ESTIMATION AND CONTROL OF t)l'EIlATE TLME OF RELAYS 109 



when proposed. This can be avoided, with no loss in generahty, by trans- 

 forming each relay into an equi^^alent single relay in a local cii-cuit. Then 

 the foregoing methods for single relays can be applied to each in lurn. 



This procedure is the common device of breaking iij) a complicated 

 problem into parts, each of which can be solved by familiar methods. 

 Two assumptions are made. The first is that each winding is a lumped 

 two terminal nc^twork. At any instant the same current is in (>v(ny turn 

 of each relay and there is no propagation time involved. This holds for 

 the times invoh-ed in electromagnets. The second assumption is that, 

 when the relays have different op(M'at(> times, the current reduction 

 caused by the motional impedance of the first one to operate does not 

 significantly extend the operate times of the later ones. In the following 

 transformations, the winding turns, ciu'rents, and hence magnetomotive 

 forces, are kept constant. 



Identical Relays 



If the two relays are identical they have some impedance Z(p) which 

 is the same for both. Part of this is the dc resistance and the other part 

 is the ac effect, proportional to the tiu'ns squared. By the extension of 

 Ohm's Law to ac circuits, when a potential source is applied to the two 

 identical devices in series, exactly one-half the source appeal's across 

 either device at any time after application, forming a virtual constant 

 potential point. Thus if a battery of Ea volts is applied, exactly one-half 

 the battery appears across each, including the effect of eddy currents. 

 Now if the voltage across a coil is known, then the response is uniquely 

 determined, knowing just the relay characteristics and the voltage, 

 disregarding the mechanism of how the voltage is applied. For this 

 situation the voltage is in a most convenient form, represented by ex- 

 actly one-half the battery. The operate time can easily be determined 

 for either relay with this information, as the effects of eddy currents and 

 motion are included in the data. The coil constant is already known and 

 the power is one-half the total power. 



General Case 



This procedure can be generalized to include any division of dc re- 

 sistance, different turns, and diffei-ent magnetic structures providing, 

 for the latter case, that eddy currents can be ignored. The justification 

 for this will be considered later. 



The basic problem is: gi\'en the battery voltage E^ , relay No. 1 of .Vi 

 turns, a 1 turn inductance Li , and resistance T^i , in series with relay No. 



