TRANSMISSION LINE EQUATIONS 14,? 



Two similar expressions may be obtained in which the initial current i(o) 

 instead of v(o) appears on the right. If the line is terminated in its char- 

 acteristic impedance at x = 0, v(o) = —Zoi(o), and the voltages and currents 

 produced by the impressed field are 



v{o) = -i (r + eir\\ - Zot) 



(1.41) 



i{o) - i Z7'(r + dir\\ - Zot) 



As in §1.4 these expressions may be computed by the first and second 

 methods described in §1.3. For example, the application of the second 

 method to the relation (1.39) which must exist between v{o) and i{o) in an 

 infinite line gives 



v{o) = i: N(yl) [^ i{o) - -i- (x - ^ t)1 (1.42) 



where i\^(7r) is the square matrix (1.23). 



1.6 Outline of Proofs 



The proof of the results which have been stated is divided into three parts. 

 In the first part it is shown that if F is a matrix such that (a) its square is 

 ZV and (b) every element in the matrix e~^ approaches zero as x —^ «: , 

 then the expressions for i'(.v) and i(x) involving F and Zo satisfy the trans- 

 mission line equations. In the second part of the proof it is shown that if 

 certain requirements are met F as obtained by the first method satisfies the 

 conditions (a) and (b) and hence the expressions for v{x) and i{x) given by 

 the first method are correct. The third part of the proof discusses a general 

 procedure which may be used to prove the equations which constitute the 

 second method. 



Both the second and the third parts of the proof are based upon the solu- 

 tion of the transmission line equations which is sketched in Appendix I. 

 This solution assumes that the propagation constants of the normal modes 

 of propagation are unequal, and our proofs are limited accordingly. How- 

 ever, considerations of continuity seem to show that the first method is 

 valid even when two or more propagation constants are equal. Under the 

 same circumstances the second method suggests the use of the confluent 

 form of Sylvester's theorem. 



1.7 Relations Obtained by Considering An Infinite Line 



We suppose that we are going to deal with transmission lines possessing 

 the non-singular, symmetrical impedance and admittance matrices Z and Y . 

 We further suppose that, by some means or other, we have determined a 

 matrix F which satisfies the two conditions; (a) the square of F is 



F2 = ZY, (1.43) 



and (b) every element in the matrix e""" approaches zero as x -^ oo , 



« F.D.C. §3.10. 



