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Frequency pulling in a quartz oscillator refers to the phenomenon where the oscillator's resonant frequency deviates from its nominal value due to external factors, such as changes in load capacitance or stray capacitance. This deviation is typically expressed in parts per million (ppm) or parts per billion (ppb).
The frequency pulling effect arises from the interplay between the quartz crystal's inherent properties and the external circuit elements. The crystal's resonant frequency is determined by its mass, shape, and elastic properties. However, when connected to an external circuit, the presence of parasitic capacitances and inductances can influence the overall resonant frequency.
One of the primary causes of frequency pulling is the load capacitance, which represents the capacitance between the crystal's electrodes and the external circuitry. As the load capacitance increases, the oscillator's resonant frequency decreases. This effect is more pronounced in parallel-mode crystals, where the load capacitance directly affects the crystal's equivalent inductance.
Other factors that can influence frequency pulling include:
Stray capacitance: Parasitic capacitances arising from wiring and circuit layout can also contribute to frequency pulling.
Operation in the parallel mode allows a greater range of frequency pulling as that mode is more influenced by the external loading capacitor. For series mode deliberate changes in the sustaining amplifier phase shift works better.
Temperature variations: Temperature changes can affect the crystal's elasticity and resonant frequency.
Mechanical stress: Mechanical vibrations or shock can temporarily alter the crystal's resonant frequency.
At non-resonant frequencies a crystal has quite a high impedance. At series resonance it has impedance R, a few 10's of ohms. At parallel resonance it has a very high impedance and resonant voltage magnification. The series and parallel resonant frequencies are close but not identical.
The equivalent circuit of a quartz crystal resonator consists of an inductor (L) representing the mass of the crystal material, a capacitor (C1) representing the crystal's compliance, a resistor (R) representing energy losses, and another capacitor (Co) representing the capacitance between the crystal's electrodes. An additional external capacitance, called the load capacitance (CL), is crucial for the crystal to vibrate at its resonant frequency. The value of CL, typically between 20pF and 30pF, is specified in the crystal's datasheet and significantly affects the crystal's resonant frequency, especially in its parallel mode of operation.
The frequency shift (Δf) caused by the load capacitance can be calculated using the formula:
Δf = fs * (C1^2 / (2(C0 + CL)))
where fs is the crystal's series resonant frequency.
The average pullability of the crystal, expressed as the frequency shift per picoFarad change in load capacitance, can be calculated using the formula:
ppm/pF = C1 * 10^-6 / (2(C0 + CL))^2
To achieve these calculations, knowledge of the shunt capacitance (C0), motional capacitance (C1), and load capacitance (CL) is essential. The limits of Δf depend on the crystal's quality factor (Q), which is related to the values of the electronic components in the equivalent circuit and the load capacitance.
The frequency the oscillator operates at can also be changed by altering the phase shift of the sustaining amplifier. Ie. introducing lags or leads in the response of the amplifier at the frequency of operation.
Example Circuits:
Using 3 terminal ceramic filter/resonator.
Sustaining amplifier phase shift control of frequency. Changing the bias on the diode alters the lag or lead of the amplifer.
Chart for the phase shift frequency pulling circuit above.
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