220 BELL SYSTEM TECHNICAL JOURNAL 



Observe ilial wlien t = o the value of T is also zero and C= —log T„. 

 Hence 



7-=r„(i-r'') (8) 



Equation 8 shows that the transient relation between temperature 

 rise T and time is exponential and ultimately the temperature rise 

 is T=T„. 



The final tenii)erature 7„, reached 1)\- the winding may be deter- 

 mined b>- writing equation 7 in the form 



"^" = 4^6 (21) 



where E I represents the constant wattage applied to the winding 

 and 4.186 is the Joule equivalent. If the room temperature is Tr 

 and the ultimate temperature rise ?"„, it is evitlent that the final 

 temperature of the winding is 



Tj^Tm-^-Tr, 



EI 

 ^^=086p + ^'- 



B\' introducing a new constant A'l which re[)resents the ability of 



the structure to dissipate heat and also includes the factor -;— ,-/,i 



4.1bb 



we have ' 



Tf=^ + Tr, (9) 



in which .4 i is the area of the winding but docs not include the ends. 



The value of A'l can be readily determined by obtaining an experi- 

 mental curve between E I and Tm. This is obtained by gradually 

 increasing E I but holding the wattage constant for each value long 

 enough for the final temperature rise to take place. The value of 

 Tm is calculated by observing the change in resistance of the winding. 



The constant current and constant voltage characteristics are 

 determined in a similar manner with the important e.xception that 

 the quantity of heat Q supplied per second is not constant but varies 

 in accordance with the change in resistance with temperature. Thus 

 for constant current conditions 4.180 Q dl = P R clt and for constant 



voltage conditions 4.186 Q dl = ^ dl, where R= — ^.ri'-i — for cen- 

 tigrade degrees and Ro is taken at 0° C. 



' For single spool relays Ki = 50 to 60, and for double spool relays A'i = 35 to 50. 



