Transcript Modeling of Inverters Based on Piezoelectric Transformers

CCFL Inverters based on
Piezoelectric
Transformers: Analysis
and Design Considerations
Prof. Giorgio Spiazzi
• Dept. Of Information Engineering – DEI
• University of Padova
Power Electronics Group - PEL
1
Outline
• Characteristics of Cold Cathode
Fluorescent Lamps (CCFL)
• Review of piezoelectric effect
• CCFL inverters based on piezoelectric
transformers
• Design considerations
• Modeling
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2
Cold Cathode Fluorescent Lamp
(CCFL)
• CCFL is a mercury vapor discharge light
source which produces its output from a
stimulated phosphor coating inside glass lamp
envelope.
• Closely related to “neon” sign lamps first
introduced in 1910 by Georges Claude in Paris
• Cold cathode refers to the type of electrodes
used: they do not rely on additional means of
thermoionic emission besides that created by
electrical discharge through the tube
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3
Cold Cathode Fluorescent Lamp
(CCFL)
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4
Cold Cathode Fluorescent Lamp
(CCFL)
• The phosphors coating the lamp tube inner
surface are composed of Red-Green-Blue
fluorescent compounds mixed in the
appropriate ratio in order to obtain a good
color rendering when used as an LCD display
backlight
Energy conversion efficiency:
Ultraviolet light
Visible light
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5
Cold Cathode Fluorescent Lamp
Lamp v-i characteristic:
Lamp
length
Lamp voltage primarily depends on length and is fairly constant
with current, giving a non-linear characteristic. Lamp current is
roughly proportional to brightness or intensity and is the
controlled variable of the backlight supply.
6
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Cold Cathode Fluorescent Lamp
• These lamps require a high ac voltage for
ignition and operation.
• A sinusoidal voltage provides the best
electrical-to-optical conversion.
• There are four important parameters in
driving the CCFL:
–
–
–
–
strike voltage
maintaining voltage
frequency
lamp current
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7
Cold Cathode Fluorescent Lamp
• Operating a CCFL over time results in degradation
of light output. Typical life rating is 20000 hours to
50% of the lamp initial output
• The light output of a CCFL has a strong
dependency on temperature
Percentage of light output as a function
of lamp temperature
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8
Cold Cathode Fluorescent Lamp
Lamp display housing:
• Stray capacitances to ground cause a
considerable loading effect that can
easily degrade efficiency by 25%
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9
Current-fed
Self-Resonant
Royer
Converter
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10
High voltage
transformer
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11
Ballast
capacitor
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12
Self resonant
inverter
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13
Control of
supply current
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14
Lamp current
measurement
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15
Dimming
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16
Magnetic and Piezoelectric
Transformer Comparison
Magnetic transformer characteristics
•
•
•
•
Low cost
Multiple sources
Single-ended or balanced output
Wide range of load conditions (output power easily
scaled)
• Secondary side ballasting capacitor required
• Reliability affected by the high-voltage secondary
winding
• EMI generation (stray high-frequency magnetic field)
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17
Magnetic and Piezoelectric
Transformer Comparison
Piezoelectric transformer characteristics
•
•
•
•
•
•
•
•
•
•
Inherent sinusoidal operation
High strike voltage (no need of ballasting capacitor)
No magnetic noise
Small size
High cost (but decreasing)
Must be matched with lamp characteristics
Reduced power capability
Single-ended output (balanced output are possible)
Few sources
Unsafe operation at no load (can be damaged)
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18
Magnetic and Piezoelectric
Transformer Comparison
Size comparison
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19
Piezoelectric Effect
• The piezoelectric effect was
discovered in 1880 by Jacques and
Pierre Curie:
– Tension and compression applied to
certain crystalline materials generate
voltages (piezoelectric effect)
– Application to the same crystals of an
electric field produces lengthening or
shortening of the crystals according to
the polarity of the field (inverse
effect)
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Group - PEL
20
Piezoelectric Effect
• In the 20th century metal oxide-based piezoelectric
ceramics have been developed.
• Piezoelectric ceramics are prepared using fine
powders of metal oxides in specific proportion
mixed with an organic binder. Heating at specific
temperature and time allows to attain a dense
crystalline structure
• Below the Curie point they exhibit a tetragonal or
rhombohedral symmetry and a dipole moment
• Adjoining dipoles form regions of local alignment
called domains
• The direction of polarization among neighboring
domains is random, producing no overall polarization
• A strong DC electric field gives a net permanent
polarization (poling)
21
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Piezoelectric Effect
Polarization
Polarization using
a DC electric field
Residual
polarization
Polarization axis
Random orientation
of polar domains
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22
Piezoelectric Effect
Residual
polarization
P
E
Residual
polarization
S
E
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Effect of electric field E
on polarization P and
corresponding
elongation/contraction of
the ceramic material
Relative increase/decrease
in dimension (strain S) in
direction of polarization
23
Piezoelectric Effect
Poling voltage
Generator and motor actions
of a piezoelectric element
Disk after
polarization
(poling)
Disk
compressed
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Disk
stretched
Applied
Applied
voltage of
voltage of
same polarity
opposite
as poling
polarity as
voltage
poling voltage
24
Piezoelectric Effect
Polarization
S=sE.T+d.E
Actuator
behavior
D=d.T+T.E
Transducer
behavior
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Where:
S: Strain [ ]
T: Stress [N/m2]
E: Electric Field [V/m]
s: elastic compliance [m2/N]
D: Electric Displacement [C/m2]
d: Piezoelectric constant [m/V]
25
Piezoelectric Effect
Based on the poling orientation, the
piezoelectric ceramics can be design to
function in:
longitudinal mode: P is parallel to T
Has a larger d33, along the thickness
direction
when compared to the planar direction
transverse mode: P is perpendicular to T
Has a larger d31, along the planar direction
when compared to the thickness direction
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26
Piezoelectric Transformers (PT)
• In Piezoelectric Transformers,
energy is transformed from
electrical form to electrical form
via mechanical vibration.
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27
Piezoelectric Transformers (PT)
Three main categories
• Longitudinal vibration mode
– Transverse actuator and Longitudinal
transducer Rosen-type or High-Voltage PT
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Piezoelectric Transformers (PT)
Three main categories
• Thickness vibration mode
– Longitudinal actuator and Longitudinal
transducer Low-Voltage PT
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Piezoelectric Transformers (PT)
Three main categories
• Radial vibration mode
– Transverse actuator and Transverse
transducer (radial shape preferred)
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30
Equivalent Electric Model
R
L
Io
C
+
Ui
-
+
iL
Co
Ci
Uo
RL
-
1:N
Rosen-type
R 0.756199 
L 2.464173mH
C 3.57nF
N 35.89
Ci 81.6216nF
Co 23.85pF
length=30mm
width=8mm
thickness=2mm
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Thick. Vibr. mode
1.44 
27H
254pF
0.47
2.305nF
8.911nF
length=20mm
width=20mm
thickness=2.2mm
Radial Vibr. mode
6.89991 
7.93842mH
269.171pF
0.908
4.60799nF
1.62414nF
radius=10.5mm
thickness1=0.76mm
thickness2=2.28mm
31
Voltage Gain
Load resistance: 1M, 100k, 10k,
5k, 1k, 500
Resonance frequencies
Rosen-type Piezoelectric Transformer sample
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32
Input Impedance
Load resistance: 1M, 100k, 10k, 5k,
1k, 500
Rosen-type Piezoelectric Transformer sample
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33
Half-Bridge Inverter for PT
C2
+
UDC
S1
C1
iinv
+
ui
iL
PT
Lamp
S2
-
Half-Bridge inverter
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34
Soft-Switching Condition
Half-bridge
inverter
ui
tr
UDC
t
T/2
j/ w
U1
iinv
Fundamental
components
t
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35
Coupling Networks
TABLE I.
PIEZOELECTRIC TRANSFORMER FUJI T2508A: ROSEN TYPE MODEL
PARAMETER VALUES
R = 5.37 
C = 10.22 nF
Ci = 150.4 nF Co = 15.48 pF
L = 0.699 mH
UA
S2
C2
R
+
+
UA
S1
C1 uinv
-
Half-Bridge
inverter
Coupling
network
+
n21 = 50.749
+
ui
L
C
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+
iL
Co
Ci
-
Uo
PT Rosen-type Model
Zg
io
1:n21
Lamp
36
Coupling Networks
Series inductor
Ls
CN1
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• It is not always possible to find
a value for input inductor that
guarantees both power transfer
and soft switching requirements
• Less circulating energy as
compared to parallel inductor
• Non linear control
characteristics can lead to
large signal instabilities
37
Effect of Coupling Inductor Ls on
Voltage Conversion Ratio
MPT = UoRMS/UiRMS, Mi = UiRMS/UinvRMS, Mg = Mi  MPT
f1
70
f2
[dB]
|Mgd|I =1mA
o
Udc=13V,
Ls=42H
|Mgd|I =6mA
o
Io=1mA
|Mg|{
|MPT|{
Io=6mA
|Mi|{
-10
45
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50
55
60
65
fsw [kHz]
70
75
80
38
Effect of Coupling Inductor Ls on
Input Impedance
Positive input phase
f1
60
f2
Zg
[dB]

2
Io=6mA
|Zg|
0
Io=1mA
0
45
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50
55
60
65
fsw [kHz]
70
75


80 2
39
Effect of Coupling Inductor Ls on
Voltage Conversion Ratio
• It introduces an additional voltage gain
(frequency dependent) between the RMS
value of the inverter voltage fundamental
component and the RMS value of the PT input
voltage
• It introduces more resonant peaks in the
overall voltage gain Mg (limitation in switching
frequency variation)
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40
Control Characteristics: Variable
Frequency
Io
[mARMS]
Udc = 13V
5
4
3
2
1
60
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62
63
fsw [kHz]
64
65
41
Control Characteristics: Variable dc
Link Voltage
Increasing LS value causes the gain curve
Io = f(UA) to become non monotonic
Io
[mARMS] fsw = 65kHz
5
Ls = 38H
4
3
Ls = 42H
2
CN1
1
11
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12
13
14
Udc [V]
15
16
42
Large-Signal Instability
Main converter waveforms when Udc is slowly
approaching 21V (fsw = 65kHz, Ls = 42H).
ILS = [5A/div]
Io = [10mA/div]
ui = [50V/div]
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43
Coupling Networks
Parallel inductor
CB
Lp
CN2
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• It is always possible to find a
value for input inductor that
guarantees both power
transfer and soft switching
requirements
• Higher circulating energy as
compared to series inductor
44
Effect of Coupling Network on
Voltage Conversion Ratio
CN2: Lp=20H, CB=1F, Udc=30V
f2
f1
50
|Mgd|I =1mA
[dB]
o
|Mgd|I =6mA
o
|Mg|{
|MPT|{
}Io=1mA
}Io=6mA
Io=1mA
Io=6mA
|Mi|{
0
45
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50
55
60
65
fsw [kHz]
70
75
80
45
Effect of Coupling Network on Input
Impedance
f1
60
f2
Zg
[dB]

2
Io=6mA
Io=1mA
0
|Zg|
0
45
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50
55
60
65
fsw [kHz]
70
75


80 2
46
Effect of Coupling Network on Switch
Commutations
• Differently from the series inductor
coupling network, now the inductor
current iLp has to charge and discharge
also the PZT input capacitance, that is
much higher than the switch output
capacitances, so that the positive
impedance phase is a necessary but not
sufficient condition to achieve soft
commutations
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47
Experimental Measurements
Trapezoidal PT input voltage
iLp
io
uinv
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Charge of
input
capacitance48
Control Characteristics: Variable
Frequency
Lp = 20H, CB = 1F
Io
[mARMS]
CN2
5
4
3
2
Udc = 30V
1
64
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66
68
fsw [kHz]
70
72
49
Control Characteristics: variable dc
link voltage
Lp = 20H, CB = 1F
Io
[mARMS]
CN2
5
4
3
2
fsw = 65kHz
1
10
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15
20
25
Udc [V]
30
35
50
Half-Bridge Inverter for PT
Frequency Control
• Square-wave output voltage
• Switching frequency changes in order
to control lamp current
• Attention must be paid to the
resonance frequency change with load
• Dedicated IC available
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51
Half-Bridge Inverter for PT
Frequency Control
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52
Half-Bridge Inverter for PT
Duty-cycle Control
• Constant switching frequency
• Asymmetrical output pulses
• Amplitude of fundamental input voltage component
is controlled by the duty-cycle
• Many control ICs for DC/DC converters can be
used
ui
UDC
t on
duty  cycle 
TS
U1
t
ton
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TS
53
Full-Bridge Inverter for PT
• Switching frequency control
• Duty-cycle control
• Phase-shift control
+
UDC
+
ui
S1
S4
Full-Bridge inverter
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iinv
iL
PT
Lamp
S2
S3
-
54
Full-Bridge Inverter for PT
Phase-Shift Control
• Constant switching frequency
• Amplitude of fundamental input
voltage component is controlled by
phase shifting the inverter legs
• No DC voltage applied to PT
• Dedicated control IC
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55
Resonant Push-Pull Topology
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56
Resonant Push-Pull Topology
• Variable switching frequency
• Voltage gain at PT input
• Sinusoidal driving voltage
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57
Analysis of Small-Signal Instabilities
and Modeling Approaches
Example of high-frequency V –I
characteristics
OSRAM L 18W/10
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Steady-state VRMS-IRMS
Characteristic
MATSUSHITA
FHF32 T-8 32W
Negative incremental
impedance
Positive incremental
impedance
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Modulated Lamp Voltage and Current
uo t   2 Uo  ûo  sinws t 
io t   2 Io  îo sinws t 
ûo t   Ûo sinwm t 
îo t   Îo sinwm t  m 
Incremental
impedance:
fm=200Hz
fm=2kHz
Ûo
ZL 
m
Îo
OSRAM
L 18W/10
fm=5kHz
Upper trace: iLamp [0.5A/div] Lower trace: uLamp [74V/div]
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60
Lamp Incremental Impedance
Im(ZL)
wm= 0
Approximation:
wm= 

s 
1



wz 

ZL  KL

s 
 1

w
p 

Re(ZL)
KL< 0, wz< 0
Right-half plane zero
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Lamp Model (Ben Yaakov)
Uo max  R SIo1  Uo1  R SIo1  R LIo1
Uo max
RL 
 RS
Io
Uo1
Uo2
slope  R S
slope  R L
Io1
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Io2
68
Lamp Model (Ben Yaakov)
K3
RL 
 K2
Io
K2, K3 = lamp constants
The lamp resistance is considered to be dependent
on a delayed version of RMS lamp current
 K3

Uo  R LIo  
 K 2  Io
 Iod

Small-signal perturbation:
ûo 
Uo
Io
 îo 
Io Ioq
Uo
Iod
Io Iodq
K

K
K
 îod   3  K 2  îo  3 îod  R Lq îo  3 îod
Iodq
Iodq
 Iodq

Subscript q means quiescent point
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69
Lamp Model (Ben Yaakov)
Delay:
Îod s  
1
s
1
wL
Îo s   GL s Îo s 

K3
K3 
s 

 GL Îo s
Ûo s  R Lq Îo s 
GL Îo s    1 
R Lq 
Iodq
wL 
Iodq 

 s

  R Lq  K 2  GL Îo s
 wL


s 
 1

Ûo s
wz 

Z L s 
 K 2

s 
Îo s
 1

w
p 

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ZL 0  K 2  0
K2
wz   wL
R Lq
wp  wL
70
Lamp Pspice Model (Ben Yaakov)
io2
IoRMS2
io=uo/RL
+
uo
K3
RL 
 K2
Io
-
Lamp time constant
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71
Lamp Model (Do Prado)
R L  a e b PL
PL = Lamp power
a, b positive constants
Accounts also for
the positive slope
1200
1.2
Uo [VRMS]
1100
RL [M]
1.0
1000
0.8
900
0.6
800
0.4
700
0.2
600
1
2
3
Io [mARMS]
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4
5
6
1
2
3
4
5
6
Io [mARMS]
72
Lamp Model (Do Prado)
R L  a e b PL
Small-signal perturbation:






Uoq  ûo  Ioq  îo a e b PL p̂L   Ioq  îo a e b PL e bp̂L  RLq Ioq  îo 1  b p̂L 
Subscript q means quiescent point
ûo  R Lq îo  bR LqIoq p̂L
Ûo s  R Lq Îo s  bR LqIoq P̂L s
Delay:





P̂L s  Uoq  Ûo s Ioq  Îo s GL  Uoq Îo s  IoqÛo s GL s
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73
Lamp Model (Do Prado)
s
1


1  bPLqGL s
1  bPLq
wL 1  bPLq 
Ûo s
Z L s 
 R Lq
 R Lq
s
1  bPLqGL s
1  bPLq
Îo s
1
wL 1  bPLq 

s 
 1

 wz  
Ûo s
wz 


ZL 
 R Lq  


Îo s
 wp  1  s 
wp 

If bPLq>1:
wz  wL 1 b PLq 
wp  wL 1 b PLq 
wz<0, ZL(0)<0
Negative incremental impedance and
RHP zero
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74
Lamp Pspice Model (Do Prado)
io
uo-R4io
RL
uo
PL
Lamp time constant
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75
Lamp Model (Onishi)
A o  A1e  A2Io  A 3 e  A4Io
RL 
Io
A0-A4 positive constants
 Ao  A1e  A2Iod  A3e  A4Iod 
 Io
Uo  R LIo  

Iod


Small-signal perturbation:
ûo 
Uo
Io
 îo 
Io Ioq
Uo
Iod

 îod  R Lq îo   A1A2e
 A2Ioq
 A3 A 4 e
 A 4Ioq

 R Lp îod
Io Iodq
Subscript q means quiescent point
Delay:
Îod s  GL sÎo s



s 
A I
A I
Ûo s  R Lq  1 
   A1A2e 2 oq  A3 A 4 e 4 oq  R Lp
wL 


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

 GL sÎo s

76
Lamp Model (Onishi)

s 
 1

Û s
wz 
Z L s  o
 R s 

s 
Îo s
 1

w
p 

Rs
wz   wL
R Lq
wp  wL
If RS>0:
R s  A1A 2 e
 A2Ioq
 A3 A 4e
 A 4Ioq
wz<0, ZL(0)<0
Negative incremental impedance and
RHP zero
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77
Lamp Pspice Model (Onishi)
Lamp time constant
IoRMS
Uo=RLio
RL
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78
Control Problem
Ûo s Uo s  wz
ZL s 

,
Io s  wp
Îo s
wz  0
An Impedance with a RHP zero cannot be driven
directly by a voltage source, since its current
transfer function will contain a RHP pole
Power Electronics Group - PEL
79
Series Impedance Lamp Ballast
ZB Io(s)
US(s)
+
Uo(s)
-
ZL
Io s
1
1
1 1


US s ZB 1  ZL ZB 1  TF
ZB
TF must satisfy Nyquist stability criterion
ZL
TF 
ZB
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80
Example of Instability
Series inductor coupling network
fosc  6kHz
Inverter
LS
is
ui
+
-
PT
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Lamp
81
Example of Instability
Parallel inductor + dc blocking capacitor
coupling network
ILp = [1A/div]
Io = [2mA/div]
fosc=6.45kHz
Power Electronics Group - PEL
82
Phasor Transformation [11]
A sinusoidal signal x(t) can be represented
by a time

varying complex phasor Xt , i.e.:


xt   e Xt  e jw t
S

Example: AM signal xt   2XM  x̂  coswst   

Xt   2 XM  x̂ cos   j sin 
Power Electronics Group - PEL
83
Phasor Transformation
Example: FM signal
xt   2XM coswst  x̂ 

Xt   2XM cosx̂   j sinx̂ 
diL t 
L
 uL t 
dt

 e UL t  e jw t
Inductor phasor transformation:
 




d
L e IL t  e jw t
dt



 d IL t  jwSt

jwS t
Le 
e
 jwS IL t e   e UL t  e jwSt
dt


Power Electronics Group - PEL
S
S


84
Phasor Transformation
Inductor phasor transformation:
L
+

d IL t 
uL
L
Power Electronics Group - PEL
dt
iL
diL t 
L
 uL t 
dt


 jwSL IL t   UL t 
+

IL t  L

U L t 
-
jwSL
85
Phasor Transformation
duC t 
Capacitor phasor transformation: C
 iC t 
dt
C
+

dU C t 
uC
C
dt
iC


 jwSCU C t   IC t 
+

IC t 

U C t 
-
C
1/jwSC
Power Electronics Group - PEL
86
Generalized Averaging Method [13]
A waveform x(•) can be approximated on the interval [
t-T, t ] to arbitrary accuracy with a Fourier series
representation of the form:
x t  T  s    x
t  e
k
 jkws  t  T  s 
k
x
k
s  (0, T], ws 
2
T
t  = time-dependent complex Fourier series coefficients
calculated on a sliding window of amplitude T
T
1
x k t    x t  T  s e  jkws t  T  s ds 
T0
t
1
  x   e  jkws  d
T tT
Power Electronics Group - PEL
87
Generalized Averaging Method
The analysis computes the time evolution of these
Fourier series coefficients as the window of length T
slides over the waveform x(•). The goal is to
determine an appropriate state-space model in
which these coefficients are the state variables
Classical state-space averaging theory:
t
1
x t    x   d  x
T t T
0
t 
The average value coincides with the
Fourier coefficient of index 0!
Power Electronics Group - PEL
88
Application to Power Electronics
Let’s apply the generalized averaging method to a
generic state-space model that has some periodic
time-dependence:
dx  t 
 f x t , ut 
dt
u(t) = periodic function of time with period T
Let’s compute the relevant Fourier
coefficients of both sides:
Power Electronics Group - PEL
dx t 
dt
 f x t , ut 
k
k
89
Differentiation Property
dx
k
dt
t 
dxt 

dt
 jkws x
k
t 
k
This relation is valid for constant frequency ws,
but still represents a good approximation for
slowly varying ws(t)
dx  t 
dt
dx
k
t 
dt
Power Electronics Group - PEL
 f x t , ut  k
k
  jkws x
k
t   f xt ,ut  k
90
Transform of Functions of
Variables
f xt ,ut  k  ?
A general answer does not exist unless function f
is a polynomial. In this case, the following
convolutional relationship can be used:
xy
k
 x
k i
y
i
i
where the sum is taken of all integers i.
Power Electronics Group - PEL
91
Lamp dynamic Model
A o  A 3 e A4Io
RL 
Io
Only the negative slope in the
UoRMS-IoRMS curve is modeled





y
t
uo t   GL yuo t 
io t   
A4 y t  
A

A
e
 0

3

 dyt   w i t i t   yt 
L o
o
 dt
y(t) is lamp RMS
current squared
Power Electronics Group - PEL
92
Generalized averaged lamp model
Considering that lamp voltage uo(t) and current io(t) are,
with a good approximation, sinusoidal waveforms, we can
take into account only the complex Fourier coefficients
corresponding to indexes +1 and –1 (actually only one of
the two coefficients is necessary), while for the
variable y(t), only the index-0 coefficient is considered,
since we are concerned with its dc value.

 io 1  GL y 0 uo 1  GL  y 0  uo 1

 io 1  GL y 0 uo 1  GL  y 0  uo 1
d y
0
 wL  io io 0  y 0 

 dt
2
i oi o 0  i o 1 i o  1  i o  1 i o 1  2 i o 1 i o  1  2 i o 1
Power Electronics Group - PEL
93
Non-linear large-signal lamp
model
Each complex variable is decomposed
into real and imaginary part:
uo = uo+juo
io = io+jio

yo
u
 GLuo,
io, 
A 4 yo o,
Ao  A3e

 dy o  w 2 i2  i2  y
L
o
o
o
 dt

y
0

 yo
The fundamental component
amplitude of the lamp current is:
Power Electronics Group - PEL

2 io
1
 2 i2o  i2o
94
Comparison between complete model
and fundamental component model
Step change of the lamp RMS current from 4 to 6mARMS
Lamp current Io [mARMS]
7
Fundamental component
model
6
5
Complete
model
4
0
0.2
0.4
Time [ms]
Power Electronics Group - PEL
95
Small-signal lamp model
Considering small-signal perturbations around an
operating point:
1  GLqR Sq Uo

ŷo
 îo  GLqûo  GLq
2Yo

1  GLqR Sq Uo

ŷo
 îo  GLqûo  GLq
2Yo

 dŷo  w 4G U î  U î  ŷ
L
Lq
o o
o o
o
 dt


yo
uo ,  GLuo ,
io , 
A y
Ao  A3e

 dyo  w 2i2  i2   y 
L
o
o
o
 dt
4
where:
o
R Sq   A 3 A 4 e

A 4 2 I  I
2
o
2
o

GLq 

Yo
Ao  A 3e
A 4 Yo



2I2o  I2o 
Ao  A 3e
A 4 2 I2o  I2o 
Lamp current fundamental component amplitude
2 io
1
2
i2o
 i2o
Power Electronics Group - PEL
îo 
2
2
o
I
I
2
o
I
î  Io îo
o o


4
 Io îo  Io îo
Io

96
Ballast dynamic model
 dis t  1
 dt  L Us signsinws t   ui t 
s

only the complex Fourier
 diL t   1   Ri t   u t   uo t   u t  
L
i
C

 dt
L 
n21

coefficients of indexes +1
 du t  1
are considered
 i  is t   iL t 
dt
C
i

 duC t  iL t 
 d is 1
 dt  C
1   2



j
w
i

U

j

u




s s 1
s
i 1
 du t 




dt
L

1
i
t



s 

 o 
 L  io t  
uo 1

 d iL 1
 dt
1
Co  n21





j
w
i


R
i

u


u
s L 1
L 1
i 1
C 1
 dt

L
n21



1
 d ui 1
 is 1  iL 1 


j
w
u

s
i 1
 dt
Ci
2

signsinws t  1   j
 d uC 1   jw u  iL 1

s
C 1
 dt
C
d u

1  iL 1
o 1


  jws uo 1 
 io 1 
Co  n21
 dt

97
Power Electronics Group - PEL
 d îs
ûi


î

I
w

ˆ

s s
s s
dt
Ls

 d îs
1 2




î

I
w

Û

û
ˆ

s s
s s
s
i 

dt
L


s 


1
ûo
 d îL



î

I
w


R
î

û


û
ˆ
s L
L s
L
i
C 


 dt
L
n21


ûo

1
 d îL

 ûC 
ˆ s    R îL  ûi 
 dt    s îL  IL w
L
n21


1
 dûi   û  U w

îs  îL
ˆ
s i
i s
 dt
Ci
 dû
1
 i    sûi  Ui w
îs  îL
ˆs 
 dt
Ci
 dû
î
 C   sûC  UC w
ˆ s  L
C
 dt
 dûC
îL



û

U
w

ˆ

s C
C s
dt
C


1  îL
 dûo   û  U w




î
ˆ
s o
o s
o 
 dt
Co  n21



1  îL
 dûo
 îo 
ˆs 
 dt    sûo  Uo w


Co  n21





Power Electronics Group - PEL

Ballast
small-signal
model
Complete ballast
model:
 x̂  Ax̂  Bû

 ẑ  Cx̂

û  w
ˆs

ẑ  îs
îL
Ûs
ûi ûC

T
ûo
ŷo

T
98
Large-signal and small-signal
model comparison
Us amplitude step variation (-5%)
-1
-2
Large-signal
non linear model
-3
60
Small-signal
linear model
0
100 200 300 400 500 600 700 800 900 1000
Time [s]
Power Electronics Group - PEL
Large-signal
non linear model
40
Duopk [V]
Duipk [V]
0
20
0
Small-signal
linear model
-20
-40
-60
0
100 200 300 400 500 600 700 800 900 1000
Time [s]
99
Instability analysis
Plot of the highest real part of the system
eigenvalues l as a function of the RMS lamp current
for different values of the lamp time constant
L=1/wL
max (Re[l])
2000
0
wL=120krad/s
wL=80krad/s
-2000
-4000
1
Lamp current Io [mARMS]
Unstable
wL=100krad/s
wL=60krad/s
2
3
4
5
6
7
Lamp current Io [mARMS]
Power Electronics Group - PEL
8
8
4
0
-4
-8
49.6
49.7
49.8
49.9
50
50.1
Time [ms]
100
Instability analysis
Plot of the highest real part of the system eigenvalues l
as a function of the RMS lamp current for different
values of the coupling inductor Ls (wL = 100krad/s)
2000
max (Re[l])
Unstable
0
Ls=10H
Ls=15H
Ls=20H
-2000
-4000
1
Ls=28H
2
3
4
5
6
7
8
Lamp current Io [mARMS]
Power Electronics Group - PEL
101
Conclusions
• Piezoelectric transformers represent good
alternative to magnetic transformers in
inverters for CCFL
• Different inverter topologies and control
techniques must be compared in order to find
the best solution for a given application
• Large-signal as well as small-signal
instabilities can arise due to the dynamic lamp
behavior
Power Electronics Group - PEL
102
References
1. Ray L. Lin, Fred C. Lee, Eric M. Baker and Dan Y. Chen, “Inductor-less Piezoelectric
Transformer Electronic Ballast for Linear Fluorescent Lamp” IEEE Applied Power
Electronic Conference (APEC), 2001, pp.664-669.
2.Chin S. Moo, Wei M. Chen, Hsien K. Hsieh, “An Electronic Ballast with Piezoelectric
transformer for Cold Cathode Fluorescent Lamps” Proceedings of IEEE International
Symposium on Industrial Electronics (ISIE), 2001, pp. 36-41.
3.H. Kakehashi, T. Hidaka, T. Ninomiya, M. Shoyama, H. Ogasawara, Y. Ohta, “Electronic
Ballast using Piezoelectric transformer for Fluorescent Lamps” ”IEEE Power
Electronics Specialists Conference Proc. (PESC), 1998, pp.29-35.
4.Sung-Jim, Kyu-Chan Lee and Bo H. Cho, “Design of Fluorescent Lamp Ballast with PFC
using Power Piezoelectric Transformer” IEEE Applied Power Electronic Conference
Proc. (APEC), 1998, pp.1135-1141.
5.Ray L. Lin, Eric Baker and Fred C. Lee, “Characterization of Piezoelectric
Transformers”, Proceedings of Power Electronics Seminars at Virginia Tech, Sept. 1921, 1999, pp. 219-225.
6.E. Deng, S. Cuk, “ Negative Incremental Impedance and Stability of Fluorescent
Lamps,” IEEE Applied Power Electronics Conf. Proc. (APEC), 1997. pp.1050-1056.
7.S. Ben-Yaakov, M. Shvartsas, S. Glozman, “Statics and Dynamics of Fluorescent lamps
Operating at High Frequency: Modeling and Simulation,” IEEE Trans. On Industry
Applications, vol.38, No.6, Nov./Dec. 2002, pp.1486-1492.
Power Electronics Group - PEL
103
References
8. S. Ben-Yaakov, S. Glozman, and R. Rabinovici, “Envelope simulation by SPICE compatible models of
electric circuits driven by modulated signals,” IEEE Trans. Ind. Electron., vol. 47, pp. 222–225, Feb.
2000.
9. S. Glozman, S. Ben-Yaakov, “Dynamic interaction analysis of HF ballasts and fluorescent lamps
based on envelope simulation,” IEEE Trans. Industry Application, vol. 37, Sept./Oct. 2001, pp.
1531-1536.
10.Y. Yin, R. Zane, J. Glaser, R. W. Erickson, “Small-Signal Analysis of Frequency-Controlled Electronic
Ballast“, IEEE Trans. On Circuits and Systems, - I: Fund. Theory and Applications, vol.50, No.8,
August 2003, pp.1103-1110.
11.C. T. Rim, G. H. Cho, “Phasor Transformation and its Application to the DC/DC Analyses of
Frequency Phase-Controlled Series Resonant Converters (SRC),” Trans. On Power Electronics, Vol.5.
No.2, April 1990, pp.201-211.
12.J.Ribas, J.M. Alonso, E.L. Corominas, J. Cardesin, F. Rodriguez, J. Garcia-Garcia, M. Rico-Secades, A.J. Calleja,
“Analysis of Lamp-Ballast Interaction Using the Multi-Frequency-Averaging Technique,” IEEE Power
Electronics Specialists Conference CDRom. (PESC), 2001.
13.R. Sanders, J. M. Noworolski, X. Z. Liu, G. Verghese, “Generalized Averaging Method for Power
Conversion Circuits,” IEEE Trans. On Power Electronics, Vol.6, No.2, April 1991, pp.251-258.
14.M. Cervi, A. R. Seidel, F. E. Bisogno, R. N. do Prado, “Fluorescent Lamp Model Based on the
Equivalent Resistance Variation,” IEEE Industry of Application Society (IAS) CDROM, 2002.
15.Onishi N., Shiomi T., Okude A., Yamauchi T., "A Fluorescent Lamp Model for High Frequency Wide
Range Dimming Electronic Ballast Simulation" IEEE Applied Power Electronic Conference (APEC),
1999, pp.1001-1005.
Power Electronics Group - PEL
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