242 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



For a given relay, the dimensions t and d are fixed, and only the wire 

 size and the coil depth h can be varied. For a given wire size, therefore, 

 the values of h and A^ are fixed by the resistance. As Cn and Cr are known 

 quantities, the values of Ko can be determined and plotted against R for 

 each wire size, as illustrated in Fig. 10. Such a chart applies to a particu- 

 lar relay, as the curves depend on the fixed values of I and d applying. 

 Here it can be seen that for a given resistance the finest gauge wire is 

 the cheapest. Because of this, and since power costs are minimized with 

 the highest resistance, the finest gauge wire would always give the 

 cheapest coil. However, this becomes a theoretical case. As finer and 

 finer wires are used, A = ira becomes larger, and thus N/R or NI/E 

 becomes smaller. In all relay design, an ampere-turns requirement must 

 be met. Thus a practical solution to this problem of optimum costs is to 

 choose the finest gauge wire which will meet the ampere-turns require- 

 ment. 



If the same relay is assumed as for Fig. 10, the variations of ampere 

 turns with resistance are as shown in Fig. 11. Starting with the NI re- 

 quired, the wire size can be determined from a plot like Figure 1 1 . Then 

 Ki is determined from a plot like Fig. 10. 



Now, with the gauge of wire chosen, K2 is almost a constant and can 

 be considered independent of R. Thus, upon differentiating (21), Ct will 

 be found to be a minimum when R has a value designated by 



R, = VkJK^ . (22) 



The most economical coil, resulting from use of a resistance 7?o , will have 

 a system cost of 



Co = 2VK^2 • (23) 



As can be seen by the slopes of the curves on Fig. 11, if the optimum 

 resistance, i^o , is larger than the resistance value at the desired ampere 

 turns, the NI requirement will not be met. Then either the selection of 

 Ro must be made for the next coarser wire, or the resistance value chosen 

 at just the required NI, whichever is cheaper. Conversely, if there is a 

 very large A^7 margin, a finer gauge wire could be tried. In all, two trials 

 should give the optimum values. 



A procedure has now been outlined for choosing the coil design for 

 optimum power-plus- winding cost. Curves similar to Figs. 10 and 11 

 should be draw^n for the applicable relay parameters. Then the K2 value 

 is obtained for the finest gauge meeting the ampere-turns requirement, 

 and Ro and Co found. Then, the NI value applying for Ro must be checked 

 and adjustments made as above, if necessary. 



