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**Butterworth Van Dyke Equiv. Circuit**

Left branch: Called the STATIC branch and only contains capacitances representing the quartz capacitance and external connection capacitances. Gives a current component 90o out of phase with an applied voltage. Right branch: Called the MOTIONAL branch and represents the acoustic resonances in the quartz and its load. At series resonance the L and CS reactances cancel, leaving only R, representing the losses in the quartz and its load. Gives a current in-phase with an applied voltage. When R is large, this in-phase current is small. The current from the static branch is no longer negligible and will distort the zero phase frequency to give errors in the resonant frequency. The effect of CP must be eliminated or compensated to ensure that the resonant frequency is not distorted.

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**Effect of Static Capacitance on Phase Behavior of Current**

The standard oscillator can be set to provide positive feedback for oscillation at some fixed phase. We show three cases here where the resonant frequency is identical at 5 MHz in each, but where the BVD resistance varies from 50 to 1250 ohms. For low R (high Q), the frequency for zero phase is virtually identical with that of the 5 MHz intrinsic resonance. But as R increases, the zero phase frequency increases, and for resistances above 800 ohms, no zero of phase is possible and oscillation ceases. However, by designing to oscillator to provide positive feedback at 60o, oscillation can be maintained even for 1250 ohms! The frequency of oscillation is NOT that of the intrinsic 5 MHz resonance, giving rise to frequency changes unrelated to the motional changes in the resonator and film.

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**Measurement Techniques**

Standard Oscillator Frequency only Impedance Analysis All BVD Elements Compensated Phase Locked Osc. Frequency and Resistance Dissipation Frequency and Decay time

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**Influence of Size of Static Capacitance on Frequency Deviation**

Larger values of static capacitance give rise to larger discrepancies between the intrinsic 5 MHz resonance and the oscillation frequency of a zero phase oscillator. In order to compare with real systems, we test the behavior of oscillation frequencies using the well-defined glycerol-water system. For glycerol-water mixtures, we can theoretically predict the frequency changes and compare those with observed values over a large resistance range.

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**Comparing The Techniques In Glycerol-Water Solutions**

All techniques, except the uncompensated oscillator, yield the same results for freq change. They all also agree with theoretical predictions. Only the impedance analyzer and the compensated phase lock osc. were able to provide resistance data. They are also in agreement with each other and with theory. Capacitance compensation is critical!

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**A modeled response of an oscillator adjusted to 65 degrees phase.**

The dashed blue line shows the frequency changes that would occur, using a model for the oscillator which has been adjusted to oscillate at a phase shift of 65 degrees. This is not intended to provide an explanation of the observed frequency discrepancies, but only to show that such a source of error is possible and reasonable.

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Capacitor: Let us consider the following circuit consisting of an ac voltage source and a capacitor. The current has a phase shift of + /2 relative to.

Capacitor: Let us consider the following circuit consisting of an ac voltage source and a capacitor. The current has a phase shift of + /2 relative to.

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