1 Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [1] Power Electronics Laboratory Department of Electrical and Computer Engineering Ben-Gurion University of the Negev P.O. Box 653, Beer-Sheva 84105, ISRAEL Phone: +972-8-646-1561; Fax: +972-8-647-2949; Email: sby@ee. bgu.ac.il; Website: www.ee.bgu.ac.il/~pel Gordon Seminar, Tel-Aviv University, June 2006 Shmuel (Sam) Ben-Yaakov Power Electronics of Piezoelectric Elements Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [2] 1. Introduction • Piezoelectricity • Brief overview of Piezoelectric devices • Actuators • Vibrating vans • Motors • Micro-PowerGenerators/Dampers • Transformers • Miscellaneous OUTLINE Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [3] OUTLINE (Cont.) 2. Models of Piezoelectric devices 3. Drivers 4. Rectifiers 5. PT based CCFL Ballasts • The stability issue • Envelope Simulation • Thermal effects
67
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1
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [1]
Power Electronics LaboratoryDepartment of Electrical and Computer Engineering
Ben-Gurion University of the NegevP.O. Box 653, Beer-Sheva 84105, ISRAEL
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [11]
Features and Applications:
• Small deflection (µm range)• Static and dynamic applications• Light deflection • Positioning, no friction or backlash • Valve control
L
LStack of piezoelectric elements
Actuators
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [12]
XY Positioning
200 nm span
5
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [13]
Serial BimorphParallel Bimorph
•Same idea as bi-metal•Large deflection, mm range
Bimorph Benders
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [14]
Electricalterminals
ppx −∆
VanBimorphs Piezoelectric element
Base
Vibrating Van
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [15]
Bi-Morph Actuators and Vibrating Vans
• Large deflection • Light Choppers• Remote operation• Valve control • Fan
Features and Applications:
6
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [16]
• Nanomotion Ltd. Israel
piezo ceramicelectrode A
electrode B
electrode C B
A
common
Ellipticmovement
S1 S2
stator
piezo actuatordriver
VAC
stage
Piezoelectric motors
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [17]
Operation Demo
Nano_motor.avi
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [18]
Nanomotion’s NanoLens
7
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [19]
Features and Applications:
• Linear motion• Circular motion• Sub-micron motion and positioning• Small size• Vacuum compatible• Camera lenses• HD drive• Microelectronics manipulators
Piezoelectric Motors
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [20]
MASS
Mechanical Vibration
Micro-Power Generators (Mechanical to Electrical energy harvesting)
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [21]
Silicon beamPiezoelectric
element
Vout
63Ni radioisotopeemitter
[12]
Micro-Power Generator
8
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [22]
MIT’s Piezo Tennis Shoe
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [23]
Shaking table
Piezoelectric element
Beam
Vin
Dampers
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [24]
Sports Active Damping Patent
9
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [25]
Ski DampingCEDRAT TECHNOLOGIES & SKI ROSSIGNOL
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [26]
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [27]
Features and Applications:
• Active suspension• Skis• Motorcycles• Remote energy sources• Tennis shoes• Structures
Micro-Power Generators and Dampers
10
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [28]
Piezoelectric Transformers
Vin VoPT RL
Vin
Vo
[15]
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [29]
Transformer examples
P
P
T
Vin
Vout
• Radial mode[16]
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [30]
VP
VS
primarypart
secondarypart
x
xdisplacement
potentialVS
supportpoint
poling
• High voltage gain
[15-18, 70]
Rosen Type Piezoelectric Transformer
11
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [31]
• Higher voltage transfer ratio
Vin
A
A
Rosen Type Piezoelectric TransformerMultilayer
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [32]
Step Down
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [33]
Characteristics of Piezo transformersAdvantages– Potentially low costs– Compact size– High efficiency – Ability to work at high frequency– Good insulation capability– No windings, i.e. no magnetic fields
Disadvantages– Resonant device (frequency and load dependent)– Low Power– Cost
12
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [34]
1. Fluorescent lamp driver for laptop backlight (commercial)
• Trading switching losses with conduction losses[21, 29]
Voltage-Fed Half Bridge Inverter with a Parallel Matching Inductor LP
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [105]
Series Inductor
PTLin
• Simple method for obtaining soft switching• May attenuate or boost PT input voltage• May change overall frequency response• Best dealt by simulation • A coupling capacitor will eliminate DC on PT
[25, 29, 48, 50, 62, 63, 69, 74]
36
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [106]
Input Impedance of the PT
Fr equency
100KHz 150KHzp( I ( L2) ) - p( V( PTI NAC) )
0d
- 100d
100d
SEL>>
V( OUTAC) / V( PTI NAC)
0. 50
1. 00
0. 05
• Not always capacitive around the operating frequency
inductive Fr equency
100. 0KHz43. 6KHzp( I ( L4) ) - p( V( PTTAC) )
0d
50d
100dV( out ) / V( PTTAC)
0
2. 5
5. 0
7. 0
SEL>>
Vout/Vin
Phase
PT “A” PT “B”
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [107]
Fr equency
100KHz 120KHz 140KHz 150KHzV( OUTAC) / V( I NPUTAC)
0
1. 0
2. 0
P( - I ( V6) )- 100d
0d
100d
SEL>>
V( OUTAC) / V( PTI NAC)0
1. 0
2. 0
• Examination by small signal (AC) simulation
Overall Vout/Vin
PT response
Phase of series inductor current
1mH 1.6mH
2.4mH
PT Operating Region
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [108]
Ti me
920. 0us 930. 0us914. 3us 935. 0usV( I NPUTTRAN)
50V
- 10VSEL>>
V( PTTRANS)0V
25V
50VI ( L4)
- 40mA
0A
40mA
Inductor current
PT voltage
HB commutation
• Large-signal time-domain (TRAN) simulation
37
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [109]
Ti me
672. 00us 674. 00us 676. 00usV( I NPUTTRAN)
0V
20V
40V
52V
Green=1mHRed=1.6mHBlue= 2.4mH
• HB commutation
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [110]
Diode Clamped Resonant Snubber
Q1
Q2
Lr
Cin
VCC
DQ1
DQ2
PTC1
C2
D1
D2 Cex
• Forces soft commutation [39, 44]
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [111]
Lin=10m; Cex=1n
Single layer, RosenType, ELS-60 Eleceram Co.
Ti me
700. 0us 710. 0us 720. 0us 730. 0usV( I NPUTTRAN)
0V
200VV( PTTRANS)
- 200V
0V
200V
SEL>>
I ( L104)- 100mA
0A
100mAV( OUTTRAN)
- 500V
0V
500V
Output voltage
Inductor current
PT input voltage
HB commutation
38
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [112]
0Itt0Itt
Q43
D31
>−
>−0Vtt
offQtt
inC10
20
>−
−−
Advantages:1. Only one switch2. Low cost3. Better EMI suppression
Disadvantage:1. Small operational range2. High voltage stress
L Lr
Cin C'o
Rm
Cr R'oVin V'oDQ
VCin
IDIQ
IIN PT
Control
[33-35, 54]
Class E Inverter
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [113]
Piezoelectric Transformers inHigh Voltage Application
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [131]
S 2
S 1 L bC f
D 3
D 1
D 2
PZT
C s e
R L
D 4
C 1
C 2
+
Inverter
C
[39, 40, 61]
HV DC Output
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [132]
.
2000
3000
4000
5000
6000
1000
0
VLmax
(V )L fr
[V ,
V]
L
[R , MOhm]0 5 10 15 20
L[39, 40]
45
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [133]
1.004
1.008
1.012
1.000
1.016
1 10 100 1000 10000[RL, kOhm]
[ω∗]
• Need for frequency tracking[40, 45]
Frequency of Maximum Output
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [134]
The Proposed Frequency Tracking Method
Current iD2 in part of cycle θ is proportional to irThe phase of the current iD2 is the same as one of irThe trailing edge of the iD2 level detection wave may be used as Phase Reference
Phase Reference
D2
D1
Cf RL
VCo VL
Co
Cr RmLrVin
ir irN
VCoN
iD2 leveldetection
iD2
ir
θ ϑ
ϑ
ϑ
[44, 46]
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [135]
Experimental Setup for Resonant Frequency Tracking
370uH10k
4.7k 4.7k
D1
D2CL RL
PTVin Vout
p(Vin) p(Id)
Phasedetector
39k
1n
VCO
4.2k
1.5k3.3k
1k15v
15v 15v
FFQ
QIR2110driver
0.22u
10.2n
10k4.7
10k4.7
CD4046A
VLF
f
Highvoltage
V1V4 V2
V3
BN
6.6n
[44]
46
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [136]
The System With PLL Control
[44]
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [137]
for Micro-Power Generators and Dampers
[82]
Resonant Rectifier
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [138]
Vibrating Piezo - Electrical Model
os)opt(L C
Rω
=1
RL
PoutoCini RL
( )22
1 Los
L)rms(inout
RC
RIP
ω+=
Ropt
Piezo Source Load
47
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [139]
Capacitive Source Problem
oCini LC
inVouti
LVLI
1BD 2BD
3BD 4BD
RL
inV
inI
outi
1t 2t 3t
)pk(inV
πω+
πω−
π==Los
DosP
Lout RC
VCI
Ii 21
42
IL
LLos
DosP
LL RRC
VCI
)R(P ⋅⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
πω
+
πω−
π=
2
21
42
)VV(C
R
L
Dos
)opt(L 21
12 +ω⋅
π=
DB2 ‘ DB3 DB1 ‘ DB4
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [140]
What if Co=0 ?
ini LC
inVouti
LVLI
1BD 2BD
3BD 4BD
RL
inV
outiIL
π=
πω+
πω−
π==
=
P
Los
DosP
LoutI
RC
VCI
Ii
oC
221
42
0
LP
LL RI)R(P ⋅⎟⎠⎞
⎜⎝⎛π
=22
∞≈)opt(LR
ini
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [141]
Earlier solutions and their limitations
oCini RLL
os LC
1=ω
Disadvantage
Large inductance
Passive and active solutions
EmulatorDisadvantages
1.Large in size
2.Difficult to tune
3.High sensitivity
4.External source
48
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [142]
Proposed Resonant Rectifier Circuit
oCini LC LR
inVouti
1D 2D
resL
derV
cV
1sw 2sw
COMP.
resi
dtd
LVLI
1BD 2BD
3BD 4BD
Capacitive Source
Differentiator
Comparator
Diode Bridge
Inductor & Switches
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [143]
Principles of operation
cV
derV
resi
inV
outi
1t 2t 3t 4t 5t
oCini LC LR
inVouti
1D 2D
resL
derV
cV
1sw 2sw
COMP.
resi
dtd
LVLI
Why the commutation was not completed during t3~t4 ?
1. Co Voltage droped during t2~t3.
2. Power loss during t3~t4.
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [144]
Losses Calculation
2
r
r
)(2
os2
D)pk(in)loss(R
1
e1Cf)VV(Pr
r
r
⎟⎟⎠
⎞⎜⎜⎝
⎛ωα
+
−⋅⋅⋅−=
ωα
π−
2
r
r
)(
D)pk(inosD)loss(D
1
e1)VV(CfV2Pr
r
⎟⎟⎠
⎞⎜⎜⎝
⎛ωα
+
+⋅−⋅⋅⋅=
ωα
π−
Q)pk(in)loss.(Comp IV2P ⋅≈
s)pk(in)n(gss)pk(in)p(gs)loss(Gate fVQfVQP ⋅⋅+⋅⋅≈
LD)loss(Bridge IV2P ⋅≈Bridge losses
Gate drive losses
Comparator losses
Diode losses
Resistance losses
49
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [145]
oCexV LR
inV
1D 2D
resL
derV
SV+
SV−
nM pMsensR
hysRderC
derR
COMP. cV
1SD2SD
1SC2SC
LC1BD 2BD
3BD 4BD
inR
LV
+
−
ini
Experiment with dummy current source
• Schottky diodes 1N5817
• Ultra low power IC(MAX921, Maxim, USA)
• MOSFET (VP0104, VN0104)
200V floating source
Rin=100KΩ
Rsens=1KΩ
Co=330nF
Lres=1mH
CL=1µF
fs=185Hz
DifferentiatorCurrent source
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [146]
LR
inV
1D 2D
resL
derV
SV+
SV−
nM pM
hysRderC
derR
COMP. cV
1SD2SD
1SC2SC
LC1BD 2BD
3BD 4BD
LV
+
−
Actuator Transducer
exV
Longitudinally piezoelectric bimorph van element
RBL-1-006 model, Piezo Systems, Inc, USA
Experiment with Piezoelectric Generator
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [147]
Experimental Circuits
oCini LC
inVouti
LVLI
1BD 2BD
3BD 4BD
RL
2.Proposed rectifier.
3.Proposed rectifier with external supplies.
a. Higher VS reduces Rds(on).b. Supplies the (small) power
consumption of the comparator circuitry.
1.Reference circuit.
oCexV LR
inV
1D 2D
resL
derV
SV+
SV−
nM pMsensR
hysRderC
derR
COMP. cV
LC1BD 2BD
3BD 4BD
inR
LV
+
−
1SD2SD
1SC2SC
ini
External
50
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [148]
Results
251%3.16mW3.51KΩ3.34VRESONANT RECTIFIER WITH EXTERNAL SUPPLIES VS=±4V
142%1.79mW2.75KΩ2.2VRESONANT RECTIFIER
-1.26mW2.1KΩ1.6VSTANDARD RECTIFIER
GAIN (%) COMPARED WITH STANDARD
RECTIFIER
OUTPUT POWER
OPTIMAL LOAD
RESISTANCE
OUTPUT VOLTAGE (DC)CIRCUIT TOPOLOGY
230%1.23mW11.43KΩ3.75V
118%0.636mW5.19KΩ1.818V
-0.537mW5.89KΩ1.779V
RESONANT RECTIFIER WITH EXTERNAL SUPPLIES VS=±4V
RESONANT RECTIFIER
STANDARD RECTIFIER
GAIN (%) COMPARED WITH STANDARD
RECTIFIER
OUTPUT POWER
OPTIMAL LOAD
RESISTANCE
OUTPUT VOLTAGE (DC)CIRCUIT TOPOLOGY
Dummy current source
Piezoelectric Generator
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [149]
Experiment Waveforms(with external PS)
inin iV ⋅
ini
inV
1t 2t 3t 4t 5t
inV
derV
5V/div
2mA/div
2V/div
2V/div
Input Voltage
Input Current
Input Voltage
Instantaneous input power
Derivative Signal
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [150]
The Resonant Rectifier
The operation of the new rectifier is based on self commutation of capacitor voltage
A considerable portion of the losses are due to forward voltage drop of the bridge diodes.
The resonant rectifier exhibits a substantial improvement compared to the conventional rectifier.
Additional improvement could be achieved by replacing the diode bridge by a synchronous rectification scheme.
51
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [151]
PTInverter VoVin= 5V Lamp
[31, 47-58]
PT Based CCFL Ballasts
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [152]
Peak voltage
[47]
Cold cathode Fluorescent lamp (CCFL)Drivers
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [153]
Requirements:• Power handling capability• High ignition voltage ~1000V• Sufficient energy to pass the peak voltage• High operating voltage ~ 600V
The Issue of Dynamic Stability
Lamp Current
Lamp Voltage[49, 50, 78]
52
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [154]
Dynamic changes of V-I characteristics of a fluorescent lamp operating at high frequency
P1
P2
B
FastCurrent
ChangesSlow Power
Changes
I(lamp) [Arms]
V(lamp) [Vrms] FastCurrent
Changes
AReq1
Req2
Vs
Static V-A line
• Linear approximation of V-I curve [78, 83]
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [155]
eq1 R
)lamp(VG =
)I(fR )rms(lampeq =
RiCi match the dynamic response of lamp
21 )lamp(iE ≡
)p(vE2 ≡
+-
+-
rms
Lamp Model
R
lamp isq p
Ci
Ri
E2E1G1
Lamp
[83]
SPICE Compatible Fluorescent Lamp Model
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [156]
1fjf
1R
Rfjf
sR)Lf(incZ
0
Ls
eq
0
L
+
−=
eqINCL
sINCLRZfForRZ0fFor
→∞→−→→
• “Right-Half Complex-Plane” Zero [83]
Incremental impedance of fluorescent lamps
53
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [157]
Frequency100Hz 10KHz 1.0MHz
db
-50
-45
-40
0o
-200o
-100o
| Yinc |
Phase
Negative incremental resistance at low modulating frequency [83]
Incremental Admittance of Experimental Lamp Obtained by Simulation
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [158]
Modulation frequency 200Hz
Negative incremental resistance
Excitation frequency 50kHz
measuredresponse
simulatedresponse
I
V
I
V
[83]
Response to modulation
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [159]
Modulation frequency 5kHzExcitation frequency 50kHz
Positive incremental resistance
IV
measuredresponse
simulatedresponse
IV
[83]
Response to a modulation
54
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [160]
1A
-1A
200V
-200V
measuredresponse
simulatedresponse
I
V
I
V
[83]
Response to a power step
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [161]
Vlamp
Zlamp
Vex +-
IlampZballast
1
Vex
Zballast
Zlamp
VlampIlampBallast
Zballast Zlamp
Vex
lampballast
ZZ
1LoopGain =
Ballast-Lamp InteractionFeedback Model
[78]
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [162]
Stability Criteria for Carrier Driven SystemsWhat is Z ?
Lamp(Non-Linear
NegativeResistance
Load)
LsCs
Vin fc
ssrc CL2π
1ff =>>
ssrc CL2π
1ff =≈
Stable
Unstable
55
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [163]
For the Lamp-Ballast system:The relevant impedance is the INCREMENTAL IMPEDANCE under the specific carrier excitation
( )ex
exminc ∆I
∆VfZ =
LsCs
Vex
Iex∆ Iex
∆Vex
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [164]
How does an incremental impedance (Zinc) behave ?Example & Intuitive Observation
• Yinc(fm) will have a peak when fm = |fr – fc| • Envelope analysis
f r fc
fm fm
( ) ( )( ) ( )t2sint2sinA1tV cmmex ff ππ+=
LsCs
Rs
AM ModulatedSignal
Yinc fm( )
fm sweep
Vex
Y ofResonantNetwork
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [165]
• Fast large signal simulation(as compared to TRAN simulation)
• Very fast small signal simulation(as compared to TRAN simulation)
• Large (TRAN) and Small Signal (AC) compatible
• Can be implemented on any modern circuit simulator
For details see [77-79] and Appendix A
SPICE Compatible Envelope Simulation
56
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [166]
[77-79]
Envelope Simulation Primer
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [167]
Power System Driven by a Modulated Signal
Modulator-Driver
Reactivenetwork Load
uc(t)
um(t) )t(uout)t(u
The need for Envelope Simulation
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [168]
( ) ( ) ( )( ) ]etjItIRe[ti tf2j21
cπ+=
A Primer to Envelope Simulation
Any analog modulated signal (AM, FM or PM) can be described by the following expression:
The Current in the network excited by u(t):
( ) ( ) ( ) ( ) ( )( ) ( )( ) ]etjUtURe[
tf2sintUtf2costUtutf2j
21
c2c1
cπ−
=π+π=
57
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [169]
( ) ( ) ( )
( ) ( ) ( )⎪⎩
⎪⎨
⎧
π+=
π−=
]tLIf2dt
tdIL[j]tV[j
tLIf2dt
tdILtV
1c2
2
2c1
1
( ) ( ) ( )tItIti 22
21 +=
( ) ( ) ( )tVtVtv 22
21 +=
Phasor Analysis
Inductance
LiL
L
LI2
I1
I2(t)ωcL
I1(t)ωcLIm
Re
+ -
+-
V1
V2
( ) ( )( ) ( ) ( )
( ) ( ) ]e)tLIf2jdt
tdIjL
ttIf2dt
tdILRe[(]etjVtVRe[
tf2j1c
2
2c1tf2j
21
c
c
π
π
π+
+π+=−
( ) ( )dt
tdiLtv =
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [170]
Phasor Analysis
Capacitance
Resistance
VC C
C
CV2
V1ωcCV2ωcC
Im
Re
V1I1
I2
RR
RIm
Re I1
I2
V1
V2
( ) ( )dt
tdvCti =
( ) ( )tRitv =
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [171]
Splitting the Networkinto Two Cross-Coupled Components -
Imaginary and Real
LoadNetworkSource inV
( )tu
outV
58
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [172]
imaginary circuitcomponent
real circuitcomponent
coupling
inre
inim
outre
outim
U1
U2
22 V(outim)V(outre) + outV
Real Load Component
Imaginary Load Component
Splitting the Networkinto Two Cross-Coupled Components -
Imaginary and Real
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [173]
Example: Piezoelectric Transformer Driven by FM Signal (SPICE)
Vin
Excitation
Ro
LoadVoutFMVoVin
Rectifier
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [174]
Example: Piezoelectric Transformer Driven by FM Signal (SPICE)
Vin
Excitation
Ro
LoadVoutFMRectifier
Co
Lr
+-
Cr Rm1:n
Vo/n
Vo
I(Lr)/n
Ci
Equivalent cirquit of the PiezoelectricTransformer
I(Lr)
59
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [175]
Example: Piezoelectric Transformer Driven by FM Signal (SPICE)
Vin
Excitation
Ro
LoadVoutFMCo
Lr
+-
Cr Rm1:n
Vo/n
Vo
I(Lr)/n
Ci
Equivalent cirquit of the PiezoelectricTransformer
I(Lr) ReqCeq
Equivalentreplacementof rectifier
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [176]
Example: Piezoelectric Transformer Driven by FM Signal (SPICE)
Vin
Excitation
Ro
LoadVoutFMCo
Lr
+-
Cr Rm1:n
Vo/n
Vo
I(Lr)/n
Ci
Equivalent cirquit of the PiezoelectricTransformer
I(Lr)
( ) ( )( )∫π+π= dttuk2tf2cosAtu mfcc
( )tf2sinA)t(u mmm π= - Harmonic modulating signal
ReqCeq
Equivalentreplacementof rectifier
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [177]
( ) ( )( ) ( )( )( ) ( )tf2sintf2sinsinA
tf2costf2sincosAtu
cmc
cmcππβ
−ππβ=
Example: Piezoelectric Transformer Driven by FM Signal (SPICE)
Vin
Excitation
Ro
LoadVoutFMCo
Lr
+-
Cr Rm1:n
Vo/n
Vo
I(Lr)/n
Ci
Equivalent cirquit of the PiezoelectricTransformer
I(Lr)
( ) ( )( )tf2costf2cosAtu mcc πβ−π=m
mffAkwhere =β
ReqCeq
Equivalentreplacementof rectifier
60
Prof. S. Ben-Yaakov, Power Electronics of Piezoelectric Elements, June 2006 [178]