MODE OF VIBRATION

The vibration modes of the quartz crystal Units are grouped into flexure, extensional, face shear and thickness shear modes. The schematics of the vibration modes and the plate cuts usually used are listed in Table 1. Fundamental mode and overtone modes can be operated in any kinds of resonators. Fundamental mode is most often used, but for the thickness type devices the overtone modes are often used as well, as shown in Table 1.

Table.1 Vibration Mode and Cut Angle.

FREQUENCY-TEMPERATURE CHARACTERISICS

Most of the quartz products are used as an electrical circuit component for frequency selection and/or frequency control, so the frequency-temperature characteristic of the devices is the most important parameter. For the usually used quartz crystal cuts, their frequency-temperature characteristics are shown in Fig.2. AT-cut is the most popular crystal cut in the quartz devices. Fig.3 shows the frequency-temperature characteristics of the AT-cut crystal operating in thickness-shear mode, with the cut angle deviation as a parameter. It is shown that AT-cut quartz has excellent frequency stability over a wide temperature range since the first-and second-order of the temperature coefficients go to zero in this range and the temperature coefficient is only dominated by a third-order function of the temperature deviation.

Fig. 2 Frequency-temperature characteristics of various quartz cuts.

Fig. 3 AT - cut frequency-temperature characteristics.

EQUIVALENT CIRCUIT OF A CRYSTAL RESONATOR

Fig.4 shows the schematic of a resonator and its symbol. The electrical properties of the unloaded resonator can be approximately expressed in Butterworth-Van Dyke (BVD) equivalent circuit as shown in Fig.5 when operating near a resonance zone. By using the four parameters shown in Fig.5, the major electrical properties of a crystal resonator and of a oscillator consisting of the resonator are described as follows.

(1) Nominal Frequency and Its Tolerance or Calibration Accuracy

The center frequency of a crystal resonator is typically specified in megahertz (MHz)or kilohertz (kHz). There is an amount of frequency deviation from the nominal frequency at ambient temperature (referenced to 25oC) for a real device. The tolerance of the center frequency deviation, as a parameter of the device, is specified with a maximum value, expressed in percent(%) or parts per million(ppm).

(2) Frequency-Temperature Stability

Frequency-Temperature stability is indicated by the amount of frequency variation from the value at the standard amblient temperature (25oC, usually), caused by the operating temperature change. This parameter is specified by a curve showing the frequency variation (expressed in % or ppm) versus the temperature deviation from the standard temperature (25oC). The temperature stability of a quartz device depends on the type of cut, the mode of vibration, and the dimension of the quartz blank. Besides, the deviation value is associated with the operating temperature range, the load capacitance and the drive level of the resonator.

(3) Resonance Frequencies

In literatures and product descriptions, there are three pairs of resonance frequencies, i,e., the "series resonance frequency" and "parallel resonance frequency", ( fs and fp ) , the "resonance frequency" and "anti-resonance" frequency, ( fr and fa ), and the "maximum and minimum total admittance located" frequencies, ( fm And fn ). All of them can be obtained from the lumped equivalent circuit parameters as given in Fig.5. The definitions and relationship of the resonance frequency pairs can be clearly expressed in a complex admittance diagram given in Fig.6.

Fig.6 Complex Admittance of Resonator

Fig.7 Plot of resistance for a quartz crystal

Fig.8 Plot of reactance vs frequency for a quartz crystal

The series and parallel resonance frequencies, fs and fp are determined by taking the input electrical conductance (real part of the admittance)and resistance (real part of the electric input impedance) in maximum, respectively, as shown in Fig.7. The resonance frequency, fr and anti-resonance frequency, fa , are given by the two roots where the susceptance (imaginary part of the input electric admittance) equals to zero, as shown in Fig.8. The resonance frequency and anti-resonance frequency fr and fa are the frequencies of principal interest in two terminal applications. For evaluating the equivalent circuit of a resonator, however, the characteristic frequencies, fs and fp are more important. They are given by

Where, C1and L1are the motional capacitance and motional inductance, respectively, and C0 is the static capacitance appearing in shunt branch.

(4) Motional Capacitance C1 and Motional Inductance L1

These two parameters are definitely related by the series resonance frequency, fs, as given in Eq.(1a), and fs is a very sure parameter in resonator design and in characterization. Only the value of C1 is specified in industry standard and L1 can be obtained from

The value of C1 is very small in comparison with capacitances usually used in oscillation circuits and can be evaluated from the material and geometry parameters of the crystal plate and electrodes.

(5) Static Capacitance (in Shunt)

The shunt capacitance, C0 , is a static capacitance, which is present whether the device is oscillating or not. The value of C0 can be measured at very low frequency (less than or about 1.0 MHz), and theoretically is given by

where, A is the electrode area, d is the thickness of the blank, and is the dielectric constant of the corresponding crystal cut.

(6) Quality Factor-Q

As a resonator, quality factor-Q value is a very important parameter. In specification, unloaded and loaded Q values are specified. The unloaded Q, or mechanical Q, can be expressed by

where, R1 is the resistance appearing in the series branch. The loaded Q value depends on the loaded circuit.

(7) Equivalent Series Resistance (ESR)

The resistance R1 appearing in the series branch (fig.5) can be measured at series resonance frequency, where the effects of C1 and L1 are cancelled each other and the effective result of the branch is a resistive. R1 represents the mechanical loss in the crystal unit and the holder.

(8) Load Capacitance

Load capacitance, CL, is the amount of capacitance that the oscillator exhibits when looking into the circuit through the two terminals of the resonator. The load capacitance is formally in either series or parallel with the resonator. For parallel load case the existence of CL will affect the parallel resonance frequency and the parallel-load resonance frequency, fL,is given by

This parameter is necessary to be specified.

(9)Pullability

In a parallel-load capacitance oscillator, the oscillation frequency depends on the load capacitance, CL as shown in Eq.(5). The frequency change (in ppm) as a function of the Load Capacitance change (in pF) is a specification. In certain applications where the variation of resonance frequency is mandatory (VCXO, for example), pullability has to be specified.

(10) Negative Resistance "-R"

Negative resistance is introduced to describe the electric property of an oscillator circuit, This is the amount of resistance that the oscillator circuit exhibits when looking into the circuit through the terminals of the resonator. One of the basic oscillation conditions demands the amplifier have to supply enough gain to compensate the loss in the resonator. From resonator point of view, the load has to exhibit enough "negative resistance" to compensate the resistance of the resonator. This is an important parmeter in designing oscillators.

(11) Drive Level

The drive level of an resonator is the amount of power dissipation, expressed in nanowatts, microwatts or milliwatts. Operating level is the suitable power range to assure proper start and maintain a steady state oscillation. Drive level should be operated at the minimum level to avoid long-term frequency drift and crystal fracture.

AGING

Aging is the relative change of operating frequency over a specified time period and is expressed in parts per million within a specified period. This rate of frequency change is normally exponential in character. The highest aging rate occurs within the first week of aging and decreases slowly after wards. Typically, aging is computed within first 30 days and is calculated over a long-term period (one year or ten years). Aging rate depends on many
factors: seal method, integrity, manufacturing processes, material type, operating temperature, and frequency.　

STORAGE TEMPERATURE RANGE

The specification indicates the minimum and maximum temperatures in which the devices can be stored or exposed in a non-operating state. After storing or exposing the devices at the specified temperature range for a long time, all of the specifications are guaranteed over the specified operating temperature range.