M. Hagenbach's dynamometric measurements were made by the aid of a brake.
After each experiment on the electric machine, he applied the brake to the
engine which he employed, taking care to make it run at precisely the same
speed, with the same pressure of steam, and with the same expansion as
during experiment. It would certainly be better to measure the force
expended during and not after the experiment, by means of a registering
dynamometer. Moreover, M. Hagenbach writes that his measurements by means
of the brake were very much prejudiced by external circumstances; doubtless
this is the reason of the divergences between the results obtained.
About the same time Dr. Hopkinson communicated to this institution the
results of some very careful experiments made on a Siemens machine. He
measured the force expended by means of a registering dynamometer, and
obtained very high coefficients of efficiency, amounting to nearly 90 per
cent. M. Hagenbach also obtained from one machine a result only a little
less than unity. Mechanical generators of electricity are certainly
capable of being improved in several respects, especially as regards their
adaptation to certain definite classes of work. But there appears to
remain hardly any margin for further progress as regards efficiency. Force
transformed into electricity in a generator may be expressed by i [omega] M
C; [omega] being the angular velocity of rotation; M the magnetism of one
of the poles, inducing or induced, which intervenes; and C a constant
specially belonging to each apparatus, and which is independent of
the units adopted. This constant could not be determined except by
an integration practically impossible; and the product, M C, must be
considered indivisible. Even in a magneto-electric machine (with permanent
inducing magnets), and much more in a dynamo-electric machine (inducing by
means of electro-magnets excited by the very current produced) the product,
M C, is a function of the intensity. From the identity of the expressions,
i²R and i [omega] M C we obtain the relation M C = IR/[omega] which
indicates the course to be pursued to determine experimentally the law
which connects the variations of M C with those of i. Some experiments made
in 1876, by M. Hagenbach, on a Gramme dynamo-electric machine, appear to
indicate that the magnetism, M C, does not increase indefinitely with the
intensity, but that there is some maximum value for this quantity. If,
instead of working a generator by an external motive force, a current is
passed through its circuit in a certain given direction, the movable part
of the machine will begin to turn in an opposite direction to that in which
it would have been necessary to turn it in order to obtain a current in the
aforesaid direction. In virtue of this motion the electro-magnetic forces
which are generated may be used to overcome a resisting force. The machine
will then work as a motor or receiver. Let i be the intensity of the
external current which works the motor, when the motor is kept at rest. If
it is now allowed to move, its motion produces, in virtue of the laws of
induction, a current in the circuit of intensity, i1, in the opposite
direction to the external current; the effective intensity of the current
traversing the circuit is thus reduced to i - i1. The intensity of the
counter current is given, like that of the generating current, by the
equation, i1²R = i1 [omega]1 M1 C1, or i1R = [omega]1 M1 C1, the index, 1,
denoting the quantities relating to the motor. Here M1 C1 is a function of
i - i1, not of i. As in a generator the force transformed into electricity
has a value, i [omega] M C, so in a motor the force developed by
electricity is (i - i1) [omega]1 M1 C1. On account, however, of the losses
which occur, the effective power, that is the disposable power on the shaft
of the motor, will have a smaller value, and in order to arrive at it a
coefficient of efficiency, K1, must be added. We shall then have W1 = K1
(i-i1) [omega]1 M1 C1. The author has no knowledge of any experiments
having been made for obtaining this efficiency, K1. Next let us suppose
that the current feeding the motor is furnished by a generator, so that
actual transmission by electricity is taking place. The circuit, whose
resistance is R, comprises the coils, both fixed and movable, of the
generator and motor, and of the conductors which connect them. The
intensity of the current which traverses the circuit had the value, i, when
the motor was at rest; by the working of the motor it is reduced to i - i1.
The power applied to the generator is itself reduced to W-[(i-i1)[omega]
M C]/K. The prime mover is relieved by the action of the counter current,
precisely as the consumption of zinc in the battery would be reduced by the
same cause, if the battery was the source of the current. The efficiency
of the transmission is W1/W. Calculation shows that it is expressed by the
following equations:W1/W = K K1 [([omega]11 M1 C1)/([omega]1 M C)], or = K
K1 [([omega]11 M1 C)/([omega]11 M1 C1 + (i-i1) R)]; expressions in which
it must be remembered M C and M1 C1 are really functions of (i-i1). This
efficiency is, then, the product of three distinct factors, each evidently
less than unity, namely, the efficiency belonging to the generator, the
efficiency belonging to the motor, and a third factor depending on the rate
of rotation of the motor and the resistance of the circuit. The influence
which these elements exert on the value of the third factor cannot be
estimated, unless the law is first known according to which the magnetisms,
M C, M1 C C1, vary with the intensity of the current.