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Texas Instruments Understanding Inverting Buck-Boost Power Stages in Switch Mode Power Supplies (Rev. B) Application notes
Application Report
SLVA059B – March 1999 – Revised March 2019
Understanding Inverting Buck-Boost Power Stages in
Switch Mode Power Supplies
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 application report addresses the inverting buck-boost power stage only and does not
cover control circuits. Detailed steady-state and small-signal analysis of the inverting buck-boost power
stage operating in continuous and discontinuous mode is presented. The application report concentrates
on the non synchronous version (using a diode as second switch) as a synchronous inverting buck-boost
converter acts for all load currents like a non-synchronous version in continuous conduction mode.
Variations in the standard inverting buck-boost power stage and a discussion of the power stage
component requirements are included.
1
2
3
4
5
6
7
Contents
Introduction ................................................................................................................... 2
Inverting Buck-Boost Stage Steady-state Analysis ...................................................................... 3
Inverting Buck-Boost Power Stage Small Signal Modeling ........................................................... 10
Variations of the Inverting Buck-Boost Power Stage .................................................................. 17
Component Selection ...................................................................................................... 20
Summary .................................................................................................................... 24
References .................................................................................................................. 26
List of Figures
1
Inverting Buck-Boost Power Stage Schematic........................................................................... 2
2
Inverting Buck-Boost Power Stage States
3
4
5
6
7
8
9
10
11
12
13
14
15
............................................................................... 3
Continuous Mode Inverting Buck-Boost Power Stage Waveforms ................................................... 4
Boundary Between Continuous and Discontinuous Mode ............................................................. 6
Discontinuous Current Mode ............................................................................................... 7
Discontinuous Mode Inverting Buck-Boost Converter Waveforms .................................................... 8
Power Supply Control Loop Components............................................................................... 10
Inverting Buck-Boost Power Stage Gain Versus Duty Cycle ......................................................... 10
DC and Small Signal CCM PWM Switch Model ....................................................................... 11
CCM Inverting Buck-Boost Small Signal AC Power Stage Model ................................................... 12
DCM PWM Switch Model ................................................................................................. 15
DCM Inverting Buck-Boost Converter Model ........................................................................... 15
Small Signal DCM Inverting Buck-Boost Power Stage Model ....................................................... 16
Flyback Power Stage Schematic ......................................................................................... 17
Discontinuous Mode Flyback Waveforms ............................................................................... 18
Trademarks
All trademarks are the property of their respective owners.
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Introduction
1
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Introduction
The three basic switching power supply topologies in common use are the buck, boost, and inverting
buck-boost. These topologies are non-isolated, so the input and output voltages share a common ground.
There are, however, isolated derivations of these non-isolated 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 inverting buck-boost is a popular non-isolated, inverting power stage topology. Power supply
designers choose the inverting 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 an inverting buck-boost power stage is discontinuous
or pulsating due to the power switch (Q1) current that pulses from zero to IL every switching cycle. The
output current for an inverting buck-boost power stage is also discontinuous or pulsating 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 inverting 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 inverting 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, form the effective output filter. The capacitor ESR, RC, (equivalent series resistance) and the
inductor DC resistance, RL, are included in the analysis. The resistor, R, represents the load seen by the
power stage output.
a
Q1
CR1
p
VO
ia
c
+
Drive
Circuit
C
IL = ic
R
VI
RC
RL
Figure 1. Inverting Buck-Boost Power Stage Schematic
During the normal operation of the inverting 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 LC output filter is formed. It filters the train of pulses to produce a
DC output voltage.
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Inverting Buck-Boost Stage Steady-state Analysis
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2
Inverting 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 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, VGS(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 nchannel FET is its lower RDS(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 RDS(on), but usually does not require a
floating drive circuit.
In Figure 1, 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 Section 3.
2.1
Inverting 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 inverting buck-boost power stage. This is important because it shows 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 the continuous conduction mode, the inverting 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. Figure 2 shows the circuit diagram for each of the
two states.
ON State
OFF State
a
ia
p
RDS(on)
a
VO
c
ia
IO
Vd
VO
c
IO
C
+
p
C
+
IL = ic
R
VI
IL = ic
R
VI
RC
RC
RL
RL
Figure 2. Inverting Buck-Boost Power Stage States
The duration of the ON state is D × TS = TON 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, TS. The duration of
the OFF state is called TOFF. Since there are only two states per switching cycle for continuous conduction
mode, TOFF is equal to (1−D) × TS. The quantity (1−D) is sometimes called D’. Figure 3 shows these times
and the waveforms.
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IQ1
ICR1
IL Solid
IO Dashed
0
¨,L
Vc-p Solid
VO Dashed
0
TON
TOFF
TS
Figure 3. Continuous Mode Inverting Buck-Boost Power Stage Waveforms
Referring to Figure 2, during the ON state, Q1 presents a low resistance, RDS(on), from its drain to source. It
also exhibits a small voltage drop of VDS = IL × RDS(on). There is also a small voltage drop across the DC
resistance of the inductor equal to IL × RL. Thus, the input voltage, VI, minus losses, (VDS + IL × RL), is
applied across the inductor, L. CR1 is OFF during this time because it is reverse biased. The inductor
current, IL, flows from the input source, VI, through Q1 and to ground. During the ON state, the voltage
applied across the inductor is constant and equal to VI − VDS − IL × RL. Figure 2 shows the polarity
convention for the current. The inductor current increases as a result of the applied voltage when adopting
this polarity convention for the current IL. Also, since the applied voltage is essentially constant, the
inductor current increases linearly. Figure 3 shows this increase in inductor current during TON. Equation 1
shows a version of the familiar relationship and how to calculate the amount that the inductor current
increases.
di L
VL
VL L u
Ÿ 'I L
u 'T
dt
L
(1)
Equation 2 shows how the inductor current increases during the ON state.
'I L
VI
V DS
L
IL u R L
u T ON
(2)
This quantity, ∆IL(+), is referred to as the inductor ripple current. 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 (VO − Vd − IL × RL)
where the quantity, Vd, is the forward voltage drop of CR1. The inductor current, IL, 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 VO to be a negative voltage. During the OFF state, the voltage
applied across the inductor is constant and equal to (VO − Vd − IL × RL). Maintaining the same polarity
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convention, this applied voltage is negative (or opposite in polarity from the applied voltage during the ON
time), because the output voltage VO is negative. Hence, the inductor current decreases during the OFF
time. Also, since the applied voltage is essentially constant, the inductor current decreases linearly.
Figure 3 shows this decrease in inductor current during TOFF. Equation 3 shows how to calculate the
inductor current decrease during the OFF state.
Vo
'I L
Vd
IL u R L
L
u T OFF
(3)
This quantity, ∆IL(−), is also referred to as the inductor ripple current. In steady-state conditions, the
current increase, ∆IL(+), during the ON time and the current decrease during the OFF time, ∆IL(−), must be
equal. Otherwise, the inductor current has 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 VO to obtain
the continuous conduction mode inverting buck-boost voltage conversion relationship.
Equation 4 shows how to solve for VO.
VO
ª
« VI
¬«
V DS u
T ON
T OFF
Vd
IL u R L u
T OFF º
»
T OFF
¼»
T ON
Equation 5 shows how to solve for VO while substituting TS for TON + TOFF, and using D = TON / TS and
(1 − D) = TOFF / TS.
IL u R L º
ª
D
VO
Vd
« V I V DS u
»
1 D
1 D ¼
¬
(4)
(5)
Notice that in simplifying the above, TON + TOFF is assumed to be equal to TS. This is true only for
continuous conduction mode seen in the discontinuous conduction mode analysis.
NOTE: An important observation must be made here. Setting the two values of ∆IL equal to each
other is precisely equivalent to balancing the volt-seconds on the inductor. The volt-seconds
applied to the inductor is the product of the voltage applied and the time that the voltage is
applied. This is the best way to calculate unknown values such as VO or D in terms of known
circuit parameters, and this method is applied repeatedly in this paper. Volt-second balance
on the inductor is a physical necessity and must be understood at least as well as Ohms
Law.
In the above equations for ∆IL(+) and ∆IL(−), the output voltage is implicitly assumed to be constant with no
AC ripple voltage during the ON time and the OFF time. This is a common simplification and involves two
separate effects. First, the output capacitor is assumed to be large enough that its voltage change is
negligible. Second, the voltage due to the capacitor ESR is also assumed to be negligible. These
assumptions are valid because the AC ripple voltage is designed to be much less than the DC part of the
output voltage.
The above voltage conversion relationship for VO illustrates the fact that VO can be adjusted by adjusting
the duty cycle, D. This relationship approaches zero as D approaches zero and increases without bound
as D approaches 1. A common simplification is to assume VDS, Vd, and RL are small enough to ignore.
Equation 6 shows a simplification of the above equations when you set VDS, Vd, and RL to zero.
D
VO
VI
(6)
1 D
A simplified, qualitative way to visualize the circuit operation is to consider the inductor as an energy
storage element. When Q1 is on, energy is added to the inductor. When Q1 is off, the inductor delivers
some of its energy to the output capacitor and load. The output voltage is controlled by setting the on-time
of Q1. For example, by increasing the on-time of Q1, the amount of energy delivered to the inductor is
increased. More energy is then delivered to the output during the off-time of Q1, resulting in an increase in
the output voltage.
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Unlike the buck power stage, the average of the inductor current is not equal to the output current. To
relate the inductor current to the output current, referring to Figure 2 and Figure 3, note that the inductor
delivers current to the output only during the off state of the power stage. This current averaged over a
complete switching cycle is equal to the output current because the average current in the output
capacitor must be equal to zero. Equation 7 or Equation 8 calculates the relationship between the average
inductor current and the output current for the continuous mode inverting buck-boost power stage.
T OFF
I L Avg u
I L Avg u 1 D
IO
TS
(7)
or
IL
IO
Avg
1 D
(8)
Another important observation is that the average inductor current is proportional to the output current,
and since the inductor ripple current, ∆IL, is independent of output load current, the minimum and the
maximum values of the inductor current track the average inductor current exactly. For example, if the
average inductor current decreases by 2 A due to a load current decrease, then the minimum and
maximum values of the inductor current decrease by 2 A (assuming continuous conduction mode is
maintained).
The forgoing analysis is for the inverting buck-boost power stage operation in continuous inductor current
mode. Section 2.2 is a description of steady-state operation in discontinuous conduction mode. The main
result is a derivation of the voltage conversion relationship for the discontinuous conduction mode
inverting buck-boost power stage.
2.2
Inverting Buck-Boost Steady-state Discontinuous Conduction Mode Analysis
This section investigates what happens when the load current is decreased and the conduction mode
changes from continuous to discontinuous. Recall for continuous conduction mode, the average inductor
current tracks the output current, meaning if the output current decreases, then so does the average
inductor current. In addition, the minimum and maximum peaks of the inductor current follow the average
inductor current exactly.
If the output load current is reduced below the critical current level, the inductor current is zero for a
portion of the switching cycle. This is evident from the waveforms in Figure 3 as the inductor ripple current
is independent on the load current. In a non-synchronous inverting buck-boost power stage, if the inductor
current attempts to fall below zero, it just stops at zero (due to the unidirectional current flow in CR1) and
remains there until the beginning of the next switching cycle. This operating mode is called discontinuous
conduction mode. A power stage operating in discontinuous conduction mode has three unique states
during each switching cycle as opposed to two states for continuous conduction mode. Figure 4 shows the
inductor current condition where the power stage is at the boundary between continuous and
discontinuous mode. This is where the inductor current just falls to zero and the next switching cycle
begins immediately after the current reaches zero. Figure 4 shows the absolute values of IO and IO(Crit).
Note that there are absolute values because IO and IL have opposite polarities.
IL Solid
|IO| Dashed = |IO(Crit)|
0
¨,L(max)
TON
TOFF
TS
Figure 4. Boundary Between Continuous and Discontinuous Mode
Further reduction in output load current puts the power stage into discontinuous conduction mode.
Figure 5 illustrates this condition. The discontinuous mode power stage frequency response is different
from the continuous mode frequency response and is given in Section 3. Also, the input to output
relationship is different, as the following derivations show.
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IL Solid
|IO| Dashed
0
¨,L
DTS
D2TS
D3TS
TS
Figure 5. Discontinuous Current Mode
To begin the derivation of the discontinuous conduction mode inverting buck-boost power stage voltage
conversion ratio, recall that there are three unique states that the converter assumes during discontinuous
conduction mode operation. The ON State is when Q1 is ON and CR1 is OFF. The OFF State is when Q1
is OFF and CR1 is ON. The IDLE state is when both Q1 and CR1 are OFF. The first two states are
identical to those of the continuous mode case. Figure 2 shows the circuits that are applicable except that
TOFF ≠ (1−D) × TS. The remainder of the switching cycle is the IDLE state. In addition, the DC resistance of
the output inductor, the output diode forward voltage drop, and the power MOSFET ON-state voltage drop
are all assumed to be small enough to omit.
The duration of the ON state is TON = D × TS where D is the duty cycle, set by the control circuit and
expressed as a ratio of the switch ON time to the time of one complete switching cycle, TS. The duration of
the OFF state is TOFF = D2 × TS. The IDLE time is the remainder of the switching cycle and is given as
TS − TON − TOFF = D3 × TS. Figure 6 shows these times with the corrresponding waveforms. Without going
through the detailed explanation as before, the equations for the inductor current increase and decrease
are given below. Equation 9 shows how to find the inductor current increase during the ON state.
VI
VI
'I L
u T ON
u D u T S I PK
(9)
L
L
The ripple current magnitude, ∆IL(+), is also the peak inductor current, Ipk, because in discontinuous mode,
the current starts at zero each cycle. Equation 10 gives the inductor current decrease during the OFF
state.
VO
VO
'I L
u T OFF
u D 2 u TS
(10)
L
L
As in the continuous conduction mode case, the current increase, ∆IL(+), during the ON time and the
current decrease during the OFF time, ∆IL(−), are equal. Therefore, these two equations can be equated
and solved for VO to obtain the first of two equations to be used to solve for the voltage conversion ratio.
T ON
D
VO
VI u
VI u
T OFF
D2
(11)
The output current (the output voltage VO divided by the output load R). It is the average over one
switching cycle of the inductor current during the time when CR1 conducts (D2 × TS).
VO
I PK
IO
u D2
R
2
Equation 13 shows the result of substituting the relationship for IPK (∆IL(+) ) into Equation 12.
VO
§ VI
·
1
IO
u 1 u ¨¨
u D ¸¸ u D 2
R
2
© L
¹
VO
VI u D u D 2 u T S
R
2uL
(12)
(13)
(14)
Now, there are two equations. The one for the output current (VO divided by R) just derived and the one
for the output voltage, both in terms of VI, D, and D2. Solve each equation for D2 and set the two equations
equal to each other. Using the resulting equation, an expression for the output voltage, VO, can be
derived.
Equation 15 calculates the discontinuous conduction mode inverting buck-boost voltage conversion
relationship.
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VO
VI u
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D
K
where
K
•
K is defined as
2uL
R u TS
(15)
The above relationship shows one of the major differences between the two conduction modes. For the
discontinuous conduction mode, the voltage conversion relationship is a function of the input voltage, duty
cycle, power stage inductance, the switching frequency, and the output load resistance. For continuous
conduction mode, the voltage conversion relationship is only dependent on the input voltage and duty
cycle.
IQ1
ICR1
IL Solid
IO Dashed
¨,L
VC-P Solid
VO Dashed
0
D x TS
D3 x TS
D2 x TS
TS
Figure 6. Discontinuous Mode Inverting Buck-Boost Converter Waveforms
In typical applications, the inverting buck-boost power stage is operated in either continuous conduction
mode or discontinuous conduction mode. For a particular application, one conduction mode is chosen and
the power stage is designed to maintain the same mode. Section 2.3 gives inductance relationships for
the power stage that allow it to operate in only one conduction mode, given ranges for input and output
voltage and output load current.
2.3
Critical Inductance
The previous analyses for the inverting buck-boost converter have been for continuous and discontinuous
conduction modes of steady-state operation. The conduction mode of a converter is a function of input
voltage, output voltage, output current, and the value of the inductor. An inverting buck-boost converter
can be designed to operate in continuous mode for load currents above a certain level, usually 5% to 10%
of full load. Usually, the input voltage range, the output voltage, and load current are defined by the
converter specification. This leaves the inductor value as the design parameter to maintain continuous
conduction mode.
The minimum value of inductor to maintain continuous conduction mode can be determined by the
following procedure.
1. Define IO(Crit) as the minimum output current to maintain continuous conduction mode, normally referred
to as the critical current. Figure 4 shows this value. Since you are working toward a minimum value for
the inductor, it is more straightforward to perform the derivation using the inductor current. Equation 16
calculates the minimum average inductor current to maintain continuous conduction mode.
'I L
I L min avg
I O(crit)
(16)
2
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2. Calculate L such that the above relationship is satisfied. To solve the above equation, either
relationship, ∆IL(+) or ∆IL(−) can be used for ∆IL. Note that either relationship for ∆IL is independent of
the output current level. Here, ∆IL(+) is used. The worst case condition (giving the largest Lmin) is at
maximum input voltage because this gives the maximum ∆IL. Equation 17 substitutes Equation 2 into
Equation 16 and is then solved for Lmin.
T ON min
1
L min t u V I max
V DS I L u R L u
2
I O crit
(17)
Equation 17 can be simplified by ignoring minor parasitic resistances and diode voltage drops. After
substituting TON(min) with Equation 6 and rearranged, Equation 18 is the result.
V O u TS
V I(max)
u
L min t
2 u I O crit
V O V I max
(18)
Using the inductor value you just calculated guarantees continuous conduction mode operation for output
load currents above the critical current level, IO(crit).
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Inverting Buck-Boost Power Stage Small Signal Modeling
This section moves from a detailed circuit oriented analysis approach to more of a system-level
investigation of the inverting buck-boost power stage. This section also presents techniques to assist you
in accurately modeling the power stage as a component of the control loop of an inverting buck-boost
power supply. The three major components of the power supply control loop are the power stage, the
pulse width modulator, and the error amplifier. Figure 7 shows these components in a block diagram.
VI
VO
Power Stage
Error
Amplifier
Duty Cycle
d(t)
Pulse-Width
Modulation
Error
Voltage VE
Reference
Voltage Vref
Figure 7. Power Supply Control Loop Components
Modeling the power stage presents one of the main challenges to the power supply designer. A popular
technique involves modeling only the switching elements of the power stage. An equivalent circuit for
these elements is derived and is called the PWM Switch Model where PWM means pulse width
modulated. This approach is presented here.
Figure 7 shows the two inputs for the power stage: the input voltage and the duty cycle. The duty cycle is
the control input. This input is a logic signal, which controls the switching action of the power stage and
hence the output voltage. Most power stages have a nonlinear voltage conversion ratio versus duty cycle.
Figure 8 shows a graph of the steady-state voltage conversion ratio for an inverting buck-boost power
stage operating in continuous conduction mode as a function of steady-state duty cycle, D.
The nonlinear characteristics are a result of the switching action of the power stage switching
components, Q1 and CR1. It is observed in the Simplified Analysis of PWM Converters Using the Model of
the PWM Switch: Parts I and II, available in Section 7, that the only nonlinear components in a power
stage are the switching devices – the remainder of the circuit consists of linear elements. It is also shown
in the Simplified Analysis of PWM Converters Using the Model of the PWM Switch: Parts I and II that a
linear model of only the nonlinear components can be derived by averaging the voltages and currents
associated with these nonlinear components over one switching cycle. The model is then substituted into
the original circuit for analysis of the complete power stage. Thus, a model of the switching devices is
given and is called the PWM switch model.
10
Buck-Boost Power Stage Gain
9
8
7
6
5
4
3
2
1
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.8
0.9
1
Duty Cycle
Figure 8. Inverting Buck-Boost Power Stage Gain Versus Duty Cycle
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The basic objective behind modeling power stages is to represent the non-linear behavior of power stages
as linear about an operating point. You want linearity so that you can apply the many analysis tools
available for linear systems. Referring to Figure 8, if you choose the operating point of D = 0.7, a straight
line can be constructed that is tangent to the original curve at the point where D = 0.7. This is an
illustration of linearization about an operating point, a technique used to derive the PWM switch model.
Qualitatively, you can see that if the variations in duty cycle are kept small, a linear model accurately
represents the nonlinear behavior of the power stage being analyzed.
Since a power stage can operate in one of two conduction modes such as continuous conduction mode
(CCM) or discontinuous conduction mode (DCM), there is a PWM switch model for the two conduction
modes. The DCM PWM switch model is derived here. The CCM PWM switch model is derived in the
Understanding Buck Power Stages in Switchmode Power Supplies Application Report.
3.1
Inverting Buck-Boost Continuous Conduction Mode Small-Signal Analysis
To model the inverting buck-boost power stage operation in CCM, use the CCM PWM switch model
derived in the Understanding Buck Power Stages in Switchmode Power Supplies Application Report. The
PWM switch model is inserted into the power stage circuit by replacing the switching elements. Figure 9
shows the CCM PWM Switch model, which is useful for determining the DC operating point of a power
stage and for finding AC transfer functions of a power stage.
Referring to Figure 1, the power transistor, Q1, and the catch diode, and CR1, are drawn inside a dashedline box. These are the components that are replaced by the PWM switch equivalent circuit. Terminal
labels a (active), p (passive), and c (common) are used for the PWM switch model.
Vap Ö
d
D
c
a
±
+
ia
ic
1
D
I C dÖ
p
Figure 9. DC and Small Signal CCM PWM Switch Model
The a terminal is the terminal connected to the active switch. The p terminal is the terminal of the passive
switch. The c terminal is the terminal that is common to both the active and passive switches. The three
commonly used power stage topologies all contain active and passive switches, and the above terminal
definitions can be used. In addition, substituting the PWM switch model (that you derive) into other power
stage topologies also produces a valid model for that particular power stage. To use the PWM switch
model in other power stages, substitute the model in Figure 9 into the power stage in the appropriate
orientation.
In the PWM switch model in Figure 9 and subsequent occurrences of the model, the capital letters indicate
steady-state (or DC) quantities dependent on the operating point of the circuit under study. The lowercase
letters indicate time varying quantities and can indicate a quantity with a DC component and an AC
component. The lowercase letters with a caret (hat) indicate the small AC variations of that particular
Ö
variable. For example, D represents the steady-state duty cycle, d represents small AC variations of the
duty cycle, and d or d(t) represents the complete duty cycle including any DC component and AC
variations.
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a
Vap Ö
d
D
I C dÖ
+
±
ia
1
+
p
VO
VI
D
C
c
L
R
IL = ic
RC
RL
Figure 10. CCM Inverting Buck-Boost Small Signal AC Power Stage Model
An example DC analysis is given to illustrate how simple power stage analysis becomes with the PWM
Ö
switch model. For DC analysis, d is zero, L is a short and C is an open. Equation 19 shows the result of
a simple loop equation.
Vcp
VI
V O 0 o V cp
VI V O u D
D
(19)
By using the following relationships:
IO
IL
1 D
VO
IO
R
(20)
(21)
And Vcp + Vo - IL × RL = 0, Equation 22 shows the result solved for Vcp.
V cp
§
V O u ¨1
¨
©
·
¸
R u 1 D ¸¹
RL
(22)
Equation 23 shows the result of setting the two loop equations solved for Vcp equal. It is solved for VO to
get the steady-state voltage conversion relationship for the inverting buck-boost power stage operating in
CCM, taking the inductor DC resistance, RL into account.
D
1
VO
VI u
u
RL
1 D
1
2
Ru 1 D
(23)
The above equation is usually expressed as a ratio of the output voltage, VO, to the input voltage, VI, and
is usually called M. Equation 24 shows this ratio.
VO
D
1
M
u
RL
VI
1 D
1
2
Ru 1 D
(24)
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When RL = 0, as assumed earlier, M is equal to the steady-state I/O transfer function previously
calculated. With the PWM switch parameters (Vap and Ic) determined from the DC analysis, an AC analysis
can be performed. For AC analysis, the following transfer functions can be calculated: open-loop line-tooutput, open-loop input impedance, open-loop output impedance, and open-loop control-to-output. The
control-to-output, or duty-cycle-to-output, is the transfer function most used for control loop analysis. To
determine this transfer function, use the results from the DC analysis for operating point information. This
information determines the parameter values of the dependent sources, see Equation 25 and Equation 26
as an example.
V ap V I V O V I u 1 M
and
IC
(25)
IL
IO
VO
1 D
M u VI
Ru 1 D
Ru 1 D
(26)
The above two equations are then used with loop equations to derive the duty-cycle-to-output voltage
transfer function. Set the input voltage equal to zero since you only want the AC component of the transfer
function. Without going through all the details, the following equations show that the transfer function can
be put as in Equation 27.
vÖ O
s
dÖ
G do
§
¨1
¨
u©
s · §
¸ u ¨1
& z1 ¸¹ ¨©
1
s ·
¸
& z2 ¸¹
s2
s
&O u4
& o2
where
•
•
G do |
•
S z2 |
•
&O |
•
3.2
1 D
2
1
RC u C
S z1
Q|
VI
1 D
2
uR
Du L
1 D
LuC
1 D uR
L
C
(27)
Inverting Buck-Boost Discontinuous Conduction Mode Small-Signal Analysis
The section discusses modeling the power stage when it is operating in discontinuous conduction mode
(DCM). This mode is quite different from the continuous conduction mode just covered. Start with the
derivation of the PWM switch model for DCM for the inverting buck-boost power stage. This derivation can
also be found in the Fundamentals of Power Electronics paper. This reference is available in Section 7.
The waveforms, that are averaged, are the voltage across Q1, vac, the voltage across CR1, vcp, the current
in Q1, ia, and the current in CR1, ip. Figure 6 shows the waveforms.
Some basic relationships that are used repeatedly, are stated first. Equation 28 and Equation 29 give the
terminal currents averaged over one switching cycle.
i pk
ia
ud
2
(28)
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i pk
ip
2
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u d2
where
•
variables in brackets (for example 〈ia〉) represent quantities that are averaged over one switching cycle
(29)
Since the average over one switching cycle of the voltage across the inductor is zero, the following
average voltage relationships hold:
v ac
v cp
VI
ic u RL
VO
(30)
ic u RL
(31)
Since the value of ic × RL is small compared to VI and VO, ignore these values for modeling the power
stage to make the formulae much easier to manipulate. During the time period d × TS, the current ia starts
at a value of zero and ends at the value of ipk. Since the voltage across the inductor during this time is
constant and equal to VI = 〈vac〉, Equation 32 holds.
'i a
i pk
i pk
Ÿ v ac
VI L
Lu
Lu
't
d u TS
d u TS
(32)
Similarly, during the time period d2 × TS, the current ip starts at a value of ipk and ends at zero. Also since
the voltage across the inductor is equal to −VO = 〈vcp〉, Equation 33 holds.
'i p
i pk
i pk
Ÿ v cp
VO L
Lu
Lu
't
d 2 u TS
d 2 u TS
(33)
With Equation 28, Equation 29, Equation 32, and Equation 33, begin with the derivation of the input side
(vac side) of the PWM switch model. Solve Equation 32 for ipk and use VI = 〈vac〉, then substitute into
Equation 28 to get Equation 34.
ia
VI u
d 2 u TS
2uL
(34)
Note that the average current flowing into terminal a is proportional to the input voltage, VI. Define an
effective resistance as Equation 35.
VI
2uL
Re
2
ia
d u TS
(35)
2
I
As the input looks like an equivalent resistance, you can talk about an apparent input power of V / Re,
which is used next.
To begin the derivation of the output side (vcp side), start with Equation 33, solve Equation 32 for ipk, use
VI = 〈vac〉, and then substitute back into Equation 33. Equation 36 is the result.
v cp
v ac u d
d2
(36)
Solve the above equation for d2 and substitute into Equation 29 and also use ipk from and substitute it into
Equation 32. Equation 37 is the result after some rearranging.
ip
v ac
2
u d 2 u TS
v cp u 2 u L
(37)
Finally, use 〈vac〉 = VI and substitute in the above equation. Equation 38 is the result and gives the output
side relationship.
i p u v cp
V I2
u
d 2 u TS
V I2
2uL
Re
(38)
This equation shows that the average output current times the average output voltage is equal to the
apparent input power.
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Implement the above input and output relationships into an equivalent circuit model. This model is useful
for determining the DC operating point of a power supply. The input port is simply modeled with a resistor,
Re. The output port is modeled as a dependent power source. This power source delivers power equal to
that dissipated by the input resistor, Re. Figure 11 shows the constructed equivalent circuit.
p
a
Re
p(t)
c
Figure 11. DCM PWM Switch Model
To illustrate discontinuous conduction mode power supply steady-state analysis using this model, examine
the inverting buck-boost converter. The analysis proceeds like the CCM case. The equivalent circuit is
substituted into the original circuit. The inductor is treated as a short circuit and the capacitor is treated as
an open circuit. Figure 12 shows the DCM inverting buck-boost converter model schematic, ignoring the
inductor DC resistance, RL, and the equivalent series resistance of the output capacitor, RC.
p
a
VO
Re
p(t)
+
C
VI
R
c
L
Figure 12. DCM Inverting Buck-Boost Converter Model
The apparent power dissipated in Re is determined as Equation 39.
PRe
V I2
Re
(39)
The dependent power source delivers the above amount of power to the output load resistor, R. Calculate
the voltage gain as a function of D by equating the two powers from Equation 38 and Equation 39 as
shown in Equation 40 through Equation 42.
V I2
V O2
Re
R
Vo
r
VI
(40)
R
Re
(41)
Choose the negative root and substitute Re by Equation 35.
R
Re
R
2uL
Du
R u TS
2uL
2
D u TS
(42)
Equation 43 gives the voltage conversion relationship for the DCM inverting buck-boost.
Vo
VI
Du
R u TS
2uL
(43)
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Recall from earlier that:
2uL
K
R u TS
(44)
and making the substitution in the above equation, the result is identical to that obtained by balancing the
inductor volt-seconds in the inverting buck-boost steady-state discontinuous conduction mode analysis
section.
The steady-state voltage conversion relationship for the DCM inverting buck-boost becomes Equation 45.
VO
D
VI
K
(45)
To derive the small signal model, the circuit in Figure 12 is perturbed and linearized following the
procedure similar to the one in the CCM derivation. Refer to the Fundamentals of Power Electronics to
see the detail of derivation. Figure 13 shows the resulting small signal model for the inverting buck-boost
power stage operating in DCM.
vÖ O
2 u VI
D u Re
+
vÖ I
u dÖ
0 u vÖ O
2
u vÖ I
Mu Re
2 u VI
D u Mu Re
u dÖ
M2 uR e
Re
C
R
Figure 13. Small Signal DCM Inverting Buck-Boost Power Stage Model
Equation 46 gives the duty-cycle-to-output transfer function for the inverting buck-boost power stage
operating in DCM.
vÖ O
1
G do u
s
dÖ
1
&p
where
•
G do
•
D
•
•
M
K
VO
D
Mu
K
VO
VI
2uL
R u TS
and
2
RuC
The expression for Gdo can be simplified to Equation 47.
•
G do
16
&p
VI u
(46)
R u Ts
2uL
(47)
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4
Variations of the Inverting Buck-Boost Power Stage
4.1
Flyback Power Stage
A transformer-coupled variation of the traditional inverting buck-boost power stage is the flyback power
stage. This power stage operates like the traditional inverting buck-boost power stage, except that the
single winding inductor is replaced with a two (or more) winding coupled inductor. The power switch, Q1 in
Figure 14, applies the input voltage to the primary side (LPRI) of the coupled inductor. Energy is stored until
Q1 is turned off. Energy is then delivered to the output capacitor and load resistor combination from the
secondary side (LSEC) of the coupled inductor through the output diode CR1. This power stage provides
electrical isolation of the input voltage from the output voltage. Besides providing electrical isolation, the
isolation transformer can step-down (or step-up) the input voltage to the secondary. The transformer turns
ratio can be designed so that reasonable duty cycles are obtained for almost any input voltage and output
voltage combination, thus avoiding extremely small or extremely high duty cycle values.
The flyback power stage also eliminates two characteristics which sometimes make the standard inverting
buck-boost power stage unattractive, that is the output voltage is opposite in polarity from the input voltage
and the power switch requires a floating drive. Besides providing isolation, the coupled inductor secondary
can be connected to produce an output voltage of either positive or negative polarity. In addition, since the
power switch is in series with the primary of the coupled inductor, the power switch can be connected so
that the source is ground referenced instead of connecting the drain to the input voltage as in the standard
inverting buck-boost power stage.
The flyback power stage is popular in 48-V input telecom applications and 110-V AC or 220-V AC off-line
applications for output power levels up to approximately 50 W. The exact power rating of the flyback
power stage, of course, is dependent on the input voltage and output voltage combination, its operating
environment, and many other factors. Additional output voltages can be generated by simply adding
another winding to the coupled inductor along with an output diode and output capacitor. Obtaining
multiple output voltages from a single power stage is another advantage of the flyback power stage.
Figure 14 shows a simplified schematic of the flyback power stage with a drive circuit block included. In
Figure 14, the secondary winding of the coupled inductor is connected to produce a positive output
voltage. The power switch, Q1, is an n-channel MOSFET. The diode, CR1, is usually called the output
diode. The secondary inductance, LSEC, and capacitor, C, make up the output filter. The capacitor ESR,
RC, (equivalent series resistance) is not included. The resistor, R, represents the load seen by the power
supply output.
CR1
VO
Vb
LPRI
LSEC
+
C
Va
VI
R
Q1
Drive
Circuit
Figure 14. Flyback Power Stage Schematic
Figure 15 shows the important waveforms for the flyback power stage operating in DCM.
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IQ1
ICR1 Solid
IO Dased
Vb Solid
VO Dashed
0
Va
0
D x TS
D3 x TS
D2 x TS
TS
Figure 15. Discontinuous Mode Flyback Waveforms
Equation 48 calculates the simplified voltage conversion relationship for the flyback power stage operating
in CCM (ignoring parasitics).
N
D
V O VI u s u
Np 1 D
where
•
•
Ns is the number of secondary side turns
Np is the number of primary side turns
(48)
Equation 49 calculates the simplified voltage conversion relationship for the flyback power stage operating
in DCM (ignoring parasitics).
N
D
V O VI u s u
Np
K
where
K
•
K is defined as
2 u L SEC
R u TS
(49)
Equation 50 calculates the simplified duty-cycle-to-output transfer function for the flyback power stage
operating in CCM.
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vÖ O
s
dÖ
§
s · §
s ·
¨1
¸ u ¨1
¸
¨
¸
¨
N S © & z1 ¹ © & z2 ¸¹
G do u
u
NP
s
s2
1
1
&o u4
&o2
where
•
G do |
& z1
•
& z2 |
&o |
Q|
1 D
2
1
RC u C
•
•
VI
1 D
2
uR
D u L SEC
1 D
2
uR
L SEC u C
1 D uR
L SEC
•
(50)
C
Equation 51 calculates the simplified duty-cycle-to-output transfer function for the flyback power stage
operating in DCM.
vÖ O
1
G do u
Öd
s
1
&p
where
•
•
G do
VI u
&p
2
RuC
NS
u
NP
R u TS
2 u L SEC
(51)
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Component Selection
This section discusses the function of each of the main components of the inverting buck-boost power
stage. The electrical requirements and applied stresses are given for each power stage component.
The completed power supply, made up of a power stage and a control circuit, usually must meet a set of
minimum performance requirements. This set of requirements is usually referred to as the power supply
specification. Many times, the power supply specification determines individual component requirements.
5.1
Output Capacitance
In switching power supply power stages, the output capacitance stores energy in the electric field due to
the voltage applied. Thus, qualitatively, the function of a capacitor is to maintain a constant voltage.
The value of the output capacitance of an inverting buck-boost power stage is generally selected to limit
the output voltage ripple to the level required by the specification. The series impedance of the capacitor
and the power stage output current determine the output voltage ripple. The three elements of the
capacitor that contribute to its impedance (and output voltage ripple) are the equivalent series resistance
(ESR), equivalent series inductance (ESL), and capacitance (C). The following gives guidelines for output
capacitor selection.
Use Equation 52 for continuous inductor current mode operation to determine the amount of capacitance
needed as a function of the output load current, IO, switching frequency, fS, and desired output voltage
ripple, ∆VO. This equation assumes all the output voltage ripple is due to the capacitance of the capacitor
because the output capacitor supplies the entire output load current during the power stage ON state.
I O Max u D Max
Ct
f S u 'V O
where
•
IO(Max) is the maximum output current and DMax is the maximum duty cycle
(52)
Use Equation 53 for discontinuous inductor current mode operation to determine the amount of
capacitance needed. This equation assumes all the output voltage ripple is due to the capacitance of the
capacitor.
§
2uL
I O Max u ¨ 1
¨
R u TS
©
Ct
f S u 'V O
·
¸
¸
¹
(53)
However, in many practical designs, to get the required ESR, a capacitor with much more capacitance
than calculated in Equation 53 must be selected. For continuous inductor current mode operation, use
Equation 54 to find the ESR needed to limit the ripple to ∆VO. This equation assumes there is enough
capacitance such that the ripple due to the capacitance can be ignored.
'V O
ESR d
§ I O Max
'I L ·
¨
¸
¨ 1 D Max
2 ¸¹
©
(54)
Use Equation 55 for discontinuous inductor current mode operation to find the ESR needed to limit the
ripple to the peak-to-peak ripple, ∆VO. This equation assumes there is enough capacitance such that the
ripple due to the capacitance can be ignored.
'V O
ESR d
'I L
(55)
Ripple current flowing through the ESR of a capacitor causes power dissipation in the capacitor. This
power dissipation causes a temperature increase internal to the capacitor. Excessive temperature can
seriously shorten the expected life of a capacitor. Capacitors have ripple current ratings that are
dependent on ambient temperature and must not be exceeded. Referring to Figure 3, the output capacitor
ripple current is the inductor current, IL, minus the output current, IO. Use Equation 56 to find the RMS
value of the ripple current flowing in the output capacitance (continuous inductor current mode operation).
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I C RMS
IO u
D
1 D
(56)
ESL can be a problem by causing ringing in the low MHz-region but can be controlled by choosing low
ESL capacitors, limiting lead length (PCB and capacitor), and replacing one large device with several
smaller ones connected in parallel.
There are four capacitor technologies: low-impedance aluminum, polymer or solid tantalum and multi-layer
ceramic capacitors (MLCCs) are suitable for low-cost commercial applications. Low-impedance aluminum
electrolytics are the lowest cost and offer high capacitance in small packages, but ESR is higher than the
other three. Polymer electrolytics have become popular for the power-supply industry in recent years.
These capacitors offer the best of both worlds – a low ESR that is stable over the temperature range and
high capacitance in a small package. MLCCs or solid tantalum chip capacitors are probably the best
choice if a surface-mounted device is needed. These two types offer a low ESR that is relatively stable
over the temperature range, high ripple-current capability, low ESL, surge-current testing, and a high ratio
of capacitance to volume. If MLCCs are used, it is important to review the DC bias effect as they loose
capacitance when DC voltage is applied.
5.2
Output Inductance
In switching power supply power stages, the function of inductors is to store energy. The energy is stored
in their magnetic field due to the current flowing. Thus, qualitatively, the function of an inductor is usually
to attempt to maintain a constant current or equivalently to limit the rate of change of current flow.
The value of output inductance of a inverting buck-boost power stage is generally selected to limit the
peak-to-peak ripple current flowing in it. In doing so, the mode of operation of the power stage, continuous
or discontinuous, is determined. The inductor ripple current is directly proportional to the applied voltage
and the time that the voltage is applied, and it is inversely proportional to its inductance.
Many designers prefer to design the inductor themselves, but that topic is beyond the scope of this report.
However, the following discusses the considerations necessary for selecting the appropriate inductor. See
the TI-Unitrode Magnetics Design Handbook for details on inductor design.
In addition to the inductance, other important factors that need to be considered when selecting the
inductor are its maximum DC or peak current and maximum operating frequency. Using the inductor within
its DC current rating is important to ensure that it does not overheat or saturate. Operating the inductor at
less than its maximum frequency rating ensures that the maximum core loss is not exceeded, resulting in
overheating or saturation.
Magnetic component manufacturers offer a wide range of off-the-shelf inductors suitable for DC/DC
converters, some of which are surface mountable. There are many types of inductors available; the most
popular core materials are ferrites and powdered iron. Bobbin or rod-core inductors are readily available
and inexpensive, but care must be exercised in using them, because they are more likely to cause noise
problems than are other shapes. Multilayer inductors are becoming popular nowadays. Be aware that
these might cause noise problems like Bobbin or rod-core inductors. Custom designs are also feasible,
provided the volumes are sufficiently high.
Current flowing through an inductor causes power dissipation due to the DC resistance from the inductor.
This power dissipation is easily calculated. Power is also dissipated in the core of the inductor due to the
flux swing caused by the AC voltage applied across it, but this information is rarely directly given in
manufacturer’s data sheets. Occasionally, the maximum operating frequency of the inductor and applied
volt-seconds ratings give you some guidance regarding core loss. The power dissipation causes a
temperature increase in the inductor. Excessive temperature can cause degradation in the insulation of
the winding and cause increased core loss. Care must be exercised to ensure all the maximum ratings
from the inductor are not exceeded. Use to find the loss in the inductor.
2
Pinductor
I L RMS
u R Cu
PCore
where
•
RCu is the winding resistance
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Power Switch
In switching power supply power stages, the function of the power switch is to control the flow of energy
from the input power source to the output voltage. In an inverting buck-boost power stage, the power
switch (Q1 in Figure 1) connects the input to the inductor when the switch is turned on and disconnects
when the switch is off. The power switch must conduct the current in the output inductor while on and
block the difference between the input voltage and output voltage when off. Also, the power switch must
change from one state to the other quickly in order to avoid excessive power dissipation during the
switching transition.
The type of power switch considered in this report is a power MOSFET. Other power devices are available
but in most instances, the MOSFET is the best choice in terms of cost and performance (when the drive
circuits are considered). The two types of MOSFET available for use are the n-channel and the p-channel.
P-channel MOSFETs are popular for use in inverting buck-boost power stages because driving the gate is
simpler than the gate drive required for an n-channel MOSFET. Use Equation 58 to find the power
dissipated by the power switch, Q1.
2
§ IO ·
1
u V I V O u ¨¨
PD Q1
I L RMS u r DS on u D
¸¸ u t r t f u f S Q Gate u V GS u f s
2
©1 D ¹
where
•
•
tr and tf are the MOSFET turn-on and turn-off switching times
QGate is the MOSFET total gate charge
(58)
After selecting p-channel or n-channel, other parameters to consider while selecting the appropriate
MOSFET are the maximum drain-to-source breakdown voltage, V(BR)DSS and the maximum drain current,
ID(Max).
The selected MOSFET must have a V(BR)DSS rating greater than the maximum difference between the input
voltage and output voltage, and some margin must be added for transients and spikes. The selected
MOSFET must also have an ID(Max) rating of at least two times the maximum power stage inductor current.
However, many times this is not sufficient margin and the MOSFET junction temperature must be
calculated to make sure that it is not exceeded. Use Equation 59 to estimate the junction temperature.
T J T A PD u R 4JA
where
•
•
22
TA is the ambient or heatsink temperature
RΘJA is the thermal resistance from the MOSFET chip to the ambient air or heatsink
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Component Selection
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5.4
Output Diode
The output diode conducts when the power switch turns off and provides a path for the inductor current.
Important criteria for selecting the rectifier include: fast switching, breakdown voltage, current rating, lowforward voltage drop to minimize power dissipation, and appropriate packaging. Unless the application
justifies the expense and complexity of a synchronous rectifier, the best solution for low-voltage outputs is
usually a Schottky rectifier. The breakdown voltage must be greater than the maximum difference between
the input voltage and output voltage, and some margin must be added for transients and spikes. The
current rating must be at least two times the maximum power stage output current. Normally the current
rating is much higher than the output current because power and junction temperature limitations
dominate the device selection.
The voltage drop across the diode in a conducting state is primarily responsible for the losses in the diode.
The power dissipated by the diode can be calculated as the product of the forward voltage and the output
load current. The switching losses which occur at the transitions from conducting to non-conducting states
are very small compared to conduction losses and are usually ignored. Use Equation 60 to find the power
dissipated by the rectifier.
PD Diode
VD u I O u 1 D
where
•
VD is the forward voltage drop of the catch rectifier
Use Equation 61 to estimate the junction temperature.
T J T A PD u R 4JA
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(61)
23
Summary
6
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Summary
This application report described and analyzed the operation of the inverting buck-boost power stage. The
two modes of operation, continuous conduction mode and discontinuous conduction mode, are examined.
Steady-state and small-signal are the two analyses performed on the inverting buck-boost power stage.
The flyback power stage is presented as a variation of the basic inverting buck-boost power stage. The
main results of the steady-state analyses are summarized below.
Use Equation 62 to find the voltage conversion relationship for CCM.
IL u R L º
ª
D
VO
Vd
« VI V DS u
»
1 D
1 D ¼
¬
(62)
or a slightly simpler version:
D
1
VO
VI u
u
RL
1 D
1
Ru 1 D
(63)
2
which can be simplified to:
D
VO
VI u
1 D
(64)
Use Equation 65 to find the relationship between the average inductor current and the output current for
the continuous mode inverting buck-boost power stage.
IO
I L Avg
1 D
(65)
Use Equation 66 to find the discontinuous conduction mode inverting buck-boost voltage conversion
relationship.
D
VO
VI u
K
where K is defined as K
24
2uL
R u TS
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Summary
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The major results of the small-signal analyses are summarized below. Use Equation 67 to find the smallsignal duty-cycle-to-output transfer function for the inverting buck-boost power stage operating in CCM.
§
s · §
s ·
¨1
¸ u ¨1
¸
¨ & z1 ¸ ¨ & z2 ¸
vÖ O
©
¹
©
¹
s G do u
2
Öd
s
s
1
& o u 4 &o2
where
•
•
G do |
1 D
2
1
RC u C
s z1
•
s z2 |
•
&o |
Q|
VI
1 D
2
uR
DuL
1 D
LuC
1 D uR
L
(67)
C
Use Equation 68 to find the small-signal duty-cycle-to-output transfer function for the inverting buck-boost
power stage operating in DCM.
vÖ O
1
G do u
Öd
s
1
&p
•
where
•
G do
VO
D
VI
K
and
2
(68)
RuC
The requirements for the inverting buck-boost power stage components based on voltage and current
stresses applied during the operation of the inverting buck-boost power stage are presented in Section 5.
For further study, several references are given.
•
&p
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25
References
7
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References
1. Texas Instruments, Designing with the TL5001 PWM Controller Application Report (SLVA034)
2. Texas Instruments, Designing Fast Response Synchronous Buck Regulators Using the TPS5210
Application Report (SLVA044)
3. Texas Instruments, Understanding Buck Power Stages in Switchmode Power Supplies Application
Report (SLVA057)
4. Texas Instruments, TPS63700 DC-DC Inverter Datasheet (SLVS530)
5. Texas Instruments, AN-2264 LMR70503 Demo Board User's Guide (SNVU155)
6. V. Vorperian, R. Tymerski, and F. C. Lee. (1989). Equivalent Circuit Models for Resonant and PWM
Switches. IEEE Transactions on Power Electronics, Vol. 4, No. 2, pp. 205-214.
7. R. W. Erickson. (1997). Fundamentals of Power Electronics. New York: Chapman and Hall.
8. V. Vorperian. (1990). Simplified Analysis of PWM Converters Using the Model of the PWM Switch:
Parts I and II, IEEE Transactions on Aerospace and Electronic Systems. Vol. AES-26, pp. 490-505.
9. E. van Dijk, et al. (1995). PWM-Switch Modeling of DC-DC Converters, IEEE Transactions on Power
Electronics. Vol. 10, No. 6, pp. 659-665.
10. G. W. Wester and R. D. Middlebrook. (1973). Low-Frequency Characterization of Switched DC-DC
Converters, IEEE Transactions an Aerospace and Electronic Systems. Vol. AES-9, pp. 376-385.
11. R. D. Middlebrook and S. Cuk. (1977). A General Unified Approach to Modeling Switching-Converter
Power Stages, International Journal of Electronics. Vol. 42, No. 6, pp. 521-550.
12. E. Rogers. (1998). Control Loop Modeling of Switching Power Supplies, Proceedings of EE Times
Analog & Mixed-Signal Applications Conference. San Jose, CA.
26
Understanding Inverting Buck-Boost Power Stages in Switch Mode Power
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