238 BELL SYSTEM TECHNICAL JOURNAL 



Equation (15.44) indicates that the capacitance, Ct, determines the oper- 

 ating frequency between coi and 002 of any given crystal. The capacitance, 

 however, may include reflected reactances from associated circuits or pos- 

 sibly from circuits unintentionally coupled to the oscillator. Normally, 

 crystals are adjusted to frequency for a specified value of Ct. This makes 

 it of interest to measure the magnitude of Ct over the range of the manual 

 oscillator tuning adjustments, as well as over the frequency range of the 

 oscillator. Fig. 15.9 shows the circuit capacitance, Ct, plotted as a function 

 of tuning of the plate circuit of a Tuned-Plate oscillator. Tuning of the 

 plate circuit is expressed in terms of percentage of maximum grid current. 



15.80 Circuit Analysis Involving the Accuracy of P.I. 



Measurements 



The method of P.I. measurement just described involved a number of 

 unverified approximations. These approximations under the majority of 

 conditions will be proven to be justified and the resulting expressions for 

 percentage error will be obtained. 



It is necessary to apply a method of analysis that is most readily adaptable 

 to the crystal circuit. The analysis described in this section involves the 

 use of Conformal representation as a means of determining (1) the behavior 

 of the equivalent crystal circuit, (2) the error resulting in P.I. from assum- 

 ing operation at the resonant frequency rather than the frequency for mini- 

 mum impedance, and (3) the comparison of frequency of oscillation in the 

 P.I. meter with other oscillator circuits. 



Generally, the variations of reactance. A', and resistance, R, of the equiv- 

 alent circuit (Fig. 15.1) are plotted as a function of frequency, and the 

 analysis of the impedance, R + ^'A^, between the resonant frequency, wi, 

 and anti-resonant frequency, aj2, are handled in precisely the same way as 

 any linear passive element. This was essentially the procedure used to 

 derive the equations in section 15.60. The analysis required to evaluate the 

 errors leads to rather an elaborate study; however, in Section 15.81, it will 

 be shown that it is very helpful analytically if the impedance of the crystal 

 is plotted in the form of a circle diagram, that is, with the ordinate repre- 

 senting reactance, A, and the abscissa representing resistance, R. 



15.81 Conjormal Representation 



Conformal Representation or Mapping is a convenient tool which for this 

 application enables the physical operating condition to be expressed quanti- 

 tatively from its graphical counterpart. Physical interpretation also 

 makes it possible to draw many other conclusions that by other methods 

 prove clumsy and laborious. 



