320 Mr. G. A. Campbell on Loaded 



the loaded line, including the attenuation coefficient and 

 velocity of propagation*. 



To determine the line impedance (K), observe that the 

 impedance is periodic and apply formula (14) to the circuit, 

 beginning at the middle of one coil and extending to the 

 middle of the next coil : — 



\Rd 



(»/+*)_, 



k 



e -2yd 



1_ ~~ Ud 



1-f 



:-(f + K) 



'_*-*& 



7 ti(i rr 



or 



K = v *' + R -r + Rdk coth V d 



='V( 



l+i'tobK'Xl+g™,!,'/) .(19) 



which completely determines the loaded line impedance at 

 the middle of a load. The impedance at any other point 

 might be found by the same method. 



Substituting the values of V and K given by (18) and 

 (19) in equation (11) we have the formula for the current at 

 any load, or substituting in (12), (13), (13 a), or (17) we 

 have the value of the receiving current, but the substitution 

 can best be made after numerical values are obtained. 



The method which has been employed in deducing (18) 



* For a simpler proof of equation (18), short circuit the loaded Hue at 

 the middle of the coil B and consider the ratio of the current at A to 

 the current at B. As section A-B may then be considered, either, (1) 

 as a uniform line of constants k, y, and length d, terminating in an im- 

 pedance Hd/2, or, (2) as a portion of a uniform line of constants K, r, 

 and length d, terminating in a short circuit, two values for the ratio of 

 the current at A to the current at B may he obtained, and the two 

 equated give a relation between the two sets of line constants. The two 

 expressions are found on making the proper substitutions in (11) to be 

 identically the right and left hand members of equation (18). 



In this proof it is to be noticed that the loaded line is short-circuited 

 at the middle of a load in order that it shall act like a short-circuited 

 uniform line with the constants K, T; if the line is not short-circuited 

 at the middle of a load (or at the middle of a line section) a wave tra- 

 verses on reflexion a line which is not throughout of uniform periodic 

 structure. For the suggestion leading to this proof I am indebted to 

 Dr. A. E. Kennelly. (July 1902.) 



