DIELECTRIC CONSTANTS AND POWER FACTORS 117 



z 



/i 



Now, the power factor of "Z/' is the same as that of -^ — , as long as 



Z( 



l/I 



— is substantially a resistance, and since 



we have 



. . sinh 2a/ + i sin 2be ... 



tanh {ai + thf) = / — -* , (6) 



cosh 2a( -\- cos 26^ 



power factor Z( = p.f. = . (7) 



sin 2b( 



Substituting eq. (5) in (7), 



sinh ^{d - d') 



p.f. = -^ , (8) 



sin ^ (A^ + t) 



A 



which is the power factor of the loaded line segment in terms only of meas- 

 urable lengths. 



This does not complete the theory, however. We are interested in the 

 power factor of the dielectric itself and it is evident that except for very 

 short dielectric segments, the variation of the standing electrical field along 

 the dielectric segment will result in a calculated power factor smaller than 

 the true one. We also wish to determine the dielectric constant. 



The impedance of the dielectric line segment, open circuited at the far 

 end, can be written as 





Z( = . ' '^ .-. (9) 



tanh 



where "a" is the attenuation per unit length and "e" is the dielectric constant. 

 Hence tanh (c^ + ibi) = ■\/~t tanh [a-\-i — — ^ j / and 



sinh lat 

 sm — e — i 



