WAVE PROPAGATION OVER CONTINUOUSLY LOADED ]VIRES 193 



In this case, the propagation constant is uniquely determined from 

 a knowledge of the admittance between the loaded wires and of their 

 series impedance. The "internal impedance" of the loaded wires 

 comprises the larger part of this impedance. For the purpose of 

 engineering design work, it is convenient to have at hand approximate 

 formulas for the "internal impedance." 



The exact expression for the impedance is given by the last of 

 equations (2). \A'hen the magnetic sheath is thin, as compared to 

 the radius of the copper wire, certain approximations can be made. 

 These are explained in Appendix A. The result is the following 

 formula for the "internal impedance": 



where F = wjULob 



Z i _\ — orF + ioiCi 

 -^ + lull 



^7rX,X,M-.>/'^ + Mi&(^^-^log^ 



(5) 



II = 2ir''-\,n.af \\t{\, - Xi) + Xi^l +2^/ 



/? /? 

 R = p ,'p = d.-c. resistance of one of the pair of bi-metallic 



conductors, 



Ri = , ,., = d.-c. resistance of the inner part of the conductor 

 TrKib- . 



(the wire), 



i?2 = ^ , ., 7T- = d.-c. resistance of the outer part of the 



irXMi- — 0~) , / , , 1 X 



conductor (the sheath), 



Lo = 2^2 log T = low-frequency inductance contributed by the 



sheath, 



b = radius of the wire, 



a = outside radius of the sheath, 



t = a — b = thickness of the sheath, 

 ^1, Ml = conductivity and permeability of the wire, 

 X2, M2 = conductivity and permeability of the sheath, 



CO = Itt times the frequency, 



