Quartz Crystal
Quartz, composed of silicon and oxygen (silicon dioxide), exhibits piezoelectric properties.
This device generates an electrical potential when pressure is applied on the surfaces of the crystal.
Inversely, when electrical potential is applied to the surfaces of a crystal, mechanical deformation or vibration is generated. These vibrations occur at a frequency determined by the crystal design and oscillator circuit.
Figure 1 Quartz Crystal Structure Figure 2 Cross Section of Crystal Perpendicular to Axis ZZ
Cutting Angle & Vibration Mode
Cut angles differ depending upon the applications (Oscillation frequencies and electrical characteristics). The crystals used in crystal units are normally in the form of plates or elements cut from synthetic crystal. The Fig. 3 shows main cutting angles and Fig. 4 shows vibration modes, frequency range and capacity rations (typical values). Take the most popular AT-cut crystal wafer for example, it is in a plane which makes an angle of 35º15º to the Z-axis and the wager thickness is approximately 0.06mm in the case of 25MHz fundamental-wave thickness-shear vibration.
Figure 3 Cutting Angles and Vibration Mode
Figure 4 Basic Characterizations
Frequency Tolerance / Stability over Temperature
The Frequency Tolerance of a crystal is defined as the initial deviation of the crystal frequency as compared to the absolute at 25°C. The Frequency Stability over temp is defined as the frequency deviation compared to the measured frequency at 25°C OVER the defined operating temperature range (I.E. 0°C to +70°C). Stability tolerance is sometimes expressed as a percentage of frequency deviation rather than as Parts Per Million (PPM) The conversion is as follows:
.01 % = 100 PPM
.005% = 50 PPM
.001 % = 10 PPM etc.
The stability tolerance of a crystal needs to be specified, along with the operating temperature range. For instance, a crystal may be specified as having a frequency stability tolerance of ±50 PPM over an operating temperature of -45°C to + 85°C, and having a frequency tolerance of ±50 PPM at +25°C.
Frequency ---- Temperature Characteristics
Characteristics of an AT-cut the frequency-temperature characteristics of an AT-cut crystal unit most generally used at present are expressed by cubic curves (See Fig.5). A crystal plate is cut at an angle at which a required frequency tolerance is obtained in the given operating temperature range. Actually, however there can be some dispersion in apparent cutting angle due to the result of cutting and polishing accuracy in the successive processes. Therefore, it is necessary to raise processing accuracy.
Figure 5 Temperature Characters of Popular Cuts
Equivalent circuit
The equivalent circuit depicts electrical activity of a quartz crystal unit operating at its natural resonant frequency. The shunt capacitance (Co) represents the capacitance of the crystal electrodes plus the capacitance of the holder and leads. R1, C1, and L1 compose the "motional arm" of the crystal, and are referred to as the motional parameters. The motional inductance (L1) represents the vibrating mass of the crystal unit. The motional capacitance (C1) represents the elasticity of the quartz, and the resistance (R1), represents bulk losses occurring within the quartz.
Quartz crystal can be modeled as a series LRC circuit in parallel with a shunt capacitor. Figure 6 shows this generic circuit model.
Figure 6 Generic Crystal Model (Fundamental Mode)
Equivalent Series Resistance (ESR)
The equivalent series resistance is the resistive element (R1) of the quartz crystal equivalent circuit (see Equivalent Circuit below) This resistance represents the equivalent impedance of the crystal at natural resonant frequency (series resonance) ESR is measured by a Crystal Impedance (CI) meter.
ESR values are generally stated as maximum values expressed in ohms. The ESR values vary with frequency, mode of operation, holder type, crystal plate size, electrode size, and mounting structure. It is worth noting that the ESR value at a given frequency for an AT- strip crystal design is generally higher than that of the standard (round blank) design. This becomes more significant at lower frequencies. When transitioning from a series resonant through-hole HC-49/U type crystal to a smaller surface mount type utilizing an AT-strip crystal, some consideration may be given to the difference in the ESR values produced by different cuts.
fS = (Series) frequency = 
Load Capacitance
The load capacitance CL is a factor for determining the "conditions" of a crystal unit when used in the oscillation circuit. In an ordinary oscillation circuit, the crystal unit is used in a range where it functions as an inductive reactance. In such usage, the oscillation circuit operates as a capacitive reactance. In other words, when the oscillation circuit is seen from both terminals of the crystal unit, the oscillation circuit can be expressed as a series circuit of a negative resistance -R and a capacitance CL. At that time this capacitance is called the load capacitance. The relationship between load capacitance and oscillation frequency is not linear. When the load capacitance is small, the amount of frequency variation is large, and when the load capacitance is increased, frequency variation lowers. If the load capacitance is lessened in the oscillation circuit to secure a large allowance for the oscillation frequency, the frequency stability will be greatly influenced even by a small change in the circuit. The load capacitance can be chosen from standard values specified in the catalog.
Resonance Mode
Crystals have two modes of resonance: parallel and series. All crystals exhibit both resonance modes. The oscillator circuit is calibrated for one or the other, but not both. For applications requiring no tighter than 100ppm frequency accuracy, this spec is usually not an issue. However, if you are attempting to control frequency (or time) to within 100ppm, the resonance-mode specification becomes important. The difference to the crystal vendor is in which mode the crystal is calibrated during manufacturing. The crystal vendor sets up an oscillator circuit with the crystal in a customer-specified series resonance or parallel resonance and calibrates the crystal. Figure 7shows crystal impedance behavior versus frequency as well as the relative location of each resonance mode.
Figure 7 Crystal Impedance Vs Frequency Series vs. Parallel
"Series" resonant crystals are intended for use in circuits which contain no reactive components in the oscillator feedback loop. "Parallel" resonant crystals are intended for use in circuits which contain reactive components (usually capacitors) in the oscillator feedback loop. Such circuits depend on the combination of the reactive components and the crystal to accomplish the phase shift necessary to start and maintain oscillation at the specified frequency. Basic depictions of two such circuits are shown below (Fig 8).
Figure 8 Series Vs Parallel Circuits
Overtone Crystals
Because of the physical properties and geometry of an AT cut quarts blank, a crystal can vibrate at many frequencies. The lowest frequency is called the fundamental frequency and can be supplied up to about 45MHz. Higher frequencies (to over 200MHz) are achieved by operating the crystal at odd overtones, 3rd, 5th, 7th and 9th tuning the circuit so that the crystal oscillates at its designed overtone frequency (Fig.9)
Overtone crystals are specially processed for plane parallelism and surface finish in order to enhance their performance at the required overtone frequency. The overtone frequency is higher than the equivalent harmonic multiple of the fundamental by approximately 25MHz per overtone.
Figure 9 Overtone Responses of Quartz Crystal
Spurious Response
It is also possible for a crystal to vibrate at a frequency that is not related to its fundamental or overtone frequencies. Such undesired frequencies are referred to as spurious responses. The manufacturing processes are designed to minimize (not eliminate) the spurious responses and maximize the crystal activity at the desired frequency. The circuit designer should further guard against spurious responses by ensuring that the oscillator feedback circuit achieves its highest gain at the desired operating frequency.
Drive Level
Drive Level- the amount of power dissipated by the oscillating crystal unit. Usually expressed in terms of milliwatts (mW), and is usually specified in terms of current through the resonator or power dissipated by the resonator. The latter is preferable. The drive level of a crystal is a function of the reactance of the input and output capacitance of the inverter or microprocessor and all other external components including the crystal. To calculate drive level, "ohm's law" for power is used. Drive level should be held to a minimum to avoid problems with stability, aging, nonlinear coupled modes and other nonlinear effects (Fig 10).
Figure 10 Maximum Driving Current Vs Vibration Mode
Pullability
The Pullability if a crystal refers to a crystal operating in the parallel mode and is a measure of the frequency change as a function of load capacitance. Pullability is important to the circuit designer who whish to achieve several operating frequencies with a single crystal by means of switching various values of load capacitance (Fig 11).
Figure 11 Frequency Vs Load CapacitanceAgain
A change in the frequency and/or the resistance of a quartz crystal unit with the passage of time. Attributable to the relaxation of strain in the resonator and to mass transfer mechanisms within the resonator package due to contamination. Other factors include drive level, ambient temperature, wire fatigue, and frictional wear.
These factors are minimized by design considerations, including the mechanical design of the mounting structure, and by the design and control of certain manufacturing processes. Most of the aging effects of a crystal occur within the first 60 days of operation leading to slower aging characteristics through the first year. The integrity of the hermetic characteristics of the crystal package is a major factor in determining how well a crystal will age.
Negative Resistance
Te negative resistance; also refer to as degree of oscillation allowance that can be used for judging the quality of the circuit side oscillation motility. The used of a circuit with an insufficient negative resistance may lead to such an unexpected trouble as the quartz crystal unit falling to initiate oscillation even when power has been switched on.
Characteristics of Frequency Vs. Load Capacitance
For many applications there are requirements to pull the crystal frequency by using a load reactive element. This may be necessary in order to trim out the manufacturing tolerance or in phase locked loop and frequency modulation applications. In most applications the load reactive element in capacitive and therefore only this case is now considered(Fig 12).
Useful Equations:
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Equations
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Definitions
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fS = (Series) frequency =
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C0 = Static Capacitance in farads
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CL = Load capacitance =
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C1 = Motional capacitance in farads
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Co = Shunt capacitance =
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CL = Load capacitance in farads
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C1 = Motional capacitance =
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f = Nominal frequency in Hz
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L1 = Motional inductance =
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fL = Anti-resonant frequency in Hz
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R1 = Series resistance =
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fS = Series resonant frequency in Hz
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Q = Quality factor =
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L = Inductance into Henrys
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fL - fS = f =
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PL = Pullability (ppm/pF)
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PL = Pullability =
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Q = Quality factor
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R1 = Series resistance in ohms
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