This document discusses small signal analysis based closed loop control of a buck converter. It first provides background on buck converters and their use. It then describes performing small signal analysis to linearize the system for control purposes. Different control design methods like bode plot analysis and Zeigler-Nichols tuning are examined. The proportional and integral gains of a PI controller are derived using these frequency domain techniques. Simulation results show the output voltage is regulated as desired with the PI controller despite input voltage and load disturbances.

1. Small signal analysis based closed loop control of buck converter
K.V.V.S.R.Chowdary*, S Devi Prasad†
* Assistant Professor, School of Electrical Engineering, KIIT University Bhubaneswar, India, E-mail: rchowdaryfel@kiit.ac.in,
†
M.Tech Research Scholar, School of Electrical Engineering KIIT University Bhubaneswar,
Email:sdeviprasad555@gmail.com.
Keywords: Buck converter, PI controller,bode plot, Zeighler
Nicholes method.
Abstract
From the past several years with the advent of renewable
sources prognosis of DC-DC converter had a remarkable
growth in the fields of power electronic application. Buck
converter closed loop control is popular because it is easy to
get feedback. In this paper we have implemented the control
technique based on small signal analysis of Dc buck
converter. The implementation of control technique with the
DC-DC buck converter is analyzed using tuning methods like
frequency domain and Ziegler Nichols for better performance
and transient response. The perturbation of input voltage
source and load has applied to the model and obtained the
desired output.
1 Introduction
The Buck converter are mostly used in electronic
circuit design and low power rated supplies. According to
the study there are more than 450 model of DC-DC converter.
For development of DC-DC converters technique this
research is vital important. Experts, scientist, researcher from
different parts of the world has spend more than 25 years in
this research area[8].
The closed loop control of DC-DC converter ensures
not only the desired output response but also maintain input
power quality control [6]. In embedded technology
applications like DSP, FPGA requires fast and better response
[12].In closed loop control of Buck converter two important
attributes are regulating the out-put voltage and output
voltage is insensitive to the disturbances[3]. This paper
constitutes small signal mode representation of buck
converter. Then estimated PI controller design parameters
from bode plot and also used ZN-Method. These methods are
explained in the following sections. The controller parameters
are verified by considering the changes in the input voltage
and load resistances. All the above mentioned statements are
provided with the simulation results.
2 Buck Converter
Dc converters in functionality resembles transformer for
DC Circuits. This is also known as voltage step-down DC-
DC converter because the output voltage is less than the input
voltage. It constitutes input DC voltage source(V) , High
switching frequency switch(S), diode(D), inductor(L), filter
capacitor(C), and output resistance(R). The power circuit
diagram is shown figure 2.1.
Buck converter operates in two modes of
configurations. In mode 1 at time t=0 the switch 'S' is switch
is turn ON. There will be the rise of input current which flows
through inductor (L), filter capacitor(C), and resistor(R). In
mode 2 at time t=t1 the switch 'S' is switched off. The energy
stacked in the inductor delivers to the resistor with the help of
conducting freewheeling diode (D). The current which is
flowing from the inductor continually go through L, C, load
and diode D. The inductor current continues to fall until the
switch 'S' ON again in the next cycle.
Figure2.1 Buck Converter
Small Signal Analysis:
Buck converter is a non linear system to make the system
linear; we are using small signal analysis. Small-signal
simulation is most frequently used technique in the branch of
electrical engineering. This is used to estimate the behavior
of non linear device with linear equation [1]. Small signal
analysis will give the better perception about the inherent
features of the system for closed loop control of output
voltage.
In this model we considered both voltage and duty ratio are
inputs and the output is desire output voltage. The small-
signal ac inductor loop equivalent circuit equation and the
small signal ac capacitance loop equivalent circuit is shown in
figure 2.2 and figure 2.3 respectively. The analysis of a small-
signal ac model at a quiescent operating point (I, V) of buck
converter is very important to make the system linear [12].

2. The small signal equivalent circuit of buck converter is shown
in the figure 2.4.
Figure 2.2 Small signal ac inductor loop equivalentcircuit
Figure 2.3 Small Signal ac capacitance loop equivalent circuit
Figure 2.4 Small signal equivalent circuit of buck converter
L {di͂(t)/dt} = D. Ṽg(t) + d(t).Vg - Ṽ(t) (1)
C{dV͂ (t)/dt} = i͂(t) - {dV͂ (t)/R} (2)
The Equation 1 and Equation 2 are came from the small
signal ac inductor loop equivalent circuit and small signal ac
capacitance loop equivalent circuit which is show in figure
2.3 and figure 2.4 .
Transfer function:
The capacitor C and the output resistor R in figure are
referred to the primary side of 1:1 transformer. Now, we need
to find out the transfer function of the buck converter. Since,
there are two independent ac inputs (Vg(s),d(s)),the voltage
at the output ac variations v(s), can be formulated in term of
arising from these two input are So, the transfer function can
be defined as
Gvd(s)=VgR /(LCR S2
+LS + R) (3)
Gvg(s) = D.R /(LCR S2
+LS + R) (4)
Figure 2.5Block Diagram of controller with buck converter
Design of transfer function of buck converter
Design vales: Vg=12 volt, R=5Ω, L= 145.8 µH , C
= 200 μF Frequency = 25 kHz saw tooth signal carrier varies
from 0 to 1[16]. So, according to the given parameters open
loop system function of the buck converter is given by
GOL(s)= Gvd(s)H(s)(1/Vm)
= VgR /(LCR S2
+LS + R)
The step response of the transfer function is shown in the
figure 2.6.and values are given below
Rise Time : 9.2475e-05
Settling Time : 0.0015
Settling Min : 8.4763
Settling Max : 18.4834
Overshoot : 54.0284
Undershoot : 0
Peak : 18.4834
Peak Time : 2.5180e-4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10
-3
0
2
4
6
8
10
12
14
Open loop Step Response
Time (seconds)
Magnitude(V)
Figure2.6 . Step response of the transfer function

3. 3 CONTROLLER DESIGN
The PI is one of the families of PI controller which is most
flexible and simple method. For turning up of PI controller,
the value of proportional (KP) and integral (KI) needs to be
estimated. The PI controller depends upon the past and
present value. The proportional constant depends on existing
error; the integral constant depends on the aggregation of past
error.
There are numerous types of methods available in literature to
select proportional gain (KP) and Integral gain (KI) for the PI
controller [13]. They are like Ziegler-Nichols, Time domain
design, Frequency domain design. To apply any of these
methods, the common requirement is to have the open loop
transfer function of the system. Out of these methods,
frequency response method is quite popular because of its
ease and effectiveness. The transfer function of the buck
converter is extracted from the small signal model. So by
assigning the values to the transfer function, we can design
the PI controller by frequency domain or bode plot. From
bode plot we can determine the magnitude and the phase
margin Determining these values we can easily determine the
value of proportional gain (KP) and Integral gain (KI)[10].
Bode plot
From bode plot we observe phase margin at the cross
over frequency gain (7º). It may cause the closed loop system
to be unstable as the integral controller adds an additional
phase lag. It is recommended to have a phase margin above
for the stable closed loop system. So, it is prefer to take the
cross over frequency gain more than 45º .If we take the new
cross over frequency gain at 130º then we will get amagnitude
26.2 and the new gain cross over frequency which is denoted
as jω.The proportional constant and Integral constant found
from the figure3.1. As follows
KP = 10-(|G(jω)|/20)
KP = 10-(26.2/20)
= 0.049
KI = (ω/10) KP
KI = 15.4 X 100 X 0.049 = 75.4259.
Figure 3.1. Bode plot of GOL(s) showing frequency and magnitude (dB) at
50o phase margin
Now whatever process we are establishing is to ensure the
desired response at the output. For verifying the desired
response we constituted the model with the above design
values. Following figure 3.2 indicates Simulink based Model
of buck converter with PI controller
Figure 3.2. Simulink based Model of converter with PI controller
The output voltage of the DC buck converter is maintained to
be constant. In the above figure input, reference and output
measured voltages are shown. So for that we need compare to
the reference voltage. After comparing with the reference
voltage, we will get the error signal. This error signal is fed to
the PI controller. From the bode plot we obtain the PI
controller parameters i.e. Value of Kp and KI. The response
signal from the PI controller is given to the saturation which
is later compared with the repeated sequence. So, finally the
pulse is generated and given to the gate terminal of the switch
(MOSFET). Simulation result for the model shown in the
figure3.3.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
0
2
4
6
8
10
12
Time(sec)
OUTPUTVOLTAGE(v)
Figure 3.3. Response of buck converter output voltage using bode plot
Ziegler Nichole method
Assume KI and Kd is zero. And we have to ascertain
the value of ultimate gain (Ku) and ultimate period (Tu). By
substituting Ku and Tu we can find the value of Kp and KI as
per the given table. From the transfer function we will take
the characteristic equation .Using the characteristic equation
and applying to the Routh-Hurwitz criterion, the value of the
Ku and Tu .After getting the ultimate gain and ultimate period
values and putting in the table, the proportional constant and
integral constant are determined.

4. Table 1 Determination of Kp,Ki, Values
Table 2 Representation of Characteristic Equation
The characteristic equation from the transfer function is
According to the Routh Hurwitz criterion by using
characteristic equation, the value of ultimate gain(Ku) and
ultimate period (Tu) are determined.
Taking the even equation
2.916 X 10-8
S2
+ 5 + Kcr = 0 (5)
Put S = jω
So, the equation (5) will be
2.916 X 1.458 X 10-4
(jω)2
+ 5 + Kcr = 0
Since 5 + Kcr >0 , So the value of Kp lies between 0 to 5 i.e
0< Kcr <5. If the maximum value Kcr =5 then the value of ω =
5.85606 X 103
Since ω = 2Πf =2Π/ Tu , So Tu = 1.071 X 10-3
.
Substituting the value in the table2,
we will get the value of the Kp = 2.5 and the value of KI =
2521.Putting the value of Kp and the value of KI the transient
response is observed .The output voltage is shown in the
figure 3.4.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
0
1
2
3
4
5
6
7
8
9
10
Time(sec)
OUTPUTVOLTAGE(V)
Figure3.4. Response of buck converter output voltage using ZN method
By the visual inspection from the figure 3.3, 3.4 we can
understand that the transfer function which we considered
from small signal analysis can be validated. Also it is clear
that the response is fairly smooth incase of ZN method (figure
3.3).
4 PERTURBATION OF INPUT VOLTAGE
For the model shown in the figure7 if there is perturbation in
input voltage. This consideration is quite realistic. For
observing the output response of the model we have
considered controlled source. Hence from the figure4.1 it is
clear that the constant voltage is maintained even though
input voltage or reference value is varied. This response
plotted by considering the parameters obtained from the
bode plot & Z- N method.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
2
4
6
8
10
12
14
Time
Figure 4.1.Response of output voltage for Perturbation in the input voltage
using
Now for the same model perturbation in load is applied
response was plot in the figure 4.2. Variation in the load has
been attained with the help of variable resistance. Figure 4.2
clearly depicts that even though load resistance is varied then
also fairly flexible control obtained.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
2
4
6
8
10
12
14
Time
Figure4.2.Response of output voltage for Perturbation in the load
Conclusion
In this paper we have implemented the control technique
based on small signal analysis of DC-DC buck converter. The
output voltage is steady at 5V even though the input varies
from 5V to 12V.That means ,one can apply any voltage
between 5V to 12V as input, the output will be certainly reach
to 5V in steady state. The variation of input voltage source
and load has applied to the model and the obtained desired
output. It is also observed that ZN method provides optimum
settling time and nearly minimum rise time with fairly smooth
response. From the results obtained in disturbance rejection
mode the ZN step response method provides smooth response
than Frequency domain method.

5. Acknowledgements
I would be greatly Thankful to KIIT University,
School of Electrical Engineering for allowing to me
carry out this work. .
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