110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195-1: 



and relay operation, it is necessary: (1) to be able to estimate the operate 

 and release times of any relay for which this information is needed in 

 circuit studies, and (2) to develop relays capable of meeting specific- 

 timing requirements, both fast and slow. 



The essential basis for such design and estimation is a dynamic theory 

 of the operation of electromagnets, of sufficient accuracy for engineering 

 use. The theory presented here is applicable to electromagnets in gen- 

 eral, although the applications discussed are those relating to relays. 

 The theory presented is approximate, partly because of the difficulty of 

 providing a more exact treatment, and partly because simplicity and 

 generality are more important for engineering purposes than accuracy 

 in estimation, which is in any case subject to correction by measured 

 results. The theoretical relations are used alone only in preliminary 

 estimation and in the initial stages of development, as discussed in Part 

 I of this article. In advanced development and in the modification and 

 application of existing structures, as discussed in Part II, the theory is 

 used as a guide in the correlation and extrapolation of observed per- 

 formance. 



The dynamics of electromagnets involve the concurrent and inter- 

 related phenomena of field development and armature motion, and are 

 accordingly governed by the two differential force equations respectively 

 applying, each containing a coupling term expressing their reaction on 

 each other. These basic equations are formally identical with those of 

 electromechanical transducers, such as loud-speakers, but the treatment 

 has little else in common. The operation of an electromagnet is the tran- 

 sient change from one state of ecjuilibrium to another, as distinguished 

 from the sustained low amplitude oscillations of the transducer case. 



The coupling terms represent the effect of field energy changes on the 

 coil voltage and on the force causing armature motion respectively. In 

 the steady state, the field energy is a function of magnetomotive force 

 and of armature position alone. In a transient state, eddy currents in 

 the magnetic members aft'ect both field energy and the pattern of the 

 field. In developing an approximate theory, it is assumed that these 

 effects are confined to the total effective mmf, and that the pattern of 

 the field is the same as that of the static field: i.e., that the field energy 

 associated with any portion of the structure, such as an air gap, is fixed 

 by the flux linkages of the coil and by the armature position. The limita- 

 tions on the analysis imposed by this sim])lifying assumption are dis- 

 cussed at the end of the next section. 



