144
IEEE POWER ELECTRONICS LETTERS, VOL. 3, NO. 4, DECEMBER 2005
Interpreting Small Signal Behavior of the
Synchronous Buck Converter at Light Load
Julian Yan Zhu
Abstract—The pulse-width-modulation (PWM) buck converter
with synchronous rectifiers operating at light load is usually modeled by its continuous conduction mode (CCM) model. However,
the actual power-stage small-signal control-to-output response
shows a different behavior from what the traditional CCM model
predicts, specifically, more damping around the double-pole frequency, instead of more resonance. This paper presents a modified
small-signal light-load model for a synchronous buck converter.
The developed model accurately predicts the actual small-signal
behavior of a PWM converter at light load. The derived averaged
switch model for light load can also be used for the small-signal
models of the other basic PWM converters operating in CCM at
light load.
Fig. 1. Bode plots of the power-stage transfer function of a synchronous buck
converter.
Index Terms—Light load, PWM converter, small-signal model.
I. INTRODUCTION
ULSE-WIDTH-MODULATION (PWM) buck converters
are usually modeled by the well-known small-signal
averaging models that have existed for many years [1], [2].
The models were developed for continuous and discontinuous
conduction operation at heavy load and light load respectively.
Among them, the continuous conduction mode (CCM) model
is more often used for its wide applications and easy control
design. The model predicts a control-to-output voltage transfer
function with a double-pole resonance. Its dc gain changes
with the input voltage, and its resonance become stronger with
an increased phase lag around the resonant poles at light load.
Therefore, the feedback control loop should be designed carefully to ensure stability over the entire range of input voltage
and load.
In recent years, the synchronous buck converter has been
widely adopted by industry by replacing the rectifier diode with
a complementary switching MOSFET. It can achieve high efficiency by taking advantage of the low conduction loss of the
MOSFET. Moreover, it is able to operate in CCM even at light
load, and its operation allows the inductor current to reverse,
making it possible to turn on the top MOSFET with zero voltage.
It was called a quasi-square-wave converter in the early 1990s
[3]–[5].
The CCM model described in Middlebrook’s paper [1] is usually used to design the control loop of the synchronous buck
converter. However, the actual measurement of the power-stage
transfer function shows a result different from what people expect from the model. The resonance is smoothed out when the
load current decreases, as Fig. 1 shows. This is against the theory
P
Manuscript received May 31, 2005; revised November 17, 2005. Recommended by Associate Editor J. Sun.
The author is with the Power Business Unit, Linear Technology Corporation,
Milpitas, CA 95035 USA (e-mail: jzhu@linear.com).
Digital Object Identifier 10.1109/LPEL.2005.863605
that says the resonance should be stronger at light load. We tend
to believe that the model could not be wrong, as it has been published in textbooks [6], [7] for more than twenty years. But our
measurement also seems to be correct. Thus, the reasons that
cause this discrepancy have to be ascertained so that a proper
control loop can be designed based on the right model. So far,
few papers have addressed this issue.
In past years, some papers discussed the models of
quasi-square-wave converters [8]–[10]. However, reference
[9] addresses the variable-frequency control model, which
is not applicable to our constant-frequency PWM control.
Reference [10] uses sampled-data modeling techniques to develop a small-signal model for the boost converter. It involves
complicated mathematical modeling and the calculation of
discrete-time small-signal model matrices. Therefore, it is not
convenient for engineering design. Although its results also
show the damping effect at light load, it does not explain the
reason. Reference [8] derives the steady state and small-signal
ac model for various nonisolated dc-dc converters. Its models
include the switching transition times, which are longer than
the traditional PWM hard-switching converters. However, its
ac model still predicts a more severe resonance at light load. So
the above phenomenon still cannot be explained. As such, it is
necessary to derive a light-load small-signal model for the real
situation.
This paper derives a new model for the synchronous buck
converter at light-load operation. The model predicts the lightload small-signal behavior that the experimental result exhibits.
It includes the nonideal device parameters that are usually ignored in the model.
II. SMALL SIGNAL MODEL OF SYNCHRONOUS
BUCK CONVERTER
The CCM model for the traditional PWM buck converter
is shown in Fig. 2. The small-signal control-to-output transfer
function is described in (1). It can be seen that the resonance is
affected by load resistor R and the inductor dc resistor
.A
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ZHU: INTERPRETING SMALL SIGNAL BEHAVIOR
Fig. 2.
145
CCM ac small-signal circuit model for a buck converter.
Fig. 4.
Small-signal model of the synchronous buck converter in Fig. 3.
Fig. 5.
Inductor current at very light load.
Fig. 3. Synchronous buck converter including parasitic components.
high R incurs strong resonance. The experimental measurement
shows that the control-to-output small-signal behavior of the
synchronous buck converter under heavy load can be predicted
well by this model. It is understandable, since the synchronous
buck converter works exactly as a traditional buck converter.
However, the light-load behavior is different, as shown in Fig. 1.
The following sections will derive a modified small-signal lightload model based on the PWM switch-averaging model:
amount of valley current is shifted into the Schottky diode. After
a short deadtime, , the top MOSFET M1 is turned on again,
the diode stops conducting, and the inductor current increases.
During the conducting time of the diode, a relatively high resistance appears in the current path. After averaging this resistance
can be written in (3).
over the whole switching cycle,
(3)
(1)
In order to evaluate the influence from the circuit parasitic
components at light load, the synchronous buck converter is
is the
drawn in Fig. 3, including parasitic components.
of the top MOSFET M1 and trace resistance,
is the equivis the
alent series resistance in the rectifier current path.
ESR of the input capacitors,
is the dc resistance of the inductor, and
is the equivalent series resistance of the output
capacitors.
The small-signal model including these parasitic components
is derived in Fig. 4. Its control-to-output transfer function is
modified as in (2), shown at the bottom of the page. It can be
,
and
all contribute to an equivalent seseen that
, which adds to the damping of the
ries resistor together with
model. Assuming that the device temperatures are around 25
at all the load conditions,
,
and
do not change significantly with the output current, while
does. This is because
is combined with
of the bottom MOSFET M2 and
the ESR of the Schottky diode, where the ESR of the Schottky
diode increases considerably when the forward current is small.
It can be found from most diode datasheets.
At light load, the valley current is close to zero before it goes
to negative. When the MOSFET M2 is turned off, this small
is normally very small compared to the switching
Although
cycle (from tens of nanoseconds to a hundred nanoseconds), the
can easily increase from tens of
to hundreds
averaged
of
, because the ESR of the diode can go up to a few ohms
when the forward current drops below 1A. The increased
damps the resonance at light load.
When the load current reduces further, the inductor current
goes negative, as shown in Fig. 5. At the turn-off points of the
top MOSFET and the bottom MOSFET, the inductor current
charge and discharge the junction capacitors of the MOSFETs
ramps up and down
and diodes. The switch node voltage
linearly during the switching intervals, which become longer
than those under heavy load. The small-signal behavior changes
accordingly. In reality, the inductor current is not always able to
up to , The resultant
will have a rising edge
charge
in Fig. 5.
Based on the waveforms in Fig. 5, a new model is derived
as follows. Assume the transition intervals are very small and
., The inductor current can then be treated as a constant current during the intervals. According to the three-terminal-switch average model [2], the averaged terminal current
and voltage can be written in (4) and (5):
(4)
(5)
(2)
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146
IEEE POWER ELECTRONICS LETTERS, VOL. 3, NO. 4, DECEMBER 2005
TABLE I
PARASITIC COMPONENTS OF THE DEVICES AND CIRCUIT
Fig. 6. Small-signal model of the synchronous buck converter at light load.
On the other hand, the input current is also affected by the
input voltage disturbance with an additional current component.
When the switching intervals approach zero, the model is reduced to the original without transition intervals in Fig. 4. The
complete model parameters are derived and given in the Appendix.
In (4) and (5):
III. EXPERIMENTAL VERIFICATION
In the above formulas,
,
,
is the junction capacitance
, respectively, and
of M1, M2 and the Schottky diode during
,
,
is the junction capacitance of M1, M2 and the
Schottky diode during
, respectively.
The small-signal model can be derived in (6) and (7) by applying a small perturbation around the steady-state operating
points to (4) and (5) as follows:
(6)
(7)
Letting
and
, the equivalent small small-signal
ac model is shown in Fig. 6.
It can be seen that this light-load operation includes an additional lossless resistance
in the small-signal ac equivalent
circuit. This resistance causes more damping to the resonance
at light load. The value of
is derived in (8), and the control-to-output voltage transfer function can be modified as (9),
shown at the bottom of the page, where
:
(8)
The developed model was established for an 8 24 V input,
3.3 V/15 A output synchronous buck converter. The converter
operates at 525 kHz. The output inductor and capacitor are
and
, respectively. The top and
chosen as 1
bottom MOSFETs are HAT2168H and HAT2165H, respectively. A Schottky diode (B340) is also added in parallel with
the synchronous bottom FET. Other parameters of the converter
are summarized in Table I.
Based on the above model, a MathCAD program is written
to calculate the transfer functions of the power and control circuits. The calculated control-to-output transfer functions for different load currents are plotted in Fig. 7, in comparison with the
measured ones (using a Venable frequency analyzer). It can be
seen that the calculated transfer functions correlate well with the
measured results both at light load and heavy load. The transfer
functions based on the original model are also plotted in Fig. 8,
which shows significant discrepancies to the measured results
at light load.
It is noted that the light-load small-signal behavior
observed above actually causes less phase lag, which is good
for the stability of the system fortunately. This explains the
fact that the compensation is commonly designed at heavy
load for the worst case consideration, and thus helps
us better understand the light-load small-signal model of
PWM converters. Also, the developed model is helpful
to the appropriate compensation loop design for some
special converters operating at critical conduction mode or
quasi-square-wave mode.
IV. CONCLUSIONS
This letter presents a modified light-load small-signal ac
model for a synchronous buck converter. The developed model
accurately predicts the actual small-signal behavior of a PWM
converter at light load: more damping effect, contrary to
strong resonance, as the ideal model predicts. The derived
averaged-switch model for the light load can also be used to
model other synchronous PWM converters operating at light
load, such as boost, buck/boost, sepic and Cuk converters. The
(9)
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ZHU: INTERPRETING SMALL SIGNAL BEHAVIOR
147
Fig. 7. Measured power-stage small-signal response and the calculated ones with the modified model (solid line: measured; dotted line: simulated).
Fig. 8. Measured power-stage small-signal response and the calculated ones with the original model (solid line: measured; dotted line: simulated).
general modeling idea is also helpful for accurate modeling of
some converters operating in the critical conduction mode.
APPENDIX
Note:
is assumed at light load.
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[7] G. Chrysiss, High Frequency Switching Power Supplies: Theory and Design. New York: MCGraw-Hill, 1984.
[8] W. Moussa and J. E. Morris, “DC and AC characteristics of zero voltage
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[10] A. J. Forsyth and R. I. Gregory, “Small-signal modeling and control of
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