The Closed-Loop Control for Dual-Output Boost Converter
with Single Inductor
Yaw-Shih Shieh (謝曜式), Jyun-Hong Chen (陳俊宏) and Tze-Yun Sung (宋志雲)
Department of Microelectronics Engineering, Chung Hua University
No. 707, Sec. 2, Wufu Rd., Hsinchu, 30012, Taiwan, R.O.C.
Tel: 03-518-6397 Fax: 03-518-6891
Email: ysdaniel@chu.edu.tw, jyunhong.chen@delta.com.tw, bobsung@chu.edu.tw
Abstract
The power supply is a very common component in the electronic products. For the sake of reducing the size
of product, how to design a power supply providing multiple output voltage levels with the smaller volume and
light weight is an important issue. Owing to the volume and weight of the power supply being dominated by
inductors, the number of inductors should be minimized. In this paper, the topology in which a dual-output boost
converter with single inductor is developed. Due to the nonlinearity of the PWM switching circuit, it is very
difficult to analyze the behavior and performance of switching converter exactly. The study utilizes the algorithm
that the PWM switching circuit is approximated by its time-average model, to obtain the state equations of
switching converter. Based on the developed model, a closed-loop controller is proposed to regulate both voltage
outputs simultaneously. Finally, the validations of the proposed model and controller are verified by the
experiments and computer simulations.
Keywords: time-average model, dual-output boost converter, closed-loop controller
1. Introduction
The
DC
converters
are
topics of converter.
the
necessary
research
Hence, it is worth the effort to
continuously.
In
this
paper,
the
components for many modern electronic products
time-average model, based on the large-signal
such as mobile phone, LCD TV, etc.
The main
approximation during a switching period [1-4], is
requirements of the converter are small volume,
utilized to model the switching circuit of converter in
multiple output voltages with accurate levels.
order to reduce the difficulty of dealing nonlinear
As
the results, how to reduce the number of inductors
nature of switching.
and reduce the volume of inductor are the essential
model is nonlinear, at steady state, the controller can
issues.
be designed carefully to regulate the output voltages.
In this paper, the proposed model and
converter with minimizing the number of inductors is
presented.
Although the time-average
There were some closed-loop control schemes
proposed [5-8] for switching circuits.
In this paper,
Basically, the switching converter is a nonlinear
the modified macro variable method [8], with respect
system due to the nature of switching in power
to the time-average model of switching circuit, is
circuits and the MIMO system for multiple output
proposed.
voltages levels.
is considered. The time-average model of the
It is a big and interesting challenges
to model, analyze, design and implement.
There
were many researchers who contributed in those
Firstly, the single-output boost converter
switching circuit is derived and the proposed control
algorithm is thus applied.
Computer simulations
and experimental results show that the proposed
voltages, is
control algorithm can improve the performance of
converter easily.
Vo
1

Vi 1  d
Moreover, the results obtained for
(1)
single-output boost converter are extended to the case
where 0  d  1 [9].
of dual-output boost converter with the some
to determine the output voltage of converter under
modifications for different structures of switching
the
circuit.
The proposed control algorithm can
performance of converter, the closed-loop control
improve the system performance significantly, which
strategy should be used. However, equation (1) is not
is
adequate for closed-loop control. Based on averaging
proved
by
the
circuit
simulations
and
implementations.
open-loop
Equation (1) is usually used
control.
For
improving
the
the node voltages and currents of branches during a
This paper is organized as follows.
The
switching period, the large-signal time-average model
operating theorems of the single-output boost
will be found and applied on the analysis and design
converter and the time-average model are presented
of closed-loop control of boost converter [1].
in section 2.
The proposed topology of the
There are two switches in the basic boost
dual-output boost converter and the corresponding
converter, shown in Figure 1, in which power
time-average model are derived in section 3. The
transistor Q is the active switch controlled by VG and
closed-loop controls with the modified macro
diode D is the passive switch. Terminals a, p and c
variable method for single and dual outputs are show
form
in sections 4 and 5, respectively.
time-average model of switching circuit are depicted
Finally, the
a
switching
circuit.
The
variables
of
computer simulations and experimental results are
in Figure 3.
given in section 6.
shown in Figure 2, these variables can be found as
2. Single-output boost converter with
time-average model of switching circuit
According to switching conditions
follows:
(a) For 0  t  dTs
ia (t )  ico (t )
(2)
converter is examined to sure that the time-average
i p (t )  0
(3)
model is suitable for analyzing the steady-state
Vcp (t )  Vap (t )
(4)
performance of the nonlinear switching circuits.
Vac (t )  0
(5)
The time-average model for single-output boost
The basic boost converter is shown in Figure 1 where
(b) For dTs  t  Ts
VI is DC voltage source, Q is switch, power transistor,
ia (t )  0
(6)
L is inductor for energy transfer, and C, capacitor,
i p (t )  ico (t )
(7)
and R, resister, present load impedance.
Vcp (t )  0
(8)
Vac (t )  Vap (t )
(9)
Under the
steady-state condition, the timing diagram of Q is
shown in Figure 2, in which T is switching period, tON
From equations (2)-(9), the variables of switching
is the turn-on time interval of Q and d is thus the duty
circuit can be determined in term of ico and Vap as
cycle of Q.
Under the continuous-conduction mode, the
ia   d ico 
(10)
i p   d ' ico 
(11)
Vcp   d Vap 
(12)
steady-state relationship, found by voltage-second
balance equation, between the input and output
Vac   d ' Vap 
(13)
approximating
switching
circuit
consideration,
because the power transfer of each output is
where d   1  d and “  ” denotes the average
determined by the corresponding power transistor,
value of variable.
and D1 and D2 are for avoiding reverse current.
For simplicity and clarity, we will
omit “  ” in the following derivation. Because ico is
The timing diagram of power transistors are
the negative of inductor current iL and Vap is the
shown in Figure 7 where Ts is switching period, d1, d2
negative of output voltage, Vo or Vc the variables of
and d3, d1+d2+d3=1, are turn-on time intervals of Q1,
switching circuit can be expressed in terms of
Q2 and Q3, respectively. And the switching circuit
inductor
depicted by terminals a1, a2, a3 and c can be
current
and
output
voltage.
The
time-average model of switching circuit is shown in
approximated by the time-average model.
Figure 4.
The boost converter in which the
variables of switching circuit are depicted and
switching circuit is represented by time-average
defined in Figure 8. The average values of variables
model is shown in Figure 5.
are
And the state equation
is
1 d 
VI 
L   iL    
(14)



1 Vc   L 

0

 
RC 
steady state, iL  0 and Vc  0 , the

 iL   0
    1  d
Vc  
 C
At

relationship between DC voltage source and output
voltage is
ia1   d1 ico 
(16)
ia2   d2 ico 
(17)
ia3   d3 ico 
(18)
Vca2   d1 Va1a2   d3 Va3 a2 
(19)
Vca3   d1 Va1a3   d 2 Va2 a3 
(20)
Va1c   d 2 Va1a2   d3 Va1a3 
(21)
The time-average model of the switching circuit is
Vo
1

Vi 1  d
(15)
The result of equation (15) is the same as that of
equation (1).
The
It is clear that the time-average model
can analyze the switching circuit of boost converter.
3. Dual-output boost converter with
time-average model of switching circuit
As previous results that the time-average model
represents the switching circuit for single-output
shown in Figure 9.
For using state equation to analyze and design the
converter, variables in equations (16)-(21) must be
expressed in terms of states, current of inductor and
voltages of capacitors, as follows:
ico  iL
(22)
Va1a2  Vc1
(23)
Va3 a2  Vc2  Vc1
(24)
Va1a3  Vc2
(25)
boost converter, the dual-output boost converter with
Replacing the switching circuit, depicted by terminals
some modifications for the different structure of
a1, a2, a3 and c in Figure 6, by the time-average
switching circuit can be used by time-average model.
model shown in Figure 9, the time-average based
The schematic of dual-output boost converter is
dual-output boost converter is shown in Figure 10
shown in Figure 6 where VI is DC voltage source, L
and the state equation is.
is inductor for energy transfer, Q1, Q2 and Q3 are
active switches controlled by gate voltages, and Ci
and Ri, i=1,2, are the load impedances of outputs,
respectively. The switching circuit is composed by Q1,
Q2 and Q3.
Diodes D1 and D2 do not be taken into

 0
 iL  
    d2
Vc1    C
V   1
 c2   d 3
 C2
d2
L
1

R1C1

0
d3 

 VI
L   iL  
L
 
0  Vc1    0


 
1  Vc 2   0


R2 C2 



 (26)



In equation (26), d1, d2 and d3 are the duty cycles
of power transistors.
The steady-state relationships
respectively, and k is a weighting factor.
The
control law is chosen as
T    0
between DC voltage source, VI, and output voltages
(30)
where T  0 is a design parameter to determine the
are
Vc1 
Vc 2 
d 22
d2
VI
 d 32
(27)
d 22
d3
VI
 d 32
(28)
convergent rate of macro variable.
Because of the
positive value of macro variable, the control law will
force macro variable to be zero, thus x1= x1ref and x2=
x2ref as time approaches to infinite.
In the case of open-loop control, the values of
output voltages can be determined by the values of d1,
d2 and d3, in equations (27) and (28).
For the given specification of output power, x1ref
and x2ref are related as
For
x1ref 
closed-loop control, the values of d1, d2 and d3 are
x22ref
(31)
VI R
determined by controller and fed to the converter
The power delivered from DC power source has to be
described by equation (26) in order to regulate the
greater than the power absorbed by the load.
output voltages of converter.
The converter
clear that the inductor is the intermediate stage for
described in equation (26) is a MIMO nonlinear
power transfer from the DC power source to the load,
system due to the system matrix containing control
the energy stored in the inductor thus has to be large
inputs, d1, d2 and d3.
enough for the requirement of load.
equation
4. Closed-loop control of single-output
boost converter
The block diagram of the control system for the
single-output boost converter is shown in Figure 11,
where Vref is the desired output voltage, Vout is the
actual output voltage, d is the control input to the
converter and controller is a regulator that generates
d from difference of voltages, e.
The problem is to
design the controller such that Vout approaches Vref as
(31)
satisfactory
is
the
regulation
necessary
of
It is
Criterion of
condition
single-output
of
boost
converter under closed-loop control.
From equations (14), (29) and (30), the duty
cycle can be represented as
kVI e1 x2 e2 ke12  e22


RC
2T
d  1 L
kx2 e1 x1e2

L
C
(32)
where e1=x1-x1ref and e2=x2-x2ref are the errors of
inductor current and output voltage, respectively.
closely as possible.
For the nonlinearity of converter, the analysis
of system and design of controller are very difficult
and still an open problem.
In this paper, we
proposed the control scheme [7] and the macro
variable is represented as
  ( x2  x2 ref ) 2  k ( x1  x1ref ) 2 ,
(29)
For the given values of parameters, once the inductor
current and output voltage are measured and
converted via ADC(Analog to Digital Converter), d
can be determined digitally by microprocessor.
5. Closed-loop control of dual- output
boost converter
The block diagram of the closed-loop control
where states x1 and x2 are the inductor current and
system for the dual-output converter is shown in
output voltage, respectively, x1ref and x2ref are the
Figure 12 where V1ref and V2ref are the desired output
desired
voltages, V1out and V2out are the actual output voltages,
inductor
current
and
output
voltage,
d1, d2 and d3 are the duty cycles of power transistors
D
k2 x3 E1 x1 E3

L
C2
(43)
E
k 2 x2 E1
L
(44)
and e1 and e2 are the voltage error of each outputs.
The controller is a regulator which generates duty
cycles from voltage errors.
The problem is to
design the controller such that each output voltage
F
approaches its desired voltage as closely as possible.
k2V1 E1 x3 E3 k2 E12  E32


L
R2C2
2T
(45)
The results of the controller design for
E1  x1  x1ref
(46)
single-output boost converter is extended to the
E2  x2  x2ref
(47)
dual-output boost converter.
The macro variables
for dual-output converter are defined as
For the given values of parameters, once the inductor
current and output voltage are measured and
 1  ( x2  x2 ref ) 2  k1 ( x1  x1ref ) 2
(33)
 2  ( x3  x3ref ) 2  k1 ( x1  x1ref ) 2
(34)
converted via ADC(Analog to Digital Converter), d1,
d2 and d3 can be calculated from equations (37)-(39)
digitally by microprocessor.
From the specifications of each output, x1ref,
where states x1, x2 and x3 are the inductor current and
x2ref and x3ref have to meet following constrain so as to
output voltages of each output, respectively, x1ref,
ensure the power transfer.
x2ref and
x3ref are the desired inductor current and
x22ref
output voltages of each output, respectively, and k1
x1ref 
and k2 are the weighting factors of inductor currents
for each macro variable.
R1

X 32ref
R2
VI
(48)
The control laws are
6. Simulations and experimental results
chosen as
T 1   1  0
(35)
In this section we will show the computer
T 2   2  0
(36)
simulation and experimental results to verify the
The duty cycles can be found by equations (26),
usefulness of the proposed closed-loop control
algorithm.
(33)-(36) and represented as follows:
For the single-output boost converter, the
d1 
DC  FB  AF  CE  AD  EB
AD  EB
(37)
d2 
FB  DC
AD  EB
(38)
d3 
CE  AF
AD  EB
(39)
experimental results are shown in Figure 13 and
Figure 14 for different settings of x2ref = 6 v and 10 v,
respectively.
The parameters of circuit are VI = 5 v,
L = 6000 h, C = 6600f, R = 2000 , T = 0.1 and k
= 0.5. The switches Q is of IRF840 N type MOSFET.
D is of IN404 diode. The switching frequency is 10
where
A
k1 x2 E1 x1 E2

L
C1
k x E
B 1 3 1
L
kVE
x E
k E 2  E22
C 1 1 1 2 2  1 1
L
R1C1
2T
kHz. Inductor current is measured by current probe
(40)
and output voltage is measured by voltmeter.
In Figures 13 and 14, channel 1 and channel 2
(41)
are the waveforms of duty cycle and VG, respectively,
and channel 4 presents the waveform of inductor
(42)
currents at steady-state condition. The actual output
voltages are 6.02 and 10.03 v, respectively, which are
“PWM-Switch Modeling of DC-DC Converter,”
very close to the desired voltages.
For the dual-output boost converter, the values
IEEE Transactions on Power Electronic, vol. 10,
of parameters are the same as those for single-output
boost converter except for T = 0.01 and k1 = k2 = 0.5.
no. 6, pp. 659-664, 1995.
[2] C. T. Rim, G. B. Joung and G. H. Cho, “A State
In addition, Q1 is of IRF840 N type MOSFET, and Q2
Space
and Q3 are of IRF9530 P type MOSFET.
Converters,”
For x2ref = 10 v and x3ref = 8 v, the experimental
respectively.
of
Power
Non-Ideal
Electronics
DC-DC
Specialists
Conference, PESC 88, 19th Annual IEEE, vol. 2,
results are shown in Figure 15 and the computer
simulations are shown in Figures 16 and 17,
Modeling
pp. 943-950, 1998.
[3] Tang W., Lee F. C., Ridley R. B., “Small-Signal
In Figure 15, channel 1, 2 and 3
Modeling of Average Current-Mode Control,”
present waveforms of duty cycles d1, d2 and d3, and
Applied Power Electronics Conference and
channel 4 presents the waveform of inductor current.
Exposition,
The actual output voltages are 9.24 and 7.18 v.
1992, pp. 747-755, 1992.
The
simulations of the first and second output voltages
APEC 92, Conference Proceedings
[4] Ridley R. B., ”A New, Continuous-Time Model
for Current-Mode Control,” IEEE Transactions
are shown in Figures 16 and 17, respectively.
According to parasitic resistances and turn-on
on Power Electronics, vol.6, no.2, pp. 271-280,
voltage drops in the power transistors and diodes, the
differences of results between the circuit simulations
1991
[5] Javier Sebastian and Javier Uceda, The Double
Converter ： A Fully Regulated Two Output
and hardware implementation are occurred.
DC-DC Converter,” IEEE Transactions on Power
7. Conclusions
Electronic, vol. PE-2, no. 3, pp. 239-246, 1987.
This paper proposes a control algorithm, based
on
modified
macro
variable,
by
which
[6] Marian K. Kazimierczuk, Nehru Sathappan and
the
Dariusz Czarkowski, “Voltage-Mode-Controlled
dual-output boost converter, using single inductor,
can be regulated under closed-loop control.
PWM
Besides,
Converter
with
A
IEEE National, vol. 1, pp. 413-419, 1993.
[7] Enrico Santi, Antonello Monti, Donghong Li,
switching circuit of dual-output boost converter
adequately.
DC-DC
Proportional Controller,” Proceedings of the
because the nature of switching circuit is nonlinear, a
time-average model is used to approximate the
Buck
Karthik
Proddutur,
and
Roger
A.
Although the approximated model is
Dougal, ”Synergetic Control for DC-DC Boost
still nonlinear system with multiple outputs, the
Converter ： Implementation Options,” IEEE
proposed control algorithm is suitable to regulate
Transactions on Industry Applications, vol. 39,
converter practically.
Finally,
from
no. 6, pp. 1803-1813, 2003.
computer
simulations
and
[8] Y. He and F. L. Luo, “Study of Sliding-Mode
experimental results, the proposed control scheme is
Control for DC-DC Converters,” International
verified.
Conference on Power System Technology, pp.
1969-1974, 2004.
References
[1] Edwin Van Dijk, Herman J. N. Spruijt, Dermont
M.
O’Sullivan
and
J.
Ben
Klaassens,
[9]
Mohan,
Undeland
and
Robbins,
Power
Electronics, 2nd edition, John Wiley and Sons,
New York, 1995.
dVC
L
[10] Hong-Ming Chen, “Dual-output boost converter
+
- +
iL
using single inductor,” These for the master
VI
+
diL
VC
C
R
VO
-
degree of institute of electrical engineering,
-
Chung Hua University, 2004.
[11] “AIC1721/1721D 150mA/300mA Low Drop
Out
Linear
Regulator
With
1%
Figure 5 The single-output boost converter with
time-average model
Output
Accuracy,” Analog Integrations Corporation,
iL
Oct 2003.
Q2
c
L
a2
D1
+
iC1
R1
C1
+ VL Q1
VI
[12] “AIC1340 High Performance, Triple-Output,
iO1
+
Q3
a1
Auto-Tracking Combo Controller,” Analog
a3
D2
iC2
iO2
C2
R2
Integrations Corporation, Oct 2003.
iL
L
+ VL VI
D
c
iQ
VO2
-
Figure 6 Dual-output boost converter
p
+ VD - iD
Q
VG
VO1
+
iC
iO
C
R
Q1
d1 TS
VO
Q2
a
-
d2 TS
Q3
d3 TS
TS
iL
Figure 1 Single-output boost converter
i1
i2
VG
i0
i0
t
0
tON
T
0
t
T
dT
Figure 7 The waveforms of switching signals
of Q1, Q2 and Q3 in figure 6
Vca 2
Figure 2 The waveform of switching signal of Q in
Figure 1
a2
ia 2
ico
d2 TS
c
d3 TS
ia 3
Va1 a2
Vcp
d1 TS
d′T
c
p
ico
ia1
Va1c
ip
a3
Vca3
Va1a3
dT
Vac
Vap
a1
ia
a
Figure 3 The topology of switching circuit of
single-output boost converter
Figure 8 The topology of switching circuit of
dual-output boost converter
d1V a1a2 +d3Va3 a2
ico
ic
dVap
c
+ -
p
a2
+ -
a3
d1V a1a3 +d2V a2a3
d1 ic
dic
+ -
c
Vap
a
Figure 4 The time-average model of switching circuit
of single-output boost converter
a1
Figure 9 The time-average model of switching circuit
of dual-output boost converter
-
d2 iL
L
+
+
d1VO1+d3(VO1-VO2)
VO1
C1
R1
iL
VI
d1iL
-
d3 iL
+
+
d1VO2+d2(VO2-VO1)
C2
R2
VO2
-
Figure 10 The dual-output boost converter with
time-average model
Vref +
e
controller
d
converter
Figure 15 Experimental results of dual-output
converter with parameters x1ref = 16 mA,
x2ref = 10 v, x3ref = 8 v, VI = 5 v, T = 0.01
and k1 = k2 = 0.5
Vout
-
Figure 11 The block diagram of the closed-loop
control of single-output boost converter
V1ref +
V2ref +
d1
-
eO1
eO2
controller
d2
d3
Vout1
converter
Vout2
-
Figure 16 Simulations of the first output of
dual-output converter with parameters
x1ref = 16 mA, x2ref = 10 v, x3ref = 8 v,
VI = 5 v, T = 0.01 and k1 = k2 = 0.5
Figure 12 The block diagram of the closed-loop
control of dual-output boost converter
路示意圖
Figure 17 Simulations of the second output of
dual-output converter with parameters
x1ref = 16 mA, x2ref = 10 v, x3ref = 8 v,
VI = 5 v, T = 0.01 and k1= k2 = 0.5
Figure 13 Experimental results of single-output
converter with parameters x1ref = 3.6 mA,
x2ref = 6 v, VI = 5 v, T = 0.1 and k = 0.5
Figure 14 Experimental results of single-output
converter with parameters x1ref = 10 mA,
x2re f = 10 v, VI = 5 v, T = 0.1 and k = 0.5