IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 1, JANUARY 2012
641
Comparison of Wide- and High-Frequency
Duty-Ratio-to-Inductor-Current Transfer Functions
of DC–DC PWM Buck Converter in CCM
Nisha Kondrath and Marian K. Kazimierczuk
Abstract—Wide- (WF) and high-frequency (HF) small-signal models are
presented for pulsewidth-modulated dc–dc buck converter power stage.
An exact duty-ratio-to-inductor-current transfer function is derived using
the WF model and compared to an approximate transfer function derived
using the HF model for the buck converter power stage in continuous
conduction mode. The theoretical results are validated using experimental
results.
Index Terms—Current-mode control, dc–dc converters, power-stage
transfer function.
Fig. 1. Open-loop PWM dc–dc buck converter. (a) Power stage. (b) SSM
for CCM.
I. I NTRODUCTION
Modeling of the power stages of pulsewidth-modulated (PWM)
dc–dc converters has been presented in several publications [1]–[6].
An accurate modeling of the power stage is essential as the powerstage transfer functions are required in the correct modeling of closedloop functions [7]–[16]. It is commonly believed that current-mode
control converts the power stage into an inductor. A first-order dutycycle-to-inductor-current transfer function was presented in [7]. This
letter presents a wide-frequency (WF) second-order transfer function
for the power stage, which is valid for the frequency range 0 <
f < fs /2 and validates the model with experimental results. The
results also show that while the power stage is inductive in the highfrequency (HF) range, the inductive model does not represent the
whole frequency range as commonly believed. In this letter, the powerstage model for the buck converter derived using energy-conservation
approach [1] has been selected to derive the WF and HF models
to obtain the corresponding duty-cycle-to-inductor-current transfer
functions for the buck converter operating in continuous conduction
mode (CCM).
The objectives of this letter are to present WF, low-frequency (LF),
and HF small-signal models (SSMs) of PWM dc–dc buck converter
power stage for CCM, to compare the resultant duty-cycle-to-inductorcurrent transfer functions, and to validate the presented theory using
experimental frequency responses.
II. SSM OF B UCK C ONVERTER
An open-loop PWM dc–dc buck converter is shown in Fig. 1(a).
The circuit components are the following: switching components S1
and D1 , passive components L and C, and load resistance RL . Let
VI , D, IL , and VO be the dc components, and let vi , d, il , and vo
be the small-signal components of input voltage, duty cycle, inductor
Manuscript received March 12, 2010; revised June 24, 2010 and
December 21, 2010; accepted March 3, 2011. Date of publication March 28,
2011; date of current version October 4, 2011.
N. Kondrath is with the Department of Electrical and Computer Engineering,
University of Minnesota Duluth, Duluth, MN 55812 USA (e-mail: nkondrat@
d.umn.edu).
M. K. Kazimierczuk is with the Department of Electrical Engineering, Wright State University, Dayton, OH 45435 USA (e-mail: marian.
kazimierczuk@wright.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2011.2134053
Fig. 2. SSM of a buck converter operating in CCM to derive duty-cycle-toinductor-current transfer function Tpi . (a) For WF range. (b) For LF range.
(c) For HF range.
current, and output voltage, respectively. Let iL = IL + il be the total
inductor current, vO = VO + vo be the total output voltage, and io be
the small-signal component of the load current.
The SSM of the PWM dc–dc buck converter for CCM [1] is shown
in Fig. 1(b). This model relies on the principle of energy conservation
and thereby takes into account the inductor current ripple. It is valid for
magnitudes of small-signal components at least one order lower than
the corresponding dc components and for frequencies no greater than
half the switching frequency fs , i.e., f ≤ fs /2 [1]–[3], [7], [11], [13].
In the figure, rC is the capacitor equivalent series resistance (ESR),
and the resistance r = DrDS + (1 − D)RF + rL , where rDS is the
MOSFET ON-resistance, RF is the diode forward resistance, and rL is
the inductor ESR. There are three input quantities that would affect the
inductor current in the converter: 1) duty cycle; 2) input voltage; and
3) output current. This letter analyzes the duty-cycle-to-inductor-current
transfer function, which is the power-stage transfer function relevant to
current-mode-controlled PWM dc–dc buck converter in CCM.
III. D UTY-R ATIO - TO -I NDUCTOR -C URRENT T RANSFER
F UNCTION OF B UCK C ONVERTER
In this section, the WF duty-cycle-to-inductor-current transfer function of a buck converter in CCM is derived, including parasitics and
MOSFET delay. Also, HF and LF transfer functions of the converter
are derived.
A. WF Transfer Function
Fig. 2(a) shows the SSM to derive the duty-cycle-to-inductorcurrent transfer function for a WF range, i.e., 0 ≤ f ≤ fs /2. This is
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 1, JANUARY 2012
obtained by reducing the input voltage and output current to zero in
the buck converter SSM shown in Fig. 1(b). From the figure
VI d(s)
Z1 (s) + Z2 (s)
il (s) =
(1)
where
Z1 (s) = r + sL
Z2 (s) = RL 1
sC
+ rC
(2)
RL (1 + sCrC )
.
=
1 + sC(RL + rC )
(3)
Thus, the WF duty-cycle-to-inductor-current transfer function is obtained as
il (s) Tpi (s) =
d(s) v
VI
Z1 (s) + Z2 (s)
=
i =io =0
VI
=
RL (1+sCrC )
r + sL + 1+sC(R
L +rC )
s + ωzi
= Tpix 2
s + 2ζω0 s + ω02
(4)
where
Tpix =
VI
L
1
C(RL + rC )
RL + r
ω0 =
LC(RL + rC )
C[RL rC + rC r + RL r] + L
and ζ = .
2 LC(RL + rC )(RL + r)
ωzi =
Fig. 3. Bode plots of duty-cycle-to-inductor-current transfer function Tpi for
CCM using MATLAB.
(5)
Including the MOSFET delay [3]
IV. S IMULATION R ESULTS
Td (s) = e−std ≈ −
s−
s+
2
td
2
td
(6)
the duty-ratio-to-inductor-current transfer function becomes
Tpid (s) = Tpi (s)Td (s) = −Tpix
s−
s + ωzi
2
2
s + 2ζω0 s + ω0 s +
2
td
2
td
.
(7)
B. LF Transfer Function
At dc and LFs, i.e., for 0 ≤ f ≤ fzi , the SSM of the buck converter
in Fig. 2(a) reduces to the SSM shown in Fig. 2(b). Therefore,
at LFs, il (s) = VI d(s)/(RL + r). Thus, the duty-ratio-to-inductorcurrent transfer function reduces to
il (s) TpiLF (s) =
d(s) v
=
i =io =0
VI
RL + r
(8)
which indicates that the power stage is resistive at dc and LFs. Also,
Z2LF = RL .
C. HF Transfer Function
At HFs, i.e., for f0 ≤ f ≤ fs /2, the inductor dominates all of the
other components, and hence, the buck converter SSM in Fig. 2(a)
reduces to the SSM shown in Fig. 2(c). From Fig. 2(c), il (s) =
VI d(s)/(sL). Therefore, at HFs, the duty-cycle-to-inductor-current
transfer function is
TpiHF (s) =
which indicates that the power stage with current-mode control can be
modeled as an inductor at HFs.
il (s) d(s) v
i =io =0
=
1
VI
= Tpix
sL
s
(9)
The selected buck converter design operating in CCM had the following parameters: VI = 35 V, VO = 7 V, IO = 0.7 A, D = 0.208,
L = 254 μH, rL = 36 mΩ, C = 68 μF, rC = 520 mΩ, RL = 10 Ω,
rDS = 0.4 Ω, RF = 0.1 Ω, VF = 0.7 Ω, and fs = 100 kHz. The
MOSFET delay was measured as td = 0.55 μs.
Bode plots illustrating the duty-cycle-to-inductor-current transfer
function for buck converter operating in CCM for WF and HF ranges
are shown in Fig. 3(a) and (b). The WF transfer function is a lowpass filter transfer function with one zero and two complex conjugate
poles. For the selected design, the corner frequency of the zero is
fzi = 222.5 Hz, the corner frequency of the poles is f0 = 1.184 kHz,
and the dc gain is Tpio = 10.73 dBA. The HF transfer function is
an integrator function. This indicates that the power stage behaves as
an inductor at HFs. The WF Bode plots without delay converge to
the HF Bode plots at HFs. However, the MOSFET delay component
introduces additional phase at HFs.
V. E XPERIMENTAL R ESULTS
An experimental setup to measure the duty-cycle-to-inductorcurrent transfer function under small-signal conditions for an openloop buck converter is shown in Fig. 4. The switching components
were IRF530 power MOSFET and MUR820 power diode. IR2110
was used to drive the high-side MOSFET. The operating point, passive
component values, and the parasitic values were the same as given
in Section IV. LM357N was used as a comparator, with a ramp signal
vramp of an amplitude of 4 V at the inverting input and a dc voltage Vdc
at the noninverting input, to provide the input to the driver. To obtain a
vGS waveform with D = 0.208, the dc voltage was VDC = 0.638 V.
The pulsewidth modulator consisting of the comparator introduces a
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 1, JANUARY 2012
643
The WF exact transfer function Tpi is a second-order low-pass filter
function with a zero. The frequency of the zero is lower than the corner
frequency of the complex poles. The power stage SSM reduces to an
inductor at HFs. The HF approximate transfer function TpiHF is a firstorder integrator function due to cancellation of the effect of zero by
one of the poles. It reduces to the form TpiHF (s) = Tpix /s, where
Tpix = VI /L for buck converter.
From the simulations as well as the measured plots, it can be seen
that the power stage of PWM dc–dc
converters can be modeled as an
√
inductor at HFs, from f0 ≈ 1/ LC to fs /2. In this frequency range,
the duty-cycle-to-inductor-current transfer function |Tpi | ≈ VI /(ωL)
and the phase φT pi = −90◦ for the buck converter. However, for low
frequencies from 0 to f0 , a complete WF SSM should be considered.
Fig. 4.
Experimental setup to measure Tpi for the buck converter in CCM.
R EFERENCES
Fig. 5. Experimental Bode plots of duty-cycle-to-inductor-current transfer function for the buck converter in CCM measured with HP 4194A
Impedance/Gain-Phase Analyzer. The lower limit of the axes are −30 dBA and
−140◦ , and the upper limit of the axes are 20 dBA and 60◦ with 5 dBA/div
and 20◦ /div for |Tpi | dBA and φTpi (◦ ), respectively.
gain [2], [3] of 20 log(1/4) = −12 dB. The small-signal vosc with
an amplitude of 0.05 V with f = 10 Hz to 100 kHz was supplied by
the Hewlett-Packard (HP) 4194A Impedance/Gain-Phase Analyzer. A
Pearson model 411 wide-bandwidth ac current probe was used to sense
the small-signal inductor current. The output voltage vtest of the probe
was given to the test channel of the HP 4194A Impedance/Gain-Phase
Analyzer. The probe frequency response was measured using a simple
resistive load of RL = 1 Ω and the same dc current I = 0.7 A as in the
buck converter design. The effect of the probe frequency response on
the measured Tpi response was compensated using the compensation
function of the 4194A network analyzer.
Fig. 5 shows the experimental Bode plots of the duty-cycle-toinductor-current transfer function for the buck converter in CCM
measured using HP 4194A Gain-Phase Analyzer under small-signal
conditions. A gain of 12 dBA must be added to the magnitude response
to account for the duty-cycle modulator gain. A positive phase was
added to LF phase response by the input capacitor used to supply the
analyzer input. The resultant Bode plots are almost identical to the WF
Bode plots with delay shown in Fig. 3.
VI. C ONCLUSION
The WF and HF small-signal models which can be used to derive the
duty-ratio-to-inductor-current transfer function of the buck converter
have been presented. The duty-cycle-to-inductor-current transfer functions of the buck converter for WF range as well as for HF range have
been derived and compared. Experimental results have been presented
to validate the theoretical results.
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