PERFORM. I XCE fXDEX OF QUARTZ PLATES 



243 



The coordiuales of these two points represent values of II'; now (15.52) may 

 be solved for Z. 



a - yW 



Substituting in values for «, f3, 7 and 8 given by (15.59), we find 



^^'-R.^- '^•^^ = ' T^i^^r^ = -'' (r, + x.r + Rl ^''-''^ 



The real component of Z must be zero since the function of Z is coincident 

 with the I'-axis. By substituting values of Ro and A'o from (15.66) and 

 (15.67), we find 



_ (C02 — Wl) V/^ 



also 



M [1 - Vl - {v/<^y] 



(co2 — coi) v/o^ 



M 



1 + 



('J 



1 



Subtracting (15.71) from (15.70) to get Aco we have 



Aco 



M 



A 



where 



A 



^ Vl - (v/a)- - Vl + (v/ay 

 (ri/ay 



(15.70) 

 (15.71) 



(15.72) 

 (15.73) 



W2 — COl 



Now the lim .4=1. When- = ^, then Aw = 



W<r-0 0- M M 



A 



15.82 Circuit Analysis Involving Crystals 



The same procedure could be followed for the impedance, Zi , in Fig. 

 15.3; however, the impedance expression conforming to (15.52) may be 

 written directly if a more general expression is derived for impedances 

 added in parallel or series. 



Paralleling the impedance, W, with an impedance, T, (Fig. 15.11) modifies 

 the constants in (15.52) but not its form providing T is essentially constant 

 between coi and wo . For parallel impedances the impedance equation 

 becomes, 



WT a + ^Z ^ a' + ^'Z 



~ y + 5'Z (15.74) 



W ^ T 



^ + "r +^+r 



