226 ADVANCED ELECTRICITY AND MAGNETISM. 



The distortionless line. Let / be the current and E the 

 voltage in a pure wave which enters a transmission line. Then 

 the rate at which energy is lost in heating the wires of unit length 

 of the line is R W P, where R w is the resistance of the two wires 

 in unit length of the line. Let RI be the leakage resistance 

 between wires of unit length of line, then E/Ri is the leakage 

 current from wire to wire because of imperfect insulation, and 

 E?/Ri is the rate at which energy is lost in unit length of the 

 line because of imperfect insulation. Now R W I 2 is the rate 

 of loss of magnetic energy and E*/Ri is the rate of loss of electric 

 energy in unit length of line, and if R W P = E 2 /Ri, then magnetic 

 energy and electric energy are lost at the same rate; but electric 

 and magnetic energies are equal at the start, therefore they re- 

 main equal and the wave remains pure if R W P = E 2 /Ri. On a 

 line where this condition is satisfied there is no wave distortion, 

 and such a line is therefore called a distortionless line. The 



condition is best expressed by substituting the value -~ for 



E 2 



-- according to equation (9) of Art. 116. This gives R w Ri 



772 7" 



= = so that the condition of distortionless transmission is : 

 J. C 



RwRi = (0 



where R w is the resistance of the wire* in unit length of the line, 

 RI is the leakage resistance of unit length of the line, L is the 

 inductance per unit length of line and C is the capacity per unit 

 length of line. 



123. Reduction of wire loss of energy by loading a line. 

 A telephone line of which the inductance is increased by the 

 insertion of coils of wire at intervals along the line as indicated 

 in Fig. 162 is called a loaded line. The presence of these coils 

 increases the resistance of the line slightly and also its capacity. 



* Including both wires. 



