The Small Load Approximation Revisited

Abstract

The chapter contains two extensions of the SLA. The first (Sect. 6.1) generalizes the SLA to arbitrary resonator shapes and modes of vibration. The load impedance in this formulation is a 3rd-rank tensor. The formalism shows that the statistical weight in area-averaging is the square of the local amplitude. The second extension (Sect. 6.2) is a perturbation analysis, applied to the model of the parallel plate. The perturbation is carried to 3rd order. The 3rd-order result fixes an inconsistency obtained when treating viscoelastic thin films in air with the conventional SLA.

Glossary

Variable

Definition

$\left\langle . \right\rangle$

Average

[.]

As a superscript: perturbation order

A

Effective area of the resonator plate

A

ω²-operator

c _ijkl

Stiffness tensor

comp

As an index: related to compressional waves

c _D

Stiffness tensor at constant electric displacement (Eq. 6.1.33)

cr

As an index: exerted by the crystal

d _f

Film thickness

diss

As an index: caused by dissipative processes

d _q

Thickness of the resonator

${\hat{\mathbf{D}}},{\mathbf{D}}$

Electric displacement

e

As an index: electrode

${\hat{\mathbf{E}}},{\mathbf{E}}$

Electric field

f _n

Resonance frequency at overtone order n

f _r

Resonance frequency

f ₀

Resonance frequency at the fundamental (f ₀ = Z _q /(2m _q ) = Z _q /(2ρ _q d _q ))

G _q

Shear modulus of AT-cut quartz (G _q  ≈ 29 × 10⁹ Pa)

h

One of the tensors quantifying piezoelectric coupling (Eq. 6.1.33)

k

Wavenumber

h _q

Half the thickness of the resonator plate

I

Second moment of area

J̃

Shear compliance

L

Width of a resonator plate

liq

As an index: liquid

m _f

Mass per unit area of a film

m _q

Mass per unit area of the resonator (m _q  = ρ _q d _q  = Z _q /(2f ₀))

n

Overtone order

n

n _i, Surface normal (a vector)

OC

As an index: resonance condition of the unloaded plate under open-circuit conditions. With no current into the electrodes (more precisely, with vanishing electric displacement everywhere), piezoelectric stiffening is fully accounted for by using the piezoelectrically stiffened shear modulus

q

As an index: quartz resonator

r, r _i

Position, a vector and its components

ref

As an index: reference state of a crystal in the absence of a load

r _S , r _S,i

Position on the resonator surface, a vector and its components

S

As an index: Surface

S

Infinitesimal strain tensor (Eq. 6.1.33)

T

Temperature

T

Stress tensor (Eq. 6.1.33)

û

Displacement (a vector)

v̂

Velocity (a vector)

Z̃ _L,ijk

Load impedance in tensor form

Z̃ _mot

Impedance to the left of the transformer in the Mason circuit (Fig. 4.10)

Z _q

Acoustic wave impedance of AT-cut quartz (Z _q  = 8.8 × 10⁶ kg m⁻² s⁻¹)

Γ

Imaginary part of a resonance frequency

δ_αβ

Kronecker δ (δ_αβ = 1 if α = β, δ_αβ = 0 otherwise)

δ(.)

Dirac δ-function

ε ⁻¹ _S

Inverse dielectric permittivity at constant strain (a tensor) (Eq. 6.1.33)

Δ

As a prefix: A shift induced by the presence of the sample

${\hat{\upsigma }}_{S,ij}$

Stress tensor

η _liq

Viscosity

μ

Non-dimensional mass (Eq. 6.2.9)

ρ

Density

ξ _liq

Nondimensional shear-wave impedance of the bulk liquid (Eq. 6.2.9)

ω

Angular frequency

ω _r

Angular resonance frequency (–ω ² _r is the eigenvalue of the ω²-operator)

ζ

Non-dimensional measure of the inverse square shear-wave impedance (Eq. 6.2.9)