**1. Introduction**

482 Acoustic Waves – From Microdevices to Helioseismology

[43] O.Tigli, and M.E.Zaghloul, A Novel Saw Device in CMOS: Design, Modeling, and Fabrication*, IEEE Sensors journal*, vol. 7, No. 2, February 2007, pp.219-227. [44] K.Nakamura, A Simple Equivalent Circuit For Interdigital Transducers Based On The

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Acoustic devices like Bulk Acoustic Wave (BAW) resonators and filters represent a key technology in modern microwave industry. More specifically, BAW technology offers promising performance due to its good power handling and high quality factors that make it suitable for a wide range of applications. Nevertheless, harmonics and 3IMD arising from intrinsic nonlinear material properties (Collado et al., 2009) and dynamic self-heating (Rocas et al., 2009) could represent a limitation for some applications.

Driven by the need for highly linear devices, there is a demand for further development of accurate models of BAW devices, capable of predicting the nonlinear behavior of the device and its impact on a circuit. Many authors have attempted to model the nonlinearities of BAW devices by using different approaches, mostly involving phenomenological lumped element models. Although these models can be useful because of their simplicity, they are mainly limited to narrow-band operation and they usually cannot be parameterized in terms of device-independent parameters (Constantinescu et al., 2008). Another approach consists of extending all the material properties on the constitutive equations to the nonlinear domain and introducing the nonlinear relations to the model implementation, which leads to several possible nonlinear sources increasing model complexity (Cho et al., 1993; Ueda et al., 2008). On the other hand, (Feld, 2009) presents a one-parameter nonlinear circuit model to account for the intrinsic nonlinearities. Such a model does not include the self-heating mechanism and can underestimate the 3IMD by more than 20 dB.

In this work, a model that uses several nonlinear parameters to predict harmonics and 3IMD distortion is presented. Its novelty lies in its ability to predict the nonlinear effects produced by self-heating in addition to those due to intrinsic nonlinearities in the material properties. The model can be considered an extension of the nonlinear KLM model (originally proposed by Krimholtz, Leedom and Matthaei) (Krimholtz et al., 1970) to include the thermal effects due to self-heating caused by viscous losses and electrode losses. For this purpose a thermal domain circuit model is implemented and coupled to the electro-acoustic model, which allows us to calculate the dynamic temperature variations that change the material properties. In comparison to (Rocas et al., 2009), this work describes the impact that electrode losses produce on the 3IMD, presents closed-form expressions derived from the

Sources of Third–Order Intermodulation Distortion

the temperature-dependent material properties.

ω*2 =* ω*0 +* Δ

dissipated power on the signal amplitude

*ω / 2* and

**2.2 Self-heating**

If ω*1* = ω*0 -* Δ

ω

in Bulk Acoustic Wave Devices: A Phenomenological Approach 485

Third-order intermodulation distortion due to dynamic self-heating is a well known process in microwave power amplifiers (Camarchia et al., 2007; Parker et al., 2004; Vuolevi et al., 2001) but has received less attention in passive devices (Rocas et al., 2010). What makes it different from the 3IMD caused by intrinsic nonlinearities is its dependence on the envelope frequency of the signal. For the particular case of a two-tone experiment, in which the envelope is a sinusoid, the thermal generation of 3IMD has a low-pass dependence on the

Recent results of two-tone 3IMD tests in BAW resonators as a function of the tones spacing reveal the important impact of self-heating effects in thin-Film Bulk Acoustic Resonators (FBAR) (Collado et al., 2009; Feld, 2009; Rocas et al., 2008) and Solidly Mounted Resonators (SMR) (Rocas et al., 2009). Heating produced by viscous damping in the acoustic domain and by ohmic loss in the electric domain produce local temperature oscillations which affect

occurs as a result of electric and acoustic losses, and the quadratic dependence of the

Δ Δ

*P V d t V t* ω

2 2

 ω

ω

2

*K c TK c cT cT cK* = +Δ +Δ +Δ (8)

 ω

∝ −+ + (5)

 ω

10 20 cos cos

222 121 0

ωω

ω

It is important to point out that the temperature variation at the envelope frequency (

ω

2 0 12 0

These frequency components produce temperature variations on the device at the same

( ) ( ) ( ). *K ZP*

 ωω

*1*) is the most relevant for the generation of spurious signals because of the low-pass filter

frequency changes of the material properties, and consequently, generate undesired 3IMD. In addition to being able to calculate the temperature oscillations, we also need to determine how these oscillations influence the device performance. For the specific case of BAW devices, there is consensus in assuming that the detuning of BAW devices with temperature is due to the variation of multiple material properties with temperature (Lakin et al., 2000; Ivira et al., 2008; Petit et al., 2007). We reflect this in our model by adding a temperature-

01 2 (, ) *D DD D D*

ω

1 cos(2 ) cos(2 ) 2

+Δ +

*V t t VV t*

111 cos(2 ) 222

∝++ −Δ +

*PVVV t t d*

ω

cos( ).

ω

*VV t*

gives rise to several frequency components of the dissipated power:

2

1 2

power and the thermal impedance as (Parker et al., 2004)

frequencies. These temperature variations *K(*

character of the thermal impedance *Zth(*

dependent term to the stiffened elasticity in (1)

+ Δ

*ω / 2* are the input signals for a two-tone test, dissipation

 ω 2

*)* can be written in terms of the dissipated

= *th d* (7)

*)*. These slow temperature oscillations induce low

(6)

Δ*ω =* ω*2*

envelope frequency due to the slow dynamics related with heating effects.

circuit model and validates the model with extensive measurements that confirm the necessity to include dynamic self-heating to accurately predict the generation of spurious signals in BAW devices.

### **2. Nonlinear generation mechanisms**

The origin of nonlinearities in BAW resonators has been controversial and there still exists no consensus (Nakamura et al., 2010). However, recent results point to several underlying causes which combine in different ways to give rise to a wide range of nonlinear effects (Rocas et al., 2009). We summarize the nonlinear effects of a stiffened elasticity, and then address the nonlinearity due to self-heating caused by viscous losses and electrode losses. We develop a circuit model to describe self-heating effects, and compare the measured results with simulations. Closed-form expressions for a simple one-layer BAW device model are then extracted to better understand the nonlinear generation mechanisms.

### **2.1 Nonlinear stiffened elasticity**

Nonlinear elasticity has been proposed as the predominant contribution to the measured second harmonics and as a potential source of the observed 3IMD products (Collado et al., 2009) in two-tone experiments.

The approach described in (Collado et al., 2009) starts by considering a nonlinear stressstrain relation under electric field described by a nonlinear stiffened elasticity *cD(T)* in the form of the polynomial

$$\mathcal{L}^D(T) = \mathcal{c}\_0^D + \Delta \mathcal{c}\_1^D T + \Delta \mathcal{c}\_2^D T^2 \tag{1}$$

where *T* is the stress. As detailed in (Collado et al., 2009), (1) translates into a nonlinear distributed capacitance *Cd(v)* in the equivalent electric model of the acoustic transmission line (Auld, 1990), in which the voltage *v* is equivalent to force. In the equivalent electric model (1) transforms into:

$$\mathbf{C}\_d(\upsilon) = \mathbf{C}\_{d,0} + \Delta \mathbf{C}\_1 \upsilon + \Delta \mathbf{C}\_2 \upsilon^2. \tag{2}$$

Equation (2) leads to the nonlinear acoustic Telegrapher's equations which can be used to obtain the maximum voltage amplitude occurring along a resonating transmission line as shown in (Collado et al., 2009; Collado et al., 2005). When the device is driven by two tones at frequencies ω*l* and ω*2*, standing waves with maximum force amplitudes *V*ω*1* and *V*ω*2* are trapped in the line. Then, as detailed in (Collado et al., 2009), the nonlinear capacitance (2) is responsible for generating 3IMD signals that result from adding the contributions due to Δ*c1D* and Δ*c2D*:

$$V\_{a\mathbf{o}\_{12}} = A\_1 Q\_L V\_{a\mathbf{i}\_1}^2 V\_{a\mathbf{i}\_2}^"\Delta \mathbf{C}\_1^{-2} \tag{3}$$

$$V\_{a\mathbf{q}\_2} = A\_2 Q\_l V\_{a\mathbf{q}\_l}^2 V\_{a\mathbf{q}\_2}^"\Delta\mathcal{C}\_2\tag{4}$$

where ω*12 = 2*ω*1* ω*2*, *QL* is the loaded quality factor and *A1* and *A2* are constants that depend on the geometry of the device and on its materials. Identical results would be obtained for the 3IMD at *2*ω*2* ω*1* (which we will denote as ω*21*).

### **2.2 Self-heating**

484 Acoustic Waves – From Microdevices to Helioseismology

circuit model and validates the model with extensive measurements that confirm the necessity to include dynamic self-heating to accurately predict the generation of spurious

The origin of nonlinearities in BAW resonators has been controversial and there still exists no consensus (Nakamura et al., 2010). However, recent results point to several underlying causes which combine in different ways to give rise to a wide range of nonlinear effects (Rocas et al., 2009). We summarize the nonlinear effects of a stiffened elasticity, and then address the nonlinearity due to self-heating caused by viscous losses and electrode losses. We develop a circuit model to describe self-heating effects, and compare the measured results with simulations. Closed-form expressions for a simple one-layer BAW device model

Nonlinear elasticity has been proposed as the predominant contribution to the measured second harmonics and as a potential source of the observed 3IMD products (Collado et al.,

The approach described in (Collado et al., 2009) starts by considering a nonlinear stressstrain relation under electric field described by a nonlinear stiffened elasticity *cD(T)* in the

where *T* is the stress. As detailed in (Collado et al., 2009), (1) translates into a nonlinear distributed capacitance *Cd(v)* in the equivalent electric model of the acoustic transmission line (Auld, 1990), in which the voltage *v* is equivalent to force. In the equivalent electric

Equation (2) leads to the nonlinear acoustic Telegrapher's equations which can be used to obtain the maximum voltage amplitude occurring along a resonating transmission line as shown in (Collado et al., 2009; Collado et al., 2005). When the device is driven by two tones

trapped in the line. Then, as detailed in (Collado et al., 2009), the nonlinear capacitance (2) is responsible for generating 3IMD signals that result from adding the contributions due to

> 12 1 2 2\* 2 *V AQV V C*

12 1 2 2 \* *V AQV V C*

on the geometry of the device and on its materials. Identical results would be obtained for

ω*21*).

 ω = Δ 1 1 *L* ω

 ω = Δ 2 2 *L* ω

*2*, *QL* is the loaded quality factor and *A1* and *A2* are constants that depend

ω

ω

*1* (which we will denote as

*2*, standing waves with maximum force amplitudes *V*

2

2

01 2 ( ) *D DD D c T c cT cT* = +Δ +Δ (1)

,0 1 2 ( ) . *C v C Cv Cv d d* = +Δ +Δ (2)

ω*1* and *V*ω*2* are

(3)

(4)

are then extracted to better understand the nonlinear generation mechanisms.

signals in BAW devices.

**2. Nonlinear generation mechanisms**

**2.1 Nonlinear stiffened elasticity**

2009) in two-tone experiments.

form of the polynomial

model (1) transforms into:

ω*l* and ω

ω*2* ω

at frequencies

Δ*c2D*:

Δ*c1D* and

where ω*12 = 2*ω*1* ω

the 3IMD at *2*

Third-order intermodulation distortion due to dynamic self-heating is a well known process in microwave power amplifiers (Camarchia et al., 2007; Parker et al., 2004; Vuolevi et al., 2001) but has received less attention in passive devices (Rocas et al., 2010). What makes it different from the 3IMD caused by intrinsic nonlinearities is its dependence on the envelope frequency of the signal. For the particular case of a two-tone experiment, in which the envelope is a sinusoid, the thermal generation of 3IMD has a low-pass dependence on the envelope frequency due to the slow dynamics related with heating effects.

Recent results of two-tone 3IMD tests in BAW resonators as a function of the tones spacing reveal the important impact of self-heating effects in thin-Film Bulk Acoustic Resonators (FBAR) (Collado et al., 2009; Feld, 2009; Rocas et al., 2008) and Solidly Mounted Resonators (SMR) (Rocas et al., 2009). Heating produced by viscous damping in the acoustic domain and by ohmic loss in the electric domain produce local temperature oscillations which affect the temperature-dependent material properties.

If ω*1* = ω*0 -* Δ*ω / 2* and ω*2 =* ω*0 +* Δ*ω / 2* are the input signals for a two-tone test, dissipation occurs as a result of electric and acoustic losses, and the quadratic dependence of the dissipated power on the signal amplitude

$$P\_d \approx \left( V\_1 \cos\left( \left( \alpha \theta\_0 - \frac{\Delta \theta}{2} \right) t \right) + V\_2 \cos\left( \left( \alpha \theta\_0 + \frac{\Delta \theta}{2} \right) t \right) \right)^2 \tag{5}$$

gives rise to several frequency components of the dissipated power:

$$\begin{aligned} P\_d &\approx \frac{1}{2}V\_1^2 + \frac{1}{2}V\_2^2 + \frac{1}{2}V\_1^2 \cos(2\,a\_0t - \Delta\omega t) + \\ &\frac{1}{2}V\_2^2 \cos(2\,a\_0t + \Delta\omega t) + V\_1V\_2 \cos(2\,a\_0t) \\ &+ V\_1V\_2 \cos(\Delta\omega t). \end{aligned} \tag{6}$$

These frequency components produce temperature variations on the device at the same frequencies. These temperature variations *K(*ω*)* can be written in terms of the dissipated power and the thermal impedance as (Parker et al., 2004)

$$K(\alpha) = Z\_{\iota\iota}(\alpha) P\_{\iota}(\alpha). \tag{7}$$

It is important to point out that the temperature variation at the envelope frequency (Δ*ω =* ω*2* ω*1*) is the most relevant for the generation of spurious signals because of the low-pass filter character of the thermal impedance *Zth(*ω*)*. These slow temperature oscillations induce low frequency changes of the material properties, and consequently, generate undesired 3IMD.

In addition to being able to calculate the temperature oscillations, we also need to determine how these oscillations influence the device performance. For the specific case of BAW devices, there is consensus in assuming that the detuning of BAW devices with temperature is due to the variation of multiple material properties with temperature (Lakin et al., 2000; Ivira et al., 2008; Petit et al., 2007). We reflect this in our model by adding a temperaturedependent term to the stiffened elasticity in (1)

$$\mathcal{L}^D(T, K) = \mathcal{c}\_0^D + \Delta \mathcal{c}\_1^D T + \Delta \mathcal{c}\_2^D T^2 + \Delta \mathcal{c}\_K^D K \tag{8}$$

Sources of Third–Order Intermodulation Distortion

parasitic resistances *PρΔω*:

while keeping the tones spacing

P

0

**2.2.3 Circuit model with self-heating effects**

electrodes

(m

W)

in Bulk Acoustic Wave Devices: A Phenomenological Approach 487

contribution to the 3IMD can be calculated by the use of (10) and the power dissipated in the

= Δ *T L K th*

12 1 \* *V AQ C P Z V*

Whereas the parasitic resistance and distributed conductance can be obtained from the measured scattering parameters, that is, they produce distinguishable measurable effect, examination of (13) and (14) looks like both self-heating mechanisms produce the same experimental observable so they may not be distinguishable. This is true if a two-tone experiment at a fixed frequency is performed, but the two effects have different frequency

the dissipation due to ohmic losses is different than that produced by viscous losses, as shown in Fig 1. This information is extremely useful to validate the model with 3IMD

the electrodes and the piezoelectric layer are thin and made of good thermal conductors, so that the thermal impedance between those layers is negligible, as will be verified with the

Resonance

Max. Stress

Antiresonance

1.8 1.9 2 2.1 2.2 2.3 2.4 Frequency (GHz)

Fig. 1. Simulations of the dissipated power, for an input power of 20 dBm, due to acoustic

A circuit model implementation to reproduce thermal effects should be capable of predicting dynamic temperature variations. To achieve this, we extend the nonlinear KLM

> 2 , *p d th th*

*K KP C z k tk* ∂ ∂ ρ

viscous damping (solid line) and electrode electric losses (dashed line)

model (Collado et al., 2009) to include the thermal domain (Rocas et al., 2009). The procedure starts with the one dimensional heat equation along the *z* direction:

2

ρ Δ Δ ω ωω (14)

*0* of the 2 tones is swept

. This is because

Δω

1.2

0.96

P

 (

viscosity

m

W)

0.72

0.48

0.24

0

= − ∂ ∂ (15)

ω

constant. This happens because the frequency pattern of

ρ ,ω

dependence that can be distinguished if the central frequency

Δω

measurements by looking at the frequency dependence of the 3IMD. Note that (13) and (14) keep the same definition of thermal impedance *Zth,*

temperature simulations shown in Section III.B.2.

0.8

0.64

0.48

0.32

0.16

where *K* represents the temperature, the equivalent capacitance is

$$\mathbf{C}\_{d}(\upsilon,\mathbf{K}) = \mathbf{C}\_{d,0} + \Delta \mathbf{C}\_{1}\upsilon + \Delta \mathbf{C}\_{2}\upsilon^{2} + \Delta \mathbf{C}\_{\mathbf{K}}\mathbf{K}\_{\prime} \tag{9}$$

where each of the nonlinear terms Δ*C1*, Δ*C2* and Δ*CK* are related to their counterparts Δ*c1 D* , Δ*c2D*, Δ*cKD* respectively, as detailed in Appendix I.

The term Δ*CK* generates 3IMD, whose maximum voltage *V*ω*12* can be found in a similar way as the contribution of Δ*C1* in (3) and Δ*C2* in (4) (see details in Appendix I):

$$V\_{a\_{l2}} = A\_{l'} Q\_{l} \Delta C\_{K} P\_{d \Lambda \alpha \nu} Z\_{t \hbar \Lambda \alpha \nu}^{"} V\_{a\_{l} \nu} \tag{10}$$

where *AT* is a constant that depends on the device geometry and material parameters, *QL* is the loaded quality factor, *Zth,*Δω is the thermal impedance (7) evaluated at Δ*ω*, and *Pd,*Δω is the Δω frequency component of the dissipated power in (6). Equation (10) describes the 3IMD signal due to self-heating effects, inside the acoustic transmission line, in terms of the dissipated power. As detailed in the following sub-sections, the dissipated power is due to both electric and acoustic loss, thus both effects contribute to the 3IMD in (10).

### **2.2.1 3IMD due to viscous losses**

Viscosity is introduced in the model as a complex elasticity (Auld, 1990), which translates into a shunt resistance *Rd,η* in series with the shunt capacitance *Cd* in a transmission line implementation. Appendix II details a model transformation to go from the original *Rd,η* to an equivalent model in which the viscosity is implemented as a conductance *Gd* in parallel with the capacitance *Cd*. The equivalent model allows for an easier extraction of the closedform expressions.

The instantaneous dissipated power due to viscous damping at each position *z* along the transmission line of length *l* (thickness of the piezoelectric layer) is

$$\frac{\partial P\_{d,\Lambda\alpha}(z)}{\partial z} = G\_d V\_{a\mathfrak{q}} V\_{a\mathfrak{q}} \cos^2 \left(\frac{\pi z}{l}\right)\_{l'} \tag{11}$$

which can be integrated along *l* to obtain the total dissipated power

$$P\_{d, \alpha \nu} = \frac{1}{2} I G\_d V\_{\alpha \natural} V\_{\alpha\_2}^\* \,. \tag{12}$$

Equation (12) can be combined with (10) to obtain the peak 3IMD voltage (*V*η*,*ω*12* ) due to the viscous damping

$$V\_{\eta, a\_{\Omega\_2}} = \frac{1}{2} A\_I l G\_d Q\_L \Delta C\_K Z\_{th\Lambda\alpha\rho}^{"\*} V\_{a\_{\parallel}}^2 V\_{a\_{\parallel}}^{"\*} \tag{13}$$

### **2.2.2 3IMD due to loss in the electrodes**

There is certain agreement in considering ohmic losses as a significant dissipation mechanism (Thalhammer et al., 2005) in addition to the viscous damping. As it will be discussed in section II.B.3, electrodes losses are introduced in the circuit model as parasitic series resistances at the input and at the output ports, and their values are determined by fitting the model to the measured scattering parameters in the linear regime. Their

2

*C2* in (4) (see details in Appendix I):

ω ω

Δ

12 1 \* , *V AQ C P Z V*

where *AT* is a constant that depends on the device geometry and material parameters, *QL* is

Viscosity is introduced in the model as a complex elasticity (Auld, 1990), which translates into a shunt resistance *Rd,η* in series with the shunt capacitance *Cd* in a transmission line implementation. Appendix II details a model transformation to go from the original *Rd,η* to an equivalent model in which the viscosity is implemented as a conductance *Gd* in parallel with the capacitance *Cd*. The equivalent model allows for an easier extraction of the closed-

The instantaneous dissipated power due to viscous damping at each position *z* along the

1 2 , 2 ( ) cos *d d P z z GV V z l*

> 1 2 \*

ω

 ω=

12 1 2

2 *V A lG Q C Z V V*

There is certain agreement in considering ohmic losses as a significant dissipation mechanism (Thalhammer et al., 2005) in addition to the viscous damping. As it will be discussed in section II.B.3, electrodes losses are introduced in the circuit model as parasitic series resistances at the input and at the output ports, and their values are determined by fitting the model to the measured scattering parameters in the linear regime. Their

= Δ *T d L K th*Δ

π

\* 2\*

ω ω

 ω

= ∂ , (11)

. (12)

η*,*ω

(13)

*12* ) due to the

ω ω

1 2 *P lG V V d d* Δω

∂ Δ

 frequency component of the dissipated power in (6). Equation (10) describes the 3IMD signal due to self-heating effects, inside the acoustic transmission line, in terms of the dissipated power. As detailed in the following sub-sections, the dissipated power is due to

= Δ *T L K d th* Δ Δ

 ω

is the thermal impedance (7) evaluated at

,0 1 2 (, ) , *C vK C Cv Cv CK d d* = +Δ +Δ +Δ *K* (9)

ω

*CK* are related to their counterparts

*12* can be found in a similar way

(10)

*ω*, and *Pd,*

Δωis the

Δ

Δ*c1D* ,

where *K* represents the temperature, the equivalent capacitance is

*cKD* respectively, as detailed in Appendix I.

*C1* in (3) and

Δω

Δ

Δ*C1*, Δ*C2* and

Δ

both electric and acoustic loss, thus both effects contribute to the 3IMD in (10).

*CK* generates 3IMD, whose maximum voltage *V*

ω

transmission line of length *l* (thickness of the piezoelectric layer) is

which can be integrated along *l* to obtain the total dissipated power

,

η ω

**2.2.2 3IMD due to loss in the electrodes**

ω

,

Equation (12) can be combined with (10) to obtain the peak 3IMD voltage (*V*

1

where each of the nonlinear terms

Δ*c2D*, Δ

Δω

The term

Δ

as the contribution of

form expressions.

viscous damping

the loaded quality factor, *Zth,*

**2.2.1 3IMD due to viscous losses**

contribution to the 3IMD can be calculated by the use of (10) and the power dissipated in the parasitic resistances *PρΔω*:

$$V\_{\rho,a\_{l2}} = A\_T Q\_L \Delta C\_K P\_{\rho \Delta \rho} Z\_{\text{th} \Delta \rho} \stackrel{"\text{}}{V}\_{a\_l} \tag{14}$$

Whereas the parasitic resistance and distributed conductance can be obtained from the measured scattering parameters, that is, they produce distinguishable measurable effect, examination of (13) and (14) looks like both self-heating mechanisms produce the same experimental observable so they may not be distinguishable. This is true if a two-tone experiment at a fixed frequency is performed, but the two effects have different frequency dependence that can be distinguished if the central frequency ω*0* of the 2 tones is swept while keeping the tones spacing Δω constant. This happens because the frequency pattern of the dissipation due to ohmic losses is different than that produced by viscous losses, as shown in Fig 1. This information is extremely useful to validate the model with 3IMD measurements by looking at the frequency dependence of the 3IMD.

Note that (13) and (14) keep the same definition of thermal impedance *Zth,*Δω. This is because the electrodes and the piezoelectric layer are thin and made of good thermal conductors, so that the thermal impedance between those layers is negligible, as will be verified with the temperature simulations shown in Section III.B.2.

Fig. 1. Simulations of the dissipated power, for an input power of 20 dBm, due to acoustic viscous damping (solid line) and electrode electric losses (dashed line)

### **2.2.3 Circuit model with self-heating effects**

A circuit model implementation to reproduce thermal effects should be capable of predicting dynamic temperature variations. To achieve this, we extend the nonlinear KLM model (Collado et al., 2009) to include the thermal domain (Rocas et al., 2009).

The procedure starts with the one dimensional heat equation along the *z* direction:

$$\frac{\partial^2 \mathbf{K}}{\partial z^2} = \frac{\rho \mathbf{C}\_p}{k\_{th}} \frac{\partial \mathbf{K}}{\partial t} - \frac{P\_d}{k\_{th}} \tag{15}$$

Sources of Third–Order Intermodulation Distortion

in Fig. 4.

in Bulk Acoustic Wave Devices: A Phenomenological Approach 489

temperature rise is used to modify the distributed acoustic capacitance *CNL(T,K)*, as shown

*∆z*

*CNL(T,K) Gd*

Fig. 4. Implementation of a section of the piezoelectric layer with the acoustic and thermal domains coupled by the generated heat at *Gd* and the temperature *K*. *Ld* is the acoustic

Thermal Domain Extension

... ...

**Rth,1/2 Rth,Si/2 Rth,Si/2**

**Cth,1 Cth,Si**

**Tamb Tamb**

**∆z**

Piezoelectric Layer (Thermal Domain)

... ...

**Zair Zair**

Electro-Acoustic Conversion

Fig. 5. Complete circuit model with thermo-acoustic model of the piezoelectric layer, top and bottom layers, and lossy electrodes. Electric losses, in the electrodes, and viscous losses, in the piezoelectric layer, produce dissipation that is coupled to the thermal domain to reproduce temperature rise. The temperature rise is used to change the material properties On the other hand, the parasitic electrodes losses are implemented by use of a lumped resistor at the input and output of the modeled device as shown in Fig. 5. As done for the viscosity, the dissipation in each resistor is coupled to the thermal model as a heat source. In

Rin Rout

*Ld/2 Pd*

*Rth/2*

*Cth*

*+ F+dF -*

Bottom Layers

Bottom Layers (Acoustic Domain)

*+ K+dK -*

*Ld/2*

Acoustic Domain

Thermal Domain

*Rth/2*

*+ F -*

Top Layers (Thermal Domain)

**Rth,1/2**

**Rrad**

**Rconv**

Top Layers (Acoustic Domain)

distributed inductance *Ld* = *ρ·A·Δz.*

Nonlinear KLM model

*+ K -*

where the equivalent distributed parameters can be identified as the volumetric heat capacitance

$$\mathbf{C}\_{d,th} = \rho \mathbf{C}\_p \tag{16}$$

and the thermal resistance

$$R\_{d,th} = \frac{1}{k\_{th}}\tag{17}$$

with *Cp* and *kth* being the material-specific heat capacity and thermal conductivity, respectively.

With the above-mentioned distributed parameters, a thermal distributed model can be constructed as a cascade of sections of series resistances and shunt capacitance, where each section corresponds to a specific thickness and area. Figure 2 shows a segment with *Rth= Rd,th·Δz/A* and *Cth= Cd,th·A·*Δ*z*, where *A* is the area of the cross-section perpendicular to the *z* direction. In such a thermal equivalent circuit the equivalents of voltage and current are the temperature and heat respectively.

Fig. 2. Implementation of a *Δz* section of thermal equivalent circuit

The thermal model of a multilayer SMR can be implemented as a cascade of the previously described sections for each material, as shown in Fig. 3. The boundary conditions are the ambient temperature, modeled as a voltage source under the substrate, and the parallel combination of the radiation and convection resistances, terminated with a voltage source at ambient temperature on the upper side of the device (Larson et al., 2002).

As it can be seen from Fig. 3, the thermal impedance seen from any point along the line has a low-pass filter behavior, which means that for faster variations of the heat source, smaller temperature variations are produced.

The piezoelectric layer is implemented as a cascade of cells, in which the dissipated power due to viscous damping is directly coupled to its correspondent thermal cell. A current source is used because current is the analogue of heat in the thermal domain. The

where the equivalent distributed parameters can be identified as the volumetric heat

*C C d th*, = ρ

1

*th*

,

*R*

*d th*

with *Cp* and *kth* being the material-specific heat capacity and thermal conductivity,

With the above-mentioned distributed parameters, a thermal distributed model can be constructed as a cascade of sections of series resistances and shunt capacitance, where each section corresponds to a specific thickness and area. Figure 2 shows a segment with *Rth=*

direction. In such a thermal equivalent circuit the equivalents of voltage and current are the

The thermal model of a multilayer SMR can be implemented as a cascade of the previously described sections for each material, as shown in Fig. 3. The boundary conditions are the ambient temperature, modeled as a voltage source under the substrate, and the parallel combination of the radiation and convection resistances, terminated with a voltage source at

Cth,2

Fig. 3. Thermal model of the upper and lower materials' stacks with boundary conditions As it can be seen from Fig. 3, the thermal impedance seen from any point along the line has a low-pass filter behavior, which means that for faster variations of the heat source, smaller

The piezoelectric layer is implemented as a cascade of cells, in which the dissipated power due to viscous damping is directly coupled to its correspondent thermal cell. A current source is used because current is the analogue of heat in the thermal domain. The

Rth/2

...

Rth,1/2 Rth,2/2 Rth,Si/2 Rth,Si/2

Rth,2/2

Cth,Si Tamb

Rth/2

Cth

Z direction

Fig. 2. Implementation of a *Δz* section of thermal equivalent circuit

Rth,1/2

Cth,1

ambient temperature on the upper side of the device (Larson et al., 2002).

*p* (16)

*k* = (17)

*z*, where *A* is the area of the cross-section perpendicular to the *z*

capacitance

respectively.

and the thermal resistance

*Rd,th·Δz/A* and *Cth= Cd,th·A·*

temperature and heat respectively.

Z direction

temperature variations are produced.

Rrad

Rconv Tamb

Δ

temperature rise is used to modify the distributed acoustic capacitance *CNL(T,K)*, as shown in Fig. 4.

Fig. 4. Implementation of a section of the piezoelectric layer with the acoustic and thermal domains coupled by the generated heat at *Gd* and the temperature *K*. *Ld* is the acoustic distributed inductance *Ld* = *ρ·A·Δz.*

Fig. 5. Complete circuit model with thermo-acoustic model of the piezoelectric layer, top and bottom layers, and lossy electrodes. Electric losses, in the electrodes, and viscous losses, in the piezoelectric layer, produce dissipation that is coupled to the thermal domain to reproduce temperature rise. The temperature rise is used to change the material properties

On the other hand, the parasitic electrodes losses are implemented by use of a lumped resistor at the input and output of the modeled device as shown in Fig. 5. As done for the viscosity, the dissipation in each resistor is coupled to the thermal model as a heat source. In

Sources of Third–Order Intermodulation Distortion

for simplicity.

Ma

**3. Experimental results**

g

(traces) for a wide range of separation between tones

nitude of I3MD(d

B

m)

in Bulk Acoustic Wave Devices: A Phenomenological Approach 491

sweep the central frequency *ω0* in a 600 MHz range around the resonance frequency, which is 2.18 GHz. By doing this, we can distinguish the 3IMD produced by viscous self-heating from that produced by electric self-heating by analyzing the resulting frequency dependence. In the former case, we do not connect the dissipation in the electrodes to the thermal domain (Fig.6a), whereas in the latter case we do not connect the dissipation in the piezoelectric layer to the thermal domain (Fig.6b). The 3IMD frequency dependences are a direct consequence of the frequency dependences of the dissipated power. More specifically, a minimum at the anti-resonance frequency appears in Fig. 6.b because there is minimum current flowing through the electrodes at anti-resonance, which can be used in experimental

In the second set of simulations we keep the central frequency constant at 2.18 GHz and we change the separation between tones from 100 Hz to 1 MHz. This allows us to reproduce the low-pass filter behavior of the thermal impedance. Figure 7 shows the results of the second set of simulations for a wide range of separation between tones when the self-heating effects are due to viscous losses, where it is clear the low-pass filter behavior of the temperature induced effects. A very similar plot was obtained for electrode losses, which is not shown

> Separation between tones (Hz) 102 103 104 105 106

Fig. 7. Magnitude and phase of equation (13) (circles) and simulations with the circuit model

Figures 6 and 7, in addition to giving useful qualitative information about the 3IMD generation due to the self-heating mechanism, show that the formulation of equations (13) and (14) is in very good agreement with the simulations, so that these expressions can be used for a better understanding of the temperature-induced 3IMD in BAW resonators.

Four state-of-the-art rectangular Solidly-Mounted Resonators (SMR) from a commercial manufacturer, with different areas summarized in Table 1, have been measured. The resonators have a 1.25 μm thick aluminum nitride layer and a W - SiO2 Bragg mirror

(alternating layers of W and SiO2), and show quality factors around 1800.

Δω*/2*π

> Phase of 3IMD

(degrees)

= 220 Hz and

In the first set of simulations we keep the tones spacing constant at

measurements to identify different sources of self-heating effects.

fact, dissipation in the input and output resistors is coupled to the correspondent top and bottom thermal sections that model the electrodes. The complete model can be seen in Fig. 5, where a cell of the piezoelectric layer like that in Fig. 4, is highlighted in red.

In the figure above, the electric-acoustic conversion box includes those elements of the KLM model whose purpose is the electro-acoustic signal conversion (Krimholtz et al., 1970). Additionally, the material layers above and below the piezoelectric are shown as simplified blocks for clarity.

### **2.2.4 Comparison of formulation and nonlinear simulations**

We use the circuit model of Fig. 5, with only a piezoelectric layer, to check the accuracy of the formulation described in the previous section. The circuit model has been simulated, reproducing a two-tone experiment, with Harmonic Balance techniques by use of a commercial CAD software. A simple model is implemented making use of 100 cells to reproduce a 1.25 μm thick and 2.33·10-8 m2 piezoelectric layer with a quality factor of 1800. The electrodes losses and viscous losses are coupled to a low-pass thermal impedance.

Fig. 6. Comparison of the magnitude and phase of *2ω1-ω2* calculated with equation (13) (circles) in Fig.6a (viscous losses, no electrode losses) and equation (14) (circles) in Fig.6b (electrode losses, no viscous losses), vs. simulation with the circuit model (solid lines)

fact, dissipation in the input and output resistors is coupled to the correspondent top and bottom thermal sections that model the electrodes. The complete model can be seen in Fig. 5,

In the figure above, the electric-acoustic conversion box includes those elements of the KLM model whose purpose is the electro-acoustic signal conversion (Krimholtz et al., 1970). Additionally, the material layers above and below the piezoelectric are shown as simplified

We use the circuit model of Fig. 5, with only a piezoelectric layer, to check the accuracy of the formulation described in the previous section. The circuit model has been simulated, reproducing a two-tone experiment, with Harmonic Balance techniques by use of a commercial CAD software. A simple model is implemented making use of 100 cells to reproduce a 1.25 μm thick and 2.33·10-8 m2 piezoelectric layer with a quality factor of 1800. The electrodes losses and viscous losses are coupled to a low-pass thermal impedance.

> Phase of 3IMD

0

(a)

(degrees)

0

(b)

Phase of 3IMD

(degrees)

1.8 1.9 2 2.1 2.2 2.3 2.4

1.8 1.9 2 2.1 2.2 2.3 2.4

Frequency (GHz)

Fig. 6. Comparison of the magnitude and phase of *2ω1-ω2* calculated with equation (13) (circles) in Fig.6a (viscous losses, no electrode losses) and equation (14) (circles) in Fig.6b (electrode losses, no viscous losses), vs. simulation with the circuit model (solid lines)

Frequency (GHz)

where a cell of the piezoelectric layer like that in Fig. 4, is highlighted in red.

**2.2.4 Comparison of formulation and nonlinear simulations**

blocks for clarity.

M

M

a

nitu

de of 3IMD

 (

d

B

m)

g

a

nitu

de of 3IMD(

d

B

m)-80

g

In the first set of simulations we keep the tones spacing constant at Δω*/2*π = 220 Hz and sweep the central frequency *ω0* in a 600 MHz range around the resonance frequency, which is 2.18 GHz. By doing this, we can distinguish the 3IMD produced by viscous self-heating from that produced by electric self-heating by analyzing the resulting frequency dependence. In the former case, we do not connect the dissipation in the electrodes to the thermal domain (Fig.6a), whereas in the latter case we do not connect the dissipation in the piezoelectric layer to the thermal domain (Fig.6b). The 3IMD frequency dependences are a direct consequence of the frequency dependences of the dissipated power. More specifically, a minimum at the anti-resonance frequency appears in Fig. 6.b because there is minimum current flowing through the electrodes at anti-resonance, which can be used in experimental measurements to identify different sources of self-heating effects.

In the second set of simulations we keep the central frequency constant at 2.18 GHz and we change the separation between tones from 100 Hz to 1 MHz. This allows us to reproduce the low-pass filter behavior of the thermal impedance. Figure 7 shows the results of the second set of simulations for a wide range of separation between tones when the self-heating effects are due to viscous losses, where it is clear the low-pass filter behavior of the temperature induced effects. A very similar plot was obtained for electrode losses, which is not shown for simplicity.

Fig. 7. Magnitude and phase of equation (13) (circles) and simulations with the circuit model (traces) for a wide range of separation between tones

Figures 6 and 7, in addition to giving useful qualitative information about the 3IMD generation due to the self-heating mechanism, show that the formulation of equations (13) and (14) is in very good agreement with the simulations, so that these expressions can be used for a better understanding of the temperature-induced 3IMD in BAW resonators.

### **3. Experimental results**

Four state-of-the-art rectangular Solidly-Mounted Resonators (SMR) from a commercial manufacturer, with different areas summarized in Table 1, have been measured. The resonators have a 1.25 μm thick aluminum nitride layer and a W - SiO2 Bragg mirror (alternating layers of W and SiO2), and show quality factors around 1800.

Sources of Third–Order Intermodulation Distortion

3IMD (d

frequency-dependent 3IMD level

B

m)

simulations (dashed lines) of *ω1* and *ω2* are also presented

Δc D 1

Δc D 2

Output Pow

er (d

B

m)

0

20

all the devices.

in Bulk Acoustic Wave Devices: A Phenomenological Approach 493

et al., 2010). In fact, as done in (Collado et al., 2010), the second harmonic is used to extract a unique unitless value *ΔcD1* = 10.5 of the stiffened elasticity that fits the second harmonic for

ω ω1 2 ,

1 ∆c

D 2

∆c D

the *ΔcD1* and *ΔcD2* contributions to the 3IMD (*2ω1-ω2* and *2ω2-ω1* overlap). Both

sufficient to explain the measurements. Measurements (squares and triangles) and

1.9 1.94 1.98 2.02

Frequency (GHz)

102 103 104 105 106 107

Tones Spacing (Hz)

Fig. 9. 3IMD measurements (squares are *2ω1-ω2*, circles are *2ω2-ω1*) and circuit simulations of

the *ΔcD1* and *ΔcD2* contributions to the 3IMD (*2ω1-ω2* and *2ω2-ω1* overlap), for several separations between tones. The intrinsic contributions cannot reproduce the envelope

Fig. 8. 3IMD measurements (squares are *2ω1-ω2*, circles are *2ω2-ω1*) and circuit simulations of

measurements and simulations are done for *Δf* = 220 Hz. The intrinsic nonlinearities are not

From (8) it is clear that several sources, characterized by Δ*cD1*, Δ*cD2* and Δ*cDK*, can generate 3IMD. Therefore, we follow a step-by-step procedure that includes several experiments to determine which nonlinear source is designed responsible for each observable at each experiment by use of the circuit model:

## **3.1 Linear modeling of the devices under test**

Broadband one-port S-parameters measurements of the devices have been performed after an on-wafer OSL calibration. The measurements have been done at a power level of -10 dBm to ensure the linear regime and are used to fit the linear parameters of the circuit model. The only differences between the devices are the resonator area and electrode losses.

Electric losses due to the resistivity of the electrodes are modeled as lumped parasitic resistances, so their values are dependent on the resonator area and have a broadband effect on the linear device response. Table 1 summarizes the device areas and electric resistances.

On the other hand, acoustic losses due to viscosity only have an observable effect at those frequencies where there is substantial electro-acoustic coupling, that is around resonance and anti-resonance. By using the model transformation in Appendix II, the acoustic losses that fit all devices can be described with the same material viscosity value *η* = 0.033 N·s·m-2, what verifies the validity of the linear model.

Table 1. Tested devices

### **3.2 Nonlinear characterization**

The nonlinear behavior of a BAW resonator arises from different contributions due to intrinsic nonlinear material properties and self-heating mechanisms.

### **3.2.1 Intrinsic nonlinearities**

Intrinsic nonlinearities due to the stiffened elasticity, as shown in (1), predominate above other intrinsic nonlinear material contributions. As a consequence, the second harmonic has the same frequency dependence as the mechanical stress in the piezoelectric layer (Collado

3IMD. Therefore, we follow a step-by-step procedure that includes several experiments to determine which nonlinear source is designed responsible for each observable at each

• We first adjust the linear model to the measured S-parameters of the devices, so that the electric and viscous losses can be quantified. The procedure consists of a fine tuning of

• Second harmonic measurements are performed along the frequency range of interest to

Δ*cD1*.

• Third-order intermodulation distortion measurements with sweeping the tone spacing are conducted to quantify the frequency dependence of the thermal impedance, and set

Broadband one-port S-parameters measurements of the devices have been performed after an on-wafer OSL calibration. The measurements have been done at a power level of -10 dBm to ensure the linear regime and are used to fit the linear parameters of the circuit model. The

Electric losses due to the resistivity of the electrodes are modeled as lumped parasitic resistances, so their values are dependent on the resonator area and have a broadband effect on the linear device response. Table 1 summarizes the device areas and electric resistances. On the other hand, acoustic losses due to viscosity only have an observable effect at those frequencies where there is substantial electro-acoustic coupling, that is around resonance and anti-resonance. By using the model transformation in Appendix II, the acoustic losses that fit all devices can be described with the same material viscosity value *η* = 0.033 N·s·m-2,

Resonator Area (m2) Electric Resistance Rs (Ω)

The nonlinear behavior of a BAW resonator arises from different contributions due to

Intrinsic nonlinearities due to the stiffened elasticity, as shown in (1), predominate above other intrinsic nonlinear material contributions. As a consequence, the second harmonic has the same frequency dependence as the mechanical stress in the piezoelectric layer (Collado

intrinsic nonlinear material properties and self-heating mechanisms.

A1 6.41e-8 0.28 A2 4.88e-8 0.37 A3 2.33e-8 0.42 A4 1.25e-8 1

only differences between the devices are the resonator area and electrode losses.

(Łepkowski et al., 2005) because this contribution cannot be independently extracted

*cD2* also contributes to the 3IMD generation. We use the literature value in

Δ

*cKD* accordingly.

Δ*cD1*, Δ*cD2* and

Δ

*cDK*, can generate

From (8) it is clear that several sources, characterized by

experiment by use of the circuit model:

extract the intrinsic nonlinear parameter

**3.1 Linear modeling of the devices under test**

what verifies the validity of the linear model.

Table 1. Tested devices

**3.2 Nonlinear characterization**

**3.2.1 Intrinsic nonlinearities**

the temperature coefficient of stiffened elasticity

the material properties.

Δ

from measurements.

• The term

et al., 2010). In fact, as done in (Collado et al., 2010), the second harmonic is used to extract a unique unitless value *ΔcD1* = 10.5 of the stiffened elasticity that fits the second harmonic for all the devices.

Fig. 8. 3IMD measurements (squares are *2ω1-ω2*, circles are *2ω2-ω1*) and circuit simulations of the *ΔcD1* and *ΔcD2* contributions to the 3IMD (*2ω1-ω2* and *2ω2-ω1* overlap). Both measurements and simulations are done for *Δf* = 220 Hz. The intrinsic nonlinearities are not sufficient to explain the measurements. Measurements (squares and triangles) and simulations (dashed lines) of *ω1* and *ω2* are also presented

Fig. 9. 3IMD measurements (squares are *2ω1-ω2*, circles are *2ω2-ω1*) and circuit simulations of the *ΔcD1* and *ΔcD2* contributions to the 3IMD (*2ω1-ω2* and *2ω2-ω1* overlap), for several separations between tones. The intrinsic contributions cannot reproduce the envelope frequency-dependent 3IMD level

Sources of Third–Order Intermodulation Distortion

is currently under investigation.

Output

nonlinear contributions

P

o

wer (

d

B

m

)

20

0

2 - ω ω1 2 2 - ω ω2 1

3IMD (d

B

m)

in Bulk Acoustic Wave Devices: A Phenomenological Approach 495

all contributions, which match the measurements for all the tone separations. The model reproduces very well the measured 3IMD values except the asymmetry between lower and higher 3IMD that appears for envelope frequencies around 6 MHz. This asymmetry is considered to be a consequence of a cancellation between different 3IMD contributions and

> Tones Spacing (Hz) 102 103 104 105 106 107

Fig. 11. Measurements (stars, triangles, squares and circles are *ω1*, *ω2*, *2ω1-ω2* and *2ω2-ω1* respectively) and simulations (thin dotted line is *ω1* and *ω2*, solid line is *2ω1-ω2* and *2ω2-ω1* overlapped) of the fundamentals and the 3IMD for different tones spacing for A1. The dotted line, the dash-dot line and the dashed line are simulations of the 3IMD contribution due to *ΔcDK*, *ΔcD2* and *ΔcD1* respectively. The solid line is the 3IMD simulation with all the

1.88 1.92 1.96 2 2.04

Viscosity

Electrodes

Frequency (GHz)

ω ω1 2 ,

All Δc D 1 Δc D 2 Δc D K

10 -3

10 -2

10 -1

(a)

Dissipation

 (W)

The parameter *ΔcD1* is responsible for second harmonic generation, which in turn mixes with the fundamental frequencies *ω1* and *ω2*, and gives rise to 3IMD. On the other hand, *ΔcD2* directly generates a certain level of 3IMD distortion. We use the literature value of *ΔcD2 =* - 1·10-10 N-1m2 (Łepkowski et al., 2005).

With the above-mentioned values of intrinsic nonlinearity, simulations of 3IMD are performed obtaining values which are below the measured levels, as shown in Fig. 8. This shows that other contributions exist. Figure 9 shows 3IMD measurements at different envelope frequencies, centered at the frequency where the 3IMD is maxima. The measurements reveal a strong dependence with the envelope frequency that cannot be accounted for intrinsic nonlinearities. The dependence of the 3IMD level on the envelope frequency suggests that the 3IMD is dominated by a thermal effect.

### **3.2.2 Self-heating**

The thermal model, as presented in section II.B.3, is implemented by using the literature values of thermal conductivity and specific heat for each layer. The materials stack is composed of more than ten layers. Dissipation on the electrodes and viscous losses are coupled to the thermal domain by means of current sources. By using the model we can determine the temperature distribution along the materials stack, even at different envelope frequencies as shown in Fig. 10. Note that the temperature is almost the same in the piezoelectric layer and the electrodes, which validates the hypothesis in Section II.B.2, of negligible thermal impedance between the electrodes and the piezoelectric layer, to obtain the closed-form expressions.

Fig. 10. Simulations of the distribution of the *z* temperature variation, inside the resonator, calculated with the circuit model. Simulations are done for resonator A1 at *Pin* = 30 dBm, for different separation between tones

A value of *ΔcDK/cD0*= -15 ppm/K is found to fit the 3IMD best for all resonators. Figure 11 shows the intrinsic and self-heating 3IMD contributions independently as well as the sum of

The parameter *ΔcD1* is responsible for second harmonic generation, which in turn mixes with the fundamental frequencies *ω1* and *ω2*, and gives rise to 3IMD. On the other hand, *ΔcD2* directly generates a certain level of 3IMD distortion. We use the literature value of *ΔcD2 =* -

With the above-mentioned values of intrinsic nonlinearity, simulations of 3IMD are performed obtaining values which are below the measured levels, as shown in Fig. 8. This shows that other contributions exist. Figure 9 shows 3IMD measurements at different envelope frequencies, centered at the frequency where the 3IMD is maxima. The measurements reveal a strong dependence with the envelope frequency that cannot be accounted for intrinsic nonlinearities. The dependence of the 3IMD level on the envelope

The thermal model, as presented in section II.B.3, is implemented by using the literature values of thermal conductivity and specific heat for each layer. The materials stack is composed of more than ten layers. Dissipation on the electrodes and viscous losses are coupled to the thermal domain by means of current sources. By using the model we can determine the temperature distribution along the materials stack, even at different envelope frequencies as shown in Fig. 10. Note that the temperature is almost the same in the piezoelectric layer and the electrodes, which validates the hypothesis in Section II.B.2, of negligible thermal impedance between the electrodes and the piezoelectric layer, to obtain

z

Fig. 10. Simulations of the distribution of the *z* temperature variation, inside the resonator, calculated with the circuit model. Simulations are done for resonator A1 at *Pin* = 30 dBm, for

A value of *ΔcDK/cD0*= -15 ppm/K is found to fit the 3IMD best for all resonators. Figure 11 shows the intrinsic and self-heating 3IMD contributions independently as well as the sum of

0 5 10 15 20 25 30

Temperature Rise (K)

60 Hz

10 KHz

DC

frequency suggests that the 3IMD is dominated by a thermal effect.

AlN

Top electrode

SiO2

Bottom electrode

SiO2 W

> W ...

Silicon

different separation between tones

1·10-10 N-1m2 (Łepkowski et al., 2005).

**3.2.2 Self-heating**

the closed-form expressions.

all contributions, which match the measurements for all the tone separations. The model reproduces very well the measured 3IMD values except the asymmetry between lower and higher 3IMD that appears for envelope frequencies around 6 MHz. This asymmetry is considered to be a consequence of a cancellation between different 3IMD contributions and is currently under investigation.

Fig. 11. Measurements (stars, triangles, squares and circles are *ω1*, *ω2*, *2ω1-ω2* and *2ω2-ω1* respectively) and simulations (thin dotted line is *ω1* and *ω2*, solid line is *2ω1-ω2* and *2ω2-ω1* overlapped) of the fundamentals and the 3IMD for different tones spacing for A1. The dotted line, the dash-dot line and the dashed line are simulations of the 3IMD contribution due to *ΔcDK*, *ΔcD2* and *ΔcD1* respectively. The solid line is the 3IMD simulation with all the nonlinear contributions

Sources of Third–Order Intermodulation Distortion

**4. Conclusion**

*2*ω*1* ω*2* (and *2*

the nonlinear system baseline level, which is around -60 dBm.

dissipation through the substrate, will also be investigated.

ω*1* and ω

*nl*

12

ω

**5. Appendix I – 3IMD equations**

*1*), when

ω

**6. Appendix II - Model transformation**

ω*2* ω

where KΔω =Zth (Δω) Pd (Δω) .

elasticity in the acoustic transmission line, so further research is needed.

in Bulk Acoustic Wave Devices: A Phenomenological Approach 497

Figure 12 shows measurements and simulations of the 3IMD about the frequency range of interest, for each resonator, by use of the circuit model. In this experiment, the two tones are swept around the resonating frequencies keeping the separation between tones constant at 100 Hz. The results show good agreement between simulations and measurements above

The dashed and dotted lines in Fig. 12 show the simulated dissipated power due to electrodes losses and viscosity respectively, which have different frequency dependences according to the maximum electric current and mechanical stress, respectively. 3IMD measurements for resonators A3 and A4 in Fig. 12 show a peak at the antiresonant frequency that is underestimated by the simulations. The frequency dependence in that range points to a possible electric-field contribution to the 3IMD. This contribution is below the system nonlinear baseline level for resonators A1 and A2, and the area scaling has not been successfully reproduced by use of an electric-field dependent permittivity or stiffened

The role of self-heating and material nonlinearities in the generation of 3IMD in bulk acoustic wave devices has been evaluated through measurements, models and equations. Self-heating is found to have a very significant contribution to 3IMD and thus thermal considerations are critical in the device design. The presented circuit model implementation offers the possibility to predict 3IMD in BAW resonators, given their materials stack and geometry. With such information one can use the resonator model to accurately predict 3IMD in filters. Further research will be performed to investigate the relation between the electric-field contribution to 3IMD and the cancellation shown in the measurements. The development of a 3D equivalent thermal model, to take into account complex heat

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* are at resonance:

*K I z z j CK V z l*

\*

*d eq L*

1 1 2 1

Δ = −

, \* 12

ωΔ

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

2

12 1

,0

*K L d th*

*C V jQ P Z V C C Q*

 ωΔ Δω ω

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

( ) 1

1

ω ω cos

π

0,

 −

*v*

(19)

2

*j W*

∂ = Δ ∂ (18)

Fig. 12. Measurements (filled circles are *2ω1-ω2*, empty circles are *2ω2-ω1*) and simulations (solid line is *2ω1-ω2*, dash-dot line is *2ω2-ω1*) of the 3IMD for resonators A1, A2, A3 and A4 in Fig.12.a, Fig.12.b, Fig.12.c and Fig.12.d respectively. The figures also show the dissipation in the electrodes (dashed line) and in the piezoelectric layer due to viscous losses (dotted line)

Figure 12 shows measurements and simulations of the 3IMD about the frequency range of interest, for each resonator, by use of the circuit model. In this experiment, the two tones are swept around the resonating frequencies keeping the separation between tones constant at 100 Hz. The results show good agreement between simulations and measurements above the nonlinear system baseline level, which is around -60 dBm.

The dashed and dotted lines in Fig. 12 show the simulated dissipated power due to electrodes losses and viscosity respectively, which have different frequency dependences according to the maximum electric current and mechanical stress, respectively. 3IMD measurements for resonators A3 and A4 in Fig. 12 show a peak at the antiresonant frequency that is underestimated by the simulations. The frequency dependence in that range points to a possible electric-field contribution to the 3IMD. This contribution is below the system nonlinear baseline level for resonators A1 and A2, and the area scaling has not been successfully reproduced by use of an electric-field dependent permittivity or stiffened elasticity in the acoustic transmission line, so further research is needed.


**7. Conclusions**

The main research problem that was considered in the paper was an insufficient means of assessing the degree of the implementation of lean maintenance. To find the solution of identified problem, artificial intelligence methods such as decision trees and the rough set theory were used. When analyzing the results presented, it should be noted that the models generated with the rough set theory achieved much better results than in decision trees. The decision rules generated by DT showed better values for all indicators for the classifier for the class of 30–50%. However, better values for the class of 70–85% were achieved for RST, mainly for LEM2 algorithm. The number of rules generated by the LEM2 algorithm is the smallest compared to the other algorithms. This shows that a large number of rules is not needed to get good prediction results in the investigated problem.

The resulting indicators for testing the quality of classifiers confirmed the high usefulness of the generated decision rules, both those using decision trees and the rough set theory. The developed dependencies allow us to assess which results a given company can expect after the implementation of specific lean maintenance methods and tools, and which lean maintenance methods and tools should be used to achieve the intended goals. These dependencies may be the basis for determining the directions and effects of implementing lean maintenance in manufacturing companies. Additionally, an expert system in the form of a software application, developed on the basis of the generated dependencies (decision rules), allows for the selection of appropriate actions in order to obtain the best results after implementing lean maintenance.

The presented studies can be used by enterprises to build and organize maintenance processes, to select an appropriate action strategy, but above all, to improve already implemented activities in this area. Although the research was conducted in a limited area, it was based on common assumptions, principles, and objectives of implementing the lean maintenance concept in the enterprise. Therefore, the presented solutions are useful for practical use by all production companies for forecasting and assessing the effectiveness of implementing lean maintenance methods and tools, regardless of the region.

Moreover, the positive results obtained during the conduct of the described study lead to the conclusion that the activities in these areas should be continued. In particular, there ought to be studies considering the assessment of the effectiveness of using other methods and tools recommended in the literature within lean maintenance implementation; the possibility of extending functionality designed in a computer application; and the use of other methods of data exploration for generating decision rules and comparing their classification quality.

**Author Contributions:** K.A. gave the theoretical and substantive background for the provided research and conceived and designed the experiments; L.P. made an experimental verification of the proposed approach; A.G. provided technical guidance and gave critical review for this paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.


*3.1. Traffic and Trajectory Analysis*

In the first step of the evaluation, an analysis of the traffic data was carried out in order to determine appropriate research horizon and measure the lengths of the travelled trajectories. Figures 7–9 present in detail the results of the validation trials and effects described above.

**Figure 7.** Flight trajectory results obtained under the simulation conditions (green) and compared with real traffic reference values for flows of 40 aircraft per hour (blue) and 20 aircraft per hour (orange).

In Figure 7, the vertical axis presents the average distance flown within the radius of 100 NM from the EDDM reference point for arriving aircraft. The horizontal axis presents the results of validation trials executed by the five ATCOs (C1–C5) testing the two traffic scenarios, differing with distribution of 3D-FMS and 4D-FMS flights, where 30 and 60 correspond, respectively, to the S30 scenario and the S60 scenario. The results obtained during the simulation have been marked as a green line. They can be directly compared with real traffic data for a flow of 40 aircraft per hour (marked in blue—R1) and for a flow of 20 aircraft per hour (marked in orange—R2). These reference flown distances were calculated as averages based on 10 h of arrival traffic at EDDM extracted from the OpenSky database for both.

Treating that as a reference, it can be observed that even with a smaller number of total arrivals in R1 and R2, the introduction of innovative airspace structure, new FMS procedures and ATCOs supporting systems resulted in a reduction in flight distance for all ATCOs and all scenarios. In each case, the simulation results were lower than all reference values.

Figure 8 presents the distances flown by aircraft as cumulative occurrence curve divided into 5 NM lengths, where again the green line refers to the simulation results, and the blue and orange lines present the data of the reference scenarios. Within the simulation scenarios, the numbers of flights covering shorter distances were slightly higher than those in the reference scenarios. This was particularly evidenced by the first two peaks observed in Figure 8, which are substantially higher, representing well over half of scheduled flights (sum of 68.3%) that arrived at the airport in the range of 100–115 NM, in contrast to 36.9% (orange) in one reference. In addition to that, the real traffic data show that a significant amount of flights (corresponding to 25% of occurrences) needed a distance of 125 NM to reach the airport.

**Figure 8.** Cumulative occurrence curve for flight-distance results obtained in validation trials (green) and real traffic reference values for flows of 40 aircraft per hour (blue) and 20 aircraft per hour (orange).

The last set of results is related to the number of approach operations performed. This situation is reflected in the results presented in Figure 9. The figure displays a comparison between number of approaches executed in two simulation scenarios, where different distributions of 3D-FMS and 4D-FMS operations are analysed. The blue bars represent the numbers for the S30 scenario, and the orange bars correspond to the S60 scenario. The bars display that for each ATCO, a greater number of landed aircraft was recorded during the S60 scenario in comparison to the S30 scenario.

### *3.2. Mental Workload*

Alongside the capacity analysis, the mental workload analysis was conducted. Therefore, Figure 10 shows the mean ISA ratings depending on the share of 4D-FMS equipped aircraft (30% vs. 60% vs. 80%). The results were averaged for all participants and assessment times.

**Figure 10.** Mean ISA ratings in the different scenarios (S30 vs. S60 vs. S80) summarised over all participants and assessment times. Error bards represent standard deviations.

Mean ISA ratings were the highest in the S30 scenario, followed by the S60 scenario and then the S80 scenario. For the S30 and S60 scenarios, respectively, ISA ratings fell between 2 (relaxed) and 3 (comfortable), indicating a slightly lower than mid-level mental workload. For the S80 scenario, mean ISA ratings were below 2 (relaxed), pointing towards mental underloading. It is worth noting that the S80 scenario was slightly shorter than the others runs, hence the lower sample size.

Moreover, the results from NASA-TLX were assessed. Figure 11 shows the mean raw NASA-TLX scores for scenarios S30 and S60. The mean global score and all mean sub-scores were higher in the S30 scenario than in the S60 scenario, indicating higher overall workload in the S30 scenario than in the S60 scenario on a descriptive level. This is in line with the ISA ratings. Standard deviations were especially high for the sub-scales frustration and performance.

**Figure 11.** Mean raw TLX scores for the scenarios (S30 vs. S60 vs. S80). Error bars represent standard deviations.


**2.1 Introduction**

The three major classes of financial risks and their corresponding definitions are defined as follows:

Note that [8] argued that the probability of an operational risk event increases with many personnel and with a greater transaction volume. The latter is also based on the study by [9] who investigated the effect of bank size on operational loss amounts and deducted that, on average, for every unit increase in bank size, operational losses are predicted to increase by approximately a fourth of a root of that. Note that there are different classes of operational losses that the financial industry must be aware of; see [2]:

Category (i) has been argued that it is not feasible/implausible in the financial industry, with (ii) and (iii) are unimportant and can often be both prevented. However, category (iv) tends to cause the most devastation loses, with the best example being the 1995 Barings Bank's collapse (also portrayed in the movie "Rogue Trader"). Consequently, banks must be extremely cautious of these types of losses as they tend to cause bankruptcy in many financial institutions. Low-frequency/high-severity operational losses can be extreme in size when they are compared to the rest of the data. If you construct a histogram of the loss distribution, the low-frequency/high-severity operational losses events would be placed in the far-right end, which often referred to as "tail event". Due to operational loss, data exhibit such tail events. We say that the data are heavy-tailed.

In different fields that use "Data Science" techniques (e.g. insurance, banks, etc.), different types of distributions are used to model data due to the different products that are offered by several financial institutions. These financial institutions are

increasingly measuring and managing their exposure to different types of risks; see [4]. A proper evaluation of risk in any financial institution is an uncertainty problem that may easily lead to a bankruptcy of that firm and consequently is a major concern for national and international financial regulatory bodies.

It is important to mention that risk data from different products offered by financial institutions (e.g. micro-insurance, re-insurance, investment, savings, stock exchange, etc.) are distributed differently; see for instance [10]. Consequently, a thorough understanding of a variety of distributions is a must for an inspiring data scientist. According to [2], the most applied basic distributions in quantifying operational risk are those that are skewed to the right (right-tailed). There are two main ways to categorize the right-tailed loss distributions, i.e. parametric and nonparametric approaches. More specifically, in this chapter, we will consider the following most common parametric loss distributions: (i) exponential, (ii) gamma, (iii) Weibull, (iv) Pareto, (v) Burr, and (vi) log-normal. It is worth mentioning that these are not the only existing loss distributions, for example, a combination of two of the above, i.e. the composite Weibull-Pareto distribution in the context of risk is discussed in [11].

The next subsections discuss the following: Section 2.2 provides some distributional properties of the considered parametric loss distributions. More importantly, Section 2.2 provides the literature review of some publications that applied the considered distributions in the context of risk analysis. Next, Section 2.3 gives a brief discussion on nonparametric loss distributions (seldomly used), and Section 2.4 discusses some well-known methods of quantifying risk. Section 2.5 discusses other types of risks that use the LDA. Finally, Section 2.6 gives some concluding remarks.

### **2.2 Parametric loss distributions**

A summary of some publications that discussed parametric loss distribution's application in operational risk is provided in **Table 1**. This table was constructed with an effort to easily identify which type of loss distributions is discussed in these separate publications. The corresponding loss distribution function properties are listed in **Table 2** with the expressions adopted from [12, 13] and Chapter 6 of [2].

Note that when the different parameters in **Table 2** are varied, the distributions tend to vary significantly, especially in the tail area. The latter will be illustrated in detail in the next section.

## *2.2.1 Pareto distribution*

The Pareto distribution is a very heavy-tailed distribution that takes on positive values, and its parameter *α* is used to determine the size of the tail heaviness. The Pareto distribution tail is monotonically decreasing, and this means that the tail decreases as *x* increases and it becomes is thicker for values of *x* closer to zero. To derive the Pareto distribution, assume that a variate *x* follows an exponential distribution with mean *β*1 ; furthermore, suppose that *β* follows a gamma distribution, therefore the *x* follows a Pareto distribution; see [17]. Note that when *α*< 1, a very heavy tail is encountered with the mean and variance being infinite. This means that losses of infinite sizes are theoretically possible. The extreme heaviness of the Pareto distribution tail makes it ideal for modeling losses of high magnitudes; see [2]. There are also different versions of the Pareto distribution that are used in risk analysis. The most popular of those variations is being the generalized Pareto distribution (GPD).

*Quantifying Risk Using Loss Distributions DOI: http://dx.doi.org/10.5772/intechopen.108856*

#### **Table 1.**

*A summary of publications discussed in this chapter and their classification according to the type of loss distribution.*

The GPD is especially good while modeling data greater than a high threshold, also known as estimation of tails of extreme losses.

## *2.2.2 Burr distribution*

The Burr distribution is heavy-tailed, and it is skewed to the right; see [19]. The Burr distribution is a special case of GPD described in subsection 2.2.1. It has three parameters which gives it more flexibility over the traditional Pareto distribution. The Burr distribution has an additional parameter *γ* and when *γ* ¼ 1, it reduces to a Pareto distribution. One of the well-known uses of the Burr distribution is modeling natural catastrophes and as a result, it is a popular distribution or model for use in the insurance industry for pricing of premiums; see [2, 22]. The family of Burr distributions goes back to 1941, and it is sometimes referred as the extended Pareto or beta prime distribution. All the PDFs of the loss distributions in the Burr family have a monotonically decreasing, right-skewed tails; see [15]. The Burr distribution is well recognized in probability theory with many applications in agriculture, biology, etc., see [20].

### *2.2.3 Gamma distribution*

The gamma distribution is a light-tailed distribution which is skewed to the right. The gamma distribution is a two-parameter distribution that is a generalization of the exponential distribution. It is a two-parameter probability distribution, where *x* is a

random variable, *β* is a scale parameter, *α* is the shape parameter, and *Γ*ð Þ∙ is a gamma distribution; see **Table 2**. The gamma distribution is said to be the generalization of the exponential distribution because for *γ* ¼ 1 it becomes the exponential with parameter *λ* ¼ 1*=β*, and it is usually used to model time between events; see [2]. According to [7], the gamma distribution is one of the most important loss distributions in risk analysis because it forms the base for creating many of the popular distributions we have. Exponential distribution is described by a density *f* and distribution *F* given in **Table 2** above, where *λ* represents the "failure" rate. The distribution is tractable and has unique mathematical properties, e.g. the failure distribution is described by a single parameter known as the mean time to failure, denoted by *θ*, also that the failure rate is defined by knowing the mean life, i.e. *λ* ¼ 1*=θ* . The exponential distribution can be used to model the time elapsed until the next event (e.g. accident); see [23].

The exponential PDF has a monotone decrease and an exponentially decreasing and light tail. This means that when it is applied in risk analysis, the event of high losses is given an almost zero probability; see [2]. Due to this property, [14] stated that the exponential distribution is not used very much in operational losses, but the constant decrease of the tail is useful for modeling lifetime data of items which have a constant failure rate. The exponential distribution has attractive and easily understandable mathematical properties; thus, it is mostly used in risk analysis for developing other models.

## *2.2.4 Weibull distribution*

Another generalization of the exponential distribution is the Weibull distribution, and it has two parameters (see **Table 2**) compared to the one parameter of the exponential distribution. The Weibull distribution has a light tail which is skewed to the right. The additional parameter allows the Weibull distribution to have more flexibility as well as heavier or lighter tail than the exponential distribution. That is, the Weibull distribution has a lighter tail than exponential distribution if *α* <1, equals to the exponential distribution if *α* ¼ 1 and has a heavier tail than exponential distribution if *α*> 1; see [2]. Furthermore, [2] stated that in risk analysis, the heavy-tailed Weibull distribution is a popular model as it has been shown to be optimal for modeling asset returns as well as used in reinsurance.

### *2.2.5 Log-normal distribution*

A log-normal distribution is a moderately heavy-tailed distribution that is skewed to the right. The distribution is derived by taking the natural logarithm of the data and fitting it to the normal distribution. The distribution is right-tailed and takes on only positive *x* values. The log-normal distribution, like the normal distribution, has parameters *μ* and *σ* (see **Table 2**). The distribution is useful for modeling of claim sizes. The thick tail and right skewness properties make it fit many situations. The lognormal can also resemble the normal distribution if the *α* is very small, and this property is not always desirable for analyzing risk; see [16].

### **2.3 Nonparametric loss distribution**

In the nonparametric loss distributions (e.g. empirical distribution function), all the data on the certain risk type is considered. In other words, we do not have to

estimate any parameters as all the data are available (which is hardly ever the case); see [2, 16]. According to [7], the CDF of the empirical distribution, *Fn*ð Þ *x* , is given by:

$$F\_n(\boldsymbol{\pi}) = \frac{1}{n} \# \{ \boldsymbol{i} : \boldsymbol{\pi}\_{\boldsymbol{i}} \le \boldsymbol{\pi} \} \tag{1}$$

where # denotes the number of observations ≤ *x*, and *n* is the total number observations in the sample.

Some of the advantages of nonparametric loss distributions are as follows:

Below are some of the disadvantages:

• Less efficient to compute and may provide inaccurate results, especially when the underlying distribution is known.

### **2.4 Risk quantification**

According to [18, 24], LDA is widely used to quantify operational risk; moreover, both [18, 24] showed that when quantifying operational risk, the PDF for an occurrence and the frequency for that occurrence are approximated firstly for a certain risk type or business line then later for the institution. The process of deriving these probability distributions is done in three steps: firstly, the loss severity distribution is derived; secondly, the loss frequency distribution is also derived; and lastly, the aggregate loss distribution is found by compounding the severity and frequency loss distributions.

The Value-at-Risk (VaR), which is a combination of expected and unexpected losses, is used when approximating the PDFs, and [18, 24] stated that the Capital-at-Risk (CaR) given in Eq. (2) is just the VaR, and this value is computed for a certain risk type cell and a certain occurrence type:

$$\text{CaR } (i, j; a) = \text{EL } (i, j) + \text{UL } (i, j; a) \tag{2}$$

Note that in Eq. (2), we use the indices *i* and *j* to denote a given business line and a given event type, *EL i*ð Þ , *j* is the expected loss, and *UL i*ð Þ , *j*; *α* is the unexpected loss at significance level *α*.

Another method of quantifying risk is the internal measurement approach (IMA). According to [18], when using IMA, the business type and the event type risk are both quantified using

$$\text{CaR } (i, j) = \text{EL } (i, j) \times \chi \text{ } (i, j) \times \text{RPI } (i, j) \tag{3}$$

where *γ* is the scaling factor and RPI is the risk profile index.

LDA is of great importance when computing regulatory capital, and as noted by [18], even though LDA is such a great tool, it also has its downside. This is due to a lack of data *Quantifying Risk Using Loss Distributions DOI: http://dx.doi.org/10.5772/intechopen.108856*

and even if a bank keeps large amounts of losses data, they may still be unrepresentative of potential extreme losses. Three of some popular approaches under LDA is the extreme value theory (EVT), VaR, and IMA which are briefly discussed in Section 3.

Risks need to be measurable so that they can be evaluated and examined. It is ideal to have a high-quality historical data that can be subjected to in-depth statistical analysis. Numerous quantification models tend to lack high-quality statistical data. The types of risk determine which quantification technique to use. In finance, the main methods to quantify risk are as outlined below; see [2, 3]:

• Dynamic financial analysis

This simulates the enterprise's overall risks as well as their interactions. Typically, forecast balance sheets and projected income statements are produced as outputs using cashflows.

• Financial Conditions Reports (FCR)

The Financial Conditions Report (FCR) displays both the current state of solvency and potential future developments. The volume and profitability of new business as well as any special characteristics it might have would typically be projected.

• Quantitative methods

Quantitative methods are employed for risks in insurance and underwriting, markets, and economies, such as interest rate, basis risk, and market fluctuations. Time series and scenario analysis might be included, as well as the fitting of statistical models and subsequent calculation of risk metrics like VaR.

• Credit risk models

Instead of measuring the risk in a credit portfolio, these models assess the credit risk of a single entity (business or person). These may be quantified as well as subjectively, and counterparty risk is one of them. A credit risk model's job is to take the state of the overall economy and the circumstances surrounding the company under consideration as inputs and provide a credit spread as an output. In this context, structural and reduced form models are the two main groups of credit risk models. Based on the value of a company's assets and obligations, structural models are used to assess the likelihood that a default will occur.

• Asset Liability Modeling (ALM)

This approach, which is common in the insurance industry and primarily measures liquidity and capital requirements, might be used by various types of financial companies.

• Scenario analysis

Operational hazards and other risks that are challenging to measure, such as legal risk, regulatory risk, agency risk, moral hazard, strategic risk, political risk, and reputational risk, are often covered under these.

• Sensitivity testing

It is used to change each parameter separately and measure how much the outputs of the model fluctuate or are sensitive to different variables.

### **2.5 Other risks**

Another risk that is quantified using LDA is credit risk, which is the estimation of expected and unexpected loss from credit defaults; see [10]. Note that [21] used the inverse Gaussian distribution to quantify credit risk and used Copula functions and Laplace transformation to run an algorithm that quantifies the corresponding probabilities. In this chapter, our focus is not on credit risk. Hence, a reader who might be interested in how the probability distribution of defaults is quantified can go through the articles by [10, 18].


*2.1. Data Modelling*

In the data modelling we will apply the Wepro model to synthesise the residential electricity load profile at the city level through the household profile generators namely LPG and ALPG.