Chapter-1

Introduction

1.1 Overview

Electromagnetic waves find such extensive use in the field of electronics, that the term

“wave”, in a general context, refers to an electromagnetic wave. However, the very high velocity

with which an electromagnetic wave travels results in large wavelengths. Hence, construction of

devices whose dimensions are of the order of a wavelength becomes increasingly difficult. On the

other hand, waves such as acoustic waves, by virtue of their relatively less velocity, have smaller

wavelengths for a given frequency. This property can be extensively exploited in the construction

of micron sized devices which can operate at frequencies of the order of GHz.

A Surface Acoustic Wave (SAW) is an acoustic wave, travelling along the surface of a

material exhibiting elasticity, with displacement amplitude that typically decays exponentially

with depth into the substrate, so that they are confined to within roughly one wavelength. A

mechanical disturbance created in an elastic medium travels along it in the form of an elastic wave.

Such mechanical disturbances can be created in piezoelectric materials through an electrical

excitation. Hence, electrical energy given at the input port of a piezoelectric material will then get

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converted into acoustic energy and propagate as an acoustic wave throughout the length of the

material. It is finally reconverted into its electrical form, by means of suitable transducers. Hence,

the benefits offered by acoustic waves by virtue of their shorter wavelength are exploited, while

the input given to the input transducer and output taken from the output transducer remain in

electrical form.

Surface Acoustic Wave (SAW) Devices involve the conversion of electrical energy into

acoustic energy and of acoustic energy to electrical form. However, the representation of both

forms of the signal in a single form allows the comparison of the properties of the two signals,

thereby enabling the analysis of the system performance. Representation of both electrical and

acoustic signals in the electrical form is a better option than their representation in acoustic form,

as the former gives wider insights into the electrical characteristics of the model. Also, the

representation of both the electrical and acoustic signals in electrical form provides for the

simultaneous viewing of the signals in both forms and thereby, facilitates the comparison of the

signal characteristics like amplitude and phase, before and after the energy conversion. This marks

the necessity for the development of an equivalent model, which represents the acoustic signal

generated, in its equivalent electrical terms. Equivalent circuit models of SAW devices are used

for such representation of the mechanical signals generated, in terms of an electrical signal, such

that the characteristics of the former are adequately accounted for by the latter. However, these

models can produce accurate results only over a fixed range of the device’s mechanical properties

such as the electromechanical coupling coefficient, k2.

1.2 Objective

The conversion of electrical energy into acoustic energy at the input, and from acoustic

form to electrical form at the output of a SAW device is accomplished by the use of Inter Digitated

Transducers (IDTs). Hence, the primary goal of an equivalent model is the development of an

electrical circuit, whose output is similar in characteristics, to the acoustic signal generated by the

IDTs. The equivalent models, hence, aim at the development of electrical circuits, whose

functionalities most closely resemble that of an IDT, and using this circuit as the basic building

block in developing the SAW device model.

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The objective of the project is to develop the two variants of the Mason’s Equivalent circuit

model for SAW device, the Crossed-Field Model and the In-Line Field Model. The energy

conversion efficiency of each of these models is determined for different values of the mechanical

parameters of the material with which the SAW device is made. The range of values of k2 for

which each of these models gives accurate duplication of the results of the original model, is to be

determined, so that a judgment on the particular model to be used for a given material can be made,

based on the results.

1.3 Problem Statement

a. To develop P-Spice models of the two variants of the Mason’s equivalent circuit model,

namely the Crossed-Field model and the In-Line field model.

b. To calculate the energy conversion efficiency of each of these models for different values

of the electromechanical coupling coefficient, k2, by comparing the values of k2 used in the

development of the model and those obtained through simulation.

c. To determine the range of values of k2 over which each of these models produces accurate

results and to thereby, suggest the suitable equivalent model, for a given material.

4

Chapter-2

Literature review

2.1 SAW Propagation

A travelling longitudinal acoustic wave is associated with an electric field, which is set up

along the direction of propagation of the wave, with the field intensity characterized by the acoustic

intensity and the wave frequency [1]. Propagation of an acoustic wave through a piezoelectric

material results in an AC electric field. A phase difference between the field developed and the

wave results and energy is transferred from the wave to the electrons, thereby causing attenuation

of the wave. A Surface Acoustic Wave Hybrid System combines the electrical properties of

piezoelectric materials and semiconductors, making possible, an external bias controlled

attenuation and velocity of the Surface Acoustic wave [2].

A Quasi 2-Dimensional Quantum well is formed on the surface of the piezoelectric, the

conductivity of which is governed by the electric boundary condition [3]. A conductive surface on

the piezoelectric material results in attenuation of the wave and an associated decrease in its

velocity [4]. The attenuation and the velocity change can be made adjustable, by means of

dynamically varying the conductivity of the 2-D electron system on the surface of the piezoelectric.

5

Hence, a voltage-controlled Surface Acoustic Wave Hybrid device can be constructed, with the

conductivity of the surface adjusted by means of an external bias voltage [5].

2.2 Equivalent Circuit models of SAW devices

Equivalent circuit models of SAW devices need to be developed, so that the acoustic

signals can be expressed in terms of an electrical signal, whose characteristics such as amplitude

and phase provide insights into those of the acoustic signal. Thus, direct comparison of an acoustic

signal with an electrical signal is possible, while the characteristics of the original signal are

retained. In this regard, several equivalent circuit models have been developed, each with its own

benefits and limitations [6]. However, all these models are similar to each other in that the ultimate

goal of any model is the electrical equivalent of an IDT finger, several of which constitute to an

IDT. The input and output IDTs, along with the electrical equivalent of a sensitive propagation

path, form the electrical model for a SAW device [7].

Several models have been devised and developed for modelling and analyzing SAW

devices, each with its own benefits and limitations. Widely known methods include the Impulse

model, Mason’s Equivalent Circuit Model, Coupling-of-Mode model, P-Matrix model, Angular

Spectrum of Waves model and Scattering Matrix Approach [8]. Mason’s equivalent circuit model

has two variants: the In-line field model and the Crossed-field model, with the difference between

the two being that the crossed-field model is more suited for SAW devices made of materials with

small and moderate values of k2, whereas the in-line field model is more suited for high values of

k2. All the models listed above, with the exception of the impulse model, include second order

effects such as reflection, dispersion, charge distribution etc. and are hence, widely used for the

modelling of SAW devices [9] [10].

2.2.1 Advantages of Equivalent Models

Equivalent circuits find extensive use in Electronics and are frequently used for the analysis

of electrical circuits. The following advantages of using equivalent circuit models for the electrical

form representation of signals in other forms underline their significance [11].

1. A very powerful set of intellectual tools are available for the understanding of electrical

circuits. Hence, this knowledge can be used for the better analysis and understanding of

devices and systems in which the signals of other forms are involved. By developing

6

electrical equivalents of these systems, their behavior can be understood by making use of

the tools available for electrical circuits.

2. The equivalent circuit approach has distinct advantages over the direct wave equations

approach, which is based on physical or chemical equations analysis, in that the electrical

equivalent circuits give homogeneity to the analysis techniques.

3. Electrical circuit approach is intrinsically correct form an energy point of view.

4. A further advantage of electrical circuit model is that it permits efficient modelling of the

interaction between the electric and non-electric components of a system, as both the

electrical and mechanical portions of a system are represented by the same means.

2.2.2 Challenges involved in using equivalent models

1. Care must be taken to make sure whether the boundary conditions are compatible with

those used in the original system.

2. All the parameters of the original model have to be adequately accounted for and their

significance in the original model has to be accurately replicated by the equivalent model.

3. All sensitive elements in the original model are to be identified and means to equivalently

represent these elements have to be developed.

2.3 SAW Parameters:

2.3.1 Center frequency:

The center frequency of SAW devices is determined by the period of IDT fingers and the

acoustic velocity [12]. The equation that determines the operation frequency is

𝑓𝑜 =𝑣𝑆𝐴𝑊

𝜆 (2.1)

where is the wavelength, determined by the periodicity of the IDT and vSAW is the acoustic wave

velocity. In general,

= p = finger width*4 (2.2)

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2.3.2 Electromechanical Coupling Coefficient, k2

The measure of the efficiency of a given piezoelectric substance in converting a given electric

signal into mechanical energy associated with a Surface Acoustic Wave is called the

Electromechanical Coupling Coefficient, K2 or k2.

K2= (Stored mechanical energy/ input electrical energy) (2.3)

The value of k2 can be determined as the ratio of the acoustic energy at the output of the input

transducer to the electrical energy at the input of the input transducer. However, it can also be

determined as the ratio of the electrical energy at the output of the output transducer and the

acoustic energy at the input of the output transducer [13]. In either case, this ratio of energies can

be equivalently calculated as the ratio of the squares of the respective amplitudes. Also, if the

propagation path is designed not to be sensitive, the electromechanical coupling coefficient can

directly be determined as the ratio of the output and input electrical signal amplitudes.

The electromechanical coupling coefficient can also be calculated by determining the

velocities in case of a free surface and of a short-circuit surface. The value of k2 can be determined

as

𝐾2 = 2𝑉𝑜−𝑉𝑠

𝑉𝑜 (2.4)

where Vo is the wave velocity in case of a free surface and Vs, in the case of a short-circuited

surface.

8

Chapter-3

Surface Acoustic Waves

3.1 Generation of Surface Acoustic Waves

Surface Acoustic Waves or Rayleigh waves have a longitudinal shear and a vertical shear

component that can couple with any media in contact with the surface. The propagation of a SAW

on an unbounded elastic surface is associated mechanically with a time-dependent elliptical

displacement of the surface structure. One component of this physical displacement is parallel to

the SAW propagation axis, while the other is normal to the surface. Surface particle motion is

predominantly in the plane containing these two axes, with the wave motions being 900 out of

phase with one another in the time domain, making the displacement component maximum at a

given instant and zero at the other. The amplitude of the surface wave is small compared with the

wavelength and decreases exponentially with the distance from the surface. The penetration depth

of the wave into the substrate varies inversely with frequency.

In engineering applications, surface acoustic waves are generated via electromechanical or

magneto-mechanical transduction using piezoelectric and magneto-restrictive materials. The most

9

efficient way of generating and detecting SAWs on a piezoelectric surface is through the use of

Inter Digital Transducers (IDTs).

Fig.3.1: Shear components in Rayleigh waves

. An Inter Digital Transducer is a device that consists of two inter-locking comb-shaped

arrays of metallic electrodes. SAWs generated using IDTs can be used in the design of analog

electrical filters operating at selected frequencies between 10 MHz and 1 GHz or above.

Fig-3.2 Basic two-port SAW Device, fabricated on a piezoelectric substrate

An associated electrical signal exists for a SAW on a piezoelectric substrate, which allows

electrostatic coupling via a transducer. The wave can be electro-acoustically accessed and tapped

at the substrate surface and its velocity is approximately 104 times less than that of electromagnetic

waves. The SAW wavelength is of the same order of magnitude as line dimensions produced by

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photolithography and the lengths for both short and long delays are achievable on reasonably small

sized substrates.

Piezoelectric materials generate an electric charge in response to applied mechanical stress.

The extent to which a piezoelectric substance can cause an energy conversion characterizes the

material. The measure of the efficiency of a given piezoelectric substance in converting a given

electric signal into mechanical energy associated with a Surface Acoustic Wave is called the

Electromechanical Coupling Coefficient, K2.

K2= (Stored mechanical energy/ input electrical energy) (3.1-a)

or

K2= (Stored electrical energy/ input mechanical energy) (3.1-b)

The electromechanical coefficients of the same material can be changed by changing the

orientation of the device, by way of changing the crystal cut. The tabulation, Table-3.1 gives the

electromechanical coupling coefficients, free surface velocities and metallized surface velocities

for surface acoustic waves in LiNbO3 for various cuts and propagation directions.

Table-3.1: K2 values for surface acoustic waves in LiNbO3 for various cuts and propagation

directions

3.2 Propagation of Surface Acoustic Waves:

A travelling longitudinal acoustic wave passing through an insulated solid conductor gives

rise to an electric field along the direction of propagation of the wave, which is proportional to the

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acoustic intensity and the frequency of the wave. The travelling longitudinal acoustic wave,

characterized by the wave vector ‘q’, will interact with the conduction electrons, giving rise to

absorption and induced emission, involving momentum transfer to an electron. The electrons

transfer the net momentum thus gained, in part to the thermal lattice vibrations, but an unbalance

of the thermal equilibrium remains, which leaves a predominant forward motion of the electrons.

If the material is electrically insulated, charges accumulate at its boundaries, giving rise to an

electric field, which counterbalances the forward motion.

When an acoustic wave travels through a piezoelectric material, an AC electric field is set

up, due to the periodically strained regions. Conduction electrons react to this, leading to a spatial

redistribution of the carriers. Electrons bunch up at the minima of the potential energy, which are

periodic due to the acoustic wave. Due to a difference in velocity between electrons and acoustic

wave, there is a phase difference between the field and the wave, which causes an energy transfer

from the wave to the electrons and a resultant attenuation of the acoustic wave.

The combination of the electronic properties of semiconductor hetero-junctions and the

acoustic properties of piezoelectric materials yields Surface Acoustic Wave Hybrid Systems. A

2-dimensional quantum well is formed on the piezoelectric material, whose conductivity can be

controlled by the bias of the semiconductor material. The semiconductor and the piezoelectric

hence form a system, whose effective conductivity is controlled by both semiconductor and

piezoelectric properties. Conductivity of the 2-dimensional electron system in the quantum well,

which can be modified via the field effect, modifies the velocity of the SAW.

An elastic wave is known to propagate as a stiffened elastic wave or electro-elastic wave

for most directions in a piezoelectric crystal and the phase velocity of the stiffened wave is greater

than the velocity of the unstiffened wave in the same direction. The effect which causes this

velocity increase is called Piezoelectric Stiffening which also influences the orientation of the

unstiffened normal mode axes. The propagation velocity of the Surface Acoustic Wave on strong

piezoelectric materials greatly depends on the electric boundary condition: a conductive surface

prohibits the effect of piezoelectric stiffening, thus reducing the velocity of the Surface Acoustic

Wave.

The interaction of a SAW and Quasi 2-Dimensional Electron System (Q2DES) on the

surface of a piezoelectric substance results in a change of both the phase velocity and attenuation.

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Both phase velocity and attenuation change with the electromechanical coupling coefficient, Keff2,

which is very small in normal semiconductor materials. Hence, the maximum change in phase

velocity and attenuation are small. A combination of the electric properties of GaAs and a strong

piezoelectric such as LiTaO3 or LiNbO3 solves this problem.

Controlling the carrier concentration in the semiconductor using external bias and thereby,

the conductivity, the propagation velocity and attenuation of the SAW can be controlled. Electric

field originating from the SAW gets coupled with the carriers present in the semiconductor,

resulting in induced currents. Since power is transferred from the SAW, it attenuates.

Fig.3.3: Semiconductor SAW Hybrid Structure

However, the residual air gap between the semiconductor and the piezoelectric limits the

reproducibility of the coupling between electron system and the surface acoustic wave. Also, sheet

conductivity is not easily tunable. This can be prevented by using the Epitaxial Lift-Off (ELO)

Technique to transfer thin GaAs/AlGaAs structure, containing a Q2DES by lifting it off the

substrate. By eliminating the air gap between the semiconductor and the piezoelectric by way of

depositing the layers one above the other, the velocity of the surface acoustic wave can be

controlled more efficiently by external bias, because of the quantum well on the piezoelectric

coming in direct contact with the semiconductor.

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Fig.3.4: Schematic Geometry of a typical Hybrid Device.

3.3 Applications of Surface Acoustic Waves:

The applications of Surface Acoustic Waves include, but are not limited to:

SAW devices can generally be designed to provide quite complex signal processing

functions within a single package containing only a single piezoelectric substrate with

superimposed thin film input and output Interdigital transducers.

SAW devices have very good repeatability in performance from device to device.

They find extensive application in mobile and space-borne communication systems,

since they can be implemented in small, rugged, light and power-efficient modules.

High frequency devices can be fabricated using relatively inexpensive

photolithographic techniques, as against the expensive electron-beam lithography

process.

SAW devices are used as MEMS sensors.

They also find application in non-destructive testing.

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Chapter-4

Mason’s Equivalent Circuit Model

4.1 Introduction to the Mason’s model

The basic block of a SAW device that makes an equivalent model necessary is that of

the IDT, which converts electrical energy onto acoustic form and acoustic energy into acoustic

form. Hence, all the equivalent models of SAW devices aim at developing electrical

equivalents for IDTs. This is achieved by developing the electrical circuitry that produces

electrical signals which are similar in all respects to the acoustic signals that are generated by

the IDTs. In this regard, the development of an equivalent model has to take into account, all

the parameters that influence the output waveform in any way. The amplitude and phase

characteristics of the signal predicted by the equivalent model, in particular, must match those

of the acoustic signal generated by the IDTs.

An IDT is made up of several small blocks called “fingers”, each of which contributes

to the final cause of inter-converting electrical and acoustic energies. Hence, the Mason’s

model gives the necessary electrical circuit to develop an IDT finger. Several IDT fingers will

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Fig 4.1 Periodic sections of an IDT

together make up an IDT. The circuit for the fingers in both the input and output IDTs is

essentially the same, since the functioning of the IDTs does not depend on whether the input

signal is electrical or acoustic. Hence, once an IDT finger has been developed, several such

fingers can be connected properly to give rise to an IDT. The input IDT and the output IDT,

separated by a transmission line, which corresponds to the sensitive propagation path, give the

complete electrical circuit of a SAW Delay-line.

4.2 Variants of the Mason’s model

The major limitation of all equivalent models is that the same model cannot produce

results with same accuracy over all the ranges of the physical, mechanical or electrical

parameters of the original model. In this regard, any model is inherently limited in performance

by the system parameters and hence, the same model cannot be used under all conditions to

represent the original system. The equivalent models used to represent SAW devices are no

different in this sense, as the same model cannot be used for different values of the

electromechanical coupling coefficient, k2. Hence, two variants of the Mason’s model have

been developed, namely, the Crossed-Field model and the In-line Field model. These two

models differ from each other in that the Crossed-field model produces accurate results for

small and moderate values of k2, whereas the In-line field model does so, for larger values of

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k2. In the Crossed-Field model, the electric field is normal to the acoustic propagation vector,

while in In-line field model, the electric field is parallel to the propagation vector.

Fig 4.2 Electric field and acoustic wave propagation direction in Mason’s equivalent models

4.2.1 Crossed-Field Model

The Crossed-Field model is used to develop the electrical circuit for an IDT finger, by

using frequency dependent resistance blocks, whose resistance is minimum for the center

frequency of the SAW device and very high for remaining frequencies. Thus, the input energy

propagates only for the frequencies in close proximity to the resonant frequency, ‘fo’ of the

SAW device.

4.2.2 In-line Field Model

The second type of the Mason’s equivalent circuit is the In-line Field model. It differs

from the Crossed-field model in that this model gives more accurate results for higher values

of k2. In terms of the electrical equivalent circuit developed, the In-line field model differs from

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Fig 4.3 Crossed-Field Model

the Crossed-field model, in that the former consists of two negative capacitors in each IDT

finger. The function of these negative capacitors is to decrease the effective static capacitance

in the circuit. The physical significance of these capacitors is that a decrease in the static

capacitance corresponds to a decrease in the accumulated charge in the SAW device, which in

turn implies more movement of charge in the device. Hence, for the materials in which a finite

amount of mechanical stress corresponds to greater electrical current flow, the model serves as

a better equivalent model, as the higher energy conversion factor is depicted in the form of a

negative capacitor. Hence, the In-line field model is considered a better option than the

Crossed-Field model, for those SAW devices, which are made of materials with large values

of k2.

Fig 4.4 In-Line Field model

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4.3 Propagation path

The propagation path between the input and output IDTs, along which the acoustic

wave propagates is a sensitive element, as it results in attenuation and phase shift of the acoustic

signal generated by the input IDT. Hence, the equivalent model developed must represent this

path in electrical terms, such that the amplitude and phase variations of the acoustic wave,

caused by the propagation path, are replicated in the electrical signal. The propagation path can

be modelled as a transmission line, which is made up of two series impedances and a shunt

impedance, as shown in the figure 4.5

Fig 4.5 Transmission line representation of the propagation path

4.4 Mathematical Modelling of the Mason’s model:

4.4.1 Mathematical modelling of IDT fingers

The electrical signal is given at the electrical port, while the acoustic signal propagates

through the acoustic ports, for the input IDT. For an output IDT, the electrical signal can be

taken from the electrical ports. Each finger receives the acoustic signal from its preceding

finger and gives the processed acoustic signal to the next finger, thus resulting in the

propagation of acoustic energy. The input acoustic port of the first finger of the input IDT,

however, does not receive any input signal and is hence to be terminated with a matched

impedance, so that reflections arising from an unbalanced terminal are avoided. Similarly, the

output acoustic port of the last finger of the output IDT is also to be terminated using a matched

load. This matched load is determined as

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𝑍𝑜 =1

𝐶𝑠𝑓𝑜𝑘2 (4.1)

where Cs is the static capacitance of each of the fingers, fo is the resonant frequency of

the SAW device and k2 is the electromechanical coupling coefficient.

The Crossed-field model consists of two types of frequency dependent impedance

blocks, namely the series impedance block and the shunt impedance block. The impedance of

each of these blocks is a function of frequency and is minimum in magnitude for the resonant

frequency.

The series impedance is given by

𝑍𝑠𝑒𝑟 = 𝑗𝑅𝑜tan(𝛼) (4.2)

where 𝛼 is given by

𝛼 =𝜋2𝑓

𝜔𝑜=

−𝑗𝜋𝑠

𝜔𝑜 (4.3)

Hence, the series impedance is given by

𝑍𝑠𝑒𝑟(𝑠) = 𝑗𝑅𝑜tan(−𝑗𝑠

4𝑓𝑜) (4.4)

The shunt impedance is given by

𝑍𝑝𝑎𝑟 = −𝑗𝑅𝑜

𝑆𝑖𝑛(2𝛼) (4.5)

The shunt impedance in the s-domain can thus be expressed as

𝑍𝑝𝑎𝑟(𝑠) = −𝑗𝑅𝑜

sin(−𝑗𝑠

2𝑓𝑜) (4.6)

The most important difference between the Crossed-field model and the in-line field

model, in terms of the electrical components that make up the circuit, is that the negative

capacitances that are present in the in-line field model are short-circuited in the Crossed-field

model. The magnitude of the negative capacitance is equal to that of the capacitance, Co.

However, the shunt capacitances do exist in both the models and can be determined as

𝐶𝑜 =𝐶𝑠

2 (4.7)

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4.4.2 Mathematical Modelling of the Propagation path

The series impedance of the propagation path is mathematically given by

𝑍𝑃_𝑠𝑒𝑟 = 𝑗𝑍𝑜tan(𝛾), (4.8)

where

𝛾 =5𝜋𝑓

𝑓𝑜 (4.9)

The series impedance of the propagation path can hence be expressed in the s-domain as

follows

𝑍𝑃𝑠𝑒𝑟(𝑠) = 𝑗𝑍𝑜tan(−5𝑗𝑠

2𝑓𝑜) (4.10)

Similarly, the shunt impedance is given by

𝑍𝑃_𝑠ℎ = −𝑗𝑍𝑜

sin(𝛾) (4.11)

Hence, the shunt impedance can be expressed in the s-domain as

𝑍𝑃𝑠ℎ(𝑠) = −𝑗𝑍𝑜

sin(−5𝑗𝑠

𝑓𝑜) (4.12)

4.5 Hierarchical Development of SAW device

The SAW device can be developed hierarchically from each of the blocks described

above. The IDT fingers constitute the IDTs and the IDTs, connected by the transmission line,

which depicts the sensitive propagation path, constitute the SAW device. The number of fingers

that make up the IDT determines the energy conversion efficiency of the IDT. However, the

improvement achieved in the energy conversion gets saturated as the number of IDTs is

increased beyond 10. Hence, for most simulation purposes, an IDT made up of five fingers

gives acceptable results. The electrical signal is fed to the electrical ports of all the fingers, as

shown in the figure, while the acoustic signal propagates through the fingers.

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Fig 4.6 Hierarchical development of IDT from fingers

The SAW delay-line model can be developed as shown in the figure-.

Fig 4.7 Electrical equivalent of SAW device

Vi and Vo represent the input and the output electrical signals respectively. The

impedance Zo is necessary at the input acoustic port of the first finger of the input IDT and at

the output acoustic port of the last finger of the output IDT, so as to prevent reflections of the

signals. If these reflections are not inhibited, the acoustic signal that propagates through the

fingers gets corrupted by these reflections cumulatively. The input and the output IDTs are

connected to each other through the transmission line, which serves as the propagation path, as

shown in the figure 4.7.

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Chapter-5

P-Spice model of SAW device using Mason’s model

5.1 Frequency Dependent Impedance in P-Spice

The importance of the frequency dependent impedance blocks in developing the

Mason’ equivalent circuit models has been explained in Chapter-4. However, frequency

dependent impedance blocks are not directly available in P-Spice to incorporate into the model.

Hence, circuits, whose functionality- in terms of the current carrying characteristics- depends

on the frequency of operation, have to be used equivalently as frequency dependent current

blocks. Since the current is a function of frequency of operation for a given input voltage, the

impedance can be said to vary as a function of the operation frequency. Hence, if the resonant

frequency is chosen as the operating frequency, it is possible to devise circuits, the impedance

offered by which is minimum at the resonant frequency, and is very high at other frequencies.

The component in the P-Spice component libraries which can be used equivalently as

a frequency dependent impedance block is the GLAPLACE component. This component is a

current controlled device, in which the current flowing through the circuit involving this

component as a series element, is dependent on the frequency. The mathematical function that

describes the variation of the impedance as a function of the frequency is to be given as the

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description of the block in the XForm section. The reciprocal of the s-domain form of the

equation that describes the dependence of the impedance on frequency can be given as the

XForm input, so that the frequency response of the circuit is as desired. The circuit in which

the GLaplace component is to be used is shown in the figure.

Fig 5.1 Frequency dependent impedance realized using GLaplace block

The XForm inputs for the IDT fingers and of the propagation path are mathematically

described below.

The XForm expression of the series impedance block of the IDT fingers is given by

𝑋𝐹𝑜𝑟𝑚 =1

𝑍𝑠𝑒𝑟(𝑠)=

−𝑗

𝑅𝑜tan(−𝑗𝑠

4𝑓𝑜) (5.1)

Similarly, the expression for the shunt impedance block of the IDT fingers is given by

1

𝑍𝑝𝑎𝑟(𝑠)=

𝑗sin(−𝑗𝑠

2𝑓𝑜)

𝑅𝑜 (5.2)

The XForm expressions of the series and shunt impedances of the propagation path can

also be developed accordingly.

1

𝑍𝑃𝑠𝑒𝑟(𝑠)= −

𝑗

𝑍𝑜tan(−5𝑗𝑠

2𝑓𝑜) (5.3)

gives the expression for the series impedance in the propagation path and

24

1

𝑍𝑃𝑠ℎ(𝑠)=

𝑗sin(−5𝑗𝑠

𝑓𝑜)

𝑍𝑜 (5.4)

gives the expression for the shunt impedance in the propagation path.

5.2 Development of SAW Delay-line model

5.2.1 P-Spice model of the Crossed-Field model

The Crossed-Field model has been discussed in chapters 3 and 4 and this section

describes the development of a Spice model of this model. The simulation procedure adopted

in developing one of the most important blocks in the model, namely the frequency dependent

impedance blocks, has been discussed in the section 5.1. This section deals with the

interconnections between these blocks and the coupling mechanism by which the signals

propagate.

The Crossed-Field developed on P-Spice tool has been depicted in the figure below.

Fig 5.2 P-Spice model of the Crossed Field model

The Imp_ser_CF blocks represent the series impedance blocks and the Imp_par_CF blocks

correspond to the shunt impedance blocks. A transformer with a 1:1 coupling is used for

coupling the electrical equivalent of the acoustic signal received from the previous finger, to

the capacitive circuit, which consists of the capacitors Co1 and Co2. The processed signal is then

25

coupled into the output section of the IDT finger, through a 1:1 coupled transformer. The P-

Spice circuit developed to realize a practical transformer is shown in the figure.

Fig 5.3 P-Spice realization of a practical transformer

The Crossed-Field block thus developed can be incorporated into the hierarchical

design for the IDT, by interconnecting several blocks of fingers, developed as described above.

5.2.2 P-Spice model of the In-Line Field model

The procedure adopted in developing the In-line field model is similar to that used in

the development of the Crossed-Field model. The circuit developed for the In-Line field model

is shown in the figure.

Fig 5.4 P-Spice model of the In-line field model

26

P-Spice tool has a built-in negative capacitance handling capability and hence, the

negative capacitance can be directly used, by giving a negative value to the normal capacitor.

This rules out the difficulty in developing a negative capacitance by making use of other

techniques like using a constant current source in loop with a negative capacitance. The results

produced by the negative capacitance component available in P-Spice tool are very accurate,

as against the erroneous results produced by the other techniques used to generate a negative

capacitance.

5.2.3 Development of propagation path

The electrical equivalent of the propagation path has been discussed in Chapter 4. This

section discusses the P-Spice realization of this electrical circuit. The circuit developed to

represent the propagation path is as shown in the figure.

Fig 5.5 Realization of the propagation path

The mathematical equations governing the frequency responses of the impedance

blocks has been earlier presented. The input port of the delay line is connected to the output

port of the input IDT; from which it receives the acoustic signal. The delay line results in the

modification of the acoustic signal characteristics, like the amplitude and phase. The output

IDT receives this processed acoustic signal from the output port of the propagation path and

converts the acoustic energy into electrical form.

27

5.2.4 Hierarchical development of SAW Delay Line system

A bottom-up approach or top-down approach can be adopted in designing a SAW

device. In either approach, the bottom-most level of the hierarchy is the IDT finger and the

level immediately above is that of the IDT. The interconnections between fingers so as to result

in an IDT is shown in the figure.

Fig 5.6 Hierarchical development of an IDT

The top-level view of the SAW device, as developed in the simulation tool has been presented

in the figure.

Fig 5.7 Top-level view of the circuit realization of SAW device

The time domain analysis of the circuit developed as described above gives the signals

at various points in the SAW device in electrical form. The results obtained and discussion on

the results is presented in Chapter-6.

28

Chapter-6

Results and Discussion

The electromechanical coupling coefficient, k2, can be determined as the ratio of the

electrical energy given at the input and the acoustic energy at the output of the input IDT.

Alternatively, it can also be evaluated as the ratio of the acoustic energy at the input port of the

output IDT and the electrical energy at its output port. In terms of the amplitudes of the signals,

the value of k2 can be determined as the ratio of the squares of the amplitudes of the respective

signals.

A different method has been chosen in this project for the calculation of the value of k2.

This approach is based on the principle that both the input and output IDTs are identical with

regard to their energy conversion efficiency. Hence, the overall ratio of the amplitude of the

output electrical signal to the input electrical signal directly gives the value of k2, provided no

attenuation of the signal occurs in the propagation stage. Hence, providing a short circuit to the

signal at the output of the input IDT, so as to reach the input port of the output IDT directly

without having to flow through the propagation path, allows the calculation of k2 as a direct

ratio of amplitudes, instead of the squares of amplitudes. However, it has to remembered that

this is only correct from a simulation point of view, as in reality, the attenuation resulting from

the propagation path is inevitable. The model for the direct evaluation of k2 is shown in the

figure 6.1

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Fig 6.1 Model for the direct evaluation of k2

The results obtained have been described and discussed in the sections that follow.

6.1 Results for Crossed-Field Model

The simulation was run, by developing the model and its components for the values of

k2 from 0.01 to 0.15 and the value of k2 obtained from the simulation results was compared to

the value expected in each case. Few such simulations have been presented in the figures 6.2

(a-d).

Fig 6.2-a k2=0.014

30

Fig 6.2-b k2=0.04

Fig 6.2-c k2=0.05

Fig 6.2-d k2=0.096

Fig 6.1 Results for the Crossed-Field Model

31

The results obtained here highlight the general trend that the Crossed-Field Model gives

more accurate results for values of k2 below 0.09. However, the percentage of error introduced

increases, as the value of k2 is increased beyond this value.

6.2 Results for the In-line Field Model

The technique adopted in evaluating the performance of the In-Line field model is

similar to the one used for the Crossed field model. The components in the model developed

on the tool were modified, so as to ideally give an electromechanical coupling of values ranging

from 0.01 to 0.15 and in each case, the value predicted by the model was determined, from the

simulation results. The results of few simulations have been presented in the figures 6.3 a-d.

Fig 6.3-a k2=0.04

Fig 6.3-b k2=0.075

32

Fig 6.3-c k2=0.12

Fig 6.3-d k2=0.15

Fig 6.3 Results of the In-line field model

From the results obtained for the In-Line field model, it is evident that the model is not

accurate for small values of k2, while it is reasonably accurate for larger values of the

electromechanical coupling coefficient.

33

6.3 Comparison of the Crossed-Field model and the In-Line model

In the sections 6.1 and 6.2, the performances of the Crossed field model and the In-Line

field model were respectively outlined. This section presents a comprehensive comparison of

the performances of the two models, with regard to their energy conversion efficiencies.

Comparing the percentage of error introduced by each model for a given value of k2

gives an insight into which of the two models is a better choice for that particular value of k2.

This error percentage was determined for different values of k2 and the comparison of the two

models, with respect to this percentage is presented in the figure 6.4

Fig 6.4 Comparison of the Crossed-Field model and the In-line field model, with regard to

energy conversion efficiency

From figure 6.4, the range of values of k2 for which each model is more effective than

the other can be known. However, the most important result that can be extracted from this

analysis is the Crossover point, the value of k2 for which both the models are equally effective

and to the either side of which, one model performs better than the other. The Crossover point

has been found out to be 0.1286. Hence, the Crossed Field Model is the better choice, for

materials whose electromechanical coefficients are less than 13%, whereas the In-line Field

model is better for SAW devices made of piezoelectric materials, with a k2 value above 13%.

0

2

4

6

8

10

12

14

16

0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 . 1 2 0 . 1 4 0 . 1 6

K2

OU

TPU

T

K2 INPUT

COMPARISON OF K2 VALUES

Crossed-Field In-line field

34

Chapter-7

Conclusions and Future Work

The results of the simulations performed using P-Spice tool have been presented in

Chapter-6. This chapter covers the conclusions that can be drawn from the results obtained and

the possible extensions of the work that has been performed as a part of the project.

From the results, it can be concluded that equivalent models for SAW devices, like

most equivalent models, are limited in their performance by the physical or the electrical

parameters of the system. This conclusion can be drawn in accordance with the results obtained

which indicate that the Crossed-Field model gives better results for small and moderate values

of k2, whereas the In-Line model performs better for devices made of materials with larger k2

values. From this work, it has been found out that the Crossed-Field model is a better choice

for the electrical form representation of a SAW device system for values of k2 below 12.9%,

while the In-line field model gives more accurate results for values of k2 above this threshold.

According to literature, the maximum value of k2 that can be achieved using present

technology for a piezoelectric material which can be used to develop a SAW device is 10.2%

and this is possible by the use of YX-128o cut crystals. Hence, from this viewpoint, the work

that has been presented in this report serves as a justification for the use of the Crossed-Field

model for the representation of SAW devices, considering the present advancements in

35

technology. However, this does not rule out the significance of the In-Line field model, as it is

likely that the electromechanical coupling coefficient of materials can increase to values as

high as 15% in the next 20 years. Hence, the In-Line field model can potentially be a useful

equivalent model to represent SAW systems made of the novel devices that are likely to be

developed in the future. The In-Line field model can be used for the electrical form

representation of Hybrid Systems, in which the conductivity of the SAW device is manipulated

by the use of other materials in contact with piezoelectric material. Also, the In-Line field

model is significant from a research perspective, as research on MEMS involves simulation

work, which analyses the behavior of materials, which can give an electromechanical coupling

as high as 20%. Hence, both the Crossed field model and In-line field model have their own

respective benefits and limitations, as described in this study.

A future extension of the work presented here can be with regard to the development

of a hybrid circuit to represent an IDT finger, which can bring together, the advantages of each

of the Mason’s equivalent models. An alternative that is worth exploring is changing the

configuration of an IDT, viewed as the serial connection of the IDT fingers. A novel

configuration can be devised, in which an IDT is developed as the combination of fingers, some

made up of the Crossed-field model, and the others, developed using the In-line field model.

Such a hybrid equivalent circuit can help achieve an accurate replication of the signals

corresponding to the original system, in their electrical form, over a wider range of the device’s

mechanical parameters.

36

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[8] Bhattacharyya, Suneet Tuli, and S.Majumdar, “SPICE Simulation of Surface Acoustic

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

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[11] W.P.Mason, Physical Acoustics, Vol 1A, Academic Press, New York 1964.

37

[12] Trang Hoang, “SAW Parameters Analysis and Equivalent Circuit of SAW Device”, Pg.

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