Post on 23-Jun-2020
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Energy Harvesting from Ambient Vibrations
Frederic Giraud
L2EP – University Lille1
November 27, 2012
Frederic Giraud Master E2D2 University Lille1 - L2EP
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Table of contents
1 IntroductionWhat is Energy Harvesting ?Generator TechnologiesSummary
2 Modelling of a piezoelectric energy harvesterPresentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
3 An Example of inverterIntroductionSSHI: Synchronized Switch Harvesting on Inductor
Frederic Giraud Master E2D2 November 27, 2012 2 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
What is Energy Harvesting ?
We extract energy from an ambient and free source:
Frederic Giraud Master E2D2 November 27, 2012 3 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
What is Energy Harvesting ?
We extract energy from an ambient and free source:
−Wind
−Light
−Vibrations
−Thermal
−...
SOURCE
Frederic Giraud Master E2D2 November 27, 2012 3 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
What is Energy Harvesting ?
We extract energy from an ambient and free source:
−Wind
−Light
−Vibrations
−Thermal
−...
SOURCE
︷ ︸︸ ︷
EnergyConversion
Frederic Giraud Master E2D2 November 27, 2012 3 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
What is Energy Harvesting ?
We extract energy from an ambient and free source:
−Wind
−Light
−Vibrations
−Thermal
−...
SOURCE
︷ ︸︸ ︷
EnergyConversion
LOAD
Frederic Giraud Master E2D2 November 27, 2012 3 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
What is Energy Harvesting ?
We extract energy from an ambient and free source:
−Wind
−Light
−Vibrations
−Thermal
−...
SOURCE
︷ ︸︸ ︷
EnergyConversion
LOAD
P1 P2
Frederic Giraud Master E2D2 November 27, 2012 3 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
What is Energy Harvesting ?
We extract energy from an ambient and free source:
−Wind
−Light
−Vibrations
−Thermal
−...
SOURCE
︷ ︸︸ ︷
EnergyConversion
LOAD
P1 P2
We talk about Energy Harvesting or also energy scavenging
when the converted power is small, typically less than 1W.
Frederic Giraud Master E2D2 November 27, 2012 3 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
What is Energy Harvesting ?
We extract energy from an ambient and free source:
−Wind
−Light
−Vibrations
−Thermal
−...
SOURCE
︷ ︸︸ ︷
EnergyConversion
LOAD
P1 P2
We talk about Energy Harvesting or also energy scavenging
when the converted power is small, typically less than 1W.
η = P2P1
= 1− P1−P2P1
−→ Losses in the energy converter should be
as small as possible.
Frederic Giraud Master E2D2 November 27, 2012 3 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Objectives: Sensors Network
www.perpetuum.com
Frederic Giraud Master E2D2 November 27, 2012 4 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Objectives: Sensors Network
www.perpetuum.com
Gutierriez,A Heterogeneous Wireless IdentificationNetwork for the Localization of Animals Based onStochastic Movements
Frederic Giraud Master E2D2 November 27, 2012 4 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Objectives: Sensors Network
www.perpetuum.com
Gutierriez,A Heterogeneous Wireless IdentificationNetwork for the Localization of Animals Based onStochastic Movements
http://www.rfwirelesssensors.com, 2012
Roundy et Al.:A study of low level vibrations as a powersource for wireless sensor nodes.
Frederic Giraud Master E2D2 November 27, 2012 4 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Objectives: Power, just where you need it
http://enocean.com
Wireless
Reduce Cost, and isreconfigurable
Better Waste Cycle(Information fromEnocean)
Frederic Giraud Master E2D2 November 27, 2012 5 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Objectives: Power, just where you need it
http://enocean.com
Wireless
Reduce Cost, and isreconfigurable
Better Waste Cycle(Information fromEnocean)
Innowattech’s systems produces power with vehicles
http://www.innowattech.com
Frederic Giraud Master E2D2 November 27, 2012 5 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Objectives: Power, just where you need it
http://enocean.com
Wireless
Reduce Cost, and isreconfigurable
Better Waste Cycle(Information fromEnocean)
Innowattech’s systems produces power with vehicles
http://www.innowattech.com
The economist – April 28th 2007
Frederic Giraud Master E2D2 November 27, 2012 5 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Objectives: Marketing Purpose
Experience: Same Remotecontroller energized by humanpower, but with different packaging.
”No need for Batteries”
”Green” ”Fun”
Frederic Giraud Master E2D2 November 27, 2012 6 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Objectives: Marketing Purpose
Experience: Same Remotecontroller energized by humanpower, but with different packaging.
”No need for Batteries”
”Green” ”Fun”
54% of people Will choose the firstone because it is eco-friendly.(Jansen, Human power empirically explored)
Frederic Giraud Master E2D2 November 27, 2012 6 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Objectives: Marketing Purpose
Experience: Same Remotecontroller energized by humanpower, but with different packaging.
”No need for Batteries”
”Green” ”Fun”
54% of people Will choose the firstone because it is eco-friendly.(Jansen, Human power empirically explored)
Several projects are born fromthis fact: Metis Produces energyfrom dancers’ movements.
In Toulouse, the system VIHA
proposes Smart Tiles to energiesstreet lights.
Frederic Giraud Master E2D2 November 27, 2012 6 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
The energy converter
−Wind
−Light
−Vibrations
−Thermal
−...
SOURCE LOAD
Frederic Giraud Master E2D2 November 27, 2012 7 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
The energy converter
−Wind
−Light
−Vibrations
−Thermal
−...
SOURCE LOAD
︷ ︸︸ ︷
EnergyConversion
ElectricityElec.
Frederic Giraud Master E2D2 November 27, 2012 7 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
The energy converter
−Wind
−Light
−Vibrations
−Thermal
−...
SOURCE LOAD
︷ ︸︸ ︷
EnergyConversion
ElectricityElec.
Generator
Frederic Giraud Master E2D2 November 27, 2012 7 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
The energy converter
−Wind
−Light
−Vibrations
−Thermal
−...
SOURCE LOAD
︷ ︸︸ ︷
EnergyConversion
ElectricityElec.
Generator Inverter
Frederic Giraud Master E2D2 November 27, 2012 7 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Solar Harvesters:
Frederic Giraud Master E2D2 November 27, 2012 8 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Solar Harvesters:
This sensor measures IntraOccular Pres-sure http://cymbet.com
Handbag to recharge electronicdevices http://www.neubers.de
Frederic Giraud Master E2D2 November 27, 2012 8 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Solar Harvesters:
This sensor measures IntraOccular Pres-sure http://cymbet.com
Handbag to recharge electronicdevices http://www.neubers.de
Solar Energy Harvester Evaluation Kit http://ti.com
Power and current asa function of voltage:
I,P
V
Frederic Giraud Master E2D2 November 27, 2012 8 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Solar Harvesters:
This sensor measures IntraOccular Pres-sure http://cymbet.com
Handbag to recharge electronicdevices http://www.neubers.de
Solar Energy Harvester Evaluation Kit http://ti.com
Power and current asa function of voltage:
I,P
VMPPT strategies,require Energymanagement of thesystem
Frederic Giraud Master E2D2 November 27, 2012 8 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Thermoelectric
A Peltier module from http://www.tellurex.com
VDC
RDC IDC
Frederic Giraud Master E2D2 November 27, 2012 9 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Thermoelectric
A Peltier module from http://www.tellurex.com
VDC
RDC IDC
IDC
VDC
IDC
P
T1
Frederic Giraud Master E2D2 November 27, 2012 9 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Thermoelectric
A Peltier module from http://www.tellurex.com
VDC
RDC IDC
IDC
VDC
IDC
P
T1 T2
Frederic Giraud Master E2D2 November 27, 2012 9 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Thermoelectric
A Peltier module from http://www.tellurex.com
VDC
RDC IDC
IDC
VDC
IDC
P
T1 T2
Frederic Giraud Master E2D2 November 27, 2012 9 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Thermoelectric
A Peltier module from http://www.tellurex.com
VDC
RDC IDC
IDC
VDC
IDC
P
T1 T2
=
=
Frederic Giraud Master E2D2 November 27, 2012 9 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Thermoelectric
A Peltier module from http://www.tellurex.com
VDC
RDC IDC
IDC
VDC
IDC
P
T1 T2
=
=
Temperature Gradient (residential). Lindsay Miller,http://uc-ciee.org
plumbing application from http://www.nextreme.com
Frederic Giraud Master E2D2 November 27, 2012 9 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vB
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vB=
∼
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vB
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vBt
e,vB, i
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vBt
e,vB, i
1
1) Diode turns ON
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vBt
e,vB, i
1 2
1) Diode turns ON2) di
dt= 0 because e − vB = 0
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vBt
e,vB, i
1 2
3
1) Diode turns ON2) di
dt= 0 because e − vB = 0
3) i = 0, diode turns OFF
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vBt
e,vB, i
1 2
3〈i〉
1) Diode turns ON2) di
dt= 0 because e − vB = 0
3) i = 0, diode turns OFF
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vBt
e,vB, i
1 2
3〈i〉
1) Diode turns ON2) di
dt= 0 because e − vB = 0
3) i = 0, diode turns OFF vB
〈i〉, P
Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vBt
e,vB, i
1 2
3〈i〉
1) Diode turns ON2) di
dt= 0 because e − vB = 0
3) i = 0, diode turns OFF vB
〈i〉, P
P = 〈vB i〉 = vB〈i〉Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vBt
e,vB, i
1 2
3〈i〉
1) Diode turns ON2) di
dt= 0 because e − vB = 0
3) i = 0, diode turns OFF vB
〈i〉, P
P = 〈vB i〉 = vB〈i〉Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Magnetic
stoppercoilstopper
φx
e = −N dφdt
= N dφdx
dxdt
v
e(t)
L i
vB=
∼t
e,vB, i
1 2
3〈i〉
1) Diode turns ON2) di
dt= 0 because e − vB = 0
3) i = 0, diode turns OFF vB
〈i〉, P
P = 〈vB i〉 = vB〈i〉Frederic Giraud Master E2D2 November 27, 2012 10 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Equivalent electrical circuit
vim
im is a current proportionalto the deformation speed
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Equivalent electrical circuit
vim
im is a current proportionalto the deformation speed
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Equivalent electrical circuit
vim
im is a current proportionalto the deformation speedComparison Magn. Piezo
Magnetic Piezo.Voltage source Current
sourceInductive capacitiveLarge Stroke Small
StrokeRemote action
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Equivalent electrical circuit
vim
im is a current proportionalto the deformation speedComparison Magn. Piezo
Magnetic Piezo.Voltage source Current
sourceInductive capacitiveLarge Stroke Small
StrokeRemote action
Roundy, A piezoelectric vibrationbased generator for wireless
electronics (2004)
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Equivalent electrical circuit
vim RL
im is a current proportionalto the deformation speedComparison Magn. Piezo
Magnetic Piezo.Voltage source Current
sourceInductive capacitiveLarge Stroke Small
StrokeRemote action
Roundy, A piezoelectric vibrationbased generator for wireless
electronics (2004)
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Equivalent electrical circuit
vim RL
im is a current proportionalto the deformation speedComparison Magn. Piezo
Magnetic Piezo.Voltage source Current
sourceInductive capacitiveLarge Stroke Small
StrokeRemote action
Roundy, A piezoelectric vibrationbased generator for wireless
electronics (2004)
ω
P
ω0
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Equivalent electrical circuit
vim RL
im is a current proportionalto the deformation speedComparison Magn. Piezo
Magnetic Piezo.Voltage source Current
sourceInductive capacitiveLarge Stroke Small
StrokeRemote action
Roundy, A piezoelectric vibrationbased generator for wireless
electronics (2004)
ω
P
ω0
RLopt
PMax
RL
P
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Energy ManagementEquivalent electrical circuit
vim RL
im is a current proportionalto the deformation speedComparison Magn. Piezo
Magnetic Piezo.Voltage source Current
sourceInductive capacitiveLarge Stroke Small
StrokeRemote action
Roundy, A piezoelectric vibrationbased generator for wireless
electronics (2004)
ω
P
ω0
RLopt
PMax
RL
P
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Piezoelectric
An electronic lighter,http://freepatentsonline.com
Piezoelectric crystals
Cantilever beam
w(t) = Wsin(ωt)
Energy ManagementEquivalent electrical circuit
vim RL
=
∼
im is a current proportionalto the deformation speedComparison Magn. Piezo
Magnetic Piezo.Voltage source Current
sourceInductive capacitiveLarge Stroke Small
StrokeRemote action
Roundy, A piezoelectric vibrationbased generator for wireless
electronics (2004)
ω
P
ω0
RLopt
PMax
RL
P
Frederic Giraud Master E2D2 November 27, 2012 11 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
There is no ”one fit all” solution
Each solution may be efficient in a certain range of Power.Meanwhile, shrinking Chips consumption come at a time when energy harvesting becomes efficient and practical
(source: IDtechex.com)
Frederic Giraud Master E2D2 November 27, 2012 12 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
What is Energy Harvesting ?Generator TechnologiesSummary
Table of contents
1 IntroductionWhat is Energy Harvesting ?Generator TechnologiesSummary
2 Modelling of a piezoelectric energy harvesterPresentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
3 An Example of inverterIntroductionSSHI: Synchronized Switch Harvesting on Inductor
Frederic Giraud Master E2D2 November 27, 2012 13 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Coordinates and assumptions
moving Case
Bender
Frederic Giraud Master E2D2 November 27, 2012 14 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Coordinates and assumptions
moving Case
Bender
The bender is attached to a vibrating andrigid case,
Frederic Giraud Master E2D2 November 27, 2012 14 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Coordinates and assumptions
moving Case
BenderM
The bender is attached to a vibrating andrigid case,
A mass M is attached to increase powerharvesting,
Frederic Giraud Master E2D2 November 27, 2012 14 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Coordinates and assumptions
moving Case
BenderM
w(t)
The bender is attached to a vibrating andrigid case,
A mass M is attached to increase powerharvesting,
w(t) is the deflection of the beam,
Frederic Giraud Master E2D2 November 27, 2012 14 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Coordinates and assumptions
moving Case
BenderM
w(t)
ℜ
The bender is attached to a vibrating andrigid case,
A mass M is attached to increase powerharvesting,
w(t) is the deflection of the beam,
We define ℜ a fixed reference frame,
Frederic Giraud Master E2D2 November 27, 2012 14 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Coordinates and assumptions
moving Case
BenderM
w(t)
ℜℜ′
The bender is attached to a vibrating andrigid case,
A mass M is attached to increase powerharvesting,
w(t) is the deflection of the beam,
We define ℜ a fixed reference frame,
And ℜ′ a reference frame affixed to thevibrating case.
Frederic Giraud Master E2D2 November 27, 2012 14 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Coordinates and assumptions
moving Case
Bender
−→F p→M
Mw(t)
ℜℜ′
The bender is attached to a vibrating andrigid case,
A mass M is attached to increase powerharvesting,
w(t) is the deflection of the beam,
We define ℜ a fixed reference frame,
And ℜ′ a reference frame affixed to thevibrating case.
−→F p→m is the force of the Bender onto themass M.
Frederic Giraud Master E2D2 November 27, 2012 14 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Coordinates and assumptions
moving Case
Bender
−→F p→M
Mw(t)
ℜℜ′
The bender is attached to a vibrating andrigid case,
A mass M is attached to increase powerharvesting,
w(t) is the deflection of the beam,
We define ℜ a fixed reference frame,
And ℜ′ a reference frame affixed to thevibrating case.
−→F p→m is the force of the Bender onto themass M.
The case is supposed to be controlled inposition, and we have yc = Asin(ωt) theamplitude of the case’s vibration.
Frederic Giraud Master E2D2 November 27, 2012 14 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Coordinates and assumptions
moving Case
Bender
−→F p→M
Mw(t)
ℜℜ′
The bender is attached to a vibrating andrigid case,
A mass M is attached to increase powerharvesting,
w(t) is the deflection of the beam,
We define ℜ a fixed reference frame,
And ℜ′ a reference frame affixed to thevibrating case.
−→F p→m is the force of the Bender onto themass M.
The case is supposed to be controlled inposition, and we have yc = Asin(ωt) theamplitude of the case’s vibration.
Gravity is neglected, as well as Inertiamomentum of the bender,
Frederic Giraud Master E2D2 November 27, 2012 14 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Coordinates and assumptions
moving Case
Bender
−→F p→M
Mw(t)
ℜℜ′
vim
i
The bender is attached to a vibrating andrigid case,
A mass M is attached to increase powerharvesting,
w(t) is the deflection of the beam,
We define ℜ a fixed reference frame,
And ℜ′ a reference frame affixed to thevibrating case.
−→F p→m is the force of the Bender onto themass M.
The case is supposed to be controlled inposition, and we have yc = Asin(ωt) theamplitude of the case’s vibration.
Gravity is neglected, as well as Inertiamomentum of the bender,
Actuator electrical convention.
Frederic Giraud Master E2D2 November 27, 2012 14 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M: Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Mw(t) = f +MAω2sin(ωt) = f + facc
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Mw(t) = f +MAω2sin(ωt) = f + facc
Electrical Behaviour: Capacitive
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Mw(t) = f +MAω2sin(ωt) = f + facc
Electrical Behaviour: Capacitive
v = 1Cb
∫(i − im)dt
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Mw(t) = f +MAω2sin(ωt) = f + facc
Electrical Behaviour: Capacitive
v = 1Cb
∫(i − im)dt
Piezoelectric effect
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Mw(t) = f +MAω2sin(ωt) = f + facc
Electrical Behaviour: Capacitive
v = 1Cb
∫(i − im)dt
Piezoelectric effect
im derives from deflection: im = Nw
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Mw(t) = f +MAω2sin(ωt) = f + facc
Electrical Behaviour: Capacitive
v = 1Cb
∫(i − im)dt
Piezoelectric effect
im derives from deflection: im = Nw
while v produces an internal forcefp = Nv
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Mw(t) = f +MAω2sin(ωt) = f + facc
Electrical Behaviour: Capacitive
v = 1Cb
∫(i − im)dt
Piezoelectric effect
im derives from deflection: im = Nw
while v produces an internal forcefp = Nv
Material’s behaviour
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Mw(t) = f +MAω2sin(ωt) = f + facc
Electrical Behaviour: Capacitive
v = 1Cb
∫(i − im)dt
Piezoelectric effect
im derives from deflection: im = Nw
while v produces an internal forcefp = Nv
Material’s behaviour
The material is elastic: fs = Ks
∫wdt
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Mw(t) = f +MAω2sin(ωt) = f + facc
Electrical Behaviour: Capacitive
v = 1Cb
∫(i − im)dt
Piezoelectric effect
im derives from deflection: im = Nw
while v produces an internal forcefp = Nv
Material’s behaviour
The material is elastic: fs = Ks
∫wdt
With some friction inside:fs = Ks
∫wdt + Ds w
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Equations
Dynamic of the mass M:
M d2
dt2(w(t) + Asin(ωt)) = Fp→M = f
Mw(t) = f +MAω2sin(ωt) = f + facc
Electrical Behaviour: Capacitive
v = 1Cb
∫(i − im)dt
Piezoelectric effect
im derives from deflection: im = Nw
while v produces an internal forcefp = Nv
Material’s behaviour
The material is elastic: fs = Ks
∫wdt
With some friction inside:fs = Ks
∫wdt + Ds w
These actions are opposite to the PEeffect: f = fp − fs
Glossary
M the mass
f , the force onto M
A, vibration’s amplitude
ω, vibration’s pulsation
facc is the inertial force
i current of the device(actuator convention)
im motional current
fp inside piezo force
N Piezoelectric forcefactor (depends ongeometry)
fp Piezo internal force
fs Material internal elasticforce
Ks equivalent stiffness(depends on geometry)
Ds Viscous coefficient
Frederic Giraud Master E2D2 November 27, 2012 15 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
EMR of the system
v =1
Cb
∫
(i − im)dt
︸ ︷︷ ︸
Frederic Giraud Master E2D2 November 27, 2012 16 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
EMR of the system
v =1
Cb
∫
(i − im)dt
︸ ︷︷ ︸
i
vSE
Frederic Giraud Master E2D2 November 27, 2012 16 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
EMR of the system
v =1
Cb
∫
(i − im)dt
︸ ︷︷ ︸
i
v im
v
SE
︷ ︸︸ ︷
fp = Nv, im = Nw
Frederic Giraud Master E2D2 November 27, 2012 16 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
EMR of the system
v =1
Cb
∫
(i − im)dt
︸ ︷︷ ︸
i
v im
v
SE
︷ ︸︸ ︷
fp = Nv, im = Nw
w
fp
w
f
w =1
M
∫
(f − facc )dt
︸ ︷︷ ︸
Frederic Giraud Master E2D2 November 27, 2012 16 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
EMR of the system
v =1
Cb
∫
(i − im)dt
︸ ︷︷ ︸
i
v im
v
SE
︷ ︸︸ ︷
fp = Nv, im = Nw
w
fp
w
f
w =1
M
∫
(f − facc )dt
︸ ︷︷ ︸
︷ ︸︸ ︷
f = fp − fs︷ ︸︸ ︷
fs = Ks
∫
wdt + Ds w
w
fs
Frederic Giraud Master E2D2 November 27, 2012 16 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
EMR of the system
v =1
Cb
∫
(i − im)dt
︸ ︷︷ ︸
i
v im
v
SE
︷ ︸︸ ︷
fp = Nv, im = Nw
w
fp
w
f
w =1
M
∫
(f − facc )dt
︸ ︷︷ ︸
facc
w
︷ ︸︸ ︷
f = fp − fs︷ ︸︸ ︷
fs = Ks
∫
wdt + Ds w
w
fs
SM
Frederic Giraud Master E2D2 November 27, 2012 16 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
EMR of the system
v =1
Cb
∫
(i − im)dt
︸ ︷︷ ︸
i
v im
v
SE
︷ ︸︸ ︷
fp = Nv, im = Nw
w
fp
w
f
w =1
M
∫
(f − facc )dt
︸ ︷︷ ︸
facc
w
︷ ︸︸ ︷
f = fp − fs︷ ︸︸ ︷
fs = Ks
∫
wdt + Ds w
w
fs
SM
pe = v .i is the output power,
Frederic Giraud Master E2D2 November 27, 2012 16 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
EMR of the system
v =1
Cb
∫
(i − im)dt
︸ ︷︷ ︸
i
v im
v
SE
︷ ︸︸ ︷
fp = Nv, im = Nw
w
fp
w
f
w =1
M
∫
(f − facc )dt
︸ ︷︷ ︸
facc
w
︷ ︸︸ ︷
f = fp − fs︷ ︸︸ ︷
fs = Ks
∫
wdt + Ds w
w
fs
SM
pe = v .i is the output power, pm = facc w is the mechanical input power
and both should be < 0
Frederic Giraud Master E2D2 November 27, 2012 16 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
EMR of the system
v =1
Cb
∫
(i − im)dt
︸ ︷︷ ︸
i
v im
v
SE
︷ ︸︸ ︷
fp = Nv, im = Nw
w
fp
w
f
w =1
M
∫
(f − facc )dt
︸ ︷︷ ︸
facc
w
︷ ︸︸ ︷
f = fp − fs︷ ︸︸ ︷
fs = Ks
∫
wdt + Ds w
w
fs
SM
pe = v .i is the output power, pm = facc w is the mechanical input power
and both should be < 0
v
i e
i
SE Ω
T
Tr
Ω
SM Comparison with a DC motor
Things are not so differerent.
Frederic Giraud Master E2D2 November 27, 2012 16 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
EMR of the system
v =1
Cb
∫
(i − im)dt
︸ ︷︷ ︸
i
v im
v
SE
︷ ︸︸ ︷
fp = Nv, im = Nw
w
fp
w
f
w =1
M
∫
(f − facc )dt
︸ ︷︷ ︸
facc
w
︷ ︸︸ ︷
f = fp − fs︷ ︸︸ ︷
fs = Ks
∫
wdt + Ds w
w
fs
SM
pe = v .i is the output power, pm = facc w is the mechanical input power
and both should be < 0
v
i e
i
SE Ω
T
Tr
Ω
SM Comparison with a DC motor
Things are not so differerent.
Frederic Giraud Master E2D2 November 27, 2012 16 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Asumption
facc = MAω2sin(ωt) for harmonic oscillations
Frederic Giraud Master E2D2 November 27, 2012 17 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Asumption
facc = MAω2sin(ωt) for harmonic oscillations
complex notation: x is the complex phasor of x(t), meansx(t) = ℑ(x)
Frederic Giraud Master E2D2 November 27, 2012 17 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Asumption
facc = MAω2sin(ωt) for harmonic oscillations
complex notation: x is the complex phasor of x(t), meansx(t) = ℑ(x)since oscillations are harmonic, we will write:x = Xe jωt
Frederic Giraud Master E2D2 November 27, 2012 17 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Asumption
facc = MAω2sin(ωt) for harmonic oscillations
complex notation: x is the complex phasor of x(t), meansx(t) = ℑ(x)since oscillations are harmonic, we will write:x = Xe jωt
for steady state operation, X is constant, leading todxdt
= jωXe jωt
Frederic Giraud Master E2D2 November 27, 2012 17 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Asumption
facc = MAω2sin(ωt) for harmonic oscillations
complex notation: x is the complex phasor of x(t), meansx(t) = ℑ(x)since oscillations are harmonic, we will write:x = Xe jωt
for steady state operation, X is constant, leading todxdt
= jωXe jωt
|x | = |X | = X
Frederic Giraud Master E2D2 November 27, 2012 17 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Asumption
facc = MAω2sin(ωt) for harmonic oscillations
complex notation: x is the complex phasor of x(t), meansx(t) = ℑ(x)since oscillations are harmonic, we will write:x = Xe jωt
for steady state operation, X is constant, leading todxdt
= jωXe jωt
|x | = |X | = X
for example, f acc = MAω2e jωt and |f acc | = MAω2.
Frederic Giraud Master E2D2 November 27, 2012 17 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Asumption
facc = MAω2sin(ωt) for harmonic oscillations
complex notation: x is the complex phasor of x(t), meansx(t) = ℑ(x)since oscillations are harmonic, we will write:x = Xe jωt
for steady state operation, X is constant, leading todxdt
= jωXe jωt
|x | = |X | = X
for example, f acc = MAω2e jωt and |f acc | = MAω2.
v = −RLi
Frederic Giraud Master E2D2 November 27, 2012 17 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Asumption
facc = MAω2sin(ωt) for harmonic oscillations
complex notation: x is the complex phasor of x(t), meansx(t) = ℑ(x)since oscillations are harmonic, we will write:x = Xe jωt
for steady state operation, X is constant, leading todxdt
= jωXe jωt
|x | = |X | = X
for example, f acc = MAω2e jωt and |f acc | = MAω2.
v = −RLi
Frederic Giraud Master E2D2 November 27, 2012 17 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Asumption
facc = MAω2sin(ωt) for harmonic oscillations
complex notation: x is the complex phasor of x(t), meansx(t) = ℑ(x)since oscillations are harmonic, we will write:x = Xe jωt
for steady state operation, X is constant, leading todxdt
= jωXe jωt
|x | = |X | = X
for example, f acc = MAω2e jωt and |f acc | = MAω2.
v = −RLi
v RLim
i
Frederic Giraud Master E2D2 November 27, 2012 17 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Asumption
facc = MAω2sin(ωt) for harmonic oscillations
complex notation: x is the complex phasor of x(t), meansx(t) = ℑ(x)since oscillations are harmonic, we will write:x = Xe jωt
for steady state operation, X is constant, leading todxdt
= jωXe jωt
|x | = |X | = X
for example, f acc = MAω2e jωt and |f acc | = MAω2.
v = −RLi
v RLim
i And RL ≪ 1Cbω
, yieldsv ≃ −RLim
Frederic Giraud Master E2D2 November 27, 2012 17 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
ω
P2
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
ω
P2
ω0
P2max
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
ω
P2
ω0
P2max
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
P2Max = (MAω2)2
2N2RL
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
P2Max = (MAω2)2
2N2RL
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
RL
P2max
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
P2Max = (MAω2)2
2N2RL
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
RL
P2max
We can harvest as muchpower as we want
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
P2Max = (MAω2)2
2N2RL, Wmax = MAω
N2RL
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
RL
P2max
We can harvest as muchpower as we want
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
P2Max = (MAω2)2
2N2RL, Wmax = MAω
N2RL
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
RL
P2max ,Wmax
We can harvest as muchpower as we want
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
P2Max = (MAω2)2
2N2RL, Wmax = MAω
N2RL
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
RL
P2max ,Wmax
We can harvest as muchpower as we want, at theexpense of largedisplacement of thebender’s tip
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
P2Max = (MAω2)2
2N2RL, Wmax = MAω
N2RL
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
RL
P2max ,WmaxTechnological limit for Wmax
We can harvest as muchpower as we want, at theexpense of largedisplacement of thebender’s tip
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
P2Max = (MAω2)2
2N2RL, Wmax = MAω
N2RL,
vmax = MAω2
N
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
RL
P2max ,WmaxTechnological limit for Wmax
We can harvest as muchpower as we want, at theexpense of largedisplacement of thebender’s tip
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
P2Max = (MAω2)2
2N2RL, Wmax = MAω
N2RL,
vmax = MAω2
N
P1Max = 12 faccωWMax = (MAω2)2
N2RL= P2Max
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
RL
P2max ,WmaxTechnological limit for Wmax
We can harvest as muchpower as we want, at theexpense of largedisplacement of thebender’s tip
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For an ideal generator (Ds = 0)
Mw = f + f acc
f = Nv − Ksw
v = −RLim = −RLNw
Mw + N2RLw + Ksw = f acc−→ RL acts as a damping
P2 = − 12RL|im|2
|im| = N |w | = ω.N.|facc|√(Ks−Mω
2)2+(N2RLω)2
P2 = − 12
N2RLω2|facc |
2
(Ks−Mω2)2+(N2RLω)2
P2Max = (MAω2)2
2N2RL, Wmax = MAω
N2RL,
vmax = MAω2
N
P1Max = 12 faccωWMax = (MAω2)2
N2RL= P2Max
ω
P2
ω0
P2max RL ց
The vibrations should occur atgenerator’s resonance frequency
ω0 =√
KsM
RL
P2max ,WmaxTechnological limit for Wmax
Assumtionnot valid
We can harvest as muchpower as we want, at theexpense of largedisplacement of thebender’s tip
Frederic Giraud Master E2D2 November 27, 2012 18 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
P2 = − 12RL|im|2
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
P2 = − 12RL|im|2
P2 = − 12
RLN2ω
2|f acc |2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
P2 = − 12RL|im|2
P2 = − 12
RLN2ω
2|f acc |2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
P1 = − 12ℜ(f acc .w
∗)
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
P2 = − 12RL|im|2
P2 = − 12
RLN2ω
2|f acc |2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
P1 = − 12ℜ(f acc .w
∗)
P1 = − 12
(Ds+RLN2)ω2|f acc |
2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
P2 = − 12RL|im|2
P2 = − 12
RLN2ω
2|f acc |2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
P1 = − 12ℜ(f acc .w
∗)
P1 = − 12
(Ds+RLN2)ω2|f acc |
2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
ω
|w |
ω
P2, P1
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
P2 = − 12RL|im|2
P2 = − 12
RLN2ω
2|f acc |2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
P1 = − 12ℜ(f acc .w
∗)
P1 = − 12
(Ds+RLN2)ω2|f acc |
2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
ω
|w |
ω0
Wmax
ω
P2, P1
ω0
P1max
P2max
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
P2 = − 12RL|im|2
P2 = − 12
RLN2ω
2|f acc |2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
P1 = − 12ℜ(f acc .w
∗)
P1 = − 12
(Ds+RLN2)ω2|f acc |
2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
ω0 should still be the vibration’s pulsation.
ω
|w |
ω0
Wmax
ω
P2, P1
ω0
P1max
P2max
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
P2 = − 12RL|im|2
P2 = − 12
RLN2ω
2|f acc |2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
P1 = − 12ℜ(f acc .w
∗)
P1 = − 12
(Ds+RLN2)ω2|f acc |
2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
ω0 should still be the vibration’s pulsation.
ω
|w |
ω0
Wmax
Lost
ω
P2, P1
ω0
P1max
P2max
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
P2 = − 12RL|im|2
P2 = − 12
RLN2ω
2|f acc |2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
P1 = − 12ℜ(f acc .w
∗)
P1 = − 12
(Ds+RLN2)ω2|f acc |
2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
ω0 should still be the vibration’s pulsation.
ω
|w |
ω0
Wmax
Lost
ω
P2, P1
ω0
P1max
P2max
P1 P2
Generator’s Power Losses
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
f = Nv − Ksw − Dsw
w =f acc
(Ks−Mω2)+jω(Ds+N2RL)
P2 = − 12RL|im|2
P2 = − 12
RLN2ω
2|f acc |2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
P1 = − 12ℜ(f acc .w
∗)
P1 = − 12
(Ds+RLN2)ω2|f acc |
2
(Ks−Mω2)2+ω
2(Ds+N2RL)2
ω0 should still be the vibration’s pulsation.
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL,
P2max = 12
N2RL|f2acc |
(Ds+N2RL)2
ω
|w |
ω0
Wmax
Lost
ω
P2, P1
ω0
P1max
P2max
P1 P2
Generator’s Power Losses
Frederic Giraud Master E2D2 November 27, 2012 19 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL
P2max = 12
N2RL|f acc |2
(Ds+N2RL)2
Frederic Giraud Master E2D2 November 27, 2012 20 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL
P2max = 12
N2RL|f acc |2
(Ds+N2RL)2
RL
|w |
RL
P2, P1
Frederic Giraud Master E2D2 November 27, 2012 20 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL
P2max = 12
N2RL|f acc |2
(Ds+N2RL)2
When RL ց, P1 ր, which is good, but thelosses increase as well: it exists an optimalvalue of RL which optimizes P2 for a givenvibration amplitude.
RL
|w |
Lost
RL
P2, P1
Frederic Giraud Master E2D2 November 27, 2012 20 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL
P2max = 12
N2RL|f acc |2
(Ds+N2RL)2
When RL ց, P1 ր, which is good, but thelosses increase as well: it exists an optimalvalue of RL which optimizes P2 for a givenvibration amplitude. It can be shown:
RLopt =Ds
N2 ,
P2opt =|f acc |
2
8Ds
P1opt =|f acc |
2
4Ds= 2.P2opt
RL
|w |
Lost
RL
P2, P1
Frederic Giraud Master E2D2 November 27, 2012 20 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL
P2max = 12
N2RL|f acc |2
(Ds+N2RL)2
When RL ց, P1 ր, which is good, but thelosses increase as well: it exists an optimalvalue of RL which optimizes P2 for a givenvibration amplitude. It can be shown:
RLopt =Ds
N2 ,
P2opt =|f acc |
2
8Ds
P1opt =|f acc |
2
4Ds= 2.P2opt
RL
|w |
Lost
RL
P2, P1
RLopt
Frederic Giraud Master E2D2 November 27, 2012 20 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL
P2max = 12
N2RL|f acc |2
(Ds+N2RL)2
When RL ց, P1 ր, which is good, but thelosses increase as well: it exists an optimalvalue of RL which optimizes P2 for a givenvibration amplitude. It can be shown:
RLopt =Ds
N2 ,
P2opt =|f acc |
2
8Ds
P1opt =|f acc |
2
4Ds= 2.P2opt
|vmax | = NRLω0|w | = NRL|facc |(Ds+N2RL)
RL
|w |
Lost
RL
P2, P1
RLopt
Frederic Giraud Master E2D2 November 27, 2012 20 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL
P2max = 12
N2RL|f acc |2
(Ds+N2RL)2
When RL ց, P1 ր, which is good, but thelosses increase as well: it exists an optimalvalue of RL which optimizes P2 for a givenvibration amplitude. It can be shown:
RLopt =Ds
N2 ,
P2opt =|f acc |
2
8Ds
P1opt =|f acc |
2
4Ds= 2.P2opt
|vmax | = NRLω0|w | = NRL|facc |(Ds+N2RL)
RL
|w |
Lost
RL
P2, P1
RLopt
RL
|v |
Frederic Giraud Master E2D2 November 27, 2012 20 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL
P2max = 12
N2RL|f acc |2
(Ds+N2RL)2
When RL ց, P1 ր, which is good, but thelosses increase as well: it exists an optimalvalue of RL which optimizes P2 for a givenvibration amplitude. It can be shown:
RLopt =Ds
N2 ,
P2opt =|f acc |
2
8Ds
P1opt =|f acc |
2
4Ds= 2.P2opt
|vmax | = NRLω0|w | = NRL|facc |(Ds+N2RL)
RL
|w |
Lost
RL
P2, P1
RLopt
RL
|v |
Frederic Giraud Master E2D2 November 27, 2012 20 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL
P2max = 12
N2RL|f acc |2
(Ds+N2RL)2
When RL ց, P1 ր, which is good, but thelosses increase as well: it exists an optimalvalue of RL which optimizes P2 for a givenvibration amplitude. It can be shown:
RLopt =Ds
N2 ,
P2opt =|f acc |
2
8Ds
P1opt =|f acc |
2
4Ds= 2.P2opt
|vmax | = NRLω0|w | = NRL|facc |(Ds+N2RL)
voltage is not so high.
RL
|w |
Lost
RL
P2, P1
RLopt
RL
|v |
Frederic Giraud Master E2D2 November 27, 2012 20 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real generator (Ds 6= 0)
Wmax = |facc |(Ds+N2RL)ω0
, P1max = 12
|f acc |2
Ds+N2RL
P2max = 12
N2RL|f acc |2
(Ds+N2RL)2
When RL ց, P1 ր, which is good, but thelosses increase as well: it exists an optimalvalue of RL which optimizes P2 for a givenvibration amplitude. It can be shown:
RLopt =Ds
N2 ,
P2opt =|f acc |
2
8Ds
P1opt =|f acc |
2
4Ds= 2.P2opt
|vmax | = NRLω0|w | = NRL|facc |(Ds+N2RL)
voltage is not so high.
RL
|w |
Assumption
not valid
Lost
RL
P2, P1
RLopt
Assumption
not valid
RL
|v |
Assumption
not valid
Frederic Giraud Master E2D2 November 27, 2012 20 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Example
Device’s properties
N = 0.012N/V , Ks = 6300N/m, Cb = 300nF , Ds = 0.17Ns/m,M = 1g
Calculate for A = 0.1mm
the best working frequency,
the harvested P2 power in the bestcase,
the power of the source P1 in suchbest case,
the optimal resistor RL,
the deflection amplitude of the bender,
the voltage for this working point.
Validation
Is v = −RLim a validassumption?
Frederic Giraud Master E2D2 November 27, 2012 21 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Answers
The best working frequency is given by
f0 =12π
√Ks
M= 1
2π
√63001.10−3 = 400Hz
Frederic Giraud Master E2D2 November 27, 2012 22 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Answers
The best working frequency is given by
f0 =12π
√Ks
M= 1
2π
√63001.10−3 = 400Hz
For the best case, P2 is given by
P2 =|f acc |
2
8Ds= (1.10−3.1.10−4.(2π.400)2)2
8.0,17 = 464mW
Frederic Giraud Master E2D2 November 27, 2012 22 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Answers
The best working frequency is given by
f0 =12π
√Ks
M= 1
2π
√63001.10−3 = 400Hz
For the best case, P2 is given by
P2 =|f acc |
2
8Ds= (1.10−3.1.10−4.(2π.400)2)2
8.0,17 = 464mW
For the best case P1 = 2.P2 = 928mW
Frederic Giraud Master E2D2 November 27, 2012 22 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Answers
The best working frequency is given by
f0 =12π
√Ks
M= 1
2π
√63001.10−3 = 400Hz
For the best case, P2 is given by
P2 =|f acc |
2
8Ds= (1.10−3.1.10−4.(2π.400)2)2
8.0,17 = 464mW
For the best case P1 = 2.P2 = 928mW
The optimal resistor RLopt is given byRLopt =
Ds
N2 = 0,170,0122
= 1180Ω
Frederic Giraud Master E2D2 November 27, 2012 22 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Answers
The best working frequency is given by
f0 =12π
√Ks
M= 1
2π
√63001.10−3 = 400Hz
For the best case, P2 is given by
P2 =|f acc |
2
8Ds= (1.10−3.1.10−4.(2π.400)2)2
8.0,17 = 464mW
For the best case P1 = 2.P2 = 928mW
The optimal resistor RLopt is given byRLopt =
Ds
N2 = 0,170,0122
= 1180Ω
The deflection Wmax is given by
Wmax = |facc |(Ds+N2RL)ω0
= (1.10−3.1.10−4.(2π.400)2)2.0,17.2π.400 = 738µm!
Frederic Giraud Master E2D2 November 27, 2012 22 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Answers
The best working frequency is given by
f0 =12π
√Ks
M= 1
2π
√63001.10−3 = 400Hz
For the best case, P2 is given by
P2 =|f acc |
2
8Ds= (1.10−3.1.10−4.(2π.400)2)2
8.0,17 = 464mW
For the best case P1 = 2.P2 = 928mW
The optimal resistor RLopt is given byRLopt =
Ds
N2 = 0,170,0122
= 1180Ω
The deflection Wmax is given by
Wmax = |facc |(Ds+N2RL)ω0
= (1.10−3.1.10−4.(2π.400)2)2.0,17.2π.400 = 738µm!
The voltage is then given byvmax = NRLω0|w | = 0, 012.1180.2π.400.738.10−6 = 26V
Frederic Giraud Master E2D2 November 27, 2012 22 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Answers
The best working frequency is given by
f0 =12π
√Ks
M= 1
2π
√63001.10−3 = 400Hz
For the best case, P2 is given by
P2 =|f acc |
2
8Ds= (1.10−3.1.10−4.(2π.400)2)2
8.0,17 = 464mW
For the best case P1 = 2.P2 = 928mW
The optimal resistor RLopt is given byRLopt =
Ds
N2 = 0,170,0122
= 1180Ω
The deflection Wmax is given by
Wmax = |facc |(Ds+N2RL)ω0
= (1.10−3.1.10−4.(2π.400)2)2.0,17.2π.400 = 738µm!
The voltage is then given byvmax = NRLω0|w | = 0, 012.1180.2π.400.738.10−6 = 26V
ZCb = 1Cbω
= 12π.400.300.10−9 = 1990Ω ≈ RL, −→ Calculations
are NOT valid!
Frederic Giraud Master E2D2 November 27, 2012 22 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
rLeq ≃ RL and kLeq ≃ 0, no problem,just use the assumption
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
rLeq ≃ RL and kLeq ≃ 0, no problem,just use the assumption
ω0 ≫ 1RLCb
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
rLeq ≃ RL and kLeq ≃ 0, no problem,just use the assumption
ω0 ≫ 1RLCb
ω
rLeq |dB , kLeq |dB
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
rLeq ≃ RL and kLeq ≃ 0, no problem,just use the assumption
ω0 ≫ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
rLeq ≃ RL and kLeq ≃ 0, no problem,just use the assumption
ω0 ≫ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
rLeq ≃ RL and kLeq ≃ 0, no problem,just use the assumption
ω0 ≫ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
rLeq ≃ RL and kLeq ≃ 0, no problem,just use the assumption
ω0 ≫ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
rLeq ≪ RL and kLeq is high, an otherstudy is needed
Frederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
For a real Generator, Ds 6= 0 and v 6= −RLim
v RLim
i
We can show:v = − RL
1+jωRLCbim = −jωNRL
1−jωRLCb
1+(RLCbω)2wm
and we write: v = −jωNrLeqw − Nkeqw
2 Cases to consider, since ω should be ω0:
ω0 ≪ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
rLeq ≃ RL and kLeq ≃ 0, no problem,just use the assumption
ω0 ≫ 1RLCb
ω
rLeq |dB , kLeq |dB
ω01
RLCb
rLeq ≪ RL and kLeq is high, an otherstudy is needed
((Ks + N2keq)−Mω2) + jω(Ds + N2rLeq)w = f accFrederic Giraud Master E2D2 November 27, 2012 23 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1
rL ≃ RL ; keq ≃ 0 rL < RL ; keq ր
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1Rb2
rL ≃ RL ; keq ≃ 0 rL < RL ; keq ր
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1Rb2
rL ≃ RL ; keq ≃ 0 rL < RL ; keq ր
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1Rb2
rL ≃ RL ; keq ≃ 0 rL < RL ; keq ր
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1Rb2Rb3
rL ≃ RL ; keq ≃ 0 rL < RL ; keq ր
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1Rb2Rb3
rL ≃ RL ; keq ≃ 0 rL < RL ; keq ր
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1Rb2Rb3
rL ≃ RL ; keq ≃ 0 rL < RL ; keq ր
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1Rb2Rb3Rb4
rL ≃ RL ; keq ≃ 0 rL < RL ; keq ր
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1Rb2Rb3Rb4
rL ≃ RL ; keq ≃ 0 rL < RL ; keq ր
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Rule of the Thumb
Let Rb defined as RbCbω0 = 1, ω0 <1
RLCb→ RL < Rb
ω
P2/P2max
ω0
RL
P2/P2Max
RLopt
Rb1Rb2Rb3Rb4
rL ≃ RL ; keq ≃ 0 rL < RL ; keq ր
Frederic Giraud Master E2D2 November 27, 2012 24 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Partial Conclusion
If Ropt < Rb
A resistor can recover the power optimally. But the study was notvalid for high RL. What happens for RL ≫ Rb?
Frederic Giraud Master E2D2 November 27, 2012 25 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Partial Conclusion
If Ropt < Rb
A resistor can recover the power optimally. But the study was notvalid for high RL. What happens for RL ≫ Rb?
If Ropt > Rb
The power is not well extracted because P2 < P2max . Thishappens for high damped mechanisms.A simple resistor cannot recover the power optimally (and this isdue to Cb).
Frederic Giraud Master E2D2 November 27, 2012 25 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
What happens if Ropt < Rb and RL ≫ Rb?
resonance is shifted
rLeq = RL
1+(RLCbω)2≃ RL
(RLCbω)2
keq = RLω2RLCb
1+(RLCbω)2≃ 1
Cb
((Ks + N2keq)−Mω2) + jω(Ds + N2rLeq)w = f acc leads to:
ω′0 =
√Ks+N2keq
M=
√
Ks+N2
Cb
M
Frederic Giraud Master E2D2 November 27, 2012 26 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
What happens if Ropt < Rb and RL ≫ Rb?
resonance is shifted
rLeq = RL
1+(RLCbω)2≃ RL
(RLCbω)2
keq = RLω2RLCb
1+(RLCbω)2≃ 1
Cb
((Ks + N2keq)−Mω2) + jω(Ds + N2rLeq)w = f acc leads to:
ω′0 =
√Ks+N2keq
M=
√
Ks+N2
Cb
M
Another optimal resistor
The optimal power is harvested is N2rLeq = Ds , leading to:N2RL
(RLCbω′0)
2 = N2
RLC2b
Ks+N2Cb
M
= Ds
RLopt2 = N2
Ds
1
C2bKsM
(1+ N2
KsCb)=
R2b
RLopt
1
1+ N2
KsCb
Frederic Giraud Master E2D2 November 27, 2012 26 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
What happens if Ropt < Rb and RL ≫ Rb?
RLopt
RL(log)
P2/P2Max
Frederic Giraud Master E2D2 November 27, 2012 27 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
What happens if Ropt < Rb and RL ≫ Rb?
RLopt
RL(log)
P2/P2Max
RLopt2
Frederic Giraud Master E2D2 November 27, 2012 27 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Conclusion
For the realistic case, there is one or two resistive loads whichallow to extract the maximum of power,
RLopt
RL(log)
P2/P2Max
RLopt2
Frederic Giraud Master E2D2 November 27, 2012 28 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Conclusion
For the realistic case, there is one or two resistive loads whichallow to extract the maximum of power,
There exist an optimal frequency which may vary if RL is veryhigh,
RLopt
RL(log)
P2/P2Max
RLopt2
ω
P2/P2max
ω0
Frederic Giraud Master E2D2 November 27, 2012 28 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Conclusion
For the realistic case, there is one or two resistive loads whichallow to extract the maximum of power,
There exist an optimal frequency which may vary if RL is veryhigh,
For highly damped structure, a simple resistor is not optimal.
RLopt
RL(log)
P2/P2Max
RLopt2
ω
P2/P2max
ω0 ≃ N2
Cbω0
Ds
P2/P2Max
Frederic Giraud Master E2D2 November 27, 2012 28 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Conclusion
For the realistic case, there is one or two resistive loads whichallow to extract the maximum of power,
There exist an optimal frequency which may vary if RL is veryhigh,
For highly damped structure, a simple resistor is not optimal.
→ we always want to harvest the maximum of power!
RLopt
RL(log)
P2/P2Max
RLopt2
ω
P2/P2max
ω0 ≃ N2
Cbω0
Ds
P2/P2Max
Frederic Giraud Master E2D2 November 27, 2012 28 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
Presentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
Table of contents
1 IntroductionWhat is Energy Harvesting ?Generator TechnologiesSummary
2 Modelling of a piezoelectric energy harvesterPresentation of the systemEMR of the systemPower Extraction on a load resistor RL from harmonic oscillation
3 An Example of inverterIntroductionSSHI: Synchronized Switch Harvesting on Inductor
Frederic Giraud Master E2D2 November 27, 2012 29 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL=
∼
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
vL = −RLim which is > 0again, and
v = vL = −RLim
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
vL = −RLim which is > 0again, and
v = vL = −RLim
time
T0 = 1
2π
√
KsM
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
vL = −RLim which is > 0again, and
v = vL = −RLim
time
T0 = 1
2π
√
KsM
The current is rectified, but we still havev = −RLim: Power harvesting depends on RL
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
vL = −RLim which is > 0again, and
v = vL = −RLim
time
T0 = 1
2π
√
KsM
The current is rectified, but we still havev = −RLim: Power harvesting depends on RL
Moreover, it is still not optimal if Ropt > Rb.
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
vL = −RLim which is > 0again, and
v = vL = −RLim
time
T0 = 1
2π
√
KsM
The current is rectified, but we still havev = −RLim: Power harvesting depends on RL
Moreover, it is still not optimal if Ropt > Rb.
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
vL = −RLim which is > 0again, and
v = vL = −RLim
time
T0 = 1
2π
√
KsM
The current is rectified, but we still havev = −RLim: Power harvesting depends on RL
Moreover, it is still not optimal if Ropt > Rb.This is why, some want to compensate for Cb:
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
vL = −RLim which is > 0again, and
v = vL = −RLim
time
T0 = 1
2π
√
KsM
The current is rectified, but we still havev = −RLim: Power harvesting depends on RL
Moreover, it is still not optimal if Ropt > Rb.This is why, some want to compensate for Cb:
v
imRL
vL
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
vL = −RLim which is > 0again, and
v = vL = −RLim
time
T0 = 1
2π
√
KsM
The current is rectified, but we still havev = −RLim: Power harvesting depends on RL
Moreover, it is still not optimal if Ropt > Rb.This is why, some want to compensate for Cb:
v
imRL
vL
This is a bad solution because it works only forω0 (what if the frequency shifts?), and theInductor is large (because ω0 usually is small)
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Introduction
v
imRL
im > 0v
imRL
vL
vL = RLim which is > 0and v = −vL = −RLim
im < 0v
imRL
vL
vL = −RLim which is > 0again, and
v = vL = −RLim
time
T0 = 1
2π
√
KsM
The current is rectified, but we still havev = −RLim: Power harvesting depends on RL
Moreover, it is still not optimal if Ropt > Rb.This is why, some want to compensate for Cb:
v
imRL
vL
This is a bad solution because it works only forω0 (what if the frequency shifts?), and theInductor is large (because ω0 usually is small)−→Non linear Techniques
Frederic Giraud Master E2D2 November 27, 2012 30 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Why Synchronized
Frederic Giraud Master E2D2 November 27, 2012 31 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Why Synchronized
SE v
i
vL
iL
α
i
v im
v
w
fp
w
f
facc
w
w
fs
SM
Frederic Giraud Master E2D2 November 27, 2012 31 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Why Synchronized
SE v
i
vL
iL
α
i
v im
v
w
fp
w
f
facc
w
w
fs
SM
Frederic Giraud Master E2D2 November 27, 2012 31 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Why Synchronized
SE
H bridge
v
i
vL
iL
α
i
v im
v
w
fp
w
f
facc
w
w
fs
SM
Frederic Giraud Master E2D2 November 27, 2012 31 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Why Synchronized
SE v
i
vL
iL
α
i
v im
v
w
fp
w
f
facc
w
w
fs
SM
Strategy: v = −Ropt im leads tofpN= −Ds
N2Nw or, fp = −Dsw .
Frederic Giraud Master E2D2 November 27, 2012 31 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Why Synchronized
SE v
i
vL
iL
α
i
v im
v
w
fp
w
f
facc
w
w
fs
SM
Strategy: v = −Ropt im leads tofpN= −Ds
N2Nw or, fp = −Dsw .
Frederic Giraud Master E2D2 November 27, 2012 31 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Why Synchronized
SE v
i
vL
iL
α
i
v im
v
w
fp
w
f
facc
w
w
fs
SM
fpref
Strategy: v = −Ropt im leads tofpN= −Ds
N2Nw or, fp = −Dsw .
Frederic Giraud Master E2D2 November 27, 2012 31 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Why Synchronized
SE v
i
vL
iL
α
i
v im
v
w
fp
w
f
facc
w
w
fs
SM
fpref
vref
Strategy: v = −Ropt im leads tofpN= −Ds
N2Nw or, fp = −Dsw .
Frederic Giraud Master E2D2 November 27, 2012 31 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Why Synchronized
SE v
i
vL
iL
α
i
v im
v
w
fp
w
f
facc
w
w
fs
SM
fpref
vrefiref
Strategy: v = −Ropt im leads tofpN= −Ds
N2Nw or, fp = −Dsw .This shows that v should be controlled.
Frederic Giraud Master E2D2 November 27, 2012 31 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Why Synchronized
SE v
i
vL
iL
α
i
v im
v
w
fp
w
f
facc
w
w
fs
SM
fpref
vrefiref
Strategy: v = −Ropt im leads tofpN= −Ds
N2Nw or, fp = −Dsw .This shows that v should be controlled.SSHI does this withefficiency.
Frederic Giraud Master E2D2 November 27, 2012 31 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations
+v
i
t = 0
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations
+v
i
t = 0
t
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations
+v
i
t = 0
t
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
SSHI
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
SSHI
Piezo
v
im K
iI
RLvLCF
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
SSHI
Piezo
v
im K
iI
RLvLCF
t
v , iI , im, vL
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
SSHI
Piezo
v
im K
iI
RLvLCF
t
v , iI , im, vL
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
SSHI
Piezo
v
im K
iI
RLvLCF
t
v , iI , im, vL
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
SSHI
Piezo
v
im K
iI
RLvLCF
t
v , iI , im, vL
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
SSHI
Piezo
v
im K
iI
RLvLCF
t
v , iI , im, vL
on off on off on off on off
Switching is synchronized on w , or
im = Nw .
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
SSHI
Piezo
v
im K
iI
RLvLCF
Operating point: calculate vL from im
1st Harmonic assumption: P = 12
V 2L
RL≃ 1
2
4VLImπ
, VL = 4πRL Im
t
v , iI , im, vL
on off on off on off on off
Switching is synchronized on w , or
im = Nw .
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
How it works
LC oscillations Switched inductor
+v
i
t = 0
t
v , i+v
it
v , i
SSHI
Piezo
v
im K
iI
RLvLCF
Operating point: calculate vL from im
1st Harmonic assumption: P = 12
V 2L
RL≃ 1
2
4VLImπ
, VL = 4πRL Im
SSHI controls v and synchronises it, but doesn’t impose fp = −Ds w .
t
v , iI , im, vL
on off on off on off on off
Switching is synchronized on w , or
im = Nw .
Frederic Giraud Master E2D2 November 27, 2012 32 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Conclusion
Performances
SSHI can extract energy more efficiently than a resistor whendamping is important,
Ds
P2/P2Max
SSHI
RL
Frederic Giraud Master E2D2 November 27, 2012 33 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Conclusion
Performances
SSHI can extract energy more efficiently than a resistor whendamping is important,
But Power extraction still depends on the load,
Ds
P2/P2Max
SSHI
RL
Frederic Giraud Master E2D2 November 27, 2012 33 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
Conclusion
Performances
SSHI can extract energy more efficiently than a resistor whendamping is important,
But Power extraction still depends on the load,
Needs to measure bender’s deflection w(t).
Ds
P2/P2Max
SSHI
RL
Frederic Giraud Master E2D2 November 27, 2012 33 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
General conclusion
In this presentation, applications of Energy Harvesting were shown.The modelling of a piezoelectric generator has shown that thepower source needs an adaptation:
in frequency,
in load.
to maximize the harvested poer.The key energy management rules were presented through theanalysis of the EMR of the system. A typical power electroniccircuit was also presented, but the bibliography shows a lot ofexample.
Frederic Giraud Master E2D2 November 27, 2012 34 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
References I
S Adhikari, M I Friswell, and D J Inman, Piezoelectric energy harvesting from broadband random vibrations,
Smart Materials and Structures 18 (2009), no. 11, 115005.
R. G Ballas, H. F Schlaak, and A. J Schmid, Closed form analysis of piezoelectric multilayer bending
actuators using constituent equations, The 13th International Conference on Solid-State Sensors, Actuatorsand Microsystems, 2005. Digest of Technical Papers. TRANSDUCERS ’05, vol. 1, IEEE, June 2005,pp. 788– 791 Vol. 1.
R. Djugum, P. Trivailo, and K. Graves, A study of energy harvesting from piezoelectrics using impact forces,
The European Physical Journal Applied Physics 48 (2009), no. 1, 11101.
Daniel Guyomar, Gal Sebald, Sbastien Pruvost, Mickal Lallart, Akram Khodayari, and Claude Richard,
Energy harvesting from ambient vibrations and heat, Journal of Intelligent Material Systems and Structures20 (2009), no. 5, 609 –624.
Aman Kansal, Jason Hsu, Sadaf Zahedi, and Mani B. Srivastava, Power management in energy harvesting
sensor networks, ACM Transactions on Embedded Computing Systems 6 (2007), 32–es.
Elie Lefeuvre, Adrien Badel, Claude Richard, and Daniel Guyomar, Piezoelectric energy harvesting device
optimization by synchronous electric charge extraction, Journal of Intelligent Material Systems andStructures 16 (2005), no. 10, 865 –876.
H. Lhermet, C. Condemine, M. Plissonnier, R. Salot, P. Audebert, and M. Rosset, Efficient power
management circuit: From thermal energy harvesting to above-IC microbattery energy storage, IEEEJournal of Solid-State Circuits 43 (2008), no. 1, 246–255.
Frederic Giraud Master E2D2 November 27, 2012 35 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
References II
G. A. Lesieutre, G. K. Ottman, and H. F. Hofmann, Damping as a result of piezoelectric energy harvesting,
Journal of Sound and Vibration 269 (2004), no. 3-5, 991–1001.
D. Niyato, E. Hossain, M. M Rashid, and V. K Bhargava, Wireless sensor networks with energy harvesting
technologies: a game-theoretic approach to optimal energy management, IEEE Wireless Communications 14(2007), no. 4, 90–96.
Shad Roundy, Paul Kenneth Wright, and Jan M. Rabaey, Energy scavenging for wireless sensor networks:
with special focus on vibrations, Springer, 2003.
G. Sebald, E. Lefeuvre, and D. Guyomar, Pyroelectric energy conversion: Optimization principles, IEEE
Transactions on Ultrasonics, Ferroelectrics and Frequency Control 55 (2008), no. 3, 538–551.
Qing-Ming Wang and L. E Cross, Constitutive equations of symmetrical triple layer piezoelectric benders,
IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 46 (1999), no. 6, 1343–1351.
Frederic Giraud Master E2D2 November 27, 2012 36 / 40
IntroductionModelling of a piezoelectric energy harvester
An Example of inverter
IntroductionSSHI: Synchronized Switch Harvesting on InductorConclusion
End of the presentation
Questions?
Frederic Giraud Master E2D2 November 27, 2012 37 / 40