Parameter Definitions

Permittivity ε

The permittivity, or relative dielectric constant, ε, is a measure of the polarizability of the material. The directionality of the permittivity is expressed by tensor components, whereby the same component indexes are used as for th electric field and dielectric displacement.

For example,

• ε₃₃^T
  describes the permittivity value in the polarization direction (direction 3) when an electric field is applied also in the polarization direction, under conditions of constant mechanical stress (T = 0: “free” permittivity)
• ε₁₁^S
  is the electric field and dielectric displacement in direction 1, perpendicularto to the polarization direction at constant deformation S = 0: clamped permittivity)
   

Piezoelectric charge constant d_ij

The piezoelectric charge or strain constant d is a measure of the electric charge induced in response to a mechanical stress, or the achievable mechanical strain when an electric field is applied (T = constant). For example,

• d₃₃
  is the charge density developed per mechanical stress, or, alternatively, strain developed per unit of electric field strength, all in the polarization direction.
   

Piezoelectric voltage constant g_ij

The piezoelectric voltage constant g defines the ratio of the electric field strength E to the effective mechanical stress T. If one divides the respective piezoelectric charge constants dij by the corresponding permittivity value one gets the corresponding g_ij constant For example,

• g₃₁
  describes the electric field induced in direction 3 by a mechanical stress acting in direction 1.
   

Elastic constant s_ij

The elastic constant or compliance s is a measure of ij the ratio of the relative deformation S to the mechanical stress T. Because it depends on the interaction of mechanical and electrical energy, the electrical boundary conditions must be taken into consideration.

For example,

• S₃₃^E
  describes the ratio of the mechanical strain in direction 3 to the mechanical stress in direction 3, with constant electric field (for E = 0: short circuit)
• S₅₅^D
  is the ratio of a shear strain to the effective shear stress with constant dielectric displacement for D = 0:open circuit)
   

Note:

Young’s modulus Y_ij, which is often used in the English speaking world, is the reciprocal of the compliance constant.

Frequency constant N_i

The frequency constant N corresponds to half the speed of the sound wave propagating in the ceramic body (with the exception N_P, the planar oscillation). The indexes identify the corresponding direction of oscillation, for which the respective dimension A determines the (series) resonant frequency f_R: ( N = f_R * A ).

For example,

• N₃
  describes the frequency constant for the longitudinal oscillation of a slim rod polarized in the longitudinal direction
• N₁
  is the frequency constant for the transverse oscillation of a slim rod polarized direction 3
• N_P
  is the frequency constant of the surface (planar) oscillation of a round disk
• N_t
  is the frequency constant of the thickness oscillation of a thin disk polarized in the thickness direction.
   

Mechanical quality factor Q_m

The mechanical quality factor Q_m characterizes the m “sharpness of the resonance” of a piezoelectric body (resonator) and is primarily determined from the 3 dB bandwidth of the series resonance of the resonating system. The reciprocal value of the Q-factor is the ratio of resistance to reactance, the mechanical loss factor, tanδ.

Coupling factor k

The Coupling factor k is a measure of the effectiveness of the piezoelectric effect (not the efficiency, as it is frequently incorrectly called!). It describes the ability of a piezoelectric material to transform electrical energy into mechanical energy and vice versa. Mathematically, the size of the coupling factor is determined by the square root of the ratio of stored mechanical energy to the total energy applied. At resonance, k is a function of the form of oscillation of the piezoelectric body.

For example,

• k₃₃
  is the coupling factor for the longitudinal oscillation
• k₃₁
  is the coupling factor for the transverse longitudinal oscillation
• k_p
  is the coupling factor for the radial oscillation (planar) of a round disk
• k_t
  is the coupling factor for the thickness oscillation of a plate
• k₁₅
  is the coupling factor for the thickness shear oscillation of a plate
   

Dynamic behavior

The electromechanical behavior of a piezoelectric body in oscillation can be represented by an electrical equivalent circuit.

C₀ + C₁ is the capacitance of the dielectric. The series circuit, consisting of C, L, and R, models the change in the mechanical properties of elastic deformation, effective mass (inertia) and mechanical losses resulting from internal friction. This description of the oscillatory circuit can only be used for frequencies in the vicinity of the lowest natural resonant frequency, however.

Most piezoelectric material parameters are determined by means of impedance measurements on special test bodies at resonance (see page 17, Comments to Table). The diagram illustrates a typical impedance curve.

The piezoelectric parameters are determined / calculated from the series and parallel resonances. These correspond closely to the impedance minimum f_m and maximum f_n.

Oscillation states or modes are determined by the geometry of the body, its mechanoelastic properties and the direction of polarization.

The most important oscillation states of common resonators are shown with the corresponding constants in the following illustration.

Oscillation modes of piezoelectric components

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Equivalent circuit of a piezoelectric resonator

Typical impedance curve