12th WSEAS International Conference on CIRCUITS, Heraklion, Greece, July 22-24, 2008
Small-Signal Models of Some Basic PWM Converters
ELENA NICULESCU*), DORINA-MIOARA PURCARU*) and MARIUS-CRISTIAN NICULESCU**)
Department of Electronics and Instrumentation*) and Department of Automation and Mechatronics**)
University of Craiova
Al. I. Cuza Street, No. 13, Craiova, RO-200585
ROMANIA
Abstract: - The paper presents the full-order small-signal models of some basic PWM converters (Cuk, Sepic and Zeta)
with discontinuous inductor current mode (DCM). The derivation procedure is based on the small-signal PWM switch
model in DCM and converter topologies derived from their original topologies. The small-signal characteristics of
these converters are presented in symbolic forms. Such models can be directly used in conjunction with a
computational environment as Matlab, MathCAD, and Mathematica to analyze the small-signal low-frequency
dynamics of the converters.
Key-Words: - Full-order small-signal model, PWM switch model in DCM, Fourth-order PWM converters.
current and average diode current etc.
- Every basic converter topology has its corresponding
structure with identical operating behaviour that clearly
highlights the three-terminal PWM switch [8].
- The waveforms of inductor currents and voltages have
the same shape in both operating modes (CCM and
DCM).
- All three basic converter topology can be design to
achieve low-ripple input or output current either with
coupled inductors or with separate inductors [1]-[8].
In this paper, the AC model of the pulse-width
modulation switch in discontinuous conduction mode is
used in the modeling process of the PWM Cuk, Sepic,
and Zeta converters operating in DCM. As has been
presented in [9], the AC model of the PWM switch in
DCM represents the small-signal characteristics of the
nonlinear part of the converter, which consists of the
active and passive switch pair. This averaged technique
is easily applied to all three second-order PWM
converter, i.e. buck, boost and buck-boost converter and
Cuk converter where the PWM switch is clearly
highlighted. Using the model of the PWM switch in
DCM, the PWM Cuk converter in DCM is analyzed for
DC and small-signal low-frequency characteristics in
[9]. Its small-signal transfer functions confirm the fact
that the Cuk converter operating in DCM is a fourthorder system. The other fourth-order converters namely
Sepic and Zeta converters in CCM were completely
analyzed in [10] as converters belonging to boost and
buck family, respectively. Also, the AC (small-signal
low-frequency) characteristics of some versions of Sepic
converter in CCM or DCM are derived in [11]- [14].
Our work aims to carry out the small-signal dynamic
model of the PWM Cuk, Sepic and Zeta converters in
DCM in the terms of line-to-output voltage, control-to-
1 Introduction
Basic PWM converter topologies, such as buck, boost,
buck-boost, Cuk, Sepic, Zeta and their versions with an
isolation transformer, are most widespread in dc-to-dc
applications. A basic fourth-order PWM converter
consists of an active and passive switch pair, and two
inductors (coupled or separate) and two capacitors. The
advantages and disadvantages of this converter type as
low input and/or output current ripples and need an extra
capacitor with large ripple-current-carrying capability
are widely presented in literature [1]-[10].
The most common operating modes of these PWM
converters are the continuous inductor current mode
(CICM or CCM) and discontinuous inductor current
mode (DICM or DCM). It is well known that CCM and
DCM are not the only possible operating modes in
converter topologies with more than two reactive
elements [1]. The PWM converters are nonlinear
dynamic systems with structural changes over an
operation cycle. Two networks are repeatedly switched
by the action of the converter switches in CCM, while
in DCM three networks are periodically switched [1],
[2], [6].
The three basic fourth-order PWM converters
namely Cuk, Sepic, and Zeta are step up/down
converters and only Cuk topology is of invertingpolarity type. Despite their topological differences,
these three converters exhibit many common features
summarised as follows:
- They are converters that share the same expression not
only for the dc voltage conversion ratio in both CCM
and DCM, but also for the length of the decay interval
of inductor currents, the boundary between CCM and
DCM, the average inductor currents, the average switch
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12th WSEAS International Conference on CIRCUITS, Heraklion, Greece, July 22-24, 2008
converter highlights the PWM switch with its three
terminals denoted a (active), p (passive) and c
(common) as it is shown in Fig. 2. However, in the
derived converter topology, if a common reference
potential of the input and output is required, the control
of the power transistor has to be realized potential free
with respect to the potential reference. We must mention
that the PWM derived converters in Fig. 2 serve here
exclusively for a clear evidence of the PWM switch in
the converter structure.
output voltage, load-to-input current and control-to-input
current transfer functions, and output impedance and
input admittance. Such a model in symbolic form is
appropriate for a computer-aided analysis of this
converter. Certainly, the dynamic model of which
derivation is based on an averaging method is subject to
the usual limitation of linear models [15] – [19].
The paper is organized as follows. The original and
derived converter topologies are presented in Section II.
The model of PWM switch in DCM, small-signal
equivalent circuits of derived converters in DCM and
their full-order dynamic models in a symbolic form are
given in Section III. The Section IV concludes the paper.
Generally, the usual notations and conventions are
employed in this paper. For instance, the constant
switching frequency is denoted fs and the distinct time
intervals are denoted d1Ts, d2Ts and d3Ts, with d1+ d2 = 1
for CCM and d1+ d2 + d3 = 1 for DCM respectively. For
a voltage-mode control, the duty cycle d1 is the control
variable. The time interval d1Ts is the time during which
the transistor is on and the diode is off. The time
interval d2Ts is the time during which the transistor is off
and the diode is on, and the third interval d3Ts is the time
during which both the transistor and diode are off. The
capital letters represent the large-signal dc values, while
the lower-case letters represent the time-varying
variables. The lower-case letters with “^” above them
denote the small-signal s-domain expressions of the
corresponding time-varying variables.
2 Original
Topologies
and
Derived
Fig. 1. The original topologies of the basic fourthorder PWM converters: a. Cuk converter; b. Sepic
converter; c. Zeta converter
Converter
In order to point out the PWM switch, the original
circuit is transformed into an equivalent circuit named
derived topology herein. The derived topology of each
converter is obtained by changing only the place of the
energy storage capacitor C1 by moving it into the return
line. The function of the original topology is retained
when the capacitor is moved. The schematic diagrams
of the original topologies of the converters are given in
Fig. 1 and their derived topologies in Fig. 2,
respectively.
The same state equations describe a converter pair
consisting of original and derived converters and
operating either in CCM or DCM as it can be seen at
first glance from the switched circuits of the two
topologies over a switching period. The analysis of the
original and derived topologies in CCM and DCM
shows an identical behaviour of each converter pair too.
So, studying only one of these converters, we obtain the
characterization of both original and derived converters.
Among the two converters, i. e. original and derived
converters, only the schematic diagram of the derived
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Fig. 2. The derived topologies of the basic fourthorder PWM converters: a. Cuk converter; b. Sepic
converter; c. Zeta converter
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The DC characteristics of the basic fourth-order
PWM converters in both CCM and DCM are largely
presented in literature. For DC analysis, a lot of
simplified and detailed analysis methods could be
applied to converter inclusively those based on the PWM
switch model. Also, the well-known method of the statespace averaging yields the same results as previously
mentioned method. For the sake of brevity, this shall not
be treated in more detail here and we will give some DC
results of a simplified analysis of separated-inductor
converter in which the effect of the parasitic elements of
circuit was neglected. Only the quantities of interest for
the model process together with their significations
and/or definitions are given below as follows:
- DC voltage conversion ratio, Md=VO/VI = ±D1/D2;
-Parameter of conduction through an equivalent inductor
with inductance Le=L1//L2, Ke=2Lefs / R;
- D2= K e ; IS=(Md)2VI / R; ID=MdVI / R..
Fig. 3. Small-signal model of the PWM switch in
DCM
In the modeling process of the power processor that
is open-loop converter, we consider the small-signal
disturbances of the line voltage, load current and duty
cycle, i.e. v̂i , î d and d̂1 , as input variables, and their
results in the disturbances of the output voltage and input
current, i.e. v̂ o and îi , as output variables, as it is
schematic shown in Fig. 4.
The effect of inductor coupling can be included in the
equivalent inductance of converter and the above
formulae remain unchanged.
3 Small-Signal Equivalent Circuits of the
PWM Converter in DCM
Substitution of the ac model of the PWM switch in DCM
into the equivalent circuit of the derived converters
allows us to develop the ac characteristics of the openloop PWM converters in DCM in a symbolic form.
The following two equations describe the ac (smallsignal low-frequency) model of the three terminal PWM
switch in DCM as it is demonstrated in [9]:
îa = g i v̂ ac + k i d̂1
(1)
î p = g f v̂ ac + k o d̂ 1 − g o v̂ cp
(2)
where
g i = I a / Vac
k i = 2 I a / D1
g f = 2 I p / Vac
(3)
(4)
(5)
k o = 2 I p / D1
(6)
g o = I p / V pc .
(7)
Fig. 4. Variables considered in modelling of the
power processor
The small-signal low-frequency dynamic model of a
PWM converter in DCM will be described by the
following equation:
v̂ o (s )  Gu11 (s ) Gu12 (s ) v̂i (s )  G d 1 (s )
 î (s )  = G (s ) G (s ) î (s ) + G (s ) d̂ 1 (s ) .
u 22
 i   u 21
 d   d 2 
(8)
The above equation contains all ac characteristics of an
open-loop PWM converter, namely:
- The line-to-output voltage transfer function (audio
susceptibility)
v̂ (s )
;
(9)
Gu11 (s ) = o
v̂i (s ) îd =0
Equations (1) and (2) correspond to the equivalent
circuit model shown in Fig. 3.
For a given DC operating point of the converter, the
currents Ia and Ip, and the voltages Vac and Vcp have the
following expressions: Ia = IS = (Md)2VI/R; Ip = ID =
MdVI/R; Vac = VI; Vcp = VO= MdVI. The equations (3) – (7)
yield the model parameters corresponding to respective
operating point: gi=(Md)2/R; gf=2Md/R; go=1/R;
ki=2(Md)2VI/RD1; ko=2MdVI/RD1.
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d̂1 = 0
- The output impedance
v̂ (s )
Gu12 (s ) = Z o (s ) = o
;
î d (s ) v̂i =0
(10)
d̂1 = 0
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Fig. 5. Small-signal equivalent circuits of PWM derived converter in DCM:
a. Cuk converter; b. Sepic converter; b. Zeta converter
- The load-to-input current transfer function (the load
current disturbance is îo = −î d )
- The control-to-output voltage transfer function
G d 1 (s ) =
v̂ o (s )
d̂ 1 (s ) v̂i =0
;
(11)
Gu 22 (s ) =
îd = 0
- The input admittance
Gu 21 (s ) = Yin (s ) =
îi (s )
;
v̂i (s ) îd = 0
=−
î d (s ) v̂i = 0
d̂1 = 0
îi (s )
îo (s ) v̂i =0
;
(13)
d̂1 = 0
- The control-to-input current transfer function
G d 2 (s ) =
(12)
d̂1 = 0
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îi (s )
îi (s )
d̂ 1 (s ) v̂i = 0
.
(14)
îd = 0
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Under small-signal disturbances in average variables of
converter at a given DC operating point (Ip, Vac, D1), the
substitution of the small-signal equivalent circuit model
of the PWM switch of Fig. 3 into each PWM derived
converter results in the small-signal equivalent circuits in
Fig. 5. Using the equations in s domain written on each
equivalent circuit and after some algebra, we get the
expressions of the above mentioned open-loop transfer
functions and input and output impedance as follows:
M d 1 + b2 s 2
Gu11 (s ) =
=
1 + a1s + a 2 s 2 + a 3 s 3 + a 4 s 4
,
(15)
M d 1 + b2 s 2
D(s )
(
(
given in [10] and [12], one observes the same forms and
polynomial degrees of the homonym transfer functions
and input and output impedances. To illustrate this
behavioral similarity, the frequency characteristics of the
output-to-line voltage and control-to-output voltage
transfer functions of a PWM Zeta converter operating in
DCM and CCM are given in Fig. 6 and 7, respectively.
)
)
1 + c1 s + c 2 s 2 + c3 s 3
,
D(s )
(16)
1 + e1 s + e 2 s 2 + e3 s 3
,
D (s )
(17)
Gu12 (s ) = Z o (s ) = Z o (0)
Gu 21 (s ) = Yin (s ) = Yin (0 )
Gu 22 (s ) = −
f2s2
,
D (s )
(18)
1 + m1 s + m 2 s 2 + m3 s 3
,
(19)
D (s )
1 + n1 s + n 2 s 2 + n3 s 3
G d 2 (s ) = −G d 2 (0)
.
(20)
D(s )
The DC voltage conversion ratio, output resistance
and input conductance, and polynomial coefficients
appearing in equations (15) – (20) are function on the
circuit element parameters and DC quantities of
operating point of converter.
The denominator D(s) of the open-loop transfer
functions shows that the PWM converter in DCM is of
fourth order:
(21)
D(s ) = 1 + a1 s + a 2 s 2 + a 3 s 3 + a 4 s 4 .
For a properly designed fourth-order converter, the
denominator consists of two quadratic factors whose
resonances are well separated and almost entirely
damped by the load. The parasitic resistances have
almost no effect on the two resonant frequencies of D(s)
and contribute very little to the damping of the
resonances under normal loading conditions. Therefore,
we can write [6]:

s
s 2 
s
s2 
D(s ) = 1 +
+ 2 1 +
+ 2  .
(22)
ω 01Q1 ω 01 
ω 02 Q2 ω 02 

Under the assumption of moderate to high quality factors
and well-separated resonances, we can find out the
relationships between the coefficients ai and the
resonance pulsations and quality factors, respectively
[6].
Comparing the AC model of PWM converters in
DCM and that of the same converter in CCM as it is
G d 1 (s ) = −G d 1 (0 )
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Fig. 6. The frequency characteristics of the outputto-line voltage (Gu11 – black line) and control-to-output
voltage (Gud1 – gray line) transfer functions of the
PWM Zeta converter in DCM.
Fig. 7. The frequency characteristics of the outputto-line voltage (AU – black line) and control-to-output
voltage (Fd - gray line) transfer functions of the PWM
Zeta converter in CCM.
The modeling procedure is presented for an ideal nocoupled inductor converter, but it can be applied to an
isolated converter and coupled-inductor converter
including the parasitic elements.
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[8] J. W. Kolar, H. Sree, N. Mohan and F. C. Zach,
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4 Conclusion
Full-order dynamic models of the Cuk, Sepic and Zeta
converters in discontinuous conduction mode are
presented in this paper. The derivation procedure is
based on the small-signal PWM switch model in DCM
and derived converter topologies. These derived
converter topologies are identical with the basic
converter topologies concerning their operational
behaviour and serve exclusively for modeling process
herein. The small-signal characteristics of the PWM
converters in DCM in terms of audio susceptibility,
control-to-output voltage transfer function, load-to-input
current transfer function, control-to-input current transfer
function, input admittance and output impedance are
obtained in a similar way to the small-signal properties
of the linear amplifiers. The derived models describe the
PWM converters in DCM as fourth-order systems
allowing us to predict the small-signal low-frequency
dynamic behaviour of a Cuk, Sepic or Zeta converter in
DCM and to embed it within a feedback control loop.
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