E12 - 2
2011 International Conference on Electrical Engineering and Informatics
17-19 July 2011, Bandung, Indonesia
Small Signal Averaged Model of DC Choppers for
Control Studies
Amir Hassanzadeh #1, Mohammad Monfared*2, Saeed Golestan§3, Reza Dowlatabadi #4
#
Sama technical and vocational training college, Islamic Azad University, Mashhad Branch
Mashhad, Iran
1
hassanzadeh82@gmail.com
reza.dowlatabadi@gmail.com
4
*
Department of Electrical Engineering, Ferdowsi University of Mashhad
Mashhad 91775-1111, Iran
2
§
m.monfared@ieee.org
Islamic Azad University, Abadan Branch
Abadan 63178-36531, Iran
3
s.golestan@ieee.org
Abstract— Todays, DC to DC power converters, known as DC
choppers, are widely employed in a variety of applications such
as power supplies, spacecraft power systems, hybrid vehicles and
DC motor drives. Among these converters, Buck, Boost, BuckBoost, Cuk and Sepic have found the most industrial
applications. This paper determines the small signal averaged
state-space models of these converters and provides an excellent
understanding of the frequency domain behaviour and a tool to
design and analyse the feedback control loops for them. The
theoretical results are validated by simulations in PSIM
software.
Keywords— DC chopper, small signal averaged model, control.
I. INTRODUCTION
DC choppers are widely used in power electronic
applications to serve as DC to DC power converters. Several
works have been conducted in order to determine the small
signal averaged state-space models of these converters [1-7].
Provided that the natural frequencies of the converter, as well
as the frequencies of variations of the converter inputs are
much lower than the switching frequency, then the small
signal averaged model is a valid representative of converter
performance in response to small AC variations about the
equilibrium operating point. Based on the control theory, such
a general model lets derive the expressions for the converter
control to output transfer function, input to output voltage
transfer function, input impedance, and output impedance, etc.
Based on this information, the controller design and
performance analysis can be easily accomplished using the
well-known control techniques such as Root loci and Bode
plot [6 ,7].
In this work, the small signal averaged state-space models
of open loop DC choppers such as Buck, Boost, Buck-Boost,
Cuk, and Sepic are obtained. Then, the small signal transfer
functions are readily derived from this AC small signal
models. The most important transfer function is the variation
978-1-4577-0751-3/11/$26.00 ©2011 IEEE
in output voltage in response to the variation in the duty cycle.
Simplified equations for crossover frequency and phase
margin are proposed. Each converter is simulated in PSIM to
validate the theoretical analysis. This paper is a reference for
valid AC models for various DC choppers which provides an
excellent understanding of the frequency domain behaviour
and a tool to design and analyse the feedback control loops for
these converters.
II. SMALL SIGNAL AVERAGED MODEL
The small signal averaged state-space method is a
generalized analysis tool which is readily applicable to either
simple circuits or complex structures [6, 7]. The linear
averaged time-invariant models achieved by using this method
are relatively simple, but a lot of mathematical efforts are
needed to derive the final results. To obtain such models, the
step-by-step procedure proposed in [6 ,7] is adopted to our
problem.
A. Buck Chopper
The schematic of the Buck converter is shown in Fig. 1.
Fig. 1 Buck converter
For the converter of Fig. 1, there are two distinct switching
states as follows:
1) The switch S conducts during the duty cycle (0 < t < DT):
PM   arctan(
4 LER 2 C  2 L2
)
2(1  E )CR 2  L
(7)
B. Boost Chopper
The modelling procedure is the same as the steps followed
for the Buck chopper. So, as a matter of simplicity, just the
final results are reported. The Boost converter is shown in Fig.
3. The small signal averaged state-space model is as follows:
 dvˆc   1
D '
 IL 
 dt   
 C 
  vˆc   0 
RC
C


(8)
 dˆ
     1  eˆ  
Vc 
iˆL   
 diˆL   D '



0
L
 dt    L
 L 

And the small signal transfer function of the output voltage
variation due to the duty cycle change is:
I
D'
Vc
 L s
vˆc ( s )
C
LC
(9)

s
D '2
dˆ ( s )
2
s 

RC LC
where D΄ = 1 – D. Comparing the Bode plots of Fig. 4, shows
that the analytical results are in good agreement with the
experiments.
20
Fig. 3 Boost converter
0
-20
40
3
30
4
10
10
0
pahse (deg)
The gain margin is infinite and the simplified relations for
the crossover frequency, ω0, and the phase margin, PM, are as
bellow:
E
1
(6)
0 
 2 2
LC 2 R C
Mag (dB)
Mag (dB)
vc
 dvc 1
 dt  C (iL  R )
(1)

 diL  1 (e  v )
c
 dt L
where D and T are the duty cycle and the switching period,
respectively.
2) The switch is off during the rest of the switching period
(DT < t < T):
vC
 dvc 1
 dt  C (iL  R )
(2)

 diL  1 (v )
C
 dt L
Based on equations (1) and (2), the small signal averaged
state-space model is obtained as bellow, in which the variables
with a hat are small AC variations about the equilibrium
operating point:
 dvˆc   1
1
0 
 dt   
  vˆc  0 

   RC C      D  eˆ   E  dˆ
(3)
 
 diˆL   1
 iˆL   
0

L
L
 dt   L

By applying the Laplace transform to (3) the following
small signal transfer function of the output voltage variation
due to the duty cycle change is obtained:
E
vˆc ( s )
LC
(4)

dˆ ( s ) s 2  s  1
RC LC
The Bode plots of the transfer function of equation (4) and
the experimental circuit simulated in PSIM are compared in
Fig. 2 which confirms the validity of the obtained analytical
results.
20
10
0
-50
3
4
10
10
-100
3
pahse (deg)
-150
4
10
10
Frequency (Hz)
Fig. 2 Bode plot of control to output (Buck converter)
(-): analytical (--): experimental
300
200
100
3
This converter has two poles as:
1
1
1
P1,2  


2 RC
(2 RC ) 2 LC
4
10
10
Frequency (Hz)
(5)
Fig. 4 Bode plot of control to output (Boost converter)
(-): analytical (--): experimental
This converter has two poles and a right half-plane zero as:
C. Buck-Boost Chopper
The Buck-Boost converter is shown in Fig. 5. The small
signal averaged state-space model and the transfer function of
control to output are as equations (13) and (14).
 dvˆc   1
D '
 IL 
 dt   
 C 
 vˆc   0 
RC
C


(13)
 dˆ
     D  eˆ  
Vc  E 
iˆL   
 diˆL   D '




0

L

 dt   L
 L 

I
D'
 L s
(Vc  E )
vˆc ( s )
C
LC

(14)
s
D '2
dˆ ( s )
2

s 
RC LC
The analytical Bode plot is validated by the experimental
results in Fig. 6. Obviously, the frequency domain
performance is similar to the Boost converter.
Fig. 5 Buck-Boost converter
Similar to the Boost converter, this converter has two poles
and a right half-plane zero as:
D '(Vc  E )
1
1
D '2


, Z
(15)
2
2 RC
LC
LI L
(2 RC )
The simplified equations for the crossover frequency and
the phase margin are:
I
(16)
0  L
C
L


IL


 LI L 2
R
)  arctan 
PM  arctan(
(17)
2
2 
CD '(Vc  E )
  LI L  CD ' 


P1,2  
40
Mag (dB)
30
20
10
0
3
4
10
pahse (deg)
D 'Vc
1
1
D '2
P1,2  


, Z
(10)
2
2 RC
LC
LI L
(2 RC )
The simplified equations for the crossover frequency and
the phase margin are:
I
(11)
0  L
C
L


IL


 LI L 2
R
)  arctan 
PM  arctan(
(12)
2
2 


'
CD 'Vc
LI
C
D
L




10
300
200
100
3
4
10
10
Frequency (Hz)
Fig. 6 Bode plot of control to output (Buck-Boost converter)
(-): analytical (--): experimental
D. Cuk Chopper
The Cuk converter is shown in Fig. 7.
Fig. 7 Cuk converter
Equations (18) and (19) are the small signal averaged statespace model and the transfer function of control to output,
respectively. The Bode plots are depicted in Fig. 8.
Experimental waveforms agree with the analytical results.
The Cuk converter has four left half-plane poles and two
right half-plane zeros.
D'
D
 dvˆc1   0
 
0
 IL1  IL2 
 dt  
C1
C1
 C 


0  


1
ˆ

v




c1
1
1
0  0
 dvˆc2 

0
0



 vˆ


 dt  
RC2
C2   c2 





  1  eˆ  Vc1
 d̂ (18)


ˆ
ˆ


D
'
di
i
  
 L1  
L1

0
0
0


L
L


1
 1
 dt 
L

 iˆL2  0  V

  1
   c1


1
 diˆL2   D
L


0
0


 2

 dt 
L2
 L2

vˆc 2 (s)

dˆ (s)
Vc1
DL1
 C1LV
1 c1  2
 D'2  s  D'2  I L1  I L2  s  D '


D2  2  L2 L1D2 
 C1C2 L1 L2  4  C1L1 L2  3  C1 L1
 D'2  s   RD'2  s   D'2  C2 (L2  L1 D' 2 )  s   R  RD'2  s 1








(19)
These converters must first store the energy in the inductor
during a certain time before dumping it into the output
capacitor during the rest of the switching period. If the duty
cycle quickly changes in response to a perturbation, the
inductor naturally limits the current slew rate and the output
voltage drops. The presence of right half-plane zeros means a
serious limit on the available loop bandwidth.
Mag (dB)
40
20
0
3
4
10
10
III. CONCLUSIONS
Thanks to their advantages, Buck, Boost, Buck-Boost, Cuk
and Sepic choppers have found a lot of industrial applications.
In this paper, the small signal averaged state-space models of
these converters are obtained which provide an excellent
understanding of the frequency domain behaviour and a tool
to design and analyse the feedback control loops for these
converters. The theoretical results are validated by simulations
in PSIM software.
pahse (deg)
300
200
100
0
-100
3
4
10
10
Frequency (Hz)
Fig. 8 Bode plot of control to output (Cuk converter)
(-): analytical (--): experimental
50
40
Mag (dB)
E. Sepic Chopper
The Sepic converter is shown in Fig. 9.
30
20
10
0
3
4
10
10
Fig. 9 Sepic converter
pahse (deg)
600
The small signal averaged state-space model and the
transfer function of output to control are as equations (20) and
(21) and the Bode plot are depicted in Fig. 10.
D'
D
 IL1  IL2 
 dvˆc1   0
0
 
 C 
 dt  
C1
C1
1



0  


vˆ
 dvˆc2   0  1 D' D'   c1  0   IL1  IL2 
 


 dt  
C2  ˆ
RC2 C2 C2  vˆc2    

 1  eˆ 



d (20)
 iˆL1    Vc1 Vc2 
 diˆL1   D'  D' 0
0


L




 1
 dt 
L
L1

 iˆL2  0   L1

  1
  

D'
Vc1 Vc2 
 diˆL2   D

0
0 


 dt   L
L2
 2

 L2

The Sepic chopper has four left half-plane poles and three
right half-plane zeros.
The apparent feature for all converters under the study is
the resonance and Boost, Buck-Boost, Cuk and Sepic
choppers, as indirect energy transfer converters, suffer from
the presence of right half-plane zeros. The physical origin of
the right half-plane zero is the indirect energy transfer in boost
operation.
400
200
3
4
10
10
Frequency (Hz)
Fig. 10 Bode plot of control to output (Sepic converter)
(-): analytical (--): experimental
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[2]
[3]
[4]
[5]
[6]
[7]
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 RC1 L1 L2  I L1  I L 2  s 3  D ' RC1 (Vc1  Vc 2 )  L1  L2  s 2  RDL1  I L1  I L 2  s  (2 DD '2  D '3  D ' D 2 ) R Vc1  Vc 2  
vˆc 2 ( s )

dˆ ( s ) C1C2 L1 L2 Rs 4  C1 L1 L2 s 3  R  L1 (C2 D 2  C1 D '2 )  L2 (C1  C2 ) D '2  s 2  ( D 2 L1  D '2 L2 ) s  R ( D '4  2 DD '3  D 2 D '2 )
(21)