A Steady-state Analysis of PWM SEPIC Converter

6 Pages·1.571 MB·English

by  Niculescu E., Purcaru D., Niculescu M.

Proceedings of the 10th WSEAS International Conference on CIRCUITS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp217-222)

A Steady-state Analysis of PWM SEPIC Converter

ELENA NICULESCU, DORINA-MIOARA PURCARU and M. C. NICULESCU
Departments of Electronics and Instrumentation, and Mechatronics
University of Craiova
Al. I. Cuza Street, No. 13, Craiova, RO-200585
ROMANIA

Abstract: - The paper presents a unified steady-state analysis of PWM SEPIC converter with coupled inductors
and separate inductors for both continuous and discontinuous conduction modes. The results of this simplified
analysis allow to determine the operating point of converters and further the small-signal low-frequency
parameters of linear model, and to design the converter too. The expressions of all steady-state properties of
converter take the effects of inductor coupling into account. The converter with separate inductors is treated as
a particular case of the more general coupled-inductor case.

Key-Words: - Coupled and non-coupled inductor PWM SEPIC Converter, Steady-state analysis

1 Introduction make a simplified analysis of coupled-inductor
SEPIC converter avoiding magnetics theory.
As it is well known, the PWM converters are

nonlinear dynamic systems with structural changes
Nomenclature
over an operation cycle: two for the continuous
conduction mode (CCM) and three for the iL1 = current through L1
discontinuous conduction mode (DCM). Coupled i = ripple component of i
1 L1
inductor techniques can be applied to PWM SEPIC I = initial and final condition of i
L01 L1
converter operating either with CCM or DCM to
I = average value of i
achieve low-ripple input current [1], [2], [3]. L1D 1
The paper provides a complete steady-state IL1 = average value of iL2
characterization for coupled-inductor PWM SEPIC i = current through L
L2 2
converter with both CCM and DCM. The converter i = ripple component of i
2 L2
with separate inductors is treated as a particular case
I = initial and final condition of i
of the more general coupled-inductor case. In L02 L2
I = average value of i
Section II, the steady-state analysis of PWM SEPIC L2D 2
converter with CCM is made. The expressions of I = average value of i
L2 L2
average input and output currents, output voltage and I = average output current
O
voltage across energy storage capacitor, which take
i = current through switch
the effects of inductor coupling into account, are S
i = current through diode
derived here. The same steady-state properties of D
PWM SEPIC converter with DCM are derived in V = dc input voltage
I
Section III. The boundary equation is given in v = output voltage
O
Section IV. Simplified mathematical models for the
v = voltage across L
steady-state behavior of converter with CCM and L1 1
v = voltage across L
DCM have been obtained as result of this analysis. L2 2
The analysis is based on the equations written on V = average voltage across C
C1 1
the circuit for each time interval corresponding to D = switch-on duty cycle
1
states of switches (transistor and diode) and the low-
D = diode-on duty cycle
ripple capacitor voltages, and the waveforms of 2
D = switch and diode off ratio
electrical quantities corresponding to operating mode 3
of the converter. The analysis and modelling of f = switching frequency
s
SEPIC converter has surfaced in many forms over L = inductance of filtering inductor L
1 1
the years [1] – [5]. The purpose of this paper is to
Proceedings of the 10th WSEAS International Conference on CIRCUITS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp217-222)
L = inductance of filtering inductor L and diode, average voltage across the energy storage
2 2
capacitor C .
L = mutual inductance 1
M
As it can be seen from the inductor current
k = coupling coefficient
c waveforms shown in Fig. 3a and d, the ripple
n = turn ratio
component i respectively i is superposed over the
1 2
C = capacity of energy storage capacitor C
1 1 dc component I respectively I that represent
L01 L02
C = capacity of output filtering capacitor C
2 2 the initial and final condition of inductor current in
R = load resistance
L respectively L :
1 2
h = conversion efficiency of converter
i =i +I (1)
M = dc voltage conversion ratio L1 1 L01
i =i +I (2)
L2 2 L02

2 Steady-state Analysis of SEPIC
Converter with CCM
The diagram of conventional PWM SEPIC converter
with coupled inductors is given in Fig. 1a. The
topology shown in Fig. 1b is another structure of
PWM SEPIC converter with coupled inductors and
identical operating behaviour as that in Fig. 1a. This
one clearly highlights the PWM switch and the
possibility to use the PWM switch model with all its
benefits [6]. As compared to the conventional SEPIC
topology, the control of the switch has to be realized
potential free with respect to the potential reference

in the second topology.
Fig.2 Equivalent circuits for: a. two coupled
Assume that the converter operates in continuous
inductors; b. SEPIC topology with coupled inductors
conduction mode and that the two inductors are

magnetically coupled together to reduce the input
In order to determine the average values of the
current ripple and increase the power density. The
current components, we need to find di / dt and
equivalent circuit of two coupled inductors is given 1
in Fig. 2a, and the waveforms of inductor currents di / dt firstly. The input and output voltages as
2
and voltages are shown in Fig. 3, where D =1-D . well as the voltage across the energy storage
2 1
capacitor C are supposed without ripple. Based on
1
the equivalent circuit of two coupled inductors in
Fig. 2a, we have
di di
v =L 1 +L 2 (3)
L1 1 dt M dt
di di
v =L 1 +L 2 . (4)
L2 M dt 2 dt
Solving for the unknowns di / dt and di / dt, we
1 2
find
di v L v L
1 = L1 2 - L2 M (5)
dt L L -L2 L L -L2
1 2 M 1 2 M
di v L v L
2 = L2 1 - L1 M (6)
dt L L -L2 L L -L2
1 2 M 1 2 M
Fig.1 Two SEPIC PWM converter topologies with After some algebra, the previous equations can be
coupled inductors written as
di v -k v / n
1 = L1 c L2 (7)
The target of this analysis is that to find the dt (1-k2)L
c 1
expressions of dc voltage conversion ratio, average
di v -k nv
inductor currents, average currents through switch 2 = L2 c L1 (8)
( )
dt 1-k2 L
c 2
Proceedings of the 10th WSEAS International Conference on CIRCUITS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp217-222)
For 0£t £DT , substituting the inductor voltages
vL1 1 s

v =V and v =V into (7) and (8) yield:
D T D T L1 I L2 C1
1 s 2 s
di V -k V / n
1 = I c C1 (9)
( )
VI dt 1-kc2 L1
a . di V -k nV
2 = C1 c I . (10)
0 T t ( )
s dt 1-k2 L
VO c 2
We proceed similarly for DT £t£T when
1 s s
v =V -V -V and v =-V , and we obtain:
L1 I C1 O L2 O
iL1 di V -V -V +k V / n
1 = I C1 O c O (11)
( )
dt 1-k2 L
c 1
I di V +k n(V -V -V )
b . L01 2 =- O c I C1 O . (12)
0 t dt (1-kc2)L2

Since the total rise of i and respectively i
1 2
i1 D1TsVI (cid:1) during 0£t£D1Ts must to be equal to the total fall
L
IL1D 1e of the same currents during D1Ts £t£Ts, we have:
(cid:6)di (cid:3) (cid:6) di (cid:3)
c . 0 t D1Ts(cid:4)(cid:5)dt1(cid:1)(cid:2) =(1-D1)Ts(cid:4)(cid:5)- dt1(cid:1)(cid:2) (13)
0£t£D1Ts D1Ts£t£Ts
vL2 (cid:6)di (cid:3) (cid:6) di (cid:3)
D1Ts(cid:4)(cid:5) dt2(cid:1)(cid:2) =(1-D1)Ts(cid:4)(cid:5)- dt2(cid:1)(cid:2) . (14)
0£t£D1Ts D1Ts£t£Ts
V The substitution of (9), (10) and (11), (12) into
I
(13) and (14) yields the steady-state relationships
d.
0 t between the dc input voltage and the average output

V voltage, and the average voltage across the energy
O
storage capacitor C1:
DV
V =V and V = 1 I .
i C1 I O 1-D
L2 1
From these results, the dc voltage conversion ratio of

coupled-inductor PWM SEPIC converter with CCM
I
L02 is obtained as

e. V D
0 t M = O = 1 (15)
V 1-D
I 1
DTV
1 s I (cid:1) and
i
2 L2e V =V . (16)
C1 I

I The above results have been obtained even for the
L2D
f. separate inductor case. So, the magnetically coupling

0 t
of inductors do not modifies the dc voltage

conversion ratio that only depends on the duty cycle
iD
i of PWM converter and also the relationship between
S

the voltages V , V and V . On the other hand, the
C1 O I
relationship (16) shows that the voltages across the

two inductors are equal during each distinct time
g.
interval that allows to couple the inductors in order
0 t
to reduce the input or output current ripple [1], [2].

Using these new results, (9), (10) and (11), (12)
Fig.3 The waveforms of currents and voltages in
become:
PWM SEPIC converter with CCM
di V
1 = I (17)
dt L
1e
Proceedings of the 10th WSEAS International Conference on CIRCUITS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp217-222)
di V (cid:6) (cid:3)
2 = I (18) I = D1VI (cid:4) D1 - 1 (cid:1) (29)
dt L2e L01 R (cid:4)(cid:5)(1-D )2 K (cid:1)(cid:2)
1 1m
and
DV (cid:6) 1 1 (cid:3)
di1 =-VO (19) IL02 = 1R I (cid:4)(cid:5)1-D - K (cid:1)(cid:2). (30)
dt L 1 2m
1e
We can now to find the average currents of transistor
di V
2 =- O (20) switch and diode as follows:
dt L
2e D2V M2V
where we denoted two effective inductances: I =D (I +I )= 1 I = I (31)
( ) S 1 L1 L2 R(1-D )2 R
1-k2 L 1
L = c 1 (21) DV MV
1e 1-k / n I =(1-D )(I +I )= 1 I = I . (32)
( c ) D 1 L1 L2 R(1-D ) R
1-k2 L 1
L = c 2 . (22) The results of this simplified analysis completely
2e 1-k n
c characterise the steady-state behaviour of the
The equations (17) to (20) show that coupled coupled-inductor PWM SEPIC converter in
inductors fed by the same voltage waveforms will continuous conduction mode. For the separate
exhibit as two equivalent inductors, which only inductor case, it is enough to set k at zero into the
c
depend on the physical inductor structure. The same
above results. As consequence, L fiL , L fiL ,
1e 1 2e 2
equations give the theoretical conditions to cancel
K fiK and K fiK .
the inductor current ripples that is L ޴ or 1m 1 2m 2
1e
L ޴. So, the analysis results from the coupled
2e
inductor SEPIC converter can be extended to non- 3 Steady-state Analysis of SEPIC
coupled inductor SEPIC converter and vice versa.
Converter with DCM
Furthermore, two coupled inductors exhibit as two
For SEPIC PWM converter as well as for Cuk PWM
equivalent inductors with the effective inductances
converter, a discontinuous conduction mode (DCM)
given by (21) and (22), and shown in Fig. 2b.
means that the third time interval of operation cycle
Therefore, Fig. 2b serves exclusively for a clear
is nonzero, not that either inductor current is
representation of (17) and (20).
discontinuous. Coupled inductor techniques can be
The average values of the inductor current ripples
applied to PWM SEPIC converter with DCM to
can be directly found from the waveforms of the
achieve low-ripple input current [1].
inductor currents given in the Fig. 3 as:
Assume that the output current is now further
DT V DV
I = 1 s I = 1 I (23) decreased to enable the converter to enter into
L1D 2L k R
1e 1m discontinuous conduction mode. The waveforms of
DT V DV inductor currents and voltages corresponding to this
I = 1 s I = 1 I (24)
L2D 2L k R operating mode are shown in Fig. 4.
2e 2m
Three distinct time intervals appears there,
where we denoted two parameters of conduction
namely DT , D T and D T with D +D +D =1
through the inductors as follows: 1 s 2 s 3 s 1 2 3
for a constant switching frequency. For DCM,
K =2L f / R (25)
1m 1e s
besides of expressions of average inductor currents,
K =2L f / R. (26)
2m 2e s average input and output currents, average voltage
Next, we determine the components IL01 and across the energy storage capacitor C and average
1
IL02 of inductor currents. For this purpose, we use currents transistor and diode, we have to determine
the equations of the dc current through the load and the parameter D that fixes the decay interval of
2
the efficiency as follows:
inductor currents and the dc voltage conversion ratio
V
O =I =(1-D )(I +I +I +I ) (27) implicitly.
R D 1 L1D L01 L2D L02 Concerning the waveforms of inductor currents, it
V2 can see that the shapes of the currents in Fig. 4 are
O =hV (I +I ). (28) similar to those shown in Fig. 3. Only the scales are
R I L1D L01
different and the currents I and I are equal
Using the assumption of 100% efficiency, the L01 L02
and have an inverse sense:
following results are obtained:
I =-I =I . (33)
L01 L02 L0
The currents through inductors can be expressed as
Proceedings of the 10th WSEAS International Conference on CIRCUITS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp217-222)
i =i +I (34) For the first two time intervals DT and D T ,
L1 1 L0 1 s 2 s
i =i -I . (35) (9), (10) and (11), (12) keep their validity. For the
L2 2 L0
third time interval (D T ), the voltages across
3 s
v inductors are zero: v =v =0. Taking into account
L1 D T D T D T L1 L2
1 s 2 s 3 s that the average voltages across the inductors over a
switching period are zero, the following relationships
result:
V
I D
V =V = 2 V . (36)
a. C1 I D O
0 Ts t 1
V The above relationships yield the dc voltage
O
conversion ratio

V D
iL1 M = O = 1 (37)
V D
I 2
and
IL0 V =V . (38)
C1 I
b . The remark regarding the relationship (38) that is
0 t the same as (18) for the converter with CCM keeps

i DT V its validity for the converter with DCM. The
1 1 s I
determination of the conversion ratio M needs the
L
1e
value of parameter D too. It is obvious that the
IL1D 2
conditions of ripple cancellation from the inductor

c.
current found for CCM remain unchanged for DCM.
0 t
Also, the two effective inductances and two

parameters of conduction through the inductors hold
v
L2 their expressions.

Using the waveforms of inductor currents, the

formula of average value of ripple component of
V
I
inductor currents can be written as:

d. D (D +D )V D (D +D )V
0 t I = 1 1 2 I = 1 1 2 I (39)
VO L1D 2L1e fs K1mR
D (D +D )V D (D +D )V
IL2D = 1 21L f 2 I = 1 K1 R2 I . (40)
i 2e s 2m
L2 The calculation of these components of inductor

currents needs to find the parameter D firstly. In
2
e. 0 t order to find the expression of the component IL0,
- IL0 we use again the relationship of the converter
efficiency, that is
DT V V I =hV (I +I ). (41)
1 s I O O I L1D L0
i2 L For assumption of 100% efficiency, the above
2e
equation yields
I
L2D DV (cid:12) D D (cid:9)
f. 0 t IL0 = 1R I (cid:10)(cid:10)(cid:11)K21m - K12m (cid:7)(cid:7)(cid:8). (42)
As it can be seen from the waveforms of currents
iS iD in the transistor and diode, shown as iS and
respectively i in Fig. 4g, the total rise of i during
D S

0£t£DT should be equal to the total fall in i
g. 1 s D
0 t during D1Ts £t£(D1+D2)Ts. This means that we
have
Fig.4 The waveforms of the currents and voltages in
PWM SEPIC converter with DCM
Proceedings of the 10th WSEAS International Conference on CIRCUITS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp217-222)
(cid:6)di di (cid:3) steady-state currents such as the average input and
DT (cid:4) 1 + 2(cid:1) = output currents, and the average currents of transistor
1 s(cid:5)dt dt (cid:2)
0£t£D1(cid:215)Ts (43) and diode are found. For the dc voltage conversion
(cid:6) di di (cid:3) ratio and the average voltage across the energy
D T (cid:4)- 1 - 2(cid:1) storage capacitor, we found the same formula as in
2 s(cid:5) dt dt (cid:2)
D1(cid:215)Ts£t£(D1+D2)(cid:215)Ts the separate inductor case because we neglected the
and after substituting the corresponding slopes of equivalent series resistance of capacitors. The
inductor currents and using (38), we find the dc steady-state analysis developed here supplies the
voltage conversion ratio in (37) once more. formula of equivalent inductances in the coupled-
The average currents of transistor and diode result inductor case that are entailed by a dynamic analysis
as follows: based on the small-signal equivalent circuit of
D M2V converter. This simplified analysis yields the same
IS = D +1D (IL1D +IL2D)= R I (44) results as one based on the large-signal PWM switch
1 2 [6].
D MV
I = 2 (I +I )= I . (45) Formally, the dc conversion ratio of PWM SEPIC
D D +D L1D L2D R converter with coupled-inductors and operating in
1 2
Now, we proceed to determine the formula for the DCM has the same formula as in the separate-
parameter D2. For this purpose, we use the dc load inductor case. But, the parameter D2 takes the
current combining (39), (40) with (45). As result, the coupling effect into account, by means of the
parameter D is given by the formula parameter of conduction through an equivalent
2
inductor with the inductance L . The average input
D = K (46) em
2 em and output currents, and the average currents of
where the quantity
transistor and diode as well as the boundary equation
K =2L f / R (47)
em em s include the coupling effect too. So, the steady-state
represents the parameter of conduction through an operating point and the parameters of small-signal
equivalent inductor with the inductance low-frequency model of converter depend on the
L =L // L . coupling parameter.
em 1e 2e

References:
4 Boundary between the Continuous
[1] J. W. Kolar, H. Sree, N. Mohan and F. C. Zach,
and Discontinuous Conduction Modes
Novel Aspects of an Application of ‘Zero’-
For a complete characterization of the PWM SEPIC
Ripple Techniques to Basic Converter
converter operating, we find now the equation of the
Topologies, IEEE PESC’97, Vol. 1, pp. 796-803
boundary between CCM and DCM for the coupled-
[2] J. Chen and C. Chang, Analysis and Design of
inductor case. It is well known that a converter will
SEPIC Converter in Boundary Conduction Mode
change the operating mode when the following
for Universal-line Power Factor Correction
equality is satisfied:
Applications, in Proc. IEEE PESC’0, 2001.
D2 =1-D1. (48) [ 3 ] S. Ben-Yaakov, D. Adar and G. Rahav, A SPICE
This means that the parameter K reached its compatible Behavioral Model of SEPIC
em
critical value Converters, IEEE PESC’96, Vol. 2, pp. 1668-
K =(1-D )2. (49) 1673
emcrt 1 [4] W. M. Moussa, Modeling and performance
The same equation can be derived starting from (33).
evaluation of a DC/DC SEPIC converter,
All these results characterize the PWM SEPIC
APEC’95, Vol. 2, pp. 702–706
converter with separate inductors, case in which we
[5] D. Adar, G. Rahav and S. Ben-Yaakov, A
have to set k at zero.
c unified behavioral average model of SEPIC
converters with coupled inductors, IEEE
PESC’97, Vol. 1, pp. 441- 446
5 Conclusion [6] V. Vorperian, Simplified analysis of PWM
A steady-state analysis that completely characterizes converters using model of PWM switch, Parts I
the behavior of PWM SEPIC converter with coupled (CCM) and II (DCM), IEEE Trans. on Aerospace
or separate inductors and operating in CCM or DCM Electronic Systems, Vol. 26, 1990, pp. 497-505
was made in this paper. Taking the effects of
inductor coupling into account, the expressions of all