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

ANALYSIS OF SMALL-SIGNAL MODEL OF A PWM DC-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

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 on Final 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

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 converter. 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 converter operating in continuous-conduction-mode (CCM) are derived. For the pre-chosen design, 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 performed 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. iii

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 iv

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 v

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 T p 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 T p for a buck-boost...................... 11 2.5 Open-loop phase Bode plot of input control to output voltage transfer function T p for a buck-boost with and without 1µs delay................ 11 2.6 Discrete point open-loop magnitude Bode plot of input control to output voltage transfer function T p for a buck-boost.................... 12 2.7 Discrete points open-loop phase Bode plot of input control to output voltage transfer function T p for a buck-boost...................... 12 2.8 Small-signal model of the buck-boost to determine the input to output voltage transfer function M v the input to output function................ 13 2.9 Open-loop magnitude Bode plot of input to output voltage transfer function M v for a buck-boost............................... 16 2.10 Open-loop phase Bode plot of input to output transfer function M v for a buckboost....................................... 16 vi

2.11 Open-loop magnitude Bode plot of input to output transfer function M v for a buck-boost.................................... 17 2.12 Open-loop phase Bode plot of input to output transfer function M v for a buckboost....................................... 17 2.13 Open-loop magnitude Bode plot of input impedance transfer function Z i for a buck-boost.................................... 19 2.14 Open-loop phase Bode plot of input impedance transfer function Z i for a buckboost....................................... 19 2.15 Discrete points open-loop magnitude Bode plot of input impedance transfer function Z i for a buck-boost........................... 20 2.16 Discrete points open-loop phase Bode plot of input impedance transfer function Z i for a buck-boost............................. 20 2.17 Small-signal model of the buck-boost for determining output impedance Z o. 21 2.18 Open-loop magnitude Bode plot of output impedance transfer function Z o for a buck-boost................................... 24 2.19 Open-loop phase Bode plot of output impedance transfer function Z o for a buck-boost.................................... 24 2.20 Discrete points open-loop magnitude Bode plot of output impedance transfer function Z o for a buck-boost........................... 25 2.21 Discrete points open-loop phase Bode plot of output impedance transfer function Z o for a buck-boost............................. 25 2.22 Open-Loop step response due to step change in input voltage v i........ 28 2.23 Open-Loop step response due to step change in load current i o........ 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 vii

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-loop small-signal voltage controlled buckboost....................................... 40 3.5 Magnitude Bode plot of modulator and input control to output voltage transfer function T mp for a buck-boost.......................... 43 3.6 Phase Bode plot of modulator and input control to output voltage transfer function T mp for a buck-boost.......................... 43 3.7 Magnitude Bode plot of the input control to output voltage transfer function T k before the compensator is added for a buck-boost.............. 44 3.8 Phase Bode plot of the input control to output voltage transfer function T k 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 T c for a buck-boost.. 52 3.11 Phase Bode plot of the controller transfer function T c 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 T cl for a buck-boost.................................... 55 viii

3.15 Phase Bode plot of the input control to output transfer function T cl for a buckboost....................................... 55 3.16 Magnitude Bode plot of the input control to output transfer function T cl for a buck-boost.................................... 56 3.17 Phase Bode plot of the input control to output transfer function T cl for a buckboost....................................... 56 3.18 Magnitude Bode plot of the input to output voltage transfer function M vcl for a buck-boost................................... 58 3.19 Phase Bode plot of the input to output voltage transfer function M vcl for a buck-boost.................................... 58 3.20 Magnitude Bode plot of the input to output voltage transfer function M vcl for a buck-boost................................... 59 3.21 Phase Bode plot of the input to output voltage transfer function M vcl for a buck-boost.................................... 59 3.22 Magnitude Bode plot of the input impedance transfer function Z icl for a buckboost....................................... 63 3.23 Phase Bode plot of the input impedance transfer function Z icl for a buck-boost. 63 3.24 Magnitude Bode plot of the input impedance transfer function Z icl for a buckboost....................................... 64 3.25 Phase Bode plot of the input impedance transfer function Z icl for a buck-boost. 64 3.26 Magnitude Bode plot of the output impedance transfer function Z ocl for a buck-boost.................................... 66 3.27 Phase Bode plot of the output impedance transfer function Z ocl for a buck-boost. 66 3.28 Magnitude Bode plot of the output impedance transfer function Z ocl for a buck-boost.................................... 67 3.29 Phase Bode plot of the output impedance transfer function Z ocl for a buck-boost. 67 ix

3.30 Closed Loop step response due to step change in v i............... 69 3.31 Closed Loop step response due to step change in i o............... 71 3.32 Closed Loop step response due to step change in v r.............. 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 x

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.

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 dissipation and it is impossible to generate voltages higher than the supply voltage. Switchingmode 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 1

(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. 2

2 Open-Loop Buck-Boost Derived small-signal open-loop transfer functions for the input control to ouput voltage transfer fuction T p, audio susceptibilitym v, input impedance Z i and output impedance Z o. Using 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 v i, duty cycle d and load current i o. 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 buckboost 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: V I = 48 V, D = 0.407, V F = 0.7V, r DS = 0.4 Ω, R F = 0.02 Ω, L = 334 mh, C = 68 µf, r C = 0.033 Ω, and R L = 14 Ω. 3

v sd V d SD d v i I d L Di l D ( v i o v) C R L v o i o Z 1 L i l r C r Z 2 Figure 2.1: Small-signal model of buck-boost. Z o i o v o M v v o v o v i v o d T p Figure 2.2: Block diagram of buck-boost. 4

i Z 2 v i = 0 v sd i o = 0 V d SD d I d L Di l Dv sd = Dv o C R L v o Z 1 L i l r C r Z 2 Figure 2.3: Small-signal model of buck-boost to determine T p the input control to output voltage 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 T p. The derivation using Fig 2.3 of T p is below starting from first principles of KCL and KVL. Finding the transfer function of the plant T p i Z2 = v z 2 Z 2 = v o Z 2 (2.1) Using KCL i l i Z2 I l d Di l = 0 i l (1 D)i Z2 I l d = 0 (2.2) i l Z 1 V SD d Dv sd = v o (2.3) v sd = v o (2.4) i l Z 1 = v o V SD d Dv o 5

i l = v o(1 D)v sd d Z 1 (2.5) Substituting values v o (1 D)V SD d Z 1 (1 D) v o Z 2 I L d = 0 (2.6) v o (1 D) 2 V SDd(1 D) v o I L d = 0 Z 1 Z 1 Z 2 [ (1 D) 2 v o 1 ] [ = d I L V ] SD(1 D) Z 1 Z 2 Z 1 T p v o(s) d(s) v o =i o =0= ( ) I l V SD(1 D) Z ( 1 ) = (1 D) 2 Z 1 Z 1 2 ( ) I l 1 V SD(1 D) I L (1 D) 2 Z 1 Z 2 (2.7) I l = I D (1 D) = I o (1 D) = v o (1 D)R L (2.8) Using KVL ri L Dv sd v F v o = 0 (2.9) v sd v I v F v o = 0 v sd = v I v F v o (2.10) Substituting values [ r v o (1 D)R L ] D(v I v F v o )v F v o = 0 (2.11) 6

[ r v o (1 D)R L ] Dv F o v F v o = Dv I [ r v o v I D v F 1D (1 D)R l v o ( vf v o )] = v I D [ ( r v o (1 D) 1 v )] F = v I D (1 D)R L v o ( v sd = v I v F v o = v o 1 v F v ) ( I = I L R L (1 D) 1 v F 1 ) v o v o v o M V DC v sd I L = 1 D [ ( R L (1 D) 1 v )] F v o r T p (s) = ( ) T p (s) v I o L 1 V SD(1 D) d I l v i =i o =0 = (1 D) 2 Z 1 Z 2 T p (s) = ] I L [Z 1 (1 D) v sd I L ] I L [Z 1 (1 D) v sd I L Z 1 (1 D) 2 Z 2 Z 2 = Z 1 Z 2 (1 D) 2 I L [Z 1 (1 D) v sd I L ]Z 2 Z 1 (1 D) 2 Z 2 (2.12) Z 1 = r sl (2.13) Z 2 = R ( L rc sc 1 ) R L r C sc 1 (2.14) 7

v sd I L = 1 D [ ( R L (1 D) 1 v ) ] F r v o (2.15) I L = v o (1 D)R L (2.16) T p = I L [Z 1 (1 D) v sd I L ]Z 2 Z 1 (1 D) 2 Z 2 (2.17) DenT p = (r sl)(1 D) 2 R ( L rc sc 1 ) R L r C sc 1 DenT p = ( R L r C 1 ( )(r sl)(1 D) 2 R L r c 1 ) sc sc = R L r rr C r 1 sc slr L slr C L ( (1 D)2 R L r C R ) L C sc = scr L r scr C r r s 2 CLR L s 2 LCr C sl(1 D) 2 (scr L r C R L ) = s 2 C[ r(r L r C )(1 D) 2 ] R L r C L s r(1 D)2 R L LC(R L r C ) LC(R L r C ) (2.18) [ NumT p = (sc) 1 LC(R L r C ) ]I L [ RL ( rc 1 sc ) R L r C 1 sc ] [ R L r C 1 ][ Z 1 (1 D) v ] sd sc I L = I L (R L (scr C 1)(r sl)) 1 D ( [R L (1 D) 2 1 v )][ ] F v o r 1 LC(R L r C ) 8

( V o = (R L (scr C 1)) (1 D)R L V o NumT p = ((1 D)R L r C ) 1 LC(R L r C ) ) [ (L) s 1 ( R L (1 D) 2 (1 v )] F )r(1 2D) DL v o ( s 1 )[ s 1 ( Cr C DL (R L(1 D) 2 1 v )] F )r(1 2D) v o (2.19) ζ = C[ r(r L r C )(1 D) 2 ] R L r C L 2 LC(R L r C )[r(1 D) 2 R L ] r(1 D) ω o = 2 R L LC(R L r C ) (2.20) (2.21) ω zn = 1 Cr C (2.22) ω zp = 1 ( [R L (1 D) 2 1 v ) ] F r(1 2D) DL v o (2.23) V o T p v o d v i =i o =0 = (1 D)(R L r C ) (sω zn )(sω zp ) s 2 2ζ ω o sω 2 o (2.24) V o T px = (1 D)(R L r C ) (2.25) = V o r C (1 D)(R L r C ) T po = T p (0) = V o R ω zn ω zp (1 D)(R L r C ) ωo 2 ( ) ( Cr 1 1 C DL R L (1 D) 2( ) ) 1 v F v o r(1 2D) r(1 D) 2 R L LC(R L r C ) 9

= T po = V o r C (1 D)(R L r C ) V o D(1 D) ( R L (1 D) 2( ) ) 1 v F vo r(1 2D) CDLr C r(1 D) 2 R L LC(R L r C ) R L(1 D) 2( ) 1 v F v o r(1 2D) r(1 D) 2 R L (2.26) The Bode plot of T p is shown in Fig 2.4 and 2.5. The input control to output voltage transfer function T p 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 T p is also found using discrete 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 magnitude 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 Z icl, 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 T p. 10

50 40 30 T p (db V) 20 10 0 10 20 10 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.4: Theoretical open-loop magnitude Bode plot of input control to output voltage transfer function T p for a buck-boost. 180 150 t d = 0 t d = 1µ s 120 90 φ Tp ( ) 60 30 0 30 60 90 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.5: Open-loop phase Bode plot of input control to output voltage transfer function T p for a buck-boost with and without 1µs delay. 11

50 40 30 T p (db V) 20 10 0 10 20 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.6: Discrete point open-loop magnitude Bode plot of input control to output voltage transfer function T p for a buck-boost. 180 150 120 90 φ Tp ( ) 60 30 0 30 60 90 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.7: Discrete points open-loop phase Bode plot of input control to output voltage transfer function T p for a buck-boost. 12

d = 0 i o = 0 i i v sd i Z 2 Di l = Dv sd D ( v i o v) C v i R L v o Z 1 L i l r C r Z i Z 2 Figure 2.8: Small-signal model of the buck-boost to determine the input to output voltage transfer function M v 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 i o = 0 is shown in Fig 2.8. Using this model to derive equations for the input voltage to out voltage transfer function M v, also known as audio susceptibility. Again, using first principles to start the derivation process. From KCL i l Di l i Z2 = 0 (2.27) Using KVL i Z2 = v o Z 2 (2.28) i l = i Z 2 (1 D) = v o (1 D)Z 2 (2.29) i l Z 1 v o Dv sd = 0 13

i l Z 1 = v o D(v i v o ) i l Z 1 = Dv i (1 D)v o v o Z 2 (1 D) Z 1 = Dv i (1 D)v o Dv i = (1 D)v o [1 ] Z 1 Z 2 (1 D) 2 = D (1 D) M v (s) v o(s) v i (s) D d=i o =0 = [ ] (1 D) 1 Z 1 Z 2 (1 D) 2 [ 1 1 Z = D (1 D)2 1 (1 D) Z 2 (1 D) 2 1 (1 D) Z 1 Z 2 ] M v = (1 D)DR Lr C L(R L r C ) sω zn s 2 2ζ ω o sω 2 o (2.30) M vx = (1 D)DR Lr C L(R L r C ) (2.31) M vo = M v (0) = (1 D)DR Lr C L(R L r C ) ω zn ω 2 o = (1 D)DR Lr C L(R L r C ) 1 Cr C r(1 D) 2 R L LC(R L r C ) 14

Fig: 2.9 and 2.10 show the theoretical Bode plots of M v. M vo = (1 D)DR L r(1 D) 2 R L (2.32) 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 Z i 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 Di l i l i Z2 = 0 i Z2 = Di l i l i Z2 = (1 D)i l (2.33) From KVL Z 1 i l D(v i v o )v o = 0 Z 1 i l Dv io (1 D) = 0 i l Z 1 Dv i i l (1 D)(1 D)Z 2 = 0 i l Z 1 Dv i i l (1 D) 2 Z 2 = 0 Dv i = i l (Z 1 (1 D) 2 Z 2 ) 15

10 0 10 20 M v (db V) 30 40 50 60 70 80 90 10 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.9: Open-loop magnitude Bode plot of input to output voltage transfer function M v for a buck-boost. 180 150 120 φ MV ( ) 90 60 30 0 10 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.10: Open-loop phase Bode plot of input to output transfer function M v for a buckboost. 16

10 0 10 20 M v (db V) 30 40 50 60 70 80 90 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.11: Open-loop magnitude Bode plot of input to output transfer function M v for a buck-boost. 180 150 120 φ MV ( ) 90 60 30 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.12: Open-loop phase Bode plot of input to output transfer function M v for a buckboost. 17

Z i = v i = (Z 1 (1 D) 2 Z 2 ) i i D 2 = 1 [ D 2 r sl r(1 ] D)2 R L (1 D)2 LC(R L r C ) = (r sl)(r L r C sc 1 )R L(r C sc 1 )(1 D)2 D 2 (R L r C sc 1 ) [ = 1 rrl rr C sc r slr L slr C C L R ] Lr C (1 D) 2 (1 D) 2 R L sc D 2 R L r C sc 1 = 1 D 2 LC(R L r C )s 2 [ C(r(r C R L )R L r C (1 D) 2 )L ] sr L (1 D) 2 r scr L scr C 1 Z i = 1 D 2 (LC(R L r C )) ( ) s 2 C(r(R Lr C )R L r C (1 D) 2 L s R L(1 D) 2 r LC(R L r C ) LC(R L r C ) sc(r L r C )1 where ω rc = 1 C(R L r C ) (2.34) Figs: 2.13 and 2.14 show the theoretical Bode plots of Z i. Z i = L D 2 s 2 2ζ ω o sω 2 o sω rc (2.35) As shown above the Magnitude of Z i decreases quickly with an increase in D. Figs 2.15 and 2.16 show the discrete point Bode plots for Z i. 18

180 160 140 120 Z i (db V) 100 80 60 40 20 0 10 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.13: Open-loop magnitude Bode plot of input impedance transfer function Z i for a buck-boost. 90 60 30 φ Zi ( ) 0 30 60 10 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.14: Open-loop phase Bode plot of input impedance transfer function Z i for a buckboost 19

180 160 140 120 Z i (Ω) 100 80 60 40 20 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.15: Discrete points open-loop magnitude Bode plot of input impedance transfer function Z i for a buck-boost. 90 60 30 φ Zi ( ) 0 30 60 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.16: Discrete points open-loop phase Bode plot of input impedance transfer function Z i for a buck-boost. 20

v sd i b i t i Z2 d = 0 v i = 0 i o = 0 Di l Dv sd = Dv o C R L v o v t Z 1 L i l r C r Z 2 Z o Figure 2.17: Small-signal model of the buck-boost for determining output impedance Z o. 2.1.4 Open-Loop Output Impedance Solving for the transfer function of output impedance Z o. Output impedance Z o of the buckboost small-signal model is shown in Fig 2.17 where all three inputs d, v i, and i o equal 0. A test voltage v t with a current of i t is applied to the output of the model. The ratio of v t to i t determines Z o. Using KVL KCL Z 1 i l Dv t v t = 0 (2.36) i l = (1 D)v t Z 1 (2.37) i b i Z2 i t = 0 i b = (1 D)i l (2.38) 21

(1 D)i l = i t i Z2 (1 D) 2 v t Z 1 = i t v t Z 2 v t ( c 1 Z 2 ) = i t (2.39) = Z o = v t i t = 1 = 1 Z 2 (1 D)2 Z 1 ( R (r sl) L (r C 1 = r sl R L(r C 1 (r sl) [ R L ( rc 1 sc = 1 LC(R L r C ) Z o = sc ) R L r C 1 sc sc ) R L r C sc 1 Z 1 Z 1 Z 2 (1D) 2 ) (1 D) 2 (r sl) ( ( R L rc 1 )) 1 LC(R L r C ) sc )] RL (1 D) 2( r C 1 sc s 2 CrR Lr C LR L LCR L r C s rr L LCR L r C s 2 2ζ ω o sωo 2 ( s r L ) ( s 1 Cr C ) s 2 2ζ ω o sω 2 o ) ω rl = r L (2.40) ω zn = 1 Cr C (2.41) 22

Z o v t = R Lr C i t (R L r C ) (sω rl )(sω zn ) s 2 2ζ ω o sω 2 o (2.42) Figs: 2.18 and 2.19 show the Bode plots for Z o. As D increases so does the magnitude of Z o at low frequencies. Figs: 2.18 and 2.19 show the discrete points PSpice simulated Bode plots for Z o. 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 v o due to a step change of 1 Volt in input voltage v i. The total input voltage is given by equation 3.40 where u(t) is the unit step function and V I (0 ) is the input voltage before applying the step voltage. v I (t) = V I (0 ) V I u(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 v i (t) = V I u(t) = v I (t) V I (0 ) (2.44) Changing from time domain to s-domain 23

7 6 5 Z o (db V) 4 3 2 1 0 10 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.18: Open-loop magnitude Bode plot of output impedance transfer function Z o for a buck-boost. 60 30 0 φ Zo ( ) 30 60 90 10 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.19: Open-loop phase Bode plot of output impedance transfer function Z o for a buckboost 24

7 6 5 Z o (Ω) 4 3 2 1 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.20: Discrete points open-loop magnitude Bode plot of output impedance transfer function Z o for a buck-boost. 60 30 0 φ Zo ( ) 30 60 90 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 2.21: Discrete points open-loop phase Bode plot of output impedance transfer function Z o for a buck-boost 25

v i (s) = L{v i (t)} (2.45) where the step change in s-domain is v i (s) = V I s (2.46) Therefore the transient response due to a step change in v i becomes v o (s) = v (s) s (2.47) = V I M vo ω 2 0 ω zn sω zn s(s 2 2ζ ω o sωo) 2 = V sω zn IM vx s(s 2 2ζ ω o sωo) 2 (2.48) Returning from s-domain to time domain v o (t) = L{v o (s)} (2.49) producing the magnitude of the transient response is = V I M vo 1 1 2ζ ω o ω zn ( ωo ω zn ) 2 e σt sin(ω dt φ) (2.50) 1 ζ 2 Where φ = tan 1 ω d ) ω zn (1 ζω o ω zn ( ) 1 ζ tan 1 2 ζ (2.51) The total output voltage response is v o (t) = V(0 )v o (t) t 0 (2.52) 26

The maximum overshoot defined in equation where v o ( ) is the steady state value of the output voltage. S max = v omax v o ( ) v o ( ) (2.53) Obtaining the derivative for equation and setting it equal to zero produces the time instants at which the maximum of v o occurs v omax = V I M vo 1 1 2ζ ω o ω zn ( ωo ω zn ) 2 e πζ (2.54) 1 ζ 2 Therefore the maximum overshoot is S max = 1 2ζ ω o ω zn ( ωo ω zn ) 2 e πζ 1 ζ 2 (2.55) The maximum relative transient ripple of the total output voltage can be defined as δ max = v omax v o ( ) v o ( ) (2.56) where v o ( ) is defined as the steady state value of the output voltage. Given the measured values of the circuit are: V I = 48 V, D = 0.407,V F = 0.7 V, r DS =.4 Ω, R F = 0.02 Ω, L = 334 mh, C = 68 µf, r C = 0.033 Ω, and R L = 14 Ω. These values lead to a maximum overshoot, S max = 35.67% and a relative transient ripple δ max = 1.05%. The step change due to v i is shown in Fig: 2.22. 27

28 28.1 28.2 28.3 v O (V) 28.4 28.5 28.6 28.7 28.8 0 1 2 3 4 5 t (ms) Figure 2.22: Open-Loop step response due to step change in input voltage v i. 2.2.2 Open-Loop Response due to Load Current Step Change Response of v o due to a step change of.1 Amp in i o. The total load current is given by equation 3.40 where u(t) is the unit step function and I o (0 ) is the input current before applying the step current. I o (t) = I o (0 ) I o u(t) (2.57) Step change in the time domain is i o (t) = i o (t) I o (0 ) = I o u(t) (2.58) 28

Changing from time domain into s-domain the step change becomes i o (s) = I o s (2.59) The transient component of the output voltage is v o (s) = Z o (s)i o (s) = I or L r C R L r C (sω zn )(sω rl ) s(s 2 2ζ ω o sω 2 o) = I o Z ox (sω zn )(sω rl ) s(s 2 2ζ ω o sω 2 o) (2.60) Switching back from s-domain to time domain v o (t) = L 1 {v o (s)} (2.61) The total output voltage is v o (t) = V(0 )v o (t) (2.62) Again, the maximum relative transient ripple of the total output voltage can be defined as δ max = v omax v o( ) v o( ) where v omax is the steady state value of the total output voltage. Given the measured values of the circuit are: V I = 48 V, D = 0.407, V F = 0.7 V, r DS = 0.4 Ω, R F = 0.02 Ω, L = 334 mh, C = 68 µf, r C = 0.033 Ω, and R L = 14 Ω. These values lead to a maximum overshoot, S max = 155.75% and a relative transient ripple δ max = 0.708%. The step change due to i o is shown in Fig: 2.23. 29

28 28.05 28.1 v O (V) 28.15 28.2 28.25 28.3 28.35 0 1 2 3 4 5 t (ms) Figure 2.23: Open-Loop step response due to step change in load current i o. 2.2.3 Open-Loop Response due to Duty Cycle Step Change The step response of v o 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. d T (t) = D d T u(t) (2.63) The time domain step change in the duty cycle is d(t) = d T (t) D = d T u(t) (2.64) 30

leading to the s-domain version which is d(s) = d T s (2.65) The transient response of the output voltage of the open-loop buck-boost in s-domain is v o (s) = T p (s)d(s) = d T T p (s) s Switching back from s-domain to time domain = d T T po ω 2 o ω zn ω zp (sω zn )(s ω zp ) s(s 2 2ζ ω o sω 2 o ) (2.66) v o (t) = L 1 {v o (s)} The total output voltage is v o (t) = V(0 )v o (t) Again, the maximum relative transient ripple of the total output voltage can be defined as δ max = v omax v o( ) v o( ) where v omax is the steady state value of the total output voltage. Given the measured values of the circuit are: V I = 48 V, D = 0.407, V F = 0.7 V, r DS = 0.4 Ω, R F = 0.02 Ω, L = 334 mh, C = 68 µf, r C =.033 Ω, and R L = 14 Ω. These values lead to a maximum overshoot, S max = 36.01% and a relative transient ripple δ max = 10.16%. The step change due to d is shown in Fig: 2.24. 31

28 30 32 v O (V) 34 36 38 40 42 0 1 2 3 4 5 t (ms) 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: V I = 48 V, D = 0.389, L = 334 mh, C = 68 µf, r C = 0.033 Ω, r L = 0.32Ω and R L = 14 Ω. An International Rectifier IRF150 power MOSFET is selected, which has a V DSS = 100V, I SM = 40 A, r DS = 55 mω, C o = 100pF, and Q g = 63nC. Also, an International Rectifier 10CTQ150 Schottky Common Cathode Diode is selected with a V R = 100V, I F(AV) = 10A, V F = 0.73V and R F = 28mΩ. The duty cycle for the MOSFET changes from 0.407 to 0.389 to obtain the correct output of 28 V as predicted using MatLab. Also the switching frequency for V p which controls the duty cycle is 100kHz 32

D L C v i V I V p r L r C R L i o v o Figure 2.25: Open-loop buck-boost model with disturbances. V out (V) 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 0 1 2 3 4 5 6 7 8 t (ms) Figure 2.26: Open-loop buck-boost response without disturbances. allowing for a fast response time. The disturbances to the system are v i, i o 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

D L C v i V I V p r L r C R L v o 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 between 27.33V and 27.55V, the average value of steady state is 27.425V. The maximum overshoot is S max = 21.74% and settling time within one percent is 2ms which contains the ripple of steady state value. The relative maximum overshoot is δ 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 S max = 88% and settling time within one percent is 1.4ms which contains the ripple of steady state value. The relative maximum overshoot is δ max = 0.779%. 34

27.4 27.6 27.8 28 V out (V) 28.2 28.4 28.6 28.8 29 5 6 7 8 9 10 t (ms) Figure 2.28: Open-Loop step response due to step change in input voltage using PSpice. D L C V I R L i o V p r r L C v o Figure 2.29: PSpice model of Open-Loop buck-boost with step change in load current. 35

27.8 27.9 28 V out (V) 28.1 28.2 28.3 28.4 28.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t (ms) 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 V p is opened and the switch to V p2 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 the output voltage is contained between the bounds of 33.86 V and 34.2 V. The settling time is within one percent, which contains the ripple, of steady state value is 2.5ms. 36

D V I L C v R o L V p r r L C V p 2 Figure 2.31: PSpice model of Open-Loop buck-boost with step change in duty cycle. 27 28 29 30 V out (V) 31 32 33 34 35 0 1 2 3 4 5 6 t (ms) Figure 2.32: Open-Loop step response due to step change in duty cycle using PSpice. 37

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. v r, v e, v c and v f are all ac components of the reference voltage, error voltage, output voltage of controller and and feedback voltage respectively. T p is the small-signal control to output transfer function of the non-controlled buck-boost. M v is the open-loop input to output voltage transfer function. Z o is the open-loop output impedance. T m 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. T c 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 v r, v i, and i o respectively and the output is v o. The output voltage is expressed in transfer functions as v o (s) = T ct m T p 1βT c T m T p v r M v 1βT c T m T p v i Z o 1βT c T m T p i o 38

v GS v I v i V I L C R L i o v o v R v E Z i V v c V d T v AB v F Z f v t v F β R A R B v o Figure 3.1: Closed loop circuit of voltage controlled buck-boost with PWM. i i Z 2 v sd D ( v i o v) v i I d L Di l V d SD C R L v o i o Z 1 L i l r C r Ζ icl Ζ ocl v R v f v e v T c T d β v f R B R A v o Figure 3.2: Closed loop small-signal model of voltage controlled buck-boost. 39

Z o i o M v v i v o v o v e T m v c T c d T p v o v o v R v f α β Figure 3.3: Block diagram of a closed-loop small-signal voltage controlled buck-boost. Z o i o v o v i v o M v v o 1 1 T v o A v r Figure 3.4: Simplified block diagram of a closed-loop small-signal voltage controlled buckboost. Where A = T c T m T p and T = βa PWM transfer function is expressed = A 1T v r M v 1T v i Z o 1T i o = T cl v r M vcl v i Z ocl i o (3.1) 40

Where V T m is peak value of the triangular pulse. T m d(s) v c (s) = 1 V T m (3.2) Combining T m and T p together, the control to output transfer function is produced giving a new function T mp, given by T mp (s) = T m (s)t p (s) = v o r C V T m (1 D)(R L r C ) (sω zn )(s ω zp ) s 2 2ζ ω o sω 2 o (3.3) T mpx = v o r C V T m (1 D)(R L r C ) (3.4) T mpo = T m T po = T po V T m = v o r C ω zn ω zp V T m (1 D)(R L r C )ω 2 o (3.5) Figs: 3.5 and 3.6 show the Bode plots of T mp. To find the compensator value find T k T k v f v c vi =0 = βt m T p = βt mp βv o r C = V T m (1 D)(R L r C ) (sω zn )(sω zp ) s 2 2ζ ω o sω 2 o (3.6) T kx = βv o r C V T m (1 D)(R L r C ) (3.7) βv o r C ω zn ω zp T ko = V T m (1 D)(R L r C )ωo 2 (3.8) 41

Changing from s-domain into jω domain gives T k = T ko ( ) 2 ( ) 2 1 f f zn 1 f f zp ( ( ) ) 2 2 ( ) 2 1 f f o 2ζ f f o (3.9) ( ) ( ) f f φ Tk = tan 1 tan 1 tan 1 f zn f zp ( ) 2ζ f f o ( ) 2 1 f f o when f f o < 1, (3.10) ( ) ( ) f f φ Tk = 180 tan 1 tan 1 tan 1 f zn f zp ( ) 2ζ f f o ( ) 2 1 f f o when f f o > 1. (3.11) Figs: 3.7 and 3.8 show the Bode plots of T k. 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 Z ocl. 5. To reduce sensitivity of the closed-loop gain T cl to component values over a wide frequency range. 6. To reduce the input to output noise transmission. 42

40 30 20 T mp (db) 10 0 10 20 30 40 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 3.5: Magnitude Bode plot of modulator and input control to output voltage transfer function T mp for a buck-boost. 180 150 120 90 φ Tmp ( ) 60 30 0 30 60 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 3.6: Phase Bode plot of modulator and input control to output voltage transfer function T mp for a buck-boost. 43

10 0 10 T k (db) 20 30 40 50 60 10 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 3.7: Magnitude Bode plot of the input control to output voltage transfer function T k before the compensator is added for a buck-boost. 0 30 60 90 φ Tk ( ) 120 150 180 210 240 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 3.8: Phase Bode plot of the input control to output voltage transfer function T k before the compensator is added for a buck-boost. 44

R bd C 2 R 3 C 3 C 1 R 2 h 11 v F R 1 vc v R 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 90 phase shift at all frequencies. This causes stability issues which is negated by the lead part 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 = Z i = h 11 R 1 (R 3 1 sc 3 ) R 1 R 3 1 sc 3 ) ) h 11 (R 1 R 3 sc 1 3 R 1 (R 3 sc 1 3 R 1 R 3 1 sc 3 = sc 3h 11 R 1 sc 3 h 11 R 3 h 11 R 1 R 3 sc 3 R 1 sc 3 R 1 sc 3 R 3 1 = C 3(h 11 (R 1 R 3 )R 1 R 3 ) C 3 (R 1 R 3 ) s h 11 R 1 C 3 (h 11 (R 1 R 3 )R 1 R 3 ) 1 s C 3 (R 1 R 3 ) 45

( = h 11 R ) 1R 3 s h 11 R 1 C 3 (h 11 (R 1 R 3 )R 1 R 3 ) (R 1 R 3 ) 1 s C 3 (R 1 R 3 ) (3.12) Z f = ( ) 1 sc 2 R 2 sc 1 1 R 2 1 sc 1 1 sc 2 = 1 sc 2 R 2 1 C 1 C 2 s 2 R 2 1 sc 1 1 sc 2 = 1 C 2 ( ) s R 1 2 C ( 2 ) (3.13) s s C 1C 2 R 2 C 1 C 2 and h 11 = R AR B R A R B. 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 = A v (s) v 1 c(s) v f (s) = Z f C = 2 Z i h 11 R 1R 2 R 1 R 2 R 1 R 3 C 2 (h 11 (R 1 R 3 )R 1 R 3 ) ( ) s R 1 2 C ( 2 s s C 1 C ) 2 R 2 C 1 C 2 h s 11 R 1 C 3 (h 11 (R 1 R 3 )R 1 R 3 ) s 1 C 3 (R 1 R 3 ) ( )( ) s R 1 1 2 C 2 s C 3 (R 1 R 3 ) ( s s C 1C 2 h R 2 C 1 C 2 )(s 11 R 1 C 3 (h 11 (R 1 R 3 )R 1 R 3 ) ) (3.14) Because v r = 0, v e = v r v f = v f, the voltage transfer function of the integral lead controller is T c v c v e = v c(s) v f (s) = B (sω zc1)(sω zc2 ) s(sω pc1 )(sω pc2 ) (3.15) where B = R 1 R 3 C 2 (h 11 (R 1 R 3 )R 1 R 3 ) 46

( ω zc1 = s 1 ) R 2 C 2 ( ) 1 ω zc2 = s C 3 (R 1 R 3 ) ω pc1 = ( s C ) 1 C 2 R 2 C 1 C 2 ( ) h 11 R 1 ω pc2 = s C 3 (h 11 (R 1 R 3 )R 1 R 3 ) Assume that ω zc1 = ω zc2 = ω zc and ω pc1 = ω pc2 = ω pc. Therefore K = ω pc1 ω zc1 = ω pc2 ω zc2 = ω pc ω zc = C 1 C 2 R 2 C 1 C 2 1 R 2 C 1 = h 11 R 1 C 3 (h 11 (R 1 R 3 )R 1 R 3 ) 1 C 3 (R 1 R 3 ) = (h 11 R 1 )(R 1 R 3 ) h 11 (R 1 R 3 )R 1 R 3 = 1 C 2 C 1 (3.16) This leads to the voltage transfer function of the controller to be T c v c = B (sω zc) 2 v e s(sω pc ) 2 = B (sω zc) 2 s(sω pc ) 2 = B(1 ω s zc ) 2 K 2 s(1 ω s pc ) 2. For s = jω, the magnitude and phase shift of T c is T c ( jω) = B(1 ω ω zc ) 2 K 2 s(1 ω ω pc ) 2 = B ω m K = 1 ω m C 2 (R 1 h 11 ) 47

and φ Tc = π 2 2tan 1 ω ω zc ω ω pc 1 ω2 ω zc ω pc Design of Integral Lead Controller For stability reasons a gain margin GM 9dB, a phase margin PM 60, and the cutoff frequency f c = 2kHz is chosen. The values of the buck-boost are V I = 48 V, D = 0.407,V F = 0.7 V, r DS = 0.4 Ω, R F = 0.02Ω, L = 334 mh, C = 68 µf, r C = 0.033 Ω, and R L = 14 Ω. The maximum value of phase in T c occurs ω c = ω m = Kω zc = ω pc K = K R 2 C 1 = h 11 R 1 KC3 (h 11 (R 1 R 3 )R 1 R 3 ) Therefore the maximum phase shift possible can be described by φ m = φt c ( f m ) π ( ) K 1 2 = 2tan 1 2 K Solving for K leads to ( ) 1sin φm2 K = ) = tan 1 sin( 2( φm φm2 4 π ) 4 Therefore, φ m = π 4tan 1 ( K) Assuming V T m = 5V, the reference voltage is V R = D nom V T m =.407(5) = 2.035V 48

The feedback network transfer function βis β = V F = V R = V o V o R A = 2 R A R B 28 = 0.0714 Assuming R B = 910Ω, R A is ( ) ( ) 1 1 R A = R B β 1 = 910.0714 1 = 11.83kΩ = 12kΩ If R B = 910Ω, R A = 12kΩ then h 11 = R AR B R A R B = 846Ω. h 22 can be neglected because R A R B is so much larger then R L. Utilizing the cutoff frequency f c the phase φ Tk and φ m are and ( ) ( ) φ Tk = 180tan 1 fc tan 1 fc tan 1 f zn f zp φ m = PM φ Tk 90 = 153.9. 2ζ f c f o 1 ( fc fo ) 2 = 183.9 This leads to ( K = tan 2 φm 4 π ) = 76.42. 4 Knowing K, f zc and f zp are calculated f zc = f c K = 228.779Hz 49

and f zp = f c K = 17.484kHz The magnitude of T k and T c are used to calculate B T k = T ko ) 2 ( ) 2 1( fc f zn 1 fc f zp [ ( ) ] 2 2 ( ) 2 1 fc fo 2ζ fc f o = 0.1945 Therefore T c = 1 T k = 1.1954 = 5.141 B = ω c K T c = 4.9374x10 6 rad/s Values of compensator are calculated. Assume R 1 = 100kΩ and using the equations above C 2 = T k( f c ) =.1535nF.15nF. ω c (R 1 h 11 ) R 3 = R 1[R 1 h 11 (K 1)] (K 1)(R 1 h 11 ) = 475Ω 470Ω C 1 = C 2 (K 1) = 11.313nF 12nF R 2 = K ω c C 1 = 57.97kΩ 56kΩ C 3 = R 1 h 11 = 6.95nF 6.8nF Kωc [R 1 R 3 h 11 (R 1 R 3 )] 50

The pole and zero frequencies of the control circuit with standard resistor and capacitor values are f zc1 = 1 2πR 2 C 1 = 236.84Hz f zc2 = 1 2πC 3 (R 1 R 3 ) = 232.96Hz ( ) C1 f pc1 = f zc1 1 = 19.184kHz C 2 and f pc2 = R 1 h 11 2πC 3 [R 1 R 3 h 11 (R 1 R 3 )] = 17.881kHz. Figs: 3.10 and 3.11 show the Bode plots of T c. 3.1.2 Loop Gain of System Loop gain of the system is T(s) v f v e vi =i o =0 = βt c T m T p = T c T k βbv o r C T(s) = V T m (1 D)(R L r C ) ( (sωzc ) 2 ) (sω zn )(s ω zp ) s(sω pc ) 22 2ζ ω o sωo 2) ( (sωzc ) 2 ) (sω zn )(s ω zp ) T(s) = T x s(sω pc ) 22 2ζ ω o sωo 2) (3.17) where βbv o r C T x = V T m (1 D)(R L r C ) (3.18) 51

70 60 50 T c (db V) 40 30 20 10 0 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 3.10: Magnitude Bode plot of the controller transfer function T c for a buck-boost. 90 60 30 φ Tc ( ) 0 30 60 90 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 3.11: Phase Bode plot of the controller transfer function T c for a buck-boost. 52

Figs: 3.12 and 3.13 show the Bode plots of T. 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 T cl v o v r io =v i =0 = T ct m T p 1βT c T m T p = 1 β T 1T = 1 β T 1T T cl = 1 β βbv o r C [ (sωzc ) 2 ] (sω zn )(s ω zp ) V T m (1 D)(R L r C ) s(sω pc ) 2 (s 2 2ζ ω o sωo 2) T cl = T x β ( (sω zc ) 2 (sω zn )(s ω zp ) s(sω pc ) 2 (s 2 2ζ ω o sω 2 o) ) (3.19) Figs: 3.14 and 3.15 show the Bode plots of T cl. Figs: 3.16 and 3.17 show the discrete point Bode plots of T cl. 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 M vcl = 1 M vcl v o v t vr =i o =0 = M v 1T (1 D)DR L r C sω zn L(R L r C ) s 2 2ζω o sωo 2 βbv o r C V T m (1 D)(R L r C ) ( ) (sωzc ) 2 (sω zn )(s ω zp ) s(sω pc ) 2 (s 2 2ζω o sωo 2) sω M zn ( vx (s = 2 2ζω o sωo) 2 s(sωpc ) 2 (s 2 2ζ ω o sωo 2)) s(sω pc ) 2 (s 2 2ζ ω o sωo 2)T x(sω zc ) 2 (sω zn )(s ω zp ) 53

30 20 10 T (db) 0 10 20 30 40 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 3.12: Magnitude Bode plot of the loop gain transfer function T for a buck-boost. 0 30 60 90 120 φ T ( ) 150 180 210 240 270 300 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 3.13: Phase Bode plot of the loop gain transfer function T for a buck-boost. 54

25 20 15 T cl (db V) 10 5 0 5 10 15 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 3.14: Magnitude Bode plot of the input control to output transfer function T cl for a buck-boost. 180 150 120 90 60 φ Tcl ( ) 30 0 30 60 90 120 10 1 10 2 10 3 10 4 10 5 f (Hz) Figure 3.15: Phase Bode plot of the input control to output transfer function T cl for a buckboost. 55