Modeling the Resonator as a Parallel Plate

Abstract

After an introduction to complex resonance frequencies, the chapter provides a thorough discussion of the acoustic impedance, the acoustic wave impedance, and other types of impedances with relevance to either the QCM itself or to related problems. The load impedance (the complex ratio of the amplitudes of periodic stress and periodic velocity, both evaluated at the resonator surface) is, what the QCM measures on a fundamental level. The description continues with three separate but equivalent ways of modeling the viscoelastic plate and its resonances. All three formulations have their benefits and drawbacks. Building on these models, it is proven that the complex frequency shift is proportional to the complex load impedance, which is the essence of the small load approximation. The load impedance can be averaged over area and time. The last section deals with samples, which themselves are small resonators with their own resonance frequency. In the presence of such “coupled resonances”, the frequency shift may be positive or negative, depending on whether the resonance frequency of the coupled resonator is smaller or larger than the resonance frequency of the crystal.

Glossary

Variable

Definition (Comments)

〈.〉

Average (over area or time, as indicated by an index)

A

(Effective) area of the resonator plate (see Sect. 7.4)

B _el

Electric susceptance (B _el  = Im($\tilde{Y}_{el}$))

B _off

Offset of the susceptance in a data trace of B _el (f) (fit parameter in Eq. 4.5.30)

$\tilde{c}$

Speed of (shear) sound $\left( \tilde{c} = \left( {\tilde{G}/\uprho} \right)^{ 1/ 2} \right)$

C ₁

Motional capacitance

$\overline{C}_{ 1}$

Motional capacitance of the four-element circuit (Piezoelectric stiffening and the load have been taken into account)

d

Thickness of a layer

d _q

Thickness of the resonator (d _q  = m _q /ρ _q  = Z _q /(2ρ _q f ₀))

dC

Infinitesimal capacitance (Fig. 4.8a)

dL

Infinitesimal inductance (Fig. 4.8a)

D

Dissipation factor (D = 1/Q = 2Γ/f _r )

E

Electric field

E _tot

Total kinetic energy involved in a collision (see text above Eq. 4.2.16)

e ₂₆

Piezoelectric stress coefficient (e ₂₆ = 9.65 × 10⁻² C/m² for AT-cut quartz)

f

As an index: film

f

Frequency

f̃

Complex resonance frequency ($\tilde{f} = f_{r} + {\text{i}}\Gamma$)

f̃ _d

Damped resonance frequency ($\tilde{f}_{d} = \left( {f_{r}^{2} -\Gamma ^{2} } \right)^{1/2} +\; {\text{i}}\Gamma$, also: “ringing frequency”)

f̃ _r

Undamped resonance frequency

f _s

Series resonance frequency (f _r  = f _s by definition in this book)

f ₀

Resonance frequency at the fundamental

f ₀ = Z _q /(2m _q ) = Z _q /(2ρ _q d _q )

Might also have been called f ₁; we follow the literature in calling it f ₀

$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{F}$

Force

$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{F}_{ex}$

An external force driving a resonator

$\tilde{G}$

Shear modulus

G _e

Electric conductance (G _el  = Re($\tilde{Y}_{el}$))

G _max

Conductance on the peak of the resonance (fit parameter in Eq. 4.5.30)

G _off

Offset of the conductance in a data trace of G _el (f) (fit parameter in Eq. 4.5.30)

G _q

Shear modulus of AT-cut quartz (G _q  ≈ 29 × 10⁹ Pa, often called µ _q in the literature. G _q  is the piezoelectrically stiffened modulus (Sect. 5.3))

$\tilde{J}$

Shear compliance

h

Half the thickness of a plate or layer, half the length of a cable

H

Magnetic field

Î

Electric current

$\tilde{k}$

Wavenumber ($\tilde{k} = {\tilde{\upomega }/}\tilde{c}$)

k _t

Piezoelectric coupling coefficient (k ² _t  = e ₂₆/(ε _q ε₀ G _q )²)

liq

As an index: liquid

$\bar{L}_{j}$

Transfer matrix for layer j (See Eq. 4.4.2)

L ₁

Motional inductance

$\bar{L}_{ 1}$

Motional inductance of the four-element circuit (Piezoelectric stiffening and the load have been taken into account)

mot

As a index: motional (Applies to either the motional branch of the four-element circuit (Fig. 4.12b) or to the left-hand side of the transformer in the Mason-circuit (Fig. 4.10))

M

Mass

m _q

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

M _R

Mass of a Resonator

M _red

Reduced mass of a coupled two-body system (1/M _red  = 1/M _A  + 1/M _B )

n

Overtone order

n _right , n _left

Number of turns of a transformer (Eq. 4.5.7)

n, ñ

Refractive index ($\tilde{n} = {\tilde{\upvarepsilon }}_{r}^{ 1/ 2}$)

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

P

As an index: Particle (Sect. 4.6.3)

p _tot

Total momentum involved in a collision (see text above Eq. 4.2.16)

q

As an index: quartz resonator

Q

Q-factor (Q = 1/D = f _r /(2Γ))

ref

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

$\tilde{r}$

Amplitude reflection coefficient (reflectivity, for short)

Not to be confused with the power reflectance coefficient

Defined with regard to displacement (not stress)

$\tilde{r}_{q,\,S}$

Reflectivity evaluated at the resonator surface (see text above Eq. 4.4.6)

R

As an index: Resonator

R ₁

Motional resistance

$\overline{R}_{1}$

Motional resistance of the four-element circuit (piezoelectric stiffening and the load have been taken into account)

$\overline{S}_{i,\,j}$

Transfer matrix the interface between layer i and j (see Eq. 4.4.2)

S

As an index: Surface

t

Time

t̃

Amplitude transmission coefficient

û

(Tangential) displacement

Û

Electric voltage

v̂

Velocity (v̂ = iωû)

w

Half-band full-width of a resonance (w = 2Γ)

x(t), x̃(t)

Some time-harmonic function (real or complex) $\begin{gathered} \left(x t \right) = x_{0} { \cos }(\upomega t + {\upvarphi }), \hfill \\ \tilde{x}\left( t \right) = \hat{x} { \exp }({\text{i}}\upomega\,t)\hfill \\ \end{gathered}$

x̂

The function’s amplitude ($\hat{x} = x_{0} { \cos }({\upvarphi }) - {\text{i}}x_{0} { \sin }({\upvarphi })$)

z

Spatial coordinate perpendicular to the surface

Ỹ _el

Electric admittance ($\tilde{Y}_{el} = 1/\tilde{Z}_{el}$)

Z̃

Acoustic wave impedance (Table 4.1, mostly a shear-wave impedance)

$\tilde{Z}_{ac}$

Acoustic impedance (Table 4.1)

Z̃ _el

Electric impedance (Table 4.1)

Z̃ _EM

Electromagnetic wave impedance of plane waves (Table 4.1)

Z̃ _L

Load impedance (Table 4.1)

Z̃ _L,ijk

Load impedance in tensor form (Table 4.1)

Z _LCR

Electric impedance of an LCR-circuit (Eq. 4.5.15)

Z̃ _liq

Acoustic wave impedance of a liquid ($\tilde{Z}_{liq} = \left( {{\text{i}}\upomega \uprho _{liq}\upeta_{liq} } \right)^{ 1/ 2}$)

Z̃ _m

Mechanical impedance (Table 4.1)

Z _P

Characteristic impedance of a particle coupled to the main resonator (Z _P  = (κ _P M _P )^1/2, Sect. 4.6.3)

Z _q

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

Z _R

Characteristic impedance of a resonator (Z _R  = (κ _R M _R )^1/2, Sect. 4.6.3)

$\tilde{Z}_{W,\,et}$

Electric wave impedance (Table 4.1)

Z ₁₁, Z ₁₂

Elements of the transmission line in Fig. 4.18

γ _D

Damping factor (has units of Hz, γ _D  = 2πΓ)

Γ

Imaginary part of a resonance frequency

δf

A difference in frequency from the resonance frequency (Eq. 4.1.29)

δ _L

Loss angle (tan(δ _L ) = G″/G′ = J″/J′ often called tan(δ) in rheology)

δ _q

Loss angle of the quartz plate (can be an effective parameter capturing losses other than the intrinsic viscous dissipation)

Δ

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

ε

A small quantity (In Taylor expansions)

${\tilde{\upvarepsilon }}$

Dielectric permittivity (${\tilde{\upvarepsilon }} = {\tilde{\upvarepsilon }}_{r}\upvarepsilon_{0}$)

ε _q

Dielectric constant of AT-cut quartz (ε _q  = 4.54)

${\tilde{\upvarepsilon }}_{r}$

Relative dielectric permittivity (Also: “dielectric constant”)

ε₀

Dielectric permittivity of vacuum (ε₀ = 8.854 × 10⁻¹² C/(Vm))

φ

Phase

ϕ

Factor converting between mechanical and electric quantities in the Mason circuit (ϕ = Ae ₂₆/d _q )

${\tilde{\upeta }}$

Viscosity (${\tilde{\upeta }} = \tilde{G}/\left( {{\text{i}}\upomega} \right)$)

η _q

The “elastic viscosity” of AT-cut quartz, defined as η _q  = G _q ″/ω. η _q is roughly independent of frequency (G _q ″ is not). η _q depends on the defect density

κ _R

Spring constant of a Resonator

${\tilde{\upmu}}$

Magnetic permeability

μ₀

Magnetic permeability of vacuum (μ₀ = 4π × 10⁻⁷ Vs/Am)

${\upmu}_r, \tilde{\upmu}_r$

Relative magnetic permeability

ρ

Density

ρ _q

Density of crystalline quartz (ρ _q  = 2.65 g/cm³)

σ

(Tangential) stress

σ _S

Tangential stress at the resonator surface (Also: “traction”)

ξ _R

Drag coefficient of a Resonator (sometimes called “friction coefficient”, not to be confused with the friction coefficient in tribology)

ω

Angular frequency

$\upomega_{0}$

Undamped angular resonance frequency (ω₀ = (κ _R /M _R )^1/2)

ω _LC

Undamped resonance frequency of an LCR-circuit (ω _LC  = (LC)^−1/2)

ω _P

Particle resonance frequency of a coupled resonance (ω _P  ≈ (κ _P /M _P )^1/2)

${\tilde{\upomega }}_{r}$

Angular resonance frequency (${\tilde{\upomega }}_{r} = 2\uppi\tilde{f}_{r} = 2\uppi \left( {f_{r} + {\text{i}}\Gamma } \right)$)