**3.1 Why Equivalent Circuit model is chosen?**

Actual devices exist in a three-dimensional physical continuum. Their behaviour is governed by the laws of physics, chemistry, biology, and electronics. From a general point of view, the analysis of devices can be carried out by using some equations of laws of physics, chemistry … For example; the analysis of piezoelectric resonators or transducers and their application to ultrasonic system can be solved by using the wave equation [36],[37]. But through analysis, equivalent electrical circuit representations of devices can be extracted. So, they can be readily expressible with Equivalent Electric Circuit. Below is the presentation of advantages and disadvantages of equivalent circuit.

## **Advantages:**


### **Disadvantages:**


In many systems, both commercial and industrial, pressure measurement plays a key role. Since pressure is a normal stress (force per unit area), pressure measurement can be done by using piezoelectric material which can convert stress into voltage. Equivalent circuits such as Mason's model [36] provide a powerful tool for the analysis and simulation of piezoelectric transducer elements. Most of the analogous circuits which have appeared in the literature implement transducers as the circuit elements. This model simulates both the coupling between the mechanical and electrical systems and the coupling between the mechanical and acoustical systems [39]. The mechanical, electrical and acoustic parts of piezoelectric transducer can be varied and analysed about behaviour by implementing equivalent circuits on computer tools such as Ansoft®, Spice, ADS, etc. For IDT composing of N periodic sections, Smith et al [41] developed the equivalent circuit model based on Berlincourt et al [40] work about equivalent circuit for Length Expander Bar with parallel electric field and with perpendicular electric field and based on the equivalent circuit for electromechanical transducer presented by Mason [36]. "Smith model" henceforth will be used to indicate this model. From this model, some models for SAW device in literature have been implemented. However, these models would include only IDTs [42], [43]. In SAW pressure sensor, one of sensitive parts is propagation path. It should be included in the model. The hybrid model based on Smith model for SAW pressure sensor which includes the IDTs and propagation path have been constructed.

Actual devices exist in a three-dimensional physical continuum. Their behaviour is governed by the laws of physics, chemistry, biology, and electronics. From a general point of view, the analysis of devices can be carried out by using some equations of laws of physics, chemistry … For example; the analysis of piezoelectric resonators or transducers and their application to ultrasonic system can be solved by using the wave equation [36],[37]. But through analysis, equivalent electrical circuit representations of devices can be extracted. So, they can be readily expressible with Equivalent Electric Circuit. Below is the




In many systems, both commercial and industrial, pressure measurement plays a key role. Since pressure is a normal stress (force per unit area), pressure measurement can be done by using piezoelectric material which can convert stress into voltage. Equivalent circuits such as Mason's model [36] provide a powerful tool for the analysis and simulation of piezoelectric transducer elements. Most of the analogous circuits which have appeared in the literature implement transducers as the circuit elements. This model simulates both the coupling between the mechanical and electrical systems and the coupling between the mechanical and acoustical systems [39]. The mechanical, electrical and acoustic parts of piezoelectric transducer can be varied and analysed about behaviour by implementing equivalent circuits on computer tools such as Ansoft®, Spice, ADS, etc. For IDT composing of N periodic sections, Smith et al [41] developed the equivalent circuit model based on Berlincourt et al [40] work about equivalent circuit for Length Expander Bar with parallel electric field and with perpendicular electric field and based on the equivalent circuit for electromechanical transducer presented by Mason [36]. "Smith model" henceforth will be used to indicate this model. From this model, some models for SAW device in literature have been implemented. However, these models would include only IDTs [42], [43]. In SAW pressure sensor, one of sensitive parts is propagation path. It should be included in the model. The hybrid model based on Smith model for SAW pressure sensor which includes

chemical equations approach (such as direct wave equations approach).


**3. Equivalent circuit for SAW delay line based on Mason model** 

presentation of advantages and disadvantages of equivalent circuit.

elements are presented clearly by Warren P.Mason [57], [58].

those used in the original derivation of the equivalent circuit [58].

microwave network theory, integrated circuit etc.

the IDTs and propagation path have been constructed.

**3.1 Why Equivalent Circuit model is chosen?** 

**Advantages:** 

**Disadvantages:** 

Another equivalent model is based on the Coupling-Of-Modes (COM) theory. An excellent recent review of COM theory used in SAW devices was written by K.Hashimoto [10]. Based on the COM equations, as the force and voltage analogy can be used, the relationships between the terminal quantities at the one electrical port and two acoustic ports for an IDT have been done. K.Nakamura [44] introduced a simple equivalent circuit for IDT based on COM approach that is developed in section 4.

In conclusion, the equivalent-circuit model is chosen because it can allow fast design. This allows the designer to determine the major dimensions and parameters in number of fingers, fingers width, aperture, delay line distance, frequency response, impedance parameters and transfer characteristics of SAW device.

## **3.2 Equivalent circuit for IDT including N periodic sections**

Based on Berlincourt et al [39] about equivalent circuit for Length Expander Bar with parallel electric field and with perpendicular electric field and based on the equivalent circuit for electromechanical transducer presented by Mason [36], Smith and al [41] have developed the equivalent circuit for IDT composed of N periodic sections of the form shown in Figure 12.

Fig. 12. Interdigital transducer diagram

One periodic section as shown in Figure 13 (a) can be presented by analogous onedimensional configurations: "crossed-field" model as in Figure 13 (b), and "in-line" model as in Figure 13 (c). In "crossed-field" model, the applied electric field is normal to the acoustic propagation vector; while in "in-line field" model, the electric field is parallel to the propagation vector.

The important advantage of two one-dimensional models is that each periodic section can be represented by equivalent circuit of Mason, as shown in Figure 14 for "crossed-field" model and Figure 15 for "in-line field" model. The difference between these two equivalent circuits is that in "crossed-field" model, the negative capacitors are short-circuited. Where:

$$
\alpha = \frac{\theta}{4} = \frac{\pi}{2} \frac{\alpha}{a\_b} \tag{12}
$$

SAW Parameters Analysis and Equivalent Circuit of SAW Device 455

Fig. 15. Mason equivalent circuit for one periodic section in "in-line field" model

11 0

= −

*jG <sup>y</sup>*

=

*y jG g*

13 0

= −

*y jG tg*

0 0

*G*

ω

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

*g g C C*

 ω

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

0 0 0 0

α

α

<sup>=</sup> − −

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

ω

α

0 0

α

cot

α

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

0

cascade and electrically in parallel as represented in Figure 16.

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

G0=R0-1, R0 is expressd by (13): - for the "crossed-field" model:


12 0

13 0

= −

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

33

= −

One periodic section can be represented by the 3-port network [y] matrix. The [y] matrix of one periodic section for 2 models as follows (see Appendix, section Appendix 1), with

0

α

α

sin(4 )

cot (4 )

α

 α

2

0

*G*

ω

ω

 α 0

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

(14)

2

(15)

1

 α

α

cot (2 )

(2 4 )

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

*G C*

 α

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

1

<sup>−</sup>

sin(2 )

 α

sin(2 ) cot cot (2 ) 2

33 0 0

ω

= +

*y j C G tg*

With periodic section transit angle

$$
\theta = 2\pi \frac{\omega}{a\_0}$$

$$
R\_0 = \frac{2\pi}{a\_0 \mathcal{C}\_s k^2} \tag{13}$$

R0 is electrical equivalent of mechanical impedance Z0 [59] k: electromechanical coupling coefficient C0=Cs/2 with Cs : electrode capacitance per section ω0 is center angular frequency

Fig. 13. Side view of the interdigital transducer and 2 analogous one-dimensional configurations (a) Actual model, (b) "crossed-field" model, (c) "in-line field" model

Fig. 14. Mason equivalent circuit for one periodic section in "crossed-field" model

Fig. 15. Mason equivalent circuit for one periodic section in "in-line field" model

One periodic section can be represented by the 3-port network [y] matrix. The [y] matrix of one periodic section for 2 models as follows (see Appendix, section Appendix 1), with G0=R0 -1, R0 is expressd by (13):


454 Acoustic Waves – From Microdevices to Helioseismology

2 ω

0 2 0 2 *s*

ω

*C k* π

θ π

*R*

Fig. 13. Side view of the interdigital transducer and 2 analogous one-dimensional configurations (a) Actual model, (b) "crossed-field" model, (c) "in-line field" model

Fig. 14. Mason equivalent circuit for one periodic section in "crossed-field" model

R0 is electrical equivalent of mechanical impedance Z0 [59]

C0=Cs/2 with Cs : electrode capacitance per section

0

<sup>=</sup> (13)

ω<sup>=</sup>

With periodic section transit angle

k: electromechanical coupling coefficient

ω0 is center angular frequency

(a)

(b)

(c)

$$\begin{aligned} y\_{11} &= -j\mathbf{G}\_0 \cot \mathbf{g}(4\alpha) \\ y\_{12} &= \frac{j\mathbf{G}\_0}{\sin(4\alpha)} \\ y\_{13} &= -j\mathbf{G}\_0 \mathbf{t} \mathbf{g}\alpha \\ y\_{33} &= j(2\alpha \mathbf{C}\_0 + 4\mathbf{G}\_0 \mathbf{t} \mathbf{g}\alpha) \end{aligned} \tag{14}$$


$$\begin{aligned} y\_{11} &= -jG\_0 \cot g\alpha \Big(\frac{G\_0}{a\mathcal{C}\_0} - \cot g(2\alpha)\Big) \Big| 2 - \frac{\left(\frac{G\_0}{a\mathcal{C}\_0} - \frac{1}{\sin(2\alpha)}\right)^2}{\left(\frac{G\_0}{a\mathcal{C}\_0} - \cot g(2\alpha)\right)^2} \Big| \\ y\_{12} &= jG\_0 \frac{\cot g\alpha \Big(\frac{G\_0}{a\mathcal{C}\_0} - \frac{1}{\sin(2\alpha)}\Big)^2}{2\left(\frac{2G\_0}{a\mathcal{C}\_0} - \cot g\alpha\right)\left(\frac{G\_0}{a\mathcal{C}\_0} - \cot g(2\alpha)\right)} \\ y\_{13} &= -jG\_0 \frac{tg\alpha}{1 - \frac{2G\_0}{a\mathcal{C}\_0}tg\alpha} \\ y\_{33} &= \frac{j2a\mathcal{C}\_0}{1 - \frac{2G\_0}{a\mathcal{C}\_0}tg\alpha} \end{aligned} \tag{15}$$

In IDT including N periodic sections, the N periodic sections are connected acoustically in cascade and electrically in parallel as represented in Figure 16.

SAW Parameters Analysis and Equivalent Circuit of SAW Device 457

SAW, it is necessary to construct the model for propagation path. Based on the equivalent circuit for electromechanical transducer presented by Mason [36], equivalent circuit of

> 2 *l v* ω

Due to the piezoelectric effect, an RF signal applied at input IDT stimulates a micro-acoustic wave propagating on its surface. These waves propagate in two directions, one to receiving IDT and another to the medium. The approximations as follows are assumed to construct



Based on these two approximations, the [Y] matrix representation of IDT in section 3.2, and propagation path representation in section 3.3, the SAW delay line can be expressed as

= (18)

γ

propagation path is presented as in Figure 17.

with v is SAW velocity, l is propagation length.

**3.4 Equivalent circuit for SAW delay line** 

the equivalent circuit for SAW delay line:

characteristic admittance Y0 to one terminal of IDT.

Fig. 18. Equivalent circuit of SAW delay line, based on Mason model

expressed as the no-loss transmission line.

equivalent circuit as in Figure 18.

Where

Fig. 17. Equivalent circuit of propagation path, based on Mason model

Fig. 16. IDT including the N periodic sections connected acoustically in cascade and electrically in parallel

Matrix [Y] representation of N-section IDT for two models, "crossed-field" model and "inline" model are in (16) and (17), respectively (the calculation development is presented in Appendix, section Appendix 1):


$$\begin{aligned} Y\_{11} &= -jG\_0 \cot g(4N\alpha) \\ Y\_{12} &= \frac{jG\_0}{\sin(4N\alpha)} \\ Y\_{13} &= -jG\_0 tg\alpha \\ Y\_{33} &= jN(2a\mathcal{C}\_0 + 4G\_0 tg\alpha) \end{aligned} \tag{16}$$


$$\begin{aligned} Y\_{11} &= -\frac{Q\_{11}}{Q\_{12}}\\ Y\_{12} &= \frac{1}{Q\_{12}}\\ Y\_{13} &= -jG\_0 \frac{tg\alpha}{1 - \frac{2G\_0}{a\alpha C\_0}tg\alpha} \\ Y\_{33} &= \frac{j2\alpha b\mathcal{N}C\_0}{1 - \frac{2G\_0}{a\alpha C\_0}tg\alpha} \end{aligned} \tag{17}$$

It was shown in the literature that the crossed field model yielded better agreement than the experiment when compared to the in-line model when K is small. In section 2, K is always smaller than 7.2%. Besides, in section stated above, the "crossed-field" model is simpler than "in-line field" model in term of equations of all element of [Y] matrix. Consequently, the "crossed-field" model is selected henceforth for the calculating, modeling the devices.

### **3.3 Equivalent circuit for propagation path**

The delay line SAW device could be used for pressure sensor application. The sensitive part of this kind of device will be the propagation path. To model the pressure sensor using

Fig. 16. IDT including the N periodic sections connected acoustically in cascade and

11 0

= −

*jG <sup>Y</sup>*

=

13 0

= −

11

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

=

= −

12

*Y*

33

**3.3 Equivalent circuit for propagation path** 

= −

*Y jG tg*

12

Matrix [Y] representation of N-section IDT for two models, "crossed-field" model and "inline" model are in (16) and (17), respectively (the calculation development is presented in

0

*N*

α

α

sin(4 )

*Y jG g N*

cot (4 )

α

 α (16)

(17)

(2 4 )

0 0 0

α

α

0 0

It was shown in the literature that the crossed field model yielded better agreement than the experiment when compared to the in-line model when K is small. In section 2, K is always smaller than 7.2%. Besides, in section stated above, the "crossed-field" model is simpler than "in-line field" model in term of equations of all element of [Y] matrix. Consequently, the "crossed-field" model is selected henceforth for the calculating, modeling the devices.

The delay line SAW device could be used for pressure sensor application. The sensitive part of this kind of device will be the propagation path. To model the pressure sensor using

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

−

ω

α

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

electrically in parallel

Appendix, section Appendix 1): - In "crossed-field" model:


SAW, it is necessary to construct the model for propagation path. Based on the equivalent circuit for electromechanical transducer presented by Mason [36], equivalent circuit of propagation path is presented as in Figure 17.

Fig. 17. Equivalent circuit of propagation path, based on Mason model Where

$$\mathcal{Y} = \frac{\alpha l}{2v} \tag{18}$$

with v is SAW velocity, l is propagation length.

### **3.4 Equivalent circuit for SAW delay line**

Due to the piezoelectric effect, an RF signal applied at input IDT stimulates a micro-acoustic wave propagating on its surface. These waves propagate in two directions, one to receiving IDT and another to the medium. The approximations as follows are assumed to construct the equivalent circuit for SAW delay line:


Based on these two approximations, the [Y] matrix representation of IDT in section 3.2, and propagation path representation in section 3.3, the SAW delay line can be expressed as equivalent circuit as in Figure 18.

Fig. 18. Equivalent circuit of SAW delay line, based on Mason model

SAW Parameters Analysis and Equivalent Circuit of SAW Device 459

*k V* = ω

11 12 ( ) ( ) ( ) *dA x j x j x jK A x jK e A x j e V dx*

12 11 ( ) () () *dA x j x j x jK e A x jK A x j e V dx*

K11 and K12 are coupling coefficients, sum of the coupling coefficient coming from the

2

piezoelectric perturbation and that coming from the mechanical perturbation.

δ

δ

is the constant associated with the convention from electrical to SAW quantities,

0

β

in the +x direction, and the magnitudes h1 and h2, respectively.

*q* δ

ββ

**4.2 Equivalent circuit for IDT based on COM theory** 

= −*k k* , with 0

Where the subscripts 1 and 2 indicate the elementary waves with wavenumbers 0 1 *k* +

1 2 11 12

1 11 12

1

From the equations (25) and (26), the particle velocities at the both ends of the IDT can be

1 2 *v h* (0) *ph q*

1 2 *v* (0) *ph h q*

1 2 ( ) *j NL j NL v NL e h e ph q V* β

*K*

11 12

ζ

ζ

( ) 1 1

 β

 δ

*<sup>p</sup> <sup>K</sup>* β δ

(x) obey the following coupled-mode equations [60]:

2

δ

<sup>2</sup> *<sup>k</sup> L* π

ζ<sup>+</sup> −− − =+ + (25)

ζ<sup>−</sup> − − = ++ (26)

2 2

, () =± + − *K K* (27)

− − <sup>=</sup> (28)

*K K* <sup>=</sup> + + (29)

*V* <sup>+</sup> =+ + (30)

*V* <sup>−</sup> = ++ (31)

ζ<sup>+</sup> <sup>−</sup> =± + + (32)

( ) 12 0 1 2 ( ) *<sup>j</sup> <sup>x</sup> <sup>j</sup> <sup>x</sup> jk x v x h e ph e q V e*

( ) 12 0 1 2 ( ) *<sup>j</sup> <sup>x</sup> <sup>j</sup> <sup>x</sup> jk x v x phe he q V e* β

 β

 β

/ *SAW* (21)

 δ ζ

 δ ζ

= (24)

βand

+ − =− − + (22)

− − + − = +− (23)

Where k is the wave number

The amplitude A+(x) and A-

ζ

0 2 *k* + β

expressed as:

+

−

Where V is the voltage applied to the IDT,

The solution to (22) and (23) can be expressed as

Section 3 gives the equivalent circuit of SAW delay line, including IDT input, IDT output and propagation path. All of calculation developments are presented in appendix, section 2. In this appendix, a new equivalent circuit of IDT including N periodic section plus one finger, which we call it "N+1/2", also are developed and presented. Another representation of SAW delay line is [ABCD] matrix representation which also proposed in appendix, section Appendix 4. [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.
