PERFORMANCE INDEX OF QUARTZ PLATES 249 



order to have a common Z plane. In this way, transformation of operating 

 points such as the "anti-resonant frequency" operating point (P4) for the 

 Zab impedance circle, could be subtracted from the "minimum impedance 

 frequency" operating point (Pi) of the P.I. meter impedance circle in the 

 common Z-plane. As in section 15.81, the frequency difference represents 

 the arithmetic difference between P4 and Pi in the Z-plane in terms of 



"^ (w — coo). As an example, look at the calculated curve in Fig. 15.6. 



This curve was computed for the case when R^ is negligible. The deriva- 

 tion of (15.51) given in section 15.92 precisely follows the procedure just 

 described. 



It is of interest to note that (15.27) may also be derived by Conformal 

 means. It is more laborious than the usual circuit equations of section 

 15.2; however, it does provide a check of the methods used. 



15.84 Errors of Other Approximations 



Further consideration of P.I. meter errors leads to the assumptions made 

 in the derivation of (15.27). The derivation of (15.27) assumed that the 

 resistor, Ra, was non-reactive. While actually it can be made essentially 

 noninductive, we have neglected the effect of the input capacitance of the 

 attenuator that is shunted across its terminals. The error from neglecting 

 this capacitance in (15.28) is given by the following expression 



Per Cent Error = 100 [l - ^ ^^^ JT ^V ^ (l^-^^) 



Where Qa equals the reactance of the shunt capacitance of the attenuator, 

 Cu , divided by the magnitude of the calibration resistor, Ra . 



It is interesting to note in the derivation of (15.28) that Ra was assumed 

 to be very much less than Xca which introduces an error of. 



.89) 



Per Cent Error = 100 1 - i/i 4. f— T (15 



15.90 Derivation of Circuit Equations 



15.91 Derivation of Equation (15.42) 



Other equations used in this paper may best be developed from Fig. 

 15.3. By analysis of the input impedance, Zi , the basis for the development 

 of (15.42) is as follows: 



