TRANSIENT SOLUTIONS OF ELECTRICAL NETWORKS 117 



D. General Method for Determining Reflections 

 The method for obtaining the successive current reflections of the 

 Hne given in the preceding section, is laborious to carry out in com- 

 pHcated cases and hence it is desirable to obtain a simple method for 

 determining the reflections. Such a method is the expansion of the 

 expression for the impedance by the binomial theorem in order to 

 get the successive reflections. In the above example "the current i is 



i = E/{R-\-ja>L). 



We note that the expression for the impedance is approached by that 

 of a short circuited line when the R and L of the line are finite and the 

 capacity and leakance approach zero. Hence 



R+ ju^L Ro tanh P Rq{1 - g-^^) 

 Now the expansion of 

 1 



1 - e- 



= 1 + g-2^ + e~'P + 



Hence 



i ->~ (1 + e-2^)(l + e-2^ + e-^^ + . . .) 

 = -f [1 + le-'^P + 2e-'^ + • • • + 2£'-2("-i)^ + ••■], 



which is the expression for the reflections given by equation (23). 



In all the following problems it will be found that a similar process 

 for obtaining the reflections can be followed. It is evident that any 

 method which gives an expression for the current in the form 



i = £[ao + aie-2^'-^ + a^e-^i^^ + • • • + a„e--"^"^ + • • •] (29) 



will give the reflections, for if we take the real part of this expression 

 we have 



i = £o[ao cos (w/ + ^) + ai cos [w(/ — 2D) + 0] + • • • 



+ On cos [co(/ - 271D) + 0] + • • •]• 



Each term represents a current which adds to the original current 

 after a time of delay 2D, 4D, • • • 2wZ), and hence the wth terms repre- 

 sents the wth reflection. Therefore any method, such as the above, 

 which gives the current in the form of equation (29), will give the 

 reflections. 



