V. M. Khrushchev.

Theory of Repulsion Motors.

PART II¹.

In this part, which had earlier been intended in the form of two separate articles, the author confines himself only to the study of the Thomson motor, considering that the application of the conclusions to other types of repulsion motors, after what was said in the first part, presents no difficulty.

1. Commutation.

The question of commutation is one of the most difficult questions in electrical engineering. If with direct current one cannot speak of a successful solution of the question in a quantitative respect, then all the more can there be no question of it with alternating current—where it is complicated by the phenomenon of static induction. Therefore, abandoning in advance any attempt at a quantitative solution of the question, the author here confines himself chiefly only to an investigation, in a qualitative respect, of the influence of the currents of the short-circuited coil on the operation of the motor, insofar as this is permitted by the point of view on the action of the motor adopted by him.

a) Electromotive forces of the short-circuited coil.

The short-circuited coil of a Thomson motor is penetrated by two magnetic fluxes, $\Phi_{no}$ and $\Phi_{np}$, as a result of which, during rotation of the motor, four electromotive forces are induced in it:

1) The electromotive force due to the pulsation of the longitudinal flux

\[ E'_{no} = 4{,}44 \cdot \sim_1 \cdot w_k \cdot 2l \cdot B_{no} \cdot \frac{D}{2} \cdot 10^{-3} \cdot \int_{\frac{\pi}{2}-\lambda}^{\frac{\pi}{2}} \sin \alpha \cdot d\alpha = \]

\[ = 4{,}44 \cdot \sim_1 \cdot D \cdot l \cdot w_k \cdot B_{no} \cdot \sin \lambda \cdot 10^{-8}, \ldots\ldots (1) \]

lagging in phase by an angle $\frac{\pi}{2}$ behind the longitudinal flux $\Phi_{no}$;

2) Electromotive force of rotation in the longitudinal field

\[ E''_{n\delta} = \frac{1}{\sqrt{2}}\cdot B_{n\delta}\cdot \cos\lambda \cdot w_k \cdot 2l \cdot \pi \cdot D \cdot \infty_2 \cdot 10^{-8} = \]

\[ = 4{,}44 \cdot \infty_2 \cdot D \cdot l \cdot w_k \cdot B_{n\delta}\cdot \cos\lambda \cdot 10^{-8}, \ldots\ldots (2) \]

in phase with the longitudinal field;

3) Electromotive force from the pulsation of the transverse flux

\[ E'_{np} = 4{,}44 \cdot \infty_1 \cdot D \cdot l \cdot w_k \cdot B_{np}\cdot \cos\lambda \cdot 10^{-8}, \ldots\ldots (3) \]

lagging in phase by an angle \(\frac{\pi}{2}\) behind the transverse flux \(\Phi_{np}\).

4) Electromotive force of rotation in the transverse field

\[ E''_{np} = 4{,}44 \cdot \infty_2 \cdot D \cdot l \cdot w_k \cdot B_{np}\cdot \sin\lambda \cdot 10^{-8}, \ldots\ldots (4) \]

in phase with the transverse field.

Passing to the diagram of these electromotive forces, it is necessary to note that the short-circuited coil is coupled electromagnetically both with the longitudinal and with the transverse rotor coils; therefore, in order to establish the phases of the electromotive forces of the short-circuited coil, it is necessary to consider the sequence of actions of the longitudinal and transverse fluxes in these couplings with it.

Let \(O\Phi_{n\delta}\) (Fig. 1) represent the vector of the longitudinal flux; then the vector of the electromotive force induced in the short-circuited coil by the pulsation of this flux will be \(OE'_{n\delta}\), lagging by an angle \(\frac{\pi}{2}\) behind \(O\Phi_{n\delta}\). By simple reasoning we can come to the conclusion that the vector of the electromotive force in this coil from the pulsation of the transverse flux—\(OE'_{np}\)—leads in phase by an angle \(\frac{\pi}{2}\) the vector of the secondary current \(OJ_2\). Let us consider the case when \(\angle r = 0\) and the angle between the secondary and magnetizing currents is equal to \(\pi\); as a result of the fact that, under these conditions, the transverse coil assists the rotation, the transverse and longitudinal fluxes are arranged in space as shown in Fig. 2. Since they both thread the short-circuited coil from the front side to the rear side and pulsate simultaneously, the electromotive forces induced in this coil by their pulsation will be directed in one direction and in phase. Above we saw that the vector \(OE'_{n\delta}\) lags in phase

Fig. 1.

at an angle \(\dfrac{\pi}{2}\) from \(O\Phi_{no}\); consequently the vector \(OE'_{np}\) must be laid off at an angle \(\dfrac{\pi}{2}\) to the vector of the secondary current \(OJ_2\), in the direction of advance. Applying the usual rules for determining the direction of the electromotive forces of rotation, it is not difficult to see that \(E''_{no}\) tends to create a flux opposite to \(\Phi_{no}\), while \(E'''_{np}\)—acting in the same direction as \(\Phi_{np}\); therefore the vector \(OE''_{no}\) must be laid off at an angle \(\pi\) to \(O\Phi_{no}\), and the vector \(OE'''_{np}\) at an angle \(\pi\) to \(OJ_2\). Fig. 3 presents the diagram of the electromotive forces induced in the short-circuited coil of a repulsion motor for a speed less than synchronous and for a small brush-shift angle; the resultant of all these electromotive forces is represented by the vector \(OE_p\).

Fig. 2.      Fig. 3.

Let us consider a motor devoid of secondary resistance, armature reaction, and phase shift between the magnetizing current and the magnetic flux; the voltage diagram of the rotor of such a motor will consist of four vectors representing the electromotive forces \(E_{nc.1}\), \(E_{vr.2}\), \(E_{nc.2}\), \(E_{vr.1}\) (see p. 7 of Part I), which, as is seen from the diagram, are connected by the following equation:

\[ E^2_{nc.1} + E^2_{vr.2} = E^2_{nc.2} + E^2_{vr.1}, \]

Substituting the values \(E_{nc.1}\), \(E_{vr.2}\), \(E_{nc.2}\), and \(E_{vr.1}\), we obtain:

\[ 4{,}44 \cdot \infty_1 \cdot D^2 \cdot l \cdot \frac{w_2}{\tau} \cdot 10^{-8} \cdot B_{no} \cdot \sqrt{\cos^2 \lambda + v^2 \sin^2 \lambda} = \]

\[ = 4{,}44 \cdot \infty_1 \cdot D^2 \cdot l \cdot \frac{w_2}{\tau} \cdot 10^{-8} \cdot B_{np} \cdot \sqrt{\sin^2 \lambda + v^2 \cos^2 \lambda}, \]

whence

\[ B_{np} = B_{no}\sqrt{\frac{\cos^2 \lambda + v^2 \sin^2 \lambda}{\sin^2 \lambda + v^2 \cos^2 \lambda}} \ . . . . . . . \ (5) \]

This formula shows that at low speeds and small brush-shift angles \(B_{np}\) may be considerably greater (even several times) than \(B_{no}\); thus the induction in the iron in repulsion motors varies within wide limits and may reach very large values, which of course has a great

affects the operation of the motor and is one of the reasons for the discrepancy between the theoretical characteristics and those obtained experimentally.

At synchronism, from formula 5 we have:

\[ B_{pr}=B_{nd}; \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (6) \]

since then

\[ E_{ns.1}=E_{vr.1}, \quad \text{and} \quad E_{ns.2}=E_{vr.2}, \]

the vector of the transverse flux lags by an angle \(\frac{\pi}{2}\) behind the longitudinal one; consequently, for a motor devoid of secondary resistance and leakage reaction, at synchronism a circular rotating field is obtained. Applying condition 6) to the expressions of the electromotive forces induced in the short-circuited coil, we obtain the equalities:

\[ E'_{nd}=E''_{pr} \quad \text{and} \quad E'_{pr}=E''_{nd}, \]

and the diagram of these electromotive forces is represented by four vectors (Fig. 4, dotted lines) \(OE''_{pr}\), \(OE'_{nd}\), \(OE'_{pr}\), and \(OE''_{nd}\), pairwise equal and directed in opposite directions, so that their resultant is equal to zero and, consequently, commutation takes place without a commutating electromotive force.

In the presence in the motor of a phase displacement between the magnetizing current and the magnetic flux, of secondary resistances and of leakage reaction, \(B_{pr}\) at synchronism is less than \(B_{nd}\), and the vector of the transverse flux lags behind the longitudinal one in phase by an angle greater than \(\frac{\pi}{2}\), as a result of which the rotating field obtained is no longer circular, but elliptical. The diagram of the electromotive forces of the short-circuited coil for this case is shown in Fig. 4 (solid lines); their resultant is no longer equal to zero, but is represented by the vector \(OR\).

Fig. 4.

However, this diagram too is far from reality; the point is that precisely in the commutation zone the form of the field may deviate considerably from a sinusoid, and therefore the electromotive forces of the short-circuited coil may have values very different from those calculated by formulas 1—4.

Using formula 2a of Part I, we can express the induction in

zone of commutation, with a sinusoidal distribution of the field \(B\), through the induction under a trapezoidal \(B_m\), as follows:

\[ B = B_c \cdot \sin(90-\alpha) = \frac{\pi^2 \beta_4\left(1-\frac{2}{3}\beta_4\right)}{4\cos\alpha} \cdot B_m \cdot \sin(90-\alpha) = \]

\[ = \frac{\pi}{4}\cdot \beta_4\left(1-\frac{2}{3}\beta_4\right)B_m, \]

where \(\alpha\) is the angle of displacement of the brushes from the field amplitude. The expression standing before \(B_m\), for large values of \(\alpha\), is much less than unity; therefore the electromotive force of rotation, calculated on the assumption of a trapezoidal field \(E''_m\), is considerably greater than that for the sinusoidal field \(E''_c\); thus, for a transverse field, when

\[ \lambda = 18^\circ \;(\alpha = 72^\circ)\quad E''_m = 2.34\,E''_c. \]

For small values of \(\alpha\), however, \(E''_m\) differs little from \(E''_c\), for example when

\[ \alpha = 18^\circ\quad E''_m = 1.08\,E''_c. \]

As for the electromotive forces of pulsation, they, as could be expected in advance, differ little from one another for either distribution of the fields; thus, when

\[ \lambda = 18^\circ\quad E'_c = 1.02\,E'_m. \]

Increasing in Fig. 4 the vector \(OE''_{np}\) by a factor of 2.34, we obtain a diagram with the resultant vector \(OE_p\), now directed in an entirely different direction. It must be noted, however, that this diagram too represents only a rather rough approximation to reality, since the vector \(OE''_{nd}\), both in phase and in magnitude, also differs from that shown in Fig. 4; but in view of the fact that the complex form of the longitudinal field presents great difficulties for a more exact determination of this vector, and also because our aim is only the qualitative aspect of the phenomenon, on which comparatively small changes in the phase and magnitude of the vector \(OE''_{nd}\) have no essential influence, we shall not enter into the details of their determination, as having no direct bearing on our purpose.

When \(\lambda = 0^\circ\) (at synchronism), only one electromotive force of rotation is induced in the short-circuited coil in the longitudinal field, \(E''_0\), while when \(\lambda = 90^\circ\) there is only the electromotive force of pulsation \(E'_{90}\), the two being equal in magnitude:

\[ E''_0 = E'_{90} = 4.44 \cdot c_1 \cdot D \cdot l \cdot w_k \cdot B_{nd} \cdot 10^{-8}. \]

V. M. Khrushchov.

Both \(E''_0\) and \(E'_{90}\) make angles with the current smaller than \(90^\circ\).
For \(\lambda = 10^\circ — 50^\circ\), the resulting electromotive forces acting are, in magnitude, considerably smaller than \(E''_0\) and \(E''_{90}\), and in phase lag behind the rotor current by angles greater than \(90^\circ\).

With the rotor stationary, again taking as equal to zero the secondary resistance, the leakage reactance, and the angle \(\gamma\), from formula 5 we have:

\[ B_{np} = B_{nd}\cdot \operatorname{ctg}\lambda, \]

since \(E_{br.1}\) and \(E_{br.2}\) are equal to zero, the transverse flux \(\Phi_{np}\) lags in phase by the angle \(\pi\) behind the longitudinal \(\Phi_{nd}\). Consequently, in the short-circuited coil there will act only two electromotive forces, \(E'_{nd}\) and \(E'_{np}\) (Fig. 5, dotted lines), coinciding in phase and giving the resultant:

\[ E_{p.k} = 4{,}44 \cdot \omega_1 \cdot D \cdot l \cdot w_k \cdot B_{nd} \cdot \cos c\lambda \cdot 10^{-8}. \]

In the presence of secondary resistance and leakage reactance,

\[ B_{np} < B_{nd}\operatorname{ctg}\lambda, \]

the vector \(O\Phi_{np}\) lags behind the vector \(O\Phi_{nd}\) already by an angle \(\alpha\) smaller than \(180^\circ\) (Fig. 5); therefore the vector \(OE'_{np}\) leads \(OE'_{nd}\) by the angle \(\pi-\alpha\); their resultant is represented by the vector \(OE_{p.k}\).

Fig. 5.

Fig. 6.

With an increase from \(0^\circ\) of the shift angle of the brushes, \(E'_{nd}\) increases, while \(E'_{np}\) at first increases, passes through a maximum, and then decreases; as a result, their resultant \(E_{p.k}\) also has a maximum. Curve I of Fig. 6 represents \(E_{p.k}\) as a function of \(\lambda\); from this curve it is seen that \(E_{p.k}\) reaches a maximum at a small angle

displacing the brushes \((\lambda = 18^\circ—19^\circ)\), so that starting the motor at these angles is inconvenient with respect to commutation.

At an infinitely high speed the secondary current is equal, in phase and in magnitude, to the magnetizing current; the longitudinal and transverse fluxes coincide in phase, and \(B_{np}\) from formula 5 is equal to

\[ B_{np} = B_{nb} \cdot tg\,\lambda . \]

The electromotive forces induced in the short-circuited coil by the pulsation of the longitudinal and transverse fluxes \(E'_{nb}\) and \(E'_{np}\) (Fig. 7) are directed in opposite directions and are equal in magnitude, so that they cancel each other. The electromotive forces due to rotation in the longitudinal and transverse fields are directed in one direction and give the resultant:

Fig. 7.

Fig. 8.

\[ E_p = 4{,}44 \cdot \infty_2 \cdot D \cdot l \cdot w_k \cdot B_{np} \cdot sec\,\lambda \cdot 10^{-8}, \]

which is infinitely large.

With decreasing speed the resultant electromotive force decreases, reaches a minimum at a speed slightly less than synchronous, and again increases with further decrease of the speed; Fig. 8 represents \(E_p\) as a function of speed.

c) The current of the short-circuited coil and its influence on the primary circuit.

The resultant electromotive force \(E_p\) produces in the short-circuited coil a current \(i_p\), which at the beginning of the short circuit is zero, then increases, passes through a maximum, and then begins to decrease under the influence of the ever-increasing resistance of the circuit; since this maximum will be the greater, the greater at the given moment is the instantaneous value of the resultant electromotive force, it follows that the current of the short-circuited coil \(i_p\) is in phase with it. The current \(i_p\) is superposed on the commu-

the current of the coil being commutated, \(i_k\); if the resultant electromotive force is in phase with the current of that part of the rotor from which the coil passes over (the transverse coil), then the current \(i_p\) delays commutation, and therefore, as is known, exerts a demagnetizing action on the primary circuit; if the resultant electromotive force is opposite in phase to the current being commutated, then the current \(i_p\) assists commutation and consequently exerts a magnetizing action on the primary circuit. If, finally, it leads or lags in phase behind the current being commutated by an angle \(\frac{\pi}{2}\), then the current \(i_p\) produces over-commutation in some parts of the period and under-commutation in others. Since the short-circuited coil passes from the transverse into the longitudinal position, and the transverse current is opposite in phase to the longitudinal one, then, when the phase of the vector of the resultant electromotive force and that of the secondary-current vector differ by an angle \(\pi\), the current of the short-circuited coil produces in the primary circuit an apparent decrease of the secondary-current vector; when their phases coincide, an apparent increase; and when the phase of \(OE_p\) leads or lags behind \(OJ_2\) by an angle \(\frac{\pi}{2}\), the current \(i_p\) produces in the primary circuit an apparent leading or lagging of the secondary-current vector. Thus, in reality, the vector of the primary current is the geometric sum not of two vectors, but of three: the magnetizing current, the secondary current, and the current of the short-circuited coil.

We see that at infinitely great speed \(E_p\) is infinitely great; however, the current of the short-circuited coil \(i_p\) in this case has a finite value, since its apparent resistance depends on speed. Thus

\[ i_p = \frac{4.44 \cdot D \cdot l \cdot w_k \cdot B_m \cdot \sec \lambda \cdot 10^{-8}} {k \cdot L_k} \]

where \(k\) is a certain constant, and \(L_k\) is the coefficient of leakage induction of the coil. This current of the short-circuited coil differs in phase from the longitudinal flux by an angle \(\pi\), and from the secondary current by an angle somewhat greater than \(\pi\); consequently it counteracts commutation and produces an apparent decrease of the secondary current, i.e. causes the appearance of a wattless primary current.

With decreasing speed, the resultant electromotive force, as well as the angle \(\mu\) between it and the secondary current (see Fig. 9), decrease; but since \(\pi > \mu > \frac{\pi}{2}\), the current \(i_p\), on the one hand, counteracts commutation and therefore reduces the primary current, the primary and secondary powers, and the torque, while on the other hand—increas-

...causes a shift of the primary current away from the voltage. With a further decrease in speed, the resultant electromotive force continues to decrease, while the angle \(\mu\) begins again to increase; at a speed slightly below synchronous it is equal to \(\pi\), and \(E_p\) reaches a minimum; at a speed of approximately \(0.75^{1}\) of synchronous, \(\mu = \dfrac{3\pi}{2}\), and then \(\mu > \dfrac{3\pi}{2}\). Correspondingly, up to a speed slightly below synchronous, the current \(i_p\) decreases the power factor; after that it increases it and, up to a speed \(\sim 0.75\) of synchronous, decreases the primary current, power, and torque, while at a lower speed it increases them.

Fig. 9. Geometrical locus (I) of the end of the vector of the resultant electromotive force \(E_p\) with variation of the rotational speed.

Of interest is the influence exerted on the reaction of the short-circuited coil by the form of the transverse field. With a sinusoidal field, as we have seen, at synchronism \(E_p\) makes with \(J_2\) an angle less than \(\dfrac{\pi}{2}\); consequently it assists commutation and increases the motor current, so that the speed at which the generator action of the short-circuited coil changes into motor action is above synchronous. With a trapezoidal field it is already considerably below synchronous. Thus, the strong field of the transverse coil, which creates in the commutation zone a trapezoidal field, greatly worsens the power factor and commutation of repulsion motors. It seems to the author that, for their improvement, the most correct means would be to place on the stator a special compensating winding carrying the rotor current, or simply a short-

\(^{1}\) For the given motor.

closed, which would weaken, in the commutation zone, the field of the transverse coil.

With the rotor stationary, the current of the short-circuited coil is equal to

\[ i_{p.k}=\frac{E_{p.k}}{\sqrt{r_k^2+(2\pi\cdot\infty_1\cdot L_k)^2}}, \]

and in phase it lags behind \(E_{p.k}\) (see Fig. 5) by the angle

\[ \rho=\operatorname{arctg}\frac{2\pi\cdot\infty_1\cdot L_k}{r_k}. \]

As a result, the reaction of the current \(i_{p.k}\) in the primary circuit is expressed by an increase of the primary current and by a decrease of its displacement from the voltage, whereby the power and torque increase.

2. Construction of working diagrams from experimental data.

Let us turn to the diagram of successive inversions (see Part I, Fig. 12, Table I); the angle \(\varepsilon\) between the straight line \(K'p\), representing the geometrical locus of the vectors of total apparent resistances, and the axis of ordinates \(AY\), is expressed by formula 19 of Part I; if in it we neglect the terms \(x'_2\sin 2\gamma\) in the numerator and \(r'_2\sin 2\gamma\) in the denominator, then we obtain the following expression:

\[ \varepsilon = \operatorname{arctg} \frac{ x'_3\left(1+\frac{c_2}{c_1}\right)\cos\gamma+x'_2 }{ x'_3\left(1+\frac{c_2}{c_1}\right)\sin\gamma-r'_2 } = \operatorname{arctg} \frac{(x'_3+x'_4)\cos\gamma+x'_2}{(x'_3+x'_4)\sin\gamma-r'_2} \ldots (7) \]

but

\[ x'_3=\frac{c_1}{c_2}x'_4=x'_4\tg^2\lambda \quad\text{and}\quad x'_3+x'_4=\frac{x'_4}{\cos^2\lambda}. \]

The reaction of the longitudinal coil, reduced to the primary circuit, is \(-x'_{n0}\), and consequently also \(x'_4=x'_{n0}\cos\gamma\), and does not depend on the angle of brush displacement; whereas the secondary reaction of the stray field and the secondary resistance, reduced to the primary circuit, when the angle of brush displacement changes, change with the transformation coefficient. If by \(x''_2\) and \(r''_2\) we denote the values of \(x'_2\) and \(r'_2\) at the angle of brush displacement \(\lambda=0\), then, for \(\lambda\) different from zero,

\[ x'_2=\frac{x''_2}{\cos^2\lambda} \quad\text{and}\quad r'_2=\frac{r''_2}{\cos^2\lambda}. \]

Thus, substituting into formula 7 the values obtained for \((x'_3+x'_4)\), \(x'_2\) and \(r'_2\), and reducing by \(\cos^2\lambda\), we obtain

\[ \varepsilon=\operatorname{arctg}\frac{x'_4\cos\gamma+x''_2}{x'_4\sin\gamma-r''_2}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ (7a) \]

an expression independent of the angle of brush displacement. Consequently, the straight lines of the total apparent resistances for all angles of brush displacement make one and the same angle with the axis of ordinates, and the centers of the circles of total conductivities of the repulsion motor, for all angles of brush displacement, lie on one straight line.

Fig. 10.

We shall now obtain the geometric locus along which an arbitrary point \(T'\) of the line of total apparent resistances \(K'_p\) moves when \(\lambda\) varies (Fig. 10). Let us take a new system of coordinates with origin at the point \(A'\) and with the axis of abscissas \(A'S'\), where \(A'S'\) is the diameter from \(A'\) of the circle \(K''_p\) (Fig. 12, Part I). From the right triangle \(A'T'D\),

\[ (T'D)^2=(A'T')^2-(A'D)^2, \]

substituting into this expression the value of \((T'A')^2\) (see Part I, p. 25),

\[ (T'A')^2=\frac{1}{(T''A')^2} =\frac{Z'^2_{2v}\,x'^2_4}{x'^2_4+Z'^2_{2v}+2x'_4\left(X'_{2v}\cos\gamma+R'_{2v}\sin\gamma\right)} =\frac{Z'^2_{2v}x''^2_4}{4\rho'''^2} \]

and the value of \((A'D)\) (see Part I, p. 23, formula 18),

\[ A'D=\frac{x'_4\left(x'_3+x'_2\cos\gamma+r'_2\sin\gamma\right)}{2\rho'''}, \]

After several transformations we obtain

\[ DT'=\frac{x'_4\left(x'_3 c_2 v+r'_2\cos\gamma-x'_2\sin\gamma\right)}{2x_{\mathrm{III}}}, \]

and, after reducing the numerator and denominator by \(\frac{1}{\cos^2\lambda}\) and substituting the value of \(c_2\),

\[ DT'=\frac{x'_4\left(vx'_4\sin2\lambda+r''_2\cos\gamma-x''_2\sin\gamma\right)} {2\left(x'_4+x''_2\right)};\ .\ .\ .\ .\ (8) \]

for \(v=0\)

\[ DT'_{v=0}=DK'=\frac{x'_4\left(r''_2\cos\gamma-x''_2\sin\gamma\right)} {2\left(x'_4+x''_2\right)}, \]

an expression independent of the angle of displacement of the brushes. Thus, the geometrical locus of the end of the vector of the total resistance of the motor at \(v=0\), as the angle of displacement of the brushes varies, is a straight line \(K'p_k\), parallel to \(A'S\) and separated from it by the distance \(DK'\), determined by expression (8).

Reducing in the expression \(A'D\) the numerator and denominator by \(\frac{1}{\cos^2\lambda}\), we obtain

\[ A'D=\frac{x'_4\left(x'_4\sin^2\lambda+x''_2\cos\gamma+r''_2\sin\gamma\right)} {2\left(x'_4+x''_2\right)}; \]

for \(\lambda=90^\circ\)

\[ A'D_{\lambda=90^\circ}= \frac{x'_4\left(x'_4+x''_2\cos\gamma+r''_2\sin\gamma\right)} {2\left(x'_4+x''_2\right)} \simeq \frac{x'_4}{2}, \]

for \(\lambda=0^\circ\)

\[ A'D_{\lambda=0^\circ}= \frac{x'_4\left(x''_2\cos\gamma+r''_2\sin\gamma\right)} {2\left(x'_4+x''_2\right)}. \]

These values \(A'D_{\lambda=0^\circ}\) and \(A'D_{\lambda=90^\circ}\) determine the position of the extreme points \(L\) and \(M\) of the above-mentioned geometrical locus.

Let us pass to a new system of coordinates with origin at the point \(L\) and with the axis of abscissas along the line \(LM\) on \(DT'_{v=0}\); the coordinates of the point \(T'\) in this system are represented by:

\[ \xi=\frac{x_4'^2\sin^2\lambda}{2\left(x'_4+x''_2\right)} \quad \text{and} \quad \eta=\frac{v x_4'^2\sin2\lambda}{2\left(x'_4+x''_2\right)};\ .\ .\ .\ .\ .\ (9) \]

for \(\lambda=0\), \(\xi=0\) and \(\eta=0\)

for

\[ \lambda=90^\circ,\quad \xi=\frac{x_4'^2}{2\left(x'_4+x''_2\right)}, \]

and \(\eta=0\), independently of \(v\).

Excluding from the expressions for the coordinates of the point \(T'\) the angle \(\lambda\), we obtain the equation

\[ \eta^{2}+v^{2}\xi^{2}=\frac{v^{2}x_{4}^{\prime\,2}}{2\left(x'_4+x''_2\right)}\,\xi,\ldots\ldots\ldots(10) \]

which represents the equation of an ellipse, with diameter \(LM\) equal to

\[ \frac{x_{4}^{\prime\,2}}{2\left(x'_4+x''_2\right)} \]

and with the origin of coordinates at the point of its intersection with this diameter. For \(v=1\), equation (10) becomes the equation of the circle \(K'p_c\)

\[ \eta^{2}+\xi^{2}=\frac{x_{4}^{\prime\,2}}{2\left(x'_4+x''_2\right)}\,\xi \]

with radius

\[ \rho=\frac{x_{4}^{\prime\,2}}{4\left(x'_4+x''_2\right)}. \]

Thus, the geometrical locus of the end of the vector of the motor’s total resistance at synchronism, with variation of the brush displacement angle, is the circle \(K'p_c\), described on \(LM\) as on a diameter.

Inverting the straight line \(LM\) with respect to the point \(A\), we obtain the circle \(Kp_k\), passing through the origin of coordinates \(A\); the point \(M\) on the straight line \(Kp_k\) corresponds on the circle \(Kp_k\) to \(K_{90}\), and the point \(L\) to \(K_0\). Inverting the circle \(K'p_c\) with respect to the point \(A\), we obtain the circle \(Kp_c\), with its center on the line \(AN\), passing through the points \(K_{90}\) and \(K_0\).

Thus we obtain the following important conclusion: the geometrical locus of the end of the vector of the motor’s total conductance, at its stationary position and at synchronism, with variation of the brush displacement angle, consists of the circles \(Kp_k\) and \(Kp_c\).

The angle

\[ LC'K'=\operatorname{arctg}\frac{LK'}{C'K} =\operatorname{arctg}\left( \frac{x_{4}^{\prime\,2}\sin^{2}\lambda}{2\left(x'_4+x''_2\right)} : \frac{x_{4}^{\prime\,2}\sin 2\lambda}{2\left(x'_4+x''_2\right)} \right)=\lambda; \]

this gives us the possibility, for the given circle \(K'p_c\), by drawing from the point \(M\) the straight line \(MC'\) at the angle \(\lambda\) to \(ML\), to determine the position of the point \(C'\), the perpendicular from which to \(ML\) gives the point \(K'\), and the segment \(C'K\)—the scale of speed.

The speed \(v_0\), at which the delivered power is equal to zero, is determined by expression 32 (Part 1, p. 36); if in it we substitute the values \(c_1\) and \(c_2\), express \(x'_3\) through \(x'_4\), and reduce by \(\dfrac{1}{\cos^{2}\lambda}\), then we obtain for \(v_0\) the expression

\[ v_0=\frac{x''_2+x'_1\cos\gamma}{r''_2+x'_4\sin\gamma}\operatorname{ctg}\lambda. \]

V. M. Khrushchev.

Substituting it into formula (9), we obtain the expression for the coordinates of the end of the vector of the motor’s total resistance, for which the delivered power is equal to zero,

\[ \xi=\frac{x_4' \sin^2 \lambda}{2(x_4' + x_2')} \]

and

\[ \eta= \frac{x_2''+x_4' \cos \gamma}{r_2''+x_4' \sin \gamma} \cdot \frac{x_4'^2}{2(x_2''+x_4')} \cos^2 \lambda; \]

eliminating the angle \(\lambda\) from the last expressions, we have

\[ \eta= \frac{x_2''+x_4' \cos \gamma}{r_2''+x_4' \sin \gamma} \cdot \frac{x_4'^2}{2(x_2' + x_4')} - \frac{x_2''+x_4' \cos \gamma}{r_2''+x_4' \sin \gamma}\,\xi \]

— the equation of the straight line \(Kp_1'\), representing the locus of the end of the vector of total resistance at \(W_2'=0\) as the angle of brush displacement varies.

For \(\xi=0\)

\[ \eta \cong \frac{x_4'^2}{2(r_2''+x_4' \sin \gamma)} \]

and for

\[ \xi=\frac{x_4'^2}{2(x_2' + x_4')} - \eta =0, \]

thus this straight line passes through the point \(M\) and makes with the axis of ordinates the angle \(\mu\)

\[ \angle \mu = \operatorname{arctg} \frac{r_2''+x_4' \sin \gamma}{x_2''+x_4' \cos \gamma} \qquad\qquad (11) \]

Inverting this straight line \(Kp_1'\) with respect to the point \(A\), we obtain the circle \(Kp_1\), passing through the points \(A\) and \(K_{90}\), with its center on the line \(AP\) at an angle \(\mu\) to \(AS\). This circle \(Kp_1\) represents the locus of the current vector for which the delivered power is equal to zero, as the angle of brush displacement varies.

The center \(O_1\) of this circle evidently represents the point \(D\) (Fig. 17, p. 32, Part I), obtained more accurately than on p. 33 of Part I. However, for the relations existing in repulsion motors between \(r_2''\), \(x_2''\), \(x_4'\), and \(\gamma\), the angle \(PAY\) is very close to \(2\gamma\), \(\angle O_1K_{90}A \cong \gamma\); consequently \(K_{90}O_1\) is parallel to \(AY\), and the simple construction indicated on p. 33 of Part I for the point \(D\) is applicable also to finding the center of the circle \(Kp_1\).

The theoretical considerations just presented make it possible very quickly and easily to construct the working diagram of a repulsion motor for any angle of brush displacement, having performed only the usual tests—

...by short-circuit and no-load. This construction is shown in the following example.

Example. In a repulsion motor of the Thomson type, manufactured by Westinghouse, of 6 horsepower, the brushes were set at the angle \(\lambda = 0^\circ\), and at a voltage of 34 volts the current was measured as 67.4 amperes and the power as 1110 watts; reduced to the normal voltage, the data will be as follows:

\[ P_k = 110, \qquad J_k = 218, \qquad \cos \varphi_k = 0.484 \]

Then the brushes were lifted, and the following were measured:

\[ P_0 = 110, \qquad J_0 = 19\ \text{amp.}, \qquad W_0 = 198\ \text{watts} \]

therefore

\[ \cos \varphi_0 = 0.0947. \]

Fig. 11.

From these data the current vectors \(AJ_k\) and \(AJ_0\) are constructed on the diagram of Fig. 11; through the points \(J_k\), \(J_0\), and \(A\) the circle \(Kp_k\) is drawn; by reversing the vectors \(AJ_k\) and \(AJ_0\), the points \(L\) and \(M\) are obtained; on the straight line \(LM\), as on a diameter, the circle \(Kp'\) is constructed, which is reversed into \(Kp_c\). From \(A\), parallel to the straight line \(LM\), \(AS\) is drawn—the geometric locus of the centers of the working diagrams when \(\lambda\) is varied. From the point \(J_0\)

a straight line has been drawn parallel to the axis of ordinates and the circle \(Kp_1\) constructed; if (by formula 10) the angle \(\mu\) is calculated and from it the center of the circle \(Kp_1\) is constructed, it turns out that, for the given motor, it lies at a distance of only \(0.05\) mm from the first center.

When, in this way, all the auxiliary constructions have been made, we can obtain the working diagram, for example for the angle \(\lambda = 18^\circ\). For this purpose, from the point \(M\), at an angle of \(18^\circ\) to \(ML\), we draw the straight line \(MC'\), until it intersects the circumference \(Kp_c\) at the point \(C'\); from it we drop a perpendicular to \(LM\), and through the point \(K'\) we draw the straight line \(AK'\) until it meets the circumference \(Kp_k\) at the point \(K\). The circumference \(Kp\), passing through \(K\) and \(A\) and having its center on the line \(AS\), represents the required working diagram of the motor. The current with the rotor stationary, \(AK'\), from this diagram is equal to 114 amp. The synchronism point \(C\) is obtained as the intersection of the circumferences \(Kp\) and \(Kp_c\); the current at synchronism, \(AC\), is equal to 56 amp. The point \(O\), at which \(W_2' = 0\), is determined by the intersection of the circumferences \(Kp\) and \(Kp_1\), but since this intersection is directly very difficult to find precisely, the construction indicated on p. 33 of Part I was applied, namely: the line \(OO_1\), joining the centers of the circumferences \(Kp\) and \(Kp_1\), is prolonged to its intersection with the circumference \(Kp_1\) at the point \(b\), and from it upward the segment \(bd\), equal to \(bA\), is laid off; then the line of delivered power is represented by the straight line \(Kd\). The line of the rotating moment is constructed as a straight line passing through \(A\) and perpendicular to \(OO_1\).

3. Experimental investigation of the repulsion motor.

The working diagram has been constructed by us with a whole series of assumptions departing from reality, as a result of which the motor characteristics obtained from it, as can be expected in advance, differ from those obtained experimentally. Undoubtedly, the first place among all the factors producing this difference belongs to the reaction of the short-circuited coil, the effect of which is so strong that the difference between the theoretical and experimental results may almost entirely be attributed to the action of this reaction, especially if only the qualitative aspect of the phenomenon is considered. The aim of the present experimental investigation, carried out in the electrotechnical laboratory of the Tomsk Technological Institute, was to obtain such data as could be directly compared with the theoretical; obviously, for this the experiment had to be arranged so that losses due to friction were excluded.

A repulsion motor of the Thomson type, manufactured by Westinghouse, was subjected to testing.

Its principal data are as follows: 6 h.p., 110–160 volts, 960 rev. per min., 50 periods, 3 pairs of poles.

Stator: internal diameter — 307 mm,
  length — 120 mm,
  air gap — 1 mm,
  number of slots — 72,
  number of turns — 144.

The stator winding is continuous, so that the field produced by it is triangular.

Rotor: diameter — 305 mm,
  number of slots — 62,
  number of turns — 124.

Commutator: diameter — 195 mm;
  length — 65 mm,
  number of segments — 124.

The motor was connected to a transformer with a variable number of secondary turns, by which its voltage was regulated. In the motor circuit an ammeter, voltmeter, and wattmeter were inserted, and by means of a belt drive it was coupled to a direct-current dynamo-machine, which was loaded by lamp rheostats. In the circuit of the dynamo-machine there were likewise inserted an ammeter, voltmeter, and wattmeter, and in its inductor circuit a regulating rheostat and an ammeter. At a constant brush displacement angle \(\lambda = 18^\circ\), the load of the dynamo-machine was varied and, correspondingly, the speed of the repulsion motor changed; the greatest speed that was attained was equal to 1525 rev. per min.; from this speed down to 800 rev. inclusive, the motor voltage was maintained equal to 110 volts, while at lower speeds it was reduced in order not to allow currents dangerous to the motor and instruments. In this way a series of instrument readings was observed. Then the voltage of the repulsion motor was disconnected, and the dynamo-machine was connected to the voltage; and, measuring the power absorbed by it as a motor, after subtracting the ohmic losses, the friction losses of the entire unit plus the iron losses of the dynamo-machine were determined.

In Fig. 12, as functions of speed, are shown: curve I—the primary current of the tested motor, curve II—the primary power, and curve III—the power factor; and in Fig. 13, curve IV—the delivered power, curve V—the torque, and a series of points circled under curve VI—the coefficient of useful effi-

action. All this represents the result of the test described above; the current and the primary power up to a speed of 800 rev. are taken directly from the experiment, while the values given below, reduced to normal voltage—the delivered power, as the sum of the observed delivered power of the dynamo-machine + the ohmic losses in it + the losses due to friction and in the iron—below 800 rev., also reduced to normal voltage, the torque, the power factor, and the efficiency—are calculated. In the same Figs. 12 and 13, the dotted lines show

Fig. 12. Experimental and theoretical: primary current (I, I′), primary power (II, II′), and power factor (III, III′) as functions of speed.

the curves of current (I′), primary power (II′), power factor (III′), delivered power (IV′), torque (V′), and efficiency (VI′) as functions of speed, obtained from the working diagram for \(\lambda = 18^\circ\) (Fig. 11). As can be seen from Figs. 12 and 13, the experimental efficiency over the entire curve differs very little from the theoretical one, despite the fact that the experimental primary and delivered powers, at speeds more or less different from the synchronous speed, differ greatly from the theoretical ones; this is explained

because most of the energy of the short-circuited coil is transferred into the primary circuit, and only a small part of it is lost in the coil. The speed at which the reaction of the short-circuited coil produces only a demagnetizing action (the experimental power factor is equal to the theoretical one, point \(a\) of Fig. 12), and likewise that at which its generator action passes into motor action (the experimental powers are equal to the theoretical ones, points \(b\) of Fig. 12 and \(c\) of Fig. 13), are further reduced in comparison with those which we obtained on p. 9 from the diagram of

Fig. 13. Experimental and theoretical: output power \((IV, IV')\), rotating moment \((V, V')\), and efficiency \((VI')\), as functions of speed.

Fig. 9; this may be explained, on the one hand, by saturation of the iron, and on the other by the fact that the vector \(OE_{nd}''\) in reality, owing to the difference of the form of the longitudinal field from a sinusoid, lags behind \(O\Phi_{nd}\) by an angle smaller than \(\pi\) and is greater in magnitude than was assumed in constructing the diagram of Fig. 9. Together with this, the curves of Figs. 12 and 13 clearly show how unfavorable the conditions of commutation are even at speeds below synchronous, not to speak of those exceeding it. Therefore, from the point of view of commutation, the normal number of revolutions of a repulsion motor must be considerably below the synchronous speed.

In Fig. 14 are presented the curves: I—of the power factor, II—of the efficiency, as functions of the angle of brush displacement

under synchronism, obtained experimentally in exactly the same way as the curve in Figs. 12, 13 and I′, (dotted line)—the theoretical curve—of the power factor—from a series of diagrams. As may be seen from these curves, the experimental power factor, with the exception of very small angles of brush displacement, is less than the theoretical one, and its maximum, under the influence of the reactions of the short-circuited coil, is shifted toward still smaller angles of brush displacement; as regards the efficiency, it can be determined from the diagrams more or less accurately only for brush-displacement angles less than \(45^\circ\); in this part its curve is so close to the experimental one that it could not even be plotted in Fig. 14. It must, however, be noted that several points of the experimental curve of the efficiency were obtained somewhat above the theoretical one; this is explained by the inaccuracy of the experiment, which consisted in assuming equality of the friction losses under load and during no-load running of the unit. The remaining curves—of current, powers, and torque—are not given; in general they retain the form of the theoretical ones (Figs. 23 and 27 of Part I).

Fig. 14. Experimental and theoretical: power factor (I, I′) and efficiency (II) as functions of the angle of brush displacement.

As a result of our investigation, we can draw the following conclusions:

1) The operating diagram of a repulsion motor cannot have the same significance as, for example, the diagram of an asynchronous motor, since it does not take into account the reaction of the short-circuited coil, which strongly distorts the curves obtained from the diagram and representing current, power, torque, and power factor as functions of speed and of the brush displacement angle; however, its influence on the efficiency curve is small.

2) The conditions of commutation, owing to the great deviation, in the commutation zone, of the form of the transverse field from a sinusoid, are extremely unfavorable in repulsion motors; nevertheless, in this respect, in the author’s opinion, certain improvements are possible.

3) The most favorable speeds with respect to commutation must be considered those considerably below synchronous speed, while the most advantageous speeds with respect to efficiency and power factor are those above synchronous speed; therefore, as the normal speed of the motor, one must choose a speed close to synchronous.

4) The most advantageous brush displacement angles at synchronism, in the sense of efficiency, power factor, and commutation, as has been proved experimentally and theoretically, must be considered small brush displacement angles \((12^\circ—22^\circ)\), within which the normal power of the motor should lie.

5) The conditions of commutation require that starting the motor into operation be carried out at brush displacement angles greater than the normal ones.

6) The reaction of the short-circuited coil reduces, in comparison with the theoretical value, the power factor at speeds still lower than synchronous; improvement of the commutation conditions should have a favorable effect in this respect as well.

Addenda to Part I.

To § 7, p. 30.

Equating to unity \(\cos B_2BB_4\) in the formula expressing the ratio \(\dfrac{AB_2}{AB}\) (which we are entitled to do, since \(\angle B_2BB_4\) is usually very small) and substituting the value

\[ ab = AB \cdot z_1 y_1, \]

(see p. 29), we obtain

\[ AB = \frac{AB_2}{1 + z_1 y_a}. \]

Carrying out similar transformations with the expression for the angle \(B_2AB_4\), we obtain

\[ \angle B_2AB_4 = \arcsin \frac{-\cos(\gamma + \psi_1)}{\dfrac{1}{y_a z_1} + 1}. \]

To § 14, p. 51.

The efficiency coefficient may be represented in the form

\[ \eta = \frac{W'_2}{W'_2 + V_c} = \frac{E'_2 J'_2 \cos \psi_2} {E'_2 \cdot J'_2 \cos \psi_2 + J_2^{\prime 2} \left( r_1 + r'_2 + \frac{c_1}{c_2} r_a \right) + J_a^2 (r_1 + r_a)} = \]

\[ = \frac{E'_2 \cos \psi_2} {E'_2 \cos \psi_2 + J'_2 R_1 + \dfrac{J_a^2}{J'_2} R_2}. \]

Taking \(E'_2\) and \(\cos \psi_2\) as constants, we obtain the condition for a maximum of the efficiency coefficient as a function of speed:

\[ J_2^{\prime 2} R_1 = J_a^2 R_2, \]

from which we see that the maximum efficiency coefficient is obtained at small currents. In reality, under the influence of the reaction of the short-circuited coil, it shifts toward larger currents.

The maximum of the efficiency coefficient at synchronism as a function of the angle of displacement of the brushes is obtained under the condition:

\[ \frac{J_a^2 R_2}{J_2'^2} + R_1 + \frac{\partial R_1}{\partial \lambda}\cdot \frac{J_2'}{\dfrac{\partial J_2'}{\partial \lambda}}, \]

or, substituting the value

\[ J_2' = \frac{E_2'}{Z_{2v}'}, \]

we have:

\[ J_2'^2 R_1 = J_a^2 R_2 + 2E_2'^2 \frac{\dfrac{\partial R_1}{\partial \lambda}} {\dfrac{\partial Z_{2v}'}{\partial \lambda}} \simeq J_a^2 R_2 + 2J_{45}'^{\,2}\cdot (r_2' + r_a), \qquad \ldots \ (11) \]

where \(J_{45}'\) is the rotor current at \(\lambda = 45^\circ\). This formula shows that the angle of displacement of the brushes at which the efficiency coefficient reaches a maximum is less than \(45^\circ\), since \(r_2'\) in repulsion motors is always greater than \(r_1\), and that this angle is the smaller, the greater the rotor resistance and the iron losses. However, formula (11) should be regarded as a very rough approximation to reality.

The curve of the efficiency coefficient as a function of the angle of displacement of the brushes, in addition to a positive maximum, also has a negative maximum. This can most simply be proved by the fact that, for some \(\lambda\), namely for that at which the speed determined by formula 32 of Part I is equal to the synchronous speed, the efficiency coefficient is equal to zero. This angle of displacement of the brushes is close to \(90^\circ\), and therefore this phenomenon has no practical significance.

V. Khrushchev.

Tomsk
May, 1913.

Misprints Noted in Part I

Table of Contents

1. See Izv. T. T. I., 1913, vol. XXIX. ↩