In this study, the performance of a direct current (DC)–DC buck converter is analyzed in the presence of nonidealities in passive components and semiconductor devices. The effect of these nonidealities on the various design issues of a DC–DC buck converter is studied. An improved expression for duty cycle is developed to compensate the losses that occur because of the nonidealities. The design equations for inductor and capacitor calculation are modified based on this improved expression. The effect of the variation in capacitor equivalent series resistance (ESR) on output voltage ripple (OVR) is analyzed in detail. It is observed that the value of required capacitance increases with ESR. However, beyond a maximum value of ESR (
r
_{c,max}
), the capacitor is unable to maintain OVR within a specified limit. The expression of
r
_{c,max}
is derived in terms of specified OVR and inductor current ripple. Finally, these theoretical studies are validated through MATLAB simulation and experimental results.
I. INTRODUCTION
Direct current (DC)–DC converters are widely used as power supply in various applications
[1]

[4]
. Several applications, such as aerospace, military, chemical refineries, and mines, require highly regulated, optimally designed, and compact power supplies. For such applications, an optimum design of inductance and capacitance for buck converter is reported in
[5]
,
[6]
. In all practical DC–DC converters, power loss occurs because of the internal resistances of inductors, capacitors, and nonideal switching devices. Most power electronic textbooks and papers have neglected some or all these nonideal elements while analyzing and designing DC–DC converters
[7]

[11]
. However, this condition is unacceptable for an accurate and welldesigned power supply because these nonidealities affect the desired values of the duty cycle, inductor, and capacitor of a DC–DC converter
[12]
,
[13]
. For example, the expression for the duty cycle of an ideal buck converter is
where
V
_{o}
is the output voltage, and
V
_{g}
is the input voltage.
However, this relation does not remain valid for a nonideal buck converter. Owing to the parasitic resistances of inductor, capacitor, diode, and switch, practical DC–DC converters have power losses. Therefore, the actual duty cycle of a nonideal DC–DC buck converter should be greater than the ideal duty cycle given in Equ. (1) to compensate these losses. An improved expression for this actual duty cycle is derived in this study.
Similarly, the equivalent series resistance (ESR) of an output capacitor plays an important role in the design of capacitors
[14]
. As the ESR increases, the output voltage ripples (OVR) also increases, thereby reducing the effectiveness of filter capacitors. Moreover, high ESR may lead to instability and increase the power loss in converters
[15]
. Therefore, arbitrary selection of ESR is inadvisable for a precisely designed DC–DC buck converter. However, the ESR of a capacitor is a parasitic parameter that cannot be avoided by capacitor manufacturers. Nonetheless, power supply designers always prefer a capacitance with low ESR. No analytical solution is available in the literature to evaluate the upper limit of ESR that can be used without exceeding the specified OVR at a particular switching frequency. In this study, a formula for the maximum value of ESR is developed through an indepth analysis of capacitor voltage ripples to select a capacitor with proper ESR.
The remainder of this paper is organized as follows to resolve the abovementioned issues for a buck converter: First, a detailed analysis of a nonideal buck converter is presented. Second, improved relationships for duty cycle, inductance, and capacitance design are proposed. Third, the analytical expression for maximum allowable ESR for specified OVR is proposed. Finally, MATLAB simulations and experimental results are obtained to confirm these theoretical studies.
II. ANALYSIS OF A NONIDEAL DC–DC BUCK CONVERTER
The power circuit of a nonideal DC–DC buck converter is shown in
Fig. 1
. All symbols have a standard meaning, as indicated in
Table I
. The buck converter is operating in a continuous current conduction mode with duty cycle
D
and switching frequency
f
(or switching period
T
)
[7]
. The voltage and current equations of the buck converter during the switchon period (0 <
t
≤
DT
) and the switchoff period (
DT
<
t
≤
T
) are expressed below:
Nonideal DC–DC buck converter.
BUCK CONVERTER SPECIFICATIONS
BUCK CONVERTER SPECIFICATIONS
Mode 1
: Switchon (0 <
t
≤
DT
)
Mode 2
: Switchoff (
DT
<
t
≤
T
)
 A. SteadyState Analysis
For steadystate analysis, the time variables are substituted by their respective DC values as
[8]
i_{L}
(
t
) =
I_{L}
,
v_{g}
(
t
) =
V_{g}
,
v_{c}
(
t
) =
V_{c}
v_{o}
(
t
) =
V_{o}
.
According to the principle of inductor voltsecond balance, in the steadystate, the average inductor voltage must be equal to zero.
Similarly, according to the principle of capacitor charge balance, in the steadystate, the average capacitor current must be equal to zero.
The average output voltage of the buck converter is
By substituting Equs. (2) and (5) into Equ. (8), we obtain
Substituting Equs. (3) and (6) into Equ. (9) obtains
Similarly, substituting Equs. (4), (7), and (9) into Equ. (10) gives
Replacing Equ. (12) into Equ. (11) provides the relationship for average output voltage as
where
D
^{ꞌ}
= 1 −
D
.
Equ. (14) depicts that the output voltage of the buck converter depends on nonidealities and load resistance.
The output voltage variation versus duty cycle plot is shown in
Fig. 2
at a constant input voltage and different load resistances. For a particular duty cycle, the nonideal buck converter produces an output voltage less than that of the ideal buck converter. The output voltage decreases further as the load resistance is decreased.
Fig. 3
shows the output voltage variation for different values of input voltages at constant
R
. For a particular duty cycle, the difference between the output voltage of ideal and nonideal buck converter becomes larger as input voltage increases. Therefore, the metal oxide semiconductor fieldeffect transistor (MOSFET) switch should be kept ON for an extended time to achieve the same output voltage in the presence of nonidealities. In other words, the actual duty cycle must be greater than the ideal duty cycle given by Equ. (1).
Output voltage variation with duty cycle at various R and constant V_{g}.
Output voltage variation with duty cycle at various V_{g} and constant R.
An improved expression of actual duty cycle is derived by solving Equ. (14) as
where
D
_{ideal}
=
V
_{o}
/
V
_{g}
is the duty cycle for an ideal buck converter.
The modified duty cycle
D
in Equ. (15) and the ideal duty cycle
D
_{ideal}
in Equ. (1) are plotted with the desired output voltage (
V_{o}
), as shown in
Fig. 4
. It verifies that the actual duty cycle should be greater than the ideal duty cycle to obtain the desired output voltage.
Duty cycle variation with output voltage.
 B. Inductor Current Ripple (ICR) and Inductor Design
Let
x
_{L}
be the ICR factor and
Δi
_{L}
be the ripple current such that
Δi
_{L}
=
x
_{L}
I
_{L}
.
The magnitude of ripple current
Δi
_{L}
in the steadystate is
Substituting
I
_{L}
from Equ. (12),
Replacing
Δi
_{L}
=
x
_{L}
I
_{L}
=
x
_{L}
V
_{O}
/
R
and rearranging Equ. (17),
Equ. (18) provides the design value of inductance for the desired ICR in terms of converter parameters and nonideal elements. The required inductance is calculated for ideal and nonideal buck converters and is plotted with respect to duty cycle in
Fig. 5
, which depicts clearly that in the nonideal case, the required inductance value is more than the ideal case. The difference becomes significant at low duty cycles.
Inductance variation with duty cycle.
 C. OVR and Capacitor Design
The capacitor current and voltage ripple waveforms in the steadystate are shown in
Fig. 6
. If the ESR of output capacitor is considered, then the OVR
Δv
_{o}
consists of the following two components:
Capacitor current and voltage ripple waveforms.
1. Voltage ripple caused by capacitor (
Δv
_{c}
)
2. Voltage ripple caused by the presence of ESR (
Δv
_{rc}
)
Analysis during switchon
(0 <
t
≤
DT
): The capacitor current dynamics can be given as
[16]
The voltage ripple contribution by the capacitor itself is
where
Δv
_{c}
(0) is the initial voltage across the capacitor at
t
= 0. The voltage ripple contribution by the ESR of the capacitor is
Therefore, the total voltage ripple during the switchon period is
The minimum value of
Δv
_{o}
(
t
) occurs at time
t
_{1}
and is given by
Analysis during switchoff
(
DT
<
t
≤
T
): The capacitor current dynamics can be given as
The voltage ripple contribution by the capacitor itself is
where
Δv
_{c}
(
DT
) is the initial voltage across the capacitor at
t
=
DT
. In the steadystate,
Δv
_{c}
(
DT
) =
Δv
_{c}
(
0
).
The voltage ripple contribution by the ESR of the capacitor is
Therefore, the total voltage ripple during the switchoff period is
The maximum value of
Δv
_{o}
(
t
) occurs at time
t_{2}
and is given by
1) Voltage Ripples:
The total peaktopeak OVR is
Substituting the values from Equs. (23) and (29) into Equ. (31) and simplifying them gives
Letting
r
_{c}
= 0 gives the OVR for the ideal capacitor as available in the literature.
We can also find the contribution of capacitor voltage ripple and ESR voltage ripple in the peaktopeak OVR.
The ripple voltage contribution by the capacitor is
Δ
v_{c}
= Δ
v_{c}
(
t
_{2}
)Δ
v_{c}
(
t
_{1}
).
The ripple voltage contribution by ESR is
Δ
v_{rc}
= Δ
v_{rc}
(
t
_{2}
)Δ
v_{rc}
(
t
_{1}
).
By substituting the values from Equs. (24) and (30), we have
The three voltage ripple variations defined in Equs. (32)(34) are plotted with respect to ESR, as shown in
Fig. 7
. From this figure, with an increase in ESR,
Δv
_{rc}
increases at a faster rate than
Δv
_{c}
decreases, thereby causing a net increase in
Δv
_{o}
. However, as the value of
r
_{c}
increases beyond
r
_{c,max}
,
Δv
_{rc}
becomes higher than
Δv
_{o}
, which is practically impossible. This result implies that the capacitor is no longer able to keep OVR within the specified limit for
r
_{c}
>
r
_{c,max}
. The exact relation for this
r
_{c,max}
is derived later.
Variation in voltage ripples with ESR.
2) Output Capacitor Design:
Let the specified maximum OVR be
Δv
_{om}
. The capacitor is designed such that the following condition must be satisfied:
By substituting
Δv
_{o}
from Equ. (32) and simplifying it, gives
Equ. (36) is a quadratic constraint in
C
. It is solved to generate the minimum value of the capacitor
C
as
This equation provides the value of minimum capacitance required for
r
_{c}
≤
r
_{c,max}
.
3) Derivation for Maximum Permissible ESR:
The additional term
in Equ. (32) appears because of the presence of ESR. As ESR increases, OVR also increases. If the ESR of capacitor is not selected properly, then it may increase the total OVR beyond the maximum permissible value. Therefore, the relationship for the maximum permissible value of ESR for the specified OVR and switching frequency should be determined. In Equ. (37),
C
_{min}
must be a real quantity that satisfies the following condition:
On simplifying,
Therefore, the maximum permissible value of ESR (
r
_{c,max}
) for specified OVR (
Δv
_{om}
)and ICR(
Δi
_{L}
) is given by
If the ESR value is greater than the
r
_{c,max}
in Equ. (40), then the capacitor will not be able to keep the steadystate OVR in the specified limit. This condition is verified by the experimental results in the subsequent section.
Substituting Equ. (40) into Equ. (37) obtains the minimum value of capacitor at
r
_{c}
=
r
_{c,max}
as
From Equ. (32), for an ideal capacitor (
r
_{c}
= 0), the minimum value of
C
is
Equ. (41) provides the minimum capacitance required for the worstcase ESR (
r
_{c,max}
).
Fig. 8
shows the variation in minimum capacitance value as a function of ESR. The capacitor value increases with an increase in ESR. At
r
_{c}
=
r
_{c,max}
, the capacitor value becomes double that of with
r
_{c}
= 0. For
r
_{c}
>
r
_{c,max}
, the capacitance value is a complex value. However, as shown in
Fig. 8
, MATLAB simulation plots only the real part.
Value of the minimum capacitance at different ESR.
Substituting the values of
Δi
_{L}
from Equ. (17) into Equ. (40),
This relation depicts that for specified OVR, the maximum permissible value of ESR (
r
_{c,max}
) is proportional to switching frequency. Therefore, as the switching frequency of the converter increases, the power supply designer is allowed to use a high ESR capacitor without violating the OVR constraint.
Fig. 9
shows the variation in
r
_{c,max}
with frequency. If the switching frequency is 50 kHz, then the designer may use a capacitor with an ESR of 0.6 Ω. As the frequency increases, the required capacitor value decreases.
Maximum permissible ESR (r_{c,max}) vs. frequency.
III. SIMULATION AND EXPERIMENTAL RESULTS
For MATLAB simulation, the buck converter parameters given in
Table I
are used. For given specifications, according to the relation proposed in Equ. (15), the actual duty cycle is calculated as 0.6415, which is greater than the ideal duty cycle (=0.6). For this actual duty cycle, the inductance value is calculated as 490 μH using Equ. (18). The maximum value of ESR(
r
_{c,max}
) and the minimum capacitance (
C
_{min}
) are calculated as 0.2398 Ω and 50 μF, respectively, using Equs. (40) and (41) for the OVR and ICR specified in
Table I
.
The simulated output voltage for ideal and actual duty cycles is shown in
Fig. 10
. The results confirm that for ideal duty cycle
D
= 0.6, the output voltage is 11.2 V; for actual duty cycle
D
= 0.6415, the output voltage is settled to 12 V (as desired).
Simulated output voltage response for ideal and nonideal duty cycles.
We consider four cases for four different values of ESR to investigate the effect of ESR variation on OVR. In each case, the total OVR
Δv
_{o}
, the ripple caused by capacitor
Δv
_{c}
and the ripple caused by ESR
Δv
_{rc}
are evaluated and plotted for two switching cycles, as discussed below.
Case 1
(
r
_{c}
= 0): This is the case for an ideal capacitor. However, such a capacitor cannot be obtained in practice. The voltage ripples are shown in
Fig. 11
(a). The voltage ripple Δ
v_{o}
is the same as Δ
v_{c}
. The peaktopeak OVR is 0.06 V.
Simulated voltage ripple waveforms for different ESR values (a) r_{c} = 0, (b) r_{c} = 0.1 Ω (r_{c} < r_{c,max}), (c) r_{c} = 0.2398 Ω (r_{c} = r_{c,max}), and (d) r_{c} = 0.4Ω (r_{c} > r_{c,max}).
Case 2
(
r
_{c}
= 0.1 Ω <
r
_{c,max}
): The voltage ripples are displayed in
Fig. 11
(b). Owing to the presence of ESR, the total peaktopeak OVR is increased to 0.07 V. However, this value is smaller than the maximum allowed OVR (0.12 V).
Case 3
(
r
_{c}
=
r
_{c,max}
= 0.2398 Ω): The voltage ripples are shown in
Fig. 11
(c). The ESR voltage ripples increase further, such that the peaktopeak OVR reaches 0.12 V, which is equal to the maximum allowed OVR.
Case 4
(
r
_{c}
= 0.4 Ω >
r
_{c,max}
): As shown in
Fig. 11
(d), the voltage ripples caused by ESR highly increase. The peaktopeak OVR is 0.19 V, which is greater than the allowed OVR limit (0.12 V). Thus, the desired performance of the buck converter is degraded.
A hardware prototype of buck converter is set up to verify the theoretical studies and simulation results, as shown in
Fig. 12
. A ferrite core inductor of 490 μH (with 0.5 Ω ESR) and an electrolytic capacitor of 50 μF (with 0.1 Ω ESR) are used. The other key components are listed in
Table II
. The experimental results are shown in
Fig. 13
.
Fig. 13
(a) shows that the output voltage is 11.2 V with a duty cycle of 0.6. If the duty cycle is maintained at 0.642, as obtained from the proposed formula in Equ. (15), then the output voltage is 12 V.
Figs. 13
(b) and
13
(c) show the OVR for
r
_{c}
<
r
_{c,max}
and
r
_{c}
>
r
_{c,max}
, respectively. For
r
_{c}
= 0.1 Ω, the OVR is 75 mV, which is within the maximum specified limit (120 mV). For
r
_{c}
= 0.4 Ω, the OVR is 190 mV, as shown in
Fig. 13
(c). This value is greater than the maximum specified limit (120 mV). Therefore, the experimental results validate the theoretical and simulation results. The dependency of OVR on switching frequency is also experimentally validated for different values of the capacitor ESR.
Fig. 14
(a) shows that with
r
_{c}
= 0.1 Ω, the OVR is less than 0.12 V for a frequency greater than 20 kHz.
Fig. 14
(b) shows that with
r
_{c}
= 0.35 Ω, the OVR is within the specified limit for a frequency above 30 kHz. Similarly,
Fig. 14
(c) depicts that if the converter operates at a switching frequency greater than or equal to 50 kHz, then the OVR is within the desired limit even with a large ESR value of 0.5 Ω. Hence, if the converter operates at a higher frequency, the OVR may remain within the specified limit even with a higher ESR value. The OVR depends on ESR and converter frequency in an opposite manner.
Experimental setup for the DC–DC buck converter.
KEY COMPONENT LIST
Experimental results: (a) output voltage response with ideal and proposed duty cycles, (b) OVR with r_{c} = 0.1 Ω, and (c) OVR with r_{c}= 0.4 Ω.
Experimental results of output voltage variation with frequency with (a) r_{c} = 0.1 Ω, (b) r_{c} = 0.35 Ω, and (c) r_{c} = 0.5 Ω.
IV. CONCLUSION
In this study, the duty cycle formula for a nonideal DC–DC buck converter is improved considering parasitic elements. The effect of these parasitics on inductor and capacitor designs is analyzed. Analyses show a significant difference in inductor and capacitance values with the inclusion of nonideal components. The ESR of output capacitor contributes significant ripples to output voltage. For specified OVR and switching frequency, the maximum allowable value of this ESR is derived. The simulation and experimental results indicate that an ESR beyond this maximum value results in unwanted OVR. Therefore, this analysis may be interesting and useful for a power electronic engineer to design a highprecision power supply. The proposed analysis, with suitable modifications, can be generalized to design other types of DC–DC converters.
BIO
Man Mohan Garg received his B.E. (Electrical Engineering) from the M.B.M. Engineering College Jodhpur, India, in 2008. He received his M.Tech in Electrical Drive and Power Electronics from the Indian Institute of Technology (IIT) Roorkee, India, in 2010. He has completed his Ph.D from Electrical Engineering Department of IIT Roorkee, India. His current research interests include design, modeling and control of DC–DC power converters, photovoltaic integration, and dc microgrid.
Mukesh Kumar Pathak received his B.E. degree in Electrical Engineering from the LD Engineering College, Ahmedabad, India in 1986. He obtained his M.Tech and Ph.D. degrees from the IIT Delhi, India. He joined the NIT, Kurukshetra, India as a lecturer in 1987. In 1989, he joined the NIT, Hamirpur, India, where he served until 2007 as an associate professor. He is currently working as an associate professor in the Electrical Engineering Department of IIT Roorkee, India. His research interests include control of power converters and electrical drives, and power quality improvement using FACT devices.
Yogesh Vijay Hote received his B.E. in Electrical Engineering from the Govt. College of Engineering, Amaravati, India, in 1998, his M.E. in Control Systems from the Govt. College of Engineering, Pune, India, in 2000, and his Ph.D. from Delhi University in 2009. He worked at the Netaji Subhas Institute of Technology New Delhi from May 2001 to February 2011. He is currently working as an assistant professor at the Department of Electrical Engineering of IIT Roorkee, India. His field of research includes robust control, robotics, numerical analysis, and power electronic converters.
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