. The Bell System technical journal . llations again to assume asteady value it is necessary for the amplitude a to change a sufficientamount to cause a to become zero. Thus in () we put 8a equal tozero and solve for the required amplitude change. This may then be elimi-nated from () resulting in the final expression 1 dA dd _ 1 dA dd80. = -4aFa^ AJ^W ^^ ^2 ^j) ^dAdd_ 1 ^ cl0A doj da A da dco which gives the frequency change Sco in terms of the change of the inde-pendent variable 8V. Frequency Stability of Conventional Oscillator In applying this equation to the oscillator circ
. The Bell System technical journal . llations again to assume asteady value it is necessary for the amplitude a to change a sufficientamount to cause a to become zero. Thus in () we put 8a equal tozero and solve for the required amplitude change. This may then be elimi-nated from () resulting in the final expression 1 dA dd _ 1 dA dd80. = -4aFa^ AJ^W ^^ ^2 ^j) ^dAdd_ 1 ^ cl0A doj da A da dco which gives the frequency change Sco in terms of the change of the inde-pendent variable 8V. Frequency Stability of Conventional Oscillator In applying this equation to the oscillator circuit, Fig. , we mustfirst set up the conditions for oscillations. The ijl(3 equation is IjlXi X2 Rg ^^ ^ aXsRpRo - X1X2X3] - [RpX^iXi + X3) + RgXiiX. + Xs)] ()The oscillating conditions /x/3 = 1 requires XgRpRg = X1A2X3 194and BELL SYSTEM TECHNICAL JOURNAL fjiXiX^Rg + RpX2{X, + X3) + RgX,{X2 + Xs) = 0 ()It will be assumed that the following relations exist: M = MV), Rg = /2(a), Xs = Xi + X2 + X3 = Mco), i?p = a constant. Fig. —Equivalent oscillator circuit analyzed for frequency stabilityThen we obtain from () AdV ~ fjidV IdA ^ J_dRg[A da Rg da [_ i M=i_ ^1 r A do: Xi dw |_ dV 1 + 1 + + 1 de _ _\ BRg XoRt J da ~ ] ] X2 -\- Xs _ _ 11X2 J da Rg da nXiXz i?pX2 + Rg(X2 + X3) nRpXi Rp{X2 + X,) +RgXiiRp X\ + T j_aX2r X2 dcjo [_ 1_ 6X3 rXs{RpX2+RgX{)l ^3 5w [_ llRp X1X2 J ■] () dddo} fiX iX2\_ Xi d(ji + (X3 - X) ^ /I A3 000 J X2 5co PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 195 By substitution of these values in () and disregard of Xg in comparisonwith all other As the equation for frequency stability is obtained as 1 ^M V V V 7 - TT> A1A2A3 ^ - ^^^ () dV From this we learn that the values of the reactances Xi , A2, and X3should be small and the values of Rp and Rg large to give small changes inCO when V is varied. These variables are more or less limited, however, bythe conditions necessary for sustained oscillations according to equa
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