KENNELLY. — EQUIVALENT CIRCUITS OF COMPOSITE LINES. 39 



From which 



B" = B f (\ + \/ 1 - 5? J ohms. (28) 



P ' \ R " = Bf ohms. (29) 



r/g = B'Y = B' P " = B f B g = z 2 (ohms) 2 (30) 



6 = La = tanh- a |/^ hyps. (31) 



The last two formulas serve to evaluate z and 6 for any single line, 

 when the sending-end impedances of that line (B f and B g ) have been 

 correctly measured. 



Looped or Metallic-Return Single Circuits. 



If we consider single metallic circuits, like those of wire-telephony, 

 or of single-phase power-transmission, 



Let r„ = the linear resistance (ohms per loop km.). 



g u — the linear leakance (mhos per loop km.). 

 /,, = the linear inductance (henrys per loop km.). 

 c„ = the linear capacitance (farads per loop km.). 



Then r„ = 2 r ohms per km. 



g„ = g/2 mhos per km. 



/,, = 2l henrys per km 



c u = c/2 farads per km. _, 



(32) 



where r, g, I, and c are the corresponding linear constants per wire km. 

 Substituting in equations (1), (2), and (3), we have 



a„ = a hyps per loop km., (33) 



0„ = 6 hyps, (34) 



and z„ = 2z ohms. (35) 



That is, the attenuation-constant, and the angle subtended by the 

 looped line, are respectively identical with the attentuation-constant 

 and angle subtended by one wire only operated to zero potential. The 

 surge-impedance of the metallic circuit is double the surge-impedance 

 of one wire to ground, or zero potential. The voltage impressed upon 

 the loop is, however, double the voltage impressed on each wire singly 



