**5. Appendix I – 3IMD equations**

At each elemental section, and following a similar process than that described in (Collado et al., 2009), the nonlinear capacitance acts as an infinitesimal nonlinear current generator at *2*ω*1* ω*2* (and *2*ω*2* ω*1*), when ω*1* and ω*<sup>2</sup>* are at resonance:

$$\frac{\partial I\_{\text{ul}\_{\text{ul}\_{\text{in}}}}(z)}{\partial z} = \frac{1}{2} \dot{f} \partial\_{\text{l}z} \Delta C\_{\text{K}} K\_{\text{A}o}^{\*} V\_{o\text{u}} \cos\left(\frac{\pi z}{l}\right) \tag{18}$$

where KΔω =Zth (Δω) Pd (Δω) .

Therefore the broadband energy balance all over the acoustic transmission line leads to

$$V\_{\rm a\_{2}} = -j\frac{1}{2}Q\_{\rm L}\frac{\Delta C\_{\rm K}}{C\_{\rm d,0}}P\_{\rm d\Delta o}Z\_{\rm ib\,\Delta o}^{\ast}V\_{\rm a\_{1}}\left(\frac{1}{j\frac{C\_{\rm q}Q\_{\rm L}}{2\mathcal{W}\_{0,v}}-1}\right) \tag{19}$$

### **6. Appendix II - Model transformation**

Losses are introduced as a complex elasticity by means of the viscous damping coefficient *η*:

Sources of Third–Order Intermodulation Distortion

in Bulk Acoustic Wave Devices: A Phenomenological Approach 499

<sup>1</sup> <sup>50</sup>

<sup>=</sup>

*m*

+ +

*jX j C*

<sup>1</sup> Re

<sup>1</sup> Re

*Q*

*and Systems for Communications*, pp. 599-603, 26-28 May 2008

*Transactions on Microwave Theory and Techniques*, submitted.

*and Techniques*, vol. 57, no. 12, Dec. 2009,pp. 3019-3029

*Appl. Supercond.* vol. 15, No. 1, March 2005, pp. 26-39.

*in m*

where *Zin* is the input impedance of the device and *Xm* is the series reactive term of the KLM model (Krimholtz et al., 1970). *Q0* in (28) represents the unloaded quality factor, that is

> *<sup>S</sup>* <sup>=</sup>ωτ

Auld B. A., Acoustic Fields and Waves in Solids (Krieger, Malabar, Florida), Vol. I, 1990 Camarchia V., Cappelluti F., Pirola M., Guerrieri S. D., Ghione G. 2007. Self-Consistent

Cho Y., Wakita J. 1993. Nonlinear equivalent circuits of acoustic devices. *Ultrasonics Symposium, 1993. Proceedings, IEEE 1993*, pp. 867-872 vol. 2, 31 Oct-3 Nov 1993 Constantinescu F., Nitescu M., Gheorghe A. G. 2008. New Nonlinear Circuit Models for

Collado C., Rocas E., Padilla A., Mateu J., O'Callaghan J. M., Orloff N. D., Booth J. C., Iborra

Collado C., Rocas E., Mateu J., Padilla A., O'Callaghan J. M. 2009. Nonlinear Distributed

Collado C., Mateu J. and O'Callaghan J. M. 2005. Analysis and Simulation of the Effects of

Feld D. A. 2009. One-parameter nonlinear mason model for predicting 2nd & 3rd order

Feld D. A., Parker R., Ruby R., Bradley P., Shim D. 2008. After 60 years: A new formula for

*Z jX j C*

 − − 

11 0 2 <sup>11</sup> 1 *S*

Electrothermal Modeling of Class A, AB, and B Power GaN HEMTs Under Modulated RF Excitation. *IEEE Transactions on Microwave Theory and Techniques*, vol.

Power BAW Resonators. *ICCSC 2008. 4th IEEE International Conference on Circuits* 

E., Aigner R. 2010. First-order nonlinearities of bulk acoustic wave resonators. *IEEE* 

Model for Bulk Acoustic Wave Resonators. *IEEE Transactions on Microwave Theory* 

Distributed Nonlinearities in Microwave Superconducting Devices. *IEEE Trans.* 

nonlinearities in BAW devices. *2009 IEEE International Ultrasonics Symposium (IUS)*,

computing quality factor is warranted. *2008 IEEE International Ultrasonics* 

\*

(30)

− (31)

\*

0

ω

0

1

ω

By circuit analysis of the KLM circuit model, it can be found that *β* is

β

obtained from S-parameters using (Feld et al., 2008)

55, no. 9, Sept. 2007, pp. 1824-1831.

pp. 1082-1087, 20-23 Sept. 2009

*Symposium*, pp. 431-436, 2-5 Nov. 2008

**8. References** 

$$c \to c + \eta \frac{\partial}{\partial t} \tag{20}$$

The inverse damping coefficient can also be understood as the conductance per unit length *Gd=η-1*. With that, the acoustic telegrapher equations, making use of the analogy between the acoustic and electric domains, can be written as:

$$\frac{\partial V}{\partial z} = -L\_d jcol\tag{21}$$

and

$$\frac{\partial I}{\partial z} = -\frac{1}{Ac^{\circ} + j\alpha A\eta} j\alpha V. \tag{22}$$

The shunt admittance of the acoustic transmission line implementation, given by (22) and in which *A·cD=Cd-1*, is a shunt capacitance in series with a resistance. To transform this to be a capacitance in parallel with the loss term, we introduce eq. 8 in eq. 22 and expand the shunt admittance in as a Taylor series. The result is a conductance value in parallel with a nonlinear capacitance of the form:

$$\mathbf{C}\_{d}(\upsilon, K) = \mathbf{C}\_{d,0} + \Delta \mathbf{C}\_{1}\upsilon + \Delta \mathbf{C}\_{2}\upsilon^{2} + \Delta \mathbf{C}\_{K}K \tag{23}$$

whose terms are related with the material linear and nonlinear properties as follows:

$$\mathbf{G} = \alpha \mathbf{\hat{C}}\_{d,0} \, ^2 A \boldsymbol{\eta} \tag{24}$$

$$
\Delta \mathbf{C}\_1 = \frac{\mathbf{c}\_1^D}{\left(A \mathbf{c}\_0^D\right)^2} \tag{25}
$$

$$
\Delta \mathbf{C}\_2 = -\frac{c\_2^D}{A^3 \left(\mathbf{c}\_0^D\right)^2} \tag{26}
$$

$$
\Delta \mathbf{C}\_1 = -\frac{\mathbf{c}\_K^D}{A \{\mathbf{c}\_0^D\}^2} \tag{27}
$$

### **7. Appendix III - Broadband loaded quality factor**

The loaded quality factor can be defined as (Russer, 2006)

$$Q\_{\perp} = \frac{Q\_0}{1 + \beta} \tag{28}$$

where *β* relates the dissipated power in the acoustic resonator *Pres*, that is the acoustic transmission line, and the externally dissipated power *Pext* as follows:

$$
\mathcal{B} = \frac{P\_{\text{ext}}}{P\_{\text{res}}}.\tag{29}
$$

By circuit analysis of the KLM circuit model, it can be found that *β* is

$$\beta = \frac{\text{Re}\left(\frac{1}{\left(50 + jX\_w + \frac{1}{j\alpha \mathcal{C}\_0}\right)^\circ}\right)}{\text{Re}\left(\frac{1}{\left(Z\_{\text{in}} - jX\_m - \frac{1}{j\alpha \mathcal{C}\_0}\right)^\circ}\right)}\tag{30}$$

where *Zin* is the input impedance of the device and *Xm* is the series reactive term of the KLM model (Krimholtz et al., 1970). *Q0* in (28) represents the unloaded quality factor, that is obtained from S-parameters using (Feld et al., 2008)

$$Q\_0 = \alpha \sigma \frac{\left| \mathbf{S}\_{11} \right|}{1 - \left| \mathbf{S}\_{11} \right|^2} \tag{31}$$

### **8. References**

498 Acoustic Waves – From Microdevices to Helioseismology

*cc*

*z*

whose terms are related with the material linear and nonlinear properties as follows:

2 2 *G CA* =ω

> *<sup>c</sup> <sup>C</sup> Ac*

*<sup>c</sup> <sup>C</sup> A c*

*<sup>c</sup> <sup>C</sup> A c*

**7. Appendix III - Broadband loaded quality factor**  The loaded quality factor can be defined as (Russer, 2006)

transmission line, and the externally dissipated power *Pext* as follows:

1 2 <sup>0</sup> ( ) *D K D*

0 1 *<sup>L</sup> <sup>Q</sup> <sup>Q</sup>*

> . *ext res P P* β

where *β* relates the dissipated power in the acoustic resonator *Pres*, that is the acoustic

β

 η

1 1 2 <sup>0</sup> ( ) *D D*

2 2 3 2 <sup>0</sup> ( ) *D D*

acoustic and electric domains, can be written as:

nonlinear capacitance of the form:

and

+→ η

The inverse damping coefficient can also be understood as the conductance per unit length *Gd=η-1*. With that, the acoustic telegrapher equations, making use of the analogy between the

> *d <sup>V</sup> <sup>L</sup> <sup>j</sup> <sup>I</sup>*

ω

<sup>1</sup> . *<sup>D</sup> <sup>I</sup> <sup>j</sup> <sup>V</sup> z Ac j A*

ω η

The shunt admittance of the acoustic transmission line implementation, given by (22) and in which *A·cD=Cd-1*, is a shunt capacitance in series with a resistance. To transform this to be a capacitance in parallel with the loss term, we introduce eq. 8 in eq. 22 and expand the shunt admittance in as a Taylor series. The result is a conductance value in parallel with a

ω

2 ,0 1 2 (, ) *C vK C Cv Cv CK d d* = +Δ +Δ +Δ *<sup>K</sup>* (23)

*t*

(20)

<sup>∂</sup> = − ∂ (21)

*<sup>d</sup>*,0 (24)

Δ = (25)

Δ =− (26)

Δ =− (27)

<sup>=</sup> <sup>+</sup> (28)

= (29)

<sup>∂</sup> = − ∂ + (22)

∂ ∂

Auld B. A., Acoustic Fields and Waves in Solids (Krieger, Malabar, Florida), Vol. I, 1990


**22** 

*Japan* 

Takahiko Yanagitani *Nagoya Institute of Technology* 

**Shear Mode Piezoelectric Thin Film Resonators** 

Acoustic microsensor technique, well known as QCM (Quartz crystal microbalance) or TSM (Thickness shear mode) sensor, is an effective method to detect small mass loading on the sensor surface. This sensor can be operated even in liquid by using shear mode resonance. Therefore, shear mode piezoelectric film resonators are attractive for liquid microsensor

Shear wave has some unique features compared with the longitudinal wave, for example, it has extremely low velocity in the liquid. Longitudinal wave velocity in the water is 1492.6 m/s, whereas, shear wave velocity in the water is 20-60 m/s at 20-200 MHz (Matsumoto et al., 2000). Therefore, shear mode vibrating solid maintains its vibration even in the liquid, because the difference of acoustic impedance which determines the refection coefficient of

> *l s l s Z Z Z Z* <sup>−</sup> Γ =

( ) ( ) 1 2 *Z R jX c j* =+ = + ρ

*R* and *X* represent the real part and imaginary part of the acoustic impedance and

represent mass density, stiffness constant and viscosity in the medium, respectively.

2 α

<sup>−</sup> *jcj*

2 2 22

*v*

=+

is attenuation factor (B. A. Auld, 1973).

 ωη

( ) <sup>2</sup>

*<sup>v</sup> <sup>X</sup>*

ω α

ρ ρω

2 2 22

*v*

ωα

ωη

+ (1)

(2)

(3)

<sup>=</sup> <sup>+</sup> (4)

ρ, *c* and η

**1.1 Shear mode bulk acoustic wave devices and sensors** 

solid / liquid interface is very large in the case of shear wave. The complex refection coefficient Γ of the interface is given as

where Zs and Z*l* are the complex acoustic impedance of solid and liquid.

ω

*v*

*<sup>v</sup> <sup>R</sup>*

ω α<sup>=</sup> <sup>+</sup> ,

ρ ω

technique such as biosensors and immunosensors.

Complex acoustic impedance can be written as

Acoustic wave equation gives dispersion relation of

α

According to (2) and (3), acoustic impedance gives

where *v* is velocity and

**1. Introduction** 

