There are many ways to cut crystal resonators from a piece of quartz. Therefore, the natural frequency of the mechanical system is independent of electrode area and proportional to the inverse of the thickness: In a cubic quartz resonator, the electrodes are usually positioned across the narrowest dimension. The spring modulus of the same resonator is proportional to the product of the electrode area and the inverse of the thickness.ģ. Where A is the electrode area and T is the thickness ( Fig.
The effective mass of a quartz resonator with electrode metallization on opposing faces of the narrowest dimension is proportional to the product of the electrode area and the electrode spacing (the narrowest dimension or thickness).
Which is of the form d 2Y/dt 2 + (ω0/QF)*dY/dt + Y*ω0 2 = 0.īecause the electrical and mechanical models are assumed equivalent, the natural frequency of the mechanical system must equal the natural frequency of the electrical system. Equating these forces (with no external driving force) gives: Frictional loss is assumed proportional to the velocity of the dashpot’s plunger and the dashpot’s friction constant (D). Hooke’s Law (F = K*Y) provides the spring force, where K is the spring modulus and Y is displacement from equilibrium.
The mechanical model of a crystal oscillator is a simple compliance (spring)-inertia (mass)-damping (dashpot) system.Ī simple linear model equates two forces: spring force and frictional force with mass multiplied by acceleration (Newton’s Second Law). In the simple mechanical model-mass, spring, dashpot-the forces applied to the crystal (ignoring gravity) accelerate the mass (F = ma) ( Fig. This yields the well-known result for LCR circuits: the natural frequency ω0 is the square root of the inverse of the product of the inductance and capacitance.
#Crystal filter design for am operation series#
L1: Determined by C1 and the operating frequency (motional inductance)įor a series LCR circuit with no driving voltage, summing the voltages across the elements produces:. R1: 10 Ω to 150 Ω (equivalent series resistance, ESR). C1: 2 fF to 20 fF (motional capacitance). The fundamental resonant mode of a quartz crystal can be modeled as an LCR network shunted by a capacitor.įor crystals operating in the fundamental mode with a 5-MHz to 30-MHz frequency range, typical values of the circuit elements are: The shunt capacitance represents the physical capacitance formed by both the parallel plate capacitance of the electrode metallization and the stray package capacitance.ġ. The series LCR branch, often called the motional arm, models the piezoelectric coupling to the mechanical quartz resonator. Quartz crystals are modeled electrically as a series LCR branch in parallel with a shunt capacitance ( Fig. (Topics relevant in other types of radio systems, such as crystal oscillator phase noise, aren’t limiting factors in ISM radios and aren’t covered.) Table Of Contents This information is based on experience from more than a decade designing ISM-band (industrial, scientific, medical) radios. These include load capacitance negative resistance startup time frequency stability versus temperature drive-level dependency crystal aging frequency error and spurious modes. This article covers the primary design considerations for fundamental-mode oscillators using AT-cut crystals. Crystals are widely used in oscillators, timebases, and frequency synthesizers for their high quality factor (QF) excellent frequency stability tight production tolerances and relatively low cost. Their piezoelectric properties (voltage across the crystal deforms it deforming the crystal generates a voltage) allow them to be the frequency-determining element in electronic circuits. Appropriately cut quartz crystals can be used as high-quality electromechanical resonators.
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