basic dc dc conveter

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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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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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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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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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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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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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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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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


12/15


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


13/15


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.


14/15


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


15/15


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.

REFERENCES

[1] D.W. Hart, Introduction to Power Electronics. EnglewoodCliffs, NJ:

Prentice-Hall, 1994.[2] R. Erickson, Synthesis of switch-mode converters, in Proc. IEEE

PESC, 1983, pp. 922.[3] D. Maksimovic and S. Cuk, General propertiesand synthesisof PWM

DC-to-DC converters, in Proc. IEEE PESC, 1989, pp. 515525.[4] D. Maksimovic, On topological assumptions on PWM convertersA

re-examination, in Proc. IEEE PESC, 1993, pp. 141147.[5] D. C. Hopkin and D. W. Root, Synthesis of a new class of converters

that utilize energy recirculation, in Proc. IEEE PESC, 1994, pp.

11671172.[6] E. E. Landsman, A unifying derivation of switching dc-dc converter

topologies, in Proc. IEEE PESC, 1979, pp. 239243.[7] M. H. Rashid, Ed., Power Electronics Handbook. New York: Aca-

demic, 2001, ch. 17.2.2, by Luo, F.L..

[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.

basic dc dc conveter

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