242 



BELL SYSTEM TECHNICAL JOURNAL 



4. The condition which must exist when the crystal reactance is zero 

 between wi and C02 occurs when o- = rj or M = 2. 



It follows that the error of measuring, say the series resistance Ri , by 

 varying the frequency for maximum transmission and assuming true 

 resonance when the crystal is between two low non-inductive resistors is 

 associated with the difference between the length of the two vectors — P3 

 and — Pi . The per cent error caused by the crystal capacitor, Co, by 

 this method of measurement of Ri is given as 



Per cent error of R\ = 100 



100 



1 - 



Ps 



■i 



■y/a^ + r?^ — a- 

 0" — Vo-- — T]-. 



(15.63) 



If 0- and T] are substituted in (15.63), we find 



Per cent error of Ri = 100 



1 + 



1 - 



/, 



_4_ 



1 + 



/ 



' + w^ 



(15.64) 



This difference in amplitude was caused by the difference in frequency 

 between resonance and minimum impedance. This frequency difference 

 may be determined by transforming the points Pi and P3 into the Z plane 

 by (15.52) and subtracting them arithmetically. 



In order to express the coordinates of any point in the U'-plane by its 

 real and imaginary components let 



W = i?o + jXo 

 Now the coordinates of P3 may be expressed as, 



i..=.[l-^l- ('-/]; 



Xn = 



(15.65) 



(15.66) 



The coordinates of Pi are similarly given by 



1 



R, = a I - 



Xo = —7] 



1 + 

 1 



m 



1 + 



&\ 



(15.67) 



