Modelling and control of a Buck converter
Bachelor thesis
by
Shun Yang
Karlskrona, Blekinge, Sweden
Date: 2011-06-22
This thesis is presented as part of Degree of Bachelor of Science in Electrical
Engineering
Blekinge Institute of Technology
School of Engineering
Department of Electrical Engineering
Supervisor: Anders Hultgren
Examiner: Jörgen Nordberg
i
Modelling and control of a Buck converter
Bachelor thesis
at
Blekinge Institute of Technology
By
Shun Yang 870924-6030
Examiner: Jörgen Nordberg. Supervisor: Anders Hultgren
Karlskrona, Blekinge, Sweden
Date: 2011-06-22
ii
Acknowledgement
First and foremost I offer my sincerest gratitude to my supervisor, Anders
Hultgren, who has supported me throughout my thesis with his patience and
knowledge whilst allowing me the room to work in my own way. I attribute the level
of my Bachelor’s degree to his encouragement and effort and without him this thesis,
too, would not have been completed or written. One simply could not wish for a better
or friendlier supervisor.
In my daily work I have been blessed with a friendly and cheerful group mate --Gaojun Chen. He has provided good arguments about controller theory and helps me
regain some sort of fitness: healthy body, healthy mind. He is a good companion and
has had the good grace to pester me much less than average with computer questions.
He provided an experienced ear for my doubts about writing a thesis.
The Department of Electrical Engineering has provided the support and
equipment I have needed to produce and complete my thesis. My deepest gratitude
also extend to all lecturers of at Blekinge Institute of Technology (BTH), for their
guidance, ideas, and support in completing this final year’s thesis project. This project
had opened my eyes on solving real problems and I was able to relate them with what
I’ve been studied in BTH during the past 3 years.
Finally, I thank my parents for supporting me throughout all my studies at
University and providing financial fund to help me to complete my bachelor
education.
iii
Abstract
DC/DC buck converters are cascaded in order to generate proper load voltages.
Rectified line voltage is normally converted to 48V, which then, by a bus voltage
regulating converter also called the line conditioner converter, is converted to the bus
voltage, e.g. 12V. A polynomial controller converter transforms the 12V into to a
suitable load voltage, a fraction of or some few voltages. All cascaded converters are
individually controlled in order to keep the output voltage stable constant. In this
presentation focusing on the polynomial controller converter implemented as
Ericsson’s buck converter BMR450. In this paper modeling, discretization and control
of a simple Buck converter is presented.
For the given DC-DC-Converter-Ericsson BMR 450 series, analyzing the
disturbance properties of a second order buck converter controllers by a polynomial
controller.
The project is performed in Matlab and Simulink.
The controller properties are evaluated for measurement noise, EMC noise and for
parameter changes.
Keyword: polynomial controller, second order, buck converter
iv
Table of Contents
Acknowledgement ........................................................................................................................... iii
Abstract ............................................................................................................................................ iv
List of Tables ................................................................................................................................... vi
List of Figures ................................................................................................................................. vii
List of Symbols .............................................................................................................................. viii
Chapter 1.
Introduction ............................................................................................................... 1
Chapter 2.
Modeling of Buck Converter..................................................................................... 3
Chapter 3.
Polynomial Controller Design................................................................................. 11
Chapter 4.
Simulation on MATLAB ........................................................................................ 15
Chapter 5.
Results and Conclusion ........................................................................................... 19
References ....................................................................................................................................... 30
Appendix ......................................................................................................................................... 31
Appendix A Simulink Models ..................................................................................................... 31
Appendix B MATLAB Code ....................................................................................................... 33
Appendix C Interesting Phenomenon found in the simulation .................................................... 38
Appendix D Transfer function ..................................................................................................... 42
v
List of Tables
Table 1
Table 2
Table 3
Table 4
Table 5
Table 6
Parameters of Buck Converter .................................................................................. 2
Parameters of system ............................................................................................... 10
Polynomial coefficients of system........................................................................... 10
Pole Placement of Polynomial Controller ............................................................... 14
Coefficients of Polynomial controller ..................................................................... 14
Parameters of system ............................................................................................... 27
Appendix Table 1
Appendix Table 2
Appendix Table 3
Appendix Table 4
Capacitor’s Resistor equals Inductor’s ................................................... 38
Capacitor’s Resistor is bigger than Inductor’s ........................................ 39
Capacitor’s Resistor is smaller than Inductor’s ...................................... 40
Capacitor’s Resistor equals Inductor’s ................................................... 40
vi
List of Figures
Fig. 1 Overall system circuit .................................................................................................. 3
Fig. 2 PWM Wave .................................................................................................................. 4
Fig. 3 PWM controlled input voltage dVg ............................................................................. 4
Fig. 4 Circuit with Voltage Source ......................................................................................... 5
Fig. 5 Circuit without Voltage Source .................................................................................... 7
Fig. 6 Controller with Buck Converter System [2] .............................................................. 11
Fig. 7 The shadow region inside the unit cycle is stable for system .................................... 13
Fig. 8 Buck Converter System with polynomial Controller [2] ........................................... 15
Fig. 9 System block .............................................................................................................. 15
Fig. 10 Polynomial controller block..................................................................................... 16
Fig. 11 PWM with polynomial controller ............................................................................ 16
Fig. 12 Relational operator................................................................................................... 17
Fig. 13 Switch ...................................................................................................................... 17
Fig. 14 Saturation condition ................................................................................................. 18
Fig. 15 Discrete model without saturation, system output and controller output................. 19
Fig. 16 Continue model, system output and controller output ............................................. 20
Fig. 17 Simulation result, without disturbance, without noise ............................................. 21
Fig. 18 Noise Magnitude ...................................................................................................... 22
Fig. 19 Output Voltage with noise ........................................................................................ 22
Fig. 20 L=0.9μH, C=150μF , Simulation result, with disturbance, without noise ............... 23
Fig. 21 +10%, L=0.99μH, C=165μF, Simulation result, with disturbance, without noise ... 24
Fig. 22 -10%, L=0.81μH, C=135μF, Simulation result, with disturbance, without noise .... 25
Fig. 23 Simulation result, with disturbance, with noise ....................................................... 26
Fig. 24 RC=50mΩ, C=500μF, Simulation result, with disturbance, without noise ............... 26
Fig. 25 Output Voltage before and after disturbance when RL =2mΩ, RC=50mΩ, C=500uF
......................................................................................................................................... 27
Fig. 26 Output Voltage filtered noise when RL =2mΩ, RC=50mΩ, C=500uF ...................... 28
Fig. 27 Output Voltage with noise when RL =2mΩ, RC=50mΩ, C=500uF .......................... 29
Appendix Fig. 1 Continuous Time Model without Saturation ............................................. 31
Appendix Fig. 2 Discrete Time Model without Saturation .................................................. 31
Appendix Fig. 3 Discrete Time Model with Saturation ....................................................... 32
Appendix Fig. 4 PWM Model with Controller .................................................................... 32
Appendix Fig. 5 Output Voltage before and after disturbance when RL = 2mΩ and RC=2mΩ
......................................................................................................................................... 38
Appendix Fig. 6 Output Voltage before and after disturbance when RL < RC ...................... 39
Appendix Fig. 7 Output Voltage before and after disturbance when RL >RC ....................... 40
Appendix Fig. 8 Output Voltage before and after disturbance when RL =10mΩ and
RC=10mΩ ........................................................................................................................ 41
vii
List of Symbols
%
V
L
C
R
d
DC
fs
iL
yref
PWM
Percentage
Volt
Inductor
Capacitor
Resistor
Duty Cycle
Direct Current
Switching Frequency
Switch/Inductor Current
Reference Voltage
Pulse Width Modulation
viii
Chapter 1. Introduction
This chapter describes the thesis background, objectives, scopes, and summary.
In the chapter, it briefs the description of the buck converter and the voltage-mode
controller as well as the objectives and the scopes. At the end, outline of this thesis is
given in this chapter.
1.1 Thesis Background
Direct current to direct current (DC-DC) converters are power electronics circuits
that converts direct current (DC) voltage input from one level to another. DC-DC
converters are also called as switching converters, switching power supplies or
switches. DC-DC converters are important in portable device such as cellular phones
and laptops.
Why DC-DC converter is needed? Just image that when wanting to use a device
with low voltage level, if connect the device such as laptop or charger directly to the
rectified supplied from the socket at home, the device might not functioning properly
or it might be broken due to overcurrent or overvoltage. The voltage level needs to be
converted to suitable voltage level for the equipment to function properly. In this
project, the configuration of DC-DC converter chosen for study was buck
configuration. Buck converter converts the DC supply voltage to a lower DC output
voltage level.
The control method chosen to maintain the output voltage from the buck
converter was polynomial controller. Polynomial controller technique compares the
actual output voltage with the reference voltage. The difference between both voltages
will drive the control element to adjust the output voltage to the fixed reference
voltage level. This is called as voltage regulation.
1.2 Thesis Objectives
The main objective of this project is to design a buck converter controller based
on the theory for discrete polynomial controllers. A basic introduction can be found in
[1].The converter control signal is implemented as a Pulse Width Modulated signal
making the system into a switched system. It is then interesting to find the nonlinear
system properties by simulations. The control design should be evaluated by means of
accuracy in output voltage level given some different kind of disturbances. The
disturbances are load current changes, measurement noise and parameter variations.
1.3 Thesis Scopes
The scopes of this thesis are:
i. Study the operation of buck converter.
ii. Design the mathematical system model.
iii. Design the polynomial controller.
iv. Simulation of buck converter and controller circuit using Simulink and
MATLAB in order to test the properties of the system.
v. Get the conclusion from simulation result.
1
The parameters used in the project are as below, see Table 1.
Table 1
Parameters of Buck Converter
Topology
DC-DC Buck Converter
Inductance
L = 0.9μH
Capacitance
C = 150μF / 500μF
Resistance of Inductor
RL = 2mΩ / 10 mΩ
Resistance of Capacitor
RC = 2mΩ / 10mΩ / 50mΩ
DC Input Voltage
Vg = 12V
Reference Voltage (Expected DC Output Voltage)
yRef = 3.3V
Wanted Inductor Current
iI = 0-20A
Sample Frequency
fs = 330kHz
Sample interval
h = 3μs
1.4 Outline of Thesis
This thesis consists of 5 chapters. In the first chapter, it discusses thesis
background, objectives, and scopes. In Chapter 2, building of mathematical system
model and theories on buck converter are displayed.
The polynomial controller design is showed in Chapter 3 while the simulation
implementation on Matlab and Simulink are offered in Chapter 4. In Chapter 5, it
discusses the simulation result and conclusions obtained upon successfully
completing thesis project. Finally, the last part in the thesis provides the references
and appendices used in the thesis project.
2
Chapter 2. Modeling of Buck Converter
A Buck converter consists of a transistor and diode that applies the supply voltage
on an inductor capacitor, LC, circuit. The output voltage is the voltage across the
capacitor.
The input voltage u on the LC circuit is controlled by pulse width modulation,
PWM. i.e. part of a cycle time, the voltage applied on the LC circuit is Vg and the rest
of the cycle time the input voltage is zero. The range of duty cycle, d=[0, 1], is the
relative time of the cycle time that the supply voltage is connected to the circuit, i.e.
the input is Vg.
The duty cycle d is the control signal to the Buck converter. A circuit diagram
can be seen in Fig. 1.
Switch
RL
L
node 1
Vg
RC
node 2
iI
C
Fig. 1
Overall system circuit
The system need to be modeled to get the relationship between the input signal
and output.
The Buck converter system is a switched system. It consists of two switching
states: firstly, when the switch is connected to node 1, charging status; secondly, when
the switch is connected to node 2, discharging status.
The switched system can be modeled as an averaged system. The averaged
system describes the switched system up to about one tenth of the PWM frequency. In
this chapter deriving the two system models of these two switching state, and then
take the average of the two systems according to the duty cycle.
PWM is the method of choice to control modern power electronics circuits. The
basic idea is to control the duty cycle of a switch such that a load sees a controllable
average voltage. To achieve this, the switching frequency (repetition frequency for the
PWM signal) is chosen high enough that the LC filter cannot follow the individual
switching events.
In this case, the sampling interval is 3μs. So the sample frequency is 330 kHz, see
Fig. 2.
3
1
0.9
0.8
PWM wave
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
Time
Fig. 2
2
2.5
3
-5
x 10
PWM Wave
The Fig. 2 shows PWM signals for saw-tooth wave.
With pulse-width modulation control, the regulation of output voltage is achieved
by changing the duty cycle of the switch, keeping the frequency of operation constant.
Duty cycle refers to the ratio of the period for which the voltage source is kept
connected to the cycle period. A clearer understanding can be acquired by the Fig. 3
Discrete polynomial controlled Buck, switched continuous model with saturation
controlled input voltage, controller output, PWM wave
4
dVg/3
d
pwm
3.5
3
2.5
2
1.5
1
0.5
0
0.5
1
1.5
2
Time
Fig. 3
2.5
-5
x 10
PWM controlled input voltage dVg
When the controller output (green line) d, is bigger than the saw-tooth waveform
(red line), the switch will be connected to node 1, which is charging status and the
input voltage(blue line) is 12V. Otherwise, the switch will be connected to node 2,
discharging status and the input voltage is 0V.
4
The frequency of the repetitive waveform with a constant peak, which is shown
to be a saw-tooth, establishes the switching frequency. This frequency is kept constant
in a PWM control.
2.1 Build model with voltage source
At first, the voltage source is included in the circuit when the switch is connected
to node 1, see Fig. 4.
RL
L
RC
Vg
C
Fig. 4
iI
Circuit with Voltage Source
The simple Buck converter can be modeled as Equation Chapter 2 Section 1
 iU

 iC
 iR
 C
 iRL

 uL
u
 I
 0 0 0 0
 
 0 0 0 0
 0 0 0 0

 0 0 0 0
  1 1 1 1
 
  0 1 1 0

1 0   uU 

 
1 1  uC 
1 1  uRC 


1 0   uRL 
0 0   iL 

 
0 0   iI 
(2.1)
Where u is the voltage to the converter and i is the current.
The network model can be described in the simple form:
 iC   0 0 0 1   uC   0 1
  
 
 

 iRC    0 0 0 1    uRC    0 1   uU 
i   0 0 0 1 u   0 0  i 
I 
 RL  
 RL 
 u   1 1 1 0   i   1 0 
 L
L 
(2.2)
Modeling the two energy storing components it has:
 iC   0 1   uC   0 1  uU   0 0   uRC
 
   
   

 uL   1 0   iL   1 0   iI   1 1  uRL



(2.3)
The last term can be eliminated using
 iRC

 iRL
  0 1  uC   0 1  uU 
  
   
 
0
1

  iL   0 0   iI 

and
5
(2.4)
 uRC

 uRL
  RC
  
  0
0   iRC

RL   iRL



(2.5)
0   0 1  uU 
 

RL   0 0   iI 
(2.6)
So:
 uRC

 uRL
  RC
  
  0
0   0 1  uC   RC
 

RL   0 1  iL   0
Then it can simplify the equation:
 iC   0 1   uC   0 1  uU   0 0   0  RC   uU 
 
   
    

 
0   iI 
 uL   1 0   iL   1 0   iI   1 1  0
1
0
  uC   0 1   uU 

   
 
 1  RC  RL   iL   1 RC   iI 
(2.7)
As the output voltage is the sum of capacitor voltage and resistor voltage, where
resistor voltage is the product of current and resistor, then it has
u0  uC  uRC
(2.8)
uRC  RC iL
(2.9)
u 
u0  uC  uRC  1 RC   C 
 iL 
(2.10)
 iC   0
1
  uC   0 
   
      uU
 uL   1  RC  RL   iL   1

u  1 R  uC 

C 
 0 
 iL 

Including the constitutive equations, the system is modeled by
d
 
 dt uC   0


 d i   1
 L  
 dt   L
1


0

 uC  
C
   
RC  RL   iL   1



L 
 L
1
C   uU 
 
RC   iI 

L 
(2.11)

 u0   1 RC   uC   0 0   uU 
 
   
 
 iL   0 1   iL   0 0   iI 
(2.12)
(2.13)
The ABC-differential equation system is given as
 x  A1 x  B1u

 y  C1 x
6
(2.14)

 0
A1  
 1

 L
1


C

RC  RL 


L 

 0
B1  
 1

 L
(2.15)
1
C

RC 

L 

(2.16)
 1 RC 
C1  

0 1 
(2.17)
Where u is the input of system which is Vg in the diagram and y is the output of
the system.
2.2 Build model without voltage source
When the switch is connected to node 2, the voltage source is not included in the
circuit.
Then the circuit is as below, see Fig. 5.
RL
L
RC
C
Fig. 5
 iC

 iRC
i
 RL
 uL

 uI
iI
Circuit without Voltage Source
 0 0 0
 
 0 0 0
 0 0 0
 
  1 1 1
 1 1 0
 
7
1 1  uC 

 
1 1  uRC 
1 1   uRL 

 
0 0   iL 
0 0   iI 
(2.18)
 iC   0 0 0 1   uC
  
 
 iRC    0 0 0 1    uRC
i   0 0 0 1 u
 RL  
 RL
 u   1 1 1 0   i
 L
L
Modeling the two energy storing components:
  1
  
   1  i
 0  I
  
 0 

 iC   0 1   uC   0 0   uRC
 
   

 uL   1 0   iL   1 1  uRL
 iRC

 iRL
  1
    iI
 0 
  0 1  uC   1
  
      iI
0
1

  iL   0 

 uRC

 uRL
  RC
  
  0
0   iRC

RL   iRL
(2.20)
(2.21)



 uRC   RC 0   0 1  uC   RC

  

   
 uRL   0 RL   0 1  iL   0
 0 RC   uC    RC 

 iI
   
 0 RL   iL   0 
(2.19)
(2.22)
0   1
i

RL   0  I
(2.23)
Then it can simplify the equation:
 iC   0 1   uC   1
 
       iI 
 uL   1 0   iL   0 
 0 0   0 RC   uC   0 0    RC 
 iI
   



 1 1  0 RL   iL   1 1  0 
(2.24)
1
0
  uC   1 

      iI
 1  RC  RL   iL   RC 
Including the constitutive equations, the system is modeled as below:
d
 
 dt uC   0


 d i   1
 L  
 dt   L
1

 1


 uC   C 
C
   
 iI
RC  RL   iL   RC 




L 
 L 
(2.25)
The ABC-differential equation system is given as
d
 
 dt uC   0


 d i   1
 L  
 dt   L
1
1


0  


 uC 
C
C  uU 
   
 
RC  RL   iL  
RC   iI 


0

L 

L 
(2.26)
x  A2  x  B2  u
(2.27)
8
 u0   1 RC   uC 
 
 
 iL   0 1   iL 
(2.28)
y  C2  x
(2.29)
2.3 Get average system
The time averaged system can be derived as the weighting averaged of the two
models.
The averaged system Saverage, system S1 with voltage source and system S2
without voltage source. Assuming the duty cycle d, and then it has
Saverage  d  S1  1  d   S2
(2.30)
As it can be observed that ABCD matrix of the two systems, that ACD matrices
are the same while the B matrix are different.
So only need to take the average value of B matrix.
B  d  B1  1  d   B2
(2.31)
The B matrix can be written as:
B  d  ( B11  B12 )  1  d   ( B21  B22 )
(2.32)
 0 0
 0 1/ Cc 
where B11  
 and B12   0 R / L 
1/ L 0 
c


0 0
 0 1/ Cc 
B21  
 and B22   0 R / L 
0 0
c


Then the calculation is as below:
  0 0   0 1/ Cc    Vg 
  0 0  0 0   0 1/ Cc    Vg 
Bu   
    (1  d )
  0 R / L  I  d  


1/
L
0
 
c
 0 

  0 0  0 0   0 Rc / L    I 0 
 0
 
 Vg / L

 0

 Vg / L
0   0  I 0 / Cc  
  0 0   0  I 0 / Cc  
d  

  (1  d )

0   0 I 0 Rc / L  
  0 0   0 I 0 Rc / L  
(2.33)
0
 0  I 0 / Cc 
d 

0
 0 I 0 Rc / L 
 I 0 / Cc   d 
 0

 
 Vg / L Rc / L   I 0 
Then the averaged system is as below:
 x  Ax  Bu

 y  Cx
Writing the continuous-time system in a detailed way:
9
(2.34)
d
 
 dt uC   0


 d i   1


L 
 dt   L
1
C
R  RL
 C
L


0
  uC  
   
  iL   Vg



 L
I0 
Cc   d 
 
Rc   I 0 

L 
(2.35)
Since B is the averaged result of the switched system and the input voltage and
the capacitor, resistor and inductor values are fixed. So B will be decided by the duty
cycle d, the value of B will change when duty cycle d changes, so arranging d be the
control signal.
As the averaged system is a continuous time system, it needs to be transferred it
into a discrete time system. So taking samples from the continuous time system by a
sampling interval h.
The discretization is performed using Matlab, see Appendix B. The zero order
holder is assumed on the input signal, when performing the discretization.
The discrete-time transfer function. Given the system
 x  Fx  Gu

 y  Cx
(2.36)
The transfer function is given by
Y
1
(2.37)
 C  zI  F   G
U
As it is mentioned in chapter one, the suitable parameters for the first performed
simulation as below, see Table 2.
H  z 
Table 2
Parameters of system
Topology
Values
Inductance
L=0.9μH
Capacitance
C=150μF
Resistance of Inductor
RI=2mΩ
Resistance of Capacitor
RC=2mΩ
Sampling interval
h=3μS
PWM sampling interval
hPWM=30ns
And then the system coefficients as below, see Table 3.
Table 3
Polynomial coefficients of system
System Polynomial coeffecients
Values
a1
-1.9209
a2
0.9868
b1
0.4746
b2
0.3157
10
Chapter 3. Polynomial Controller Design
The controller with buck converter system is as below, see Fig. 6.
V z
Controller
Yref  z 
System
1
C z
Kr
U z
H z  
B z
A z 
Y z
D z 
Fig. 6
Controller with Buck Converter System [2]
It gets the reference voltage as input of the controller and u as the output of
controller, while u is the input of buck converter system and y is the output of buck
converter system. Equation Chapter 3 Section 1
More importantly, yref serves as the feedback of the system and it will be
compared with the reference voltage. And by the comparing, the controller will adjust
to make the output voltage close to the reference voltage. If the output voltage is
higher than the reference voltage, then the controller will make the switch open to
discharge the circuit; while the output voltage is lower than the reference voltage, the
controller will close the switch to charge for the circuit.
The controller is a discrete transfer function, designed in order to get a certain
poles in the closed system. (The described design method is adopted from
Schmidtbaer, [2])
As in the second order discrete system, transfer function is [2]
H  z 
Y  z
Bz
b  z 1  b2  z 2
 1

1
2
U  z  1  a1  z  a2  z
A z 
(3.1)
The transfer function can be expressed as a difference equation [2]
y  k   a1  y  k  1  a2  y  k  2   b1  u  k  1  b2  u  k  2 
(3.2)
A controller is introduced as a general difference equation of the same order as
the system’s difference equation [2]
u  k   Kr  yref  k   d0  y  k   d1  y  k  1  d 2  y  k  2  c1  y  k  1
(3.3)
The controller can be given as a Z-transform [2]
U  z 
D z
d0  d1  z 1
Kr
K
Y z 
Y z 
 Y  z   r Yref  z 
1
1 ref  
1  c1  z
1  c1  z
C z
C z
11
(3.4)
As Y  z   U  z   H  z   U  z  
Y  z  
B z
, the closed system is then given as [2]
A z 
B z D z
B  z   Kr
Y  z  
Yref  z 
A z   C  z 
A z  C  z 
Y  z
B  z   Kr

Yref  z  A  z   C  z   B  z   D  z 
(3.5)
(3.6)
When z=1, Kr can be designed in order to get the correct stationary gain, [2]
K r  D 1 
C 1  B 1 P 1

A 1
B 1
(3.7)
This implies that the output signal Y stationary will be equal to the input value
Yref(z).
For polynomial controller design, looking at the system first.
From the system, the transfer function is H  z  
B z
A z 
Then start the designation of polynomial controller, which refer to determine the
polynomials C(z) and D(z).
3.1 Get transfer function from system
From the system, when taking samples from continuous time model to get the
discrete model. Since the system is second order system. The transfer function
H(z)=B(z)/A(z) have second order, where polynomials B(z) and A(z) have second
order in the converter system.
As in chapter 2, it has A  z   1  a1  z 1  a2  z 2 and B  z   b1  z 1  b2  z 2
3.2 Design pole placement
For pole placement P(z), where P(z)=A(z)C(z)+B(z)D(z), in order to keep the
system stable, which means P(z)=0 has its roots located inside the unit cycle, see Fig.
7 [1].
12
Z-plane
Im
1
1 Re
Fig. 7
The shadow region inside the unit cycle is stable for system
As it can be seen from Fig. 7, z=0.5, z=0.5, and z=0.5are in the shadow region.
Then as the example, it can be chosen as:
P( z)  (1  0.5  z 1 )  (1  0.5  z 1 )  (1  0.5  z 1 )
(3.8)
3.3 Get polynomial coefficients of C(z) and D(z)
When choosing the order of the C and D polynomial, it gives the closed system
an arbitrarily pole placement. As the rules of polynomial controller given in
Scmidtbaer, the polynomial C(z) and D(z), where C(z) and D(z) satisfy nC=nB-1=1
and nD=nA-1=1. So polynomial C(z) and D(z) have first order. And the order of
polynomial P(z) which is nP=nA+ nB-1=3
It has the rule that the controller should be monic so that assume
C  z   1  c1  z 1 and D  z   d0  d1  z 1
Then it has [2]
P  z   A z  C  z   B  z   D  z 
 1  a1  z 1  a2  z 2   1  c1  z 1    b1  z 1  b2  z 2    d 0  d1  z 1 
(3.9)
 1   a1  c1  b1d 0  z 1   a1c1  a2  b1d1  b2 d 0  z 2   a2c1  b2 d1  z 3
And it also has
P  z   1  q1  z 1 1  q2  z 1 1  q3  z 1   1  p1  z 1  p2  z 2  p2  z 3 (3.10)
Where q1, q2 and q3 are three chosen poles.
p1  q1  q2  q3
p2  q1  q2  q2  q3  q1  q3
p3  q1  q2  q3
Identifying the coefficients for P(z): [2]
13
(3.11)
p1  a1  c1  b1  d 0
p2  a1  c1  a2  b1  d1  b2  d 0
(3.12)
p3  a2  c1  b2  d1
By calculation, the solution is as below: [2]
b12  p3  b1  b2   a2  p2   b22   p1  a1 
c1 
a2  b12  b22  a1  b1  b2
d0 
p1  a1  c1
b1
d1 
p3  a2  c1
b2
(3.13)
In this system, A(z) and B(z) polynomials coefficients is given in chapter 2, as
showed in Table 4, and the pole placement as below, see Table 4:
Table 4
Pole Placement of Polynomial Controller
Controller Pole Placement
Values
q1
0.5
q2
0.5
q3
0.5
And C(z) and D(z) polynomials coefficients are below, see Table 5.
Table 5
Coefficients of Polynomial controller
Controller Polynomial
Values
coeffecients
c1
0.0808
d0
0.7166
d1
-0.6485
The pole placement is fixed and the sampling interval h=3e-6 which is used by
Ericsson now. When placing the poles it has to judge that the control signal do not
saturate outside 0-1 for a reasonable change in load current, see chapter 4.
14
Chapter 4. Simulation on MATLAB
After obtaining the mathematical system model and polynomial controller model,
the simulation block building in Simulink of Matlab. In this chapter, the block of
system is built first. Secondly, adding the controller block after the system block, see
Fig. 8. Finally adding the PWM with the controller.
V z
Controller
Yref  z 
System
1
C z
Kr
U z
H z  
B z
A z 
Y z
D z 
Fig. 8
Buck Converter System with polynomial Controller [2]
4.1 Build the system block in Simulink
At first, building the system block with voltage source and current source as input,
and the system output is the voltage of the load, at the same the inductor current can
be observed, see Fig. 9.
Fig. 9
System block
4.2 Build the controller block in Simulink
When taking the output voltage as the input of controller and compare it with the
reference voltage, the output voltage is regarded as a feedback.
Then continue with the controller block. The controller output is d, see Fig. 10.
15
Fig. 10
Polynomial controller block
The input of the controller is the output of the system, while the output of the
controller which will use to compare with PWM wave and then control the switch to
adjust the charging and discharging of circuit.
4.3 Build the PWM block in Simulink
In this controller, when the controller output is less than PWM value, set the
switch closed which is charging status and when the controller output is more than
PWM value, set the switch open which is the discharging status.
For the detail blocks, see Fig. 11.
???
Repeating
Sequence
1
Constant
Saturation
<=
d
dVg
Relational
Operator
Constant2
Switch
To Workspace5
0
Constant1
Fig. 11
PWM with polynomial controller
In Simulink, the relational operator is used to compare two controller output with
PWM value, see Fig. 12.
16
Fig. 12
Relational operator
And when useing switch to adjust according to the output of relational
operator, see Fig. 13.
Fig. 13
Switch
By the way, having the saturation for the controller output between 0 and 1
17
because the range of PWM value is between 0 and 1, see Fig. 14.
Fig. 14
Saturation condition
4.4 Add the noise to the Simulink model
The buck converter has some disturbance properties, such as EMC noise and
noise at the switch. And then measure and analysis the noise by changing parameters.
Since the component in Simulink of Matlab is ideal which means it is perfect,
without any noise. But in the real case, add those noises, so in order to realize the
disturbance evaluation, the noise need to be added independently.
The noise in the model is band-limited white noise.
For the completed Simulink model, please see models in Appendix A.
For the simulation code, please see Matlab code in the Appendix B
18
Chapter 5. Results and Conclusion
In chapter 4, finishing the model building in Simulink, now, run the simulation
and analysis the results. The simulation period use is 100 sample intervals which are
enough for us to observe the result.
The inductor current is much bigger than the output voltage and controller output.
In order to observe the result clearly, it shows the inductor current 10 times smaller.
5.1 Simulation results of Continuous model and discrete model
For the discrete model, the simulation result is showed in Fig. 15.
Discrete model
3.5
X= 42
Y= 3.3
3
Vout
d
Output voltage
2.5
2
1.5
1
X= 42
Y= 0.275
0.5
0
0
Fig. 15
20
40
60
80
Sampling sequence k
100
120
Discrete model without saturation, system output and controller output
As it can be observed that the output voltage of discrete system can satisfy the
required voltage.
The controller output can get stable quickly (in 16 sample intervals) and remain
stable. What’s more, the controller make the output voltage come to stable status
quickly (in 16 sample intervals) and remain the wanted output voltage (reference
voltage 3.3V). The simulation is according to the polynomial controller theory, so it is
an illustration of the theory by numerical calculations.
When designing the controller, it should not hit the saturation limits too much for
a reasonable disturbance. And the simulation result satisfies it.
For chosen pole placement of the shown controller, see Table 4.
For the continuous model, the simulation result is showed in Fig. 16
19
Continuous model
3.5
X: 0.000135
Y: 3.3
3
Vout
d
Output voltage
2.5
2
1.5
1
X= 0.000135
Y= 0.275
0.5
0
0
0.5
1
1.5
2
Time s
Fig. 16
2.5
3
3.5
-4
x 10
Continue model, system output and controller output
As the reference voltage is 3.3V, and the output voltage can achieve it very well.
What is more, the output voltage remains stable at 3.3V. It shows that the continuous
model with polynomial controller works perfectly.
The controller output can get stable quickly (in 48 μs) and remain stable. What’s
more, the controller make the output voltage come to stable status quickly (in 48 μs)
and remain the wanted output voltage (reference voltage 3.3V). It is very close to the
polynomial controller theory.
For the detail continuous and discrete models, see appendix A.
Later on in this chapter, noise is adding above the output voltage measurement.
The noise which introduced have magnitude between -0.2 and 0.2 and frequency as
33MHz. And changing the load current from 5A to 15A to check the disturbance and
observe the controller quality.
5.2 Simulation result without disturbance and without noise
The model, i.e. a switched time continuous model. This is the interesting part of
the work. It would like to investigate how good the controller works for this system
with disturbance.
For the model block, see model 4 in appendix.
For the simulation result, see Fig. 17.
20
Simulation result, without disturbance, without noise
Output voltage, Inductor current, Controller output
6
Vout
d
iL/10
5
4
X: 0.0001593
Y: 3.316
3
2
1
0
-1
0
X: 0.0001593
Y: 0.2749
0.5
1
1.5
2
Time
Fig. 17
2.5
3
3.5
-4
x 10
Simulation result, without disturbance, without noise
It can be seen that after 60μs, the output voltage get quite stable near 3.3V and
the controller output is also very stable near 0.275 = 3.3/12, which is equal to
reference voltage divide input voltage.
5.3 Simulation result without disturbance and with noise
In the real case, the circuit has some noise, such as noise at the switch according
to high frequency switching, measurement noise and so on.
Here, adding some band-limited white noise to simulate it.
The noise magnitude is between -0.2 and 0.2, and the noise frequency is 33MHz.
The frequency is a very rough estimation based on a measurement, see Fig. 18.
And the measurements are introduced by the switching, but use it the whole time.
21
Noise Magnitude
0.25
0.2
0.15
0.1
Noise
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0
1
2
3
Time
Fig. 18
-4
x 10
Noise Magnitude
The noise used in the simulation is suitable, not so small and not so big. And then
use the same kind of noise in the simulation. The noise is added with the output
voltage as the measurement noise.
And the output voltage measurement with noise can be seen in Fig. 19.
Output voltage measurement with noise, without disturnance
4.5
4
3.5
Measurement Noise
3
2.5
With Noise
2
1.5
1
0.5
0
-0.5
0
1
2
Time
Fig. 19
Output Voltage with noise
For the noise position, see appendix A.
22
3
-4
x 10
It can be find that after adding some noise, the output is still stable. The noise is
not very big but has some bad effect on the stability of controller. When the output
cannot get stable, the range is 3.0V-3.5V. It shows that the controller without noise is
more stable than the one with noise.
5.4 Simulation result with disturbance and without noise
In the model, the current source is used as the load, when the output current
suddenly changes, there is some disturbance for the output voltage.
To observe the disturbance and check whether the polynomial controller has
enough ability to turn the output voltage back to reference voltage. Setting the current
change range as 5A-15A.
See Fig. 20
Simulation result, with disturbance, without noise
Output voltage, Inductor current, Controller output
6
Vout
d
iL/10
5
4
X: 0.0001246
Y: 3.346
X: 0.0002289
Y: 3.338
3
X: 0.0001594
Y: 2.863
2
1
0
-1
0
X: 0.0001246
Y: 0.2767
0.5
1
X: 0.0002289
Y: 0.2766
1.5
2
Time
Fig. 20
2.5
3
3.5
-4
x 10
L=0.9μH, C=150μF , Simulation result, with disturbance, without noise
The stable output voltage under control is about 3.342 V which is quite close to
the reference voltage 3.3V, and it means the controller is good.
At the 60th sample interval (180μs) changing the load current from 5A to 15A, the
output voltage falls from 3.346V to 2.862Vand it recover to the stable voltage 3.338V
soon (in 10 sample intervals). The drop percentage is (3.346-2.863)/3.346 = 14%,
which is acceptable for the good performance. When the load changes suddenly the
controller can take the control of output voltage and make it recover to the stable
voltage soon.
23
When having some parameters change, to observe the change of simulation
result.
In the simulation changing the inductance and capacitance by increasing 10%.
See Fig. 21.
Simulation result, with disturbance, without noise
Output voltage, Inductor current, Controller output
6
Vout
d
iL/10
5
X: 0.0001306
Y: 3.352
4
X: 0.0002187
Y: 3.315
3
X: 0.0001593
Y: 2.908
2
1
0
X: 0.0001306
Y: 0.2706
X: 0.0002187
Y: 0.2781
-1
-2
Fig. 21
0.5
1
1.5
Time
2
2.5
3
-4
x 10
+10%, L=0.99μH, C=165μF, Simulation result, with disturbance, without noise
The stable output voltage under control is about 3.346 V which is quite close to
the reference voltage 3.3V both before and after the disturbance. It means the
controller is good.
At the 60th sample interval (180μs) changing the load current from 5A to 15A, the
output voltage falls from 3.352V to 2.908Vand it recover to the stable voltage 3.315V
soon (in 10 sample intervals). The drop percentage is (3.352-2.908)/3.352= 13%,
which is acceptable for the good performance. When the load changes suddenly the
controller can take the control of output voltage and make it recover to the stable
voltage soon.
Following this simulation change the inductance and capacitance by decreasing
10%.
See Fig. 22 for simulation result.
24
Simulation result, with disturbance, without noise
Output voltage, Inductor current, Controller output
5
4
X: 0.000127
Y: 3.336
Vout
d
iL/10
X: 0.0002234
Y: 3.348
3
X: 0.0001592
Y: 2.82
2
1
0
-1
X: 0.000127
Y: 0.2748
0
0.5
1
X: 0.000223
Y: 0.2749
1.5
2
2.5
Time
Fig. 22
3
3.5
-4
x 10
-10%, L=0.81μH, C=135μF, Simulation result, with disturbance, without noise
The stable output voltage under control is about 3.345 V which is quite close to
the reference voltage 3.3V both before and after the disturbance. It means the
controller is good.
At the 60th sample interval (180μs) changing the load current from 5A to 15A, the
output voltage falls from 3.336V to 2.82Vand it recover to the stable voltage 3.348V
soon (in 10 sample intervals). The drop percentage is (3.346-2.82)/3.346 = 15.7%,
which is acceptable for the good performance. When the load changes suddenly the
controller can take the control of output voltage and make it recover to the stable
voltage soon.
And from Fig. 21 and Fig. 22, there is no big difference for simulation result.
It means the change below 10% is tolerant.
5.5 Simulation result with disturbance and with noise
Adding the same noise like what it is done in section 5.3.
For simulation result, see Fig. 23.
25
Output voltage measurement with noise, with disturnance
4.5
4
3.5
Measurement Noise
3
2.5
2
With Noise
1.5
1
0.5
0
-0.5
0
1
2
3
Time
Fig. 23
-4
x 10
Simulation result, with disturbance, with noise
As it can be seen, after adding the noise, the output voltage is not so stable like
before. But the output voltage is close to the reference voltage totally and the
controller output is close to 0.275 but not so stable.
The simulation need to try large capacitor, and observe the controller quality. For
the simulation result, see Fig. 24.
Simulation result, with disturbance, without noise
Output voltage, Inductor current, Controller output
10
Vout
d
iL/10
8
6
4
2
0
-2
0
0.5
1
1.5
2
Time
Fig. 24
2.5
3
3.5
-4
x 10
RC=50mΩ, C=500μF, Simulation result, with disturbance, without noise
26
As it can be seen from Fig. 24, the large capacitor with large resistor has a bad
simulation result. The output voltage does not reach the stable reference voltage 3.3V,
and it can be seen by viewing the control signal d, that the system is not stable.
By changing the controller design the system can be stable. The changed design
implies that the closed system get a lower bandwidth.
5.6 With large load capacitor
For the simulation the parameters given in Table 6.
Table 6
Parameters of system
Topology
Values
Inductance
L=0.9μH
Capacitance
C=500μF
Resistance of Inductor
RI=2mΩ
Resistance of Capacitor
RC=50mΩ
Sampling interval
h=3μS
PWM sampling interval
hPWM=30ns
And the simulation result is as below, see Fig. 25.
Discrete polynomial controlled Buck, switched continuous model
Output Voltage, Inductor Current and Duty signal
4
3.5
3
Vout
d
iL/10
2.5
2
1.5
1
0.5
0
-0.5
0
0.5
1
Time s
Fig. 25
1.5
-3
x 10
Output Voltage before and after disturbance when RL =2mΩ, RC=50mΩ, C=500uF
Using a FIR filter for averaging the signal for one PWM period gives the
signal shown in Fig. 26. The FIR filter removes the ESR ripple.
27
output voltage, inductor current and duty signal
Discrete polynomial controlled Buck, switched continuous model filtered measurement
3.5
3
X: 0.0005167
Y: 3.35
X: 0.001281
Y: 3.33
X: 0.00085
Y: 3.133
2.5
2
1.5
1
0.5
0
0
0.5
1
Time s
Fig. 26
1.5
-3
x 10
Output Voltage filtered noise when RL =2mΩ, RC=50mΩ, C=500uF
From Fig. 26, it can be seen the output voltage clear that the stable output voltage
before disturbance is 3.35V and the stable output voltage after disturbance is 3.33V.
Both of them are close to the reference voltage 3.3V. And during the disturbance
the output voltage drops to the minimum value 3.133V, so the drop percentage is
(3.35-3.133)/3.35 = 6.48%. It has better control about the disturbance.
When the small capacitor has about 14% voltage drop.
After adding the noise in Fig. 18, get the simulation result as below, see Fig. 27.
28
Output voltage measurement with noise
4
3.5
Measurement Noise
3
2.5
2
With Noise
1.5
1
0.5
0
-0.5
0
0.5
1
Time
Fig. 27
1.5
-3
x 10
Output Voltage with noise when RL =2mΩ, RC=50mΩ, C=500uF
From Fig. 27, it can be seen the output voltage with noise, the noise have
magnitude between -0.2 and 0.2 and frequency as 33MHz.
5.7 Conclusions
After observing the simulation results above, conclusions can be drawn as below:
1. The controller is good enough that it can keep the output voltage at the
reference voltage level.
2. A larger capacitor has better quality to deal with disturbance because it has
less voltage drop with the current disturbance. The larger capacitor will imply
a lower bandwidth in the closed system.
3. A controller design that works for a small capacitor load can give a non-stable
system for a larger capacitor load.
4. It is possible to use pole placement in a discrete polynomial controller as
design method
29
References
Books
[1] Karl J. Åström, Björn Wittenmark; Computer controlled systems, third edition;
1997; Prentice Hall international, Inc., Upper Saddle River, New Jersey, USA
ISBN:0-13-314899-8
[2] Bengt Schmidtbauer; Analog och digital Reglerteknik; 1988; Studentlitteratur;
Lund, Sweden; ISBN: 91-44-26601-4
Websites
[3] http://en.wikipedia.org/wiki/Buck_converter
2011-02-26
[4] http://en.wikipedia.org/wiki/DC-to-DC_converter
2011-02-28
[5] http://en.wikibooks.org/wiki/Control_Systems/Polynomial_Design
2011-04-06
[6] http://www.mathworks.com/academia/student_version/
2011-04-16
[7] http://edu.levitas.net/Tutorials/Matlab/Simulink/
2011-04-26
[8] http://en.wikipedia.org/wiki/Electromagnetic_compatibility
2011-06-11
30
Appendix
Appendix A Simulink Models
Model 1:
Continuous time model without saturation, see Appendix Fig. 1.
t
dc1
Clock
To Workspace2
To Workspace1
Kr
Step1
Gain2
Add
1
Bctf(s)
cpoly(z)
Actf(s)
Discrete Filter
Transfer Fcn
yc1
To Workspace7
dpoly(z)
1
Discrete Filter2
Appendix Fig. 1
Continuous Time Model without Saturation
The model consists of the polynomial controller and a continuous system. For
Simulation result, see Fig. 16.
Model 2:
Discrete time model without saturation, see Appendix Fig. 2.
d1
To Workspace1
Kr
Step1
Gain2
Add
1
Bdtf(z)
cpoly(z)
Adtf(z)
Discrete Filter
Discrete Filter1
y1
To Workspace7
dpoly(z)
1
Discrete Filter2
Appendix Fig. 2
Discrete Time Model without Saturation
The model consists of the polynomial controller and a discrete system. For
Simulation result, see Fig. 15.
31
Model 3:
Discrete time model with saturation, see Appendix Fig. 3.
d2
To Workspace1
Saturation
1
Bdtf(z)
cpoly(z)
Adtf(z)
Discrete Filter
Discrete Filter1
Kr
Step1
Add
Gain2
y2
To Workspace7
dpoly(z)
1
Discrete Filter2
Appendix Fig. 3
Discrete Time Model with Saturation
Model 4:
PWM model with controller, see Appendix Fig. 4
pwm
Repeating
Sequence
To Workspace4
1
Constant
SaturationRate Transition
1/z
1
Kr
cpoly(z)
Step1
Gain2
Add
<=
Relational
Operator
Discrete Filter
Band-Limited
White Noise2
Add2
Switch
Vout
x' = Ax+Bu
y = Cx+Du
To Workspace7
State-Space
iL
0
To Workspace3
Constant1
Step2
d
Rate Transition1
ZOH
Add1
To Workspace1
dpoly(z)
1
Discrete Filter2
Band-Limited
White Noise1
ts
Clock
To Workspace2
dVg
To Workspace5
Appendix Fig. 4
PWM Model with Controller
The model consists of the polynomial controller and PWM switched system. For
Simulation result, see Fig. 17-27.
32
Appendix B
MATLAB Code
The Code shows how to simulate the models:
It shows the parameters setting, system building, controller designing and
simulation result plotting.
The simulation result showed is with and without disturbance and noise.
%% BMR450 Simple controller
clear all
close all
clc
%% DC-DC Converter System Design
% Circuit data
L = 0.9e-6;
% 0.99e-6 / 0.81e-6;
Cc = 150e-6;
% 165e-6 / 135e-6;
Rl = 2e-3;
% 10e-3
Rc = 2e-3;
% 10e-3 / 50e-3
Vg = 12;
% Input Voltage
I_intial = 5; % Load Current; 10
I_final = 15;
% Sampling interval for the controller
h = 1*3e-6;
% pwm signal frequency
fs = 1/h;
% Samplinginterval for the pwm generator
hpwm = h/100;
% Set point
yRef = 3.3;
I_steptime = 50*h;
% noise = 1e-10;
noise = 0;
% Model
A = [ 0
1/Cc ; -1/L
-(Rc+Rl)/L ];
% Input signals duty cycle and load current
B1 = [ 0
-1/Cc ; 1/L
B2 = [ 0
-1/Cc ; 0
Rc/L ];
Rc/L ];
% With power supply
% Without power supply
33
B = B1*Vg;
%averaged system, the control signal is duty cycle d
C = [ 1
Rc ; 0
D = [ 0
0 ; 0
1 ];
0 ];
% Model two input
AA = A;
BB = [ 0
-1/Cc ; Vg/L
Rc/L ];
CC = C;
DD = D;
% Initial values
uC0 = 0;
iL0 = 0;
% Continuous system
Gcss = ss(A,B(:,1),C(1,:),D(1,1));
[Ac,Bc,Cc,Dc] = ssdata(Gcss)
Gctf = tf(Gcss);
[Bctf,Actf] = tfdata(Gctf,'v')
% Discrete system
Gdss = c2d(Gcss,h,'zoh');
[Ad,Bd,Cd,Dd] = ssdata(Gdss)
Gdtf = tf(Gdss);
[Bdtf,Adtf] = tfdata(Gdtf,'v')
% plot of step response
figure(1)
step(Gctf)
hold on
step(Gdtf)
hold off
% polynomial coefficients, index means # delays
Bd0 = Bdtf(1);
Bd1 = Bdtf(2);
Bd2 = Bdtf(3);
Ad0 = Adtf(1);
Ad1 = Adtf(2);
Ad2 = Adtf(3);
%% Polynomial Controller Design a la Schmidtbauer
34
b1 = Bd1;
b2 = Bd2;
a1 = Ad1;
a2 = Ad2;
% Pole placement
q1 = 0.5;
q2 = 0.5;
q3 = 0.5;
%polynomial controller derivation using method from Schmidtbauer
p1 = -q1-q2-q3;
p2 = q1*q2 + q2*q3 + q1*q3;
p3 = -q1*q2*q3;
% p1 = a1 + c1 + b1*d0
% p2 = a1*c1+ a2 + b1*d1 + b2*d0
% p3 = a2*c1 + b2*d1
% polynomial coefficients
c1 = (b1^2*p3 + b1*b2*(a2-p2) + b2^2*(p1-a1))/(a2*b1^2 + b2^2 - a1*b1*b2);
d0 = (p1-a1-c1)/b1;
d1 = (p3-a2*c1)/b2;
a1
a2
b1
b2
c1
d0
d1
% C polynomial and D polynomial
cpoly = [ 1 c1 ];
dpoly = [ d0 d1 ];
% design of Stationary Gain
% Kr = D(1)+A(1)C(1)/B(1)=(d0+d1)+(1+a1+a2)(1+c1)/(b1+b2)
Kr = (d0+d1)+(1+a1+a2)*(1+c1)/(b1+b2);
%% Simulation
N = 100;
% 100 sample intervals
sim('Buckd1',N);
% Discrete time model without saturation
sim('Buckd2',N);
% Discrete time model with saturation
35
sim('Buckc1',N*h);
sim('Buckswitch',N*h);
% Continuous time model
% PWM model with saturationplot
figure(2)
stairs([y1 d1])
legend('Vout','d')
title('Discrete model')
xlabel('Sampling sequence k'),ylabel('Output voltage')
figure(3)
td=0:h:h*(length(y1)-1);
stairs(td,y1)
hold on, stairs(td,d1),hold off
legend('Vout','d')
title('Discrete model without saturation')
xlabel('Time s'),ylabel('Output voltage')
figure(4)
stairs([y2 d2])
legend('Vout','d')
title('Discrete model with saturation')
xlabel('Sampling sequence k'),ylabel('Output voltage')
figure(5)
plot(t,yc1)
dt=t(end)/(length(dc1)-1);
tt=0:dt:t(end);
hold on,stairs(tt,dc1),hold off
legend('Vout','d')
title('Continuous model')
xlabel('Time s'),ylabel('Output voltage')
figure(6)
plot(ts,Vout,ts(1:length(d)),d,ts,iL/10)
legend('Vout','d','iL/10')
grid
title('Switched continuous model with saturation')
title('Simulation result, with disturbance, without noise')
xlabel('Time'),ylabel('Output voltage, Inductor current, Controller
output')
figure(7)
plot(ts,Vout,ts(1:length(d)),d,ts,pwm)
legend('Vout','d','PWM')
36
grid
title('Switched continuous model with saturation')
xlabel('Time'),ylabel('Output voltage, Controller output, PWM wave')
figure(8)
plot(ts(1:1001),dVg(1:1001)*12,ts(1:1001),d(1:1001),ts(1:1001),pwm(1:
1001))
legend('dVg','d','PWM')
title('Discrete polynomial controlled Buck, switched continuous model
with saturation')
xlabel('Time'),ylabel('controlled input voltage, controller output, PWM
wave')
figure(9)
plot(ts(1:2001),d(1:2001),ts(1:2001),pwm(1:2001))
legend('d','PWM')
title('Controller output & PWM wave')
xlabel('Time'),ylabel('Controller output, PWM wave')
figure(10)
plot(ts(1:10000),Noise(1:10000))
title('Noise Magnitude')
xlabel('Time'),ylabel('Noise')
figure(11)
plot(ts(1:10000),Vout_N(1:10000))
legend('With Noise')
title('Output voltage measurement with noise, without disturnance')
xlabel('Time'),ylabel('Measurement Noise')
37
Appendix C
Interesting Phenomenon found in the simulation
The simulation shows some interesting phenomenon which will be showed in the
last section of chapter 5.
When having C = 150μF, L = 0.9μH, if many different combinations of resistors
with capacitor and inductor are tried, and choosing the most 4 typical combinations,
the different simulation results are showed as below.
In Appendix Table 1 the first combination is given:
Appendix Table 1
Capacitor’s Resistor equals Inductor’s
Resistor
Values
RL
2mΩ
RC
2mΩ
Output voltage, Inductor current, Controller output
Simulation result, with disturbance, without noise
Vout
d
iL/10
3.6
3.5
3.4
3.3
3.2
3.1
3
0.5
Appendix Fig. 5
1
1.5
Time
2
2.5
-4
x 10
Output Voltage before and after disturbance when RL = 2mΩ and RC=2mΩ
In Appendix Fig. 5, it can be seen that the output voltage before and after
disturbance is almost the same around 3.33V. The transfer function can be calculated
0.01963  z 2  0.01805  z  0.001586
by use of Matlab, see Appendix D, H ( z ) 
.
z 3  1.5  z 2  0.75  z  0.125
The stationary gain is -0.0196.
38
In Appendix Table 2 the second combination is given:
Appendix Table 2
Capacitor’s Resistor is bigger than Inductor’s
Resistor
Values
RL
2mΩ
RC
10mΩ
Simulation result, with disturbance, without noise
Output voltage, Inductor current, Controller output
3.55
Vout
d
iL/10
3.5
3.45
3.4
3.35
3.3
3.25
3.2
3.15
3.1
0.5
1
1.5
2
Time
Appendix Fig. 6
2.5
3
-4
x 10
Output Voltage before and after disturbance when RL < RC
In Appendix Fig. 6, it can be seen that the output voltage before disturbance (about
3.36V) is lower than the one after disturbance (about 3.41V). The transfer function
can be calculated by use of Matlab, see Appendix D,
H ( z) 
0.0188  z 2  0.02177  z  0.002519
. The stationary gain is -0.0188.
z 3  1.5  z 2  0.75  z  0.125
In Appendix Table 3 the third combination is given:
39
Capacitor’s Resistor is smaller than Inductor’s
Appendix Table 3
Resistor
Values
RL
10mΩ
RC
2mΩ
Output voltage, Inductor current, Controller output
Simulation result, with disturbance, without noise
Vout
d
iL/10
3.6
3.5
3.4
3.3
3.2
3.1
3
2.9
0.5
1
Appendix Fig. 7
1.5
Time
2
2.5
-4
x 10
Output Voltage before and after disturbance when RL >RC
In Appendix Fig. 7, it can be seen that the stable output voltage before
disturbance (about 3.33V) is bigger than the one after disturbance (about 3.27V). The
transfer function can be calculated by use of Matlab, see Appendix D,
0.01964  z 2  0.01774  z  0.001343
H ( z) 
. The stationary gain is -0.0196.
z 3  1.5  z 2  0.75  z  0.125
In Appendix Table 4 the forth combination is given:
Appendix Table 4
Capacitor’s Resistor equals Inductor’s
Resistor
Values
RL
10mΩ
RC
10mΩ
40
Simulation result, with disturbance, without noise
Output voltage, Inductor current, Controller output
3.8
Vout
d
iL/10
3.7
3.6
3.5
3.4
3.3
3.2
3.1
3
2.9
2.8
0.5
Appendix Fig. 8
1
1.5
Time
2
2.5
-4
x 10
Output Voltage before and after disturbance when RL =10mΩ and RC=10mΩ
In Appendix Fig. 8, it can be seen that the stable output voltage before and after
disturbance is almost the same around 3.35V. The transfer function can be calculated
by use of Matlab, see Appendix D, H ( z ) 
0.01881 z 2  0.02135  z  0.002538
.
z 3  1.5  z 2  0.75  z  0.125
The stationary gain is -0.0188.
From Appendix Fig. 5-8, the conclusion is:
When RL =RC, the stationary output voltage before and after the load current
disturbance will be almost the same. In that case there is no need for including an
integrator part in the controller; when RL <RC, the stationary output voltage before
disturbance will be smaller than the one after disturbance; when RL >RC the stationary
output voltage before disturbance will be bigger than the one after disturbance. In
these cases there is a need for including an integrator in the controller to stationary
remains the correct output voltage.
41
Appendix D Transfer function
The transfer function can be calculated by the Matlab command sequence
% Definition of continuous system:
Csys=ss(AA,BB,CC,DD);
% Discretizing the system:
Dsys=c2d(Csys,h,'zoh');
% Definition of the transfer function for the controller:
Rsys=tf(dpoly,cpoly,h);
% Calculation of the closed system:
Msys=feedback(Dsys,Rsys,1,1,-1);
% the closed system has two inputs (d and I0) and two outputs (u0 and IL).
% Generation of all four step responses
step(Msys)
% calculation of all zeros, poles and static gains.
[z,p,k,Ts]=zpkdata(Msys)
% calculation of all transfer functions
Hz=tf(Msys)
42