FP-DCDC

BUDAPEST UNIVERSITY OF TECHNOLOGY AND

ECONOMICS

FINAL PROJECT

APPLICATION OF THE CONTROL UNIT PWM (PULSE WIDTH MODULATION)

László Arany

2009

2

BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS

FACULTY OF MECHANICAL ENGINEERING

FINAL PROJECT

Mechatronics BSc, Integrated Engineering

Assigned to: László ARANY (H8QF9G) e-mail: [email protected] ..........................................................................

First Name Surname

Application of the control unit PWM (Pulse-width modulation)

Supervisors: Károly Zabán, Department of Automation and Applied Informatics Phone: +36-1-463-2338, e-mail: [email protected]

Final exam subjects: 1. Mechatronics 2. Power electronics and motion control 3. Analog and digital electronics

Date of issue: 9 September, 2009 Deadline for submission: 11 December, 2009

Outline:

The student has to study the technical literature of the PWM and has to study the control unit PWM of LEYBOLD DIDACTIC GMBH. The student has to build some different experiments using this model.

Tasks in detail:

1. Study the theory of the PWM 2. Study control unit PWM by the model of LEYBOLD DIDACTIC GMBH. 3. Design and build simple applications of the control unit PWM by this model 4. Write a laboratory instruction for students

.................................................................................. Research professor Dr. István Nagy

Department of Automation and Applied Informatics Confirmed by

.................................................................................. Prof. Dr. Stépán Gábor

Dean, Faculty of Mechanical engineering Final project is received by:

.................................................................................. Student

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DECLARATION

I, the undersigned, hereby declare that the Final Project submitted contains the results of my own work, assisted by my supervisor(s) and that all other results taken from the technical literature or other sources (eg. Internet) are clearly identified and referred to. I acknowledge that the results described in the Final Project can be used by the Department of the Supervisor.

Budapest, ................. month ...........day ................year

................................................. Student’s signature

4

CONTENTS

INTRODUCTION....................................................................................................................................................... 8

THEORETICAL BACKGROUND OF PULSE-WIDTH MODULATION... ..................................................... 10

I. WHAT IS PWM?................................................................................................................................................ 10

A. Basics of PWM............................................................................................................................................. 10

B. History and application of PWM .................................................................................................................. 11

C. Advantages and drawbacks of PWM............................................................................................................ 14

II. PWM METHODS ............................................................................................................................................. 15

A. Single pulse width modulation ..................................................................................................................... 15

B. Multiple pulse width modulation .................................................................................................................. 16

C. Sinusoidal pulse width modulation............................................................................................................... 17

III. PWM SWITCHING STRATEGIES ................................................................................................................ 18

A. Natural Sampled PWM................................................................................................................................. 18

B. Regular Sampled PWM ................................................................................................................................ 19

C. Optimized PWM........................................................................................................................................... 20

D. Suboptimal PWM ......................................................................................................................................... 20

IV. SUMMARY..................................................................................................................................................... 21

ESSENTIAL DC-DC CONVERTERS.................................................................................................................... 22

I. BASICS .............................................................................................................................................................. 22

A. Converter technologies ................................................................................................................................. 22

B. History of DC/DC conversion technologies ................................................................................................. 22

C. Classification of power supplies ................................................................................................................... 23

D. Efficiency and power relationships of the DC/DC converter ....................................................................... 26

II. PRINCIPLES OF PWM DC/DC CONVERTERS ............................................................................................ 27

A. Relationship among Current, Voltage, Energy, and Power .......................................................................... 27

B. Electromagnetic Compatibility ..................................................................................................................... 28

C. The topologies of DC/DC converters............................................................................................................ 29

III. THE BUCK CONVERTER ............................................................................................................................. 30

A. Circuit description ........................................................................................................................................ 30

B. Analysis of the Buck Converter for CCM .................................................................................................... 33

C. Buck converter in DCM................................................................................................................................ 35

IV. THE BOOST CONVERTER........................................................................................................................... 36


B. Analysis of the Boost Converter for CCM.................................................................................................... 38

C. Boost converter in DCM............................................................................................................................... 40

VI. THE BUCK-BOOST CONVERTER............................................................................................................... 41


B. Analysis of the Buck-Boost Converter for CCM.......................................................................................... 42

C. Buck-Boost converter in DCM ..................................................................................................................... 45

DESCRIPTION OF THE MEASUREMENT KIT................. ............................................................................... 47

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I. LEYBOLD DIDACTIC GMBH MEASUREMENT KIT .................................................................................. 47

A. Stabilized Power Supply...............................................................................................................................47

B. Reference Variable Generator....................................................................................................................... 48

C. Control Unit PWM/PFM .............................................................................................................................. 49

D. MOSFET ...................................................................................................................................................... 51

E. IGBT ............................................................................................................................................................. 51

F. Diode............................................................................................................................................................. 52

G. Panel of different loads................................................................................................................................. 52

II. OTHER INSTRUMENTS USED...................................................................................................................... 52

A. Digital Oscilloscope ..................................................................................................................................... 53

B. Analog Voltmeter and Ammeter, Digital Multimeter................................................................................... 53

C. Voltage Divider Probe .................................................................................................................................. 53

D. Current Clamp .............................................................................................................................................. 53

E. Isolation Transformer.................................................................................................................................... 53

MEASUREMENTS AND EVALUATION............................................................................................................. 54

I. THE BUCK CONVERTER................................................................................................................................ 54

A. The Minimal and the Maximal Voltage........................................................................................................ 55

B. The voltage across the diode......................................................................................................................... 55

C. The boundary between CCM and DCM ....................................................................................................... 56

D. Output voltage in case of DCM.................................................................................................................... 57

E. Output voltage in case of CCM..................................................................................................................... 58

II. BOOST CONVERTER ..................................................................................................................................... 59

A. The Minimal and the Maximal Voltage........................................................................................................ 60


C. The boundary between DCM and CCM and output voltages ....................................................................... 61

III. BUCK-BOOST CONVERTER........................................................................................................................ 62

A. The Maximal and the Minimal Voltages ...................................................................................................... 63


C. Boundary of CCM and DCM........................................................................................................................ 64

D. Output voltage in case of CCM and in case of DCM................................................................................... 65

IV. THE SINGLE PHASE POWER INVERTER.................................................................................................. 66

CONCLUSION AND ACKNOWLEDGEMENT .................................................................................................. 68

REFERENCES.......................................................................................................................................................... 70

APPENDIX A ............................................................................................................................................................ 71

APPENDIX B ............................................................................................................................................................ 83

APPENDIX C ............................................................................................................................................................ 86

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ABSTRACT

The main aim of the final project was to write a measurement guide for MSc students, mainly for

the subject “Selected Chapters of Electrical Engineering”. A deeper study of the PWM methods

were required, then the testing of the instruments of the Department. After getting acquainted

with the operation and principles of the essential DC/DC converters, measurements were done to

investigate the DC/DC converter setups using the instruments of LD DIDACTIC GMBH. These

elements are designed for educational purposes. Done with testing a measurement guide was

written, containing the required information and instructions to execute basic measurements to

investigate the basic converter types. These are the Buck converter, the Boost converter, and the

Buck/Boost converter. During the semester an additional task occurred, which was not part of

the original outline of the thesis. The single phase power inverter circuit was built of the LD

DIDACTIC GMBH elements. The task was to execute Fourier transformation on the result

waveforms, and to analyze the harmonic distortion for different values of the PWM frequency.

During the measurements I learned the operation of the digital oscilloscope.

7

ABSZTRAKT

A szakdolgozat célja az volt, hogy egy mérési útmutató készüljön MSc-s hallgatók részére,

elsősorban a “Válogatott fejezetek az elektrotechnikából” című tárgy laboratóriumi méréseihez.

A PWM technológiák és módszerek mélyebb tanulmányozásával kezdődött a project, ezután a

tanszéken található eszközök tesztelése volt a feladat. Az alapvető DC/DC konverterek

működésének és elméletének megismerése után a LD DIDACTIC GMBH által gyártott

eszközökből összerakott DC/DC konverter áramkörökkel kapcsolatos mérések következtek.

Ezeket az eszközöket speciálisan oktatási célokra tervezték. A tesztelés végeztével elkészült a

mérési útmutató, amely tartalmazza a mérés elvégzéséhez szükséges információkat és

utasításokat. A mérések három alapvető konverterre vonatkoznak: a feszültség-csökkentő

konverter, a feszültség-növelő konverter, és a feszültség-csökkentő/növelő konverter. A

szemeszter során a project bővült egy feladattal, ami nem szerepelt az eredeti kiírásban. Az

egyfázisú inverter kimenő jelét kellett analizálni a MATLAB szoftverrel. A Fourier

transzformáció után a harmonikus torzítás vizsgálatát kellett elvégezni különböző PWM

frekvenciákra. A mérések során megismerkedtem a digitális oszcilloszkóppal.

8

CHAPTER 1

INTRODUCTION

Pulse Width Modulation is a technology

dating back to the early 1960’s. Its usage was

common in analog techniques, and with years

it became widely used in digital devices, such

as microcontrollers and programmable

circuits. Pulse width modulation is mainly

used to generate a desired output voltage level

as an average of a rectangular voltage

waveform. This is used for example in DC

motor speed control, voltage regulation,

DC/DC converters and inverters, audio

amplification, sound effects, and also in

telecommunication, however, in that case the

pulse widths are carrying information, the

average output voltage is not important.

In this thesis the practical applications that are

dealt with are the DC/DC converter circuits.

Power conversion technologies play a very

important role in the operation of electrical

devices. The energy generated in power plants

is driven through several steps of power

conversion before it reaches the homes and

other consumers. The line voltage is a

alternating voltage with a frequency of 50Hz

in Europe, and a frequency of 60Hz in the

USA, and with an effective value of 230V in

Europe and an effective value of 110V in the

USA. This voltage can be used directly for

many applications and devices, but in most

cases power transformers are required. AC/AC

transformers transform the alternating current

to an alternating current of another amplitude

and/or frequency. AC/DC rectifiers convert

the AC voltage to an unregulated DC voltage.

DC/DC converters convert the unregulated

DC voltage to a regulated, stable voltage. The

output voltage can be adjustable or a fixed

value. DC/AC inverters convert the DC

voltage to an AC voltage of a certain

amplitude and frequency. In the focus of this

project are the DC/DC converters.

In the field of power electronics and drives

DC/DC conversion technology is a major

subject area, and has been going through

serious development for six decades. The most

important applications of DC/DC converters

are the computer power supplies,

programmable circuits, cellular phones, LED

power sources, solar cell battery charging

circuits, laser power supplies, and motor

control. Laboratory measurements, laser

applications, and motor speed control usually

require adjustable voltage DC/DC converters.

With the increasing popularity of wireless

technology and the effort to reduce the size of

devices have a positive effect on the

development of both PWM technologies and

DC/DC conversion. DC/DC converters are

widely used nowadays, and the application

and the market of these converters are

increasing rapidly. The quick development of

DC/DC conversion techniques resulted in a

9

large CAGR (Compound annual growth rate)

of 9% in recent years, which compares to the

AC/DC power supply market, which had a

CAGR of only about 7.5% during the same

period. The DC/DC converter market is

undergoing some serious changes as a result

of two main trends in the electronics industry:

low voltage and high power density. [1]

The LEYBOLD DIDACTIC GMBH elements

on the department are suitable for basic

measurements and for the investigation of the

operation of PWM and DC/DC converters.

With the help of the measurement guide

written it may be easier for students to get

acquainted with this rapidly developing

technology, which will probably dominate the

market of conversion technology for several

years, according to the widespread use of

computers and wireless technologies.

The number of figures is like “x.y”, where “x”

means the chapter, and “y” means the number

of the figure within that chapter. For example

Figure 3.12 means the 12th figure in chapter 3.

The equations are numbered in the same way,

Equation 2.8 for example means the 8th

equation of Chapter 2.

The citation numbers are not connected to

chapters, [13] means the 13th citation in the

whole document. Citation numbers are usually

at the title of sections. This means that the

following section is based on those

books/articles/websites, or further reading is

available in those items.

At several points of the thesis the “LEYBOLD

DIDACTIC GMBH” name is mentioned. The

company is called LD DIDACTIC today, but

the guides and instruments used were

produced with the company’s former name.

10

Figure 2.1 – Pulse width and duty cycle

CHAPTER 2

THEORETICAL BACKGROUND OF PULSE-WIDTH

MODULATION

I. WHAT IS PWM?

PWM is a method to deliver power to the load,

it is a way of delivering energy through a

succession of pulses rather than a continuously

varying (analogue) signal. Using continuously

varying signal the power losses highly depend

on the output power. At the extremes (0 or

maximal output) the efficiency is high, but it

is bottoming out at midrange where the power

dissipated as heat roughly equals to that

delivered to the load. Using PWM we can

calculate with a constant loss percentage. The

efficiency of PWM is usually above 90%.

A. Basics of PWM

Using Pulse Width Modulation we add the

maximum voltage to the load during the ON-

state, and zero during the OFF-state. The

average voltage present at the output depends

on the ratio of the on state duration and the

switching period. The switching period (Ts)

can be calculated from the switching

frequency (fs) with the following equation.

(See Figure 2.1)

ss f

T1= (2.1)

The ratio of the ON-state duration and the

switching time period is called duty cycle (or

duty ratio), and usually given in percentage.

S

ON

T

TD = (2.2)

We see that the PWM signal is a rectangular

pulse wave and modulating the pulse width we

modify the average value of the output.

Considering that vmin is the minimum and vmax

is the maximum value of the waveform )(tf ,

while D is the Duty cycle, we can calculate the

average value (v ). The switching period is TS.

dttfT

vST

S∫⋅=0

)(1

(2.3)

The wave has a value of vmax for STDt ⋅<<0

and a value of vmin for SS TtTD <<⋅ . With

these and the basic rules of integration

Equation 2.3 becomes:

11

Figure 2.2 – Average output voltage

+⋅= ∫∫

⋅

⋅

dtvdtvT

vS

S

S T

TD

TD

Smin

0

max

1 (2.4)

Executing the integration we get the following

equation:

S

SS

T

TDvTDvv

⋅−⋅+⋅⋅= )1(minmax

minmax )1( vDvD ⋅−+⋅= (2.5)

(Figure 2.2 shows this in a graph.)

The second part of the sum is not necessary in

many cases, e.g. considering DC technology,

0min =v , so Equation 2.5 becomes:

maxvDv ⋅= (2.6)

Considering these, it is evident that the

average value v is directly dependent on the

Duty cycle (D)

B. History and application of PWM [2]

PWM technology is used since the early

sixties, although it is considered to be a

relatively modern technology. At the

beginnings the simplest methods were used,

because of the ease of implementation using

analog techniques. These technologies were

hard to implement using microprocessors, and

with the transcendental equations this process

faced difficulties in computer aided design, so

new sampling processes were created.

In the late 1970’s the Optimized PWM

technology was developed, which used

numerical methods, that was easy to

implement in digital techniques, although an

analogue implementation was not possible.

Suboptimal PWM followed, and today PWM

is used in many different technologies.

Telecommunications. In the field of

telecommunications, the widths of the

different pulses generally correspond to

specific values of the data encoded at one end

as well as decoded at the other. There are

various lengths of the pulses and the

information will be sent after regular intervals.

At the receiving side with a simple RC or LC

circuit can remove the high frequency square

wave and execute the A/D conversion. By

keeping the signal digital (as the value is 0 or

1 in pu.) noise effects are minimized. Noise

can only affect the signal if it is strong enough

to change a 1 to 0 or vice versa. [2] [3]

Power delivery, motor drives. With

semiconductor switches high frequency PWM

power control systems are easy to design The

ON/OFF states of the modulation are used to

control the switches that will set the voltage

across or current through the load. The major

advantage of this system is that in ideal case

12

there is no power dissipation. The switches are

either OFF and not conducting any current, or

ON and have (ideally) no voltage drop across

them. The product of the current and the

voltage at any given time defines the power

dissipated by the switch, thus the power

dissipation of the switch is ideally zero. More

precisely, semiconductor switches such as

MOSFETs, IGBTs or BJTs are non-ideal

switches, but a relatively high efficiency can

be reached. During the transitions between on

and off states, considerable power is

dissipated in the switches, but the change of

state between ON and OFF is quite rapid

relative to typical ON or OFF times, so the

average power dissipation is quite low

compared with the power being delivered.

PWM is also often used to control the supply

of electrical power to another device such as

in speed control of electric motors,

fundamental operation audio switching

amplifiers or brightness control of light

sources and many other power electronics

applications. PWM is widely used for speed

control of DC motors, and it is because the

speed and rotation of the motor will be much

smoother than in case of continuous analogue

DC voltage control. The motor gets pulses

with high frequency, thus because of the

inertia of the rotating part the speed will be

smooth. For example, in case of a car driven

by a DC motor, if we increase an analogue

voltage linearly until the torque is high enough

to overcome the stiction, the car will start

roughly (with inappropriate speed), as the

stiction is higher than the friction. If we use

PWM, the motor will give out its maximal

torque, as the voltage is always the maximum,

and the pulse duration is changed at a constant

frequency. This will result in a much smoother

operation. From this point of view the main

advantages of the PWM are:

• High efficiency

• Wider operational range

• Longer lived motors

Using this method, the current is limited to a

safe value for the windings. PWM allows a

very linear response in motor torque even

down to low PWM% without causing damage

to the motor. Most motor manufacturers

recommend PWM control rather than the older

voltage control method. [2] [4]

Voltage regulation. A PWM controller is also

used in the efficient voltage regulators. If

voltage is switched to the load with an

appropriately tuned duty cycle, a desired level

of the voltage at the output can be

approximated. The switching noise is filtered

with the help of a capacitor and an inductor.

DC-DC converters use several methods for the

control of the semiconductor switches. PWM

is the simplest and most widely used one of

them. [2]

Audio effects and amplification. PWM is

sometimes used in sound (music) synthesis, in

particular subtractive synthesis, as it gives a

13

sound effect similar to chorus or slightly

detuned oscillators played together. (In fact,

PWM is equivalent to the difference of two

sawtooth waves.) The ratio between the high

and low level is typically modulated with a

low frequency oscillator, or LFO. In addition,

varying the duty cycle of a pulse waveform in

a subtractive-synthesis instrument creates

useful timbral variations. Some synthesizers

had a duty-cycle trimmer for their square-

wave outputs, and that trimmer could be set by

ear; the 50% point was distinctive, because

even-numbered harmonics essentially

disappeared at 50%.

In recent years there is a new type of

amplifiers on the market called “Class D”

amplifiers. The operation of these instruments

is based on the PWM technology. Its input is a

standard audio line level signal. This audio

line level signal is sinusoidal with a frequency

ranging from 20Hz to 20kHz typically. This

signal is compared with a high frequency

triangle or sawtooth wave to create a PWM

signal. This PWM signal drives the power

stage creating an amplified digital signal. A

low pass filter is applied to the amplified

signal to filter out the PWM carrier frequency

and retrieve the sinusoidal audio signal. Most

Class D amplifiers switch from about 300kHz to

2MHz.

Class D amplifiers are very efficient (usually

between 90 and 95%), they provide the best

use of energy stored in limited power sources

(such as a battery). The high efficiency

decreases heat-sink requirements, and Class D

amplifiers do not heat their neighbouring

components as much as other topologies.

Its main drawback is that the switching of the

outputs causes high EMI (Electromagnetic

interference). Another concern is that their

sound quality is not as good as Class AB

topologies.

Historically, a crude form of PWM has been

used to play back PCM digital sound on the

PC speaker, which is only capable of

outputting two sound levels. By carefully

timing the duration of the pulses, and by

relying on the speaker's physical filtering

Figure 2.3 - DC PWM Motor Speed Controller

Figure 2.4 – Class D 3000W car amplifier

14

properties (limited frequency response, self-

inductance, etc.) it was possible to obtain an

approximate playback of mono PCM samples,

although at a very low quality, and with

greatly varying results between

implementations. In more recent times, the

Direct Stream Digital sound encoding method

was introduced, which uses a generalized form

of pulse-width modulation called pulse density

modulation, at a high enough sampling rate

(typically in the order of MHz) to cover the

whole acoustic frequencies range with

sufficient fidelity. This method is used in the

SACD format, and reproduction of the

encoded audio signal is essentially similar to

the method used in class-D amplifiers. [5] [6]

C. Advantages and drawbacks of PWM

In the previous sections several advantages

and disadvantages of PWM techniques were

mentioned, let this section be a summary of

them.

Main advantages:

• High efficiency, usually above 90%

• Low power dissipation causes lesser

heating, so PWM systems do not need

complicated cooling.

• Because of the cooler operation, the

power amp requires much less heat

sink mass, thus the Class D amplifiers

that use PWM are more compact and

weigh less than other solutions.

• In motor control, PWM methods drive

the DC motor always with maximal

torque, so a smoother operation is

possible.

• With limited flowing current, PWM

control can extend the DC machine’s

lifespan.

• PWM provides wider operational

range for DC motor drives

• PWM allows very linear response in

motor torque, even at lower duty

cycles (without damaging the motor).

• Considering the control of

semiconductor switches in DC-DC

converters, PWM is one of the

simplest and most efficient methods of

control.

• In microcontroller technology (and

other digital solutions) it is a much

simpler – and therefore much cheaper

– solution to keep the signal digital

rather than convert it into a continuous

analog signal. Microcontrollers can

give out “analog” signals using PWM

with a clock and a simple counter, thus

no D/A converter is necessary (which

is relatively expensive).

• Considering telecommunication, the

noise effects are minimized, as the

noise level must be high enough to

change a 0 to 1 or vice versa to affect

the PWM signal.

15

Drawbacks:

• Considering power amplifiers, the

main disadvantage of PWM is that it

causes large EMI (Electromagnetic

Interference), which is not acceptable

in many cases.

• Another aspect is that PWM

amplifiers’ sound quality is not as

good as the other topologies’. This

should be considered only in high

quality sound technology, as the

quality in general is mainly determined

by the quality of loudspeakers.

• Abrupt changes in current (and also in

voltage) are present at high-frequency

switching, so special techniques are

required to protect the elements of the

circuit, such as shielding and filtering.

This may call for more complicated (or

expensive) circuits.

II. PWM METHODS [7] [8] [9]

In this section the basic methods of PWM

signal generation are detailed. The output

average value depends on the duration of the

ON time and the OFF time. The ratio can be

changed in three ways:

• Maintaining a constant pulse width,

and varying the number of pulses per

half cycle

• Varying the pulse width for fixed

number of pulses per half cycle

• Varying both the pulse width and

number of pulses per half cycle

A. Single pulse width modulation [7]

The basic type of voltage control method.

Using this technique, only one pulse per cycle

is present, and the width of the pulse is varied

to control the output voltage. Maximum

voltage is present if °= 180δ . Reducing the

pulse width the mean voltage decreases and

the harmonic content of the output increases.

The width of the pulse is varied by varying the

gating signals as shown in Figure 2.5. A

rectangular signal of amplitude Ar is compared

with a triangular carrier wave of amplitude Ac.

Several parameters need to be calculated,

these are shown below. [10]

Figure 2.5 – Single Pulse Width Modulation

16

The modulation index:

c

r

A

AM = (2.7)

The RMS output voltage:

πδ⋅=VVO (2.8)

Fourier analysis of the resultant waveform

yields due to symmetry:

0=na , (2.9)

and:

⋅⋅⋅

= ∑= 2

sin4

...5,3,1

δπ

n

n

Vb

N

in (2.10)

Distortion factor:

∑=

⋅=N

i

n

n

V

VDF

1

2

1

1 (2.11)

Total harmonic distortion:

∑∞

=⋅=

,...3,2

2

1

1

inV

VTHD (2.12)

B. Multiple pulse width modulation [7]

Using this method, the harmonic content is

reduced due to the several pulses in each half

cycle. The inverter output with multiple pulse

width modulation is as shown in Figure 2.6.

Modulation is achieved by a comparison of

triangular wave with a DC voltage. Switching

instants are determined by the intersection of

the two waves. By changing the amplitude of

the control signal, pulse width may be varied

thereby varying the output voltage.

The number of pulses per half cycle:

r

c

f

fp

⋅=

2 (2.13)

The pulse width varies from 0 to p

π

The RMS value of the output voltage:

πδ⋅⋅= p

VVO (2.14)

Using the following expressions

π⋅⋅=

n

VB

2 (2.15)

2δαφ += m (2.16)

the Fourier analysis of the resultant waveform

gives:

Figure 2.6 – Multiple Pulse Width Modulation

17

0=na (2.17)

( ) ( )[ ]∑∞

=

+⋅−⋅⋅⋅⋅=1

sinsin2

sinn

n nnn

Bb φπφδ

(2.18)

These equations suggest that the distortion

factor is reduced significantly compared to

single pulse modulation. Also, it may be

observed that if p is large, amplitudes of the

lower order harmonics would decrease.

Although, in the process, higher harmonics

increase, they may be filtered out easily.

C. Sinusoidal pulse width modulation [7]

In this method, pulses over a half cycle of

unequal widths are generated. Pulse width is a

sinusoidal function of the angular position of

each cycle. This is done by comparing a

sinusoidal wave against a triangular carrier

wave. This method is mainly used because of

its simplicity and the ease of implementation.

The number of pulses per half cycle is being

decided by the triangular carrier frequency to

that of the modulated sinusoid, fr.

The waveform in Figure 2.7 is a 2-level

waveform with the output changing from +V

to –V.

The fundamental component of the PWM

output waveform with N chops per quarter

cycle is:

( )[ ]∑=

+ −⋅−⋅⋅=N

ii

iVV

1

11 1cos12

4 απ

(2.19)

The waveform of Figure 2.8 is a 3-level

waveform, where the output voltage ranges

form +V to –V.

From Figure 2.7 if δm is the width of the mth

pulse:

∑=

⋅=p

m

mO VV

1 πδ

(2.20)

Due to symmetry:

0=na (2.21)

Using Equation 2.15 and Equation 2.16:

( )[ ]φπφδ +⋅−⋅⋅⋅⋅=∑=

nnnBb mp

mn sinsin

21

(2.22)

Observing the harmonic profile, the distortion

factor is significantly reduced compared to

multiple pulse width modulation. This

Figure 2.7 – Sinusoidal 2-level PWM

18

Figure 2.8 – Sinusoidal 3-level PWM

modulation type reduces all harmonics less

than or equal to 2p-1. This means that the

PWM pushes all harmonics into a high

frequency range around the switching

frequency and its multiples, around harmonics

mf, 2mf. 3mf, and so on.

The 3-level waveforms can be generated either

by combining two suitably phased 2-level

waveforms or directly. In the direct method,

pulse widths in each half cycle are modulated

according to the positive half cycle of the sine

wave. A polarity discriminator provides the

gating logic necessary to correctly apply the

PWM sequence to the switching devices of the

inverter. Natural and regular sampled PWM

types are of sinusoidal modulation.

III. PWM SWITCHING STRATEGIES [8]

There are various available strategies applied

to generate the switching edges of the PWM

waveform. These are listed and detailed in this

section.

These strategies are:

• Natural Sampled PWM

• Regular Sampled PWM

• Optimal PWM

• Suboptimal PWM

A. Natural Sampled PWM [7]

As mentioned before, it is possible to compare

a sine wave of required frequency with that of

the carrier wave, and the signals derived

therefrom can be used to trigger the power

devices of the half-bridge or the full-bridge

inverter which will result in the 2- or 3-level

waveforms.

This PWM strategy was popular in the early

1960’s because of its simplicity and easy

implementation using analog techniques.

This strategy is based on the continuous real

time comparison of the sinusoidal modulating

signal and a triangular (or saw tooth) carrier.

The instantaneous intersection of the carrier

and modulating signals determines the

switching instants by a natural selection or

sampling, called natural sampled PWM.

Natural sampled PWM has the advantage of a

simple and well defined modulation

19

procedure, which makes it suitable for

analogue implementation, but the natural-

sampling process is non-linear, and the pulse

widths of the PWM are defined by

transcendental equations, thus its use in

computer-aided design, analysis and also in

microprocessor technology faces difficulties.

This is the direct result of the non linear

sampling modulation process, making it

unsuitable for digital hardware or

microprocessor implementation.

2 level PWM. Pulse widths are proportional to

the amplitudes of the modulating signal at the

switching instants. The centres of the pulses

are neither equidistant nor uniformly spaced.

Pulse width is given by a transcendental

equation.

( ) ( )[ ]

+⋅+⋅= 12 sinsin

21

2tt

MTtp ωω (2.23)

As the waveform switches between two levels

(that is +V and –V), it is termed as 2-level

PWM. This contains carrier frequency

components.

The RMS output voltage:

2V

VO = (2.24)

3-level PWM. With proper phase-shift, we

can combine two of the 2-level PWM, so a 3-

level PWM can be obtained. Another solution

is the use of a full-bridge inverter. The main

advantage of this solution over the 2-level

PWM is that

VVO = . (2.25)

Output power for a full-bridge inverter is 4

times higher and the fundamental component

is twice that of the half-bridge inverter

(although the peak reverse blocking voltage of

each transistor and the quality of output

voltage of half-bridge and full-bridge inverters

are the same.)

The expression for the pulse-width in case of

3-level PWM:

( )12 sinsin2

ttTM

tp ωω += (2.26)

B. Regular Sampled PWM [7]

Using regular-sampled PWM techniques, the

problems of the natural-sampled PWM can be

totally eliminated. The linear sampling process

of the regular-sampled PWM allows the

samples of the modulating wave to be taken at

regularly spaced intervals. The popularity of

this method peaked in the 1970’s, it was

dominating until the end of that decade. This

improved method made the digital realization

possible.

In this method, the modulating wave does not

vary continuously during the sampling period,

it either remains constant or changes its level

at a certain distant within the sampling period,

according to which two types of regular

sampled PWM are defined.

If the modulating signal has a constant value,

then the pulse widths are proportional to the

amplitude of the modulating wave at the

20

uniformly spaced switching instants, hence the

name uniform or regular sampling. In case of

regular sampled PWM, the relationship

between the fundamental component of the

PWM waveform and the modulation index is

linear.

With the leading and trailing edges of the

resulting pulse determined by the constant

level of the modulating wave, the PWM is

symmetric sampled, the centres of the pulses

are equidistant.

With the leading and trailing edges defined by

different levels of the modulating wave, the

result is an asymmetric regular-sampled

PWM. The sampling rate is twice the carrier

frequency, the pulse widths are proportional to

the modulating signal amplitudes at the

switching instants. The centres of the pulses

are neither equidistant nor uniformly spaced.

The natural sampled and the regular sampled

PWM techniques are based on a well defined

modulation process, thus can be implemented

using analog, digital, or microprocessor

techniques.

C. Optimized PWM [7]

Analog implementation of this technique is

not possible, because it uses numerical

techniques rather than definable modulation

process. It was popular mainly in the late

1970’s.

As a result of numerical minimization search

techniques, optimized PWM signals are more

complex to generate. It allows the PWM

harmonic spectrum to be tailored to achieve a

particular performance specification.

A general PWM waveform is defined by a set

of angles, the switching angles are determined

to get required characteristics using different

optimization techniques. Optimization can be

performed with respect to a variety of

parameters (THD, elimination of particular

harmonics, etc). Choosing a particular

parameter and substituting in the general

PWM equation will yield a set of equations,

which are then solved to yield the values of

switching angles. These angles are stored in a

look-up table and PWM waveforms could be

generated. On-line computation is hard

(almost impossible) in optimal PWM.

The disadvantage of optimal PWM is that a

wave which is optimal with respect to one

parameter doesn’t need to be optimal with

respect to another. This method is more

complex and needs large memory as the

switching angles for each frequency of

operation will have to be stored separately.

D. Suboptimal PWM [7]

This method was developed to maintain a well

defined modulation process which can be

easily and efficiently implemented in

microprocessor software and still reproduces

the desirable characteristics of the optimized

PWM. Suboptimal is so termed because

approximations are made to optimized

characteristics to make the digital method of

realization possible. There are several methods

21

of suboptimal PWM, which I do not describe

in detail in this thesis, although it is to be

mentioned that difficulties are encountered in

the implementation of these methods.

IV. SUMMARY

In this chapter the basics of Pulse Width

Modulation were described. In the first section

the basic principles of generating the average

DC value with PWM, the definition of duty

cycle, the applications of PWM and the

advantages and drawbacks of PWM were

detailed. The second section presented the

basic Pulse Width Modulation methods, and

the third section the basic techniques of

generating the PWM signal were mentioned.

22

CHAPTER 3

ESSENTIAL DC-DC CONVERTERS

I. BASICS

This section gives an overview of the possible

converters, and classifies the DC-DC

converters.

A. Converter technologies

In everyday life, industry, research and

development we can find applications of the

equipments for power conversion. In the field

of power electronics conversion technique is

an important research area. The equipment can

be separated to four technologies:

• AC/AC transformers

• AC/DC rectifiers

• DC/AC inverters

• DC/DC converters

AC/AC transformers are used to transform a

voltage of a certain amplitude and frequency

to another amplitude and/or frequency level.

A rectifier is an electrical device that converts

alternating current (AC) to direct current

(DC), a process known as rectification.

Rectifiers have many uses including

components of power supplies and detectors

of radio signals.

An inverter is an electrical device that

converts direct current (DC) to alternating

current (AC); the resulting AC can be at any

required voltage and frequency with the use of

appropriate transformers, switching, and

control circuits.

A DC to DC converter is an electronic circuit

which converts a source of direct current (DC)

from one voltage level to another, usually they

are used to convert an unregulated DC voltage

to a regulated, stable voltage value. This type

of converters is the main topic of this chapter.

B. History of DC/DC conversion

technologies [1]

In the field of power electronics and drives

DC/DC conversion technology is a major

subject area, and has been going through

serious development for six decades. DC/DC

converters are widely used in industrial

applications and computer power supplies.

The quick development of DC/DC conversion

techniques resulted in a large CAGR[17]

(Compound annual growth rate) in the recent

years. The worldwide market of DC/DC

converters has grown from U.S. $3336 million

in 1995 to U.S. $5128 million in 2004. This

means a compound annual growth rate of 9%,

which compares to the AC/DC power supply

market, which had a CAGR of only about

23

7.5% during the same period. The DC/DC

converter market is undergoing some serious

changes as a result of two main trends in the

electronics industry: low voltage and high

power density.

DC/DC conversion techniques date back to the

1920’s, when the voltage conversion was

solved by a simple voltage divider (e.g.

potentiometer or rheostat). This technology

can only lower the voltage level and it works

with poor efficiency.

The multiple-quadrant chopper is the second

step in the development of DC/DC

conversion. It was taking a long time for

engineers to design an equipment to convert

DC voltage for actuators, like transformers do

in AC/AC conversion.

The Second World War delayed the

development of DC/DC converters, but after

the war communication technologies went

through a rapid development, which resulted

in the evolution of DC/DC converters.

C. Classification of power supplies [11]

All active electronic circuits, both analog and

digital require power supplies. Many

electronic systems require several different

level DC supply voltages. DC power supplies

are widely used in computers,

telecommunication, instrumentation

equipment, aerospace, medical, and defence

electronics. A DC supply is derived from a

battery or an AC utility line using transformer,

rectifier, and filter. The resultant raw DC

voltage is not constant enough and contains

high AC voltage ripple. Because of this, the

raw DC voltage is not appropriate for most

applications, thus voltage regulators are used

to make the DC voltage more constant and to

Figure 3.1 – Classification of power supplies

24

attenuate the AC ripple.

There are two main types of power supplies:

regulated and unregulated. The regulated

voltage is kept within a narrow range of 1-2%

of the desired nominal value, in spite of line

voltage, load current, and temperature

variations. Regulated DC power supplies are

called DC voltage regulators (there are also

DC current regulators).

Figure 3.1 shows the classification of

regulated power supplies.

In linear voltage regulators transistor are

operated in the active region, as dependent

current sources with relatively high voltage

drops at high currents, dissipating a large

amount of power and resulting in low

efficiency. Linear regulators are heavy and

large, but they exhibit low noise level and are

suitable for audio applications.

Switching mode converters operate the

transistors as switches, thus they dissipate

significantly lesser power than transistors

operated as dependent current sources. When

the transistors conduct high current, the

voltage drop is very low on them (0 in ideal

case), and they conduct (nearly zero or) no

current when the voltage drop is high on them.

Because of this, the conduction loss of the

switching mode converters are low, their

efficiency is usually above 80-90%. It is to be

mentioned that the switching losses reduce the

efficiency at higher frequencies, losses

increase proportionally to switching

frequency. Linear and switched capacitor

regulator circuits (except for large capacitors)

can be fully integrated and are used in low-

power and low-voltage applications, usually

below several watts and 50 V. PWM and

resonant regulators are used at high power and

voltage levels. They are small in size, light in

Figure 3.2/a – AC/DC power supply with linear regulator

Figure 3.2/b – AC/DC power supply with switching-mode voltage regulator

25

weight, and have high conversion efficiency.

Two typical power supplies are shown in

Figure 3.2. The power supply in Figure 3.2/a

contains a linear DC voltage regulator while

the power supply in Figure 3.2/b contains a

switching-mode DC voltage regulator.

The first one consists of a low-frequency step-

down power line transformer, a front-end

rectifier, a low-pass filter, a linear voltage

regulator, and a load. The nominal AC voltage

of the mains network is 110VRMS in the USA

and 230VRMS in Europe, but the actual voltage

varies within a range of ±20% of the nominal

voltage. The frequency of the AC line voltage

is 50Hz in Europe, and 60Hz in the USA

(however spacecraft applications use 20kHz,

in aircraft applications 400Hz is typical). The

line transformer reduces the relatively high

voltage of the mains network to a lower level,

usually ranging from 5 to 28VRMS. Because of

the low frequency of the AC line voltage, the

transformer is heavy and bulky. The output

voltage of the rectifier and the filter is

unregulated, and varies because of the peak

voltage of the power line varies, therefore a

voltage regulator is required between the filter

and the load.

The second power supply shown in Figure

3.2/b consists of a front-end rectifier, a low-

pass filter, an isolated DC/DC switching mode

voltage regulator, and a load. Such a circuit is

called an off-line power supply, as the AC

voltage is rectified directly from the AC

power line, thus no bulky low-frequency line-

transformer is needed. The switching-mode

voltage regulator contains a high-frequency

transformer to obtain DC isolation for the

entire supply, but since the switching

frequency is much higher than that of the AC

line frequency, the weight and size of the

transformer as well as inductors and capacitors

is reduced. The switching frequency usually

ranges from 25kHz to 500kHz (to eliminate

audio noise effects, it should be above

20kHz). A PWM switching mode voltage

regulator generates a high-frequency

rectangular voltage wave, which is rectified

and filtered. The duty cycle of the rectangular

wave is varied to control the output voltage,

therefore these voltage regulators are called

PWM DC/DC converters.

Power converters are used to convert one form

of electric energy to another. DC/DC

converters are applied to convert a regulated

or unregulated input DC voltage to an output

DC voltage of constant value, and keep its

value within a narrow range of the desired

value even if the line voltage, the load current,

or the temperature vary. This input voltage is

usually a battery or a rectified AC line

voltage. Different from the linear voltage

regulators, the output of PWM DC/DC

converters may be either lower or higher than

the input voltage, thus there are step-down and

step-up converters. Step-down converters

always decrease the input voltage value to a

lower level, while step-up converters increase

the input voltage to a higher average value

(the minimal output voltage is the input

voltage). There are so called step-up/step-

26

down converters. Their output voltage can be

either higher or lower than the output.

Considering the polarity of the output voltage,

there are inverting (opposite polarity) and

non-inverting (same polarity) DC/DC

converters. The dc–dc converters may have

common negative or common positive input

and output terminals, and may have a single

output or multiple outputs. There are fixed

output and adjustable output voltage supplies.

E.g. 1.8V fixed voltage supplies are required

in some power electronics applications (e.g.

FPGAs or CPLDs, computer power supplies),

power supplies with adjustable output voltage

are necessary for several laboratory

measurements, He-Ne lasers, motor control

and so on. In some applications,

programmable power supplies with digitally

selected output voltages are required. Power

supplies may be non-isolated or isolated,

transformers can be used to obtain dc isolation

between the input and output and between the

different outputs. The most important

requirements of power supplies are: high

efficiency, high power density, high

reliability, and low cost.

D. Efficiency and power relationships of the

DC/DC converter [11]

The following letters will be used in this

section:

II (Average value of the) input current

Ii Function of input current

IP Input power

IV (Average) input voltage

OP Output power

LSP Power loss on the converter

η Efficiency

O

LS

P

P Normalized power loss

The input currents of switching-mode DC/Dc

converters are usually pulsating. The DC

component of the input current is given by:

Figure 3.3/b – Adjustable voltage power supply

Figure 3.3/a – DC/DC converter

27

∫⋅=T

II dtiT

I0

1 (3.1)

The DC input power of a converter is:

II

T

II

T

III IVdtiT

ViVT

P ⋅=⋅⋅=⋅⋅= ∫∫00

11 (3.2)

Neglecting the AC components of the output

voltage and current (which are very small), the

output power of the DC/DC converter is

OOO IVP ⋅= , (3.3)

and the power loss on the converter is

IOLS PPP −= (3.4)

The efficiency of the DC/DC converter can be

expressed by the following equation:

O

LSLSO

O

I

O

P

PPP

P

P

P

+=

+==

1

1η (3.5)

From which:

11 −=ηO

LS

P

P (3.6)

The normalized power loss decreases if the

efficiency increases.

II. PRINCIPLES OF PWM DC/DC

CONVERTERS [11]

In this section the basic equations of the

DC/DC converters and the methods of EMI

optimization will be discussed.

A. Relationship among Current, Voltage,

Energy, and Power

The average value of current i(t) is given by

∫⋅=T

AV dttiT

I0

)(1

, (3.7)

and the RMS value of the current is

∫⋅=T

RMS dttiT

I0

2 )(1

. (3.8)

Likewise, the average value of the voltage v(t)

can be expressed by the following equation

∫⋅=T

AV dttvT

V0

)(1

, (3.9)

and the RMS value is

∫⋅=T

RMS dttvT

V0

2 )(1

. (3.10)

The instantaneous power is given by

)()()( tvtitp ⋅= . (3.11)

The energy dissipated in a component or

delivered by a source over the time interval t1

∫ ∫ ⋅==1 1

0 0

)()()(t t

dttvtidttpW . (3.12)

For periodic waveforms in steady state the

power absorbed by a component or delivered

by a source is the time-average of the

instantaneous power over a period T of the

operating frequency,

WfT

Wdttvti

TP

T

⋅==⋅⋅= ∫0

)()(1

(3.13)

28

The average charge stored in a capacitor is

zero for periodic waves in steady state,

0)(0

== ∫T

C dttiQ (3.14)

This is called the principle of capacitor charge

balance or capacitor ampere-second balance.

Thus, the average current through a capacitor

for steady-state operation is zero,

∫ ===T

CAVC dttiTT

QI

0

)( 0)(1

. (3.15)

For periodic waveforms in steady state, the

average magnetic flux linkage of an inductor

over one period is zero,

∫ ==T

L dttv0

0)(λ . (3.16)

This is called the inductor linkage balance or

inductor volt-second balance. The average

voltage across the inductor in steady state is

zero,

∫ ===T

LAVL dttvTT

V0

)( 0)(1λ

. (3.17)

The instantaneous energy stored in a capacitor

is

)(21

)( 2 tCvtw CC = , (3.18)

And in an inductor is

)(21

)( 2 tLitw LL = . (3.19)

B. Electromagnetic Compatibility

The switching of the semiconductor devices

causes current pulses at the input of the power

supplies. In technology of switching-mode

power supplies it is a very important problem

to understand and optimize the

electromagnetic compatibility. Because of the

switching of transistors and diodes PWM

converters work with rectangular waveforms,

thus with short rise time and fall time and high

dtdv/ and dtdi / . Therefore, these

waveforms exhibit a wide and strong

harmonic spectrum. PWM converters are

notorious sources of noises, radio frequency

interference and electromagnetic interference.

Suppressing the EMI is a important issue in

the design of switched-mode converters.

The mentioned strong harmonics may

interfere with electronic devices. The IEC

61000-3-2 international standard sets the

available level of harmonics.

Depending on the noise transmission there are

two main categories of noises:

• conducted noise (450 kHz to 30 MHz);

• radiated noise (30MHz to 1 GHz).

Figure 3.4 – Electromagnetic compatibility

29

There are two types of conducted noise, these

are differential-mode noise and common mode

noise. Conducted noise can be suppressed by

adding appropriate filters to reduce the power

levels of unwanted frequencies in a specific

frequency band. Metallic shields are used to

prevent radiated noise. Another solution is the

spectral modification of EMI at source. This

affects both radiated and conducted

interference. These methods spread the

spectrum of the converter waveforms, thus the

power levels at specific frequencies are

reduced below the required levels to meet the

requirements of the EMI standards, without

additional filters and shields. One solution for

the problem is random modulation of the

converter switching frequency, this produces

random jitter around normal periodic voltage

and current waveforms, thus spreading the

spectrum and reducing the spectral peaks.

Another method is the converter operation

under chaos and chaotic modulation. All

switching mode converters are strongly

nonlinear systems, the occurrence of chaos is

quite common in them. The occurring chaotic

behaviour in switching-mode converters lead

to some inherent problems in practical

applications.

The current ripple and the power level of AC

components increase under chaotic operation,

reducing efficiency, and on the other hand, the

spectrum has a higher emission floor, and it

spreads into low frequency range, resulting in

audible acoustic disturbances.

These topics are not subjects of this thesis,

thus no further details will be discussed.

C. The topologies of DC/DC converters

In switched-mode technology a lot of

topologies are used. The family of single-

ended PWM DC/DC converters contains the

following topologies:

• Buck converter

• Boost converter

• Buck/Boost converter

• Flyback converter

• Forward converter

• Cúk converter

• SEPIC (single ended primary input

converter)

• Dual SEPIC (also called Zeta or

inverse SEPIC) converter

The flyback converter is a transformer version

of the buck/boost converter, while the forward

converter is a transformer version of the buck

converter. The flyback converter and the dual

SEPIC are identical on the primary side of the

transformer. The SEPIC and the Cúk

converters are identical on the primary side of

the transformer. The SEPIC and the flyback

converters are identical on the secondary side

of the transformer. Likewise, the dual SEPIC

30

and the Cúk converters are identical on the

secondary side of the transformer.

The multiple-switch PWM DC/DC converters

are the half-bridge, the full bridge, and the

push-pull converters.

Switched-mode converters use duty-cycle

control of switching elements to block the

flow of the energy from the input to the output

and thus achieve voltage regulation. Using this

type of converters, the size of the transformer

and the energy storage components are

significantly reduced. Because of the high

switching frequency, a small transformer with

a ferrite core is appropriate. This size

reduction is very important in several

applications including aerospace, computers,

and wireless technologies. Of course, there is

a penalty paid due to the increased noise level,

which is important both at the input and at the

output of the supply due to the switching of

the semiconductor devices. Also the control

circuit required for PWM applications is much

more complicated than that of the linear

regulators.

III. THE BUCK CONVERTER [11]

In this section the circuit diagram of the buck

converter is explained. The waveforms for

both Continuous Conduction Mode and

Discontinuous Conduction Mode are

presented and derived. The voltage and

current stresses of the components are

calculated. Transfer function of the converter

is given for both modes. The boundary

between CCM and DCM is defined.

A. Circuit description

The circuit diagram of the Buck converter is

shown in Figure 3.5.

The circuit diagram of the Buck converter

contains four elements. These are a

semiconductor switch (S), a diode (D), an

inductor (L) and a filter capacitor (C). The

resistor RL represents a DC load. The

semiconductor switch is usually a power

MOSFET (Metal Oxide Semiconductor Field

Effect Transistor) because of its high speed,

however, Bipolar Junction Transistors (BJTs),

Insulated Gate Bipolar Transistors (IGBTs) or

MOSFET-controlled thyristors are also

commonly used. The diode D is called a

freewheeling (or flywheel) diode. The

switching network consisting of the transistor

and the diode “chops” the DC input voltage,

therefore this kind of converters are called

choppers. The chopper always reduces the

input voltage to a lower level, therefore it is a

step-down converter.

Figure 3.5 – The circuit diagram of the Buck

converter

31

The switch S is controlled by a PWM control

circuit with a switching frequency of T

fS

1= .

The Duty cycle of the PWM is

ONSOFFON

ONON tftt

t

T

tD ⋅=

+== , (25)

where ONt is the time when the switch is

closed and OFFt is the time when the switch is

open. Since the duty cycle D of the drive

voltage vGS varies, so does the duty ratio of

other waveforms. This permits the regulation

of the dc output voltage against changes in the

DC input voltage VI and the load resistance RL

(or the load current IO).

It is difficult to drive the transistor, because

the gate of the MOSFET is not referenced to

ground. Floating gate drive is needed for the

transistor furthermore with the input current of

the converter being discontinuous an LC filter

may be required at the input.

The buck converter can operate in continuous

and discontinuous conduction mode. In CCM

the current is continuous and flows through

the entire cycle. In DCM the current flows

only during a part of the cycle, there is a time

interval when the current is zero, and only

starts to rise at the beginning of the new cycle.

Boundary of the continuous and discontinuous

operation mode is called critical operation.

Operation in CCM. Figure 3.6 shows the

Buck converter and the equivalent circuits of

the Buck converter for CCM, both for the time

interval when the S is ON and D is OFF

(Figure 3.6/b), and for the interval when S is

OFF and D is ON (Figure 3.6/c).

At time t=0 the switch is turned on by the

driver. The voltage across the diode is

ID VV −= , thus it becomes reverse biased. The

voltage across the inductor is VL=VO-VI,

therefore the current of the inductor begins to

rise with a slope of L

VV IO −. The inductor

current Li flows through the switch, SL ii = .

Energy is transferred from the DC input to the

inductor, capacitor and the load during this

interval. At time t=DT the driver turns off the

switch. Because of the non-zero current of the

Figure 3.6/a – The Buck converter

Figure 3.6/b – Equivalent circuit for CCM – The

switch is ON and the diode is OFF

Figure 3.6/c – Equivalent circuit for CCM – the

switch is OFF and the diode is ON

32

inductor the current keeps flowing through the

load resistor in the same direction after the

switch-OFF of the transistor, as the current of

the inductor is a continuous function of time.

The inductor acting like a current source, turns

on the diode, thus the voltage across the

switch is VI, and the voltage across the

inductor is –VO. The current of the inductor

begins to decrease linearly with a slope of

L

VO− . The input voltage source is

disconnected from the circuit for this time

interval, thus no energy is delivered from the

input source to the load. During the time

interval when the switch is OFF, the energy

reservoir formed from the inductor and the

capacitor maintains the load voltage. At time

Tt = the switch is turned on again, hence

energy increases. The transistor is turned on at

a high voltage and the switch voltage

waveform is rectangular, accordingly the

PWM converters are operated at hard

switching. The input voltage is converted to a

square wave at the input of the L-C-RL circuit.

The L-C-RL circuit works as a second order

low-pass filter, and converts the square wave

into a low ripple DC voltage. The average

output voltage equals to the average of the

square wave because the average voltage in

the inductor is zero for steady state. Varying

the MOSFET gate-to-drive voltage duty cycle,

the pulse width and through it the ON-time of

the switch S can be controlled. The average

value of the PWM voltage is almost

independent of the load for CCM operation, it

depends on the duty cycle (and the input

voltage):

IO VDV ⋅= , (3.20)

Theoretically the duty ratio is varied between

0 and 100%, and thus the output voltage

between 0 and VI. Practically, due to the

resolution the actual value of D is varied

Figure 3.7 – Buck converter in CCM

33

between 5 and 95%. The DC input VI may

vary over time while the output voltage should

be kept at a fixed value, therefore the value of

D is controlled so that when VI increases D

decreases, and vice versa. This way the

average value of the square wave IVD ⋅ is

held constant. The duty cycle is controlled by

a relatively complex control circuit.

The current of the inductor contains an AC

component independent of the load current in

CCM. The DC component of the current

equals to the load current. Because the DC

output current flows through the inductor,

only one half of the B-H curve of the ferrite

core is exploited, the inductor should be

designed such that the core will not saturate. A

core with an air gap and an appropriate large

volume may be necessary.

B. Analysis of the Buck Converter for CCM

The following will be assumed in the analysis

below:

• The power MOSFET and the diode are

ideal components

• The transistor output capacitance, the

diode capacitance, and the lead

inductances are zero, and thus

switching losses are neglected.

• Passive components are linear, time-

invariant, and frequency-independent.

• The output impedance of the input

voltage source VI is zero for both dc

and ac components.

• The converter is operating in steady

state.

• The switching period T=1/fS is much

shorter than the time constants of

reactive components.

Time interval TDt ⋅≤<0 . During this

period the diode is reverse biased. The voltage

across the inductor is

dt

diLVVv L

OIL ⋅=−= . (3.21)

The current through the inductor equals to the

current flowing through the switch. It is

expressed in Equation 3.22.

In Equation 3.22 )0(Li is the initial current in

the inductor L at time t=0.

The peak inductor current becomes

)0()(

)( LOI

L iL

TDVVDTi +⋅⋅−= , (3.23)

The peak-to-peak ripple current of the

inductor L is expressed in Equation 3.24.

The diode voltage is

ID Vv −= , (3.25)

The peak value of the reverse voltage of the

diode:

IDM VV = , (3.26)

The peak value of the switch current:

2L

OSM

iII

∆+= , (3.27)

The increase of the stored magnetic energy in

the inductor during the time interval

TDt ⋅≤<0 :

34

[ ])0()(2

1 22)( LLinL iDTiLW −⋅=∆ (3.28)

At the time t=DT the switch is turned off by

the driver.

Time interval TtDT ≤< . During this

interval the semiconductor switch S is OFF

and the diode D is ON. Figure 3.6/c shows the

equivalent circuit of the Buck converter for

this interval.

The current of the inductor is not zero at the

time DT when the switch turns on, and the

current of the inductor iL is a continuous

function of time, the inductor acts like a

current source, and turns the diode on. While

the current through the switch and the voltage

across the diode is zero, the voltage across the

inductor L is:

dt

diLVv L

OL =−= , (3.29)

The current through the inductor L and the

diode D is expressed from Equation 3.30:

)()( DTiDTtL

VL

O +−− 3.31)

In the previous equation iL(DT) is the initial

condition of the inductor L at t=DT. The peak-

to-peak ripple current of the inductor L is

=−=−=∆L

DTVTiDTii O

LLL

)1()()(

Lf

DV

S

O

⋅−= )1(

(3.32)

In CCM the peak-to-peak value of the

inductor current ripple Li∆ is independent of

the load current IO and depends only on the

input voltage VI and thereby on the duty cycle

D. For a certain output voltage VO the

maximum output current ripple occurs at the

maximum input voltage, which is present at

the minimal duty cycle.

Lf

DVi

S

OL ⋅

−=∆ )1( minmax (3.33)

The switch voltage vS and the peak switch

voltage VSM are equal:

ISMS VVv == (3.34)

The peak diode and switch currents are the

same:

2L

OSMDM

iIII

∆+== (3.35)

The driver turns on the switch at t=T.

The magnetic energy stored in the inductor is

decreased during this interval. The decrease is


)0()0()0(1

00

LOI

L

tOI

L

t

LSL itL

VVidt

L

VVidtv

Lii +−=+−=+⋅== ∫∫ (3.22)

( )Lf

DDV

Lf

DVV

L

DTVViDTii

S

I

S

OIOILLL ⋅

−⋅=⋅

−=−=−=∆ )1()()0()( (3.24)

35

[ ])()(21 22

)( TiDTiLW LLoutL −⋅=∆ (3.36)

For steady state operation, the magnetic

energy decrease expressed by Equation 3.36

equals to the energy increase expressed by

Equation 3.28 ( )()( inLoutL WW ∆=∆ ).

Device Stresses for CCM. The maximum

voltage and current stresses of the

semiconductor components can be calculated.

The voltage stress:

axDMSM VVV Immaxmax == (3.37)

The current stress calculation can be seen in

Equation 3.38.

DC voltage transfer function for CCM. The

voltage transfer function of the lossless Buck

converter:

DI

I

V

VM

O

I

I

ODCV ==≡ (3.39)

DCVM ranges from 0 to 1.

The DC current transfer function:

DI

IM

I

OIDC

1== (3.40)

C. Buck converter in DCM

In this section the equations of the

Dicontinuous conduction mode will not be

derived, only the final forms are mentioned.

Letters used in this section:

VDCM DC voltage transfer function of the

converter

Sf Switching frequency

L Inductance in the circuit

D Duty cycle

LR DC load resistance

maxSMI Maximum of the peak switch

current

Figure 3.8 – Buck converter in DCM operation

36

maxDMI Maximum of the peak diode

current

max.IV Maximum of the DC input voltage

of the converter

minLR Minimum value of load resistance

maxL Maximum inductance for DCM

operation

maxBD maximum duty cycle at the

CCM/DCM boundary

DC voltage transfer function for DCM:

L

S

VDC

RD

LfM

2

811

2

++= (3.41)

for L

SVDC R

LfM

21−≤ .

Device stresses for DCM:

Lf

DVViII

S

OILDMSM

minmax.maxmaxmax

)( −=∆==

(3.42)

Maximum inductance for DCM:

S

BL

f

DRL

2

)1( maxminmax

−= (3.43)

The waveforms for DCM are shown in Figure

3.8.

IV. THE BOOST CONVERTER [11]

In this section the operation of the Boost

converter is detailed. The waveforms for both

Continuous Conduction Mode and


presented and derived. The voltage and

current stresses of the components are

calculated. Transfer functions of the converter

are given for both modes. The boundary

between CCM and DCM is defined.


The circuit diagram of the Boost converter is

shown in Figure 3.9/a.

The circuit diagram of the Boost converter

contains four elements. These are a

semiconductor switch (S), a diode (D), an

inductor (L) and a filter capacitor (C). The

resistor RL represents a DC load. The

semiconductor switch is a MOSFET in this

case. The diode D is a freewheeling diode. For

steady state the output voltage of the boost

converter is always higher than the input

voltage level, the circuit “boosts” the input

voltage to a higher level. Therefore it is a step-

up converter.


)()()()(1

DTiTDtL

VDTidt

L

VDTidtv

Lii L

OL

T

DT

OL

T

DT

LDL +−−=+−=+⋅== ∫∫ (3.30)

Lf

DVI

Lf

DVVI

IIII

S

OO

S

OaxO

LODMSM 2

)1(2

)(2

minmax

minImmax

maxmaxmaxmax

−+=−+==∆+== (3.38)

37

circuit with a switching frequency of

TfS /1= . The Duty cycle is D=tON/T

The driving of the transistor is simple because

the gate of the MOSFET is referenced to

ground.

The Boost converter can operate in continuous

and discontinuous conduction mode. The

Boost converter cannot operate at ∞=LR in

DCM because the filter capacitor has no way

to discharge. Boundary of the continuous and

discontinuous operation mode is called critical

operation.

Operation in CCM. Figure 3.9 shows the

Boost converter and the equivalent circuits of

the Buck converter for CCM, both for the time

interval when the S is ON and D is OFF

(Figure 3.9/b), and for the interval when S is

OFF and D is ON (Figure 3.9/c).



OD VV −= , thus it becomes reverse biased. The

voltage across the inductor is VL=VI, therefore

the current of the inductor begins to rise with a

slope of LVI / . The inductor current Li flows

through the switch, SL ii = .

Energy is transferred from the DC input to the

inductor, the magnetic energy increases. At

time t=DT the switch turns OFF. Because of

the non-zero current of the inductor the

current starts to flow through the diode, the

capacitor, and the resistor. The voltage across

the inductor is 0>−= OIL VVv , hence, the

inductor current decreases with a slope of

LVV OI /)( − . The diode current equals to the

inductor current. At time t=T, the switch turns

on again, and a new cycle begins.

The boost converter has poor ability to prevent

hazardous transients and failures. When a high

amplitude wave occurs in the input voltage the

phenomena of cycle-skip appears. It means

that because the input voltage is higher than

the output voltage the diode D stays on for

several full cycles. Because of this a large

current spike occurs on the diode which might

destroy it. A similar effect appears at the

initial start-up of the converter. The output

voltage is zero at the beginning, thus the

Figure 3.9/a – The Boost converter



Figure 3.9/c – Equivalent circuit for CCM – the switch

is OFF and the diode is ON

38

output voltage is lower than the input voltage

until the steady state is reached. To protect the

converter a diode needs to be connected in the

circuit. Its anode is connected to the input

source VI and its cathode is connected to the

output filter capacitor. With the capacitor and

the additional diode forming a peak rectifier,

the current flows from the input of the

converter to the output of the converter

through the additional diode. When the output

voltage is higher than the input voltage the

additional diode becomes reverse biased, and

normal converter operation begins.

B. Analysis of the Boost Converter for

CCM

The following will be assumed in the analysis

below:


ideal components









and ac components.


period the switch S is ON, the diode is reverse

biased. The voltage across the switch vS and

the diode current are zero. The voltage across

the inductor is

dt

diLVVv L

OIL ⋅=−= . (3.44)

Figure 3.10 – Boost converter in CCM

39



expressed in Equation 3.45.

In Equation 3.45 )0(Li is the initial current in

the inductor L at time t=0.


)0()( LI

L iL

TDVDTi +⋅⋅= , (3.46)


inductor L is expressed in Equation 3.47. For

fixed values of fS, L, and VO:

)21( DLf

V

dD

id

S

OL −⋅

=∆, (3.48)

Setting this derivative zero, the maximal value

of Li∆ occurs at D=0.5:

Lf

Vi

S

OL 4max =∆ (3.49)

As D increases the current ripple also

increases until D becomes 0.5, then it

decreases to zero between 0.5 and 1.


OD Vv −= , (3.50)

The average value of the inductor current IL is

equal to the dc input current I I . The peak

value of the switch current

212LOL

ISM

i

D

IiII

∆+−

=∆+= , (3.51)

The increase of the stored magnetic energy in


TDt ⋅≤<0 :

[ ])0()(2

1 22)( LLinL iDTiLW −⋅=∆ (3.52)


the driver.



and the diode D is ON. Figure 3.9/c shows the

equivalent circuit of the Boost converter for

this interval. The switch current and the diode

voltage are zero. The inductor discharges

during this interval. The inductor voltage is:

0<=−=dt

diLVVv L

OIL , (3.53)


diode D is expressed from Equation 3.54. In

this equation iL(DT) is the initial value of the

inductor current at time t=DT.

)()( DTiDTtL

VVL

OI +−− (3.55)


)0()0(1

)0(1

00

LI

L

t

IL

t

LSL itL

VidtV

Lidtv

Lii +=+=+⋅== ∫∫ (3.45)

Lf

DDV

Lf

DV

L

DTViDTii

S

O

S

IILLL ⋅

−⋅⋅=⋅

==−=∆ )1()0()( (3.47)

)()()(1

)(1

DTiTDtL

VVDTidtVV

LDTidtv

Lii L

OIL

T

DT

OIL

T

DT

LDL +−−=+−=+⋅== ∫∫ (3.54)

40

inductor L is

=−−=−=∆L

TDVVTiDTii IO

LLL

)1)(()()(

Lf

DDV

S

O

⋅−⋅⋅= )1(

(3.56)

Where )1( DVV OI −⋅= .

The voltage across the switch S is given by

SMOS VVv == (3.57)

The peak current of the diode and the switch is

given by:

212LOL

ISMDM

i

D

IiIII

∆+−

=∆+== (3.58)

This expression for the worst case is:

212maxmaxmax

max.maxmaxLOL

ISMDM

i

D

IiIII

∆+−

=∆+==

(3.59)

This time interval ends at t=T when the switch

is turned on by the driver.


decreasing during this interval. The decrease

is expressed by the following equation:

[ ])()(21 22






DC voltage transfer function for CCM. The

voltage transfer function of the lossless Buck

converter:

DI

I

V

VM

O

I

I

ODCV −

==≡1

1 (3.61)

DCVM ranges from 1 to ∞ .

The DC current transfer function:

DI

IM

I

OIDC −== 1 (3.62)

C. Boost converter in DCM


Discontinuous Conduction Mode will not be




converter



D Duty cycle



current


current


of the converter



operation


CCM/DCM boundary


41

2

211

2

Lf

RD

M S

L

VDC

++= (3.63)

for 12

3

−≥

VDC

VDC

S M

M

Lf

RL.


Lf

DViII

S

ILDMSM

maxmin.maxmaxmax =∆== (3.64)

ODMSM VVV == maxmax (3.65


≥−

<−

=

3

1

2

)1(3

1

2

)1(

2maxmaxmin

2minminmin

max

Dforf

DDR

Dforf

DDR

L

S

BBL

S

BBL

(3.66)


3.12.

VI. THE BUCK-BOOST CONVERTER

[11]

In this section the circuit diagram of the Buck-

Boost converter is explained. The waveforms

for both Continuous Conduction Mode and


presented and derived. Transfer function of

the converter is given for both modes. The

boundary between CCM and DCM is defined.

Voltage and current ripple calculations are

presented in detail.


The circuit diagram of the Buck-Boost

converter is shown in Figure 3.10/a.

The Buck-Boost converter circuit contains

four elements. These are a MOSFET switch

(S) operated as a controllable switch, a diode

(D), an inductor (L) and a filter capacitor (C).

The resistor RL represents a DC load.


circuit with a switching frequency of

TfS /1= . The Duty cycle of the PWM is

TtD ON /= .

Figure 3.11/a – The Buck-Boost converter





42

It is difficult to drive the transistor, because

the gate of the MOSFET is not referenced to

ground, the gate drive is floating.

The Buck-Boost converter can operate in

continuous and discontinuous conduction

mode. Boundary of the continuous and

discontinuous operation mode is called critical

operation.

Let us consider operation in CCM. Figure

3.10 shows the Buck-Boost converter and the

equivalent circuits of the Buck converter for

CCM, both for the time interval when the S is

ON and D is OFF (Figure 3.10/b), and for the

interval when S is OFF and D is ON (Figure

3.10/c).



)( OI VV +− thus it becomes reverse biased.

The voltage across the inductor is VI and gives

rise to a linear increase in the inductor current

with a slope of VI/L.

At time t=DT the driver turns off the switch.

The diode turns on, and the inductor drives

current through the circuit. The voltage across

the inductor is -VO. This causes the decrease

of the current with a slope of -VO/L. The

voltage across the switch is VI+VO.

At time Tt = the switch is turned on again,

hence energy increases, a new cycle begins.

B. Analysis of the Buck-Boost Converter

for CCM

The analysis will be done with the following

assumptions:


ideal components



Figure 3.12 - Boost converter in DCM operation

43







and ac components.


period the diode is reverse biased. The voltage

across the inductor is

dt

diLVv L

IL ⋅== . (3.67)



expressed in Equation 3.68. )0(Li is the initial

current in the inductor L at time t=0.


Lf

DVi

L

TDVDTi

S

IL

IL ⋅

⋅=+⋅⋅= )0()( , (3.69)


inductor L is expressed as:

Lf

DV

L

DTViDTiDTi

S

IILLL ==−=∆ )0()()(

(3.70)

Figure 3.13 – Buck-Boost converter in CCM

)0()0(1

)0(1

LO

L

T

DT

IL

T

DT

LDL itL

VidtV

Lidtv

Lii +−=+=+⋅== ∫∫ (3.68)

44


D

V

MVVVv O

VDCOOID −=

+⋅−=+−= 1

1)( ,

(3.71)

The peak value of the switch current:

212)(LOL

IOpeakLSM

i

D

IiIIII

∆+−

=∆++== ,

(3.72)

The increase of the magnetic energy stored in


TDt ⋅≤<0 :

[ ])0()(2

1 22)( LLinL iDTiLW −⋅=∆ (3.73)


the driver.



and the diode D is ON. Figure 3.10/c shows

the equivalent circuit of the Buck converter

for this interval.

The current of the inductor is not zero at the

time DT when the switch turns on, and the

current of the inductor iL is a continuous

function of time, the inductor acts like a

current source, and turns the diode on. While

the current through the switch and the voltage

across the diode is zero, the voltage across the

inductor L is:

dt

diLVv L

OL =−= , (3.74)


diode D is expressed in Equation 3.75.

In the previous equation iL(DT) is the initial

condition of the inductor L at t=DT. The peak-

to-peak ripple current of the inductor L is

=−=−=∆L

DTVTiDTii O

LLL

)1()()(

Lf

DV

S

O

⋅−= )1(

(3.76)

The switch voltage vS and the peak switch

voltage VSM are equal:

D

VVVVv O

OISMS =+== (3.77

The peak diode and switch currents are the

same:

212)(LOL

IOpeakLDM

i

D

IiIIII

∆+−

=∆++==

)0()()()(1

)(1

LS

IOL

T

DT

OL

T

DT

LDL iLf

DVTDt

L

VDTidtV

LDTidtv

Lii ++−

−=+−=+⋅== ∫∫ (3.75)

212max

max

maxmaxmax.maxmaxmax

LOLIOSMDM

i

D

IiIIII

∆+−

=∆++≈= (3.79)

Lf

DV

D

IiIIII

S

OOLIODMSM 2

)1(

12max

max

maxminmax.maxmaxmax

−+−

=∆++== (3.82)

45

(3.78)

The maximum value of the peak currents are

expressed in Equation 3.79. The maximum

DC input current occurs at Dmax while the

maximum peak-to-peak ripple current of the

inductor occurs at Dmin.

The driver turns on the switch at t=T.


decreased during this interval. The decrease is


[ ])()(21 22






Device Stresses for CCM. The maximum

voltage and current stresses of the

semiconductor components can be calculated.

The voltage stress:

minmax.maxmax D

VVVVV O

OIDMSM =+== (3.81)

The current stress calculation can be seen in

Equation 3.82.

DC voltage and current transfer function

for CCM. The DC voltage transfer function

can be calculated from the following

expression:

D

D

I

I

V

VM

O

I

I

OVDC −

==≡1

(3.83)

The DC current transfer function is

D

D

I

IM

I

OIDC

−== 1 (3.84)

C. Buck-Boost converter in DCM


Discontinuous Conduction Mode will not be




converter



D Duty cycle

Figure 3.14/a – The Buck-Boost converter





46



current


current


of the converter



operation


CCM/DCM boundary

minD The minimum duty cycle

maxSMV Maximal peak voltage of the

switch

maxDMV Maximal peak voltage of the diode


Lf

RDM

S

LVDC 2

⋅= (3.85)

for L

S

R

LfD

21−≤ .


Lf

DViII

S

ILDMSM

minmax.maxmaxmax =∆== (3.86)

OIDMSM VVVV +== max.maxmax (3.87)


S

BL

f

DRL

2

)1( 2maxmin

max

−= (3.88)


3.15.

Figure 3.15 – Buck-Boost converter in DCM

operation

47

CHAPTER 4

DESCRIPTION OF THE MEASUREMENT KIT

The practical part of the task was to execute

measurements concerning some basic

applications of PWM. These simple examples

are the three basic types of DC/DC converters.

An additional task occurred during the

semester that meant investigation of the single

phase power inverter circuit. To execute these

measurements I used the devices and

measurements of the Department. These were

mainly the elements of the LEYBOLD

DIDACTIC measurement kit, but several

other instruments were also needed. This

chapter gives an overview of the devices and

instruments that are necessary to execute these

measurements.

I. LEYBOLD DIDACTIC GMBH

MEASUREMENT KIT

This section gives an overview of the

operation and functionality of the elements

designed by LEYBOLD DIDACTIC GMBH

for educational purposes. In the next chapter

(Chapter 5 – Measurements and evaluation)

the necessary elements are listed at the

beginning of all sections.

The following list contains all the elements

used:

• Stabilized Power Supply ±15V

(No. 72686)

• Reference Variable Generator

(No. 73402)

• Control Unit PWM/PFM

(No. 735341)

• MOSFET (No. 73542)

• IGBT (No. 735346)

• Diode (No. 73502)

• Panel of different loads

(No. 73509)

A. Stabilized Power Supply [12]

The device supplies the system with the

necessary DC voltage ±15V, it has a 0V safety

socket. It is a stabilized power supply, the

output voltage ripple is low. Figure 4.1 shows

the power supply, the numbers refer to the

following:

1. Mains switch, illuminated

2. Mains fuses M 1.0

3. Safety sockets, (+15 V DC) for tapping

of the output voltage

4. Safety sockets, (0V) for tapping of the

output voltage +15 V DC

5. Safety sockets, (–15 V DC) for the

tapping of the output voltage

6. LED, for monitoring the output voltage

+15 V DC

48

7. LED, for monitoring the output voltage

–15 V DC

8. Connection socket

Technical data:

• Input voltage: 230 V AC,

50...60 Hz

• Output voltage: ±15 V DC/3 A

– stabilized, short-circuit proof

• Fuse: Mains fuse M 1.0

• Output: 8 safety sockets, 4 mm

• Connection: Mains connection

cable with earthing-pin plug

B. Reference Variable Generator [13]

The reference variable generator provides the

necessary adjustable DC voltage for the

Control Unit PWM/PFM that controls the

semiconductor switches (these are the

MOSFET and the IGBT in this case). With a

knob we can adjust the output voltage of this

reference variable generator. Its scale allows

us to change the control voltage with a

minimal value of 0.5V. There can be smaller

changes in the control voltage, but in that case

the control voltage cannot be precisely read

from the panel.

Its input voltage is ±15V and its output

voltage varies between ±10V.

The operation of this panel (the numbers

refer to Figure 4.2):

If switch S1 (1) is in the lower position, the

following signal may be tapped at output (2):

1. With a bridging plug connected to

(3): – 10 V to + 10 V

2. With a bridging plug connected to

(7): 0 V to + 10 V

If switch S1 (1) is in the upper position, a

reference voltage or reference variable may be

connected to input (6). If connections (5) and

(7) are plugged in, then positive step changes

from 0 V to the set end value are generated by

the switch (1). Positive or negative outgoing

step functions can be supplied with connection

(5) and (3). The voltage value is determined

by the setting of the reference variable

generator (4).

Figure 4.1 – Stabilized Power Supply

(No.72686)

49

Output (2) is buffered.

C. Control Unit PWM/PFM [14]

The control unit provides the control signal for

the semiconductor switches. The unit can

generate the control signal three different

ways, these are:

• Pulse Width Control

• Pulse Frequency Control

• Two-Position Frequency Control

Pulse Width Control. Using this type of

control, the width of the pulse (the ON-time -

or the OFF-time - of the switch) is varied,

while the time period T (and thus the

switching frequency f) is held constant. Pulse

Width Control is the easiest control method to

operate and can be used with every switching

controller.

Pulse Frequency Control. In this type of

control tON the pulse duration is kept constant,

while the tOFF and thus the time period T and

the switching frequency f is varied. This type

of control should be used only with low-

setting control. However, it should be noted

here that, usually, low operating frequencies

demand complex application of smoothing

elements if pulsating current is to be avoided.

Two-position Frequency Control. This type

of control is often used with load current or

load voltage closed-loop control. The

corresponding ON or OFF switching pulse is

supplied by the controller as soon as the

current or voltage actual value leaves the

acceptable tolerance range. This type of

control can only be used in conjunction with a

low setting controller.

The control unit consists of four fully

independent elements, the pulse width

modulator, the pulse frequency modulator, the

two-position controller, and the output

amplifier. The numbers used in the description

of these independent elements refer to Figure

4.3.

Pulse Width Modulator. A control voltage is

applied at control input (1). It is only sensible

to set an amplitude between 0 and 10V. This

Figure 4.2 – Reference Variable Generator

(No. 73402)

50

amplitude determines the mark-space ratio

(duty cycle) of the square-wave voltage

present at output (9). The duty cycle then has a

value of 0..90%. The pulse frequency can be

set roughly using rotary switch (4) and finely

adjusted using potentiometer (5).

Pulse Frequency Modulator. The control

voltage at (1) also determines the frequency of

the PFM. The 0V value corresponds to the

minimum frequency of 20Hz, while the 10V

value corresponds to the maximum frequency

of 20kHz. The pulse duration tON is set

roughly using rotary switch (6) and finely

adjusted using potentiometer (7). The square-

wave voltage is present at output (10)

The Two-Position Controller. The desired

value of the quantity to be controlled is

present at input (1). Usually, this is achieved

using the set point potentiometer 734 02. The

actual value of the controlled quantity is fed

back to input (2). If necessary, the actual value

must be adjusted to the required amplitude

using voltage divider 20:1 734 20 or the offset

adjust 734 19. (These instruments are not

available at the Department.)

The pulse at output (11) is switched off when

the actual value is larger than the desired value

by the amount of the hysteresis set using

potentiometer (8). When the actual value is

less than the desired value by the amount of

the hysteresis, the pulse is switched on at

output (11). The pulse settles in relation to the

time overload constants.

Output Amplifier. The input of the output

amplifier is connected with one of the three

outputs described previously via a bridging

plug. The signals are then inverted by the

upper amplifier and supplied to the output (13)

via the transformer, in the case of the lower

amplifier, the signals are transformed non-

inverted to output (15). The respective

switching states are displayed with LEDs (14)

and (16). In order to be able to transform long

pulses with a trigger pulse transformer, the

signal is chopped with approx. 60kHz and

rectified again on the secondary side. Because

of this the output voltage has a relatively

strong ripple, which is, however, not a

disturbance factor.

If the control input HNI (3) is connected to

ground, the pulses are inhibited. The voltage

at output (15) then continuously amounts to

0V. Thus output (13) is always switched on.

Figure 4.3 – The Control Unit PWM/PFM

51

For this reason output (13) is primarily used

for triggering the gate turn-off thyristor, i.e.

for the turn-off pulse of the GTO thyristor. In

order to do this output (15) and (13) must be

connected in antiparallel configuration.

D. MOSFET [15] [16]

This element is operated in switching-mode.

As a semiconductor switch it is controlled by

the Control Unit PWM (No. 735341). The

MOSFET is shown in Figure 4.4.

Essential features of the MOSFET:

• High switching capacity

• Easy parallel connection of several

transistors to increase capacity

• Extremely short switching times

• Adjustable switching time

• Linear characteristics

• Very high cut-off frequency

• High current and voltage stability

• High pulse stability (no “second

breakdown”)

• No storage time

Application of MOSFETs

MOSFETs are particularly suitable for

applications requiring fast switching at low

with very few external triggering components

and power levels. The control power depends

on the circuit output. Switched-mode power

supplies are the main application field of

MOSFETs. High cut-off power MOSFETs

(500-1000V) are used for switched-mode

power supplies, and low cut-off power

MOSFETs (100-200V) are used for DC

choppers fed with a direct voltage.

E. IGBT [15] [16]

This element is operated in switching-mode.

As a semiconductor switch it is controlled by

the Control Unit PWM (No. 735341). The

IGBT is shown in Figure 4.5.

Essential features of the IGBT:

• Relatively high switching frequency

(30kHz)

• Higher current carrying capacity

• Goon ON-state response

• Triggering requirements are very low

• Very good utilization of the chip area

Figure 4.4 – Metal-Oxide Semiconductor Field-

Effect Transistor (MOSFET)

52

• Excellent ruggedness and tolerance of

overloads

Application of IGBTs

These switches are used medium to high

power applications, such as switched-mode

power supplies, traction motor control and

induction heating. IGBTs have achieved great

commercial success in switching applications,

particularly in the field of drive technology.

F. Diode

The component is a diode which conducts

current through the circuit in the OFF-state of

the switch. It is a fast recovery diode. Its

maximal current is 11A, and the maximal

voltage stress is 1000V.

G. Panel of different loads

This panel contains three resistors of

resistance R=100Ω, a resistor of R=1000 Ω,

three capacitors (C1=4µF, C2=8µF, C3=16µF),

and two inductors of an inductance of 50mH.

These can be connected in series or in parallel.

A picture of this panel is seen in Figure 4.6.

II. OTHER INSTRUMENTS USED

Some other equipment needed to be used to

execute the measurement. These are described

below.

Figure 4.5 – Insulated Gate Bipolar Transistor

(IGBT)

Figure 4.7 – The panel of different loads

Figure 4.6 - Diode

53

A. Digital Oscilloscope

A digital scope was used to save pictures of

the waveforms. The scope pictures in this

thesis were captured by a Digilent

Technologies digital scope. It has a USB

connection, with that the pictures could be

saved directly to a pen drive.

B. Analog Voltmeter and Ammeter, Digital

Multimeter

These instruments were used to measure the

DC voltage in case of the DC/DC converters

and the effective value in case of the inverter.

C. Voltage Divider Probe

This device is used to connect high voltages to

the oscilloscope. The largest scale on the

scope is 5V/div, which is not enough to

display voltages as high as 150-180V. This

instrument divides the voltage value by 10,

thus a voltage value of 180V will be only 18V

on the scope. The device can be seen on

Figure 4.8.

D. Current Clamp

A current clamp was used to generate a

voltage waveform proportional to the current

flowing in cables, thus the Continuous and

Discontinuous Conduction Modes could be

analyzed. From the magnetic field of the

current flowing in the cable this device

generates a voltage waveform that shows the

shape of the current through the wire. This

voltage value can be connected to the scope.

The two settings are 100mV/A and 10mV/A.

Figure 4.9 shows the device working.

E. Isolation Transformer

For security reasons the scope was connected

to the mains voltage through an isolation

transformer.

Figure 4.8 – Voltage Divider

Figure 4.9 – Fluke Current Clamp

54

CHAPTER 5

MEASUREMENTS AND EVALUATION

The measurements done as a practical part of

this thesis were aimed at finding actual tasks

for the laboratory measurements of the

subject of Selected Chapters of Electrical

engineering. The devices of the department

are designed and manufactured by Leybold

Didactic GmbH. The main task was to

execute and document measurements

available with the equipment of the

department, and to suggest possible

additional elements which could improve

and widen the range of executable

measurements.

I. THE BUCK CONVERTER

The first task was to execute measurements

concerning the Buck (or step-down)

converter. The circuit diagram of the Buck

converter is shown in Figure 5.3, while

Figure 5.2 shows a photo of the actual

Leybold devices and the wiring required for

the Buck converter.

The Buck converter circuit investigated in

this measurement consists of six Leybold

elements: a stabilized power supply -15 to

+15 V (No. 72686), a reference variable

generator (No. 73402), a control unit

PWM/PFM (No. 735341), a MOSFET (No.

Figure 5.2 – Wiring of the Buck converter

Buck converterOutput voltage with respect to the duty cycle

(digital multimeter)

0

3

6

9

12

15

0% 20% 40% 60% 80% 100%

Duty Cycle [%]

Ou

tpu

t vo

ltag

e [V

]

Figure 5.1 – Buck converter output voltage

55

73542), a diode (No. 73502), and a block

containing different kinds of loads –

inductors, resistors, capacitors (No. 73509).

The detailed description of the elements used

during this measurement is found in the

previous chapter (Chapter 4 – Description of

the measurement kit).

During the measurements I used a MOSFET

as the semiconductor switch, because of its

good properties at high speeds. As load an

inductance of 50mH, a capacitance of 8µF,

and three resistances of 100Ω connected in

parallel making up a resultant resistance of

Ω3.33 & were used.

The resistor is operating as load resistor, the

capacitance and the inductor are creating an

output filter. The control unit is operated as a

PWM generator.

The measurement tasks and the measurement

results will be presented in the following

sections.

A. The Minimal and the Maximal Voltage

The first task investigating the Buck

converter was to analyze the output voltage

with an oscilloscope. During the

measurement I used a digital scope, so that

the pictures of the waveforms can be easily

saved and used.

Figure 5.3 shows the waveform of the output

voltage for different duty cycles.

It is to be mentioned, that the voltage ripple

is relatively high in the output voltage. The

minimal value of the voltage is found at

D=0%, and the maximal value can be found

at around D=95%, the maximal stable duty

cycle of the control unit PWM. The maximal

and minimal values of the voltage are:

VU 0min ≈

VU 15max ≈

Table 2 of APPENDIX B shows the output

voltage values measured with a digital

device with respect to the duty cycle.

Letters used:

Vc [V] Control signal value in volts.

D [-] Duty cycle in percentage.

Vo [V] Output voltage

In case of the Buck converter, the maximal

voltage is not a function of the PWM

frequency, its value is always slightly below

15V.

B. The voltage across the diode

The next task was to examine the diode

voltage with the digital scope with respect to

the duty cycle. In case of the Buck converter

the diode voltage is the reverse of the output

Figure 5.3 – The Buck converter

56

voltage, because the average voltage on the

inductor equals to zero for the entire period.

(See Chapter 3 – Essential DC/DC

converters)

Table 1 of APPENDIX B contains the

measured voltage values.

According to Table 1 of APPENDIX B we

can state, that the diode voltage is roughly

the same as the output voltage of the

converter.

Figure 5.4 shows the diode voltage

waveforms with respect to the duty cycle.

Figure 5.5 shows the diagram of the diode

voltage as a function of the duty cycle. The

function looks like the function of the output

voltage.

C. The boundary between CCM and DCM

The next step was to find the boundary

between CCM and DCM. The measurements

revealed that there is no Discontinuous

Conduction Mode at 500Hz and above.

Therefore the frequencies at which the task

could be executed ranged from 20 to 500Hz.

A frequency of 100Hz was chosen for the

investigation.

Figure 5.6 shows discontinuous conduction

mode, the boundary between the two modes,

and continuous conduction mode.

Figure 5.4 – Diode voltage (The duty cycle

decreases from the top to the bottom.)

57

D. Output voltage in case of DCM

The point of this measurement was to check

the output voltages with respect to the duty

ratio for Discontinuous Conduction Mode,

and then compare it to the results of the

calculation of output voltage with the help og

the duty cycle. In the next step these results

are compared to the measurements done at a

higher frequency, where no Discontinuous

Conduction Mode is present. The output

voltage is calculated with the original

equation of the output voltage

ICCMc VDV ⋅=− , (5.1)

Table 3 of APPENDIX B shows the

measured voltage Vm, the Duty cycle D, and

the calculated output voltage Vc.

Figure 5.7 shows the graph of the calculated

and the measured voltage values. The

measured values are significantly lower than

the calculated in case of DCM.

Table 3 of APPENDIX B shows that the

error reaches 7% at some voltage levels,

which is not acceptable. The error in optimal

case is around 1-2%. The equation for the

CCM is not suitable for frequencies at which

DCM is present.

Buck converterDiode voltage with respect to the duty cycle (digital

multimeter)

0

3

6

9

12

15

0% 20% 40% 60% 80% 100%

Duty Cycle [%]

Dio

de

volt

age

[V]

Figure 5.5 – Diode voltage in buck converter

Figure 5.6 – Buck converter - DCM, the boundary, and

CCM at 200Hz PWM frequency

58

In case of DCM, the output voltage is also a

function of the load current and the

inductance.

The output voltage can be calculated from

Equation 5.2.

12

1

2 +⋅⋅

⋅⋅=

TVD

ILVV

I

OIO (5.2)

E. Output voltage in case of CCM

The measurements showed that there is no

DCM above around 500Hz. In case of the

DC/DC converters the applied frequency of

the PWM is far above this value, usually

between 25 and 500kHz. It is in most cases

above 20kHz to eliminate audible noise

effects. Therefore the operation in CCM is

more common in DC/DC converter

applications. Lower frequencies are applied

e.g. in motor control, where because of the

inertia of the rotor (and the inductance of the

DC motor) higher frequencies are not

acceptable.


measured and the calculated voltage values

with respect to the duty cycle. The table also

contains the calculated errors in percentage.

Figure 5.8 shows the graph of the calculated

and the measured voltages.

Examining the error percentage values, it can

be noticed, that in the range of D=45..75%

the output voltage is more precise, than in

the case of DCM. The relative error in this

section is around 2% (or even lower), while

in the case of DCM the relative error is

constantly around 5% (or above).

Nonetheless it must be acknowledged, that

an undoubtedly large error occurs in both

cases for some values of the duty cycle. The

Buck converter (Measured and calculated voltage for CCM)

0

1,5

3

4,5

6

7,5

9

10,5

12

13,5

15

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Duty cycle [%]

Vol

tage

[V]

Measured voltage Calculated voltage

Figure 5.8 – Measured and calculated voltage for

CCM

Buck converter (Measured and calculated voltage for DCM)

0,0

3,0

6,0

9,0

12,0

15,0

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Duty cycle [%]

Vo

ltag

e [V

]

Measured voltage Calculated voltage

Fig. 5.7 – Measured and calculated voltage for DCM

59

reason of this might be the inner errors of the

devices used, and the instability of the input

voltage, or inappropriate value of the filter

capacitance.

II. BOOST CONVERTER

The second setup was the Boost converter

circuit. The Boost (or step-up) converter’s

output voltage is always greater or equal to

the input voltage. The circuit diagram of the

Boost converter is shown in Figure 5.9. A

photo of the assembly of the Leybold devices

and the wiring is shown in Figure 5.10. The

suggested wiring from the Leybold Didactic

guide is found in Appendix B/2.

The following elements of the Leybold

measurement kit were used in this

measurement:


(No. 72686)


(No. 73402)


(No. 735341)


• Diode (No. 73502)


(No. 73509)


during this measurement is found in Chapter

4 – Description of the measurement kit.

Figure 5.10 – Wiring of the Boost converter

Figure 5.11 - Output voltage of the Boost

converter

(C=28µF, L=50mH, R=1000Ω, fPWM=20Hz

60

MOSFET was used during this measurement

as well. The load consisted of an inductance

of 50mH, a capacitance of 28µF, and a load

resistor of 1000Ω. These values were

suggested by the guide of the PWM device.

The resistor is operating as load resistor, the

capacitance and the inductor are creating an

output filter. The control unit is operated as a

PWM generator. A voltage divider probe is

used in this measurement, because of the

high voltages. The oscilloscope’s maximal

scale is 5V/scale, thus the voltage value

connected to the scope is divided by 10 with

this instrument.

A. The Minimal and the Maximal Voltage

The first task was to examine the output

voltage waveform with the help of a digital

scope. One of the waveforms is shown in

Figure 5.11.

The second part of this was to find the

minimal and maximal value of the output

voltage. Theoretically, the minimal value of

the output voltage equals to the input

voltage, and this value is found at D=0. The

maximal value of the output voltage is very

high, the voltage transfer function of the

boost converter for D=1 is infinity in theory.

Practically, the maximal value can be

measured, but its value is highly frequency-

dependent. The efficiency of the boost

converter is poor above D=0.9, and the

maximal voltage is usually found at

D=0.95..0.97.

Table 5 of APPENDIX B shows the maximal

voltages for different frequencies.

Figure 5.12 shows the maximal voltage and

the voltage at D=1 with respect to the

frequency on a logarithmic and on a linear

scale. At low and high frequencies the

maximal voltages are lower than in the case

of frequencies between 100Hz and 2000Hz.


The aim of this section was to analyze the

voltage waveform across the diode. In case

of the Boost converter during the ON state of

the switch the diode is reverse biased, during

the OFF state of the semiconductor switch

the diode conducts current in the circuit.

Figure 5.14 shows the waveforms of the

diode voltage.

The voltage stress of the diode is high in this

type of converters.

Figure 5.9/a – The Boost converter

61

C. The boundary between DCM and CCM

and output voltages

In case of the Boost converter there is no

DCM above 200Hz for the applied loads.

The next task was to measure the output

voltage with respect to the duty cycle for the

two modes of operation. As the filter

capacitance has a smoothing effect on the

output voltage, I measured for two different

capacitors.

Measurements were executed both for a

frequency where there is DCM, and for

another frequency, where there is no DCM.

(These frequencies are 50Hz for DCM, and

Boost converterOutput voltage for different frequencies and capacitances

0

30

60

90

120

150

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Duty cycle [%]

Vo

ltag

e [V

]

f=50Hz, C=0.008mF

f=50Hz, C=0.028mF

f=20kHz, C=0.008mF

f=20kHz, C=0.028mF

Figure 5.13 – Boost converter – Output voltage at two different frequencies and two different capacitors.

Boost converterMaximal votlage with respect to the frequency

0

20

40

60

80

100

120

140

160

180

0 2500 5000 7500 10000 12500 15000 17500 20000

Frequency [Hz]

Vo

ltag

e [V

]

Voltage at D=1 Maximal voltage (D=0.95..0.98)

Figure 5.12/a – Maximal voltage and voltage at D=1

for different frequencies (Linear scale)

Boost converterMaximal votlage with respect to the frequency

0

20

40

60

80

100

120

140

160

180

10 100 1000 10000 100000

Frequency [Hz] (Logarithmic scale)

Vo

ltag

e [V

]

Voltage at D=1 Maximal voltage (D=0.95..0.98)

Figure 5.12/b – Maximal voltage and voltage at D=1 for different frequencies (Logarithmic scale)

62

20kHz for only CCM). The results can be

seen in Table 7 of APPENDIX B.

In Table 7 of APPENDIX B there is a row

marked with “max”. This means the maximal

reachable voltage at an uncertain value of the

control voltage (and with that an uncertain

value of the duty cycle). This duty cycle

values are usually between 95% and 98%.

Figure 5.13 shows the diagrams of the

voltage values presented in Table 7 of

APPENDIX B. It can be seen, that at 50Hz

(so at lower frequencies) the capacitance of

the capacitors has greater effect on the output

voltage than in the case of higher PWM

frequencies. Thus operation at higher

frequencies reduce the size of the inductors

and capacitors required, therefore the

converters are smaller, which is one of the

most important requirements today.

III. BUCK-BOOST CONVERTER

The final practical task was the

measurements of the Buck-Boost (or step-up

step-down) converter. Figure 5.15 shows the

circuit diagram of the Buck-Boost converter,

and Figure 5.16 is a picture of the devices

and the wiring.

For the Buck-Boost converter circuit the

following Leybold devices are necessary:


(No. 72686)

Figure 5.14 – Diode voltage (The duty cycle

decreases from the top to the bottom.)

Figure 5.15 – The Buck-Boost converter

63


(No. 73402)


(No. 735341)


• Diode (No. 73502)


(No. 73509)


during this measurement is found in the

previous chapter (Chapter 4 – Description of

the measurement kit).

The semiconductor switch is a MOSFET in

this case as well. As a load an inductance of

50mH, a capacitance of 8µF, and a resistance

of 1000Ω were used. The resistor is

operating as load resistor, the capacitance

and the inductor are creating an output filter.

The control unit is operated as a PWM

generator.

Because of the high output voltage an

additional instrument, a voltage divider

probe is necessary, so that only ten percent

of the output voltage is connected to the

digital scope. (Its maximal scale of 5V is not

able to display voltages this high, there for

the voltage divider probe divides the output

voltage by 10).

A. The Maximal and the Minimal

Voltages

In the case of the Buck-Boost converter, the

output voltage can be higher or lower than

the input voltage. The transfer function for

D=0 is 0, and for D=1 infinity in theory. In

practice, the minimal value is zero, and the

output voltage can be relatively high:

VU 0min =

VU 160max =

The efficiency of the converter is poor if

D>85%..90%.

The maximal voltage is the function of the

PWM frequency in the case of the Buck-

Boost converter. Table 9 of APPENDIX B

Figure 5.16 – Wiring of the Buck-Boost converter

64

shows the voltage values at D=100% and the

maximal voltages with respect to the duty

cycle. The maximal voltages usually occur

between D=95% and D=98%.


Figure 5.17 shows the waveforms of the

voltage across the diode for different duty

cycles. In the ON-state of the switch the

diode is reverse biased, and in the OFF-state

of the switch the diode conducts the current

of the inductor. A high voltage stress occurs

in this kind of converters.

Table 6 of APPENDIX B shows the voltage

values across the diode for three different

PWM frequencies. The maximal voltage is

usually found at D=0.95..0.98, in Table [-] of

APPENDIX B the expression “max” means

this uncertain value of the duty cycle, where

the diode voltage is maximal

C. Boundary of CCM and DCM

The task was to find the boundary between

Continuous and Discontinuous Conduction

Modes for each frequency. In case of the

Buck-Boost converter there is no CCM

below 100Hz, and no DCM above 2kHz.


boundary between CCM and DCM for each

frequency. In Table [-] of APPENDIX B the control

voltage is proportional to the duty cycle,

Figure 5.17 – Diode voltage for different Duty

cycles

65

Uc=8V means a duty cycle of 80%. To

understand the working of the Leybold

Didactic control unit PWM see Chapter 4 –

Description of the measurement kit.

D. Output voltage in case of CCM and in

case of DCM

The point of this section is to examine the

load voltage for the two types of operation.

For DCM operation a PWM frequency of

50Hz was chosen, a PWM frequency of

20kHz was suitable for measurements in

Continuous Conduction Mode. Table 10 of

APPENDIX B shows the measurement data

acquainted at 50Hz for two different values

of the capacitance. Table 10 of APPENDIX

B also contains the measurement data for

20kHz and for two different values of the

capacitance.

Figure 5.19/a-b shows the voltages for both

50Hz and 20kHz in diagrams. The

calculation of the output voltage for CCM:

D

DVV IO −

⋅=1

(5.3)

Where VO is the output voltage, VI is the

input voltage, and D is the duty cycle.

In case of DCM the output voltage can be

calculated from

Lf

RDVV

S

LIO ⋅⋅

⋅⋅=2

, (5.4)

where VI is the input voltage, VO is the

output voltage, D is the duty cycle, fS is the

switching frequency, and L is the inductance.

As seen in Figure 5.18 the measured voltages

are not accurate, the voltage values are

higher than the calculated voltage below

D=85%, and much lower above this duty

Buck-Boost converterMaximal votlage with respect to the frequency

0

20

40

60

80

100

120

140

160

180

0 2500 5000 7500 10000 12500 15000 17500 20000

Frequency [Hz]

Vo

ltag

e [V

]

Voltage at D=1 Maximal voltage

Figure 5.18/b – Maximal voltage and voltage at D=1

for different frequencies (Linear scale)

Buck-Boost converterMaximal votlage with respect to the frequency

0

20

40

60

80

100

120

140

160

180

10 100 1000 10000 100000

Frequency [Hz] (Logaritmic scale)

Vo

ltag

e [V

]

Voltage at D=1 Maximal voltage

Figure 5.18/a – Maximal voltage and voltage at D=1

for different frequencies (Logarithmic scale)

66

cycle value. The capacitance has a larger

effect on the output voltage at lower

frequencies.

IV. THE SINGLE PHASE POWER

INVERTER

The single phase power inverter circuit is

shown in Figure 5.20

The circuit requires an inductor (L=50mH), a

capacitor (C=8µF), a resistor (R=33Ω), two

MOSFETs, the Control Unit PWM and a

signal generator. The input of this circuit is a

sinusoidal DC voltage with its value varying

between 0 and 10V. The frequency of this

voltage is 50Hz. The inverter circuit has an

output of ±15V AC. The input voltage is

generated by a function generator, Figure

5.21/a shows the input voltage. (The CSV

file was saved by the digital scope, then it

was saved in MATLAB as a figure.)

The task was to analyse the amplitude of

different harmonics in the response of the

inverter. The output voltage values were

saved in a CSV file (comma separated

values), and then imported into MATLAB.

In MATLAB the Fourier transform of the

signals was executed. Measurements were

done at different frequencies with different

kind of loads (ohmic, ohmic-capacitive,

ohmic-capacitive-inductive).

Figure 5.22 shows the output voltage

(effective value) as a function of the PWM

frequency on a logarithmic scale. The graph

also contains the DC part of the voltage.

Buck-Boost converter Measured voltages at 50Hz and calculated voltage

0,0

50,0

100,0

150,0

200,0

250,0

300,0

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Duty cycle [%]

Vo

ltag

e [V

]

Measured voltage (C=0.008mF) Measured voltage (C=0.028mF)

Calculated voltage

Figure 5.19/a – Measured voltages at 50Hz PWM

frequency and the calculated voltage

Buck-Boost converter Measured voltages at 20kHz and calculated voltage

0

50

100

150

200

250

300

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Duty cycle [%]

Vo

ltag

e [V

]

Measured voltage (C=0.008mF) Measured voltage (C=0.028mF)

Calculated voltage

Fig. 5.19/b – Measured voltages at 20kHz PWM

frequency and calculated voltage.

Figure 5.20 – Simple single phase power inverter

67

APPENDIX C contains several figures of

output voltages and the amplitude spectrum

of them.

In general we can say that with a full load

(inductance, capacitance, resistance) the

amplitude spectrum is smoother, higher

harmonics are decreased. Also, at higher

PWM frequencies the amplitude of higher

harmonics are decreasing.

Figure 5.21/a – Input voltage of the inverter

Figure 5.21/b – Amplitude spectrum of the input voltage of the iverter

Output voltage as a function of the PWM frequency

-6,00

-4,00

-2,00

0,00

2,00

4,00

6,00

8,00

10,00

10 100 1000 10000 100000

Pulse Width Modulation frequency [Hz]

Lo

ad v

olt

age

(ou

tpu

t vo

ltag

e) [

V]

Figure 5.22 – Effective value of the output voltage and the DC component of it

68

CHAPTER 6

CONCLUSION AND ACKNOWLEDGEMENT

The assigned task. The aim of the project

was to execute basic measurements with

several DC/DC converters as an application

of Pulse Width Modulation, and to write a

measurement guide for MSc students.

All of the assigned tasks were executed

successfully. CHAPTER 5 contains the

measurement results, the measurement guide

can be found in APPENDIX A.

An additional task occurred during the

semester. It was the investigation of the level

of different harmonics at various frequencies

of the pulse-width modulation in case of the

single phase power inverter. Several

measurements were executed concerning this

inverter.

Several errors must be mentioned. The

measurements were not always easy to

execute, as the current and voltage ripples

had very high values, and noises had a large

effect on the measurement results. The

boundary between continuous conduction

mode and discontinuous conduction mode

was very complicated to find, especially in

case of the boost and buck-boost converters,

where higher voltage values occurred.

During these measurements a voltage divider

probe had to be used, thus further error

possibilities occurred resulting in an unstable

high-noise signal on the oscilloscope. With

these affecting the measurement, the analysis

of the critical operation was hard and

inaccurate in case of the boost and buck-

boost converters.

Future development possibilities.

According to a measurement guide written

by LD DIDACTIC GMBH, with additional

elements of the measurement kit the results

of these measurements can be smoothened.

In addition, with these new elements several

other circuits can be measured, for example

the Flyback converter, the Forward

converter, the Half-bridge, and the Full-

bridge converters. Other measurements can

be executed concerning DC/AC inverter

circuits and AC/DC rectifier circuits. With

these elements the range of the executable

measurements of this subject can be

widened, and the elements can be used in

other subjects, for example the subject

Power Electronics of the BSc students.

According to this development idea, we

requested a price list of these elements so

that with these prices the possibilities of the

department become clear.

These elements and their prices are listed

below:

69

Name Cat. No Net Price [HUF]

Required quantity

Rectifier 735 065 53200 1 Power Transformer 735 105 171600 1 Electrolyte capacitor 735 095 75480 1 RMS meter 727 10 353100 1 Fuse, three-fold, super-fast 735 18 54040 1 Interference suppression filter 735 190 94460 1 Isolation amplifier 735 261 489220 1 Transformer 45/90 726 80 249980 1

Sum 1541080

Voltage divider probes, current clamps and

other accessories used during the

measurements were not designed by LD

DIDACTIC GMBH. It is to be considered,

that these measurements might be more

precisely executed with the original

instruments designed by the manufacturer of

the components of the measurement kit.

However, the prices of these instruments are

relatively high, so it would require a large

investment to improve the quality of the

results, the sum of the net price of the listed

devices is 1,541,080 Hungarian Forints.

I would like to thank the department for the

instruments I used and Károly Zabán for his

help during the execution of the

measurements and his support during the

whole semester. Without his assistance the

completion of my work would not have been

possible.

70

REFERENCES

[1] Fang Lin Luo; Hong Ye, “Essential DC/DC converters” ISBN 0-8493-7238-0 ©2006 by Taylor & Francis Group, LLC, pp.1-2

[2] http://www.hydrogencarkits.net/pwm-controllers/

[3] http://en.wikipedia.org/wiki/Pulse-width_modulation

[4] http://www.myo-p.com/

[5] John Guy, “Class D amplifier FAQ” © 2009 National Semiconductor Corporation, originally posted to: http://www.audiodesignline.com/howto/212000761

[6] Jun Honda; Jonathan Adams, “Class D Audio Amplifier Basics”, Application Note AN-1071, International Rectifier, 5 August 2005, http://www.irf.com

[7] M K Venkatesha; KA Krishamurthy, ”Study and design of new inverter/converter thyristor circuits for various purposes”, a thesis submitted to the University of Mysore, 10 April 1997, Chapter 2 – ”Review Of Various PWM Techniques” pp.22-51

[8] Dorin O. Neacsu ”Power Switching Converters Medium and High Power”, ISBN 0-8247-2625-1 ©2006 by Taylor & Francis Group, LLC, pp.75-151

[9] Jaroslav Dudrik, Juraj Oetter ”High-Frequency Soft-Switching DC-DC Converters for Voltage and Current DC Power Sources”, Acta

Polytechnica Hungarica, Vol. 4, No. 2, 2007, pp.29-44

[10] Graham Holmes; D Grahame Holmes; Thomas A Lipo, “Pulse Width Modulation for Power Converters: Principles and Practice”, IEEE Computer Society Press, ISBN 0471208140, 31 January 2004, pp.57-61

[11] Marian K. Kazimierczuk, “Pulse-width Modulated DC-DC Power Converters” ISBN 978-0-470-77301-7 (HB) ©2008 John Wiley & Sons, Ltd

[12] LD DIDACTIC Instruction Sheet: http://www.leybold-didactic.de/ga/7/726/72686/72686E.PDF

[13] LD DIDACTIC Instruction sheet: http://www.leybold-didactic.com/ga/7/734/73402/73402e.pdf

[14] LD DIDACTIC Instruction sheet: http://www.leybold-didactic.de/ga/7/735/735341/735341de.pdf

[15] LD DIDACTIC

”Power Electronics: Switched-mode power supplies, power-factor correction and inverters”, T12.2.2.2, ID:565 402

[16] Dorin O. Neacsu ”Power Switching Converters Medium and High Power”, ISBN 0-8247-2625-1, ©2006 by Taylor & Francis Group, LLC, Chapter 2 – ”High-Power Semiconductor Devices”, pp.19-37

[17] http://www.investopedia.com/terms/c/cagr.asp

71

BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS Faculty of Electrical Engineering and Informatics Department of Automation and Applied Informatics

Group of Electrical Engineering

Measurement guide

INVESTIGATION OF DC-DC CONVERTERS

72

Compulsory literature:

1. dr. JÁRDÁN, Rafael Kálmán: Topics for POWER ELECTRONICS &MOTION CONTROL I., p. 37-38 and p. 40-47. http://get.bme.hu/edu/subjects/BMEVIAUA017/Lecture%20Notes%20PE1/PE-1-Lecture%20Notes.pdf

2. Descripton titled „Investigation of DC-DC converters

3. Leybold Didactic instruction sheet

1. Aim of the measurement:

The point of the measurement is to get acquainted with the operation of the 3 main types of DC-DC converters. We measure the output voltage and the current in the circuit.

2. Theoretical background:

Converters are electric devices that convert a certain type of electric property into another. Power converters are divided into 4 main types:

• AC-DC converters (choppers): convert alternating current to direct current

• DC-AC converters (inverters): convert direct current to alternating current

• DC-DC converters: modifies a certain voltage level of a DC supply to another voltage

level

• AC-AC converter: modifies the amplitude or frequency of an AC supply

During this measurement we will get acquainted with the 3 basic types of DC-DC converters:

• Buck converter: the output voltage is lower than the input voltage

• Boost converter: produces voltage increase, output voltage is higher than the input

voltage

• Buck/Boost converter: the output voltage can be higher as well as lower than the input

voltage

The most important component of all three types is a semiconductor switch which can be

GTO, IGBT or MOSFET. During the laboratory we will use MOSFET and IGBT. We control

the switches with PWM (Pulse Width Modulation). The main point of PWM:

We compare a control signal to a reference signal (e.g. triangle wave). This way we get a

series of constant cycle time pulses. If the control signal is higher than the reference signal,

than the value of the pulse is 1, if the control signal is lower, this value is 0. The ratio of the

73

duration of the switch ON state and the switching period TS=1/fS is the duty cycle. This value

can be manipulated through setting the control voltage value.

Figure 1 – Generation of the PWM signal

Figure 2 – PWM signal

Duty cycle: OFFON

ONON

tt

t

T

tD

+==

The panel Nr. 735 341 can generate the switching control signal two more ways beyond

PWM: control by frequency modulation and two-position frequency control. These methods

will not be used during this measurement.

74

Buck converter:

Buck converters decrease the input voltage level; the output voltage is always lower than the input. Fig. 3 shows the buck converter in both OFF and ON state of the switch.

Figure 3/a – The Buck converter in the switch-ON state of the switch

Figure 3/b – The Buck converter in the switch-OFF state of the switch

As you can see in this figure if we turn off the switch, the energy stored in the coil drives current through the diode. If the switch is turned off for enough time, the current falls to zero and only rises again when we turn on the switch. This is called Discontinuous Conduction Mode (DCM). If we do not wait till the current fully stops and turn on the switch earlier, then we are talking about Continuous Conduction Mode (CCM).

Fig. 4 shows the characteristics for both continuous and discontinuous conduction mode. The connection between the input and the output voltage in case of CCM:

io UDU ⋅=

−oU Output voltage

75

−D Duty cycle

−iU Input voltage

Boost converter

Boost converters produce higher voltage on the output than the input voltage. Fig. 5 shows the boost converter both ON and OFF state.

Figure 5/a – The Boost converter in the ON-state of the switch

Figure 5/b – The Boost converter in the OFF-state of the switch

Just as in the case of the Buck converter we can talk about Continuous and Discontinuous Conduction Mode (Fig. 6)

The connection between the input and the output voltage in case of Continuous Conduction Mode:

⋅=

OFFio t

TUU

76

−oU Output voltage

−iU Input voltage

−T Switching period

−OFFt OFF state duration in a cycle

Buck-Boost converters

In case of the Buck-Boost converter the output voltage can be both higher and lower than the input voltage. Fig. 7 shows the Buck-Boost converter in both ON and OFF state

Figure 7/a – The Buck-Boost converter in the ON-state of the switch

Figure 7/b – The Buck-Boost converter in the OFF-state of the switch

Just as in the case of the Buck and the Boost converters we can talk about Continuous and Discontinuous Conduction Mode (Fig. 8).

The connection between the input and the output voltage in case of CCM:

77

⋅=

OFF

ONio t

tUU −oU Output voltage

−iU Input voltage

−ONt ON state duration

−OFFt OFF state duration

3. Review Questions

1. What kind of converter types do you know? What do we use them for?

2. List the basic types of DC/DC converters.

3. What is the point of PWM? What do we use it for?

4. What is duty cycle? How can we change its value?

5. Draw the circuit of the Buck converter. How does it work? What is the relation between the

output voltage and the input voltage?

6. What do we call Continuous and Discontinuous Conduction Mode? Is the output voltage

smaller or greater than the input voltage?

7. What is the relation between the output voltage and the duty cycle in case of the Buck

converter?

8. Draw the circuit of the Boost converter. How does it work? Is the output voltage smaller or

greater than the input voltage?

9. What is the relation between the output voltage and the duty cycle in case of the Boost

converter?

10. Draw the circuit of the Buck-Boost converter. How does it work? Is the output voltage

smaller or greater than the input voltage?

11. What is the relation between the output voltage and the duty cycle in case of the Buck-

Boost converter?

78

4. Measurement Tasks

Always have your measurement setups checked by your measurement leader.

1. Build a circuit of a semiconductor switch (MOSFET) and a resistor connected in series. Examine the control signal of the semiconductor switch and the voltage drop on the resistor with an oscilloscope. What happens if we change the voltage level of the control signal? Calculate the duty cycle for three different voltage levels leaving the PWM signal’s frequency at a constant value.

The frequency of the PWM signal: f=…………

=1U =1D

=2U =2D

=3U =3D

2. Now change the frequency of the PWM signal. What happens? Calculate the duty cycle for three different frequency values leaving the voltage of the control signal at a constant level.

Voltage level of the control signal: U=…………

=1f =1D

=2f =2D

=3f =3D

3. Set up the Buck converter circuit shown in Fig. 3 (the PWM generator is connected like it is shown in appendix 1). Ω= 1000R FC µ28= mHL 500= Examine the voltage drop on the output with an oscilloscope. What do you experience if you change the duty cycle? The voltage varies between two values:

=minU

=maxU

4. Examine the voltage drop on the diode with the help of an oscilloscope. What happens when you change the duty cycle?

5. Find the voltage value at which the system is on the boundary between Continuous and Discontinuous Conduction Mode using 100Hz frequency PWM signal.

U100Hz=

fPWM=

79

6. Set the frequency to a level at which there is discontinuous conduction mode. Now measure the output voltage for ten different duty cycles. In these same measurement points calculate the output voltage with the duty cycle and the input voltage.

=PWMf ……………

Vm [V] D [%] V calc [V]

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

7. Repeat the tasks mentioned in point 6 for a frequency level at which there is no discontinuous conduction mode. What happens?

=PWMf ……………


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

8. Set up the Boost converter circuit shown in Fig. 5 (the PWM generator is connected like it is shown in appendix 2). Ω= 1000R FC µ28= mHL 500= Examine the voltage drop on the output with an oscilloscope. What do you experience if you change the duty cycle? The voltage varies between two values:

=minU

=maxU

9. Examine the voltage drop on the diode with the help of an oscilloscope. What happens if you change the duty cycle?

80


U100Hz=

fPWM=


=PWMf ……………


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

12. Repeat the tasks mentioned in point 11 for a frequency level at which there is no discontinuous conduction mode. What happens?

=PWMf ……………

Vmin [V] D V calc [V]

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

81

13. Set up the Buck-Boost converter circuit shown in Fig. 7 (the PWM generator is connected like it is shown in appendix 3). Ω= 1000R FC µ28= mHL 500= Examine the voltage drop on the output with an oscilloscope. What do you experience if you change the duty cycle? The voltage varies between two values:

=minU

=maxU

14. Examine the voltage drop on the diode with the help of an oscilloscope. What happens when you change the duty cycle?


U100Hz=

fPWM=


=PWMf ……………

Vmin [V] D V calc [V]

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

82

17. Repeat the tasks mentioned in point 11 for a frequency level at which the conduction mode is not discontinuous. What happens?

=PWMf ……………

Vmin [V] D Vcalc [V]

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

18. Simulate the converters using the Electronics Workbench software.

83

Table 3 – Buck converter - Measured and calculated voltage values for DCM

Vm [V] D [-] V c [V] Error [%]

0,0 0% 0,00 0,0% 0,1 5% 0,75 4,3% 0,6 10% 1,50 6,0% 1,2 15% 2,25 7,0% 1,9 20% 3,00 7,3% 2,7 25% 3,75 7,0% 3,5 30% 4,50 6,7% 4,2 35% 5,25 7,0% 4,9 40% 6,00 7,3% 6,0 45% 6,75 5,0% 6,5 50% 7,50 6,7% 7,3 55% 8,25 6,3% 8,0 60% 9,00 6,7% 8,8 65% 9,75 6,3% 9,5 70% 10,50 6,7%

10,1 75% 11,25 7,7% 11,0 80% 12,00 6,7% 12,0 85% 12,75 5,0% 12,5 90% 13,50 6,7% 13,5 95% 14,25 5,0% 14,0 100% 15,00 6,7%

APPENDIX B

MEASUREMENT DATA

Table 1 –Buck converter Diode voltage with respect to the duty cycle (digital multimeter)

Vc [V] D[%] V D [V]

0,0 0% 0,00 0,5 5% 1,22 1,0 10% 4,45 1,5 15% 7,20 2,0 20% 9,15 2,5 25% 10,69 3,0 30% 11,82 3,5 35% 12,66 4,0 40% 13,26 4,5 45% 13,68 5,0 50% 13,98 5,5 55% 14,23 6,0 60% 14,40 6,5 65% 14,58 7,0 70% 14,70 7,5 75% 14,79 8,0 80% 14,84 8,5 85% 14,90 9,0 90% 14,88 9,5 95% 14,84

10,0 100% 14,82

Table 2 – Buck converter - Output voltage with respect to the duty cycle (digital multimeter

Vc [V] D [%] Vo [V]

0,0 0% 0,00 0,5 5% 1,12 1,0 10% 4,36 1,5 15% 7,06 2,0 20% 9,17 2,5 25% 10,66 3,0 30% 11,81 3,5 35% 12,65 4,0 40% 13,25 4,5 45% 13,65 5,0 50% 14,00 5,5 55% 14,22 6,0 60% 14,42 6,5 65% 14,57 7,0 70% 14,69 7,5 75% 14,77 8,0 80% 14,84 8,5 85% 14,89 9,0 90% 14,87 9,5 95% 14,84

10,0 100% 14,81

Table 4 – Buck converter - Measured and calculated voltage values for CCM

Vm [V] D [%] V c [V] Error [%]

0 0% 0,00 0,00% 2 5% 0,75 8,33%

2,75 10% 1,50 8,33% 3,4 15% 2,25 7,67%

4 20% 3,00 6,67% 4,6 25% 3,75 5,67% 5,3 30% 4,50 5,33% 5,9 35% 5,25 4,33% 6,5 40% 6,00 3,33% 7,1 45% 6,75 2,33% 7,8 50% 7,50 2,00% 8,4 55% 8,25 1,00% 9,1 60% 9,00 0,67% 9,7 65% 9,75 0,33%

10,5 70% 10,50 0,00% 11 75% 11,25 1,67%

11,5 80% 12,00 3,33% 12 85% 12,75 5,00% 13 90% 13,50 3,33%

13,5 95% 14,25 5,00% 14 100% 15,00 6,67%

84

Table 5 – Maximal output voltage of the Boost converter for different frequencies

fPWM [Hz] V D=1 [V] Vmax [V]

20 125 122 30 122 120 50 129 142

100 131 155 200 147 160 250 144 160 300 143 161 500 141 161

1000 140 160 2000 137 159 3000 133 157 5000 130 152

10000 123 146 20000 115 126

Table 7 – Ooutput voltage of the Boost converter for different frequencies and capacitors

Table 6 – Buck-Boost converter – Voltage across the diode with respect to the duty cycle for different

frequencies

Letters used: D Duty cycle in percentage.

VD Diode voltage in volts.

fPWM PWM frequency in Hz.

VD

D [%] fPWM= 300Hz

fPWM= 2kHz

fPWM= 20kHz

0% 0,0 0,0 0,0 5% 1,3 2,4 9,1

10% 4,4 3,0 11,5 15% 7,2 3,7 13,5 20% 11,0 4,5 15,0 25% 14,5 5,5 16,5 30% 17,8 6,7 18,5 35% 21,0 8,2 20,5 40% 24,2 9,7 23,0 45% 27,5 12,0 26,0 50% 30,5 14,5 29,5 55% 34,0 17,5 32,0 60% 38,0 21,5 37,0 65% 40,0 26,0 43,0 70% 44,0 32,0 50,0 75% 46,0 40,0 59,0 80% 48,0 52,0 69,0 85% 64,0 68,0 84,0 90% 90,0 93,0 98,0 95% 135,0 135,0 115,0

100% 142,0 130,0 95,0

max 160,0 160,0 115,0

Table 8 – Buck-Boost converter – The boundary between CCM and DCM

fPWM [Hz] UC [V]

100 Hz and below

no CCM

200 8,5

250 8,0

300 8,0

500 7,5

1000 6,0

2000 Hz and above

no DCM

85

Table 9 – Buck-Boost converter - Maximal voltage and voltage at D=1 with respect to the

PWM frequency

fPWM [Hz] VD=1 [V] V max [V]

20 107 121 25 110 121 30 110 126 50 122 148

100 140 155 200 140 158 250 133 158 300 128 157 500 121 155

1000 118 154 2000 114 149 2500 111 148 3000 108 146 5000 104 139

10000 93 127 20000 92 126

Letters used:

fPWM[Hz] The PWM frequency

VD=1 The voltage value at D=1

Vmax The maximal output voltage.

(D=0.95..0.98)

Table 10 – Buck-Boost converter - Maximal voltage and voltage at D=1 with respect to the PWM frequency

VM

fPWM=50Hz fPWM=20kHz D [%]

C=8µF C=28µF C=8µF C=28µF

VC-

CCM

0% 0,0 0,0 0,0 0,0 0,0 5% 0,0 0,0 8,0 8,1 0,8

10% 0,3 3,5 9,6 9,6 1,7 15% 7,1 8,3 11,0 11,0 2,6 20% 14,5 16,5 12,5 12,0 3,8 25% 20,0 25,0 14,0 13,5 5,0 30% 27,0 32,5 15,5 15,0 6,4 35% 34,0 41,0 17,5 17,0 8,1 40% 40,0 47,0 19,5 19,5 10,0 45% 45,0 54,0 22,0 22,0 12,3 50% 51,0 61,0 25,0 25,0 15,0 55% 56,0 67,0 28,5 28,5 18,3 60% 61,0 74,0 32,5 32,5 22,5 65% 67,0 80,0 38,0 39,0 27,9 70% 72,0 86,0 45,0 45,0 35,0 75% 76,0 92,0 54,0 54,0 45,0 80% 82,0 99,0 64,0 64,0 60,0 85% 86,0 105,0 78,0 79,0 85,0 90% 91,0 110,0 99,0 100,0 135,0 95% 96,0 120,0 125,0 130,0 285,0

100% 120,0 145,0 110,0 105,0 - max 128,0 155,0 135,0 135,0 -

Letters used:

fPWM[Hz] The PWM frequency

86

APPENDIX C

Voltage waveforms and amplitude spectrums for different PWM frequencies in case of a full load (capacitor, inductor, resistor)

Documents

FP-DCDC