Proceedings

of the Tomsk Technological Institute

of Emperor Nicholas II.

Vol. 14. 1909. No. 2.

II.

A. A. Potebnya.

Commutator Motors of Single-Phase Alternating Current.

First article.

1–36.

Commutator Motors for Single-Phase Alternating Current.

In recent years, single-phase-current motors with a commutator and an armature constructed according to the type of the armatures of direct-current machines have been attracting the general attention of specialists. Abroad, interest in the appearance of these motors on the market was at one time shared even by the general press. This interest is explained by the circumstance that, for the time being, only from the application of these motors can one expect a satisfactory solution to the pressing question of electrical engineering and railway practice: the transition to electric traction on lines of great length.

In fact, although direct-current motors in their modern construction may be regarded as almost an ideal machine, there is a substantial obstacle to the further development of their application for traction purposes: direct current can be supplied directly from stations only for tramway networks of a comparatively limited district. For large networks and for lines of great length it is necessary to generate, at the stations, alternating current of high voltage and to transmit it to transformer substations. By converting, at the substations, the high-voltage alternating current into direct current for feeding the motors, we obtain, even for urban roads, a loss of 25–30% from the station busbars to the motor brushes. This loss, together with the depreciation charges on the cost of substations and feeder cable, and the additional expenses (care of rotary transformers), raises the cost of energy at the working conductor, in comparison with its cost at the central station, by 60% and even by 80%*).

Although it is possible that, in the event of success with mercury transformers, which have recently been under development (Westinghouse), these figures will change in a favorable direction, one can hardly expect any particular advantage from their application, since, at a considerable, in all probability, price, these devices have a very limited lifetime (1500 working hours).

*) F. Eichberg. Ueber Einphasenbahn. Z. f. E. 1904.

The advantages of the direct application of alternating current for traction purposes are obvious. However, the mere fact of eliminating rotating transformers is not sufficient to adopt every system that directly utilizes alternating current.

Synchronous motors are excluded, as being incapable of starting from rest under load, since they develop torque only in synchronism. Likewise, single-phase asynchronous motors are of little use because of the small magnitude of the initial torque. Three-phase current has found application on several railways, with the voltage in the working conductors reaching 3000 and even (Berlin–Zossen) 10000 volts. This system, despite the high qualities of motors with a rotating field, also has substantial defects: still insufficiently large starting torque and the difficulty of regulating speed, since the cascade connection used for this purpose gives, with two motors, only two economical speeds, which even on public-service lines proves insufficient. Even if this latter difficulty is removed by installing in the train itself an economical frequency transformer, there still remains the need for two working conductors of different potentials, which increases the cost of the line and greatly complicates the arrangement of switches.

The single-phase commutator motor is not an invention of recent years; it is as old as the direct-current motor. Indeed, the very fact that a direct-current motor does not change its direction of rotation when the direction of the supply current is changed points to the possibility of feeding it with alternating current of any frequency.

Furthermore, as early as 1887, E. Thomson pointed out that the ordinary armature of a direct-current dynamo, placed in an alternating field and short-circuited through brushes shifted by a certain angle from the neutral line, begins to rotate, developing a considerable torque. These facts gave rise to the proposal of a whole series of designs for single-phase motors with a commutator, of which, however, not one proved viable and appeared on the market. The chief obstacle in the inventors’ path was the difficulty of avoiding sparking. This difficulty seemed so insurmountable that for a time attempts to obtain a practically suitable single-phase motor with a commutator were almost abandoned. In one of the best works on electric traction, La Traction électrique by A. Blondel and F. R. Dubois, published in 1898,

the authors exclude these motors from the number of types applicable to traction on the grounds that they cannot be built for powers greater than 4–5 horsepower. Other authors, such as Steinmetz in his book Alternating Current Phenomena, even while setting forth in detail the theory of commutator motors, regarded them as a theoretically interesting apparatus having no practical significance, in the nature of a reaction motor. L. B. Atkinson alone, in England, who patented in 1895–1898 a whole series of systems of commutator motors, showed, in a paper before the Institution of Civil Engineers in London*) all the high qualities of the machines that interest us, and pointed out the full possibility of reducing sparking at the commutator to harmless limits.

At the same time there appeared on the market, and obtained some distribution, the motors of E. Arnold (the Wagner & Co. works in St. Louis) and of Déri (the Austrian Union Company). These motors were built up to 20 horsepower, but both Arnold and Déri use the commutator only during starting; thereafter their motors operate with a short-circuited armature, like ordinary asynchronous single-phase motors.

Despite the above-mentioned prejudice of the majority of engineers against commutator motors, it became necessary to turn to them when a real need arose to use alternating current for traction purposes. The moment that marked the final victory of the system under consideration may be regarded as the paper by engineer B. G. Lamme**) before a meeting of the American Institute of Electrical Engineers. In this paper, concerning the construction and operation by means of single-phase current of the Washington–Baltimore–Annapolis line, 46 miles long, i.e. about 74 km, Lamme pointed to a fact that attracted particular attention, namely that commutator single-phase motors have characteristics fully satisfying the conditions of traction, in general very similar to the characteristics of series-wound direct-current motors, and automatically vary their speed with changes in load. Atkinson had also pointed out the possibility of regulating the speed within wide limits without a significant lowering of the efficiency.

It was recognized by all those present at the paper that, if the designers succeeded in coping with sparking, the applica—

*) Min. of Proceed. of the Inst. of Civ. Eng. XXXII (1898), p. 113.
**) Trans. of the Amer. Inst. of Electr. Eng. 1902, p. 1231.

... of these motors will be a serious advance in railway practice.

From that time onward there begin to appear, and continue to appear to this day, numerous theoretical investigations of commutator motors. Unfortunately, we still have very few experimental design data.

Most of the large electrical-engineering firms have taken up the development of their own types of single-phase commutator motors, and in doing so they have already succeeded everywhere in attaining a large starting torque, good power factor and efficiency, and an almost complete absence of sparking at the commutator.

It may now be said with confidence that the commutation of alternating current, provided certain conditions are observed, presents no greater difficulties than that of direct current. Everything depends on the voltage applied to a section, which can always be made sufficiently small. It is somewhat more difficult to deal with the currents induced in a short-circuited section by alternating current, but these currents too can be brought within harmless limits by reducing the number of turns per section and increasing its resistance by raising the resistance in the commutator connections and by using carbon brushes, and by placing the section being commutated under the proper magnetic conditions. Likewise, simple means have been found for increasing $\cos \varphi$.

The purpose of the present article is, on the basis of the studies that have appeared in print, to give a theory of the principal types of single-phase commutator motors and the foundations for their calculation. The occasion for its preparation was the desire to provide guidance in the design of alternating-current railways to students carrying out projects under the author’s supervision in the fifth year of the Mechanical Department of the Tomsk Technological Institute. This explains the detail, perhaps excessive from the point of view of most readers, of some of the derivations admitted by the author. For the same reasons the author gives preference to the simpler methods of German, and partly American, engineers over the more elegant and mathematically precise investigations of French electrical engineers.

1. Systems of commutator motors.

At the present time, as is always the case during the period of development of some mechanism and the elaboration of its practical forms, a whole series of circuits and constructions of single-phase commutator motors is being proposed. For the present it is difficult to judge, owing to the lack of ...

of experimental data, to speak of the real value of these systems and to predict which motor will outlive the others, driving them in the end from the market. It seems unnecessary to enumerate all the patented or described systems and designs, especially since this has been done for most of them, with a certain critical appraisal, in Ohnos’s article, published in translation in the journal Elektrichestvo for 1904 (pp. 163 and 188). We shall confine ourselves to the investigation of the basic circuits and their simplest combinations.

a) Shunt motor. Fig. 1. In this motor different currents flow through the armature and inductor windings. Since the self-induction of the magnet winding is very large, the current flowing through it lags behind the voltage at the terminals by almost 90°. In the armature, when running idle, there also flows an almost wattless current. Therefore both currents will be in phase, and the motor starts from rest with a strong turning moment. With an increase in speed, the lag of the current in the armature decreases, while in the inductor winding it remains as before. Therefore the phase difference between the field and the working current increases and the moment decreases. In order to develop sufficient power the motor takes an inordinately large current, which lowers the efficiency. The power factor will be very small. These circumstances apparently make the shunt motor completely unsuitable for practice.

Fig. 1.

b) Series motor. In motors of this type one and the same current passes through the inductor and armature windings and, consequently, the field and the working current are always in phase. The circuit is shown in Fig. 2. The turning moment is obviously always proportional to the product of the instantaneous values of the field and the current, or, with a slight saturation of the iron, proportional to the square of the current. The power factor in this motor is always considerably less than unity, since the self-inductions of the inductor and armature windings, adding together, produce a phase difference between voltage and current, which

Fig. 2.

impossible to eliminate completely. We can, however, reduce it to some extent by compensating the armature field, which is not needed for producing torque, by means of an additional winding placed at right angles to the main armature winding and calculated so that the field of this winding is equal in magnitude and opposite in direction to the armature field, as shown in Fig. 3. But the fields of the inducing winding F and of the compensating winding C combine, forming a resultant field whose axis makes an acute angle with the line of the brushes; this field can obviously be created by a single winding occupying an intermediate position between windings F and C. It is clear that we shall obtain the same result in the circuit of Fig. 2 by turning the brushes through a certain angle from the neutral line, as shown by the dotted line. It proves very expedient, instead of connecting windings F and C in series, to short-circuit winding C. The diagram of the compensated series motor obtained in this way is shown in Fig. 4: in the short-circuited winding C, the armature field induces a current, which in turn creates a field that almost completely compensates the armature field.

Fig. 3.

Fig. 4.

c) Motor with short-circuited brushes, or repulsion motor. Fig. 5 shows the diagram of the repulsion motor proposed by Atkinson. Winding C, acting as a pri-

Fig. 5.

the primary winding of the transformer, induces a current in the armature winding. The field of winding F acts on this current, producing a strong torque. Since both windings give a resultant field, the axis of which is inclined to the line of the brushes, they, as in the series motor (Figs. 2 and 3), can be replaced by a single winding with an axis making a certain acute angle with the line of the brushes. Fig. 6 shows the diagram of Thomson’s repulsion motor (chronologically earlier than Atkinson’s motor) with one winding. Thomson’s motor has an important advantage in simplicity of construction; the advantages of Atkinson’s motor consist in the possibility of regulating the armature current by shunting the compensating winding C, and in the possibility of changing the direction of rotation of the armature by commutating the current in one of the field windings.

Fig. 6.

At the same time Latour in France and Winter and Eichberg in Germany proposed a modification of the scheme of Fig. 5, shown in Fig. 7. The winding F, with an axis perpendicular to the line of the short-circuited brushes, whose field serves to produce the torque, is eliminated, and its field is replaced by the field produced by the primary current flowing in the armature through the brushes FF′, whose line is perpendicular to the line of the short-circuited brushes CC′. The Winter-Eichberg or Latour motor is quite incorrectly called by many electricians a compensated series motor: the field created by the current flowing into the armature winding through the brushes FF′, connected in series with the field winding, is not compensated; the field winding C compensates the field created by the current circulating in the armature through the short-circuited brushes (secondary); therefore we shall adopt for this motor the name proposed by Danielson: compensated repulsion motor.

Fig. 7.

Short-circuiting the armature of a repulsion motor by means of two pairs of brushes (Fig. 8) is recommended as a means of avoiding sparking at the commutator when starting from rest, especially for high-power motors ). Deri makes use of two pairs of brushes for speed regulation, which presents certain difficulties in the ordinary repulsion motor (of the Thomson type) *). He attains this aim by fixing a pair of diametrically opposite brushes so that their lines coincide with the axis of the field \(OO'\) (Fig. 9), and by moving the other two brushes around the circumference of the commutator, the angle embraced by each pair of short-circuited brushes \(\alpha\) varying from \(0\) to \(180^\circ\).

Fig. 8.

Besides the ordinary repulsion motor, E. Thomson proposed another repulsion motor of the type shown in Fig. 10. In this motor, instead of 4 narrow brushes, connected metallically to one another in pairs, two broad brushes are used, each embracing about a quarter of the commutator (with two poles). The short-circuit currents are at the same time useful acting currents. The drawback of this motor is the great friction of the brushes.

Fig. 9.

The motors of Figs. 8 and 10 have much in common, since each pair of short-circuited brushes corresponds to one broad brush. In these motors all the armature conductors not falling within the angles embraced by a pair of short-circuited brushes, or by one broad brush, remain without current, and consequently are lost for the production of torque. By introducing connecting wires closed upon the short-circuited brushes or broad

Fig. 10.

) M. Latour. Die zweite Form des Repulsionsmotor. E. T. Z. 1904. 952.
*) K. Sehneztler. Ein neuer Rep. Motor und sein Vorausher. E. T. Z. 1905. 72,

brushes into the primary circuit, as shown in Figs. 11 and 12, we eliminate this drawback and, at the same time, obtain compensated motors without resorting to additional brushes. In the connection of Fig. 11, usually \(\alpha=\frac{\pi}{2}\). The circuit of Fig. 12 was patented by Lundell.

The B. Pibl’s and Co. works in Edinburgh builds compensated repulsion motors according to the simplified circuit of Fig. 13. The simplification consists in replacing four brushes by three. The method of operation and the relation of the fields require no explanation.

Other numerous circuits of single-phase commutator motors, representing combinations of the forms listed, we shall not give.

Fig. 11.

2. Magnetic field of commutator motors.

The resultant field of a commutator motor is composed of the stator field and the rotor field; both, in the presence of compensation, are composed respectively of two fields: the main and compensating field in the stator, or the main field and the short-circuit field in the rotor.

The form of these fields depends on the arrangement of the stator and rotor windings, and on the distribution of the stator iron. In motors with pure series excitation, in which it is desirable to reduce the transverse magnetizing action of the armature, the projecting poles are made as in ordinary direct-current machines. In those types where it is desirable to reduce the magnetic reluctance in the path of the lines of the compensating—

Fig. 12.

Fig. 13.

A. A. Potebnya.

...fields, to distribute the iron more evenly over the entire surface of the armature, and to arrange the stator similarly to the stator of an ordinary single-phase induction motor.

In the case of separate projecting poles, the stator winding is concentrated; if leakage is neglected, then one may imagine that all turns of the inductor winding lie on the contour of the pole tip. The curve of the magnetomotive force of Fig. 14 will be a rectangle, shown by a dotted line, where \(\tau\) is the pole pitch and \(\lambda\) is the pole arc in angular quantities. In the case of a two-pole motor \(\tau=\pi\). Owing to magnetic leakage, the field curve differs from the curve of the magnetomotive force, and has the form shown in the drawing by the solid line.

Fig. 14.

When the field is produced by a winding uniformly distributed over the entire length of the pole pitch, the curve of magnetomotive forces is the triangle of Fig. 15. With uniformly distributed stator iron (without projecting poles), the field will also have the same form, if the saturation of the iron is not great. The curve of Fig. 15 is constructed on the assumption that the uniformly distributed inductor winding, or the armature winding, producing the field under consideration, is replaced by a uniformly distributed layer of copper, through which a current \(I=in\) flows, where \(i\) is the current flowing in each conductor, and \(n\) is the number of conductors lying side by side and carrying current in one direction. If the resistance of the iron is neglected, then the induction \(B_a\) in the interiron space at a distance \(a\) from the middle of such a layer (in the case of the armature field this will be the middle of the distance between adjacent brushes of the same designation) is determined by the equation

Fig. 15.

\[ \tag{1} 1{,}6\,B_a \delta = 2J \cdot \frac{a}{\pi}, \]

where \(\delta\) is the air gap in cm. From the equation and the drawing it is seen that the neutral line of the field passes through the middle of a row of conductors carrying current in one direction, or divides in half the distance between the brushes.

Fig. 16.

When the armature winding with continuous iron covers not its entire surface, but only some part of it corresponding to the arc \(\nu\tau\) (\(\nu\) is a proper fraction), or if the brushes, the number of which is greater than the number of poles, divide the armature winding into parts producing different fields, as in the motors whose diagrams are shown in Figs. 8, 9, 10, 11, and 12, then the field, if the reluctance of the iron is neglected, will have the form of Fig. 16.

Fig. 17.

In reality, owing to the saturation of the iron, especially of the teeth, the field curve approaches a sinusoid; besides,

Fig. 18.

some deforming action is introduced by the presence of depressions between the teeth. The armature field undergoes an even greater deformation in motors with projecting poles. This deformation depends on the position of the brushes relative to the pole tips. In Figs. 17, 18, 19, and 20 are shown the deformations of the triangular armature field for the position of the brushes on the neutral line and for three other positions. In a similar manner the field will also be deformed in the case of a trapezoidal distribution of the magnetomotive force.

Fig. 19.

Fig. 20.

However the field curves may be deformed, we may with certainty say that the density of the magnetic flux in the inter-iron space \(B\) is a periodic function of the angle \(\alpha\) (in electrical degrees), measured along the circumference of the armature from the neutral line of this field, and, moreover, both halves of the field curve are symmetric with respect to the axis of abscissas (angles). Therefore, in the most general case, we may represent the dependence between \(B\) and \(\alpha\) by a Fourier series in which all even terms are equal to zero,

\[ \tag{2} B=\sum_{0}^{\infty} a_{2p+1}\cdot \operatorname{Cos}(2p+1)\alpha +\sum_{0}^{\infty} b_{2p+1}\cdot \operatorname{Sin}(2p+1)\alpha, \]

where

\[ a=\frac{1}{\pi}\int_{-\pi}^{+\pi} B\operatorname{Cos}(2p+1)\alpha\,d\alpha, \]

\[ b=\frac{1}{\pi}\int_{-\pi}^{+\pi} B\operatorname{Sin}(2p+1)\alpha\,d\alpha. \]

Let us find this expansion for a trapezoidal field. From Fig. 16 it is seen that, for values of \(\alpha\)

\[ \text{from }-\pi\text{ to }-\left(1-\frac{r}{2}\right)\pi \qquad B=-B_{max}\frac{2(\pi+\alpha)}{r\pi}, \]

\[ \text{from }-\left(1-\frac{r}{2}\right)\pi\text{ to }-\frac{r}{2}\pi \qquad B=-B_{max}, \]

\[ \text{from }-\frac{r}{2}\pi\text{ to }+\frac{r}{2}\pi \qquad B=+B_{max}\frac{2\alpha}{r\pi}, \]

\[ \text{from }+\frac{r}{2}\pi\text{ to }+\left(1-\frac{r}{2}\right)\pi \qquad B=+B_{max}, \]

\[ \text{from }+\left(1-\frac{r}{2}\right)\pi\text{ to }+\pi \qquad B=+B_{max}\frac{2(\pi-\alpha)}{r\pi}. \]

Representing the coefficients \(a_{2p+1}\) and \(b_{2p+1}\) in the form of sums, or integrals, each of which extends over an interval with the same law of variation of \(B\):

\[ \frac{1}{\pi}\int_{-\pi}^{+\pi} = \frac{1}{\pi} \left\{ \int_{-\pi}^{-\left(1+\frac{r}{2}\right)\pi} + \int_{-\left(1+\frac{r}{2}\right)\pi}^{-\frac{r}{2}\pi} + \int_{-\frac{r}{2}\pi}^{+\frac{r}{2}\pi} + \int_{+\frac{r}{2}\pi}^{\left(1-\frac{r}{2}\right)\pi} + \int_{\left(1-\frac{r}{2}\right)\pi}^{\pi} \right\}, \]

where under the integral signs there stands the expression \(B\Cos(2p+1)\alpha\,d\alpha\), or respectively \(B\Sin(2p+1)\alpha\,d\alpha\), and substituting the above values of \(B\), we find

\[ a_{2p+1}=0,\qquad b_{2p+1} = -\frac{8B_{max}}{r\pi^2(2p+1)^2} \Sin(2p+1)\frac{r\pi}{2}, \]

and, consequently, finally the induction in the inter-iron space, in the case of a trapezoidal distribution of the magnetic-flux density, is represented by the series

\[ B= \frac{8B_{max}}{r\pi^2} \left\{ \Sin\frac{r\pi}{2}\Sin\alpha + \frac{1}{9}\Sin 3\frac{r\pi}{2}\Sin 3\alpha + \frac{1}{25}\Sin 5\frac{r\pi}{2}\Sin 5\alpha +\ldots \right\}. \tag{3} \]

For a triangular field, from this we obtain, putting \(r=1\),

\[ B= \frac{8B_{max}}{\pi^2} \left\{ \Sin\alpha - \frac{1}{9}\Sin 3\alpha + \frac{1}{25}\Sin 5\alpha -\ldots \right\}. \tag{4} \]

It is still simpler to obtain the series representing \(B\) in the case of a rectangular field (Fig. 14, dotted line); noting that for values of \(\alpha\)

\[ \text{from } -\pi \text{ to } -\frac{\pi+\lambda}{2} \qquad B=0, \]

\[ \text{from } -\frac{\pi+\lambda}{2} \text{ to } -\frac{\pi-\lambda}{2} \qquad B=-B_{max}, \]

\[ \text{from } -\frac{\pi-\lambda}{2} \text{ to } +\frac{\pi-\lambda}{2} \qquad B=0, \]

\[ \text{from } \frac{\pi-\lambda}{2} \text{ to } \frac{\pi+\lambda}{2} \qquad B=+B_{max}, \]

\[ \text{from } \frac{\pi+\lambda}{2} \text{ to } \pi \qquad B=0, \]

we find

\[ a_{2p+1} = -\frac{B_{max}}{\pi} \int_{-\frac{\pi+\lambda}{2}}^{-\frac{\pi-\lambda}{2}} \cos(2p+1)\alpha\,d\alpha + \frac{B_{max}}{\pi} \int_{\frac{\pi-\lambda}{2}}^{\frac{\pi+\lambda}{2}} \cos(2p+1)\alpha\,d\alpha =0, \]

\[ \begin{aligned} b_{2p+1} &= -\frac{B_{max}}{\pi} \int_{-\frac{\pi+\lambda}{2}}^{-\frac{\pi-\lambda}{2}} \sin(2p+1)\alpha\,d\alpha + \frac{B_{max}}{\pi} \int_{\frac{\pi-\lambda}{2}}^{\frac{\pi+\lambda}{2}} \sin(2p+1)\alpha\,d\alpha \\ &= (-1)^p \frac{4B_{max}}{\pi(2p+1)} \sin\frac{(2p+1)\lambda}{2} \end{aligned} \]

and finally

\[ (5)\quad B=\frac{4B_{max}}{\pi} \left\{ \sin\lambda\sin\alpha -\frac{1}{3}\sin\frac{3\lambda}{2}\sin 3\alpha +\frac{1}{5}\sin\frac{5\lambda}{2}\sin 5\alpha -\ldots \right\}. \]

3. Fundamental equations of equilibrium of the electromotive forces.

We shall consider separately the action of each of the fields, both that of the inductor and that of the armature, on the given winding. Let

\(B\) — the instantaneous value of the induction of the field under consideration,

\(i=J\sin(\omega t-\psi)\) the instantaneous value of the current exciting the field,

\(B_k\) — the \(k\)-th term in the expansion of the function \(B\),

\(b,\ b_k\) are the values of the functions \(B\) and \(B_k\) corresponding to \(i = 1\) ampere,

\(\Phi\) is the flux linking one element of the winding under consideration,

\(\Phi\) is the value of \(\Phi\) corresponding to \(i = 1\) ampere,

\(D\) is the diameter of the armature, \(l\) the length of the active part of the side of the winding element in cm.

Then

\[ B = B_1 \operatorname{Sin} \alpha + B_3 \operatorname{Sin} 3\alpha + B_5 \operatorname{Sin} 5\alpha + \ldots . \]

If it is assumed that the magnetic reluctance remains constant, then all \(B_k\) are proportional to the current \(i = J \operatorname{Sin}(\omega t - \varphi)\), and

\[ B = b \cdot J \operatorname{Sin}(\omega t - \varphi) = \sum b_k \operatorname{Sin} k\alpha \cdot J \operatorname{Sin}(\omega t - \varphi). \]

Further,

\[ \Phi = \frac{Dl}{2} \int B\,d\alpha = \frac{Dl}{2} \int b\,d\alpha \cdot J \operatorname{Sin}(\omega t - \varphi), \]

\[ \Phi = \frac{Dl}{2} \int b\,d\alpha, \]

where the integral is taken between the limits corresponding to the position of the sides of the winding element, and

\[ \Phi = J \cdot \Phi \cdot \operatorname{Sin}(\omega t - \varphi). \]

The e.m.f. developed in the winding element when the flux \(\Phi\) changes is

\[ e' = -\frac{d\Phi}{dt} = -\left(\frac{\partial \Phi}{\partial \alpha}\cdot\frac{d\alpha}{dt} + \frac{\partial \Phi}{\partial t}\right). \]

\[ \frac{d\alpha}{dt} = \omega_1 \]

is the angular velocity of the relative displacement of the field and the conductors in electrical degrees. If \(\Omega\) is the angular velocity of the armature, then obviously

\[ \omega_1 = p\Omega, \]

where \(p\) is the number of pairs of field poles. The velocity \(\omega_1\) can be expressed in terms of the rate of change of the field \(\omega = 2\pi\sim\), where \(\sim\) is the number of periods:

\[ \omega_1 = s\omega; \]

\(s\) is an abstract number, positive or negative. It is not difficult to see that

\[ e' = -\left\{\omega_1 \frac{\partial \Phi}{\partial \alpha}\operatorname{Sin}(\omega t - \varphi) + \omega\Phi\operatorname{Cos}(\omega t - \varphi)\right\} J. \]

The total instantaneous e.m.f. of the winding under consideration will be

\[ \tag{6} e=-\left\{\operatorname{Sin}(\omega t-\varphi)\,\omega_1\sum\frac{\partial\Phi}{\partial\alpha} +\operatorname{Cos}(\omega t-\varphi)\,\omega\sum\Phi\right\}J, \]

where the summation is carried out over all consecutively connected elements of the winding. Obviously, the sums \(\sum \frac{\partial\Phi}{\partial\alpha}\) and \(\sum\Phi\), for a given position of the brushes, are constant quantities.

Let us apply formula (6) to the derivation of the basic equations of the motor systems listed above.

I. Series motor. The air gap is penetrated by two fluxes:

\[ \Phi_i \]

—the field produced by the inductor, and

\[ \Phi_a \]

—the field produced by the armature winding.

Both fields are stationary in space. Since the inductor winding is stationary, in the formula for the e.m.f. induced in the inductor winding one must take \(\omega_1=0\).

If \(e_i\) is the e.m.f. of the inductor winding, \(e_a\) is the e.m.f. of the armature winding, the symbol \(\sum_i\) denotes summation over the elements of the inductor winding, and \(\sum_a\) that over the armature, then

\[ e_i=-\omega J\left(\sum_i\Phi_i+\sum_i\Phi_a\right)\operatorname{Cos}(\omega t-\varphi), \]

\[ e_a=-\omega_1J\left(\sum_a\frac{\partial\Phi_i}{\partial\alpha} +\sum_a\frac{\partial\Phi_a}{\partial\alpha}\right)\operatorname{Sin}(\omega t-\varphi) \]

\[ -\omega J\left(\sum_a\Phi_i+\sum_a\Phi_a\right)\operatorname{Cos}(\omega t-\varphi). \]

The voltage loss due to the ohmic resistance \(R\) of the consecutively connected inductor and armature windings is

\[ e_r=JR\operatorname{Sin}(\omega t-\varphi). \]

Consequently, if \(e_k\) is the voltage at the motor terminals,

\[ e_k-e_r+e_a+e_i=0, \]

whence

\[ e_k = E_k \operatorname{Sin}\omega t = J(M\omega_1 + R)\operatorname{Sin}(\omega t-\varphi) + JN\omega \operatorname{Cos}(\omega t-\varphi), \tag{7} \]

where, for brevity, \(M\) and \(N\) denote constants depending on the construction of the motor and on the position of the brushes,

\[ \begin{aligned} M &= \sum_a \frac{\partial \Phi_i}{\partial \alpha} + \sum_a \frac{\partial \Phi_a}{\partial \alpha},\\ N &= \sum_i \Phi_i + \sum_i \Phi_a + \sum_a \Phi_i + \sum_a \Phi_a. \end{aligned} \tag{7 bis} \]

II. Repulsion motor. In motors of this type (Figs. 5, 6, 8) there also act two fields, but excited by two different currents: the field of the stator, excited by the primary current

\[ i_1 = J_1 \operatorname{Sin}(\omega t-\varphi_1), \]

and the field of the rotor, excited by the secondary current

\[ i_2 = J_1 \operatorname{Sin}(\omega t-\varphi_2). \]

The e.m.f. of the inductor winding (stator), according to the preceding, is

\[ e_i = -\omega J_1 \sum_i \Phi_i \cdot \operatorname{Cos}(\omega t-\varphi_1) -\omega J_2 \sum_i \Phi_a \operatorname{Cos}(\omega t-\varphi_2). \]

The e.m.f. in the armature winding (rotor) is

\[ \begin{aligned} e_a ={}& -\omega_1 J_1 \sum_a \frac{\partial \Phi_i}{\partial \alpha} \,\operatorname{Sin}(\omega t-\varphi_1) -\omega_1 J_2 \sum_a \frac{\partial \Phi_a}{\partial \alpha} \,\operatorname{Sin}(\omega t-\varphi_2)\\ &-\omega J_1 \sum_a \Phi_i \cdot \operatorname{Cos}(\omega t-\varphi_1) -\omega J_2 \sum_a \Phi_a \operatorname{Cos}(\omega t-\varphi_2). \end{aligned} \]

As was said above, in the motors under consideration the iron of the inductor is distributed uniformly; consequently the armature field is symmetrical with respect to the brush line; therefore

\[ \sum_a \frac{\partial \Phi_a}{\partial \alpha} = 0; \]

it is likewise obvious that

\[ \sum_i \Phi_a = \sum_i \Phi_i . \]

Reasoning as before, we obtain the fundamental equations of the motor

\[ \begin{aligned} E_k \operatorname{Sin}\omega t &= \omega J_1 L_1 \operatorname{Cos}(\omega t-\varphi_2) + \omega J_2 M \operatorname{Cos}(\omega t-\varphi_2) + J_1 R_1 \operatorname{Sin}(\omega t-\varphi_1),\\ 0 &= \omega J_1 M \operatorname{Cos}(\omega t-\varphi_1) + \omega L_2 \operatorname{Cos}(\omega t-\varphi_2) + J_2 R_2 \operatorname{Sin}(\omega t-\varphi_2)\\ &\quad + \omega_1 J_1 N \operatorname{Sin}(\omega t-\varphi_1), \end{aligned} \tag{8} \]

where \(R_1, R_2\) are the resistances of the windings of the inductor and armature, and, for brevity,

\[ \sum_i \Phi_i = L_1,\qquad \sum_a \Phi_a = L_2, \]

\[ \sum_i \Phi_a = \sum_a \Phi_i = M,\qquad \sum_a \frac{\partial \Phi_i}{\partial \alpha} = N. \tag{8 bis} \]

From the meaning of the derived equations (8) it is clear that the order of summation must be chosen so that \(L_1\), \(L_2\), and \(M\) are positive. Since the product \(\omega_1 N\) must necessarily be \(>0\), the sign of \(\omega_1\) (the direction of rotation) will be determined by the sign of \(N\), the summation of which is carried out in the same order as that of \(M\).

III. Compensated repulsion motor (Figs. 7, 11, and 12).

In addition to the field of the inductor

\[ B_i = b_i J \operatorname{Sin}(\omega t-\varphi_1) \]

and the field of the armature, excited, as in the preceding motor, by the current passing through the short-circuited brushes,

\[ B_{a2} = b_{a2} J_2 \operatorname{Sin}(\omega t-\varphi_2), \]

the current passing through the series-connected brushes creates still another field, as in the series motor,

\[ B_{a1} = b_{a1} J_1 \operatorname{Sin}(\omega t-\varphi_1). \]

Reasoning as before, we find:

the electromotive force in the inductor winding

\[ e_i = -\omega J_1 \left\{\sum_i \Phi_i + \sum_i \Phi_{a1}\right\} \operatorname{Cos}(\omega t-\varphi_1) -\omega J_2 \sum_i \Phi_{a2}\operatorname{Cos}(\omega t-\varphi_2); \]

E.m.f. between series-connected brushes

\[ e_{a1}=-\omega_1 J_1 \left\{ \sum_{a1}\frac{\partial \Phi_i}{\partial \alpha} + \sum_{a1}\frac{\partial \Phi_{a1}}{\partial \alpha} \right\} \operatorname{Sin}(\omega t-\zeta_1) -\omega_1 J_2 \sum_{a1}\frac{\partial \Phi_{a2}}{\partial \alpha} \operatorname{Cos}(\omega t-\zeta_2) \]

\[ -\omega J_1 \left\{ \sum_{a1}\Phi_i+\sum_{a1}\Phi_{a1} \right\} \operatorname{Cos}(\omega t-\zeta_1) \]

\[ -\omega J_2 \sum_{a1}\Phi_{a2}\operatorname{Cos}(\omega t-\zeta_2); \]

E.m.f. between short-circuited brushes

\[ e_{a2}=-\omega_1 J_1 \left\{ \sum_{a2}\frac{\partial \Phi_i}{\partial \alpha} + \sum_{a2}\frac{\partial \Phi_{a1}}{\partial \alpha} \right\} \operatorname{Sin}(\omega t-\zeta_1) -\omega_1 J_2 \sum_{a2}\frac{\partial \Phi_{a2}}{\partial \alpha} \operatorname{Sin}(\omega t-\zeta_2) \]

\[ -\omega J_1 \left\{ \sum_{a2}\Phi_i+\sum_{a2}\Phi_{a1} \right\} \operatorname{Cos}(\omega t-\zeta_1) \]

\[ -\omega J_2 \sum_{a2}\Phi_{a2}\operatorname{Cos}(\omega t-\zeta_2). \]

The symbol \(\sum_{a1}\) denotes summation over all conductors lying between the series-connected brushes, through which the current \(J_1\) flows; the symbol \(\sum_{a2}\) denotes summation over the conductors between the short-circuited brushes, carrying the current \(J_2\).

Since in the motors under consideration the iron is made continuous, as in the preceding type, in order to ensure the existence of flux from the short-circuited part of the winding, we have

\[ \sum_{a1}\frac{\partial \Phi_{a1}}{\partial \alpha} = \sum_{a2}\frac{\partial \Phi_{a2}}{\partial \alpha} =0. \]

Denoting further

\[ \sum_i \Phi_i = L_1,\quad \sum_{a1} \Phi_{a1}=L_2,\quad \sum_{a2}\Phi_{a1}=L_2^{1}, \]

\[ \sum_i \Phi_{a1}=\sum_{a1}\Phi_i=M_1,\quad \sum_i \Phi_{a2}=\sum_{a2}\Phi_i=M_2, \]

\[ \tag{9 bis.} \sum_{a1}\Phi_{a2}=\sum_{a2}\Phi_{a1}=M, \]

\[ \sum_{a1}\frac{\partial \Phi_{a2}}{\partial \alpha} = \sum_{a2}\frac{\partial \Phi_{a1}}{\partial \alpha} =N,\quad \sum_{a1}\frac{\partial \Phi_i}{\partial \alpha}=N_1,\quad \sum_{a2}\frac{\partial \Phi_i}{\partial \alpha}=N_2, \]

we shall represent the equations of equilibrium of the e.m.f.’s for this type of motors:

\[ e_k+e_i+e_{a1}-iR_i=0, \]

\[ e_{a1}-iR_a=0, \]

in the form

\[ \tag{9} \begin{aligned} E_k\operatorname{Sin}\omega t &=\omega J_1(L_1+2M_1+L_2)\operatorname{Cos}(\omega t-\varphi_1) +\omega J_2(M+M_2)\operatorname{Cos}(\omega t-\varphi_2)\\ &\quad+\omega_1J_1N_1\operatorname{Sin}(\omega t-\varphi_1) +\omega_1J_2N\operatorname{Sin}(\omega t-\varphi_2) +R_iJ_1\operatorname{Sin}(\omega t-\varphi_1),\\ 0 &=\omega J_1(M+M_2)\operatorname{Cos}(\omega t-\varphi_1) +\omega J_2L_2^{1}\operatorname{Cos}(\omega t-\varphi_2)\\ &\quad+\omega_1J_1(N+N_2)\operatorname{Sin}(\omega t-\varphi_1) +R_aJ_2\operatorname{Sin}(\omega t-\varphi_2); \end{aligned} \]

the coefficients entering into the derived equations evidently depend on the position of the brushes.

From the equations obtained, as particular cases, the equations for motors of other schemes can easily be derived by determining the corresponding coefficients.

All the derived equations (7, 7 and 9) represent an algebraic dependence between sinusoidal terms of the same periodicity. Therefore, if the sinusoids entering into these equations are represented in the form of vectors, then the algebraic sums and differences are replaced by the geometrical sums and differences of these vectors. With such a geometrical interpretation of the equations, their simple and visual investigation by graphical means becomes possible.

4. Diagram of a series motor.

In the diagram we shall reckon the angles between the vectors and the axis in the direction opposite to the motion of the clock hand.

If we lay off the vector (Fig. 21), coinciding in phase with the current,

\[ O A = (M \omega_1 + R) J = (M s \omega + R) J \]

to the right from point \(O\) along the horizontal, then the vector

\[ A B = J N \omega \]

is laid off upward from point \(A\) at a right angle to \(O A\). The hypotenuse \(O B\) will then represent, in magnitude and direction, the vector of the terminal voltage.

Fig. 21.

Since the terminal voltage is constant in magnitude regardless of the load, it is very convenient to take its vector as the basis of the diagram; then the circle constructed on this vector as on a diameter (Fig. 22) will be the geometric locus of point \(B\) for varying \(J\) and \(s\), i.e., for varying load.

Fig. 22.

The diagram constructed in this way has a great similarity to Heyland’s diagram for asynchronous motors, and, just like the latter, makes it possible to construct the entire characteristic of the motor operation in terms of the quantities in the following manner:

a) The measure of the current \(J\) is the segment \(B A\), since its length is proportional to the product of the current \(J\) by the constant factor \(\omega N\).

b) The segment \(OA\) is a measure of the power factor \(\cos \varphi\), since the direction \(OA\) is the direction of the current vector, \(AB\) is perpendicular to it, \(\angle AOB=\varphi\), and consequently

\[ OA = OB \cdot \cos \varphi = \text{Const.}\cos \varphi . \]

c) The power absorbed by the motor is

\[ W_1=\frac{1}{2}E_k J\cos\varphi; \]

this quantity is proportional to the product of the segments

\[ AB \times OA = BO \times AF, \]

or proportional to the area of the triangle \(ABO\).

Let the point \(C\) divide the segment \(OA\) into the parts \(AC\) and \(CO\), related as \(R : s\omega M\). Then \(AC=RJ\), on the scale of e.m.f.; if this segment is multiplied by the segment \(AB\) on the current scale, we obtain the area of the triangle \(ABC\), representing the power

\[ W_2=\frac{1}{2}RJ^2, \]

lost in heating the windings of the inductor and the armature on the same scale on which the area of the triangle \(BAO\) represents the power \(W_1\).

Hence it is clear that the mechanical power of the motor \(W_3\), if the losses due to hysteresis, Foucault currents, and friction are neglected, will be equal, on the same scale, to the difference of the areas of the triangles \(BAO\) and \(BAC\), i.e. to the area of the triangle \(BCO\), or to the product \(BO \times CH\).

But \(BO=\text{Const}\), and therefore, on one and the same scale, there will be

\[ W_1=AF,\qquad W_2=CH. \]

When the load changes, the point \(C\) will remain on the circle \(BCO\), described from the center \(S_1\), which lies at the intersection of the perpendicular \(SS_1\), erected to the straight line \(OB\) from its midpoint \(S\), and the inclined line \(BS_1\), drawn to the same straight line at the angle \(\delta\), if

\[ \tg \delta=\frac{AC}{AB}=\frac{R}{\omega N}. \]

However, one can find the segment representing, on the power scale, the magnitude \(W_2\), without resorting to the construction of the second circle.

Let us draw the current vector \(BA_0\), corresponding to \(\varphi_0=\frac{\pi}{2}-\delta\). Then, from the similarity of the triangles \(BAC\) and \(BfC\), we have:

$$ \frac{fF}{BF}=\frac{AC}{AB}, $$

or \(fF \times AB = BF \times AC\); substituting here \(AB = BO \sin \varphi\), \(BF = AB \sin \varphi\), we obtain \(fF \times OB = AB \times AC\). But the product \(AB \times AC = W_2\), and consequently the power \(W_2\) is represented by the segment \(fF\) on the same scale on which the quantities \(W_1\) and \(W_2\) are represented by the segments \(AF\) and \(CH\). Hence

$$ W_2 = CH = AF - fF = Af. $$

d) The measure of the number \(s\), determining the speed of rotation of the motor, is the ratio

$$ \frac{OC}{AB}=\frac{s\omega MJ}{\omega JN}=\frac{M}{N}\,s=\mathrm{Const.}\,s. $$

But

$$ OC=AO-AC=AB(\operatorname{ctg}\varphi-\operatorname{tg}\hat{\delta}), $$

and therefore

$$ s=\mathrm{Const.}(\operatorname{ctg}\varphi-\operatorname{tg}\hat{\delta}). $$

This quantity is represented by the segment \(MM_1\) of any vertical between the fixed straight line \(BA_0\) and the direction of the vector \(AB\). The number \(s=0\) when the vectors \(BA\) and \(BA_0\) coincide, and consequently \(BA_0=J_0\), i.e. the current at the moment of starting the motor into operation.

e) The torque \(T\) may be represented either by the segment \(BF\), or by the segment \(fF\); indeed

$$ T=\frac{W_2}{s\omega}; $$

but

$$ W_2=CH;\quad s=\mathrm{Const.}\,\frac{OC}{AB}, $$

therefore

$$ T=\mathrm{Const.}\,\frac{CH \times AB}{OC}, $$

and since

$$ \frac{CH}{OC}=\frac{BF}{AB}, $$

$$ T=\mathrm{Const.}\,BF=\mathrm{Const.}\,fF. $$

f) Finally, let us find the length representing the electrical efficiency coefficient. The ohmic loss, in fractions of the absorbed energy, is the ratio

$$ \frac{R J^2}{W_1}=\frac{fF}{AF}=\operatorname{tg}\delta\cdot\frac{BF}{AF}=\text{Const.}\operatorname{tg}\varphi. $$

This quantity is represented by the segment \(nn'\) of any straight line \(nn'\) parallel to the axis \(BO\), between the ordinate axis \(BK\) and the vector \(BA\). Since at \(s=0\) all the energy supplied to the motor goes into heating the windings, let the segment \(nn''\) be taken as unity; then the electrical efficiency \(\eta\) is represented by the difference between this segment and the segment \(nn'\)

$$ \eta=n'n''. $$

To make use of the diagram constructed, we must also find the scales for measuring the resulting segments.

The scales of the coefficients, power, and efficiency are already known: in the first case the base of the diagram \(BO\) is taken as unity, in the second—the segment \(nn''\).

The scale for measuring the current is found from the consideration that

$$ BA_0=J_0, $$

the current at no load, and the numerical value of the latter is found by putting \(s=0\) in the basic equation,

$$ BA_0=J_0=\frac{E_k}{\sqrt{R^2+\omega^2 N^2}}\ \text{amperes}. $$

Having calculated \(J_0\), it is not difficult to find the scale of powers, since the segment \(A_0F_0\) represents the power absorbed by the motor with the armature stalled, going entirely into heating the windings,

$$ A_0F_0=\frac{1}{2}RJ_0^2\ \text{watts}. $$

We find the scale of the number \(s\) by constructing the angle \(\varphi_1\), at which \(s=1\), and determining it from the condition

$$ \operatorname{tg}\varphi_1=\frac{\omega N}{R+\omega M}. $$

The mechanical power \(W_2\) is expressed by means of the current and the constants of the motor as follows

$$ W_2=\frac{1}{2}J^2s\cdot\omega\cdot M\ \text{watts}. $$

Hence the torque \(T\) in kg-m will be

$$ T=\frac{2.736}{75}J^2M. $$

Substituting here \(J_0\) in place of \(J\), we find the torque at starting, represented by the segment \(A_0F_0\) or \(BF_0\). Consequently, the scale of torques is determined from the condition

\[ A_0F_0 \text{ or } BF_0=\frac{75}{2.736}J_0^2 M \ \text{kg·m}. \]

With the aid of the vector diagram constructed, it is easy to construct a diagram in rectangular coordinates (Fig. 23), taking \(J\) as the independent variable. The curves obtained show complete similarity between the motor under consideration and a direct-current motor with series excitation.

Fig. 23

The diagrams obtained experimentally differ somewhat from those constructed theoretically by the indicated method: first, because we did not take into account the change of magnetic permeability at saturation, and second, because we neglected all losses except ohmic ones. But in general the character of the curves remains the same, as is seen from Fig. 24, which shows curves taken from a Westinghouse motor of 100 horsepower at 225 volts and 25 periods, the coefficient of efficiency being taken together with the gear transmission. Curve \(T\) is the curve of tractive efforts, proportional to the torques.

The losses omitted by us can be approximately introduced into the diagram. Hysteresis losses fall into two parts: the loss due to reversal of magnetization of the iron of the inductor and armature as a consequence of the alternating current, and the losses due to reversal of magnetization of the iron of the armature alone during rotation. The dependence of the losses of the second category, as well as the losses from

...of Foucault currents on speed and load can be found only from experiment. We shall neglect these losses. Losses of the first category, as is known, are proportional to the 1.6 power of the maximum induction and—consequently—to approximately the 1.6 power of the current \(J\). If, partly for simplicity and partly in order to take account of the discarded losses, we replace the power 1.6 by the second power, then the hysteresis loss is \(=\mathrm{Const.}\,J^2\), and therefore we can introduce the hysteresis loss by a corresponding increase of \(R\).

Fig. 24.
In the diagram: left scales—miles/hour and values \(1.0, 0.8, 0.6, 0.4, 0.2\); right scales—pounds and HP; horizontal scale—amperes. Curves are marked \(v\), \(W_2\), \(J\), and \(\varphi b\).

Friction losses according to Heibach can be introduced in the following manner: compute, for normal operation of the motor, the friction loss \(W_f\), and represent it on the power scale by the segment \(W_f=f'h\); set it upward from the straight line \(BA_0\). At a speed equal to zero, i.e. at the point \(A_0\) of the diagram, this segment becomes zero.

If we assume that this loss is proportional to the load, and draw a straight line through the points \(A_0\) and \(h\), then at any load the friction loss will be represented by the ordinate segment between the straight lines \(A_0B\) and \(A_0h\).

5. Diagram of the repulsion motor.

In Fig. 25 are shown the polygons of vectors representing equations (8); \(OJ_1\) and \(OJ_2\) are the directions of the vectors of the primary and secondary currents. Laying off in the proper directions their vectors,

\[ OA=R_1J_1,\qquad AB=\omega L_1J_1,\qquad BC=\omega MJ_2, \]

components of e.m.f. included in the first equation of group (8), we obtain the voltage polygon of the armature winding \(OABCO\). The resultant vector \(OC\) represents the voltage \(E_k\) at the terminals. In the same manner let us construct the e.m.f. polygon of the rotor winding \(OPQRO\), corresponding to the second equation (8), where

\[ OP = R_2 J_2,\qquad PQ = \omega L_2 J_2,\qquad QR = \omega M J_1,\qquad OR = s\omega N J_1. \]

In order to make use of the polygons found for investigating the properties of the repulsion motor in the same way as we made use of the triangle of Fig. 21 for the series motor, we must be able to construct, starting from some vector constant in magnitude and direction, the geometric locus of the ends of all the component vectors of the polygons.

Of the eight vectors entering into both polygons, only the vector of the terminal voltage \(E_k\) will have a magnitude independent both of the load and of the brush position. It is therefore most convenient to take it as the basis of the diagram.

Fig. 25.

Let us find the geometric locus of the point \(B\); in Fig. 26 let us take the point \(O\) as the origin of coordinates, the direction of the terminal-voltage vector \(OC\) as the \(X\)-axis, and the direction \(OC'\), leading \(OC\) by \(90^\circ\) (rotation of the vectors counterclockwise), as the \(Y\)-axis. Then, introducing the notation (Fig. 25)

\[ \angle COJ_1 = \zeta_1,\qquad \angle COJ_2 = \zeta_2,\qquad \angle OBA = \delta_1,\qquad \angle OPQ = \delta_2, \]

where \(\zeta_1, \zeta_2\) are the angles of lag of the primary and secondary currents behind the terminal voltage, and

\[ \delta_1 = \operatorname{arctg}\left(\frac{R_1}{\omega L_1}\right),\qquad \delta_2 = \operatorname{arctg}\left(\frac{R_2}{\omega L_2}\right), \]

the coordinates of the point \(B\) may be represented in the form:

\[ x = \frac{AB}{\cos \delta_1}\sin(\zeta_1+\delta_1),\qquad y = \frac{AB}{\cos \delta_1}\cos(\zeta_1+\delta_1) \]

or

\[ x = E - BC\cdot \sin \zeta_2,\qquad y = -BC\cos \zeta_2. \]

From triangle \(OQR\)

\[ QR = OQ \cos OQR, \]

but

\[ OQ=\frac{PQ}{\cos \delta_2}, \qquad \angle OQR=\pi-(\varphi_2-\varphi_1+\delta), \qquad \frac{PQ}{BC}=\frac{L_2}{M}, \qquad \frac{QR}{AB}=\frac{M}{L_1}, \]

and therefore

\[ -\,BC \cdot \cos(\varphi_2-\varphi_1+\delta) = AB\,\frac{M^2}{L_1L_2}\cos\delta_2 . \]

Eliminating \(BC\), \(AB\), \(\varphi_1\), and \(\varphi_2\) from the equation obtained and the four expressions for \(x\) and \(y\), we find the equation of the required geometric locus:

\[ \tag{10} \left[M^2\cos\delta_1\cos\delta_2-L_1L_2\cos(\delta_1+\delta_2)\right](x^2+y^2) \]
\[ +\,E_kL_1L_2\left[\cos(\delta_1+\delta_2)x-\sin(\delta_1+\delta_2)y\right]=0. \]

Fig. 28.

The curve found is a circle passing through the origin of coordinates. The equation of the normal to this circle at the origin of coordinates is

\[ \sin(\delta_1+\delta_2)x+\cos(\delta_1+\delta_2)y=0. \]

The distance from the origin of coordinates to the point \(F\) of intersection of the circle with the axis of abscissas (Fig. 26) is found by putting \(y = 0\) in equation (10):

\[ OF = \frac{Ek}{ 1-\dfrac{M^2 \Cos \delta_1 \Cos \delta_2}{L_1 L_2 \Cos(\delta_1+\delta_2)} }. \]

Hence follows a simple method of constructing this circle. Compute \(OF\), lay off this magnitude, on the voltage scale, in the positive direction from the origin of coordinates, i.e. from the origin of the vector of the terminal voltage; from the middle of the segment \(OF\) erect a perpendicular; then draw the straight line \(OO_1\) at the angle \(FOO_1 = -(\delta_1+\delta_2)\). The center of the circle is the point of intersection of the perpendicular and the inclined line.

The position of the point \(B\) on the circle \(OBF\) completely determines all the vectors of the e.m.f. of the inductor, which form the polygon \(OABCO\). In order to connect with the position of the point \(B\) also the polygon of the e.m.f. of the armature, denoted in Fig. 25 by the letters \(OPQRO\), we multiply all its component vectors by the constant factor

\[ \frac{L_1}{M}. \]

Then the vector \(QR\) becomes equal to the vector \(AB\); superposing them, we shall observe that at the same time the directions of the vectors \(OA\) and \(OR\), and of the vectors \(PQ\) and \(BC\), also coincide. Thus we obtain the polygon \(ABNMA\) (Fig. 26), where

\[ AB=\omega L_1 J_1,\qquad AM=s\omega L_1 \frac{N}{M} J_1, \]

\[ MN=\frac{R_2 L_1}{M}J_2,\qquad BN=\frac{\omega L_1 L_2}{M}J_2. \]

Let us now see how, using the construction made, to find segments proportional to the speed, the currents in the windings of the inductor and armature, the absorbed and developed power, the torque, the power factor, and the efficiency.

a) Speed. Dividing \(AM\) by \(AB\), we obtain

\[ s\frac{N}{M}=\frac{AM}{AB}=\tg ABM. \]

But it is not hard to see that

\[ \angle ABM=\angle G_0OG, \]

and consequently

\[ s\cdot\frac{N}{M}=\tg G_0OG, \]

or, introducing the notation \(\beta = \operatorname{arctg}\dfrac{N}{M}\),

\[ s=\frac{\operatorname{tg}G_0GG}{\operatorname{tg}\beta}. \tag{11} \]

Hence the following construction of the quantity \(s\): on the \(Y\)-axis take an arbitrary point \(S\) and from this point erect the perpendicular \(SS_1\); continue the vector \(BC\) until it intersects the circle at the point \(G\); the segment \(Ss\), from the foot of the perpendicular to its intersection with the straight line \(OG\), is proportional to the number \(s\), and, in the proper scale,

\[ s=Ss. \]

We shall find the scale for measuring \(s\) by observing that \(s=1\) when \(G_0OG=\beta\). Therefore, drawing the straight line \(Os_1\) so that \(\angle SOs_1=\beta\), we obtain the segment \(Ss_1=1\). From formula (11) it is clear that, when

\[ \beta=\frac{\pi}{2},\quad s=0, \]

i.e. motion is impossible.

b) The primary current \(J_1\) may, in magnitude, be represented by the vector

\[ OB=\sqrt{R_1^2+\omega^2L_1^2}\,J_1; \]

in direction this vector leads the current by the angle

\[ \frac{\pi}{2}-\delta_1. \]

c) The secondary current is represented by the vector

\[ BC=\omega M J_2. \]

d) The power factor \(\operatorname{Cos}\varphi_1\) is found by drawing the straight line \(OT\) so that \(\angle SOT=\delta_1\), and constructing on the diameter of arbitrary length \(OT\) a semicircle. If \(OT=1\), then \(Of_1=\operatorname{Cos}\varphi_1\).

e) The power absorbed by the motor is

\[ W_1=E_kJ_1\operatorname{Cos}\varphi_1. \]

Since \(J_1=\mathrm{Const}.\,OB\), then

\[ W_1=\mathrm{Const}.\,OB\operatorname{Cos}\varphi_1. \]

If we draw the straight line \(OJ\) so that \(\angle FOJ=\delta_1\), and drop onto it from \(B\) the perpendicular \(BW\), then \(\angle OBW=\varphi_1\), \(OB\operatorname{Cos}\varphi_1=BW\), and consequently, in the proper scale,

\[ W_1=BW. \]

f) We shall find the mechanical power, neglecting losses due to friction, hysteresis, and Foucault currents. The electrical efficiency of the armature winding is

\[ \eta_1=\frac{E_k\operatorname{Cos}\varphi_1-R_1J_1}{E_k\operatorname{Cos}\varphi_1}=\frac{aC}{OC}=aC, \]

...if \(OC = 1\). The power transmitted to the armature winding is the product of the electromotive force developed in the armature under the action of the pulsations of the resultant field, by the current \(J_2\), and by the cosine of the angle of phase difference of this electromotive force and the current \(J_2\). This electromotive force is balanced by the electromotive force of armature rotation (vector \(AM\)) and by the electromotive force expended on ohmic resistance (vector \(AN\)), and consequently is represented by the vector \(AN\). Therefore the electrical efficiency of the armature winding is

\[ \eta_2 = \frac{AN \cdot \operatorname{Cos} ANM - MN} {AN \operatorname{Cos} ANM} = \frac{hM}{hN}, \]

if \(Ah\) is the perpendicular dropped from \(A\) to the direction \(MN\).

If, further, we drop a perpendicular from \(C\) to \(AB\), and through the point \(m\) of intersection of this perpendicular with the straight line \(BM\) draw the straight line \(mn \parallel AB\), then

\[ \frac{hM}{hN} = \frac{AM}{AL} = \frac{Qm}{Qc} = \frac{an}{aC}, \]

and the total electrical efficiency of the motor will be

\[ \eta = \eta_1 \eta_2 = an. \]

Then the mechanical power is

\[ W_2 = \eta W_1 = \operatorname{Const.}\, OB . Cs\, \eta_1 . an = \operatorname{Const.}\, OB . mQ. \]

g) Rotating moment

\[ T = \operatorname{Const.}\, \frac{W_2}{s} = \operatorname{Const.}\, OB\, \frac{\operatorname{tg} ABM}{mQ}, \]

and consequently

\[ T = \operatorname{Const.}\, OB . PB. \]

The point \(P\) is evidently the point of intersection of \(OB\) and the circle \(OPC\), constructed on \(OC\) and subtending the angle

\[ \frac{\pi}{2} + \delta_1. \]

The quantities \(W_2\) and \(T\) in our diagram are represented by products of two segments; \(PB\) represents the rotating moment referred to a unit of primary current.

To facilitate the construction, we note that

\[ \smile KO = 2\delta_1,\qquad \smile m_1G = 2\delta_2. \]

Considering the diagram, it is not difficult to see that, as the rotating moment changes from zero to the maximum, developed at

at starting from rest, point \(B\) moves along the circle from point \(P_0\) to point \(B_0\). The point \(P_0\) is the point of intersection of the circles \(OBF\) and \(OPC\) and—therefore—corresponds to \(PB=0\), i.e. to no-load running. When point \(B\) reaches the position \(B_0\), \(G\) will coincide with \(G_0\), and the segment \(Ss\), representing the speed, will become zero: the point \(B_0\) corresponds to \(\omega_1=0\), i.e. to starting from rest, or to a short-circuited armature.

By following on the diagrams the change of the torque \(T\) as a function of the change of the current \(J_2\) in the armature winding, we shall see that this current, under all loads, deviates comparatively little from its mean value, and that the decrease of the torque as the speed increases depends not on the decrease of the secondary current, but on the change of the phase-difference angle of the currents \(J_1\) and \(J_2\).

It remains to find the scales for measuring the vectors \(J_1\), \(J_2\), and \(T_r\), i.e. the segment \(PB\), representing the torque referred to the unit of primary current.

Consider the triangle \(OB_0C\), in which \(OB_0=J_{20}\)—the current at starting from rest, and \(\angle OB_0C=\delta_1+\delta_2\).

From this triangle,

\[ OC^2=OB_0^2+CB_0^2-2\,OB_0\cdot CB_0\cos(\delta_1+\delta_2), \]

\[ CB_0=OB_0\cos(\delta_1+\delta_2)-OC\cdot\cos OCG_0, \]

\[ \frac{OB_0}{2}=OO_1\cos O_1OB_0=OO_1\cos OCG_0, \]

whence

\[ OB_0=\frac{2\cdot OO_1\cdot OC}{\sqrt{OC^2+4OO_1^2\sin^2(\delta_1+\delta_2)}}, \]

\[ CB_0=\frac{OC\bigl(2\cdot OO_1\cdot\cos(\delta_1+\delta_2)-OC\bigr)} {\sqrt{OC^2-4OO_1^2\sin^2(\delta_1+\delta_2)}}. \]

But from the equation of the circle

\[ OO_1=\frac{E_kL_1L_2}{2\left(M^2\cos\delta_1\cos\delta_2-L_1L_2\cos(\delta_1+\delta_2)\right)}, \]

and since

\[ OC=E_k,\quad OB_0=J_{10}\sqrt{R_1^2+L_1^2\omega^2},\quad CB_0=J_{20}\omega M, \]

then

\[ J_{10}= \frac{E_k L_1 L_2} {\sqrt{(R_1^2+L_1^2\omega^2)(M^2\operatorname{Cos}\delta_1\operatorname{Cos}\delta_2+L_1L_2)[M^2\operatorname{Cos}\delta_1\operatorname{Cos}\delta_2-(1+2\operatorname{Cos}(\delta_1+\delta_2))L_1L_2]}} , \]

\[ J_{20}= \frac{E_k\{2L_1L_2\operatorname{Cos}(\delta_1+\delta_2)-M^2\operatorname{Cos}\delta_1\operatorname{Cos}\delta_2\}} {M\omega\sqrt{(M^2\operatorname{Cos}\delta_1\operatorname{Cos}\delta_2+L_1L_2)[M^2\operatorname{Cos}\delta_1\operatorname{Cos}\delta_2-(1+2\operatorname{Cos}(\delta_1+\delta_2))L_1L_2]}} . \tag{12} \]

Since for \(s=0\) all the energy, if all losses except the ohmic ones are neglected, goes to heating the windings, then

\[ W_{10}=B_0 W_0=R_1J_{01}^{2}+R_2J_{20}^{2}. \]

Knowing the scales for measuring the primary power, the efficiency coefficient, and the speed, we shall find \(W_1\) from the diagram for some definite speed, for example—the synchronous one, and determine the torque from the relation

\[ T \text{ at synchronism}= \frac{W_1 \text{ at synchr.}\times \eta \text{ at synchr.}}{\omega}. \]

Having thus determined all the necessary scales, from the circular diagram it is easy to obtain the diagram in rectangular coordinates.

Fig. 27.

On Fig. 27 such a diagram is shown for a Gen. El. Co. motor of 60 h.p. for 500 volts and 25 periods. (Slichter El. Wd. 1904).

6. Diagram of a compensated repulsion motor.

The problem of the graphical investigation of the equilibrium equations of the e.m. forces of this motor is much more complicated than the preceding ones. In those latter cases it is sufficient to know the position of one of the vertices of the polygon of the primary e.m. forces, at a given terminal voltage, in order to construct the whole polygon; in the case of a compensated motor it is necessary to know the position of three vertices of the polygon, and their loci will not be circles, but more complicated curves.

For brevity, in equations (9) let us denote

\[ \omega (L_1 + 2M_1 + L_2) = a,\quad \omega N_1 = b_1,\quad \omega (M + M_2) = c,\quad \omega N = d, \]

\[ \omega (N + N_2) = m,\quad \omega L'_2 = n. \]

Graphically these equations will be represented by the polygons \(O A_1 A B C D O\) and \(O N P Q O\) (Fig. 28), where

\[ O A_1 = R_1 J_1,\quad A_1 A = aJ_1,\quad A B = cJ_2,\quad B C = sdJ_2,\quad C D = sbJ_1, \]

\[ O D = E_k,\quad QD = smJ_1,\quad OQ = cJ_1,\quad PN = R_2J_2,\quad NO = nJ_2. \]

We shall now eliminate from the polygon \(O A B C D\) the vectors that depend on \(J_2\), using the polygon \(O N P Q\). For this, on the sides

Fig. 28.

\(AB\) and \(CD\), as on hypotenuses, construct the right triangles \(AB'B\) and \(CC'B\), similar to the triangle \(OPQ\). Then, if we denote

\[ \tg \delta_1=\tg OAA_1=\frac{R_1}{a},\quad \tg \delta_2=\frac{R_2}{n}, \]

we have, since

\[ OP=\frac{n}{\Cos \delta_2}J_2, \]

\[ AB'=\frac{c_2}{n}\Cos \delta_2 J_1,\qquad BB_1=s\frac{cm}{n}\Cos \delta_2 J_1, \]

\[ C'B=s\frac{dc}{n}\Cos \delta_2 J_1,\qquad CC'=s_2\frac{dm}{n}\Cos \delta_2 J_1. \]

To obtain the equations of the sought geometric loci, we must represent the coordinates of each of the vertices of the polygon, choose one of the sides as the base of the diagram, and take the sum of the projections of the sides of the polygon \(OAB'BC'CDO\) on the coordinate axes; in doing so each coordinate will be expressed in two ways, and from the four equations thus obtained \(s\), \(\mathcal{E}_1\), and \(J_1\) must be eliminated. The equations found in this way are very complicated; in order to simplify the diagrams, we proceed as follows.

Let us divide all the vectors of the polygon \(OAB'BC'C'CD\) by \(J\). Then we shall obtain a polygon similar to the former one, but all of whose vectors have the dimension of resistance; such a polygon can therefore be called the diagram of apparent resistances. After the indicated division the vectors \(OA\) and \(AB'\) will be constant, and consequently the triangle \(OAB'\), by virtue of the constancy of the angles \(\delta_1\) and \(\delta_2\), will be an unchanging figure. We shall place this triangle at the base of the diagram, drawing it in an arbitrary scale of resistances. If, when the load changes, we leave its vertices fixed, then the point \(B\) will move along the straight line \(B'B\), perpendicular to the fixed vector \(AB'\), while the geometric loci of the points \(C\) and \(D\) will be certain curves, which we shall now determine.

Let us take the point \(B'\) as the origin of coordinates; as the \(X\)-axis, the direction of the straight line \(B'A\) (from the point \(B'\) to the right), and as the \(Y\)-axis, the straight line \(B'B\) (upward), perpendicular to it. Then the coordinates of the point \(C\) are

\[ x=C'C=s^2\frac{dm}{n}\Cos \delta_2,\quad y=B'C'=s\frac{(m-d)c}{n}\Cos \delta_2; \]

eliminating \(s\), we obtain:

\[ x=y^2\frac{nmd}{(m-d)^2c^2\Cos \delta_2}. \tag{13} \]

the equation of a parabola with vertex at \(B'\), whose axis coincides with the \(X\)-axis.

The coordinates of the point \(D\):

\[ x=s^{2}\frac{m d}{n}\operatorname{Cos}\delta_{2}+s l\operatorname{Sin}\delta_{2}, \]

\[ y=s\left(b+\frac{(m-d)c}{n}\right)\operatorname{Cos}\delta_{2}, \]

whence, eliminating \(s\), we have

\[ \tag{14} x=y^{2}\frac{m d n}{\left[b n+(m-d)c\right]^{2}\operatorname{Cos}\delta_{2}} +y\frac{b n}{b n+(m-d)c}\operatorname{tg}\delta_{2} \]

—also the equation of a parabola passing through the point \(B'\), with its axis parallel to the \(X\)-axis. For very small \(R_{2}\) the axes of both parabolas coincide.

Having constructed the fundamental triangle \(AB'O\) of the diagram of apparent resistances and both parabolas, for each position of the point \(B\) or \(D\) it is not difficult to construct the entire polygon in which the vectors important to us are: \(AB\), proportional to the ratio \(\dfrac{J_{2}}{J_{1}}\), and \(OD\), proportional to the ratio \(\dfrac{E_{k}}{J_{1}}\). It is best to proceed from the point \(B\), whereby we set the magnitude \(s\). The scale \(s\) is determined from the following considerations.

From Fig. 28 b

\[ s\frac{m}{c}=\operatorname{tg}POQ, \]

or, passing to Fig. 28 a,

\[ s=\frac{c}{m}\operatorname{tg}BAB'. \]

Consequently \(s=1\), when \(\operatorname{tg}BAB'=\dfrac{m}{c}\), or, introducing the notation

\[ \beta=\operatorname{arc\ tg}\frac{m}{c}, \]

\[ s=1,\quad \text{when } BAB'=\beta . \]