Buck–boost converter
This article is about the type of switched-mode power supply. For the autotransformer, seebuck–boost trans-former.
The buck–boost converter is a type ofDC-to-DC
con-The basic schematic of an inverting buck–boost converter. verterthat has an output voltage magnitude that is either greater than or less than the input voltage magnitude. It is equivalent to aflyback converterusing a single inductor instead of a transformer.[1]
Two different topologies are called buck–boost converter. Both of them can produce a range of output voltages, from an output voltage much larger (in absolute magni-tude) than the input voltage, down to almost zero. The inverting topology The output voltage is of the
oppositepolaritythan the input. This is a switched-mode power supplywith a similar circuit topology to theboost converterand thebuck converter. The output voltage is adjustable based on theduty cycle
of the switching transistor. One possible drawback of this converter is that the switch does not have a terminal at ground; this complicates the driving cir-cuitry. Another drawback is of any consequence if the power supply is isolated from the load circuit (if, for example, the supply is a battery) because the sup-ply and diode polarity can simsup-ply be reversed. The switch can be on either the ground side or the supply side.
Abuck (step-down) convertercombined with aboost (step-up) converter
The output voltage is typically of the same polarity of the input, and can be lower or higher than the input. Such a non-inverting buck-boost converter may use a single inductor which is used for both the buck inductor and the boost inductor,[2][3][4] sometimes called a “four-switch buck-boost converter”,[5] it may use multiple inductors but only a single switch as in the SEPIC and Ćuk
topologies.
The rest of this article describes the inverting topology.
1 Principle of operation
V i V V o LD
S
C
R
L
I L I S I D V DFig. 1: Schematic of a buck–boost converter.
On-State
Off-State
Fig. 2: The two operating states of a buck–boost converter: When the switch is turned on, the input voltage source supplies current to the inductor, and the capacitor supplies current to the resistor (output load). When the switch is opened, the inductor supplies current to the load via the diode D.
The basic principle of the buck–boost converter is fairly simple (see figure 2):
• while in the On-state, the input voltage source is di-rectly connected to the inductor (L). This results in accumulating energy in L. In this stage, the capaci-tor supplies energy to the output load.
• while in the Off-state, the inductor is connected to the output load and capacitor, so energy is trans-ferred from L to C and R.
2 1 PRINCIPLE OF OPERATION
Compared to thebuckandboostconverters, the charac-teristics of the buck–boost converter are mainly:
• polarity of the output voltage is opposite to that of the input;
• the output voltage can vary continuously from 0 to
−∞ (for an ideal converter). The output voltage ranges for a buck and a boost converter are respec-tivelyVito 0 andVito∞.
1.1
Conceptual overview
Like the buck and boost converters, the operation of the buck-boost is best understood in terms of the inductor’s “reluctance” to allow rapid change in current. From the initial state in which nothing is charged and the switch is open, the current through the inductor is zero. When the switch is first closed, the blocking diode prevents current from flowing into the right hand side of the circuit, so it must all flow through the inductor. However, since the inductor doesn't like rapid current change, it will initially keep the current low by dropping most of the voltage pro-vided by the source. Over time, the inductor will allow the current to slowly increase by decreasing its voltage drop. Also during this time, the inductor will store en-ergy in the form of a magnetic field.
1.2
Continuous mode
T D.T T Off T On t t t V L V O V D 0 0 0 V oltage Current Switch state I min I av I max V i I L 0n 0n 0ff V i -V O V O I S I DFig 3: Waveforms of current and voltage in a buck–boost con-verter operating in continuous mode.
If the current through the inductor L never falls to zero during a commutation cycle, the converter is said to op-erate in continuous mode. The current and voltage wave-forms in an ideal converter can be seen in Figure 3. Fromt=0tot=D T , the converter is in On-State, so the
switch S is closed. The rate of change in the inductor current (IL) is therefore given by
d IL d t =
Vi
L
At the end of the On-state, the increase of IL is therefore:
∆ILOn = ∫ D T 0 d IL = ∫ D T 0 Vi L d t = ViD T L
D is the duty cycle. It represents the fraction of the com-mutation period T during which the switch is On. There-fore D ranges between 0 (S is never on) and 1 (S is always on).
During the Off-state, the switch S is open, so the inductor current flows through the load. If we assume zero voltage drop in the diode, and a capacitor large enough for its voltage to remain constant, the evolution of IL is:
d IL d t =
Vo
L
Therefore, the variation of IL during the Off-period is:
∆ILOff = ∫ (1−D)T 0 d IL= ∫ (1−D)T 0 Vo d t L = Vo(1− D) T L As we consider that the converter operates in steady-state conditions, the amount of energy stored in each of its components has to be the same at the beginning and at the end of a commutation cycle. As the energy in an in-ductor is given by:
E = 1 2L I
2 L
it is obvious that the value of IL at the end of the Off state must be the same with the value of IL at the beginning of the On-state, i.e. the sum of the variations of IL during the on and the off states must be zero:
∆ILOn+ ∆ILOff= 0
Substituting ∆ILOnand ∆ILOffby their expressions yields:
∆ILOn+ ∆ILOff=
ViD T
L +
Vo(1− D) T
L = 0
This can be written as:
Vo
Vi
= −D
1− D
This in return yields that:
D = Vo
1.4 Limit between continuous and discontinuous modes 3
From the above expression it can be seen that the polar-ity of the output voltage is always negative (because the duty cycle goes from 0 to 1), and that its absolute value increases with D, theoretically up to minus infinity when D approaches 1. Apart from the polarity, this converter is either step-up (a boost converter) or step-down (a buck converter). Thus it is named a buck–boost converter.
1.3
Discontinuous mode
Fig 4: Waveforms of current and voltage in a buck–boost con-verter operating in discontinuous mode.
In some cases, the amount of energy required by the load is small enough to be transferred in a time smaller than the whole commutation period. In this case, the current through the inductor falls to zero during part of the pe-riod. The only difference in the principle described above is that the inductor is completely discharged at the end of the commutation cycle (see waveforms in figure 4). Al-though slight, the difference has a strong effect on the output voltage equation. It can be calculated as follows: Because the inductor current at the beginning of the cycle is zero, its maximum value ILmax(at t = D T ) is
ILmax=
ViD T
L
During the off-period, IL falls to zero after δ.T:
ILmax+
Voδ T
L = 0
Using the two previous equations, δ is:
δ =−ViD Vo
The load current Iois equal to the average diode current (
ID). As can be seen on figure 4, the diode current is equal
to the inductor current during the off-state. Therefore, the output current can be written as:
Io= ¯ID=
ILmax
2 δ
Replacing ILmax and δ by their respective expressions
yields: Io=− ViD T 2L ViD Vo =−V 2 i D2T 2L Vo
Therefore, the output voltage gain can be written as:
Vo
Vi
=−ViD 2T 2L Io
Compared to the expression of the output voltage gain for the continuous mode, this expression is much more complicated. Furthermore, in discontinuous operation, the output voltage not only depends on the duty cycle, but also on the inductor value, the input voltage and the output current...
1.4 Limit between continuous and
discon-tinuous modes
-6 -5 -4 -3 -2 -1 0 1 0 0.05 0.1 0.15 0.2 0.25 Normalized voltage Normalized current continuous discontinuous D=0.8 D=0.6 D=0.4 D=0.2 D=0.0Fig 5: Evolution of the normalized output voltage with the nor-malized output current in a buck–boost converter.
As told at the beginning of this section, the converter op-erates in discontinuous mode when low current is drawn by the load, and in continuous mode at higher load current levels. The limit between discontinuous and continuous modes is reached when the inductor current falls to zero exactly at the end of the commutation cycle. with the no-tations of figure 4, this corresponds to :
4 2 NON-IDEAL CIRCUIT
D + δ = 1
In this case, the output current Iolim (output current at
the limit between continuous and discontinuous modes) is given by:
Iolim = ¯ID=
ILmax
2 (1− D)
ReplacingILmaxby the expression given in the
discontinu-ous mode section yields:
Iolim =
ViD T
2L (1− D)
AsIolimis the current at the limit between continuous and
discontinuous modes of operations, it satisfies the expres-sions of both modes. Therefore, using the expression of the output voltage in continuous mode, the previous ex-pression can be written as:
Iolim = ViD T 2L Vi Vo (−D)
Let’s now introduce two more notations:
• the normalized voltage, defined by|Vo|=VoVi . It
cor-responds to the gain in voltage of the converter; • the normalized current, defined by|Io|=T ViL Io. The
term T Vi
L is equal to the maximum increase of the
inductor current during a cycle; i.e., the increase of the inductor current with a duty cycle D=1. So, in steady state operation of the converter, this means that|Io|equals 0 for no output current, and 1 for the
maximum current the converter can deliver. Using these notations, we have:
• in continuous mode,|Vo|=−1−DD ; • in discontinuous mode,|Vo|=−2D2|Io|;
• the current at the limit between continuous and dis-continuous mode isIolim=Vi T2L D(1−D)=2Iolim|Io|D(1−D).
Therefore the locus of the limit between continuous and discontinuous modes is given by 1
2|Io|D(1−D)=1
.
These expressions have been plotted in figure 5. The dif-ference in behavior between the continuous and discon-tinuous modes can be seen clearly.
2
Non-ideal circuit
-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalised Voltage Duty Cycle Ideal inductor (R L =0) Non-ideal inductor (R/R L =100) Inductor no ideal (R/R L =10)Fig 6: Evolution of the output voltage of a buck–boost converter with the duty cycle when the parasitic resistance of the inductor increases.
2.1 Effect of parasitic resistances
In the analysis above, no dissipative elements (resistors) have been considered. That means that the power is trans-mitted without losses from the input voltage source to the load. However,parasitic resistancesexist in all circuits, due to theresistivityof the materials they are made from. Therefore, a fraction of the power managed by the con-verter is dissipated by these parasitic resistances. For the sake of simplicity, we consider here that the in-ductor is the only non-ideal component, and that it is equivalent to an inductor and a resistor in series. This assumption is acceptable because an inductor is made of one long wound piece of wire, so it is likely to exhibit a non-negligible parasitic resistance (RL). Furthermore, current flows through the inductor both in the on and the off states.
Using the state-space averaging method, we can write:
Vi= ¯VL+ ¯VS
where V¯L and V¯S are respectively the average voltage
across the inductor and the switch over the commuta-tion cycle. If we consider that the converter operates in steady-state, the average current through the inductor is constant. The average voltage across the inductor is:
¯ VL= L ¯ dIL dt + RL ¯ IL = RLI¯L
When the switch is in the on-state, VS=0. When it is
off, the diode is forward biased (we consider the contin-uous mode operation), thereforeVS=Vi−Vo . Therefore,
the average voltage across the switch is:
¯
5
The output current is the opposite of the inductor cur-rent during the off-state. the average inductor curcur-rent is therefore:
¯
IL= −Io 1− D
Assuming the output current and voltage have negligible ripple, the load of the converter can be considered purely resistive. If R is the resistance of the load, the above ex-pression becomes:
¯
IL= −Vo (1− D)R
Using the previous equations, the input voltage becomes:
Vi= RL −Vo
(1− D)R+ (1− D)(Vi− Vo) This can be written as:
Vo
Vi
= R −D
L
R(1−D)+ 1− D
If the inductor resistance is zero, the equation above be-comes equal to the one of the ideal case. But when RL in-creases, the voltage gain of the converter decreases com-pared to the ideal case. Furthermore, the influence of RL increases with the duty cycle. This is summarized in fig-ure 6.
3
See also
• Ćuk converter • SEPIC converter
4
References
[1] The Flyback Converter- Lecture notes - ECEN4517 -
De-partment of Electrical and Computer Engineering - Uni-versity of Colorado, Boulder.
[2] “Non-inverting Buck-Boost Regulator” (p.9)
[3] ST AN2389: “An MCU-based low cost non-inverting
buck-boost converter for battery chargers”
[4] Motorola Semiconductor. “Application note AN954: A
Unique Converter Configuration provides step-up/down functions”. 1985. "... a unique step-up/down configura-tion can be created ... which still employs a single inductor for the voltage transformation.”
[5] Haifeng Fan. “Wide VIN and High-Power Challenges
with Buck-Boost Converters”. 2015.
5 Further reading
• Daniel W. Hart, “Introduction to Power Electron-ics”, Prentice Hall, Upper Saddle River, New Jersey USA, 1997ISBN 0-02-351182-6
• Christophe Basso, Switch-Mode Power Supplies: SPICE Simulations and Practical Designs. McGraw-Hill.ISBN 0-07-150858-9.
6 6 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
6
Text and image sources, contributors, and licenses
6.1
Text
• Buck–boost converter Source:https://en.wikipedia.org/wiki/Buck%E2%80%93boost_converter?oldid=727340276Contributors: Glenn,
Giftlite, DavidCary, Filceolaire, Vadmium, TedPavlic, Hooperbloob, DavideAndrea, Krishnavedala, SmackBot, Bmearns, Vadim tr, Mak17f, Chris the speller, Bluebot, Oli Filth, Chendy, Nakon, CyrilB, Yves-Laurent, Mikiemike, CmdrObot, Nagarroth, Zyl.tw, Gra-hamDavies, RadioGuyTed, Lulo.it, The Original Wildbear, Broadbot, Anishsane, Mortivik, Hello9393, Wdwd, ClueBot, Ejay, Addbot, Semiwiki, Dmazor, Yobot, Ptbotgourou, Materialscientist, TheAMmollusc, A.amitkumar, LucienBOT, RedBot, Miracle Pen, Alph Bot, Ponydepression, ClueBot NG, Helpful Pixie Bot, Silvrous, Mattande, Melonkelon, Kingoffrog88 and Anonymous: 92
6.2
Images
• File:Buck-boost_continuous_discontinuous.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8f/Buck-boost_
continuous_discontinuous.svgLicense: GFDL Contributors:
• Buck-boost_continuous_discontinuous.pngOriginal artist:Buck-boost_continuous_discontinuous.png:User:CyrilB
• File:Buck_boost.png Source:https://upload.wikimedia.org/wikipedia/commons/1/10/Buck_boost.pngLicense: Public domain
Contribu-tors: Transferred fromen.wikipediato Commons byTrixtusingCommonsHelper. Original artist:Mak17fatEnglish Wikipedia
• File:Buckboost_chronogram.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/45/Buckboost_chronogram.svgLicense:
CC-BY-SA-3.0 Contributors: No machine-readable source provided. Own work assumed (based on copyright claims). Original artist: No machine-readable author provided.CyrilB~commonswikiassumed (based on copyright claims).
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