SWITCHED NETWORK FOR TIME MULTIPLEX SYSTEMS 1409 



C 



f 



de2 

 ~dt 



dir 

 dt 



-ir(t)A(t), 

 [e. - c-Mt), 



c^« = aoA(o. 



(3) 

 (4) 

 (5) 



Differentiating the middle equation and eliminating de^/dt and dez/dt 

 we get f or ^ / < r : 



d ir 



^zVA(o + -J h(/) - c,mm 



(0) 



in which we used the notation bit) for the Dirac function and the knowl- 

 edge that 



dA{t) 

 di 



= 5(0 - bit - r). 



(7) 



Equation (6) represents the behavior of the resonant circuit of Fig. 2 

 for the following initial conditions: 



i(0+) = 0, 

 (/tV(0 + ) cM - e^m 



dt 



I 



(8) 



(9) 



In Appendix I it is shown that the resulting current ir{t) is, for the in- 

 terval ^ / < T, 



where 



m = ch(o) - cM]si{t), 



IT . TTl 1 . , r r\ ^ , 



— - sm — = - 0)0 Sill coof tor < / < r 

 Si( ) =\'^r r 2 



elsewhere 



(10) 



(11) 



with 



■K 



Wo 



(C 



(12) 



Thus the zeroth approximation to the exact ir{t) is given for the interval 

 ^ / ^ r by 



iM = C[em - f3(0)]si(/). 



(13) 



