References
CROLS, J., and STEYAERT, M.s.J.: Switched-opamp: An approach to
realize full CMOS switched-capacitor circuits at very low power
supply voltages, IEEE J. Solid-State Circuits, 1994, SC-29, (8),
pp. 936-942 F E R R I , G , COSTA. A., and BASCHIROTTO, A.: A 1.2V
rail-to-rail switched buffer. Proc. ICECS, Lisbon, POrtUgdl, 1998,
pp. 4548 BRANDT, B., FERGUSON, P.F., and REBESEHINI, M.: Analog
circuit design for AX ADCs in NORSWORTHY, s.R., SCHREIER, R., and
TEMES,G.C. (Eds.): Delta-sigma data converters (IEEE Press, 1997),
Chap. 11 BASCHIROTTO, A., and CASTELLO, R.: A 1 V 1.8-MHz CMOS
switched-opamp SC filter with rail-to-rail output swing, IEEE .I.
Solid-State Circuits, 1997, SC-32, ( 1 2), pp. 1979-1986
Unified model of PWM switch including inductor in DCM
Sung-Soo Hong
The conventional small-signal circuit model in the discontinuous
conduction mode (DCM) shows large discrepancies at high
frequencies. A new unified small-signal circuit model of the
pulsewidth modulation (PWM) switch including inductor in the DCM is
proposed to overcome the inaccuracy of the conventional
small-signal circuit model.
Introduction: Several techniques have been proposed for
modelling pulsewidth modulation (PWM) converters operating in the
discon- tinuous conduction mode (DCM). The models can be classified
into the reduced order and full order models, according to whether
the inductor current dynamics are eliminated. The reduced order
models can be useful in the control loop design with small
bandwidth. However, large discrepancies are observed at high
frequencies, especially in the phase responses. Since the full
order models in [l, 21 include the inductor current dynamics, these
are more accurate than the reduced order models. The model in [ 11
is more convenient because it obtains small-signal circuit models
of PWM converters by replacing only switching part with its
small-signal circuit model. However, the model in [1] still shows
large discrepancies at higher frequencies [2]. Even though the
model in [2] is more accurate than the model in [l], a circuit
model has not been presented.
The purpose of this work is to develop a new unified small-sig-
nal circuit model of the PWM switch including inductor operating in
the DCM, which is accurate at high frequencies.
Small-signal circuit modelling: In this Section, a new unified
small- signal circuit model of the PWM switch including inductor is
developed, which is useful for the analysis of different PWM con-
verters. Since the small-signal model is derived by perturbing
aver- aged relationships, the equations describing the averaged
motion of PWM converters are required.
r _ _ _ - _ _ _ _ _ _ _ _ _ _ - - _ _ _ _ _
i, :c m b n - p; P I I
Fig. 1 Boost converter example
Even though the averaged model of the PWM switch including
inductor in [3] has been developed for the buck converter, this
model can be applied to buck-boost and boost converters. Fig. 1
shows the notations for the boost converter example. Regardless of
converter types, the following equation holds [3]:
ubp = dlvap + (1 - d l - d2)vcp (1) Meanwhile, in order to
describe converter behaviour fully, a
duty-ratio constraint is needed. The following accurate
duty-ratio
constraint has been presented in [2]:
Perturbations of eqns. 1 - 3 result in
Substituting 2, from eqn. 6 into eqns. 4 and 5, and applying the
volt-second balance equation, results in
i j b p = k,,6,, + k,,6,, + rcT, + k,dJ1 ( 7 )
where
2LFs r, = ~ D2
The resultant small-signal circuit model of PWM inductor in the
DCM is shown in Fig. 2.
P Fig. 2 Unijkd small-signal circuit model of PWM switch
including inductor
DC analysis: The boost converter is analysed here. For a given
output current Z,, the following equation is satisfied:
IP = -I, (17)
10 ELECTRONICS LETTERS 7th January 1999 Vol. 35 No. 1
Since the inductor is considered as a short circuit under DC
condi- tions, the following equation holds:
V u p I p = V C & (18)
From eqns. 17 and 18, and Kirchoffs current law, the following
equation can be derived:
(19) v,plo IC = -
V * C vac and 1, = -- 1fcpIo
The volt-second balance relation for the inductor gives
VucD, = V c p D 2 (20)
From eqns. 3 and 20, the duty ratios can be obtained as
This process can be easily followed for buck and buck-boost con-
verters. All the results are summarised in Table 1.
Table 1: DC values
Buck I Boost I Buck-Boost I
Model validutions: The accuracy of the circuit model proposed in
this Letter is verified by comparison with the mathematical model
presented in [2], through computer simulations in the case of the
boost converter. The parameters are: V, = 5V, V, = 18.35V, L = SW,
C = 40p.F, R = 20Q F, = 100kHz.
In [2], only the following numerical result is given for the
con- trol-to-output transfer function without deriving the
analytical expressions of the small-signal model from the averaged
model:
(22) v^o(s ) - 21.82(1 - ~/281609)
- z(s) (1 + ~/2965.57)(1+ ~/761503)
1 2 3 4 5 10 10 10 10 10
f , Hz E33 Fig. 3 Comparison of control-to-output transfer
function employing pro- posed circuit model with mathematical model
in [2]
- proposed circuit model . . . . . . . . . . . mathematical
model
The frequency responses of eqn. 22 and the control-to-output
transfer function employing the proposed circuit model are shown in
Fig. 3. As can be seen in the Figure, there is no discrepancy
between the frequency responses of the mathematical model in [2]
and the proposed circuit model. This means that the proposed
small-signal circuit model of the PWM switch including inductor in
the DCM is more accurate than the small-signal circuit models in
[l].
Conclusions: Conventional small-signal circuit models in the DCM
show large discrepancies at high frequencies. The model in [2] is
more accurate than the conventional models. However, an accu- rate
circuit model has still not been reported. In this Letter, a uni-
fied small-signal circuit model of a PWM switch including inductor
has been developed, which is more accurate than the pre- vious
circuit model. The accuracy of the proposed model has been verified
by showing that there is no discrepancy between the fre- quency
responses of the mathematical model and the proposed model.
0 IEE 1999 Electronics Letters Online No: I9990069 Sung-Soo Hong
(Satellite Research Laboratory, Injormation and Communication
Research Center, Hyundai Electronics Industries Co. Ltd., San
136-1, Ami-ri, Bubal-rub, Ichon-si, Kyoungki-do, 467-701,
Korea)
4 November 1998
References
1 VORPERIAN, v.: 'Simplified analysis of PWM converters using
model of PWM switch, Part 11: Discontinuous conduction mode', IEEE
Trans. Aerosp. Electron. Syst., 1990, 26, (3), pp. 497-505
2 SUN, J. , MITCHELL, D., GREUEL, M . , KREIN, P., and BASS, R.:
'Modeling of PWM converters in discontinuous conduction mode - A
reexamination'. IEEE Power Electron Spec. Conf. Rec., 1998, pp. 6
15-622
3 HONG, s.s., IO, B.R., and YOUN, M.J.: 'Duty cycle generator
for average model of buck converter with current-mode control -
using analog behavioral modeling of PSPICE', IEEE Trans. Power
Electron., 1996, PE-11, (6), pp. 785-795
Adaptive multiwavelet prefilter
Yang Xinxing and Jiao Licheng
In conventional wavelet transforms, prefdtering is not necessary
due to the lowpass property of a scaling function. This is no
longer true for multiwavelet transforms. A novel adaptive
multiwavelet prefilter is presented which is designed to overcome
this problem. Simulation results show that the approximation
accuracy of the prefiltering algorithm is satisfactory.
Introduction: In the last 10 years, single wavelet transforms
have been widely developed and applied owing to their
time-frequency local property and fast transform algorithm (Mallat
algorithm). Certain properties of wavelets such as orthogonality,
compact sup- port, linear phase, and high approximatiodvanishing
moments of the basis function, are found to be useful in image
compression applications. Unfortunately, all real-valued scalar
(uni-) wavelets (with only one scaling function and one mother
wavelet) can never possess all the above properties simultaneously.
To overcome these drawbacks, more than one scaling function and
mother wavelet (so-called multiwavelets) need to be used. Recently,
new theories concerning multiwavelet have been developed [1 ~ 31.
Although, multiwavelets have many advantages over single wave- lets
in theory, unlike scalar wavelets, Mallat's multiresolution
decomposition and reconstruction algorithms cannot be employed
directly. Therefore it is necessary to initialise multiwavelets in
order that proper discrete decomposition algorithms can be devel-
oped. Recently, a few prefilter design methods have been pre-
sented [4 - 61. In this Letter, a novel adaptive multiwavelet
prefilter is designed to carry out discrete decomposition. Simula-
tion results show that the approximation accuracy of the prefilter-
ing algorithm is satisfactory and that the design method is
simple.
ELECTRONICS LElTERS 7th January 1999 Vol. 35 No. I 11
