Discrete-Time Model for PWM Converters in
Discontinuous Conduction Mode
Mohammed S. Al-Numay
King Saud University/Electrical Engineering Department, Riyadh, Saudi Arabia
authors suggested a new multi-frequency averaging
(MFA) model, that is able to estimate the average
values of state variables at low switching frequency.
One disadvantage of such models is the existence of
two different average models for CCM and DCM which
would lead to the loss of the main advantage of SSA
models. The boundary between these two modes is
dependent upon the ripple current in the inductor, and
this information is lost in the averaging process. A
small-signal frequency domain representation of PWM
converters for DCM to be used for analysis is provided
in [4]. This model is not suitable for time simulation of
PWM converters.
Abstract— A new discrete-time model for pulse-width
modulated (PWM) converters operating in the discontinuous
conduction mode (DCM) which leads to the exact discretetime mathematical representation of the averaged values of
the output signal is proposed.
This model can also provide the averaged values of
other internal signals with little increase in simulation
time. The use of piecewise linear (PL) iteration method
dramatically reduces the simulation time, while introducing
a little simulation approximation. It is compared to other
existing models with respect to accuracy and simulation
speed through a numerical example of boost converter. This
method gives the exact one-cycle-average (OCA) values of
signals at switching instants if PL iteration is not used and,
by far, more accurate than other methods if PL iteration is
not used. Numerical simulations demonstrate the superiority
of the proposed method in terms of accuracy and speed.
Sampled-data modeling techniques, on the other hand,
provide the most accurate and most natural means to
represent the behavior of PWM systems since these
systems inherently operate in a clocked and cyclical
fashion and because they are well suited to the design of
digital controllers. Sampled-data models allow us to focus
on cycle-to-cycle behavior, ignoring intracycle ripples.
This makes them effective in general simulation, analysis
and design. Early discussion of sampled-data models
for switchmode power converters were presented in [5]
and [6] and later extended in [7] and [8]. An algorithm
to increase the speed of such simulation methods is presented in [9]. These models predict the values of signals at
the beginning of each switching period, which most of the
times represent peaks or valleys of the signals rather than
average values. To better understand the average behavior
of the system, a discrete-time model for the OCA signals
was presented in [10]. These models were derived for
CCM and can not be used for DCM.
It should be mentioned that the exact behavior of
any switched system can be simulated using the exact
continuous-time model in hand. Although best accuracy
can be obtained using such model, this model is extremely
slow as compared to even SSA models not to mention the
fast discrete-time models. It is clear from the literature
that there is a need for accurate and fast algorithm to
be used for simulation, analysis and design of switched
systems operating in DCM.
In this paper, a new sampled-data model for DCM
PWM converters is introduced. Besides accuracy and
speed, the motivation for the new model is based on
the fact that, in many power electronic applications, it is
the average values of the voltage and current rather than
their instantaneous values that are of greatest interest. To
I. I NTRODUCTION
PWM converters are widely used for operating switch
controlled systems. These systems are usually operated
in two modes of operation, namely: continuous and discontinuous conduction modes. The DCM of operation
typically occurs in dc/dc converters at light load. For lowpower applications, many designers prefer to operate in
the DCM in order to avoid the reverse recovery problem
of the diode. DCM operation has also been considered
a possible solution to the right-half plane (RHP) zero
problem encountered in buck-boost and boost derived
topologies. In single-phase ac/dc converters with active
power factor correction (PFC), the input inductor current
becomes discontinuous in the vicinity of the voltage zero
crossing; some PFC circuits are even purposely designed
to operate in DCM over the entire line cycle in order to
simplify the control. Proper analytical models for DCM
operation of PWM converters are therefore essential for
the analysis and design of converters in a variety of
applications (see [1] and references therein).
Averaging methods are sometimes used to produce
approximate continuous-time models for PWM systems
by neglecting the switching period of the switches and
the sampling period of the microprocessor controller. A
recently published paper [1] has developed a new statespace averaged model for PWM converters operating
in the discontinuous conduction mode (DCM SSA).
This model is a generalization of the well-known state
space average (SSA) model for continuous conduction
mode (CCM) [2]. To improve accuracy, a recent paper
[3] presented some of the issues involved in applying
frequency-selective averaging to modeling the dynamic
behavior of switched systems operating in CCM. The
1-4244-0121-6/06/$20.00 ©2006 IEEE
800
EPE-PEMC 2006, Portorož, Slovenia
where the input nonlinearities A(d1 , d2 ), B(d1 , d2 ),
C(d1 , d2 ) and D(d1 , d2 ) are given by
accommodate for such applications, the proposed model
provides the discrete-time response of the OCA signals
of DCM PWM converters. This model (DCM OCA)
is compared to most related existing models through
numerical comparative example of boost converter.
A(d1 , d2 )
1
Φ3 Φ2 Φ1
D(d1 , d2 ) := C1 Γ∗1 + C2 (Φ∗2 Γ1 + Γ∗2 )
+ C3 (Φ∗3 (Φ2 Γ1 + Γ2 ) + Γ∗3 ).
A. System Description
(1)
where u ∈ Rm is the input vector, x ∈ Rn is the
state vector, and y ∈ Rp is the output vector. The
system switches between three topologies, (A1 , B1 , C1 ),
(A2 , B2 , C2 ), and (A3 , B3 , C3 ), with switching intervals
determined by
Φ∗i (t) :=
τ2
τ3
:=
:=
kT ≤ t < kT + d1k T
kT +
kT +
d1k T ≤ t < kT + (d1k + d2k )T
(d1k + d2k )T ≤ t < kT + T
Γ∗i (t) :=
(3)
(4)
(5)
(14)
(15)
(16)
an equivalent (but standard form) representation of the
OCA large-signal model is given by:
x∗k+1
yk∗
= A∗ (d1k , d2k )x∗k + B ∗ (d1k , d2k )uk
= C ∗ x∗k
where
⎡
A∗ (d1 , d2 ) := ⎣
⎡
The signal, y ∗ (t) is used to develop a new discrete-time
model for DCM PWM converters. This model provides
the basis for discrete-time simulation of the averaged
value of any state in the DCM PWM system, even during
transient non-periodic operating conditions.
B (d , d ) := ⎣
∗
1
2
C ∗ (d1 , d2 ) :=
A(d1 , d2 ) 0n×p
C(d1 , d2 )
B(d1 , d2 )
1
(18)
(19)
⎤
⎦
(20)
0p×p
⎤
⎦
(21)
2
D(d , d )
0p×n
Ip×p
(22)
Note that not only the OCA values of output signal will
be available but also the values of the signals (without
averaging) at the beginning of every switching period as
well.
B. Discrete-Time Model
It is desired to compute, without approximation, the
evolution of all system variables at the sampling instants,
t = kT assuming three different topologies for the
system. Since the state and output equations (1)–(2)
are piecewise-linear with respect to time t, the desired
discrete-time model can be obtained symbolically. Using
the notation, xk := x(kT ) and yk∗ := y ∗ (kT ), the result
is the OCA large signal model
= A(d1k , d2k )xk + B(d1k , d2k )uk
= C(d1k , d2k )xk + D(d1k , d2k )uk
(13)
Note that the averaging operation adds “sensor” dynamics
to the system; as a consequence, the large-signal model
(7)–(8) is not in standard state-space form. By defining
the augmented state vector x∗ ∈ Rn+p such that
⎤
⎡
xk+1
⎦
x∗k+1 := ⎣
(17)
1
2
1
2
C(dk , dk )xk + D(dk , dk )uk
where T is the switch period, (d1k + d2k ) ∈ [0, 1] are
the switch duty ratios, and k is the discrete-time index.
All auxiliary inputs will be assumed to be piecewise
constants, i.e. u(t) = uk for all t ∈ [kT, (k + 1)T ).
This assumption is not necessary and is made for convenience only; more general cases would only require more
complex notations. The OCA representation of the output
signal [10] is given by
1 t
∗
y(τ )dτ.
(6)
y (t) :=
T t−T
xk+1
∗
yk+1
eAi t
t
eAi τ bi dτ
0
1 t
Φi (τ )dτ
T 0
t
1
Γi (τ )dτ.
T 0
Φi (t) :=
Γi (t) :=
:=
(12)
The arguments d1 T , d2 T , and (1−d1 −d2 )T for (Φ1 , Φ∗1 ,
Γ1 and Γ∗1 ), (Φ2 , Φ∗2 , Γ2 and Γ∗2 ) and (Φ3 , Φ∗3 , Γ3 and
Γ∗3 ), respectively are omitted from the above equations
for notation simplicity. where
(2)
τ1
(9)
B(d , d ) := Φ3 (Φ2 Γ1 + Γ2 ) + Γ3
(10)
1 2
∗
∗
∗
C(d , d ) := C1 Φ1 + C2 Φ2 Φ1 + C3 Φ3 Φ2 Φ1 (11)
II. P ROPOSED M ODEL (DCM OCA)
The DCM PWM converter can be described by
⎧
t ∈ τ1
⎨ A1 x(t) + B1 u(t) ,
A
x(t)
+
B
u(t)
,
t ∈ τ2
ẋ(t) =
2
2
⎩
A3 x(t) + B3 u(t) ,
t ∈ τ3
⎧
t ∈ τ1
⎨ C1 x(t) ,
C2 x(t) ,
t ∈ τ2
y(t) =
⎩
C3 x(t) ,
t ∈ τ3
:=
2
C. Duty-ratio Computation
Unlike duty ratio d1 ,the duty ratio d2 is not known
in advance and should be computed at the zero-crossing
of the current signal during every switching period. A
single variable nonlinear function can be solved for d2
using any standard nonlinear equation solver. For Matlab
simulations, a built-in nonlinear equation solver (fzero)
can be used to solve for d2 . To reduce the simulation time,
(7)
(8)
801
the Piecewise Linear (PL) iteration method used in [11]
for tracking control is adopted here to solve the nonlinear
equation for the duty ratio d2 .
Consider a PWM converter operating in the DCM. It
is desired to find the value of d2 corresponding to the
boundary between topologies 2 and 3. From the beginning
of the first topology to the end of the second topology,
the original signals (without averaging) are governed by
the state space equation
xk+1 = F (d1k , d2k )xk + G(d1k , d2k )uk
iL
L
=
=
d2
and rearrangement of the equation gives
0 = C0 Φ2 (d2k T )(Φ1 (d1k T )xk + Γ1 (d1k T )uk )
(28)
A3 =
γσ
δσ
=
−βσ(j) x0 + δσ(j) uk
ασ(j) x0 + γσ(j) uk
(35)
− L1
1
− RC
0
0
1
0 − RC
B2 =
B3 =
C2 =
0
1
L
0
0
C3 =
0
0
1
1
(37)
(38)
The DCM SSA model for boost converter is given [1] by:
2x1 x2
du
−
(39)
L
dT (u − x1 )
x2
x1
−
(40)
ẋ2 =
C
RC
and the conventional discrete-time model (CDTM) is
given by
ẋ1
for σ = 1, 2, . . . , nσ , where W = 1/nσ is the width of
the uniform segments and nσ is the number of segments
used for linearization. The domain over which the input
nonlinearities are approximated is d2 ∈ [0, 1]. The coefficients of the approximation are related to the original
model by
βσ
1
C
C0 Φ2 (d2 T ) ≈ ασ d2 + βσ , (σ − 1)W ≤ d2 ≤ σW (29)
C0 Γ2 (d2 T ) ≈ γσ d2 + δσ , (σ − 1)W ≤ d2 ≤ σW (30)
C0 Φ2 (σW ) − C0 Φ2 ((σ − 1)W )
W
= C0 Φ2 (σW ) − ασ W
C0 Γ2 (σW ) − C0 Γ2 ((σ − 1)W )
=
W
= C0 Γ2 (σW ) − γσ W.
0
A2 =
Γ1 (d1k T )uk
+
is nothing
The term, x :=
but the states vector evaluated at the end of the first
interval of switching period, i.e. at, kT + d1k T . This
quantity does not depend on d2 and can be computed
ahead of time, at each switching interval, without iteration. Hence, it will be assumed constant while formulating
the PL procedure. The nonlinear scalar functions to be
approximated by PL functions are
=
(j+1)
To compare existing models with DCM OCA model,
consider the boost converter circuit shown in Fig. 1. The
input is u = Vg and state variables are x1 = iL and
x2 = vC . The same parameter values used in [3] for boost
converter are used here except the value of R which is
increased to force the converter to operate in the DCM.
These are: R = 20 Ω, L = 100 µH, C = 4.4 µF, Vg =
5 V, T = 100 µs, and D = 0.5.
The boost converter is defined by
1 0
0
A1 =
C1 = 0 1 (36)
B1 = L
1
0 − RC
0
+ C0 (Φ2 (d2k T )Γ1 (d1k T ) + Γ2 (d2k T ))uk (27)
ασ
boost converter
III. N UMERICAL E XAMPLE
= C0 Φ2 (d2k T )Φ1 (d1k T )xk
Φ1 (d1k T )xk
R
where j is the iteration index. The accuracy of PL iteration
depends on the number of segments (nσ ). The precalculation time will increase as nσ increases but the number
of iterations will slightly increase as nσ increases. This
PL iteration process is used in the simulation programs
for discrete-time models to reduce the simulation time.
to be the nonlinear equation equals to zero at the boundary
between topologies 2 and 3 (commonly current signal).
Then, at every switching period, the nonlinear equation
to be solved for d2 is given by
0
C
-
The iteration equation is then given by
(23)
Φ2 (d2k T )Φ1 (d1k T )
(24)
2
1
2
Φ2 (dk T )Γ1 (dk T ) + Γ2 (dk T ). (25)
+ C0 Γ2 (d2k T )uk .
+
vC
Fig. 1.
Note that this is the same state space model for CCM
with (1 − d1k ) replaced by d2k . The value of d2k at every
switching interval, k is to be calculated. Define the scalar
function
y 0 = C0 xk+1
(26)
0
dT
Vg
where the input nonlinearities F (.) and G(.) are given by
F (d1k , d2k )
G(d1k , d2k )
(1-d)T
=
xk+1 = A(d1k , d2k )xk + B(d1k , d2k )uk
(41)
where the input nonlinearities A(d1 , d2 ) and B(d1 , d2 ) are
defined in (9) and (10).
All simulations were performed using Matlab 7 on a
personal computer (Pentium 1.6 GHz) running Microsoft
Window XP. Results of switched, DCM SSA, and CDTM
for the boost converter are shown in Fig. 2. The DCM
OCA model is also added to the plot to show its accuracy
as compared with its continuous counterpart DCM SSA
model as well as other models. In the figures, the current
and voltage signals are represented by:
(31)
(32)
(33)
(34)
802
3
Steady−state error (%)
3
Current (A)
2
1
0
−1
0
0.1
0.2
0.3
Time (ms)
0.4
2.25
1.5
0.75
0
70
0.5
16
120
130
8
Steady−state error (%)
Voltage (V)
90
100 110
Switching Time (µs)
Fig. 3. current steady-state error for DCM SSA (◦) and DCM OCA
with PL ()
13
10
7
4
80
0
0.1
0.2
0.3
Time (ms)
0.4
0.5
4
2
0
70
Fig. 2. simulation comparison of various models for boost converter
operating in DCM
80
90
100
110
Switching Time (µs)
120
130
Fig. 4. voltage steady-state error for DCM SSA (◦) and DCM OCA
with PL ()
− : switched and DCM SSA
◦ : CDTM
: DCM OCA
It should be noted that no approximation is made in
deriving the new discrete-time model. Consequently, the
steady-state average values predicted by the DCM OCA
model are more accurate than the ones obtained by the
DCM SSA method for this example. The only approximation in the proposed model is in the computation of d2 if
the PL method is used. In this example the results of the
DCM OCA model are computed using Matlab nonlinear
equation solver and also using the PL method. The steadystate values are iL = 1.16402 A and vC = 10.79134 V
for DCM SSA, iL = 1.14667 A and vC = 10.43337 V
for the DCM OCA without PL approximation and iL
= 1.133 A and vC = 10.38242 V for the DCM OCA
with PL approximation (nσ = 20). The accuracy of the
DCM SSA method decreases as the switching frequency
decreases, while the accuracy of the proposed model does
not depend on the switching frequency. Figs. 3 and 4 show
the steady state errors of the DCM SSA and DCM OCA
(with PL) methods as functions of switching time (T ).
The normalized simulation times required by each method
are summarized in Table I. Note that the steady-state
values can be exactly computed from the state equations
at equilibrium, that is
x̄∗ = (I − A∗ )−1 B ∗ .
6
S IMULATION T IMES
FOR
TABLE I
B OOST C ONVERTER
Method
IN
DCM
Normalized
Simulation Time
Switched
41
DCM SSA
0.77
CDTM with PL
0.83
DCM OCA w/o PL
11.20
DCM OCA with PL
1.00
the memory size required for storing the data used for
online computation but, at the same time, will increase
the simulation time.
IV. C ONCLUSION
This paper proposed a new model which provides the
discrete-time response of the OCA value of the output
signal in DCM PWM converters. This model is used
as a simulation model for PWM converters operating
in the DCM. It is compared to existing models through
a numerical example of boost converter. The proposed
model provides the most accurate OCA values while the
DCM SSA model predicted the next accurate average
values. The discrete-time OCA model [10] is a special
case of the proposed mode, and hence a combination
of these two models may be used to simulate converters
operating in both modes of operation.
(42)
As expected, the same values iL = 1.14667 A and vC
= 10.43337 V are obtained by solving this equation at
d1 = 0.5. It should be mentioned that a similar piecewise linearization technique is used for CDTM for fair
comparison. The algorithm presented in [9] will reduce
803
ACKNOWLEDGMENT
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This work was supported in part by the Research
Center, College of Engineering, King Saud University,
Riyadh, Saudi Arabia under grant number 4/426.
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