Lee Julie JoAnn

8/23/2019 Lee Julie JoAnn

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ANALYSIS OF SMALL-SIGNAL MODEL OF A PWMDC-DC BUCK-BOOST CONVERTER IN CCM.

A thesis submitted in partial fulfillment

of the requirements for the degree of

Master of Science in Engineering

By

Julie J. Lee

B.S.EE, Wright State University, Dayton, OH, 2005

2007

Wright State University


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WRIGHT STATE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

August 17, 2007

I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY

SUPERVISION BY Julie J. Lee ENTITLED Analysis of Small-

Signal Model of a DC-DC Buck-Boost Converter in CCM

BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF Master of Science in Engineering

Marian K. Kazimierczuk, Ph.D.Thesis Director

Fred D. Garber, Ph.D.Department Chair

Committee onFinal Examination

Marian K. Kazimierczuk, Ph.D.

Kuldip S. Rattan, Ph.D.

Ronald Riechers, Ph.D.

Joseph F. Thomas, Jr., Ph.D.

Dean, School of Graduate Studies


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Abstract

Lee, Julie J. M.S. Egr., Department of Electrical Engineering, Wright State University, 2007.

Analysis of Small-Signal Model PWM DC-D Buck-Boost Converter in CCM.

The objective of this research is to analyze and simulate the pulse-width-modulated (PWM)

dc-dc buck-boost converter and design a controller to gain stability for the buck-boost con-

verter. The PWM dc-dc buck-boost converter reduces and/or increases dc voltage from one

level to a another level in devices that need to, at different times or states, increase or decrease

the output voltage.

In this thesis, equations for transfer funtions for a PWM dc-dc open-loop buck-boost con-

verter operating in continuous-conduction-mode (CCM) are derived. For the pre-chosen de-

sign, the open-loop characterics and the step responses are studied. The converter is simulated

in PSpice to validate the theoretical analysis. AC analysis of the buck-boost converter is per-

formed using theoretical values in MatLab and a discrete point method in PSpice. Three

disturbances, change in load current, input voltage, and duty cycle are examined using step

responses of the system. The step responses of the output voltage are obtained using MatLab

Simulink and validated using PSpice simulation.

Design and simulation of an integral-lead (type III) controller is chosen to reduce dc error

and gain stability. Equations for the integral-lead controller are given based on steady-state

and AC analysis of the open-loop circuit, with a design method illustrated. The designed

controller is implemented in the circuit, and the ac behavior of the system is presented.

Closed loop transfer fuctions are derived for the buck-boost converter. AC analysis of the

buck-boost converter is studied using both theoretical values and a discrete point method in

PSpice. The step responses of the output voltage due to step change in reference voltage, input

voltage and load current are presented. The design and the obtained transfer functions of the

PWM dc-dc closed-loop buck-boost converter are validated using PSpice.

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Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Open-Loop Buck-Boost 3

2.1 Transfer Functions for Small-Signal Open-Loop Buck-Boost . . . . . . . . . 3

2.1.1 Open-Loop Input Control to Output Voltage Transfer Function . . . . 3

2.1.2 Open-Loop Input Voltage to Output Voltage Transfer Function . . . . 13

2.1.3 Open-Loop Input Impedance . . . . . . . . . . . . . . . . . . . . . . 15

2.1.4 Open-Loop Output Impedance . . . . . . . . . . . . . . . . . . . . . 21

2.2 Open-Loop Responses of Buck-Boost using MatLab and Simulink . . . . . . 23

2.2.1 Open-Loop Response due to Input Voltage Step Change . . . . . . . 23

2.2.2 Open-Loop Response due to Load Current Step Change . . . . . . . 28

2.2.3 Open-Loop Response due to Duty Cycle Step Change . . . . . . . . 30

2.3 Open-Loop Responses of Buck-Boost Using PSpice . . . . . . . . . . . . . 32

2.3.1 Open-Loop Response of Buck-Boost . . . . . . . . . . . . . . . . . 32

2.3.2 Open-Loop Response due to Input Voltage Step Change . . . . . . . 34

2.3.3 Open-Loop Response due to Load Current Step Change . . . . . . . 34

2.3.4 Open-Loop Response due to Duty Cycle Step Change . . . . . . . . 36

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3 Closed Loop Response 38

3.1 Closed Loop Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.1 Integral-Lead Control Circuit for Buck-Boost . . . . . . . . . . . . . 42

3.1.2 Loop Gain of System . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.3 Closed Loop Control to Output Voltage Transfer Function . . . . . . 53

3.1.4 Closed Loop Input to Output Voltage Transfer Function . . . . . . . 53

3.1.5 Closed Loop Input Impedance . . . . . . . . . . . . . . . . . . . . . 57

3.1.6 Closed Loop Output Impedance . . . . . . . . . . . . . . . . . . . . 62

3.2 Closed Loop Step Responses of Buck-Boost . . . . . . . . . . . . . . . . . . 65

3.2.1 Closed Loop Response due to Input Voltage Step Change . . . . . . 65

3.2.2 Closed Loop Response due to Load Current Step Change . . . . . . . 68

3.2.3 Closed Loop Response due to Reference Voltage Step Change . . . . 71

3.3 Closed Loop Step Responses using PSpice . . . . . . . . . . . . . . . . . . . 73

3.3.1 Closed Loop Response of buck-boost . . . . . . . . . . . . . . . . . 73

3.3.2 Closed Loop Response due to Input Voltage Step Change . . . . . . 74

3.3.3 Closed Loop Response due to Load Current Step Change . . . . . . 76

3.3.4 Closed Loop Response due to Reference Voltage Step Change . . . . 79

4 Conclusion 81

4.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Appendix A 83

References 86

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List of Figures

2.1 Small-signal model of buck-boost. . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Block diagram of buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Small-signal model of buck-boost to determine Tp the input control to output

voltage transfer function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Theoretical open-loop magnitude Bode plot of input control to output voltage

transfer function Tp for a buck-boost. . . . . . . . . . . . . . . . . . . . . . 11

2.5 Open-loop phase Bode plot of input control to output voltage transfer function

Tp for a buck-boost with and without 1s delay. . . . . . . . . . . . . . . . 11

2.6 Discrete point open-loop magnitude Bode plot of input control to output volt-

age transfer function Tp for a buck-boost. . . . . . . . . . . . . . . . . . . . 12

2.7 Discrete points open-loop phase Bode plot of input control to output voltage

transfer function Tp for a buck-boost. . . . . . . . . . . . . . . . . . . . . . 12

2.8 Small-signal model of the buck-boost to determine the input to output voltage

transfer function Mv the input to output function. . . . . . . . . . . . . . . . 13

2.9 Open-loop magnitude Bode plot of input to output voltage transfer function

Mv for a buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.10 Open-loop phase Bode plot of input to output transfer function Mv for a buck-

boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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2.11 Open-loop magnitude Bode plot of input to output transfer function Mv for a

buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.12 Open-loop phase Bode plot of input to output transfer function Mv for a buck-

boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.13 Open-loop magnitude Bode plot of input impedance transfer function Zi for a

buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.14 Open-loop phase Bode plot of input impedance transfer function Zi for a buck-

boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.15 Discrete points open-loop magnitude Bode plot of input impedance transfer

function Zi for a buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.16 Discrete points open-loop phase Bode plot of input impedance transfer func-

tion Zi for a buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.17 Small-signal model of the buck-boost for determining output impedance Zo. 21

2.18 Open-loop magnitude Bode plot of output impedance transfer function Zo for

a buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.19 Open-loop phase Bode plot of output impedance transfer function Zo for a

buck-boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.20 Discrete points open-loop magnitude Bode plot of output impedance transfer

function Zo for a buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.21 Discrete points open-loop phase Bode plot of output impedance transfer func-

tion Zo for a buck-boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.22 Open-Loop step response due to step change in input voltage vi. . . . . . . . 28

2.23 Open-Loop step response due to step change in load current io. . . . . . . . 30

2.24 Open-Loop step response due to step change in duty cycle d. . . . . . . . . . 32

2.25 Open-loop buck-boost model with disturbances. . . . . . . . . . . . . . . . 33

2.26 Open-loop buck-boost response without disturbances. . . . . . . . . . . . . 33

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2.27 PSpice model of Open-Loop buck-boost with step change in input voltage. . 34

2.28 Open-Loop step response due to step change in input voltage using PSpice. . 35

2.29 PSpice model of Open-Loop buck-boost with step change in load current. . . 35

2.30 Open-Loop step response due to step change in load current using PSpice. . 36

2.31 PSpice model of Open-Loop buck-boost with step change in duty cycle. . . . 37

2.32 Open-Loop step response due to step change in duty cycle using PSpice. . . 37

3.1 Closed loop circuit of voltage controlled buck-boost with PWM. . . . . . . . 39

3.2 Closed loop small-signal model of voltage controlled buck-boost. . . . . . . 39

3.3 Block diagram of a closed-loop small-signal voltage controlled buck-boost. . 40

3.4 Simplified block diagram of a closed-loopsmall-signal voltage controlled buck-

boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Magnitude Bode plot of modulator and input control to output voltage transfer

function Tmp for a buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6 Phase Bode plot of modulator and input control to output voltage transfer

function Tmp for a buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.7 Magnitude Bode plot of the input control to output voltage transfer function

Tk before the compensator is added for a buck-boost. . . . . . . . . . . . . . 44

3.8 Phase Bode plot of the input control to output voltage transfer function Tk

before the compensator is added for a buck-boost. . . . . . . . . . . . . . . 44

3.9 The Integral Lead Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.10 Magnitude Bode plot of the controller transfer function Tc for a buck-boost. . 52

3.11 Phase Bode plot of the controller transfer function Tc for a buck-boost. . . . 52

3.12 Magnitude Bode plot of the loop gain transfer function T for a buck-boost. . . 54

3.13 Phase Bode plot of the loop gain transfer function T for a buck-boost. . . . . 54

3.14 Magnitude Bode plot of the input control to output transfer function Tcl for a

buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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3.15 Phase Bode plot of the input control to output transfer function Tcl for a buck-

boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.16 Magnitude Bode plot of the input control to output transfer function Tcl for a

buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.17 Phase Bode plot of the input control to output transfer function Tcl for a buck-

boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.18 Magnitude Bode plot of the input to output voltage transfer function Mvcl for

a buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.19 Phase Bode plot of the input to output voltage transfer function Mvcl for a

buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.20 Magnitude Bode plot of the input to output voltage transfer function Mvcl for

a buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.21 Phase Bode plot of the input to output voltage transfer function Mvcl for a

buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.22 Magnitude Bode plot of the input impedance transfer function Zicl for a buck-

boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.23 Phase Bode plot of the input impedance transfer function Zicl for a buck-boost. 63

3.24 Magnitude Bode plot of the input impedance transfer function Zicl for a buck-

boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.25 Phase Bode plot of the input impedance transfer function Zicl for a buck-boost. 64

3.26 Magnitude Bode plot of the output impedance transfer function Zocl for a

buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.27 Phase Bode plot of the output impedance transfer function Zocl

for a buck-boost. 66

3.28 Magnitude Bode plot of the output impedance transfer function Zocl for a

buck-boost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.29 Phase Bode plot of the output impedance transfer function Zocl for a buck-boost. 67

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3.30 Closed Loop step response due to step change in vi. . . . . . . . . . . . . . . 69

3.31 Closed Loop step response due to step change in io. . . . . . . . . . . . . . . 71

3.32 Closed Loop step response due to step change in vr. . . . . . . . . . . . . . 73

3.33 Closed loop buck-boost model with disturbances. . . . . . . . . . . . . . . . 74

3.34 Closed loop buck-boost response without disturbances. . . . . . . . . . . . . 75

3.35 PSpice model of Closed Loop buck-boost with step change in input voltage. 76

3.36 Closed Loop step response due to step change in input voltage using PSpice. 77

3.37 PSpice model of Closed Loop buck-boost with step change in load current. . 77

3.38 Closed Loop step response due to step change in load current using PSpice. . 78

3.39 PSpice model of Closed Loop buck-boost with step change in duty cycle. . . 79

3.40 Closed Loop step response due to step change in reference voltage using

PSpice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

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Acknowledgements

I would like to thank my advisor, Dr. Marian K. Kazimierczuk, for his guidance and input on

the thesis development process.

I also wish to thank Dr. Ronald Riechers and Dr. Kuldip S. Rattan for serving as members

of my MS thesis defense committee, giving the constructive criticism necessary to produce a

quality technical research document.

I would also like to thank the Department of Electrical Engineering and Dr. Fred D. Garber,

the Department Chair, for giving me the opportunity to obtain my MS degree at Wright State

University.


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1 Introduction

1.1 Background

Trends in the current consumer electronics market demand smaller, more efficient devices.

With the increasing use of electronic devices on the market, a demand of low power and low

supply voltages is ever increasing. The key for power management is balancing need for

less power and lower supply voltages with maintaining operational ability. Many electronic

devices require several different voltages and are provided by either a battery or a rectified

ac supply line current. However, the voltage is usually not the required, or the ripple voltage

could be to high. Voltage regulator methodology is a constant dc voltage despite changes in

line voltage, load and temperature.

Voltage regulator can be classified into linear regulators and switching-mode regulators.

Some drawbacks of linear regulators are poor efficiency, which also leads to excess heat dis-

sipation and it is impossible to generate voltages higher than the supply voltage. Switching-

mode regulators can be separated into the following categories: Pulse-Width Modulated (PWM)

dc-dc regulators, Resonant dc-dc converters, and Switched-capacitor voltage regulators. The

PWM dc-dc regulators can be divided into three important topologies: buck converter, boost

converter, and buck-boost converter. The buck-boost converter is chosen for analysis.

The PWM dc-dc buck-boost converter reduces and increases dc voltage from one level to

a another [1]-[5]. A buck-boost converter can operate in both continuous conduction mode

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(CCM), which is the state discussed, and discontinuous conduction mode (DCM) depending

on the inductor current waveform. In CCM, the inductor current flows continuously for the

entire period, whereas in DCM, the inductor current reduces to zero and stays at zero for the

rest of the period before it begins to rise again.

1.2 Thesis Objectives

The objectives of this thesis are as follows:

1. To analyze and simulate the dc-dc buck-boost converter for open-loop.

2. To design a control circuit for the buck-boost converter.

3. To analyze and simulate the dc-dc buck-boost converter for closed-loop.

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2 Open-Loop Buck-Boost

Derived small-signal open-loop transfer functions for the input control to ouput voltage

transfer fuction Tp, audio susceptibilityMv, input impedance Zi and output impedance Zo. Us-

ing the transfer functions finding the AC analysis of the transfer fuction by finding the Bode

plots. Step responses of the system are found due to a step change in input voltage vi, duty

cycle d and load current io.

2.1 Transfer Functions for Small-Signal Open-Loop

Buck-Boost

2.1.1 Open-Loop Input Control to Output Voltage Transfer

Function

A small-signal open-loop buck-boost model is shown in Fig 2.1. A block diagram of a buck-

boost converter is shown in Fig 2.2. The MOSFET and diode are replaced by a small-signal

model of a switching network (dependent voltage and current sources), the inductor is replaced

by a short and the capacitor is replaced by an open circuit. The pre-chosen measured values

of the circuit are: VI = 48 V, D = 0.407, VF = 0.7V, rDS = 0.4 , RF = 0.02 , L = 334 mH,

C= 68 F, rC = 0.033 , and RL = 14 .

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Z1

Dil

I dL

V dSD

D v v( i o)

il

RL

io

rC

vo

+

vsd+

vi

+d

L

r

C

Z2

+

+

Figure 2.1: Small-signal model of buck-boost.

Zo

Mv

d

+

+

io

vi

vo vo

Tp

vo

vo

Figure 2.2: Block diagram of buck-boost.

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Dil

I dL

V dSD

il

rC

vsd+

Z1

d

L

r

C

RL vo

+

vi = 0

io = 0

iZ2

Dvsd Dvo=

Z2

+

+

Figure 2.3: Small-signal model of buck-boost to determine Tp the input control to output volt-

age transfer function.

The dependent sources are related to duty cycle. Setting the other two inputs to zero relates

the control input to the output. This transfer function due to duty cycle affecting the output is

Tp. The derivation using Fig 2.3 ofTp is below starting from first principles of KCL and KVL.

Finding the transfer function of the plant Tp

iZ2 =vz2Z2

=vo

Z2(2.1)

Using KCL

il + iZ2Il dDil = 0

il (1D) + iZ2 Ild= 0 (2.2)

ilZ1VSDdDvsd = vo (2.3)

vsd = vo (2.4)

ilZ1 = vo +VSDdDvo

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il =vo(1D) + vsdd

Z1(2.5)

Substituting values

vo(1D) +VSDdZ1

(1D) + voZ2

ILd= 0 (2.6)

vo(1D)2Z1

++VSDd(1D)

Z1+

vo

Z2ILd= 0

vo

(1D)2

Z1+

1

Z2

= d

IL VSD(1D)

Z1

Tp vo(s)d(s)

|vo=io=0=

Il VSD(1D)Z1

(1D)2Z1

+ 1Z2

= Il

1 VSD(1D)IL

(1D)2 + Z1Z2

(2.7)

Il =ID

(1D) = Io

(1D) =vo

(1D)RL (2.8)

Using KVL

rILDvsd + vF vo = 0 (2.9)

vsd vI vF + vo = 0 vsd = vI + vF vo (2.10)

Substituting values

r

vo(1D)RL

D (vI+ vFvo) + vF vo = 0 (2.11)

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r

vo(1D)RL

+DvF o + vF vo = DvI

vo

r

(1D)Rl vIDvF

vo+ 1 +D

vF

vo

= vID

vo

r

(1D)RL + (1D)

1 vFvo

= vID

vsd = vI+ vF

vo =

vo1

vF

vo vI

vo = ILRL(1D)1

vF

vo 1

MV DC

vsd

IL=

1

D

RL(1D)

1 +

vF

|vo| + r

Tp(s) vod|vi=io=0 =

IL

1 VSD(1D)

Il

(1D)2 + Z1

Z2

Tp(s) =IL

Z1 (1D) vsdIL

Z1Z2

+ (1D)2

Tp(s) =IL

Z1 (1D) vsdIL

Z1+(1D)2Z2

Z2

=IL

Z1 (1D) vsdIL

Z2

Z1 + (1D)2Z2 (2.12)

Z1 = r+ sL (2.13)

Z2 =RL

rc +

1sC

RL + rC+

1sC

(2.14)

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vsd

IL=

1

D

RL(1D)

1 +

vF

|vo|

+ r

(2.15)

IL =vo

(1D)RL (2.16)

Tp =IL

Z1 (1D) vsdIL

Z2

Z1 + (1D)2Z2 (2.17)

DenTp = (

r+

sL) + (

1

D)

2 RL rc + 1sCRL + rC+ 1sC

DenTp =

RL + rC+

1

sC

(r+ sL) + (1D)2RL

rc +

1

sC

= RLr+ rrC+ r1

sC+ sLRL + sLrC+

L

C+ (1D)2

RLrC+

RL

sC

= sCRLr+ sCrCr+ r+ s2CLRL + s2LCrC+ sL + (1D)2 (sCRLrC+RL)

= s2 +C

r(RL + rC) + (1D)2RLrC

+L

LC(RL + rC)s +

r+ (1D)2RLLC(RL + rC)

(2.18)

NumTp = (sC) 1

LC(RL + rC)

ILRL rc + 1

sC

RL + rC+1

sC

RL + rC+

1

sC

Z1 (1D)vsd

IL

= IL (RL(sCrC+ 1) (r+ sL)) 1D

RL(1D)2

1 +

vF

|vo| + r

1

LC(RL + rC)

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= Vo(1D)RL (RL(sCrC+ 1))

1

LC(RL + rC)

(L)

s 1

DL

RL(1D)2(1 + vF|vo|) + r(12D)

NumTp = Vo((1D)RL + rC)

s +

1

CrC

s 1

DL(RL(1D)2

1 +

vF

|vo|) + r(12D)

(2.19)

=C

r(RL + rC) + (1D)2RLrC

+L

2

LC(RL + rC) [r+ (1D)2RL](2.20)

o = r+ (1D)2RLLC(RL + rC)

(2.21)

zn = 1CrC

(2.22)

zp =1

DL

RL(1D)2

1 +

vF

|vo|

+ r(12D)

(2.23)

Tp vod|vi=io=0 = Vo(1D) (RL + rC)

(s +zn)(s +zp )s2 + 2os +2o

(2.24)

Tpx = Vo(1D)(RL + rC) (2.25)

Tpo = Tp(0) =VoR

(1

D) (RL + rC)

znzp

2o

=VorC

(1D)(RL + rC)

1CrC

1

DL

RL(1D)2

1 + vF|vo|

+ r(12D)

r+(1D)2RLLC(RL+rC)

9


21/98

=VorC

(1D) (RL + rC)

RL(1D)2

1+vF|vo|

+r(12D)

CDLrC

r+(1D)2RLLC(RL+rC)

Tpo =Vo

D(1D)

RL(1D)2

1 + vF|vo|

+ r(12D)

r+ (1D)2RL

(2.26)

The Bode plot ofTp is shown in Fig 2.4 and 2.5.

The input control to output voltage transfer function Tp has a non-minimum phase system

due to the right hand plane zero. The complex pole of the system is dependent on duty cycle,

D. The Bode plots for input control to output voltage transfer function Tp is also found usingdiscrete points. Discrete points were used rather than sweeping the circuit because PSpice

sweeps are only accurate for linearized circuits. Therefore, a sinusoidal source was inserted,

and the magnitude of ripples in either the voltage or the current are used to determine the mag-

nitude of the function. Phase of the function was found my determining the time difference

between the two signals of interest. Distingushing the ripple from the noise was a challange

to be overcome. The answer was to boost the signal voltage but still maintain the small-signal

condition of the system. Therefore, for this thesis, a magnitude value of ten or less volts for

the test voltage is considered a small-signal. Most test voltages did not need to exceed five

volts to distinguish between the noise and ripple. The only one that required a higher value

is Zicl , which is caused by the MOSFET being placed in series with the sinusoidal voltage

source. In addition the MOSFET has a floating node associated with it. Figs 2.6 and 2.7 show

the discrete point Bode plots of the control input to output voltage transfer function Tp.

10


22/98

100

101

102

103

104

105

20

10

0

10

20

30

40

50

f (Hz)

|Tp

|(dBV

)

Figure 2.4: Theoretical open-loop magnitude Bode plot of input control to output voltage

transfer function Tp for a buck-boost.

101

102

103

104

105

90

60

30

0

30

60

90

120

150

180

f (Hz)

T

p()

td

= 0

td

= 1 s

Figure 2.5: Open-loop phase Bode plot of input control to output voltage transfer function Tpfor a buck-boost with and without 1s delay.

11


23/98

101

102

103

104

105

20

10

0

10

20

30

40

50

f (Hz)

|Tp

|(dBV

)

Figure 2.6: Discrete point open-loop magnitude Bode plot of input control to output voltage

transfer function Tp for a buck-boost.

101

102

103

104

105

90

60

30

0

30

60

90

120

150

180

f (Hz)

T

p()

Figure 2.7: Discrete points open-loop phase Bode plot of input control to output voltage trans-

fer function Tp for a buck-boost.

12


24/98

Z1

Dil

il

r

vi

C

+

L

r

C=

Z2

vsd+

RL vo

+

+

Zi

Z2ii

i

D v v(i o

)

Dvsd

d= 0

io = 0

Figure 2.8: Small-signal model of the buck-boost to determine the input to output voltage

transfer function Mv the input to output function.

2.1.2 Open-Loop Input Voltage to Output Voltage Transfer

Function

A small-signal model of a buck-boost converter with inputs d= 0 and io = 0 is shown in Fig

2.8.

Using this model to derive equations for the input voltage to out voltage transfer function Mv,

also known as audio susceptibility. Again, using first principles to start the derivation process.

From KCL

il Dil + iZ2 = 0 (2.27)

iZ2 =vo

Z2(2.28)

il = iZ2(1D) =

vo

(1D)Z2(2.29)

Using KVL

ilZ1 + vo +Dvsd = 0

13


25/98

ilZ1 = vo +D(vivo)

ilZ1 = Dvi + (1D)vo

voZ2(1D)Z1 = Dvi + (1D)vo

Dvi =

(1

D)vo1 + Z1

Z2(1D)2

Mv(s) vo(s)vi(s)

|d=io=0 = D

(1D)1 + Z1

Z2(1D)2

= D(1D)

1

1 + Z1Z2(1D)2

= D(1D)(1D)

2

1

(1D) + Z1Z2

Mv =(1D)DRLrC

L(RL + rC)

s +zns2 + 2os +2o

(2.30)

Mvx =(1D)DRLrC

L(RL + rC)(2.31)

Mvo = Mv(0) =

(1D)DRLrC

L(RL + rC)

zn

2o

= (1D)DRLrCL (RL + rC)

1CrC

r+(1D)2RLLC(RL+rC)

14


26/98

Mvo = (1D)DRLr+ (1D)2RL (2.32)

Fig: 2.9 and 2.10 show the theoretical Bode plots ofMv.

Using PSpice to determine certain points of interest gives the following Bode plot shown in

figures 2.11 and 2.12.

2.1.3 Open-Loop Input Impedance

Finding the input impedance Zi for the open-loop buck-boost converter circuit. Using Fig

2.8 to derive the open-loop impedance of the buck-boost small-signal model.

Using KCL

Dil il iZ2 = 0

iZ2 = Dil il

iZ2 = (1D)il (2.33)

From KVL

Z1il +D(vi vo) + vo = 0

Z1il +Dvio (1D) = 0

ilZ1 +Dvi il(1D)(1D)Z2 = 0

ilZ1 +Dvi il (1D)2Z2 = 0

Dvi = il(Z1 + (1D)2Z2)

15


27/98

100

101

102

103

104

105

90

80

70

60

50

40

30

20

10

0

10

f (Hz)

|Mv

|

(dBV

)

Figure 2.9: Open-loop magnitude Bode plot of input to output voltage transfer function Mv for

a buck-boost.

100

101

102

103

104

105

0

30

60

90

120

150

180

f (Hz)

MV

()

Figure 2.10: Open-loop phase Bode plot of input to output transfer function Mv for a buck-

boost.

16


28/98

101

102

103

104

105

90

80

70

60

50

40

30

20

10

0

10

f (Hz)

|Mv

|

(dBV

)

Figure 2.11: Open-loop magnitude Bode plot of input to output transfer function Mv for a

buck-boost.

101

102

103

104

105

0

30

60

90

120

150

180

f (Hz)

MV

()

Figure 2.12: Open-loop phase Bode plot of input to output transfer function Mv for a buck-

boost.

17


29/98

Zi =vi

ii=

(Z1 + (1D)2Z2)D2

=1

D2

r+ sL +

r+ (1D)2RLLC(RL + rC)

(1D)2

=(r+ sL)(RL + rC+

1sC) +RL(rC+

1sC)(1D)2

D2(RL + rC+1

sC)

=1

D2

rRL + rrC+

rsC+ sLRL + sLrC+

LC +RLrC(1D)2 + (1D)2RLsC

RL + rC+1

sC

=1

D2

LC(RL + rC) s2 +

C(r(rC+RL) +RLrC(1D)2) +L

s +RL(1D)2 + r

sCRL + sCrC+ 1

Zi =1

D2 (LC(RL + rC))

s2 + C(r(RL+rC)+RLrC(1D)

2+LLC(RL+rC)

s + RL(1D)2+r

LC(RL+rC)

sC(RL + rC) + 1

where

rc =1

C(RL + rC)(2.34)

Zi =L

D2s2 + 2os +

2o

s +rc(2.35)

Figs: 2.13 and 2.14 show the theoretical Bode plots ofZi.

As shown above the Magnitude of

|Zi

|decreases quickly with an increase in D.

Figs 2.15 and 2.16 show the discrete point Bode plots for Zi.

18


30/98

100

101

102

103

104

105

0

20

40

60

80

100

120

140

160

180

f (Hz)

|Zi|

(db

V)

Figure 2.13: Open-loop magnitude Bode plot of input impedance transfer function Zi for a

buck-boost.

100

101

102

103

104

105

60

30

0

30

60

90

f (Hz)

Z

i()

Figure 2.14: Open-loop phase Bode plot of input impedance transfer function Zi for a buck-

boost

19


31/98

101

102

103

104

105

0

20

40

60

80

100

120

140

160

180

f (Hz)

|Zi

|()

Figure 2.15: Discrete points open-loop magnitude Bode plot of input impedance transfer func-

tion Zi for a buck-boost.

101

102

103

104

105

60

30

0

30

60

90

f (Hz)

Z

i()

Figure 2.16: Discrete points open-loop phase Bode plot of input impedance transfer function

Zi for a buck-boost.

20


32/98

Dil

il

r

Z1

CL

r

C

Z2

+

vsd+

tv

RL vo

+

Dvsd

Dvo=

iZ2

it

Zo

ib

d= 0

vi = 0

io = 0

Figure 2.17: Small-signal model of the buck-boost for determining output impedance Zo.

2.1.4 Open-Loop Output Impedance

Solving for the transfer function of output impedance Zo. Output impedance Zo of the buck-

boost small-signal model is shown in Fig 2.17 where all three inputs d, vi, and io equal 0. A

test voltage vt with a current ofit is applied to the output of the model. The ratio ofvt to it

determines Zo.

Using KVL

Z1il Dvt + vt = 0 (2.36)

il =(1D)vt

Z1(2.37)

KCL

ib + iZ2 it = 0

ib = (1D)il (2.38)

21


33/98

(1D)il = it iZ2

(1D)2vtZ1

= it vtZ2

vt

c +

1

Z2

= it (2.39)

Zo =vt

it=

1

1Z2 + (1D)2

Z1

=Z1

Z1Z2 + (1 +D)2

=

(r+ sL)

RL(rC+

1sC)

RL+rC+1

sC

r+ sL +RL(rC+

1sC

)

RL+rC+1

sC

(1D)2

=(r+ sL)

RL

rC+

1sC

(r+ sL)

RL

rC+

1sC

+RL(1D)2

rC+1

sC=

1

LC(RL + rC)

s2 + CrRLrC+LRLLCRLrC s +rRL

LCRLrC

s2 + 2os +2o

Zo =1

LC(RL + rC)

s + rL

s + 1CrC

s2 + 2os +2o

rL =

r

L(2.40)

zn =1

CrC(2.41)

22


34/98

Zo vtit

=RLrC

(RL + rC)

(s +rL)(s +zn)

s2 + 2os +2o(2.42)

Figs: 2.18 and 2.19 show the Bode plots for Zo.

As D increases so does the magnitude of|Zo|at low frequencies.

Figs: 2.18 and 2.19 show the discrete points PSpice simulated Bode plots for Zo.

2.2 Open-Loop Responses of Buck-Boost using

MatLab and Simulink

2.2.1 Open-Loop Response due to Input Voltage Step Change

Response of output voltage vo due to a step change of 1 Volt in input voltage vi. The total

input voltage is given by equation 3.40 where u(t) is the unit step function and VI(0) is the

input voltage before applying the step voltage.

vI(t) = VI(0) +VIu(t) (2.43)

rearranging T_{p} open-loop input control to output voltage transfer function

M_{v} open-loop input to output voltage transfer function, audio suceptibility

Z_{i} open-loop input impedance transfer function

Z_{o} open-loop output impedance transfer function

vi(t) = VIu(t) = vI(t)VI(0) (2.44)

Changing from time domain to s-domain

23


35/98

100

101

102

103

104

105

0

1

2

3

4

5

6

7

f (Hz)

|Zo

|(dBV

)

Figure 2.18: Open-loop magnitude Bode plot of output impedance transfer function Zo for a

buck-boost.

100

101

102

103

104

105

90

60

30

0

30

60

f (Hz)

Z

o()

Figure 2.19: Open-loop phase Bode plot of output impedance transfer function Zo for a buck-

boost

24


36/98

101

102

103

104

105

0

1

2

3

4

5

6

7

f (Hz)

|Zo

|()

Figure 2.20: Discrete points open-loop magnitude Bode plot of output impedance transfer

function Zo for a buck-boost.

101

102

103

104

105

90

60

30

0

30

60

f (Hz)

Z

o()

Figure 2.21: Discrete points open-loop phase Bode plot of output impedance transfer function

Zo for a buck-boost

25


37/98

vi(s) = L{vi(t)} (2.45)

where the step change in s-domain is

vi(s) =VI

s(2.46)

Therefore the transient response due to a step change in vi becomes

vo(s) =v(s)

s(2.47)

= VIMvo 20

zn

s +zns (s2 + 2os +2o )

= VIMvx s +zns (s2 + 2os +2o )

(2.48)

Returning from s-domain to time domain

vo(t) = L{vo(s)} (2.49)

producing the magnitude of the transient response is

= VIMvo

1 +

1 2o

zn+

ozn

2et12 sin(dt+)

(2.50)

Where

= tan1 dzn

1 ozn

+ tan112

(2.51)The total output voltage response is

vo(t) = V(0) + vo(t) t 0 (2.52)

26


38/98

The maximum overshoot defined in equation where vo() is the steady state value of the

output voltage.

Smax = vomaxvo()vo()

(2.53)

Obtaining the derivative for equation and setting it equal to zero produces the time instants at

which the maximum ofvo occurs

vomax = VIMvo

1 +

1 2o

zn+

ozn

2e

12

(2.54)

Therefore the maximum overshoot is

Smax =

1 2o

zn+

ozn

2e12

(2.55)

The maximum relative transient ripple of the total output voltage can be defined as

max =vomax vo()

vo()(2.56)

where vo() is defined as the steady state value of the output voltage. Given the measured

values of the circuit are: VI = 48 V, D = 0.407,VF = 0.7 V, rDS = .4 , RF = 0.02 , L =

334 mH, C= 68 F, rC = 0.033, and RL = 14. These values lead to a maximum overshoot

, Smax = 35.67% and a relative transient ripple max = 1.05%.

The step change due to vi is shown in Fig: 2.22 .

27


39/98

0 1 2 3 4 5

28.8

28.7

28.6

28.5

28.4

28.3

28.2

28.1

28

t (ms)

vO

(V)

Figure 2.22: Open-Loop step response due to step change in input voltage vi.

2.2.2 Open-Loop Response due to Load Current Step Change

Response of vo due to a step change of .1 Amp in io. The total load current is given

by equation 3.40 where u(t) is the unit step function and Io(0) is the input current before

applying the step current.

Io(t) = Io(0) +Iou(t) (2.57)

Step change in the time domain is

io(t) = io(t)Io(0) = Iou(t) (2.58)

28


40/98

Changing from time domain into s-domain the step change becomes

io(s) =Io

s(2.59)

The transient component of the output voltage is

vo(s) = Zo(s)io(s) = IoRLrCRL + rC

(s +zn)(s +rl )

s (s2 + 2os +2o )

= IoZox (s +zn)(s +rl )s (s2 + 2os +2o )

(2.60)

Switching back from s-domain to time domain

vo(t) = L1{vo(s)} (2.61)

The total output voltage is

vo(t) = V(0) + vo(t) (2.62)

Again, the maximum relative transient ripple of the total output voltage can be defined as

max =vomaxvo()

vo()

where vomax is the steady state value of the total output voltage. Given the measured values of

the circuit are: VI = 48 V, D = 0.407, VF = 0.7 V, rDS = 0.4 , RF = 0.02 , L = 334 mH,

C= 68 F, rC = 0.033, and RL = 14. These values lead to a maximum overshoot, Smax =155.75% and a relative transient ripple max = 0.708%. The step change due to io is shown in

Fig: 2.23.

29


41/98

0 1 2 3 4 528.35

28.3

28.25

28.2

28.15

28.1

28.05

28

t (ms)

vO

(V)

Figure 2.23: Open-Loop step response due to step change in load current io.

2.2.3 Open-Loop Response due to Duty Cycle Step Change

The step response ofvo for a step change of 0.1 in d is given. The total duty cycle is given

by equation 3.40 where u(t) is the unit step function and D is the duty cycle before applying

the step change in the duty cycle.

dT(t) = D +dTu(t) (2.63)

The time domain step change in the duty cycle is

d(t) = dT(t)D = dTu(t) (2.64)

30


42/98

leading to the s-domain version which is

d(s) =dT

s(2.65)

The transient response of the output voltage of the open-loop buck-boost in s-domain is

vo(s) = Tp(s)d(s) =dTTp(s)

s

= dTTpo 2o

znzp

(s +zn)(szp )s(s2 + 2os +2o )

(2.66)

Switching back from s-domain to time domain

vo(t) = L1{vo(s)}

The total output voltage is

vo(t) = V(0) + vo(t)

Again, the maximum relative transient ripple of the total output voltage can be defined as

max =vomaxvo()

vo()

where vomax is the steady state value of the total output voltage. Given the measured values of

the circuit are: VI = 48 V, D = 0.407, VF = 0.7 V, rDS = 0.4 , RF = 0.02 , L = 334 mH,

C= 68 F, rC = .033 , and RL = 14 . These values lead to a maximum overshoot, Smax =

36.01% and a relative transient ripple max = 10.16%. The step change due to d is shown in

Fig: 2.24 .

31


43/98

0 1 2 3 4 5

42

40

38

36

34

32

30

28

t (ms)

vO

(V)

Figure 2.24: Open-Loop step response due to step change in duty cycle d.

2.3 Open-Loop Responses of Buck-Boost Using

PSpice

2.3.1 Open-Loop Response of Buck-Boost

A circuit showing the open-loop buck-boost is shown in Fig 2.25 and a model of this circuit is

shown in Fig 2.1. The measured values of the circuit are: VI = 48 V, D = 0.389, L = 334 mH,

C= 68 F, rC = 0.033, rL = 0.32 andRL = 14.An InternationalRectifier IRF150 power

MOSFET is selected, which has a VDSS = 100V , ISM = 40A, rDS = 55 m, Co = 100pF, and

Qg = 63nC. Also, an International Rectifier 10CTQ150 Schottky Common Cathode Diode is

selected with a VR = 100V, IF(AV) = 10A, VF = 0.73V and RF = 28m . The duty cycle for

the MOSFET changes from 0.407 to 0.389 to obtain the correct output of28 V as predictedusing MatLab. Also the switching frequency for Vp which controls the duty cycle is 100kHz

32


44/98

RL ioI

V

vi

o

Vp

v

D

+L C

rL rC

Figure 2.25: Open-loop buck-boost model with disturbances.

0 1 2 3 4 5 6 7 840

38

36

34

32

30

28

26

24

22

20

18

16

1412

10

8

6

4

2

0

t (ms)

Vout

(V)

Figure 2.26: Open-loop buck-boost response without disturbances.

allowing for a fast response time. The disturbances to the system are vi, io and d.

The output voltage of the buck-boost without any disturbances can be seen in Fig 2.26 . The

maximum overshoot is 41.25 %, settling time within five percent of steady state value is 3ms,

and settling time withing one percent of steady state value is 7ms.

33


45/98

CVp

RLI

V

vi

o

rL r

v

D

+L C

Figure 2.27: PSpice model of Open-Loop buck-boost with step change in input voltage.

2.3.2 Open-Loop Response due to Input Voltage Step Change

The PSpice circuit with step change in input voltage is shown in Fig 2.27. An additional

voltage pulse source of 1 volt was added with a delay of 10 ms, so that the circuit ran for

sufficient time to reach steady state value, and then the disturbance is activated. The output

voltage of the buck boost can be seen in Fig 2.28. The voltage ripple is .22 volts contained be-

tween 27.33V and 27.55V, the average value of steady state is 27.425V. The maximumovershoot is Smax = 21.74% and settling time within one percent is 2ms which contains the

ripple of steady state value. The relative maximum overshoot is max = 0.455%.

2.3.3 Open-Loop Response due to Load Current Step Change

The PSpice circuit with step change in load current is shown in Fig 2.29. An additional

current pulse source of .1 Amp was added with a delay of 10 ms so that the circuit ran for

sufficient time to reach steady state value, then the disturbance is activated. The output voltage

of the buck boost can be seen in Fig 2.30 . The voltage ripple is 0.22V and the output voltage

is contained between 28.36V and 28.14V. the average value of steady state is 22.25V .The maximum overshoot is Smax = 88% and settling time within one percent is 1.4ms which

contains the ripple of steady state value. The relative maximum overshoot is max = 0.779%.

34


46/98

5 6 7 8 9 1029

28.8

28.6

28.4

28.2

28

27.8

27.6

27.4

t (ms)

Vout

(V)

Figure 2.28: Open-Loop step response due to step change in input voltage using PSpice.

CVp

RL ioo

rL

r

v

D

+L C

VI

Figure 2.29: PSpice model of Open-Loop buck-boost with step change in load current.

35


47/98

0 .5 1 1.5 2 2.5 3 3.5 4 4.5 528.5

28.4

28.3

28.2

28.1

28

27.9

27.8

t (ms)

Vout

(V)

Figure 2.30: Open-Loop step response due to step change in load current using PSpice.

2.3.4 Open-Loop Response due to Duty Cycle Step Change

The PSpice circuit with step change in duty cycle is shown in Fig 2.31. An addition voltage

pulse source is added with a switch now on both Pulse generators. The first pulse generator

has its switch closed and running for the first ten ms so that it can achieve the desired steady

state value then simultaneously the switch to Vp is opened and the switch to Vp2 is closed with

the new duty cycle increased by 0.1. The output voltage of the buck boost can be seen in Fig

2.32 . The voltage ripple is 0.34V the average voltage of steady state is 34.03 V and theoutput voltage is contained between the bounds of33.86V and 34.2V . The settling timeis within one percent, which contains the ripple, of steady state value is 2.5ms.

36


48/98

CVp

RLv

o

rL

r

V

+L C

Vp2

D

I

Figure 2.31: PSpice model of Open-Loop buck-boost with step change in duty cycle.

0 1 2 3 4 5 635

34

33

32

31

30

29

28

27

t (ms)

Vout

(V)

Figure 2.32: Open-Loop step response due to step change in duty cycle using PSpice.

37


49/98

3 Closed Loop Response

3.1 Closed Loop Transfer Functions

Fig 3.1 shows the power stage of a buck-boost converter circuit with single-loop control

circuit. The control circuit is a single-loop voltage mode control. A small-signal closed-loop

model of the buck-boost is shown in Fig 3.2.

A block diagram of the closed-loop buck-boost is shown in Fig 3.3. vr , ve , vc and vf are

all ac components of the reference voltage, error voltage, output voltage of controller and and

feedback voltage respectively. Tp is the small-signal control to output transfer function of the

non-controlled buck-boost. Mv is the open-loop input to output voltage transfer function. Zo

is the open-loop output impedance. Tm is the transfer function of the pulse width modulator

(PWM). The function is the inverse of the hiegth of the ramp voltage being sent to the second

op-amp which is being used as a comparator. Tc is the transfer function for the lead-intergral

controller and T is the loop gain. The circuit is one control input and two disturbances and

one output where the independent inputs are vr, vi, and io respectively and the output is vo.

The output voltage is expressed in transfer functions as

vo(s) =TcTmTp

1 +TcTmTpvr+

Mv

1 +TcTmTpvi Zo

1 +TcTmTpio

38


50/98

VV

+

+

++

+ +

RA

RB

vo

vo

vGS

RL io

+

+

L C

R

VI

vc

Zf

Zi

vAB

vt

iv

dTvE

vF

v

vI

++

+vF

Figure 3.1: Closed loop circuit of voltage controlled buck-boost with PWM.

I dL

il

io

rC

vsd+

Z1

Dil

L

r

C

+ +

+ T T d+

+

+

vi

+

vo

+

RL

Z2

V dSD

D v v( i o)

oclicl

ii

vf

RBRAv

f vo

ve cv

vR

Figure 3.2: Closed loop small-signal model of voltage controlled buck-boost.

39


51/98

voTpTm Tc

vR

ve vc

vf

d+ vo

Zo

Mv

io

vi

vo

vo

Figure 3.3: Block diagram of a closed-loop small-signal voltage controlled buck-boost.

Zo

Mv+

+

io

vi

vo

vo

vo

vr

vo

A

1 + T

1

Figure 3.4: Simplified block diagram of a closed-loop small-signal voltage controlled buck-

boost.

= A1 + T

vr+ Mv

1 + Tvi Zo

1 + Tio = Tclvr+Mvcl viZocl io (3.1)

Where A = TcTmTp and T = A

PWM transfer function is expressed

40


52/98

Tm d(s)vc(s)

=1

VT m(3.2)

Where VT m is peak value of the triangular pulse.

Combining Tm and Tp together, the control to output transfer function is produced giving a

new function Tmp , given by

Tmp (s) = Tm(s)Tp(s) =vorC

VT m(1D)(RL + rC)(s +zn)(szp)s2 + 2os +2o

(3.3)

Tmpx = vorCVT m(1D)(RL + rC) (3.4)

Tmpo = TmTpo =Tpo

VT m=

vorCznzp

VT m(1D)(RL + rC)2o(3.5)

Figs: 3.5 and 3.6 show the Bode plots ofTmp .

To find the compensator value find Tk

Tkvf

vc|vi=0 = TmTp = Tmp

= vorCVT m(1D)(RL + rC)

(s +zn)(s +zp )

s2 + 2os +2o(3.6)

Tkx =vorC

VT m(1D)(RL + rC) (3.7)

Tko =vorCznzp

VT m(1D)(RL + rC)2o(3.8)

41


53/98

Changing from s-domain into jdomain gives

|Tk|

= Tko

1 +

f

fzn2

1 +

f

fzp 2

1 +

f

fo

22+

2ffo

2 (3.9)

Tk = tan1

f

fzn

tan1

f

fzp

tan1

2ffo

1

f

fo

2 when f

fo< 1, (3.10)

Tk =180 + tan1

ffzn

tan1

f

fzp

tan1

2f

fo

1

ffo

2 when ffo > 1. (3.11)

Figs: 3.7 and 3.8 show the Bode plots ofTk.

3.1.1 Integral-Lead Control Circuit for Buck-Boost

The following reasons explain the need for a control circuit in dc-dc power converters:

1. To achieve a sufficient degree of relative stability, an acceptable gain between 6 to 12

dB and phase margins between 45 and 90.

2. To reduce dc error.

3. To achieve a wider bandwidth and fast transient response.

4. To reduce the output impedance Zocl .

5. To reduce sensitivity of the closed-loop gain Tcl to component values over a wide fre-

quency range.

6. To reduce the input to output noise transmission.

42


54/98

101

102

103

104

105

40

30

20

10

0

10

20

30

40

f (Hz)

|Tmp

|(dB)

Figure 3.5: Magnitude Bode plot of modulator and input control to output voltage transfer

function Tm p for a buck-boost.

101

102

103

104

105

60

30

0

30

60

90

120

150

180

f (Hz)

T

mp

()

Figure 3.6: Phase Bode plot of modulator and input control to output voltage transfer function

Tmp for a buck-boost.

43


55/98

100

101

102

103

104

105

60

50

40

30

20

10

0

10

f (Hz)

|Tk

|(dB)

Figure 3.7: Magnitude Bode plot of the input control to output voltage transfer function Tkbefore the compensator is added for a buck-boost.

101

102

103

104

105

240

210

180

150

120

90

60

30

0

f (Hz)

Tk

()

Figure 3.8: Phase Bode plot of the input control to output voltage transfer function Tk before

the compensator is added for a buck-boost.

44


56/98

h

vF vc

C3R3

R1

vR

C1 R2

Rbd

C2

++

11

+

Figure 3.9: The Integral Lead Controller

Fig 3.9 shows the integral-lead controller that was chosen. This controller was chosen

because the integral part of the controller allows for high gain at low frequencies but introduces

a 90phase shift at all frequencies. This causes stability issues which is negated by the leadpart of the controller that compensates for the phase lag and more. Theoretically is should

introduce a 180 phase lead but practically produces a shift of between 150 and 160. The

impedances of the amplifier are

Zi = h11 +R1

R3 +

1sC3

R1 +R3 +

1sC3

=h11

R1 +R3 +

1sC3

+R1

R3 +

1sC3

R1 +R3 +

1sC3

=sC3h11R1 + sC3h11R3 + h11 +R1R3sC3 +R1

sC3R1 + sC3R3 + 1

=C3(h11(R1 +R3) +R1R3)

C3(R1 +R3)

s + h11+R1C3(h11(R1+R3)+R1R3)

s + 1C3(R1+R3)

45


57/98

=

h11 +

R1R3

(R1 +R3)

s + h11+R1

C3(h11(R1+R3)+R1R3)

s + 1C3(R1+R3)

(3.12)

Zf =1

sC2

R2 + 1sC1

R2 +

1sC1

+ 1sC2=

1sC2

R2 + 1C1C2s2

R2 +1

sC1+ 1sC2

=1

C2

s + 1R2C2

s

s + C1+C2R2C1C2

(3.13)and

h11 =RARB

RA +RB .

Assume infinite open-loop dc gain and open-loop bandwidth of the operational amplifier.

Therefore, from equations 3.12 and 3.13, the voltage transfer function of the amplifier is

Av(s) vc(s)vf(s)

=ZfZi

=

1C2

h11 +R1R2

R1+R2

s+ 1R2C2

s

s+C1+C2

R2C1C2

s+

h11+R1C3(h11(R1+R3)+R1R3)

s+1

C3(R1+R3)

=R1 +R3

C2(h11(R1 +R3) +R1R3)

s + 1R2C2

s + 1

C3(R1+R3)

s

s + C1+C2R2C1C2

s + h11+R1

C3(h11(R1+R3)+R1R3)

(3.14)

Because vr = 0, ve = vrvf =vf, the voltage transfer function of the integral lead controlleris

Tc vcve

= vc(s)vf(s)

= B(s +zc1)(s +zc2)

s(s +pc1)(s +pc2)(3.15)

where

B =R1 +R3

C2(h11(R1 +R3) +R1R3)

46


58/98

zc1 =

s +

1

R2C2

zc2 =

s +

1

C3(R1 +R3)

pc1 =

s +

C1 +C2R2C1C2

pc2 = s + h11 +R1C3(h11(R1 +R3) +R1R3)

Assume that zc1 = zc2 = zc and pc1 = pc2 = pc. Therefore

K=pc1zc1

=pc2zc2

=pczc

=

C1+C2R2C1C2

1R2C1

=

h11+R1C3(h11(R1+R3)+R1R3)

1C3(R1+R3)

=(h11 +R1)(R1 +R3)

h11(R1 +R3) +R1R3= 1 +

C2

C1(3.16)

This leads to the voltage transfer function of the controller to be

Tc vcve

= B(s +zc)2

s(s +pc)2= B

(s +zc)2

s(s +pc)2=

B(1 + szc )2

K2s(1 + spc

)2.

For s = j, the magnitude and phase shift ofTc is

|Tc(j)| =B(1 + zc )

2

K2s(1 + pc )2

=B

mK=

1

mC2(R1 + h11)

47


59/98

and

Tc =

2+ 2tan1

zc pc

1 + 2

zcpc

Design of Integral Lead Controller

For stability reasons a gain margin GM 9dB, a phase margin PM 60, and the cutofffrequency fc = 2kHz is chosen. The values of the buck-boost are VI = 48 V, D = 0.407,VF =

0.7 V, rDS = 0.4 , RF = 0.02, L = 334 mH, C= 68 F, rC = 0.033 , and RL = 14 .

The maximum value of phase in Tc occurs

c = m =

Kzc =pc

K=

K

R2C1=

h11 +R1KC3 (h11(R1 +R3) +R1R3)

Therefore the maximum phase shift possible can be described by

m = Tc(fm) +

2= 2tan1

K12

K

Solving for K leads to

K=1 + sin

m2

1 sin

m2

= tan2m4

+

4

Therefore,

m =+ 4tan1(

K)

Assuming VT m = 5V , the reference voltage is

VR = DnomVT m = .407(5) = 2.035V

48


60/98

The feedback network transfer function is

=VF

Vo=

VR

Vo= RA

RA +RB= 2

28= 0.0714

Assuming RB = 910, RA is

RA = RB

1

|| 1

= 910

1

.07141

= 11.83 k = 12 k

IfRB = 910, RA = 12 k then

h11 =RARB

RA +RB= 846.

h22 can be neglected because RA +RB is so much larger then RL.

Utilizing the cutoff frequency fc the phase Tk and m are

Tk = 180 + tan1fc

fzn tan1fc

fzp tan1

2fcfo

1 fcfo2 = 183.9

and

m = PMTk90 = 153.9.

This leads to

K = tan2m4

+

4

= 76.42.

Knowing K , fzc and fzp are calculated

fzc =fcK

= 228.779Hz

49


61/98

and

fzp = fc

K= 17.484kHz

The magnitude ofTk and Tc are used to calculate B

|Tk| = Tko

1 +

fc

fzn

21 +

fc

fzp

2

1

fcfo

22+

2fcfo

2 = 0.1945

|Tc| = 1|Tk| =1

.1954= 5.141

Therefore

B = cK|Tc| = 4.9374x 106 rad/s

Values of compensator are calculated. Assume R1 = 100 k and using the equations above

C2 =|Tk(fc)|

c (R1 + h11)= .1535 nF .15nF.

R3 =R1 [R1h11(K1)](K1) (R1 + h11) = 475 470

C1 = C2(K1) = 11.313 nF 12nF

R2 =

K

cC1 = 57.97 k 56 k

C3 =R1 + h11

Kc [R1R3 + h11(R1 +R3)]= 6.95nF 6.8nF

50


62/98

The pole and zero frequencies of the control circuit with standard resistor and capacitor values

are

fzc1 = 12R2C1= 236.84Hz

fzc2 =1

2C3(R1 +R3)= 232.96Hz

fpc1 = fzc1

C1

C2+ 1

= 19.184kHz

and

fpc2 =R1 + h11

2C3 [R1R3 + h11(R1 +R3)]= 17.881kHz.

Figs: 3.10 and 3.11 show the Bode plots ofTc.

3.1.2 Loop Gain of System

Loop gain of the system is

T(s) vfve|vi=io=0 = TcTmTp = TcTk

T(s) = BvorCVT m(1D)(RL + rC)

(s +zc)2(s +zn)(szp)s(s +pc)22 + 2os +2o )

T(s) = Tx

(s +zc)

2(s +zn)(szp)s(s +pc)22 + 2os +2o )

(3.17)

where

Tx = BvorCVT m(1D)(RL + rC) (3.18)

51


63/98

101

102

103

104

105

0

10

20

30

40

50

60

70

f (Hz)

|Tc

|(dBV

)

Figure 3.10: Magnitude Bode plot of the controller transfer function Tc for a buck-boost.

101

102

103

104

105

90

60

30

0

30

60

90

f (Hz)

T

c()

Figure 3.11: Phase Bode plot of the controller transfer function Tc for a buck-boost.

52


64/98

Figs: 3.12 and 3.13 show the Bode plots ofT. The controller expands the bandwidth by

moving the gain cross-over frequency by one kilohertz.

3.1.3 Closed Loop Control to Output Voltage Transfer Function

The control to output voltage closed-loop transfer function of the buck-boost is

Tcl vovr|io=vi=0 =

TcTmTp

1 +TcTmTp=

1T

1 + T=

1

T

1 + T

Tcl = 1

BVorCVT m(1D)(RL + rC)

(s +zc)2(s +zn)(szp )s(s +pc)2 (s2 + 2os +2o )

Tcl =Tx

(s +zc)

2(s +zn)(szp )s (s +pc)

2 (s2 + 2os +2o )

(3.19)

Figs: 3.14 and 3.15 show the Bode plots ofTcl . Figs: 3.16 and 3.17 show the discrete point

Bode plots ofTcl .

3.1.4 Closed Loop Input to Output Voltage Transfer Function

The input to output voltage closed-loop transfer function of the buck-boost is

Mvcl vovt|vr=io=0 =

Mv

1 + T

Mvcl =

(1D)DRLrCL(RL+rC) s+zns2+2os+2o

1 + BvorCVT m(1D)(RL+rC)

(s+zc)2(s+zn)(szp )

s(s+pc)2(s2+2os+2o )

=Mvx

s+zn(s2+2os+2o )

s(s +pc)

2(s2 + 2os +2o )

s(s +pc)2(s2 + 2os +2o ) + Tx(s +zc)2(s +zn)(szp)

53


65/98

101

102

103

104

105

40

30

20

10

0

10

20

30

f (Hz)

|T|

(db)

Figure 3.12: Magnitude Bode plot of the loop gain transfer function T for a buck-boost.

101

102

103

104

105

300

270

240

210

180

150

120

90

60

30

0

f (Hz)

T

()

Figure 3.13: Phase Bode plot of the loop gain transfer function T for a buck-boost.

54


66/98

101

102

103

104

105

15

10

5

0

5

10

15

20

25

f (Hz)

|Tcl

|(dBV

)

Figure 3.14: Magnitude Bode plot of the input control to output transfer function Tcl for a

buck-boost.

101

102

103

104

105

120

90

60

30

0

30

60

90

120

150

180

f (Hz)

T

cl

()

Figure 3.15: Phase Bode plot of the input control to output transfer function Tcl for a buck-

boost.

55


67/98

101

102

103

104

105

0

5

10

15

20

25

f (Hz)

|Tcl

|(dBV

)

Figure 3.16: Magnitude Bode plot of the input control to output transfer function Tcl for a

buck-boost.

101

102

103

104

105

120

90

60

30

0

30

60

90

120

150

180

f (Hz)

T

cl

()

Figure 3.17: Phase Bode plot of the input control to output transfer function Tcl for a buck-

boost.

56


68/98

=Mvxs(s +pc)2s +zn

s(s +pc)2(s2 + 2os +2o ) + Tx(s +zc)2(s +zn)(szp) (3.20)

Figs: 3.18 and 3.19 show the Bode plots of input to output voltage transfer function Mvcl .

Figs: 3.18 and 3.19 show the discrete point Bode plots ofMvcl .

3.1.5 Closed Loop Input Impedance

Fig 3.2 is used to derive the equations for the input impedance, and setting vr = 0,

d= voTcTm (3.21)

From the small-signal model of the buck-boost in Fig 3.2 and using KCL

ILdDil + il + iZ2 = 0

il(1D) = ILd voZ2

(3.22)

Rearranging gives

il =ILd

(1D) vo

(1D)Z2 . (3.23)

ii = Dil +ILd (3.24)

Substituting equations 3.21 and 3.23 into 3.24 provides the equation

ii = D

ILd

(1D) vo

(1D)Z2

+ILd

57


69/98

101

102

103

104

105

90

80

70

60

50

40

30

20

10

f (Hz)

|M

vcl

|

(db)

Figure 3.18: Magnitude Bode plot of the input to output voltage transfer function Mvcl for a

buck-boost.

101

102

103

104

105

360

330

300

270

240

210

180

150

120

90

f (Hz)

M

vcl

()

Figure 3.19: Phase Bode plot of the input to output voltage transfer function Mvcl for a buck-

boost.58


70/98

101

102

103

104

105

90

80

70

60

50

40

30

20

10

f (Hz)

|M

vcl

|

(db)

Figure 3.20: Magnitude Bode plot of the input to output voltage transfer function Mvcl for a

buck-boost.

101

102

103

104

105

360

330

300

270

240

210

180

150

120

90

f (Hz)

M

vcl

()

Figure 3.21: Phase Bode plot of the input to output voltage transfer function Mvcl for a buck-

boost.59


71/98

ii = D

IL(voTcTm)

(1D) vo

(1D)Z2

+IL(voTcTm)

ii =

voD

(1D)Z2 ILvoTcTm D

1D+ 1

ii = vo(1D)Z2

ILvoTcTm(1D)

ii =

D

(1D)Z2 +ILvoTcTm

(1D)

vo. (3.25)

DC analysis gives the equation

IL =Io

(1D) (3.26)

and

Io =vo

RL. (3.27)

Yicl =vi

ii|vr=0 =

Dil +ILd

vi(3.28)

Substituting equations 3.26 and 3.22 into 3.28 gives the equation

Yicl = D

(1

D)Z2

+ILTcTm(1

D)

vo. (3.29)

Using the definition ofYicl , equations 3.20 and dividing through by Tp yields the equation

=

IoTcTm(1D)2

D

(1D)Z2

vo

vi

60


72/98

=

IoTcTm(1D)2

D

(1D)Z2

Mvcl

= IoTcTm(1D)2

D

(1D)Z2Mv

1 + T

Mv

Tp=

(1D)DRLrCL(RL+rC)

s+zns2+2os+2o

Vo(1D)(RL+rC)(s+zn)(s+zp )s2+2os+2o

Mv

Tp=

(1D)2RLDLvo

=D(1D)2

LIo(szp )

Mv =

D(1D)2LIo(szp )

Tp (3.30)

=

D(1D)2TpLIo(szp )

IoTcTm(1D)2

=

T

L(szp )(1 + T)

Yicl =T

L(szp )(1 + T)

DMv

(1D)Z2(1 + T)

Yicl =

L(szp ) Tcl D

(1D)Z2Mvcl (3.31)

=L(1D)RLrC(szp )(s +zn)

D(1D)RLrC(s+zn) Tx (s+zc)2(s+zn)(szp )+D(RL+rC)(s+rc)L(szp ) (1D)DRLrCLLC) s(s+pc)2(s+zn)s(s+pc)22 +2os+2o )+Tx(s+zc)2(s+zn)(szp ) (3.32)

Zicl =L

s(s +pc)2(s2 + 2os +

2o ) + Tx(s +zc)

2(s +zn)(szp)

DTx(s +zc)2(s +zn) +D2s(s +rc)(s +pc)2(3.33)

61


73/98

NumZicl = s(s +pc)2(s2 + 2os +

2o ) + Tx(s +zc)

2(s +zn)(szp )

= (s3 +2pcs2 +pcs)(s

2 +2os +2o )+Tx(s

2 +2zcs+2

zc)(s2 +(zp +zn)szpzn)

NumZicl = s5 +s4(2o +2pc +Tx)+s

3

2o +

2pc + 4opc + T x((zp +zn) + 2zc)

+

s22pc

2o + 2o

2pc + Tx(zpzn + 2zc(znzp ) +2zc

+

s2o +

2pc + Tx(2zczpzn +zc(zp +zn))

Tx2zczpzn (3.34)

DenZicl = D2s4 + s3 [DTx + 2pc +rc] + s

2 DTx(zn + 2zc) +D2(2pc + 2rcpc)+sDTx(2zczn +

2zc) +D

2(rc2

pc)

+DTxzn2

zc (3.35)

Figs: 3.22 and 3.23 show the Bode plots of closed-loop input impedance Zicl . Figs: 3.24 and

3.25 show the discrete point Bode plots of closed-loop input impedance Zicl .

3.1.6 Closed Loop Output Impedance

The closed-loop output impedance for the buck-boost is

Zocl voio|vi=vr=0 =

Zo

1 + T

62


74/98

101

102

103

104

105

0

50

100

150

200

250

f (Hz)

|Zicl

|

()

Figure 3.22: Magnitude Bode plot of the input impedance transfer function Zicl for a buck-

boost.

101

102

103

104

105

180

150

120

90

60

30

0

30

60

90

f (Hz)

Zicl

()

Figure 3.23: Phase Bode plot of the input impedance transfer function Zicl for a buck-boost.

63


75/98

101

102

103

104

105

0

50

100

150

200

250

f (Hz)

|Zicl

|

()

Figure 3.24: Magnitude Bode plot of the input impedance transfer function Zicl for a buck-

boost.

101

102

103

104

105

180

150

120

90

60

30

0

30

60

90

f (Hz)

Zicl

()

Figure 3.25: Phase Bode plot of the input impedance transfer function Zicl for a buck-boost.

64


76/98

Zocl =

RLrC(RL+rC)

(s+rL)(s+zn)s2+2os+2o

1 + T

Zocl =Zoxs(s +pc)

2(s +zn)(s +rl )

s(s +pc)2(s2 + 2os +2o ) + Tx(s +zc)2(s +zn)(szp) (3.36)

Figs: 3.26 and 3.27 show the Bode plots of closed-loop output impedance Zocl . Figs: 3.28 and

3.29 show the certain discrete point Bode plots of closed loop output impedance Zocl .

3.2 Closed Loop Step Responses of Buck-Boost

3.2.1 Closed Loop Response due to Input Voltage Step Change

Response of output voltage vo due to a step change of 1 Volt in input voltage vi. The total

input voltage is given by equation 3.37.

vI(t) = VI(0) +VIu(t) (3.37)

vi(t) = vI(t)VI(0)

vi(s) = L{vi(t)}

vi(s) =vI

s

vo(s) = Mvcl(s)vi(s)

65


77/98

101

102

103

104

105

0

.25

0.5

.75

1

1.25

1.5

f (Hz)

|Z

ocl

|()

Figure 3.26: Magnitude Bode plot of the output impedance transfer function Zocl for a buck-

boost.

101

102

103

104

105

90

60

30

0

30

60

90

f (Hz)

Z

ocl

()

Figure 3.27: Phase Bode plot of the output impedance transfer function Zocl for a buck-boost.

66


78/98

101

102

103

104

105

0

0.5

1

1.5

f (Hz)

|Z

ocl

|()

Figure 3.28: Magnitude Bode plot of the output impedance transfer function Zocl for a buck-

boost.

101

102

103

104

105

90

60

30

0

30

60

90

f (Hz)

Z

ocl

()

Figure 3.29: Phase Bode plot of the output impedance transfer function Zocl for a buck-boost.

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vo(s) =Mvcl vI

s

vo(t) = L{vo(s)}

vo(t) = V(0) + vo(t)


normalized output voltage.

Smax =vomax vo()

vo()(3.38)

The relative maximum ripple defined in the following equation where vo() is the steady state

value of the output voltage.

max =vomax vo()

vo()

where vo()is defined as the steady state value of the output voltage. Given the measured

values of the circuit are: VI = 48 V, D = 0.407,VF = .7 V, rDS = 0.4 , RF = 0.02 , L =

334 mH, C= 68 F, rC = 0.033 , and RL = 14 . These values lead to maximum relative

transient ripple max = 0.625%. The step change due to vi is shown in Fig: ??.

3.2.2 Closed Loop Response due to Load Current Step Change

Response of output voltage vo due to a step change of 0.1 Amp in load current io

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0 2 4 6 8 1028.2

28.18

28.16

28.14

28.12

28.1

28.08

28.06

28.04

28.02

28

t (ms)

vO

(V)

Figure 3.30: Closed Loop step response due to step change in vi.

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Io(t) = Io(0) +Iou(t)

io(t) = io(t)Io(0)

io(s) = L{io(t)}

io(s) =Io

s

vo(s) = ocl (s)io(s)

vo(s) =Zocl io

s

vo

(t) = L{

vo

(s)}

vo(t) = V(0) + vo(t)


normalized output voltage.

Smax =vomax vo()

vo()(3.39)

max =vomaxvo()

vo()

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0 1 2 3 4 528

27.98

27.96

27.94

27.92

27.9

27.88

t (ms)

vO

(V)

Figure 3.31: Closed Loop step response due to step change in io.

where vo()is defined as the steady state value of the output voltage. Given the measured

values of the circuit are: VI = 48 V, D = 0.407, VF = 0.7 V, rDS = 0.4 , RF = .02 , L =

334 mH, C= 68 F, rC = 0.033 , and RL = 14 . These values lead to a maximum relative

transient ripple ofmax = 0.375%. The step change due to load current io is shown in Fig:

3.32 .

3.2.3 Closed Loop Response due to Reference Voltage Step

Change

Response of output voltage vo due to a step change of 1 volt in reference voltage vr

vR(t) = VR(0) +VRu(t)

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vr(t) = vR(t)VR(0)

vr(s) = L{vr(t)}

vr(s) =vR

s

vo(s) = Tpcl (s)vr(s)

vo(s) =Tpcl vR

s

vo(t) = L{vo(s)}

vo(t) = V(0) + vo(t)

The maximum undershoot is defined as

Smax =vomax vo()

vo()(3.40)

where vo() is defined as the steady state value of the normalized output voltage. The relative

maximum ripple defined in equation where vo()is the steady state value of the output voltage.

max =vomaxvo()

vo()

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0 1 2 3 4 528

27.98

27.96

27.94

27.92

27.9

27.88

t (ms)

vO

(V)

Figure 3.32: Closed Loop step response due to step change in vr.

Given the measured values of the circuit are: VI = 48 V, D = 0.407,VF = 0.7 V, rDS = 0.4 ,

RF = 0.02 , L = 334 mH, C= 68 F, rC = 0.033 , and RL = 14 . These values lead to

a maximum undershoot ofSmax = 28.57 % and a maximum relative transient ripple max =

9.25%. The step change due to vr is shown in Fig: ??.

3.3 Closed Loop Step Responses using PSpice

3.3.1 Closed Loop Response of buck-boost

A circuit showing the closed-loop buck-boost and control circuit is shown in Fig 3.33.

The measured values of the circuit are: VI = 48 V, D = 0.389, L = 334 mH, C = 68 F,

rC = 0.033, rL = 0.32 and RL = 14. An International Rectifier IRF150 power MOSFET

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VV

+

+

++

+ +

RA

RB

vo

vo

vGS

RL io

+

+

L C

R

VI

vc

Zf

Zi

vAB

vt

iv

dTvE

vF

v

vI

++

+vF

Figure 3.33: Closed loop buck-boost model with disturbances.

is selected, which has a VDSS = 100V , ISM = 40A, rDS = 55m, Co = 100pF, and Qg = 63nC.

Also, a International Rectifier 10CTQ150 Schottky Common Cathode Diode is selected with

a VR = 100V, IF(AV) = 10A, VF = 0.73V and RF = 28 m . The control circuit contains

a National Semiconductor LF357 op-amp. The op-amp selected is not rail to rail and has

aVmax = 18V . The voltage divider values for are RA = 12 k and RB = 910. Thecontrol circuit is shown in Fig 3.9 and contains the following values: R1 = 100k, R2 = 56 k,

R3 = 470, Rbd = 100 k, C1 = 12nF, C2 = .15nF, C3 = 6.8nF, and h11 = 846.

The output voltage of the buck-boost without any disturbances or step changes can be seen

in Fig 3.34. The relative maximum overshoot is max = 1.78%, and a settling time within two

percent of steady state value in 2.2ms.

3.3.2 Closed Loop Response due to Input Voltage Step Change

The PSpice circuit with step change in input voltage is shown in Fig 3.35. An addition

voltage pulse source of 1 volt was added with a delay of 10 ms so that the circuit ran for

sufficient time to reach steady state value before the disturbance is activated.

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0 1 2 3 4 530

2826

24

22

20

18

16

14

12

10

8

6

4

2

0

t (ms)

Vout

(V)

Figure 3.34: Closed loop buck-boost response without disturbances.

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B

vo

vo

vGS

R

RA

R

L

+

+

L C

VV

+

+

+ +

+ +

VI

vc

Zf

Zi

vR

vt

iv

dTvE

vF

vAB

vI

++

+vF

Figure 3.35: PSpice model of Closed Loop buck-boost with step change in input voltage.

The output voltage of the buck boost can be seen in Fig 3.36. The voltage ripple is 0.22 volts

contained between 28.65V and 28.3V, the average value of steady state is 28.475V.The maximum overshoot is Smax = 72% and settling time is within two percent in 5 ms which

contains the ripple of steady state value. The relative maximum overshoot is max = 0.526%.

The reason steady state did not return to 28 as predicted by MatLab is because of the non-ideal op-amps. The gain is only 667, not infinite, as shown in the MatLab model.

3.3.3 Closed Loop Response due to Load Current Step Change

The PSpice circuit with step change in load current is shown in Fig 3.37. An additional current

pulse source of 0.1 Amp was added with a delay of 10 ms so that the circuit ran for sufficient

time to reach steady state value, and then the disturbance is activated.

The output voltage of the buck boost can be seen in Fig 3.38. The voltage ripple is .35V, and

the output voltage is contained between

27.81V and

28.16V. the average value of steady

state is 27.99V .The relative maximum overshoot max = 1.07% and settling time is withintwo percent in 2.2ms which contains the ripple of steady state value.

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0 1 2 3 4 529

28.8

28.6

28.4

28.2

28

27.8

t (ms)

Vout

(V)

Figure 3.36: Closed Loop step response due to step change in input voltage using PSpice.

B

vo

vo

vGS

RL i

RA

R

o

+

+

L C

VV

+

+

++

+

VI

vc

Zf

Zi

vR

vt

dTvE

vF

vAB

++

+vF

Figure 3.37: PSpice model of Closed Loop buck-boost with step change in load current.

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0 0.5 1 1.5 2 2.528.5

28.4

28.3

28.2

28.1

28

27.9

27.8

27.7

27.6

27.5

t (ms)

Vout

(V)

Figure 3.38: Closed Loop step response due to step change in load current using PSpice.

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B

vo

vo

vGS

R

RA

R

L

+

+

L C

VV

+

+

+ +

+

VI

vc

Zf

Zi

vR

vt

dTvE

vF

vAB

++

+vF

Figure 3.39: PSpice model of Closed Loop buck-boost with step change in duty cycle.

3.3.4 Closed Loop Response due to Reference Voltage Step

Change

The PSpice circuit with step change in reference voltage is shown in Fig 3.39. The Piecewise

Linear function in PSpice is utilized to create a step function in the reference voltage.

The output voltage of the buck boost can be seen in Fig3.40 . The voltage ripple is 0.42V,

the average voltage of steady state is 30 V and the output voltage is contained between the

bounds of30.13V and 29.85V . The maximum overshoot Smax = 675%, and settling timeis within two percent in 6 ms, which contains the ripple of steady state value. The relative

maximum overshoot is max = 45%

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0 1 2 3 4 5 6 7 844

42

40

38

36

34

32

30

28

26

t (ms)

Vout

(V)

Figure 3.40: Closed Loop step response due to step change in reference voltage using PSpice.

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4 Conclusion

4.1 Contributions

The principles on operation of open-loop and closed-loop of the dc-dc buck-boost converter

is discussed. Also, design and analysis of an intergral-lead type III controller for the closed-

loop buck-boost is discussed. Equations for the transfer functions and step responses for a

selected prechosen design of a dc-dc buck-boost converter. For the selected design, the step

response and Bode plots is found using both matlab and PSpice. The observations can be

summarized as:

1. The discrete point Bode plots found coincide with the theoretical Bode plots given by

MatLab.

2. The step responses determined by PSpice are consistent with the theoretical step re-

sponses given by MatLab.

3. Stabilizing the buck-boost converter is a challenge because of the RHP zero but can be

accomplished by using a type III controller.

4. The theoretical phase shift achieved by the integral-lead is 180 but in reality only 150

to 160 can be achieved.

5. The magnitude of input to output voltage transfer function Mv is reduced by negative

feedback.

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4.2 Future Work

Improving the response time, efficiency and reducing losses is a major challenge because

of the practical limitations of a buck-boost conveter. Selecting the MOSFET, diode, and cur-

rent transformer for experimentation for small-signal applications is a good choice for future

research. Voltage mode control and Current mode control of the PWM dc-dc buck-boost. As

well as distinguish the characteristics of finding a Bode plot without using discrete points.

Also, finding a methodology for the charateristics of a small-signal model.

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Appendix A

VO DC output voltagevo AC component of the output voltage

vO Total output voltage

IO DC output current

io AC component of the output current

iO Total output current

d AC component of duty cycle

D DC component of duty cycle

VI DC input Voltage

vi AC component of the input voltage

vI Total input voltage

rDS Parasitic on-resistance of the MOSFET

rL Parasitic componet of Inductor

RF Forward resistance of the diode

VF Forward voltage drop of diode

VSD DC component of the voltage between MOSFET and Diode

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vsd AC component of the voltage between MOSFET and Diode

r Combination of parasitic componets of mosfet, diode, and inductor

C Capacitor

L Inductor

IL DC inductor current

il AC component of inductor current

iL Total inductor current

rC Parasitic component of capacitor

RL Load resistor

Z1 Impedance caused by the series combination ofrand inductor

Z2 Impedance caused by the parallel combination of load resistor and capacitor

Tp Open-loop input control to output voltage transfer function

Mv Open-loop input to output voltage transfer function, audio suceptibility

Zi Open-loop input impedance transfer function

Zo Open-loop output impedance transfer function

Tcl Closed-loop input control to output voltage transfer function

Mvcl Closed-loop input to output voltage transfer function, audio suceptibility

Zicl Closed-loop input impedance transfer function

Zocl Closed-loop output impedance transfer function

Tc Compensator transfer function

T Loop gain transfer function

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Tm Modulator transfer fuction

Tmp Modulator and open-loop input control to output transfer fuction

Tk Gain of the system before control added

A Forward gain TcTm p

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References

1. M. K. Kazimierczuk, Class notes, EE 742-Power Electronics II, Wright State University,

Winter 2006.

2. R. D. Middlebrook and S. Cuk, Advances in Switched-

Documents

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