**4. Equivalent circuit for IDT based on the Coupling-Of-Mode theory**

The Coupling-Of-Modes formalism is a branch of the highly developed theory of wave propagation in periodic structure, which has an history of more than 100 years. This theory covers a variety of wave phenomena, including the diffraction of EM waves on periodic gratings, their propagation in periodic waveguides and antennas, optical and ultrasonic waves in multi-layered structures, quantum theory of electron states in metal, semiconductors, and dielectrics…. Theoretical aspects of the wave in periodic media and applications were reviewed by C.Elachi [4], in which it included theories of waves in unbounded and bounded periodic medium, boundary periodicity, source radiation in periodic media, transients in periodic structures, active and passive periodic structures, waves and particles in crystals. An excellent recent review of COM theory used in SAW devices was written by K.Hashimoto [10].

A simple equivalent circuit for IDT based on COM approach was proposed by K.Nakamura [29]. This model would be useful to analyze and design SAW devices. Based on the COM equations, the relationships between the terminal quantities at the one electrical port and two acoustic ports for an IDT have been done.

### **4.1 COM equation for particle velocities**

Consider an IDT including N periodic sections with periodic length of L as shown in Figure 19.

Fig. 19. IDT including N periodic sections

The particle velocities v+(x) and v- (x) of the wave propagating in the +x and –x directions in the periodic structure can be expressed as follows with the time dependence exp(jωt) term:

$$
\upsilon^\*(\mathbf{x}) = A^\*(\mathbf{x}) e^{-\mathsf{A}\mathbf{x}} \tag{19}
$$

$$
\upsilon^-(\mathbf{x}) = A^-(\mathbf{x})e^{i\mathbf{x}}\tag{20}
$$

Where k is the wave number

458 Acoustic Waves – From Microdevices to Helioseismology

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

The Coupling-Of-Modes formalism is a branch of the highly developed theory of wave propagation in periodic structure, which has an history of more than 100 years. This theory covers a variety of wave phenomena, including the diffraction of EM waves on periodic gratings, their propagation in periodic waveguides and antennas, optical and ultrasonic waves in multi-layered structures, quantum theory of electron states in metal, semiconductors, and dielectrics…. Theoretical aspects of the wave in periodic media and applications were reviewed by C.Elachi [4], in which it included theories of waves in unbounded and bounded periodic medium, boundary periodicity, source radiation in periodic media, transients in periodic structures, active and passive periodic structures, waves and particles in crystals. An excellent recent review of COM theory used in SAW

A simple equivalent circuit for IDT based on COM approach was proposed by K.Nakamura [29]. This model would be useful to analyze and design SAW devices. Based on the COM equations, the relationships between the terminal quantities at the one electrical port and

Consider an IDT including N periodic sections with periodic length of L as shown in Figure 19.

the periodic structure can be expressed as follows with the time dependence exp(jωt) term:

() () *jkx v x A xe* − − = (20)

(x) of the wave propagating in the +x and –x directions in

() () *jkx v x A xe* + + <sup>−</sup> = (19)

**4. Equivalent circuit for IDT based on the Coupling-Of-Mode theory** 

multiplying the matrices of elemental networks.

devices was written by K.Hashimoto [10].

two acoustic ports for an IDT have been done.

**4.1 COM equation for particle velocities** 

Fig. 19. IDT including N periodic sections

The particle velocities v+(x) and v-

$$k = \text{op} \nmid V\_{\text{SAW}} \tag{21}$$

The amplitude A+(x) and A-(x) obey the following coupled-mode equations [60]:

$$\frac{dA^{+}(\mathbf{x})}{d\mathbf{x}} = -jK\_{11}A^{+}(\mathbf{x}) - jK\_{12}e^{j2\delta x}A^{-}(\mathbf{x}) + j\mathcal{J}e^{j\delta x}V\tag{22}$$

$$\frac{dA^{-}(\mathbf{x})}{d\mathbf{x}} = jK\_{12}e^{-j2\delta\mathbf{x}}A^{\*}(\mathbf{x}) + jK\_{11}A^{-}(\mathbf{x}) - j\zeta\mathcal{E}e^{-j\delta\mathbf{x}}V\tag{23}$$

Where V is the voltage applied to the IDT,

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

K11 and K12 are coupling coefficients, sum of the coupling coefficient coming from the piezoelectric perturbation and that coming from the mechanical perturbation.

$$
\delta \mathcal{S} = k - k\_0 \quad \text{with} \quad k\_0 = \frac{2\pi}{L} \tag{24}
$$

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

$$\boldsymbol{\upsilon}^{+}\left(\mathbf{x}\right) = \left(h\_{1}e^{-j\beta\_{1}\mathbf{x}} + p\text{h}\_{2}e^{-j\beta\_{2}\mathbf{x}} + q\mathcal{L}\mathcal{V}\right)e^{-j\mathbf{k}\_{0}\mathbf{x}}\tag{25}$$

$$\boldsymbol{\sigma}^{-}\left(\mathbf{x}\right) = \left(p\mathbf{h}\_1 e^{-j\beta\_1 \mathbf{x}} + \mathbf{h}\_2 e^{-j\beta\_2 \mathbf{x}} + q\boldsymbol{\zeta}^{\prime}\boldsymbol{V}\right) e^{j\mathbf{k}\_0 \cdot \mathbf{x}}\tag{26}$$

Where the subscripts 1 and 2 indicate the elementary waves with wavenumbers 0 1 *k* + β and 0 2 *k* + βin the +x direction, and the magnitudes h1 and h2, respectively.

$$
\beta\_1 \beta\_2 = \pm \sqrt{(\delta + K\_{11})^2 - K\_{12}^{-2}} \tag{27}
$$

$$p = \frac{\beta\_1 - \delta - K\_{11}}{K\_{12}} \tag{28}$$

$$q = \frac{1}{\delta + K\_{11} + K\_{12}} \tag{29}$$

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

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

$$
\sigma^+ \text{(O)} = h\_1 + ph\_2 + q \text{(\text{\textquotedblleft}V\text{)}}\tag{30}
$$

$$\text{tr}^{-}(\mathbf{O}) = p\mathbf{h}\_1 + \mathbf{h}\_2 + q\mathbf{\tilde{\zeta}}V \tag{31}$$

$$\boldsymbol{\upsilon}^{+}\left(\mathrm{NL}\right) = \pm \left(e^{-j\beta\_{1}\mathrm{NL}}\boldsymbol{h}\_{1} + e^{j\beta\_{1}\mathrm{NL}}\boldsymbol{p}\boldsymbol{h}\_{2} + q\boldsymbol{\zeta}^{\prime}\boldsymbol{V}\right) \tag{32}$$

SAW Parameters Analysis and Equivalent Circuit of SAW Device 461

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

1 1 2 2

*p p v q <sup>F</sup> p j p j v F*

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

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

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

*p p <sup>q</sup> p j p j*

− −

In the acoustic wave transducer using piezoelectric effect, the force and voltage analogy can be used. Therefore, the COM-based circuit of IDT as matrix in (46) can be considered as the reciprocal circuit. The reciprocity theorem states that if a voltage source E acting in one branch of a network causes a current I to flow in another branch of the network, then the same voltage source E acting in the second branch would cause an identical current I to flow in the first branch. By using this theorem in this case, replacing V and F1 together, the same

(1 ) <sup>2</sup>

η

φφ

θθ

2

φ

θ

φ

0

φ = = η

1 *<sup>p</sup> <sup>Z</sup>*

θ

β

1 1 00 0 2 2

*v F j Z jZ jZ v F*

<sup>=</sup>

*j Z jZ jZ*

*<sup>T</sup> j C jZ jZ jZ I V*

 +

00 0

1 1

*p q*β

*NL j NL* 2 ζ

2 sin2 tan 2

 ζ

22 2

2 tan 2 sin2

(1 ) *<sup>p</sup> <sup>q</sup> j p* η ζ

00 0

1 1

 φ

 θ

θθ

θθ

1 1

1 tan 2 1 sin 2

θ

 η

 β

1 sin 2 1 tan 2

θ

(1 ) (1 ) ( 2) (1 ) (1 )

η

*p p j NC q NL jp jp I V*

β

+ +

*p p <sup>v</sup> <sup>q</sup> VF F*

θ β

ββ

ζ

From these equations, the matrix as follows can be obtained:

*s*

 ζη

ζ

> ζ

ω

value I requirement leads the following equations:

From (46), (47), and (48), the matrix as in (46) becomes:

ω

Where

Where

1 sin 2 1 tan 2

 θ

= *NL* /2 (44)

≡ =− 1 2 (45)

 θ (46)

(49)

 θ

<sup>+</sup> <sup>=</sup> − (47)

= 2 *j* (48)

<sup>−</sup> = = + (50)

*C NC T s* = (52)

(51)

= + − − (43)

*p j p j*

 β

θ

$$\boldsymbol{\upsilon}^{-}\left(\mathrm{NL}\right) = \pm \left(e^{-j\beta\_{1}\mathrm{NL}}\,\mathrm{pl}\mathbf{i}\_{1} + e^{j\beta\_{1}\mathrm{NL}}\,\mathrm{il}\_{2} + q\zeta'\boldsymbol{V}\right) \tag{33}$$

The upper and lower signs in (32) and (33) correspond to the cases N=i and N=i+0.5, respectively, where i is an integer. Consequently, the total particle velocities at the two acoustical ports can be expressed as:


$$
\upsilon\_1 = \upsilon^+(0) + \upsilon^-(0) = (1+p)(h\_1 + h\_2) + 2q\zeta'V \tag{34}
$$


$$v\_2 = -\left[\upsilon^+(NL) + \upsilon^-(NL)\right] = \mp \left[ (1+p)(e^{-j\beta\_1 \mathcal{M}} h\_1 + e^{j\beta\_1 \mathcal{M}} h\_2) + 2q\zeta V\right] \tag{35}$$

The two forces at two acoustic ports are considered to be proportional to the difference of v+ and v- . For the simplicity, these forces can be expressed as follows:

$$F\_1 = \upsilon^+(0) - \upsilon^-(0) = (1 - p)(h\_1 - h\_2) \tag{36}$$

$$F\_2 = \upsilon^+ (\text{NL}) - \upsilon^- (\text{NL}) = \pm \left[ (1 - p)(e^{-j\beta\_1 \text{NL}} h\_1 - e^{j\beta\_1 \text{NL}} h\_2) \right] \tag{37}$$

From these equations, h1 and h2 are the terms of F1 and F2 as follows:

$$\mathcal{H}\_1 = \frac{e^{j2\beta\_1 \mathcal{N} \mathcal{L}}}{(1 - p)(e^{j2\beta\_1 \mathcal{N} \mathcal{L}} - 1)} F\_1 \mp \frac{e^{j\beta\_1 \mathcal{N} \mathcal{L}}}{(1 - p)(e^{j2\beta\_1 \mathcal{N} \mathcal{L}} - 1)} F\_2 \tag{38}$$

$$h\_2 = \frac{1}{(1-p)(e^{j\frac{2\beta\_1 \text{NL}}}-1)} F\_1 \mp \frac{e^{j\beta\_1 \text{NL}}}{(1-p)(e^{j\frac{2\beta\_1 \text{NL}}}-1)} F\_2 \tag{39}$$

The current I at the electrical ports can be expressed as:

$$\begin{split} I &= \eta \int\_{0}^{\mathrm{NL}} \left[ (1+p)(h\_{1}e^{-j\beta\_{1}x} + h\_{2}e^{-j\beta\_{2}x}) + 2q\zeta V \right] dx + j\alpha \mathrm{NC}\_{s}V\\ &= j\eta \left\{ (1+p) \left[ \frac{h\_{1}}{\beta\_{1}}(e^{-j\beta\_{1}\mathrm{NL}} - 1) + \frac{h\_{2}}{\beta\_{2}}(e^{j\beta\_{1}\mathrm{NL}} - 1) \right] \right\} + 2q\zeta\eta \mathrm{NL}V + j\alpha \mathrm{NC}\_{s}V \end{split} \tag{40}$$

where η is the constant associated with the convention from SAW to electrical quantities, therefore associated with the coupling factor K.

Cs is the capacitance for one electrode pair.

By substituting equations (38) and (39) in (34), (35) and (40), the following equations can be obtained:

$$I = (j \alpha \text{NC}\_s + 2q \zeta \text{\(\gamma\)} V + \frac{\eta(1+p)}{j\beta(1-p)} F\_1 \mp \frac{\eta(1+p)}{j\beta(1-p)} F\_2 \tag{41}$$

$$v\_1 = 2\eta \mathcal{L}V + \frac{1+p}{1-p} \frac{1}{j \tan 2\theta} F\_1 \mp \frac{1+p}{1-p} \frac{1}{j \sin 2\theta} F\_2 \tag{42}$$

$$v\_2 = \mp 2q\zeta' V \mp \frac{1+p}{1-p} \frac{1}{j \sin 2\theta} F\_1 + \frac{1+p}{1-p} \frac{1}{j \tan 2\theta} F\_2 \tag{43}$$

Where

460 Acoustic Waves – From Microdevices to Helioseismology

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

The upper and lower signs in (32) and (33) correspond to the cases N=i and N=i+0.5, respectively, where i is an integer. Consequently, the total particle velocities at the two

1 1 <sup>2</sup> *vv v* (0) (0) (1 )( ) 2 *p h h q*

2 1 <sup>2</sup> ( ) ( ) (1 )( ) 2 *j NL j NL v v NL v NL p e he h q V*

The two forces at two acoustic ports are considered to be proportional to the difference of v+

2 1 <sup>2</sup> ( ) ( ) (1 )( ) *j NL j NL F v NL v NL p e he h* + − <sup>−</sup>

1 1 1 1

<sup>112</sup> 2 2 (1 )( 1) (1 )( 1) *j NL j NL j NL j NL e e h FF p e p e*

1 1 2 12 2 2

(1 )( 1) (1 )( 1)

*h h j p e e <sup>q</sup> NLV <sup>j</sup> NC V*

By substituting equations (38) and (39) in (34), (35) and (40), the following equations can be

(1 ) (1 ) ( 2) (1 ) (1 ) *<sup>s</sup> p p <sup>I</sup> <sup>j</sup> NC <sup>q</sup> NL V <sup>F</sup> <sup>F</sup>*

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

+ +

θ

*p p <sup>v</sup> <sup>q</sup> VFF p j p j*

η

β

1 tan 2 1 sin 2

 β

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

 ζ

*j NL j NL <sup>e</sup> h FF p e p e*

. For the simplicity, these forces can be expressed as follows:

From these equations, h1 and h2 are the terms of F1 and F2 as follows:

2

β

1

The current I at the electrical ports can be expressed as:

β

ω

ζ

therefore associated with the coupling factor K. Cs is the capacitance for one electrode pair.

0

η

η

*NL*

β

β

1 2

*I p h e h e q V dx j NC V*

 β

*jx jx*

1 2 1 2

> ζη

1 2

β

−

β

(1 )( ) 2

− −

=+ + + + 

1 1

*j NL j NL*

 β

= + −+ − + +

(1 ) ( 1) ( 1) 2

acoustical ports can be expressed as: - Particle velocity at port 1 (x=0):


and v-

where η

obtained:

( ) 1 1

 β

ζ<sup>−</sup> <sup>−</sup> =± + + (33)

ζ*V* + − = + =+ + + (34)

> β

1 1 <sup>2</sup> *Fv v* (0) (0) (1 )( ) *p h h* + − = − =− − (36)

1 1

= − =± − − (37)

 β

 β<sup>=</sup> − −− − (38)

1

β

*j NL*

 β<sup>=</sup> − −− − (39)

ω

*s*

 ζη

1 2

 θ

 η

> β

= + − − (42)

*jpjp*

+ + =+ + − − (41)

*s*

(40)

ω

 β ζ

1 1

β

+ − <sup>−</sup> =− + = + + + (35)

β

$$
\theta = \beta \text{NL} \mid 2 \tag{44}
$$

$$
\beta \equiv \beta\_1 = -\beta\_2 \tag{45}
$$

From these equations, the matrix as follows can be obtained:

$$
\begin{bmatrix} I \\ v\_1 \\ v\_2 \end{bmatrix} = \begin{bmatrix} j\rho \text{NC}\_s + 2\eta \zeta \eta \text{NL} \end{bmatrix} \qquad \begin{array}{c} \eta (1+p) \\ \overline{j\beta (1-p)} \qquad \mp \frac{\eta (1+p)}{j\beta (1-p)} \\\ 2q\zeta \qquad \qquad \frac{1+p}{1-p} \frac{1}{j\tan 2\theta} \quad \mp \frac{1+p}{1-p} \frac{1}{j\sin 2\theta} \begin{bmatrix} V \\ F\_1 \\ F\_2 \end{bmatrix} \\\ \mp 2q\zeta \qquad \qquad \mp \frac{1+p}{1-p} \frac{1}{j\sin 2\theta} \quad \frac{1+p}{1-p} \frac{1}{j\tan 2\theta} \end{bmatrix} \tag{46}
$$

In the acoustic wave transducer using piezoelectric effect, the force and voltage analogy can be used. Therefore, the COM-based circuit of IDT as matrix in (46) can be considered as the reciprocal circuit. The reciprocity theorem states that if a voltage source E acting in one branch of a network causes a current I to flow in another branch of the network, then the same voltage source E acting in the second branch would cause an identical current I to flow in the first branch. By using this theorem in this case, replacing V and F1 together, the same value I requirement leads the following equations:

$$2q\zeta = \frac{\eta(1+p)}{j\beta(1-p)}\tag{47}$$

$$\text{l.} \eta = \text{2} \,\text{j}\mathcal{Z} \tag{48}$$

From (46), (47), and (48), the matrix as in (46) becomes:

$$
\begin{bmatrix} I \\ v\_1 \\ v\_2 \end{bmatrix} = \begin{bmatrix} j\alpha\mathbb{C}\_{\mathbb{T}} + \frac{\phi^2}{j2\theta\mathbb{Z}\_0} & \frac{\phi}{j2\theta\mathbb{Z}\_0} & \mp \frac{\phi}{j2\theta\mathbb{Z}\_0} \\ \frac{\phi}{j2\theta\mathbb{Z}\_0} & \frac{1}{j\mathbb{Z}\_0\tan 2\theta} & \mp \frac{1}{j\mathbb{Z}\_0\sin 2\theta} \\ \mp \frac{\phi}{j2\theta\mathbb{Z}\_0} & \mp \frac{1}{j\mathbb{Z}\_0\sin 2\theta} & \frac{1}{j\mathbb{Z}\_0\tan 2\theta} \end{bmatrix} \begin{bmatrix} V \\ F\_1 \\ F\_2 \end{bmatrix} \tag{49}
$$

Where

$$Z\_0 = \frac{1-p}{1+p} = \frac{1}{q\beta} \tag{50}$$

$$
\phi = \eta \text{NL} = \text{2 j\"\" NL} \tag{51}
$$

$$\mathbf{C}\_{T} = \mathbf{N} \mathbf{C}\_{s} \tag{52}$$

SAW Parameters Analysis and Equivalent Circuit of SAW Device 463

Then, by expressing h1 and h2 in terms of F1 and F2 based on equations (53) and (54), the v1

*e e vhh F F e e*

+

21 2 2 2 1 2

Using the relation between complex number and trigonometry, the v1 and v2 can be

1 12 ' ' '' 0 0 1 1 tan 2 sin 2 *v FF*

2 12 '' ' ' 0 0 1 1 sin 2 tan 2 *v FF*

Consequently, the equivalent circuit for propagation path can be represented by the π-

Based on Mason model, the equivalent circuit of propagation path was presented in Figure 17, which has star form. In Figure 21, the circuit has triangle form. By using triangles and stars transformation theory published by A.E. Kennelly, equivalent circuit of propagation in these two figures is the same. Consequently, the approachs that are based on Mason model

Based on section 4.2 and 4.3, equivalent circuit of SAW delay line based on COM theory is

In this model, some parameters must to be calculated or extracted. SAW velocity v, piezoelectric coupling factor K could be calculated from section 2. The periodic length L (or

The parameters K11 and K12 are coupling coefficients. They are sum of the coupling coefficient coming from the piezoelectric perturbation and that coming from the mechanical perturbation, and their equations for calculation are complicated [55]. Exact equations for

θ

θ

β

<sup>−</sup> +

*e e v he he F F*

1 2 1 1

 β

2

 β

 β

 θ

 θ*jZ* = − <sup>+</sup> (64)

= += − − − (62)

*jZ* = − (63)

=+= − − − (61)

 β

2 1 1 1

*jl j l*

*j l j l*

*e e* β

*j l j l j l j l*

2 112 2 2 1 2

*jkl jkl*

*jZ*

*jZ*

Fig. 21. Equivalent circuit of propagation path based on COM theory

and COM theory can get the same equivalent circuit of propagation path.

**4.4 Equivalent circuit for SAW delay line based on COM theory** 

wavelength λ) is determined by design and fabrication.

β

β

and v2 become as follows:

expressed as follows:

circuit of Figure 21:

presented in Figure 22.

<sup>2</sup> *F v NL v NL* () () + − = − (60)

Consequently, the simple equivalent circuit obtained for IDT with N electrode pairs is shown in Figure 20:

Fig. 20. Equivalent circuit IDT based on COM theory

### **4.3 Equivalent circuit for propagation path based on COM theory**

In SAW devices, the propagation path should be taken into account. It is necessary to determine the equivalent circuit for a propagation path of distance l between 2 IDTs. This propagation path is a uniform section of length l, with a free surface or a uniformly metallized surface. In this case, K11=K12=0, and β=δ.

Consequently, from equations (38) and (39), h1 and h2 can be expressed as:

$$h\_1 = \frac{e^{j^{12}\beta l}}{e^{j^{12}\beta l} - 1} F\_1 - \frac{e^{j\beta l}}{e^{j^{12}\beta l} - 1} F\_2 \tag{53}$$

$$\mathcal{H}\_2 = \frac{1}{e^{j2\beta l} - 1} F\_1 - \frac{e^{j\beta l}}{e^{j2\beta l} - 1} F\_2 \tag{54}$$

And, the particle velocities are expressed as:

$$
\upsilon^{+}(\mathbf{x}) = \hbar\_1 e^{-j(k\_0 + \beta\_1)\mathbf{x}} = \hbar\_1 e^{-j\mathbf{k}\mathbf{x}} \tag{55}
$$

$$\text{tr}^{-}\left(\mathbf{x}\right) = \text{h}\_{2}e^{j(\mathbf{k}\_{0}+\beta\_{1})\mathbf{x}} = \text{h}\_{2}e^{j\mathbf{k}\mathbf{x}}\tag{56}$$

If the v1, v2, F1, and F2 are defined as:

$$v\_1 = v^\*(0) + v^-(0)\tag{57}$$

$$v\_2 = -[v^+(NL) + v^-(NL)]\tag{58}$$

$$F\_1 = v^+(0) - v^-(0) \tag{59}$$

Consequently, the simple equivalent circuit obtained for IDT with N electrode pairs is

shown in Figure 20:

Fig. 20. Equivalent circuit IDT based on COM theory

metallized surface. In this case, K11=K12=0, and β=δ.

And, the particle velocities are expressed as:

If the v1, v2, F1, and F2 are defined as:

**4.3 Equivalent circuit for propagation path based on COM theory** 

Consequently, from equations (38) and (39), h1 and h2 can be expressed as:

2

β

1

β

In SAW devices, the propagation path should be taken into account. It is necessary to determine the equivalent circuit for a propagation path of distance l between 2 IDTs. This propagation path is a uniform section of length l, with a free surface or a uniformly

> <sup>112</sup> 2 2 1 1 *j l j l j l j l e e h FF e e* β

2 12 2 2

0 1 ( ) 1 1 ( ) *<sup>j</sup> k x jkx v x he he* <sup>+</sup> −+ − β

0 1 ( ) 2 2 ( ) *j k x jkx v x he he* <sup>−</sup> <sup>+</sup>β

<sup>1</sup> *vv v* (0) (0)

<sup>1</sup> *Fv v* (0) (0)

*j l j l <sup>e</sup> h FF e e*

 β

*j l*

β

= − − − (53)

= − − − (54)

= = (55)

= = (56)

+ − = + (57)

+ − = − (59)

<sup>2</sup> *v v NL v NL* [ ( ) ( )] + − =− + (58)

 β

 β

1 1

$$F\_2 = \upsilon^+ \text{(NL)} - \upsilon^- \text{(NL)}\tag{60}$$

Then, by expressing h1 and h2 in terms of F1 and F2 based on equations (53) and (54), the v1 and v2 become as follows:

$$\mathbf{h}\mathbf{v}\_1 = h\_1 + h\_2 = \frac{e^{j^2\beta l} + 1}{e^{j^2\beta l} - 1} F\_1 - \frac{2e^{j\beta l}}{e^{j^2\beta l} - 1} F\_2 \tag{61}$$

$$\upsilon v\_2 = h\_1 e^{-j\ell l} + h\_2 e^{j\ell l} = \frac{2e^{j\beta l}}{e^{j^2 \beta l} - 1} F\_1 - \frac{e^{j^2 \beta l} + 1}{e^{j^2 \beta l} - 1} F\_2 \tag{62}$$

Using the relation between complex number and trigonometry, the v1 and v2 can be expressed as follows:

$$w\_1 = \frac{1}{jZ\_0'\tan 2\theta} F\_1 - \frac{1}{jZ\_0'\sin 2\theta} F\_2 \tag{63}$$

$$v\_2 = -\frac{1}{jZ\_0^{'}\sin 2\theta^{'}}F\_1 + \frac{1}{jZ\_0^{'}\tan 2\theta^{'}}F\_2\tag{64}$$

Consequently, the equivalent circuit for propagation path can be represented by the πcircuit of Figure 21:

Fig. 21. Equivalent circuit of propagation path based on COM theory

Based on Mason model, the equivalent circuit of propagation path was presented in Figure 17, which has star form. In Figure 21, the circuit has triangle form. By using triangles and stars transformation theory published by A.E. Kennelly, equivalent circuit of propagation in these two figures is the same. Consequently, the approachs that are based on Mason model and COM theory can get the same equivalent circuit of propagation path.

### **4.4 Equivalent circuit for SAW delay line based on COM theory**

Based on section 4.2 and 4.3, equivalent circuit of SAW delay line based on COM theory is presented in Figure 22.

In this model, some parameters must to be calculated or extracted. SAW velocity v, piezoelectric coupling factor K could be calculated from section 2. The periodic length L (or wavelength λ) is determined by design and fabrication.

The parameters K11 and K12 are coupling coefficients. They are sum of the coupling coefficient coming from the piezoelectric perturbation and that coming from the mechanical perturbation, and their equations for calculation are complicated [55]. Exact equations for

SAW Parameters Analysis and Equivalent Circuit of SAW Device 465

Figure 24 shows the effects of O11 on S21(dB) of SAW device N=50, vSAW=5120m/s, λ=8μm, K=0.066453 when O12=0. So, O11 coefficient shifts the center frequency of SAW device, the positive value of O11 reduces the center frequency f0 of device, the negative on will increase the f0.

Fig. 24. Effect of O11 on S21(dB), N=50, vSAW=5120m/s, λ=8μm, K=0.066453, O12=0

its effect can be ignored. In conclusion, in our work, value of K11 and K12 are 0.

Fig. 25. Comparison between Hydrid model and COM model (O11=O12=0)

**and COM thoery** 

The effect of K11 and K12 could be explained by their measurement method [61]. K11 could be derived from the measurement of frequency response, therefore the usefulness of its calculation could be limited. Meanwhile, K12 can be extracted from FEM. It is shown in literature that K12 depends on the thickness of finger with respect to the wavelength. In our work, the ratio thickness/wavelength (its maximum value is 300nm/8μm) is too small that

**5. Comparison of equivalent circuit of SAW device based on Mason model** 

Figure 25 presents the comparison between hybrid model and COM model in that O11= O12=0, distance between 2 IDTs is 50λ. These models could be the same, except that a

≈1.58 dB

K11 and K12 were given by Y.Suzuki et al [55], but it seems so complex that their usefulnesses could be limited. However, from this work of Y.Suzuki et al [55], we propose the K11 and K12 could be expressed as follows:

$$\mathbf{K}\_{\rm II} = \mathbf{O}\_{\rm II} \mathbf{K}^2 \mathbf{k}\_{\rm I} \tag{65}$$

$$\mathbf{K}\_{12} = \mathbf{O}\_{12} \mathbf{K}^2 \mathbf{k}\_0 \tag{66}$$

Where k0 is stated by (24) 0 <sup>2</sup> *<sup>k</sup> L* π= and K is piezoelectric coupling factor.

O11 is so-called self-coupling constant of finger, and O12 is so-called coupling constant between fingers. O12 could also presents the reflective wave between two fingers.

Fig. 22. Equivalent circuit of SAW delay line based on COM theory

Fig. 23. Effect of O12 on S21(dB), N=50, vSAW=5120m/s, λ=8μm, K=0.066453, O11=0

Figure 23 shows the effects of O12 on S21(dB) of SAW device N=50, vSAW=5120m/s, λ=8μm, K=0.066453 when O11=0. S21 is the transmission coefficient in the scattering matrix representation [28].

K11 and K12 were given by Y.Suzuki et al [55], but it seems so complex that their usefulnesses could be limited. However, from this work of Y.Suzuki et al [55], we propose the K11 and K12

 K11= O11K2k0 (65) K12= O12K2k0 (66)

O11 is so-called self-coupling constant of finger, and O12 is so-called coupling constant

between fingers. O12 could also presents the reflective wave between two fingers.

= and K is piezoelectric coupling factor.

could be expressed as follows:

Where k0 is stated by (24) 0

representation [28].

(a)

<sup>2</sup> *<sup>k</sup> L* π

Fig. 22. Equivalent circuit of SAW delay line based on COM theory

Fig. 23. Effect of O12 on S21(dB), N=50, vSAW=5120m/s, λ=8μm, K=0.066453, O11=0

Figure 23 shows the effects of O12 on S21(dB) of SAW device N=50, vSAW=5120m/s, λ=8μm, K=0.066453 when O11=0. S21 is the transmission coefficient in the scattering matrix Figure 24 shows the effects of O11 on S21(dB) of SAW device N=50, vSAW=5120m/s, λ=8μm, K=0.066453 when O12=0. So, O11 coefficient shifts the center frequency of SAW device, the positive value of O11 reduces the center frequency f0 of device, the negative on will increase the f0.

Fig. 24. Effect of O11 on S21(dB), N=50, vSAW=5120m/s, λ=8μm, K=0.066453, O12=0

The effect of K11 and K12 could be explained by their measurement method [61]. K11 could be derived from the measurement of frequency response, therefore the usefulness of its calculation could be limited. Meanwhile, K12 can be extracted from FEM. It is shown in literature that K12 depends on the thickness of finger with respect to the wavelength. In our work, the ratio thickness/wavelength (its maximum value is 300nm/8μm) is too small that its effect can be ignored. In conclusion, in our work, value of K11 and K12 are 0.

## **5. Comparison of equivalent circuit of SAW device based on Mason model and COM thoery**

Figure 25 presents the comparison between hybrid model and COM model in that O11= O12=0, distance between 2 IDTs is 50λ. These models could be the same, except that a

Fig. 25. Comparison between Hydrid model and COM model (O11=O12=0)

SAW Parameters Analysis and Equivalent Circuit of SAW Device 467

Fig. Appendix.2. Mason equivalent circuit for one periodic section in "in-line field" model

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

1 11 12 13 1 2 21 22 23 2 3 31 32 33 3

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.

(Appendix.2)

(Appendix.3)

(Appendix.1)

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

*i yyye i yyye i yyye* <sup>=</sup>

One periodic section can be expressed by the 3-port network as follows:

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

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

*i y y ye i y yye i y yye* =− − <sup>−</sup>

*i yy ye i y y ye i y yye* = − <sup>−</sup>

small difference in the peak value of S21 (dB) occurs. This difference could be explained by using "crossed-filed" model instead of actual model as in Figure 13.
