SIMULATION STUDY OF THERMAL ELECTRIC CIRCUIT
By
Dafa Yan

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering—Master of Science
2014

ABSTRACT
SIMULATION STUDY OF THERMAL ELECTRIC CIRCUIT
By
Dafa Yan
Thermoelectric generator works under different working conditions, especially different
temperature and load resistance. In order to improve its efficiency, the circuit was in place between
thermoelectric generator and load resistance using a buck converter. Compared with the direct
connection case the buck converter improves efficiency, especially with small load resistance.
Power output and current ripple are the central focus for evaluating this buck converter.
Regarding thermoelectric generator characteristic, PMW (pulse-width-modulation) mode is
selected for regulating buck converter switch. The tool box selected for building simulation model
of buck converter is Simscape in matlab/Simulink. To verify the model, differential equation
mathematic model and state-space simulation model are built. To configure the parameters of the
buck converter, all the possible data is collected by running simulation model with different sets
of parameters. After configuration, the optimal parameters for the buck converter is obtained. A
method of selecting optimal parameter set is introduced for the given requirement for the circuit
performance.
With a set of optimal circuit parameters such as capacitor and inductor values, a controller of PWM
signal is to be designed for optimizing the circuit system efficiency, where the extreme seeking
control was used. With the controller, it proves that even with different load resistance, the buck
converter can delivered maximum power with optimized the duty cycle of PWM.

ACKNOWLEDGEMENTS
It has been two years since I enrolled in Michigan State University. Every slightly progress
I made here comes from continual support from my professors and friends. I would like to first
and foremost thank my academic advisor, Professor George Zhu. Even before we met each other,
he had already began to inspire me for a perspective future through email communication. I would
never forget the moment that he welcomed me in his office for my first time at MSU. I was so
worried and confused about my future as a new student at that time. His kindness and patience
filled me full of hope and confidence. With his encouragement and direction at the beginning of
my harsh life in MSU, I overcame many difficulties and set up my mind to become a control
engineer. During the academic life at school, he is always nice and cared about me. He pushed me
when I was lazy and encouraged me when I got depressed. He helped me plan my future and taught
me how to communicate and behave as a good engineer. I always feel full of courage and
confidence with his standing behind me. I know this is not only for now, but forever. What I
learned from him is not only the knowledge about engineering but also the truth in life. I will
thank him forever.
I would like to thank my committee members, professor Choi and professor Schock. I
always recalled the direction from professor Choi in nonlinear control course. Every time I stepped
into his office seeking for help about the course project, he always directed me patiently and
answered my question. I would like to thank professor Schock for his kindness to let me sit in his
engine class. The engine knowledge I learned in his lecture is truly necessary in my career.
I would like to thank my friends in our lab. I have to say that my life is enriched and much
better because of them. Yingxu Wang is the first friend I made in Michigan State University. He

iii

is always the first one who comes to my mind when I need help. Jie Yang is always avid to help
me and support me as a reliable senior. Shupeng Zhang taught me control knowledge continually.
With his help, I started to find that I have enthusiasm of learning control. I will never forget the
studying time I spent with Yifan Men. We were partners of conducting this research at the
beginning. The research life we shared together was happy and harvest. Tao Zeng treated me as
one of his best friends. He is always willing to share my problems both in life and study selflessly.
Ruitao Song is the one I truly appreciate for this thesis. He really helped me a lot on state-space
modeling and extreme seeking control. Without his help, I could never have my thesis completed
well.
After all, I would like to thank my parents. Their supporting is my foundation of finishing
master education in America. Everything they did for me and sacrificed for me is unforgettable. I
will memorized their cultivation and supporting forever and ever.

iv

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................................ vii
LIST OF FIGURES ..................................................................................................................... viii
KEY TO SYMBOLS AND ABBREVIATIONS .......................................................................... xi
Chapter 1 Introduction .....................................................................................................................1
1.1.Thermoelectric Overview .......................................................................................................1
1.1.1. History of Thermoelectric Generator ............................................................................3
1.1.2. Application of Thermoelectric Generators in Modern World .......................................4
1.2. Motivation and Strategy of Circuit Optimization .................................................................6
1.2.1. Original Circuit Research and Background ...................................................................7
1.2.2 Motivation of Power output Optimization ...................................................................13
1.2.3 Strategy of Circuit Optimizition ...................................................................................17
Chapter 2 Literature Review .........................................................................................................20
2.1. State-Space Average Method .............................................................................................20
2.2. Buck Converter Transfer Function .....................................................................................22
2.3. Pulse Frequency Modulation ..............................................................................................23
2.4. Pulse Width Modulation .....................................................................................................24
2.5. PID Controller ....................................................................................................................25
2.5.1. Fuzzy logic Control .....................................................................................................26
2.5.2 LQR Approach .............................................................................................................26
2.6 Sliding Mode Controller .....................................................................................................27
2.7. Conclusion ..........................................................................................................................27
Chapter 3 Analytical Analysis and Simulation .............................................................................28
3.1. Mathematic Analysis of Circuit Model ..............................................................................28
3.1.1. Mathematic Model of Primarily Improved Circuit .....................................................29
3.1.2 Mathematical Model of Buck Converter ......................................................................37
3.1.3. Average Technique Simulation model ........................................................................45
v

3.2 Simulation with Simscape ...................................................................................................47
3.2.1 Simulation Model of Primarily Improved Circuit ........................................................48
3.22 Simulation Model of Buck Converter ...........................................................................51
3.3 Results Comparing ..............................................................................................................53
3.3.1 Comparing Primarily Improved Circuit Model and Analytical Solution .....................53
3.3.2 Comparing of Buck Converter State Space Model and Simscape Model ....................55
3.3.3. Comparing of Primarily Improved Model and Buck Converter Model ......................58
3.4. Conclusion ..........................................................................................................................60
Chapter 4 Simulation Study and Configuration ............................................................................61
4.1. Strategy of Configuring Circuit ..........................................................................................62
4.2. Load Resistance .................................................................................................................65
4.3. General Simulation .............................................................................................................67
4.4. Detail Simulation ................................................................................................................71
4.5. Conclusion ..........................................................................................................................73
Chapter 5 Controller Design .........................................................................................................75
5.1 Relationship between Power Output and Duty Cycle .........................................................75
5.2. Extreme Seeking Control ..................................................................................................77
5.3. Simulation Model with Controller .....................................................................................78
5.4. Simulation Result ...............................................................................................................80
5.5. Conclusion ..........................................................................................................................81
Chapter 6 Conclusion and Future Work ........................................................................................82
6.1 Conclusion ...........................................................................................................................82
6.2. Future Work .......................................................................................................................83
BIBLIOGRAPHY ..........................................................................................................................84

vi

LIST OF TABLES

Table 1-1 Parameters of original circuit (1) .....................................................................................9
Table 1-2 Parameters of original circuit (2) ...................................................................................12
Table 1-3 Parameters of improved circuit (1) ................................................................................16
Table 3-1 Parameters of improved circuit (2) ................................................................................35
Table 3-2 Parameters of improved circuit (3) ................................................................................53
Table 3-3 Parameters of buck converter (1) ..................................................................................57
Table 2-1 Parameters of buck converter (2) ..................................................................................58
Table 4-1 Parameters of buck converter (3) ..................................................................................65
Table 4-2 General simulation parameters ......................................................................................67
Table 4-3 Detail simulation parameters .........................................................................................71
Table 4-4 Selected parameters .......................................................................................................74
Table 5-1 Parameters of buck converter (4) ..................................................................................75
Table 5-2 Simulation parameters of controlled buck converter ....................................................80

vii

LIST OF FIGURES

Figure 1-1 Thermoelectric generator ...............................................................................................2
Figure 1-2 Multi-Mission radioisotope thermoelectric generator model .........................................5
Figure 1-3 Thermoelectric generator in exhaust gas recirculation system .....................................6
Figure 1-4 Model of original generator circuit ................................................................................7
Figure 1-5 Relationship between power output and load resistance................................................9
Figure 1-6 Simulation model of original circuit of thermoelectric generator ...............................11
Figure 1-7 Results comparing of the analytical solution and simulation result .............................12
Figure 1-8 Improved circuit with capacitors and switch ...............................................................13
Figure 1-9 Simulation model of improved circuit ........................................................................15
Figure 1-10 Power outputs comparing ..........................................................................................17
Figure 1-11 Buck converter (1)......................................................................................................19
Figure 2-1 Buck Converter (2) .......................................................................................................22
Figure 2-2 Proposed PWM Buck Converter ..................................................................................25
Figure 3-1 Open state of the primarily improved circuit ...............................................................29
Figure 3-2 Capacitors transformation ...........................................................................................31
Figure 3-3 Simplification of resistors ...........................................................................................32

viii

Figure 3-4 Simplification of close circuit .....................................................................................33
Figure 3-5 Analytical solution of primary circuit ..........................................................................36
Figure 3-6 Open case circuit .........................................................................................................37
Figure 3-7 I-V characteristic of semiconductor diode ...................................................................39
Figure 3-8 V-I characteristic of semiconductor diode ...................................................................40
Figure 3-9 Close Case Circuit ........................................................................................................41
Figure 3-10 The B matrices of state spaces ..................................................................................42
Figure 3-11 The A matrices of state spaces ...................................................................................43
Figure 3-12 The state space model ...............................................................................................44
Figure 3-13 Average technique simulation model ........................................................................46
Figure 3-14 the Simulation model of primarily improved circuit..................................................48
Figure 3-15 Simulation configuration parameters ........................................................................50
Figure 3-16 Data collection setting in Scope .................................................................................52
Figure 3-17 Buck converter simulation model .............................................................................52
Figure 3-18 Simulation and analytical results ...............................................................................54
Figure 3-19 Combination model of buck converter and state space ..............................................56
Figure 3-20 Power output of buck converter .................................................................................57
Figure 3-21 Relationship of power output and duty cycle .............................................................59

ix

Figure 4-1 Flow chart of data processing ......................................................................................64
Figure 4-2 Relationship of power output and load resistance........................................................66
Figure 4-3 Performance of circuits with different parameter sets .................................................68
Figure 4-4 Power output performance ...........................................................................................69
Figure 4-5 Load resistance current ripple performance .................................................................70
Figure 4-6 Performance of circuits with different parameters sets ................................................72
Figure 4-7 Power output performance ...........................................................................................72
Figure 4-8 Current ripple performance ..........................................................................................73
Figure 5-1 Power output performance ........................................................................................... 76
Figure 5-2 Block diagram of extreme seeking scheme ..................................................................77
Figure 5-3 Simulation model of extreme seeking controller .........................................................78
Figure 5-4 System Plant Simulation Model ...................................................................................79
Figure 5-5 Power efficiency of the Buck Converter ......................................................................80
Figure 5-6 Duty Cycle of Buck Converter .....................................................................................81

x

KEY TO SYMBOLS AND ABBREVIATIONS

Ω

Ohm

µF

Microfarad

µH

Microhenry

Am

Ampere

C

Capacitor

CCM

Continuous conduction mode

DCM

Discontinuous conduction mode

F

Farad

H

Henry

Hz

Hertz

I

Current

kHz

Kilohertz

L

Inductor

LQR

Linear-Quadratic Regulator

PFM

Pulse frequency modulation

PWM

Pulse width modulation
xi

PW

Power

PID

Proportional-integral-derivative

R

Resistance

S

Second

t

Time

T

Period

TEG

Thermoelectric generator

U

Direct current electric source

V

Voltage

xii

Chapter 1 Introduction

The thermal efficiency range of gasoline engines are between 25% and 40%. Half of the rejected
heat is carried away by the exhaust gas emission. Reusing the exhausted energy is a very important
as improving energy efficiency is urgent due to limited energy resource. A significant role is played
by thermoelectric generator that utilizes the waste exhaust energy. Load and internal resistance of
a thermoelectric generator circuit changes continuously as the exhaust gas temperature varying. In
order to maximize the power output of a thermoelectric generator, optimization of thermoelectric
generator circuit with on-line control is necessary.

1.1 Thermoelectric Generator Overview
To convert thermal energy into electricity, 100 watt skutterudite thermoelectric generator designed
and developed by MSU was applied as the foundation for studying thermoelectric circuit in this
research. In this skutterudite thermoelectric generator, two materials named N-type material and
P-type material are utilized. The characteristic of N-type material drives current toward the hot
side whereas the P-type material works in the opposite way. Electromotive force is generated while
a pair of N-type and P-type components are exposed to two sides with a temperature gradient. An
Electricity source is formed while several of these power units connect in series. The advantages
of thermoelectric generator is highlighted on its quiet working property and widely used in some
special cases. The thermoelectric generator can be applied in exhaust system of vehicles such as
exhaust pipe, EGR (exhaust gas recirculation) cooler, and even on engine wall [1].
1

Thermoelectric
generator

Internal
resistance

Internal resistance

Figure 1-1 Thermoelectric generator
2

1.1.1 History of Thermoelectric Generator
Thermoelectric generator is a widely-used electric generator due to its advantages mentioned
before. The theoretic foundation of thermoelectric generator was formed as a discovering in 1821
AD by Thomas Johann. As there was temperature gradient over the compass which was made by
two different metal materials, there was tiny current going through the compass. The deflection of
compass caused by electromagnetism was the proof that there was current generated by the
temperature gradient over two type metals when they were connected.
In 1834 AD, the phenomenon that heat would be generated at the junction of two types of metals
while a current was amounted over it was discovered by a French physicist Jean Charles Athanase
Peltier. By reversing the current, the cooling effect would also be detected. The relationship
between current and the heat absorbed or generated by the current is proportional.
William Thomson was the one who made the conclusion and the connection of these two theories
based on thermodynamic theory. The theory, named as Thomson effect, states that the heat
absorbed or created by thermocouple is proportional to both current and temperature gradient over
the thermocouple.
After the theory foundation has been set, numerous experiments had been done in order to study
thermoelectric generator. Three classical applications were introduced as they showed the
advantages of using a thermoelectric generator comparing with other electric generators [2].

3

1.1.2 Application of Thermoelectric Generators in Modern World
Regarding the material and working principle of thermoelectric generators, the benefits of using
them is obvious. Especially considering the wide range of application and outstanding efficiency,
a commensurate emphasis is given to thermoelectric generator in science research and commercial
utilization. Furthermore, the reversibility of thermoelectric couple is also a prominent
characteristic which will also have a great effect when it is applied for absorbing heat.
A well-known application example of thermoelectric generator is the Multi-Mission Radioisotope
thermoelectric generator designed by NASA for operating on Mars and vacuum of space. Longlived, reliable and stable power source are needed in space exploration missions as two hazards,
radiations and vacuum are concluded by the working conditions of the outer space. These
requirements are met successfully by this newly designed power source device, Multi-Mission
Radioisotope thermoelectric generator. The two components of this thermoelectric generator are
plutonium-238 heat source and thermoelectric couple. By converting the heat generated by
plutonium’s decay into electricity, the thermoelectric generator can support the spacecraft to work
in Mars. In its first working year in Mars, this power system have supported the spacecraft to drive
on Mars for more than 1.6 kilometers and send back 190 gigabits data. It also fired more than
75000 laser shots [3].

4

Figure 1-2 Multi-Mission radioisotope thermoelectric generator model
Wearable devices are also a typical application of thermoelectric generator. Google glass is the
specific product which can utilize the heat from human body with a thermoelectric generator.
Organic material is preferred by human heat thermoelectric generators. The flexibility and
compatibility of this kind of material are the crucial reasons for being chosen as the material
applied on human skin [4].
This research focuses is placed on the thermoelectric generator which is supposed to be apply in
automotive system, especially to the gas exhaust system. Considering the efficiency and economy
requirements of automotive industries, thermoelectric generator is unreplaceable. Other than these,
the effect on reducing noise and saving space also contributes to the usage of thermoelectric
generator. As an example of the benefits from using thermoelectric generator in vehicles, the
application in BMW X6 SUVs is persuasive. The thermoelectric generator is designed to be
integrated in the radiator of the exhaust gas recirculation system rather than the traditional design
5

method that configures thermoelectric generator as a separate module in the exhaust system
underneath vehicle. According the customer testing, fuel consumption is reduced as well as 250
watts electricity was generated by the thermoelectric generator. In the meantime, there is also
additional heat supplying to passenger compartment by the thermoelectric generator being
configured in this way [5]. Thermoelectric generator is forecasted to play a significant role in
energy saving in future.

Figure 1-3 Thermoelectric generator in exhaust gas recirculation system

1.2

Motivation and Strategy of Circuit Optimization

The circuit conducting power generated from thermoelectric couple to load is also a core part of
thermoelectric generator. Motivation and strategy of circuit designing are introduced in this
chapter. Basic tool for researching the circuit is Matlab/Simulink. The goal circuit is been designed

6

as a buck converter which are used to regulate the load resistor voltage while the value of resistance
charges constantly.

1.2.1. Original Circuit Research and Background
Electromotive force can be provided by the circuit source which is connected in series with the
thermoelectric couple units. This parameter depends on temperature gradient of circumstance and
property of thermoelectric couple. It can be treated as a constant value that cannot be impacted by
structure or component parameters of generator circuit. As the working temperature of
thermoelectric generator are quite high, the internal resistance of the power source cannot be
ignored. An original thermoelectric generator circuit can be simplified as a circuit consisted with
one source resistance, one load resistance and a constant voltage source. The load resistance value
could vary while the electrical load changes. To demonstrate the original structure well, a circuit
is shown in Figure 1-4.

Rl

Rt
Ut

Figure 1-4 Model of original generator circuit
7

The source of the circuit is the thermoelectric couple that provides the electricity caused by
temperature gradients over the thermoelectric unit. It is presented as Ut. The internal resistance of
the power source is modeled by the resistor Rt. The load resistor is presented by Rl. As the load
resistance varies all the time as the electric load changes, the power output changes value as well.
The values of power source and internal resistance are set as constants as they are uncontrollable
parameters of thermoelectric generator. Based on ohm’s law and other related electric principles
and definition, the power output can be derived as following:

 Ut 
I 
;
R

R
l 
 t
2
P  I  Rl ;

(1.1)

2

 Ut 
P
  Rl ;
 Rt  Rl 
Based on equation (1.1), the system power output depends on the load resistance. In this case, load
resistance is the solely variable. The power output is the dependent variable. In order to figure out
the changing curve of the power output, derivative of the power output with respect to the load
resistance is necessary.
2

 Ut 
d (
  Rl )
R

R
dP
t 
  l
;
dRl
dRl

(1.2)

dP U t 2 ( Rt 2  Rl 2 )

;
dRl
( Rt  Rl ) 2
From the equation (1.2), conclusion can be made easily that the power output approaches its peak
value as the load resistance equal to internal resistance of the thermoelectric generator. The power

8

output increasing as the load resistance augmenting before the value approach the internal
resistance.
With the circuit demonstrated as show in this figure 1-4, the power output as a function of the load
resistance is based on the experimental data, and the parameters are assumed as shown in table 11.
Table 1-1 Parameters of original circuit (1)
Ut (V)
16

Rt (Ω)

Rl ( Ω)
0 to 2

0.5

Using Matlab to plot the relationship between power output and load resistance, the peak power
output, augment and decreasing ratio are demonstrated as below.
0.014

0.012

Power Output

0.01

0.008

0.006

0.004

0.002

0

0

0.2

0.4

0.6

0.8

1
Load Resistance

1.2

1.4

1.6

1.8

2

Figure 1-5 Relationship between power output and load resistance
From the plot, it is clearly shown the peak of the power output occurs when the load resistance is
equal to the internal resistance of the circuit.

9

Aiming at comparing the result of the math model of original circuit which consists only basic
components as load resistance, internal resistance and electricity source, Matlab/Simulink is used
as the core tool for developing circuit model and simulation circuit. In this thermoelectric circuit
study, Simscape tool box is used as the modeling tool. Figure 1-6 is the Simscape model of the
electric circuit of thermoelectric generator.

10

Figure 1-6 Simulation model of original circuit of thermoelectric generation

11

With some similar process of simulating the circuit result, the relationship curve of analytical and
simulation solutions can be acquired and compared with each other. In the process of the
comparing, it can easily generate the conclusion that both solutions are identical and match each
other very well. Table 1-2 is the parameters used to run for modeling.
Table 1-2 Parameters of original circuit (2)
Ut (V)
16

Rt (Ω)

Rl ( Ω)
0 to 1

0.5

Figure 1-7 is the result of comparing of two curves from math model and simulation model.

0.014
Analytical Result
Simulation Result
0.012

Power Output

0.01

0.008

0.006

0.004

0.002

0

0

0.1

0.2

0.3

0.4

0.5
Load Resistance

0.6

0.7

0.8

0.9

1

Figure 1-7 Results comparing of the analytical solution and simulation result
From Figure 1-7, it can be seen that the simulation result matches the analytical result perfectly.
This is quite a verification process for simulation with Matlab/Simulink. In order to optimize the
circuit for the best system perform when the load resistance changes, a controlled switch and two

12

capacitors are introduce in to this system as a buck converter, which play the role of optimizing
the system efficiency.

1.2.2 Motivation of Power output Optimization
With the goal to optimize circuit, a circuit has been designed with controlled the duty cycle. The
new circuit consists a switch and two capacitor. The two capacitors are paralleled used for storing
energy while the power cannot be delivered to the load resistor directly. Duty cycle is used to
control the circuit for optimal efficiency. With the controlled duty cycle, even with the changing
load resistance value, the circuit can optimized for maximum power. The research involves with
studying the control circuit and focuses on configuring the parameters. Mathematical model and
matlab Simscape model will also be compared to figure out the simulation accurate. The improved
method of the circuit is demonstrated following in figure 1-8.

Rt

Ut

Ct

Ct

Rl

Figure 1-8 Improved circuit with capacitors and switch
13

This circuit has two working states when the switch opens and closes. The first state is open state.
In this working state, the capacitor Ct keeps being charged by the power source. In the meantime,
the capacitor Cl supports load resistor Rl alone. In the close state, the switch closes while the whole
system is connected. In this state, the power source and capacitor Ct charge load resistor. The
percent of close state over the given period is duty cycle. The period is sum of the time open and
closed and is determined by control frequency. In our research, 0.0001s is chosen as the period
that is equivalent to 10 kHz control frequency. The percent of duty cycle is the control parameter.
By adapting duty cycle, the system can be controlled for optimal efficiency with vary load. There
are several advantages of the improved circuit compared with the original one without switch and
capacitors. With the controlled system, it is possible to deliver more power to the load resistor with
low load resistance. This can be proved by simulation results to be introduced later. The simulation
of the circuit is shown in figure 1-9.

14

Figure 1-9 Simulation model of improved circuit

15

By simulating the model of improved circuit, the data of power output with different load
resistances can be collected. In order to compare with the power output of the orignial circuit, the
same parameter of power source and resistors are chosen for simulation. The parameters of
capacitors and controlled duty cycle are list in table 1-3 abd they were optimized in charter 4.
Table 1-3 Parameters of improved circuit (1)
Ut (V)
16

Ct (F)
400e-6

Cl (F)
10e-6

Rt (Ω)
0.5

Rl ( Ω)
0.05 to 0.6

The power outputs and load resistance values relationship shown in figure 1-10 demonstrates the
power output can be improved over the orignal one.

16

-3

13

x 10

Improved Circuit
Orignal Circuit

12
11

Power Output

10
9
8
7
6
5
4
0

0.1

0.2

0.3

Load Resistance

0.4

0.5

0.6

0.7

Figure 1-10 Power outputs comparing
From the figure, it can be detected easily that part of the blue curve is above red curve which means
that the power output is enhanced in this section. This means even with controling duty cycle, the
power output of improved circuit is higher than the original circuit with the load resistance is
smaller than 0.6Ω. This motivates the development of the controlled circuit.

1.2.3 Strategy of Circuit Optimizition
By investigating the improved circuit, there are some problems needed to be solved in the
improved circuit. The main problem is that at the instant of the switch closing, the current through
the swith approaches infinity high. Under the parameters of Table 1-3, simulation results show that
the current through the switch approaches 66100 A. the high current could lead to damaged swith
17

or reduce life. Therefore, the simulation results shown in figure1-10 us to study the power output
ability. In the following simulation work, the improved circuit model with two capacitors and an
inductor is also going to be used as a power output varification.
To solve this problem, buck converter are introduced and utilized in this research as the optimized
circuit format.
In order to decrease the current pulse generated by the capacitors at the close moment, an inductor
is added between two capacitors. The characteristic of inductor is opposing the change of current
which flows through it. It effectively prevent the current changing abruptly. However, as soon as
the switch opens, the current flowing though the inductor stops suddently. Due to the characteristic
of inductor, the voltage of inductor is the changing rate of current flowing though it self. This
abruptly changing of current can make the voltage of the inductor at this time approachs infinity,
which is also not faverate in real world application. To cope with this potential problem and also
aim at improving power output, a diode is added. The cathod terminal connected between the
switch and inductor. The anode is connect to the grounding loop side. With this design strategy,
the diode performs as disconnection when the switch is close for the reason that the electric
potential over cathode is higher than that over anode because of electric source. The current cannot
flow through the diode. The electric source keep charging the capacitors, inductor, internal resistor
and load resistor. This state are not affected by the diode added. In the opening case, the diode
perform as a wire with slight resistance. The advantages of the new circuit are two folds: the
inductor can store certain energy that can contribute to the load resistance power output when the
circuit is open; and the current impulse and voltage impulse are significantly reduced. This circuit
is a buck converter format. However, as the internal resistance is considerately large, this circuit

18

cannot be classified as a traditional buck converter. The figure 1-11 demonstrated how the final
optimized circuit built.
L

Rt
Cl

Ct

Ut

Figure 1-11 Buck converter (1)

19

Rl

Chapter 2 Literature Review

The buck converter is widely used in industry. In this chapter, some outstanding articles about
modeling buck converter, buck converter control design and buck converter application are
summarized and discussed. These articles are used as the background knowledge to make this
thesis easy to read.

2.1. State-Space Average Method
Buck converter is a variable structure system as it has two working status. One working status is
closed status. When the switch of the buck converter is closed, the power source charges the other
components of the buck converter. The power is delivered from the source power through the
switch to the storage electric components and to the load resistor as well. The other working status
is open status. In this status, the switch opens and the power is delivered from the storage
components to the load resistor. The electric power source is isolated from the load resistor. With
this structure feature, a buck converter can be defined as a nonlinear system [6].
To build the mathematic model of the buck converter, regular method of building state-space
equations cannot be adopted. The state-space average method can be used as a replacement. The
central idea of state-space average method is building the system state-space equations in different
work states, respectively. Then, the next step is combining the two state-spaces models in to one
state-space model by multiplying their corresponding cycle ratio. The cycle ratio depends on the
duty cycle.
20

For example, regarding the buck converter shown in Figure 1-10, the state-space of the open
working state can be modeled as
 x(t )  A1 x(t )  B1u(t );

 y (t )  C1 x (t )  D1u(t );

(2.1)

and the state-space of close working state can be modeled as
 x(t )  A2 x (t )  B2u(t );

 y (t )  C2 x(t )  D2u(t );

(2.2)

The duty cycle time of the buck converter is presented as d and the associated period is T. Thus,
the duty cycle of the close working can be calculated as

2 

d
;
T

(2.3)

And the cycle of open working state is

1  1 

d
;
T

(2.4)

The new state space model can be generated as
 A  1  A1  2  A2 ;
B    B    B ;

1
1
2
2

C  1  C1  2  C2 ;

 D  1  D1  2  D2 ;

(2.5)

With this state-space average method, a linear state-space model can be obtained even though the
target buck converter is a nonlinear system [6]. All the state-space matrices above are for
demonstrating. So the specific matrices are not shown here.
21

However, this method has some disadvantages for building the system mathematic model and it is
not adopted in this research. The significant disadvantage is that the response of the linear form
state-space model is much different from the actual response of the physical system which
oscillating a lot with the oscillated duty cycle input. This makes considerable error comparing with
real buck converter system, especially regarding the power output. In this research, the way to
build mathematic model is different from this one. The system state-spaces model are not
combined into a new. Simscape simulation model and real-time switching state-space model are
used in this research. They yields accurate response, compared with the state-space average model
method. This will be discussed in Chapter 3 with model comparing.

2.2. Buck Converter Transfer Function
In article [7], Optimum Design of Buck Converter Controller using LQR Approach, a simplified
approach of obtaining the buck converter transfer function was introduced. The sample buck
converter is shown as following.
Rl

L

Rt
Cl
Ut

Rc

Figure 2-1 Buck converter (2)

22

R

In this article, the way to build the overall buck converter transfer function is building the transfer
function of the system in closed working state and using it as the overall transfer function. The
effective voltage is the source voltage multiplies the duty cycle. The result transfer function is

G( s) 

R  R  Rc  C  s
;
( LC ( R  Rc )) s  ( L  RlCR  RlCRc  RRcC )  s  ( R  Rl )
2

(2.6)

This transfer function is proved to work relatively well [7] for relatively small internal resistor Rt.
In this research, the internal resistance is not small and cannot be neglected. On the other hand, the
buck converter designed in our research contains another capacitor Ct close to the power source
for storing energy when the switch is open. The effect of this capacitor cannot be ignored also. For
these reasons, building transfer function in this way is not suitable for our research.

2.3. Pulse Frequency Modulation
In many fields of industries, stable constant power sources are required. Linear supply device can
meet this goal with the disadvantage that some energy is wasted when it is delivered through some
passive elements. Switch regulators have advantages on this aspect rather than linear supply
regulator. Switch regulators can be classified into two types: PFM (Pulse Frequency Modulation)
and PWM (Pulse Frequency Modulation) [8].
PFM is selected frequently in the applications that needs to be considered to provide very low
stand-by currents. The PFM is applied in the application with light load also. In this model, the
way to control the buck converter switch is controlling the frequency of the switch with a fixed
duty cycle width [9].

23

There are several features with the PFM when it is applied to a buck converter. The switching
frequency is usually low compared with other pulse modulations. It will yields relatively larger
voltage ripple due to this reason. PFM buck converter is also relatively hard to be assessed as the
light load needs discontinuous conduction model.
In our research, the PFM is not selected for controlling target buck converter due to high stand-by
current of the thermoelectric generator. In order to reduce the ripple, a high frequency switch
control is preferred in this research.

2.4. Pulse Width Modulation
PWM (pulse width modulation) is the method selected in this research. There are many reports
and researches about PWM. The way of controlling buck converter by PWM is to turn on and of
the switch with varying duty cycle at a fixed frequency.
PWM buck converter performs quite well when it working with high load. The power output is
high and the ripple is reasonable. However the light load is not the working range of PWM. The
ordinary working frequency of the PWM is 200 KHz t0 2 mHz. With this frequency, the energy
lost in switch part is quite high and cannot be neglected when the load is a small. However in this
research, the resistance of the switch is small enough to be neglected as it compared with the load
resistance.
Some design methods are also studied in order to find a modulation that combined the advantages
of PFM and PWM.

A study about applying a high efficiency PWM with a novel DCM

(discontinuously-conducting-mode) on buck converter reports that the buck converter can be
controlled to perform well under both high and light load. The size of controller and the
performance of the buck converter are both outstanding as shown in Figure 2-2. The block IL
24

detector and Mode -control circuit do not work with the current through the inductor is positive.
Once the IL detector detects the current is under zero, these two blocks will switch the buck
converter from regular mode to DCM mode [10].

Figure 2-2 Proposed PWM Buck Converter
(A High-Efficiency PWM DC-DC Buck Converter with a Novel DCM Control under LightLoad, Chu-Hsiang Chia)

2.5. PID Controller
As the most widely used controller, PID (proportional-integral-derivative) controller is also used
in buck converter area. There are many researches about applying PID controller into the buck

25

converter. To improve the control performance, some advanced theories are applied in the
research.

2.5.1. Fuzzy logic Control
Fuzzy logic based PID controller was also used to control the buck converters when the
performance of PID controller cannot meet the requirement in some aspects. By employ fuzzy
logic based PID controller, the performance can be improved. The process of fuzzy logic contains
two parts: fuzzification and defuzzification. The first part can be demonstrated in this way:
converting probabilistic data into fuzzy value. The second part of defuzzification is applying the
membership functions on the input data and yielding output in a form of specific value. Regarding
this scheme, the specific mathematic model of target system is not necessary. This characteristic
of fuzzy logic method is very feasible for building controller for some nonlinear system like buck
converter [11]. According the research in [11], the result of the fuzzy logic based PID controller is
far better than a conversional PID controller for controlling buck converter. The relative research
report in [12] also introduced fuzzy logic PID controller in a concisely experimental way.

2.5.2 LQR Approach
LQR (linear-quadratic regulator) is widely used in controller design. In research report [7], LQR
are used for figuring out the feedback signal to tune PID gains in a buck converter. By building
the cost function of the buck converter system, the optimum gain vector is acquired. With this
vector, the PID gains can be tuned. These three gains are used for tracking the reference signal
with reduced error. These gains forms the optimum gain vector.

26

2.6 Sliding Mode Controller
Sliding mode control is a control method for nonlinear system. When it is applied to a structure
variable system buck converter, there are several advantages of using sliding mode method to
design controller. The sliding mode controller is insensitive to external disturbance for its outstand
robustness. It also has fast response ability and good transient performance.
The way to select variables for buck converter system is different from the conventional way to
build differential equations of the circuit system. The variables selected are not the physical values
as the voltages of capacitors or currents through the inductors. The variables are selected as the
output voltage error, differential of this voltage error and its integral [13].
Regarding selecting the sliding surface that trajectories are going to converge to, research reported
in [14] made a good explanation on this part. Only getting the inequities which defined the range
of sliding coefficient is not enough for getting a good coefficient. To figure out the coefficient,
Ackermann’s Formula for designing static controllers is adopted. Regarding specific problem on
designing controller for buck converter, the bandwidth of the desired frequency response in
conjunction with the existence condition are the key elements.

2.7. Conclusion
As a widely used switching regulator, buck converter is combined with many advanced control
theories. Sample data feedback [15], optimal control theory [16] and double-frequency buck
converter [17] are all relative technologies about buck converter control. The control theory
adopted in this research is extreme seeking control. This is going to be introduced further in
Chapter 5.

27

Chapter 3 Analytical Analysis and Simulation

To study the circuit and the associated control design for the buck converter, developing simulation
model is necessary. Especially in this research, some models are not able to be realized in real
experiments but it can be simulated in software. The parameters and strategies, even for some
extreme cases, can be easily acquired or verified by simulations without building the physical
system. In this chapter, the emphasis is focused on developing both analytical model and Matlab
simulation model for the circuit. The modeling accuracy is validated by comparing their simulation
results.

3.1. Mathematic Analysis of Circuit Model
In this section, the mathematic analysis of circuit model is introduced and demonstrated. In fact,
in chapter 1, the mathematic method of original circuit has already been show and the result is
utilized to optimize circuit into buck converter format. There are two mathematic models
introduced in this section. The first one is analytical model of the improved circuit. This circuit is
consisted with electric source, internal resistor, load resistor, switch and two capacitors as shown
in figure 1-8. The second one is the optimized circuit with added inductor and diode in the
improved circuit as shown in figure 1-11. The advantages of the mathematic analysis of circuit
models are: the circuit can be processed more accurately and efficiently; controller design can be
conducted more logically and clearly. With the foundation of mathematic model developed in this
section, verification simulations model is more readily and impressively.

28

3.1.1. Mathematic Model of Primarily Improved Circuit
In order to conduct the analysis of the primarily improved circuit mathematically, it is necessary
to analyze the circuit as the first step. As mentioned in Chapter 1, there are two working states of
the circuit within one period. The first state is the open case. The second state is named as closed
case. In figure 3-1, the open case work state is shown, the whole circuit is divided into two parts.
The left part is consisted by the power source, internal resistor and capacitor C t. As the electric
source Ut and internal resistor Rt are defined as constants, the whole part A circuit can be seen as
a standard capacitor charging circuit. In the meantime, the part B circuit can be seen as a capacitor
discharging circuit.

Rt
Cl

Ct

Rl

Ut

A

B

Figure 3-1 Open state of the primarily improved circuit
Regarding the part A and part B circuits, building mathematical solution requires to clarify initial
conditions of these two circuits. To research this goal, the whole system is analyzed over a control

29

period. Note that at the steady-state initial voltages of capacitors at the instant of the switch opening
are equal to the voltages at the end of the last closed case. Regarding the closed case, two capacitors
are connected in parallel. The voltages of parallel capacitors are equal according to the principle
of electric circuit. Further considering this circuit structure, a conclusion can be made that the
initial voltages of two capacitors are equal to each other. U0 are used to represent for the initial
voltages of capacitors.

(U  U 0 )  e
I Ct  t
Rt



t
Rt Ct

;

(3.1)

t

U Ct (t ) 

1
 I C ( )d  U 0 ;
Ct 0 t

(3.2)

These two equations (3.1) and (3.2) are the mathematic equations of charging capacitors which
has initial voltage from the part A circuit. The solution of these two equations are shown below.

UCt (t )  U t  U 0  e

t
( Rt Ct )

 Ute

t
( Rt Ct )

;

(3.3)

In the meantime, part B circuit is also working as a capacitor being discharged. The equation of
this circuit is shown like this.
t

I Cl 

U 0 Rl Cl
e
;
Rl

(3.4)

t

U Cl (t ) 

1
 I C ( )d  U 0 ;
Cl 0 L

(3.5)

(3.4) and (3.5) are two equations that describes the process of discharging capacitor in part B
circuit. The solution of these set of equations are shown below.
30

UCt (t )  U 0  e

t
Rl Cl

(3.6)

;

With equation (3.3) and (3.6), the capacitor voltage as a function of time t during opening case can
be obtained easily. Based on these equations, capacitor voltage can be obtained for the opening
case working state.
Regarding the closed case working state, the same method can be applied to the circuit. The initial
condition of the circuit comes first to be considered. Because of the parallel connection, a new
capacitor can be the equivalent of these two capacitors. Figure 3-2 shows this transformation.

Ct

Cl

C

Figure 3-2 Capacitors transformation
The combined capacitance of new capacitor can be calculated as shown below.

Ct  Cl  C;

(3.7)

At the instant of closing the switch, the two capacitor’s voltage has to be equal as they are parallel.
However the electric charges of these two capacitors cannot be transferred instantaneously. So the
electric charge energy in two capacitors are conserving in this instant. According to the
conservation of charges, the voltage Uc of new formed capacitor C can be get as the following
equations. Assumes that the time of open case work state is t1.

31

UCt (t1 )  Ct  UCl (t1 )  Cl  U c  C;

U c (0) 

U Ct (t1 )  Ct  U Cl (t1 )  Cl
Ct  Cl

(3.8)

(3.9)

;

The next step is simplifying the resistors of the circuit. According to Thevenin’s resistance theorem,
when a single resistance can replace a set of parallel resistors, and the voltage source is also treated
as a short circuit. In this way, the new resistor can be formed to replace the parallel resistors as
shown in Figure 3-3.

Rt

Rl

Ut

U

Rt

Rl

U

R

Figure 3-3 Capacitors transformation
With these following equations, the resistance are worked out.

R

Rt  Rl
;
Rt  Rl

(3.10)

Also according to the Thevenin’s voltage source theorem, the new voltage can be calculated as
following.

32

U

Rl
 Ut;
Rt  Rl

(3.11)

By transforming capacitors and simplifying resistors, a new circuit is generated as the Figure 3-4
shown below.

Rt

Ut

R

Ct

Cl

Rt

U
t

C

Replacement

Close Case Circuit
Figure 3-4 Simplification of close circuit

According to this new formed circuit and its initial voltage, the voltage of the capacitor C for the
closed case can be acquired easily by solving following equations.

[U  U c (0)]  e
IC 
Rt



t
RC

;

(3.12)

t

U C (t ) 

1
 I C ( )d  U c (0);
C 0

(3.13)

After solving the set of equations above, the voltages capacitor of C for the closed case can be
shown as follows.

33

t

t

U c (t )  U c (0)  e RC  U  e RC  U c (0);

(3.14)

As a periodic system, the voltages of capacitors also varies periodically. Also as the discussion
about the initial voltages of open case, the voltages of capacitors at the end of closed case are the
same as the initial voltages of open case. T and t1 are defined as the period and length of open case.
Thus, the duty cycle is T-t1.With time length t1 and T-t1, the closed case end voltage and open case
initial voltage can be expressed as follows.
UC (T  t1 )  U0;

(3.15)

By solving this equation, expression of U0 can be obtained as well. Substituting the expression of
U0 into equation (3.3) and (3.14), all capacitor voltages during a period can be presented with only
one variable t1 (duty cycle), as the length of open case.
Next, regarding figuring out the relationship between power output and duty cycle, is to be studied,
along with the power expression of load resistance is the next. As the capacitor C l is parallel
connected with load resistor, the voltage of Cl can be used as the voltage of Rl. In this way, the
expression of load resistor power output can be derived as following.
2
U l (t ) 

P
;
1

P2

Rl

2
U c (t ) 


;

Rl

KT  t  KT  t1

(3.16)

KT  t1  t  ( K  1)T

(3.17)

As the goal is to control the duty cycle, it is important to figure out the transient power output and
the power output over a period as a function of the duty cycle. For this reason, the work done over

34

a period is described in following. It is also calculated in two steps as there are both open and
closed cases.
t1

W1   Pdt
1 ;

(3.18)

0

W2  

T t1

0

(3.19)

P2dt;

The work done over one period is shown below.

W  W1  W2 ;

(3.20)

Note that the work is only a function of t1. By controlling t1, the work of W can be regulated.
In order to study the relationship curves, a matlab script is developed to simulate these equations.
It can be seen in Appendix. Table 3-1 shows the parameters used in simulations. By running this
program, a curve is generated to show the relationship between duty cycle and power output. The
parameters are selected based by try and error.
Table 3-1 Parameters of improved circuit (2)
Ut (V)
16

T (s)
0.0001

Ct (F)
150e-6

Cl (F)
20e-6

Rt (Ω)
0.5

Rl ( Ω)
0.2

According to the parameters offered from table 3-1, running the program generates the curve
shown in Figure 3-4. From this figure, it is obviously that while the duty cycle is between 30 to 60
percent, the power output over a peroid approachs to its peak. To calculate this peaking power
output and the corresponing duty cycle, a new matlab script is developed. The duty cycle that can
give out most power output is at 53 percent of the whole period. Within the original circuit, the
same resistance of load resistor can only generate around 0.01024 J work in 0.0001s, which is
35

smaller than the peak power calculated by manually at 0.0113J. One observation is that there is
considerable power output at the 0 percent duty cycle point where the power out put supposed to
be zero. Considering the primarily improved circuit studied, there is a untoleratable high current
flow though the switch at closing,causomg the voltage reset on both capacitors. This might be the
reason why there is power output even if the duty cycle approaches zero. The current impulse
delivers significantly power to the Cl and load resistor at the time of switch closing. This makes
the circuit cannot be used in practice.
0.012

0.01

Power Output

0.008

0.006

0.004

0.002

0

0

0.2

0.4

0.6
Duty Cycle

0.8

Figure 3-5 Analytical solution of primary circuit

36

1

1.2
x 10

-4

3.1.2 Mathematical Model of the Buck Converter
In this section, the target is to study the final optimized circuit, a buck converter circuit, which is
feasible to be built in real experiment. The structure of the optimized circuit is different from the
primarily improved circuit with added inductor and diode in to it. With the same method, the
differential equations cannot be founded as easy as the primarily circuit for the reason that the buck
converter contains nonlinear component rather than primarily improved circuit. To deal with this
issue, developing state space model is a good way. Figure 1-11 presents the buck converter to be
studied.
To build the state space model of the buck converter circuit, two work states need to be separated.
Similar to the primarily improved circuit, there are two work states, closed state and open state. In
order to keep researching consistently, the open state is first to be studied. Figure 3-6 is the open
state buck converter. There are also two parts in this circuit, part A and part B.
L

Rt
Ct

Rl

Cl

Rd

Ut

Part A

Part B
Figure 3-6 Open case circuit

The part A is almost the same as the one in the primarily improved circuit. However the part B
circuit is much different from the former one. The voltages of capacitor Ct and Cl are defined as x1
37

and x3 as well the current through the inductor L is x2. As there is only one capacitor in part A
circuit, it is quite simple to build its state space based on Kirchhoff law.

Ut  I Rt  Rt  x1;

(3.21)

1
 x1;
Ct

(3.22)

x1 

Based on the circuit equation above, the state space of part A is shown below.

x1  

1
1
x1  
U0
Rt  Ct
Rt  Ct

(3.23)

Regarding the part B circuit, the diode is a nonlinear component of the circuit which increases the
complication of the building state space model. In this circuit, the diode is modeled to allow a
current flow in one direction. In this case, the diode has the same effect of a nonlinearly varying
resistor that depends on the voltage over the diode. Once the voltage is less than a certain voltage,
the resistance approaches infinity. Otherwise, the resistance approaches zero.
To make the resistance of diode even more accurate, the voltage-current characteristic curve was
sketched based on the Physicist’s model shown below.
I  I 0 (exp{

eV
}  1);
kT

(3.24)

In this equation, I is the current through the diode; V is the voltage over the diode. T is the
temperature in Kelvin; e is electron charge 1.602*10-19C; k is Boltzmann’s constant
1.38*10-23 J/K.

38

With selected parameters in the above equation, a curve of voltage current (V-I) for the diode
model can be sketched and shown below.

Current/Amp

Voltage/V

Figure 3-7 I-V characteristic of semiconductor diode
By changing the coordinates, here comes the V-I characteristic of semiconductor diode can also
be shown in Fig 3-8. The tangents of the V-I imply the resistance value corresponding to the current.
According to Fig 3-8, the differential resistance of this diode varying for infinity large to ignorable
small with the current increasing from negative value to considerable positive value. The method
adopted in this research is to collect the data of resistance under different voltage values. Based on
this data set, a look-up table can be developed. Based on this table, the curve of diode resistance
can be founded. The state space simulation are developed using this block structrue.

39

Voltage/V

Current/Amp

Figure 3-8 V-I characteristic of semiconductor diode
However, before implementing the state space model into Simulink, Rd are defined as the
changeable resistance of diode.
The differential equations of part B can be shown below.
L  il  Rd  il  UCl  0;

il  Cl  U Cl 

U Cl
Rl

;

(3.25)

(3.26)

The state space of this circuit is shown below.

L  x2   Rd  x2  x3 ;
x2  Cl  x3 

40

x3
;
Rl

(3.27)

Combining equations of (3.23) and (3.27), the state space model is shown below.
 x1  
1
  
   Rt  Ct
 x2  
  0
  
 x3  
   0
  

  x1   1 
    Rt  Ct 

   
1   x2   0 

   
 Ut
L    

1   x3   0 

  

Rl  Cl 
   


0

0

Rd
L
1
Cl



(3.28)

After modeling the circuit for open state, the closed state circuit can be modeled in the same
way. The close state circuit is shown as in figure 3-9.
L

Rt
Ct

Cl

Rd

Rl

Ut

Figure 3-9 Closed state circuit
In this case, because of the power source is connected to the main circuit, the voltage and current
relationship can be modeled as following.
I Rt  I Ct  I L

(3.29)

I Rt  Rt  UCt  U0;

(3.30)

I L  I Cl 

U Cl
Rl

41

;

(3.31)

Denoting the x1, x2 and x3 in the same way as the open state, the state space model of closed
circuit can be founded as follows.
 x1  
1
  
   Rt  Ct
 x2   1
 
   L
 x3  
   0
  



  x1   1 
    Rt  Ct 

   
1   x2   0 

   
 Ut
L    

1   x3   0 

  

Rl  Cl 
   


1
Ct

0

0
1
Cl

(3.33)

Matlab/Simulink is selected as the tool for simulation study in this research. To implement the
models in (3.28) and (3.33), the crucial part of the system is to model the switching part. Duel to
the buck converter design, there is no voltage collision at switch. The voltages and currents related
to the inductor and capacitors are transient. With this conclusion, the system can be treated as that
the time responses at the ending of last working state is the initial conditions of the system in the
current working state. The time length of each system working state is the duty cycle multiple to
the period.

Figure 3-10 The B matrices of state spaces

42

Figure 3-10 is implementation of the B matrix for two states, where the input port 4 is the dutycycle control signal to turn on and off switch. The pulse signal can be either zero or one to close
and open respectively. Once the control signal is zero, port 3 is selected as the system output; the
system outputs closed B matrix. Port 1 stands for the power source voltage that is a constant is.

Figure 3-11 The A matrices of state spaces
The same strategy is used for matrix A model, it will also be controlled by the same pulse signal
for matrix B case. By using a switch block that provides one or zero control signals, open and close
systems can be altered easily. After building the subsystem models of the whole system, the matrix
A Simulink is showed below.

43

Part 1
Part 2
Part 4

Part 3

Part 5

Figure 3-12 The state space model
Regarding the whole system model, it can be separated into five parts as shown in Fig 3-12. Part
1 is to provide the parameters of the whole system, where the matrix elements, input voltage and
duty cycle are provide by a matlab script. With this method, modifying simulation parameters are
easy and clear. Part 2 is the matrix selection part which is introduced in Figures 3-10 and 3-11.
Part 3 is the integration for the whole state space model. Part 4 stands for the diode model. As the
diode is a nonlinear component in the whole system, the better way is to use the V-I curve to model
the diode resistance. In this function, current though the diode is a variable and the resistance is a
dependent variable. For this part, the lookup table block is adopted. With the current of inductor
inputting into the lookup table, the value of corresponding resistance is the output. Part 5 is the
power output calculation model. Based on the definition of power output over a peroid, this model
provides accumulated the work and reset it in a period. With this function, the power outputs
generated during every period can be calculated.

44

3.1.3. Average Technique Simulation model
Average Technique was introduce in Chapter 2. For verifying the simulation performance of the
developed model, a Simulink model is also developed as shown in Figure 3-13. The state-space
model can be obtained by substituting (3.28) and (3.33) into (2.5).

45

Figure 3-13 Average technique simulation model

46

3.2 Simulation with Simscape
After deriving the analytic solutions of primarily improved circuit and final optimized circuit buck
converter, simulation based upon the matlab/Simulink Simscape toolbox is conducted. Simscape
toolbox provides a simulation environment of modeling physic systems as such as hydraulic,
mechanic and electric systems. According to the introduction from Matlab, Simscape toolbox can
be used to design control system and study system-level performance. Since Simscape supports ccode generation, the designed model can be deployed in different environments.

3.2.1 Simulation Model of Primarily Improved Circuit
In this section, the process and strategy of building simulation model for the primarily improved
circuit model is discussed. Within the Simscape toolbox, the circuit can be implement according
to the figure 1-7. The capacitors, resistors, constant voltage resource and switch are base function
blocks in Simcape. In order to simulate the model, a numerical solver needed to be configured in
the Simscape toolbox. Regarding the circuit, a electrical reference is also need to the circuit. Square
pulse block is used to control the switch. In order to connect the blocks from basic toolbox with
the simscape blocks, two transfer blocks are required which are also from the utility content of
simscape toolbox. The sensor block in the Simscape is used to output the measured signal

47

Figure 3-14 the Simulation model of primarily improved circuit

48

Regarding the model simulation parameters, the fixed-step was selected as the solver type since it
was challenge to make variable-step simulation. Adopting variable-step solver led to high
simulation error and cannot generate stable power output. Also, the variable-step solver was very
slow compared to the fixed-step solver. In this case, the fixed-step type solver is a good option.
The order of solver was set to 14 in simulations. It uses a combination of Newton’s method and
extrapolation from current value to compute the model’s state at the next time step as introduced
in Matlab. Both the accuracy and the efficiency satisfy the requirement of this model. Since the
period of the pulse which control the switch is 0.0001s, the simulation time is set as 0.01s that
covers 100 cycles. It is long enough for simulation to approach steady state. The time step size
was set as 0.000001s. Which calculates 100 times in one period. The accuracy and speed of
simulating are both assured by this method. In order to improve the accuracy, the extrapolation
was selected as 4. The other parameters are also determined by try and error. The parameters are
shown in figure 3-15.

49

Figure 3-15 Simulation configuration parameters
The method of collecting data is introduced in this paragraph. The main data wanted is the power
output over one period and the current ripple on the lead resistance and switch. The high current
ripple. Regarding the power output, it has already discussed in the section that describes the state
space. Part of the method is continued using here in the simscape simulation model. According to
the definition of power generated by circuit, voltage over a component multiplies its current is the
power output. Applying this principle on load resistor Rl, the power output signal can be generated
by multiplying the voltage and current on it. In order to figure out the amount of work done by the
load resistor, the power output has to be accumulated over the peroid. For this reason, the integral
block is used here. The type of integral is set as rising type which makes sure the work can be reset
each period. The signal of reset time comes from the pulse block as it is the period for the whole
50

system. To record the data, a variable is set in the scope for the power output signal. This variable
is recorded into the workspace. The same recording method can also be applied to the current
ripple data collection. By the definition of ripple, the difference of maximum current through the
switch and minimum current are calculated by programed script.

Figure 3-16 Data collection setting in Scope

3.22 Simulation Model of Buck Converter
Development of the final optimized circuit model in Simscape tool box is quite same as the process
for the primarily improved model. The main different is the structure of the circuit. Comparing
with the primary one, an inductor and a diode added in to the circuit. The diode is required to be
configured. The default resistance of diode block 0.3 Ω is much large than expected value.

51

Figure 3-17 Buck converter simulation model

52

3.3 Results Comparing
To assure the accuracy of model work, running different models, developed using different
approaches, and comparing the simulation results are quite necessary. In this section, the goal of
conduct simulations using different models is to validate the accuracy of the models and decide
the model to be used. The first couple of models to be compared are the primarily improved circuit
model with its analytical model, and then followed by the buck converter models. The accuracy
and efficiency of state space simulation model and Simscape simulation model are the target to be
analyzed. The final comparing was focused on the performance of buck converter model and
primarily improved circuit model. The power output and efficiency are to be compared in this
section. In order to keep simulation results choose to the realistic experiment requirements, the
electric source is set to 16 volt and the internal resistance is 0.5 Ω.

3.3.1 Comparing Primarily Improved Circuit Model and Analytical Solution
The power output of the circuit is the key variable to be compared between these two models. The
selected simulation parameters are listed in the following table. The parameters were selected
randomly. In order to make the comparing result more general, several sets of parameters were
used.
Table 3-2 Parameters of improved circuit (3)
Ct (volt)

Cl (volt)

Rl (Ω)

Duty Cycle (%)

300e-6

40e-6

0.2

1 to 100

53

-3

12

x 10

Simulation Result
Analytical Solution

11
10

Power Output

9
8
7
6
5
4
3
0

0.2

0.4

0.6
Duty Cycle Length

0.8

1

1.2
-4

x 10

Figure 3-18 Simulation and analytical results
In Figure 3-18, the blue curve stands for the relationship between power output and the duty cycle
time of the analytical solution; and the red curve is from the Simscape simulation model. From the
figure, it is obviously two curves almost matches with a slightly gap. The gap could be caused by
the integral solver. As the simulation in Matlab is realized by solving the continuous differential
equations step-by-step. However the difference between these two methods is fairly smaller. The
simulation model is accurate enough for study. As the beginning of the simulation, there is a
considerate value of power output with the duty cycle approaches zero. This could be duel to the
voltage pulsation between Ct and Cl at switch. Which causes the energy stored in Capacitor Ct
transfers to Cl instantly. According to the trend of the curve, with the increasing of duty cycle, the
power output increases. The main point to be emphasized is that at certain duty cycle, the power
output is larger than the power output of the circuit without control. The Power output of original
54

circuit supposed to be 0.01041 W. The max power output of the primarily improved circuit is
almost 0.0113W. This indicate that the output power could be increased by for controlling the duty
cycle.

3.3.2 Comparing of Buck Converter State Space Model and Simscape Model
This section, is to verify the accuracy and performance of the Simscape simulation model. The
method verifying to is also compared with the target analytical model. The state space analytical
model of the target buck converter was also implemented into Simulink with basic mathematic
tool box. There are advantages for the comparing model since the two simulation models can be
run simultaneously, which makes it easy to compare simulation results. The Combination model
are shown in figure 3-19.

55

Figure 3-19 Combination model of buck converter and state space

56

Two models shares the same pulse to control the switch for the whole system. The benefit of
sharing the same pulse is that two system models can run synchronously. Another advantage is
that it is easy to compare the output signals with one scope. The parameters for configuring the
circuit model are selected by try-and-error based on realistic experiment data.
Table 3-3 Parameters of buck converter (1)
Ct (F)

Cl (F)

L (H)

Rl (Ω)

Duty Cycle (%)

250e-6

30e-6

30e-6

0.3

80

0.014
0.012

Power Output

0.01
0.008
0.006
0.004
0.002
0
5

5.5

6

6.5

7

7.5
Simulation Time

8

8.5

Figure 3-20 Power output of buck converter (1)

57

9

9.5

10
-3

x 10

The curve in Figure 3-20 shows the relationship between power output and simulation time, where
two curves are from different models and matches very well. The accuracy of the Simscape
simulation model is verified by the state space analytical solution model. According to the
parameters selected for this simulation model, the power output of the corresponding original
circuit is around 0.012w. With the simulation of 80% duty cycle, the power output approaches
0.0125 w. It can be predicted that with further, power output could be future improved.

3.3.3. Comparing of Primarily Improved Model and Buck Converter Model
The difference between two circuit models is that there are an inductor and diode added to buck
converter model comparing with the primarily improve model. The inductor reduces the current
ripple between the two capacitors. The diode is a passivity component which supposed to have not
much energy consuming. Decreasing the inductance reduces the difference between two circuit
systems. The parameters are selected as following.
Table 3-3 Parameters of buck converter (2)
Ct (F)

Cl (F)

L (H)

Rl (Ω)

Duty Cycle (%)

250e-6

30e-6

10e-6

0.3

1 to 90

58

0.014
0.012

Power Output

0.01
0.008
0.006
0.004
Primarily Improved Circuit
Buck Converter

0.002
0
0

10

20

30

40

50
Duty Cycle

60

70

80

90

100

Figure 3-21 Relationship of power output and duty cycle
From the figure above, the buck converter is able to generate higher energy than the primarily
improved circuit the duty cycles is higher than 50 percent. However when the duty cycles is lower
than 50 percent, the primarily improved circuit model can generate more energy than the buck
converter model. The power output of the buck converter approaches zero as duty cycle. Note that
the simulation of the primarily improved circuit is not accurate when the duty cycle is low. The
reason is that the primarily improved model transfers energy from capacitor Ct to Cl unrealistically
since the voltages of Ct and Cl have to be equal at an instant once the switch is closed. This action
could lead to a significantly high current between Ct and Cl. When the duty cycle is low, the
capacitor Ct is charged longer than the case with high duty cycle. As a result, the capacitor Ct also
stores more energy. And this energy will be transferred to Cl in an instant for primarily improved
59

circuit. However in the buck converter, due to the effect of inductor, the current cannot change
rapidly. So the primarily improved circuit has advantage in the efficiency of power output under
low duty cycle. However the extremely high current generated at the instant of the closing switch
is not realistic and cannot be realized. It could make the switch component be damaged due to over
current. Considering the high duty cycle cases, the buck converter circuit model can deliver even
more energy as the inductor can also stores certain energy.

3.4. Conclusion
In this chapter, analytical simulation models of primarily improved circuit and final optimized
circuit are developed. Differential equations and state space method are developed to verify the
performance and accuracy of the Simscape simulation models. Through simulation study, it was
found that the final optimized circuit significant advantage over the primarily improved circuit. As
a result, the buck converter is selected as the finalized circuit due to the fact that it is feasible for
implementation. By observing the result of the simulation model, it can be concluded that it is
necessary to control the buck converter circuit to maximize the power output. These conclusions
drawn based upon the circuit before the parameters are optimized. The parameter optimization will
be addressed in the next chapter.

60

Chapter 4 Simulation Study and Configuration

The buck converter designed in the last chapter contains three by components which needs to be
configured. They are inductor L, capacitor Ct close to electric source and capacitor Cl close to the
load resistance. In order to study the characteristic of the target buck converter and design a
controller for it, these three parameters needs to be optimized.
The goal of optimizing the circuit parameters is to assure the good circuit performance under
various operational condition. The two performance target considered are the power output and
the current ripple. Increasing the power output of this circuit under different load resistance is the
motivation of this research. The duty cycle control will be used to deliver the maximum power.
On the other hand, reducing ripple is also necessary in order to deliver high quality DC power.
The circuit parameters can also impact current ripples. By analyzing the circuit, the switcher is a
high duty-cycle component used as the control actuator of the system. When the switch is closed,
high current ripples occurs duel to voltage difference between two capacitors. The energy
transferred through the switch is considerably high. For this reason, study is focused on the current
though the switcher.
The goal of selecting the circuit parameters is to maximize high power output with the help of duty
cycle control for a reasonable ripple level. Making a good trade-off of power output and current
ripple by selecting a set of circuit parameters is the goal of this chapter.

61

4.1. Strategy of Configuring Circuit
The performances of the circuit mentioned at the beginning of Chapter 4 is the standard for
selecting circuit parameters. With the controller, the power output can be approach maximized.
The optimized a circuit parameters with closed-loop are to be found numerical simulations results.
The first step is to determine the range and increment of possible parameters. The parameters
contain duty cycle, load resistance, capacitor values and inductor value. The duty cycle and load
resistance are parameters depending on working conditions and control range. The other three
parameters are to be determined in this chapter. The controlled duty cycles will be used to
maximize the power output. With the determined the range and increment of the duty cycle, a
vector can be selected for the possible duty cycles. For example, if the range of feasible duty cycle
is between 1 percent and 100 percent, the duty cycles vector can be selected as T=[10 20 30 40 50
60 70 80 90 100]. Where the 10 percent is the increment of the duty cycle. The optimized duty
cycle lead to maximum power output is probably not concluded in this vector. However the optimal
cycle. With smaller increment of duty cycle, the optimal duty cycle could be approached closely.
The range of the duty cycle is determined to be from 1 per cent to 100 per cent at first. The analysis
that the circuit will not deliver satisfied power with small duty cycles. In order to fine a set of
optimal parameters for the circuit, the load resistance variation is also needed to be considered.
The same approach to duty cycle can be applied for load resistance validation. Assuming that the
load resistance ranges from 0.1 Ω to 1Ω, the vector of varying load resistance can be expressed as
Rl=[0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1]. The Range of other three parameters can be acquired in
the similar way.

62

The second step is to collect the simulation data of power output and current ripple with running
simulation model. By programming matlab script, the simulation model can be controlled to run
under different parameters and the associated simulation data can be also collected. The collected
data can be arrange to form matrixes and saved in matlab work space.
The third step is to process data and compare results. The process of data is introduced in Figure
4-1. The process is to study the effect of control using a matlab script. The optimal control can be
found for the maximum power output. The next step is to calculate the summation of the maximum
power output values the set of parameters. The sum of power output with different load resistance
for a given set of parameters Ct, Cl and L can be found. With the corresponding parameters of the
power output, the associated ripple can be easily and obtained. After analyzing data, the results
can be found. Usually, the curves and surfaces of the power output, ripple as function of the varying
parameters can be used to find the optimal parameters.

`

63

Set 1: Ct Cl L

......
..

Set 2: Ct Cl L

Simulation

R1
T1

0.0092

T2

0.0094

T3
T4

0.0120
0.0111

T5

0.0071

R2

R3

R4

R5 ......

......

........

Maximizing

R1
Tx

R2

R3

R4

R1

R5 ......

Ripplex

0.012 .....

Summation

R2

R3

R4

R5 ......

32 .....

Summation

Ripple Performance

Power Output Performance

Plotting Curve and comparing

Evaluation the set of Parameters

Figure 4-1 Flow chart of data processing

64

In summary, a set of matlab scripts are programed for simulating the circuit system with given
contoal parameter. With them the expected power output and the corresponding ripples can be
obtained Simscape buck converter.

4.2. Load Resistance
For finding the optimal circuit parameters, the load resistor is one of the components needed to be
studied. The load resistor varies with connected electric device. Because the research is focused
on simulation, the load resistance is assumed to be constant.
As the internal resistance thermal electric generator is defined as a constant 0.5Ω, the range of load
resistance need to be studied will be in the neighborhood of internal resistance. Recall that in
Chapter 1, it shows that the power output can be increased by switch control in Figure 1-9. In
Figure 1-9, the power output of the controlled circuit is higher than the original circuit when the
load resistance is smaller than 0.3Ω. Therefore, it is important to control the switch when the load
resistance changes. Simulation study was conducted to find the optimal duty cycle. The parameters
used for simulation is listed as following.

Table 4-1 Buck converter parameters (3)
Ct (F)

Cl (F)

L(H)

Rl(Ω)

Duty Cycle (%)

350e-6

20e-6

20e-6

0.1:0.1:1.5

10:10:100

After running this model and processing the simulate data, the relationship between maximum
power output and load resistance is obtained in Figure 4-2. Comparing it with the curve of original

65

circuit, it can be seen that the power output is enhanced when the load resistance is smaller than
0.5Ω.
0.014

Buck Converter
Orignal Circuit

0.013

Power Output

0.012
0.011
0.01
0.009
0.008
0.007

0.1

0.2

0.5

Load Resistance

1

1.5

Figure 4-2 Relationship of power output and load resistance
From the figure, it is clear that when the load resistance is smaller than the internal resistance the
power output can be increased through duty cycle control, and when point of the load resistance is
greater than the internal resistance the power output cannot be improved. After running the
Simulink mode with other parameters, the conclusion remains the same.

66

4.3. General Simulation
The central goal of parameter optimization is to find a set of parameters that yield reasonable
current ripple and power output. The first step is to find out how these parameters affect the circuit
performance. In this section, some general simulations are conducted to figure out the relationship
between the circuit performance and the parameters. The internal resistance and the power source
are fixed at 0.5Ω and 16 V, respectively.
If the parameters are not mentioned specially, the values defined in chapter 3 are used. The range
of the following parameters are selected according to the design limits and implemetation
consideration.
Table 4-2 General simulation parameters
Ct (µF)

Cl (µF)

L(µH)

Rt (Ω)

Duty Cycle (%)

50:100:450

8:30:158

8:30:158

0.1:0.1:0.6

10:10:95

The range of these parameters covered the possible combination for the target buck converter. The
results are shown below.

67

Current Ripple

15

10

5

0
0.0112

0.0114

0.0116

0.0118

0.012
Work per peroid

0.0122

0.0124

0.0126

0.0128

Figure 4-3 Performance of circuits with different parameter sets
The Figure 4-3 shows the performance (ripple and power output) of circuit with different sets of
parameters. Each point in the figure stands for a pair of power output and current ripple with the
unique set of parameters. It is obvious that the asterisks close to the lower right corner in the
coordinates represents for the most desirable performance and hence, the parameter sets. In this
case the current ripple is low, and the power output is high. In this figure, the asterisks inside the
red circle are to be selected. There might be a tradeoff between the two performance indexes:
current ripple and power output. The selection with emphasis the balance of power output and
current ripple to be discussed later when the detail simulations are available. There is a certain
tradeoff between the current ripple and peak power output shown in the green circle.
There are three parameters to be selected and we are going to study their effects to system
performance individually.

68

Work per period

0.013

0.0125

0.012

0.0115

0.011
1.6

2
1.4

1.2

1

0.8

0.6

1
0.4

0.2

0

0
-4

-4

x 10

Cl

L

x 10

Figure 4-4 Power output performance
In Figure 4-4, it can be seen that there are six surfaces in the figure. Using the matlab coded script
in appendix, the six surface stands for six values of capacitor Ct. On each surface, the maximum
power output is varying depending on the variable Cl and L, where the target parameter going to
be discussed is Ct. Form the figure, the conclusion can be made that with the Ct increasing, and the
power output of the circuit increases. And the rate of increment reduces as the Ct increases. With
the capacitance of Ct increases the capability of Ct to store energy improves also. This implies that
the power delivered from Ct to Cl is also increasing.
However, the capability of power delivering from Ct to Cl is also depends on other factors, too.
Once the capacitance become too large, the power output cannot be further enhanced as shown in
Figure 4-4.

69

Current Ripple

15

10

5

0
1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

1.6

1.4

1.2

0.8

1

0

0.2
-4

x 10

-4

x 10

0.6

0.4

L
Cl

Figure 4-5 Load resistance current ripple performance
The Figure 4-5 presents the relationship between current ripple and Capacitor Cl and inductor L.
Different surfaces stand for different Ct value. As it is shown in Figure 4-5, with the Ct increasing,
the current ripple increases. The increment rate of current ripple decreases. Regarding Ct, a tradeoff
needs to be made that the ripple and current has the same trend with the increment of Ct. The
possible value is around 200 µF.
After studying the trends regarding Ct, we are going to take a look at the relationship between the
performance and parameters Cl, L as well. For each surface in Figure 4-4 and figure 4-5, the front
views show the performance variation with the L. In Figure 4-4, the power output changes
significantly with the L at the beginning. After L approaches to 80 µH, the output power remains
unchanged, so either does the Cl cannot impact. In Figure 4-5, the current drops very quickly as L
increases. After the L approaches 80 µH, the current hardly changes also. It is clearly that the
70

inductor L will not change the system performance if it is greater than 80 µH. As shown in Cl
direction, the power output is decreases with the Cl increases. When Ct is larger than 100 µF, the
impact of Ct is minimum. It can be concluded that the ripple increases as Cl goes up. Therefore,
the value of Cl small to be less than 100 µF.

4.4. Detail Simulation
After conducting the case simulations, the specific range of Cl and L are determined. This section
conducts further simulations to determine Cl and L. The parameters are exists in Table 4-3.
Table 4-3 Detail simulation parameters
Ct (µF)

Cl (µF)

L(µH)

Rt (Ω)

Duty Cycle (%)

400

6:3:30

6:3:30

0.1:0.1:0.6

10:10:95

The following figures shows the performance of the circuit for all selected parameter combinations.

71

22
20
18

Current Ripple

16
14
12
10
8
6
4
2
0.0125

0.0125

0.0125

0.0125
0.0126
Power Output

0.0126

0.0126

0.0126

Figure 4-6 Performance of circuits with different parameter sets

0.0127

Work per period

0.0127
0.0127
0.0126
0.0125
0.0125
0.0124
4
3
2
1
0

-5

x 10

3.5

2.5

3

2

1.5

0.5

1
-5

L

Cl

Figure 4-7 Power output performance

72

x 10

25

Current Ripple

20
15
10
5
0
3.5

3

2.5

2

1.5

1

0.5

3.5

3

2.5

2

1.5

0.5

1
-5

-5

x 10

L

Cl

x 10

Figure 4-8 Current ripple performance
Regarding selecting Cl, the trends shown in Figures 4-7 and 4-8 indicate that the smaller Cl
provides good power output with reduced ripple. In this research, the final Cl is chose as 10 µF.
The parameter needed to be determined is L. Different from Cl, the power output and current ripple
is a trade-off function of the inductor value L. To make a reasonable tradeoff of the power output
and current ripple, the L is selected as 16µH. At this value, the circuit yields reasonable power
output with low ripple at the same time.

4.4. Conclusion
In this chapter, the effects of parameters of the buck converter are studied in simulations. The three
main parameters are capacitors Ct, Cl and the inductor L. The method of selecting these parameters
are based upon numerical simulation results using the developed Simulink model. After studying
the parameters sensitivity to the power output and current ripple, it is concluded that C l should be

73

small and Ct and L have a trade-off relationship with respect to power output and current ripple.
The following table lists the selected parameter for the buck converter circuit.
Table 4-4 Selected parameters
Ct

Cl

L

400 µF

10 µF

16 µH

74

Chapter 5 Controller Design

As the center piece of the thermoelectric generator circuit, the buck converter is going to work
under different load. Controlling the buck converter to adapt to the varying load resistance for the
maximum power output is the central target for controlling the duty-cycle.

5.1 Relationship between Power Output and Duty Cycle
Before designing the controller for the buck converter, we are going to study the relationship
between power output and duty cycle. With this relationship, the control strategy can be developed.
Figure 5-1shows the simulation results with load resistance defined in Table 5-1. Note that each
simulation response is associated with a unique load resistance.
Table 5-1 Parameters of buck converter (4)
Ct (µF)

Cl (µF)

L(µH)

Rt (Ω)

Duty Cycle (%)

400

10

16

0.05:0.05:0.6

10:1:90

75

0.014

0.012

Power Output

0.01

0.008

0.006

0.004

0.002

0
10

20

30

40

50

60
Duty Cycle

70

80

90

100

Figure 5-1 Power output performance
Twelve simulations were conducted with the load resistance varying from 0.05Ω to 0.6Ω. From
Figure 5-1, it is obvious that all the response curves have only one peak point. That is the maximum
power output point. The power output increases as the duty cycle goes up until it approaches
maximum. After the peak, the power goes down.
For this reason, the extreme seeking control theory is used to a good fit for the buck converter duty
cycle control to maximize the output power.
76

5.2. Extreme Seeking Control
In this section, we are going to introduce extreme seeking control theory. Extreme Seeking is used
in the field that the system transfer function is unknown or very complicated. It can be used as a
method to optimize steady-state performances for certain system with guarantee stability [18]. It
was widely used in wind energy [19] and ABS system in vehicle [20].
In this research, the target buck converter is a periodic system with a PWM controlled switch. In
this case, the controller can be designed using a deterministic extreme seeking scheme as shown
in following block diagram in Figure 5-2.

Plant

𝑘
−
𝑠

−

𝑠
𝑠+ℎ

sin(ωt)

a∙sin(ωt)

Figure 5-2 Block diagram of extreme seeking scheme
The signal sinusoid is the perturbation applied to the system. The output of the plant goes through
a high-pass filter to obtain the derivative of that output. Note that the derivative provide the
information of system response to the perturbation. Then, the derivative signal is multiplied with
the white perturbation and the result is integrated to form the extreme seeking control signal with
77

gain k. Thus, the extreme value, power output, is achieved when the derivative approaches zero
[21].

5.3. Simulation Model with Controller
With the extreme seeking scheme shown in the last section, it is simple to build a simulation model
with the extreme seeking controller as shown in Figure 5-3. In order to implement the associated
duty cycle control, a new pulse input model is to make it controllable. The real-time power output
signal is selected as the feedback signal. The Simscape mode is used as the plant model, shown in
figure 5-4.

Figure 5-3 Simulation model of extreme seeking controller

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Figure 5-4 System Plant Simulation Model

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5.4. Simulation Result
After building the extreme seeking controller system model, several simulations were conducted
to verify the performance of the extreme seeking.
Table 5-2 Simulation parameters of controlled buck converter
Ct (µF)

Cl (µF)

L(µH)

Rt (Ω)

400

10

16

0.2

140
120

Power (w)

100
80
60
40
20
0
0

0.05

0.1

0.15

0.2

Time (s)

Figure 5-5 Power efficiency of the Buck Converter

80

0.25

95
90

Duty cycle

85
80
75
70
65
60
55
0

0.05

0.1

0.15

0.2

0.25

time (s)

Figure 5-6 Duty Cycle of Buck Converter
In figure 5-5, the power output of the buck converter is stable at around 120 W. Compared with
the circuit without control, the power is much higher.
The duty cycle curve in Figure 5-6 also shows that the extreme seeking controller converges after
0.2s, and the optimized duty cycle is stabilized at around 60. This also matches the result shown
in Figure 5-1 that the best duty cycle is around 60 with the load resistance of 0.2Ω.

5.5. Conclusion
By using extreme seeking scheme, a controller is designed for the buck converter to optimize the
power output in real-time and the simulation results shows that the control converges in 0.2s.

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Chapter 6 Conclusion and Future Work

The load resistor of the thermoelectric generator varies as lead changes. This could lead to degrade
performance of the thermoelectric generator. In order to overcome this problem. A buck converter
circuit is designed, modeled, and optimized to maximize the thermoelectric generator power output.
An extreme seeking controller is also designed for the buck converter to maximize the power
output under different load resistant level.

6.1 Conclusion
As a switch regulation device, buck converter is proved to have high efficient and small size
compared with the linear supply source. In this research the buck converter applied in
thermoelectric generator, and a PMW signal was used to regulate circuit for maximal power output
as the load resistance changes.
In order to optimize the buck converter, matlab Simscape tool box is used to develop simulation
model of target buck converter. The Simscape model is verified the model with differential
equation analytical model, the state-space simulation model. The sensitivities of different
parameters, such as the capacitors and inductor in the buck converter, are obtained through
intensive simulation study, and the optimal set of parameters. The main performance indexes used
to evaluate parameter sets are current ripple and power output. With the help if the extreme seeking
control scheme, the buck converter can be controlled in real-time with maimal power output when
the load resistance changes.

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6.2. Future Work
The next step is to develop a real physical buck converter for the thermoelectric generator to verify
the modeling and control results presented in this thesis, the extreme seeking controller need to be
verified next.

83

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