778 EEPORT— 1903. 



6. Parallel Working of Alternators.^ By B. Hopkinson. 



The hunting of alternating-current machines is considered as a case of oscilla- 

 tion about a state of steady motion. The oscillations may be forced oscillations, 

 produced by uneven turning moment in tlie engines, or free oscillations, such as are 

 set up on switching a machine into parallel when slightly out of phase. The 

 importance of the forced oscillations depends largel}' on the relation between 

 the period of the cause producing them and the period of the free oscillations. 

 Let ^', be the maximum angular phase-displacement of a dynamo driven by a 

 periodically uneven turning force as compared with a wheel rotating uniformly at 



the same speed, the dynamo being disconnected from the bus-bars. Let ^ be the 



o 



i^ 

 period of the variation in turning moment, ^ the natural period of swing of tlie 



o 



dynamo. Then the maximum phase-displacement when the dynamo is connected 

 to bus-bars supplied with alternating E.M.F. of constant amplitude and perio- 

 dicity is 



The usual rule as regards flywheel effect is that ^', shall not exceed a certain 

 value, varying for different designs. This is equivalent to limiting ^„ provided 

 that 8 is small compared with S', or that the natural period of swing is long com- 

 pared with the forced period. With the weight of flywheel required to satisfy the 

 ordinary rules of design for proper parallel working, this condition is generally 

 fulfilled. These rules, therefore, as a rule, directly limit the phase angle of swing 

 in ordinary working. "WHiat is wanted, however, is limitation, not of the angle, 

 but rather of the fluctuation in the rate at which energy is given to the bus-bars to 

 wl)ich that angle corresponds. This depends on the self-induction of the machine. 

 Hence the very considerable difi'erences in the angular deviation permitted by 

 diflerent designers ; a machine with large self-induction will stand a bigger angular 

 deviation than will one with small self-induction. 



Similar considerations apply to a synchronous motor or a converter connected 

 to mains in which there is a periodical fluctuation of E.M.F., owing to uneven 

 velocity in the supply generators. In this case, however, as it is unusual to put 

 any flywheel on the motor, the danger of approximate equality between S and S' 

 is much greater. Probably many cases of hunting of rotaries might be better and 

 more cheaply cured by putting flywheels on the rotaries than by excessive require- 

 ments as to even turning moment in the generator. 



The importance of the free oscillations depends on whether they are damped 

 out or not. The equation of motion is^ 



M'fU 6:^1-1-0^ = 

 dt' dt ^ 



I being the phase displacement from the state of steady motion, and M the 

 moment of inertia of the machine on a suitable scale, The solution of this 

 equation is 



h 

 | = loC- 2M sin {U^ri). ■ 



where S = a/ --, if h is small, and r^ is the period of the oscillation. Most oscil- 

 lating systems possess viscosity, in which case b is positive, and the oscillations 



' Published in extenso in the Electrician, September 18, 1903. 



