# Oscillator Circuitry for QCM

Design of suitable interface circuits for QCM sensors requires special circuit configurations which are not to be obtained by simply modifying standard applications. In order to design suitable circuit, the fundamental equivalent circuit of piezoelectric resonator has to be considered.

The properties of quartz crystal close to a resonance frequency can be expressed by parameters of equivalent circuit, where RS is dynamic resistance (friction damping of quartz slice, acoustic damping, acoustic damping of ambient), LS is dynamic inductance (oscillating mass of the quartz), CS is dynamic capacitance (elasticity of the oscillating body), C is static parallel capacitance (capacitance between quartz electrodes). Mechanical properties of quartz are represented by the components of a series equivalent circuit (RS, LS, CS). With the static parallel capacitance C as the only electrical parameter, the mentioned components determine the basic frequency of an oscillating circuit. Considering, a series resonance frequency fS and a parallel resonance fP are given by Serial resonance is determined only by mechanical properties of quartz crystal whereas parallel resonance is particularly influenced by parasite capacity of crystal output pins and by all capacities, which are connected to output and caused frequency shift from serial resonance to parallel resonance. Obviously, the frequency of parallel resonance has higher instability caused by external parasite influences than frequency of serial resonance. The instability rate can be determined by ratio of serial and parallel resonance.

Newer developments of quartz oscillators resulted from steadily increasing requirements to frequency stability in measurement technique (frequency and time measurement) as well as in communications engineering, where operation in vacuum or protective gas dominates. Several circuits are commonly used for design of crystal oscillators, such as crystal oscillators with parallel resonance, crystal oscillators with serial resonance and Colpits oscillators. The frequency of Colpits type is in the range between parallel and serial resonance of quartz crystal. The 20 % change of parallel capacity C represents relative change 2x10-5 for parallel resonance and 4x10-9 for serial resonance. For Colpits oscillator, relative change is in the interval between these values.

Considering the values of relative change, serial resonance oscillator represents circuitry of the highest possible stability of working frequency. Other solutions give significantly worse results. Fig. 3 shows fundamental quasi-linear model. Quartz crystal Q and resistor R forms band-pass with serial resonant circuitry and zero-phase shift. To satisfy the phase condition for oscillation, the feedback circuitry of oscillator is formed by noninverting amplifier of suitable amplification whose value meets the amplitude condition for oscillation.

The frequency instability of quartz crystal oscillator is affected by instabilities of the components in quasi-linear model and, further, by instabilities associated with the nonlinear solution of amplitude condition for the oscillation. The instability of amplifier phase-transfer function represents the highest influence and its effect decreases with an increasing value of transition frequency (GBW). Considering our case, it is necessary to use an amplifier with a transition frequency of 500 MHz. Thus, the effect is reduced to the level of quartz crystal itself. Furthermore, it appears that in this case the instability of frequency is three orders lower in the magnitude than for the oscillator with parallel resonance and at least an order lower in the magnitude than for Colpits circuitry.

Also, a solution of the amplitude condition plays key role in the frequency instability. Two approaches exist. In the first approach, the amplitude of output signal is nonlinearly stabilized by the saturation of amplifier. However, the amplifier gets into a certain mode for a time, where its parameters are not guaranteed and stable. Also, influences of power instability and local thermal instability increases in the amplifier. Further, a relative high power is 210 P. SEDLAK, J. MAJZNER, J. SIKULA, K. HAJEK, NOISE MEASUREMENT SETUP FOR QUARTZ CRYSTAL… on a quartz crystal itself and increasing its thermal instability. The second approach uses inertial stabilization of the amplitude, which ensures the linear mode operation of the amplifier and the limiting power on a quartz crystal to the manufacturer's guaranteed level.

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