In this paper, a fast transient buck converter using hysteresis PWM control is presented. The proposed control method is based on hysteresis control of the capacitor C voltage. This offers a faster transient response to meet the challenges of the power supply requirements for fast dynamic input and load changes. It also provides better stability and solves the compensation problem of the error amplifier in conversional voltage PWM control. Finally, the steadystate and dynamic operation of the proposed control method are analyzed and verified by simulation and experimental results.
I. INTRODUCTION
Nowadays, dcdc converters play an important role in industry, the airspace industry, portable systems, distributed generation, electrical devices, electrical machines, power factor correction and voltage regulation
[1]
,
[2]
,
[10]

[12]
. Thus, highquality power supplying converters that can accomplish lowcost and highefficiency performance are in demand
[3]
. High accuracy, fast transient response and enhancing driving capability are expected when a converter operates in the pulse width modulation (PWM) mode
[4]
,
[5]
. Buck converters, as one of the widely used converters, have been investigated with different structures and control methods
[3]

[16]
. The widely used PWM control takes several switching cycles, due to its dutycycle limitation, to recover the output voltage in response to step changes in the loading current
[6]
. For the popular PWM control buck converter operating in continuous conduction mode (CCM), a complicated compensation network is needed to ensure stable operation due to the existence of complex poles in the loop gain transfer function
[7]
. Due to changes in the duty cycle and load condition, a change in the open loop transfer function will bring many difficulties for designing the optimum values of the parameters in the phase compensator. To avoid these difficulties in design, current mode control is applied together with voltage feedback to improve the stability and dynamic performance
[9]
. In this case, many electric components including a current sensing resistor are needed. Several approaches have been reported in the literature. Hysteresis dc–dc converters have a fast transient response since the hysteresis dc–dc converter does not require an error amplifier or external compensation components, which may slow down the transient response. Thus, the system size and chip die area are smaller than those of conventional PWM voltagemode and currentmode dc–dc converters because of the removal of the external compensation components
[3]
,
[8]
.
An improved hysteresis PWM control method using a triangle waveform modulated by output voltage feedback and input voltage feedforward was proposed. The proposed control method only uses a comparator with hysteresis, a feedback resistor, and a feedforward resistor. Thus, the number of components in the control circuit are obviously reduced. In addition, since an error amplifier is not used, the stability of the converter is improved. When the input voltage is changed, it has a smaller peak value and a faster regulation speed than the conventional control method
[5]
,
[10]
.
The operations of the control method are described in Section 2, the steadystate and dynamic characteristics of the proposed converter are analyzed in Section 3, and the fast transient response and good regulation were verified by the experiment and PSIM simulation in Section 4.
II. OPERATING PRINCIPLES
The configuration of a buck converter with a synchronous rectifier employing the proposed control scheme is shown in
Fig. 1
. The control circuit consists of a comparator
U
with hysteresis and a feedback resistor
R_{f}
. Based on the hysteresis characteristic curve of comparator
U
, the key waveforms of comparator
U
in one switching cycle are shown in
Fig.2
. The output voltage is returned to capacitor
C
for the triangular wave generator through resistor
R_{f}
. The feedforward voltage
V_{1}
is connected to capacitor
C
through the feedforward resistor
R
. The feedforward voltage
V_{1}
is also returned to comparator
U
with hysteresis through resistor
R_{2}
. The control circuit operates as follows. When the output voltage or input voltage becomes large, the charging current of capacitor
C
in the on mode period increases and the discharging current of capacitor
C
in the off mode period decreases. Thus, the on time duration of the pulse decreases, and the off time duration increases. In addition, when the input voltage
Vi
becomes large, the threshold voltage levels
V_{H}
of comparator
U
become large. Thus, the change of the input voltage can directly affect the threshold voltage
V_{H}
to rapidly regulate the duty cycle, not through the output voltage change caused by the input voltage. Therefore, the switching duty was changed, and the output voltage can be regulated. It is worth noting that the feedforward voltage
V_{1}
has two key roles in the control circuit. On the one hand it can quickly reflect the changes of the input voltage with a
(1D)Ts
time delay under the worst condition. Therefore, it has the feedforward control effect.
D
is the duty ratio and
T_{s}
is the cycle time. On the other hand, it has two kinds of values based on the switch state, that is the input voltage
Vi
and low level voltage. Thus, the threshold levels of comparator
U
were generated and vary with voltage
V_{1}
. As a result, it can achieve better response results than the conventional PWM control scheme.
III. OPERATION ANALYSIS
 A. Steadystate Analysis
To simplify the analysis, all of the circuit elements are assumed to be ideal. In
Fig.1
,
V_{f}
is the voltage across capacitor
C
,
V_{L}
and
V_{H}
are the threshold levels of comparator
U
, and
V_{U}
is the output voltage of comparator
U
. The switching cycle starts at instant t=0.
( i ) state 1: 0<
t
<
T_{ON}
When the output signal
V_{U}
is high level, the following equations are obtained:
By solving the above equations under the initial condition of
V_{f} (0)
=
V_{L}
, the next equation is obtained:
Circuit diagram of the proposed converter.
Key waveforms of the comparator.
From equation (2),
V_{f}
increase exponentially from
V_{L}
to
R_{p}
k
V_{i}
/
R
+
R_{p}
V_{o}
/
R_{f}
. Since
V_{f}
must be greater than
V_{H}
to invert the state of the comparator, the next constraint is obtained:
By setting
V_{f}
=
V_{H}
and
t
=
T_{ON}
in equation (2),
T_{ON}
is obtained:
If
V_{H}

V_{L}
<<
R_{p}
k
V_{i}
/
R
+
R_{p}
V_{o}
/
R_{f}

V_{H}
, equation (4) can be approximated as follows:
( ii ) state 2:
T_{ON}
<
t
<
T_{s}
When output signal
V_{U}
is low level, the following equations are obtained:
By solving the above equations under the initial condition of
V_{f}
(
T_{ON}
)=
V_{H}
, the next equation is obtained:
From equation (7),
V_{f}
decreases exponentially from V
_{H}
to
R_{p}
V_{o}
/
R_{f}
. Since
V_{f}
must be lower than
V_{L}
to invert the state of the comparator, the next constraint is obtained:
From the equation (7),
T_{OFF}
is obtained by setting
V_{f}
=
V_{L}
and
t
=
T_{S}
:
If
V_{H}

V_{L}
<<
V_{L}

R_{p}
V_{o}
/
R_{f}
, equation(9) can be approximated as follows:
By combining (4) and (9), the duty ratio
D
=
T_{ON}
/(
T_{ON}
+
T_{OFF}
) is:
where,
k
_{1}
=
R_{f}
/
R
.
And cycle
Ts
is :
By combining (5), (10) and
D
=
T_{ON}
/(
T_{ON}
+
T_{OFF}
) the output voltage is obtained as follows:
Where,
In addition, the output voltage can be written as:
 B. Dynamic Analysis
To further analyze the dynamic characteristics of the buck converter circuit, the small signal circuit model is obtained by applying PWM switching modeling techniques.
Fig.3
shows the smallsignal equivalent circuit of a buck converter with the proposed control scheme. In
Fig.3
,
K_{1}
is the gain of the feedforward circuit,
K_{2}
is the gain of the feedback circuit,
r_{s}
is the
ON
resistance of the switch, and
r_{c}
is the series equivalent resistance of capacitor
C
. Based on
Fig.3
, the smallsignal transfer functions can be obtained. Each openloop transfer function is defined and derived as follows.
The input to output transfer functions are expressed as:
Small signal equivalent circuit.
The duty cycle to the output voltage transfer function is expressed as:
It is well known that the output impedance of the converter has a resonance peak due to the LC resonance of the filter circuit. This resonance causes large undershoot and overshoot voltages on the output during load changes. The output impedance transfer function is expressed from the small signal analysis as:
Where,
Δv_{o}, Δv_{i}, Δd and Δi
are small perturbations in the output voltage
v_{o}
, the input voltage
v_{i}
, the switch duty ratio
d
and the output current
i_{o}
.
In addition, by combining equation (5), equation (10), and
K_{1}
=
Δd
/
Δvi
,
K_{2}
=
Δd
/
Δv_{o}
, the next equation can be written as:
The audio susceptibility
G(s)
is then expressed as follows:
Where,
T
=
K
_{2}
G
_{d}
.
Fig. 4
shows a control block diagram of the circuit proposed in this paper. As can be seen from the block diagram, the feedforward pass from the input to the duty control eliminates the effect of input voltage variations. Hence, the output voltage is not influenced by fluctuations of the input voltage.
Fig. 5
shows the results of the frequency responses of the open loop feedback control transfer function taking the proportion coefficient
k
as a parameter.
Fig.6
shows the feedforward control transfer function taking the proportion coefficient
k
as a parameter. The frequency responses of the gain and phase show that the control loop is steady and has a good phase margin with the
k
increases. The frequency responses of the gain and phase show that the gain, in both the low and high frequencies, decreases with the increase in the coefficient constant
k
. The phase characteristics are the same here because no differences are observed among any of the values of
k
.
Control block diagram of the buck converter employing the proposed control scheme.
Frequency responses of the open loop feedback control taking k as a parameter.
Frequency responses of the open loop feedforward control taking k as a parameter.
Fig. 7
shows a comparison of the results of the frequency responses of the audio susceptibility
G(s)
. It can be seen in this figure that the gain of the audio susceptibility
G(s)
is well decreased by employing the proposed feedforward control method. The feedforward control has an advantage especially when the change of an external disturbance is rapid because its information is transferred to the controller without delay. On the other hand, the feedforward path is an open loop and the control result is not corrected by this method. Therefore, it is important to design the parameters (like the proportion coefficient
k
) of the feedforward path properly in order to obtain an excellent control characteristic. The value of
K_{1}
in the audio susceptibility
G(s)
without feedforward control is zero. As a result, the feedforward control has a lower loop gain. This can be shown by equation (20). Here, the phase characteristics are the same.
Frequency responses of G(s).
Fig.8
shows the frequency responses of
G(s)
taking
k
as a parameter. The resonance peak of
G(s)
is well increased with an increase in
k
while the resonant frequency is reduced. All of them have good gain characteristics by employing the proposed feedforward control method.
Frequency characteristics of G(s) for k.
For the value of
R_{f}
, it is not easy to determine the proper value analytically. The frequency responses of
G(s)
, taking
R_{f}
as a parameter, are shown in
Fig.9
. It is obvious that the resistance
R_{f}
should be smaller for increasing the resonant frequency. It is a tradeoff taking the value of
R_{f}
or
k
between the gain of
G(s)
and the resonant frequency.
Note that the above calculations were carried out by introducing the internal resistance and
ESR
of the output capacitor.
Frequency characteristics of G(s) for R_{f}.
According to the open loop transfer function (Equation 21), the open loop root locus can be shown as
Fig.10
. It can be seen that the curvature of the root locus is a circle. The open loop transfer function has two conjugate complex poles and one zero. Actually, the two poles are far from the imaginary axis and strong damping is obtained. The separation point is farther from the imaginary axis and the root locus bends to the left. Thus, fast attenuation oscillation processes and fast steady features are achieved. In addition, a small overshoot can be achieved. In this paper the pole point is 1000 *(2.5730 + 9.9881
i
) and 1000 *(2.5730  9.9881
i
).
Root locus of the open loop transfer function.
Fig. 11
(a) shows a Nyquist diagram of the transfer function. It can be seen that point (1, 0
i
) is not surrounded by the Nyquist plot. In addition, there are no pole points of the open transfer function in the right half plane. The stability of the buck converter control system is obtained in terms of the Nyquist stability criterion.
Fig.11
(b) shows the part of the amplification region at point (1,0
i
) of the Nyquist plot. It can be seen that point (1,0
i
) is not surrounded or passed through by the Nyquist plot.
(a) Nyquist diagram of the transfer function. (b) The amplification region at the point (1,0i).
If the duty cycle can be obtained, the output voltage can also be obtained by equation
v_{o}
=
v_{i}
*
d
. According to equation (11), the parameters have been given except for the value of
k
. The accurate value of
k
can be calculated after solving a nonlinear equation by the Numerical algorithm. Here, the calculated exact value range of
k
is 0.471<
k
<0.475, and the maximum error of the output voltage was ±0.175%.
To keep the system in the steady state, the value of
k
is very important. Here, the range of
k
can be calculated through equation (20). The characteristic root of the system characteristic equation (1+
T
(
s
)) cannot but has strict negative real parts. Based on that, the range of
k
can be k>0 according to the Routh criterion.
There are two loops in the control system. The feedforward control realizes the correction of the input voltage changes and the direct regulation of the duty cycle. Especially, the feedback control obtains rapid regulation of the duty cycle when the load is changed. The input to the output equation can be written as:
Where,
T
=
K
_{2}
G
_{d}
.
To achieve a result where the output voltage variation has a minor effect on the output voltage, feedforward correction with full compensation can be used in the control system. The equation can be written as:
Here, if
G_{v}
–
K_{1}G_{d}
=0, feedforward correction with full compensation can be achieved. Finally, the value range of
k
should be 0.4951<
k
<0.4967, and the maximum error was ± 0.3%.
IV. SIMULATION AND EXPERIMENTAL RESULTS
To verify the proposed control method, a simulation model based on
Fig. 1
has been built in
PSIM
and a prototype has been designed. The circuit parameters are shown in
Table I
. The conventional PWM controller is a TL5001 PWM controller. The regulator makes use of the PI compensation network.
DESIGN PARAMETERS OF THE PROPOSED CONVERTER
DESIGN PARAMETERS OF THE PROPOSED CONVERTER
Fig. 12
(a) shows the relation between the output current and the output voltage. As can be seen in
Fig. 12
(a), the variations of the output voltage are very small. The simulation values of the output voltage are in good agreement with the experimental values.
Fig. 12
(b) shows the relation between the input voltage and the output voltage. It can be seen that the variation of the output voltage is very small even when the input voltage varies significantly.
(a) Relation of the output current and output voltage (b) Relation of the input voltage and output voltage
It is observed that no steadystate error appears in the output voltage when there are changes of the input voltage and load current even when the controller does not employ a highgain operational amplifier.
Fig. 13
(a) and
Fig. 13
(b) show the responses of the proposed controller and conventional controller undergoing 8V to 5V and 5V to 8V input voltage step changes. To see the output clearly, the conventional control output voltage (
V_{o}
1) is made by adding 0.2V. It can be seen that the proposed controller has a smaller peak value and a faster transient recovery time than the conventional controller.
(a) Simulated response to a 8V to 5V input voltage step change. (b) Simulated response to a 5V to 8V input voltage step change.
Fig. 14
(a) and
Fig. 14
(b) show the simulated and experimental responses of the proposed controller and conventional controller undergoing 5A to 2A and 2A to 5A load current step changes, respectively. It can be seen that the steadystate error becomes very small and that the employment of the proposed controller reduces the peak value of the output voltage.
(a) Simulated and experimental response to a 5Ato 2A load current step change. (b) Simulated and experimental response to a 2A to 5A load current step change.
Fig. 15
(a) and
Fig. 15
(b) give the performance of the hysteresis band change when the input voltage changes. It shows that the feedforward control has an effect on the hysteresis band to achieve a fast dynamic response. The proposed control and the constant hysteresis band control are shown in red and blue, respectively. It can be seen from these results that the transient voltages for the variation of the input voltage are improved by introducing the proposed control.
(a) 5V to 7.5V input voltage step change. (b) 7.5V to 5V input voltage step change.
Fig. 16
(a) and
Fig. 16
(b) show the gain and phase frequency characteristics of the control to the output transfer function, respectively. From the bode plots of the feedforward control to the output and the feedback control to the output transfer function, the dynamic characteristic is approved well.
(a) Bode plots of feedforward control to output transfer function.(b) Bode plots of feedback control to output transfer function
V. CONCLUSIONS
A new hysteresis control method with voltage feedforward and feedback has been proposed and analyzed in this paper. The proposed control method is based on the hysteresis control of the capacitor
C
voltage. It offers a faster transient response to meet the challenges of the power supply requirements for fast dynamic input and load changes. In addition, it provides better stability and solves the compensation problem of the error amplifier in the conversional voltage PWM control. The proposed control method only uses a comparator with hysteresis, a feedback resistor and a feedforward resistor. Therefore, the number of components in the control circuit are obviously reduced. In addition, since an error amplifier is not used, the stability of the converter is better. When the input voltage is changed, it has a smaller peak value and a faster regulation speed than the conventional control method. The proposed control method achieves a good regulation and transient response in the buck converter for input and load transients. The dynamic characteristics of the buck converter circuit and the small signal circuit model are obtained by applying PWM switching modeling techniques. The steadystate and dynamic operation of the proposed control method are analyzed and verified by simulation and experimental results. The transient performance of the output voltage for large variations in the input voltage is improved by introducing feedforward control by using proper values for
k
in the feedforward path. The gain of the audio susceptibility
G(s)
is significantly decreased by employing the proposed feedforward control method.
In order to investigate the dynamic performances of the proposed PWM controller, a small signal analysis was carried out by introducing the averaging technique. The frequency responses of the gain and phase show that the open loop control is steady and has a good phase margin when
k
increases. The frequency responses of the gain and phase show that the gains in both the low and high frequency decrease with the increase of the coefficient constant
k
. The resonance peak of
G(s)
is well reduced with an increase of
k
, while the resonant frequency is also reduced. All of these obtain a good gain by employing the proposed feedforward control method. The resistance
R_{f}
should be smaller for increasing the resonant frequency. There is a tradeoff when taking the value of
R_{f}
or
k
between the gain of
G(s)
and the resonant frequency.
It is worth mentioning that the feedforward voltage
V_{1}
has two key roles in the control circuit. On the one hand, it can quickly reflect the changes of the input voltage with a (
1D
)
T_{s}
time delay under the worst condition. As a result, has the feedforward control effect.
D
is the duty ratio and
T_{s}
is the cycle time. On the other hand, it has two kinds of values based on the switch state, that is the input voltage
V_{i}
and the low level voltage. Thus, the threshold level of comparator
U
was generated and varies with voltage
V_{1}
. The proposed method can achieve better response results than the conventional PWM control scheme.
Acknowledgements
The authors would like to acknowledge the financial support of the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning and Shanghai talent development fund (Grant No2012024). and Leading Academic Discipline Project of Shanghai Municipal Education Commission (J51303).
BIO
YongXiao Liu was born in China, in 1988. He will receive his M.S. in Electrical Engineering from the Shanghai University of Electric Power, Shanghai, China, in 2014. He is a Student Member of the China Power Supply Society. His current research interests include power electronics and control.
JinBin Zhao (M’06) was born in China, in 1972. He received his M.S. and Ph.D. in Electrical Engineering from Oita University, Oita, Japan, in 2002 and 2005, respectively. He worked as a Researcher at the R&D Headquarters of the Origin Electric Co., Ltd, Japan, from 2005 to 2011. He is currently a Professor at the Shanghai University of Electric Power, Shanghai, China. His current research interests include the control of power converters, softswitching power converters, inverters, distributed power systems, powerfactor correction and electric drive systems. He currently holds three U.S. patents, five Japanese patents and five Chinese patents. He has published 40 technical papers in journals and conference proceedings. Dr. Zhao is a Member of the Institute of Electrical and Electronics Engineers (IEEE) and the Institute of Electrical Engineers of Japan (IEEJ), the Institute of Electronics Information and Communication Engineers of Japan (IEICE) and a Senior Member of the China Power Supply Society.
KeQing Qu was born in China, in 1970. He received his Ph.D. in Electrical Engineering from Shanghai University, Shanghai, China, in 2004. He is currently an Associate Professor at the Shanghai University of Electric Power, Shanghai, China. He is also a Master Instructor. As a visiting scholar, he studied in Germany with full funding by the national foundation for study, from 2009 to 2010. His current research interests include power electronic conversion, and new energy generation and its application to power systems. He holds one Chinese patent, one monograph and has more than 30 academic theses, 12 of which were included in the international Engineering Index.
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