Quartz Crystal Resonators-Brief overview

Resonator Design

Many different parameters have an influence on the final resonator properties. Thickness and diameter of the element , electrode diameter, electrode material but also holders, sealing method etc.

Crystal elements can be manufactured plano-parallel or contoured ( with bevels, plano-convex or bi-convex ) .

Contouring is necessary to prevent edge effects . A radius of curvature can be manufactured on one or both sides of the crystal element to trap the energy in the center of the resonator. Trapping can also be performed through mass loading.

Fundamental mode and overtone mode

High frequency crystals vibrate in the thicknessshear vibration, which can be excited in fundamental or odd overtone modes.

The motional capacitance C 1n of an overtonecrystal decreases with the order n of the overtone and is approximately given by

Therefore the ratio C O /C 1 is much larger for overtone crystals than for crystals operating in fundamental mode and the pulling range is reduced by a factor of approximately n 3 . Crystals used in VCXOs, where wide pulling range is required, therefore operate in fundamental mode ...

Unwanted Response and Inharmonics ( spurious modes )

All crystal resonators produce for each overtone a main mode which is a thickness shear vibration and also unwanted responses, which are inharmonic thickness shear modes above the resonance frequency. Besides the commonly used thickness shear C-mode another thickness shear mode named B -mode exsists. It has a higher frequency and commonly lower motional resistance than the C-mode but a larger temperature coefficient . Sometimes it becomes necessary to filter this mode for the oscillator to work on the C-mode. Further unwanted modes are shear-, flexure-, thickness-and twist vibrations, which can appear above and below the required resonance frequency. With correct oscillator design the unwanted modes rarely cause problems. Unwanted modes close to the resonance frequency affect the start up behaviour of oscillators, or cause shifting to the wrong frequency during operation. Other undesired effects are frequency and resistance dips over temperature caused by unwanted modes. Spurious modes are generally specified as the ratio of resonance resistance of the inharmonic modes to the main mode resistance. KVG must have detailed information about the test circuit (e.g. pi-network or measurement bridge) and about the frequency range of the spurious modes .

Equivalent electrical circuit

Near to the resonance frequency the crystal unit is represented by an electrical two pole shown in figure8.

C O : shunt capacitance (capacitance between the electrodes, crystal holder, leads and case) C 1 : motional capacitance (represent mechanical elasticity) L 1 : motional inductance (represent mechanical inertia) R 1 : motional resistance (represents mechanical losses) Figure 8: Equivalent electrical circuit Figure 9 shows the response in amplitude and phase vs frequency around resonance.

The resonance frequency is given by:

Pulling

With a load capacitance in series or parallel to the crystal the resonance frequency is shifted according to :

and the resistance at resonance becomes:

Frequency-Temperature characteristics

The temperature characteristics of AT-and SC-cuts crystals are described by a 3rd order parabola. It is then possible to describe the relative change of frequency:

and T i is the inflection temperature. The frequency-temperature characteristic is primaraly determined by the the cut angle. For a given cut the parameter which changes the most with θ angle is A i .

C i is almost constant and T i varies between +25°C and +35°C for the AT-cut and between +85°C and +95°C for the SC-cut , depending on the dimensions of the crystal element.

Since the inflection point of the SC-cut is close to 90° it is very suitable for ovenized oscillators because a TOP around 80° leads to very low dependency of frequency against temperature. (Note the different scale between Fig. 13 and 14) Moreover SC-cut crystals are less sensitive to mechanical and thermal stress and provide lower aging and higher Q compared to the AT-cut. Varying the drive level will change the resonance and phase curves. Typically the dependance is linear with the power.

The effect is of the order of some 10 -9

/µWatt and is typically lower for the SC-cut than for the AT-cut. It can be a problem when drive level fluctuates or drifts over time.

Crystals should be used at the level of drive for which they were designed. Higher drive levels excite unwanted modes of vibration, cause serious degradation of the frequencytemperature characteristic, accelerate aging and can shift the frequency due to overheating of the resonator.

The test drive level of KVG for standard crystals is 0.1mW.

Quartz Crystal Resonators

KVG Quartz Crystal Technology

Thermal Hysteresis

Hysteresis can occur when the crystal is subjected to temperature cycling. After going through a temperature cycle as in fig. 18 , the difference between the frequency at the beginning and the end of the cycle can be higher than 1 ppm . It is typically a problem in TCXO applications where the external temperature can vary in a quite large range. By correct design of the crystal hysteresis can be minimized . Figure 18: Temperature cycle Activity Dips Figure 19: Activity dips

Dips can cause problems in crystals for VCXOs and TCXOs. A dip is characterized by deviation from the third order frequency-temperature curve.

It is caused by the excitation of unwanted modes through mechanical coupling.

Generally these modes have a strong temperature dependance so they appear as perturbations at discrete temperatures .

Dips are influenced by resonator design , drive level and oscillator circuit conditions.