1408 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



where 



A(0 = E Ht - kT) - u{t - kT - t)], (2) 



+00 



z 



fc= — 00 



with u{t) = 1 for ^ > 0, and u{t) = for t < 



This system of linear time varying equations may be broken up into 

 three sub-systems /« , R and h . It is this subdivision that suggests a 

 successive approximation scheme that will be shown to converge to the 

 exact solution. 



The zeroth approximation is obtained as follows: when the switch is 

 closed, i.e., A(^) = 1, the resonant current ir is much larger than the cur- 

 rents in and i,/. Thus, during the switch closure time, in and in' are neg- 

 lected with respect to ir in (l.b) and (l.d). Hence when A(0 = 1 the 

 system R may be solved for ir{t), €2(1) and es(t) in terms of the initial 

 conditions. The resulting function €2(1) and given function io{t) are then 

 the forcing functions of the system /a. The other function es{t) is the 

 forcing function of the system /b. Under these assumptions, the periodic 

 steady-state solution corresponding to an applied current io{t) = /oc'" 

 is easily obtained. 



The zeroth approximation will be distinguished by a subscript "0". 

 Thus iro{t) is the (steady state) zeroth approximation to the exact solu- 

 tion ir{t). 



The first approximation will be the solution of the system (1), pro- 

 vided that during the switch closure time the functions in{t) and in'{t) 

 in (l.b) and (l.d) are respectively replaced by the known functions 

 ino{t) andino'it). And, more generall}^, the {k + l)th approximation will 

 be the solution of (1) provided that during the switch closure time, the 

 functions in{t) and in(t), in (l.b) and (l.d), are respectively replaced by 

 the known solutions for ^„(0, and in'it) given by the Ath approximation 

 It will be shown later that this successive approximation scheme con- 

 verges. Let us first describe a simple method for obtaining the zeroth 

 approximation. 



IV. THE ZEROTH APPROXIMATION 



4.1 Introduction 



The problem is to obtain the steady-state solution of (1) under the 

 excitation io{t) = /oe'"'. Using the approximations indicated above, 

 during the switch closure time (that is when A(^) = 1) the system R 

 becomes 



