R. W. Erickson Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder ECEN 5807! Modeling and Control of Power Electronics Systems Instructor: Robert Erickson • • • Offices: ECOT 356, ECEE 1B55 Email: [email protected] Office hours: Course web site: • • http://ecee.colorado.edu/~ecen5807 Announcements, course materials, assignments D2L course site • • Log into https://learn.colorado.edu with your identikey Course lectures, submission of assignments, solutions, grades, discussion forum Textbook: • ECEN 5807, Spring 2015 Erickson and Maksimovic, Fundamentals of Power Electronics, 2nd edition 1 Power Electronics Courses at the University of Colorado, Boulder undergraduate ECEN 3250 Circuits 3 ECEN 2260 Circuits 2 ECEN 2250 Circuits 1 graduate ECEN 4797/5797 Intro to Power Electr. ECEN 4517/5517 Power Electronics Lab ECEN 4827/5827 Analog IC Design ECEN3170 Energy Conversion ECEN 4167/5737 Energy Conversion 2 Prerequisite for either ECEN 5807 or ECEN 5817: ECEN 5797 2 Professional certificate in power electronics ECEN 5817 Res. and Soft-Sw. Tech ECEN 5837 Mixed-Signal IC Design ECEN 5737 AC Drives ECEN 5017 Pwr Elect for Electric Drive Vehicles Coursera (MOOC): Introduction to Power Electronics ECEN 5807, Spring 2015 ECEN 5807 Modeling and Control of Power Electronics Systems Power management Graduate certificate in electric drivetrain technology Assignments Weekly homework sets, 50% of total grade Midterm exam (open book/notes, take home), 17% of grade Final exam (comprehensive, open book/notes, take home), 33% of grade All assignments and due dates will be posted on course web site You must scan your homework or exam into a pdf • • • • Black-and-white, no color or grayscale 200-400 dpi is sufficient Upload a single pdf file to the Dropbox in D2L Due dates are typically at the start of Friday lecture; no late assignments will be accepted and the Dropbox will close automatically Due date and time for CAETE students and on-campus students are the same ECEN 5807, Spring 2015 3 Required Software A version of Spice • You can download LTSpice for free. See course Vitals page for a link • Examples in class will use LTSpice. PSpice files will also be linked on website. MATLAB/Simulink • Available in most department labs • You can buy the student version from Mathworks • We will use: Matlab, Simulink, and the Control Systems Toolbox. The student version of Matlab includes all of these and much more. ECEN 5807, Spring 2015 4 Topics 1. Simulation and averaged switch modeling • CCM, DCM, and other examples • Simulation 2. Techniques of design-oriented analysis, with application to switched-mode converter systems • Middlebrook’s feedback and extra-element theorems • Input filter design • Writing complex transfer function expressions by inspection 3. Current-programmed control of PWM converters 4. Introduction to digital control of PWM converters 5. Rectifiers • Rectifier harmonics in power systems • Low-harmonic rectifiers and power factor correction converters ECEN 5807, Spring 2015 5 1. Simulation and Averaged Switch Modeling • Additional notes, Section 7.4, Chapter 11, and Appendix B • Averaged switch modeling is another approach to derive the averaged model of a PWM converter. • Well suited to Spice modeling of PWM converters • We will use this approach to model CCM, DCM, and current-programmed converters • Also useful for incorporation of ac losses (switching loss, core loss) into averaged models of PWM converters • Computer simulation of small-signal transfer functions • • • ECEN 5807, Spring 2015 Objectives of simulation Spice models Simulink models 6 Averaged Switch Modeling and Simulation ECEN 5807, Spring 2015 7 2. Techniques of Design-Oriented Analysis Chapter 10, Appendix C, and supplementary notes on website Null double injection methods for analysis of complex analog systems • Converter applications Input filter design Exact analysis of a fifth-order converter system • Middlebrook’s extra element theorem How to easily determine the effect of an added element on a circuit transfer function, without starting the analysis all over again • The n extra element theorem How to write complicated transfer functions by inspection, in rational form • Middlebrook’s feedback theorem How to easily construct the loop gain and transfer functions of a complex feedback circuit ECEN 5807, Spring 2015 8 Q1 Q1 Q1 Q1 Now multiply out and average over one period: 0 = VQ1 IQ1 + hṽQ1 (t)ĩQ1 (t)i 0 = VQ1 IQ1 + hṽQ1 (t)ĩQ1 (t)i nd average over one period: The transistor “consumes” at DC,Theorem Middlebrook’s Extra power Element )ĩQ1 (t)i The transistor “consumes”power poweratatthe DC,switching and “generates” and “generates” power the switching frequency. The at transistor functions as an nsumes” power at DC, frequency.inverter. The transistor functions as an Appendix C inverter. V I = hṽ (t)ĩ (t)i ower at the switching Q1 Q1 Q1 Q1 nsistor functions as an Vfunction How a transfer modified hṽQ1is(t) ĩQ1 (t)i by addition of an extra element Z(s): Q1 IQ1 = G(s) 0 1 Z (s) N BBB 1 + CCC ! (t)i 0 1 Q1 BBB CCC ZN (s) vout (s) Z(s) B C BBB CC CCC ! BBB 1 + = G(s) CCC ZD (s) CC B B vout (s) B Z(s) v (s) B Z(s)!1 [email protected] 1 CC+ CA in BBB 0 1 = G(s) C ZN (s) C BB CCC Z(s) B Z (s) vin (s) D CCC Z(s)!1 B ! BBB 1 + @1 + A BBB Z(s) CCC Z(s) BBB CCC D (s) C !1 B [email protected] 1 + ZSimple CA methods to find ZN(s) and ZD(s) using null double injection Z(s) to design circuits so that the extra element doesn’t significantly change G(s): How k kZ( j!)k kZN ( j!)k kZ( j!)k kZD ( j!)k kZ( j!)k kZD ( j!)k Design-oriented result: construct Bode plots of above equations, and use to shape Z(s) ECEN 5807, Spring 2015 9 Input ﬁlter design Input ﬁlter design Input Filter Design Gvd H(s) H(s) vg vg + – + – Input filter Input filter Gvd Zo(s) Zo(s) Zi(s) Zi(s) 40 dB 40 dB 30 dB 30 dB 20 dB Converter v Converter v T(s) Gvd Gvd Gvd 20 dB 10 dB 10 dB 0 dB 0 dB – 10 dB d Gvd Gvd 0˚ Gvd 0˚ – 180˚ – 10 dB – 180˚ – 360˚ T(s) Controller d Controller 100 Hz 1 kHz f 100 Hz 1 kHz • Filter canseriously seriously degrade control system behavior f • Input filter can degrade controlconverter system and cause instability • behavior Filter can seriously degrade converter control system behavior – 360˚ – 540˚ 10 kHz – 540˚ 10 kHz • Use extra element theorem to derive conditions which ensure that converter • Usedynamics Extra Element derive conditions ensure that input filter does not disrupt dynamics of areTheorem not affected by inputthat ﬁlter • control Use system extra element theorem to derive conditions which ensure that converter arefilter not affected by input ﬁlter damping •dynamics Must design input ﬁlter having adequate • Must design input having adequate damping 10 • Must design input ﬁlter having adequate damping ECEN 5807, Spring 2015 degrade converter transfer functions Design of Input Filters that Do Not Degrade Converter Transfer Functions Design criteria derived via Extra Element Design criteriatheorem: derived via Extra Element Two-section damped input ﬁlter design: damped input filter design: Two-section Theorem: 30 dB ZD ZN R2 n2 L 2 0.65 2.9 H R1 1.9 n1L1 15.6 H 20 dB fo 10 dB L2 5.8 H Cascaded sections 1 and 2 vg + – Section 1 alone L1 31.2 H C2 11.7 F 0 dB -10 dB Z( j ) > Z N ( j ) -20 dB 1 kHz ECEN 5807 : Introduction ECEN 5807, Spring 2015 10 kHz 100 kHz 9 11 Z( j ) > Z D( j ) C1 6.9 F Write the line-to-output transfer function by inspection Write the line-to-output transfer function by inspection Example: buck-boost with input ﬁlter Example: buck-boost with input filter Lf 1:D (V g – V)d (t) D' : 1 + – vg (t) + – L Rf Cf I d(t) I d(t) Cb Solution: use n extra element theorem Solution: use n extra element theorem ECEN 5807, Spring 2015 12 10 C R • Chapter 12 iL(t) + Q1 • Control A very popular method for 3. Current3.Programmed Current-Programmed Control v(t) C D1 Buck converter L turns off when its • Chapter 12 current i (t) is equal to the s • A very popular method for • A very popular method for controlling PWM converters controlling converters control (t) is(t) is equal to a • Transistor turnsPWM offinput when itsiccurrent Q1 vg(t) + – Clock Rf 0 Measure switch current v(t) C D1 signal ic(s) •control Transistor turns off when its • Simpler dynamics, more • Simpler dynamics, more robust compensator current is(t) is equal to the robust control input icompensator c(t) S Q is(t) + Rf – Clock R 0 Analog is(t)Rf comparator • Simpler dynamics, robust compensator Latch S Q R Current-programmed controller Analog comparator ic(t)Rf Control signal more ic(t) Ts + – m1 Latch Current-programmed controller Compensator Compensator vref m1 Switch current is(t) 0 v(t) Transistor 0 status: Transistor status: vref on Clock turns Conventional output voltage controller Conventional output voltage controller ECEN 5807 : Introduction ECEN 5807, Spring 2015 ntroduction Control signal Switch ic(t) current is(t) v(t) + – Control input R – Ts + – (t)Rf • Transistor • Chapter 12 iL(t) + is(t) (t)Rf controlling PWM converters R – is(t) ure tch ent ntrol input L is(t) dTs dTs Ts on off Comparator turns off Clock turns Comparator turns transistor on off transistor transistor on transistor 11 11 13 Ts off t t Buck converter example Effect of current programming on converter transfer functions! Buck converter example Comparison of control-to-output Comparison of control-to-output transfer functions transfer functions Averaged switch model used in Averaged model used in spice simulations PSPICEswitch simulations 40 dB 1 G G + – 12 V Gvc 5 0 dB 2 CCM-DCM1 Vg Gvd 3 iL 35 H 1 20 dB L 2 4 iLOAD RL + 0.05 100 F fs = 200 kHz L = 35 – 3 4 d Gvd 0˚ CPM –90˚ Gvc 10 Hz 100 Hz 1 kHz 10 kHz –180˚ 100 kHz f ECEN 5807, Spring 2015 Xcpm d –60 dB 14 control current Rf iL vc + – Ei 1 2 v(1)–v(3) + E1 + – – v(3) E2 + – v 10 –20 dB –40 dB R C Xswitch Rf = 1 fs = 200 kHz L = 35 Va = 0.6 V Digitally Controlled Buck Converter! Digitally Controlled BuckModel Converter Simulink Simulink Model CoPEC • The buckconverter converter • Buck block is same as in continuous-time block is the same assystem in• the continuousNote the parts of the system that model digital controller, including: timethesystem • A/D converter • Note the parts of the system• that model compensator Discrete-time the digital controller • Digital PWM including: – A/D converter – Discrete-time compensator, and – Digital PWM Digital PWM A/D converter Discrete-time compensator ECEN 5807, Spring 2015 15 5 odern rectiﬁers, power system harmonics, 4. Modern Rectifiers, Power System Harmonics, and LowHarmonic Rectifiers and low harmonic rectiﬁers • The traditional peak-detection rectifier injects very large harmonic currents into the ac power line. • At substantial power levels, this type of rectifier is not allowed Harmonic amplitude, percent of fundamental 100% 100% 91% 80% THD = 136% Distortion factor = 59% 73% 60% 52% 40% 32% 19% 15% 15% 13% 9% 20% 0% 1 3 5 7 9 11 13 15 17 19 Harmonic number ECEN 5807, Spring 2015 16 The Ideal Rectiﬁer The Ideal Rectifier Modeling the basic Modeling the basic functions of idealfunctions converters of ideal converters DC-DC Dc-dcconverter: converter: AC-DC Ac-dcrectifier: rectiﬁer: DCtransformer transformer dc “Loss-free resistor” iac(t) 1 : M(D) Vg + – R + + V vac(t) – – Ideal rectifier (LFR) 2 p(t) = vac / Re Re(vcontrol) + v(t) dc output vcontrol 17 i(t) – ac input ECEN 5807, Spring 2015 “loss-free resistor” Controlling a dc-dc converter to behave as an ideal rectiﬁer Controlling a DC-DC Converter to Behave as an Ideal Rectifier dc-dc converter ig(t) 1 : M(d(t)) + iac(t) vac(t) i(t) + vg(t) v(t) – – d(t) ig controller vg Controller varies d(t) as necessary, to cause ig(t) to be proportional to vg(t) Controller varies d(t) as necessary, to cause ig(t) to be proportional to vg(t) ECEN 5807, Spring 2015 18 C R Next Lecture Begin with circuit averaging and averaged switch modeling Assignment: read Sections 7.4 and 7.5 ECEN 5807, Spring 2015 19

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