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Application Report

SLUA059A - March 1999 – Revised November 2002

Understanding Buck-Boost Power Stages in Switch Mode

Power Supplies

Everett Rogers System Powe

ABSTRACT

A switching power supply consists of the power stage and the control circuit. The power stage

performs the basic power conversion from the input voltage to the output voltage and includes

switches and the output filter. This report addresses the buck-boost power stage only and

does not cover control circuits. Detailed steady-state and small-signal analysis of the

buck-boost power stage operating in continuous and discontinuous mode is presented.

Variations in the standard buck-boost power stage and a discussion of power stage

component requirements are included.

Contents

1 Introduction 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Buck-Boost Stage Steady-State Analysis 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 Buck-Boost Steady-State Continuous Conduction Mode Analysis 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Buck-Boost Steady-State Discontinuous Conduction Mode Analysis 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Critical Inductance 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Buck-Boost Power Stage Small Signal Modeling 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 Buck-Boost Continuous Conduction Mode Small-Signal Analysis 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Buck-Boost Discontinuous Conduction Mode Small-Signal Analysis 16. . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Variations of the Buck-Boost Power Stage 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 Flyback Power Stage 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Component Selection 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 Output Capacitance 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2 Output Inductance 26. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3 Power Switch 27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4 Output Diode 28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Summary 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 References 31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Trademarks are the property of their respective owners.

SLUA059A

Understanding Buck-Boost Power Stages in Switch Mode Power Supplies

1 Introduction

The three basic switching power supply topologies in common use are the buck, boost, and

buck-boost. These topologies are nonisolated, i.e., the input and output voltages share a

common ground. There are, however, isolated derivations of these nonisolated topologies.

The power supply topology refers to how the switches, output inductor, and output capacitor

are connected. Each topology has unique properties. These properties include the

steady-state voltage conversion ratios, the nature of the input and output currents, and the

character of the output voltage ripple. Another important property is the frequency response

of the duty-cycle-to-output-voltage transfer function.

The buck-boost is a popular nonisolated, inverting power stage topology, sometimes called

a step-up/down power stage. Power supply designers choose the buck-boost power stage

because the output voltage is inverted from the input voltage, and the output voltage can be

either higher or lower than the input voltage. The topology gets its name from producing an

output voltage that can be higher (like a boost power stage) or lower (like a buck power stage)

in magnitude than the input voltage. However, the output voltage is opposite in polarity from

the input voltage. The input current for a buck-boost power stage is discontinuous or pulsating

due to the power switch (Q1) current that pulses from zero to I

every switching cycle. The

output current for a buck-boost power stage is also discontinuous or pulsating. This is

because the output diode only conducts during a portion of the switching cycle. The output

capacitor supplies the entire load current for the rest of the switching cycle.

This report describes steady state operation of the buck-boost converter in continuous-mode

and discontinuous-mode operation with ideal waveforms given. The duty-cycle-to-output-

voltage transfer function is given after an introduction of the PWM switch model.

Figure 1 shows a simplified schematic of the buck-boost power stage with a drive circuit block

included. The power switch, Q1, is an n-channel MOSFET. The output diode is CR1. The

inductor, L, and capacitor, C, make up the effective output filter. The capacitor ESR, R

(equivalent series resistance) and the inductor DC resistance, R

, are included in the

analysis. The resistor, R, represents the load seen by the power stage output.

CR1

Drive

Circuit

= i

Figure 1. Buck-Boost Power Stage Schematic

During normal operation of the buck-boost power stage, Q1 is repeatedly switched on and

off with the on- and off-times governed by the control circuit. This switching action gives rise

to a train of pulses at the junction of Q1, CR1, and L. Although the inductor, L, is connected

to the output capacitor, C, only when CR1 conducts, an effective L/C output filter is formed.

It filters the train of pulses to produce a DC output voltage.

SLUA059A

Understanding Buck-Boost Power Stages in Switch Mode Power Supplies

2 Buck-Boost Stage Steady-State Analysis

A power stage can operate in continuous or discontinuous inductor current mode.

Continuous inductor current mode is characterized by current flowing continuously in the

inductor during the entire switching cycle in steady-state operation. Discontinuous inductor

current mode is characterized by the inductor current being zero for a portion of the switching

cycle. It starts at zero, reaches a peak value, and returns to zero during each switching cycle.

The two different modes are discussed in greater detail later and design guidelines for the

inductor value to maintain a chosen mode of operation as a function of rated load are given.

It is very desirable for a converter to stay in one mode only over its expected operating

conditions because the power stage frequency response changes significantly between the

two different modes of operation.

For this analysis, an n-channel power MOSFET is used and a positive voltage, V

GS(ON)

, is

applied from the Gate to the Source terminals of Q1 by the drive circuit to turn ON the FET.

The advantage of using an n-channel FET is its lower R

DS(on)

but the drive circuit is more

complicated because a floating drive is required. For the same die size, a p-channel FET has

a higher R

DS(on)

but usually does not require a floating drive circuit.

The transistor Q1 and diode CR1 are drawn inside a dashed-line box with terminals labeled

a, p, and c. This is explained fully in the Buck-Boost Power Stage Modeling section.

2.1 Buck-Boost Steady-State Continuous Conduction Mode Analysis

The following is a description of steady-state operation in continuous conduction mode. The

main goal of this section is to provide a derivation of the voltage conversion relationship for

the continuous conduction mode buck-boost power stage. This is important because it shows

how the output voltage depends on duty cycle and input voltage or conversely, how the duty

cycle can be calculated based on input voltage and output voltage. Steady-state implies that

the input voltage, output voltage, output load current, and duty-cycle are fixed and not

varying. Capital letters are generally given to variable names to indicate a steady-state

quantity.

In continuous conduction mode, the buck-boost converter assumes two states per switching

cycle. The ON State is when Q1 is ON and CR1 is OFF. The OFF State is when Q1 is OFF

and CR1 is ON. A simple linear circuit can represent each of the two states where the

switches in the circuit are replaced by their equivalent circuit during each state. The circuit

diagram for each of the two states is shown in Figure 2.

= i

DS(on)

ON State

= i

OFF State

Figure 2. Buck-Boost Power Stage States

SLUA059A

Understanding Buck-Boost Power Stages in Switch Mode Power Supplies

The duration of the ON state is D × T

= T

where D is the duty cycle, set by the control

circuit, expressed as a ratio of the switch ON time to the time of one complete switching cycle,

. The duration of the OFF state is called T

OFF

. Since there are only two states per switching

cycle for continuous conduction mode, T

OFF

is equal to (1–D) × T

. The quantity (1–D) is

sometimes called D’. These times are shown along with the waveforms in Figure 3.

∆I

OFF

CR1

Solid

Dashed

c-p

Solid

Dashed

Figure 3. Continuous Mode Buck-Boost Power Stage Waveforms

Referring to Figure 2, during the ON state, Q1 presents a low resistance, R

DS(on)

, from its

drain to source and exhibits a small voltage drop of V

× R

DS(on)

. There is also a small

voltage drop across the dc resistance of the inductor equal to I

× R

. Thus, the input voltage,

, minus losses, (V

+ I

× R

), is applied across the inductor, L. CR1 is OFF during this

time because it is reverse biased. The inductor current, I

, flows from the input source, V

through Q1 and to ground. During the ON state, the voltage applied across the inductor is

constant and equal to V

– V

– I

× R

. Adopting the polarity convention for the current I

shown in Figure 2, the inductor current increases as a result of the applied voltage. Also, since

the applied voltage is essentially constant, the inductor current increases linearly. This

increase in inductor current during T

is illustrated in Figure 3.

SLUA059A

Understanding Buck-Boost Power Stages in Switch Mode Power Supplies

The amount that the inductor current increases can be calculated by using a version of the

familiar relationship:

+ L

å DI

The inductor current increase during the ON state is given by:

()) +

) I

This quantity, ∆I

(+), is referred to as the inductor ripple current. Also notice that during this

period, all of the output load current is supplied by the output capacitor, C.

Referring to Figure 2, when Q1 is OFF, it presents a high impedance from its drain to source.

Therefore, since the current flowing in the inductor L cannot change instantaneously, the

current shifts from Q1 to CR1. Due to the decreasing inductor current, the voltage across the

inductor reverses polarity until rectifier CR1 becomes forward biased and turns ON. The

voltage applied across L becomes (V

– V

– I

× R

) where the quantity, V

, is the forward

voltage drop of CR1. The inductor current, I

, now flows from the output capacitor and load

resistor combination through CR1 and to ground. Notice that the orientation of CR1 and the

direction of current flow in the inductor means that the current flowing in the output capacitor

and load resistor combination causes V

to be a negative voltage. During the OFF state, the

voltage applied across the inductor is constant and equal to (V

– V

– I

× R

). Maintaining

our same polarity convention, this applied voltage is negative (or opposite in polarity from the

applied voltage during the ON time), because the output voltage V

is negative. Hence, the

inductor current decreases during the OFF time. Also, since the applied voltage is essentially

constant, the inductor current decreases linearly. This decrease in inductor current during

OFF

is illustrated in Figure 3.

The inductor current decrease during the OFF state is given by:

(*) +

* V

* I

OFF

This quantity, ∆I

(–), is also referred to as the inductor ripple current.

In steady state conditions, the current increase, ∆I

(+), during the ON time and the current

decrease during the OFF time, ∆I

(–), must be equal. Otherwise, the inductor current would

have a net increase or decrease from cycle to cycle which would not be a steady state

condition. Therefore, these two equations can be equated and solved for V

to obtain the

continuous conduction mode buck-boost voltage conversion relationship:

Solving for V

* V

OFF

* V

* I

) T

OFF

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