HYBRID MODELLING OF THE SECOND-ORDER PWM
CONVERTERS WITH DCM
ELENA NICULESCU
Department of Electronics and Instrumentation
University of Craiova
Al. I. Cuza Street, No. 13, Craiova
ROMANIA
Abstract: - A hybrid technique to derive the characteristic coefficients of the second-order PWM converters
with discontinuous conduction mode (DCM) is developed. The starting point is the large-signal averaged model
of the three-terminal PWM switch and a new duty-ratio constraint, and the end result is the canonical full-order
(ac small signal) model of converter. The ac small-signal model of elementary PWM converters expressed in
the terms of the characteristic coefficients represents the canonical full-order (ac small signal) model of
converter. All the output characteristic coefficients are complex functions and their singularities appear in the
input-to-output voltage and control-to-output voltage transfer functions of the power stage: the second highfrequency pole (buck, boost and buck-boost converters) and the right-half-plane (RHP) zero (boost and buckboost converters). Derivation of the characteristic coefficients of converters allows using the full-order
canonical model (ac small signal) in any form: mathematical or with equivalent circuit or functional block
diagram. All these three equivalent forms of the full-order model can correctly capture the fast dynamics
associated with the inductor current and predict the converter dynamics up to around one third of the switching
frequency.
Key-Words: - Averaged PWM switch model, PWM converters with DCM, characteristic coefficients
1 Introduction
Proper mathematical models of the PWM converters
with discontinuous conduction mode (DCM)
operation are essential for the analysis and design of
converters in a variety of applications.
DCM operation can occurs in dc/dc converters at
light load or it can be preferred by designers in order
to avoid the reverse recovery problem of the diode.
Also some designers consider the DCM operation as
o possible solution to the right-half-plane (RHP)
zero problem that encounters in boost and buckboost topologies [1], [2].
The major results concerning the modelling of
the PWM converters with DCM have been presented
in [3]. There the authors made a re examination of
all existing averaged models and proposed new fullorder averaged models in both mathematical and
circuit forms. These new models can correctly
predict the small-signal responses up to one third of
the switching frequency and are more accurate than
all previous performed models. The duty ratio
constraint that defines the parameter d 2 is
considered to be the key of an accurate prediction of
high-frequency behaviour of PWM converter. This
parameter gives the conducting time of the diode
( d 2 Ts ) that is the decay interval of the inductor
current [2].
Despite of their limitations concerning the highfrequency
phenomena
characterisation,
the
averaging techniques are simple and more likely to
give tractable mathematical and circuit models. The
derivation of an averaged model of the PWM
converters with DCM involves two major steps: the
establishing the constraint concerning the parameter
d 2 and the averaging process of the variables. The
major differences among the various models of a
PWM converter with DCM appear in the frequency
responses of converter. From this point of view, the
dynamic models of converters can be either reducedorder or full-order models. Let be the second-order
PWM converter case (buck, boost, buck-boost
converters). In the reduced-order models, the inputto-output voltage and control-to-output voltage
transfer functions have a single low-frequency pole
in the left-half-plane (LHP). In contrast with these,
the singularities of the transfer functions from the
full-order models are two poles in the LHP, one at
low-frequency and other at high-frequency, and one
zero in the RHP (boost and buck-boost converters).
The following general notations and symbols are
used in the paper: the small-signal variations of the
quantities are written with lowercase letters with a
tilde
above
them,
while
their
Laplace
transformations will be written as explicit functions
of the complex frequency s . The average values of
currents and voltages are written with lowercase
letters with a line above them. The symbol with
circle is used for an independent source (of current
or voltage) and that with rhombus for a controlled
source. Other notations: i L - inductor current; i pk peak inductor current; i I - absorbed current; i J injected current; v I - input (or line) voltage; v C capacitor voltage; v O - output voltage; R - load
resistance of the stage; C - capacity of output
capacitor; M - continuous voltage conversion ratio
( M  VO / VI ); f s - switching frequency; K conduction
parameter
through
inductor
( K  2L / RTs ). The three distinct time intervals that
characterise the DCM operation of a second-order
converter are highlighted in the Fig. 1 where: d1 is
the transistor-on duty ratio, d 2 is the diode-on duty
ratio and d 3 is the transistor and diode-off duty ratio.
For constant switching frequency, d1  d 2  d 3  1 .
In particularly, the paper shows that the full-order
canonical model with characteristic coefficients of
switching cell with DCM can be obtained by means
of hybrid modelling procedure having as starting
point the three-terminal PWM switch cell of
Vorperian [4]. Using a new duty-ratio constraint the
full-order models of second-order PWM converters
with DCM have been derived. The paper is
organised as follows. In Section 2, the large-signal
averaged models of second-order PWM converters
for DCM operation are reported. In Section 3, after a
short review of characteristic coefficient derivation,
the full-order dynamic models described by
characteristic coefficient are developed. Finally, a
comparison between the models performed and the
full-order models of PWM converters developed by
means of the modified state-space averaging method
is made.
2 Large-signal averaged models for
DCM operation
In order to perform large-signal averaged models for
PWM converters with DCM, some efforts have been
made on the modelling direction of the PWM
switching cell. For instance, in the IAC method, the
large-signal behaviour of the PWM switching cell is
described by help of the average absorbed and
injected currents: i I and i J (Fig. 2). The relations
i I  i I d1 , d 2 , v I , vO 
(1)
i J  i J d1 , d 2 , v I , vO 
(2)
ipk
iL
0 d1Ts
d2Ts
d3Ts
Ts
define these functions that are derived from the
current waveforms [6]. For the second-order PWM
converters, these functions are given in the Table 1.

Fig. 1. Inductor current waveform of second-order
PWM converter with DCM
The state-space averaging (SSA) method and the
injected-absorbed-current (IAC) method are
modelling techniques dedicated to switching dc-dc
converter representation by means of continuous
approximate models [2], [6]. Both original SSA and
IAC methods lead to reduced-order models for a
PWM converter with DCM. In this operating mode,
one considers that the inductor does not carry any
information from cycle to cycle, because the
inductor current resets to zero in every switching
cycle (Fig. 1).
iI
iJ
PWM
switching
cell
vI


C
R
vO

d1
Fig. 2. Representation of PWM switching cell in
IAC method
Table 1. Average injected and absorbed currents
iJ
d1 d1  d 2 v I  v O Ts
2L
In the relation (5), the sign plus corresponds to buck
and buck-boost converters and the sign minus
corresponds to boost converter.
Taking into account that the actual average value
of the inductor current for DCM operation is by the
form [3]
d1d 2 Ts v O
2L
d1d 2 Ts v I
2L
iL 
Buck-
d12 Ts v I
boost
2L
d1d 2 Ts v I
2L
the duty-ratio constraint has the general expression
iI
Buck
d 12 Ts
v I  v O 
2L
Boost
This modelling method uses a volt-second
balance relation of the inductor to define the dutyratio constraint. After replacing the parameter d 2
with its expression given in the Table 2, the
functions of the absorbed and injected currents
become:


d1  d 2   d1 d1  d 2 Ts v az
i pk
2
2L
2Li L
 d1 .
d1Ts v az
d2 
a
i I  i I d1 , v I , vO 
(3)
i J  i J d1 , v I , vO  .
(4)
,

ia
(8)
L
c

S
(7)
D
iL

z
ip
p
a.
Table 2. The duty-ratio constraint
d2
a

d1 v I  v O  / v O
Buck
iV c
L
Boost
d 1 v I / v O  v I 
+
-
Buck-Boost
d1 v I / v O
 p b.
An averaged switch model that can replace the
PWM switch cell with DCM operation has been
established. The switch PWM cell includes the
switch, the diode and the inductor as shown in Fig.
3a, and that is common in different topologies [3],
[4]. The averaged model for DCM operation of this
three-terminal PWM switch cell is shown in Fig. 3b
where:
iV  
d1 i L
d1  d 2
v V  d1v ap  1  d1  d 2 v zp .
(5)
(6)
iL z


vV
Fig. 3. a. Three-terminal PWM switch cell;
b. averaged model of the PWM switch cell
Replacing the three-terminal PWM switch cell by
its averaged model into the elementary PWM
converter diagrams, the equivalent circuits from Fig.
4 are obtained. This equivalent circuits together
expressions of current and voltage controlled
sources from the Table 3, and with the duty-ratio
constraint given in (8), represent the large-signal
averaged models of the second-order PWM
converters with DCM.
Table 3. Voltages of averaged model of PWM
switch cell
Buck
Boost
Buck-Boost
v ap
vI
vC
vI  vC
v zp
vC
vI  vC
vC
v az
vI  vC
vI
vI
+
-
L
iV c

iL z


+
-
vI
vV
+
v
- C
C
 p
R

a.
z

+
-
iL L
c

+-
iV
vI
a
vV p

+
-
C

vc
R

+- vI
Buck-Boost
iI
iV
iL
iV
iJ
iL
iL  iV
iL  iV
iV c

+ L
iL
vV p


v
+ c
C
 z
R

c.
Fig. 4. Large-signal averaged models of secondorder PWM converters: a. buck; b. boost; c. buckboost
Table 4. State-output averaged relations
vC
The Kirchhoff’s Laws applied on the three
equivalent circuits from the Fig. 4 together the stateoutput relations and the duty-ratio constraint (8) lead
to the following large-signal models of the secondorder PWM converters:
1) Buck converter
2i L v C
d i L d1 v I


dt
L
d1Ts v I  v C 
(9)
dv C i L v C


dt
C CR
(10)
d 2 T v  v C 
iI  1 s I
2L
(11)
iJ  iL
(12)
vO  vC
(13)
2) Boost converter
di L d1v C 2i L


dt
L
d1Ts

b.
a
Boost
vO
In order to derive the characteristic coefficients
from the large-signal averaged models of converters,
the state-output averaged relations are written too,
considering three output quantities: the absorbed and
injected currents, and the output voltage. From the
equivalent circuits given in the Fig. 4, the relations
summed in the Table 4 are obtained.
a
Buck
 vC 
1 

v I 

(14)
dv C i L d12 Ts v I v C



dt
C
2LC
CR
(15)
iI  iL
(16)
d12 Ts v I
iJ  iL 
2L
(17)
vO  vC
(18)
3) Buck-Boost converter
d i L d1 v I  v C  2i L v C


dt
L
d1Ts v I
(19)
3. Take the Laplace transform of (24) and (25), and
it obtains
dv C i L
v


 C
dt
C
2LC
CR
(20)
~
 iI (s)   A I
~   
 iJ (s) A O
d 2T v
iI  1 s I
2L
(21)
d 2T v
iJ  iL  1 s I
2L
(22)
vO  vC .
(23)
d12 Ts v I
Using the standard linearization techniques, a
steady-state (dc) model and a small-signal model can
be derived from the equations (9)-(23).
3 AC small-signal models
3.1. Review of characteristic coefficient
derivation
Modelling switching dc-dc PWM converters with
the help of the injected-absorbed-current (IAC)
method is quite simple. There, the obtaining smallsignal model described through characteristic
coefficients can be summed up in three steps [5], [6].
1. The absorbed and injected currents are
averaged over one cycle and putted in the forms (1)
and (2). The parameter d 2 is eliminated by help of
the duty-ratio constraint given in the Table 2, and
the functions of the absorbed and injected currents
become (3) and (4).
2. The linear model of the cell is found by
evaluating small increments of (3) and (4):
i
i
i
d i I  I dd 1  I dv O  I v I
d1
v O
v I
(24)
i J
i
i
dd 1  J dv O  J v I
d1
v O
v I
(25)
di J 
In the small-signal assumption, the small increments
are considered to be equal to the ac components of
quantities to which the Laplace transform can be
~
~
~
vI ;
applied: di I  iI ; d i J  iJ ; dd1  d1 ; dv I  ~
~
dv O  v O . The partial derivatives are evaluated for
a given operating point.
 BI
 BO
~
 d1 (s) 
C I  ~

v O (s) .


CO  ~

 v I (s) 
(26)
Relation (26) represents the canonical dynamic
(ac small-signal) model of any converter. Its form
rest unchanged regardless converter topology,
controlled quantity and control type. Moreover, this
mathematical linear dynamic model can be directly
transposed into an equivalent circuit with fixed
topology or into functional block diagram [6], [7].
There, six characteristic coefficients are highlighted,
namely: three input ( A I , B I , C I ) and, respectively,
three output averaged characteristic coefficients
( A O , B O , C O ). All these coefficients are real for
second-order PWM converters with DCM [6].
Finally, from Fig. 2, it can be shown that ~v O s 
~
is related to iJ s  . After replacing the injected
current with its (26) relation, the two transfer
functions of converter can be derived, namely [8]:
- line-to-output voltage transfer function
~
v s 
RC O
;
H OI s   ~O

v I s  ~d 0 1  sCR  RBO
1
(27)
- control-to-output voltage transfer function
~
v s 
RA O
.
H OC s   ~O

d1 s  ~v 0 1  sCR  RB O
I
(28)
The dynamic model of second-order PWM
converters with DCM, given by the previous transfer
functions is a reduced-order model, like as that
obtained through original SSA method. These
transfer functions describe a first-order system,
because all three output characteristic coefficients
are real even in the non-ideal converter case.
3.2. Derivation of characteristic coefficients
for full-order model
After perturbation and linearization steps applied on
the large-signal average models, given by equations
(9)-(23), the steady-state (dc) models and ac smallsignal models of elementary PWM converters are
obtained. The steady-state (dc) models have the
same forms as those shown in [3] and they not reply
here.
The dynamic model (ac small signal) described
by the linearized state equations has the general
form:
d
dt
~
~
 iL 
 iL 
vI 
~

A

B
~ 
~ 
~
,
 d1 
 v C 
 v C 
(29)
where the matrices 2  2 A and B are:
a
A 1
a 3
a2
b
, B 1

a4
b 3
b2 
.
b 4 
The components of the matrices are function on the
switching frequency, duty ratio, line voltage, load
resistance, inductance, capacity, dc voltage
conversion ratio or zero as follows.
1)Buck converter:
a1  
D
1
2M
; a2   1 ; a3  ;
C
D1Ts 1  M 
LM
a4  
D
V
1
; b1  1 ; b 2  I ; b 3  0 ; b 4  0 .
CR
L
L
2)Boost converter:
a1  
D1
1
2M  1
; a2  
; a3  ;
LM  1
C
D1Ts
D M2
2MVI
1
; b1  1
; b2 
;
a4  
LM  1
CR
L
D 2T
DTV
b3   1 s ; b4   1 s I .
2LC
LC
3)Buck-Boost converter:
a1  
a4  
D
1
2M
; a2   1 ; a3  ;
C
D1Ts
LM
D 2  M 
2VI 1  M 
1
; b1  1
; b2 
;
CR
L
L
D 2T
DTV
b3   1 s ; b4   1 s I .
2LC
LC
Apply the Laplace transform to state equations
and state-output relations. After solving the first
equation set, the state-output relations become:
~
 d (s) 
~
 iI (s)   A I s   B I s  C I s   ~ 1 
~   
  v O (s) . (30)
 iJ (s) A O s   B O s  C O s   ~

 v I (s) 
The following results have been found for the
elementary PWM converters with DCM:
1) Buck converter. All the input characteristic
coefficients are real and the output characteristic
coefficients are complex:
D12 Ts
D1Ts VI 1  M
; BI  C I 
;
AI 
2L
L
A O s  
K AO
K
K
; BO s   BO ; C O s   CO ;
s  sP
s  sP
s  sP
V
D
D
K AO  I ;
K BO  1 ;
K CO  1 ;
L
LM
L
2M
.
sp 
D1Ts 1  M 
2) Boost converter. All the characteristic
coefficients are complex:
A I s  
K
K AI
K
; B I s   BI ; C I s   CI ;
s  sP
s  sP
s  sP
s
s
K
; BO s   BO ;
A O s   K AO ZAO
s  sP
s  sP
s
s
2MVI
; K AI 
;
C O s   K CO ZCO
L
s  sP
K BI  K BO 
D1
D M2
; K CI  1
;
LM  1
LM  1
K AO 
D 2T
D1Ts VI
2M  1
; K CO  1 s ; s p 
;
2L
L
D1Ts
s ZAO 
22M  1
2
; s ZCO 
.
D1Ts
D1Ts M  1
3)Buck-Boost converter. The input characteristic
coefficients are real or null, and the output
characteristic coefficients are complex:
AI 
D12 Ts
D1Ts VI
; BI  0 ; CI 
;
2L
L
4
2
; s ZCO 
.
D1Ts
D1Ts
K OI
;
s  s p1 s  s p 2



K OC
H OC s  
.
s  s p1 s  s p 2



(31)
(32)
H OC s  
K OI s ZCO  s 
;
s  s p1 s  s p 2
(33)
K OC s ZAO  s 
.
s  s p1 s  s p 2
(34)








(36)

The model with characteristic coefficients was
chosen because its mathematical form or the
topology of synthesized equivalent circuit is
invariable. This feature greatly simplifies and
accelerates comparative investigations of different
configurations ranging from single cells to closedloop switching regulators with input filters and
cascaded connections of regulators [6], [10]. Once
found the expressions of these coefficients, all
small-signal properties of converter can be
computed: input impedance, output impedance,
input-to-output voltage and control-to-output voltage
transfer functions of open or closed-loop regulator.
4 Model verification
2) For boost and buck-boost converters
H OI s  

(35)
b 4 s  a 3 b 2  a 1b 4

s 2  a 1  a 4 s  a 1a 4  a 2 a 3

Using the output characteristic coefficients, the
input-to-output voltage and control-to-output voltage
transfer functions are by the following forms:
1) For buck converter
H OI s  

.
 a b  a 3b 2

 b 4  1 4
 s 
b4

  K OC s ZAO  s 
Ls 
s  s p1 s  s p 2
D12 Ts
s ZAO 
s  a 1  a 4 s  a 1a 4  a 2 a 3
;
 a b  a 3 b1

 b 3  1 3
 s 
b3

  K OI s ZCO  s 
Ls 
s  s p1 s  s p 2
H OC s  
s
s
DTV
; K AO  1 s I ;
C O s   K CO ZCO
L
s  sP
D1
2M
; K CO 
; sp 
;
2L
D1Ts
LM
b 3s  a 3 b1  a 1b 3
2

s
s
K
; BO s   BO ;
A O s   K AO ZAO
s  sP
s  sP
K BO 
H OI s  
The input-to-output voltage and control-to-output
voltage transfer functions can be directly obtained
from the dynamic model of PWM converter given
by the equation (29). Following this way, the
transfer functions are expressed in the terms of the
components of matrices A and B, having the general
forms:
The full-order models of the second-order PWM
converters performed in the terms of the
characteristic coefficients have been compared with
the new full-order averaged models presented in [3].
These later models are derived through modified
SSA method with the new duty-ratio constraint. The
comparison is made at large and small-signal level.
So, the averaged large-signal models obtained in this
paper (Section 2) are identical with these given by
the relation (26)  (31) from [3]. The simulation
results compared with the measured frequency
response from [3] shows that the full-order dynamic
models derived by means of the presented procedure
and those taken as reference term [3] dovetail.
Therefore, the full-order models of second-order
PWM converters with DCM described by help of
characteristic coefficients derived by this way is
accurate up to around one third of the switching
frequency.
5 Conclusion
A hybrid technique to model the second-order PWM
converters with discontinuous conduction mode
(DCM) is developed. The starting point is the largesignal averaged model of the three-terminal PWM
switch and a new duty-ratio constraint, and the end
result is the canonical full-order (ac small signal)
model of converter described by means of the
characteristic coefficients.
The ac small-signal model of elementary PWM
converters expressed in the terms of the
characteristic coefficients represents the canonical
full-order (ac small signal) model of converter. All
the output characteristic coefficients are complex
functions and their singularities appear in the inputto-output voltage and control-to-output voltage
transfer functions of the power stage: the second
high-frequency pole and the right-half-plane (RHP)
zero (boost, buck-boost). Besides the singularities of
the characteristic coefficients, which are complex
functions, appear in the transfer function of the
PWM converter. As it can be seen from the
expressions (31)  (34) of the transfer functions and
those of the characteristic coefficients, the highfrequency pole of transfer functions is very closed of
the pole s P , and the RHP zeroes from C O s  and
A O s  are recovered as RHP zeroes in transfer
functions.
Using the old duty-ratio constraints given in the
Table 2, the reduced-order models result that do not
include the second high-frequency pole or the RHP
zero.
The full-order canonical model (ac small signal)
of second-order PWM converters with DCM,
described by the characteristic coefficients, can
correctly capture the fast dynamics associated with
the inductor current and predict the converter
dynamics up to around one third of the switching
frequency. This model can be used in any form:
mathematical or with equivalent circuit or functional
block diagram.
References:
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in
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