Third order converters with current output for driving LEDs
FELIX A. HIMMELSTOSS
Faculty of Electronic Engineering and Entrepreneurship
University of Applied Sciences Technikum Wien
Hoechstaedtplatz 6, 1200 Wien
AUSTRIA
Abstract: - Light emitting diodes LEDs are highly efficient in changing electrical energy into light. The
applications are lightning, but also medical treatments, and disinfection. After a short discussion of the simplest
converter which is based on the Buck converter several third order current converters, which are suitable to
drive an LED load, are treated. The Buck converter with input filter, the output current Boost converter, and the
output current Cuk converter are shortly treated, but for the other topologies simple control techniques are
given. These converters are the current output converter based on the Zeta converter, two converters with a
quadratic term of the duty cycle in the voltage transformation ratio, the quadratic step-down converter with
current output and the D square divided by one minus d current output converter, which is a step-up-down
converter, and two converters which function only for a limited duty cycle, the (2d-1)/d step-down and the (2d-
1)/(1-d) step-up-down output current converters. The dynamics of example converters is shown with the help of
LTSpice simulations.
Key-Words: - DC/DC converters, current output, Buck-, Boost-, Zeta-, quadratic-, limited duty cycle converter,
control, simulation, modelling
Received: November 19, 2021. Revised: October 19, 2022. Accepted: November 23, 2022. Published: December 31, 2022.
1 Introduction
Beside the main purpose of efficient illumination,
LEDs are used also for disinfection (UVC-LEDs cf.
[1]), and for medical treatments (IR-LEDs cf. [2]).
Most driver circuits which use switched mode
converters have a capacitor in parallel to the output
to smooth the current through the load. The
characteristics of LEDs are temperature-dependent
and change therefore. If the converter controls only
the output voltage, the current changes and the light
stream decreases, or what is even dangerous the
current increases and the light stream and the losses
and the temperature of the devices increase, too.
This can lead to an overload of the LEDs and to
their destruction. In this paper we show converters
which have a current output; the output voltage
depends therefore on the load and changes by a
constant output current. Now it is easy to avoid an
overload of the LEDs. The basic converters are
extensively treated in the text books [3-5]. A
comprehensive review over LED drivers can be
found in [6]. Many topologies for DC to DC
converters can be found e.g. in [7, 8].
The easiest concept is a first order converter
derived from the Buck converter. Fig. 1.a shows a
Buck converter without output capacitor. Input and
output have the same reference point. A
disadvantage is the floating switch. In many LED
applications, however, it will not be necessary to
have the output referred to the same ground (e.g. in
battery applications). Therefore, one can change the
position of the active and the passive switches
(Fig.1.b). Now the active switch is no more floating
and can be easily controlled.
Fig. 1. Buck derived current converter: (a) with
floating, (b) with grounded active switch
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To control the current a two-level controller with
hysteresis is a very effective solution. In [1] the
frequency of the controller (and therefore the
switching frequency of the converter) is calculated
according to
1
212
ULI
IRVUIRV
fLEDLEDLEDLED
(1)
where VLED and RLED describe the load with simple
characteristics, U1 is the input voltage, I2 the output
current in the mean, ΔI the value of the hysteresis,
and L the value of the coil. Change of the
parameters (temperature dependence of the load,
tolerance of the inductor) influence directly the
frequency and lead to a very robust control. The
basic frequency can be chosen for the working point
by the hysteresis ΔI, which directly fixes the current
ripple of the load current.
2 Third order converters with current
output
In this section an overview about some converters
with current output which can be used to drive an
LED-load are presented. They are treated with a
diode as second semiconductor device. To reduce
losses, the diode can be exchanged by a second
active switch, which needs, however, an additional
driver and a precise control. When the necessary
control electronics (and maybe also the active
switches) are implemented in an integrated circuit,
this leads to a very effective system.
In [9] third order current converters for LED-
loads which have also a continuous input current are
described. These converters are treated here only
shortly. These converters are very useful because
the input current is continuous and therefore only a
small capacitor is necessary to avoid the influence
of the parasitic inductance at the input.
2.1 Output current Buck with input filter
To avoid a pulsating input current, an additional LC
input filter can be connected in front of the
converter according Fig. 1 as shown in Fig. 2 for the
version according to Fig. 1.a.
U1
L2
D
L1
S1
C
U2
Fig. 2. Buck converter with input filter
When the input voltage is applied, an inrush current
occurs to charge the capacitor C. The ringing
occurring can be omitted by connecting a diode in
series to L1. The capacitor is in this case charged up
to nearly the double of the input voltage. This
additional diode increases the loss of the converter
and should be applied only when an additional
reverse polarity protection is desired. The control
can be done again with a two-level controller. More
details about the inrush current, the start-up, a state-
space model and a calculation of the transfer
function between the output current and the duty
cycle can be found in [9].
2.2 Output current Boost converter
The Boost converter has a continuous input current
and by connecting an additional inductor L2 at the
output, the current at the output is also continuous
(Fig. 3).
Fig. 3. Output current Boost converter
In [9] the converter is described more in detail
(inrush current, soft-start, nonlinear and linear state
space model, transfer function of the output current
in dependence on the duty cycle). Furthermore, a
nonlinear control concept is shown. With this
converter loads which need a higher voltage than the
input voltage can be supplied.
2.3 Output current Cuk converter
From its basics the Cuk converter has the possibility
of a continuous input and output current, one only
has to remove the output capacitor to obtain a
current converter. With this topology the load can
require a higher or a lower voltage than the input
voltage available. The inrush current charges the
capacitor to a voltage of nearly double the input
voltage, when a voltage source is applied to the
input connectors, the ringing (compared to the Buck
with input filter), however, is omitted by the diode
which disables a current in the back direction. In [9]
the inrush, the soft-start, a control law, a large and a
small signal model and the transfer function are
given.
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Fig. 4. Output current Cuk converter
2. Output current Zeta converter
The advantage of this converter shown in Fig. 5 is
the fact that the position of the transistor prevents an
inrush current. So the start-up can be done by slowly
increasing the duty cycle until it reaches the
necessary value. In Fig. 6 the start-up and steps of
the duty cycle are shown which result in some
pronounced ringing. Fig. 7 depicts the current
through the coils, the input and the output voltages
and the pwm-signal which controls the active
switch. The parameters of the converter can be
taken from the appendix Fig. A.1.
a.
b.
Fig. 5 Output current Zeta converter: (a) with
floating switch, (b) with low-side switch
Fig. 6. Output current Zeta converter, start-up and
duty-cycle steps: duty cycle (brown), currents
through L1 (red), L2 (black)
Fig. 7. Output current Zeta converter, steady state:
currents through L1 (red), L2 (black); input voltage
(blue), output voltage (green), control signal
(brown)
In the stationary case the output voltage and the
voltage across the capacitor are equal. The voltage
across the output of the converter can be calculated
with the help of the voltage-time balance across L1
)1(
1dUdU C (2)
to
12 1U
d
d
UU C
. (3)
The charge balance of the capacitor can be given as
)1(
1
_
2
_
dIdI LL . (4)
2..1 Nonlinear control
The derivation of the control law for the current
through L1 starts from the desired value of the
current through the load IL2ref to obtain the reference
value for the current through L1
1
,2
1
2
,2,2,1 1U
U
I
U
U
I
d
d
II C
refLrefLrefLrefL
. (5)
With a two-level controller the current through the
coil L1 can now be controlled and the current over-
shots are limited. Instead of the output voltage,
which has a ripple, the smoother voltage across the
capacitor is used. The simulation program is shown
in the appendix Fig. A.1. The current through the
coil L1 is measured by the current controlled voltage
source H1. The two-level controller is realized with
the comparator U2. The control law which produces
the desired value for the two-level controller is
calculated by the arbitrary voltage source B1. In a
practical realization this calculation would be done
by a micro-controller. Fig. 8 shows the results. The
current through L1 reacts immediately after steps in
the desired output current and also to a step in the
input voltage.
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Fig. 8. Output current Zeta converter: input voltage
(green), voltage across the capacitor (blue); current
through L1 (red); current through L2 (load current,
black)
Time constant of the system is T=1.8 ms and the
gain is one. The system can therefore be described
by the PT1 according to
10018.0
1
2
2
sI
I
refL
L . (6)
2.5 Quadratic step-down converter with
current output
When a high gap between the input voltage and the
lower load voltage has to be bridged, a series
connection of two Buck converters can be used. The
first one produces an intermediate voltage and has
an output capacitor, and the second one is a current
Buck converter according to Fig. 1. The complete
converter has therefore two inductors and one
capacitor as all converters in this chapter. The
output voltage of this converter is equal to the
square of the duty cycle (when both switches are
controlled with the same duty cycle) multiplied by
the input voltage. The converter can be easily
controlled by a two-level controller, which leads to
a very robust system.
Another possibility to achieve a high step-down
ratio of the square of the duty cycle is based on [10].
The converter shown in Fig. 9.a has only one low-
side switch. It should be mentioned that the diode
D1 can be exchanged by a second low-side switch
which is controlled with the same signal as S1. If a
connection between input and output is desired, the
position of diode and transistor has to be exchanged
as shown in Fig. 9.b. The duty cycle must start with
small pulses which are slowly increased to charge
the capacitor and so avoid an inrush current.
a.
b.
Fig. 9. Quadratic step-down converter with current
output: (a) with low-side switch, (b) with high-side
switch
The voltage-time balances across L1 and L2 can be
given by
L1:
)1(
1dUdUU CC (7)
L2:
)1(
22 dUdUUC (8)
which leads to
1
UdUC and 1
2
2UdU . (9)
The charge balance of the capacitor can be written
according to
)1(
1
_
2
_
1
_
dIdII LLL . (10)
This leads to the current through L1 in dependence
on the duty cycle and the output current to
2
_
1
_
LL IdI . (11)
The input current of the converter is pulsating and
has the mean value
2
_
2
_
LIN IdI . (12)
To avoid the pulsating input current, an LC filter as
in Fig. 2 could be attached. A detailed description of
the feedforward control of the d-square converter
with output capacitor can be found in [11].
2.6 d2/(1-d) current output converter
The d2/(1-d) converter is again based on [10] and is
depicted in Fig. 10. For the steady state one gets for
the voltage-time balance across L1
)1(
1dUdUC (13)
and for the voltage-time balance across L2
)1(
22 dUdUU C . (14)
This leads to
1
2
2
1
U
d
d
dUU C
. (15)
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The charge balance of the capacitor can be written
according to
)1(
1
_
2
_
dIdI LL . (16)
Fig. 10. d2/(1-d) current output converter
2.6.1 Feedforward control
Including the model of the load into (15)
1
2
2
1
U
d
d
IRVU LEDLEDLED
(17)
leads to a quadratic equation of the duty cycle
0
11
2
U
IRV
d
U
IRV
dLEDLEDLEDLEDLEDLED (18)
which results with the abbreviation
1
2U
IRV
kLEDLEDLED
(19)
In (20)
k
k
k
kkkkkd 2
11
2
12 22 .
Fig. 11. d2/(1-d) current output converter, up to
down: capacitor voltage (blue), input voltage
(green); current through L1 (red); current through L2
(black)
The used simulation circuit is shown in the appendix
Fig. A.2. The comparator U1 transfers the calculated
duty cycle with the help of a saw-teeth generator
into a pwm signal. The duty cycle is calculated with
(20) by the arbitrary voltage source B2. At the
beginning the capacitor has to be charged (with the
signal soft-start) to avoid overcurrent, because the
voltage at the capacitor starts with zero and
therefore the demagnetization voltage is too low to
avoid the overcurrent. The duty cycle starts with
zero and increases slowly. Fig. 11 shows the voltage
across the capacitor, the input voltage, and the
currents through the inductors. Each change in the
input voltage and in the desired load current leads to
a ringing. The soft-start happens within the first
20 ms.
2.7 (2 d-1)/d current output converter
The circuit can be obtained from [12] when the
second capacitor is deleted and is shown in Fig 12.
Fig. 12. (2 d-1)/d current output converter
Starting from the voltage-time balance across L1 one
gets
)1(
1dUdUC . (21)
The mean value of the voltage across the capacitor
must be
21 UUUC . (22)
The mean value of the output voltage can therefore
be calculated according to
12
12 U
d
d
U
. (23)
From this equation one can recognize that the duty
cycle must be greater than 0.5 (the duty cycle is per
definition between zero and one), otherwise the
voltage would change its direction (this is not
possible in this circuit, except when the diode is
exchanged by a second MOSFET or current
bidirectional switch). The circuit is a step-down
converter.
For the capacitor the charge balance is given by
)1(
2
_
1
_
dIdI LL . (24)
When the converter is controlled by a two-level
controller for the current through L1, the reference
value for L1 can be calculated from the desired
output current (the current through L2) and from the
measured input voltage and the voltage across the
capacitor according to
1
2
_
2
_
1
_
)
1
(U
U
I
d
d
IdI C
refLrefLrefL
. (25)
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Another possibility to control the converter is a
feedforward controller. The control law for the duty
cycle can be calculated from
1
12 U
d
d
IRV LEDrefLEDLED
(26)
according to
2
_
1
1
2L
LEDLED IRVU
U
d
. (27)
Fig. 13. (2 d-1)/d current output converter
feedforward controlled, up to down: desired load
current; current through L1 (red), input voltage
(green), voltage across the capacitor (blue); load
current (black).
Fig. 13 shows a feedforward controlled converter.
The inrush current effect is not shown. Steps in the
reference value and of the input voltage lead to
ringing especially of the current through L1. To
avoid reference-value-step-ringing, an integrator for
the command input can be used.
A disadvantage of this step-down converter is the
inrush current, when the input voltage is connected.
The inrush current can be described by the integral-
differential equation
iRVdti
Cdt
di
LLU LEDLED
t
0
211
1
)( . (28)
Laplace transformation leads to
)()(
11
)()( 21
1sIRsI
s
C
ssILL
s
VU
LED
LED
(29)
which results in
2
21
2
21
2
21
211
4
)(
1
2
)/(
)(
LL
R
LLCLL
R
s
LLVU
sI
LEDLED
LED
.(30)
The conjugate complex pole pair leads to a damped
ringing with the damping factor δ, and the angular
frequency ω of
21
2LL
RLED
,
2
21
2
21 4
)(
1
LL
R
LLC
LED
. (31)
Using these abbreviations results and
21
1
LL
VU
KLED
(32)
in (30) leads to
2
2
2
2
1
)(
s
K
s
K
sI (33)
and in the time domain according to
tt
K
i
sin)exp( . (34)
The ringing stops, when the current reaches zero
again (the load has a diode characteristics)
inrushinrush TTi
sin0)( (35)
which result to
ms6.1
1962
4
)(
1
0arcsin
1
2
21
2
21
LL
R
LLC
T
LED
inrush
. (36)
a.
b.
Fig. 14. (2 d-1)/d current output converter, up to
down: (a) input voltage (green), voltage across the
capacitor (blue); current through L2 (black), current
through L1 (red); (b) inrush current (red) calculated
according to (34)
The simulation shows a good correspondence with
the calculated result (Fig. 14). Keep in mind that the
parasitic resistances of the inductors and the diode
were idealized, but included in the simulation. The
results are sufficient. Fig. 14.a depicts the input
voltage, the charging of the capacitor, and the
current through the coils. Fig. 14.b shows the
calculated current (calculated by LTSpice with an
arbitrary current source).
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2.8 (2 d-1)/(1-d) current output converter
The circuit can again be obtained from [12] when
the second capacitor is deleted.
Fig. 15. (2d-1)/(1-d) current output converter
The voltage-time balances are
)1(
1dUdU C (37)
)1(
1
2
_
2
_
dUIRV
dUIRV
L
LEDLED
C
L
LEDLED
(38)
and lead to a voltage transformation ratio for the
output voltage (the voltage across the load) and to
the capacitor voltage
d
d
U
U
1
12
1
2 d
d
U
UC
1
1
. (39)
The converter is a step-up-down converter. In the
stationary case the duty cycle must be between 0.5
and 1.
For the capacitor the charge balance is given by
)1(
1
_
2
_
dIdI LL . (40)
With the desired value of IL2 one gets for the current
through L1
1
,2,21 1U
U
I
d
d
II C
refLrefLL
. (41)
This signal is used as the reference value for a two-
level controller.
Fig. 16 shows the load characteristics. The obtained
parameters are VLED=15.4 V, RLED=1.6 Ω (linearized
around 1.5 A).
Fig. 16. Characteristics of the load
2.8.1 Feedforward control
From (39) one can calculate the control law for the
feedforward controller resulting in
2
_
1
2
_
1
2L
LEDLED
L
LEDLED
IRVU
IRVU
d
. (42)
Fig. 17. (2d-1)/(1-d) current output converter, up to
down: current through L1 (red); reference value
(brown); input voltage (green), voltage across the
capacitor (blue), output current (black)
To avoid the inrush current one has to start the duty
cycle from zero, because the capacitor has to be
charged slowly. During the on-time the input
voltage lies across the coil L1, and during the off-
time the voltage of the capacitor, which is zero at
the beginning, lies across the coil, too. Therefore,
the demagnetization time must be high at the
beginning and the magnetization time short.
Fig. 17 shows the dynamics of the converter with
soft-start and reference value step with a two-level
converter. No ringing occurs.
3 Conclusion
LEDs are very efficient means to transform
electrical energy into light (UV, visible light, and
IR). The here treated converters can be controlled
with standard linear controllers as shown in the
textbooks for control engineering. An example to
design linear controllers for a Buck converter with
the help of the free simulation tool LTSpice is
shown in [13]. Here in this paper feedforward
control and a nonlinear control concept with a two-
level converter are applied. When the error of a
feedforward control is not tolerable, a small linear
controller which only has to compensate the small
difference between the desired current and the
actual value has to be included. The converters 2.1,
2.5, 2.7 can be used for higher input voltages
compared to the load voltage, the type 2.3 can be
used for lower input voltages and the converters 2.3,
2.4, 2.6 are able to handle lower and higher input
voltages compared to the necessary load voltage.
The question concerning the inrush current and the
soft-start of all types are discussed. It should be
mentioned that the current through the capacitor is
pulsating. For aerospace, traction and other reliable
applications, ceramic or film capacitors should be
used instead of electrolytic capacitors [14].
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Appendix
In the appendix two examples of the simulation circuits (LTSpice) are shown. The parameters for the other converters
have about the same values.
Fig. A.1. Output current Zeta converter: simulation circuit for the nonlinear control
Fig. A.2. d2/(1-d) current output conrter: simulation circuit for the nonlinear control
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