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Experimental and Analytical Characterization of a Transducerfor Energy Harvesting Through Electromagnetic Induction
Daniel J. Domme
Thesis submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Masters of Science
in
Mechanical Engineering
Chris R. Fuller, Co-chair
Andrew J. Kurdila, Co-chair
Marty Johnson
April 18, 2008
Blacksburg, Virginia
Keywords: Energy Harvesting, Electromagnetic Transduction, Switched Systems
c©Copyright 2008, Daniel J. Domme
Experimental and Analytical Characterization of a Transducer for Energy
Harvesting Through Electromagnetic Induction
Daniel J. Domme
ABSTRACT
Advances in mechatronics have renewed interest in the harvesting and storage of ambi-
ent vibration energy. This work documents recent efforts tomodel a novel electromagnetic
transducer design that is intended for use in energy harvesting. The thesis details methods
of experimental characterization as well as model validation. Also presented are methods
of state space and parametric modelling eforts. In addition, this thesis presents equivalent
electrical circuit models with a focus on switched pulse-width-modulated topologies that
seek to maximize harvested energy.
Acknowledgments
I would like to use this space to thank everyone who has contributed to my education,especially the administration of the Virginia PolytechnicInstitute and State University, foroffering me the chance to pursue and obtain a quality education in the field of mechanicalengineering.
I’d particularly like to thank the faculty I have worked withon this project and others. Iam very grateful to Drs. Chris R. Fuller and Andrew J. Kurdilafor offering the opportunityto work on the research documented here, as well as providingthe necessary direction tomaintain focus on what has been a much larger undertaking than any of my undergraduateexperience. I would also like to thank Drs. Marty E. Johnson and James P. Carneal bothfor introducing me to research in the Vibrations and Acoustics Labs of Virginia Tech andfor their continued entertainment of my inquisitions regarding the sciences of acoustics,vibrations, and signal processing.
I would also like to thank the other young people with whom I have worked in closequarters here at the Vibration and Acoustics Laboratories for providing much-needed assis-tance and distractions during the day. This group includes Jesse Bullinger, Thomas Funke,Philip Gillett, Elizabeth Hoppe, Caroline Hutchison, Kamal Idrisi, Minhyung Lee, DanMennitt, Ben Smith, Mark Sumner, Dr. Alessandro Toso, and Brent Gold. I would espe-cially like to thank Brent for his continued partnership in helping to defend the honor ofthe great American sport of baseball, which finds itself withfewer and fewer friends thesedays. I must also thank Ms. Gail Coe, who not only provided help regarding the office tech-nology, but also an unending supply of fuel that allowed me tomaintain focus on especiallylong work days.
I’d also like to acknowledge the friendship of a number of Virginia Tech undergrad-uates, some of whom have already moved on, especially John Araya, Collin Cole, ChrisGustin, Jason Schroedl, Nicole Shiner, Russell Tolley, Cory Waugh, and Emari Yokota, justto name a very few. Their ability to save me from the mundane has proved to be a greatservice, the value of which they will likely never fully realize.
Finally, I’d like to thank my father, my grandfather, and my late grandmother for theirceaseless love, encouragement, and support. From the timeswe read children’s books onthe sofa to the times I doubtlessly confused you with detailsof my engineering studies,you’ve offered an overwhelming support of my education and my well-being.
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Contents
1 Introduction 1
1.1 Problem Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Passive Vibration Absorbers. . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Active Vibration Control. . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Switching Power Electronics and Averaging Methods. . . . . . . . 4
1.2.4 Semi-Active / Switched Vibration Absorbers. . . . . . . . . . . . 7
1.2.5 Energy Harvesting. . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Overview of Energy Harvesting Architecture. . . . . . . . . . . . . . . . 10
1.4 Motivation and Strategy for Current Research. . . . . . . . . . . . . . . . 12
1.5 Research Outline and Summary. . . . . . . . . . . . . . . . . . . . . . . 13
2 Open Loop System Description 14
2.1 Review of Electromagnetic Fundamentals. . . . . . . . . . . . . . . . . . 14
2.2 System Configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 System Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Transducer Characterization 22
3.1 Experimental Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Experimental Characterization Results. . . . . . . . . . . . . . . . . . . . 24
3.3 State Space System Identification. . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Physics-Based System Identification. . . . . . . . . . . . . . . . . . . . . 35
iv
3.4.1 Transfer Function Derivation. . . . . . . . . . . . . . . . . . . . . 35
3.4.2 Parameter Generation and Results. . . . . . . . . . . . . . . . . . 37
3.5 Experimental Power Analysis. . . . . . . . . . . . . . . . . . . . . . . . . 39
4 The Energy Harvesting System 44
4.1 Equivalent Circuit Development. . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Validation of the Equivalent Circuit Model. . . . . . . . . . . . . . . . . . 47
4.3 Averaging Analysis of the Energy Harvesting System. . . . . . . . . . . . 52
4.3.1 Introduction to Switched Systems and Averaging. . . . . . . . . . 52
4.3.2 State Matrices of the Energy Harvesting Prototype. . . . . . . . . 56
4.3.3 Validation of Averaging Methodology. . . . . . . . . . . . . . . . 59
4.3.4 MOSFETs and Switching Limitations. . . . . . . . . . . . . . . . 61
4.3.5 Averaging Methodology Validation Results. . . . . . . . . . . . . 62
5 Conclusions and Future Work 64
5.1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.1 Transducer Characterization. . . . . . . . . . . . . . . . . . . . . 64
5.1.2 Passive Energy Harvesting. . . . . . . . . . . . . . . . . . . . . . 65
5.1.3 Switched Systems and Averaging Methods. . . . . . . . . . . . . 66
5.2 Future Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A Singular Value Decomposition 68
Bibliography 70
v
List of Figures
1.1 Harvesting Architecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Device Schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Small Prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Schematic Diagram of Magnetic Repulsion. . . . . . . . . . . . . . . . . 19
2.4 Comparison of Linear and Nonlinear Magnetic Forces. . . . . . . . . . . 21
3.1 Experiment Schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Experiment Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Sample of Recorded Time Signals for Small Prototype. . . . . . . . . . . 25
3.4 Sample of Recorded Time Signals for Large Prototype. . . . . . . . . . . 25
3.5 Effect of the Hann Window in the Time Domain. . . . . . . . . . . . . . . 27
3.6 Effect of the Hann Window in the Frequency Domain. . . . . . . . . . . . 28
3.7 Transfer Functions for the Small Prototype. . . . . . . . . . . . . . . . . . 29
3.8 Transfer Functions for the Large Prototype. . . . . . . . . . . . . . . . . . 30
3.9 Coherence Plots for the Small Prototype. . . . . . . . . . . . . . . . . . . 30
3.10 Coherence Plots for the Large Prototype. . . . . . . . . . . . . . . . . . . 31
3.11 Comparison of Average Transfer Functions. . . . . . . . . . . . . . . . . 32
3.12 Nonparametric Model Comparison. . . . . . . . . . . . . . . . . . . . . . 35
3.13 Parametric Model Comparison - Small Prototype. . . . . . . . . . . . . . 38
3.14 Parametric Model Comparison - Large Prototype. . . . . . . . . . . . . . 39
3.15 Passive Harvesting Experimental Setup. . . . . . . . . . . . . . . . . . . 40
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3.16 Rectification Effect, Before DC Correction. . . . . . . . . . . . . . . . . . 41
3.17 Rectification Effect, After DC Correction. . . . . . . . . . . . . . . . . . 41
3.18 Transducer Power Output. . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1 Review of Harvesting Architecture. . . . . . . . . . . . . . . . . . . . . . 44
4.2 Equivalent Circuit Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Transfer Function Comparison. . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Preliminary Rectifier Circuit. . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 Simulated Oscilloscope Output. . . . . . . . . . . . . . . . . . . . . . . . 50
4.6 Comparison of Power Calculations. . . . . . . . . . . . . . . . . . . . . . 51
4.7 Diode Voltage Comparison. . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.8 Switching Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.9 Generic Load Implementation. . . . . . . . . . . . . . . . . . . . . . . . 57
4.10 Switching Shunt Capacitor Circuit. . . . . . . . . . . . . . . . . . . . . . 60
4.11 Results of Averaging Validation. . . . . . . . . . . . . . . . . . . . . . . 63
vii
List of Tables
2.1 Prototype Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 Prototype Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1 Force–Voltage Analogy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
viii
Chapter 1
Introduction
This chapter discusses the current status of energy harvesting efforts, as well as a brief sum-
mary of some current findings to date. It also introduces the objectives and methodology
of the research presented within the thesis.
1.1 Problem Description
In today’s energy-conscious world, there is a powerful impetus to develop alternative sources
of energy. Electrical systems operate remotely despite theconstraints associated with tra-
ditional power sources. For example, there exists a demand for systems such as wireless
sensor networks. Such systems would be powered by self-contained sources, requiring no
outside source of electrical power. This has stimulated interest in the research of energy
harvesting, in which research interest has grown rapidly within recent years.
Enthusiasm in the emerging field of mechatronics has driven several energy transducer
designs. For example, a significant amount of energy harvesting research has been con-
ducted with a focus on piezoelectric materials, which generate a voltage upon the applica-
tion of mechanical stress. Results thus far have been promising, as the materials generate
a high voltage useful for manipulation in electrical circuitry. An alternative to the piezo-
electric approach is the method of employing electromagnetic induction. By passing a
1
conductive coil through a magnetic field, a current is produced. This approach offers an
alternative to piezoelectric energy harvesting, which maybe limited by some design factors
depending on the intended application, such as higher frequencies of resonance associated
with piezoceramics.
As novel designs for energy harvesting systems have emerged, so too have analysis
methods for their study. As will be discussed, another area of significant research effort has
been the area of electrical switching and averaging. For years, pulse-width modulation has
been an effective means of switching between two different electrical systems for a desired
result, such as the maximization of transferable power. It is possible that implementation
of this technique may prove to be beneficial for use in energy harvesting applications.
This thesis develops modeling methods based on averaging techniques that extend these
approaches for energy harvesting transducers.
1.2 Literature Review
Historically, passive vibration absorbers have been the default means to reduce unwanted
vibration, as design is relatively straightforward and there is no need for a control power
source. As the requirements for vibration cancellation become more demanding, however,
the technique of active vibration control—the applicationof a controlled force to negate
vibration—must be considered. Active control offers vastly superior reductions in vibra-
tion for sensitive environments. Yet, a key drawback to active vibration control is the
assumption of a very large source of energy available for thecontrol force. Semi-active
vibration control arose as a result of the introduction of anenergy budget and entails using
electrical switching techniques to minimize the power required for such vibration control.
One can view recent energy harvesting research and development as a logical next step
in the progression from passive, to active, to semi-active vibration attenuation techniques.
Finally, the natural progression of vibration technology led to energy harvesting, wherein
the vibration absorption system requires no external energy to function.
2
This section summarizes some research efforts that are related to all of the aforemen-
tioned fields of vibration absorption and energy harvesting, as well as the field of switching
power electronics.
1.2.1 Passive Vibration Absorbers
Passive vibration absorbers are often the first choice amongmethods of reducing unwanted
vibration. Examples of such devices include traditional vibration dampers, shock ab-
sorbers, and so forth. Examples of such systems are providedin most elementary vibration
textbooks, such as Inman’sEngineering Vibration[30]. Though it is the oldest of vibration
reduction technologies, passive vibration absorption continues to be the subject of various
research efforts. For example, Pun and Liu [59] explored thepotential of a piecewise linear
vibration absorber in a system subject to narrow band harmonic loading.
Advances in electromechanical systems that comprise passive vibration absorbers is
particularly relevant to this thesis. Hagood and von Flotow[26] investigated the use of
piezoelectric materials to convert the kinematic energy ofvibration into dissipated electric
energy, rather than the thermal energy of friction. Davis and Lesieutre [14] presented a
method for predicting the damping performance of resistively shunted piezoceramics based
on a variation of the modal strain energy approach. In addition, Nguyen and Pietrzko [52]
performed finite-element simulations of an adaptive piezoelectric shunt damper with an
R–Lcircuit.
Alternatively, Behrens et al. [8] documented the process ofelectromagnetic, rather than
piezoelectric, shunt damping. The primary transduction was the same in principle, convert-
ing kinetic energy to electric energy.
3
1.2.2 Active Vibration Control
Active vibration control is a method of reducing vibration via the application of force in
an equal and opposite fashion to the forces imposed by the undesired vibration. Of course,
the body of literature that constitutes this field is vast. In1978, Jacquot [31] developed a
method to determine the parameters of a dynamic vibration absorber to eliminate vibrations
in sinusoidally forced Bernoulli-Euler beams. More recently, electromechanical systems
have emerged as a common research theme. Kim et al. [33] showed that the use of active
piezoelectric actuators on panels can greatly reduce the level of transmitted noise through
the panel. Lee et al. [41] performed similar experimentation, combining the use of both
active and passive piezoelectric shunt damping. Fleming and Moheimani [24] described
an active case of piezoelectric shunt damping wherein the shunt impedance was controlled
to affect the electrical resonance. Hu et al. [29] discussedthe design of an active vibra-
tion controller with hysteresis compensation and direct strain rate feedback control, which
guarantees the global stability of the overall system. In his dissertation, Papenfuss [56]
described the synthesis of an implementable control strategy for wideband active vibration
control.
1.2.3 Switching Power Electronics and Averaging Methods
In the last section, the increasingly important role of electromechanical systems in vibra-
tion attenuation has been stressed. During roughly the sametime, a significant effort was
underway to model certain power systems that rely on switching. With the evolution of
more advanced electrical systems, it is possible to apply the use of pulse width modulation
(PWM), the modulation of a switching duty cycle, to vibration absorbers. There has been
extensive study of the nature of switched systems and averaging analysis for the prediction
of behavior of PWM power distribution systems.
Witsenhausen [77] analyzed continuous time systems with states that are part contin-
uous and part discrete. Middlebrook andCuk [49] described a method wherein switched
4
state space systems could be described by averaging techniques. The method replaces the
state space descriptions of two switched circuit networks with their average, resulting in
a single continuous state space equation description. Vorperian [73] investigated the anal-
ysis of PWM DC-DC converters in discontinuous conduction mode, and concluded that
the transition from continuous to discontinuous conduction mode does not reduce the order
of the average converter model. Witulski and Erickson [78] extended the application of
the average state space modeling technique to apply to resonant switch converters, current
programmed mode, and others. They also present a general canonical model to be used in
such cases.
Kazimierczuk and Bui [32] documented the analysis, design,and experimentation of
Class E resonant DC-DC converters with an inductive impedance inverter, showing that
lossless converter operation can be obtained over the full range of potential load resistances.
Liu and Sen [46] later proposed new converter topologies with half-wave and full-wave
controlled current rectifiers. Their design featured a constant switching frequency, zero
voltage switching, and the ability to operate at a no-load condition.
Krein and Bass [35] derived a method of estimating actual waveforms or models given
averaged waveforms or models by using the KBM (Krylov-Bogoliubov-Miltropolsky) av-
eraging method, which presented in Nayfeh [51], in conjunction with a ripple correction
series or model correction terms. Krein et al. [36] also demonstrated the technique with
DC-DC converters and concluded that the averaged model neednot be restricted to small
signals or linear approximations. Sanders and Verghese [63] introduced an averaging anal-
ysis method applicable to switched circuits whose non-switch elements may be nonlinear.
Sun and Grotstollen [67] developed two uniform averaging procedures for discontinuous
systems and applied them to averaged modeling of switching power converters. In later
papers [68] [69], they presented a symbolic computation package for averaged modeling,
which enables a user to generate the averaged model for a power converter using elemen-
tary inputs and with minimum knowledge about the operation of the converter itself.
5
Chen et al. [9] proposed a novel current-controlled PWM technique for performing
the switching control for electromagnetic shaker armatureexcitation. Sun and Bass [65]
documented a method of extending the KBM averaging method toserve as a new means
to model current-mode controlled PWM DC-DC converters. In aseparate paper [64], they
presented a systematic method for ripple analysis of switching power converters based on
the averaged models, using correction terms expressed as analytical functions of the same
variables as used in the averaged model. Mitropolsky [50] presented an averaging method
featuring asymptotic decomposition, which utilizes theory of continuous transformation
groups and the Campbell-Hausdorff formula. Zhou et al. [79]presented a hybrid control
strategy based on loss analysis to optimize design of a low-voltage, low-power DC-DC
converter.
Bass and Sun [4] investigated the effects of ripple estimation at high ripple magnitudes
and concluded that both MFA (Multi-frequency averaging) and the KBM method improve
the accuracy of the averaging model. Han et al. [27] proposeda DC power distribution
system for consumers in remote areas by using a resonant fly-back converter with thyristor
switches, and a PWM voltage-source inverter. Verhulst [72]presented examples of analysis
of weakly nonlinear partial differential equations, focusing on averaging methods.
Kurdila and Feng [38] investigated the stability of modeling switch-shunted piezoelec-
tric systems, concluding that the incorporation of electrical states and the dynamics thereof
in the modeling process offered superior performance. Kurdila et al. [37] showed that the
open-circuit-short-circuit piezoelectric state-switched system can be viewed as a hybrid
Witsenhausen system, and that its stability can be addressed by the use of multiple Lya-
punov functions. Sun and Choi [66] revealed that switching instability can be predicted
by properly defined state space averaged models when combined with a properly defined
duty-ratio constraint. Kurdila et al. [39] showed that the averaging techniques for the anal-
ysis of switched power systems could also be used to model characteristics of switched
piezostructural systems. Lin and Rixen [45] employed piezoelectric patches in conjunction
with a self-switching circuit and a resistive circuit for vibration suppression.
6
1.2.4 Semi-Active / Switched Vibration Absorbers
Semi-active vibration absorbers are active vibration absorbers that employ switching con-
trol systems and are designed for use with more modest, limited power supply.
Dosch [18] described the design of a piezoceramic proof massactuator that incorporates
displacement amplification to efficiently achieve low resonant frequency. Wang [74] out-
lined results of research that developed and evaluated semi-active piezoelectric absorbers
with active constrained layers on beams. Davis and Lesieutre [15, 16] developed a solid-
state semi-active piezoceramic vibration absorber with a tunable natural frequency achieved
by shunting a piezoelectric inertial actuator with a capacitive electrical impedance. The
control system monitored the voltage and current produced by the device to estimate a vi-
bration frequency and adjust the net discrete shunt capacitance. Clark [10] used semi-active
control laws to show the state-switching concept produced vibration suppression in both a
simple spring-mass system and flexural motion of a cantilever beam.
Wang et al. [75] analyzed the electromechanical coupling mechanism in piezoelec-
tric cantilever bimorph and unimorph actuators. Clark [11]used semi-active control to
switch the shunt circuit of a piezoelectric actuator between open-circuit (high stiffness)
and short-circuit (low stiffness) states. Tang et al. [70] reviewed the then-current state of
semi-active vibration absorbers and concluded that integrating the piezoelectric materials
with other smart materials such as shape memory alloys, magnetorheological fluids, and
electric-active polymers can lead to the creation of more powerful and effective semi-active
vibration absorber configurations. Corr and Clark [13] compared the state-switching tech-
nique to a synchronized switching technique and concluded that the latter offers superior
performance.
Lesieutre et al. [44] presented the design of a piezoceramicproof mass actuator and
detailed the dual unimorph approach to motion amplification. Adachi et al. [1] developed
a new design method of the hybrid piezoelectric damping system, featuring active excita-
7
tion with series tuned R–L shunting circuits, with shunt resistors and active control gains
designed from filtered LQ control theory. Ramaratnam et al. [60] introduced the method
of switching the stiffness of a piezoceramic actuator on a cantilever beam depending on
the position of the beam tip with respect to the equilibrium,which maximized vibration
supression. Lefeuvre et al. [42] presented a technique to increase semi-active vibration
absorption with piezoelectrics by artificially increasingthe voltage amplitude delivered by
the piezoelectric patches and thereby reinforcing the electromechanical coupling.
1.2.5 Energy Harvesting
Energy harvesting by definition is a wide-ranging research area, but this section shall ad-
dress the energy harvesting research performed with vibration transducers similar to those
discussed in previous sections. Energy harvesting is a similar process to previously de-
scribed methods of vibration absorption, except that the converted electrical energy is
meant to be stored for later consumption rather than freely dissipated.
Kymissis et al. [40] documented the application of three separate energy harvesting
devices for the application of parasitic energy harvestingin shoes. The devices tested
included a rotary electromagnetic generator, a unimorph strip made from piezoceramic
composite material, and a stave made from a multilayer laminate of PVDF foil. Ottman et
al. [54] described an adaptive piezoelectric energy harvesting circuit that consists of an AC-
DC rectifier with an output capacitor, an electrochemical battery, and a switch-mode DC-
DC converter that controls the energy flow into the battery. Roundy et al. [61] investigated
electrostatic converters fabricated using silicon MEMS technology and concluded that an
output power density of 116µW/cm3 is possible from a vibration source of 2.25 m/s2 at
120 Hz. Flynn and Sanders [25] calculated the stress-limited maximum power density for
a PZT-5H sample to be 330 W/cm at 100 kHz. Their research also concluded that it is
possible to achieve efficiencies on the order of 83%.
Ottman et al. [55] presented a method of harvesting electrical energy from piezoelectric
8
materials and revealed that as the level of excitation increased, the optimal duty cycle for a
step-down converter operating in discontinuous conduction mode becomes relatively con-
stant. The step-down converter harvested energy at over 3 times the rate of direct charging
of a battery. Lesieutre et al. [43] detailed a standalone energy harvesting circuit that can
charge a battery at low excitation levels and using the battery to run a DC-DC step-down
converter at higher levels, providing more energy for storage. Poulin et al. [57] compared
electromagnetic energy harvesting devices to piezoelectric devices and concluded that both
methods of harvesting have similar equivalent circuits andsimilar power generation profile
shapes. However, due to the fundamental differences of physics, there are large differences
in key characteristic values such as output voltage and current, force and displacement
levels, output impedance, and resonant frequency. Amirtharajah and Chandrakasan [2]
constructed a prototype DSP system powered by its own generator and low power voltage
regulator. Power generated was on the order of 400µW, potentially enough for the system
to be powered entirely from ambient vibrations. Cornwell etal. [12] presented a method for
improving PZT power output from a vibrating structure by using a tuned auxiliary struc-
ture to maximize the mechanical energy available to harvest, stating that even an improper
tuning would result in an increase in available mechanical energy.
Du Plessis et al. [19] stated that packaged piezoelectric resonant beam harvesters can be
used to effectively power distributed sensor nodes for practical machinery health monitor-
ing applications. Dutoit et al. [21] presented design considerations related to the mechan-
ical performance of piezoelectric energy harvesters and noted that as a result of ambient
vibration source characterization, significant ambient vibration energy exists in the range
of 100–300 Hz. Ammar and Basrour [3] presented a nonlinear switching method, synchro-
nized switch harvesting on inductor (SSHI), and proposed analternative switching method,
synchronized switch harvesting on capacitor (SSHC), to serve as a suitable method for use
with electromagnetic generators. Beeby et al. [5] comparedmacro- and micro-scale electro-
magnetic generators and concluded that reduction in electromagnetic generator size would
lower output voltage and power levels to impractical levels. They also noted, however,
9
that electromagnetic generators should have a high number of coil turns to maximize flux
linkage and operate at maximum inertial mass displacement.Beeby et al. [7] compared
piezoelectric energy harvesters, electromagnetic generators, and electrostatic generators,
concluding that they could power levels on the order ofµW to mW, which are usable in
some modern circuits.
Dutoit and Wardle [20] designed and optimized a low-level MEMS piezoelectric en-
ergy harvester, capable of producing 313µW/cm3 (normalized by the device volume) at
0.38 V peak-to-peak from a base acceleration of 2.5 m/s2 at 150 Hz. Horowitz et al. [28]
presented the design of an acoustic energy harvester for aeroacoustic applications that em-
ploys a micro-machined piezoelectric diaphragm and that could potentially exhibit a power
density of 250µW/cm2 at a 149 dB sound pressure level. Priya et al. [58] documentedthe
development of a three-layered energy harvesting system that harvests energy from temper-
ature and light gradients in addition to piezoelectric cantilevers for use with wireless sensor
networks. Beeby et al. [6] developed a small electromagnetic energy harvester optimized
for a low level vibration of 0.06g, delivering 46µW to a resistive load of 4 kΩ at a reso-
nant frequency of 52 Hz and a power density of 307µW/cm3. The generator delivers 30%
of the total power dissipated in the generator to electricalpower in the load. Erturk and
Inman [23] analyzed the solution of non-harmonic base excitation in the case of clamped-
free Euler-Bernoulli beams, as many piezoelectric energy harvesting applications employ
cantilever beams with PZT treatments.
1.3 Overview of Energy Harvesting Architecture
The summary in section1.2.5details many different harvesting strategies. It is important
to emphasize that a general architecture for energy harvesting systems has emerged over
time. This architecture is depicted schematically in Figure1.1. Roughly speaking, the har-
vesting system comprises three components. The transducercomponent is the subsystem
that converts mechanical energy to electrical energy. The storage element is a component,
10
such as a battery or supercapacitor, that stores electricalenergy for later use. These two
components are the key elements of a passive energy harvesting system which was the first
topology for energy harvesting to appear in the literature.However, the newer semi-active
harvesting approach introduces switching system component, thereby introducing highly
efficient signal conditioning. This topology is a relatively recent development, with the
earliest documentation from the literature review indicating an emergence in 2002 [54].
Figure 1.1: Diagram showing the components of the energy harvesting systemFor passive harvesting, the switching converter and external control are absentor replaced with simple rectification.
Examples of passive energy harvesting systems include those outlined in reviewed liter-
ature from Beeby et al. [5], Kymissis et al. [40], Roundy et al. [61], and others. The primary
advantages of eliminating a switching converter system from the energy harvesting system
are the resulting simplicity of the circuit design and that there is no requirement of external
control. The transducer, if producing an alternating signal, simply requires rectification to
a DC signal, which can be produced via passive elements. However, passive elements can
incur more power loss, resulting in lower efficiency. This can prove to be significant for
low-power transducers.
Examples of semi-active energy harvesting methods are given in Ammar and Bas-
11
rour [3], Lesieutre et al. [43], and Ottman et al. [55], amongothers. These systems employ
high-efficiency switching converters that modify the voltage and current levels of the input
to suit the energy storage topology and storage device physics. However, these systems
require more complex electronic elements as well as an external controller. In some sys-
tems, the external power source can be eliminated if the system can eventually generate
enough energy to power its own switching system in a bootstrapping design. For most sys-
tems, however, an external power source is required. The presence of the power electronics
in the semi-active configuration warrants the review of switching systems and averaging
methods in section1.2.3.
1.4 Motivation and Strategy for Current Research
The aim of this thesis is to make two distinct contributions to the scientific literature:
• The thesis is to document the characterization and modelingof a novel electromag-
netic transducer system.
• The thesis presents a unified modeling methodology based on averaging techniques
that are applicable to semi-active harvesting.
These models are necessary because the inclusion of any signal conditioning may affect the
performance of the transducer itself due to mutual couplingeffects between the electrical
and mechanical systems. The scope of the thesis willnot include a detailed consideration
of the physics of storage elements or specific switching systems for the case of semi-active
harvesting. For example, Ottman et al. [55] considers a conventional buck converter as the
switching component in Figure1.1. Any number of conventional converters could be con-
sidered for the presented transducer. It is also important to emphasize that the contribution
of this thesis is not the introduction of averaging methods per se. Rather, it is the devel-
opment of these methods so that they are applicable for studying electromagnetic energy
12
harvesting. The modeling and averaging techniques to be reviewed in this thesis should
allow for an accurate description of the coupled system uponaddition of specific switching
systems and storage elements for energy harvesting.
1.5 Research Outline and Summary
This thesis details research in two primary areas: (1) the modeling and experimental char-
acterization of a novel electromagnetic energy harvestingdevice and (2) the development
of an averaging analysis technique for the study of semi-active energy harvesting via elec-
tromagnetic transducers. Chapter 1 provides an overview ofthe current state of research
regarding energy harvesting and related power electronics. Chapter 2 reviews the princi-
ples of electromagnetic energy harvesting and details a novel design for an electromagnetic
energy harvesting device. Chapter 3 details the efforts made to successfully characterize
the device whose design was given in Chapter 2. Chapter 4 provides an overview of the
energy harvesting simulation and analysis performed in theelectrical domain. Chapter 5
summarizes the findings presented in this thesis and also provides a perspective on possible
future research.
13
Chapter 2
Open Loop System Description
This chapter will review the basic physical principles behind electromagnetic energy har-
vesting, as well as introduce the working configuration for the prototype energy harvesting
device.
2.1 Review of Electromagnetic Fundamentals
The field of electromagnetics first emerged upon the discovery of Maxwell’s equations—
originally printed in 1861 [48]—which related electric field, magnetic field, electric charge,
and electric current. The most important of these relationships, for the purposes of dis-
cussing electromagnetic energy harvesting, are Faraday’sLaw of induction,
∇× E = −∂B
∂t(2.1)
and the Ampere-Maxwell equation
∇× B = µ0J + µ0ǫ0∂E
∂t. (2.2)
In Equation (2.1), E is the vector representation of the electric field,B is the magnetic
flux density, andt is time. Likewise, in Equation (2.2), µ0 is the permeability of free space,
14
4π × 10−7 NA2 , J is the current density, andǫ0 is the permittivity of free space,8.854 ×
10−12 Fm
. These two laws introduce the principles, respectively, that a changing magnetic
field will induce an electric current, and a changing electric field will generate a magnetic
field.
In the case of a solenoid or a linear electromagnetic generator, the principle of induction
expressed in Equation (2.1) can be applied to the case of a magnetic field passing througha
conductive coil. The resulting epression is the most commonexpression of Faraday’s Law,
and is expressed in terms of the magnetic flux and coil turns as
Vemf = −NdΦ
dt(2.3)
[71] whereN is the number of coil turns, andΦ is the magnetic flux in Webers. His-
torically, this expression was determined by Faraday empirically and not directly from
Equation (2.1). Using this principle, Chen et al. [9] proposed a dynamic model of an elec-
tromagnetic shaker wherein it is revealed that the that the generated voltage in the case of
a permanent magnet moving through a coil can be expressed as
Vemf = Γx (2.4)
wherex is the velocity of the permanent magnet andΓ is the flux linkage, defined by the
magnetic field strength multiplied by the total length of thecoil (Γ = Bl). The same model
also presents an expression for the back-emf force that results from the effect described in
Equation (2.2),
f = Γio (2.5)
whereio is the induced current, in Amperes. These two relationshipscan also be found in
Poulin et al. [57] and are the primary equations of electromagnetic coupling that were used
15
in the research presented in this work.
2.2 System Configurations
The electromagnetic generator system schematic is given inFigure2.1. The three masses
are permanent magnets oriented with opposing poles facing each other. This design, pro-
posed by Dr. Chris Fuller, provides a natural restoring force due to magnetic repulsion.
Unlike springs, this method of dual magnetic repulsion eliminates the loss of energy due to
spring friction and is not subject to spring fatigue failure.
N
S
N
S
N
S
kc
Г i
y
x
Figure 2.1: Free-body diagram of the electromagnetic prototype design: onesuspended magnet inducing a current as a result of motion through a movingcontinuous coil.
Two distinct prototypes existed for testing. Shown in Figure2.2, both house permanent
neodymium magnets and coil windings in accordance with the device schematic given in
Figure2.1. A summary of their dimensions is provided in Table2.1. They also feature
ventilation holes so that there is no compression of air associated with the magnet motion,
16
allowing for a greater displacement magnitude and thus greater electrical output.
Figure 2.2: Picture of the large prototype (left) and the small prototype (right)that were tested for electrical performance.
Table 2.1: Prototype PropertiesSmall Large
Diameter 1/2 in 1 1/2 inHeight 1 1/16 in 3 7/8 in
Mass 10.0 g 312.5 gInternal Resistance 5 Ω 2.5ΩInternal Inductance 0.765µH 3.014 mH
There are several reasons for testing two similar prototypes of different sizes. First, as
summarized in Poulin et al. [57], the output characteristics—including current and voltage
levels and dynamic system response—of energy harvesting devices based on electromag-
netic induction depend entirely upon the physical characteristics of the device itself. That
is, while generalizations of high current, low voltage, andlow natural frequency can be
made, specific expected values would depend entirely on the system characteristics.
17
Therefore, the key reason for testing two different prototypes is to gain a sense of scal-
ability. The respective size differences offer a means to quantify the tradeoff between the
harvesting device size and the desired output characteristics of low resonant frequency and
high output energy.
A secondary reason for testing two prototypes is to investigate the role of the energy
losses inherent in the energy harvesting circuitry. For example, piezoelectric energy har-
vesters have been suggested that use diode bridge rectification, which causes an a usually
insignificant drop in the output voltage. However, for the low-voltage output of electro-
magnetic energy harvesters, the diode drop could completely eliminate the system voltage
output altogether. The testing of both a large and a small prototype offers a comparison to
determine a minimum device size for overcoming energy harvesting losses.
2.3 System Nonlinearity
It should be noted that the force of magnetic repulsion acting on the central mobile mag-
net is nonlinear. This was a significant concern in preparingthe analysis and experiments.
Defrancesco and Zanetti [17] conducted experiments that supported the claim that the re-
pulsive force of two axial magnets can be expressed as
F = k
(
1
(r + h)2 +1
(r + 2d)2 −2
(r + d)2
)
(2.6)
wherek andh are constants,r is the distance between the two magnet faces, andd is the
axial length of the magnets. In this expression, each term isrelatively easy to comprehend,
as it assumes the axial magnets can be represented by magnetic dipoles. Equation (2.6)
sums the four resulting forces. The constantk is a scaling factor. The first term in the
sum represents the repulsive force of the two similar poles that are closest to each other.
The constanth exists to prevent the expression for this force from becoming infinite when
r is zero and the magnet faces are in contact. The second term represents the repulsion
18
force of the two similar poles that are farther separated, hence the addition of the2d term
in the denominator—these two poles are that much farther apart. The final term in the sum
represents the pair of attractive forces between the pairs of opposite poles, explaining the
opposing sign as well as the doubled numerator.
To judge whether the entire system may be represented as a linear system (i.e., to say
that the restorative force on the center magnet is aroughly linear function of the form
F = kx for small displacement), a linearization analysis must be performed on the forces
of magnetic repulsion. There exist two forces of magnetic repulsion—one from either end
of the device—as shown in Figure2.3. The expressions for those forces are given by
FL = k
(
1
((r0 + x) + h)2 +1
((r0 + x) + 2d)2 −2
((r0 + x) + d)2
)
(2.7)
and
FR = k
(
1
((r0 − x) + h)2 +1
((r0 − x) + 2d)2 −2
((r0 − x) + d)2
)
. (2.8)
and the difference of these forces is the net expression for the nonlinear force on the central
magnet,
Fnl = FL − FR. (2.9)
Figure 2.3: Schematic diagram showing the two forces of magnetic repulsionacting on the central magnet.
19
Standard procedures of linearization imply investigatingthe first two terms of the Taylor
series expansion of the nonlinear force, so that the linearized force is given as
Flin = Fnl (x0) +∂Fnl
∂x
∣
∣
∣
∣
x=x0
(x − x0) . (2.10)
It should be relatively straightforward that asx is zero, the first term of the expansion,
FL − FR, is also zero. This leads to the resulting linear expression
Flin =
(
−4k
(r0 + h)3 −4k
(r0 + 2d)3 +8k
(r0 + d)3
)
x (2.11)
As a check, some arbitrary variables were used for the equation constants (r = 10,
k = 1, h = 1, andd = 2) and the original nonlinear force was plotted as a function of
displacement,x in MATLAB . The results are presented in2.4. The figure shows that it may
be reasonable, depending on the size and position of the magnets and assuming a relatively
small central magnet displacement, to assume a linear magnetic repulsion force and thus a
linear dynamic system. The assumption of linearity was upheld throughout the process of
system characterization, and as will be evident in the next chapter, the assumption appears
to be valid based on quality of the results.
20
−5 0 5−5
−4
−3
−2
−1
0
1
2
3
4
5x 10
−3
Free Magnet Displacement, x
For
ce, F
FNL
(x) dFNL
(x)/dx Flin
(x) dFlin
(x)/dx
Figure 2.4: Plot showing the true (solid) and linear approximations (dotted) ofthe restorative magnetic repulsion forces of the system. The blue lines are therestorative forces themselves, while the red lines are their respective derivatives.
21
Chapter 3
Transducer Characterization
This chapter summarizes the characterization of the open loop system that was introduced
in the previous chapter.
3.1 Experimental Setup
The experiment described in the following section was carried out to characterize the be-
havior of both the large and small prototypes in response to mechanical excitation. By
providing a measured excitation input to the system and measuring the output voltage, the
frequency-domain response of the open loop system could be measured and compared to
system modeling efforts. A schematic of the experiment setup is provided in Figure3.1.
Figure 3.1: Schematic showing each device used in the execution of experi-mentation.
A picture of the experimental test rig used is given in Figure3.2. The large white
22
shaker, a Ling LNG V408 shaker, was used as the primary excitation source, while the
energy harvesting prototype being tested is shown mounted in the aluminum fixture. The
mounting block was designed with two distinct parts—a large1-1/4” block of aluminum
designed to attach to the shaker, and a flange plate that secures the prototype. The thickness
of the block provided some physical separation between the transducer and the shaker. The
shaker is itself a large electromagnetic device, and the thickness of the mounting block is
intended to reduce potential interference between the magnetic fields within two devices.
Figure 3.2: Experiment setup showing the small energy harvesting prototypemounted to an electromagnetic shaker.
Rather than a traditional function generator, a laptop sound card was used in conjunc-
tion with a Rane MA-6 multi-channel amplifier to drive the shaker. This technique allows
for a wide range of custom input excitation functions to be played as .wav files. For exam-
ple, the excitation profile used in this characterization experiment was a slow chirp signal
that varied from 5 to 200 Hz over the span of five minutes to achieve a sinusoidal frequency
sweep. The chirp was generated in MATLAB and saved as a .wav file for later use in the
experiment. The reason for this is because the standard method of acquiring data in the
23
Vibration and Acoustics Lab involves using a customized LabVIEW virtual instrument,
which does not include a sine sweep excitation. This manual sine sweep method was used
in order to maximize the input energy over the frequency range of interest. The average
energy of white noise excitation proved to be too low to produce any discernible electrical
response with the available sensor suite.
Two time-domain signals were collected during the experiment—both the output of an
accelerometer affixed to the mounting block and the voltage output of the energy harvesting
prototype. The signals were recorded at a sampling rate of 1024 Hz, which provides a
Nyquist frequency of 512 Hz, which is well above the maximum input signal frequency
of 200 Hz. Trial excitation runs determined a reasonable estimate of the voltage levels to
be expected during testing. Based upon these maximum outputs, input channel gains of 5
and 10 were applied to the prototype output and accelerometer measurement, respectively.
These gains were selected to maximize the dynamic range and to minimize loss due to
signal digitization while also avoiding clipping.
3.2 Experimental Characterization Results
A sample of the time-domain signals collected are given in Figures3.3and3.4for the small
and large prototypes, respectively. An important note is that the voltages do not exceed the
limit voltages of the data acquisition system, so there should be no data corruption due to
clipping. In addition, it should be noted that the magnitudeof the response of the smaller
prototype increases very gradually before the resonant peak, resulting in the possibility of
a poor estimate of the response at frequencies below about 15Hz, due to a low signal-to-
noise ratio.
In order to determine the relation of the input acceleration(x) to the output prototype
voltage (y), the transfer functionHxy is defined as the ratio of the cross power spectral
density,Pxy, and the input power spectral density,Pxx. That is,
24
0 50 100 150 200 250 300 350−0.4
−0.2
0
0.2
0.4
Acc
el V
olta
ge
Recorded Time−Domain Signals
0 50 100 150 200 250 300 350
−0.5
0
0.5
Time, s
Pro
toty
pe V
olta
ge
Figure 3.3: Raw time-domain data records of the accelerometer output (top)and prototype output (bottom) for the small prototype.
0 50 100 150 200 250 300 350
−0.4
−0.2
0
0.2
0.4
0.6
Acc
el V
olta
ge
Recorded Time−Domain Signals
0 50 100 150 200 250 300 350−2
−1
0
1
2
Time, s
Pro
toty
pe V
olta
ge
Figure 3.4: Raw time-domain data records of the accelerometer output (top)and prototype output (bottom) for the large prototype.
25
Hxy =Pxy
Pxx
. (3.1)
The two power spectral densities are estimated via Welch’s periodogram method, which is
described in [76]. The modified periodogram is calculated byapplying a window function
to the time-domain data, and computing the discrete Fouriertransform. In all cases of
power spectrum estimation for these experiments, a 1024-point Hann window function
with 50% overlap was used, as well as an FFT length of 4096 points, which provides for a
0.25-Hz frequency resolution. The Hann function is given as
w(n) = 0.5
(
1 − cos
(
2πn
N − 1
))
(3.2)
whereN is the window length—in this case, 1024 points—andn is an integer ranging from
0 toN − 1.
A demonstration of this method is given in Figure3.5. The unprocessed data has
nonzero values at the ends of the time domain data, which causes an increase in the es-
timated frequency content of the signal around the true signal frequency content, a phe-
nomenon commonly referred to as leakage. The windowed data remedies this problem by
focusing on the content in the center of the data set. The modified data set is obviously
mostly composed of the data at the center of the set rather than the data at the ends, mean-
ing that information is lost at those ends for each data sample. To reduce the loss, the next
data set begins 50% after the start of the previous data set, thus overlapping power spectrum
estimates in time.
In the frequency domain, the reduction of leakage should be apparent at low frequency
levels. A comparison of unwindowed and windowed transfer functions is given in Figure
3.6. Notice that below about 10 Hz, the unwindowed transfer function noise level is similar
to the rest of the transfer function, a distortion induced byleakage. Windowing tends to
drive that effect toward zero, as shown in the low-frequencyrange of the windowed plot,
which exhibits negligible content.
26
0 200 400 600 800 1000
−0.1
−0.05
0
0.05
0.1
0.15Unwindowed Periodogram
Res
pons
e D
ata
0 200 400 600 800 1000
−0.1
−0.05
0
0.05
0.1
0.15Periodogram with Hann Window
Sample
Res
pons
e D
ata
Figure 3.5: Comparison of unwindowed periodogram (top) and the same datamultiplied by a Hann window (bottom) for one 1024-point datavector from thesmall prototype experiment.
The final transfer function estimates were determined by a method of averaging. For
each prototype, ten experimental time-domain data sets were collected, and the correspond-
ing transfer functions were calculated. The plots of each transfer function can be seen in
Figures3.7and3.8for the small and large prototypes, respectively. In addition, the coher-
ences of the data are given in Figures3.9 and3.10. It should be noted that the coherence
of the small prototype is significantly lower, possibly due to the lower output levels of the
device. However, the results of transfer function estimation indicate that the test was highly
repeatable.
As there were ten recorded transfer functions for each prototype, it was necessary to
determine average transfer functions for each prototype. The average transfer functions
27
0 50 100 150 200−50
−40
−30
−20
−10
Win
dow
ed T
F M
ag, d
B
Windowing Effect Comparison
0 50 100 150 200−50
−40
−30
−20
−10
Non
win
dow
ed T
F M
ag, d
B
Frequency, Hz
Figure 3.6: Magnitude and phase comparison of Hann-windowed transfer func-tion (top) and unwindowed transfer function (bottom) for the small prototype.
were calculated from the averaged power spectra. This was done in order to compare the
average output to the average input. Two techniques were used to calculate the average
transfer functions, and both methods yielded the same result, which provides confidence in
the accuracy of processing the experimental data.
The first method of transfer function estimation determinesthe transfer functions of ten
different recorded sets of data and averages them together.That is,
Havg =
1N
N∑
i=1
Pxy,i
1N
N∑
i=1
Pxx,i
=
N∑
i=1
Pxy,i
N∑
i=1
Pxx,i
(3.3)
whereN is 10 andi is the index of each individual power spectrum. As before, x represents
the input acceleration, and y represents the output prototype voltage. The second method
combines the signals end-to-end into a single vector, then performs the estimation as in
28
0 50 100 150 200−80
−60
−40
−20
0
TF
Mag
nitu
de, d
B
Accleration−Voltage Transfer Function, Small Prototype
0 50 100 150 200−10
−5
0
5
10
15
Frequency, Hz
Pha
se A
ngle
, rad
Figure 3.7: Magnitude and phase plots of each of the ten experimental transferfunctions that relate input acceleration (m/s2) to output voltage (V) of the smallprototype. The offsets in the phase plot are equivalent to±2π, indicating nearlyidentical phase.
29
0 50 100 150 200−60
−50
−40
−30
−20
TF
Mag
nitu
de, d
B
Accleration−Voltage Transfer Function, Large Prototype
0 50 100 150 200−2
−1
0
1
Frequency, Hz
Pha
se A
ngle
, rad
Figure 3.8: Magnitude and phase plots of each of the ten experimental transferfunctions that relate input acceleration (m/s2) to output voltage (V) of the largeprototype.
0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Coherence, Small Prototype
Frequency, Hz
Coh
eren
ce
Figure 3.9: Coherence plots of each of the ten experimental transfer functionsof the small prototype.
30
0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Coherence, Large Prototype
Frequency, Hz
Coh
eren
ce
Figure 3.10: Coherence plots of each of the ten experimental transfer functionsof the large prototype.
Equation (3.1), only using the combined signals given by
x = x1, x2, . . . , xN, y = y1, y2, . . . , yN (3.4)
The averaged transfer functions are compared in Figure3.11. It is important to note
that in the subplot of phase angle, a 180-degree phase shift has been added to the transfer
function of the large prototype, as the experiments were conducted with the wires attached
to the data acquisition system in the opposite configurationas the experiments of the small
prototype.
3.3 State Space System Identification
The first attempt at modeling the system assumed a general state space model of the system
and sought to approximate that model numerically. This was achieved using MATLAB ’s
System Identification Package. Specifically, the package uses a parameter estimation algo-
31
0 50 100 150 200−120
−100
−80
−60
−40
−20
TF
Mag
nitu
de, d
B
Average of Accleration−Voltage Transfer Function
LargeSmall
0 50 100 150 2000
2
4
6
8
Frequency, Hz
Pha
se A
ngle
, rad
Figure 3.11: Comparison of the average transfer functions of both the smalland the large prototype.
rithm in the functionpem.m to solve the optimization problem associated with identifica-
tion of the device.
The identification problem seeks to find the matricesA,B,C,D of the equation
x(t) = Ax(t) + Bu(t) + Ke(t)
y(t) = Cx(t) + Du(t) + e(t). (3.5)
that yield the minimum norm of the output error,
VN (θ) =1
N
N∑
t=1
(y (t) − y (t|θ))2 (3.6)
wherey is the system response,θ is the vector of system parameters, and the discrete-
time period is described ast = 1, . . . , N . The algorithm used in MATLAB is derived
from the prediction error minimization technique described in section 9.3 of Ljung [47],
32
which describes the method of fitting parameterized models to experimental data. In basic
terms, this algorithm creates an expression for the performance of a model based on a set
of parameters by analyzing the mean squared error between the model and the data. The
function then iteratively solves for the minimum of the error expression and generates the
state matrices corresponding to the optimal parameter vector.
To determine the required model order and corresponding length of the parameter vec-
tor, an analysis of the system’s singular values was performed. A brief overview of the
method of singular value decomposition is given in AppendixA. If the singular value de-
composition of the matrixA results in the existence of especially low singular values,it
would indicate that the dimensions ofA are larger than is necessary to describe the system.
While there was no sharp decline of singular values to indicate dominance of a particu-
lar system order, the results of the third-order model (minimum singular value 0.01973)
seemed to be a more accurate choice than a fourth-order model(minimum singular value
0.00557).
It should also be noted that the prediction error minimization algorithm does not com-
pute the continuous model directly, but instead solves for adiscrete state space system, of
the form
x(kT + T ) = Ax(kT ) + Bu(kT ) + Ke(kT )
y(kT ) = Cx(kT ) + Du(kT ) + e(kT ). (3.7)
In order to solve for the continuous model of equation3.5, the functiond2c.m must be
used. This function performs a zero-order hold on the inputs. That is, the control inputs are
assumed piecewise constant over the sampling period. The conversion produces a continu-
ous system whose zero-order hold discretization coincideswith the solved discrete system.
The use of these algorithms—prediction error minimizationand conversion to contin-
uous form via zero-order hold—on the small prototype experimental data yielded the state
matrices
33
A =
−3.7351 706.21 57.743
−94.106 −32.138 1702.3
0.82395 1.6521 161.29
B =
3.465
1.6592
0.28287
C =[
0.42204 −0.63731 0.52759]
D = 0
K =−→0
x(0) =
−0.0028765
−0.00099759
−0.00016948
.
The resulting state space model is presented in Figure3.12. The output matches the ex-
perimental data poorly, leading to the suspicion that a parametric model developed from the
equations of motion of the system would offer better performance. It should be emphasized
that the lack of performance may not be due to the software andformulation, but rather a
lack of user expertise with the software package. In any event, the model that follows in-
corporates the physics of the transducer. As such, it shouldbe amenable to modeling the
data.
34
0 50 100 150 200
10−4
10−2
100
TF
Mag
nitu
de
Accleration−Voltage Transfer Function
0 50 100 150 200−300
−200
−100
0
100
Frequency, Hz
Pha
se A
ngle
, deg
Figure 3.12: Magnitude and phase comparison of experimental averaged trans-fer function (blue) and nonparametric model (red) for the small prototype.
3.4 Physics-Based System Identification
This section describes the derivation and accuracy of a physics-based model generated from
the system schematic diagram shown in Figure2.1.
3.4.1 Transfer Function Derivation
By using the schematic diagram model in Figure2.1 on page16, the following equations
of motion were established:
mx + c(x − y) + k(x − y) = −Γi
Γ(x − y) − Ldi
dt− Ri − ZLi = 0
(3.8)
wherem is he central magnet mass,c andk are the damping and spring constants, respec-
35
tively, Γ is the flux linkage,L andR are the transducer internal inductance and resistance,
respectively, andZL is the load impedance. When we consider that there is no current flow
due to the open loop, the first equation is set to zero, and the second equation becomes
Vout = Γ(x − y). (3.9)
Performing a Laplace transformation on the first equation and rearranging, we get
[ms2 + cs + k]X(s) = [cs + k]Y (s) (3.10)
or, solving forX(s),
X(s) =[cs + k]Y (s)
ms2 + cs + k. (3.11)
Similarly, the second equation can also be converted to the Laplace domain as
Vout(s) = Γs[X(s) − Y (s)]. (3.12)
These two Laplace equations can be combined, yielding
Γs[cs + k] − Γs[ms2 + cs + k]
ms2 + cs + kY (s) = Vout(s) (3.13)
or simply
Vout(s)
Y (s)=
−Γms3
ms2 + cs + k. (3.14)
Furthermore, the transfer function can be converted to an equivalent function using accel-
eration as the input instead of displacement.
36
(
Vout(s)
Y (s)
) (
s2Y (s)
Y (s)
)
=
(
s2Vout(s)
Y (s)
)
(3.15)
This reveals that dividing the original transfer function by s2 will yield the acceleration
transfer function. The expression can be simplified to yield
Vout(s)
Y (s)=
−Γms
ms2 + cs + k. (3.16)
This equation can be modified further when the Laplace variable s is replaced withjω, so
that the function can determine the relationship to acceleration input rather than displace-
ment. That is,
Vout(jω)
Y (jω)=
−jωΓ
−ω2 + jωc
m+
k
m
. (3.17)
3.4.2 Parameter Generation and Results
Equation3.17reveals that there are three parameters necessary for characterizing the open
loop system—Γ, c/m, andk/m. The last parameter is easily recognizable as the squared
natural frequency. In order to determine this parameter, itis only necessary to read the
appropriate peak natural frequency from the experimental data plot. Using the experimental
transfer functions, the natural frequency of the data seemsto be around 42.35 Hz, or 266.1
rad/s for the small prototype and 10.87 Hz, or 68.33 rad/s forthe large prototype. Squaring
these values leads to a value ofk/m = 70805 s−2 for the small prototype andk/m =
4668.4 s−2 for the large prototype.
For the remaining two parameters, the easiest means of determining the parameters
was to determine the error between the parametric model and the experimental model over
a range of possible parameter values and focusing on the combination ofΓ andc/m that
37
produced the lowest error. That error was defined by the equation
ε =
[∣
∣
∣
∣
Vout(jω)
Y (jω)
∣
∣
∣
∣
− |H(jω)|
]2
. (3.18)
The corresponding values resulting in error minimization areΓ = 0.68039 N/A andc/m =
40.072 s−1 for the small prototype andΓ = 3.5044 N/A andc/m = 53.376 s−1 for the large
prototype.
Figure3.13shows a comparison between the average transfer function determined from
the experimental data and the parametric model transfer function with the given parameters
for the smaller prototype. Figure3.14shows the comparison between the experimental and
parametric model transfer functions for the large prototype
0 50 100 150 200−60
−40
−20
0
TF
Mag
nitu
de, d
B
Accleration−Voltage Transfer Function
0 50 100 150 2000
2
4
6
8
Frequency, Hz
Pha
se A
ngle
, rad
Figure 3.13: Magnitude and phase comparison of experimental averaged trans-fer function (blue) and parametric model (red) for the smallprototype.
It should be noted that for the small prototype, the transferfunction magnitudes are
quite similar above 20 Hz, and the phase shows a similar shiftaround the natural frequency.
Below 20 Hz, the magnitude plots do not match up particularlywell, though this could be
38
0 50 100 150 200−60
−50
−40
−30
−20
TF
Mag
nitu
de, d
B
Accleration−Voltage Transfer Function
0 50 100 150 200−2
−1
0
1
2
Frequency, Hz
Pha
se A
ngle
, rad
Experimental DataParametric Model
Figure 3.14: Magnitude and phase comparison of experimental averaged trans-fer function (blue) and parametric model (red) for the largeprototype.
an equipment limitation. The amplifier used was designed foraudio applications, and the
response of such equipment is usually poor below 20 Hz. Despite this minor inconsistency,
it would appear that the derived parametric model is appropriate for describing the behavior
of the electromagnetic device. It also implies that any nonlinearities imposed by magnetic
repulsion are minimal.
3.5 Experimental Power Analysis
This section discusses experimentation to determine the power output of the large trans-
ducer using a passive energy harvesting strategy. Only the large transducer was subjected
to experimentation because output of the smaller transducer was too small to induce a sig-
nificant voltage across the diode bridge in the rectificationprocess.
The experimental setup uses the same configuration as outlined in section3.1, except
39
that the output lead wires of the transducer are connected toa breadboard prototype circuit
that contains a full-wave bridge rectifier of 1N5820 Schottky diodes, chosen for their high
efficiency and ready availability. The rectified voltage signal then passes through a 2.5Ω
resistor (made by placing 4 resistors with a value of 10Ω each in parallel.) This was done
to achieve maximum power transfer through a matched real impedance, as the internal
resistance of the transducer is also 2.5Ω. A circuit diagram of the experiment is given in
Figure3.15. For comparative purposes, a second set of measurements wastaken with the
diode bridge removed while the load remained 2.5Ω. In addition, two different excitation
levels were tested at numerous frequencies. In each case, two signals were measured—the
accelerometer output and the voltage across the load resistor.
Figure 3.15: Diagram showing the experimental circuit. The voltage source inthis diagram represents the large transducer. Output was the power measuredacross the resistors.
An example of one of the results for the rectified case is presented in Figure3.16. It
should be noted that the output of the load resistor is displayed with an offset bias due to
the fact that the signal was collected in an AC coupling mode.Therefore, a restorative DC
bias was introduced in post-processing in order to bring theminimum value of the signal
up to zero, and all further analysis considers this effect. An example of the DC correction
made to Figure3.16is given in Figure3.17.
Results are summarized in Table3.1, and a graphical representation of the power output
is given in Figure3.18. From these results, it is clear that most of the available power before
40
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time, s
Out
put M
agni
tude
12 Hz, 0.11 V, Rectified
Ouput PowerAccelerationOutput Voltage
Figure 3.16: Experimental result before a post-processing correction to reintro-duce a lost DC bias to the output voltage signal.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time, s
Out
put M
agni
tude
12 Hz, 0.11 V, Rectified
Ouput PowerAccelerationOutput Voltage
Figure 3.17: Experimental result after the DC bias correction, which sets theminimum of the output signal equal to zero.
41
rectification is lost in the process of conversion to a DC signal. As a result, the performance
of a passive harvester depends greatly on the properties of the diodes used for the full wave
rectifier.
Table 3.1: Prototype Properties
Frequency Peak Unrectified Rectified Efficiency mW(Hz) Acceleration (m/s2) Power (mW) Power (mW) m/s2
12 17.23 51.2 4.4 .1738 0.51618 36.84 84.9 5.5 .1461 0.33724 36.28 45 2 .0978 0.12150 38.38 8.6 0 0 0100 39.13 2.1 0 0 0
10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
3.5
Frequency, Hz
Nor
mal
ized
Out
put P
ower
, mW
/(m
/s2 )
Passive Energy Harvesting Performance
RectifiedUnrectified
Figure 3.18: Comparison of the normalized power output of the transducerbothwith and without rectification.
As shown in Table3.1, the rectification results in nearly 83 percent power loss inthe
best case. At 50 and 100 Hz, the diode losses result in total loss of transferred power.
42
Therefore, the effectiveness of a passive energy harvesting topology is determined by the
characteristics of the diodes used for rectification.
However, our results are in rough correlation with thepiezoceramicdevice presented
in Ottman et al. [55], which exhibits a power output with a similar order of magnitude in
the case of passive energy harvesting, though excitation levels and resonant frequencies
are different. Despite the difference in the physics of the underlying transducers, three
conclusions can be drawn. First, the power output of our device, if on the same order of
magnitude as the device presented in Ottman et al. [55], could meet the power needs of
many electronic systems when excited at a sufficient level and at the mechanical resonance
of the transducer. For example, Ottman et al. [55] shows that9.45 mW output is achieved
for a PZT bimorph that is approximately1.81 × 1.31 × 0.01 in. Care must be taken in
comparing these approaches since the underlying physics are quite different. Second, the
frequency of operation for the two devices is complementaryin that theoptimalfrequency
of operation for the bimorph presented in Ottman et al. [55] is near the resonant frequency
of 61-64 Hz. Third, and just as importantly, the energy harvesting effort may benefit from
the inclusion of an active switching system—possibly self-powered—in order to increase
the percentage of available power that can be harvested.
43
Chapter 4
The Energy Harvesting System
The overall energy harvesting architecture is presented again in Figure4.1. Chapters 2
and 3 have only considered the transducer component of the overall harvesting system.
The next step would be to study different switching system topologies to maximize power
transfer in the case of semi-active harvesting, since the overall efficiency of the harvesting
process is highly dependent on the switching circuit. Recall, for example, that Ottman et
al. [55] achieved a power transfer using semi-active harvesting that was more than three
times the output power of a passive harvesting method.
Figure 4.1: A review of the diagram showing the components of the energyharvesting system with a semi-active switching system.
44
However, while the modeling of transducers and switching pulse-width-modulated con-
verters have both been well-studied, there exists the need for a unified model that incorpo-
rates both of these components. Existing models for switching systems consider ideal
sources that are invariant with respect to the behavioral characteristics of the electronics of
the switching system. In the case of energy harvesting, the input cannot be considered an
ideal source, as there exists a tight coupling effect between the properties of the transducer
and the properties of the switching converter. That is, the performance characteristics of
both components are mutually dependent on each other. This leads to the need of a unified,
coupled model of both the transducer and the switching system.
The first step is to develop an equivalent circuit of the device which is required for the
averaging processes to follow. Second, it is necessary to derive the averaged equations that
will allow for simulations of different switching topologies while retaining the coupling
effects between the transducer and the switching system. Finally, a simulation is required
to verify that the averaged equations serve as a good approximation for the true behavior
of the harvesting system.
4.1 Equivalent Circuit Development
As stated in section 4–4 of Ogata [53], a mechanical system may be represented by an
equivalent electrical system that obeys a governing set of ordinary differential equations. It
is especially beneficial to model the entire electromechanical system including the energy
harvester as a single electrical system, because of the necessity to attach the device to an
energy harvesting circuit. An equivalent circuit allows for the entire system to be simulated
with electrical simulation software such as Multisim, partof the National Instruments Cir-
cuit Design Suite. Such simulations, in considering the entire energy harvesting system in
a single electrical domain, retain the inherent coupling ofthe transducer to the rest of the
electrical system.
45
In order to translate the model to Multisim, we must review the force–voltage analogy
between mechanical and electrical systems, also found in Ogata [53] and shown in Table
4.1.
Table 4.1: Force–Voltage AnalogyMechanical Systems Electrical Systems
Forcef VoltagevMassm InductanceLDamping Coefficientc ResistanceRSpring Constantk Reciprocal Capacitance1/CDisplacementx ChargeqVelocity x Currenti
Starting with equation (3.8) and recognizing the simple analogies given above, most
variables in the equations are easily represented by standard electrical components, with
exception of two terms:−Γi andΓ(x− y). However, realizing that current is analogous to
velocity, these two expressions can be represented by current-controlled voltage sources,
which are components available in Multisim. Therefore, a complete equivalent circuit
representation of the device is given in Figure4.2.
Figure 4.2: Equivalent circuit model of the harvesting device.
In the first (left-hand) current loop, the current may be defined as analogous to the
46
velocity variablex. The input to the system is the current source, which represents the
base excitation velocity,y. According to Kirchoff’s Current Law, this would mean that
the total current in the second loop—the loop in the middle, which contains the resistor
and capacitor in series—isx − y. In addition, the same loop contains a current-controlled
voltage source. Thus, by using Kirchoff’s Voltage Law, we can set the sum of the voltages
equal to zero, or
Ldi1dt
+ R (i1 − iin) +1
C
∫
(i1 − iin) dt = −Γi2 (4.1)
which is directly analogous to the first part of equation (3.8).
For the third (right-hand) current loop, most of the circuitcomponents can be deter-
mined directly from the second part of equation (3.8), except for the velocity–dependent
voltage term. However, since the velocity is directly analogous to current, as per Table4.1,
the voltage may be represented by another current–controlled voltage source. This results
in the equation
Γ (i1 − iin) = Ldi2dt
+ Ri2 + Zi2 (4.2)
which is directly analogous to the second part of equation (3.8).
In addition, it should be noted that the input for this circuit is a current source, which is
analogous to a velocity term (y). In order to obtain acceleration as an input, for comparison
to the transfer function obtained in the previous chapter, an inductor is placed in series with
the current source. The voltage across this inductor shouldyield the derivative of that of
the current source, thereby yielding a measurement of acceleration.
4.2 Validation of the Equivalent Circuit Model
While the previous section presented the development of theequivalent circuit of the trans-
ducer, This section serves to both verify the performance ofthe equivalent circuit model
and to demonstrate the performance of a simulation that incorporates the equivalent circuit
47
in a realistic harvesting architecture.
To validate the equivalent circuit model, a simple AC sweep test was performed in
Multisim, the results of which are shown in Figure4.3. (Note that the phase of both models
actually overlap. The phase variable in Multisim is restricted to−π ≤ φ ≤ π.) This test
sweeps the frequency of the input source and measures any variable expression. For the
transfer function, the expression is simplyV (10)/V (7), orVout/Yin. The equivalent circuit
performance for the small prototype is thus identical to theperformance of the parametric
model. This process was repeated for the large prototype, and the results were likewise
identical to the results of the parametric model. A plot is not given for this case because
the results of the parametric model and the equivalent circuit model are indistinguishable
in graphical form.
0 50 100 150 200
−100
−80
−60
−40
TF
Mag
nitu
de, d
B
Accleration−Voltage Transfer Function
0 50 100 150 200−5
0
5
10
Frequency, Hz
Pha
se A
ngle
, rad
Figure 4.3: Transfer functions of experimental data (blue), the parametricmodel (red), and the Multisim model (green) for the small prototype.
In order to verify that the equivalent circuit will perform as expected in a simulation of
an energy harvesting system, the equivalent circuit was modified to include a diode bridge
48
to simulate the case of passive energy harvesting, as was demonstrated in section3.5. As
in that section, only the model for the large prototype was used, as the characteristics
of the smaller prototype are insufficient to produce a significant voltage across the diode
bridge. Likewise, the rectified output was passed through a small 2.5 Ω resistor (which
is equivalent to the internal resistance of the prototype) to determine the expected power
output. Both the current and the voltage through the resistor were measured, allowing for
power calculations. The modified circuit is shown in Figure4.4.
Figure 4.4: Circuit diagram showing the incorporation of a Schottky diodebridge and a matched load resistance.
It should be noted that the diodes used in this circuit are International Rectifier 5EQ100
Schottky diodes, chosen for examination based on the claim of low power loss and high
circuit efficiency for switching power supplies and resonant power convertors. The 1N5820
diode used in the experiment documented in section3.5 was unavailable in the Multisim
diode model library.
The power is calculated by the equationP (t) = V (t)I(t). Both the voltage and current
were measured on the Multisim virtual oscilloscope, the output of which can be seen in
Figure4.5, along with the resulting power calculation. The graph clearly shows a sharp
magnitude drop at the peaks of the first few oscillations. This may be due to a simulation
of AC coupling, which removes the DC component of the input signals on the virtual
oscilloscope, similar to the effect that required correction in section3.5. To attempt to find
49
a second solution for power, the same power calculation was subjected to a time-domain
transient analysis, which exhibited the expected DC offset.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
VoltageCurrentPower (V*I)
Figure 4.5: Oscilloscope output showing load voltage, current and power forthe preliminary periods of simulation.
The Multisim documentation states that for the circuit’s transient analysis, “each input
cycle is divided into intervals, and a DC analysis is performed for each time point in the
cycle. The solution for the voltage waveform at a node is determined by the value of that
voltage at each time point over one complete cycle.” Assumptions made for this technique
are that “DC sources have constant values; AC sources have time-dependent values. Ca-
pacitors and inductors are represented by energy storage models. Numerical integration is
used to calculate the quantity of energy transfer over an interval of time.”
The resulting calculation of power delivered to the load is compared to the virtual oscil-
loscope output in Figure4.6, and the results exhibit a similar waveform to the result given
in section3.5. The calculated peak power output eventually reaches the value of 30.8 mW,
and the corresponding average power is 10.7 mW. Given that there was an input of 30 m/s2,
the resulting power per input acceleration is 0.356 mW/(m/s2), which is similar to the re-
50
sults presented in Table3.1, although the specific Schottky diodes used in the rectifier are
different.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Time, s
Pow
er, W
Power CalculationScope Power
Figure 4.6: Comparison of transient analysis of power (blue) to oscilloscopeoutput (red) for the first cycles of system oscillation.
To calculate the efficiency of this diode rectification, voltage levels both with and with-
out the presence of a full-wave rectifier diode bridge were calculated, and the result is
given is Figure4.7. The result shows a loss of 0.74 V, or 0.37 V per active diode, which
corresponds to a total efficiency of 4.7 percent.
In summary, the behavior of the equivalent circuit exhibitsa high degree of similarity
to the experimental system described in section3.5. However, the primary purpose of
this section was not to verify the experimental results of passive rectification. Rather, the
purpose is to verify that the equivalent circuit transducermodel behaves as expected and to
illustrate how it is possible to adapt a harvesting topologyto the equivalent circuit that was
developed in the last section, arriving at a complete electrical model.
51
0 0.05 0.1 0.15 0.2 0.25 0.3−1.5
−1
−0.5
0
0.5
1
1.5
Time, s
Out
put M
agni
tude
I V
inV
out
Figure 4.7: Comparison of system voltage levels with (blue) and without(red)the presence of the 5EQ100 diode bridge.
4.3 Averaging Analysis of the Energy Harvesting System
This section introduces the concept of switching electrical systems and averaging tech-
niques, as well as applying the techniques to the energy harvesting prototype model to
derive appropriate state space matrices. As a final step, themethodology is validated by
comparing the results of the averaging technique to a simulation of a switched system.
4.3.1 Introduction to Switched Systems and Averaging
As reviewed research shows, there is often an increased output of electrical energy from
semi-active vibration control systems when the load includes a switching shunt circuit. In
order to analyze the effects of the switched system as applicable to the prototypes under
review, we must first understand the basics of electrical switching systems and averaging
analysis.
The presence of a periodic switch in the load of a circuit introduces two distinct systems
controlled by a square-wave switching signal, defined by theequation
52
h(t) ,
1, t ∈ [nTs, (n + D) Ts]
0, t ∈ [(n + D) Ts, (n + 1)Ts](4.3)
whereTs is the period of oscillation, or switching period,D is the duty cycle, defined
as the percentage of the period wherein the switch is activated, andn is any integer. A
representation of this function can be seen in Figure4.8. The regulation of the duty cycle
is a form of pulse-width modulation, or PWM.
Figure 4.8: Time-domain representation of the switching function.
While the switch is activated, the electrical system can be represented in state space as
x(t) = A1x(t) + B1u(t)
y(t) = C1x(t) + E1u(t). (4.4)
Meanwhile, when the switch is deactivated, the system is represented with a different set
of state matrices,
x(t) = A2x(t) + B2u(t)
y(t) = C2x(t) + E2u(t). (4.5)
However, we can combine these two separate systems into a single representation,
53
x(t) = A(t)x(t) + B(t)u(t)
y(t) = C(t)x(t) + E(t)u(t). (4.6)
where the time-dependent state matrices are defined as
A(t) , A1h (t) + A2 (1 − h (t))
B(t) , B1h (t) + B2 (1 − h (t))
C(t) , C1h (t) + C2 (1 − h (t))
E(t) , E1h (t) + E2 (1 − h (t))
. (4.7)
At this point, it would technically be possible to analyze the state space system, but
the time-dependence of the state matrices provides considerable drawbacks. The system
is nonlinear, and numerical calculations would require extensive processing. Because the
frequencies of switching are typically much higher than those of the system oscillation, a
useful solution to these problems is to use the average values of the state matrices for the
system and perform analysis with the resulting state matrices.
The process of averaging is discussed in detail in Sanders and Verhulst [62], which
provides the fundamental averaging equations. Given the perturbation equation
dx
dt= εf(t, x) + ε2g(t, x, ε), x(t0) = x0 (4.8)
one can consider the averaged form of the equation
dx
dt= εf 0(x), x(t0) = x0, (4.9)
with
54
f 0(x) =1
T
∫ T
0
f(t, x) dt.
In these equations,x represents the average ofx. The first step in the averaging technique
is to introduce a change of variables in order to eliminate the time variable,t. We will
choose a new domain based on the switching period,Ts. That is,
t(τ) = ετ, τ = Ts. (4.10)
Thus, ast—now a function rather than a variable—increases by the period,Ts, the value of
τ will increase by one. We can further convert the variable of the derivative via the chain
rule,
dφ(τ)
dτ=
∂
∂τφ(t(τ)) ·
∂t
∂τ= ε
∂
∂τφ(t(τ)) (4.11)
whereφ denotes a function andφ is defined as
φ(τ) , φ(t(τ)).
Using this change of variables, we can rewrite equation (4.6) as
dx(τ)
dτ= ε
(
A(τ)x(τ) + B(τ)u(τ))
y(τ) = C(τ)x(τ) + E(τ)u(τ)
(4.12)
and use the principle of equations (4.8) and (4.9) to determine the averaged state space
equation,
55
d¯x(τ)
dτ= ε
(
¯A¯x(τ) + ¯
B¯u(τ))
¯y(τ) = ¯C¯x(τ) + ¯
E¯u(τ)
. (4.13)
The averaged matrices are found easily, as
A =1
Ts
∫ Ts
0
A(t, x) dt = DA1 + (1 − D)A2.
Extending this averaging technique to all the state matrices and converting the variable of
interest back tot from τ ,
dx(t)
dt= Ax(t) + Bu(t)
= (DA1 + (1 − D)A2) x(t) +(
DB1 + (1 − D)B2
)
u(t)
y(t) = Cx(t) + Eu(t)
= (DC1 + (1 − D)C2) x(t) + (DE1 + (1 − D)E2) u(t)
. (4.14)
These averaged state space equations are precisely the sameform as given in Erickson [22].
4.3.2 State Matrices of the Energy Harvesting Prototype
As it has been shown that the state matrices can be represented by their respective averages,
we can begin the derivation of the state matrices from equation (3.8) while using a generic
load impedance, shown in Figure4.9. We can rewrite equation (3.8) as
mx + c(x − y) + k(x − y) = −Γi
Γ(x − y) − Ldi
dt− Ri = vL
(4.15)
For any switching load, we can express the load voltage,vL, as
56
Figure 4.9: Partial circuit diagram showing the use of a generic load impedance.
vL =1
ZL
∫
i dt ⇒ i = vL · ZL. (4.16)
We can now substitute this expression for current into both parts of equation (4.15), result-
ing in
x +c
m(x − y) +
k
m(x − y) = −
ΓvLZL
m
Γ(x − y) − LZLvL − RZLvL − vL = 0
. (4.17)
The second of these equations can be rewritten as
vL +R
LvL +
1
LZL
vL =Γ
LZL
(x − y). (4.18)
We are now ready to begin derivation of the state space matrices. If the state and input
vectors are defined as
57
x =
x1
x2
x3
x4
=
x
x − cm
y
vL
vL + ΓLZL
y
u = [y]
(4.19)
then the derivation of the state space matrices works out nicely as the derivative of each
term in the state vectorx is calculated in terms of first-order variables,
x1 = x = x2 +c
my (4.20)
x2 = x −c
my = −
k
mx +
k
my −
c
mx −
ΓZL
mvL
= −k
mx1 +
k
my −
c
m
(
x2 +c
my)
−ΓZL
m
(
x4 −Γ
LZL
y
)
= −k
mx1 −
c
mx2 −
ΓZL
mx4 +
[
k
m−
( c
m
)2
+Γ2
mL
]
y
(4.21)
x3 = vL = x4 −Γ
LZL
y (4.22)
and
x4 = vL +Γ
L − ZL
y = −1
LZL
vL −R
LvL +
Γ
LZL
(
x2 +c
my)
= −1
LZL
x3 −R
L
(
x4 −Γ
LZL
y
)
+Γ
LZL
(
x2 +c
my)
=Γ
LZL
x2 −1
LZL
x3 −R
Lx4 +
(
RΓ
L2ZL
+Γc
mLZL
)
y
. (4.23)
Finally, we can assemble the matricesA andB of the first state space equation,
x = Ax + Bu
58
as the following:
A(t) =
0 1 0 0
− km
− cm
0 −ΓZL
m
0 0 0 1
0 ΓLZL
− 1LZL
−RL
B(t) =
cm
km−
(
cm
)2+ Γ2
mL
− ΓLZL
RΓL2ZL
+ ΓcmLZL
. (4.24)
These two state matrices are time dependent because the value of ZL changes with
time. The next section validates the averaging technique byexamining its performance in
the case of a switching shunt capacitor.
4.3.3 Validation of Averaging Methodology
As mentioned previously, the derivation of the state space matrices in equation (4.24) are
applicable to any load. To validate the accuracy of the averaging method, we can con-
sider the simple case of a switching shunt capacitor, shown in Figure4.10. Note that the
impedance of the load changes with the state of the switch. Based on the averaging meth-
ods presented so far, we can approximate the solution to thissimple system by adopting the
average value of the shunt impedance.
First, let us investigate the case of the average switching shunt impedance (denoted here
by Zsh instead ofZL.)
59
Figure 4.10: Circuit diagram showing the incorporation of a switching shuntcapacitor.
Zsh ,1
N
∫ N
0
Zsh dτ
=1
N
N∑
n=1
[
∫ (n−1)+D
n−1
(
C0 + Ch(τ))
dτ +
∫ n
(n−1)+D
(
C0 + Ch(τ))
dτ
]
=1
N
N∑
n=1
[(
C0 + C)
D]
+1
N
N∑
n=1
[C0 (1 − D)]
= C0 + CD
(4.25)
In addition, when considering the shunt impedance terms that appear in the denominators
of expressions within the state matrices, the average of thereciprocal impedance must be
used:
60
Z−1sh ,
1
N
∫ N
0
1
Zsh
dτ
=1
N
N∑
n=1
[
∫ (n−1)+D
n−1
1(
C0 + C)dτ +
∫ n
(n−1)+D
1
C0dτ
]
=1
N
N∑
n=1
[
D(
C0 + C) +
1 − D
C0
]
=D
(
C0 + C) +
1 − D
C0
(4.26)
In these equations,C0 denotes the unswitched capacitance,C denotes the capacitance that
is added when the switch is closed, andD is the duty cycle.These values are incorporated
into the state matrices of equation (4.24), as follows:
A =
0 1 0 0
− km
− cm
0 −ΓZsh
m
0 0 0 1
0ΓZ−1
sh
L−
Z−1
sh
L−R
L
B =
cm
km−
(
cm
)2+ Γ2
mL
−ΓZ−1
sh
L
RΓZ−1
sh
L2 +ΓcZ−1
sh
mL
. (4.27)
4.3.4 MOSFETs and Switching Limitations
Before analyzing the results of the averaging method, it would be useful to review the
characteristics of metal-oxide-semiconductor field-effect transistors, or MOSFETs, when
used as switches. The results in the next section include a sample MOSFET switch in
addition to the averaged model and and ideal switch, as MOSFETs are the best means of
reproducing an ideal controlled switch in the physical realm.
61
MOSFETs are essentially voltage-controlled power switching devices. They consist of
two electrical terminals—the source and the drain—and a gate terminal, which controls the
flow of electrons. When no positive voltage is applied to the gate, there is no current flow,
making this situation the equivalent of an open switch. However, as a positive voltage is
applied to the gate node, it creates a channel through which electrons can flow from the
source terminal to the drain terminal. This is the equivalent of a closed switch. Therefore,
a square-wave switching signal voltage, as in Figure4.8, may be applied to the MOSFET
gate terminal in order to open and close the channel of electron flow.
The primary drawback to working with MOSFETs is that one mustaccount for an
equivalent resistance,Ron when the transistor is conducting. Most MOSFET switches are
selected based on both this on-resistance and the charge that must be supplied to the gate
drive circuit, which gives an indication of the maximum attainable switching speed.
4.3.5 Averaging Methodology Validation Results
Figure4.11, taken from a paper in review by Kim, Domme, Kurdila, Fuller and Stepa-
nyan [34], shows the performance of the averaged model compared to the corresponding
switching system simulation. The simulation uses the circuit shown in Figure4.10 and
employs a relatively low switching frequency of 200 Hz with an 80 percent duty cycle (that
is, D = 0.8). As can be seen from the figure, the averaged model very closely resembles
the simulation in the case of an ideal switch, even at low frequency. The MOSFET real-
ization of the switch does show a more noticeable difference, as it is not ideal. (As a note
of interest, the voltage exhibited by this simulation is greater than the power delivered to a
matched resistive impedance, as presented in section4.2, which serves as a good demon-
stration of the coupling effect between the transducer and the load impedance, as described
previously.)
The paper presents the theory that the averaged model becomes more accurate as the
switching frequency increases. Given an oscillation frequency of 12 Hz, however, the
62
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−5
−4
−3
−2
−1
0
1
2
3
4
5
Time [s]
Out
put [
V]
AveragingMULTISIM(MOSFET)MULTISIM(SWITCH)
Figure 4.11: Figure from Kim et al. [34] that compares the performance of theaveraging model (blue) to the simulated performance of bothan ideal switch(red) and a MOSFET switch (green).
switching frequency of 200 Hz is only one order of magnitude higher. The low level of
error exhibited in Figure4.11corresponding to this difference between the frequency of
oscillation and the switching frequency leads to the conclusion that model agreement may
improve even further at higher switching frequencies.
In conclusion, this figure shows that the averaging method derived in this chapter is an
accurate predictor of the performance of a switched electrical system. Use of this averaging
technique will allow for an extensive review of different semi-active switching topologies
that may enhance the energy harvesting capabilities of the reviewed electromagnetic trans-
ducer.
63
Chapter 5
Conclusions and Future Work
This chapter draws conclusions based on the content of the previous chapters and outlines
the possibilities for future work related to the findings that have been presented.
5.1 Conclusions
The object of this study was to characterize the performanceof a novel electromagnetic
transducer design and to develop a modeling method using averaging techniques that ex-
tends the approach to electromagnetic energy harvesting transducers.
5.1.1 Transducer Characterization
One of the primary concerns during analysis of the physics ofelectromagnetism and mag-
netic repulsion was whether the transducer system could be modeled as a linear system.
The analysis presented earlier concluded that a linear model would be sufficient for small
displacements of the central magnet, though equation constants regarding the magnet size
and magnetic field effects could not be measured. However, characterization efforts reveal
that a linear model provides excellent agreement with experimental data.
The functional form of the transfer function relating the output voltage to the input
64
acceleration excitation is given by equation (3.17). The response of this function proved to
be a better match to describe the experimental data than a “black-box” state space model,
as described earlier in the thesis.
Investigation of equation (3.17) leads to the conclusion that overall voltage output can
be increased by an increase in coil turns, which increases the overall flux linkage,Γ. This
may be a design consideration to consider in future prototyping efforts, especially if a
passive harvesting topology will be used, which would require significant levels of trans-
ducer output voltage. This principle is the most likely reason why the output of the small
transducer prototype was insufficient for the purposes of energy harvesting. It is unknown,
however, whether an increase in the flux linkage will affect the internal damping of the
transducer. In summary, the electrical output is limited byboth the flux linkage and the
damping coefficient, which encompasses both the mechanicaldamping and the electrical
damping due to induction.
5.1.2 Passive Energy Harvesting
Results from section3.5show that a majority of the available power output from the trans-
ducer is lost when using a full-wave rectifier to convert the signal to a direct current. Max-
imum recorded efficiency is only 17 percent and the transducer exhibited an effective op-
erating range of 12 - 24 Hz, in accordance with the resonant frequency of the transducer.
However, given an input acceleration of 3.75g, the transducer was able to produce an aver-
age of 5.5 mW of power, which could possibly power a small sensor or electronic system,
which is consistent with the conclusions presented in Ottman et al. [55] for a piezoelectric
transducer.
Given that such low efficiency was recorded using high-efficiency, low-voltage diodes,
it would be useful to study the capabilities of switching systems. Possibilities include elim-
inating diode rectification by using a switching AC-DC converter, or by using a switching
converter to boost voltage levels prior to rectification. Ineither case, the passive energy har-
65
vesting paradigm provides very little power transfer, which suggests alternatives in either
rectifier design or switching AC-DC conversion.
In summary, based on the findings of this thesis and the results given in Ottman et
al. [55], it would be worthwhile to further investigate the capabilities of the transducer
for energy harvesting using a different means of AC-DC conversion. Results of such a
study may reveal that the transducer could be a suitable energy harvesting device for a
high-vibration environment at frequencies near the transducer resonance.
5.1.3 Switched Systems and Averaging Methods
Switched systems have been extensively studied in the field of power electronics. Much
more recently, during the past six years, they have become a crucial component in a general
energy harvesting architecture. In contrast to conventional power electronics applications,
the switching component is fundamentally coupled to the transducer physics. New methods
for the analysis of the energy harvesting system in Figure1.1are required.
Moreover, switching is a likely means to increase the transferred power from the given
electromagnetic transducer. In order to predict the behavior of switched systems, it is useful
to consider the averaged state matrices, which describe thesystem independent of time. The
resulting state space problem is less computationally intensive than a time-dependent case,
thereby offering quick approximations of the performance of the switched system.
The thesis has carried out the derivation of a system of averaged equations for the given
electromagnetic transducer, as well as the derivation of anequivalent circuit that employs
the physics-based model of the transducer. It has been shownthat the averaged system does
provide a reasonable approximation of the full circuit simulations obtained in Multisim. As
a result, future simulations that determine the effects of some of the numerous switching
topologies available may be quickly and easily performed, either using mathematical meth-
ods or electrical simulation software.
66
5.2 Future Work
The next logical step in studying the energy harvesting capabilities of the presented electro-
magnetic transducer is to review the literature of power electronics to ascertain a reasonable
switching system topology that would increase the power transfer when compared to the
passive harvesting architecture. The modeling efforts documented in this thesis will allow
for a straightforward simulation of a switched system by using averaging techniques, and
the equivalent circuit model presented should accurately describe the system behavior. The
averaging method appears to do a good job in retaining the coupling effects between the
mechanical and electrical components of the transducer system.
After a documentation of the performance of a semi-active topology, further research
could include a performance review of both passive and semi-active harvesting with the
use of different storage elements, such as batteries or supercapacitors. In addition, further
prototypes may be designed for use with different operatingfrequency ranges, depending
on the expected vibration environment profile.
67
Appendix A
Singular Value Decomposition
Singular value decomposition (SVD) is a technique in linearalgebra that breaks a matrix
A into two square orthonormal sets of vectors weighted by a diagonal matrix of singular
values. That is,
A = UΣV T (A.1)
whereU andV are the matrices of left- and right-singular vectors, respectively, andΣ is the
diagonal matrix of singular valuesσ1 . . . σN. The matrixU comprises the linearly inde-
pendent unit vectors that describe the system, much like thefamiliar concept of structural
modes in the realm of vibration analysis.
The singular values of the system are the scalars that determine the influence of each
vector inU . If any particular singular values are of a significantly smaller magnitude than
the other singular values, the influence of the corresponding left-singular vectors may be
insignificant to the total system response. As a result, the system could be reasonably
approximated by a system of lower order.
This reasoning is used in the model order determination process of section3.3. For
each model ordern, the corresponding SVD of the system matrixA is carried out, and the
lowest singular valueσmin is reported. That is, the state space equation could be rewritten
68
as
x(t) = UΣVTx(t) + Bu(t) + Ke(t)
y(t) = Cx(t) + Du(t) + e(t). (A.2)
In this expression, the matrix productUΣ contains the scaled orthonormal system vectors.
If the minimum singular value along the diagonal ofΣ is especially low, it would indicate
that the model ordern is unnecessarily high, and that the corresponding scaled system
vector is insignificant relative to the net system output. The system could therefore be
accurately described using a reduced order.
69
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