piezo power gen

240 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 2, APRIL 2005

On Low-Frequency Electric Power GenerationWith PZT Ceramics

Stephen R. Platt, Shane Farritor, and Hani Haider

Abstract—Piezoelectric materials have long been used as sensorsand actuators, however their use as electrical generators is less es-tablished. A piezoelectric power generator has great potential forsome remote applications such as in vivo sensors, embedded MEMSdevices, and distributed networking. Such materials are capableof converting mechanical energy into electrical energy, but devel-oping piezoelectric generators is challenging because of their poorsource characteristics (high voltage, low current, high impedance)and relatively low power output. In the past these challenges havelimited the development and application of piezoelectric genera-tors, but the recent advent of extremely low power electrical andmechanical devices (e.g., MEMS) make such generators attractive.

This paper presents a theoretical analysis of piezoelectric powergeneration that is verified with simulation and experimentalresults. Several important considerations in designing such gen-erators are explored, including parameter identification, loadmatching, form factors, efficiency, longevity, energy conversionand energy storage. Finally, an application of this analysis ispresented where electrical energy is generated inside a prototypeTotal Knee Replacement (TKR) implant.

Index Terms—Piezoelectric materials, piezoelectricity, powergeneration, PZT ceramics.

I. INTRODUCTION AND RELATED WORK

MECHANICAL stresses applied to piezoelectric materialsdistort internal dipole moments and generate electrical

potentials (voltages) in direct proportion to the applied forces.These same crystalline materials also lengthen or shorten in di-rect proportion to the magnitude and polarity of applied electricfields.

Because of these properties, these materials have long beenused as sensors and actuators. One of the earliest practical ap-plications of piezoelectric materials was the development of thefirst SONAR system in 1917 by Langevin who used quartz totransmit and receive ultrasonic waves [1]. In 1921, Cady firstproposed the use of quartz to control the resonant frequency ofoscillators. Today, piezoelectric sensors (e.g., force, pressure,acceleration) and actuators (e.g., ultrasonic, micropositioning)are widely available.

The same properties that make these materials useful for sen-sors can also be utilized to generate electricity. Such materialsare capable of converting the mechanical energy of compression

Manuscript received August 4, 2003; revised October 11, 2003. This workwas supported by the Christina M. Hixon Fund.

S. R. Platt and S. Farritor are with the Department of Mechanical En-gineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0656 USA(e-mail: [email protected]; [email protected]).

H. Haider is with the Department of Orthopeadic Surgery, Univer-sity of Nebraska Medical Center, Omaha, NE 68198-1080 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TMECH.2005.844704

into electrical energy, but developing piezoelectric generators ischallenging because of their poor source characteristics (highvoltage, low current, high impedance). This is especially true atlow frequencies and relatively low power output.

These challenges have limited the use of such generators pri-marily because the relatively small amount of available regu-lated electrical power has not been useful. The recent adventof extremely low power electrical and mechanical devices (e.g.,microelectromechanical systems or MEMS) makes such gener-ators attractive in several applications where remote power isrequired. Such applications are sometimes referred to as powerscavenging and include in vivo sensors, embedded MEMS de-vices, and distributed networking.

Several recent studies have investigated piezoelectric powergeneration. One study used lead zirconate titanate (PZT) wafersand flexible, multilayer polyvinylidene fluoride (PVDF) filmsinside shoes to convert mechanical walking energy into usableelectrical energy [2], [3]. This system has been proposed for mo-bile computing and was ultimately able to provide continuously1.3 mW at 3 V when walking at a rate of 0.8 Hz.

Other projects have used piezoelectric films to extract elec-trical energy from mechanical vibration in machines to powerMEMS devices [4]. This work extracted a very small amount ofpower ( W) from the vibration and no attempt was made tocondition or store the energy. Similar work has extracted slightlymore energy ( W) from machine and building vibrations[5].

Piezoelectric materials have also been studied to generateelectricity from pressure variations in microhydraulic systems[6]. The power would presumably be used for MEMS but thiswork is still in the conceptual phase. Other work has used piezo-electric materials to convert kinetic energy into a spark to deto-nate an explosive projectile on impact [7]. Still other work hasproposed using flexible piezoelectric polymers for energy con-version in windmills [8], and to convert flow in oceans and riversinto electric power [9].

A recent medical application has proposed the use of piezo-electric materials to generate electricity to promote bone growth[10]. This work uses an implanted bone prosthesis containing apiezoelectric generator configured to deliver electric current tospecific locations around the implant. This device uses unreg-ulated (high voltage) energy and it is not clear if the techniquehas advanced beyond the conceptual phase.

The above studies have all had some success in extractingelectrical power from piezoelectric elements. However, manyissues such as efficiency, conditioning and storage have not beenfully addressed. This paper presents an extensive theoreticalanalysis of piezoelectric power generation that is not presented

1083-4435/$20.00 © 2005 IEEE

PLATT et al.: ON LOW-FREQUENCY ELECTRIC POWER GENERATION WITH PZT CERAMICS 241

in the above studies. The analysis is verified with simulation andexperimental results. Several important considerations in de-signing such generators are explored, including parameter iden-tification, load matching, form factors, efficiency, longevity, en-ergy conversion, mechanical amplification, and energy storage.Finally, an application of this analysis is presented where elec-trical energy is generated inside an orthopedic Total Knee Re-placement (TKR) implant [11].

The theoretical analysis is applicable to a wide range of forceinput waveforms with various frequencies. The experimental re-sults are limited to relatively high forces ( kN) and low fre-quencies ( Hz.). This domain of experiments is chosen be-cause of the practical limitations of experimental measurementsand because it is most applicable to orthopedic implants and sev-eral other applications (e.g., [3], [5], [6], [9]).

The experimental results were obtained by subjecting piezo-electric elements to various cyclic mechanical loads using asingle-axis Mini-Bionix MTS 858 test machine. This machineis equipped with a PID controller and is capable of applyingarbitrary force or displacement waveforms to the piezoelectricsamples. Forces are measured using a calibrated load cell, anddisplacements are measured using an onboard linear variabledifferential transformer (LVDT). A PC running a custom Lab-view application communicating to a 12-bit National Instru-ments 6025E PCI DAQ card was used to generate force wave-forms and to record all force, displacement, and voltage mea-surements.

II. ELECTROMECHANICAL MODEL

A. Piezoelectrics

Piezoelectric materials convert energy between the mechan-ical and electrical domains. A linear isothermal model that de-scribes the relationship between mechanical stress and strainand electric field and electrical displacement (charge per unitarea) was first proposed by Voight [12] and further developed byCady [13], Mason [14], and Jaffe and Berlincourt [15], amongothers. These constitutive relationships, and the techniques tomeasure piezoelectric material properties, were standardized inthe IEEE Standard on Piezoelectricity [16].

The linear constitutive relationships are typically representedby the tensor equations

(1)

where is the mechanical strain, is the electric displace-ment, is the mechanical stress, is the electric field, isthe elastic compliance in a constant electric field, is the piezo-electric strain constant, and is the dielectric constant underconstant stress.

The entire piezoelectric coupling is contained in the straincoefficients . If , the material is not piezoelectric andthere is no coupling between the electric and mechanical fields.A larger value indicates stronger electromechanical coupling.

The piezoelectric strain coefficient is the ratio of strain to ap-plied field or charge density to applied stress. Stated differently,

measures the charge created by a given force in the absence

of an electric field (short circuit electrical conditions), or thedeflection caused by an applied voltage in the absence of an ap-plied force (stress-free mechanical conditions).

In general, a material has three orthogonal coordinate axesalong which the electric field can be applied. For isotropic mate-rials, the displacement will be parallel to the applied field. How-ever, for anisotropic materials like piezoelectric ceramics, thisis not necessarily the case. Therefore, the dielectric permittivity

is a 3 3 matrix and the electric field and electric displace-ment are 3 1 vectors.

Similarly, a longitudinal stress applied to a piezoelectric rodwill not only cause the rod to become shorter and thicker,but it also tends to rotate about its longitudinal axis due toinduced shear strains that do not exist in isotropic materials.To completely describe this behavior requires 21 independentstress and strain coefficients [17]. Thus, the stress and strainparameters in (1) are 6 1 matrices, and is a 6 6 compliancematrix.

Many piezoelectric ceramics are manufactured or used inways such that one or two of the piezoelectric strain coefficientsdominate and the other interactions can be neglected. Therelationships expressed in (1) then reduce to a pair of coupledequations that can be written as

(2)

where the subscripts and indicate the direction of the gen-erated field in response to an applied force. By convention, theaxis labeled “3” is aligned with the poling axis. The current workuses a parallel compression generator where applied forces arealong the poling axis of the ceramic so that .

B. Electrical/Mechanical Model

It is convenient to develop analytical models that can pre-dict the response of these piezoelectric elements under differentinput conditions and for various material properties. A commonapproach for describing coupled electromechanical systems isto use lumped parameter models [18].

The principal assumption made in lumped parameter mod-eling is that the physical dimensions of the system are muchsmaller than the characteristic length scales of the governingelectromagnetic, mechanical, or acoustic phenomena [18]. Thisallows the spatial and temporal variations of the system to be de-coupled. The ideal piezoelectric coupling described by (2) canbe rewritten as

(3)

where for simplicity the subscripts are not explicitly included,is the thickness of the material, and is the change in thick-ness. Assuming a sinusoidal steady-state operating condition,(3) can be differentiated with respect to time, transformed to theLaplace domain, and rearranged to yield

(4)

(5)


Fig. 1. Representation of piezoelectric coupling.

wherevelocity (m/s);

current (C/s);short circuit compliance (m/N);free capacitance (C/V);

volts (V);force (N).

The relationships described by (4) and (5) can be representedby the circuit shown in Fig. 1 [18]. In this circuit, the couplingbetween the mechanical and electrical domains is modeled asan ideal transformer (derivation in the Appendix ).

The transformer ratio can be thought of as the ratio of elec-trical output to mechanical input (V/N), and it can be determinedfrom the piezoelectric strain constant and the physical com-pliance of the element [18]

(6)

It is important to recognize that the mechanical and elec-trical properties of piezoelectric materials change with load. Forstress-free conditions (i.e., ), the charge gener-ated by an applied potential is given by

(7)

On the other hand, under blocked conditions (i.e., displace-ment constrained to be zero), the generated charge is given by

(8)

where is the electromechanical coupling factor. Thus, theblocked capacitance is times smaller than the free capac-itance. Similarly, the mechanical compliance under open circuitelectrical conditions is times smaller than the short cir-cuit compliance.

The circuit shown in Fig. 1 is an idealized representation ofthe piezoelectric effect, and it is most accurate only at low fre-quencies. Because this equivalent circuit neglects internal en-ergy storage and loss mechanisms, several additional factorsmust be added to model the dynamic performance of piezoelec-tric elements at higher frequencies [18].

The mechanical properties of a real piezoelectric elementcan be modeled by a simple spring, mass, and damper system,shown in Fig. 2. The spring represents the ideal compliance,the mass is an energy storage element, and the damper accountsfor internal mechanical losses.

Fig. 2. Lumped parameter model of mechanical system.

Using Newton’s second law and the free body diagram shownin Fig. 2, the equation of motion for this system is given by

(9)

whereexternal force (N);effective mass (kg);damping (N s m);stiffness compliance (N/m);mass displacement (m);mass velocity (m/s);mass acceleration (m s ).

Taking the Laplace transform of (9) yields the system transferfunction given by

(10)

Because mechanical impedance is defined as the ratio of theforce to the velocity of an element, these elements can be incor-porated into the ideal representation of piezoelectric couplingas shown in Fig. 3(a). Reflecting the mechanical elements ofthe lumped parameter model to the electrical side as shown inFig. 3(b) yields equivalent electrical circuit element values givenby

ohms (11)

kg henries (12)

farads (13)

Volts (14)

farads (15)

The extra term , shown in Fig. 3(b), accounts for theimpedance of a load attached to the electrical output of thepiezoelectric element. Here the load is shown as purely resis-tive, although that is not always necessarily the case.

This model is still missing several factors such as dielectricloss. It is, in effect, a simplified linear representation of whatis generally a much more complex nonlinear system. Piezo-electric ceramics have many different resonant modes, each of


Fig. 3. Piezoelectric circuit model.

which would be characterized by a unique set of mechanical pa-rameters. Hence, this model is likely valid only at frequenciesbelow the first resonant mode. Furthermore, piezoelectric mate-rial constants are not constants at all as they depend in a non-linear fashion on conditions such as mechanical and electricalboundary conditions, and the frequency and amplitude of theexcitation force profile. These effects must be considered whenusing this model to predict piezoelectric response under con-ditions significantly different from those for which the modelparameters are determined.

III. EXPERIMENTAL FIT OF MODEL PARAMETERS

All of the piezoelectric parameters in (11)–(15) can in prin-ciple be determined from the various piezoelectric material con-stants and physical dimensions of the element. There are, how-ever, a number of concerns with this approach. The impedancemeasurements used to determine piezoelectric parameters areusually small signal excitations at frequencies around 1 kHz.Therefore, these small-signal values may not be valid for high-load low-frequency applications. Furthermore, because the pa-rameters are known to change with external loading conditions,the values of the equivalent circuit elements can also be expectedto depend on the electrical and mechanical boundary conditions.

For these reasons, the values of the parameters used inthis model are experimentally determined. The response of apassive electrical network such as the one depicted in Fig. 3(b)can be written in terms of the circuit elements and input signal.If the values of the circuit elements are unknown, they canbe determined by measuring the system response to a knowninput. By modeling the behavior of a piezoelectric transduceras an analogous equivalent electrical circuit, the physicalconstants of the material can be determined by measuring itselectrical response to a known input and using (11)–(15) torelate the equivalent electrical parameters back to the physicalmechanical properties.

Under a sinusoidal input force profile, the amplitude of thevoltage across the resistive load shown in Fig. 3(b) is given by

(16)

TABLE IPARAMETER FIT

The amplitude of the input voltage , the parallelelectrical impedance of the piezoelectric and the load

, and the total circuit impedance. In terms of the circuit parameters in Fig. 3(b), the indi-

vidual impedances are given by

(17)

(18)

(19)

Equation (16) can be expanded and expressed in terms ofthe blocked electrical capacitance of the piezoelectric material( ), the electrical equivalents ( , , and ) of themechanical properties ( , , and ), and the transformerratio . The unknown values of the four electrical elements andthe transformer ratio are then determined by performing a non-linear least squares fit to the measured output voltages from apiezoelectric element under a known sinusoidal input force pro-file for various values of the load resistor, .

This approach was used to determine the parameters for acommercially available piezoelectric element. This rectangular(10 mm 10 mm 18 mm) element is composed of multiple( ) wafers of lead zirconate titanate (PZT) ceramic (theadvantage of using an element of this form is discussed in Sec-tion IV).

The inset in Fig. 4 shows output voltage time history for a1-Hz sinusoidal input force of amplitude 800 N. The averagesteady-state amplitude of the voltages measured across variousload resistors are shown as squares. The curve represents themodel fit.

The solid curve in Fig. 4 is the output predicted by (16) usingthe best fit parameters listed in the first column of Table I. Thesecond column of Table I lists the values obtained for the lumpedparameters calculated using the physical dimensions of the ele-ment and the piezoelectric constants for the PZT type 5 H mate-rial used in this element. There is excellent agreement betweenthe two sets of values except for the mechanical damping coef-ficient which is not significant at these frequencies.


Fig. 4. Voltage across the resistive load and the model fit.

This electromechanical model can be expected to providesome qualitative insight into the behavior of the piezoelectricelements under conditions in the neighborhood of that forwhich the model parameters are determined. Once verified,the model can then be used with other simulation tools to helpoptimize power conditioning and storage strategies withoutthe need to physically incorporate the piezoelectric elementsduring the early design stages.

IV. RAW POWER

A. Time History and Impedance Matching Plot

The instantaneous raw electrical power dissipated in theresistive load shown in Fig. 3(b) is given by . Thetotal power generated depends on the resistive load and appliedstress. The main graph in Fig. 5 shows the average powergenerated for a 1-Hz 800-N amplitude sine wave force inputas a function of the resistive load. The squares are the experi-mentally determined values, and the solid curve is the outputpredicted by the electromechanical model. The inset in Fig. 5shows representative time-series results for the instantaneousvoltage (middle plot) and power (bottom plot) delivered to a14.8-k resistive load. The horizontal line is the average powerdelivered to the load ( 1.08 mW.)

These results show the importance of matching the load tomaximize the power generated. At low values of resistance novoltage is produced (short circuit) and no power is generated. Athigh resistances (open circuit) no current flows and no power isgenerated.

B. Form Factors: Stacks and Transformers

As described in the mathematical model, the size and shapeof the piezoelectric element affect the amount of power gener-ated. First, for a given load, the mechanical stress (and strain) isrelated to the cross sectional area. The voltage produced is re-lated to the stress, and the power (for a resistive load) is relatedto the square of the voltage. Second, for a given stress the chargedisplacement is related to the volume of the element [ and in(7)].

The mechanical design of a piezoelectric element also hasimportant implications on the electrical source characteristics.Consider the two commercially available forms of PZT piezo-electric ceramic shown in Fig. 6. The element on the left is ahomogenous monolithic cylinder 1.0 cm in diameter and 2.0cm long (Sensor Technology™). This form has a low capaci-tance ( 47 pF) and produces a very high open circuit voltage( 10 000 V). The second element is manufactured using a pro-prietary technique by stacking multiple ( 145) PZT wafers.Each wafer has a pair of electrodes and they are assembledmechanically in series but electrically in parallel (Piezo Sys-tems™). The wafers are polarized along their thickness so thatthey exhibit a piezoelectric effect only in this direction. Thesestacks are typically used as actuators, and the multiple layershave the effect of increasing the effective capacitance (1–10 F)and decreasing the peak open circuit voltage ( 30 V). The stackis 1.0 cm square and 1.8 cm long.

The effect of this mechanical design is shown in Fig. 7. Thecurve on the left shows the experimental and simulation resultsfor the stack (similar to Fig. 6, right). The curve on the right


Fig. 5. Power as a function of resistive load: 1-Hz 800-N sine wave.

Fig. 6. Form factors: monolithic and stacks.

shows simulation results for a monolithic element of exactlythe same material, volume, and cross-sectional area (similar toFig. 6, left). Only simulation results are shown because a mono-lithic cylinder of the same material, volume and cross sectionalarea was not commercially available. However, the general char-acteristics of cylindrical monolithic elements were experimen-tally verified using the Sensor Technology PZT type 4 elementin a series of experiments and simulations as described in [11].

It is important to notice that under the same force loadingconditions, both elements produce the same power into matchedresistive loads ( 1.1 mW), although the matched load is muchlower ( k ) for the stack than the cylinder ( G ).

Fig. 7. Average power delivered to R for the stack (left) and the monolithicelement (right).

Stacking several elements mechanically in series (sharingthe same stress; strain is additive) and electrically in parallel


(sharing the same voltage; current is additive) has the effectof stepping down peak voltages and stepping up the effectivecapacitance compared to the monolithic element. This is ineffect a “mechanical transformer.” A similar “mechanical trans-former” effect could be obtained by using other mechanicalgeometries such as rows of piezoelectric film on the surface of abeam in bending. The same electrical effect can be obtained byusing an electrical transformer to reduce the output voltage ofthe monolithic element (although, because of the high voltages,arcing would be a problem).

These results show that: 1) it is critical to match the circuitload with the type of generating material, and 2) stacked andmonolithic elements of the same geometry produce the samepower into matched loads. These results also indicate that thestacked element is preferred because of the lower matching elec-trical load required. The stacked element reduces the outputvoltage to a more manageable level, and the higher capacitanceof this form is important for energy transfer (discussed in Sec-tion VI). For these reasons, the piezoelectric stacks are used inthe remainder of this work.

V. EFFICIENCY OF POWER CONVERSION

The network depicted in Fig. 3(b) can be used to explore theoverall efficiency of converting mechanical power into electricalpower. Because the voltage and current will be in phase acrossthe load resistor, the average power delivered to the load is givenby

(20)

The most conservative estimate of the efficiency for powertransfer to the load is given by the ratio of the average outputpower to the magnitude of the total apparent input power

(21)

where the root mean square (rms) amplitude of the input currentis given by

(22)

The surface of Fig. 8 shows numerical results calculated ac-cording to (21) for the overall electromechanical efficiency forthe raw power produced by the piezoelectric stack. Again, thesurface represents the simulation results and the squares and as-sociated lines represent individual experiments. The input forceprofile was sinusoidal with 800-N amplitude, and the efficiencywas calculated for a range of frequencies and load resistors,using the best fit model parameters as listed in Table I.

The results of this evaluation indicate that significant efficien-cies are obtained only along the narrow ridge of load resistorvalues for which the source and load impedances match (as inSection V). As the input frequency increases, the maximum effi-ciency occurs at smaller load resistor values. This is the behaviorexpected for an essentially capacitive device; as the excitationfrequency increases, the source impedance, ,will decrease.

Fig. 8. Efficiency as a function of frequency and resistive load.

There is also a trend toward higher efficiency as the excitationfrequency increases. The efficiency peaks as would be expectedat the resonant frequency of about 34 kHz. Note, however, thatany conclusions drawn from these trends must be qualified by therestrictionsandlimitationsof themodelused(asdiscussed inSec-tion II). In particular, the behavior of piezoelectric ceramics isgenerally nonlinear, and this is especially true at and around theresonant frequency. Furthermore, the electromechanical modelparameters are not constant as the input frequency increases.

Given the restrictions at higher frequencies, the validity ofthe model was verified by comparing the predicted efficiency tothat actually observed, for lower frequencies. For a sinusoidalinput force with rms amplitude , the total apparent inputmechanical power, , can be calculated directly as

(23)

where is the rms amplitude of the velocity, determinedfrom the measured displacement of the element. The outputpower and efficiency are calculated as usual using (20) and (21).

Measurements were made for a range of input frequenciesand resistive loads. Fig. 9 shows the predicted and actualefficiencies for 1, 5, and 10 Hz, and the solid lines and circlesshown on Fig. 8 show the experimentally determined efficien-cies superimposed on the complete numerically calculateddataset. The agreement between the model and experimentallymeasured data points is excellent and serves to further validatethe electromechanical model at these frequencies.

VI. ENERGY CONVERSION AND STORAGE

A. Introduction

Efficiently transforming the raw power from a piezoelectricgenerator into useful power is not straightforward. The resultspresented in Sections V and VI show that the low-frequencyproperties of this type of generator are characterized by highlycapacitive source impedance, and low-current–high-voltage ac(bipolar with large ripple) electrical output. This is clearly notappropriate for electrical components and sensors without sig-nificant signal conditioning. This section describes some impor-tant issues in conversion and storage of the generated energy.


Fig. 9. Electromechanical efficiency.

B. Source Capacitance

One method to smooth electrical signals and store energy isto include a capacitor (and rectifier to convert the bipolar piezo-electric output into a unipolar signal) in parallel with the resis-tive load. The capacitor stores energy when is high andreleases energy when is low. Such a capacitor might alsorepresent other forms of storage such as a battery that is chargedby the piezoelectric generator.

A difficulty with this approach is that the capacitance of thepiezoelectric ( ) makes the direct transfer of energy into astorage capacitor ( ) inefficient. Consider the simple modelshown in Fig. 10.

Here the piezoelectric is charged (compressed) to create aninitial potential and an initial charge . The initial systemenergy (assuming no initial energy on the storage capacitor

) is given by

(24)

After the switch is closed both capacitors will have the samevoltage and the charge will distribute according to the mag-nitude of the capacitors. The final stored energy, assuming nolosses, is given by

(25)

Therefore, the final system energy is reduced. These lossescan occur in the arcing of the switch or as resistive losses inthe wires. If half the energy is lost and the energytransferred to the storage capacitor is

(26)

Fig. 10. Energy transfer to a storage capacitor.

Because piezoelectric materials at low frequencies are essen-tially purely capacitive devices, energy transfer losses can beminimized by reducing the size of the storage capacitor. How-ever, in this case, no energy is transferred. Maximum energy istransferred when and the maximum value is .

This is a very simplified model of energy transfer from apiezoelectric generator, but it exemplifies the inefficiency in-herent in the process. Other studies have addressed this problemby allowing the piezoelectric to build to a peak voltage beforeclosing the switch [3]. This increases ( ) and, therefore,increases energy transferred to the storage capacitor. However,efficiency is still determined by matching capacitances.

C. RC Loading

To quantify the inefficiencies associated with a direct energytransfer approach, three piezoelectric stacks were connected toa load consisting of a storage capacitor in parallel with a re-sistive load . Full-wave bridge rectifiers were also insertedbetween the piezoelectric elements and the load to convert thebipolar piezoelectric output into a unipolar signal. The piezo-electric stack combination was subjected to a 2600-N amplitudeinput force, and the power delivered to a resistive load by thegenerator was determined for a wide range of storage capaci-tors and electrical loads.

Fig. 11 shows the design space with simulation (mesh)and laboratory results for cross sections of constant load re-sistance ( k ) and constant storage capacitance( F).

For low values of the storage capacitor, the maximum powerdelivered to a matched resistive load approaches the raw powerresults (Section IV) with the difference being due to losses inthe rectifiers. Small values of in effect reduce the system toa rectified version of the raw power experiment and maximizethe power generated. However, a small does not store muchenergy and will have little effect on the level of output ripple inthe time domain.

Increasing the value of the storage capacitor reduces theamount of ripple in the time domain but also reduces theenergy transfer efficiency. Using a 10 F capacitor reduces theefficiency of this circuit to about 40% compared to the totalraw power delivered by three piezoelectric elements to a purelyresistive load. Furthermore, examination of the time-seriesdata shows that the voltage across the load still has significant( 100%) ripple. The level of ripple is only significantly re-duced for larger values of , but then energy transfer efficiencyrapidly approaches zero.

The surface in Fig. 11 shows that maximum energy is deliv-ered to the load resistor when the storage capacitor is small and


Fig. 11. Power generated versus R and C .

the load resistor matches the impedance of the source. Any com-bination of load resistor and storage capacitor chosen to mini-mize the level of ripple will move the operating point into re-gions of low energy transfer. These results illustrate some of thedifficulties associated with low frequency piezoelectric powergenerators. While direct transfer of energy from a piezoelectricelement is possible, especially for high capacitance stacks likethe ones used in this work, the highest efficiencies will only berealized with power conditioning systems optimized to addressthe capacitive nature of piezoelectric elements.

D. Electrical Conditioning

The above analysis is important because the efficient use ofa storage capacitor (or battery) is important to supply energywhen the piezoelectric element has low potential. However, acapacitor alone may not provide sufficient signal conditioning.Therefore, a combination of a capacitive storage device and avoltage regulator may be required.

Voltage regulators are well-established devices used to createan output voltage that is constant and independent of the loadand supply variations (e.g., [19]). They appear to the source asconstant-current loads and are very effective at reducing rippleby eliminating voltage peaks.

The proper combination of storage capacitor and voltage reg-ulator, as shown in the application in Section IX, can be usedto condition the piezoelectric output to produce a more useablepower supply.

VII. LONGEVITY

It is well known that repeated mechanical and electrical cy-cling of piezoelectric ceramics results in a progressive degra-dation in performance as a function of time [20]–[22]. Duringtypical operation, a piezoelectric device experiences repeatedreversals of the electric drive field and/or mechanical load, alongwith cyclic variations in temperature. These effects contribute toperformance degradation through mechanisms such as the loss

of mobility in domain walls, depoling, the formation of micro-cracks, and the failure of the electrode-ceramic interface [23].

The degradation in the performance of piezoelectric ceramicsis often characterized in terms of logarithmic functions of time.The actual changes in performance depend on application spe-cific details such as the amplitude, frequency, and duration ofthe applied electrical and mechanical loads, as well as the par-ticular composition of the PZT material [24].

Limited data currently exist on the performance characteris-tics of piezoelectric ceramics subject to the low frequency, highmechanical loading conditions that are likely to exist in manypotential applications for piezoelectric power generation. In anattempt to characterize the long-term performance of piezoelec-tric elements under these conditions, the raw electrical powerdelivered to a resistive load was monitored for approximately1.5 million cycles. A single Piezo Systems™ piezoelectric stackwas subjected to a 440-N amplitude composite input force pro-file containing several low ( 20 Hz) frequency componentswhile its electrical output was monitored across a 29.5-k re-sistive load. The applied force profile is the expected loading inthe application discussed in Section IX. The force profile wasperiodic at a frequency of 1 Hz so the number of cycles corre-sponds directly to time measured in seconds. The average powerdelivered to the resistive load, shown in Fig. 12, was calculatedevery 30 s for the first 60 min, and then every 30 min thereafter.

Several features are immediately apparent. Little or no degra-dation in electrical output is obvious during approximately thefirst cycles. Subsequently, the output decreases approxi-mately logarithmically per decade of time. It was also observed,but not shown in this plot, that a pause in the experiment (30 minor more) can cause significant recovery of the piezoelectric gen-erator [11]. This experiment suggests that: 1) this piezoelectricgenerator would create useful power for millions of cycles, and2) the exact amount of decay in performance is application de-pendent.

The results shown in Fig. 12 are not general—the longevityof the generator depends on several application specific factors


Fig. 12. Power as a function of time.

Fig. 13. Mechanical amplification examples.

(e.g., loading stress, force profile, excitation frequency, temper-ature variations). The results do verify that the logarithmic pro-gressive degradation of piezoelectric materials is also seen whenthese materials are used as power generators.

VIII. MECHANICAL AMPLIFICATION

The electrical energy stored by a fixed volume of piezoelec-tric material can be written using (2) as

(27)

where is the volume of the element and is the squareof the applied stress.

Equation (27) implies that more energy is available from alarger element. This is similar to stating that a larger batteryhas more energy. However, (27) also dictates that the avail-able energy is proportional to the square of the applied stress.The implication is that not only is material selection impor-tant in maximizing available power, the details of form factorand force application are as well. More power may be availableby choosing a force amplification scheme applied to fewer orsmaller piezoelectric elements, rather than attempting to maxi-mize the volume of material.

Fig. 13 shows two techniques that can be used to increasethe power generation for a given load profile. In the left figurea locking mechanism is used to amplify the axial force in eachlink and thereby increase the strain in the piezoelectric. In thefigure on the right, the relatively low resonance of a mechanicalcantilever can be used to increase the strain delivered to a piezo-electric film on the surface of the cantilever (similar to [5]).

Fig. 14. Self-powered TKR model and test setup. (a) Components of theimplant model. (b) Test setup.

It should be noted that while the power generation is increasedwith higher strain, the longevity of the piezoelectric material isdecreased.

IX. POWER GENERATION IN TOTAL KNEE REPLACEMENT

IMPLANTS

The principles of piezoelectric power generation describedabove are now applied to the design of a self-powered TotalKnee Replacement (TKR) implant. Sensors encapsulated withinimplants could provide in vivo diagnostic capabilities such asthe monitoring of implant duty (i.e., walking) cycle, detectingabnormally asymmetric or high forces, sensing misalignmentand early loosening, and early detection of wear. Early diagnosisof abnormalities is critical to minimize patient harm. Becauseimplants are expected to last more than 20 years, the lack of along-term self-contained power supply is a major limitation toimplant sensors.

A simplified design of a self-powered instrumented implant isshown in Fig. 14. This design is not intended to fully address thecomplex details of implant design, but rather to demonstrate thefeasibility of piezoelectric power generation. Many additional


Fig. 15. Electrical design.

constraints (e.g., wear, bio-fouling, off axis loading) must beconsidered before clinical implementation is possible.

A typical “fixed bearing” TKR implant has three major com-ponents. The femoral component is a highly polished hard sur-face that is attached to the femur. The tibial tray is attached tothe tibial bone and supports a low friction polyethylene bearingsurface. The tray has a stem that extends into the bone to helpsupport moments and nonaxial forces. Relative motion betweenthe femoral and bearing surfaces allows the knee to function.Fig. 14(a) shows a tibial tray, similar to traditional TKR de-signs, that has been modified by making it deeper and includingthree stems to accommodate three piezoelectric stacks. Force isapplied by the femoral component to the bearing surface anda distribution plate is used to transfer the load to the three ele-ments. The elements are approximately located so an axial forceis evenly distributed in each element. However, in a real im-plant, the forces are multi-axial and the contact points betweenthe femoral component and the bearing move with any combi-nation of flexion, anterior–posterior motion, and axial rotation.Therefore, the forces would not always be equally distributed.The electrical design discussed below will accommodate forvarious force distributions. Fig. 14(b) shows the implant modelmounted on an anatomically correct synthetic tibia and femur(Sawbones™).

Output voltage was regulated with an active voltage regu-lator, as shown in the electrical circuit in Fig. 15. The outputof each of the three piezoelectric elements was rectified usingfull-wave bridge rectifiers. This unipolar output was then fil-tered and stored with a 10- F storage capacitor. This capacitoris chosen to approximately match the effective capacitance ofthe three elements.

A simple low-power MAX 666 linear regulator was used(more complex switching regulators will likely be incorporatedin future work). The output of the regulator was used to powera PIC 16LF872 microprocessor. This is a basic low-powerprocessor with both digital I/O and analog inputs.

Tests were performed where the standard 2600-N amplitudeforce profile [24] for the axial force in the knee during normalwalking was applied to the TKR model. As much as 4 mWof raw electrical power and 0.85 mW of regulated power wasgenerated in this application. The overall electrical efficiency indelivering regulated power was approximately 19%. Additionaltests were performed with the amplitude of the applied forcereduced by 50%. The top plot in Fig. 16 shows three cycles ofthe applied ISO axial force profile for normal walking [25].The middle plot shows the regulated output voltage from theMAX 666. The bottom plot shows the output status of an LEDattached to the microprocessor. The PIC microprocessor wasprogrammed to turn the LED on for a fixed period duringthe applied force cycle. This output then represents implantduty. The processor could perform many other operationssuch as measuring peak forces and/or force distributions bymonitoring the rectified voltage of the piezoelectric elements.The processor could also store the necessary data in onboardnonvolatile memory until it could be downloaded during thepatient’s next clinical visit.

Even under this reduced force condition, approximately225 W of continuous regulated power is available. This ismore than adequate to satisfy the approximately 50- W powerrequirement of the PIC 16LF872.

The experimental results presented here use the energy as it isgenerated. The generator could also be used, based on the anal-ysis presented in Section VI, to charge a battery so that energyis available when the patient is not walking.

X. CONCLUSION

Piezoelectric power generators have great potential forsome remote applications such as in vivo sensors, embeddedMEMS devices, and distributed networking. However, devel-oping a piezoelectric generator is challenging because of theirpoor source characteristics (high voltage, low current, high


Fig. 16. Regulated power output and microprocessor output.

impedance) and relatively low power output. The recent adventof extremely low power electrical and mechanical devices (e.g.,MEMS) makes such generators attractive.

This paper presents a theoretical analysis of power generationwith PZT ceramics that is verified with simulation and exper-imental results. Several important considerations in designingsuch generators are explored, including parameter identifica-tion, load matching, form factors, efficiency, longevity, energyconversion, and energy storage. Finally, an application of thisanalysis is presented where electrical energy is generated insidea prototype Total Knee Replacement (TKR) implant.

APPENDIX

That the circuit shown in Fig. 1 accurately represents thepiezoelectric coupling equations can be seen as follows. Ap-plying Kirchoff’s current law to the node on the electrical sideof the circuit yields

(28)

Applying the mechanical analog of Kirchoff’s voltage law tothe mesh on the mechanical side of the circuit yields

(29)

Using the fact that

(30)

(28) can be solved for the velocity

(31)

Substituting this result into (29) and recognizing that

(32)

yields for the current

(33)

which is the same as (5) if

(34)

Substituting (33) into (31) yields the remaining piezoelectriccoupling equation, (4).

ACKNOWLEDGMENT

The experimental work was performed at the BiomechanicsLaboratory in the Department of Orthopedic Surgery at the Uni-versity of Nebraska Medical Center. The authors would like toacknowledge the support of Dr. K. Garvin, M.D., Chairman ofthe Department of Orthopedic Surgery, UNMC.

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[7] T. G. Engel, “Energy conversion and high power pulse production usingminiature piezoelectric compressors,” IEEE Trans. Plasma Sci., vol. 28,no. 5, pp. 1338–1340, Oct. 2000.

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[25] Implants for Surgery–Wear of Total Knee-Joint Prostheses–Part 1:Loading and Displacement Parameters for Wear-Testing Machines withLoad Control and Corresponding Environmental Conditions for Test,ISO 14243-1:2002(E), Mar. 15, 2002.

Stephen R. Platt received the B.A. degree inphysics and astronomy from Williams College,Williamstown, MA, in 1983, the M.S. and Ph.D.degrees in astronomy and astrophysics from theUniversity of Chicago, Chicago, IL, in 1991, andthe M.S. degree in mechanical engineering from theUniversity of Nebraska-Lincoln in 2003.

He is currently a Research Assistant Professorin the Department of Mechanical Engineering,University of Nebraska-Lincoln. His interests in-clude biomedical sensors, surgical robotics, and

millimeter-wave detector system for observational astrophysics applications.

Shane Farritor received the B.S. degree inmechanical engineering from the University ofNebraska-Lincoln in 1992, and the M.S. and Ph.D.degrees in mechanical engineering from the Mass-achusetts Institute of Technology, Cambridge, in1998.

He is currently an Assistant Professor in the De-partment of Mechanical Engineering, University ofNebraska-Lincoln, and holds courtesy appointmentsin both the Department of Surgery and the Depart-ment of Orthopaedic Surgery at the University of Ne-

braska Medical Center, Omaha. His research interests include space robotics,surgical robotics, biomedical sensors, and robotics for highway safety.

Dr. Farritor serves on both the AIAA Space Robotics and Automation tech-nical committee and the ASME Dynamic Systems and Control Robotics Panel.

Haini Haider was born in Baghdad, Iraq, in 1960.He received the B.Eng. and Ph.D. degrees in mechan-ical engineering from the University of Sheffield,Sheffield, U.K., in 1983 and 1990, respectively.

He was on the faculty of the University ofSheffield from 1988 to 1996, where he became aLecturer in 1992. His research then encompassedfluid dynamics, mechatronics, robotics, and infor-mation technology. In 1997, he joined the facultyof University College London at the Centre ofBiomedical Engineering in Stanmore. He was the

principal mechanical and software engineer on the team of Dr. Peter Walker thatproduced the Instron-Stanmore Knee Simulator and the International StandardsOrganization (ISO) method for simulation and wear testing of knee replacementsystems. He joined the faculty of the University of Nebraska Medical Center,Omaha, in March 2000 as an Associate Professor of Biomedical Engineering atthe Department of Orthopaedic Surgery and Rehabilitation. He is the author ofover 65 publications and international conference presentations in engineeringand orthopedics biomechanics research. His current research interests involveinnovative implant design, testing, computer-aided simulation, preoperativeplanning, and image-guided and robotic surgery.

Dr. Haider is a Member of the Engineering Council (U.K.), Member of theInstitution of Mechanical Engineers (I.Mech.E) (U.K.), Member of the Amer-ican Society of Testing and Materials (ASTM) and co-chairs its committee forKnee Implant Wear Testing, Chair of the International Standards OrganizationReview Workgroup of ISO 14243-1 and 14243-2, Member of the InternationalSociety of Technology in Arthroplasty (ISTA), and Member of the OrthopaedicResearch Society (ORS). He is the winner of academic prizes including theKlinger International Research Prize, Siebersdorf, Austria.

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