論
Simulation
文
Model for Voltage-Current
Characteristics
of a Low Pressure
Discharge
Tube
under High-Frequency
Operation
Member
A voltage-current
Yoshio
characteristic
Watanabe
(Kanagawa University)
model of a low pressure discharge tube is developed for analyzing the
properties of a high-frequency operation circuit. The model is derived using the continuity equation for
electrons and the electron mobility as a function of the electron temperature. The obtained voltage-current
(V-i)
The
relation is as follows:
constants
condition.
between
in the
Using
the
the
Key words:
and
The
current
are
developed
calculation
circuit-properties.
discharge
formula
model
modulated
Simulation
model,
the
from
voltage
apply
the
discharge
waveforms
measurement
can
by
model,
determined
is good
not
only
to the
a low-frequency
are
waveforms
calculated
enough
for
measured
for several
analyzing
high-frequency
the
operation
under
cases.
the
The
standard
agreement
high-frequency
but
also
operation
to the
case
of the
component.
Voltage-current
characteristics
model,
Low
pressure
discharge,
High-frequency
operation
1.
the
Introduction
To analyze the operating characteristics
of a fluores-
discharge
voltage-current
high-frequency
operation
characteristics
is needed
characteristics
including
for the
the
under
a
low-frequency
analysis
of the
high-
cent lamp ballast, the voltage-current characteristic
model for a fluorescent lamp is needed. Several models
frequency
for a low-pressure discharge under a low-frequency (50/
60Hz)
operation
have been presented
for this
balance
purpose(1)〜(4).
quantitative
agree well
results
obtained
from the model do not
with the experimental
results due to the
ambiguity
in the
When
a fluorescent
frequency
operation
is a few
tens
lamp
circuit
kHz,
However,
the amplitude
is usually
mercial
of the discharge
Strictly
istics
power
somewhat
電 学 論A,114巻7/8号,平
when
deviate
charac-
成6年
The model
and
energy
for this purpose(1)(5).
long computation
employed
the
time and the
constants
and
electron
distribution.
acteristics.
of this paper
is to propose
a high-
In this paper, 40W (FLR 40S) and 20W
(FL 20S) fluorescent lamps are chosen as the object.
The frequency of a few tens kHz is designated as "high
frequency", since a commercial high-frequency ballast
a comon
usually operates at this frequency range.
the equiva-
2.
character-
Discharge
voltage-current
characteristic
model
a high-frequency
First,
from the linear-resistance
the model
characteristics.
equations
might be available
it takes rather
The objective
by the low-frequency.
under
circuit
continuity
frequency operation model which is simple and accurate
for analyzing the high-frequency operation circuit char-
depends on
is employed
voltage-current
lamp
Therefore,
circuit
Consequently,
the actual
of a fluorescent
characteristics.
line.
is also modulated
speaking,
operation
operating
rather
the
equation
energy
The amplitude
at low frequencies,
high-frequency
lent resistance
circuit
current.
high-
Therefore,
used for the
resistance
operation
on
However,
frequency
shows
operation
the equivalent
modulated
the commercial
lamp
characteristics.
model
is usually
of the high-frequency
teristics.
by the
of which operation
a fluorescent
good
linear-resistance
the linear-resistance
analysis
is operated
based
operation
which represents
521
the dependence
frequency
of the plasma
is investigated.
parameters
on a
As a fluorescent
lamp has a long tubular shape, the ratio of the voltage
investigated
across the positive column to the discharge voltage is
when
normally more than 80%. Thus, the discharge voltage-
the
amplitude
current characteristic of a fluorescent lamp is mainly
determined by its positive column characteristics.
The
the
low-frequency
Te
is
field,
the
E,
electron
electron
energy,
Ue
temperature),
q
section
electron
is
velocity.
under
rent, i, of 50Hz and 50kHz, respectively (see Appexdix
I). In the calculation, the polarity of i and E are
neglected since ne and Ue do not depend on the field
expressed
direction under uniform condition in the axial direction.
where
The results are shown in Fig. 1. The results show that
ne is independent of the current phase angle in the case
of 50kHz, since the diffusion time is much longer than
between
current,
since
the
current
With
increasing
change
above
and decreasing
with
results,
except
the
the
frequency
discharge
near
zeros
by
line.
ion current,
is
charge,
ne
to
is
is independent
the
the
effective
and ve
is
of
high‑frequency
the
the
electron
current
phase
ve
current.
using
cross
the
operation,
discharge
ve=μeE
is given
S
channel
changes
The
electron
ve
can
be
mobility,
μe,
as follows(5),
μe=μe0p‑1Te‑0.5……(2)
μe0 is the
and
p
is
where
filling
and Te
λ is
energy
and
a
the
is
electron
are
If
the
gas
The
species
relationship
follows(6),
Assuming
free
the
collisions,
that χ
a
be
k
is
λ is
taken
of
to
, χ
is proportional
to
, when
depends
Tek1
be
elastic
However
neglected
the
electron
consist
constant.
not
electron
and
between
collisions
also
can
is the
collision
collisions
electron
is
path, χ
electron
elastic
the
collisions
rewritten
as
mean
As
only, χ
inelastic
by
pressure.
one
mainly
constant.
Te.
per
constant.
collisions
gas
is given
rate
atom
determined
λqE)/(2√2χk)……(3)
loss
Boltzmann
.
constant
the
E
Te=(4√π
of the
characteristics
As
the
as
which
of ne can follow
of 50Hz
power
i, neglecting
discharge
proportionally
the half cycle of 50kHz under the condition investigated. On the other hand, ne changes proportionally to the
of 50Hz,
and
component
electron
the
angle
case
kHz
is modulated
commercial
current,
the
of
drift
in the
current
tens
by,
density, ne, are calculated using the continuity equations
and the energy balance equation with a sinusoidal cur-
current
separately,
than
i=qSneve……(1)
(=3/2・kTe,
and
component
is more
discharge
of the
discharge
where
electric
of the
The
The radial profile of the electron density is assumed to
be given by the Bessel function(5).
frequency
frequency
High-frequency
expressed
The
each
operation
2.1
positive column in a fluorescent lamp is assumed to be
axially homogeneous and cylindrically symmetric(5).
where
for
the
, Eq.
on
(3)
is
as follows.
Te=AE1/(1+0.5K1)……(4)
where, A is a constant. SubstitutingEqs (2) and (4)
into Eq. (1),
i=qSne(μe0p‑1A‑0.5EKf‑1)E……(5)
(a)
50Hz
where
Kf=(1+K1)/(2+K1)
should
be
uniform
as
equal
.
to 0.5
across
the
If χ
since K1=0
discharge
is
.
tube
a
constant,
Assuming
, Eq.
Kf
that
(5)
E
is
is rewritten
follows.
V=Vm(i/Im)1/Kf……(6)
where
V
is
amplitudes
Low‑frequency
The
continuity
in
E:25V/m,
Fig. 1.
Ue:0
Calculated
.3eV,
discharge
V
2.2
neglecting
(b)
the
of
the
and
,
Vm
and Im
are
the
, respectively.
component
equation
divergence
axial
voltage
i
of
for
axial
ne
flow
is
given
due
to
as
the
follows
,
uniformity
direction.
50kHz
ne:0.5×1018m‑3,
waveforms
i:0
.25A/div
of the electric
field
,E
, electron
energy,
Ue, and the electron
density ,
ne, using the continuity
equations
and the energy
balance
equation
with
a sinusoidal
discharge
where
current,
length.
νi is the
term,
the
i.
522
ionization
frequency
KgiNg+KqiNq+KriNr+KsiNs,
ambipolar
For
diffusion
which
equals
in Eq.
coefficient
the low‑frequency component
and Λ
(A4)
is the
to
the
. Da
diffusion
, ve average
T. IEE Japan, Vol. 114‑A, No . 7/8, '94
is
低気圧放電高 周波 動作特性モ デル
during
one cycle
be constant
except
modulated
Fig. 1.
of the high-frequency
near
zeros
of the current
by the low-frequency
Therefore,
low-frequency
average
discharge
current
as shown
can be replaced
characteristic,
during
to
Table 1. Measured Q1, Q2 and Q3 values for
each type of the fluorescent lamp.
while ne is
component,
ne in Eq. (7)
for the
can be taken
where
in
by ia
ia is the
one cycle of the high-
frequency.
where,
Q3=Da/Λ2.
temperature,
Eq.
The
Te.
(3),
νi is
frequency
Since
a
νi strongly
depends
Te
on
function
of
operation, νi
expression(4),
which
semi‑empirical
depends
is
is
E.
In
indicated
case
of
by
to
the
electron
as
the
simulated
similar
on
E
the
in
a
low‑
following
Guntherschulze's
laws(7).
νi=(Q1+Q2i)V……(9)
where Q1 and Q2 are constants.
The term, Q1V, re-
presents the direct ionization while the term, Q2iV,
represents the cumulative ionization.
In the case of
high‑frequency
by
the
the
peak
operation,
following
of
νi is
expression
Te
during
half
assumed
to
since
νi strongly
cycle
of
the
be
simulated
depends
on
high‑frequency.
νi=Q1+Q2ImVm……(10)
where Va is the averaged value of V during one cycle of
the high-frequency.
Combining Eq. (8) and Eq. (10),
the following equation for the low-frequency component
is obtained.
where Kv denotes the ratio of Vm/Va.
2.3
High-frequency
cent lamp
The high-frequency
operation
operation
model of a fluoresmodel represented
Fig. 2. V-i Lissajous figure of a fluorescent
operated at 47kHz.
by
V‑i
Vmand Im instead of Va and ia is much convenient for
tion
practical uses. Thus, Eq. (11) is rewritten
combining with Eq. (6).
Lissajous
and
the high-frequency
2.4
operation
back‑light
model
lamp,
tube
The
is smooth
for
during
discharge.
Measurement
40℃,
are
and
of Kv, Kf
of
which
length
outer
as expected
from
reason
why this deviation
(see Appendix II). Q1, Q2 and Q3depend on the type of
lamps and the operating temperature but do not depend
curve
during
curves
for each lamp
on the discharge current value and its waveform(4)(8)(9).
The obtained values for 40W (FLR 40S) and 20W
and shown
(FL 20S) fluorescent lamps at the operating temperature
straight
of
these
Kf is determined
電 学 論A,114巻7/8号,平
in Table
appears
operation(4)
1.
by the
成6年
following
method.
The
The
opera‑
data
for
a
lines.
lines.
523
diameter
is also
current
in the
decreasing
phase
however
and
current
occurs
with
a hysteresis
loop
current.
is not clear.
decreasing
are plotted
The
is increasing
the peak of the discharge
the
the curve
is not smooth.
curve
current,
is 4.1mm
shown
The
Using the
phase,
the
in a logarithmic
V-i
scale
in Fig. 3.
curves
somewhat
First,
discharge
around
tube
phase
a smoothed
using a chocke coil ballast
shown
47kHz
2.
from Eq. (6),
increasing
decreasing
are
Fig.
is 200mm,
The constants, Q1, Q2 and Q3 are determined from the
measurement of the discharge voltage-current waveform
40℃
under
an
V-i curve during
current
deviation
under 50Hz
measured
shown
as follows
figure.
the low-pressure
figures
at
and the
Eq. (12) gives
lamp
for both
40W
and
The Kf is obtained
The curve
upcurving
20W
show
from the gradient
for the back-light
behavior.
lamps
of
lamp
shows
When the curve
for the
5μs,
100V,
0.1A/div.
Fig. 5. Discharge voltage waveform for a 40W
lamp operated by the push-pull type inverter. IL=
0.2A (rms).
Fig. 3. V-i curves of each
during the current decreasing
lamp using
phase.
the curve
Fig. 6. Kv values for 20W and 40W lamps as a
function
of the discharge
current,
IL, in rms value.
The push-pull
type inverter
is employed.
shown
Fig. 4.
function
Kf value for 20W and 40W lamps as
of discharge
current, IL, in rms value.
a
in Fig. 5.
and the 40W lamps
the discharge
lamp
Kf value
is nearly
is approximated
obtained
Kf values
equal
for 20W
in Fig. 4 as a function
by a straight
to that
for the 40W
and
of the
40W
line,
lamps
discharge
the
lamp.
are
current,
The
since
by the
discharge
the discharge
current
approximated
Kf>0.5
indicates
neglected.
This
current,
discharge
value.
is due
collision
that
Kf slightly
inelastic
increases
to increase
with increasing
sake of simplicity,
collisions
in the
rate
of the
the discharge
hereafter,
current.
current
Kv slightly
depends
discharge
current
the
current,
dependence
waveform
is nearly
by the following
of
The Kv of both
IL.
Here-
of Kv, when the
sinusoidal,
is
formula.
Kv‑1=0.16Im+0.45(≒0.5)……(13)
can not be
with increasing
the same value.
waveform
varies
for the 20W
in Fig. 6 as a function
IL, in rms value.
on the discharge
after,
shown
IL, in rms
Kv values
are shown
current,
lamps have nearly
back-light
The obtained
current.
When a non-sinusoidal wave inverter is employed, Eq.
inelastic
(13) can not be applied to get Kv value. The Kv for a
non-sinusoidal wave inverter is obtained by following
For the
Kf is approximated
to be
0.68 and 0.59 for the 40W and 20W lamp, respectively.
procedure: First, assuming Kv=2, an approximated
voltage waveform is calculated.
Then, Kv value is
The reason
adjusting from the obtained voltage waveform.
for the difference
20W and the 40W
Kv is determined
form,
which
waveform.
is
inverter,
between
the
lamps is not clear.
from
the discharge
related
to
The discharge
the type of the operation
waveform
in Kf values
current
circuit.
the
discharge
waveform
depends
inverter,
Comparison
between
the
calculation
and the
measurement
3.1
on
voltage
by a push-pull
sinusoidal-wave
3.
wavecurrent
The discharge
of the 40W lamp operated
which is a typical
voltage
Discharge
voltage
frequency
cycle
waveform
during
a high-
The discharge voltage waveform is calculated by Eq.
type
is
(12) for a half cycle of high-frequency
524
T. IEE
Japan,
operation, where
Vol. 114‑A,
No. 7/8'94
低気圧放電 高周波動作特 性モ デル
the current waveform is given from the measurement.
The push-pull type inverter and DC input power were
employed. Therefore, the discharge current was not
modulated by a low-frequency (dIm/dt=0). The calcu-
agreement
lated and the measured voltage waveforms for the 20W
and the measured
and the 40W lamps are shown in Fig. 7. The discrepancy between the calculation and the measurement
20W lamp
operated
is a typical
non-sinusoidal
during the current increasing phase is due to the discrep-
Fig. 9. Kv=2
ancy shown in the V-i Lissajous figure. The calculated
and the measured discharge voltages in effective value
of the voltage
for the 20W and the 40W lamps are shown in Fig. 8 as
a function of the discharge current, IL, in rms value. The
calculation
ment
between
is fairly
Next,
the
calculation
case when
form is non-sinusoidal
smaller
on
(1μs,
12.5V,
Calculated
and measured
電 学 論A,114巻7/8号,平
成6年
and
in the
in the measure-
curves
AC
current
depends
of
for one cycle
amplitude
is adjusted
of the
the
voltage
cycle
power
on the
also fits the mea-
discharge
circuit
line,
the
is modulated
AC
of the high-
to 2.34.
operation
power
employed
by
line
is operated
amplitude
the
and
of the
low-frequency
the
rectification
modulation
method
(1μs,
25V,
discharge
of the
0.4A/div.)
Fig. 9. Calculated and measured discharge voltage
waveforms
for a 20W lamp operated
by the
blocking-type inverter. IL=0.37A
(rms)
1ms,
in effective
value as a function
current, IL, in rms value.
in
Similarity
0.1A/div.)
Fig. 7. Calculated and measured discharge voltage
waveform for half cycle of high frequency operation, where the current waveform is given from the
measurement.
The push-pull type inverter and DC
input power were employed. IL=0.4A
(rms).
Fig. 8.
that
a low-frequency
a commercial
discharge
0.1A/div.)
depth
lamp
are shown
the amplitude
than
to be 2.34
a high-frequency
component
20W
which
the calculation
is good, however
Envelope
When
(b)
inverter,
between
for the
inverter,
for the calculation.
The calculated
during
25V,
waveforms
by a blocking-type
waveform
is 20%
The calculated
voltage
wave
wave-
ment. This difference results from the assumption
of Kv
=2 . Using the calculated
voltage waveform
in Fig. 9,
3.2
(1μs,
measure-
current
is investigated.
discharge
sured value if Kv value
lamp
the discharge
is employed
the measurement
frequency.
40W
the
good.
the
Kv is calculated
(a)
and
50V, 0.1A/div.
Fig. 10. Calculated and measured
envelope curves
of the discharge voltage for a 40W
lamp operated
by the push‑pull type inverter.
voltage
discharge
525
in
the high-frequency
operation
inverter
on a commercial
operated
employed.
The
circuit.
calculated
The push-pull type
AC power
and the measured
curves
of the discharge
voltage
shown
in Fig. 10, where
the envelope
charge
current
ment.
The agreement
measurement
operation
waveform
the amplitude
is given
between
is fairly
model
good.
described
from
I.
Consider that the discharge tube is filled with argon
are
at 330 Pa and mercury.
of the disand the
are neglected.
the high-frequency
current
and
The ion density is equal to the electron
by
component.
Bessel function.
4.
is excited
density, ne. The plasma of the discharge is assumed to
be axially homogeneous and cylindrically symmetric.
The radial profile of the concentrations is given by the
even if
is modulated
Only mercury
ionized.
The triplet levels, 63P, are considered for
excitation level of mercury and other excitation levels
the measure-
the calculation
Thus,
envelope
lamp
by Eq. (12) is valid
of the discharge
a low-frequency
for the 40W
curve
Appendix
line was
Conclusion
The continuity equation
for Nq (63P0
density), Nr (63P1 density), Ns (63P2 density) and ne are
The voltage-current
characteristics
model
given as follows,
of a low
pressure discharge
for the high-frequency
operation is
developed.
The constants
in the model are determined
from the measured
discharge
at its operation
temperature.
effective
calculated
value
agreement
current
range.
frequency
wave
current
with
the
for each
The waveform
by
the
measurement
model
in
show
the
lamp
and the
good
practical
The model can apply not only to the high-
operation
but also
waveform
with sinusoidal
to the case
by a low-frequency
that the model has sufficient
the high-frequency
operation
and non-sinusoidal
of the modulated
component.
accuracy
circuit
discharge
It is verified
for the analysis
of
characteristics.
where,
Acknowledgment
The
author
Dr. S. Murayama
(Manuscript
to express
for his valuable
received
his appreciation
to
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
the
discussion.
Energy
July 19,'93,
ground
from
x
respectively),
τ
balance
state
level
is
to
the
density, Kxy
y
level
effective
equation
for
is
the
(x,y=g,
rate
q,
r,
imprisonment
the
electron
s,
i,
time.
energy
Ue
is as
follows.
February
4,'94)
References
(2)
is
constant
wishes
revised
(1)
Ng
where χ
is
collision
K. Wani: "Simulation model for Fluorescent Lamps", J.
Illum. Engng. Inst. Jpn. (in Japanese), 73, No. 9, 556 (1989)
N. Aoike: "Theoretical Analysis on Gas Discharge Lamp
L-C Series Ballast", ibid., 58, No. 7, 206 (1974)
S. Ozaki & K. Masumi: "Simulation of the high pressure
mercury lamp operating circuit system using equivalent conductance models", ibid., 73, No. 9, 561 (1989)
Y. Watanabe: "Simulation Model for Voltage-current Characteristics
of Low Pressure Discharge Tubes in Low
Frequency Operation", ibid., 76, No. 2, 69 (1992)
J. Polman, J. E. Werf & P. C. Drop: "Nonlinear effects in the
positive column of a strongly modulated mercury-rare gas
discharge", J. Phys. D Appl. Phys., 5, 266 (1972)
S. Takeda: Kitai Houden no Kiso (in Japanese), p. 108
(1990) Tokyo Denki Daigaku Press
G. Francis: The Glow Discharge at Low Pressure, Handbuch
der Physik XXII, p. 117 (1956) Springer-Verlag
Y. Watanabe: J. Illum. Engng. Inst. Jpn (in Japanese), 76,
No. 6, 287 (1992)
Y. Watanabe: ibid., 77, No. 10, 603 (1992)
M. A. Cayless: B. J. Appl. Phys., p. 186 (1959)
C. Kenty: J. Appl. Phys., 21, 1309 (1950)
J. F. Waymouth: Electric Discharge Lamps, p. 123 (1971)
The M. I. T. Press
is the
the
frequency,
energy
given
elastic
by
collision
Ug
difference
Eq.
(2).
loss
is the
gas
atom
between
Discharge
rate,
ν is
elastic
energy
and
to
y level. μe is
x level
current
the
is given
as
ΔVxy
follows.
i=Imsin(ωt)=0.43πR2qneμeE……(A6)
The
values
Ref.
(10)〜(12).
of
all
The calculation
constants
procedure
are
given
from
the
is as follows . The
appro-
priate initial values for the valuables are taken.
the given Ue value, rate constants
are determined
Using
Nq, Nr, Ns, ne and
using
Eqs(A1)〜(A5).
from
Eq.
cycle
of
repeated
E
(A6).
the
Ue at next
The
discharge
until
at
time
next
calculation
steady
state
time
is
current.
step
step
continued
Then
solution
are
the
is
, then
calculated
is
obtained
over
half
calculation
is
obtained.
II.
The
526
V-i
model
for a low-frequency
operation
is given
T. IEE Japan, Vol. 114‑A, No. 7/8 , '94
低気圧 放電高周波動作 特性モ デル
as follows(4).
Yoshio
Watanabe
(Member)
He received
the B. E. in 1969
and Ph. D in 1987 in Electrical
The discharge voltage and the current waveforms
are
phase,
current
where the discharge
decrease, are inserted
peak
to
trical
phase and the phase
University,
1987,
Lab.
ing
Engineering
Applied
527
Physics.
Society
he
Since April
Engineering,
of Japan
Hitachi
is a member
the
From
Central
1990, he has been
in Department
Kanagawa
and
Engineering
respectively.
joined
Professor
Dr. Watanabe
voltage sharply becomes to
in Eq. (A7).
Then Q1, Q2, Q3
成6年
1969
an Associate
values are determined.
電 学 論A,114巻7/8号,平
Kyoto
Research
taken with a choke coil ballast under 50Hz operation.
Three sets of V and i values at typical current phase,
reignition
from
of Elec-
University.
of the
Japan
Illuminat-
Society
of