Alternating Current Electric Motors - II

To turn from an example of class I. to the best example of class II. of alternating motors, I will try to discuss the novel and admirable machine invented by Mr. Tesla.

The problems involved are not very different from converter problems, and I confess that I approach them with something of ill-humor. I evolved the calorimeter method* of testing transformers, and it has been absorbed by an eminent English electrician; with Messrs. Wilkes and Hutchinson I produced† a method for obtaining simultaneous current and E. M. F. curves in the primaries and secondaries of converters. Another set of experimenters has since used the method and appropriated the credit. These are small matters, but they cause irritation. I imagine the results were somewhat too early. The editor of an electrical periodical, in reproducing one of the irregular primary curves (see the paper referred to, or Prof. Ryan’s work), remarked on the singular want of symmetry in the construction of our dynamo.

In its simplest form the Tesla motor consists of an armature rotating in a magnetic field, which is the resultant of two fields at right angles to one another, excited by two alternating currents differing in phase by \(\frac{\pi}{2}\). The resultant field rotates with a period equal to that of the exciting currents. The armature coils are short-circuited on themselves, and currents are induced in them in a direction to cause the armature to rotate with the field. Let us consider the simple case of a Tesla motor without iron, the armature of which has a large number of coils on it.

Let the two fields be as shown in Fig. 4, and let one of the armature coils make an angle \(\alpha\) with the horizontal. We will consider what takes place in this coil. We have in the first place an E. M. F. due to the induction of the two circuits 1 and 2, and in the second place a counter E. M. F. due to the motion of the coil in the fields of 1 and 2. The current is of course due to the difference of these two E. M. Fs. and the energy transformed is the product of the current and the counter E. M. F.

Let

\( i_1 \) = current in field 1.
\( i_2 \) = current in field 2.
\( I \) = maximum value of current.
\( E_1 \) = E. M. F. induced in coil from 1.
\( E_2 \) = E. M. F. induced in coil from 2.
\( e_1 \) = counter E. M. F. due to rotation in field of 1.
\( e_2 \) = counter E. M. F. due to rotation in field of 2.
\( M \) = maximum mutual induction between coil and 1 or 2.

Then if

\( i_1 = I \sin pt \), \( i_2 = I \cos pt \), and if the induction varies with the current,

\[ \left. \begin{aligned} E_1 &= M \cos\alpha \frac{di_1}{dt} = MI p \cos pt \cos\alpha \\ E_2 &= M \sin\alpha \frac{di_2}{dt} = MI p \sin pt \sin\alpha \end{aligned} \right\} = MI p \cos(pt - \alpha) = E \]

\[ \begin{aligned} e_1 &= -MI \sin pt \sin\alpha \frac{d\alpha}{dt}, \\ e_2 &= -MI \cos pt \cos\alpha \frac{d\alpha}{dt}. \end{aligned} \]

Now \(\frac{d\alpha}{dt}\) is equal to \(\frac{2\pi}{T'}\), where \(T'\) is the time of revolution of the armature; so we may put the above, \(p'\) being equal to \(\frac{2\pi}{T'}\).

\[ \begin{aligned} e_1 &= -MI p' \sin pt \sin\alpha, \\ e_2 &= -MI p' \cos pt \cos\alpha. \end{aligned} \]

Adding all these E. M. F’s. and noting that \(\alpha\) may be determined by \(\alpha = p't\) if our zero time be properly chosen,

\[ E_1 + E_2 + e_1 + e_2 = MI(p - p')\cos(p - p')t = e_0. \]

The maximum value of the current due to this E. M. F. is

\[ \frac{MI(p - p')}{\sqrt{R^2 + (p - p')^2 L^2}}, \]

where \(R\) and \(L\) refer to the coil under discussion. The total energy transformed is

\[ \frac{1}{2} \frac{M^2 I^2 p' (p - p')}{R^2 + (p - p')^2 L^2} \cos\theta, \]

where \(\tan\theta = \frac{(p - p')L}{R}\).

The energy used in heating is \(\frac{1}{2} \frac{M^2 I^2 (p - p')^2}{R^2 + (p - p')^2 L^2} R\), and the armature efficiency is \(\frac{p'}{p}\). This last is true because the applied and counter E. M. F’s. are 180 degrees different in phase, and they are therefore equally distant from the current. If we multiply the maximum values of \(E\) and \(e\) by one-half the maximum value of the current and by the cosine of \(\theta\), we will have in the first case the electrical energy applied to the armature, and in the second the electrical energy transformed into mechanical energy. The ratio of these products—the angle being the same—is simply \(\frac{p'}{p}\).

Looking at the value of the total energy transformed, we see that as \(L\) increases both the increase of the impedance and the decrease of \(\cos\theta\) will necessitate—for a given output—a decrease of \(p'\). This, of course, lowers the efficiency. As the expression for the output contains \(p'(p - p')\) it is evident that with no self-induction the maximum output is when \(p' = \frac{1}{2}p\); the efficiency being 50 per cent. If the motor has a self-induction, however, it is easily seen from the equation that the maximum output is for a value of \(p' > \frac{1}{2}p\); there is no need of giving the value.

The induced E. M. F. in the armature and the effective E. M. F. are in the same phase, while the counter-E. M. F. differs from them in phase by \(\pi\). If there were no self-induction the armature current would follow the impressed E. M. F., and the condition of affairs would be the best possible, for the maximum current would fit in with the maximum counter E. M. F., as it should. The effect of self-induction is to shift the current back from its best position and decrease the output for a given heating effect. Now, the permissible current in a machine of this kind is entirely determined by heating effects; so the self-induction should be as small as possible. The ratio of the heating effect to the output of the motor is

\[ \frac{p - p'}{p} \sec\theta. \]

For a given current \(\frac{p - p'}{p}\) is proportional to \(R\), and we see that \(\tan\theta = \frac{(p - p')L}{R}\). It goes without saying that if \(R\) is decreased, the ratio given in the equation also decreases, and the efficiency and output of the machine increases.

Before discussing the matter further let us look at the armature actions in a different way. Consider a line whose azimuth is fixed and equal to \(\alpha\), and think of the condition of the coils in this azimuth. We have an applied E. M. F. equal, as we have seen, to

\( MI p \cos(pt - \alpha) \)

and a counter E. M. F. equal to

\( MI p' \cos(pt - \alpha) \).

As far as the line is concerned we have impressed on it—\(\alpha\) is now fixed—an E. M. F. of period \(p\) instead of the period \(p - p'\) of the E. M. F. in a particular wire. But the current does not lag behind the E. M. F. by an angle \(\theta = \tan^{-1} \frac{p L}{R}\), but by an angle \(\theta = \tan^{-1} \frac{(p - p') L}{R}\), for we must remember that the wire which at any given instant has in it the maximum induced E. M. F. is rotating in the same direction as the counter E. M. F.; and when the current reaches its maximum it is closer to the counter E. M. F. by the angle through which it has rotated.

Comparing our results to those obtained in a transformer, we have an impressed E. M. F. in the moving wires which bears the ratio \(\frac{p - p'}{p}\) to the impressed E. M. F. that would exist if there were no rotation. In the same way \(\tan\theta\) is now \(\frac{(p - p') L}{R}\) instead of \(\frac{p L}{R}\), as it would be if the motor were still. Our current is

\[ \frac{MI(p - p')}{\sqrt{R^2 + (p - p')^2 L^2}}, \]

instead of

\[ \frac{MIp}{\sqrt{R^2 + p^2 L^2}}. \]

The result is the same as if we hold the motor and change its armature resistance in the ratio of \(p\) to \(p - p'\). As far as the transfer of energy from the field coils is concerned, we may reduce it to the case of two separate converters, each consisting of one pair of field coils for a primary, with a secondary made by winding one-half the armature coils at right angles to the magnetic field of the primary, the magnetic circuit being that corresponding to the pair of coils we are considering. There is no reason that the efficiency should not be high. The secondary of this converter may be considered to have in its circuit a non-inductive resistance equal to \(\frac{p}{p - p'} R\).

To take a specific case—and it is hard to find data on the subject—in the paper by Messrs. Hutchinson, Wilkes and myself referred to above, I find for a Westinghouse 10-light converter the following figures: Resistance of primary, 50 ohms; of secondary, .1 ohm; self-induction of primary, 1.96 henrys; of secondary, .0047 henrys. Under these circumstances, if our motor had the same constants we would have as a first approximation

\[ \frac{p - p'}{p} = 50 \quad (\text{I suppose 50 volts and 10 ampères}). \]

If there are 100 periods per second, \(\theta = \tan^{-1} \frac{(p - p') L}{R}\) about 30°, or the “plant efficiency” (\(= \cos\theta\)) is about 86 per cent. This being the case, \(p'\) would be less than the value assumed, and further approximations would increase the efficiency. I need not remark that the case chosen is a favorable one. The period chosen is impossible for a practical machine, and a lower one would be less advantageous.

Now let us consider what will happen if we increase all the linear dimensions of our machine. If we have a turn of wire around an iron ring, and increase the dimensions of the ring, both the self-induction and resistance will increase in the same proportion; that is, directly with the linear change. For if we suppose the permeability of the iron constant, the magnetic force due to a unit current in the wire would decrease in the ratio \(1/l\), while the area enclosed by the wire would increase as \(l^2\); the self-induction would therefore increase as \(l\), which is of course a well-known fact. But in the case of our motor, as we increase the diameter of our armature we increase the space which our wire occupies; and keeping the same thickness of winding and the same number of turns we increase the area of our wire \(l\) times and therefore decrease its resistance by the same amount. But if all the linear dimensions of the motor have increased, our thickness of wire can increase, so our resistance will again decrease by \(l\). Our self-induction has increased \(l\) times; our resistance has decreased by the same amount. If we now decrease the number of turns in a coil \(n\) times, both \(L\) and \(R\) decrease as \(n^2\). If we let the speed of our motor vary inversely as its dimensions, and the induction remains constant, then for the same E. M. F. \(n = l\), and \(L\) varies finally as \(\frac{1}{l}\) and \(R\) varies as \(\frac{1}{l^3}\). The ratio of \(L\) to \(R\) increases as our output increases, the increase being as \(l^2\). To take a specific case; in the example cited above, if we double the dimensions of the motor we had imagined as corresponding to the transformer whose data were given, then on the assumptions we have just made, if the output be increased eight (\(2^3\)) times the percentage loss in the armature will not be changed. This being the case \(p - p'\) is \(2(2\pi)\) as before; but at the same time \(\frac{L}{R}\) has four times its old value; so the tangent of the angle \(\theta\) is four times as great for the small motor. Practically we would not care to reduce our speed as much as we have imagined.

The Tesla motor is in reality very nearly a synchronous motor—not that I consider the fact much of an objection—and like all synchronous motors there are times when its coils act as dynamo coils, thereby reducing its efficiency and output. The speed may be decreased by increasing the number of poles, but it always bears an approximately fixed ratio to that of the dynamo. We cannot, however, run on one circuit a number of motors at an indefinite number of speeds.

As compared with a continuous-current motor the armature efficiency and the output for a corresponding armature will be less. The iron in both field and armature is reversed—the former at a rate corresponding to \(p\), the latter at a rate corresponding to \(p - p'\). The loss in the armature from eddy currents and hysteresis will be less than in a continuous-current motor, but the loss in the field magnets would much more than counterbalance this gain, the mass of the magnets being usually much greater than that of the armature core. At the high periods that have been used the induction that could be economically used would be comparatively low. It would be interesting to know if the jarring of the machine, with the open magnetic circuit due to the air gap, materially decreases hysteresis losses. One of the most important of the practical objections to this motor is the fact that two independent circuits must be used to magnetize the fields, so it cannot be used on ordinary lighting circuits. I will refer to this in a moment.

The great advantages of the motor lie in the fact that it has no commutator, and it permits the use of very high voltages. In the future, power will be transmitted electrically at voltages that will make machines with the commutators next to useless, while the Tesla motor will offer many advantages for such transmission.

Since bringing out his first motor, Mr. Tesla, with great industry and ingenuity, has attempted to remedy the defects in the original machine. Perhaps the most serious of these is the inability to use the simple form of motor on ordinary alternating circuits. Mr. Tesla has attempted to get the required difference of phase between the magnetizing circuits, using a single circuit from the dynamo. I have no space to describe all of these; indeed they have already been described in the technical journals. The objection I have to most of them is that they cannot give a definite phase difference for different loads on the motor. Even the plan that has been proposed of using the primary of a converter for one magnetizing circuit, with the secondary as the other circuit, the latter having a condenser connected with it to neutralize its self-induction, will not work perfectly. For, even supposing a satisfactory practical condenser could be found, it must be remembered that the self-induction of the magnetizing coil will change as the load on the motor changes, so the phase of the magnetizing currents will not differ by \(\frac{\pi}{2}\) except for some one load.

In another type of motor by the same inventor there are two sets of coils on one of the pairs of field magnet. One set is connected with the dynamo circuit; the other, in the relation of secondary to the first, is wound on the other pair of poles. A condenser or resistance will make the currents in the two circuits differ in phase by nearly \(\frac{\pi}{2}\). This plan does not differ much from the one just described; the extra transformation is performed in the motor instead of in a converter.

Still another plan of Mr. Tesla is to take advantage of the fact that magnetization soaks gradually into a solid core as current soaks into a wire. Some one, writing in Industries, has denied this; it is, however, true, and he obtains a difference in phase between his magnetic fields from this fact.

* The Electrical World, 1887. It was copied in English journals at the time.
† The Electrical World, March 31, April 28 and May 19, 1888.