n discontinuous conduc - Industrial Electronics, IEEE Transactions on IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 4, AUGUST 2000
759
Spectral Characteristics of Randomly Switched PWM
DC/DC Converters Operating in Discontinuous
Conduction Mode
K. K. Tse, Member, IEEE, Henry Shu-hung Chung, Member, IEEE, S. Y. R. Hui, Senior Member, IEEE, and
H. C. So, Member, IEEE
AbstractThis paper addresses a comparative study of the spec-
tral characteristics of four random-switching schemes that apply
to the basic pulsewidth-modulation (PWM) dc/dc converters oper-
ating in discontinuous conduction mode (DCM). They include ran-
domized pulse position modulation, randomized pulsewidth mod-
ulation, and randomized carrier frequency modulation with fixed
duty cycle and with fixed duty time, respectively. Mathematical
models that characterize the input current and output voltage of
the three basic PWM converters operating in DCM are derived. In
particular, the effectiveness of spreading the dominant switching
harmonics in the input current that normally exist in the standard
PWM scheme and the introduction of low-frequency harmonics in
the output voltage with respect to the randomness level are inves-
tigated. The validity of the models and analyses are confirmed ex-
perimentally by using a dc/dc buck converter.
Index TermsDCDC power conversion, power electronics,
pulsewidth modulation, random-switching techniques, switching
circuits.
I. I
NTRODUCTION
N
OWADAYS, switching converters have to be designed not
only to meet electrical specifications, but also to comply
with the international electromagnetic compatibility (EMC)
standards. With the introduction of the EMC directive, there
is an increasing awareness of EMC issues that highlights the
electromagnetic interference (EMI) problem of switching con-
verters [1], [2]. The pulsewidth-modulation (PWM) scheme is
a common technique that controls the power flow of switching
converters. The switching frequency is fixed and the duty cycle
of the semiconductor switches is the control parameter. Under
the steady-state operation, both the duty cycle and the switching
frequency of the gate signals are kept constant. The harmonic
powers of the input current and the output voltage concentrate
on the multiples of the switching frequency. During the last
decade, random switching, which is originated from statistical
communication theory [3], has been applied to switching
converters. Certain parameters, namely, random (stochastic)
variables, in the PWM modulator are subject to randomization.
The methodology is not merely a way to comply with the EMC
regulations, but also provides a flexible and practical approach
Manuscript received June 9, 1999; revised February 3, 2000. Abstract pub-
lished on the Internet April 21, 2000.
The authors are with the Department of Electronic Engineering, City Univer-
sity of Hong Kong, Kowloon, Hong Kong.
Publisher Item Identifier S 0278-0046(00)06831-3.
to solving acoustic noise problems in inverter-based motor
drive [4].
In this paper, the use of random-switching schemes for basic
PWM dc/dc converters operating in discontinuous conduction
mode (DCM) is presented. Under DCM, the energy stored
in the inductor is completely transferred to the output load
in every switching cycle. This feature inherently improves
the converter stability and offers the advantage of stable
closed-loop operation [5]. However, the pulsating input current
waveform consists of the harmonics at the multiples of the
switching frequency, which are the major sources of conducted
EMI [6]. With random-switching schemes, the harmonic power
concentration along the frequency spectrum will be relaxed,
and the peak level of the power spectral density (PSD) becomes
less than that of classical PWM. Discrete harmonics are
significantly reduced and the harmonic power is spread over as
continuous noise spectrum of insignificant magnitude. There
are many randomization schemes, which can spread the har-
monic energy present at one frequency across other frequencies.
Their syntheses have been addressed in many papers [7][9].
Based on the standard PWM scheme, four modulation schemes
can be categorized. They include random-pulse-position mod-
ulation (RPPM), random-pulsewidth modulation (RPWM),
and random-carrier-frequency modulation with fixed duty
cycle (RCFMFD) and with variable duty cycle (RCFMVD),
respectively. It has been shown in [11] that, among the four
schemes, RCFMFD introduces the lowest low-frequency output
noise voltage within the passband of the low-pass filter in dc/dc
converters operating in continuous conduction mode (CCM). It
is, therefore, the best choice for application to converters that
require tight output voltage regulation. However, the suitability
and applicability of the use of random modulation schemes for
converters operating in DCM have not been addressed.
A comparative study of the spectral characteristics of the
four random-switching schemes that apply to the PWM dc/dc
converters operating in DCM will be presented in this paper.
Mathematical models that characterize the input current and
output voltage of the three basic PWM converters, including
buck, boost, and buckboost converters, operating in DCM
are derived. The methodology starts with the derivations of
the spectral characteristics of a generalized current waveform
under different random switching schemes and an equivalent
transimpedance network for the three basic dc/dc converters.
Investigation emphases include the effectiveness of spreading
the dominant switching harmonics in the input current that
02780046/00$10.00 © 2000 IEEE 760
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 4, AUGUST 2000
(a)
(b)
(c)
(d)
Fig. 1.
Three PWM dc/dc converters. (a) Buck converter. (b) Boost converter. (c) Buckboost converter. (d) Theoretical waveforms of
g, i , and i .
normally exist in standard PWM scheme and the degree of
low-frequency harmonics introduced in the output voltage
with respect to the randomness level. They are quantitatively
studied with uniform probability density distribution on the
stochastic variables. The validity of the models and analyses
are confirmed experimentally by using a dc/dc buck converter.
II. R
ANDOM
S
WITCHING
S
CHEMES
Fig. 1(a)(c) shows the circuit diagrams of the three basic
PWM converters. The gate signal , inductor current
, and the
diode current
in a generic switching cycle
are illustrated in
Fig. 1(d). Under DCM, all current waveforms can be represented
by triangular pulses and expressed in a general form of
(1)
in which
for
for
elsewhere
(2)
where
time at which the th switching cycle starts;
duration of the rising slope;
duration of the falling slope;
delay time of the pulse;
,
slopes of the rising edge and the falling edge, re-
spectively.
is the initial value of the pulse at
and can be related to
in the form of
(3)
where
is a constant. In addition,
can be related to
by
(4)
where
is a constant. Moreover, the duty cycle
of
in
Fig. 1(d) in the th cycle is defined as
(5)
where
is the switching period of the th cycle.
The above equations form the basis for the derivations of the
input current
and the output voltage
of the three con-
verters.
can be expressed in the form of (2)(4). The values
of
,
,
, and
are tabulated in Table I(a). Moreover, as
shown in Fig. 2, an equivalent circuit can be used to represent
the output stage of the three converters. It consists of a current
source
, an output capacitor
, and an output load
.
and
form a transimpedance
, which is defined as the ratio be-
tween the spectral magnitude of the output voltage
and
current source
(6) TSE et al.: SPECTRAL CHARACTERISTICS OF RANDOMLY SWITCHED PWM DC/DC CONVERTERS
761
Fig. 2.
Equivalent circuit of the output stage of the three converters.
TABLE I
(a) V
ALUES OF
A, B, F ,
AND
H
FOR
i
IN THE
T
HREE
DC/DC C
ONVERTERS
.
(b) V
ALUES OF
A, B, F ,
AND
H
FOR
i
IN THE
T
HREE
DC/DC C
ONVERTERS
(a)
(b)
Depending on the converter,
can be
or
and is defined
in Table I(b), together with the values of
,
,
, and
. In
Table I,
and
are the nominal values of the input and
output voltages, respectively. The values of
for the three
converters are tabulated in Table II [5].
A. Characteristics of Different Random Modulation Schemes
As shown in Fig. 1(d)
and
under different random
switching schemes are subject to randomization of
,
or
, which can be considered as stochastic variables. RPPM is
similar to the classical PWM scheme with constant-switching
frequency. However, the pulse position is randomized within
each switching period, instead of commencing at the start of
each cycle. The delay time
controls the randomness level
in RPPM scheme. For RPWM, each pulse starts without time
delay (i.e.,
), but
is randomized. The average value
of
over the switching period is equal to the nominal duty
cycle. RCFMFD exhibits randomized switching period
and
constant duty cycle, while RCFMVD exhibits randomized
switching period
and constant pulse duration. Characteris-
tics of the stochastic variables in the four random switching
schemes are summarized in Table III.
TABLE II
V
OF THE
T
HREE
DC/DC C
ONVERTERS
[5]
TABLE III
C
HARACTERISTICS OF
S
TOCHASTIC
V
ARIABLES IN THE
F
OUR
R
ANDOM
S
WITCHING
S
CHEMES
B. Definitions of the Randomness Level
Randomness level
for each scheme is defined as follows.
For RPPM,
(7)
where the delay time
.
and
are the lower and
upper limits of the pulse delay time in each cycle.
is generally
equal to zero.
is the nominal switching period.
For RPWM,
(8)
where the pulse on-time
. Thus, the duty cycle
varies between the minimum