i

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

ii

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

iii

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.

iv

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

v

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

vi

List of Tables

Table 1 Parameters of Buck Converter .................................................................................. 2

Table 2 Parameters of system ............................................................................................... 10

Table 3 Polynomial coefficients of system ........................................................................... 10

Table 4 Pole Placement of Polynomial Controller ............................................................... 14

Table 5 Coefficients of Polynomial controller ..................................................................... 14

Table 6 Parameters of system ............................................................................................... 27

Appendix Table 1 Capacitor’s Resistor equals Inductor’s ................................................... 38

Appendix Table 2 Capacitor’s Resistor is bigger than Inductor’s ........................................ 39

Appendix Table 3 Capacitor’s Resistor is smaller than Inductor’s ...................................... 40

Appendix Table 4 Capacitor’s Resistor equals Inductor’s ................................................... 40

vii

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

viii

List of Symbols

% Percentage

V Volt

L Inductor

C Capacitor

R Resistor

d Duty Cycle

DC Direct Current

fs Switching Frequency

iL Switch/Inductor Current

yref Reference Voltage

PWM Pulse Width Modulation

1

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.

2

The parameters used in the project are as below, see Table 1.

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.

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

3

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.

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.

Fig. 1 Overall system circuit

gV

LR L

CR

C

Ii

Switch

node 1

node 2

4

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

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.

Fig. 3 PWM controlled input voltage dVg

0 0.5 1 1.5 2 2.5

x 10-5

0.5

1

1.5

2

2.5

3

3.5

4

Time

cont

rolle

d in

put

volta

ge,

cont

rolle

r ou

tput

, P

WM

wav

e

Discrete polynomial controlled Buck, switched continuous model with saturation

dVg/3

d

pwm

Fig. 2 PWM Wave

0 0.5 1 1.5 2 2.5 3

x 10-5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

PW

M w

ave

5

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.

The simple Buck converter can be modeled as Equation Chapter 2 Section 1

0 0 0 0 1 0

0 0 0 0 1 1

0 0 0 0 1 1

0 0 0 0 1 0

1 1 1 1 0 0

0 1 1 0 0 0

C C

L L

U U

C C

R R

R R

L L

I I

i u

i u

i u

i u

u i

u i

(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:

0 0 0 1 0 1

0 0 0 1 0 1

0 0 0 1 0 0

1 1 1 0 1 0

C C

L L

C C

R R U

IR R

L L

i u

i u u

ii u

u i

(2.2)

Modeling the two energy storing components it has:

0 1 0 1 0 0

1 0 1 0 1 1

C

L

RC C U

RL L I

ui u u

uu i i

(2.3)

The last term can be eliminated using

0 1 0 1

0 1 0 0

C

L

R C U

R L I

i u u

i i i

(2.4)

and

Fig. 4 Circuit with Voltage Source

gV

LR L

CR

CIi

6

0

0

C C

L L

R RC

LR R

u iR

Ru i

(2.5)

So:

0 00 1 0 1

0 00 1 0 0

C

L

R C UC C

L LR L I

u u uR R

R Ru i i

(2.6)

Then it can simplify the equation:

0 1 0 1 0 0 0

1 0 1 0 1 1 0 0

0 1 0 1

1 1

C C U UC

L L I I

C U

C L CL I

i u u uR

u i i i

u u

R R Ri i

(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

0 CC Ru u u (2.8)

CR C Lu R i (2.9)

0 1C

C

C R C

L

uu u u R

i

(2.10)

0

0 1 0

1 1

1

C C

U

C LL L

C

C

L

i uu

R Ru i

uu R

i

(2.11)

Including the constitutive equations, the system is modeled by

1 10 0

1 1

CC U

C L CL IL

du

u udt C C

d R R Ri ii

dt L L L L

(2.12)

0 1 0 0

0 1 0 0

C UC

L L I

u u uR

i i i

(2.13)

The ABC-differential equation system is given as

1 1

1

x A x B u

y C x

(2.14)

7

1

10

1 C L

CA

R R

L L

(2.15)

1

10

1 C

CB

R

L L

(2.16)

1

1

0 1

CRC

(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.

0 0 0 1 1

0 0 0 1 1

0 0 0 1 1

1 1 1 0 0

1 1 0 0 0

C C

L L

C C

R R

R R

L L

I I

i u

i u

i u

u i

u i

(2.18)

Fig. 5 Circuit without Voltage Source

LR L

CR

CIi

8

0 0 0 1 1

0 0 0 1 1

0 0 0 1 0

1 1 1 0 0

C C

L L

C C

R R

I

R R

L L

i u

i ui

i u

u i

(2.19)

Modeling the two energy storing components:

0 1 0 0 1

1 0 1 1 0

C

L

RC C

I

RL L

ui ui

uu i

(2.20)

0 1 1

0 1 0

C

L

R C

I

R L

i ui

i i

(2.21)

0

0

C C

L L

R RC

LR R

u iR

Ru i

(2.22)

0 00 1 1

0 00 1 0

0

0 0

C

L

R CC C

I

L LR L

C CC

I

L L

u uR Ri

R Ru i

u RRi

R i

(2.23)

Then it can simplify the equation:

0 1 1

1 0 0

00 0 0 0

01 1 1 1 0

0 1 1

1

C C

I

L L

C CC

I

L L

C

I

C L CL

i ui

u i

u RRi

R i

ui

R R Ri

(2.24)

Including the constitutive equations, the system is modeled as below:

1 10

1

CC

I

C L CLL

du

udt C Ci

d R R Rii

dt L L L

(2.25)

The ABC-differential equation system is given as

1 10 0

10

CC U

C L CL IL

du

u udt C C

d R R Ri ii

dt L L L

(2.26)

2 2x A x B u (2.27)

9

0 1

0 1

CC

L L

u uR

i i

(2.28)

2y C 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

1 21Saverage d S d S (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.

1 21B d B d B (2.31)

The B matrix can be written as:

11 12 21 22( ) 1 ( )B d B B d B B (2.32)

where 11

0 0

1/ 0B

L

and 12

0 1/

0 /

c

c

CB

R L

21

0 0

0 0B

and 22

0 1/

0 /

c

c

CB

R L

Then the calculation is as below:

0 0

0 0

0 0

0 1/ 0 1/0 0 0 0 0 0(1 )

0 / 0 /1/ 0 0 0 0 0

0 0 0 / 0 /0 0(1 )

/ 0 0 / 0 /0 0

0 0

c g c g

c c

c c

g c c

C V C VBu d d

R L I R L IL

I C I Cd d

V L I R L I R L

0

0

0

0

0 /

/ 0 0 /

0 /

/ /

c

g c

c

g c

I Cd

V L I R L

I C d

V L R L I

(2.33)

Then the averaged system is as below:

x Ax Bu

y Cx

(2.34)

Writing the continuous-time system in a detailed way:

10

0

0

1 00

1

CC c

C L L g cL

Idu

u dCdt C

Id R R i V Ridt L L L 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

1Y

H z C zI F GU

(2.37)

As it is mentioned in chapter one, the suitable parameters for the first performed

simulation as below, see Table 2.

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

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

11

Chapter 3. Polynomial Controller Design

The controller with buck converter system is as below, see Fig. 6.

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]

1 2

1 2

1 2

1 21

Y z B zb z b zH z

U z a z a z A z

(3.1)

The transfer function can be expressed as a difference equation [2]

1 2 1 21 2 1 2y k a y k a y k b u k b u k (3.2)

A controller is introduced as a general difference equation of the same order as

the system’s difference equation [2]

0 1 2 11 2 1r refu k K y k d y k d y k d y k c y k (3.3)

The controller can be given as a Z-transform [2]

1

0 1

1 1

1 11 1

r rref ref

D zd d z K KU z Y z Y z Y z Y z

c z c z C z C z

(3.4)

Fig. 6 Controller with Buck Converter System [2]

rK

1

C z

D z

B zH z

A z

Controller System V z

U z refY z Y z

12

As

B zY z U z H z U z

A z , the closed system is then given as [2]

r

ref

B z D z B z KY z Y z Y z

A z C z A z C z

(3.5)

r

ref

Y z B z K

Y z A z C z B z D z

(3.6)

When z=1, Kr can be designed in order to get the correct stationary gain, [2]

1 1 11

1 1r

C B PK D

A B

(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

B zH 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 1 2

1 21A z a z a z and 1 2

1 2B z b z b z

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].

13

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:

1 1 1( ) (1 0.5 ) (1 0.5 ) (1 0.5 )P z z z z (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

1

11C z c z and 1

0 1D z d d z

Then it has [2]

1 2 1 1 2 1

1 2 1 1 2 0 1

1 2 3

1 1 1 0 1 1 2 1 1 2 0 2 1 2 1

1 1

1

P z A z C z B z D z

a z a z c z b z b z d d z

a c b d z a c a b d b d z a c b d z

(3.9)

And it also has

1 1 1 1 2 3

1 2 3 1 2 21 1 1 1P z q z q z q z p z p z p z (3.10)

Where q1, q2 and q3 are three chosen poles.

1 1 2 3

2 1 2 2 3 1 3

3 1 2 3

p q q q

p q q q q q q

p q q q

(3.11)

Identifying the coefficients for P(z): [2]

Fig. 7 The shadow region inside the unit cycle is stable for system

1

1 Re

ImZ-plane

14

1 1 1 1 0

2 1 1 2 1 1 2 0

3 2 1 2 1

p a c b d

p a c a b d b d

p a c b d

(3.12)

By calculation, the solution is as below: [2]

2 2

1 3 1 2 2 2 2 1 1

1 2 2

2 1 2 1 1 2

1 1 10

1

3 2 11

2

b p b b a p b p ac

a b b a b b

p a cd

b

p a cd

b

(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:

And C(z) and D(z) polynomials coefficients are below, see Table 5.

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.

Table 5 Coefficients of Polynomial controller

Controller Polynomial

coeffecients

Values

c1

0.0808

d0

0.7166

d1

-0.6485

Table 4 Pole Placement of Polynomial Controller

Controller Pole Placement Values

q1

0.5

q2

0.5

q3

0.5

15

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.

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.

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.

Fig. 9 System block

Fig. 8 Buck Converter System with polynomial Controller [2]

rK

1

C z

D z

B zH z

A z

Controller System V z

U z refY z Y z

16

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.

In Simulink, the relational operator is used to compare two controller output with

PWM value, see Fig. 12.

Fig. 11 PWM with polynomial controller

dVg

To Workspace5Switch

Saturation

? ? ?

Repeating

Sequence

<=

Relational

Operator

d

Constant2

0

Constant1

1

Constant

Fig. 10 Polynomial controller block

17

And when useing switch to adjust according to the output of relational

operator, see Fig. 13.

By the way, having the saturation for the controller output between 0 and 1

Fig. 13 Switch

Fig. 12 Relational operator

18

because the range of PWM value is between 0 and 1, see Fig. 14.

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

Fig. 14 Saturation condition

19

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.

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

Fig. 15 Discrete model without saturation, system output and controller output

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

X= 42

Y= 3.3

Sampling sequence k

Outp

ut

voltage

Discrete model

X= 42

Y= 0.275

Vout

d

20

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.

Fig. 16 Continue model, system output and controller output

0 0.5 1 1.5 2 2.5 3 3.5

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

X: 0.000135

Y: 3.3

Time s

Outp

ut

voltage

Continuous model

X= 0.000135

Y= 0.275

Vout

d

21

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.

Fig. 17 Simulation result, without disturbance, without noise

0 0.5 1 1.5 2 2.5 3 3.5

x 10-4

-1

0

1

2

3

4

5

6

Time

Outp

ut

voltage,

Inducto

r curr

ent,

Contr

olle

r outp

ut

Simulation result, without disturbance, without noise

X: 0.0001593

Y: 0.2749

X: 0.0001593

Y: 3.316

Vout

d

iL/10

22

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.

For the noise position, see appendix A.

Fig. 19 Output Voltage with noise

0 1 2 3

x 10-4

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time

Measure

ment

Nois

e

Output voltage measurement with noise, without disturnance

With Noise

Fig. 18 Noise Magnitude

0 1 2 3

x 10-4

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25Noise Magnitude

Time

Nois

e

23

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

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.

Fig. 20 L=0.9μH, C=150μF , Simulation result, with disturbance, without noise

0 0.5 1 1.5 2 2.5 3 3.5

x 10-4

-1

0

1

2

3

4

5

6

X: 0.0001246

Y: 3.346

Time

Outp

ut

voltage,

Inducto

r curr

ent,

Contr

olle

r outp

ut

Simulation result, with disturbance, without noise

X: 0.0002289

Y: 3.338

X: 0.0002289

Y: 0.2766

X: 0.0001246

Y: 0.2767

X: 0.0001594

Y: 2.863

Vout

d

iL/10

24

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.

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.

Fig. 21 +10%, L=0.99μH, C=165μF, Simulation result, with disturbance, without noise

0.5 1 1.5 2 2.5 3

x 10-4

-2

-1

0

1

2

3

4

5

6

X: 0.0001306

Y: 0.2706

Time

Outp

ut

voltage,

Inducto

r curr

ent,

Contr

olle

r outp

ut

Simulation result, with disturbance, without noise

X: 0.0001306

Y: 3.352

X: 0.0001593

Y: 2.908

X: 0.0002187

Y: 0.2781

X: 0.0002187

Y: 3.315

Vout

d

iL/10

25

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.

Fig. 22 -10%, L=0.81μH, C=135μF, Simulation result, with disturbance, without noise

0 0.5 1 1.5 2 2.5 3 3.5

x 10-4

-1

0

1

2

3

4

5

X: 0.0001592

Y: 2.82

Time

Outp

ut

voltage,

Inducto

r curr

ent,

Contr

olle

r outp

ut

Simulation result, with disturbance, without noise

X: 0.0002234

Y: 3.348

X: 0.000223

Y: 0.2749

X: 0.000127

Y: 0.2748

X: 0.000127

Y: 3.336

Vout

d

iL/10

26

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.

Fig. 24 RC=50mΩ, C=500μF, Simulation result, with disturbance, without noise

0 0.5 1 1.5 2 2.5 3 3.5

x 10-4

-2

0

2

4

6

8

10

Time

Outp

ut

voltage,

Inducto

r curr

ent,

Contr

olle

r outp

ut

Simulation result, with disturbance, without noise

Vout

d

iL/10

Fig. 23 Simulation result, with disturbance, with noise

0 1 2 3

x 10-4

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time

Measure

ment

Nois

e

Output voltage measurement with noise, with disturnance

With Noise

27

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.

And the simulation result is as below, see Fig. 25.

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.

Fig. 25 Output Voltage before and after disturbance when RL =2mΩ, RC=50mΩ, C=500uF

0 0.5 1 1.5

x 10-3

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time s

Outp

ut

Voltage,

Inducto

r C

urr

ent

and D

uty

sig

nal

Discrete polynomial controlled Buck, switched continuous model

Vout

d

iL/10

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

28

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.

Fig. 26 Output Voltage filtered noise when RL =2mΩ, RC=50mΩ, C=500uF

0 0.5 1 1.5

x 10-3

0

0.5

1

1.5

2

2.5

3

3.5Discrete polynomial controlled Buck, switched continuous model filtered measurement

Time s

outp

ut

voltage,

inducto

r curr

ent

and d

uty

sig

nal X: 0.0005167

Y: 3.35X: 0.001281

Y: 3.33X: 0.00085

Y: 3.133

29

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

Fig. 27 Output Voltage with noise when RL =2mΩ, RC=50mΩ, C=500uF

0 0.5 1 1.5

x 10-3

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Measure

ment

Nois

e

Output voltage measurement with noise

With Noise

30

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

31

Appendix

Appendix A Simulink Models

Model 1:

Continuous time model without saturation, see Appendix Fig. 1.

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.

The model consists of the polynomial controller and a discrete system. For

Simulation result, see Fig. 15.

Appendix Fig. 2 Discrete Time Model without Saturation

y1

To Workspace7

d1

To Workspace1

Step1

Kr

Gain2

dpoly(z)

1

Discrete Filter2

Bdtf(z)

Adtf(z)

Discrete Filter1

1

cpoly(z)

Discrete FilterAdd

Appendix Fig. 1 Continuous Time Model without Saturation

Bctf(s)

Actf(s)

Transfer Fcn

yc1

To Workspace7

t

To Workspace2

dc1

To Workspace1

Step1

Kr

Gain2

dpoly(z)

1

Discrete Filter2

1

cpoly(z)

Discrete Filter

Clock

Add

32

Model 3:

Discrete time model with saturation, see Appendix Fig. 3.

Model 4:

PWM model with controller, see Appendix Fig. 4

The model consists of the polynomial controller and PWM switched system. For

Simulation result, see Fig. 17-27.

Appendix Fig. 4 PWM Model with Controller

Vout

To Workspace7

dVg

To Workspace5

pwm

To Workspace4

iL

To Workspace3

ts

To Workspace2

d

To Workspace1

Switch

Step2

Step1

x' = Ax+Bu

y = Cx+Du

State-Space

Saturation

Repeating

Sequence

<=

Relational

Operator

ZOH

Rate Transition1

1/z

Rate Transition

Kr

Gain2

dpoly(z)

1

Discrete Filter2

1

cpoly(z)

Discrete Filter

0

Constant1

1

Constant

Clock

Band-Limited

White Noise2

Band-Limited

White Noise1

Add2

Add1

Add

Appendix Fig. 3 Discrete Time Model with Saturation

y2

To Workspace7

d2

To Workspace1

Step1

Saturation

Kr

Gain2

dpoly(z)

1

Discrete Filter2

Bdtf(z)

Adtf(z)

Discrete Filter1

1

cpoly(z)

Discrete FilterAdd

33

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 Rc/L ]; % With power supply

B2 = [ 0 -1/Cc ; 0 Rc/L ]; % Without power supply

34

B = B1*Vg; %averaged system, the control signal is duty cycle d

C = [ 1 Rc ; 0 1 ];

D = [ 0 0 ; 0 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

35

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

36

sim('Buckc1',N*h); % Continuous time model

sim('Buckswitch',N*h); % 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')

37

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')

38

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:

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

by use of Matlab, see Appendix D, 2

3 2

0.01963 0.01805 0.001586( )

1.5 0.75 0.125

z zH z

z z z

.

The stationary gain is -0.0196.

Appendix Fig. 5 Output Voltage before and after disturbance when RL = 2mΩ and RC=2mΩ

0.5 1 1.5 2 2.5

x 10-4

3

3.1

3.2

3.3

3.4

3.5

3.6

Time

Outp

ut

voltage,

Inducto

r curr

ent,

Contr

olle

r outp

ut

Simulation result, with disturbance, without noise

Vout

d

iL/10

Appendix Table 1 Capacitor’s Resistor equals Inductor’s

Resistor Values

RL 2mΩ

RC 2mΩ

39

In Appendix Table 2 the second combination is given:

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,

2

3 2

0.0188 0.02177 0.002519( )

1.5 0.75 0.125

z zH z

z z z

. The stationary gain is -0.0188.

In Appendix Table 3 the third combination is given:

Appendix Fig. 6 Output Voltage before and after disturbance when RL < RC

0.5 1 1.5 2 2.5 3

x 10-4

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

3.5

3.55

Time

Outp

ut

voltage,

Inducto

r curr

ent,

Contr

olle

r outp

ut

Simulation result, with disturbance, without noise

Vout

d

iL/10

Appendix Table 2 Capacitor’s Resistor is bigger than Inductor’s

Resistor Values

RL 2mΩ

RC 10mΩ

40

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,

2

3 2

0.01964 0.01774 0.001343( )

1.5 0.75 0.125

z zH z

z z z

. The stationary gain is -0.0196.

In Appendix Table 4 the forth combination is given:

Appendix Table 4 Capacitor’s Resistor equals Inductor’s

Resistor Values

RL 10mΩ

RC 10mΩ

Appendix Fig. 7 Output Voltage before and after disturbance when RL >RC

0.5 1 1.5 2 2.5

x 10-4

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

Time

Outp

ut

voltage,

Inducto

r curr

ent,

Contr

olle

r outp

ut

Simulation result, with disturbance, without noise

Vout

d

iL/10

Appendix Table 3 Capacitor’s Resistor is smaller than Inductor’s

Resistor Values

RL 10mΩ

RC 2mΩ

41

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, 2

3 2

0.01881 0.02135 0.002538( )

1.5 0.75 0.125

z zH z

z z z

.

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.

Appendix Fig. 8 Output Voltage before and after disturbance when RL =10mΩ and RC=10mΩ

0.5 1 1.5 2 2.5

x 10-4

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

Time

Outp

ut

voltage,

Inducto

r curr

ent,

Contr

olle

r outp

ut

Simulation result, with disturbance, without noise

Vout

d

iL/10

42

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)