4. Frequency-Temperature Characteristic
Fig 3 shows the Frequency-Temperature Characteristic for a thickness-shear mode AT-CUT crsytal with the angle of cut as a parameter, since the AT-CUT Frequency-Temperature Characteristic is approximate to an equation of the third degree, it exhibits excellent frequency stability over a wide temperature range.

Note - Broken line represents loci of temperatures at which Af/AT=0
Fig. 3 : Frequency / temperature curves generalized (AT-cut)

5. Equivalent Circuit Parameters of Crystal Unit
A Crystal Unit consists of a quartz Resonator Blank with metal plating as electrode. This plating, as shown in Fig 4, is located on both sides of the crystal blank and is connected to insulated lead on the crystal package. The Crystal Unit exhibits a piezoelectric response between the two crystal electrodes as expressed in the simplified equivalent circuit shown in Fig 5. This simplied equivalent circuit is very useful to explain the basic concepts governing the crystal's behaviour and performance. Co represent the Shunt (static) Capacitances and is the sum of the capacitance between the electrodes and capacitances added by the wire leads and holder. The R1, L1, C1 elements are referred to as the motional parameters of the crystal Unit, the mechanical elasticity of the quartz is represented as a Motional Capacitance C1, the vibrating mass of the crystal is equivalnet to a Series Motional Inductance L1 and the mechanical losses of the crystal occur as an Equivalent Series resistance (ESR) R1.
Figure 4 : Inside view of crystal unit & schematic symbol
Figure 5 : Crystal unit simplified equivalnet circuit

6. Series Resonance Frequency (fr)
Crystal unit can be calibrated to two slightly different frequencies (see figure 6:fr and fa). The Lower one which is called Series Resonance Frequency (fr) is one of the two frequencies of the crystal unit alone under specified confoditions, at which the electrical impedance of the crystal unit is resistive (see figure 6). Figure 6 : Plot oif reactance VS frequency for a crystal unit

7. Anti Resonant (Parallel Resonant)
Frequency (fa) : The higher one which is called anti resonant or parallel resonant frequency (fa) is one of the two frequencies of a crystal unit alone, under specified conditions, at which the electrical impedance of the crystal unit is resistive (see figure 6)

8. Load Resonance Frequency (fL)
Series Resonance or Parallel Load Resonance Crystal Units are physically the same crystal, but are calibrated to slightly different frequencies. When a crsytal is placed into an oscillator circuit, they oscillate together at a tuned frequency. This frequency is dependent upon the crystal design and the amount of load capacitance (Cl) if any, the oscillator circuit presents to the crystal. Load capacitance is composed of a combination of the circuits discrete load capacitance, stray board capacitance, and capacitance from semiconductor Miller Effects. The presence of load capacitor shifts the working frequency of the crystal by an amount depending upon the value of CL and the value of C0 and C1. The figure 7 show the load resonance frequency (fL) according to series and parallel connections respectively. For a given value of load capacitance (CL), these Load Resonance Frequency (fL) is :

9. Load Capacitance (CL)
A standard oscillator with CL, parallel to the crystal as shown in figure 8. The total capacitance external to the crystal is called load capacitance. The crystal manufacturer needs to know the value of CL in order to adjust to the specified frequency, the load capacitance is given by : 10. Pullability
The pullability of a crystal describes how the operating frequency may be changed by varying the load capacitance (CL). An approximation of the pulling limits or load resonance frequency offset can be obtained from the formula :

No CL is connected to this the crystal
CL series to the crystal :fa is uneffected, but fr moves up to a frequency fL
CL parallel to the crystal : resonance frequency fr is not affected but the antiresonance fa shifts down to the frequrecny fL.

Figure 7 : Load resonant frequency (fL) when CL is connected to crystal unit
Figure 8 : Load capacitance (CL) parallel connected to the crystal unitFractional load resonance frequency offset DL can be easly calculated
Frequency pulling range A f L1, L2 between two load capacitance CL1 and CL2 can be obtained using the formula Fractional pulling range is as following :

11. Pull-Sensitive (S)
Pull-Sensitivy is defined as the incremental fractional frequency change for an incremental change in the load capacitance, the formula is :

12. Quality Factor (Q)
The "Q" of a crystal unit is the Quality Factor of the motional arm at resonance. The maximum stability that can be attained by the crystal unit is directly related to Q, the higher the Q the smaller the bandwith and the steeper the reactance slope (fr-fa), see figure 6. External circuit reatctance value changes have less on a high Q crystal (less pullability) than lower Q devices :The formula of Quality Factor is :
13. Nominal Frequency
The frequency assigned by the specification of the crystal unit.

14. Working Frequency (fw)
The operational frequency of the crystal unit together with its associated circuits.

15. Frequency Tolerance or Calibration Accuracy
The amount of frequency deviation from a specified nominal frequency at ambient temperature (referenced at 25ºC). This parameter is specified with a maximum and minimum frequency deviation, expressed in parts per million (ppm).

16. Frequency Stability
The amount of frequency deviation from the ambient temperature frequency. This deviation os associated with a set of operating conditions including : operating temperature range, load capacitance and drive level. This parameter is specified with a maximum and minimum frequency deviation expressed in ppm.
The frequency stability is determined by the following factors : type of quartz cut, angle of quartz cut, mode of vibration, and mechanical dimensions of quartz blank.

17. Operating Temperature Range
The range of temperature as measured on the enclosure over with the crystal unit must function though not necessary within the specified tolerances.

18. Operable Temperature range
The range of temperature as meassured on the enclosure over which the crystal unitmust function though not necessary within the specified tolerances.

19. Storage Temperature
The minimum and maximum temperature that the crystal unit can be stored or exposed to when in a non-oscillation state.
After exposing or storing the crystal unit at the maximum or minimum temperatures for a lenght of time, all of the the operating specifications are guaranteed over the specified operating temperature range.

20. Reference Temperature
The temperature at which certain crystal measurement are made, for controlled temperature units, the reference temperature is the mid point of the controlled temperature range. For non-controlled temperature units, the refrence temperature is normally 25ºC ¹ 2ºC

21. Equivalent Series Resistance (ESR)
The resistive element of crystal unit, the ESR measurement is made only at the series resonant frequency fr, not at some predetermined parallel resonant frequency (fL), the ESR can be represented by a formula :Generally, the lower the resistance value of crystal, the more active it is and less drive is required to activate it. How ever, lower values need additonal process to achieve it causing increase of unit price.

22. Load Resonance Resistance (RL)
the resistance of the crystal unit in series with a stated external capacitance at the load resoannce frequency fL.

23. Drive Level
rated drive level is the amount of power dissipation in the crystal, usually expressed in milliwatts or microwatts, it is fuction of the driving or excitation current flowing through the crystal,therefore in special cases, the level of drive may be specified in terms of crystal current or voltage.
Drive level can be calculated in the following formula :Where I stands for current to pass in the quartz crystal Re for effective resistance of quartz crystal and Re = R1 (1+C0/CL)2. Operating the crystal unit at drive levels which are too high or too low can result in improper performance, let say if the drive lvel (P) exceed the specified level, oscillation frequency will shift , crystal activity dips, excessive aging, or in extreme cases, physical failure of the resonator. This occurs because of excessive level of power causes stress for the crystal and causing temperature rise. If excessive drive level of power is applied to the quartz crystal unit, this may deteriorate or damage its characteristics. On the other hand, if the drive level is too low, the crystal unit may fail to oscillate or have degraded phase nois performance (this is exhibited to a greater degree in SC-Cut crystals).
Maximum power is the most power the crystal unit can dissipate while still maintaining operation with all electrical parameter guaranteed. Drive level should be maintened at the minimum level necessary to assure proper start up and steady state vibration, thus avoiding poor aging characteristic and crystal damage.
In addition, the maximum specified equivalent resistance of the crystal is affected by and is measured at a predetermined drive level. Therefore it is important to understand the effect of drive level on crystal performance and to operate the resonator at a suitable drive level.

24. Aging
The changes in operating frequency over a certain period of time, normally expressed as a maximum value of frequency change in parts per million (ppm) per year or per day.

25. Spurious or Unwanted Modes
Vibration at frequencies which are not fundamental or overtone modes are called Spurious or Unwanted modes. Spurious Modes is specified in terms of minimum resisitance of the spurious mode, or in terms of minimum resistance ration between the spurious mode and the main mode, or also in terms of the trasmission response in dB. When specifying spurious mode suppression it should be noted, not only can it add dramatically to the crystal cost, but it sometimes can not be done without affecting other parameter such as C1 and R1. Therefore if spurs are of your concern, please talk to us before finalying your specification.