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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 1, JANUARY 2008 387
Basic DC-to-DC ConvertersBarry W. Williams
AbstractThis paper generically classifies and unifies basic
single-switch, single-inductor, dc-to-dc converters. The boost andbuck/boost flyback converters are shown to be supplementary,each being part of the same three-port terminal transfer func-tion. Realizing one of these converters inherently results in theother as a fully functional composite output port. Three-portexamination of the forward converter reveals it too combines twodifferent converter voltage/current transfer functions. General-ized three-port analysis is used to specify the requirements of arealizable universal dc-to-dc converter theoretically capable ofproducing isolated output dc and ac voltage and current in therange .
Index TermsPulsewidth modulation (PWM).
NOMENCLATURE
DC input voltage or input port.
, DC output voltage or output port.
, DC input, output current.
, DC output current from output port to load ,
.
, Output load resistance seen from input port .
Converter switch on-state duty cycle, .
Transfer function, relationship between output
port (or ) and the input port (or ).
Change in inductor current.
I. INTRODUCTION
THE three basic, single switch-diode-inductor, nonisolated,
dc-to-dc converters have been presented and analyzed ex-
tensively in the literature and texts [1]. The forward (step-down
or buck) converter, efficiently decreases a given input dc
voltage; the step-up (boost) converter increases a given input
dc voltage; while the inverting step-up/down (buck/boost)
converter inverts the polarity of a given input dc voltage and
can increase or decrease the input voltage magnitude. These
three converters form the backbone and bases of a vast prolif-eration of dc-to-dc converters, all purporting to offer various
advantageous features and attributes.
Generalized attempts to generate a framework for the sys-
tematic synthesis of PWM converters are in fact restrictive,
being based on two-port, single-input, single-output assump-
tions [2][5]. Publication [2] presents a general 2 2 state
Manuscript received January 4, 2007; revised April 19, 2007. Recommendedfor publication by Associate Editor C. Canesin.
The author is with the Department of Electronic and Electrical Engineering,University of Strathclyde, Glasgow G1 1XW U.K. (e-mail: [email protected]).
Digital Object Identifier 10.1109/TPEL.2007.911829
matrix converter form, with capacitors and inductors but
a restriction is that the output capacitor bypasses a single loadresistor (10). Seven classes of converter are presented, most
with more than one inductor, capacitor and/or switch. In [3]
constraints are that there is a single load resistor (Definition 1,
assumption 1, part 2), that the output capacitor and load resistor
are in parallel (Property 5), and only one converter can evolve
from one matrix representation (synthesis procedure). Similarly
[4], [5] only consider the converter as a two-port network to
generate new topologies, although in [5] the output is fed back
to the input to enhance low current performance, but no use is
made of the voltage across the fed back element. Similarly in
[6], a two-port canonical cell for the common three converters
was introduced, with the output capacitor reconfigured as afeedback element between the output and the input. Such
two-port topological restrictions do not form the basis of the
general analysis to be presented.
In this paper, the three so-called basic dc-to-dc converters
are theoretically reassessed in order to generically reclassify
them. The analysis presented assumes continuous inductor cur-
rent conditions and lossless components. All variables are dc
unless otherwise stated.
II. BASIC DC-TO-DC CONVERTERS
Fig. 1 shows two possible generalized dc-to-dc converter
functional circuit blocks having different dc supply polarity ref-erence nodes, along side their ac variac (a variable auto-trans-
former) equivalents. Functional duality will show that the
dc-to-dc converter is a dc equivalent to an ac auto-transformer
but hitherto singularly without the contiguous dynamic linear
output voltage range of the ac variac. To reduce the necessary
circuit complexity and permutations, without loss of generality,
a three-port network with three terminal nodes is used. This is
the minimum number of ports if the output voltage is to be
different from the input voltage . Each part of Fig. 1 shows
that for one input voltage generator, , (whether ac or dc) two
interdependent (one degree of freedom) realizable output con-
nections are possible ( , ). The basic concepts introduced
are equally applicable to a negative input terminal or positive
input terminal reference node, as shown in Fig. 1. To reduce
further analysis permutations that will emerge, the reference
configuration is taken as shown in Fig. 1(a), specifically, the
negative terminal, , of the input voltage is taken as the
reference.
In Fig. 1(a), two voltage outputs ports (and one input port)
exist, with outputs, and , where the
transfer function is real and . For conve-
nience, without loss of generality, let the output port referenced
with respect to the input voltage reference node be
(1)
0885-8993/$25.00 2007 IEEE
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388 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 1, JANUARY 2008
Fig. 1. Generic three-port representation of dc-to-dc converters: (a) with the negative voltage input node as reference; (b) with the positive voltage input node asreference; and ac auto-transformers where
f ( ) = N = N
; (c) shown at a step-down tap with the ac supply neutral as reference; and (d) shown at a step-up tapwith the ac supply live as reference.
and let the other output port, the composite output , be
(2)
therein satisfying Kirchhoffs voltage law
(3)
Parts (c) and (d) of Fig. 1 show the ac variac equivalent cir-
cuit to the generic dc converter blocks. Note that the variac, as
shown, is a linear three-port system, with ports that are bidi-
rectional (reciprocity), provided Faradays voltage equation is
observed, namely 4.44 . where is the
number of turns, is the flux density, is cross-sectional area,
and is frequency.
Fig. 2(a) shows a linear transfer function for output ,
and Kirchhoffs voltage components complying with
. The range of has three distinct regions separated
by two distinct boundary values, namely boundaries at
0 and 1. Thus, the output transfer functions can be
analyzed for various combinations, involving at least one of
these two boundary conditions. No one basic converter, from
one output port can provide the entire required output voltage
range, .
A. Two Boundariesat and 1:
Single-Inductor Based Converters
To fulfill the voltage range , with single-inductor,
dc-to-dc converters, three different distinct, observable, func-
tional regions in Fig. 2 for need investigating, namely:
1) 0: A negative output voltage relative to the inputis functionally fulfilled, albeit nonlinearly, by the step-up/
down inverting (buck/boost) converter with the nonlinear
transfer function [1]
(4)
hence 0 1 1, where is the switchon-state duty cycle.
2) 0 1: An output voltage less than the input voltage
(but greater than zero) is fulfilled, linearly, by the forward
(buck) converter with a linear transfer function given by [1]
(5)
where 0 1.
3) 1: The third region of required operation involves
the output voltage being greater than the input voltage,
which is fulfilled, nonlinearly, by the step-up (boost) con-
verter function, namely [1]
(6)
hence 0 1 1, which is the recip-
rocal of the 0 case, 2.Ai.
Fig. 2(b) shows both the reference output and the com-
posite output voltage .
From Kirchhoffs voltage law, , since is
a fixed input dc supply, only one degree of freedom remains,
thus only one of and can be independent. If the output
voltage requirement is fulfilled by the converter giving , then
the default composite voltage at is given by
(7)
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WILLIAMS: BASIC DC-TO-DC CONVERTERS 389
Fig. 2. Voltage transfer functions of the two output ports from the generic converter in Fig. 1(a) showing: (a) the vector components comprising the output v and(b) the two outputs,
v
andv
.
Examination of this equation, along with the virtual or
composite generators requirements for in each of the three
regions, (which together, without overlap, cover all possible
output voltages for between ), follows.a) 0: An output voltage greater than the input
is functionally fulfilled, albeit nonlinearly, by the step-up
converter with the transfer function as follows:
(8)
where is a boost converter, whence
0 1 1.
That is, if is produced by an inverting step-up/down
converter then the composite voltage resulting at port is
as if it were being driven by a unique boost converter (with
the identical switch duty cycle and one circuit topology);and vice versa between the two output ports.
b) 0 1: An output voltage less than the
input voltage, linearly decreasing from to 0 over the
range, does not exist uniquely amongst the usual basic
single-switch, transformerless, single-inductor, dc-to-dc
converters. If the output voltage is provided by a
step-down converter then the voltage induced at is
(9)
where the composite port voltage transfer function is
1 , whence 0 1 1.
That is, if is produced by a step-down converter thenthe composite voltage at is as if it were being driven
by a unique converter with an output voltage given by (9);
and vice versa.
c) 1: The third region of necessary operation in-
volves the composite output voltage being less thanzero, and is fulfilled, nonlinearly, by the inverting step-up/
down converter. As previous inferred, the output voltage
can be provided by a step-up converter, thus:
(10)
where the voltage transfer function is 1 :
a buck/boost converter, with the correct relative polarity
and 0 1 1.
That is, if is produced by a boost converter then the
composite voltage resulting at is as if it were beingdriven by a unique buck/boost converter; and vice versa.
Note, the identical vice versa result occurred when con-
sidering the converter operational range 0. That
is, the same converter can provide the required voltages
for two regions: 0 and 1.
The three operational regions, satisfying Kirchhoffs voltage
law, can be realized by the two, three-port networks shown in
Fig. 3(a) and (b).
Two observations can be made from these results:
The boost and the buck/boost flyback converters are sup-
plementary, as seen from , whence
. That is, both nonlinear functions
are realized with the same physical converter circuit, (butat different ports), as shown in Fig. 3(a).
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390 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 1, JANUARY 2008
Fig. 3. Generic representation of three independent output voltage converters: (a) dc-to-dc single-inductor flyback converter covering the range 0 f ( ) 1; (b)dc-to-dc single-inductor forward converter covering the range 0 f ( ) 1; and (c) two-inductor flyback voltage converter covering the voltage range for all f ( ) .
The forward converter can produce a supplementary (com-
posite) output voltage with the linear transfer function, as shown in Fig. 3(b). The supplementary
converter output need not uniquely exist since the forward
converter output, , can always be topologically realized
with respect to the input voltage zero or positive terminals,
as shown in Fig. 1(a) and (b). That is, the forward converter
can be configured to produce the transfer function voltage
at either port, (but not both simultaneously).
Mathematically, it can be concluded that there need only
be two mutually exclusive, generic single-inductor dc-to-dc
converters, providing contiguous output voltages, covering the
voltage range . The forward converter produces positive
voltages less than the input voltage, while a flyback converter
(either boost or buck/boost) can fulfill the requirements for the
remaining output voltage range. The output voltage polarity
with respect to the input , can readily be configured by using
the appropriate circuit reference base as shown in Fig. 1(a)
and (b).
B. One Boundaryat 0:
Single and Two Inductor Based Converters
DC-to-dc converters that utilize the complexity of an extra in-
ductor and capacitor (two-inductor converters), can be assessed
as to whether they offer any advantages in providing the nec-
essary output voltage range . Only one such transformer-
less single-switch converter offers a different transfer functionto those offered by the basic, single-inductor converters [7].
In Fig. 2(a), the linear function for output , can be divided
into two (as an alternative to three) different distinct, observable,functional output voltage regions around 0, namely:
1) 0: As previously considered for this region in Sec-
tion II-A, a negative output voltage relative to the input
is functionally fulfilled, albeit nonlinearly, by the single-in-
ductor inverting step-up/down (buck/boost) flyback con-
verter with the voltage transfer function [1]:
(11)
which produces a composite port voltage
(12)
where is the boost converter transfer function.
That is, if is produced by an inverting step-up/down
converter then the composite voltage at port is as if it
were being driven by a unique boost converter; and vice
versa.
2) 0: The second region of needed operation, the
positive region, involves the output voltage being
greater than zero, which is fulfilled, nonlinearly, by the
noninverting step-up/down (buck/boost) two-inductor
converter voltage function, namely [7]
(13)
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WILLIAMS: BASIC DC-TO-DC CONVERTERS 391
Fig. 4. Generic representation of dc-to-dc converters: (a) dc-to-dc converter with the negative voltage input node as reference; (b) dc coupling reconnection of
output capacitor C ; (c) output voltage determined by composite converter transfer function v ; and (d) output v indirectly decoupled at v by C through the input
voltage sourceE
.
where 1 , hence 0
1 1.
The 0 region involves the output voltage being
less than the input voltage , (as well as greater than ),
and is fulfilled, nonlinearly, by an inverting step-up/down
converter. If the output voltage is provided by a step-up/
down converter [7], then
(14)
where the voltage transfer function , as shown
in Fig. 3(c), is 1 2 1 hence
0 1 2 1.
That is, if is produced by a noninverting buck/boost
converter then the composite voltage resulting at is as if
it were being driven by a unique converter, with a voltage
transfer flyback function as given by (14); and vice versa.
The two regions, satisfying Kirchhoffs voltage law, can
be realized by the two-inductor, three-port network shown
in Fig. 3(c), along with the three-port network for the basic
inverting step-up/down converter in Fig. 3(a), contiguous,
without overlap.
Two observations can be made from these results.
The boost and the buck/boost two-inductor, converters
are supplementary, as seen from , whence
. That is, both
nonlinear functions are realized with the same converter
circuit, (but at different ports), as shown in Fig. 3(c).
The two converter outputs in Fig. 3(c) form partially over-
lapping functions, that is, the converter voltage outputs do
not perform mutual exclusive functions.
C. One Boundaryat 1:
Single and Two Inductor Based Converters
A third alternative output range possibility is to consider tworegions about 1. The output range can be covered by the
single-inductor flyback step-up converter in Fig. 3(a) and the
two-inductor flyback converter using port in Fig. 3(c), such
that (when reconfigured for port in each case)
for
for (15)
III. THREE-PORT CONVERTER CIRCUIT REALIZATIONThe theoretical approach adopted has assumed that the re-
quired converter can inherently and topologically fulfill the re-
quired two circuit output function conditions ( , ), without
any circuit topological, magnetic, frequency or time domain
contradictions or limitations. To this end, the various topolog-
ical reconnections at the supply nodes in Fig. 4 exploit ac and
dc circuit analysis fundamentals.
DC circuit theory: The dc output capacitor can be readily
connected across either the output or output (or both)
since dc-wise, the input dc voltage source acts as infinite
series capacitance [7]. The capacitor therefore indirectly de-
couples the composite output port. Furthermore, the capacitancecan be split between the two output ports, (therein being effec-
tively in parallel) for better decoupling, if desired. The effective
transferred port reactance (capacitance: 1 ) is re-
lated to the voltage transfer function , squared, between the
two ports.
AC circuit theory: AC-wise, the dc input voltage source
appears as a short circuit, so connection of the output capacitor
or resistor to the positive reference supply terminal or neg-
ative terminal is the same in terms of ac circuit analysis. Thus
at any time, either or both and can be used as current de-
coupled, inter-dependant voltage sources .
For each converter circuit in Fig. 4, during all component
translations, the switch, diode, and inductor connections arefixed relative to one another and the input port, . Comparison
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392 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 1, JANUARY 2008
Fig. 5. Circuit permutations of the basic single-switch, single-inductor, flyback converter showing the translation from a conventional boost mode of operationto the conventional buck/boost circuit. The equivalent ac auto-transformer has been inserted at an appropriate stage of the transformations. Moving laterally fromone circuit to the next involves the reconnection of one terminal of the output capacitor or the load resistor.
with the ac auto-transformer (or transformer) can aid in the
terminal operation understanding of the various converters.
IV. SINGLE-INDUCTOR FLYBACK CONVERTER
Fig. 5 shows various output capacitor and load resistor
, permutations of the basic single-inductor flyback converter.
Translation from one output circuit configuration to the next in-
volves just one node connection translation of the supply end of
the load resistor or the supply end of the dc output capacitor. The various circuit voltage and current equations shown can
be derived by utilizing:
1) the switch and diode are in series across the output ,
hence the sum of their average voltages add to :
the average voltage across the switch is , and is
due to the supported voltage when the diode conducts,
1 ;
the average (reverse) voltage across the diode is and
is due to the supported (blocked) voltage when the
switch conducts, ;
2) also, in steady-state, the average voltage acrossthe inductor
andthe average capacitor current, areboth zero. Since the
average inductor voltage is zero, the average switch voltageis theinputvoltage, , (being in parallel, dc-wise).
Kirchhoffs current law and the network transfer function
yield the circuit currents (assuming power invariance., viz.,
.) for which [1]
and (16)
Inherently (by Kirchhoffs voltage law) the composite buck/
boost transfer function is fulfilled. The current and voltage
transfer functions assume output power flow from the appro-priate port, or 0.
The load circuit connection translation process stage where
load operation changes from boost at to buck/boost at , is
the ideal point to observe the visual similarity of this dc flyback
converter to the ac auto-transformer. In Fig. 5, the equivalent
ac auto-transformer arrangement is inserted at this translation
point. It can be shown that the dc converter is effectively (three-
port-wise) a dc auto-transformer with complete property du-
ality with the ac auto-transformer. For example, load impedance
transference in the turns ratio squared, gives .
It will be observed that one ac variac can produce the complete
output voltage range while two dc converters (flyback
and forward) are needed to match the same voltage range, albeitcontiguously with two different converters.
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WILLIAMS: BASIC DC-TO-DC CONVERTERS 393
Fig. 6. Circuit translation permutations of the basic single-switch, single-inductor, forward converter, with all ports involved in power transfer. The equivalent acauto-transformer has been inserted at an appropriate stage of the transformations. Moving laterally from one circuit to the next involves the reconnection of one
terminal of the output capacitor or the load resistor.
Both the ac and dc auto-transformer models are analyzed on
the basis that operation is lossless, that is, the power delivered to
the load is equal to the input power. DC terms are shown in the
following analysis, which is equally valid for ac (transformer)
parameters. A continuous inductor current conduction mode
(commonly termed CCM) and a resistive load are assumed.
From Kirchhoffs current law, the input current can be ex-
pressed as
(17)
The first term, , is that part of the input current
transferred to the output.
The second term, , involves that part of the input cur-
rent (which is the remainder of the input current) used to
magnetically produce the output current.
The voltage/current transfer functions rearrange to give:
and
and (18)
which show that for 0 1
the output voltage is always greater than the inputvoltage ;
the input current is always greater than the output cur-
rent ,
therein preserving power transfer balance.
Notice that the output voltage varies nonlinearly with duty
cycle, such that the flyback converter acts as a nonlinear auto-
transformer, unlike the forward converter, which has a linear
dependence. The conventional ac auto-transformer, as a variac,
usually has a linearly distributed winding giving a linear duty
cycle (position) dependent output voltage.
V. SINGLE-INDUCTOR FORWARD CONVERTER
Fig. 6 shows various output port circuit and permuta-
tions for the basic forward (step-down voltage) dc-to-dc con-
verter where translation from one circuit to the next involves
just one output terminal connection translation. Similar to the
flyback converter, the various circuit voltage and current equa-tions shown can be derived by utilizing:
1) the switch and diode are in series across ;
2) the average voltage across the inductor and the average
capacitor current, are both zero. Since the average inductor
voltage is zero:
the output voltage is the average voltage across the
switch 1 , which supports voltage when thediode conducts, 1 ;
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394 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 1, JANUARY 2008
the output voltage is the average (reverse) voltage
across the diode , which supports (reverse) voltage
when the switch conducts, .
Kirchhoffs current law and the network transfer functionyield the circuit currents, for which [1]
(19)
The current transfer functions imply that load current flows,0. Inherently with a unidirectional output current converter,
the composite 1 transfer function is produced, but (unlike the
flyback converter) this linear function cannot uniquely or solelybe derived or operational, as considered in Section VII-B.
The equivalent ac auto-transformer arrangement is inserted in
Fig. 6 which shows output circuit translation processes. Duality
exists and the dc forward converter is effectively a dc step-down
auto-transformer. For example, load impedance transference in
the turns ratio squared, gives . As previously men-
tioned, one ac auto-transformer can develop the complete output
voltage range while the forward converter is one of twodc converters needed to achieve the same output voltage range,
albeit contiguously.
Both the ac and dc auto-transformer step-down models are
analyzed on a resistive load basis and assuming that operation
is lossless. From Kirchhoffs current law, the output current can
be expressed in terms of the input current
(20)
The first term on the right hand side, , is the powerassociated with the current drawn by the output from the
input (through the magnetic component thereby producingthe second term on the right hand side).
The second term on the right hand side is the power
produced (current) by magnetic (transformer or inductor)
action, due to the primary current through the magnetic
component.
Canceling the input current component, this power equation
reduces to
(21)
which shows that
the output voltage is always less than the input voltage
; the input current is always less than the output current
, therein preserving power transfer balance.
Notice that the output voltage varies linearly with switch
duty cycle, such that the forward converter acts as a linear auto-
transformer, unlike the flyback converter, which is nonlinear inthis aspect.
The forward converter is not tenable in a purely 1
output voltage operational mode. Specifically, for a single-switch forward converter, must exist in the circuit, across
the output port , such that 1 is satisfied,
as shown in Section VII-B. Note that if the forward converter
switch is operated with the duty cycle 1 , then the output
transfer function is 1 , and the composite output port transferfunction is , which cannot exist uniquely.
VI. TWO-INDUCTOR FLYBACK CONVERTER
Fig. 7 shows various permutations of the two-inductor fly-back converter where translation from one output circuit con-
figuration to the next involves just one node connection transla-tion of the supply end of the load resistor or the supply end
of the output capacitor . Circuit voltage and current equations
utilize the fact that the average voltage and current for each in-ductor and capacitor is zero, respectively, and:
since the average inductor voltage is zero, the averageswitch voltage is the input voltage, , (being in parallel,
dc-wise);
the average (reverse) voltage across the diode is the averageoutput voltage ;
the average internal capacitor voltage is the output voltage.
Kirchhoffs current law and the network transfer functionyield the circuit currents (assuming power invariance) for
which [1]
(22)
The current and voltage transfer functions assume output
power flow from the appropriate ports.In Fig. 7, the equivalent ac auto-transformer arrangement is
inserted. It can be shown that the dc flyback converter is effec-tively (three-port-wise) a dc auto-transformer, where duality ex-
ists between the two. Again, a resistive load is assumed and both
the ac and dc auto-transformer models are analyzed on the basis
that operation is lossless.
A. Mode
From Kirchhoffs current law, the output current can be ex-
pressed in terms of the input current in the voltage step-down
mode, 1/2
for (23)
The first term, , on the right hand side is power associ-ated with the current drawn by the output from the input
(through magnetic component which produces the second
term on the right hand side).
The second term on the right hand side is the power pro-duced (current) by magnetic (transformer or inductor) ac-
tion, due to the primary current (first term) through the
magnetic component.The same equation shows that the output voltage is less
that the input voltage , for 1/2.
B. Mode
In the step-up mode 1/2, the input current can be ex-
pressed as
for (24)
The first term is that part of the input current transferred to
the output.
The second term involves that part of the input current used
to magnetically produce the output current (which is theremainder of the input current).
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WILLIAMS: BASIC DC-TO-DC CONVERTERS 395
Fig. 7. Circuit permutations of the single-switch, two-inductor flyback converter showing the translation from conventional boost mode of operation ( 1/2mode shown). The equivalent ac auto-transformer (shown in a step-down mode) has been inserted at an appropriate stage of the transformations. Moving laterallyfrom one circuit to the next involves the reconnection of one terminal of the output capacitor or the load resistor.
The voltage transfer equation shows that the output voltage
is greater that the input voltage , for 1/2, whence the
input current is greater than the output current.
Again, notice that, for any duty cycle , the output voltage
varies nonlinearly with duty cycle, such that the two-inductor
flyback converter acts as a nonlinear auto-transformer.
VII. TWO OUTPUT PORT OPERATION
Both basic types of three-port converters (flyback and for-
ward converters) can deliver power in a two output mode ,, as illustrated in the parts of Fig. 8. It will be shown that the
single-switch forward converter output voltage (or any con-
verter acting in a step-down mode where ), relies on
power being drawn from in order to delivery power, at the
correct voltage, from port .
A. Single-Inductor Flyback Converter
Operation of the single-inductor flyback, three-port converter
is considered in terms of energy conservation:
power
(25)
That is, as shown in Fig. 8(a), as expected by Kirchhoffs
current law, the transformed current requirement of each output
port is seen as a parallel circuit current (and resistance) require-
ment at the input voltage source .
The effective equivalent resistance seen by the supply
nodes is
(26)
In terms of effectively one equivalent resistor across port
, this is seen by the input port as
(27)
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Fig. 8. Circuit permutations of the basic single-switch converters: (a) basic single-inductor flyback converter; (b) basic single-inductor forward converter; and (c)basic two-inductor flyback converter; with all ports involved in power transfer.
In terms of effectively one equivalent resistor across port
, the input sees
(28)
Note , the square of which is
involved in the equivalent resistance transfer term for transfersbetween the output ports in (27) and (28).
From Fig. 8(a), Kirchhoffs current law at the load common
node is
Since the converter power flow is unidirectional (in its single-
switch form) 0, is always satisfied by the
two inter-dependent flyback converters, save for discontinuous
inductor current operation.
From (26), when 0, and . Only
dissipates power, , the minimum possible power for .
Since and equal 0, the power dissipated in the load resistoris zero.
No upper limit exists on the power that may be delivered to
either or , although from (26), equal powers are delivered
to the loads when , provided 0.
If the flyback converter output is bidirectional, by using a
full bridge leg as in Fig. 9(a), then discontinuous inductor cur-
rent operation is avoided, therein maintaining the correct output
voltages even during light load conditions. Additionally, trans-
former reciprocity is fulfilled. When the boost port becomes
the source port, the reciprocal output (the former input) is a
step-down voltage, as is port . Note that if the composite
buck/boost port is made the input, then, consistent with an ac
variac, the previous input is a buck/boost port. The two switchesin the bridge leg are operated in a complementary mode, as in
voltage source inverter bridges, and is called synchronous rec-
tification in smps terminology. Current reversal is seamless and
the voltage/current transfer functions are solely determined by
the duty cycle, at all output current levels.
B. Single-Inductor Forward Converter
Similar to the flyback converter, for the forward converter,
from power invariance
(29)
That is, as shown in Fig. 8(b), the transformed currentrequirement of each output port is seen as a parallel circuit
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WILLIAMS: BASIC DC-TO-DC CONVERTERS 397
Fig. 9. Bidirectional and unidirectional converter circuit permutations for:(a) flyback converters; (b) forward converters; and (c) two-inductor flybackconverters.
current (and resistance) requirement at the input voltagesource .
The equivalent total effective load resistance seen by the
input supply is derived to be
(30)
In terms of one effectively equivalent resistor or
across port or port , respectively, the input port experi-
ences
(31)
(32)
Note 1 , the square of which is involved
in the equivalent resistance transfer term for transfers between
the output ports and in (31) and (32). Thus the load resis-
tors are seen to act in parallel, the resistance of which is scaled
by the port voltage ratios squared when transferred. That is, the
transferred resistance is dependent of the duty cycle, . This is
as would be the case with an ac transformer, or specifically, an
auto-transformer complying with the proposed duality between
the dc converter and the ac auto-transformer.
From Fig. 8(b), the left circuit, when the output is refer-
enced with respect to the input supply negative terminal, Kirch-
hoffs current law at the load common node is
Since the converter power flow is unidirectional (in its single-
switch form), then 0 thence is a necessary and
sufficient condition for output voltage regulation. That is, it is
a condition of the forward converter, referenced with respect to
the negative supply input, that the sink current from the com-
posite port not exceed the sourced current to conventional for-
ward converter port. Thus to obtain a regulated output voltage
, must exist when 1; specifically for
which yields
(33)
The same identity holds for the circuit version where the
output is referenced with respect to the input supply posi-
tive terminal. Note, the nonexistence of satisfies (33), since
.
When and 0, , and the
converter is redundant. The output voltages are determined by
standard resistor divider equations, and the series current in eachresistor load is . Thus, the minimum total power
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398 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 1, JANUARY 2008
transfer of to the loads, occurs when 1
.
By using a full bridge leg as in Fig. 9(b), the bidirectional con-
verter formed alleviates the load resistance (current) constraint
on port and avoids discontinuous inductor current operation
during light load conditions. Also, transformer reciprocity can
be fulfilled where the port becomes the source, the reciprocityoutput (the former input) is a boost voltage and port be-
comes a buck/boost output. Using either or as an input
creates a flyback converter. As with the flyback converter, the
two switches in the bridge leg are operated in a complemen-
tary mode, giving seamless current reversal and voltage transfer
functions that are determined solely by the duty cycle for all
passive load conditions.
C. Two-Inductor Flyback Converter
Operation of the two-inductor flyback, three-port converter is
considered in terms of energy conservation
(34)
That is, as shown in Fig. 8(c), the transformed current require-
ment of each output port is seen as a parallel circuit current (and
resistance) requirement at the input voltage source .
The effective equivalent resistance seen at the supply port
is
(35)
In terms of effectively one equivalent resistor or
across port or port , respectively, the input port sees
(36)
(37)
Note 1 2 , the square of which is in-
volved in the equivalent resistance transfer term for impedance
transfers between the output ports in (36) and (37).
From Fig. 3(c), Kirchhoffs current law at the load common
node is
When 0 the current 0, in which case
is a requirement since for a unidirectional converter,
0, is a necessary requirement. From
That is, when 1/2, is a (forward converter mode)
constraint, or
That is
(38)
When 1/2, the output voltage is always greater than the
input voltage , in which case converter current flow is unidi-
rectional (in its single-switch form) and 0,
is always satisfied by the two effective flyback converters, save
for discontinuous inductor current operation.
From (35), when 1/2, whence 0 and
. Only dissipates power, . Since 0, the
power dissipated in is zero, and for 1/2, the current
reverses (relative to that direction for 1/2). For 1/2, no
upper limit exists on the power that may be dissipated in either
or , although equal powers are delivered to the loads when
1 2 . The magnitude of the output voltages are
equal when 1/3, 1/3 .
If the flyback converter output is bidirectional, by using a full
bridge leg as in Fig. 9(c), then discontinuous inductor current
operation is avoided and the correct output voltages are main-
tained during light load conditions. Also no restrictions are im-
posed on the output currents when 1/2. Additionally, trans-
former reciprocity is fulfilled. The two switches in the bridge leg
operate in a complementary mode, as in voltage source inverter
bridges, thereby giving seamless current reversal and voltage
transfer functions that are solely determined by the duty cycle.
VIII. UNIVERSAL DC-TO-DC/AC CONVERTER
The universal, transformerless, dc-to-dc converter is one
where for any given finite input voltage , any desired output
voltage (whether ac or dc) can be attained seamlessly, in
the voltage/current range . The ac auto-transformer is
hypothetically capable of achieving such an ac specification,
as shown in Fig. 10(a). Fig. 8 shows the flyback and forward
converters operating in modes where power is being drawn
from both output ports, and . Obviously, no one of these
converters is viable as a universal dc-to-dc converter; which is
depicted by the three-port block shown in Fig. 10(b).
Either output port ( or ) can be considered as the ref-erence, since it has been shown in Figs. 68, that the function
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WILLIAMS: BASIC DC-TO-DC CONVERTERS 399
Fig. 10. Theoretical three-port universal converters: (a) the ac auto-transformeras a variac and (b) the dc-to-dc converter.
of either port can be readily reconfigured to be the composite
output.
At all times, Kirchhoffs voltage law must be satisfied, that
is . This equation implies a second constraint,
that both output ports ( , ) must be able to produce a voltage
covering the whole voltage range since when one port pro-
duces an infinitely large voltage, the other port must produce a
virtually equal (differing by ) and opposite polarity voltage.
If this voltage range were to be achieved with a single-switch
converter, by varying its on-state duty cycle, from 0 pu to 1 pu,
seamlessly and monotonically but not necessarily linearly,
0 implies (or ) and 1 implies (or ), then
the transfer function (one possibility) for one port is.
(39)
which produces 0 at 1/2. Then by Kirchhoffs voltage
law, the composite port voltage is
(40)
As with any converter, only one of the two transfer functions
need be realizable, since the other can form a composite outputthat singularly need not deliver power.
Derivation of such a voltage transfer function, as in (39) and
(40), from the operational mechanisms of the dc converter in-
volves satisfying an inductor volt-second integral (Faradays
equation, as with an ac transformer), of the form, assuming con-
tinuous inductor current
(41)
where
when the switch is on, energy is transferred from the source
to the inductor ;
in the switch off-state, the formed series circuit must in-
volve the opposing output voltage , in order to transfer
the inductor energy to the load circuit.
Since a suitable, realistic, transfer function is known, the de-
pendence of the off-state voltage loop, involving , can be ex-
amined. Substituting from (41) into (39) gives
(42)
from which , which is not restricted to be a constant, is
or (43)
Real roots exist: 1/2 3/4. No matter what transfer func-
tion is assumed, when the output is infinitely large 1 or
0 then and an infinite instantaneous voltage
(power) requirement from is predicted. Only another (a
second) circuit inductor could fulfill such a voltage requirement.
It will be observed for the boost and buck/boost supplemen-
tary converters, the output voltage loops formed when energy
is transferred to the load circuit , are, and
, which do not involve any series ele-
ments other than the output capacitor, and the input voltage inthe case of the boost converter. That is, all the energy transfers
losslessly to the output circuit.
No universal, transformerless, single-switch, single-inductor,
three-node/port converter exists to fulfill this specification.
Two single-inductor, bidirectional, flyback converters (a total
of four switches therein forming a four-port network) can ful-
fill the universal ac/dc variac converter function as shown in
Fig. 11(a). The two flyback converters are configured as buck/
boost converters so that transformer versions can readily pro-
vide galvanic isolation if desired [8]. As shown, one converter
is controlled with a duty cycle while the other converter is
controlled by the complement, 1 . The bidirectional
switches in each fundamental converter remain the complementof their respective principal converter switch.
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400 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 1, JANUARY 2008
Fig. 11. Universal four-port converters: (a) and (b) dc-to-dc converters; (c) the four quadrant controller; and voltage transfer functions; and (d) isolated three-phase
load version with a capacitor common star connection.
The output voltage from converter is as for the standard
buck/boost converter, namely
(44)
The output voltage from buck/boost converter , when
controlled with a duty cycle 1 is
(45)
The output voltage between the two flyback converter
outputs is
(46)
This voltage is twice that associated with (39), since the
effective supply voltage is doubled by this two-converter
push-pull bridge arrangement. The output voltage is zero when1/2, and appropriately large in magnitude for 0 and
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WILLIAMS: BASIC DC-TO-DC CONVERTERS 401
1. Each output capacitor and supports a mean
dc voltage of , the input voltage, at zero output. Being
based on the buck/boost converter, transformer operation is
possible, as shown in Fig. 11(b), giving a galvanic isolated dc
voltage output. Both dc and ac output voltages/currents can
be produced, the maximum output fundamental frequency de-
pending on the bridge switching frequency (being significantlygreater than the bridge output frequency). The magnetizing
flux frequency is at the switching frequency, with inductor
core fluxes being reset every switching cycle. The inductors
and (and transformers and ) can be magnetically
coupled. Due to practical circuit imperfections (switch and
diode onstate voltages, etc.), the output voltage deviates from
the ideal transfer function at high duty cycles. The transformer
turns ratio is based on the practical maximum output voltage
required, which is adjusted to occur for a duty cycle of 0.95
(and 0.05due to the complementary operation of the second
converter). Inverse encoding of the control transfer function,
as with a codec, can linearize the output voltage over the
desired range, with zero output voltage at a 50% duty cycle.A third leg can be added in parallel, as shown in Fig. 11(d), to
form a three-phase converter, and together with a three-phase
transformer, can provide galvanic isolation. The primary can
be star or delta connected.
As shown in Fig. 11(c), two bidirectional forward converters
similarly configured produce the well known inverter H-bridge
linear output voltage transfer function 2 1. The common
transformer isolated variation (along with its flux reset limi-
tations) is an extension. Note the two output inductors and dc
capacitors can be reduced to one and one ac (effectively
being in series and parallel, respectively). The output second
order filter cut-off frequency 1 , determinesthe upper output frequency from the bridge (the bridge can
produce an ac or dc output voltage/current). This assumes
that the bridge switching frequency is well in excess of the
filter cutoff frequency. Using an smps design approach, is
selected on the basis of the desired inductor ripple current,
and is determined by the input voltage , according to the
equation . This will be related to
the maximum output current which occurs at maximum
load current and frequency according to
when
IX. CONCLUSION
The basic nonisolated, minimum component (switch, diode,
and inductor), boost and buck/boost converters can be realized
without any constraints, with the one unified flyback converter
circuit, although from different circuit ports. The forward con-
verter can not only realize the usual output voltage proportional
to duty cycle function, but can also produce an exploitable lin-
early decreasing output function, at the composite port, albeit
with defined load current constraintsunless a bidirectional ar-
rangement is used.
The two fundamental dc-to-dc converters (flyback and for-ward converters) can be viewed as dcac variacs, where the
usual ac transformer rules of the turns-ratio define input and
output voltage and current, as well as reciprocity (when in a re-
versible form) and impedance matching and impedance trans-
ference between windings in the turns ratio squared, all apply.
Unlike the ac variac, because of conflicting circuit re-
quirements, no one basic dc-to-dc converter can singularly
produce from one port, the complete output voltage range .
Two pushpull operated single-inductor, reversible flyback
converters realize the ac variac function. When transformer
isolated flyback converters are used, an infinitely variable acdc
single or three phase transformer function is realized.
Whether ac or dc, the variac, transformer, and dc-to-dc con-verters all comply with Faradays equation.
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that utilize energy recirculation, in Proc. IEEE PESC, 1994, pp.
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[8] S. B. Kaer and F. Blaabjerg, A novel single-phase inverter for theac-module with reduced low-frequency ripple penetration, in Proc.10th EPE Conf. Power Electron. Appl., Toulouse, France, 2003, [CDROM].
Barry W. Williams received the B.E. degree (withhonors) from Adelaide University, Adelaide, Aus-tralia, in 1976 and the Ph.D. degree from Cambridge
University, Cambridge, U.K., in 1980.Since July 2005, he has been Professor of elec-
tricalengineeringat StrathclydeUniversity, Glasgow,U.K. After graduating from Cambridge University,he lectured at Imperial College, London, U.K., forsix years, and then held the Chair of Electrical Engi-neering at Heriot-Watt University, Edinburgh, U.K.,from 1986 to 2005.