Available via license: CC BY 4.0
Content may be subject to copyright.
Manuscript received 07/24/2023; first revision 08/28/2023; accepted for
publication 10/10/2023, by recommendation of Editor Telles Brunelli Lazzarin.
http://dx.doi.org/10.18618/REP.2023.4.0019
The Charging and Discharging of Reverse Biased
Schottky Diode Voltage-Dependent Capacitor
Ivo Barbi, Life Fellow, IEEE
Solar Energy Research Center, Federal University of Catarina, Florianópolis-SC, Brasil
e-mail: ivobarbi@gmail.com
Abstract – Transient analysis of RC circuit with
voltage step source, constant resistor and nonlinear
voltage-dependent capacitors with the capacitance given
by
( ) 1
c o c bi
C v C v V
, where
o
C
and
bi
V
are constant
and
c
v
is the voltage across the capacitor, was carried out.
The analysis was restricted to this class of nonlinear
capacitors, because they represent with acceptable
approximation the capacitances of reverse-biased Schottky
diodes, which are very important and usually used in
several static power converters practical applications. The
charging and discharging are described by explicitly
solvable nonlinear first-order differential equations. The
theoretical analysis results are verified by numerical
examples with computer simulation and also by
experimentation in the laboratory.
Index Terms – RC circuit, transient analysis, voltage-
dependent capacitor, Schottky diode.
I. INTRODUCTION
It is well known that a Schottky diode biased in the reverse
direction exhibits a voltage-variable capacitance and that this
characteristic can be useful in a number of applications, which
includes voltage tuning of an LC resonant circuit, voltage-
controlled oscillators, parametric amplifiers and frequency
multipliers.
Nonlinear voltage-dependent capacitances are also
inherently present in silicon, GaN and SiC power
semiconductor devices, where they can have a significant
impact on the operation of high frequency switched-mode
power converters. They can affect the switching times,
commutation losses and the converter dynamics and can limit
the maximum operation frequency of these converters.
In [1] the study on the charging and discharging of two
types of nonlinear semiconductor capacitors through linear
resistor is reported: one having a capacitance increasing
exponentially with the voltage across it and the other, the
space-charge capacitor that obey the function
sinh
c o c c
C v C v v
. In [2] a solution for the large
signal transient response of a reverse biased p-n junction
diode to a step voltage is presented, but only during the
charging period. Reference [3] presents an analysis of the
large signal step response of a junction capacitor considering
charging and discharging through a linear resistance. It is
shown that the behavior of an abrupt or a linearly graded
junction capacitor is different from that of a conventional and
voltage-independent capacitor. In [4], using a closed-form
solution of the static Poisson’s equation in the depletion layer,
the large signal step response of a single-diffused junction is
studied and it is shown that it deviates from that of either a
linearly graded or abrupt junction. However, none of these
articles present an explicit solution to the nonlinear
differential equations that describe the transient behavior of
the voltage across the capacitors.
Reference [5] has shown how the optimal solution for
charging linear and nonlinear capacitor is computed, including
the optimal charging voltage for a MOSFET gate capacitor,
which capacitance is given by
( ) 1
c o c bi
C v C v V
.
Reference [6] also analyzes RC circuits containing a voltage-
dependent capacitor, expanding and generalizing the analysis
presented in [5],
However, as the objective of these studies was to find how
the input voltage should change with respect to time to
minimize the resistor losses, no explicit expressions for the
voltage across the capacitor as a function of time was
provided, when the circuit is supplied by an input voltage
step.
In the research reported herein we investigate the transient
behavior of RC networks containing nonlinear voltage-
dependent capacitors. The analysis was restricted to basic
circuits containing capacitors given by
( ) 1
c o c bi
C v C v V
, which represents the reverse biased
Schottky diode voltage-dependent capacitance. With this class
of capacitors, as demonstrated in this paper, it is possible to
find a closed form solution for the voltage across the nonlinear
capacitors, the energy converted into heat in the series
resistor, the energy stored in the capacitor, and the charging
and discharging times.
II. THE SCHOTTKY DIODE VOLTAGE-
DEPENDENT CAPACITANCE
Let the voltage-dependent capacitor be that shown in Fig. 1.
Fig. 1. Voltage-dependent capacitor.
The current flowing through the capacitor is given by
()
C
C
dQ v
idt
(1)
where
()
C
Qv
denotes the electric charge on the capacitor
plates, which is given by
( ) ( )
C C C
Q v C v v
(2)
We can rewrite (1) as
()
CC
C
C
dQ v dv
idv dt
(3)
which substituted in (3) yields [7]
()C
CC
dv
i C v dt
(4)
The voltage-dependent depletion-layer capacitance of a
reverse-biased Schottky diode is given by [8]
() 1
o
c
c
bi
C
Cv vV
(5)
where
o
C
is the zero-bias junction capacitance and
bi
V
is the
built-in voltage of the Schottky diode.
The measured
CV
characteristics of the Schottky diode
STPS 10L25 from STMicroelectronics are shown in Fig. 2
[7].
Fig. 2. Measured C-V characteristics of STPS10L25 [7].
From the characteristics shown in Fig. 2, we can extract
the parameters of equation (6), which are
4.5
o
C nF
and
0.326
bi
VV
.
III. CHARGING TRANSIENT OF THE VOLTAGE-
DEPENDENT CAPACITOR
Let us consider the circuit shown in Fig. 3. At
0t
the
voltage source
1
V
is connected and the capacitor, with no
initial voltage, begins to charge.
Fig. 3. Circuit for charging the voltage-dependent capacitor through the linear
resistor R.
As follows from Kirchhoff’s voltage law
1CC
V Ri v
(6)
Since
()C
CC
dv
i C v dt
(7)
we have
1
() CC
c
dv v
V
Cv dt R R
(8)
Substitution of (5) in (8) yields
11
cCC
bi c
Vvdv v
V
dt R R
Vv
(9)
Thus,
1c bi c
C
o bi
V v V v
dv
dt RC V
(10)
From (10) we obtain
1
C
c bi c o bi
dv dt
V v V v RC V
(11)
Therefore,
1
C
c bi c o bi
dv dt K
V v V v RC V
(12)
where
K
is the constant of integration.
It is well known that
o bi o bi
dt t
RC V RC V
(13)
Integrating the first term of (12) we find
1
11
1
11
ln
1
11
ln
bi c bi
C
c bi c bi
bi c bi
V v V V
dv
V v V v V V
V v V V
(14)
After appropriate algebraic manipulation we obtain
1
11
1
11
1ln 11
bi c bi
C
c bi c bi
bi c bi
V v V V
dv
V v V v V V
V v V V
(15)
In order to determine
K
, we substitute (14) and (15) in
(13) and set
0t
and
00
c
v
. Hence,
1
1
1
11
1ln 11
bi bi
bi
bi bi
V V V
KVV
V V V
(16)
Substitution of (13), (15) and (16) in (12) yields
1
1
1
1
1
1
11
ln
1
11
ln
11
ln
1
11
ln
bi c bi
bi
bi c bi
bi bi
bi o bi
bi bi
V v V V
VV
V v V V
V V V t
V V RC V
V V V
(17)
Rearranging the terms of (17) we find
1
11
1
1
11
11
ln 11
11
bi c bi
bi bi c bi
o bi
bi c bi
bi bi
V v V V
t V V V v V V
RC V
V v V V
V V V
(18)
Let
be defined by
1bi
o bi
VV
RC V
(19)
Substituting (19) in (18) we find
11
11
1 1 1 1
1 1 1 1
bi c bi bi bi
t
bi c bi bi bi
V v V V V V V
e
V v V V V V V
(20)
Solving (20) for
c bi
vV
we obtain
1 1 1
1
11
t
bi bi bi
tt
bi bi bi
bi c tt
bi bi bi bi
V V V V V V e
V e V V V e
Vv V e V V V V V e
(21)
From (21) we find
2
1 1 1
1
11
t
bi bi bi
tt
bi bi bi
c bi
tt
bi bi bi bi
V V V V V V e
V e V V V e
vV
V e V V V V V e
(22)
Equation (22) represents the behavior of the voltage across
the voltage-dependent capacitor of the electric circuit shown
in Fig. 3.
Let
,
c
v
and
t
be defined by (23), (24) and (25),
respectively.
1
bi
V
V
(23)
1
c
c
v
vV
(24)
and
o
t
tRC
(25)
Substituting (23), (24) and (25) in (22) and rearranging the
terms we obtain
2
1
1
1 1 1 1
11
t
ct
e
v
e
(26)
Fig. 4 shows the plot of the normalized voltage
c
v
across
the voltage-dependent capacitor of the circuit shown in Fig. 3,
as function of the normalized time
t
, for different values of
.
Fig. 4. Normalized transient voltage
c
v
across the voltage-dependent
capacitor in Fig. 3 as function of the normalized time
t
, taking
as a
parameter for: (a)
0.015
, (b)
0.030
and (c)
0.060
.
As shown in Fig.4, the higher the value of
, which
implies a lower value of the power supply voltage
1
V
, the
longer the time required for the capacitor to fully charge,
which occurs when
1c
vV
. This behavior is explained due to
the nonlinearity caused by the voltage-dependent capacitance
of the capacitor.
A. Charging Time
Substituting (24) and (25) in (22) and solving for
t
e
we
find
1 1 1
1 1 1
c
t
c
v
ev
(27)
Substitution of (23) in (19) yields
1
o
t
tRC
(28)
Substituting (25) in (28) we get
1
tt
(29)
Substitution of (29) in (27) gives
11 1 1
1 1 1
tc
c
v
ev
(30)
Solving (30) for
t
we find
1 1 1
1
, ln 1 1 1
c
c
c
v
tv v
(31)
Equation (31) directly gives the normalized charging time
t
as a function of the normalized voltage
c
v
across the
capacitor terminals, where
is the parameter of the equation.
As follows from (26), the voltage across the capacitor
asymptotically approaches the steady-state value, which is
equal to the input voltage
1
V
, and the transient duration is
theoretically infinite.
Let us define the rise time
r
t
as the time required for the
capacitor voltage to rise from
10%
to
90%
of its final value,
which is equal to
1
V
. Hence, the normalized voltage
c
v
rises
from
0.1
to
0.9
. The normalized rise time
r
t
is then
,0.9 ,0.1
r
t t t
(32)
Fig. 5. presents the normalized rise time as function of
,
given by (32). As shown in this figure, the rise time increases
with increasing
, or with decreasing the voltage
1
V
.
Fig. 5. Plot of the normalized rise time
r
t
versus
.
B. Energy Stored in the Capacitor
The instantaneous power in the capacitor is
c c c
p v i
(33)
Substitution of (5) in (33) gives
c
c c c
dv
p C v v dt
(34)
The energy stored in the capacitor is given by
c
c c c c
tt
dv
E p dt C v v dt
dt
(35)
Hence
c
c c c c
V
E C v v dv
(36)
Substitutions of (5) in (36) yields
c
o bi
c c c
Vbi c
CV
E v dv
Vv
(37)
Integrating (37) in the interval
1
0,V
we find
332
1
1
3
22
3
bi
c o bi bi bi
bi
V
E C V V V V
VV
(38)
C. Energy Supplied by the Voltage Source
The instantaneous power in the voltage source
1
V
is
11c
p V i
(39)
Substitution of (5) in (39) gives
11 c
c
dv
p V C v dt
(40)
The energy supplied by the voltage source is given by
1 1 1 c
c
tt
dv
E p dt V C v dt
dt
(41)
Hence
11
ccc
V
E V C v dv
(42)
Substitutions of (5) in (42) yields
11
c
o bi
c
Vbi c
CV
E V dv
Vv
(43)
Integrating (43) in the interval
1
0,V
and rearranging the
terms we find
1 1 1
2o bi bi bi
E C V V V V V
(44)
Equation (44) gives the total energy transferred from the
voltage source
1
V
to the pair
RC
during the charging of the
voltage-dependent capacitor.
D. Energy Dissipated in the Resistor
The energy
R
E
dissipated in the resistor
R
is the difference
between the energy supplied by the voltage source
1
V
and the
energy stored in the capacitor, and is defined by
1Rc
E E E
(45)
Substitution of (38) and (44) in (45), after appropriate
algebraic manipulation, yields
1 1 1
22
3
R o bi bi bi bi bi bi
E C V V V V V V V V V
(46)
E. Ratio of Energy Dissipated in the Resistor and the Energy
Stored in the Capacitor
Let us define
as the ratio of the energy
R
E
dissipated in
the resistor to the energy
c
E
stored in the capacitor. Thus,
R
c
E
E
(47)
Substituting (38) and (46) in (47) and rearranging the terms,
we find
11
11
2bi bi bi
bi bi bi
V V V V V
V V V V V
(48)
Substitution of (23) in (48) yields
21
11
(49)
Fig. 6. Ratio of the energy dissipated in the resistor to the energy stored in the
voltage-dependent capacitor, as function of
.
According to equation (49), the ratio of the energy
dissipated in the resistor
R
to the energy stored in the
capacitor is independent of the values of
R
and
o
C
, being
dependent only on the ratio of
1
bi
VV
.
A plot of
versus
, given by equation (49), is shown in
Fig. 6, which shows that, in opposition of what occurs in the
charging of a constant capacitance capacitor, the energy
dissipated in the resistor is always larger than the energy
stored in the capacitor. This phenomenon is attributed to the
voltage-dependence of the capacitor.
IV. ANALYSIS OF THE DISCHARGING TRANSIENT
OF THE VOLTAGE-DEPENDENT CAPACITOR
Since we are dealing with nonlinear systems, the principle
of superposition will not hold. Therefore, the time dependence
of charging voltage will differ from that of the corresponding
discharge voltage, and the two cases must be considered
separately.
The circuit that we shall consider is shown in Fig. 7.
Fig. 7. Circuit for discharging the voltage-dependent capacitor through a
linear resistor.
The current
C
i
is given by
()Cc
CC
dv v
i C v dt R
(50)
Substitution of (5) in (50) yields
o bi CC
bi c
CV dv v
dt R
Vv
(51)
Hence,
C
c bi c o bi
dv dt
v V v RC V
(52)
and
C
c bi c o bi
dv dt K
v V v RC V
(53)
where K is the constant of integration.
Integrating both terms of (53) we obtain
1 1 1 1 1
ln ln
bi bi c bi bi c bi
o bi
V V v V V v V
tK
RC V
(54)
Making
0t
and
0
co
vV
in (54) we obtain the constant
of integration
K
, which is given by
1 1 1 1 1
ln ln
bi bi o bi bi o bi
KV V V V V V V
(55)
Substituting (55) in (54) and rearranging the terms, we find
ln
c bi bi
c bi bi
o
o bi bi
o bi bi
v V V
v V V t
RC
V V V
V V V
(56)
Let us define the normalized quantities
t
,
1
and
c
v
as
o
t
tRC
(57)
1bi
o
V
V
(58)
and
c
c
o
v
vV
. (59)
Substitution of (57), (58) and (59) in (56) and appropriate
algebraic manipulation yields
2
1 1 1 1 1 1
1
1 1 1 1
11
11
t
ct
e
ve
(60)
where
c
v
is the normalized voltage across the voltage-
dependent capacitor, during the discharge transient.
Fig. 8. Decay transient of the normalized voltage across the voltage-
dependent capacitor versus normalized time, for: (a)
10.010
, (b)
20.030
and (c)
10.060
.
The plot of equation (60), as shown in Fig. 8, indicates that
the decay time of the voltage across the capacitor depends on
the parameter
1bi
o
VV
. The higher the value of
1
, which
means lower value of the initial voltage
o
V
across the
capacitor, the longer the decay time. This characteristic is due
to the voltage-dependence of the capacitance. It should be
noted that if the capacitance were independent of the capacitor
voltage, the decay time would no longer be dependent on the
capacitor initial voltage.
A. Discharging Time
From (60) we can write
1 1 1 1 1 1
1
1 1 1 1
11
11
t
c
t
ev
e
(61)
We can find an explicit expression for the normalized time
f
t
by solving (61) and replacing
c
v
by
1c
v
which yields
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1
ln 1 1 1
c
f
c
v
tv
(62)
Equation (62) directly gives the normalized discharge time
f
t
as a function of the normalized voltage across the
capacitor terminals, where
1
is the parameter of the equation.
As follows from (62), the voltage
c
v
across the capacitor
asymptotically approaches the steady-state value, which is
equal to zero, and the transient duration is theoretically
infinite.
Let us define the fall time
f
t
as the time required for the
capacitor voltage to fall from
90%
to
10%
of its initial value,
which is equal to
1
V
. Hence, the normalized voltage
c
v
falls
from
0.9
to
0.1
. The normalized fall time
f
t
is then
1 1 1
,0.1 ,0.9
f f f
t t t
(63)
Curves computed from (63) for several values of
1
are
presented in Fig. 9.
Fig. 9. Plot of the normalized fall time
f
t
versus
1
.
B. Ratio Between the Normalized Rise and Fall Times
To compare the normalized times
r
t
and
f
t
, both are
plotted in Fig. 10, where we note that the fall time is always
larger than the rise time. These curves show clearly how the
nonlinearity due to the voltage-dependence of the capacitance
results in strong differences between charging and discharging
times.
Fig. 10. Plot of the normalized fall times
f
t
and
r
t
, and the ratio
between them, versus
.
V. EXPERIMENTAL RESULTS
In order to experimentally verify the validity of the
theoretical analysis results, an experiment was carried out in
the laboratory with the experimental circuit shown in Fig. 11.
Fig. 11. Experimental
RC
circuit using the Schottky diode STPS10L25 as a
voltage-dependent capacitor.
The Schottky diode STPS10L25 from STMicroelectroncis
was used as the voltage-dependent capacitor, which was
associated in series with a resistor
R
. The parameters of the
diode are
0.326
bi
VV
and
4.5
o
C nF
[7].
A rectangular voltage
1
V
with appropriate frequency,
amplitude and duty-cycle was generated from the Tektronix
AFG 1022 function generator, which internal impedance is
equal to
50
. The voltages
C
v
and
1
V
were measured using
the MDO 3014 Tektronix oscilloscope.
The measured charging voltages
expC
v
and
1
V
versus time,
with
3Rk
for
110VV
and
15VV
, are shown in Figs.
12 and 13, respectively, and the corresponding discharging
voltages are plotted in Figs. 14 and 15. In the same figures,
the theoretical voltage across the capacitance of the diode
Ctheor
v
is represented by solid black symbols.
As figures show, the experimental and theoretical results
are nearly identical and the differences can be attributed to the
internal impedance of the function generator and the diode
parasitic parameters, such as leakage current and series
resistance, not included in the analysis.
It should be noted that, as predicted theoretically, the
experimental charging and discharging times are sensitive to
the value of the voltage
1
V
. The time interval for the capacitor
voltage to reach
1
0.9
C
vV
is equal to
10 s
when
110VV
and
16 s
when
15VV
. The discharging times are
6s
when
10
o
VV
and
8s
for
5
o
VV
.
Fig. 12. Experimental results during the capacitor charging when
110VV
.
(a) Voltage
1
V
generated by the function generator
2V div
. (b)
Experimental voltage
expc
v
across the Schottky diode
2V div
. (c)
Theoretical voltage
ctheor
v
across the voltage-dependent capacitor
2V div
.
Fig. 13. Experimental results during the capacitor charging when
15VV
.
(a) Voltage
1
V
generated by the function generator
1V div
. (b)
Experimental voltage
expc
v
across the Schottky diode
1V div
. (c)
Theoretical voltage
ctheor
v
across the voltage-dependent capacitor
1V div
.
Fig. 14. Experimental results during the capacitor discharging when
110VV
. (a) Voltage
1
V
generated by the function generator
2V div
.
(b) Experimental voltage
expc
v
across the Schottky diode
2V div
. (c)
Theoretical voltage
ctheor
v
across the voltage-dependent capacitor
2V div
.
Fig. 15. Experimental results during the capacitor discharging when
110VV
. (a) Voltage
1
V
generated by the function generator
2V div
.
(b) Experimental voltage
expc
v
across the Schottky diode
2V div
. (c)
Theoretical voltage
ctheor
v
across the voltage-dependent capacitor
2V div
.
VI. CONCLUSIONS
In this paper we have presented theoretical and
experimental study on the step function charging and
discharging of Schottky diode nonlinear voltage-dependent
capacitors, through a linear resistance. The circuits are
described by first-order nonlinear differential equations,
which closed form exact solution are presented.
From the performed study, we can draw the following
conclusions: (a) the energy loss in the series resistor during
charging is larger than the total energy transferred to and
stored in the nonlinear capacitor, (b) the fall time is larger than
the rise time, and (c) both the rise time and fall time are
sensitive to the voltage value, decreasing with increasing
voltage. These results indicate how the nonlinearity due to the
voltage-dependence capacitances causes substantial
differences between charging and discharging behavior.
A similar analysis can be extended to other voltage-
dependent capacitors, such as ceramic capacitor, conventional
P-N junction diodes capacitors and input and output
capacitances of the MOSFET. The theoretical analysis results
were verified by computer simulation and laboratory
experimentation.
The results obtained in this study can be directly applied to
determine the time intervals of the gate source and drain
source voltages of power semiconductors during the
commutation in static converters.
In the continuation of this research, parasitic inductances
present in the switching loops of static converters will be
included, which will allow the determination of the values of
voltage peaks across the diodes at the moment they are turned
off.
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BIOGRAPHY
Ivo Barbi (Life Fellow, IEEE) was born in Gaspar, Brazil. He
received the B.S. and M.S. degrees in electrical engineering
from the Federal University of Santa Catarina (UFSC),
Florianópolis, Brazil, in 1973 and 1976, respectively, and the
Dr. Ing. degree in electrical engineering from the Institut
National Polytechnique de Toulouse (INPT), Toulouse,
France, in 1979. He founded the Brazilian Power Electronics
Society (SOBRAEP), the Brazilian Power Electronics
Conference (COBEP), in 1990, and the Brazilian Power
Electronics and Renewable Energy Institute (IBEPE), in 2016.
He is currently a Researcher with the Solar Energy Research
Center and a Professor Emeritus in electrical engineering with
UFSC. Prof. Barbi received the 2020 IEEE William E. Newell
Power Electronics Award. He served as an Associate Editor
for the IEEE TRANSACTIONS ON INDUSTRIAL
ELECTRONICS and the IEEE TRANSACTIONS ON
POWER ELECTRONICS for several years. He is currently
Editor for the electrical engineering area of the SCIENTIFIC
REPORTS JOURNAL (Springer-Nature) and Associate Editor
of the JOURNAL OF CONTROL, AUTOMATION AND
ELECTRICAL SYSTEMS.
