638 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



and rewrite (1.12) as 



Now, if the impressed current <ri is zero, |8 must have a value /3i such 

 that 



Also, the characteristic impedance of the Hne, K, is 



K = +VZ7C 



In terms of these quantities 



and the circuit admittance F(aj, 0) is 



(2.15) 



F(co,^)=-J 



Here iC and jSi are positive quantities. We note that this admittance is 

 capacitive for /3 < /Si , that is, for waves with a phase velocity greater 

 than the natural phase velocity of the circuit, and inductive for /3 > j8i , 

 that is for waves with a phase velocity less than the natural phase velocity 

 of the cucuit. This is easily explained. For small values of /3 the wavelength 

 of the impressed current is long, so that current flows into and out of the 

 circuit at widely separated points. Between such points the long section 

 of series inductance has a higher impedance than the shunt capacitance to 

 ground; the capacitive effect predominates and the circuit impedance is 

 capacitive. However, for large values of (3 current flows into and out of the 

 circuit at points close together. The short section of series inductance be- 

 tween such points provides a path of lower impedance than that through 

 the capacitances and ground; the inductive impedance predominates and 

 the circuit is inductive. Thus, for fast waves (J3 small) the circuit is capaci- 

 tive and for slow waves (jS large) the circuit is inductive. 



We can, then, immediately make one observation. For a lossless circuit, 

 any increasing or decreasing wave must have a phase velocity less than 

 the natural phase velocity of the circuit. 



We can make another observation as well; if the circuit has loss, F(co, /3) 

 will have a real component, and from (2.14) all the waves must have an 

 imaginary component of j8, that is, they must be increasing or decreasing. 



