PRAP : Piezoelectric Resonance Analysis Program

The TE mode occurs in specimens with the following geometries

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample impedance is described by the following equation

\[ {\bf Z} = \frac{t}{i \omega {\boldsymbol \epsilon}^{S}_{33} A } \Bigg[ 1 - \frac{{\boldsymbol k}_{t}^2 tan ( {\frac{\omega}{4 {\boldsymbol f}_p }} ) } {\frac{\omega}{4 {\boldsymbol f}_p }} \Bigg] \]

\[ {\bf f_p} = \sqrt{\frac{{\bf c}_{33}^D }{4 \rho t^2} } \]

By fitting this equation to the observed resonance, the following material properties can be determined

k_t, c^D₃₃, c^E₃₃,e₃₃, h₃₃, ε^S₃₃.

The LE mode occurs in specimens with the following geometries

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample impedance is described by the following equation

\[ {\bf Z} = \frac{t}{i \omega {\boldsymbol \epsilon}^{S3=0}_{33} A } \Bigg[ 1 - \frac{{{\boldsymbol k^{2}_{33}}} tan ( {\frac{\omega}{4 {\boldsymbol f}_p }} ) } {\frac{\omega}{4 {\boldsymbol f}_p }} \Bigg] \]

\[ {\bf f_p} = \sqrt{\frac{1}{4 {\bf s}_{33}^D \rho t^2} } \]

By fitting this equation to the observed resonance, the following material properties can be determined

k₃₃, s^D₃₃, s^E₃₃, d₃₃, g₃₃, ε^T₃₃

The TS mode occurs in specimens with the following geometry

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample impedance is described by the following equation

\[ {\bf Z} = \frac{t}{i \omega {\boldsymbol \epsilon}^{S}_{11} A } \Bigg[ 1 - \frac{{\boldsymbol k}_{15}^2 tan ( {\frac{\omega}{4 {\boldsymbol f}_p }} ) } {\frac{\omega}{4 {\boldsymbol f}_p }} \Bigg] \]

\[ {\bf f_p} = \sqrt{\frac{{\bf c}^{D}_{55} }{4 \rho t^2} } \]

By fitting this equation to the observed resonance, the following material properties can be determined

k₁₅, c^D₅₅, c^E₅₅,s^D₅₅, s^E₅₅, e₁₅,h₁₅, d₁₅, g₁₅, ε^S₁₁, ε^T₁₁

The length shear mode occurs in specimens with the following geometry

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample impedance is described by the following equation

\[ {\bf Y} = \frac{i \omega {\boldsymbol \epsilon}^{T}_{11} w_1 w_2 } {L} \Bigg[ 1 - {\boldsymbol k}_{15}^2 \{ 1 - \frac{ tan ( {\frac{\omega}{4 {\boldsymbol f}_s }} ) } {\frac{\omega}{4 {\boldsymbol f}_s }} \} \Bigg] \]

\[ {\bf f_s} = \sqrt{\frac{1}{4 {\bf s}^{E}_{55} \rho L^2} } \]

By fitting this equation to the observed resonance, the following material properties can be determined

k₁₅, s^E₅₅, d₁₅, ε^T₁₁

The radial mode occurs in specimens with the following geometry

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample admittance is described by the following equation

\[ {\bf Y} = \frac{-i \omega {\boldsymbol \epsilon}^{P}_{33}  \pi {\phi}^2}   {4 t} \Bigg[ {\frac{2 ({\bf k^P})^2 } {1 - {\sigma}^P - {\bf J( \sqrt{ \frac{\omega^2 {\phi}^2 \rho} {4 {\bf c}_{11}^P} } )   } } } \Bigg] \]

\[ {\bf J(x)} \equiv \frac{\bf J_0(x) }{\bf J_1(x)} \]

\[ (\bf k^P)^2 \equiv \frac{\bf e_{13}^P} {{\bf \epsilon_{33}^P} {\bf c_{11}^P} } \]

\[ {\boldsymbol \sigma^P} \equiv \frac{- \bf s_{12}^E} {\bf s_{11}^E} \]

\[ {\bf e_{13}^P} \equiv \frac{\bf d_{13}} {{\bf s_{11}^E} + {\bf s_{12}^E}} \]

\[ {\bf c_{11}^P} \equiv \frac{\bf s_{11}^E} {({\bf s_{11}^E})^2 + ({\bf s_{12}^E})^2} \]

\[ {\boldsymbol \epsilon_{33}^P} \equiv {\boldsymbol \epsilon_{33}^T} - \frac{2 {\bf d_{13}}^2} {{\bf s_{11}^E} - {\bf s_{12}^E}} \]

By fitting this equation to the observed resonance, the following material properties can be determined

s^E₁₁, s^E₁₂, d₁₃, ε^S₃₃, k^P, k_p, ε^P₃₃, σ, c^P₁₁, e^P₁₃, s^E₆₆.

The TE mode rotated 45° about the x or y axis in a 6mm symmetry material occurs in specimens with the following geometries;

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes. In the rotated TE mode used by PRAP, the poling and specimen electrode surfaces are rotated some angle θ about the 'a' or 'b' axes of the crystal system.

The resonance observed in the sample impedance is described by the following equation

\[ {\bf Z} = \frac{t {\boldsymbol \beta_{33}^{S \theta}} }{i \omega A } \Bigg[ 1 - \frac{{\boldsymbol k}_{t}^{\theta 2} tan ( {\frac{\omega}{4 {\boldsymbol f}_p^{\theta} }} ) } {\frac{\omega}{4 {\boldsymbol f}_p^{\theta} }} \Bigg] \]

\[ {\bf f_p^{\theta}} = \sqrt{\frac{{\bf c}_{33}^{D \theta} }{4 \rho t^2} } \]

The TE mode rotated 45° about the 'a' or 'b' (X and Y) axes is of special use in constructing the complete 6mm piezoelectric matrix, as it eliminates the need for the 6mm LE mode which is generally difficult to measure accurately. The PRAP 6mm piezoelectric matrix tool recognises results for this mode when computing the complete 6mm matrix from the contents of a compound file.

By fitting this equation to the observed resonance, the following material properties can be determined

k_t^45:X,Y, c₃₃^D45:X,Y, h₃₃^45:X,Y,β₃₃^S45:X,Y

The TE mode occurs in specimens with the following geometries

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample impedance is described by the following equation

\[ {\bf Z} = \frac{t}{i \omega {\boldsymbol \epsilon}^{S}_{33} A } \Bigg[ 1 - \frac{{\boldsymbol k}_{t}^2 tan ( {\frac{\omega}{4 {\boldsymbol f}_p }} ) } {\frac{\omega}{4 {\boldsymbol f}_p }} \Bigg] \]

\[ {\bf f_p} = \sqrt{\frac{{\bf c}_{33}^D }{4 \rho t^2} } \]

By fitting this equation to the observed resonance, the following material properties can be determined

k_t, c^D₃₃, c^E₃₃,e₃₃, h₃₃, ε^S₃₃.

The LE mode occurs in specimens with the following geometries

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample impedance is described by the following equation

\[ {\bf Z} = \frac{t}{i \omega {\boldsymbol \epsilon}^{S3=0}_{33} A } \Bigg[ 1 - \frac{{{\boldsymbol k^{2}_{33}}} tan ( {\frac{\omega}{4 {\boldsymbol f}_p }} ) } {\frac{\omega}{4 {\boldsymbol f}_p }} \Bigg] \]

\[ {\bf f_p} = \sqrt{\frac{1}{4 {\bf s}_{33}^D \rho t^2} } \]

By fitting this equation to the observed resonance, the following material properties can be determined

k₃₃, s^D₃₃, s^E₃₃, d₃₃, g₃₃, ε^T₃₃

The LTE mode occurs in specimens with the following geometry

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample admittance is described by the following equation

\[ {\bf Y} = \frac{i \omega {\boldsymbol \epsilon}^{T}_{33} w_1 w_2 } {t} \Bigg[ 1 - {\boldsymbol k}_{13}^2 \{ 1 - \frac{ tan ( {\frac{\omega}{4 {\boldsymbol f}_s }} ) } {\frac{\omega}{4 {\boldsymbol f}_s }} \} \Bigg] \]

\[ {\bf f_s} = \sqrt{\frac{1}{4 {\bf s}^{E}_{11} \rho w_1^2} } \]

By fitting this equation to the observed resonance, the following material properties can be determined

k₁₃, s^E₁₁, d₁₃, g₁₃, ε^T₃₃

The TS mode occurs in specimens with the following geometry

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample impedance is described by the following equation

\[ {\bf Z} = \frac{t}{i \omega {\boldsymbol \epsilon}^{S}_{11} A } \Bigg[ 1 - \frac{{\boldsymbol k}_{15}^2 tan ( {\frac{\omega}{4 {\boldsymbol f}_p }} ) } {\frac{\omega}{4 {\boldsymbol f}_p }} \Bigg] \]

\[ {\bf f_p} = \sqrt{\frac{{\bf c}^{D}_{55} }{4 \rho t^2} } \]

By fitting this equation to the observed resonance, the following material properties can be determined

k₁₅, c^D₅₅, c^E₅₅,s^D₅₅, s^E₅₅, e₁₅,h₁₅, d₁₅, g₁₅, ε^S₁₁, ε^T₁₁

The Rotated LTE mode occurs in specimens with the following geometry

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes. In the rotated LTE mode used by PRAP, the poling and specimen electrode surfaces are rotated 45° about the 'c' axis of the crystal system.

The resonance observed in the sample admittance is described by the following equation

\[ {\bf Y} = \frac{i \omega {\boldsymbol \epsilon}^{T}_{33} w_1 w_2 } {t} \Bigg[ 1 - {({\boldsymbol k}_{13}^{45:Z})}^2 \{ 1 - \frac{ tan ( {\frac{\omega}{4 {\boldsymbol f}_s }} ) } {\frac{\omega}{4 {\boldsymbol f}_s }} \} \Bigg] \]

\[ {\bf f_s} = \sqrt{\frac{1}{4 {\bf s}^{E 45:Z}_{11} \rho w_1^2} } \]

The LTE mode rotated 45° about the 'c' (Z) axis is required to construct the complete 4mm piezoelectric matrix. The PRAP 4mm piezoelectric matrix tool recognizes results for this mode when computing the complete 4mm matrix from the contents of a compound file.

By fitting this equation to the observed resonance, the following material properties can be determined

k^45:Z₁₃, s^{E 45:Z}₁₁, d₁₃, g₁₃, ε^T₃₃

The 4mm breathing mode is the expansion mode in a square plate poled in the thickness direction.

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample admittance is described by the following equation;

\[ {\bf Y} = \frac{i \omega {\boldsymbol \epsilon} w_1 w_2 } {t} \Bigg[ 1 - {\bf k}^2 \{ 1 - \frac{ tan ( {\frac{\omega}{4 {\bf f}_s }} ) } {\frac{\omega}{4 {\bf f}_s }} \} \Bigg] \]

\[ {\bf N_s} = w_1 {\bf f_s} =  \sqrt{\frac{1} {4 \rho \bf s}}  \]

\[ \bf N_s^2 = \frac{1}{4 \rho (\bf s_{11}^E + s_{12}^E) } \Bigg[ 1 + (1 - \frac{8}{\pi^2}) (\frac{\bf s_{12}^E} {\bf s_{11}^E+s_{12}^E}) \Bigg] \]

The breathing mode is of special use in constructing the complete 4mm piezoelectric matrix as given s^E₁₁, N_s provides s^E₁₂. The PRAP 4mm piezoelectric matrix tool recognizes results for this mode when computing the complete 4mm matrix from the contents of a compound file.

By fitting this equation to the observed resonance, the following material properties can be determined;

k, s, ε, N_s, rN²_s

The stack thickness extensional resonator occurs in specimens with the following geometries;

The arrows represent the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample impedance is described by the following equation;

\[Y(\omega)=i \omega C_0 + \frac{2 N^2}{Z_{ST}}tanh \left( \frac{n \gamma}{2} \right) \]

\[C_0 = \frac{\epsilon_{33}^S A}{t_p} (1-k_t^2) \]

\[ N = \frac{A}{t_p}k_t \sqrt{\epsilon_{33}^S c_{33}^D} \]

\[ Z_{ST} = \sqrt{Z_1 Z_2 \left(2+\frac{Z_1}{Z_2}  \right)} \]

\[ \gamma = 2 asinh \left( \sqrt{\frac{Z_1}{2 Z_2}} \right) \]

\[ Z_1 = i \rho \nu^D A tan \left( \frac{\omega L}{2 \nu^D} \right) \]

\[ Z_2 = \frac{\rho \nu^D A}{i sin \left( \frac{\omega t_p}{\nu^D} \right)} + \frac{i N^2}{\omega C_0}  \]

\[ \nu^D = \sqrt{\frac{c_{33}^D}{\rho}} \]

By fitting this equation to the observed resonance, the following material properties can be determined;

k_teff, c^D_33eff, c^E_33eff, e_33eff, h_33eff, ε^S_33eff, e³³_piezo, ε^S_33piezo

The stack length extensional resonator occurs in specimens with the following geometries;

The arrows represent the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample impedance is described by the following equation;

\[Y(\omega)=i \omega n C_0 + \frac{2 N^2}{Z_{ST}}tanh \left( \frac{n \gamma}{2} \right) \]

\[C_0 = \frac{\epsilon_{33}^T A}{L} (1-k_{33}^2) \]

\[ N = \frac{A}{L} \frac{d_{33}}{s_{33}^E} \]

\[ Z_{ST} = \sqrt{Z_1 Z_2 \left(2+\frac{Z_1}{Z_2}  \right)} \]

\[ \gamma = 2 asinh \left( \sqrt{\frac{Z_1}{2 Z_2}} \right) \]

\[ Z_1 = i \rho \nu^D A tan \left( \frac{\omega L}{2 \nu^D} \right) \]

\[ Z_2 = \frac{\rho \nu^D A}{i sin \left( \frac{\omega L}{\nu^D} \right)} + \frac{i N^2}{\omega C_0}  \]

\[ \nu^D = \frac{1}{\sqrt{\rho s_{33}^D }} \]

By fitting this equation to the observed resonance, the following material properties can be determined;

k_33eff, s^D_33eff, s^E_33eff, d_33eff, g_33eff, ε^T_33eff, d³³_piezo, ε^T_33piezo

The backed or thin film thickness extensional resonator has the following geometries;;

The arrows represent the sample poling directions. The backing is much thicker than the specimen. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample impedance is described by the following equation;

\[Z(\omega) = \frac{t}{\omega A} \left \{ -\beta_{33}^S d + h_{33} \frac{u_L}{u_R} sin \left( \frac{\omega d}{\nu_p} \right) - h_{33} u_s \left[ \frac{c_{33}^D \omega}{\nu_p} \frac{u_L}{u_R} -h_{33} \right] \left[ cos \left( \frac{\omega d}{\nu_p} \right) -1 \right] \right \} \]

\[ u_s = \frac{\nu_s}{c_s \omega tan \left( \frac{\omega l}{\nu_s}  \right)} \]

\[ u_L = \frac{h_{33} \nu_p}{\omega c_{33}^D} + u_s h_{33} sin \left( \frac{\omega d}{\nu_p} \right) \]

\[ u_R = cos \left( \frac{\omega d}{\nu_p} \right)  + \frac{u_s c_{33}^D \omega}{\nu_p} sin \left( \frac{\omega d}{\nu_p} \right)  \]

By fitting this equation to the observed resonance, the following material properties can be determined;

c^D₃₃, h₃₃, ε^S₃₃, c^D_s

The backed and matched thickness extensional resonator has the following geometries;

The arrows represent the sample poling directions. The backing is usually much thicker than the specimen while the top layer, typically used to match the transducer to its projection medium, is thin compared to the piezoelectric. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes. Note that this can also be used to model the effect of thick electrodes on a thickness extensional resonator.

The resonance observed in the sample impedance is described by the following equation;

\[ Z(\omega) = Z_C -  \frac{Z_C^2 N^2}{Z_A} \]

where

\[ Z_C = \frac{1}{\omega C_0} \]

\[ C_0 = \frac{\epsilon_{33}^S A_p}{t_p} \]

\[ N = C_0 h_{33}\]

\[ Z_A = \frac{Z_T^2 + Z_T Z_B + Z_T Z_F + Z_B Z_F}{2 Z_T + Z_B + Z_F} \]

\[ Z_S = \frac{\rho_p \nu_p A_p}{i sin \left( \frac{\omega t_p}{\nu_p}  \right)} \]

\[  Z_T = i \rho_p \nu_p A_p tan \left( \frac{\omega t_p}{2 \nu_p} \right) \]

\[  Z_B = i \rho_B \nu_B A_B tan \left( \frac{\omega t_B}{\nu_B} \right) \]

\[  Z_F = i \rho_F \nu_F A_F tan \left( \frac{\omega t_F}{\nu_F} \right) \]

\[ \nu_p = \sqrt{ \frac{c_{33}^D}{\rho_p} }  \]

\[ \nu_B = \sqrt{ \frac{c_B}{\rho_B} }  \]

\[ \nu_F = \sqrt{ \frac{c_F}{\rho_F} }  \]

By fitting this equation to the observed resonance, the following material properties can be determined;

c^D₃₃, h₃₃, ε^S₃₃, c^D_t, c^D_s, k_t, v_p, v_s, v_t

The bimorph resonator has the following geometry;

The arrows represent the sample poling directions. The resonance is generated by applying a field opposite to the polling directions of the top and bottom halves. One half then contracts while the other extends which causes the bimorph to bend. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample admittance is described by the following equation;

\[ Y = \frac{i \omega L w \epsilon_{33}^T}{2h} \left[  1 + k_{13}^2 \left( \frac{3}{4} \frac{F(\Omega L)}{\Omega L} - 1 \right) \right]\]

\[  F(\Omega L) = \frac{cosh(\Omega L)sin(\Omega L) + cos(\Omega L) sinh(\Omega L) }{1 + cos(\Omega L) cosh(\Omega L) } \]

\[  \Omega = \left(  \frac{\omega^2 \rho A_0}{EI}  \right)^\frac{1}{4} = \sqrt{\frac{\omega}{a} } \]

By fitting this equation to the observed resonance, the following material properties can be determined;

k₁₃, ε^T₃₃, a, EI

a is the flexural rigidity of the bimorph while EI is the product of the bulk Young's Modulus and the area moment of inertia.

The sphere or hemisphere resonator occurs in a specimen with the following geometry;

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes. This resonance is actually a shell breathing mode resonance.

The resonance observed in the sample admittance is described by the following equation;

\[ Y = \frac{i \omega \Omega a^2}{t} \left( \epsilon_{33}^T \left( 1 - k_p^2  \right) + \frac{\epsilon_{33}^T k_p^2 \omega_s^2}{\omega_s^2 - \omega^2}  \right) \]

where

\[  \omega_s = \frac{1}{a \sqrt{\rho s_c^E} }  \]

\[ k_p = \frac{d_{31}}{\sqrt{s_c^E \epsilon_{33}^T}}   \]

and

\[ s_c^E = \frac{1}{2 (s_{11}^E + s_{12}^2 ) }  \]

By fitting this equation to the observed resonance, the following material properties can be determined;

k₁₃, ε^T₃₃, s^E₁₁+s^E₁₂

The ring resonator occurs in a specimen with the following geometry;

The arrow represents the sample poling direction. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes. This resonance is actually a ring or hoop breathing mode resonance.

The resonance observed in the sample admittance is described by the following equation;

\[ Y = \frac{i \omega \pi (r_2^2 - r_1^2) }{t} \left( \epsilon_{33}^T \left( 1 - k_{13}^2  \right) + \frac{\epsilon_{33}^T k_{31}^2 \omega_s^2}{\omega_s^2 - \omega^2}  \right) \]

where

\[  \omega_s = \frac{1}{a \sqrt{\rho s_{11}^E} }  \]

\[ k_{13} = \frac{d_{31}}{\sqrt{s_{11}^E \epsilon_{33}^T}}   \]

By fitting this equation to the observed resonance, the following material properties can be determined;

k₁₃, ε^T₃₃, s^E₁₁, d₃₁

The cylinder resonator occurs in a specimen with the following geometry;

The arrows represent the sample poling directions. The resonance is generated by applying a field between the inner and outer radius of the specimen. This radial mode then couples with a length expansion mode to form the cylinder resonance. The limits in dimension should ensure that the resonance will be free of coupling with other resonance modes.

The resonance observed in the sample admittance is described by the following equation;

\[   Y(\omega) = i \omega C \left[  \frac{\alpha_4}{\alpha_1} + k_{31}^2  \frac{\alpha_3^2 tan(KL/2) }{\alpha_1 alpha_2 (KL/2)}  \right] \]

\[  K=\frac{\omega}{c_R}, \Omega=\frac{\omega a}{c}\]

\[  c_R = c \sqrt{\frac{\alpha_2}{\alpha_1}}, \alpha_1 = 1 - (1 - \sigma^2) \Omega^2 \]

\[   \alpha_2 = 1 - \Omega^2, \alpha_3 = 1 - (1+\sigma) \Omega^2, \alpha_4 = 1 - k_{31}^2 - (1 + \sigma) (1 - \sigma - 2 k_{31}^2 ) \Omega^2 \]

\[  C = \frac{2 \pi a L \epsilon_{33}^T}{t}, c = \sqrt{\frac{1}{\rho s_{11}^E}}, \sigma = - \frac{s_{12}^E}{s_{11}^E}, k_{31}^2 = \frac{d_{31}^2}{s_{11}^E \epsilon_{33}^T}  \]

By fitting this equation to the observed resonance, the following material properties can be determined;

s^E₁₁, s^E₁₂, d₃₁, ε^T₃₃, k₃₁

The traditional circuit model used for a piezoelectric resonator is Van Dyke's circuit model.

In this circuit diagram, the circuit components are traditional real capacitance's, resistance, and inductance. This model does not account for the impedance of the piezoelectric resonator away from the centre of the fundamental resonance peak, and does not work well for lossy resonators.

The observed sample impedance is better modelled when a circuit model is constructed using complex circuit components. For thickness, length, length-thickness extensional and thickness shear resonance modes, the following circuit model is used;

This complex circuit model better accounts for the impedance of lossy resonators in the vicinity of the fundamental resonance peak frequency.

For the radial resonance mode, the following circuit model is used;

As the radial resonance is defined using the fundamental and second order resonance peak, this model works over the frequency range of the first two resonance peaks.