**6. Conclusion**

The model used for SAW pressure sensor based on delay line are presented. For usefulness and reduction of time in design process, the equivalent circuit based on COM model, in which K11, K12=0 is proposed to be used.

Acoustic wave properties in different structures of AlN/SiO2/Si, AlN/Si, and AlN/Mo/Si are analyzed. The wave velocity, coupling factor could depend on the wave propagation medium.

From analyses of these structures, the range in which there is a weak dependence of the wave velocity, coupling factor on the AlN layer thickness could be known. The SAW devices should be fabricated in this range to facilitate manufacturing.

For AlN/Si structure, this range is 3 *khAlN* ≥ .

For AlN/Mo/Si, if this kind of SAW device is fabricated in the range from 2.7 *khAlN* ≥ to facilitate manufacturing, the use of Mo layer is useless. Consequently, to take full advantage of using Mo layer in term of wave velocity and coupling factor, it should be required to control the fabrication process carefully to obtain the required AlN thickness from khAlN=1.02 to khAlN=2.7.

For AlN/SiO2/Si, this range is 5 *khAlN* ≥ for khSiO2=0.7854, for thicker SiO2 layer, this range changes based on Figure 6 and Figure 7. Besides, using SiO2 layer would reduce temperature dependence of frequency. To choose the thickness of SiO2 layer, it would consider the effect of temperature dependence and analyses of wave velocity, coupling factor.

## **7. Appendix: Development of calculation for equivalent circuit of SAW device**

### **7.1 Appendix 1. Equivalent circuit for normal IDT including N periodic sections**

Fig. Appendix.1. Mason equivalent circuit for one periodic section in "crossed-field" model

small difference in the peak value of S21 (dB) occurs. This difference could be explained by

The model used for SAW pressure sensor based on delay line are presented. For usefulness and reduction of time in design process, the equivalent circuit based on COM model, in

Acoustic wave properties in different structures of AlN/SiO2/Si, AlN/Si, and AlN/Mo/Si are analyzed. The wave velocity, coupling factor could depend on the wave propagation

From analyses of these structures, the range in which there is a weak dependence of the wave velocity, coupling factor on the AlN layer thickness could be known. The SAW

For AlN/Mo/Si, if this kind of SAW device is fabricated in the range from 2.7 *khAlN* ≥ to facilitate manufacturing, the use of Mo layer is useless. Consequently, to take full advantage of using Mo layer in term of wave velocity and coupling factor, it should be required to control the fabrication process carefully to obtain the required AlN thickness

For AlN/SiO2/Si, this range is 5 *khAlN* ≥ for khSiO2=0.7854, for thicker SiO2 layer, this range changes based on Figure 6 and Figure 7. Besides, using SiO2 layer would reduce temperature dependence of frequency. To choose the thickness of SiO2 layer, it would consider the effect of temperature dependence and analyses of wave velocity, coupling

**7. Appendix: Development of calculation for equivalent circuit of SAW device** 

Fig. Appendix.1. Mason equivalent circuit for one periodic section in "crossed-field" model

**7.1 Appendix 1. Equivalent circuit for normal IDT including N periodic sections** 

using "crossed-filed" model instead of actual model as in Figure 13.

devices should be fabricated in this range to facilitate manufacturing.

**6. Conclusion** 

medium.

factor.

which K11, K12=0 is proposed to be used.

For AlN/Si structure, this range is 3 *khAlN* ≥ .

from khAlN=1.02 to khAlN=2.7.

Fig. Appendix.2. Mason equivalent circuit for one periodic section in "in-line field" model One periodic section can be expressed by the 3-port network as follows:

Fig. Appendix.3. One periodic section represented by 3-port network, admittance matrix [y]

$$
\begin{bmatrix}
\dot{i}\_1\\ \dot{i}\_2\\ \dot{i}\_3
\end{bmatrix} = \begin{bmatrix}
y\_{11} & y\_{12} & y\_{13}\\ y\_{21} & y\_{22} & y\_{23}\\ y\_{31} & y\_{32} & y\_{33}
\end{bmatrix} \begin{bmatrix}
e\_1\\ e\_2\\ e\_3
\end{bmatrix} \tag{\text{Appendix.1}}
$$

By the symmetrical properties of one periodic section (the voltage applied at port 3 will result in stress of the same value at port 1 and 2), the [y] matrix in (Appendix.1) becomes (Appendix.2) for Figure Appendix.4 and becomes (Appendix.3) for Figure Appendix.5.

$$
\begin{aligned}
\begin{bmatrix}
\dot{i}\_{1} \\
\dot{i}\_{2} \\
\dot{i}\_{3}
\end{bmatrix} &= \begin{bmatrix}
y\_{11} & y\_{12} & y\_{13} \\
y\_{12} & y\_{11} & -y\_{13} \\
y\_{13} & -y\_{13} & y\_{33}
\end{bmatrix} \begin{bmatrix}
e\_{1} \\
e\_{2} \\
e\_{3}
\end{bmatrix} & \text{(Appendix.2)} & \begin{bmatrix}
\text{Appenidx.2}
\end{bmatrix} \\
& \text{(Appendix.2)} \\
& & \text{(resp. 2)} \\
& & \text{(resp. 2)} \\
& & \text{(resp. 2)} \\
& & \text{(resp. 2)} \\
& & \text{(resp. 2)} \\
& & \text{(resp. 2)} \\
\dot{i}\_{3} & -y\_{12} & -y\_{13} & y\_{13}
\end{bmatrix} \begin{bmatrix}
\dot{e}\_{1} \\
\dot{e}\_{2} \\
\dot{e}\_{3}
\end{bmatrix} & \text{(Appendix.3)} & \begin{array}{ccc}
\dot{e}\_{1} \\
\dot{e}\_{2} \\
\dot{e}\_{3}
\end{bmatrix} & \text{(Appenditrix.3)} \\
\begin{bmatrix}
\dot{i}\_{1} \\
\dot{i}\_{2} \\
\dot{i}\_{3}
\end{bmatrix} & \text{(Appenditrix.3)} & \begin{array}{ccc}
\dot{\mathbf{e}}\_{1} \\
\dot{i}\_{3}
\end{bmatrix} & \text{(Appenditrix.3)} \\
\text{(Appenditrix.3)} & \text{(Appenditrix.3)} & \text{(Appenditrix.3)} \\
\end{bmatrix}
\end{aligned}
$$

Fig. Appendix.4. 3-port network representation of one periodic section, with the change of sign between Y13 and Y23 to ensure that acoustic power flows symmetrically away from transducer

Fig. Appendix.5. 3-port network representation of one periodic section, with the no change of sign between Y13 and Y23

SAW Parameters Analysis and Equivalent Circuit of SAW Device 469

In IDT including N periodic sections, the N periodic sections are connected acoustically in

Fig. Appendix.6. IDT including the N periodic sections connected acoustically in cascade

Because the symmetric properties of the IDT including N section like these of one periodic section, and from (Appendix.2), (Appendix.3), Figure Appendix.4 and Figure Appendix.5,

cascade and electrically in parallel as Figure Appendix.6.

the [Y] matrices of N-section IDT are represented as follows:

Fig. Appendix.7. The [Y] matrices and the model corresponsive models

The total transducer current is the sum of currents flowing into the N sections.

( ) ( )

 y e y e y e y e y e y e ... ye ye ye ye ye ye

= −+ + −+ + + − + + −+

13 1 1 13 2 1 33 3 1 13 1 2 13 2 2 33 3 2

With m is integer number, m=1,2, …, N-1, N

3 3 1 3 2 3 N 1 3 N

−

−− −

I i i . i i

= + +… + +

Since the periodic sections are identical, the recursion relation as follows can be obtained:

e1 m=e2 m-1 (Appendix.8)

e3 N= e3 N-1= e3 N-2=...= e3 2= e3 1=E3 (Appendix.9)

i1 m=i2 m-1 (Appendix.10)

( ) ( )

13 1 N 1 13 2 N 1 33 3 N 1 13 1 N 13 2 N 33 3 N

(Appendix.11)

and electrically in parallel

Applying circuit theory, definitions of [y] matrix elements are presented:

$$\begin{aligned} y\_{11} &= \frac{\dot{i}\_1}{\mathcal{C}\_1} \Big|\_{\substack{c\_2 \to 0\\ c\_3 \to 0}} & ; \qquad y\_{12} = \frac{\dot{i}\_1}{\mathcal{C}\_2} \Big|\_{\substack{c\_1 \to 0\\ c\_3 \to 0}}\\ y\_{13} &= \frac{\dot{i}\_1}{\mathcal{C}\_3} \Big|\_{\substack{c\_1 \to 0\\ c\_2 \to 0}} & ; \qquad y\_{33} = \frac{\dot{i}\_3}{\mathcal{C}\_3} \Big|\_{\substack{c\_1 \to 0\\ c\_2 \to 0}} \end{aligned} \tag{\text{Appendix.4}}$$

And using trigonometric functions as follows:

$$\begin{aligned} \, \_t \text{tg}\,\alpha - \frac{2}{\sin(2\alpha)} &= -\cot \, g\alpha\\ \, \_t \text{tg}\,\alpha - \frac{1}{\sin(2\alpha)} &= \frac{1}{2} (\text{tg}\,\alpha - \cot \, g\alpha) \\ \, \_t \text{tg}\,\alpha \frac{3\cos(2\alpha) + 1 - \text{tg}\,\alpha \sin(4\alpha) - \text{tg}\,\alpha \sin(2\alpha)}{\text{tg}\,\alpha \sin(4\alpha) + \text{tg}\,\alpha \sin(2\alpha) - \cos(2\alpha)} &= -\text{tg}(4\alpha) \end{aligned} \tag{\text{Appendix.5}}$$

The [y] matrix can be obtained for 2 models as follows:


$$\begin{aligned} y\_{11} &= -jG\_0 \cot g(4\alpha) \\ y\_{12} &= \frac{jG\_0}{\sin(4\alpha)} \\ y\_{13} &= -jG\_0 tg\alpha \\ y\_{33} &= j(2\alpha C\_0 + 4G\_0 tg\alpha) \end{aligned} \tag{\text{Appendix.6}}$$


$$\begin{aligned} y\_{11} &= -j\mathcal{C}\_0 \cot g\alpha \Big(\frac{G\_0}{a\mathcal{C}\_0} - \cot g(2\alpha)\Big) \Bigg[ 2 - \frac{\left(\frac{G\_0}{a\mathcal{C}\_0} - \frac{1}{\sin(2\alpha)}\right)^2}{\left(\frac{G\_0}{a\mathcal{C}\_0} - \cot g(2\alpha)\right)^2} \Bigg] \\ y\_{12} &= j\mathcal{C}\_0 \frac{\cot g\alpha \Big(\frac{G\_0}{a\mathcal{C}\_0} - \frac{1}{\sin(2\alpha)}\Big)^2}{2\left(\frac{2G\_0}{a\mathcal{C}\_0} - \cot g\alpha\right)\left(\frac{G\_0}{a\mathcal{C}\_0} - \cot g(2\alpha)\right)} \\ y\_{13} &= -j\mathcal{C}\_0 \frac{tg\alpha}{1 - \frac{2G\_0}{a\mathcal{C}\_0}tg\alpha} \\ y\_{33} &= \frac{j2a\mathcal{C}\_0}{1 - \frac{2G\_0}{a\mathcal{C}\_0}tg\alpha} \end{aligned} \tag{Appendix.7}$$

2 1 3 3

= = = =

(Appendix.4)

(Appendix.5)

(Appendix.7)

*e e e e*

1 1 <sup>11</sup> <sup>12</sup> 0 0 1 2 0 0

;

= =

*y y*

*y y*

cot

 α

11 0

= −

*jG <sup>y</sup>*

=

*y jG g*

13 0

= −

*y jG tg*

0 0

*G <sup>g</sup> <sup>C</sup>*

ω

<sup>2</sup> 2 cot cot (2 )

*g g C C*

 ω

0 0 0 0

α

α

<sup>=</sup> − −

*<sup>G</sup> <sup>C</sup> y jG g <sup>g</sup> <sup>C</sup> <sup>G</sup>*

ω

α

0 0

α

cot

α

*y jG G G*

0

ω

<sup>2</sup> <sup>1</sup>

−

ω

0

0 *tg <sup>C</sup>* α

2 <sup>2</sup> <sup>1</sup>

ω

ω

*tg y jG <sup>G</sup> tg <sup>C</sup>*

12

 α

*i i*

*e e*

1 3 <sup>13</sup> <sup>33</sup> 0 0 3 3 0 0

;

= =

( )

*tg tg tg tg tg tg*

0

α

α

sin(4 )

33 0 0

ω

= +

*y j C G tg*

cot

αα

3cos(2 ) 1 sin(4 ) sin(2 ) (4 ) sin(4 ) sin(2 ) cos(2 )

+− − = − + −

cot (4 )

α

 α

2

0

*G*

ω

ω

 α 0

*<sup>g</sup> <sup>C</sup>*

(2 4 )

0 0 11 0 2 <sup>0</sup> <sup>0</sup>

 α

<sup>−</sup> = − <sup>−</sup> <sup>−</sup> <sup>−</sup>

1

 −

sin(2 )

 α

sin(2 ) cot cot (2 ) 2

 α  α

 α  α

(Appendix.6)

2

1

 α

α

cot (2 )

 αα

*e e*

*i i*

1 1 2 2

= = = =

*e e e e*

Applying circuit theory, definitions of [y] matrix elements are presented:

And using trigonometric functions as follows:

α

α

α



12 0

13 0

= −

*j C <sup>y</sup> <sup>G</sup>*

33

= − 2

*tg g*

− =−

α

sin(2 )

1 1

*tg tg g*

− =−

 αα

sin(2 ) 2

α

α

The [y] matrix can be obtained for 2 models as follows:

α

In IDT including N periodic sections, the N periodic sections are connected acoustically in cascade and electrically in parallel as Figure Appendix.6.

Fig. Appendix.6. IDT including the N periodic sections connected acoustically in cascade and electrically in parallel

Because the symmetric properties of the IDT including N section like these of one periodic section, and from (Appendix.2), (Appendix.3), Figure Appendix.4 and Figure Appendix.5, the [Y] matrices of N-section IDT are represented as follows:

Fig. Appendix.7. The [Y] matrices and the model corresponsive models

Since the periodic sections are identical, the recursion relation as follows can be obtained:

$$\mathbf{e}\_1 \mathbf{m} \mathbf{\overline{e}\_2} \mathbf{e}\_{2 \text{ m} \cdot 1} \tag{\text{Appendix.8}}$$

$$\mathbf{e}\_{3}\mathbf{v} = \mathbf{e}\_{3\,\,N\,1} = \mathbf{e}\_{3\,\,N\,2} = \dots = \mathbf{e}\_{3\,\,2} = \mathbf{e}\_{3\,\,1} = \mathbf{E}\_{3} \tag{\text{Appendix.9}}$$

$$\mathbf{i}\_{1\text{ m}} \equiv \mathbf{i}\_{2\text{ m}\cdot 1} \tag{\text{Appendix.10}}$$

With m is integer number, m=1,2, …, N-1, N

The total transducer current is the sum of currents flowing into the N sections.

$$\begin{aligned} \mathbf{I}\_{3} &= \mathbf{i}\_{31} + \mathbf{i}\_{32} + \dots + \mathbf{i}\_{3N-1} + \mathbf{i}\_{3N} \\ &= \left( \mathbf{y}\_{13}\mathbf{e}\_{11} - \mathbf{y}\_{13}\mathbf{e}\_{21} + \mathbf{y}\_{33}\mathbf{e}\_{31} \right) + \left( \mathbf{y}\_{13}\mathbf{e}\_{12} - \mathbf{y}\_{13}\mathbf{e}\_{22} + \mathbf{y}\_{33}\mathbf{e}\_{32} \right) + \dots \\ &+ \left( \mathbf{y}\_{13}\mathbf{e}\_{1N-1} - \mathbf{y}\_{13}\mathbf{e}\_{2N-1} + \mathbf{y}\_{33}\mathbf{e}\_{3N-1} \right) + \left( \mathbf{y}\_{13}\mathbf{e}\_{1N} - \mathbf{y}\_{13}\mathbf{e}\_{2N} + \mathbf{y}\_{33}\mathbf{e}\_{3N} \right) \end{aligned} \tag{Aypendist}$$

SAW Parameters Analysis and Equivalent Circuit of SAW Device 471

[] []*<sup>N</sup>*

[ ]

*<sup>N</sup> <sup>n</sup>*

− =

*n*

11 1 1 123 12 12 12 *Q X* <sup>1</sup> *I EEE QQQ*

11

12

12

1

12

cos(4 ) sin(4 ) sin(4 ) cos(4 )

cos(8 ) sin(8 ) sin(8 ) cos(8 )

cos(12 ) sin(12 ) sin(12 ) cos(12 )

α

0 <sup>12</sup> sin( 4 ) *jG <sup>Y</sup>*

*N* α

<sup>−</sup> = = <sup>−</sup>

From (Appendix.24) and (Appendix.35), Y11 and Y12 in "cross-field" model can be expressed:

11 0 *Y jG g N* = − cot ( 4 )

cos( 4 ) sin( 4 ) sin( 4 ) cos( 4 )

α

 α

> α

> > α

> > > α

<sup>=</sup> (Appendix.32)

 α

 α

 α

> α

[] [] <sup>1</sup> <sup>1</sup> 2 0

*X X K L X*

11

12

13

In "crossed-field" model, matrix [Q] can be represented in a simple form as follows:

[ ] <sup>0</sup> 0

[ ]<sup>2</sup> <sup>0</sup> 0

[ ]<sup>3</sup> <sup>0</sup> 0

[] [] <sup>0</sup> 0

*<sup>N</sup> N jR N Q K jG N <sup>N</sup>* α

**. . . . . .** etc. Consequently, matrix [Q] will be given:

*jR <sup>K</sup> jG*

*jR <sup>K</sup> jG*

*jR <sup>K</sup> jG*

α

α

α

α

α

α

<sup>−</sup> <sup>=</sup> <sup>−</sup>

<sup>−</sup> <sup>=</sup> <sup>−</sup>

<sup>−</sup> <sup>=</sup> <sup>−</sup>

To solve (Appendix.24) and (Appendix.25), matrix [Q] should be solved.

*<sup>Q</sup> <sup>Y</sup>*

<sup>1</sup> *<sup>Y</sup>*

*<sup>X</sup> <sup>Y</sup>*

The Y13 is known by (Appendix.13), so (Appendix.26) and matrix [X] don't need to be

Solving (Appendix.20) and using the boundary conditions (e0= E1, i0=I1) gives:

*Q K* = (Appendix.21)

= = (Appendix.22)

=− + − (Appendix.23)

*<sup>Q</sup>* = − (Appendix.24)

*Q*= (Appendix.25)

*<sup>Q</sup>* = − (Appendix.26)

(Appendix.27)

(Appendix.28)

(Appendix.29)

(Appendix.30)

(Appendix.31)

Where

Consequently,

solved.

By applying (Appendix.8), (Appendix.9) and boundary conditions (e11 = E1, e2N=E2), (Appendix.11) becomes:

$$\mathbf{I}\_3 = \mathbf{y}\_{13}\mathbf{e}\_{11}\mathbf{y}\_{13}\mathbf{e}\_{21}\mathbf{y} + \mathbf{N}\mathbf{y}\_{33}\mathbf{E}\_3 = \mathbf{y}\_{13}\mathbf{E}\_1\mathbf{-}\mathbf{y}\_{13}\mathbf{E}\_2 + \mathbf{N}\mathbf{y}\_{33}\mathbf{E}\_3 \tag{\text{Appendix.12}}$$

From Figure Appendix.7, the Y13 and Y33 can be expressed as:

$$\mathbf{Y}\_{13} \mathbf{\overline{y}}\_{13} \tag{\text{Appendix.13}}$$

$$\mathbf{Y}\_{33} = \mathbf{N} \mathbf{y}\_{33} \tag{\text{Appendix.14}}$$

Because the N periodic sections are connected acoustically in cascade and electrically in parallel, the model as in Figure Appendix.5 should be used to obtain the [Y] matrix of Nsection IDT.

From (Appendix.3) for one section, the i1 and i2 can be expressed

$$\mathbf{i}\_{1} = \mathbf{y}\_{11}\mathbf{e}\_{1} + \mathbf{y}\_{12}\mathbf{e}\_{2} + \mathbf{y}\_{13}\mathbf{e}\_{3},\\\mathbf{i}\_{2} = \mathbf{y}\_{12}\mathbf{e}\_{1} + \mathbf{y}\_{12}\mathbf{e}\_{2} + \mathbf{y}\_{13}\mathbf{e}\_{3} \tag{Appendix.15}$$

Equations (Appendix.15) can be represented in matrix form like [ABCD] form in electrical theory as follows:

$$
\begin{bmatrix} e\_2 \\ i\_2 \end{bmatrix} = [K] \begin{bmatrix} e\_1 \\ i\_1 \end{bmatrix} + [L] e\_3 \tag{\text{Appendix.16}}
$$

Where

$$\begin{aligned} \begin{bmatrix} K \end{bmatrix} &= \begin{bmatrix} -\frac{y\_{11}}{y\_{12}} & \frac{1}{y\_{12}}\\ \frac{y\_{11}^2 - y\_{12}^2}{y\_{12}} & -\frac{y\_{11}}{y\_{12}} \end{bmatrix} & \text{(Appendix.17) \\\\ \begin{bmatrix} L \end{bmatrix} &= \begin{bmatrix} -\frac{y\_{13}}{y\_{12}} \\ \frac{y\_{11}y\_{13} + y\_{12}y\_{13}}{y\_{12}} \end{bmatrix} & \text{(Appendix.18) \end{aligned}$$

By applying (Appendix.16) into N-section IDT as in Figure Appendix.6 and using (Appendix.9), the second recursion relation is obtained as follows:

$$
\begin{bmatrix} e\_m \\ \dot{I}\_m \end{bmatrix} = [K] \begin{bmatrix} e\_{m-1} \\ \dot{I}\_{m-1} \end{bmatrix} + [L] E\_3 \tag{\text{Appendix.19}}
$$

Where m is integer number, m=1,2, …, N-1, N

Starting (Appendix.19)(Appendix.19) by using with m=N, and reducing m until m=1 gives the expression:

$$
\begin{bmatrix} e\_N \\ \dot{I}\_N \end{bmatrix} = [Q] \begin{bmatrix} e\_0 \\ \dot{I}\_0 \end{bmatrix} + [X] E\_3 \tag{\text{Appendix.20}}
$$

Where

470 Acoustic Waves – From Microdevices to Helioseismology

By applying (Appendix.8), (Appendix.9) and boundary conditions (e11 = E1, e2N=E2),

I3= y13e1 1-y13e2 N+ Ny33E3 = y13E1-y13E2+ Ny33 E3 (Appendix.12)

Y13=y13 (Appendix.13)

 Y33= Ny33 (Appendix.14) Because the N periodic sections are connected acoustically in cascade and electrically in parallel, the model as in Figure Appendix.5 should be used to obtain the [Y] matrix of N-

 i1= y11e1+y12e2+ y13 e3, i2= -y12e1-y12e2+ y13 e3 (Appendix.15) Equations (Appendix.15) can be represented in matrix form like [ABCD] form in electrical

3

(Appendix.16)

(Appendix.17)

(Appendix.18)

(Appendix.19)

(Appendix.20)

[ ]

*K Le*

11

<sup>−</sup> <sup>=</sup> <sup>−</sup> <sup>−</sup>

2 2 11 12 11 12 12

*y y <sup>K</sup> yy y y y*

*<sup>y</sup> <sup>L</sup>*

12 12

13 12 11 13 12 13 12

*y*

<sup>−</sup>

*yy yy y*

<sup>=</sup> <sup>+</sup>

By applying (Appendix.16) into N-section IDT as in Figure Appendix.6 and using

[ ] <sup>1</sup>

Starting (Appendix.19)(Appendix.19) by using with m=N, and reducing m until m=1 gives

*m m e e*

*e e*

 = + 

 = + 

*i i*

*N N*

1 [ ] *m m*

[] [] <sup>0</sup>

0

*Q XE i i*

− −

*K LE*

3

3

*y* 1

[ ] 2 1

*e e*

*i i* = + 

[ ]

[ ]

(Appendix.9), the second recursion relation is obtained as follows:

Where m is integer number, m=1,2, …, N-1, N

2 1

From Figure Appendix.7, the Y13 and Y33 can be expressed as:

From (Appendix.3) for one section, the i1 and i2 can be expressed

(Appendix.11) becomes:

section IDT.

theory as follows:

the expression:

Where

$$\left[Q\right] = \left[K\right]^n \tag{\text{Appendix.21}}$$

$$\mathbb{E}\left[X\right] = \begin{bmatrix} X\_1\\ X\_2 \end{bmatrix} = \sum\_{n=0}^{N-1} \left[K\right]^n \left[L\right] \tag{\text{Appendix.22}}$$

Solving (Appendix.20) and using the boundary conditions (e0= E1, i0=I1) gives:

$$I\_1 = -\frac{Q\_{11}}{Q\_{12}}E\_1 + \frac{1}{Q\_{12}}E\_2 - \frac{X\_1}{Q\_{12}}E\_3 \tag{\text{Appendix.23}}$$

Consequently,

$$Y\_{11} = -\frac{Q\_{11}}{Q\_{12}}\tag{\text{Appendix.24}}$$

$$Y\_{12} = \frac{1}{Q\_{12}}\tag{\text{Appendix.25}}$$

$$Y\_{13} = -\frac{X\_1}{Q\_{12}}\tag{\text{Appendix.26}}$$

The Y13 is known by (Appendix.13), so (Appendix.26) and matrix [X] don't need to be solved.

To solve (Appendix.24) and (Appendix.25), matrix [Q] should be solved.

In "crossed-field" model, matrix [Q] can be represented in a simple form as follows:

$$\begin{bmatrix} \mathbf{K} \end{bmatrix} = \begin{bmatrix} \cos(4\alpha) & -jR\_0 \sin(4\alpha) \\ -j\mathbf{G}\_0 \sin(4\alpha) & \cos(4\alpha) \end{bmatrix} \tag{\text{Appendix.27}}$$

$$\begin{bmatrix} \mathbf{K} \end{bmatrix}^2 = \begin{bmatrix} \cos(8\alpha) & -j\mathbf{R}\_0 \sin(8\alpha) \\ -j\mathbf{G}\_0 \sin(8\alpha) & \cos(8\alpha) \end{bmatrix} \tag{\text{Appendix.28}}$$

$$\begin{bmatrix} \mathbf{K} \end{bmatrix}^{3} = \begin{bmatrix} \cos(12\alpha) & -jR\_{0}\sin(12\alpha) \\ -j\mathbf{G}\_{0}\sin(12\alpha) & \cos(12\alpha) \end{bmatrix} \tag{\text{Appendix.29}}$$

**. . . . . .** etc. Consequently, matrix [Q] will be given:

$$\begin{bmatrix} \mathbf{Q} \end{bmatrix} = \begin{bmatrix} \mathbf{K} \end{bmatrix}^{\mathbb{N}} = \begin{bmatrix} \cos(\text{N} \mathbf{4} \alpha) & -j \mathbf{R}\_0 \sin(\text{N} \mathbf{4} \alpha) \\ -j \mathbf{G}\_0 \sin(\text{N} \mathbf{4} \alpha) & \cos(\text{N} \mathbf{4} \alpha) \end{bmatrix} \tag{\text{Appendix.30}}$$

From (Appendix.24) and (Appendix.35), Y11 and Y12 in "cross-field" model can be expressed:

$$Y\_{11} = -j\mathbf{G}\_0 \cot \mathcal{g} (N \mathbf{4} \mathcal{a}) \tag{\text{Appendix.31}}$$

$$Y\_{12} = \frac{jG\_0}{\sin(\text{N}4\alpha)}\tag{\text{Appendix.32}}$$

SAW Parameters Analysis and Equivalent Circuit of SAW Device 473

Fig. Appendix.9. Equivalent circuit of "N+1/2" model IDT

Fig. Appendix.10. [Yd] matrix representation of "N+1/2" model IDT

The elements of [Yd] matrix for "crossed-field" model are given as follows:

11 0 2

*jG g N Yd*

 αα

 α

22 0

21 0

α

0

α

α

α

[ ] 11 12 13

<sup>1</sup> cot (4 ) sin (4 )(cot (2 ) cot (4 )) *Yd jG g N N g gN*

sin(2 )[cot (4 )cos(2 ) sin(2 )] cos(2 ) sin(4 ) cos(2 ) cot (4 )sin(2 )

<sup>−</sup> = − <sup>+</sup>

 αα

α

 α

sin(4 )(cos(2 ) cot (4 )sin(2 )) *Yd jG N g*

 α

αα

*N g N* α

( 2 cot (4 )sin sin(2 ))sin(2 ) 2sin sin(4 )(cos(2 ) cot (4 )sin(2 )) sin(4 )

− + <sup>+</sup> <sup>=</sup> + − <sup>+</sup>

α

*g N Yd jG g N*

α

*tg g N Yd jG tg N g N N*

*Yd Yd Yd Yd*

21 22 23 31 32 33

> α

> > αα

> > > αα

> > > > α

<sup>−</sup> <sup>=</sup> + (Appendix.40)

*<sup>N</sup>* = − <sup>+</sup> (Appendix.39)

 αα

> α

2 2

1

cot (4 )cos(2 ) sin(2 ) cos(2 ) cot (4 )sin(2 )

 αα

αα

(Appendix.35)

α

α

 α

α

 α (Appendix.36)

(Appendix.37)

(Appendix.38)

*Yd Yd Yd*

*Yd Yd Yd* <sup>=</sup>

= − <sup>+</sup>

The form of matrix [Yd] is:

12

13 0

In conclusion, matrix [Y] representation of N-section IDT is:


$$\begin{aligned} Y\_{11} &= -j\mathbf{G}\_0 \cot \mathbf{g} (4N\alpha) \\ Y\_{12} &= \frac{j\mathbf{G}\_0}{\sin(4N\alpha)} \\ Y\_{13} &= -j\mathbf{G}\_0 \mathbf{t} \mathbf{g} \alpha \\ Y\_{33} &= jN(2a\mathbf{C}\_0 + 4\mathbf{G}\_0 \mathbf{t} \mathbf{g} \alpha) \end{aligned} \tag{\text{Appendix.33}}$$


$$\begin{aligned} Y\_{11} &= -\frac{Q\_{11}}{Q\_{12}}\\ Y\_{12} &= \frac{1}{Q\_{12}}\\ Y\_{13} &= -jG\_0 \frac{t \text{g}\alpha}{1 - \frac{2G\_0}{a\alpha C\_0} t \text{g}\alpha} \\ Y\_{33} &= \frac{j2a\theta \text{NC}\_0}{1 - \frac{2G\_0}{a\alpha C\_0} t \text{g}\alpha} \end{aligned} \tag{\text{Appendix.34}}$$

Where [Q] can be calculated from (Appendix.17) and (Appendix.21).

### **7.2 Appendix 2: Equivqlent circuit for "N+1/2" model IDT**

In case IDT includes N periodic sections (like in section 3.2 plus one finger (in color red) as shown in Figure Appendix.8 that we call "N+1/2" model IDT.

Fig. Appendix.8. "N+1/2" model IDT

The equivalent circuit for this model is shown in Figure Appendix.9 and the matrix [Yd] representation is shown as in Figure Appendix.10 (letter "d" stands for **d**ifferent from model [Y] in section 3.2.


cot (4 )

α

 α (Appendix.33)

(Appendix.34)

(2 4 )

0 0 0

α

α

0 0

In case IDT includes N periodic sections (like in section 3.2 plus one finger (in color red) as

The equivalent circuit for this model is shown in Figure Appendix.9 and the matrix [Yd] representation is shown as in Figure Appendix.10 (letter "d" stands for **d**ifferent from model

<sup>2</sup> <sup>1</sup>

−

ω

α

0

*N*

α

α


sin(4 )

*Y jG g N*

33 0 0

11

12

12

2 <sup>2</sup> <sup>1</sup>

*j NC <sup>Y</sup> <sup>G</sup> tg <sup>C</sup>*

*tg Y jG <sup>G</sup> tg <sup>C</sup>*

ω

ω

1

*Q*

13 0

= −

= +

*Y jN C G tg*

ω

11 0

= −

*jG <sup>Y</sup>*

=

13 0

= −

11

*<sup>Q</sup> <sup>Y</sup> Q*

=

= −

12

*Y*

33

Where [Q] can be calculated from (Appendix.17) and (Appendix.21).

**7.2 Appendix 2: Equivqlent circuit for "N+1/2" model IDT** 

shown in Figure Appendix.8 that we call "N+1/2" model IDT.

Fig. Appendix.8. "N+1/2" model IDT

[Y] in section 3.2.

= −

*Y jG tg*

12

In conclusion, matrix [Y] representation of N-section IDT is:

(Appendix.31) and (Appendix.32):

(Appendix.24) and (Appendix.25):

Fig. Appendix.9. Equivalent circuit of "N+1/2" model IDT

Fig. Appendix.10. [Yd] matrix representation of "N+1/2" model IDT The form of matrix [Yd] is:

$$\begin{aligned} \begin{bmatrix} \,^1Yd \end{bmatrix} = \begin{bmatrix} \,^1dd\_{11} & \,^1Yd\_{12} & \,^1Yd\_{13} \\ \,^1Yd\_{21} & \,^1Yd\_{22} & \,^1Yd\_{23} \\ \,^1Yd\_{31} & \,^1Yd\_{32} & \,^1Yd\_{33} \end{bmatrix} \end{aligned} \tag{\text{Appendix.35}}$$

The elements of [Yd] matrix for "crossed-field" model are given as follows:

$$\mathrm{Yd}\_{11} = j\mathrm{C}\_{0} \left\{ \frac{1}{\sin^{2}(4Na)(\cot g(2\alpha) + \cot g(4Na))} - \cot g(4Na) \right\} \quad \text{(Appendix.36)$$

$$\mathrm{Yd}\_{12} = \frac{j\mathrm{C}\_{0}}{\sin(4Na)} \left\{ \cos(2\alpha) - \frac{\sin(2\alpha)[\cot g(4Na)\alpha\cos(2\alpha) - \sin(2\alpha)]}{\cos(2\alpha) + \cot g(4Na)\sin(2\alpha)} \right\} \quad \text{(Appendix.37)$$

$$\mathrm{Yd}\_{13} = j\mathrm{C}\_{0} \left\{ \frac{(-\mathrm{tg}\,\alpha + 2\cot g(4Na)\sin^{2}\alpha + \sin(2\alpha))\sin(2\alpha)}{\sin(4Na\alpha)(\cos(2\alpha) + \cot g(4Na\alpha)\sin(2\alpha))} + \frac{2\sin^{2}\alpha}{\sin(4Na\alpha)} - \mathrm{tga} \right\} \quad \text{(Appendix.38)$$

$$\mathrm{Yd}\_{21} = -j\mathrm{C}\_{0} \frac{1}{\sin(4Na\alpha)(\cos(2\alpha) + \cot g(4Na\alpha)\sin(2\alpha))} \quad \text{(Appendix.39)}$$

$$Yd\_{22} = jG\_0 \frac{\cot g(4Na\alpha)\cos(2\alpha) - \sin(2\alpha)}{\cos(2\alpha) + \cot g(4Na\alpha)\sin(2\alpha)}\tag{Appendix.40}$$

SAW Parameters Analysis and Equivalent Circuit of SAW Device 475

33 11 22 12 21 23 32 11 12 31

{ } ( ) 22 2

== + −+ + + (Appendix.57)

= − + − + ++ (Appendix.61)

<sup>33</sup> 11 12 13 12 *M* =+ + − − + (1 )[(1 ) ] 2 (1 ) *Y YY YY* (Appendix.62)

2

= =− + + (Appendix.58)

= =− + + (Appendix.59)

*SS S* 23 32 13 = =− (Appendix.60)

(Appendix.55)

(Appendix.56)

(1 )[(1 )(1 ) ] [ (1 ) ]

For model IDT including N identical sections, these equations can be further simplified. In

11 22 21 12 31 13 23 32 13

11 22 33 11 12 13 11 12 <sup>1</sup> *S S Y Y Y YY Y* (1 )(1 ) 2 *<sup>M</sup>*

> 12 21 12 33 13 <sup>2</sup> *SS Y Y Y* (1 ) *<sup>M</sup>*

13 31 13 11 12 <sup>2</sup> *S S Y YY* (1 ) *<sup>M</sup>*

**7.4 Appendix 4: Equivalent circuit for SAW device base on Mason model, [ABCD]** 

In SAW device, each input and output IDTs have one terminal connected to admittance G0. Therefore, one IDT can be represented as two-port network. [ABCD] matrix (as in Figure Appendix.11) is used to represent each IDT, because [ABCD] matrix representation has one interesting property that in cascaded network, the [ABCD] matrix of total network can be

**7.4.1 Appendix 4.1: [ABCD] Matrix representation of IDT** 

obtained easily by multiplying the matrices of elemental networks.

Fig. Appendix.11. [ABCD] representation of two-port network for one IDT

{ } 22 2 33 33 11 12 13 11 12 <sup>1</sup> *S Y Y Y Y YY* (1 )[(1 ) ] 2 (1 ) *<sup>M</sup>*

22 2

= = = = =−

*Y Y Y Y Y Y YY Y*

=+ + + − − + − −

*Y Y Y YY Y Y Y YY*

3

det( )

*M Y*

case of Figure Appendix.7 (b):

Therefore, Sij's take the following form

= Π+

13 31 22 21 32

*Y Y Y YY*

[ (1 ) ]

− −−

where

Where

**Matrix representation** 

$$\mathrm{Yd}\_{23} = j\mathrm{G}\_0 \frac{-\mathrm{tg}\alpha + 2\cot\mathfrak{g}(4Na)\sin^2(2\alpha) + \sin(2\alpha)}{\cos(2\alpha) + \cot\mathfrak{g}(4Na)\sin(2\alpha)}\tag{\text{Appendix.41}}$$

$$\mathbf{Y}d\_{\mathfrak{z}1} = -j\mathbf{G}\_0 \mathbf{t} \mathbf{g} \alpha \tag{\text{Appendix.42}}$$

$$Yd\_{32} = -jG\_0 \sin(2\alpha)\tag{\text{Appendix.43}}$$

$$Yd\_{33} = j\alpha \mathbb{C}\_0(2N - 1) + j\mathbb{G}\_0\{\sin(2\alpha) + (4N + 1)\text{tg}\alpha\} \tag{\text{Appendix.44}}$$

### **7.3 Appendix 3: Scattering matrix [S] for IDT**

The scattering matrix [S] of a three-port network characterized by its admittance matrix [Y] is given by [3]:

$$S = \Pi\_3 - 2Y(\Pi\_3 + Y)^{-1} \tag{\text{Appendix.45}}$$

Where Π3 is the 3x3 identity matrix.

After expanding this equation, the scattering matrix elements for a general three-port network are given by the following expressions:

$$\begin{split} S\_{11} &= \frac{1}{M} \{ (1+Y\_{33})(1-Y\_{11}+Y\_{22}-Y\_{11}Y\_{22}+Y\_{12}Y\_{21}) + \\ &+ Y\_{13} [Y\_{31}(1+Y\_{22}) - Y\_{21}Y\_{32}] + Y\_{23} [Y\_{32}(Y\_{11}-1) - Y\_{12}Y\_{31}] \} \end{split} \tag{\text{Appendix.46}}$$

$$S\_{12} = -\frac{2}{M} [Y\_{12}(1+Y\_{33}) - Y\_{13}Y\_{32}] \tag{\text{Appendix.47}}$$

$$S\_{13} = -\frac{2}{M} \left[ Y\_{13} (1 + Y\_{22}) - Y\_{12} Y\_{23} \right] \tag{\text{Appendix.48}}$$

$$S\_{21} = -\frac{2}{M} \left[ Y\_{21} (1 + Y\_{33}) - Y\_{23} Y\_{31} \right] \tag{\text{Appendix.49}}$$

$$\begin{split} S\_{22} &= \frac{1}{M} \{ (1+Y\_{33})(1+Y\_{11}-Y\_{22}-Y\_{11}Y\_{22}+Y\_{12}Y\_{21}) + \\ &+ Y\_{13} [Y\_{31}(Y\_{22}-1) - Y\_{21}Y\_{32}] + Y\_{23} [Y\_{32}(Y\_{11}+1) - Y\_{12}Y\_{31}] \} \end{split} \tag{\text{Appendix.50}}$$

$$S\_{23} = -\frac{2}{M} [Y\_{23}(1+Y\_{11}) - Y\_{13}Y\_{21}] \tag{\text{Appendix.51}}$$

$$S\_{31} = -\frac{2}{M} \left[ Y\_{31} (1 + Y\_{22}) - Y\_{21} Y\_{32} \right] \tag{\text{Appendix.52}}$$

$$S\_{32} = -\frac{2}{M} [Y\_{32}(1+Y\_{11}) - Y\_{12}Y\_{31}] \tag{\text{Appendix.53}}$$

$$\begin{split} S\_{33} &= \frac{1}{M} \{ (1 - Y\_{33}) (1 + Y\_{11} + Y\_{22} + Y\_{11} Y\_{22} - Y\_{12} Y\_{21}) + \\ &+ Y\_{13} [Y\_{31} (Y\_{22} + 1) - Y\_{21} Y\_{32}] + Y\_{23} [Y\_{32} (Y\_{11} + 1) - Y\_{12} Y\_{31}] \} \end{split} \tag{\text{Appendix.54}}$$

where

474 Acoustic Waves – From Microdevices to Helioseismology

*tg g N Yd jG g N* α

α

*Yd*33 0 = −+ + + *j*

*Yd jG tg* 31 0 = −

32 0 *Yd jG* = − sin(2 )

The scattering matrix [S] of a three-port network characterized by its admittance matrix [Y]

After expanding this equation, the scattering matrix elements for a general three-port

13 31 22 21 32 23 32 11 12 31

*Y Y Y YY Y Y Y YY*

11 33 11 22 11 22 12 21

*S Y Y Y YY YY*

<sup>1</sup> (1 )(1 )

= + −+− + +

[ (1 ) ] [ ( 1) ]

<sup>12</sup> [ ] 12 33 13 32 <sup>2</sup> *S Y Y YY* (1 ) *<sup>M</sup>*

<sup>13</sup> [ ] 13 22 12 23 <sup>2</sup> *S Y Y YY* (1 ) *<sup>M</sup>*

<sup>21</sup> [ ] 21 33 23 31 <sup>2</sup> *S Y Y YY* (1 ) *<sup>M</sup>*

13 31 22 21 32 23 32 11 12 31

*Y Y Y YY Y Y Y YY*

22 33 11 22 11 22 12 21

33 33 11 22 11 22 12 21

*S Y Y Y YY YY*

<sup>1</sup> (1 )(1 )

= − +++ − +

[ ( 1) ] [ ( 1) ]

+ +− + +−

*S Y Y Y YY YY*

<sup>1</sup> (1 )(1 )

= + +−− + + + −− + +−

[ ( 1) ] [ ( 1) ]

<sup>23</sup> [ ] 23 11 13 21 <sup>2</sup> *S Y Y YY* (1 ) *<sup>M</sup>*

<sup>31</sup> [ ] 31 22 21 32 <sup>2</sup> *S Y Y YY* (1 ) *<sup>M</sup>*

<sup>32</sup> [ ] 32 11 12 31 <sup>2</sup> *S Y Y YY* (1 ) *<sup>M</sup>*

13 31 22 21 32 23 32 11 12 31

*Y Y Y YY Y Y Y YY*

+ + − + −−

23 0

ω

**7.3 Appendix 3: Scattering matrix [S] for IDT** 

network are given by the following expressions:

{

{

{

*M*

*M*

*M*

Where Π3 is the 3x3 identity matrix.

is given by [3]:

2

 αα

− + <sup>+</sup> <sup>=</sup> + (Appendix.41)

αα*C N*(2 1) sin(2 ) (4 1) *jG Nt* <sup>0</sup> { *g* } (Appendix.44)

3 3 *S YY* 2( )<sup>−</sup> =Π − Π + (Appendix.45)

=− + − (Appendix.47)

=− + − (Appendix.48)

=− + − (Appendix.49)

=− + − (Appendix.51)

=− + − (Appendix.52)

=− + − (Appendix.53)

 α

(Appendix.42)

}

}

}

(Appendix.46)

(Appendix.50)

(Appendix.54)

(Appendix.43)

2 cot (4 )sin (2 ) sin(2 )

αα

cos(2 ) cot (4 )sin(2 )

α

α

1

$$\begin{array}{l} M = \det(\Pi\_3 + Y) \\ = (1 + Y\_{33})[(1 + Y\_{11})(1 + Y\_{22}) - Y\_{12}Y\_{21}] - Y\_{23}[Y\_{32}(1 + Y\_{11}) - Y\_{12}Y\_{31}] - \text{(Appendix.55)} \\ - Y\_{13}[Y\_{31}(1 - Y\_{22}) - Y\_{21}Y\_{32}] \end{array}$$

For model IDT including N identical sections, these equations can be further simplified. In case of Figure Appendix.7 (b):

$$\begin{aligned} Y\_{11} &= Y\_{22} \\ Y\_{21} &= Y\_{12} \\ Y\_{31} &= Y\_{13} \\ Y\_{23} &= Y\_{32} = -Y\_{13} \end{aligned} \tag{\text{Appendix.56}}$$

Therefore, Sij's take the following form

$$S\_{11} = S\_{22} = \frac{1}{M} \left\{ (1 + Y\_{33})(1 - Y\_{11}^2 + Y\_{12}^2) + 2Y\_{13}^2(Y\_{11} + Y\_{12}) \right\} \tag{\text{Appendix.57}}$$

$$S\_{12} = S\_{21} = -\frac{2}{M} \left[ Y\_{12} (1 + Y\_{33}) + Y\_{13}^2 \right] \tag{\text{Appendix.58}}$$

$$S\_{13} = S\_{31} = -\frac{2}{M} Y\_{13} (1 + Y\_{11} + Y\_{12}) \tag{\text{Appendix.59}}$$

$$S\_{23} = S\_{32} = -S\_{13} \tag{\text{Appendix.60}}$$

$$S\_{33} = \frac{1}{M} \left[ (1 - Y\_{33}) \left[ (1 + Y\_{11})^2 - Y\_{12}^{\ \ 2} \right] + 2Y\_{13}^2 (1 + Y\_{11} + Y\_{12}) \right] \tag{\text{Appendix.61}}$$

Where

$$M = (1 + \mathcal{Y}\_{33})[(1 + \mathcal{Y}\_{11})^2 - \mathcal{Y}\_{12}^{\ \ 2}] - 2\mathcal{Y}\_{13}^2(1 + \mathcal{Y}\_{12})\tag{A.pppendix.62})$$

### **7.4 Appendix 4: Equivalent circuit for SAW device base on Mason model, [ABCD] Matrix representation**

### **7.4.1 Appendix 4.1: [ABCD] Matrix representation of IDT**

In SAW device, each input and output IDTs have one terminal connected to admittance G0. Therefore, one IDT can be represented as two-port network. [ABCD] matrix (as in Figure Appendix.11) is used to represent each IDT, because [ABCD] matrix representation has one interesting property that in cascaded network, the [ABCD] matrix of total network can be obtained easily by multiplying the matrices of elemental networks.

Fig. Appendix.11. [ABCD] representation of two-port network for one IDT

SAW Parameters Analysis and Equivalent Circuit of SAW Device 477

*Nj N <sup>A</sup> tg N j N* α

ωα

α

sin(4 ) (2 cot 4)(cot(4 ) ) 1 cos(4 ) sin(4 ) *<sup>N</sup> <sup>D</sup> N CZ <sup>N</sup> <sup>j</sup> tg Nj N*

in which N is replaced by M (number of periodic sections in output IDT)

sin(4 ) (2 cot 4)(cot(4 ) ) 1 cos(4 ) sin(4 ) *out <sup>M</sup> <sup>A</sup> M CZ <sup>M</sup> <sup>j</sup> tg Mj M*

α

α

 α

 α

the [ABCD] matrix of output IDT can be easily obtained:

Consequently, the [ABCD] matrix of output IDT is:

α

α

frequency f0. By setting:

One interesting property of [ABCD] of "crossed-field" mode is:

α

This means [ABCD] matrix is reciprocal.

α

[ ] sin(4 ) cos(4 ) 1 cos(4 ) sin(4 )

0 *<sup>A</sup> <sup>B</sup>*

= + <sup>+</sup> <sup>+</sup> − − (Appendix.73)

<sup>1</sup> *C GD B*

In SAW device, the ouput IDT is inverse of input IDT. By the reciprocal property of [ABCD],

 Aoutput= Dinput (Appendix.76) Boutput= Binput (Appendix.77) Coutput= Cinput (Appendix.78) Doutput= Ainput (Appendix.79)

ωα

= + <sup>+</sup> <sup>+</sup> − − (Appendix.80)

<sup>0</sup> [ ] 1 sin(4 ) cos(4 ) 1 cos(4 ) sin(4 ) *out Mj M <sup>B</sup>*

[ ] sin(4 ) cos(4 ) 1 cos(4 ) sin(4 ) *out Mj M <sup>D</sup>*

0

ππ

*G tg M j M* α

*tg M j M* α

1 *out out out C GA B*

At the center frequency f0, the [ABCD] matrix becomes infinite since α=0.5π(f/f0)= 0.5π. However, [ABCD] elements may be calculated by expanding for frequency very near

> 0 <sup>0</sup> 2 22 2 *f f x f N*

π

α

[ ] 0 0

 α

 α

<sup>−</sup> <sup>=</sup> − − (Appendix.81)

<sup>−</sup> <sup>=</sup> − − (Appendix.82)

<sup>−</sup> = += + (Appendix.84)

=− + (Appendix.83)

αα

 α

αα

 α

[ ] 0 0

0

 α

<sup>−</sup> <sup>=</sup> − − (Appendix.71)

 α

*G*= (Appendix.72)

 α

=− + (Appendix.74)

AD-BC=1 (Appendix.75)

αα

To find the [ABCD] matrix for one IDT in SAW device, the condition that no reflected wave at one terminal of IDT, and the current-voltage relations by [Y] matrix in section are used as follows:

Fig. Appendix.12. Two-port network for one IDT

12 13 11 13 13 0

$$\begin{bmatrix} I\_1 \\ I\_2 \\ I\_3 \end{bmatrix} = \begin{bmatrix} Y\_{11} & Y\_{12} & Y\_{13} \\ Y\_{12} & Y\_{11} & -Y\_{13} \\ Y\_{13} & -Y\_{13} & Y\_{33} \end{bmatrix} \begin{bmatrix} V\_1 \\ V\_2 \\ V\_3 \end{bmatrix} \tag{\text{Appendix.63}}$$

And I1=-G0V1 (Appendix.64)

From these current-voltage relations, the V3 and I3 are given:

$$V\_3 = \frac{Y\_{11}^2 - Y\_{12}^2 + Y\_{11}\mathbf{G}\_0}{Y\_{12}Y\_{13} + Y\_{11}Y\_{13} + Y\_{13}\mathbf{G}\_0}V\_2 - \frac{\mathbf{G}\_0 + Y\_{11}}{Y\_{12}Y\_{13} + Y\_{11}Y\_{13} + Y\_{13}\mathbf{G}\_0}I\_2 \tag{\text{Appendix.65}}$$

$$I\_3 = \frac{-(Y\_{13}Y\_{12} + Y\_{13}Y\_{11} + Y\_{13}\mathbf{G}\_0)^2 + (Y\_{11}Y\_{33} - Y\_{13}^2 + Y\_{33}\mathbf{G}\_0)(Y\_{11}^2 - Y\_{12}^2 + Y\_{11}\mathbf{G}\_0)}{(\mathbf{G}\_0 + Y\_{11})(Y\_{12}Y\_{13} + Y\_{11}Y\_{13} + Y\_{13}\mathbf{G}\_0)}V\_2 - \tag{\text{Appendix.66}}$$

$$-\frac{Y\_{11}Y\_{33} - Y\_{13}^2 + Y\_{33}\mathbf{G}\_0}{Y\_{12}Y\_{13} + Y\_{11}Y\_{13} + Y\_{13}\mathbf{G}\_0}I\_2$$

From (Appendix.65) and (Appendix.66), equivalence between port 3 in Figure Appendix.12 equals to port 1 in Figure Appendix.11, and consideration of direction of current I2 in Figure Appendix.11 and Figure Appendix.12, [ABCD] matrix representation for two-port network of IDT in obtained:

$$A = \frac{Y\_{11}{1}^2 - Y\_{12}{^2} + Y\_{11}G\_0}{Y\_{12}Y\_{13} + Y\_{11}Y\_{13} + Y\_{13}G\_0} \tag{\text{Appendix.67}}$$

$$B = \frac{G\_0 + Y\_{11}}{Y\_{12}Y\_{13} + Y\_{11}Y\_{13} + Y\_{13}G\_0} \tag{\text{Appendix.68}}$$

$$\mathbf{C} = \frac{-(Y\_{13}Y\_{12} + Y\_{13}Y\_{11} + Y\_{13}G\_0)^2 + (Y\_{11}Y\_{33} - Y\_{13}^2 + Y\_{33}G\_0)(Y\_{11}^2 - Y\_{12}^2 + Y\_{11}G\_0)}{(G\_0 + Y\_{11})(Y\_{12}Y\_{13} + Y\_{11}Y\_{13} + Y\_{13}G\_0)} \tag{\text{Appendix.69}} \quad \text{(Appendix.69)}$$

$$D = \frac{Y\_{11}Y\_{33} - Y\_{13}^2 + Y\_{33}G\_0}{Y\_{12}Y\_{13} + Y\_{11}Y\_{13} + Y\_{13}G\_0} \tag{\text{Appendix.70}}$$

In case of "crossed-field" model, the [ABCD] can be further simplified:

To find the [ABCD] matrix for one IDT in SAW device, the condition that no reflected wave at one terminal of IDT, and the current-voltage relations by [Y] matrix in section are used as

> 1 11 12 13 1 2 12 11 13 2 3 13 13 33 3

And I1=-G0V1 (Appendix.64)

11 12 11 0 0 11 3 22 12 13 11 13 13 0 12 13 11 13 13 0 *Y Y YG G Y VVI YY YY YG YY YY YG*

2 22 2

From (Appendix.65) and (Appendix.66), equivalence between port 3 in Figure Appendix.12 equals to port 1 in Figure Appendix.11, and consideration of direction of current I2 in Figure Appendix.11 and Figure Appendix.12, [ABCD] matrix representation for two-port network

> 2 2 11 12 11 0 12 13 11 13 13 0

0 11 12 13 11 13 13 0

− + + + −+ −+ <sup>=</sup> + ++ (Appendix.69)

2 11 33 13 33 0 12 13 11 13 13 0

*YY YY YG*

2 22 2

*YY Y YG <sup>D</sup> YY YY YG*

*Y Y YG <sup>A</sup> YY YY YG*

*G Y <sup>B</sup>*

13 12 13 11 13 0 11 33 13 33 0 11 12 11 0 0 11 12 13 11 13 13 0 ( ) ( )( ) ( )( ) *YY YY YG YY Y YG Y Y YG <sup>C</sup> G Y YY YY YG*

In case of "crossed-field" model, the [ABCD] can be further simplified:

13 12 13 11 13 0 11 33 13 33 0 11 12 11 0 3 2 0 11 12 13 11 13 13 0

− + + + −+ −+ = − + ++

( ) ( )( ) ( )( )

*YY YY YG YY Y YG Y Y YG I V G Y YY YY YG*

− + <sup>+</sup> = − ++ ++ (Appendix.65)

− + <sup>=</sup> + + (Appendix.67)

<sup>+</sup> <sup>=</sup> + + (Appendix.68)

− + <sup>=</sup> + + (Appendix.70)

(Appendix.63)

(Appendix.66)

*I YY YV I Y Y YV I Y YYV* = − <sup>−</sup>

follows:

Fig. Appendix.12. Two-port network for one IDT

2 11 33 13 33 0

*YY Y YG <sup>I</sup> YY YY YG*

12 13 11 13 13 0

− + <sup>−</sup> + +

of IDT in obtained:

From these current-voltage relations, the V3 and I3 are given:

2

2 2

$$A = \frac{\sin(4Na\alpha) - j\cos(4Na\alpha)}{tga[1 - \cos(4Na\alpha) - j\sin(4Na\alpha)]} \tag{\text{Appendix.71}}$$

$$B = \frac{A}{G\_0} \tag{\text{Appendix.72}}$$

$$D = \frac{\sin(4Na)}{1 - \cos(4Na) - j\sin(4Na)} \left[ N(2aC\_0Z\_0\cot a + 4)(\cot(4Na) + j) + tga\right] \tag{\text{Appendix.73}}$$

$$C = -\frac{1}{B} + G\_0 D \tag{\text{Appendix.74}}$$

One interesting property of [ABCD] of "crossed-field" mode is:

$$\text{AD-BC=1} \tag{\text{Appendix.75}}$$

This means [ABCD] matrix is reciprocal.

In SAW device, the ouput IDT is inverse of input IDT. By the reciprocal property of [ABCD], the [ABCD] matrix of output IDT can be easily obtained:

 Aoutput= Dinput (Appendix.76) Boutput= Binput (Appendix.77) Coutput= Cinput (Appendix.78) Doutput= Ainput (Appendix.79)

in which N is replaced by M (number of periodic sections in output IDT) Consequently, the [ABCD] matrix of output IDT is:

$$A\_{\rm out} = \frac{\sin(4Ma)}{1 - \cos(4Ma) - j\sin(4Ma)} \left[ M(2aC\_0 Z\_0 \cot \alpha + 4)(\cot(4Ma) + j) + \text{tg}\alpha \right] \quad \text{(Appendix.80)}$$

$$B\_{\rm out} = \frac{1}{G\_0} \frac{\sin(4M\alpha) - j\cos(4M\alpha)}{t g \alpha [1 - \cos(4M\alpha) - j\sin(4M\alpha)]} \tag{\text{Appendix.81}}$$

$$D\_{\rm out} = \frac{\sin(4M\alpha) - j\cos(4M\alpha)}{\text{tg}\alpha [1 - \cos(4M\alpha) - j\sin(4M\alpha)]} \tag{\text{Appendix.82}}$$

$$C\_{out} = -\frac{1}{B\_{out}} + G\_0 A\_{out} \tag{\text{Appendix.83}}$$

At the center frequency f0, the [ABCD] matrix becomes infinite since α=0.5π(f/f0)= 0.5π. However, [ABCD] elements may be calculated by expanding for frequency very near frequency f0.

By setting:

$$\alpha = \frac{\pi}{2} \frac{f - f\_0}{f\_0} + \frac{\pi}{2} = \frac{\chi}{2N} + \frac{\pi}{2} \tag{\text{Appendix.84}}$$

SAW Parameters Analysis and Equivalent Circuit of SAW Device 479

*device device in in path path out out device device in in path path out out A B AB A B A B C D CD C D C D* <sup>=</sup>

Where [ABCD]in is calculated from (Appendix.71), (Appendix.72), (Appendix.73) and

[ABCD]out is calculated from (Appendix.80), (Appendix.81), (Appendix.82) and (Appendix.83).

[1] C.C.W.Ruppel, W.Ruile, G.Scholl, K.Ch.Wagner, and O.Manner, Review of models for

[2] L.A.Coldren, and R.L.Rosenberg, Scattering matrix approach to SAW resonators, *IEEE* 

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Films, *IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control,* Vol.36,

[ABCD]path is calculated from (Appendix.93) and (Appendix.94).

*Ultrasonics Symposium*, 1976, pp.266-271.

Vol.43, No.3, May 1996.

No.3, May 1989.

[3] R.W.Newcomb, Linear Multiport Synthesis, McGraw Hill, 1966.

*IEEE*, vol.64, No.12, December 1976, pp.1666-1698

*A AA A BC A AB C BD C device in* = +++ *path out in path out in path out in path out* (Appendix.97)

*B AA B BC B AB D BD D device in* = ++ + *path out in path out in path out in path out* (Appendix.98)

*C C A A DC A C B C DD C device in* = + ++ *path out in path out in path out in path out* (Appendix.99)

*D C A B DC B C B D DD D device in* =+++ *path out in path out in path out in path out* (Appendix.100)

(Appendix.96)

And the [ABCD] equivalent matrix of SAW device is shown in Figure Appendix.14

Fig. Appendix.14. [ABCD] matrix of SAW device

[ABCD] matrix of delay line SAW is

(Appendix.74).

**8. References** 

1994.

$$\text{Where}\tag{1}\tag{2}\tag{3}\tag{3}\tag{3}\tag{4}\tag{4}\tag{4}\tag{4}\tag{4}\tag{4}\tag{4}\tag{5}\tag{4}$$

By using the limit of some functions as follows:

$$\lim\_{\mathbf{x}\to\mathbf{0}} [\sin(4\mathbf{N}\mathbf{z})] = \lim\_{\mathbf{x}\to\mathbf{0}} [\sin(2\mathbf{x})] = 2\mathbf{x} \tag{\text{Appendix.86}}$$

$$\lim\_{x \to 0} [\cos(4Nx)] = \lim\_{x \to 0} [\cos(2x)] = 1 \tag{A} \qquad \text{(Appendix.87)}$$

$$\lim\_{x \to 0} [tg x] = \lim\_{x \to 0} [-\cot(\frac{x}{2N})] = -\frac{2N}{x} \tag{A.ppendix.88}$$

The [ABCD] matrix of input IDT is obtained:

$$A = \frac{2\mathbf{x} - \mathbf{j}}{j4N} \tag{\text{Appendix.89}}$$

$$B = \frac{1}{G\_0} \frac{2\pi - j}{j4N} \tag{\text{Appendix.90}}$$

$$\mathbf{C} = 2\pi f \mathbf{C}\_0 \mathbf{x} - 4\mathbf{N} \mathbf{G}\_0 - j \left( \pi f \mathbf{C}\_0 + \frac{4\mathbf{N} \mathbf{G}\_0}{2\mathbf{x} - j} \right) \tag{\text{Appendix.91}}$$

$$\mathbf{D} = 2\pi f \mathbf{C}\_0 \mathbf{Z}\_0 \mathbf{x} - 4\mathbf{N} - j\pi f \mathbf{C}\_0 \mathbf{Z}\_0 \tag{\text{Appendix.92}}$$

### **7.4.2 Appendix 4.2: [ABCD] matrix representation of propagation path**

Based on equivalent circuit star model of propagation path in section 3.3, [ABCD] matrix representation of propagation way can be obtained clearly:

$$A\_{path} = D\_{path} = \cos \mathcal{D} \theta \tag{\text{Appendix.93}}$$

$$B\_{\rm path} = C\_{\rm path} = j \sin \mathcal{D} \theta \tag{\text{Appendix.94}}$$

$$\text{With } \qquad \qquad \qquad \theta = \frac{\pi f l}{v} \tag{\text{Appendix.95}}$$

Where l is the length of propagation path between two IDTs.

So, [ABCD] matrix representations of input IDT, propagation way and output IDT are obtained. They are cascaded as Figure Appendix.13:

Fig. Appendix.13. Cascaded [ABCD] matrices of input IDT, propagation way and output IDT

And the [ABCD] equivalent matrix of SAW device is shown in Figure Appendix.14

Fig. Appendix.14. [ABCD] matrix of SAW device

[ABCD] matrix of delay line SAW is

478 Acoustic Waves – From Microdevices to Helioseismology

*f f x N f* π

0 0 lim[sin(4 )] lim[sin(2 )] 2 *x x N xx* α

0 0 lim[cos(4 )] lim[cos(2 )] 1 *x x N x* α

<sup>2</sup> lim[ ] lim[ cot( )]

2 4 *<sup>x</sup> <sup>j</sup> <sup>A</sup> j N*

0 1 2 4 *<sup>x</sup> <sup>j</sup> <sup>B</sup> G j N*

00 0 <sup>4</sup> 2 4

0 0 0 0 *D fC Z x N j fC Z* ≈ −− 2 4

Based on equivalent circuit star model of propagation path in section 3.3, [ABCD] matrix

*v* π θ

So, [ABCD] matrix representations of input IDT, propagation way and output IDT are

Fig. Appendix.13. Cascaded [ABCD] matrices of input IDT, propagation way and output IDT

≈ −− + <sup>−</sup>

 π

> π

> > θ

θ

*NG C fC x NG j fC*

*x N tg N x*

0 0

α → →

π

representation of propagation way can be obtained clearly:

cos2 *A D path path* = =

sin 2 *BC j path path* = =

Where l is the length of propagation path between two IDTs.

With *fl*

obtained. They are cascaded as Figure Appendix.13:

π

**7.4.2 Appendix 4.2: [ABCD] matrix representation of propagation path** 

*x x* 2

0

<sup>−</sup> <sup>=</sup> (Appendix.85)

→ → = ≈ (Appendix.86)

→ → = ≈ (Appendix.87)

= − ≈− (Appendix.88)

<sup>−</sup> <sup>≈</sup> (Appendix.89)

<sup>−</sup> <sup>≈</sup> (Appendix.90)

(Appendix.91)

(Appendix.92)

(Appendix.93)

(Appendix.94)

= (Appendix.95)

0

*x j*

2

Where 0

By using the limit of some functions as follows:

The [ABCD] matrix of input IDT is obtained:

$$
\begin{bmatrix}
\begin{bmatrix}
\boldsymbol{A}\_{\text{device}} & \boldsymbol{B}\_{\text{device}} \\
\boldsymbol{\mathcal{C}}\_{\text{device}} & \boldsymbol{D}\_{\text{device}}
\end{bmatrix} = \begin{bmatrix}
\boldsymbol{A}\_{in} & \boldsymbol{B}\_{in} \\
\boldsymbol{\mathcal{C}}\_{in} & \boldsymbol{D}\_{in}
\end{bmatrix} \begin{bmatrix}
\boldsymbol{A}\_{path} & \boldsymbol{B}\_{path} \\
\boldsymbol{\mathcal{C}}\_{path} & \boldsymbol{D}\_{path}
\end{bmatrix} \begin{bmatrix}
\boldsymbol{A}\_{out} & \boldsymbol{B}\_{out} \\
\boldsymbol{\mathcal{C}}\_{out} & \boldsymbol{D}\_{out}
\end{bmatrix} \tag{\text{Appendix.96}}
$$

$$A\_{device} = A\_{in}A\_{path}A\_{out} + B\_{in}C\_{path}A\_{out} + A\_{in}B\_{path}C\_{out} + B\_{in}D\_{path}C\_{out} \tag{\text{Appendix.97}}$$

$$B\_{device} = A\_{in}A\_{path}B\_{out} + B\_{in}\mathbb{C}\_{path}B\_{out} + A\_{in}B\_{path}D\_{out} + B\_{in}D\_{path}D\_{out} \tag{\text{Appendix.98}}$$

$$\mathbf{C}\_{dviice} = \mathbf{C}\_{in}A\_{path}A\_{out} + D\_{in}\mathbf{C}\_{path}A\_{out} + \mathbf{C}\_{in}B\_{path}\mathbf{C}\_{out} + D\_{in}D\_{path}\mathbf{C}\_{out} \qquad \text{(Appendix.99)}$$

$$D\_{device} = \mathbf{C}\_{in} A\_{path} B\_{out} + D\_{in} \mathbf{C}\_{path} B\_{out} + \mathbf{C}\_{in} B\_{path} D\_{out} + D\_{in} D\_{path} D\_{out} \quad \text{(Appendix.100)}$$

Where [ABCD]in is calculated from (Appendix.71), (Appendix.72), (Appendix.73) and (Appendix.74).

[ABCD]out is calculated from (Appendix.80), (Appendix.81), (Appendix.82) and (Appendix.83). [ABCD]path is calculated from (Appendix.93) and (Appendix.94).

### **8. References**


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**21** 

*Spain* 

**Sources of Third–Order Intermodulation** 

**A Phenomenological Approach** 

*Universitat Politècnica de Catalunya (UPC), Barcelona* 

Eduard Rocas and Carlos Collado

**Distortion in Bulk Acoustic Wave Devices:** 

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

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

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

et al., 2009) could represent a limitation for some applications.

underestimate the 3IMD by more than 20 dB.

**1. Introduction** 

