892 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



in a later section) they should be confined to a period when the separa- 

 tion between the surfaces to be welded is quite large. The ciu-rent should 

 be sharply decreased as the surfaces approach each other. 



DESIGN OF IDEAL WELDING CIRCUIT 



In the previous section it has been shown that arc duration \\-ill vary 

 over a wide range. This suggests that the system be designed in such a 

 way as to be independent of arc duration. 



The procedure ^^'ill be to start with the desired temperature versus 

 time relationship of the two opposing surfaces. From this the correspond- 

 ing current versus time relationship can be found and finally the cir- 

 cuit giving such a current distribution selected. Obviously, the "safest" 

 temperature-time relationship is one where the temperature is kept 

 constant at the desired level T. The corresponding current distribution 

 will be derived on the basis of one-dimensional heat flow. Let 



u = temperature 



T = desired temperature at surface 

 a^ = diffusivity of material 



X = distance in direction of heat flow 

 A = cross-sectional area over which heat flow occurs 

 K = heat conductivity of material 

 Vm = voltage across arc 



i = transient current 



Start mth the differential equation for one-dimensional heat flow: 



For the boundary conditions: 



u = t <0 



(6) 

 u L=o = T t> 



The solution is : 



(2 rxliaVt ., \ 



(7) 



In order to determine the heat input required we must find the gradient 

 at the surface. Expanding equation (7) : 



