132 BELL SYSTEM TECHNICAL JOURNAL 



the value of (51) is 



a{t) = e-^'7o(XO, (52) 



which is the sohition of the integral equation. 



IV. Other Types of Boundary Conditions 



The solutions obtained before are all on the assumption that no 

 energy exists in the network before the voltage is applied. Other 

 types of boundary conditions are sometimes desirable, but these can 

 all be derived from the above solutions. 



The next most important case is the case where the network has 

 come to its equilibrium value and the voltage is suddenly taken off. 

 This condition is the same as would result if a negative voltage E 

 were suddenly applied to the circuit, and hence the solution is the 

 steady state value of the current minus the current which flows on 

 application of the voltage E. 



Another type of boundary condition sometimes occurring is the one 

 where energy exists in the network when / is zero. This may be 

 taken account of by assuming that the voltage is applied before t 

 equals zero. To take account of this condition analytically, examine 

 the expansion 



E 



i = y= £[ao + aie--''^^ + a-.e^'''^^ + • • • + a„e-2»/a.i) _p . . .]_ 



According to the above assumption, the voltage is applied when 

 / = — /o, hence for n we substitute 



- J__L A 

 ^ 2D'^2D' 



The above series is then replaced by the integral 



i = eC a{f + to)e-^''^'+'o)dL (53) 



J -to 



Another boundary condition of interest occurs when the voltage 

 is taken off before an equilibrium value has been reached. If we 

 count time as starting when the voltage is taken oft, or what amounts 

 to the same thing, when a negative voltage is applied, the symbolic 

 solution takes the form 



i = Elf a{t+ to)e-^'''^'+'o^dt - f a{t)e-i'''dt'}. (54) 



*)-ta Jo 



