AVERAGED DISCRETE-TIME MODEL FOR FAST
SIMULATION OF PWM CONVERTERS IN DCM
M. S. Al-Numay
Electrical Engineering Department
King Saud University, Riyadh 11421, Saudi Arabia
alnumay@ksu.edu.sa
Abstract
Continuous and discrete-time averaged models are available for pulse width modulated (PWM)
converters operating in the continuous conduction mode (CCM). A new discrete-time model for
PWM converters operating in the discontinuous conduction mode (DCM) is proposed that leads
to the exact discrete-time mathematical representation of the averaged values of the output signal.
This model can also provide the averaged values of other internal signals with little increase in simulation time. This model is also used to simulate PWM converters operating in
both DCM and CCM when combined with the averaged CCM model in [1]. It is compared to
other models with respect to accuracy and simulation speed through a numerical example of
boost converter. This method gives the exact values of the one-cycle-average (OCA) values of
signals at switching instants. Numerical simulations show the accuracy and speed of the proposed
method as well as its advantage in simulating PWM converters operating in both CCM and DCM.
Key Words: Switched systems, sampled-data model, pulse width modulated (PWM)
converter, averaged model, discrete-time simulation.
1
1.
Introduction
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 low-power
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 [2] and references
therein). On the other hand, converters for high-power applications are designed to operated in
the CCM. Nevertheless, when these converters are operated at reduced power, DCM will appear
during parts of the line period which means a need for a model appropriate for both modes of
operation.
Energy-conservation approach to medelling PWM DC-DC converters was presented in [3].
It provides a systematic method for including parasitic resistances and offset voltage source of
power switches into averaged dynamic large signal, dc, and small-signal circuit models of PWM
converters operating in CCM. This method is can not be used for DCM of operation. Other 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 [2] has developed a new state-space 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 CCM [4]. To
2
improve accuracy, a recent paper [5] presented some of the issues involved in applying frequencyselective averaging to modeling the dynamic behavior of switched systems operating in CCM.
The 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 [6]. This model does not reflect high frequency component superimposed on the
average values of voltages and current, but it reflects instantaneous average voltages and current.
These average waveforms are essential to understand, design control circuits, and evaluate the
open- and closed-loop dynamic performance. This model is not meant for time simulation of
PWM converters.
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 [7] and [8] and later
extended in [9] and [10]. An algorithm to increase the speed of such simulation methods is presented in [11]. 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 [1]. These models were derived for CCM and can not be used for
DCM.
3
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 circuits, it is the average values of the voltage and current rather than their instantaneous values that are of greatest interest. To 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.
2.
Proposed Model
2.1
System Description
The DCM PWM converter can be described by:

A1 x(t) + B1 u(t) ,





A2 x(t) + B2 u(t) ,
ẋ(t) =





A3 x(t) + B3 u(t) ,

C1 x(t) ,





C2 x(t) ,
y(t) =





C3 x(t) ,
t ∈ τ1
t ∈ τ2
(1)
t ∈ τ3
t ∈ τ1
t ∈ τ2
t ∈ τ3
4
(2)
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:
τ1 := kT ≤ t < kT + d1k T
(3)
τ2 := kT + d1k T ≤ t < kT + (d1k + d2k )T
(4)
τ3 := kT + (d1k + d2k )T ≤ t < kT + T
(5)
where T is the switch period, (d1k + d2k ) ∈ [0, 1] are the switch duty ratios, and k is the discretetime 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 [1] is given by:
1
y (t) :=
T
∗
Z
t
y(τ )dτ
(6)
t−T
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.
2.2
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:
xk+1 = A(d1k , d2k )xk + B(d1k , d2k )uk
(7)
∗
yk+1
= C(d1k , d2k )xk + D(d1k , d2k )uk
(8)
5
where the input nonlinearities A(d1 , d2 ), B(d1 , d2 ), C(d1 , d2 ) and D(d1 , d2 ) are given by:
A(d1 , d2 ) := Φ3 Φ2 Φ1
(9)
B(d1 , d2 ) := Φ3 (Φ2 Γ1 + Γ2 ) + Γ3
(10)
C(d1 , d2 ) := C1 Φ∗1 + C2 Φ∗2 Φ1 + C3 Φ∗3 Φ2 Φ1
(11)
D(d1 , d2 ) := C1 Γ∗1 + C2 (Φ∗2 Γ1 + Γ∗2 ) + C3 (Φ∗3 (Φ2 Γ1 + Γ2 ) + Γ∗3 )
(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:
Φi (t) := eAi t
Z t
eAi τ bi dτ
Γi (t) :=
0
Z t
1
∗
Φi (t) :=
Φi (τ )dτ
T 0
Z
1 t
∗
Γi (τ )dτ
Γi (t) :=
T 0
(13)
(14)
(15)
(16)
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:

x∗k := 
xk
C(d1k−1 , d2k−1 )xk−1
+
D(d1k−1 , d2k−1 )uk−1


(17)
an equivalent (but standard form) representation of the OCA large-signal model is given by:
x∗k+1 = A∗ (d1k , d2k )x∗k + B ∗ (d1k , d2k )uk
yk∗ = C ∗ x∗k
where:
(18)
(19)

A∗ (d1 , d2 ) := 
A(d1 , d2 ) 0n×p
1
2
C(d , d ) 0p×p
6


(20)

B ∗ (d1 , d2 ) := 
C ∗ (d1 , d2 ) :=
B(d1 , d2 )
1
2
D(d , d )


0p×n Ip×p
(21)
(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.
2.3
Duty-ratio Computation
Unlike duty ratio d1 ,the duty ration d2 is not known in advance and should be computed at the
zero-crossing of the current signal in 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, the Piecewise Linear (PL) iteration method used in [12] 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
(23)
where the input nonlinearities F (.) and G(.) are given by:
F (d1k , d2k ) = Φ2 (d2k T )Φ1 (d1k T )
(24)
G(d1k , d2k ) = Φ2 (d2k T )Γ1 (d1k T ) + Γ2 (d2k T )
(25)
Note that this is the same state space model for CCM with (1 − d1k ) replaced by d2k . The value of
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d2k at every switching interval, k is to be calculated. Define the scalar function:
y 0 = C0 xk+1
(26)
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 = C0 Φ2 (d2k T )Φ1 (d1k T )xk + C0 (Φ2 (d2k T )Γ1 (d1k T ) + Γ2 (d2k T ))uk
(27)
and rearrangement of the equation gives:
0 = C0 Φ2 (d2k T )(Φ1 (d1k T )xk + Γ1 (d1k T )uk ) + C0 Γ2 (d2k T )uk
(28)
Note that the term, x0 := Φ1 (d1k T )xk + Γ1 (d1k T )uk is nothing 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:
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)
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:
ασ =
C0 Φ2 (σW ) − C0 Φ2 ((σ − 1)W )
W
(31)
βσ = C0 Φ2 (σW ) − ασ W
(32)
γσ =
(33)
C0 Γ2 (σW ) − C0 Γ2 ((σ − 1)W )
W
δσ = C0 Γ2 (σW ) − γσ W
8
(34)
Figure 1: boost converter
The iteration equation is then given by:
d2
(j+1)
=
−βσ(j) x0 + δσ(j) uk
ασ(j) x0 + γσ(j) uk
(35)
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 iteration 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.
3.
Numerical Example
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 [5] 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:
A1 =
A2 =
0
0
1
0 − RC
− L1
1
− RC
0
1
C
B1 =
B2 =
9
1
L
0
1
L
0
C1 =
C2 =
0 1
0 1
(36)
(37)
A3 =
0
0
1
0 − RC
B3 =
0
0
C3 =
0 1
(38)
The DCM SSA model for boost converter is given [2] by:
du
2x1 x2
−
L
dT (u − x1 )
x1
x2
=
−
C
RC
ẋ1 =
(39)
ẋ2
(40)
and the CDTM is given by:
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 conventional discrete-time
model (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:
− : switched and DCM SSA
−− : MFA
◦ : 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
10
3
2.5
Current (A)
2
1.5
1
0.5
0
-0.5
0
0.1
0.2
0.3
0.4
0.5
Time (ms)
0.6
0.7
0.8
0
0.1
0.2
0.3
0.4
0.5
Time (ms)
0.6
0.7
0.8
16
Voltage (V)
14
12
10
8
6
Figure 2: simulation comparison of various models for boost converter operating in DCM
11
Table 1: Simulation Times for Boost Converter in DCM
Method
Switched
DCM SSA
CDTM with PL
DCM OCA w/o PL
DCM OCA with PL
Normalized
Simulation Time
41
0.77
0.83
11.20
1.00
also using the PL method. The steady-state 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
shows the steady state errors of the DCM SSA and DCM OCA (with PL) methods as functions of
switching period (T ). The normalized simulation times required by each method are summarized
in table 1. Note that the steady-state values can be exactly computed from the state equations
at equilibrium, that is:
xˉ∗ = (I − A∗ )−1 B ∗
(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 [11] will reduce the memory size
required for storing the data used for online computation but, at the same time, will increase the
simulation time.
The DCM OCA model combined with the discrete-time model in [1] can exactly simulate
the behavior of a PWM converter operating in both CCM and DCM. By changing the duty ratio,
one can force the converter to operate in both CCM and DCM. The same example of boost
12
3
Steady-state error (%)
2.5
2
1.5
1
0.5
0
70
80
90
100
110
Switching Time (μs)
120
130
Figure 3: current steady-state error for DCM SSA (◦) and DCM OCA with PL ()
8
7
Steady-state error (%)
6
5
4
3
2
1
0
70
80
90
100
110
Switching Time (μs)
120
130
Figure 4: voltage steady-state error for DCM SSA (◦) and DCM OCA with PL ()
13
Table 2: Simulation Times for Boost Converter in CCM and DCM
Method
Switched
DCM SSA
MFA
CDTM with PL
DCM OCA w/o PL
DCM OCA with PL
Normalized
Simulation Time
45
1.2
4.8
0.85
12.2
1.00
converter is used to show the simulation results of boost converter operating in the CCM and
then in the DCM. This was achieved by operating the converter using the duty ratio defined by:

0 ≤ k < 10
 0.8 cos (1000kT )
1
dk =
(43)

0.45
k ≥ 10
Fig. 5 shows the simulation results of the signals where the four methods, including MFA,
are plotted for comparison. MFA method simulates, with good accuracy, the averages in the CCM
operation period, while the DCM SSA method approximates, with good accuracy, the averages
in the DCM operation period. The proposed method, on the other hand, is able to simulate the
OCA behavior of the system throughout the simulation time with best accuracy and excellent
simulation time. The normalized simulation times required by all methods are summarized in
table 2.
4.
Conclusion
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 for DCM and MFA model predicted the
next accurate average values for CCM. The discrete-time OCA model [1] is a special case of
14
8
7
6
Current (A)
5
4
3
2
1
0
-1
0
0.2
0.4
0.6
Time (ms)
0.8
1
1.2
0
0.2
0.4
0.6
Time (ms)
0.8
1
1.2
35
30
Voltage (V)
25
20
15
10
5
0
Figure 5: simulation comparison of various models for boost converter operating in CCM and
DCM
15
the proposed mode, and hence a combination of these two models is used to simulate converters
operating in both modes of operation.
Acknowledgement
This work was supported in part by the Research Center, College of Engineering, King Saud
University, Riyadh, Saudi Arabia under grant number 4/426.
References
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Biography
Mohammed S. Al-Numay was born in Riyadh, Saudi Arabia in 1963. He received the B.Sc.
degree (Second Class Honors) from King Saud University, Riyadh, Saudi Arabia, in 1986, the
M.Sc. degree from Michigan State University, East Lansing, MI, in 1990, and the Ph.D. degree
from Georgia Institute of Technology, Atlanta, GA, in 1997, all in electrical engineering. He is
currently an Assistant Professor in the Electrical Engineering Department at King Saud Uni-
17
versity. His research interests include discrete-time modelling, simulation and control of PWM
systems, and digital control of non-minimum phase systems.
18