Analysis Of Equivalent Circuit Modeling Engineering Essay
The technology for portable electronic devices has grown tremendously, new technology allows these devices to perform multiple applications and are smaller in size. But, battery and energy storage technology has not kept up. Energy harvesting can provide endless energy supply for the electronics lifespan as it can replace the need of a battery as power source. Ambient energy of different form can be harvested and converted into electrical energy .. The harvested energy is then used to power electronic devices thus eliminating the need for maintenance and replacement of the power source. Analysis of the piezoelectric vibration energy harvesting is performed to obtain the maximum power generated by the system. The structure that has been considered is in the form of unimorph configuration cantilever beam. Analysis in the single degree of freedom or lumped-element model is employed. The method used to implement the analysis is by equivalent circuit modeling method. It is found that a single degree of freedom unimorph cantilever beam produces maximum power of 6.03nW at resonant frequency of 45.4Hz. The optimal load resistance for maximum power generation is 85kâ„¦.
First and foremost, I would like to express my deepest gratitude and appreciation to my supervisor, Dr Kok Swee Leong for all his guidance and patience throughout the course of this project. I would also like to thank Associate Professor Abdul Rani B. Othman and En Mohd. Azlishah B Othman and all the lecturers of University Teknikal Malaysia for all the support during the period of my graduate studies.
A special thank you is dedicated to my friends in the Pesisir PUO class for all the support during ups and downs during the course of study. Above all, I gratefully acknowledge the effort of my family, especially my parents in shaping my thinking with their love, affection and guidance. Not forgetting my beloved daughter for all the support, sacrifices and every single difficulties and challenges that she had to endure throughout my studies.
Last but not the least, I thank the management and all my friends in Politeknik Ungku Omar for their moral and emotional support.
I declare that this thesis entitle "Analysis of Equivalent Circuit Modeling of Piezoelectric Energy Harvesting" is the result of my own research except as cited in the references. This thesis has not been accepted for any degree and is not concurrently submitted in candidature of any other degree.
Signature : â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.
Name : AZLINDA BINTI ADZAR
Date : â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.
LIST OF TABLES
Summary of vibrations sources
Summary of electromagnetic generators with their main characteristics
Summary of electrostatic kinetic energy harvesters
Summary of magnetostrictive energy harvesters
Comparison of different energy conversion mechanisms
Analogy between electrical and mechanical domain of lumped-element system
Geometric and material parameters of piezoelectric cantilever
The single degree of freedom lumped element parameter (Erturk & Inman, 2008)
Equivalent circuit paramaters of the unimorph piezoelectric energy harvester
Output voltage, resonant frequency with increasing resistive load
LIST OF FIGURES
p-n junction of a photovoltaic cell
Forms of PV cell
Examples of electromagnetic generators
Unimorph cantilever beam configuration
The standard axes and directions for a piezoelectric material
Orientation of electric dipole in piezoelectric material
Direct and indirect effect of piezoelectric material when voltage is applied
Single degree of freedom system developed (William & Yates, 2006)
Equivalent circuit model of piezoelectric cantilever beam
Circuit with AC current source
Circuit with AC voltage source
Maximum output voltage at short-circuit resonance frequency
Maximum output voltage at open-circuit resonance frequency
Output voltage with various resistive load
Power with various resistive load
Output voltage as a function resistive load with different frequencies
Comparison of the maximum power obtained at two different frequencies
TABLE OF CONTENTS
LIST OF TABLES
LIST OF FIGURES
Scope of Work
Outline of Report
Transduction Mechanism Of Energy Harvesting
Comparison of Energy Conversion Mechanisms
Review of Previous Work on Piezoelectric Energy Harvesting
Constitutive Equation of Piezoelectric Materials
Lumped-Element Parameter Single Degree of Freedom Electromechanical System
Equivalent Circuit Representation
Parameters of the piezoelectric energy harvester
Natural frequency of the unimorph cantilever
Paramaters of the unimorph piezoelectric energy harvester
RESULT AND DISCUSSION
Determining the short circuit and open circuit resonant frequency
Voltage and Frequency with Various Resistive Loads
Generated Power with Different Resistive Loads
Output Voltage versus Load Resistance
Power versus Load Resistance
Contributions of Project
Recommendation of Future Work
In our world today, people are living in a fast evolving technological way of life. More and more people are carrying portable electronic devices than ever before. These devices allow for power and versatility in communication and problem solving. But, as the technology for portables has grown tremendously, battery and energy storage technology has not kept up. New technology allows for these portables to become smaller, but battery size remains the same. Perhaps, sometimes the battery must be larger in order to accommodate the greater power demands by a portable device. Batteries have several disadvantages, among which are the need of replacements or periodical recharging as well as problems with their size and weight as compared to high technology electronics that requires high energy consumption and are also decreasing in size. The goal is for energy harvesting to ultimately replace finite sources of energy, such as batteries, extending the lifetime of the electronic device and eliminating the need for maintenance.
One way to defeat these limitations is to implement energy harvesting from the environment which is then used as power supply to electronic devices or as power bank to recharge a battery. The electrical energy to power the electronics is generated from ambient energy either kinetic (vibration), electromagnetic or thermal energy. In order to accomplish energy harvesting from the environment, piezoceramic materials are used as piezoelectric to convert ambient energy to electrical energy. Energy harvesting can provide endless energy supply for the electronics lifespan.
1.2 Scope of Work
The scope of this project focuses on low vibration energy harvesting by piezoelectric energy conversion mechanisms. The single degree of freedom system energy harvester is employed. The analysis on maximum power generation is made on a rectangular, unimorph configuration cantilever beam operating in 31 mode of piezoelectric operation.
The objective of this project is to analyze the piezoelectric vibration energy harvester using equivalent circuit modeling and perform a simulation of the equivalent circuit model in Multisim for maximum power generation.
1.4 Overview of the Report
This report consists of 5 chapters. Chapter 1 presents the general concept of energy harvesting and basic information of the work done. Chapter 2 covers the background and review of literature on work previously done by other researchers. The fundamental concepts of energy harvesting particularly by piezoelectric energy conversion mechanism is discussed. Chapter 3 covers the methodology adopted in the implementation of this work in order to obtain the analysis on power generation of piezoelectric energy harvester by equivalent circuit modeling method. Chapter 4 comprised of simulation data and the findings of this work is discussed. Finally in Chapter 5, the results are summarized based on the results obtained and recommendations on future work is suggested.
Energy harvesting is the conversion of ambient energy present in the environment into electrical energy. It provides the most promising solution, whereby wasted ambient energy such as light and vibration can be converted to useful electrical power and becoming an attractive alternative to conventional battery. It is identical in principle to large-scale renewable energy generation, for example, solar or wind power, but very different in scale. While large-scale power generation is concerned with megawatts of power, energy harvesting typically refers to micro- to miliwatts, which means much smaller power generation systems (Kazmierski, 2011). Among the ambient energy sources that are readily available are solar, wind and kinetic.
2.1.1 Solar Energy
Solar energy is one of the most popular renewable energy sources that has been used as an alternative energy source. Solar energy has the greatest availability as compared to other energy source as it is readily available and plentiful. Solar energy is clean and emission free since it does not contribute any pollutants or by-products that can harm the natural environment. Conversion of solar energy into electrical energy is applied in various fields such as vehicular, residential, space, aircraft and naval applications, just to name a few. Solar cells is capable to provide a power density of 15,000 Î¼W / cm3 under strong sunlight
Conversion of sunlight into electricity is done using the photovoltaic (PV) cell in a process known as photoelectric effect. The PV cell that is exposed to sunlight may reflect, absorb or allow the light to pass through it. However, only absorbed light is capable of generating electricity. Electricity generation is due to the energy from the light that is transferred to the electrons in the PV cell. Initially in their normal positions in the atom structure, the electrons move out of the semiconductor material becoming the current in an electrical circuit. A built in electric field that exist as a significant property of the PV cell acts as the force or voltage that is required to drive the current to flow through a load. The built-in electric field is induced by placing two layers of different semiconductor materials, n-type and p-type in contact with each other. The n-type and p-type semiconductor materials have excess electrons and holes, respectively. A p-n junction is created by putting two different type of semiconductor materials together, namely the n-type and p-type materials, hence the generation of electric field (Khaligh & Onar, 2010). Figure 2.1 shows the p-n junction of a photovoltaic cell.
Figure 2.1 p-n junction of a photovoltaic cell.
The basic building block of a solar electrical system is a PV cell. A PV cell could not generate as much power by working individually, typically about 2W. The output power can be increased by using multiple PV cells that are connected together to form larger units, known as modules. Modules can be connected either in series or parallel, to produce higher power. Interconnected modules are known as arrays. Figure 2.2 shows the forms of PV cells.
(a) (b) (c)
Figure 2.2 - Forms of PV cell (a) cell (b) module (c) array
from (Khaligh & Onar, 2010)
2.1.2 Wind Energy
Wind flow energy has been a sustainable energy source that serve many purposes beneficial to humans such as water pumping, grain milling, and electricity generation. The structure of wind energy harvesting is called Wind turbines. Apart from being renewable, plentiful and widely distributed; wind energy benefits the environment in the sense that it may be used as an alternative to fossil fuel based power generation. Wind energy does not contribute any greenhouse gas emissions to the environment. Wind turbines capture the kinetic energy of winds through the rotating blades that are connected to a coupled shaft and electric generator. The generator converts the mechanical energy into electrical power. Despite its usefulness in the area of energy harvesting, wind turbines are costly, its practicality and feasibility is in large scale applications (Khaligh & Onar, 2010).
2.1.3 Kinetic Energy
Kinetic energy is one of the most readily available energy source. Kinetic energy harvesting adopts the principle of the displacement of a moving part or the mechanical deformation of a certain structure in the devices designed for harvesting energy (Mateu & Moll, 2011). Since most of the vibration power generators are resonant systems, maximum power is generated when the frequency of the ambient energy matches the resonant frequency of the generator. Differences in these two frequencies contribute to a decrease in power generation. Zhu & Beeby (2011) suggested that the possible mechanical energy sources are moving human body or a vibrating structure. Frequency of the mechanical excitation of a human body less than 10Hz and more than 30Hz for vibrating structure such as machinery. In practical applications, low level vibrations has low amplitude values in the order of a few microns. In order to extract energy the use of inertial generator is needed, where it must resonate at a characteristic frequency which correspond to resonance.
Roundy (2005) measured a variety of vibration sources in the range of 0 - 200Hz, which is defined as low-level vibrations. Peak acceleration and frequency of different vibration sources is shown in Table 1.
Table 2.1 Summary of vibrations sources
Peak Acc (m/s2)
Frequency of Peak (Hz)
Base of 5 HP 3-axis machine tool
Kitchen blender casing
Door frame just after door closes
Small microwave oven
HVAC vents in office building
0.2 - 1.5
Wooden deck with people walking
External windows nest to a busy road (size 2' x 3')
Notebook computer while CD is being read
Second story floor of a wood frame office building
Conclusion made by Roundy (2005) state that maximum power can be obtained when the fundamental resonant frequency of the low level vibration energy harvester match the frequency of ambient vibration.
2.2 Transduction Mechanism Of Energy Harvesting
Electrical power is extracted by employing either only one or a combination of energy conversion mechanisms or the transduction mechanism, namely electromagnetic, electrostatic, magnetostrictive and piezoelectric.
Faraday's Law stated that electrical current is induced in any closed circuit when the magnetic flux through a surface bounded by the conductor changes. An electromagnetic generator uses permanent magnets to produce strong magnetic field and coil as the conductor. The generator works either by fixing the permanent magnet or coil to the frame while the other attached to inertial mass. Vibration causes a relative displacement and generation of electromotive force (emf). The characteristics of an electromagnetic generator is high output current level but low voltages. Examples of electromagnetic generators is shown in Figure 2.3 (Zhu and Beeby, 2011).
Figure 2.3 Examples of electromagnetic generators. (Zhu & Beeby, 2011)
Table 2.2 Summary of electromagnetic generators with their main characteristics.
(Zhu & Beeby, 2011)
Excitation level (ms2)
Williams et. al.
Ching et. al
Glynne-Jones et. al.
Koukharenko et. al.
Saha et. al.
Beeby et. al.
Klahand et. al.
Torah et. al
Wang et. al.
Electrostatic generators is based on the concept of variable capacitance driven by mechanical vibrations, where the capacitance varies between a minimum and maximum values. Conversion of mechanical to electrical energy is employed if the charge on the capacitor is constrained and as capacitance decreases, the charge move from the capacitor to a storage device or to the load. The three types of electrostatic generators are in-plane overlap, in-plane gap closing and out-of-plane gap closing as shown in Figure 2.4.
Figure 2.4 Electrostatic generators (a) in-plane overlap (b) in-plane gap closing (c) out-of-plane gap closing (Zhu & Beeby, 2011)
Table 2.3 Summary of electrostatic kinetic energy harvesters (Zhu & Beeby, 2011)
Excitation level (ms2)
Meninger et. al.
Tashiro et. al.
Mitcheson et. al.
Arakawa et. al.
Despesse et. al.
Kuehne et. al.
Yen et. al.
Sterken et. al
Lo & Tai
Hoffman et. al.
Naruse et. al.
IPO - in-plane overlap, IPGC - in-plane gap closing, OP - out-of-plane
Magnetostrictive materials is used in harvesting energy where it deforms when placed in a magnetic field and changes in the magnetic field can be induced when the material is strained. Generally, the magnetostrictive material is used in composite form of piezoelectric-magnetostrictive combination.
Table 2.4 Summary of magnetostrictive energy harvesters (Zhu & Beeby, 2011)
Excitation level (ms2)
Huang et. al.
Wang & Yuan
Dai et. al.
The piezoelectric transducer is in the form of cantilever beam where its base is clamped to the vibrating surface and the tip of the beam is allowed to move. The clamped end and the moving tip are often referred to as fixed end and free end, respectively. As the surface vibrates, the base moves with it, placing the entire beam in an accelerating reference frame. Inertial force is developed and distributed along the transducer. This causes the beam to deflect and strain is developed in the piezoelectric material. The resulting strain induces voltage in the piezoelectric layer, thus converting the mechanical energy of the vibrating surface into electrical energy.
Figure 2.5 Unimorph cantilever beam configuration
The transducer configuration in Figure 2.5 can work either in the 31 mode or the 33 mode. In the 31 mode, the polarization is in the 3 direction and strain is applied in the 1 direction meanwhile for the 33 mode, polarization and strain applied is in the same direction, which is 3 direction. The standard axes and directions for a piezoelectric material is shown in Figure 2.6 .
Figure 2.6 The standard axes and directions for a piezoelectric material
The transducer of piezoelectric energy harvesting systems performs energy conversion from mechanical domain to the electrical domain via direct piezoelectric effect. The direct piezoelectric effect is the ability of a certain ceramic materials to develop electrical polarization when mechanical strain is applied to it. Certain ceramics and crystals exhibits the piezoelectric behavior due to the polar nature of its structure. The crystal structure deforms when pressure is applied. Within each unit cell of the structure, the applied pressure caused an ion to shift thus producing a net charge displacement. Each unit cell has an electric dipole which can be re-oriented to certain directions by imposing mechanical pressure. The poling process will result in the crystal structure to be aligned in a specific manner as shown in Figure 2.7.
Figure 2.7 Orientation of electric dipole in piezoelectric material (a)before, (b)during, and (c)after the poling process, where the arrows point in the positive direction for each dipole.
The piezoelectric material is subjected to a very strong electric field to properly aligned the dipole of the crystal structure (Kaufmann). Poling is necessary for the material to take on piezoelectric properties. Poling is heating the material over the Curie Temperature which allows the molecules to move more freely and applying a large electric field which causes the crystals inside the material to align themselves in one direction. This phenomenon continues even after the electric field is taken off and the material cools.
The piezoelectric material may also exhibit a converse effect of the previously mentioned direct effect. The converse or indirect piezoelectric effect deals with conversion of electrical energy into mechanical energy. When a voltage source is applied to a ceramic element at the same polarity of poling voltage, the length of the ceramic element is elongated causing the diameter to be smaller. If a voltage source of the opposite polarity of the poling voltage is applied, the length of the ceramic is reduced and the diameter is increased. If the voltage source is in AC form, the length of the ceramic element is elongated and reduced in a cycle corresponding to the frequency of the applied voltage (Patel, xxxx).
Figure 2.8 (b) & (c) Direct and (d) & (e) indirect effect of piezoelectric material when voltage is applied (Patel).
2.2.5 Comparison Of Energy Conversion Mechanisms
Table 2.5 comparison of different energy conversion mechanisms (Zhu & Beeby, 2011)
no external voltage source
no mechanical constraints needed
high output current
difficult to integrate with MEMS fabrication process
poor performance (microscales)
low output voltage
no external voltage source
compatible with MEMS
high output voltage
no mechanical constraints needed
thin films have poor coupling
poor mechanical properties
high output impedance
low output current
easy to integrate with MEMS fabrication process
high output voltage
mechanical constraints needed
external voltage source or pre-charged electrets needed
high output impedance
low output current
ultrahigh coupling coefficient
may need bias magnets
difficult to integrate with MEMS fabrication process
2.3 Review of Previous Work on Piezoelectric Energy Harvesting
Erturk and Inman (2008) presented an analytical solution of a piezoelectric cantilever energy harvester with Euler-Bernoulli assumptions. The single degree of freedom (SDOF) modeling approach is also known as the lumped parameter modeling consider the cantilever structure as a spring-mass-damper system. The system is conveniently applied to couple the mechanical domain of the energy harvester with a simple energy harvester circuit. The approximation of this method is limited to a single vibration mode of the cantilever beam, lacking the dynamic mode shape and accurate strain distribution along the cantilever. The model of parametric study for a unimorph energy harvester is used to highlight the important features of the coupled distributed parameter system, the short circuit and open circuit behaviours.
Kasyap et. al. (2006) represented the dynamic behavior of the piezoelectric device in multiple energy domains as lumped-element model and replace them with electrical circuit elements. The model is verified experimentally by using one dimensional beam structure.
Work done by Mineto et. al. (2010) showed that a maximum electrical power is produced when piezoelectric elements that is attached to a vibrating structure that is excited at its first natural frequency. A maximum power of 2.6mW is generated due to large deflections experienced when excited at its first natural frequency of 13.4Hz at a force amplitude of 1N. Power generated at second and third natural frequency of 83.8Hz and 234.6Hz are 0.27mW and 0.063mW, respectively.
A mathematical model of unimorph piezoelectric cantilever beam with a tip mass is discussed by Fakhzan and Muthalif (2011). The model is developed on the Euler-Bernoulli bean theory assumptions. The models are implemented in MATLAB to simulate the generated voltage when the cantilever beam is excited at its natural frequencies. It is found that the harvester has three natural frequencies in the range of 0 to 1400Hz. The amplitude of the voltage generated decreases with increasing natural frequency. The maximum voltage obtained at their first to third natural frequencies for the model developed are 7.45mV at 9.61Hz, 1.34mV at 60.23Hz and 0.51mV at 168.74Hz, respectively.
Umeda et. al. (1996) were among the earliest researcher who proposed an equivalent circuit method to model the conversion of mechanical lumped element model of spring-mass-damper system into electrical energy in a piezoelectric material.
Romani et. al (2011) presented a faster and reliable technique of identifying parameters of the equivalent electromechanical circuit of a generic piezoelectric transducer. The proposed technique assumed that all forces acting on the system is expressed in scaled units by assuming a unity equivalent mass. This implies that the input force is considered numerically equal to the amplitude of input vibrations, where it can be directly measured. The parameters of the equivalent circuit were obtained and simulated in SPICE and later been compared to direct measurements in different operating conditions. The proposed models accurately agreed with the measurements data.
Equivalent circuit modeling of multi-mode piezoelectric energy harvester system is developed by Yang and Tang (2009). In this work, the parameters of the equivalent circuit models are identified theoretically and by finite element analysis for two different harvester structures, simple and complex. The simple structure is in the form of simple rectangular unimorph piezoelectric cantilever beam with a resistive load to represent the energy harvesting circuit. Meanwhile, the complex structure used is an isosceles trapezoidal unimorph piezoelectric beam with a resistive load. The equivalent circuit model for the system were established and simulated in Spice. It is found that the results of equivalent circuit modeling perfectly matched the results of theoretical and finite element analysis where maximum power is obtained at the first natural frequency. It is also noted that the backward electromechanical coupling effect somehow affect the maximum power obtained at different natural frequency with different resistive loads. This is caused by the, this proved that the equivalent circuit method is capable of taking the backward coupling effect into account as analytical and finite element analysis do.
Ajitsaria et. al. (2007) focuses on the analytical analysis based on Euler-Bernoulli and Timoshenko beam theory for voltage and power generation. The electrical equivalent circuit model is then simulated in MATLAB and Simulink. The piezoelectric beam designed for power generator is of bimorph structure. The mode of operation chosen is 3-1 mode because of higher strain and lower resonant frequencies.
Elvin and Elvin (2009) presented an equivalent circuit model for piezoelectric generators which can include any number of vibrational modes and can be easily incorporated into circuit simulator. Single degree of freedom system equations are employed and the electromechanical relations is established by using an ideal transformer or a set of current and voltage dependant sources. Analysis were made for the first the vibration modes of a cantilever unimorph harvester, equivalent circuit model showed excellent agreement with the analytical solution method. The advantage of the developed method is that the piezoelectric energy harvesting circuit can be efficiently analyzed using circuit simulator such as SPICE.
Kong et. al. (2010) proposed a new approach of impedance matching for maximum power transfer of the piezoelectric energy harvester. The distributed-parameter system were used to verify that maximum power extraction is obtained by complex conjugate and resistive impedance matching load. Comparison of both methods were made through circuit simulation, which resulted in resistive matching can be employed for circuit design when the frequency of the vibrating structure is around the resonant frequency of the piezoelectric generator. The effectiveness of the proposed method were validated experimentally. At resonant frequency of between 45.7Hz and 48.2Hz, the power generated is 6.4mW and 6.05mW for complex conjugate and resistive impedance matching, respectively. But, the complex conjugate impedance matching required the use of large value inductance of tens or several hundreds of henries, which makes the method impractical. Meanwhile, when resistive impedance matching is employed, the maximum power between the resonant frequency range is in the range of 6.04mW and 6.05mW. The power delivered to the load by resistive impedance matching is smaller than the complex conjugate matching due to the efficiency is less for larger values of reactive source components, but the efficiency is acceptably good around the resonant frequency of the system.
Phipps and Nishida (2012) proposed a simplified model of a lumped-element parameter of piezoelectric transducer beam. The simplified model sits completely in the electrical domain as compared to the conventional lumped-element parameter transducer which includes both the mechanical and electrical domains. Simplification of the model resulted in reduction of complexity but risk the robustness of the model. As a result, the simplified model is only valid for analysis at a single specific frequency which is the short circuit mechanical resonant frequency. The effective bandwidth and analysis of the non-resonant behavior of the transducer could not be determined.
Gonsalez et. al. (2010) described both piezoelectric and electromagnetic transducers of an electromechanical model energy harvester. The optimum load conditions were studied, when load tend to increase to infinity, simulating an open circuit condition the power generated become very small. When load approaches zero simulating the short circuit condition, voltage and power generated is smallest. Load optimization is important to ensure the efficiency of the harvesting system.
Roundy and Wright (2004) investigated the modeling and design of a piezoelectric vibration energy harvester to be used as power source of wireless electronics, which focused on the optimization of the generator device. A beam with uniform width is assumed and analysis are performed by using the equivalent circuit modeling. Estimation of the amount of power that can be delivered to electrical load is done by testing the model with both resistive and capacitive loads to demonstrate a realistic approximation of the actual load. The model demonstrated a power transfer of 375 Î¼W/cm3 to a resistive load and 190 Î¼W/cm3 to a capacitive load when excited at 250 m/s2 at a frequency of 120Hz.
The single degree of freedom system of a cantilever energy harvester demonstrates the direct piezoelectric effect when pressure is applied to a piezoelectric material, hence a voltage is generated. The behavior of piezoelectric materials involves the relation between mechanical and electrical domain.
The relationship between dielectric displacement, D and electric field, E under zero stress condition is
D = ÎµE (3.1)
where, Îµ is the dielectric permittivity which defined as dielectric displacement per unit electric field for constant stress with a unit of Farads per meter (F/m).
The mechanical stress, T and mechanical strain, S under zero electric field condition is related by
S = sT (3.2)
where, s is the compliance tensor which defined as strain generated per unit stress.
Since there exist a coupling of electrical and mechanical field in piezoelectric materials, the relationship that represent the electromechanical behavior is given by the linear constitutive equation of piezoelectric material (Jalili, 2010).
3.2 Constitutive Equation of Piezoelectric Materials
The linear constitutive equation of piezoelectric materials is given by
S - mechanical strain
T - mechanical stress
D - dielectric charge displacement
E - electric field strength
s - compliance ( inverse of material stiffness)
d - piezoelectric charge constant
Îµ - dielectric permittivity
When the piezoelectric material experienced deformation in the 31 mode, where the polarization is in 3 direction and the stress and strain in induced in 1 direction. The constitutive equation is derived as (Jalili, 2010)
3.4 Lumped-Element Parameter Single Degree of Freedom Electromechanical System
The lumped-element parameter system or discrete system has an essential characteristic where the physical properties of the system; mass, stiffness and elasticity are concentrated or lumped into single physical systems. The elements representing mass are rigid and elastic elements have no mass. Similar to theory of electric circuit where capacitors are considered to have no inductance, inductors have no capacitance and resistors are purely resistive.
Figure 3.1 illustrate the single degree of freedom system developed by William & Yates (2006).
Figure 3.1 single degree of freedom system developed by William & Yates (2006).
The dynamics of the single degree of freedom system representation is described by
z - spring deflection
y - input displacement
m - mass
be - electrically induced damping coefficient
bm - mechanical damping coefficient
k - spring constant
A lumped-element parameter electromechanical system consists of finite number of mass, spring, damper, inductor, capacitor and resistor interconnected together. A single degree of freedom lumped parameter system display a single mechanical resonance. The importance of single degree of freedom system are many real systems display sufficiently similar behavior to a single degree of freedom system, the understanding of real systems is improved without having to deal with complex and tedious mathematics and a distributed system within a limited frequency range often can be treated as single degree of freedom system.
3.4 Equivalent Circuit Representation
Tilmans (1997) has outlined the equivalent circuit representation of an electromechanical system. The development of equivalent circuit representation were based on the relations that exist between mechanical and electrical domain analogously. An example of the analogy is the relation of force, F displacement, x and velocity, v for a rigid mass, m is given by F = m dv/dt = m d2x/dt2 . Mathematically, these expressions is analogous to the constitutive equation of an electric inductor voltage, v = L di/dt = L d2q/dt2. The force, F functions the same as the voltage, v, the velocity, u as current, i and displacement, x as charge, q. The direct analogy of an electromechanical lumped-element system is given in Table 3.1.
Table 3.1 - Analogy between electrical and mechanical domain of lumped-element system
Elastic Compliance, 1/k
The analogy is based on the idea of equating 'across or between' variables with the 'through' variables or vice versa. An 'across' variable, voltage is analogous to 'through' variable, force, F and current, i a 'through' variable to velocity, v an 'across variable'. This implies that mechanical and electrical network topologies are not the same. A series connection in mechanical domain becomes parallel in the electrical domain, and vice versa.
The derivation of single degree of freedom equivalent piezoelectric circuit has been presented by Roundy and Wright (2004). The mechanical domain which consists of single equivalent mass, m, spring, k and damping, b is represented by an inductor, Lm, capacitor, Ck and resistor, Rb respectively. The electrical domain is represented by the capacitance of piezoelectric material, Cp and an external resistive load, RL is used to obtain the voltage generated across the piezoelectric device. The electromechanical coupling is represented by an ideal transformer. Transformer is characterized by the turns ratio, n that relates voltage on one side to the voltage on the other side.
Figure 3.2 Equivalent circuit model of piezoelectric cantilever beam
By referring to the electrical circuit shown in Figure 3.2 and applying Kirchoff's voltage law (KVL) and Kirchoff's current law (KCL) the system equations are determined. The 'across' variable on the mechanical side of the circuit is stress, Ïƒ (analogous to voltage), and the 'through' variable is strain rate, á¹ (analogous to current). The voltage sum of voltages on the mechanical part gives
Lmá¹ + Rbá¹ + + nV = Ïƒin (3.7)
i = CpV (3.8)
According to the electromechanical analogy, equation (3.6) and (3.7) are compared and the equivalent expressions for Ls, Rs and Cs are determined.
3.5 Impedance Matching
A maximum power is expected to be generated by the equivalent circuit. By maximum power transfer rule, the source impedance must be matched with the load impedance. In this case, the impedance of the mechanical domain must matched the impedance of the electrical domain.
Kong et. al. (2010) stated that The maximum power transfer occurs for a fixed AC source if the load impedance is the complex conjugate of the source impedance.
Figure 3.3 Circuit with AC current source
An AC current source shown in Figure 3.3, where is(t) = âˆš2Is sin(Ï‰t), the internal impedance is Zs = Rs + jXs and the load impedance is ZL = RL + jXL. The average power delivered to the load is:
When the load is the complex conjugate of the source impedance ZL,opt = Rs - jXs, the maximum power output is determined by the source properties only that is:
The voltage output across the source or the load is called the optimal voltage :
For circuit with a voltage source of vs(t) = âˆš2Vs sin(Ï‰t) as in Figure 3.4. The maximum power delivered to the load is
Figure 3.4 Circuit with AC voltage source
Although the conjugate matching load extracts the maximum power from the source, a direct impedance matching is usually impractical for piezoelectric energy harvesting due to the requirement of a huge inductor. An alternative approach is to use only a resistive load and try to match the source impedance. The power delivered from a current source to a load resistance of RL can be given by:
The optimal load resistance maximizing the power delivery to the load and the optimal power for the resistive load can be obtained :
3.6 Parameters of the piezoelectric energy harvester
Table 3.2 Geometric and Material Parameters of piezoelectric cantilever
Length, L (mm)
Width, b (mm)
Thickness, h (mm)
Mass density, Ï (kg m-3)
Young's Modulus, Y (GPa)
Piezo constant, d31 (pm V-1)
Dielectric constant, Îµ33 (F m-1)
3.7 Natural frequency of the unimorph cantilever
To produce maximum electrical power from piezoelectric elements which are attached to vibrating structures, the structures should be excited at their first natural frequency where they experience the largest deflections (Mineto et. al, 2010). The first mode natural frequency of the cantilever structure as presented by Kok (2010) is
3.8 Paramaters of the unimorph piezoelectric energy harvester
Table 3.3 The single degree of freedom lumped element parameter obtained from the work of Erturk & Inman (2008).
The cantilever is excited at a constant base acceleration level of 0.5g. The electrical domain representation are obtained from the parameters in Table 3.3 solved using equation (3.9 - 3.13). The equivalent circuit parameters of the energy harvester is shown in Table 3.4.
Table 3.4 Equivalent Circuit Paramaters of the unimorph piezoelectric energy harvester
The circuit as shown in Figure 3.2 is simulated in the Multisim program and analysis are made based on the results obtained .
The analysis of the piezoelectric cantilever beam is accomplished by adopting the concepts of single degree of freedom system. The primary assumptions made is that each mode of vibration of the beam is well-separated in frequency from the other modes. The frequency of the first mode of vibration is capable of generating higher output amplitude.
Based on the geometric properties of the piezoelectric cantilever, its equivalent circuit model is simulated in Multisim to obtain the response the circuit.
4.2 Determining the short circuit and open circuit resonant frequency.
In determining the resonant frequency of the circuit, a set of electrical resistive load is considered in the range of 0 to âˆž. The lowest resistance value is close to short circuit condition, whereas the largest resistance value is close to open circuit condition. The first natural frequency of the harvester is observed by setting the load resistance by using low resistive value to portray the short circuit condition. Since the use of resistive load of 0â„¦ resulted in zero output voltage and resistive load of infinity would not allow the output voltage to be determined, a range of electrical resistive load from 100â„¦ to 1Mâ„¦ is employed.
By using load resistance of 100â„¦, it is close to short circuit condition for the given harvester. The first natural frequency in short circuit condition is shown in Figure 4.1. From the plot of output voltage versus frequency, the short-circuit resonance frequency is found to be 45.4 Hz and it matches the first natural frequency of the harvester of 45.37 Hz.
Figure 4.1 maximum output voltage at short-circuit resonance frequency
The open-circuit resonance frequency of the harvester is the frequency which makes the output voltage maximum as the load resistance tends to be infinity (open-circuit condition). The value of 1Mâ„¦ is used to illustrate the load resistance that tends to be infinity. The open-circuit resonance frequency is 47.5 Hz as shown in Figure 4.2.
Figure 4.2 maximum output voltage at open-circuit resonance frequency
4.3 Voltage and Frequency with Various Resistive Loads.
The effect of output voltage and resonant frequency with different resistive loads is analyzed.
Figure 4.3 output voltage with various resistive load
The results of load, frequency and output voltage is shown in Table 4.3. An important observation is obtained from the plot in Figure 4.3, the resonant frequency moves from the short circuit resonance to the open circuit resonance frequency as the value of resistive load increases.
Table 4.1 - Output voltage, resonant frequency with increasing resistive load
Resistive Load (â„¦)
Resonant Frequency (Hz)
Output Voltage (mV)
Based on these observation, it can be concluded that the resonant frequency of a given piezoelectric cantilever harvester depends on the external resistive load. The resonant frequency can take the value in the range of the short circuit and open circuit resonant frequency.
4.4 Generated Power with Different Resistive Loads
Figure 4.4 power with various resistive load
Based on the power generated at different resistive load values, the optimum resistive load can be estimated. It is observed that maximum power is generated in between the value of 10kâ„¦ and 100kâ„¦ of resistive load. The value of optimum resistive load excited at a resonant frequency can be estimated to be in the range of 10kâ„¦ and 100kâ„¦.
4.5 Output Voltage versus Load Resistance
The output voltage with variations in load resistance for excitations at two different frequencies is shown in Figure 4.5. In both cases, the voltage increases as the load resistance increases. The output voltage for excitation at resonant frequency of 45.4Hz is around 32mV which is higher as compared to a frequency of 20 Hz, lower than resonance yields an output voltage of less than 20mV.
Figure 4.5 Output voltage as a function resistive load with different frequencies
4.6 Power versus Load Resistance
For a given excitation frequency, a certain value of load resistance that gives a maximum electrical power is obtained. This value is the optimum load resistance, Ropt. The optimum load resistance is observed in Figure 4.6 by keeping the frequency constant and the power amplitude is plotted against load resistance.
Figure 4.6 comparison of the maximum power obtained at two different frequencies
Figure 4.6 show the comparison of the maximum power obtained with excitation at the first natural frequency and a frequency lower than the natural frequency. At resonant frequency of 45.4Hz, the power is 6.03uW with an optimum resistance of 85kâ„¦. At frequency 20Hz; lower than the natural frequency of the harvester, the power is found to be 0.965 uW with load resistance of 200kâ„¦. This is in agreement with theoretical assumption that maximum power is obtained when the resonant frequency of the harvester is equal to the frequency of the base excitation.
SUMMARY & CONCLUSION
This work aims to analyze the amount power that is generated by the piezoelectric cantilever beam at low frequency vibrations by equivalent circuit modeling method. Voltage and power outputs variation with load resistance are investigated for excitations at the short circuit and open circuit resonance frequencies of the fundamental vibration mode. Optimum resistive loads of the harvester are identified for excitations at these frequencies. A lumped element single degree of freedom system of a unimorph cantilever is modeled using the equivalent circuit method and the fundamental vibration mode resonant frequency of the circuit is 45.4Hz. Analysis of power output as a function of resistive load at resonant frequency yields a generated maximum power of 6.03nW at an optimum load resistance of 85kâ„¦.
5.2 Contributions of Project
The piezoelectric cantilever beam is system that consists of two different domains, the mechanical and electrical. Analysis of the system requires both the energy domains to be coupled. By employing the equivalent circuit modeling method, the parameters in mechanical domain in analogously represented by an equivalent circuit elements. Further analysis is much easier when done in the same domain.
5.3 Future Work
A distributed parameter piezoelectric energy harvester and a bimorph configuration cantilever can be employed to generate higher output power.
Optimization of the parameters of the piezoelectric cantilever such as geometry, thickness and the total mass of the system, in order to maximize harvested energy may result in a better performance harvester.
A tip mass may also be added to the cantilever structure so that resonant frequency can be tuned for higher power generation.
Analysis based on resistive load is useful but does not give a realistic approximation of the actual electrical load of the energy harvesting circuit. The use of capacitive load can be employed to represent the electrical circuits for a more accurate representation.
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