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Modeling of CCFL using Lamp Delay and Stability Analysis of

Backlight Inverter for Large Size LCD TV

Chang-Gyun Kim1), Kyu-Chan Lee1) and Bo H. Cho2)

1) Interpower Co., Ltd.

1578-51, Silim1-dong, Gwanak-gu

Seoul, Korea 151-869

E-mail : [email protected]

2) Power Electronics System Lab.

School of Electrical Engineering

Seoul National University

Seoul, Korea, 151-742

Abstract - Cold cathode fluorescent lamps (CCFL) for LCD

TVs have a special v-i characteristic, which can make the

backlight inverter system unstable. In this paper, modeling of

both the cold cathode fluorescent lamp and the backlight

inverter is performed in terms of incremental impedance in

the frequency domain. The stability analysis of the backlightinverter and the CCFL is presented using the derived models.

It is concluded that the time delay, after which the lamp

voltage responds to the lamp current change, contributes to

the incremental impedance of the CCFL at the high

frequency region, which may cause instability of the

backlight inverter. The time delay is considered as a lamp

characteristic. The effect of the inverter design parameters on

the stability is also investigated. The lamp model and the

stability analysis results are verified from experiments with a

backlight inverter for 32-inch LCD TVs.

1. INTRODUCTION

It is a well-known fact that cold cathode fluorescent lamps

(CCFL), like most other electric discharge lamps, have a

negative incremental impedance, which can make the

inverter system unstable. In fact, instability has been

observed in experiments with CCFLs for large size LCD

TV’s, even in the open loop configuration. Mathematical

representations of this phenomenon are needed to better

understand the lamp’s behavior and to improve the backlight

inverter design.

Several studies [1-6] have been reported in the literature to

establish mathematical models. The authors of [1] presented

a model based on the physical principles inside the gas

discharge, which is too complicated to give a clear picture of

the lamp’s negative incremental impedance. Francis [2]

proposed a simple differential equation which satisfactorily

describes the first order effect of the lamp’s behavior. Various

modifications [3,4] were made to Francis’ equation in order

to increase its accuracy. Pspice models were also developed

as a tool to optimize ballast designs [5-7]. The final forms of

the models are a set of differential equations which are

shaped to closely duplicate the measured waveforms. The

ultimate goal of the above modeling approaches is to

accurately simulate the lamp’s voltage and currentwaveforms in the time domain.

However, those models cannot analyze the instability

phenomenon observed in experiments. The focus of lamp

and inverter modeling in this paper is to characterize its

salient feature and to investigate its requirement of a

backlight inverter from the circuit stability’s perspective. At

high frequencies, a lamp’s steady-state impedance can be

considered as a resistor [8]. Thus, instead of studying a

lamp’s steady-state behavior, our efforts are focused on its

incremental impedance in the frequency domain, specifically

on the high frequency behavior which causes instability.

In Section 2, modeling of the CCFL incremental impedanceis performed. The incremental impedance obtained from the

derived model is compared with measurement data using

720mm/4mm CCFLs. In Section 3, the other components of

the inverter are also modeled in terms of the incremental

impedance. In Section 4, the stability analysis is performed

using the derived models. The analysis results are verified

from experiments. The effects of the design parameters on

the stability are also presented. Conclusions are summarized

in Section 5.

2. MODELING OF LAMP INCREMENTAL IMPEDANCE

2.1 Steady-state characteristics of the CCFLs

The CCFL v-i characteristic at high frequencies

approximates a resistor [8]. However, the lamp resistance is

not a constant but a function of the lamp operating current.

The experimental data of the moving operating point is

plotted in Fig.1 and Fig.2 where v, i represents the rms value

0-7803-8975-1/05/$20.00 ©2005 IEEE. 1751

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for the sinewave operation. The equivalent resistance can be

approximated using two exponential functions as

][130][750 .. 3701500][ A I A I

Lamprms Lamprms Lamp eek R

⋅−⋅−+=Ω . (2.1)

The approximated equivalent resistance and lamp voltage

are plotted with solid lines in Fig.1 and Fig.2. There have

been two aspects to describe a lamp’s impedance. One is the

straight line from the origin to an operating point in Fig.1,which has a positive v/i slope and describes a lamp’s steady-

state behavior. Another is a curve between two operating

points, which has a negative dv/di slope and describes the

small-signal change of an operating point or its incremental

behavior.

The specifications of CCFLs used in the tests are listed in

Table 1.

TABLE 1. Specifications of CCFLs used in the experiment

LengthOuter

Diameter

Typ.

Current

Typ.

VoltageLuminance Frequency

720

mm4 mm 6.2 mA

1065 [V]

±7%

15700±13%

cd/m2

40kHz ~

60kHz

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01600

700

800

900

1000

1100

1200

1300

i Lamp.rms

[A]

V L a m p . r m s

[ V ]

‘o’ : experiments

‘ ‘ : approximation

Figure 1. The CCFL steady-state operating point in v-i plane

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0

2

4

6

8

10

12

14x 10

5

i Lamp.rms

[A]

R L a m p

‘o’ : experiments

‘ ‘ : approximation

Figure 2. The equivalent CCFL resistance

2.2 Derivation of the CCFL incremental impedance

To study a lamp’s incremental impedance, a perturbation

which is sufficiently small in magnitude is applied around an

operating point. For example, the perturbed lamp voltage and

current are

( )( ) ( )( ) )sin(ˆ,)sin(ˆ t t i I it t vV v s s ω ω +=+= , (2.2)

where ω

s is the switching angular frequency. The applied

perturbation voltage and the resulting current can be

expressed as (considering the lamp current as a phase

reference)

( ) ( ) )sin(ˆˆ,)sin(ˆˆ t it it vt v mmm ω φ ω =+= , (2.3)

where ω m is the low modulation frequency. The incremental

impedance is defined as :

mmi

v j Z φ ω ∠≡

ˆ

ˆ)( . (2.4)

The steady-state impedance is usually associated with a

large-signal operating point while the incremental impedance

is concerned with the small-signal perturbations of the

operating point. The above concepts are illustrated in Fig. 3,

where a CCFL is driven by a high frequency sinewave at the

fixed switching frequency. The amplitude of the sinewave is

modulated by a small-signal at the low varying frequency,ω m.

The incremental impedance of the lamp is defined as the

ratio of the slow varying signals which modulate the lamp

voltage and current, respectively.

At a certain modulation frequency, the lamp voltage

responds to the lamp current not immediately, but after some

time delay (T d ) as shown in Fig. 3. Observation of the

experimental waveforms at various operating conditions has

shown that there is almost a constant time delay between the

lamp voltage and the lamp current. It is concluded that this

time delay contributes to the incremental impedance at high

frequencies which may cause the instability. This time delay

is considered as a lamp characteristic.

Considering the time delay, the lamp voltage of (2.3)

becomes

( ) ))(sin(ˆˆ π ω +−⋅= d m T t vt v . (2.5)

The steady-state equivalent lamp resistance is a function of

the lamp current as shown in Fig.2. The amplitude modulated

lamp current also causes the equivalent resistance to vary

with time. If the equivalent resistance responds to the lampcurrent without a time delay, the lamp voltage valley will

occur at the same time as the lamp current peak. This means

that the equivalent resistance also has a time delay with

respect to the lamp current.

( )( ) R Lamp Lamp T t i Rt R −=)( (2.6)

R Lamp(i) is a steady-state equivalent resistance according to

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the lamp current as (2.1). T R is the delay time of the

equivalent resistance with respect to the lamp current.

Considering the envelope component only, the amplitudes

of the lamp voltage and current of Fig. 3 becomes

( ) ( )t vV t vmˆ+= (2.7)

( ) ( )t i I t imˆ+= . (2.8)

The lamp voltage valley (V min) of Fig. 3 can be expressed as

d m T

T t

m t vV +=

=4

min )( (2.9)

d m T

T t

m Lamp t it R+=

⋅=4

)()( , (2.10)

where T m is the period of the modulation frequency.

Substitution of (2.6) and (2.8) into (2.10) gives,

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ⎟ ⎠

⎞⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ⎟ ⎠

⎞⎜⎝

⎛ −++= d

m Rd

m Lamp T

T i I T T

T i I RV

4ˆ

4ˆ

min (2.11)

Assuming T d = T R, and substituting (2.3) into (2.11), then

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

⎟ ⎠

⎞

⎜⎝

⎛

++⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

⎟ ⎠

⎞

⎜⎝

⎛

+≅ d

m

m

m

m Lamp T

T

i I

T

i I RV 4sinˆ

4sinˆ

min ω ω

(2.12)

( ) ( )d m Lamp T i I i I R ω cosˆˆ ⋅++= . (2.13)

In the same manner, the voltage peak can be derived as

( ) ( )d m Lamp T i I i I RV ω cosˆˆmax −−= . (2.14)

The maximum and minimum of the lamp current ( I max and

I min) are defined as :

i I I i I I ˆ,ˆminmax −≡+≡ . (2.15)

By substituting (2.15) into (2.13) and (2.14), the minimum

and maximum of the lamp voltage becomes

( )d mT i I RV ω cosˆminmin += , (2.16)

( )d mT i I RV ω cosˆmaxmax −= , (2.17)

where

rmsrms I Lamp

I Lamp R R R R

.max.minminmax , == . (2.18)

From (2.16) and (2.17)

( ) ( ) ( )d mT i R R I R RV V ω cosˆminmaxminmaxminmax +−−=− .

(2.19)

From (2.15)

( )2

ˆ minmax I I i −= , (2.20)

Therefore, (2.19) becomes

( d mT R R

I I I

R R

I I

V V ω cos

2

minmax

minmax

minmax

minmax

minmax +−

−

−=

−

−) . (2.21)

vm

(t)

i m

(t)

i(t)

v(t)

V

I

T d

V ma x

V mi n

I ma x

I min

t = 0 t = T d +T

m /4

Figure 3. Illustration of the incremental impedance under high

frequency sinewave operation

From (2.4) and (2.21), the magnitude of the incremental

impedance becomes

i

v X ac Lamp ˆ

ˆ. ≡ (2.22)

( )d mT R R

I I I

R Rω cos

2

minmax

minmax

minmax +−

−

−= . (2.23)

The phase of the incremental impedance can be obtained

directly from the time delay (T d ). Therefore, the resulting

incremental impedance becomes

( ) ( d mac Lampm Lamp T X j Z ω π ω −∠= . ) . (2.24)

In Fig. 4, the derived model of the lamp incremental

impedance (T d =13µsec) is plotted as a function of the

modulation frequency and compared with measurement data.

A good match is observed, thus verifying the validity of the

derived model. The lamp incremental impedance, Z Lamp(s),has a negative resistance characteristic at low frequencies. As

the modulation frequency increases, the magnitude of

Z Lamp(s) increases but its phase decreases.

102

103

104

105

80

90

100

110

120Magnitude of lamp Impedance [dB]

102

103

104

105

-100

0

100

200Phase of lamp Impedance [deg]

frequency [Hz]

‘o’ : Experiments

‘ ‘ : Derived Model

Figure 4. The lamp incremental impedance in frequency domain

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3. INCREMENTAL IMPEDANCE MODELING OF RESONANT

COMPONENTS

Fig. 5 shows the power stage and its equivalent circuit in a

full-bridge resonant backlight inverter for the CCFL. L1 is an

externally added inductor. C 1 is a blocking capacitor and has

a trivial effect on the inverter operation, and therefore is

omitted in the equivalent circuit. L s is the total inductance ofthe primary side inductor and the leakage inductance of the

transformer. v s is a sinewave voltage source generated by a

DC voltage source and the switching devices. R s is the

winding resistance of the transformer, and C p represents an

externally added capacitor along with parasitic capacitances.

By testing the backlight inverter with the CCFL, instability

has been observed in the open loop arrangement. Lamp

voltage and current oscillate at a frequency of about 12kHz.

The incremental impedance of each component of the

equivalent circuit in Fig. 5 is derived to explain the instability.

By the fundamental frequency approximation, the parallel

capacitor voltage (v p) can be assumed to be a pure sinewaveat the switching frequency,

( t V t v s p p ω sin)( = ) . (3.1)

The capacitor current becomes

Cpd s p p T t I t i .sin)( −= ω , (3.2)

where

p s p p V C I ω = , (3.3)

4.

sCpd

T T −= . (3.4)

Considering the increments of the lamp voltage and current

as shown in Fig. 6,

1212 p p s p p p V V C I I −=− ω . (3.5)

The incremental impedance of C p is defined as

CpCpCp X Z φ ∠≡ , (3.6)

where

12

12

p p

p p

Cp I I

V V X

−

−≡ (3.7)

Cpd mCp T .ω φ = (3.8)

From (3.5) and (3.7), the magnitude of the incremental

impedance results in

s pCp C X ω

1

= . (3.9)

From (3.4) and (3.8), the phase of the incremental

impedance is

s

mCp

f

f ⋅−=

2

π φ , (3.10)

Lamp

Transformer

V IN

L1

1:N

C 1

C P

Lamp

Z Load

vs

Rs

Z out

Ls

C P Z

Cp

Z Ls

i s

i p

v p

+

_

Figure 5. Power stage and its equivalent circuit of the backlight inverter

under test

f s

v p

(t)

V p1

T d.Cp

i p(t)

V p2

I p2 I

p1

Figure 6. Illustration of C p voltage and current waveform to derive the

incremental impedance

where f m is the modulation frequency and f s is the switching

frequency.

In a similar manner, the incremental impedance of the series

inductor ( L s) and the series resistance ( R s) can be derived.

The related voltage and current waveforms are shown in Fig.

7. The equivalent ac source (v s) is considered as a phase

reference.

0∠= s s V v (3.11)

p p p p p p V vV v θ θ ∠=∠= 2211 , (3.12)

s s s s s s I i I i θ θ ∠=∠= 2211 , (3.13)

The amplitudes of the current ( I s1 and I s2) can be derived

from the impedance at the switching frequency.

( ) ( ) 22

2

222

1

1 ,

s s s

p s

s

s s s

p s

s

R L

vv I

R L

vv I

+

−=

+

−=

ω ω

(3.14)

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The difference between the two amplitudes of the current

becomes

( ) 12

2212

1 p s p s

s s s

s s vvvv

R L

I I −−−+

=−ω

(3.15)

From the phase diagram of Fig. 8 and using the small-signal

assumption, (3.15) can be approximated as

( )( ) Ls p p

s s s

s s V V

R L

I I θ

ω

cos112

2212 −

+≅− , (3.16)

where

1θ θ π θ +−= p Ls , (3.17)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⋅−

⋅−= −

p p s

p p

V V

V

θ

θ θ

cos

sintan

1

111 . (3.18)

θ 1 is the phase of v s-v p1.

The time delay (T d.Ls) of the series impedance is

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ = −

s

s s s Lsd

R

LT T

ω

π

1. tan

2. (3.19)

i s(t)

θ=

v p

(t)

I s2 I

s1

vs(t)

θ

ps

V p1

V p2

V S

Figure 7. Illustration of voltage and current waveforms to derive Z Ls

θ

Ls'

θ

Ls

θ

1

θ

p

v p1

v p2

vs

θ

Ls' ≅ θ

Ls

Figure 8. The phase diagram of the voltages related to the Z Ls

The phase due to the time delay (T d.Ls) at the modulation

frequency becomes

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ = −

s

s s

s

m Ls

R

L

f

f ω φ 1tan . (3.20)

From the definition of the incremental impedance, (3.16)

and (3.20), the incremental impedance of the series

inductance and the series resistance results in

( ) Ls

Ls

s s s Ls

R L Z φ

θ

ω ∠

+−=

cos

22

. (3.21)

4. STABILITY ANALYSIS AND EXPERIMENTAL RESULTS

In this section, stability analysis is performed and the

experimental results are presented. The parameters of an

example design of the inverter are listed in Table 2.

TABLE 2. The parameters of inverter design example

P Lamp f s Ls C p Rs vs v p φ p

6.6 W 50

kHz

330

mH

40

pF

945 1030

Vpk

1610

Vpk

120

deg

The load impedance and the output impedance are defined

as shown in Fig. 5 and can be expressed as

Cp Ls

Cp Ls

out Z Z

Z Z Z

+

⋅= , . (4.1) Lamp Load Z Z =

The lamp current can be expressed with the impedances as

( ) ( )

( ) ( )

( )⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +⋅

=

s Z

s Z s Z

sV s I

Load

out Load

s Lamp

1

. (4.2)

Fig. 9 shows the incremental impedance of each component( Z Ls , Z Cp , Z out , Z load ). The load impedance and the output

impedance have similar magnitudes at around 12kHz and the

phase difference is about 180˚, which implies the possibility

of instability. For further analysis of the instability, the

characteristic function (T char ) is defined as

( ) ( )

( ) s Z

s Z sT

Load

out char ≡ . (4.3)

The Nyquist plot of T char is presented in Fig. 10, which

shows that the plot encircles the (-1,0) point. That is, the

system may become unstable from the Nyquist Criteria. This

analysis explains the oscillating waveforms of the lamp

voltage and current observed in experiments.

The effects of the inverter parameters on the stability are

analyzed using the derived models and verified from

experiments. Three operating conditions are selected to

observe the effect of the switching frequency, and the

resonant components. The results are listed in Table 3.

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102

103

104

105

85

90

95

100

105

110

102

103

104

105

-200

-100

0

100

200

frequency [Hz]

m a g n i t u d e [ d B ]

p h a s e [ d e g ]

Zout

Zout

Z Ls

Z Ls

Z Cp

Z Cp

Z Load

Z Load

Figure 9. Incremental impedances of Z Ls , Z Cp , Z out and Z Load

-14 -12 -10 -8 -6 -4 -2 0-0.5

0

0.5

1

1.5

2

2.5

Real

I m a g

Figure 10. Nyquist plot of the characteristic function (T char )

Table 3.

Three operating conditions (change of f s and the resonant components)Operating

Conditions f s Ls C p Rs

case 1 50 kHz 330 mH 40 pF 945 Ω

case 2 57 kHz 330 mH 40 pF 945 Ω

case 3 50 kHz 660 mH 20 pF 945 Ω

At a higher switching frequency (case 2), the phase of Z out

increases as shown in Fig. 11(a). This makes the system

more stable, which can be seen from the Nyquist plot of Fig.

11(b). This analysis is verified from experiments as shown in

Fig. 12, which shows instability at the switching frequency

of 50kHz and stable operation at the switching frequency of

57kHz.When the resonant inductor has a larger inductance (case 3)

at the same resonant frequency, X Ls and X Cp both increase.

The magnitude of the output impedance crosses that of the

load impedance at a higher modulation frequency as shown

in Fig. 13(a). The phase of the output impedance decreases

more slowly than that of the lamp. Therefore, the phase

difference decreases, and this helps the system become stable.

The Nyquist plot of Fig. 13(b) shows that a larger L s

inductance at the same resonant frequency can change the

unstable system to a stable one, which can be verified from

experimental waveforms, as shown in Fig. 14.

5. CONCLUSION

Modeling of both the cold cathode fluorescent lamp

(CCFL) and the backlight inverter are presented in terms of

the incremental impedance. Stability analysis is performed

using the derived models. The stability analysis in this paper

can explain the instability observed in experiments. The time

delay of the lamp voltage response to the lamp current has a

great effect on the lamp incremental impedance characteristic

and also the stability. The effects of the design parameters of

the inverter are also investigated through the stability

analysis. The derived models and stability analysis results are

verified from experiments with the CCFL and the backlightinverter for large size LCD TVs.

102

103

104

105

85

90

95

100

105

110

102

103

104

105

-100

0

100

200

Zout (fs=50kHz)

Zout (fs=50kHz)

Zout (fs=57kHz)

Zout (fs=57kHz)

Z Load

Z Load

frequency [Hz]


p h a s e [ d e g ]

(a) Frequency domain analysis

-14 -12 -10 -8 -6 -4 -2 0-0.5

0

0.5

1

1.5

2

2.5

Real

I m a g

fs=50kHz

fs=57kHz

(b) Nyquist plot

Figure 11. Frequency domain and Nyquist plot according to the

switching frequency ( f s=50kHz and 57kHz)

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V L a m p

[ 1 k V / d i v ]

i L a m p [ 2 . 7 m A / d i v ]

time [50usec/div]

(a) fs = 50kHz (case 1)

V L a m p

[ 1 k V / d i v ]

i L a m p

[ 2 . 7 m A / d i v ]

time [50usec/div]

(b) fs = 57kHz (case 2) Figure 12. Experimental waveforms according to the switching

frequency ( f s=50kHz and 57kHz)

102

103

104

105

80

90

100

110

120

102

103

104

105

-100

0

100

200

frequency [Hz]


p h a s e [ d e g

]

Z out

(case 1 )

Z out

(case 3 )

Z Load

Z Load

Z out

(case 3 ) Z

out (case 1 )

(a) Frequency domain analysis

-25 -20 -15 -10 -5 0-0.5

0

0.5

1

1.5

2

2.5

3

case 1

(Ls=0.33H,

Cp=40pF)

case 3

(Ls=0.66H,

Cp=20pF) I m a g

Real

(b) Nyquist plot

Figure 13. Frequency domain and Nyquist plot according to the

resonant components ( Ls , C p) (case 1 and 3)

V L a m p

[ 1 k V / d i v ]

i L a m p [ 2 . 7 m A / d i v ]

time [50usec/div]

(a) Ls=0.33H, Cp=40pF (case 1)

V L a m p

[ 1 k V / d i v ]

i L a m p

[ 2 . 7 m A / d i v ]

time [50usec/div]

(b) Ls=0.66H, Cp=20pF (case 3) Figure 14. Experimental waveforms according to the resonant

components ( Ls , C p) (case 1 and 3)

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[4] P. R. Herrick, “Mathematical Models for high Intensity

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