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`Iii-again Makfiimqrvié
`
`Fundamentals qt
`Power Electronics
`
`Second Edition
`
`
`
`INTEL 1324
`
`INTEL 1324
`
`1
`
`

`

`Fundamentals of
`
`Power Electronics
`
`SECOND EDITION
`
`Robert W. Erickson
`
`Dragan Maksimovié
`University of Colorado
`Boulder, Colorado
`
`2
`
`

`

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`Library ol' Ccngrcaa Cataloging-ia-Pahlicalion
`
`Erickson. Robert W. {Robert Warm}. [956—
`Fundunentals of power electronics I Robert W. Erickson Dragon Marksinowic.—Jl’"f'iI ed.
`p. cm.
`Includes bibliographical references and index.
`ISBN 913-1-475L9559-1
`ISBN QTS-fl-JDfi-diifldiiufi {efloukl
`
`[N1] illlllflfl'i'li-fl-ZiDfi-dmfi
`
`i. Power electronics. i. Maksimnric. Draaan. Isor- il. Title.
`
`HUSH.” .Efllflflfl
`
`fillJflludfll
`
`
`
`oooszsfia
`
`Copyright ‘7' IDDI hy Khmer Academic Publishers. Sixth Printing 101M.
`Cover on Copyright ‘ 199! by Laccnl Technologies Inc. Mi rights reserved. Used with
`Who
`Softcover reprint of the hardcover 2nd edition lml Gill-iiJflIi-‘TZ'ifl-T
`
`All rial“: rosin-red. No part cflllis publication may be reproduced. stored in a retrieval system or
`transmitted in any form or by any:r moans. mechanical. photo-swing, recording. or otherwise,
`without the prior wrinen permission of the publisher. Kiower Academic Publishers. In] Philip
`Drive, Assinippi Polk, Not-well, Massachusetts H206!
`
`Printed on acid—flee paper.
`
`3
`
`

`

`
`
`Preface
`
`1
`
`Introduction
`
`Li
`
`1.2
`
`[.3
`
`Introduction to Power Processing
`
`Several Applications of Power Elecu'onics
`
`Elements of Power Electronics
`
`References
`
`I
`
`2
`
`Cnnvertere in Equilibrium
`
`Principles ol' Steadyr State Converter Analysis
`
`2.1
`
`2.2
`
`2.3
`2.4
`
`2.5
`
`Introduction
`
`inductor Volt-Seems] Balance. Capacitor Charge Balance. and the Smell-Ripple
`Approximation
`
`fluent Converter Example
`Cult Cmverter Example
`
`Estimating the Output Voltage Ripple in Common Containing 'I'wo-Polc
`Law-Pace Filters
`
`2.6
`
`Summary of Key Points
`
`References
`
`Problems
`
`3
`
`Steady-State Equivalent Circuit Modeling. Lame, and Efficiency
`
`3.1
`
`3.2
`
`3.3
`
`The DC ”Transformer Model
`
`Inclusion at“ inductee Copper Lose
`
`Construction of Equivalent Circuit Model
`
`:tth
`
`1
`
`1
`
`T
`
`9
`
`ll
`
`13
`
`I3
`
`15
`
`22
`2‘:
`
`3]
`
`34
`
`34
`
`35
`
`39
`
`39
`
`42
`
`45
`
`4
`
`

`

`i"
`
`o= 1—
`T, n.
`
`"gotta: (it)
`
`(2.21)
`
`The average value. or dc component. of the capacitor current must be zero in cquilibri urn-
`This should be an intuitive result. It a dc current is applied to a capacitor. then the capacitor will
`charge continuallyr and its voltage will increase without bound. Likewise. if a dc voltage is applied to an
`inductor. then the flux will increase continually and the inductor current will increase without hound.
`Equation {2.27}. called the principle of capacitor amp-second balance or capacitor charge balance. can
`be used to find the steady-state currents in a switching converter.
`
`2.3
`
`BOOST CONVERTER EXAMPLE
`
`The boost converter. Fig. 2.13m}. is another well~known switchedurntxle converter that is capable of pro-
`ducing a dc output voltage greater in magnitude than the dc input voltage. A practical realization of the
`switch. using a MOSHE-7F and diode. is shown in Fig. 2. I3tb}. Let us apply the small-ripple approxima-
`tion and the principles of inductor volt—second balance and capacitor charge balance to find the steady“-
`state output voltage and inductor current for this converter.
`1With the switch in position i. the right-hand side of the inductor is connected to ground. result-
`ing in the network of Fig. 2.l4{a). The inductor voltage and capacitor current for this subinterval are
`given by
`
`V£=vi
`v
`l-(=—§
`
`{2.23}
`
`Use of the linear ripple approximation. v a it leads to
`
`tail
`
`[h]
`
`VS
`
`V
`
`+
`
`1.!
`
`v
`
`+
`
`5
`
`

`

`and want wntcfonns.
`
`With the switch in pmiticn 2, the: inductor is connected to the output, leading to the circuit of Fig.
`2.14m). Thc inductor 1Inmltagc and capacimr currcnt are thcn
`
`{1.29}
`
`”i.”vg“’
`. .'
`1.-
`f{--I£-*fi
`
`Use cfthc Emil-ripple approximaticn. v - 1Ir"mttii,_- !. Ircudx to
`
`Equatimts [2.29) and (2.31) arc uscd tn sketch the inducmr vultagc and capacitcr current waveform cf
`Fig. 2. I5.
`
`{2.3m
`
`fig. 1.15 How cmwcrtcr wing:
`
`'1‘”
`
`min 1
`
`6
`
`

`

`5 4
`
`3 ‘
`
`2
`
`Fig. LIE Dc conversion ratio MID} ol'
`the boost converter.
`
`
`{l
`{1.2
`{1.4
`llfi
`as
`l
`
`Hi
`
`It can be inferred from the inductor voltage waveform of Fig. 2.15m that the dc output voltage
`V is greater than the input voltage til. During the first subinterval. vtlt) is equal to the dc input voltage Pr
`and positive volt-seconds are applied to the inductor. Since. in steadyetate. the total volt—seconds applied
`over one switching period must be rem. negative volt-seconds must be applied during the second sub-
`intenol. Therefore. the inductor voltage during the second subinterval. {Vs — to. must be negative.
`Hence. t' is greater than Vr
`The total volt-seconds applied to the inductor over one switching period are:
`
`J: viirldr = (mot; + (V; — V]D‘T,
`
`By equating this expression to zero and collecting terms. one obtains
`
`nth + at — err = a
`
`Solution for H attd by noting that (D + If} = l. yields the expression for the output voltage.
`
`y s gr.
`
`(1331
`
`(2-33!
`
`(2.34)
`
`The voltage conversion ratio M{D} is the ratio of the output to the input voltage of a tic-dc converter.
`Equation (2.34} predicts that the voltage conversion ratio is given by
`
`_ v _
`
`“ml—”FITILPWZLE
`
`_
`
`_
`
`(235}
`
`This equation is plotted in Fig. 2J6. At t) = {3. V: VI. The output voltage increases as It increases. and in
`the ideal case tends to infinity as 13 tends to I. So the ideal boost converter is capable of producing any
`output voltage greater than the input voltage. There are. of course. limits to the output voltage that can be
`produced by a practical boost converter. In the next chapter. component nonideatities are modeled. and it
`is found that the maximum output voltage of a practical boost converter is indm limited. Nonethelefl.
`ventI large output voltages can he produced if the nonidealities are sufficiently small.
`The dc compment of the inductor current is derived by use of the principle of camcitor charge
`balance. flaring the first suhinterval. the capacitor supplies the load current. and the capacitor is partially.r
`discharged. During the second subinterval.
`the inductor current supplies the load and, additionally.
`recharges the capacitor. The act change in capacitor charge over one switching period is found by inte-
`nrnrinn flu—ts 17
`trill atria-tandoori“ n." Ii'in. Al 1‘Jrk'l
`
`7
`
`

`

`Fig. 1.11 variation of inductor cutters dc
`
`component I with duty cycle. burst converter.
`
`l]
`
`[1.1
`
`11.4
`
`lift
`
`{1.3
`
`l
`
`D
`
`F idfldt =( - "HUT. + (t _ 3;;)o'r,
`
`Collecting terms. and equating the result to acro. leads the steady—state result
`
`-g [D+D']+lfl'={l
`
`By noting that {D + D’II = l. and by solving for the inductor current dc compo-Mitt I. one obtains
`
`.t . _V_
`HR
`
`mm
`
`{13?}
`
`{133}
`
`So the inductor current dc component I is equal to the load current, WK. divided by D’. Substitution of
`Eq. {2.34} to eliminate V yields
`
`
`h v;
`D'JR
`
`{2.39}
`
`This equation is planed in Fig. 2.1T. it can be seen that the inductor current becomes large as D
`approaches I.
`This inductor current. which coincides with the dc input concur in the boost converter. is greater
`than the load current. Physically. this must he the case: to the extent that the converter elements are ideal.
`the converter input and output pmvcrs are equal. Since the etutterter output voltage is greater than the
`input voltage. the input cunent must likewise be greater than the output current. In practice. the inductor
`current [lows through the seaticonductor forward voltage drops. the inductor winding resistance. and
`other sources of power loss. As the duty cycle apporacltes one. the inductor current becomes very large
`and these component nonidealitics lead to large power losses. In crutsequcnce. the efficiency of the boost
`converter decreases rapidly at high duty cycle.
`Nest. let us sketch the inductor current so} waveftn'rn and derive an expression for the inductor
`current ripple art. The inductor voltage waveform vL{r) has been already found (Fig, 2. l5), so we can
`sketch the inductor current waveform directly. During the first suhintcrval. with the switch in position I.
`the slope ol‘the inductor current is given by
`
`likewise, when the switch is in position 2. the slope of the inductor current waveform is
`
`Lil” . sit! - 11:.
`dt
`L _ L
`
`team
`
`8
`
`

`

`Fig.
`
`1.18 Boost converter inductor current
`
`on)
`
`waveform on}.
`
`in m p m _
`it: ‘ J1— ' “LE"
`
`{2.41}
`
`The inductor cunent waveform is sketched in Fig. 2.13. During the first suhinterval. the change in induc-
`tor current. 2M... is equal to the slope multiplied by the length of the whinlervsl. or
`
`Solution for of. leads to
`
`V
`
`2 '1 I I! 0T.
`
`V
`
`or; = Ti: or,
`
`{2.4:}
`
`{1-43)
`
`This expression can housed to select the inductor value I. such that a given value of ML is obtained.
`Likewise, the capacitor voltage v(r} waveform can be sketched, and an expression derived for
`the output voltage ripple peak magnitude or. The capacitor current waveform ittt} is given in Fig. 2-15.
`During the first subintervai, the slope of the capacitor voltage waveform Ht} is
`
`Dining the second suhinterval. the slope is
`
`M_Etfl-;£
`dl'
`_ c 'nc
`
`fl£fl=iflflfié_%
`
`[2.44]
`
`{2.45}
`
`The capacitor voltage wavefon'n is sketched in Fig. 2J9. During the first subinterval, the change itt
`capacitor voltage. — 2cm is equal to the slope multiplied by the length of the sot-interval:
`
`_ 2s..- = it“? or,
`
`{2.46}
`
`Solution for the yields
`
`Fig.
`
`1.1! Boost converter output voltage
`
`waveform Flt].
`
`9
`
`

`

`10
`
`at - 31%. or,
`
`(2.471
`
`This expression can be used to select the capacitor value if to obtain a given output voltage ripple peak
`magnitude as.
`
`2.4
`
`CUE COMRTER EXAMPLE
`
`As a mood example. consider the Cult convener of Fig. like). This converter performs a dc conver-
`sion function similar to the hock-hotel converter: it can either increase or decrease the magnitude of the
`dc voltage, and it inverts the polarity. A practical realization using a transistor and diode is illustrated in
`Fig. 220(1)).
`This converter operates via capacitive energy transfer. As illustrated in Fig. 2.21. capacitor C. is
`connected through L, to the input source while the switch is in position 2. and source energyr is stored in
`CI. 1When the switch is in position I. this energy is released through [.1 to the load.
`The inductor currents and capacitor voltages are defined. with polarities assigned somewhat
`arbitrarily. in Fig. 2.20m. in this section. the principles of inductor volt-sotond balance and capacitor
`charge balance are applied to find the dc components of the inductor contents and capacitor voltages. The
`voltage and current ripple magnitudes are also found.
`During the first subinterval. while the switch is in position I. the converter circuit reduces to
`Fig. 2.2 [{a). The inductor voltages and capacitor currents are:
`
`1"“ = VI
`v t = — v — vs
`L...
`l
`x.
`‘cl "2
`.
`.
`P1
`it“! = '1 — :3
`
`,
`{2.43}
`
`lb)
`
`it no cut converter: a with ideal switch a
`
`racticalrealiaation usin_ Mossermuaiuuc.
`
`10
`
`