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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)
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=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.
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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.
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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.
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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
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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)
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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)
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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.
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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.
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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.
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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
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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.
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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)
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(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)
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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
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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
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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
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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)
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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 .
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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)
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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.
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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)
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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 .
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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
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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.
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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%.
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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.
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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.
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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.
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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
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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 3.1: 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.
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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
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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)
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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.
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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.
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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.
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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)
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=
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)
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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)
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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
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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
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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
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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)
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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.
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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)
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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.
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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.
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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.
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=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
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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-
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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-
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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
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=
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)
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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
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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.
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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.
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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)
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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.
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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)
The maximum overshoot defined in equation where vo() is the steady state value of the
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)
The maximum overshoot defined in equation where vo() is the steady state value of the
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-