Equivalent Circuit

      A quartz crystal resonator is a mechanically vibrating system that is linked, via the piezoelectric effect, to the electrical world.  It consists of a quartz plate with metal plating (electrodes), which is located on both sides of the quartz plate and is connected to insulated leads on the crystal package.  Although the theoretical analysis of this device is a relatively complex electro-mechanical function, it can be expressed in terms of a simple equivalent circuit in the vicinity of the resonance frequencies as shown below:

         Equivalent electric circuit of quartz crystal resonator

     The C₀, called the “shunt” or static capacitance, is the capacitance due to the electrodes on the crystal plate plus the stray capacitances due to the crystal enclosure.  Shunt capacitance is present whether the crystal plate is oscillating or not (unrelated to the piezoelectric effect of the quartz).  The R, C₁ and L₁ portion of the circuit is known as the “motional arm”, which arises from the mechanical vibrations of the crystal.  R represents the equivalent motional arm resistance; C₁ represents the motional capacitance of the quartz; and L₁ is the motional inductance, a function of the mass.  The C₀ to C₁ ratio is a measure of the interconversion between electrical and mechanical energy stored in the crystal, i.e., of the piezoelectric coupling factor, k.  C₀/C₁ increases with the square of the overtone number.  When a dc voltage is applied to the electrodes of a resonator, the capacitance ratio C₀/C₁ is also a measure of the ratio of electrical energy stored in the capacitor formed by the electrodes to the energy stored elastically in the crystal due to the lattice strains produced by the piezoelectric effect.

       When working with most crystal resonators, it is only necessary to specify one motional component or the other due to the absolute correlation between C₁ and L₁ (C₁*L₁ is constant for a given series resonant frequency).  The industry standard is to specify a proper value of C₁ only.   The actual value of C₁ has physical limitations when it is realized in a quartz crystal design.  These constraints include the quartz cut, the mechanical design, the mode of operation and the nominal frequency of the crystal resonator.

       Although the equivalent circuit appears relatively simple, determining the several characteristic frequencies useful in describing resonator equivalent circuit properties is surprisingly complex.  For the most part, simple approximations are used.  The following notation is useful in determining some of these frequencies.

            F_s (Series resonance frequency) = 1 / [2p(L₁C₁)^1/2]

            F_p (Parallel Resonance frequency) = f_s[1 + 1/(2g)]

            g (Capacitance ratio) = 2pf_sC₀ / C₁

            Q (Quality factor) = 2pf_sL₁ / R₁

            M (Figure of merit) = Q / g

Series vs. Parallel Resonance

    The frequency at which oscillation occurs defined by f_s is called Series Resonance Frequency.  The reactance/impedance curve of a crystal shown below reveals where mechanical resonance occurs.

     Series resonance occurs at the point the curve crosses zero.  At this point, the crystal appears resistive in the circuit, impedance is at minimum and current flow is at maximum.  As the frequency is increased beyond the point of series resonance, the crystal appears inductive in the circuit.  When the reactance of the motional inductance and shunt capacitance cancel, the crystal is at the frequency called Anti-resonance Frequency denoted as (f_p).  At this point, impedance is maximized and the current flow is minimized.  The frequency of a crystal at f_p is inherently unstable and should be never selected as the frequency of operation for an oscillator.  The region between fs and fp is typically called “Area of Parallel Resonance”.  Parallel resonance can occur by adding load capacitance to the crystal in series resulting a positive frequency shift determined by:

Df = f_sC₁/2(C₀ + C_L)

      The unavoidable presence of C_o in the equivalent circuit produces this anti-resonance, sometimes also called parallel resonance.  In this connection, r = C₀/C₁ is an important resonator parameter because it is inversely proportional to the spacing between resonance and antiresonance and thereby determines the maximum bandwidth in filters and the tuning range in oscillators.