}=\Omega_2 p_2, \tag{40} \]

\[ \omega_{01}=\frac{2\pi n_{01} p_1}{60},\quad \omega_{02}=\frac{2\pi n_{02} p_2}{60} \]

It is evident that, by virtue of the equality of the numbers of periods of the electromotive forces in steady motion, \(\omega_{01}=\omega_{02}=\omega\), where

\[ \omega=2\pi\infty. \]

From relation (40) it is seen that the angular velocities \(\omega_1,\omega_2\) may be represented in the form

\[ \omega_1=\omega(1+x_1),\quad \omega_2=\omega(1+x_2). \]

The angular velocities denoted by us by the symbol \(\omega\) with different indices are the angular velocities with which the ray rotates in the clock diagram representing the sinusoidal wave.

If, as before, we take as the axis of real values (in the symbolic representation of electromotive forces and currents) the direction of the vector \(\mathbf{E}_1\), the electromotive force of the first alternator, then, as is clear from what was said in § 2, the electromotive force of the second alternator may be represented in the form of a vector \(\mathbf{E}_2\) of the same periodicity as \(\mathbf{E}_1\), but with variable argument \(\gamma+\xi\), where \(\gamma\) is the constant angle corresponding to the mean position (steady motion), and \(\xi\) is the variable part.

The change of the argument \(\gamma+\xi\) during an element of time \(dt\) is \(d\xi\), and is equal to the relative angular velocity of the vectors \(\mathbf{E}_1\) and \(\mathbf{E}_2\) multiplied by the element of time. But the relative angular velocity of the vectors \(\mathbf{E}_1\) and \(\mathbf{E}_2\) is evidently equal to the difference of the corresponding angular velocities of the rays in the clock diagram, and therefore

\[ d\xi=(\omega_2-\omega_1)\,dt; \]

but

\[ \omega_2-\omega_1=\omega(1+x_2)-\omega(1+x_1)=\omega(x_2-x_1), \]

whence

\[ d\xi=\omega(x_2-x_1)\,dt \]

or, introducing the notation

\[ \varphi=\omega t,\quad d\varphi=\omega\,dt, \]

\[ \frac{d\xi}{d\varphi}=x_2-x_1 \tag{41} \]

At every instant the kinetic energy of the rotating parts will be represented in the form

\[ \left. \begin{aligned} \pi_1&=\frac{1}{2}\Theta_1 g\Omega_1^{2} =\frac{1}{2}\Theta_1 g\Omega_{01}^{2}(1+x_1)^2,\\ \pi_2&=\frac{1}{2}\Theta_2 g\Omega_2^{2} =\frac{1}{2}\Theta_2 g\Omega_{02}^{2}(1+x_2)^2, \end{aligned} \right\} \tag{42} \]

where the factor \(g=9.81\) is introduced in order to express the kinetic energy in watt-seconds.

As was said, the change in the kinetic energy of the rotating parts of a machine over some time is equal to the difference between the mechanical energy supplied by the engine during the same time and the electrical energy developed by the alternator. If \(W_{1\xi}, W_{2\xi}\) are the powers of the dynamo machines at the given instant for the phase angle \(\gamma+\xi\), and \(P_1, P_2\) are the corresponding mechanical powers of the engine machines, then

\[ P_1-W_{1\xi}=\frac{d\pi_1}{dt}, \]

\[ P_2-W_{2\xi}=\frac{d\pi_2}{dt}. \tag{43} \]

From equation (42), taking into account equations (40), we have

\[ \frac{d\pi_1}{dt} = \frac{\Theta_1 g}{p_1^2}\,\omega^2(1+x_1)\frac{dx_1}{dt} \]

\[ \frac{d\pi_2}{dt} = \frac{\Theta_2 g}{p_2^2}\,\omega_2(1+x_2)\frac{dx_2}{dt}. \]

For us, only those cases of disturbance of motion are of interest in which the functioning of the machines is still possible; and for these cases \(x_1\) and \(x_2\) are such small fractions,

that, as terms into which they enter as factors, we may neglect them without great error, and, taking into account that

\[ \frac{d x_1}{d t}=\frac{d x_1}{d\varphi}\frac{d\varphi}{d t}=\frac{d x_1}{d\varphi}\omega,\quad \frac{d x_2}{d t}=\frac{d x_2}{d\varphi}\frac{d\varphi}{d t}=\frac{d x_2}{d\varphi}\omega \]

we may write

\[ \left. \begin{aligned} \frac{d\pi_1}{d t}&=\frac{\Theta_1 g}{p_1^2}\,\omega^3\,\frac{d x_1}{d\varphi}\\ \frac{d\pi_2}{d t}&=\frac{\Theta_2 g}{p_2^2}\,\omega^3\,\frac{d x_2}{d\varphi} \end{aligned} \right\} \tag{44} \]

These expressions must be substituted into equations (43). The power of the driving machine can, in general, be represented in the form

\[ P=P_o+f(\varphi), \]

where \(P_o\) is the mean power of the machine over a complete revolution, and \(f(\varphi)\) is a certain periodic function of the crank angle, and consequently also of the angle \(\varphi=\omega t\).

Next let us expand \(W_{1\xi}\) and \(W_{2\xi}\) in a Taylor series, and, owing to the smallness of \(\xi\), discard terms of the second and higher powers; we obtain:

\[ \left. \begin{aligned} W_{1\xi}&=W_1+\frac{dW_1}{d\gamma}\xi\\ W_{2\xi}&=W_2+\frac{dW_2}{d\gamma}\xi \end{aligned} \right\} \tag{45} \]

Obviously here

\[ W_1=P_{01},\quad W_2=P_{02}. \]

13. Let us first consider the case when

\[ f(\varphi)=0; \]

this occurs when the alternating-current generator is driven by a turbine, steam or water, or when a synchronous motor directly drives a centrifugal pump.

After substituting expressions (44) and (45) into (43), we obtain

\[ \omega^3 \frac{d x_1}{d\varphi} + \frac{p_1^2}{\Theta_1 g}\frac{d W_1}{d\gamma}\,\xi =0, \]

\[ \omega^3 \frac{d x_2}{d\varphi} + \frac{p_2^2}{\Theta_2 g}\frac{d W_2}{d\gamma}\,\xi =0. \]

Subtracting the first of these equations from the second, we shall have

\[ \omega^3 \frac{d}{d\varphi}(x_2-x_1) + \left( -\frac{p_1^2}{\Theta_1 g}\frac{d W_1}{d\gamma} + \frac{p_2^2}{\Theta_2 g}\frac{d W_2}{d\gamma} \right) \tag{46} \]

Since

\[ \frac{d\xi}{d\varphi}=x_2-x_1, \]

then

\[ \frac{d}{d\varphi}(x_2-x_1)=\frac{d^2 \xi}{d\varphi^2} \tag{47} \]

Differentiating equation (18) with respect to \(\gamma\), we find

\[ \left. \begin{aligned} \frac{d W_1}{d\gamma} &= \frac{E_1 E_2}{\triangle}(C\operatorname{Sin}\gamma-D\operatorname{Cos}\gamma) \\[4pt] \frac{d W_2}{d\gamma} &= \frac{E_1 E_2}{\triangle}(C\operatorname{Sin}\gamma+D\operatorname{Cos}\gamma). \end{aligned} \right\} \tag{48} \]

Substituting the values of \(\dfrac{d}{d\varphi}(x_2-x_1)\), \(\dfrac{dW_1}{d\gamma}\), \(\dfrac{dW_2}{d\gamma}\) from (47), (48) into (46), we have

\[ \frac{E_1 E_2}{\omega^3 \triangle g} \left\{ C\left(\frac{p_2^2}{\Theta_2}-\frac{p_1^2}{\Theta_1}\right)\operatorname{Sin}\gamma + D\left(\frac{p_1^2}{\Theta_1}+\frac{p_2^2}{\Theta_2}\right)\operatorname{Cos}\gamma \right\}\xi + \frac{d^2\xi}{d\varphi^2} =0 \tag{49} \]

or, denoting

\[ \frac{E_1 E_2}{\omega^3 \triangle g} \left\{ C\left(\frac{p_1^2}{\Theta_2}-\frac{p_1^2}{\Theta_1}\right)\operatorname{Sin}\gamma + D\left(\frac{p_1^2}{\Theta_1}+\frac{p_2^2}{\Theta_2}\right)\operatorname{Cos}\gamma \right\} =k, \]

\[ k\xi+\frac{d^2\xi}{d\varphi^2}=0 \tag{49 bis} \]

In integrating this equation, three cases are possible:

\[ \begin{aligned} &1)\quad k>0\\ &2)\quad k<0\\ &3)\quad k=0 \end{aligned} \]

1st case. The integral of the equation will be

\[ \xi=A_1\operatorname{Cos}\sqrt{k\varphi}+A_2\operatorname{Sin}\sqrt{k\varphi} \]

At the initial instant one has \(\varphi=0,\ \xi=0\), and therefore

\[ A_1=0 \]

and

\[ \xi=A_2\operatorname{Sin}\sqrt{k\varphi} \]

Differentiating this with respect to \(t\), we obtain

\[ \frac{d\xi}{dt}=\omega k A_2\operatorname{Cos}\sqrt{k\varphi} \]

At the initial instant \(\left(\dfrac{d\xi}{dt}\right)_0=v_0\) will represent the initial relative angular velocity of the vectors of the electromotive forces on the diagram; hence

\[ A_2=\frac{v_0}{\sqrt{k}\omega}, \]

and

\[ \xi=\frac{v_0}{\sqrt{k}\omega}\operatorname{Sin}\sqrt{k\omega t}. \]

In the case under consideration our system of two alternators will be in stable equilibrium and, under the action of an external impulse, will execute oscillations about the position of equilibrium with amplitude \(\dfrac{v_0}{\omega k}\) and with period

\[ T=\frac{1}{\infty\sqrt{k}}, \]

where \(\infty\) is the normal number of periods of the current in the established motion.

2nd case. The integral of the equation will be

\[ \xi = A_1 e^{\sqrt{-k}\,\varphi} + A_2 e^{-\sqrt{-k}\,\varphi} \]

Since for \(\varphi = 0\), \(\xi = 0\), then

\[ A_1 = - A_2 \]

and

\[ \xi = A_1 \left(e^{\sqrt{-k}\,\varphi} - e^{-\sqrt{-k}\,\varphi}\right). \]

Reasoning as in the preceding case, we find

\[ A_1 = \frac{v_0}{2\omega\sqrt{-k}} \]

and finally

\[ \xi = \frac{v^0}{2\omega\sqrt{-k}} \left(e^{\sqrt{-k}\,\varphi} - e^{-\sqrt{-k}\,\varphi}\right) \]

The negative term of the expression in parentheses tends to zero with time, while the positive term increases without bound.

This means that for \(k < 0\) the system is in unstable equilibrium, and with the slightest disturbance the angle of difference of phase of the electromotive forces increases, and parallel operation becomes impossible.

3rd case. The integral of the equation is

\[ \xi = A_1 \varphi + A_2 \]

Reasoning as in the preceding case, we find

\[ A_1 = \frac{v_0}{\omega}, \quad A_2 = 0 \]

and

\[ \xi = \frac{v_0}{\omega}\varphi \]

The equilibrium is unstable and the parallel operation of the alternators is impossible on the same grounds as in the 2nd case.

From what has been said it is evident that, for the possibility of parallel operation of two alternators (under the assumption \(P_1=\mathrm{Const}\), \(P_2=\mathrm{Const}\)), the condition \(k>0\) is necessary, or, since the quantity \(\dfrac{E_1E_2}{g}\Delta\omega\) is essentially positive, it must be

\[ D\left(\frac{p_1^2}{\Theta_1}+\frac{p_2^2}{\Theta_2}\right)\cos\gamma + C\left(\frac{p_2^2}{\Theta_2}-\frac{p_1^2}{\Theta_1}\right)\sin\gamma >0 \tag{50} \]

14. Let us now consider the question of the oscillations of two alternators operating in parallel, assuming that the function \(f(\varphi)\) is not zero, and that the alternators are provided with devices for damping oscillations. Under this assumption, from equations (43) we obtain

\[ \left. \begin{aligned} \frac{dW_1}{d\gamma}\,\xi+\frac{d\tau_1}{dt}+F_1&=f_1(\varphi)\\[4pt] \frac{dW_1}{d\gamma}\,\xi+\frac{d\tau_2}{dt}+F_2&=f_2(\varphi) \end{aligned} \right\} \tag{51} \]

where \(F_1\), \(F_2\) are the powers going to the damping of oscillations. To solve the question we must determine the form of the functions \(F_1\), \(F_2\), \(f_1(\varphi)\), and \(f_2(\varphi)\).

The powers \(F_1\) and \(F_2\) depend on the Foucault currents developing in the pole shoes, or in systems of conductors specially arranged for this purpose (as, for example, in Leblanc’s system).

In determining these quantities we may regard the alternator as an asynchronous motor. The role of the primary winding is played by the armature winding of the alternator; the role of the secondary winding is played by the massive pole shoes or the rods of the Leblanc system. At constant speed of rotation the number of lines of the armature field penetrating any circuit on the pole shoes is constant, and we may imagine the magnetic system as if rotating synchronously with the armature field. When the speed of rotation of the alternator changes, this imaginary relative motion of the magnetic system with respect to the armature field will change and will cease to be synchronous; the magnetic

the flux passing through some circuit on the pole pieces will cease to be constant; the change of this flux will evidently be proportional to the change in the speed of rotation, and will excite in the poles of the magnets, or in the rods of the Leblanc system, currents similar to the currents developing in the rotor winding of an asynchronous motor. The role of slip here will be played by the change in the speed of rotation of the armature in fractions of the mean speed.

The power of an asynchronous motor is expressed by the formula *):

\[ \mathrm{P}=\zeta^2 \frac{\mathrm{p}^2 Q_2}{32\rho l_2}\Phi^2(\omega_1-\omega)\omega \]

where \(\zeta\) is a fraction close to unity, depending on the character of the rotor winding, in the present case on the type of pole pieces or damping device; \(\mathrm{p}\) is the number of poles of the inductor, \(Q_2\) is the cross-section of the rods of the secondary winding, \(l_2\) is the mean length of one rod, \(\rho\) is the specific conductivity of the material of the secondary winding, \(\Phi\) is the resultant magnetic flux, in the present case the total magnetic flux produced by the armature field, \(\omega\) is the synchronous speed, \(\omega_1\) is the actual speed of rotation of the rotor. Putting

\[ \zeta^2 \frac{\mathrm{p}^2 Q_2}{32\rho l_2}=c^2\;(\text{constant}) \]

we shall have, for the first alternator, the power going to the damping of oscillations

\[ F_1=c_1^2\Phi_1^2 x_1(1+x_1)\omega^2, \]

or, neglecting the second degree of \(x_1\):

\[ \begin{aligned} F_1 &= c_1^2\Phi_1^2\omega^2 x_1 \\ F_2 &= c_2^2\Phi_2^2\omega^2 x_2 \end{aligned} \qquad \begin{array}{l} \\[-1.7em] \left.\begin{array}{l}\\ \\ \end{array}\right\} \end{array} \tag{52} \]

and likewise for the second alternator.

The magnitudes of the fluxes \(\Phi_1, \Phi_2\) may be expressed as functions of the quantities \(e_1, e_1', e_2, e_2', r_1, r_2, s_1, s_2\), which we consider given, fol—

) A. Voronov. Theory and Calculation of Multiphase Asynchronous Motors*, p. 14, eq. 37.

in the following manner. \(\Phi_1\) and \(\Phi_2\) are, as has been said, the magnetic fluxes arising under the action of the demagnetizing ampere-turns of the armature. According to Kapp*) these ampere-turns are

\[ X_g = k q J \operatorname{Sin} d \]

where \(q\) is the number of conductors in each coil of the armature winding, \(d\) is the angle of phase difference between the current and the electromotive force, and

\[ k = n \frac{0.57}{m} \operatorname{Sin} \frac{m\pi}{2} \]

where \(n\) is the number of phases, and \(m\) is the ratio of the pole width to the pole pitch. Hence, if by \(R_1, R_2\) we denote the mean magnetic reluctances in the path of the demagnetizing fluxes of the armatures of both alternators, the fluxes \(\Phi_1\) and \(\Phi_2\) will be

\[ \left. \begin{aligned} \Phi_1 &= n_1 \frac{0.57}{m_1}\frac{q_1}{R_1}\operatorname{Sin}\frac{m_1\pi}{2} J_1 \operatorname{Sin} d_1 \\ \Phi_2 &= n_2 \frac{0.57}{m_2}\frac{q_2}{R_2}\operatorname{Sin}\frac{m_2\pi}{2} J_2 \operatorname{Sin} d_2 \end{aligned} \right\} \tag{53} \]

The expressions \(J_1 \operatorname{Sin} d_1\), \(J_2 \operatorname{Sin} d_2\) are the projections of the current vectors on the directions of the electromotive forces. In the notation adopted by us,

\[ J_1 \operatorname{Sin} d_1 = \frac{i_1 e'_1 - i'_1 e_1}{E_1} \]

\[ J_2 \operatorname{Sin} d_2 = \frac{i_2 e'_2 - i'_2 e_2}{E_2} \]

Substituting here the values \(i_1, i_2, i'_1, i'_2\) from equations (8), and taking the direction of the vector \(E_1\) as the real axis, we obtain, neglecting changes of the angle \(\gamma\),

\[ \left. \begin{aligned} J_1 \operatorname{Sin} d_1 &= \frac{E_1}{\Delta}\left[ B_1 E_1 + (C \operatorname{Sin}\gamma - D \operatorname{Cos}\gamma) E_2 \right] \\ J_2 \operatorname{Sin} d_2 &= \frac{E_2}{\Delta}\left[ B_2 E_2 + (C \operatorname{Sin}\gamma + D \operatorname{Cos}\gamma) E_1 \right] \end{aligned} \right\} \tag{54} \]

) G. Kapp, Dynam. f. Gl. & Wechs.* 3rd ed., § 103.

Putting

\[ c_1\Phi_1\omega=G_1,\quad c_2\Phi_2\omega=G_2 \]

we shall finally have the expressions for the powers going toward damping the oscillations

\[ F_1=G_1x_1,\quad F_2=G_2x_2 \tag{55} \]

Let us now turn to determining the form of the function \(f(\varphi)\). The tangential pressure \(T\), as a periodic function of the angle \(\alpha\) of rotation of the crankshaft, may, as is known, be represented in the form of a series

\[ \begin{aligned} T={}&A_0+A_1\operatorname{Cos}\alpha+A_2\operatorname{Cos}2\alpha+A_3\operatorname{Cos}3\alpha+\cdots\\ &+B_1\operatorname{Sin}\alpha+B_2\operatorname{Sin}2\alpha+B_3\operatorname{Sin}3\alpha+\cdots \end{aligned} \]

where

\[ A_0=\frac{1}{2\pi}\int_0^{2\pi}T\,d\alpha=T_m \quad\text{(mean tang. pressure)} \]

\[ A_i=\frac{1}{2\pi}\int_0^{2\pi}T\,\operatorname{Cos} i\alpha\,dx \]

\[ B_i=\frac{1}{2\pi}\int_0^{2\pi}T\,\operatorname{Sin} i\alpha\,dx \]

If \(\Omega\) is the angular velocity, then the power will be

\[ P=T\Omega, \]

\[ P_0=T_m\Omega; \]

If \(p_1, p_2\) are the numbers of pole pairs of the alternators, then

\[ \alpha_1=\frac{\varphi}{p_2},\quad \alpha_2=\frac{\varphi}{p_1}, \]

and on the basis of all that has been said we shall have

\[ \left. \begin{aligned} f_1(\varphi)={}&A_{11}\operatorname{Cos}\frac{\varphi}{p_1} +A_{12}\operatorname{Cos}\frac{2\varphi}{p_1}+\cdots\\ &+B_{11}\operatorname{Sin}\frac{\varphi}{p_1} +B_{12}\operatorname{Sin}\frac{2\varphi}{p_1}+\cdots\\[6pt] f_2(\varphi)={}&A_{21}\operatorname{Cos}\frac{\varphi}{p_2} +A_{22}\operatorname{Cos}\frac{2\varphi}{p_2}+\cdots\\ &+B_{21}\operatorname{Sin}\frac{\varphi}{p_2} +B_{22}\operatorname{Sin}\frac{2\varphi}{p_2}+\cdots \end{aligned} \right\} \tag{56} \]

In order to investigate the phenomenon that interests us, we shall seek what influence each term of the expansion will have on the change in the quantities \(x_1\), \(x_2\), and \(\xi\). In the series (56) let us combine each two terms of the \(k\)-th order,

\[ A_{1k}\operatorname{Cos}\frac{k\varphi}{p_1} + B_{1k}\operatorname{Sin}\frac{k\varphi}{p_1} = R_{1k}\operatorname{Sin}\left(\frac{k\varphi}{p_1}+\chi_1\right) \]

\[ A_{2k}\operatorname{Cos}\frac{k\varphi}{p_2} + B_{2k}\operatorname{Sin}\frac{k\varphi}{p_2} = R_{2k}\operatorname{Sin}\left(\frac{k\varphi}{p_2}+\chi_2\right) \]

and, substituting these values into equations (51), integrate them. Since for the time being we shall consider only the \(k\)-th term separately, we may, without impairing generality, assume

\[ \chi_1=0,\quad \chi_2=0, \]

and then, having made in equations (51) the indicated substitution and having substituted in them, moreover, the quantities found for \(F_1\), \(F_2\), we find

\[ \left. \begin{aligned} \frac{dW_1}{d\gamma}\xi &+ \frac{\Theta_1 g\omega^3}{p_1^2}\frac{dx_1}{d\varphi} + G_1^2 x_1 = R_{1k}\operatorname{Sin}\frac{k\varphi}{p_1} \\ \frac{dW_2}{d\gamma}\xi &+ \frac{\Theta_2 g\omega^3}{p_2^2}\frac{dx_2}{d\varphi} + G_2^2 x_2 = R_{2k}\operatorname{Sin}\frac{k\varphi}{p_2} \end{aligned} \right\} \tag{57} \]

Putting in these equations

\[ \frac{dW_1}{d\gamma}\cdot\frac{p_1^2}{\Theta_1 g\omega^3}=b_1,\quad \frac{G_1^2 p_1^2}{\Theta_1 g\omega^3}=a_1,\quad \frac{R_{1k}p_1^2}{\Theta_1 g\omega^3}=S_{1k}, \]

\[ \frac{dW_2}{d\gamma}\cdot\frac{p_2^2}{\Theta_2 g\omega^3}=b_2,\quad \frac{G_2^2 p_2^2}{\Theta_2 g\omega^3}=a_2,\quad \frac{R_{2k}p_2^2}{\Theta_2 g\omega^3}=S_{2k}, \]

together with equations (41), we obtain a system of three linear differential equations of the first order with three unknowns \(\xi\), \(x_1\), \(x_2\):

\[ \left. \begin{aligned} \frac{dx_1}{d\varphi}+a_1x_1+b_1\xi &= S_{1k}\operatorname{Sin}\frac{k}{p_1}\varphi \\ \frac{dx_2}{d\varphi}+a_2x_2+b_2\xi &= S_{2k}\operatorname{Sin}\frac{k}{p_2}\varphi \\ \frac{d\xi}{d\varphi}+x_1-x_2&=0 \end{aligned} \right\} \tag{58} \]

The complete integral of equations (58) is the sum of some particular solution and the complete integral of the equations without the right-hand side. The complete integral of the equations without the right-hand side is

\[ \begin{aligned} x_1 &= -\left( C_1\frac{b_1}{a_1+\lambda_1}e^{\lambda_1\varphi} + C_2\frac{b_1}{a_1+\lambda_2}e^{\lambda_2\varphi} + C_3\frac{b_1}{a_1+\lambda_3}e^{\lambda_3\varphi} \right) \\ x_2 &= -\left( C_1\frac{b_2}{a_2+\lambda_1}e^{\lambda_1\varphi} + C_2\frac{b_2}{a_2+\lambda_2}e^{\lambda_2\varphi} + C_3\frac{b_2}{a_2+\lambda_3}e^{\lambda_3\varphi} \right) \\ \xi &= C_1e^{\lambda_1\varphi} + C_2e^{\lambda_2\varphi} + C_3e^{\lambda_3\varphi} \end{aligned} \tag{59} \]

where \(C_1\), \(C_2\), \(C_3\) are arbitrary constants, and \(\lambda_1\), \(\lambda_2\), \(\lambda_3\) are the roots of the characteristic equation:

\[ \begin{vmatrix} \lambda+a_1, & 0, & b_1\\ 0, & \lambda+a_2, & b_2\\ 1, & -1, & \lambda \end{vmatrix} =0 \]

or

\[ \lambda^3+(a_1+a_2)\lambda^2+\{a_1a_2-(b_1-b_2)\}\lambda+a_1b_2-a_2b_1=0\ldots \tag{60} \]

We shall seek a particular solution of equations (58) in the form

\[ \begin{aligned} x_1 &= M_1\operatorname{Sin}\frac{k}{p_1}\varphi + N_1\operatorname{Cos}\frac{k}{p_1}\varphi + P_1\operatorname{Sin}\frac{k}{p_2}\varphi + Q_1\operatorname{Cos}\frac{k}{p_2}\varphi \\ x_2 &= M_2\operatorname{Sin}\frac{k}{p_1}\varphi + N_2\operatorname{Cos}\frac{k}{p_1}\varphi + P_2\operatorname{Sin}\frac{k}{p_2}\varphi + Q_2\operatorname{Cos}\frac{k}{p_2}\varphi \\ \xi &= M\operatorname{Sin}\frac{k}{p_1}\varphi + N\operatorname{Cos}\frac{k}{p_1}\varphi + P\operatorname{Sin}\frac{k}{p_2}\varphi + Q\operatorname{Cos}\frac{k}{p_2}\varphi \end{aligned} \tag{61} \]

We shall find the coefficients \(M_1\), \(N_1,\ldots,P,Q\) by substituting expressions (61) into equations (58) and equating to zero the coefficients of \(\operatorname{Sin}\frac{k}{p_1}\varphi\), \(\operatorname{Cos}\frac{k}{p_1}\varphi\), \(\operatorname{Sin}\frac{k}{p_2}\varphi\), and \(\operatorname{Cos}\frac{k}{p_2}\varphi\) in the equations obtained. We obtain, for determining the 12 required coefficients as functions of \(a_1\), \(a_2\), \(b_1\), \(b_2\), and the number \(k\), 12 equations.

For us, only the quantity \(\xi\) is of interest; the most general expression for it will be

\[ \xi = C_1 e^{\lambda_1 \varphi} + C_2 e^{\lambda_2 \varphi} + C_3 e^{\lambda_3 \varphi} + \sum_1^k R_{1k}\operatorname{Sin}\left(\frac{k}{T_1}\varphi+\beta_1\right) + \]

\[ + \sum_1^k R_{2k}\operatorname{Sin}\left(\frac{k}{p_2}\varphi+\beta_2\right)\ldots \tag{62} \]

where \(R_{1k}\) and \(R_{2k}\) are constants easily determined from the preceding, and \(\beta_1,\ \beta_2\) are angles, which include the angles \(\chi_1\) and \(\chi_2\) previously discarded by us.

By examining expression (62) we can easily decide whether the motion will be stable or not.

It will be stable if \(\xi\), under all possible disturbances, remains within certain limits and decreases with time; and it will be unstable if \(\xi\) increases without bound with time. The sum of the trigonometric functions entering the right-hand side of equation (62) will evidently always oscillate within certain limits, and the stability of the motion will depend entirely on the sum of the three terms

\[ C_1 e^{\lambda_1 \varphi} + C_2 e^{\lambda_2 \varphi} + C_3 e^{\lambda_3 \varphi}. \]

If at least one of the roots of equation (60), \(\lambda_1,\ \lambda_2\), or \(\lambda_3\), is real and positive, then \(\xi\), at the very first, even the slightest disturbance, will begin to increase rapidly; the angle of phase difference of the electromotive forces, \(\gamma\), will at once reach the limit of stability, and further operation of the alternators connected in parallel will become impossible. If equation (60) has no positive real roots, but the real part of the complex roots is positive, then the same thing will occur, only not at once, but after a series of oscillations of increasing amplitude.

The parallel operation of two alternating-current machines will be possible only when either all the roots of equation (60) are real and negative, or when one of them is real and negative, and the other two are conjugate complex roots with negative real part.

From the theory of algebraic equations it is known that the cubic equation

\[ x^3 + p x^2 + q x + r = 0 \]

has no roots with positive real parts if

\[ pq > r . \]

For our case this condition takes the form

\[ (a_1 + a_2)\{a_1 a_2 - (b_1 - b_2)\} > a_1 b_2 - b_1 a_2 \]

or, after reduction,

\[ a_1 a_2 (a_1 + a_2) > a_1 b_1 - a_2 b_2 \ldots \tag{63} \]

This is the condition for the stability of the motion of two alternators connected in parallel. The asymmetry of the formula obtained depends on the fact that, as the origin for counting the angles \(\gamma\), we took the direction of the vector of one of the electromotive forces (in our case, that of the first alternator). If, as this origin, we had taken some other direction fixed in space, the formulas obtained would have been symmetrical, but much more complicated.

1. Of special interest is the particular case of two identical alternators, set in motion by identical engines, under the assumption

\[ E_1 = E_2, \quad \gamma = 0 . \]

This case is of interest as the one most often encountered in practice. The solution could have been obtained as a particular case of formula 63, but it is simpler to proceed directly from equations (58), where, on the basis of (48), we shall have

\[ a_1 = a_2 = a, \quad -b_1 = b_2 = b, \quad p_1 = p_2 = p; \quad S_{1k} = S_{2k} = S_k \]

Let us note that in this case the directions of the crank pins may also fail to coincide, forming between themselves an angle

\[ 2\psi = \frac{2k\pi}{p}, \]

where \(k\) is a positive integer less than \(p\). Then the equations take the form

\[ \frac{d x_1}{d\varphi} + a x_1 - b \xi = S_k \operatorname{Sin} \frac{k}{p}\varphi \]

\[ \frac{d x_2}{d\varphi} + a x_2 + b \xi = S_k \operatorname{Sin}\left(\frac{k}{p}\varphi + 2\psi\right) \]

Subtracting the first equation from the second, taking (47) into account, we have

\[ \frac{d^2 \xi}{d\varphi^2} + a \frac{d\xi}{d\varphi} + 2 b \xi = S_k \operatorname{Sin}\psi \operatorname{Cos}\left(\frac{k}{p}\varphi+\psi\right). \tag{64} \]

The complete integral of this equation is equal to the sum of the complete integral of this equation without the right-hand side and a particular solution of equation (64). The integral of the equation without the right-hand side is

\[ C_1 e^{\lambda_1 \varphi} + C_2 e^{\lambda_2 \varphi}, \]

where \(\lambda_1\), \(\lambda_2\) are the roots of the characteristic equation

\[ \lambda^2 + a\lambda + 2b = 0. \tag{65} \]

We shall seek the particular solution in the form

\[ \xi' = k_1 \operatorname{Sin}\left(\frac{k}{p}\varphi+\psi\right) + k_2 \operatorname{Cos}\left(\frac{k}{p}\varphi+\psi\right); \]

substituting this expression into equation (64) and equating to zero the coefficients of \(\operatorname{Sin}\left(\frac{k}{p}\varphi+\psi\right)\) and \(\operatorname{Cos}\left(\frac{k}{p}\varphi+\psi\right)\) in the resulting expressions, we shall have, for determining \(k_1\) and \(k_2\), two equations, solving which we find

\[ k_1 = \frac{ a \frac{k}{p} S_k \operatorname{Sin}\psi }{ 4b^2 + (a^2 - 4b)\frac{k^2}{p^2} + \frac{k^4}{p^4} }, \]

\[ k_2 = \frac{ \left(2b - \frac{k^2}{p^2}\right) S_k \operatorname{Sin}\psi }{ 4b^2 + (a^2 - 4b)\frac{k^2}{p^2} + \frac{k^4}{p^4} }; \]

therefore \(\xi'\) may be represented in the form

\[ \xi' = \sqrt{k_1^2 + k_2^2}\, \operatorname{Sin}\left(\frac{k}{p}\varphi+\psi+\beta\right) \]

where

\[ \operatorname{tg}\beta=\frac{\kappa_{2}}{\kappa_{1}}, \]

then the complete integral of equation (64) will be

\[ \xi=C_{1}e^{\lambda_{1}\varphi}+C_{2}e^{\lambda_{2}\varphi}+ \]

\[ +\sin\psi-\frac{S_{k}}{\sqrt{4b^{2}+\left(a^{2}-4b\right)\frac{\kappa^{2}}{p^{2}}+\frac{\kappa^{4}}{p^{4}}}} \sin\left(\frac{\kappa}{p}\varphi+\psi+\beta\right) \]

where

\[ \lambda_{1}=-\frac{a}{2}+\sqrt{\frac{a^{2}}{4}-2b},\qquad \lambda_{2}=-\frac{a}{2}-\sqrt{\frac{a^{2}}{4}-2b}. \]

Since \(a\) and \(b\) are always positive, equation (65) will never have positive real roots, and the motion will always be stable.

When

\[ \frac{a^{2}}{4}>2b, \]

both exponential functions in equation (66) have negative real exponents, and therefore represent a rapidly damping aperiodic motion. When

\[ \frac{a^{2}}{4}<2b \]

i.e., when the damping capacity of the dynamo-machines is not very great, the roots \(\lambda_{1}\) and \(\lambda_{2}\) will be conjugate complex quantities, and, bringing in this case expression (66) to real form and taking into account all terms of the expansion

of the tangential-pressure diagram, we shall have for the deviation \(\xi\) the expression

\[ \xi = Q e^{-\frac{a}{2}\varphi} \operatorname{Sin} \left( \sqrt{2b-\frac{a^{2}}{4}}\,\varphi+\sigma \right) + \]

\[ {}+ \operatorname{Sin}\psi \sum_{1}^{k} \frac{S_k}{ \sqrt{ 4b^{2} + \left(a^{2}-4b\right)\frac{k^{2}}{p^{2}} + \frac{k^{4}}{p^{4}} } } \operatorname{Sin} \left( \frac{k}{p}\psi+b+\beta \right), \ldots \tag{67} \]

where \(Q\) and \(\sigma\) are arbitrary constants, determined by the initial circumstances of the motion.

The summation of the terms included under the sign \(\sum\) in the right-hand side of equations (62) and (67) need not be carried out to infinity, since the values of the terms in the increasing order generally decrease rather rapidly. At which term one may stop will be indicated by the series of values of the coefficients \(S_1, S_2, \ldots\), which are easily found, provided that the diagram of the tangential pressures of the machines—the engines driving the alternators—is given.

To the oscillation represented by these sums there is added the natural oscillation of the aggregate of dynamo machines, represented by the product of a sinusoidal and an exponential function. Owing to the factor of the form \(e^{-m\varphi}\), where \(m>0\), this term tends to zero with time, and there remain only the oscillations depending on the irregularity of the running of the engine, whose periodicity is entirely independent of the periodicity of the natural oscillations of the alternators.

Condition (63) is necessary for the possibility of parallel operation of two alternators, but it is not sufficient. It is also necessary that

\[ \gamma+\xi\max. < \gamma\max. \tag{68} \]

where \(\gamma\) is the angle of phase difference of the electromotive forces of the alternators in the established motion, \(\xi\max.\) is the greatest deviation of the angle between the vectors of the electromotive forces, which

we shall obtain from equation (62) or (66), and \(\gamma \max.\) is the limiting angle of phase difference, obtained from the diagram of Fig. 3.

The greatest value \(\xi \max.\) is attained in those cases when the period of the natural oscillation of the system of two alternators coincides with the period of one of the predominant waves arising as a result of pulsation of the driving force.

We shall not dwell in detail on the investigation of the phenomenon of resonance or interference of the oscillations of alternators and driving machines, since this question has been fully clarified in the articles by Kapp,¹) Görges²) and Rosenberg³).

It remains only to draw attention to the form of equation (67), namely to the fact that \(\operatorname{Sin} x\) enters as a factor in the second term of the right-hand side of this equation. This means that if, with identical machines having the same excitation and switched in at the coincidence of phases of the electromotive forces, the directions of the crankpins of the engine machines coincide, then the term depending on the pulsation of the driving force becomes zero, and consequently the irregularity of the running of the engine machines (not speaking, of course, of accidental shocks) will have no influence on the change of the angle \(\gamma\). Hence one sees the importance of instruments making it possible, at the moment of switching in a new dynamo machine, to ascertain the coincidence of the directions of the crankpins of the machines which set the alternators in motion.⁴)

In concluding this chapter I shall once more point out the difference between the reasoning in the present article and in the cited articles by Görges and Rosenberg.

Görges considers two machines connected to a network, in which the voltage, in magnitude and phase, may be regarded as constant. Since under this assumption the sum of the works delivered to the network by both machines remains constant, then, during oscillation, the deviations of both machines from the position of equilibrium will be equal in magnitude and opposite in sign.

E. Rosenberg considers a machine rotating non-uniformly, in parallel connection with another electrically identical—

¹) Kapp. loco cit.
²) Görges. lococit.
³) Rosenberg E. T. Z. 1902, p. 425.
⁴) See description of P. Kovalev’s instrument in E. T. Z. 1900, 562.

identical with the first, equally excited, but uniformly rotating machine. In this case the voltage in the network can, of course, no longer be regarded as constant, but the character of its variation will be predicted by the proposition laid down as the basis: it will undergo oscillations equal to half the oscillations of the vector of electromotive force of the first machine.

In the proposed article no assumptions are made other than the general assumptions, already indicated in the article, of the sinusoidality of the excitation curves and the constancy of the armature reaction.

Note. Prof. Förpl¹) poses the question in the same general way in his article. However, he considers not the aggregate of two electrical machines, but only a certain mechanical analogy, abstracting from electrical phenomena and replacing the interaction of the machines by means of synchronizing currents with the tension of fictitious elastic connections. Förpl does not take into account the influence of the nonuniformity of the motion of the driving machines in his calculation.

IV. Influence of the Non-Sinusoidal Form of the Electromotive-Force Curve on the Parallel Operation of Alternators.

1. It may be said a priori that alternators with identical electromotive-force curves will be best adapted for parallel operation, since with different curves, even when the phases of the fundamental waves coincide, exchange currents corresponding to the higher-order waves of the electromotive forces will arise in the alternator circuit. In practice, however, we may encounter installations in which alternators of the most varied designs operate in parallel quite satisfactorily. As a characteristic example of such an installation one may point to the central city station in Zurich, where the old high-speed Kapp alternators with a flat ring armature, driven by turbines and giving a sharp electromotive-force curve, are connected in parallel with new large low-speed three-phase generators of the Oerlikon works, of the inductor type, driven by steam engines. These latter—

¹) Das Pendeln parallelgeschalteter Maschinen. E. T. Z. 1902, 59.

... the generators operate, when connected in parallel, as single-phase machines; for this purpose the two phases are connected in series, whereby the curve of electromotive forces becomes flat.

In the literature the question of the influence of the form of the curves of electromotive forces on the parallel operation of alternators has been touched upon rather little. I know only the article by A. Perot in C. R. 1900, “Sur l’accouplement des alternateurs au point de vue des harmoniques etc.” In that article the author argues as follows.

Whatever the forms of the curves of electromotive forces of the alternators may be, on the basis of Fourier’s theorem they can be represented in the form

\[ \begin{aligned} e_1 &= E_{10}\operatorname{Sin}\omega t + E_{12}\operatorname{Sin}(2\omega t+\varphi_{12}) + E_{13}\operatorname{Sin}(3\omega t+\varphi_{13})+\ldots,\\ e_2 &= E_{20}\operatorname{Sin}\omega t + E_{22}\operatorname{Sin}(2\omega t+\varphi_{22}) + E_{23}\operatorname{Sin}(3\omega t+\varphi_{23})+\ldots. \end{aligned} \tag{69} \]

As is evident from the written series, the phases of the fundamental electromotive forces are assumed to coincide, which obviously can always be achieved.

Let us now suppose that the fundamental electromotive force and all the higher harmonics will be identical in both machines except for one of order \(n\), existing only in the first machine. In the circuit formed by the two alternators there will then act the electromotive force \(E_n\operatorname{Sin}(n\omega t+\varphi_n)\), and if, as before, \(s_1, s_2\) are the self-induction reactances referred to the normal number of periods, then in the circuit under consideration there will circulate an exchange current which, if the ohmic resistance is neglected in comparison with the self-induction reactance, is represented by the formula

\[ \begin{gathered} i_n = J_n\operatorname{Cos}(n\omega t+\varphi_n),\\ \text{where}\qquad J_n=\frac{E_n}{n s_1+n s_2}. \end{gathered} \tag{70} \]

The simplification admitted introduces the less error the higher the order of the wave \(n\), since the ohmic resistance remains constant, while the self-induction reactance increases in proportion to the order of the wave.

This same reasoning is easily extended to the case where, in the series (69), several waves of the same order have different amplitudes or different phase angles: in this case the electromotive force in the alternator circuit corresponding to waves of the \(n\)-th order will be the difference of these waves—arithmetical, if the waves coincide in phase, and geometrical if their phases are different.

Further, the author of the article cited attempts to investigate the influence of the difference in the forms of the curves of the electromotive forces of alternators connected in parallel upon the form of the terminal-voltage curve; however, he does not fully clarify this question. We shall not dwell on this, but shall apply to practical conditions the considerations presented concerning the influence of the dissimilarity in form of the electromotive forces of both alternators upon the occurrence of circulating currents.

When a new alternator is connected to one already operating in parallel, it is brought into phase by means of phase lamps or by some other device of the same purpose, and the excitation is brought to such a degree that the electromotive forces of both alternators are equal, as judged from the readings of a voltmeter; but the voltmeter gives the effective electromotive forces, and on the busbars the effective voltages must be equalized, i.e., when alternators are connected in parallel there must exist the equality

\[ \frac{1}{T}\int_{0}^{T}\left[\sum_{1}^{\infty} E_{1k}\operatorname{Sin}(n\omega t+\varphi_{1n})\right]^{2}\,dt = \frac{1}{T}\int_{0}^{T}\left[\sum_{1}^{\infty} E_{2k}\operatorname{Sin}(n\omega t+\varphi_{2n})\right]^{2}\,dt \tag{70} \]

Since the amplitudes of the electromotive forces of all orders change in one and the same ratio when the field excitation is changed, each electromotive force may be represented as

\[ e=E\{\operatorname{Sin}\omega t+\alpha_{2}(\operatorname{Sin}2\omega t+\varphi_{2})+\ldots\} \]

where \(E\) is the amplitude of the fundamental wave, and \(\alpha_{2}, \alpha_{3}\ldots\) are proper fractions.

Then each of the integrals of equality (70) will become

\[ \mathrm{E}^2 \int_0^T \left\{\sum \alpha_n \operatorname{Sin}(n\omega t+\varphi_n)\right\}^2 dt = \sigma^2 \mathrm{E}^2 \]

where \(\sigma\) is a certain number, independent of the excitation, but dependent, owing to the deformation of the field under load, on this latter. Therefore, if in practice we wish to find the influence of the form of the electromotive-force curve on the equalizing currents in a given unit of machines, we must use the curve taken at normal load; for this purpose one may use an auxiliary winding laid parallel to the working winding.

The coefficient \(\sigma\) may be represented by a very simple formula in the following way:

\[ \sigma^2 = \int_0^T \left(\sum \alpha_n \operatorname{Sin}(n\omega t+\varphi_n)\right)^2 dt = \sum \alpha_n^2 \int_0^T \operatorname{Sin}^2(n\omega t+\varphi_n)\,dt \]

\[ + \sum \alpha_k \alpha_i \int_0^T \operatorname{Sin}(k\omega t+\varphi_k)\operatorname{Sin}(i\omega t+\varphi_i)\,dt; \]

but

\[ \int_0^T \operatorname{Sin}^2(n\omega t+\varphi_n)\,dt = \frac{1}{\omega} \int_0^{2\pi} \operatorname{Sin}^2(n\alpha+\varphi_i)\,d\alpha = \frac{\pi}{\omega} \]

\[ \int_0^T \operatorname{Sin}(k\omega t+\varphi_n)\operatorname{Sin}(i\omega t+\varphi_i)\,dt = 0 \]

and equality (69), after reduction, will take the form

\[ \mathrm{E}_1^2 \sum \alpha_{1n}^{\,2} = \mathrm{E}_2^2 \sum \alpha_{2n}^{\,2} \ldots\ldots \tag{71} \]

With the aid of equation 70, knowing the curves of the electromotive forces of both alternators and resolving them into elementary sinusoids, we easily compute the sums entering into equation (71), and, having determined the ratio \(\dfrac{\mathrm{E}_1}{\mathrm{E}_2}\), we shall find all the electromotive forces acting in the circuit of the alternators, and the equalizing currents from formula (69), for the case when the phases of the electromotive forces coincide. This will be the most favorable case, and the obtained in this way

Thus the values of the exchange currents will be minimal. In the event of a disturbance of the motion and the occurrence of oscillations, or in the event of a prolonged mismatch of phases, these values will increase. The number of terms under the summation signs is not very great, since the terms of even orders almost always vanish (the only exceptions are certain types of machines, very little used in practice, with salient-pole armatures or with very wide teeth of the armature core, as well as the newest machines of the inductor type), and since the highest order of practically noticeable waves entering into the composition of the electromotive forces rarely reaches 13, while in most cases one may stop at 7, 5, and even at 3.

Note. In the above example of the Zurich central station we have the case of a connection of machines that give a symmetrical pointed curve, having no even waves, with machines of the inductor type, whose flat unsymmetrical curve of electromotive forces contains waves of the 2nd and 4th orders.

1. Let us explain our reasoning by a particular example.

Example. Suppose we have two alternators with the same ohmic resistance and the same self-induction reactance,

\(r = 0.6,\quad s = 3.6\)

(\(s\) referred to the normal number of periods), and giving one flat curve of electromotive forces (Fig. 7a), represented by the equation

\(e = E_1[\operatorname{Sin}\varphi + 0.225\operatorname{Sin}3\varphi + 0.05\operatorname{Sin}5\varphi]\),

and another, pointed curve (Fig. 7b), represented by the equation

\[ e_2 = E_2[\operatorname{Sin}\varphi - 0.15\operatorname{Sin}3\varphi + 0.10\operatorname{Sin}5\varphi] \]

The equations of the curves of electromotive forces in this example are from the book by Steinmetz, Th. u. Ber. d. Wechs., § 223, where they are given as a characteristic example of sharply expressed flat and pointed curves.

The sums entering into equation (70) will be

\[ 1 + 0.225^2 + 0.05^2 = 1.0531 \]

and

\[ 1 + 0.15^2 + 0.10^2 = 1.0325 \]

whence

\[ \frac{E_1}{E_2} = \sqrt{\frac{1{,}0531}{1{,}0325}} =1{,}02 \]

Let, further, the electromotive force of the armature (effective) be 2500, and the normal current of each machine 75 amperes. As is seen from the numbers obtained, we shall not make a large error by taking the amplitudes of the fundamental waves as

\[ E_1=\sqrt{2}\cdot 2500 \]

\[ E_2=\sqrt{2}\cdot 2475 \]

Then the effective values of the current corresponding to the waves of different orders will be

\[ J_1=\frac{2500-2475}{2\sqrt{0{,}6^2+3{,}6^2}}=3{,}5\ \text{amperes} \]

\[ J_3=\frac{0{,}225\cdot 2500+0{,}15\cdot 2475}{2\cdot 3\cdot 3{,}6}=43{,}3 \]

\[ J_5=\frac{0{,}05\cdot 2500-0{,}1\cdot 2475}{2\cdot 5\cdot 3{,}6}=-4{,}8 \]

The instantaneous value of the exchange current will be

\[ i=\sqrt{2}\left[ 3{,}5\sin(\omega t-80^\circ\,2'15'') +43{,}3\sin(3\omega t-86^\circ49'10'') \right. \]

\[ \left. -4{,}8\sin(5\omega t-88^\circ5'25'') \right]. \]

As is seen from this expression, the angles of phase difference of the currents of the higher orders are close to \(\frac{\pi}{2}\), and therefore these currents serve little for the transfer of work from one alternator to the other, but only uselessly heat the armature; moreover, some terms (in our example the second) may acquire values of the same order as the working current.

In concluding the article, I shall point out one method of verifying the synchronism of alternators being connected in parallel. This method was proposed by Ayrton as early as the beginning of the 1890s, but has not found practical application, although, it seems to me, it has certain advantages over the methods now in use.*)

The connection diagram proposed by Ayrton is shown in Fig. 8. \(A_2, A_2\)—alternators, \(E\)—an electroscope provided on the outside with a metallic coating. This coating and the ball of the electroscope are connected with terminals of the alternators of the same name (taking into account the direction of rotation and the arrangement of the magnets). The other two terminals are connected to one another through a very large resistance \(R\) (a fairly narrow strip of paper lightly coated with graphite).

Obviously, when the phases of the electromotive forces coincide, the leaves of the electroscope and its outer coating will have, at every instant, one and the same potential.

When the phases do not coincide, the potential of the coating and of the leaves will not be the same; the latter will receive a certain charge and will diverge.

Such an instrument, especially for high-voltage machines, is much more sensitive than phase lamps.

At the Tomsk electric station, with the kind assistance of the station superintendent I. P. Belozerov, I carried out experiments on the application of the method described. The voltage on the station busbars was 2000 volts. The voltage of the phase lamps was 100 volts.

At 30 volts of voltage at the lamp terminals the lamps no longer glow, and consequently the effective difference of the electromotive forces of the two alternators at the moment of switching on may reach a magnitude of 600 volts.

In the experiments with the electroscope, the latter still gave a noticeable deflection of the leaves at a potential difference of 100 volt, and consequently the accuracy of coincidence of the phases was 6 times greater than with the ordinary method of switching on with phase

*) For a description of modern instruments indicating synchronism, see the article by W. Ritter, E. T. Z. 1900, 7.

lamps. This method requires constructive improvement, since the ordinary electroscope with aluminum leaves that was used proved to be too delicate. The leaves soon broke.

It must be said, however, that the use of the electroscope for the indicated purposes is possible only with identical curves of electromotive forces. From what was said in §§ 16 and 17 it is clear that, with different curves, equality of the potentials of the plates and leaves will not occur even when the phases of the fundamental waves coincide.

A. Potebnya.

Fig. 1.

(diagram)

Fig. 2.

(diagram)

Fig. 7a.

\[ \sin \beta - 0.225 \sin(3\beta - \pi) - 0.05 \sin(5\beta - \pi) \]

Fig. 7b.

\[ \sin \beta - 0.15 \sin 3\beta - 0.10 \sin(5\beta - \pi) \]

Fig. 8.

1. “La. Lum. El.” Vols. 45 and 46. ↩

2. “La. Lum El.” Vol. 46, p. 601. ↩

3. “E. T. Z.” 1899, p. 134. ↩

4. “E. T. Z.” 1900, p. 188. ↩

5. “E. T. Z.” 1899, p. 728. ↩

6. “E. T. Z.” 1901, p. 905. ↩