Application Note 81

AN81-1

Ultracompact LCD Backlight InvertersA Svelte Beast Cuts High Voltage Down to Size

Jim Williams, Linear Technology Corporation

Jim Phillips, Gary Vaughn, CTS Wireless Components

September 1999

INTRODUCTION

The liquid crystal display (LCD) has become ubiquitous. Itis in use everywhere, from personal computers of all sizesto point-of-sale terminals as well as instruments, autosand medical apparatus. The LCD utilizes a cold cathodefluorescent lamp (CCFL) as a light source to back light thedisplay. The CCFL requires a high voltage AC supply foroperation. Typically, over 1000 volts RMS is required toinitiate lamp operation, with sustaining voltages rangingfrom 200VAC to 800VAC. To date, the high voltage sectionof backlight “inverters” has been designed around mag-netic transformers. A great deal of effort has been directedtowards these ends, accompanied by a large volume ofdescriptive material.1 Unfortunately, as available circuitboard space continues to shrink, magnetic transformerbased approaches begin to encounter difficulty. In par-ticular, it is highly desirable to fashion laptop computerswith large area screens, leaving little room for the back-light inverter board. In many cases there is so little spaceavailable that building the inverter function inside the LCDpanel has become attractive, although to date impractical.

Limitations and Problems of Magnetic CCFLTransformers

Construction and high voltage breakdown characteristicsof magnetic transformers present barriers to implement-ing them in these forthcoming space intensive designs.Additionally, as refined as magnetic technology is, otherinverter problems associated with it also exist. Suchproblems include the necessity to optimize and calibratethe inverter for best performance with a given display type.Practically, this means that the manufacturer must, viaeither hardware or software, adjust inverter parameters toachieve optimum performance with a given display type.Changes in display choice must be accompanied by com-

, LTC and LT are registered trademarks of Linear Technology Corporation.Piezoceramic inverter circuits and techniques are covered by LTC patents issued and pending. CTSWireless Components (formerly Motorola Ceramic Products)patents, issued and pending, alsoapply.

mensurate adjustments in inverter characteristics. An-other problem area is fail-safe protection due to self-destructive transformer malfunctions. Finally, the magneticfield provided by conventional transformers can interferewith operation of adjacent circuitry. With the exception ofsize, all of these problems are addressable, althoughincurring economic and circuit/system penalties.2 What isreally needed is high voltage generating capability that isinherently better suited to coming generations of backlightinverters. Piezoceramic transformers, an arcane and littleknown technology, has been tamed and made available forCCFL inverter use. This publication summarizes results ofan extensive collaborative development effort betweenLTC and CTS Wireless Components (formerly MotorolaCeramic Products).

Piezoelectric Transformers

Piezoelectric Transformers (PZT), like magnetic devices,are basically energy converters. A magnetic transformeroperates by converting an electrical input to magneticenergy and then reconverting the magnetic energy back toan electrical output. A PZT has an analogous operatingmechanism. It converts an electrical input into mechanicalenergy and subsequently reconverts this mechanical en-ergy back to an electrical output. The mechanical transportcauses the PZT to vibrate, similar to quartz crystal opera-tion, although at acoustic frequencies. The resonanceassociated with this acoustic activity is extraordinarilyhigh; Q factors over 1000 are typical. This transformeraction is accomplished by utilizing properties of certainNote 1: See References 1 through 3 for examples.Note 2: Again, see References 1 through 3.

Application Note 81

AN81-2

ceramic materials and structures. A PZT transfomer’svoltage gain is set by its physical configuration and thenumber of layers in its construction. This is obviously verydifferent from a magnetic transformer, although some(very rough) magnetic analogs are turns ratio and coreconfiguration. Also very different, and central to anyserious drive scheme attempt, is that a PZT has a largeinput capacitance, as opposed to a magnetic transformer’sinput inductance.3

Figures 1 and 2 show PZTs; the surprisingly small size isreadily apparent. The form factor is ideal for constructingspace efficient CCFL backlight inverters. A complete, prac-tical inverter appears in Figure 3.Note 3: To call this description of PZT operation abbreviated is thekindest of verbiage. Those interested in piezoceramic theory, whethersavant or scholar, will find tutorial in Appendices A and B. Appendix Ais a brief treatment; B is considerably more detailed. Both sectionswere written by Jim Phillips of CTS Wireless Components.

AN81 F02.tif

AN81 F01.tifFigure 1. Piezoceramic Transformers Compared to a Dime for Size. 1.5 Watt (Photo Upper) and10 Watt (Photo Lower) Units Are Shown. Devices Are Narrower Than Magnetic Transformers

Figure 2. Figure 1’s Piezoceramic Transformers Rotated to ShowHeight. Dime Is Photo Center. Height Is Smaller Than Magnetic Types

Application Note 81

AN81-3

Figure 3. Complete LCD Backlight Inverter Compared to Dime for Size.Piezoceramic Transformer (Photo Left) Permits Significantly Thinner Board Size

Piezoelectric transformer technology is not new. It hasbeen employed before, in various forms.4 More familiarexamples of piezoelectrics are barbecue grill ignitors(direct mechanical input to the PZT produces an electricaldischarge) and marine sonar transducers (electrical inputproduces a pronounced sonic output). Piezoelectrics arealso used in speakers (tweeters), medical ultrasoundtransducers, mechanical actuators and fans. Piezoelectricbased backlight inverters have been attempted, but previ-ous transformer and circuit approaches could not providepower, efficiency and wide dynamic range of operation.Transformer operating regions were restricted, with com-plex and ill-performing electronic control schemes. Addi-tionally, the PZT mounting schemes employed enlargedsize, negating the PZT’s size advantage.

Developing a PZT Transformer Control Scheme

It is instructive to review the path to a practical circuit.Figure 4 treats the PZT like a quartz crystal, placing it in aPierce type oscillator.5 Self-resonance occurs and asinosoidal AC high voltage drives the lamp. This circuit hasa number of unpleasant features. Very little power isavailable, due to the circuit’s high output impedance.Additionally, the PZT has a number of other spuriousmodes besides its desired 60kHz fundamental. Changes indrive level or load characteristics induce “mode-hopping,”manifested by PZT resonance jumping to subharmonicsor harmonics. Sometimes, several modes occur simulta-neously! Operation in these modes is characterized by lowefficiency and instability. Practically, this circuit was neverintended as a serious candidate, only an exploratoryexercise. Its contribution is demonstrating that PZT self-resonance is a potentially viable path. Figure 5, a feedbackoscillator, addresses the high output impedance problemwith a totem style pair. This is partially successful, al-

though efficient totem drive devoid of simultaneous con-duction requires care. Mode-hopping persists, in this caseaggravated by the long acoustic transit time through thePZT and the wideband feedback. The sonic transit timeproduces enormous feedback phase error. Even worse,this phase error varies with line and load conditions. Thealternate feedback scheme shown senses current, asopposed to voltage. This eliminates the voltage dividerinduced loading but does nothing to address the phaseuncertainties and mode-hopping. A final problem, com-mon to all resonant oscillators, concerns start-up. Gentletapping of the PZT will usually start a reticent circuit butthis is hardly reassuring.6 Figure 6 is similar but uses aground-referred push-pull power stage, simplifying thedrive scheme. This is a better approach but phase, mode-hopping and start-up problems are as before.

Figure 7 retains the drive scheme and solves the remainingproblems. Central to circuit operation is a new transformerterminal labeled “feedback.” This connection, preciselypositioned on the transformer, provides constant-phaseresonance information regardless of operating condi-tions.Note 4: For examples, see References 4 through 11.Note 5: See References 12 through 14.Note 6: At the low voltage end, please!

PZT HIGH VOLTAGE OUT

fRESONANCE = 60kHz

AN81 F04

+V

LAMP

Figure 4. Pierce Type Circuit Sustains Resonance, ButCannot Deliver Power Efficiently. Circuit Also “Mode-Hops”Due to Transformer’s Parasitic Resonances

Application Note 81

AN81-4

When power is applied, the RC oscillator drives Q1 and Q2at a frequency outside resonance. The PZT, excited off-resonance, at first responds very inefficiently, althoughvoltage-amplified resonant waveforms appear at the feed-back and output terminals. The resonant informationpresent at the feedback terminal injection-locks the RCoscillator, pulling it to the PZT’s resonance. Now, the PZTis supplied with on-resonance drive and efficient opera-tion commences.7 The feedback terminal’s constant-phasecharacteristic is maintained over all line and load condi-tions, and the loop enforces resonance.

Figure 8, retaining the resonance loop, adds an amplitudecontrol loop to stabilize lamp intensity. Lamp current issensed and fed back to a voltage regulator to control PZTdrive power. The regulator’s reference point is variable,permitting lamp intensity to be set at any desired level. Theamplitude and resonance loops operate simultaneously,although fully independent of each other. This two loopoperation is the key to high power, wide range, reliablecontrol.

PZT

fRESONANCE = 60kHz

AN81 F06

+V

LAMP

–

+

VBIAS

∅2∅1

Figure 6. Push-Pull Version of Figure 5 Retains EfficiencyWhile Permitting Simple All N-Channel Drive. Poor PhaseCharacteristics Still Preclude Stable Loop Operation

PZT

fRESONANCE= 60kHz

R

C

RESONANCEFEEDBACK

AN81 F07

+V

+V

LAMP

Q2Q1

∅2∅1

RC OSCILLATORf ≈ 60kHz

Figure 7. Feedback Tap On Transformer Synchronizes RCOscillator, Providing Stable Phase Characteristics. LoopMaintains Fundamental Resonance Under All Conditions

Figure 9 is a detailed schematic of Figure 8’s concept. Theresonance loop is comprised of Q4 and the CMOS inverterbased oscillator. The amplitude loop centers around theLT1375 switching regulator. Figure 10 shows waveforms.Traces A and B are Q2 and Q1 gate drives, respectively.Resultant Q1 and Q2 drain responses are traces C and D.The LT1375 step-down switching regulator, respondingto the rectified and averaged lamp current, closes theamplitude loop by driving the L1-L2 junction (trace E). The4.7µF capacitor at the VC pin stabilizes the loop.8 Note thatno filtering is used—the raw LT1375 500kHz PWM outputdirectly drives the L1-L2-PZT network. This is permissiblebecause the PZT Q factor is so high that it responds onlyat resonance (again, see traces C and D).

Note 7: This is the heart and soul of a bootstrapped start-up.Note 8: This is a deceptively innocent sentence. The PZT’s acoustictransport speed furnishes an almost pure delay in the loop, makingcompensation an interesting exercise. See Appendix C, “A ReallyInteresting Feedback Loop.”

PZT

fRESONANCE = 60kHzALTERNATEFEEDBACK

AN81 F05

+V

LAMP+

–+VBIAS

Figure 5. Feedback Based Oscillator Has More Efficient Drive Stage. Poorly DefinedTransformer Phase Characteristics Cause Spurious Modes with Line and Load Variations

Application Note 81

AN81-5

PZT

fRESONANCE= 60kHz

R

C

RESONANCEFEEDBACK

AN81 F08

+V

LAMP

+V

VREF

∅2∅1

RC OSCILLATORf ≈ 60kHz

STEP-DOWN VOLTAGEREGULATOR

AMPLITUDEFEEDBACK

LAMPCURRENTSET POINT

AMPLITUDE CONTROL LOOP

RESONANCE CONTROL LOOP

Figure 8. Previous Circuit with Amplitude Control Loop Added. New Circuitry Senses Lamp Current,Accordingly Controls PZT Drive Power. Resonance and Amplitude Control Loops Do Not Interact

PZT

LT1375

VIN

C74.7µF

C20.02µF

SYNC BOOST

GND VC

VSW

FB

U1DU1C

U1BR2

10k

C20.1µF

FB

R7750k

AN81 F09

LAMP

AMPLITUDE CONTROL LOOP

BIAS SUPPLY

RESONANCE CONTROL LOOP

12VBIAS

R610k

Q42N3904

∅2∅1

Q1 Q2

L247µH

L147µH

C50.001µF

C615µF

C80.1µF

D31N5819

5V

U1F

D6

U1E

+

2.2µF+

VIN7V TO

25V

U1A

R48.2k

D1

R5750k

12VBIAS

D81N5242

Q32N3904

C100.1µF

R810k

R310k

D2

L1, L2: TOKO 646CY-470MPZT: CTS WIRELESS KPN-6000AQ1, Q2: Si9945

74C04

1N4148

1N5711 UNLESS NOTED

C347pF

R110k

D4

R95k

680Ω

INTENSITYADJUSTMENT

Figure 9. Complete Piezoceramic Transformer (PZT) Based Backlight Inverter. PZT’s Resonant Feedback SynchronizesInverter Based RC Oscillator Via Q4. Amplitude Control Loop Powers PZT Via LT1375 Switching Regulator

Application Note 81

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The feedback tap (trace F), supplying phase coherentinformation, looks like a current source to Q4 under allconditions (note trace F’s vertical scale factor). The 750kresistors in series minimize parasitic capacitance at thetransformer feedback terminal. Q4’s collector (trace G)clamps this information to a lower voltage and injection-locks the CMOS inverter based oscillator, closing theresonance loop. The oscillator insures start-up (refer totext associated with Figure 7) and effectively filters thealready narrow band resonant feedback, further insuringresonance loop fidelity under all conditions. Trace H is thePZT’s high voltage output delivered to the lamp.

In this example dimming is set with a potentiometer,although simple current summing to the LT1375 feedbackpin allows electronic control.9

Additional Considerations and Benefits

As previously mentioned, the PZT has other benefitsbesides small size. One of these is safety. A PZT cannot faildue to output shorts or opens. Short circuits knock thePZT off-resonance and it simply stops, absorbing noenergy. Open circuits do not cause arc-induced PZTfailures because the PZT cannot “arc turns” like a mag-netic transformer. However, it is always wise to sense andarrest an overvoltage condition. The PZT is capable oflarge outputs, despite its small size. Powered by a 10 voltsupply, it can easily produce 3000VRMS if uncontrolled.This mandates some form of overvoltage protection in aNote 9: See Reference 1 for information on various dimming controlschemes.

HIGH VOLTAGE IN

DISPLAY ENCLOSURE(GROUNDED)

TOFEEDBACK

C

R

AN81 F11

LAMP

PARASITIC CAPACITANCETO GROUND

( )

Figure 11. Parasitic Capacitance Absorbs Energy and Corrupts Drive Waveform Due to Finite Source Impedance vs Frequencyin Magnetic Based Inverters. Differing Amounts of Capacitance with Various Displays Cause RC Averaging Errors,Necessitating Calibration for Each Display Type. PZT’s Highly Resonant Characteristics Eliminate Calibration Requirement

A = 20V/DIV

B = 20V/DIVC = 50V/DIV

D = 50V/DIV

E = 20V/DIV

F = 500V/DIV

G = 20V/DIV

H = 2000V/DIV

HORIZ = 5µs/DIV AN81 F10

Figure 10. Waveforms for Figure 9. Traces A and B Are Q2 and Q1 Drive, Respectively. Resultant Q1 and Q2 DrainResponses Are Traces C and D. LT1375 VSW Output Is Trace E. PZT Resonance Tap Is Trace F; Q4’s Collector, Trace G.PZT High Voltage Output Is Trace H. PZT Acts As a Mechanical Filter, Producing Low Distortion Sine Waves

Application Note 81

AN81-7

production circuit. Another significant attribute is thatamplitude control loop scale factor is almost completelyindependent of load, including parasitic capacitance. Thepractical advantage is that a wide range of displays may beused with no recalibration of any kind. This is in directopposition to magnetically based inverters which requiresome form (either hardware or software based) of scalefactor recalibration when displays are changed. Under-standing why this is so requires some study.

Display Parasitic Capacitance and Its Effects

Almost all displays introduce some amount of parasiticcapacitance between the lamp, its leads and electricallyconductive elements within the display. Such elementsmay include the display enclosure, the lamp reflector orboth. Figure 11 diagrams this situation. The parasiticcapacitance to ground has two major impacts. It absorbsenergy, causing lost power. This raises overall inverterinput power because the inverter must supply both para-sitic and intended load paths. Some techniques can mini-mize the effects of parasitic capacitance loss paths butthey cannot be completely compensated.10

A second effect of parasitic capacitance, manifested inmagnetically based inverters, is much more subtle. Mag-netically based inverters have finite source impedance atfrequency, corrupting the produced sinosoid. The amountof parasitic capacitance influences the degree of corrup-tion. Different displays have varying amounts of parasiticcapacitance, resulting in varying degrees of waveformdistortion with different displays. The RC averaging timeconstant is not an RMS to DC converter and producesdifferent outputs as distortion content in its input wave-

form changes. The amplitude loop acts on the DC outputof the RC averager and the assumption is that the inputwaveform distortion content is constant. In a well de-signed magnetically based inverter this is essentially true,even as operating conditions vary. The averager’s outputerror is consistent and can be “calibrated away” by scalefactor adjustments. However, if the display type is changed,the averager is subjected to a differently distorted wave-form and scale factor adjustments are required. Thisnecessitates some form of calibration constant adjust-ment for each display type, complicating production andinventory requirements. PZT based inverters are largelyimmune to this problem because of their extraordinarilyhigh Q factor, typically over 1000. The PZT forces theoutput waveform to have a consistent amount of distor-tion, nominally zero. The PZT’s resonant mechanical filter-ing produces an almost pure sinosoidal output, even withwidely varying parasitic and intended loads. Figure 12shows PZT output voltage (trace A) and current (trace B)with a low parasitic loss display. Waveshapes are essen-tially ideal sinosoids. In Figure 13 a display with muchhigher parasitic losses has been substituted. Minor wave-form distortion, particularly in the current trace (B), isevident, although minimal. The RC averager produceslittle error vs Figure 12 and less than 0.5% lamp currentdifference occurs between the two cases. In contrast, amagnetically based inverter can easily suffer 10% to 15%lamp current differences, impacting display luminosityand/or lamp lifetime.

A = 500V/DIV

B = 10mA/DIV

HORIZ = 5µs/DIV AN81 F12

Figure 12. PZT Inverter Driving a Low ParasiticCapacitance Display. Trace A Is Lamp Voltage, Trace BLamp Current. Waveforms Are Nearly Ideal Sinosoids

A = 500V/DIV

B = 10mA/DIV

HORIZ = 5µs/DIV AN81 F13

Figure 13. Display with Higher Capacitive Loss Causes MinorDistortion. Lamp RMS Current Varies Only 0.5% vs Figure 12.

Note 10: Complete treatment of this issue is sacrificed here tomaintain focus. A more thorough investigation appears in Reference 1.

Application Note 81

AN81-8

REFERENCES

1. Williams, J., “A Fourth Generation of LCD BacklightTechnology,” Linear Technology Corporation, Appli-cation Note 65, November 1995.

2. Williams, J., “Techniques for 92% Efficient LCD Illu-mination,” Linear Technology Corporation, Applica-tion Note 55, August 1993.

3. Williams, J., “Illumination Circuitry for Liquid CrystalDisplays,” Linear Technology Corporation, Applica-tion Note 49, August 1992.

4. Rosen, C. A., “Ceramic Transformers and Filters,”Proc. Electronic Components Symposium, 1956, pp.205–211.

5. Mason, W. P., “Electromagnetic Transducers andWave Filters,” D. Van Nostrand Company, Inc., 1948.

6. Ohnishi, O., Kishie, H., Iwamoto, A., Sasaki, Y., Zaitsu,T., Inoue, T., “Piezoelectric Ceramic Transformer Op-erating in Thickness Extensional Vibration Mode forPower Supply,” Ultrasonic Symposium Proc., 1992,pp. 483–488.

7. Zaitsu, T., Ohnishi, O., Inoue, T., Shoyama, M.,Ninomiya, T., Hua, G., Lee, F. C., “Piezoelectric Trans-former Operating in Thickness Extensional Vibrationand Its Application to Switching Converter,” PESC ’94Record, June 1994, pp. 585–589.

8. Zaitsu, T., Shigehisa, T., Inoue, T., Shoyama, M.,Ninomiya, T., ““Piezoelectric Transformer Converterwith Frequency Control,” INTELEC ’95, October 1995.

9. Williams, J., “Piezoceramics Plus Fiber Optics BoostIsolation Voltages,” EDN, June 24, 1981.

10. Williams, J., Huffman, B., “DC-DC Converters, Part III.Design DC-DC Converters for Power Conservationand Efficiency,” “Ceramic Power Converter,” EDN,November 10, 1988, pp. 209–224, pp. 222–224.

11. Williams, J., Huffman, B., “Some Thoughts On DC-DCConverters,” “20,000V Breakdown Converter,” LinearTechnology Corporaton, Application Note 29, October1988, pp. 27–28.

12. Mattheys, R. L., “Crystal Oscillator Circuits,” Wiley,New York, 1983.

13. Frerking, M. E., “Crystal Oscillator Design and Tem-perature Compensation,” Van Nostrand Reinhold, NewYork, 1978.

14. Williams, J., “Circuit Techniques for Clock Sources,”Linear Technology Corporation, Application Note 12,October 1985.

Application Note 81

AN81-9

APPENDIX A

PIEZOELECTRIC TRANSFORMERS“Good Vibrations”

James R. Phillips

Sr. Member of Technical Staff

CTS Wireless Components

4800 Alameda Blvd. N.E.

Albuquerque, New Mexico 87113

(505) 348-4286

Jim.Phillips@CTSWireless1.com

Piezoelectric transformers are, in fact, not transformers.There are no wires or magnetic fields. A better analogywould actually be that of a dynamo. The piezoelectric“transformer” works exactly like a motor mechanicallycoupled to a generator. In order to understand this con-cept, one must start by understanding the basics ofpiezoelectricity.

Piezowhat?

Many materials exhibit some form of piezoelectric effect.The ones most commonly used are Quartz, Lithium Niobate,and PZT or Lead Zirconate Titanate with the transformerusing the latter. There are two piezoelectric effects, thedirect effect and the inverse effect. In the case of the directeffect, placing a force or vibration (stress) on the piezo-electric element will result in the generation of a charge.(Figure A1) The polarity of this charge depends upon theorientation of the stress when compared to the direction ofpolarization in the piezoelectric element. The polarizationdirection, for PZT, is set by poling or applying a high D.C.field in the range of 45KV/cm to the element during themanufacturing process.

The inverse piezoelectric effect is, as the name implies, theopposite of the direct effect. (Figure 2) Applying an electricfield (voltage) to the piezoelectric element results in adimensional change. (strain) The direction of the changeis likewise linked to the polarization direction. Fieldsapplied at the same polarity of the element result in a

dimensional increase, while those of opposite polarityresult in a decrease. It should be noted that an increase inone dimension in a structure would result in a decrease inthe other two through Poisson’s coupling. This phenom-enon is an important factor in the operation of the trans-former.

The piezoelectric transformer uses both the direct andinverse effects in concert to create high voltage step-upratios. (Figure 3) The input portion of the transformer isdriven by a sine wave voltage, which causes it to vibrate.(inverse effect-motor) The vibration is coupled throughthe structure to the output, which results in the generationof an output voltage. (direct effect-generator)

Alchemy and Black Magic

The piezoelectric transformer is constructed of PZT ce-ramic, but more exactly, a multilayer ceramic. The manu-facture of the transformer is similar to the manufacture ofceramic chip capacitors. Layers of flexible, unfired PZTceramic tape are printed with metallic patterns and thenaligned and stacked to form the required structure. Thestacks are then pressed, diced and fired to create the finalceramic device.

The input section of the transformer has, if fact, a multi-layer ceramic capacitor structure. (Figure 4) The metalelectrodes are patterned in such a way as to create an inter-digitated plate configuration. (section A-A) The outputsection of the transformer has no electrode plates betweenthe ceramic layers and, as a result, fires into a singleceramic structure. The end of the output section is coatedwith a conductive material, which forms the output elec-trode for the transformer.

The next step in the construction is to establish thepolarization directions in the two halves of the trans-former. The input section of the transformer is poledacross the inter-digitated electrodes resulting in a polar-ization direction aligned vertically to the thickness. Theoutput section is poled to create a horizontal or lengthoriented polarization direction. During operation, the inputis driven in thickness mode. This means that a voltage isapplied between the parallel plates of the input causing itto become thicker and thinner on alternate halves of the

Application Note 81

AN81-10

sine wave. The change in input thickness couples throughto the output section causing it to become longer andshorter, generating the output voltage. The resulting volt-age step-up ratio is then proportional to the ratio of theoutput length, which generates the voltage, to the thick-ness of the input layers, across which the drive voltage isapplied.

The Fun Part

The equivalent circuit model for the piezoelectric trans-former outwardly looks identical to that of its seriesresonant magnetic counterpart. (Figure 5) The differ-ences, however, extend past the nominal values to whatthe various components represent. The input and outputcapacitance are simply the result of having a dielectricbetween two metal plates. The effective dielectric constantof PZT runs between 400 and 5,000, depending uponcomposition. This, unfortunately, is where basic electron-ics end. The rest of the components are more complicated.The inductance, LM, is actually the mass of the trans-former. The capacitance, CM, is the compliance of thematerial or the inverse of spring rate. The compliance iscalculated from the applicable generalized beam equationand the Young’s modulus. The resistor, RM, representsthe combination of dielectric loss and the mechanical Q ofthe transformer. It is already obvious to most that trulyunderstanding this device requires background in elec-tronics, mechanics and materials, but we’re not quite doneyet.

The resonant frequency is related to the product of thecapacitance, CM and inductance, LM. This, however, rep-resents the acoustic, not electrical, resonate frequency.The transformer is designed to operate in length reso-nance. The associated motions are identical to that of avibrating string. The major difference is that the frequen-cies are in the ultrasonic range and vary, by design,between 50kHz and 2MHz. Like the string, the transformerhas displacement nodes and anti-nodes. Mechanicallyclamping a node will prevent vibration. This will reduce

efficiency in the best case and prevent operation in theworst. Mounting the transformer is crucial. It can not besimply reflowed to a PCB.

The final element in the model is the “ideal” transformerwith ratio N. This transformer actually represents threeseparate transformations. The first is the transformationof electrical energy into mechanical vibration. This is afunction of the piezoelectric constant, which is electricfield divided by stress, the stress area and the electric fieldlength. The second transformation is the transfer of themechanical energy from the input section to the outputsection and is a function of the Poisson’s ratio for thematerial. The final transformation is the transfer of me-chanical energy back into electrical energy. This is calcu-lated in a similar fashion to the input side.

A Resonant Personality

Resonant magnetic high voltage transformers have anelectrical Q of between 20 and 30. The equivalent for thepiezoelectric transformer is its mechanical Q, which ap-proaches 1,000. This is both good and bad. The ultimateefficiency can be higher, but the usable bandwidth of thetransformer is only 2.5% of that of the magnetic. Inaddition, as shown earlier, the resonant frequency isdependent upon the compliance of the material, which, inturn, is a function of the Young’s modulus. Piezoelectricmaterials have the unusual effect that Young’s moduluschanges with electrical load. In most, if not all, cases, theshift in resonant frequency over rated load is greater thanthe usable bandwidth. (Figure 6) The piezoelectric trans-former must be run at resonance to maintain efficiencyand stability. The near-resonance designs used with mag-netic transformers work poorly, if at all with piezoelectrics.Tracking oscillators are a requirement.

Rosen1 first proposed the concept of the piezoelectrictransformer in 1956. It is now evident why it took 43 yearsto get it right.

Note 1: See Reference 4.

Application Note 81

AN81-11

FORCE

AN81 FA01

PZT

AN81 FA02

PZT

VDRIVE

AN81 FA03

TRANSFORMER

VDRIVE

INPUT OUTPUT

A

CONDUCTOR PZT AN81 FA04

SECTION A-A

A

FREQUENCYRESONATE MAGNETIC

FREQUENCY

DELTA fR w/LOAD

AN81 FA06PIEZOELECTRIC TRANSFORMER

AN81 FA05

CiVIN VOUT

1:N

RL

CMLM RM

CO

Figure 1. Direct Piezoelectric Effect:Force or Vibration Results in Output Voltage

Figure 2. Inverse Piezoelectric Effect:Applied Voltage Results in Vibration or Movement

Figure 3. Transformer: Applied Voltage Resultsin Vibration Which Causes Output Voltage

Figure 4. Piezoelectric Transformer: Multilayer PZT Construction

Figure 5. Piezoelectric Transformer: Equivalent Circuit Model Figure 6. Transformer Bandwidth Comparison:Magnetic vs Piezoelectric

Application Note 81

AN81-12

APPENDIX B

PIEZOELECTRIC TECHNOLOGY PRIMER

James R. Phillips

Sr. Member of Technical Staff

CTS Wireless Components

4800 Alameda Blvd. N.E.

Albuquerque, New Mexico 87113

(505) 348-4286

Jim.Phillips@CTSWireless1.com

Piezoelectricity

The piezoelectric effect is a property that exists in manymaterials. The name is made up of two parts; piezo, whichis derived from the Greek word for pressure, and electricfrom electricity. The rough translation is, therefore, pres-sure- electric effect. In a piezoelectric material, the appli-cation of a force or stress results in the development of acharge in the material. This is known as the direct piezo-electric effect. Conversely, the application of a charge tothe same material will result in a change in mechanicaldimensions or strain. This is known as the indirect piezo-electric effect.

Several ceramic materials have been described as exhib-iting a piezoelectric effect. These include lead-zirconate-titanate (PZT), lead-titanate (PbTiO2), lead-zirconate(PbZrO3), and barium-titanate (BaTiO3). These ceramicsare not actually piezoelectric but rather exhibit a polarizedelectrostrictive effect. A material must be formed as asingle crystal to be truly piezoelectric. Ceramic is a multicrystalline structure made up of large numbers of ran-domly orientated crystal grains. The random orientation ofthe grains results in a net cancelation of the effect. Theceramic must be polarized to align a majority of theindividual grain effects. The term piezoelectric has be-come interchangeable with polarized electrostrictive ef-fect in most literature.

Piezoelectric Effect

It is best to start with an understanding of commondielectric materials in order to understand the piezoelec-tric effect. The defining equations for high permittivitydielectrics are:

CK A

tA

tAt

and Q CV QAVt

r o r= = = = → =ε ε ε ε ε

where:

C = Capacitance

A = Capacitor plate area

εr = Relative dielectric constant

εo = Dielectric constant of air= 8.85 × 10–12 farads/meter

ε = Dielectric constant

V = Voltage

t = Thickness or plate separation

Q = Charge

In addition, we can define electric displacement, D, ascharge density or the ratio of charge to the area of thecapacitor:

DQA

Vt

= = ε

and further define the electric field as:

EVt

or D E= = ε

These equations are true for all isotropic dielectrics.Piezoelectric ceramic materials are isotropic in the unpo-larized state, but they become anisotropic in the poledstate. In anisotropic materials, both the electric field andelectric displacement must be represented as vectors withthree dimensions in a fashion similar to the mechanicalforce vector. This is a direct result of the dependency of the

Application Note 81

AN81-13

ratio of dielectric displacement, D, to electric field, E, uponthe orientation of the capacitor plate to the crystal (orpoled ceramic) axes. This means that the general equationfor electric displacement can be written as a state variableequation:

D Ei ij j= ε

The electric displacement is always parallel to the electricfield, thus each electric displacement vector, Di, is equal tothe sum of the field vector, Ej, multiplied by its correspond-ing dielectric constant, εij:

D E E ED E E ED E E E

1 11 1 12 2 13 3

2 21 1 22 2 23 3

3 31 1 32 2 33 3

= + += + += + +

ε ε εε ε εε ε ε

Fortunately, the majority of the dielectric constants forpiezoelectric ceramics (as opposed to single crystal piezo-electric materials) are zero. The only non-zero terms are:

ε ε ε11 22 33= ,

Axis Nomenclature

The piezoelectric effect, as stated previously, relates me-chanical effects to electrical effects. These effects, asshown above, are highly dependent upon their orientationto the poled axis. It is, therefore, essential to maintain aconstant axis numbering scheme (Figure B1).

POLED

Z (3) #123456P

AXISXY

Z (POLED)SHEAR AROUND XSHEAR AROUND YSHEAR AROUND ZRADIAL VIBRATION

X (1)

Y (2) AN81 FB01

Figure B1

For electro-mechanical constants:

dab, a = Electrical direction; b = Mechanical direction

Electrical-Mechanical Analogies

Piezoelectric devices work as both electrical and mechani-cal elements. There are several electrical-mechanicalanalogies that are used in designing modeling the devices.

ELECTRICAL UNIT MECHANICAL UNIT

e Voltage (Volts) f Force (Newtons)

i Current (Amps) v Velocity (Meter/Second)

Q Charge (Coulombs) s Displacement (Meters)

C Capacitance (Farads) CM Compliance (Meters/Newton)

L Inductance (Henrys) M Mass (Kg)

Z Impedance ZM Mechanical Impedance

idQdt

e Ldidt

Ld Q

dt

=

= =2

2

vdsdt

f Mdvdt

Md s

dt

=

= =2

2

Coupling

Coupling is a key constant used to evaluate the “quality” ofan electro-mechanical material. This constant representsthe efficiency of energy conversion from electrical tomechanical or mechanical to electrical.

Mechanical Energy Coverted to

k2 =Electrical Charge

Mechanical Energy Input

or

Mechanical Energy Coverted to

k2 =Mechanical Displacement

Electrical Energy Input

Electrical, Mechanical Property Changes with Load

Piezoelectric materials exhibit the somewhat unique effectthat the dielectric constant varies with mechanical loadand the Young’s modulus varies with electrical load.

Dielectric Constant:

εr FREE (1 – k2) = εr CLAMPED

This means that the dielectric “constant” of the materialreduces with mechanical load. Here “Free” stands for astate when the material is able to change dimensions with

Application Note 81

AN81-14

applied field. “Clamped” refers to either a condition wherethe material is physically clamped or is driven at a fre-quency high enough above mechanical resonance that thedevice can’t respond to the changing E field.

Elastic Modulus (Young’s Modules):

YOPEN (1 – k2) = YSHORT

This means that the mechanical “stiffness” of the materialreduces when the output is electrically shorted. This isimportant in that both the mechanical QM and resonatefrequency will change with load. This is also the propertythat is used in the variable dampening applications.

Elasticity

All materials, regardless of their relative hardness, followthe fundamental law of elasticity (Figure B2). The elasticproperties of the piezoelectric material control how well itwill work in a particular application. The first concepts,which need to be defined, are stress and strain.

For any given bar of any material:

L δ

F F = APPLIED FORCEA = AREA TO WHICH FORCE IS APPLIEDδ = ELONGATION

AN81 FB02

Figure B2

Stress = σ = F/A

Strain = λ = δ/L

The relationship between stress and strain is Hooke’s Lawwhich states that, within the elastic limits of the material,strain is proportional to stress.

λ = Sσ

or, for an anisotropic material

λi = Sijσj

Note: The constant relating stress and strain is the modu-lus of elasticity or Young’s modulus and is often repre-sented by S, E or Y.

Piezoelectric Equation

It has been previously shown that when a voltage isapplied across a capacitor made of normal dielectricmaterial, a charge results on the plates or electrodes of thecapacitor. Charge can also be produced on the electrodesof a capacitor made of a piezoelectric material by theapplication of stress. This is known as the Direct Piezo-electric Effect. Conversely, the application of a field to thematerial will result in strain. This is known as the InversePiezoelectric Effect. The equation, which defines this rela-tionship, is the piezoelectric equation.

Di = dijσj

where:

Di ≡ Electrical displacement (or charge density)

dij ≡ Piezoelectric modulus, the ratio of strain to appliedfield or charge density to applied mechanical stress

Stated differently, d measures charge caused by a givenforce or deflection caused by a given voltage. We can,therefore, also use this to define the piezoelectric equationin terms of field and strain.

DiEj i

j=

σ λ

Earlier, electric displacement was defined as

Di = εij Ej

therefore,

eij Ej = dijσ

and

Ejd

Eij j

ij=

σ

which results in a new constant

gijd

Eij

ij=

Application Note 81

AN81-15

This constant is known as the piezoelectric constant andis equal to the open circuit field developed per unit ofapplied stress or as the strain developed per unit of appliedcharge density or electric displacement. The constant canthen be written as:

gfield

stressvolts meter

newtons meter

L LV t

= = =/

/

//2

∆ε

Fortunately, many of the constants in the formulas aboveare equal to zero for PZT piezoelectric ceramics. The non-zero constants are:

s11 = s22, s33, s12, s13 = s23, s44, s66 = 2(s11 – s12)

d31 = d32, d33, d15 = d24

Basic Piezoelectric Modes

See Figure B3

0

0

THICKNESS EXPANSION

THICKNESS SHEAR

FACE SHEAR

POLARIZATION DIRECTION AN81 FB03

–

+

+

–

0

0

–

+

+

–

0

0

–

+

+

–

Figure B3

Poling

Piezoelectric ceramic materials, as stated earlier, are notpiezoelectric until the random ferroelectric domains arealigned. This alignment is accomplished through a pro-cess known as “poling”. Poling consists of inducing a D.C.voltage across the material. The ferroelectric domainsalign to the induced field resulting in a net piezoelectriceffect. It should be noted that not all the domains becomeexactly aligned. Some of the domains only partially alignand some do not align at all. The number of domains thatalign depends upon the poling voltage, temperature, andthe time the voltage is held on the material. During polingthe material permanently increases in dimension betweenthe poling electrodes and decreases in dimensions parallelto the electrodes. The material can be de-poled by revers-ing the poling voltage, increasing the temperature beyondthe materials Currie point, or by inducing a large mechani-cal stress.

Application Note 81

AN81-16

Post Poling

Applied Voltage:

Voltage applied to the electrodes at the same polarity asthe original poling voltage results in a further increase indimension between the electrodes and decreases thedimensions parallel to the electrodes. Applying a voltageto the electrodes in an opposite direction decreases thedimension between the electrodes and increases the di-mensions parallel to the electrodes.

Applied Force:

Applying a compressive force in the direction of poling(perpendicular to the poling electrodes) or a tensile forceparallel to the poling direction results in a voltage gener-ated on the electrodes which has the same polarity as theoriginal poling voltage. A tensile force applied perpendicu-lar to the electrodes or a compressive force applied parallelto the electrodes results in a voltage of opposite polarity.

Shear:

Removing the poling electrodes and applying a field per-pendicular to the poling direction on a new set of elec-trodes will result in mechanical shear. Physically shearingthe ceramic will produce a voltage on the new electrodes.

Piezoelectric Benders

Piezoelectric benders are often used to create actuatorswith large displacement capabilities (Figure B4). The benderworks in a mode which is very similar to the action of abimetallic spring. Two separate bars or wafers of piezo-electric material are metallized and poled in the thicknessexpansion mode. They are then assembled in a + – + –stack and mechanically bonded. In some cases, a thinmembrane is placed between the two wafers. The outerelectrodes are connected together and a field is appliedbetween the inner and outer electrodes. The result is thatfor one wafer the field is in the same direction as the polingvoltage while the other is opposite to the poling direction,This means that one wafer is increasing in thickness anddecreasing in length while the other wafer is decreasing inthickness and increasing in length, resulting in a bendingmoment.

Loss

There are two sources for loss in a piezoelectric device.One is mechanical, the other is electrical.

Mechanical Stiffness or

Mechanical Loss: QM =Mass Resistance

Mechanical Resistance

Electrical Loss: tanδ =Effective Series ResistanceEffective Series Reactance

Simplified Piezoelectric Element Equivalent Circuit

AN81 FB05

Ci

CM LM1:N

Ri

Figure B5

Ri = Electrical resistance

Ci = Input capacitance = ε εo rA

t

εo = 8.85 × 10-12 farads/meter

A = Electrode area

t = Dielectric thickness

LM = Mass (Kg)

CM = Mechanical compliance = 1/Spring Rate (M/N)

N = Electro-mechanical Linear Transducer Ratio(newtons/volts or coulomb/meter)

This model (Figure B5) has been simplified and it ismissing several factors. It is only valid up to and slightlybeyond resonance. The first major problem with the modelis related to the mechanical compliance (CM). Complianceis a function of mounting, shape, deformation mode(thickness, free bend, cantilever, etc.) and modulus of

+ AN81 FB04

+

–

0

0

0

Figure B4

Application Note 81

AN81-17

elasticity. The modulus of elasticity is, however, anisotro-pic and it varies with electrical load. The second issue isthat the resistance due to mechanical QM has been left out.Finally, there are many resonant modes in the transform-ers, each of which has its own CM as shown in Figure B6.

SIMPLE BEAM—UNIFORM END LOAD

BEAM SECTION

F

t

W

SIMPLE BEAM—UNIFORM LOAD—END MOUNTS

MOUNT OR VIBRATION NODE

BEAM SECTION

F

t

WA

SIMPLE BEAM—UNIFORM LOAD—CANTILEVER MOUNTS

BEAM SECTION

AN81 FB07

F

t

WA

CM = 5A3

384YijI

CM = A3

8YijI

= 3A3

2YijWt3

CM = 5A3

32Wt3

I = MOMENT OF INERTIA

= Wt3

12

CM = A

AYij

LM = ρAWt

= A

Wt

Figure B7

Mechanical Compliance:

Mechanical compliance, which is the inverse of springconstant, is a function of the shape, mounting method,modulus and type of load. Some simple examples areshown in Figure B7.

The various elements that have been explained can now becombined into the design of a complete piezoelectricdevice. The simple piezoelectric stack transformer will beused to demonstrate the way they are combined to createa functional model.

Simple Stack Piezoelectric Transformer

The piezoelectric transformer acts as an ideal tool toexplain the modeling of piezoelectric devices in that itutilizes both the direct and indirect piezoelectric effects.The transformer operates by first converting electricalenergy into mechanical energy in one half of the trans-former. This energy is in the form of a vibration at theacoustic resonance of the device. The mechanical energy

AN81 FB06

Ci

CM4 LM

1:N

Ri

CM LM

CM2 LM

CM3 LM

Figure B6

Application Note 81

AN81-18

Figure B8

d2VIBRATION NODE

n = # OF THIN LAYERSnd2 = d1

AN81 FB08

d1

W

A

produced is then mechanically coupled into the secondhalf of the transformer. The second half of the transformerthen reconverts the mechanical energy into electricalenergy. Figure B8 shows the basic layout of a stacktransformer. The transformer is driven across the lowerhalf (dimension d1) resulting in a thickness mode vibra-tion. This vibration is coupled into the upper half and theoutput voltage is taken across the thinner dimension d2.

Equivalent Circuit:

The equivalent circuit model for the transformer (shown inFigure B9) can be thought of as two piezoelectric elementsthat are assembled back to back. These devices areconnected together by an ideal transformer representingthe mechanical coupling between the upper and lowerhalves. The input resistance, Ri, and the output resistance,RO, are generally very large and have been left out in thismodel. The resistor RL represents the applied load. Deter-mining the values of the various components can becalculated as shown previously.

Input/Output Capacitance:

CiInput Area

Input ThicknessAWd

o r o r= =ε ε ε ε1

Figure B9

AN81 FB09

Ci

ELECTRICAL

LOWER HALF

MECHANICAL

1:N1 1:NC N2:1

RL

CM1 LM1 RM1

CO

CM2LM2RM2

ELECTRICAL

UPPER HALF

MECHANICAL

similarly,

COutput Area

Output ThicknessnAWd

o o r o r= =ε ε ε ε2

Mechanical Compliance:

The mechanical compliance, CM, can be represented by asimple beam subjected to a uniform axial load. This isbecause the thickness expansion mode will apply uniformstress across the surface. It should be noted that the beamlength is measured with respect to the vibration node. Thevibration node is used as this is the surface which does notmove at resonance and can, therefore, be thought of as afixed mounting surface.

CBeamLength

Beam Area Y

Cd

AWY

Cd

AWY

M

M

M

=

=

=

33

11

33

22

33

Note: Even if nd2 ≠ d1 the vibration node will still be locatedin the mechanical center of the transformer.

Application Note 81

AN81-19

Mass:

LM1 = ρAWd1

LM2 = ρAWnd2 = ρAWd1

Resistance:

The resistances in the model are a function of the mechani-cal QM and Q of the material at resonance and will becalculated later.

Ideal Transformer Ratio:

The transformer ratio, N1, can be thought of as the ratio ofelectrical energy input to the resulting mechanical energyoutput. This term will then take the form of newtons pervolt and can be derived from the piezoelectric constant, g.

As before:

g ElectricalField StressVolts Meter

Newtons Meter= = /

/ 2

therefore:

1

11

gn mV m

Ng

Area of AppliedForceLength of GeneratedField

=

= =

//

or

NAW

g d1

33 1=

The output section converts mechanical energy back toelectrical energy and the ratio would normally be calcu-lated in an inverse fashion to N1. In the model, however,the transformer ratio is shown as N2 : 1. This results in acalculation for N2 that is identical to the calculation of N1.

Ng

Area of AppliedForceLength of GeneratedField

21= =

or

NAW

g d2

33 2=

The transformer 1 : NC, represents the mechanical cou-pling between the two halves of the transformer. The stacktransformer is tightly coupled and the directions of stressare the same in both halves. This results in NC ≅ 1.

Model Simplification:

The response of the transformer can be calculated fromthis model, but it is possible to simplify the model througha series of simple network conversion and end up in anequivalent circuit whose form is the same as that of astandard magnetic transformer (Figure B10).

where, due to translation through the transformer,

C N C and L L NM C M M M C22

2 2 22′ = ′ = /

but NC2 ≅ 1, therefore

C C C and L L LM M M M M M2 2 1 2 2 1′ = = ′ = =

Ci

1:N1

RL

CM1 LM1 RM1

CO

CM2LM2RM2

AN81 FB10

Ci

1:N1

1:NC

1:NC

N2:1

1:1/N2

RL

CM2′CM1 LM1 LM2′ R′

CO

Figure B10

Application Note 81

AN81-20

which allows the next level of simplification (Figure B11) therefore

Cd

WLYA W

g d

AW

Y g d

L AWdg d

A W

g dAW

NN NN

AWg d

g dAW

dd

C

= =

= =

= = =

12 2

22

33

2 2

332

12

33 332

1

133

212

2 233

212

1

2 33 1

33 2 2

1

ρ ρ

The last value we need to calculate is the motional resis-tance. This value is based upon the mechanical QM of thematerial and the acoustic resonant frequency.

Resonant Frequency

ω

ρ

ρ ρ

ρ

o

PZT

LC

d gAW

AW

Y g d

dY

dY

c Speed of Sound inPZT Y

=

=

= =

≡ =

11

2

2 11 1

1 332

33 332

12

33

133

/

/

therefore

ωo = cPZT/d1

The equation above states that the resonant frequency isequal to the speed of sound in the material divided by theacoustic length of the device. This is the definition ofacoustic resonance and acts as a good check of the model.The final derivation is the value of resistance.

QM ≡ 1/ωoRC

AN81 FB11

1:NC/N2

RLCOCi

1:N1C′ L′ R″

Figure B11

1:NC/N21:N1

RLCO

1:N

RLCOCi

C L R

AN81 FB12

Ci

CL R

Figure B12

here

L L L L AWd

CC C

C C

CC

C dAWY

M M

M M

M M

M

M

M

′ = + ′ = =

′ =

′

+ ′

= = =

1 2 1 1

1 2

1 2

12

1

1 1

33

2 2

2 2 2

ρ

Final simplification (Figure B12)

where

C C N and L L N= ′ = ′12

12

/

and, from before

NAW

g d1

33 1=

Application Note 81

AN81-21

or

R Q C

Rd Y

QY g d

AW

d g YQ AW

o M

M

M

=

=

=

1

2

2

1 33 33 332

1

12

332

33

/ ω

ρ

ρ

Note: CM and R are both functions of Y33 and Y33 is afunction of RL

It should be noted that the model is only valid for trans-formers driven at or near their fundamental resonatefrequencies. This is because the initial mechanical modelassumed a single vibration node located at the center ofthe stack which is only true when the transformer is drivenat fundamental resonance. There are more nodes whenthe transformer is driven at harmonic frequencies(Figure B13).

There are no fixed nodes at frequencies other than reso-nance. This means that the transformer must be designedwith the resonate mode in mind or phase cancellations willoccur and there will be little or no voltage gain. It is oftendifficult to understand the concept of nodes and phasecancellation, so a simple analogy can be used. In this case,waves created in a waterbed will be used to explain theeffect.

Pressing on the end of a waterbed creates a “wave” ofdisplacement that travels down the length of the bed untilit reaches the opposite end and bounces back. The waterpressure (stress) is the lowest, or negative with respect tothe water at rest, at a point just in front of the wave andhighest at a point just behind the wave. The pressures atthe crest and in the trough are at the same pressure as thebed at rest. The wave will reflect back and forth untilresistance to flow causes it to dampen out. The averagepressure over time at any point in the bed will be exactlythe same as the pressure at rest. Similarly, the averagestress in a transformer off resonance will approach zeroand there will be no net output.

σ

δ

NODE

NOTE: STRESS IS 90° OUT OF PHASE FROM DISPLACEMENTAN81 FB13

FREQUENCY WAVE LENGTH

ωo λ/2

2ωo λ

3ωo 3λ/2

Figure B13

Pressing on the end of the same bed repeatedly just afterthe wave has traveled down the length, reflected off theend, returned and reflected off the “driven” end will resultin a standing wave. This means that one half of the bed isgetting thicker as the other half is getting thinner and thecenter of the bed will be stationary. The center is the nodeand the thickness plotted over time of either end will forma sine wave. There will be no net pressure difference in thecenter, but the ends will have a pressure wave which forma sine wave 90o out of phase with the displacement. Thetransformer again works in the same manner with novoltage at the node and an AC voltage at the ends. It is fairlysimple to expand this concept to harmonics and to otherresonate shapes.

Conclusion

The number of different applications for piezoelectricceramic, and in particular PZT ceramic, is too great toaddress in a single paper. The basic principals that havebeen set forth in this primer can, however, be used to bothunderstand and design piezoelectric structures and de-vices. The ability to create devices of varying applicationsand shapes is greatly enhanced by the used of multilayerPZT ceramics.

Application Note 81

AN81-22

APPENDIX C

A Really Interesting Feedback Loop

The almost pure mechanical delay presented by the PZT’sacoustic transport presents a fascinating exercise in loopcompensation.1 Veterans of feedback loop compensationbattles will exercise immediate caution when confrontedwith a pure and lengthy delay in a loop. Neophyte design-ers will gain a lesson they will not easily forget.

Figure C1 diagrams Figure 9’s amplitude control loop, withsignificant contributions to loop transmission represented.The PZT delivers phase delayed information at about60kHz to the lamp. This information is smoothed to DC bythe RC averaging time constant and delivered to theLT1375’s feedback terminal. The LT1375 controls PZTpower with its 500kHz PWM output, closing the controlloop. The capacitor at the LT1375’s VC pin rolls off gain,nominally stabilizing the loop. This compensation capaci-tor must roll off gain-bandwidth at a low enough value toprevent loop delays from causing oscillation. Which ofthese delays is the most significant? From a stabilityviewpoint, the LT1375’s output repetition rate and thePZT’s oscillation frequency are sampled data systems.Their information delivery rate is far above the PZT’s200µs delay and the averaging time constants, and is notsignificant. The PZT delay and the RC time constant are themajor contributors to loop delay. The RC time constantmust be large enough to turn the half wave rectifiedwaveform into DC. The lumped delay of the PZT and the RCthus dominates loop transmission. It must be compen-sated by the capacitor at the LT1375 VC pin. A largeenough value for this capacitor rolls off loop gain at lowenough frequency to provide stability. The loop simplydoes not have enough gain to oscillate at a frequencycommensurate with the RC and PZT delays.2

A good way to begin to establish a value for loop roll-off isto let the loop oscillate. This is facilitated by initiallydeleting the compensation capacitor and turning the cir-cuit on. Figure C2 shows the VC pin (trace A) and FB node(trace B). The frequency of oscillation and the phaserelationship between these two signals provide valuableinsight into achievable closed-loop bandwidth.3 Loopdelay sets oscillation frequency at about 2.3kHz. Selecting

the VC value to roll off bandwidth well below this frequencyis appropriate. Figure C3 shows results with Figure 9’s4.7µF capacitor installed. Trace A, a step input, is appliedto the LT1375’s shutdown pin. When the shutdown pin isenabled, VC (trace B) slews up and the feedback pin (traceC) moves toward its control point as PZT output voltage(trace D) rises. The VC pin damping causes extended slewtime but settling is completed in 25 milliseconds withminimal overshoot. The small overshoot (past mid screen)derives from the lamp’s negative resistance characteristicand is easily handled by loop dynamics.

Some situations may require significantly faster loopresponse than simple first order compensation can pro-vide. An example is wide range dimming via PWM meth-ods. Such methods rely on rapid on-off loop cycling toachieve wide range dimming. Typically, the cycling occurswell above the flicker rate, in the 100Hz to 200Hz region.Loop settling must be very quick with respect to thisfrequency or line regulation will be poor. This is sobecause during slew the loop is out of control. If slew timeapproaches on-time, control is, by definition, poor. FigureC4 uses a feedback lead, allowing very light VC dampingwith resultant faster slew time.4 Figure C5 shows results.When the shutdown pin (trace A) is enabled, VC (trace B)slews rapidly, followed by the FB node (trace C). The PZThigh voltage envelope is trace D. Loop capture occurs in1.2 milliseconds—about 20x faster than Figure C3’s simplefirst order compensation. Note that this performancebegins to approach the limits implied by Figure C2’sinformation.5

Note 1: Perhaps this verbiage indulges drama. Conversely, nerds likeme find arcana such as this fascinating.Note 2: The high priests of feedback refer to this as “Dominant PoleCompensation.” The rest of us are reduced to more pedestriandescriptives. As such, this technique is sometimes loosely referred toas “glop comp.”Note 3: Deliberately sustaining loop oscillation is a valuable investiga-tive tool but may encounter problems in some applications. Consideraircraft flap control servos or power plant generator stabilizing loops.Note 4: Proper Bodese for this compensation technique is a “feedbackzero.”Note 5: Score one for lightning empiricism.

Application Note 81

AN81-23

60kHz HIGH VOLTAGE500kHz PWM

RC AVERAGINGTIME CONSTANT 200µs

AN81 FC01

10k

VIN

VC

VSW

0.02

FEEDBACKTERMINALCOMPENSATION

CAPACITOR HALF WAVE 60kHz

LAMP500kHz STEP-DOWN

REGULATOR(LT1375)

60kHz CARRIERWITH ≈200µs DELAY

(PZT AND DRIVE)

Figure C1. Delay Terms in Figure 9’s Amplitude Control Feedback Path.PZT’s 200µs Mechanical Delay and RC Time Constant Dominate LoopTransmission and Must Be Compensated for Stable Operation

A = 1V/DIVON 0.3VDC

LEVEL

B = 500mV/DIVON 2VDC

LEVEL

HORIZ = 200µs/DIV AN81 FB02

Figure C2. Allowing Loop to Oscillate Hints at CompensationRequirements. 2.3kHz Oscillation Frequency Appears to Derivefrom Figure C1’s Delay Terms. Trace A Is LT1375 VC Pin,Trace B Its Feedback Terminal. High Frequency Content inTrace B’s Outline, PZT Carrier Residue, Is Not Pertinent

A = 5V/DIV

B = 0.5V/DIV

HORIZ = 5ms/DIV AN81 FB03

Figure C3. Figure 9’s Loop Compensation Is Stable DespitePZT’s Mechanically Induced Delay. First Order Roll-Off fromLarge VC Pin Capacitor Stabilizes Loop. Trace A Is Step Inputat LT1375 Shutdown Pin. Traces B, C and D are VC, Feedbackand PZT Output Nodes, Respectively

C = 0.5V/DIV

D = 1000V/DIV

LT1375

0.033

0.003

2k

VSW

FB

TO INDUCTORS AND PZT

FROM 10k-0.02µFAVERAGING TIMECONSTANT

AN81 FC04

VC

Figure C4. Very Light VC Damping Augmented by aFeedback Lead Provides 20× Faster Loop Response.Technique Allows Wide Range Lamp Dimming ViaPWM without Sacrificing Line Regulation

A = 5V/DIV

B = 1V/DIV

HORIZ = 200µs/DIV AN81 FB05

Figure C5. Feedback Based Compensation Waveforms. TraceA Is Step Input at LT1375 Shutdown Pin. Traces B, C and DAre VC, Feedback and PZT Output Nodes, Respectively. Note25× Sweep Speed Increase Over Figure C3

C = 1V/DIV

D = 1000V/DIV

Information furnished by Linear Technology Corporation is believed to be accurate and reliable.However, no responsibility is assumed for its use. Linear Technology Corporation makes no represen-tation that the interconnection of its circuits as described herein will not infringe on existing patent rights.

Application Note 81

AN81-24an81f LT/TP 0999 4K • PRINTED IN USA

LINEAR TECHNOLOGY CORPORATION 1999

Linear Technology Corporation1630 McCarthy Blvd., Milpitas, CA 95035-7417(408) 432-1900 FAX: (408) 434-0507 www.linear-tech.com