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Texas Instruments TPA2013D1 Boost Converter Component Selection Application notes
Application Report
SLOA127 – March 2009
TPA2013D1 Boost Converter Component Selection
Stephen Crump .......................................................................................... Audio and Imaging Products
ABSTRACT
Loss of output power is a common problem in battery-powered systems as the battery
discharges. This problem can be overcome with a boost converter to power the audio
power amplifiers, like the converter and amplifier in TI's TPA2013D1, a single-chip
solution. Passive components used with the converter affect its operation dramatically,
so it is necessary to choose these components carefully. This application report
provides the information needed to make appropriate choices.
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5
6
7
8
Contents
Introduction .......................................................................................... 3
TPA2013D1 Configuration......................................................................... 4
Boost Circuit Operation ............................................................................ 4
Simplified Boost Circuit Input/Output Relationships............................................ 5
Boost Circuit Inductor .............................................................................. 6
Boost Circuit Capacitor ........................................................................... 10
Boost Circuit Stability ............................................................................. 15
Conclusions ........................................................................................ 21
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
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14
15
16
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18
19
20
21
Effect of Different Inductors on Output Power .................................................. 3
TPA2013D1 Circuit Configuration ................................................................ 4
Boost Converter Inductor Charge and Discharge .............................................. 5
Inductor Current Vs Inductor Value .............................................................. 6
VCC and Output Power Vs Inductor Value ....................................................... 7
Inductance and Current of Nonsaturating and Saturating Inductors ........................ 8
Effect of Different Inductors on VCC and Output Power........................................ 9
Inductor Current Limit Overshoot From Detector Delay ....................................... 9
Boost Converter Ripple Voltage Vs Output Capacitance .................................... 11
Relative Capacitance Ranges Vs Applied DC Voltage Measured for X5R and Y5V .... 12
Worst Relative Capacitance Vs Applied DC Voltage Measured for X5R and Y5V ....... 12
Boost Converter Ripple Voltage Vs Output Capacitor Voltage Rating ..................... 13
Boost Converter Stability Vs Output Capacitance and Voltage Rating ................... 14
Boost Converter Loop Response Equivalent Circuits ........................................ 15
Boost Converter Effective Output Filter Response With Nonsaturating 4.7-µH Inductor 16
Boost Converter Fixed Gain and Loop Compensation Response .......................... 17
Boost Converter Overall Loop Response With Nonsaturating 4.7-µH Inductor ........... 17
Boost Converter Effective Output Filter Response With Saturating 4.7-µH Inductor .... 18
Boost Converter Overall Loop Response With Saturating 4.7-µH Inductor ............... 19
Boost Converter Loop Compensation Response Vs Feedback Resistance and Zero
Frequency .......................................................................................... 20
Boost Converter Overall Loop Response Vs Feedback Resistance and Zero
Frequency .......................................................................................... 21
List of Tables
1
Maximum Permissible Inductor DCR Vs Output Power ...................................... 10
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2
2
Tolerance and Temperature Coefficients for Different Capacitor Materials ............... 11
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Introduction
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1
Introduction
A common problem in battery-powered systems is loss of output power as the battery discharges. Output
power must be reduced or limited to the level available at minimum battery voltage or distortion will
become unacceptable. It is possible to overcome this problem by using a boost converter, a switching
device that boosts battery output to a fixed higher voltage, to power the amplifiers. TI's TPA2013D1
combines a high-efficiency boost converter with an efficient class-D amplifier to provide a single-chip
solution to the problem.
The basics of boost circuit operation are conceptually fairly simple, but the full scope of operation includes
a number of subtleties and details that must be understood for successful circuit design. These affect
efficiency, reliability, and stability. These effects can be illustrated with a simple example.
Consider an application in which the goal is a 2.2-W output at 10% THD+N with a battery voltage of 3.6
Vdc and a boost circuit output of 5.5 Vdc. The output voltage corresponding to this goal is shown by the
green trace in Figure 1. Here the TPA2013D1 boost converter provides a consistent 5.5 Vdc to its class-D
amplifier, and the class-D amplifier produces its intended output, 2.2 Wrms at 10% THD+N. Compare this
to the red trace, made with a low-current inductor in the boost circuit, which prevents maintaining a
consistent 5.5-Vdc output. A high ripple voltage occurs at the boost circuit output, and the class-D
amplifier produces only 1.7 W at 10% THD+N, a loss of 22%.
6
Consistent VCC
2.2 Wrms Output
5
Audio Output Voltage - V
4
3
2
1
0
-1
-2
-3
-4
Sagging VCC
1.7 Wrms Output
-5
-6
0
0.5
1
t - Time - ms
1.5
2
Figure 1. Effect of Different Inductors on Output Power
The difference between the two circuits is simply the inductor. In both cases, the nominal inductor value is
4.7 µH. However, in the second case, the inductor saturates at fairly low output currents. So, it does not
permit the boost converter to transfer the energy required to reach the TPA2013D1 rated output. This
demonstrates the importance of selecting the appropriate passive components of a boost converter circuit.
This application report provides the information necessary to make the appropriate choices for the
TPA2013D1. Full details of boost converter operation can be found in the application report Understanding
Boost Power Stages in Switchmode Power Supplies (SLVA061).
This document discusses TPA2013D1 configuration, the basics of boost circuit operation and
characteristics of boost circuit passive components, their limitations, and the effects of those limitations on
circuit operation. The following subjects are discussed.
1. TPA2013D1 Configuration. The block diagram of the integrated circuit (IC) and schematic, including
the necessary supporting passive components, are considered.
2. Boost Circuit Operation. Basic theory of operation is discussed.
3. Simplified Boost Circuit Input/Output Relationships. Basic equations governing boost circuit operation
Filter-Free is a trademark of Texas Instruments.
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TPA2013D1 Configuration
4.
5.
6.
7.
2
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and requirements for supporting passive components are derived.
Boost Circuit Inductor. Requirements for this component are reviewed.
Boost Circuit Capacitor. Requirements for this component are reviewed.
Boost Circuit Stability. Requirements for stability are reviewed.
Conclusions. Conclusions are provided as a simplified guide for circuit designers.
TPA2013D1 Configuration
This document refers to the TPA2013D1 and its pin designations in describing boost circuit operation. The
designators VDD and VCC are used for input voltage and output voltage, respectively, to indicate those
quantities. The IC, including its boost converter and class-D amplifier, is connected as shown in Figure 2.
Li
Vi
Rg
Rf
Vo C
o
Ci
VDD
SW
So
FBK
VccOUT
VccIN
CLASS -D
AMPLIFIER
SDZb
SDZd
OUT+
BOOST
CONVERTER
Control
Circuits
OUT –
Si
TPA2013D1
GPIO
IN –
IN+
GAIN
GND/open/VDD
(6/15.6/20 dB)
Differential Input
Figure 2. TPA2013D1 Circuit Configuration
The boost converter section includes connections for VI or VDD and the boost switches, feedback, and
shutdown control. The basics of its operation are described in the next section. The class-D section
includes a differential input, a Filter-Free™ differential or bridge-tied output, a 3-step gain control, and an
independent VCC input.
3
Boost Circuit Operation
A boost circuit produces output voltage by charging an input inductor with current from an input voltage
source and then discharging the inductor into an output capacitor. The load for the boost circuit is
connected across the output capacitor. The inductor is switched at a high frequency by a switch and a
diode or a pair of switches. (1) The charge and discharge duty cycles typically are controlled by analog
feedback to a PWM controller that adjusts switch duty cycles to keep the output at a target voltage.
(1)
4
The basic elements and the two switch modes of the boost converter in the TPA2013D1 are illustrated in
the schematics of Figure 3. Input switch SI and output switch So are alternately closed and opened to
charge input inductor LI with current and then discharge it into output capacitor Co to create output voltage
Vo, which feeds load ZL. The feedback path and PWM generator provide the control to maintain a stable,
constant output voltage.
A switch and a diode provide the simplest circuit configuration, but in low-voltage circuits the voltage drop across a diode causes
relatively high losses and reduces efficiency. The TPA2013D1 uses a pair of switches driven in opposite phase, a configuration called
synchronous rectification, to achieve its high efficiency.
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Inductor Charge: Si closed, So open.
Current builds up in Li; Co supplies Io.
Li
Inductor Discharge: Si open, So closed.
Inductor current is discharged into Co.
Li
So
Vo
Vi
IL
So
Vo
Vi
Io
Si
Co
IL
Io
S
Co
ZL
Feedback
and PWM
Generator
ZL
Feedback
and PWM
Generator
Figure 3. Boost Converter Inductor Charge and Discharge
A number of issues must be considered before a successful design can be completed.
• Input and output voltage and current relationships.
• Requirements for passive components.
• Shortcomings of these components.
• Stability of the feedback loop.
Each of these is considered in the following text.
4
Simplified Boost Circuit Input/Output Relationships
A simplified analysis of steady-state operation of a boost circuit in which losses are ignored provides the
following input-output equations for average voltages and currents. The duty cycle of switch SI is “d”. The
second equation follows from the fact that power output must equal power input.
Output-to-input voltage ratio: (Vo / VI) = 1 / (1 – d).
Output-to-input current ratio: (Io / II) = (VI / Vo) = (1 – d).
The first equation shows how to adjust switch duty cycle to achieve a desired output voltage, an important
point in determining feedback. The second equation emphasizes that input and output currents are
inversely related to input and output voltages as in a transformer. This is an important point in selecting
the inductor.
Two other quantities that are crucial in the design of a boost circuit are peak-to-peak ripple current in the
inductor, ∆IL, and peak-to-peak ripple voltage on the capacitor, ΔVC. These can be calculated with the
following equations, which include a formula for d used in the other two equations. The switching period is
Tc .
Switch duty cycle:
d = [ (Vo – VI) / Vo]
Inductor ripple current:
ΔIL = VI × Tc × [(Vo – VI) / Vo] / LI
Capacitor ripple voltage:
ΔVC = Io × Tc × [(Vo – VI) / Vo] / Co
The ideal results presented here must be adjusted when losses and nonlinearities are considered, but the
analysis is complicated. Losses and nonlinearities are approached by examining specific cases. Several
other factors are important in boost circuit operation, including the following.
• Input current through the switches in a boost converter must be limited to avoid damaging or
destroying them. This imposes a limit on peak inductor current.
• Because of this limit on peak inductor current, inductor ripple current must be kept small relative to
inductor average current to maximize average inductor current.
• Capacitor ripple current must be kept small compared to average capacitor voltage to avoid problems
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with power supply ripple and EMC (electromagnetic compliance).
5
Boost Circuit Inductor
The inductor is typically larger and more expensive than the other boost circuit components and the most
difficult of them to specify. It is important to minimize its inductance, size, and cost. However, lower
inductance tends to reduce available output current and circuit stability. Although it is tempting to use the
smallest SMD inductors, these components are subject to saturation and resistive losses that reduce
performance even further.
5.1
Inductor Value
Inductor ripple current varies inversely with inductance, and peak inductor current ultimately must be
governed by the boost converter switch current limit. With a fixed peak value for inductor current,
increasing ripple current reduces average output current, and this reduces available output current.
The following graph (Figure 4) compares inductor current generated during one cycle into different value
inductors by a TPA2013D1 boost converter with an input of 3 Vdc, a load of 10 Ω, and an output target of
5.5 Vdc. The converter operates at frequency fc of 600 kHz, period 1.67 µs, and limits switch current at
approximately 1.5 A.
1.8
1.6
4.7 mH:
1.08 A avg.
3.3 mH:
1.03 A avg.
Inductor Current - A
1.4
1.2
1
0.8
1.8 mH:
0.88 A avg.
0.6
0.4
0.2
0
0
0.5
1
t - Time - ms
1.5
2
Figure 4. Inductor Current Vs Inductor Value
With each value of inductance, current increases until the current limit is triggered and switch SI turns off.
The inductor current then discharges until the end of the cycle, returning to its value at the start of the
cycle. By symmetry, the average inductor current must be [current limit – (peak-to-peak ripple current)/2].
Obviously, ripple current increases as inductance is reduced, and so average inductor current falls.
Note that the duty cycle of switch SI falls as inductance is reduced. This changes the input/output ratio, so
that there must be a loss in boost circuit output voltage, the power supply voltage to the class-D amplifier,
which reduces available output voltage and power. The effect is small with 4.7 µH or 3.3 µH, but larger
with 1.8 µH.
Peak current increases slightly as inductance is reduced. This is because the TPA2013D1 boost converter
filters the current signal with a time constant of nominal 100 ns to prevent triggering on noise. This sort of
filtering is typical in boost circuits. Current overshoots with low inductance because it permits the current
to rise too rapidly. The inductors used to generate the preceding graph are essentially free of saturation up
to the TPA2013D1 current limit, and still they have an effect on peak current. A later section discusses
how serious the effect of an inductor with significant saturation loss can be.
6
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A reduction in boost circuit output is important because it reduces available audio output power. The graph
in Figure 5 compares output voltages from the class-D amplifier using each of the preceding inductors, at
1 kHz and 10% THD+N into 8 Ω. The top traces show boost circuit output voltage in parallel; these are
discussed in the following text.
Output voltage, output power, and peak inductor current are almost identical with 4.7 µH and 3.3 µH,
because of the small difference in available boost circuit output current. These traces are therefore almost
identical. But with 1.8 µH, the boost circuit output current is reduced noticeably, and a loss of audio output
power occurs, about 12% compared to 4.7 µH.
6
5
Audio Output Voltage - V
4
3.3 mH:
2.07 Wrms
3
4.7 mH:
2.09 Wrms
2
1
1.8 mH:
1.82 Wrms
0
-1
-2
-3
-4
-5
-6
0
0.5
1
t - Time - mS
1.5
2
Figure 5. VCC and Output Power Vs Inductor Value
There is a slight sagging in boost circuit output voltage, class-D power supply voltage, with 4.7 µH and 3.3
µH. The power supply sag is larger with 1.8 µH, and this causes the 12% loss in audio output power.
Although this is not a serious loss in terms of audibility, it could make meeting a specification for output
power more difficult. Before an inductor with low nominal value is used, its effect on audio output power
must be understood.
5.2
Inductor Saturation
Inductor saturation can cause greater loss than a nominally low-value inductor because it can reduce
incremental inductance dramatically, in some cases to less than 1 µH. Incremental inductance decreases
as inductor current increases because of core saturation and loss of core permeability. If the loss is large
enough, peak inductor current becomes much larger than average current. Average current falls
dramatically and with it, output power. Loss of incremental inductance can even make the boost circuit
unstable. Boost circuit control loop response depends partly on the filter that is formed by LI and Co. When
the value of one of these is reduced, the unity gain frequency of the loop increases. This effect is
discussed later in the document. Inductors suitable for boost circuits retain almost all of their inductance
up to the maximum input current. Those inductors that do not retain most of their inductance to that point
are probably unsuitable.
Figure 6a compares the inductance of two inductors with a nominal value 4.7 µH versus the current in
them. Figure 6b shows current in each inductor when it is used in a TPA2013D1 boost circuit driven to its
current limit with input voltage 3.6 Vdc and output voltage target 5.5 Vdc. The first inductor, L1, loses less
than 10% of its initial value at 1.5 A, the current limit of TPA2013D1. The second inductor, L2, begins to
saturate at approximately 0.2 A and loses more than 80% of its initial value at 1.5 A. L1 produces a nearly
constant ramp like those shown in the preceding section, but L2 produces current that rises quickly to a
peak and then falls when current limit activates. Three conclusions can be drawn from inspection of the
Figure 6b graph.
• Average inductor current is significantly lower with L2 than with L1, and output power is also lower.
• Switch duty cycle is also lower with L2 than with L1, and boost circuit output voltage is also lower.
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Peak inductor current with L2 is well over the expected current limit. As noted previously, boost
converters generally require filtering in current sensing to prevent triggering on noise. The necessary
delay introduced by this filtering can permit significant overshoot in current limiting with too low an
inductance.
5
2.2
2
L1
1.8
4
Non-saturating:
1.5 A peak,
1.22 A avg.
Inductor Current - A
Inductance - mH
1.6
3
2
1.4
1.2
1
0.8
0.6
L2
1
Saturating:
1.9 A peak,
0.84 A avg.
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Inductor Current - A
1.4
1.6
a)
0
0
0.5
1
t - Time - ms
1.5
2
b)
Figure 6. Inductance and Current of Nonsaturating and Saturating Inductors
The first graph of this application report (Figure 1) compares the result of saturation in L2 to the result of
an inductor that does not saturate significantly. The graph is repeated in Figure 7 with traces added for the
boost circuit output, the power supply voltage to the class-D amplifier. Power supply ripple caused by a
loss of average inductor current during audio output peaks reduces audio output power at 10% THD+N
from 2.2 W to 1.7 W. Power supply ripple is essentially negligible with L1, but with L2 it is large enough to
cause the boost circuit output to fall more than 0.7 V during the audio output peaks.
Note: L2 is not a defective component or even a poor one. It is possible L2 could be used in relatively
low power applications with success. However, L2 is inappropriate for use with the TPA2013D1 in an
application in which the target for audio output power is one watt or more.
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6
5
VCC - Supply Voltage - V
4
Non-Saturating:
2.2 Wrms
3
Saturating:
1.7 Wrms.
2
1
0
-1
-2
-3
-4
-5
-6
0
0.5
1
t - Time - ms
1.5
2
Figure 7. Effect of Different Inductors on VCC and Output Power
In addition to the loss of audio output power, another issue must be considered, current limit overshoot.
Peak inductor current reaches 1.9 A with the saturating inductor and input voltage of 3.6 Vdc, as shown in
Figure 8a. This overshoot occurs during the 100-ns delay required for filtering against triggering on noise.
At the maximum voltage of a Li-Ion battery, nominally 4.2 Vdc, it increases to about 2.1 A, as shown in
Figure 8b. With a nonsaturating inductor, peak current is limited to nominal at both input voltages.
2.2
2.2
VDD = 3.6 V
~100 nS
2
1.8
1.4
1.2
1
0.8
1.4
1.2
1
0.8
Saturating:
2.1 A Peak
0.6
0.6
Saturating:
1.9 A Peak
0.4
0.4
0.2
0.2
0
Non-Saturating:
1.5 A Peak
1.6
Inductor Current - A
Inductor Current - A
1.8
Non-Saturating:
1.5 A Peak
1.6
VDD = 4.2 V
2
0
0.5
1
t - Time - ms
a)
1.5
2
0
0
0.5
1
t - Time - ms
1.5
2
b)
Figure 8. Inductor Current Limit Overshoot From Detector Delay
Inductor current overshoot is an issue because it can reach well above nominal current limit, as in these
cases. Repeated overcurrent at levels like these can reduce long-term reliability of the TPA2013D1 (or
any other boost converter). So, both nominal value and saturation characteristics must be considered in
choosing an inductor. Regarding these parameters, the following rule applies.
Inductance and incremental inductance used with TPA2013D1 must never be less than 2.2 µH.
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Generally, information on inductor saturation is easily available. Inductor manufacturers typically publish
data about saturation of their components in graphs of inductance versus dc current or in tables that
specify the currents at which inductor values decrease by a specific percentage, typically 10% to 35%.
Inductance also is affected by temperature, although the effect is generally not as strong as for dc current.
It is vital to review these data before selecting an inductor.
5.3
Inductor and Other Circuit Resistances
Inductor dc resistance (DCR) causes losses because of the rms currents flowing in the inductor. The
on-resistances of the switches in TPA2013D1 cause similar losses. Inductor core losses also reduce boost
circuit output power, although this effect is typically much smaller than that of DCR. The effective
resistance of the switches in the TPA2013D1 synchronous rectifier over the course of a switching cycle is
about 190 mΩ, small enough that the losses it causes are not significant even at full inductor current. But
if inductor DCR is as large as or larger than the switch resistance, the total resistance may prevent
delivering the expected output power.
The effect of circuit resistances is complicated, so no simple equation is available for it. However, the
effect can be determined at different power levels to provide a guide to maximum permissible DCR.
Table 1 provides such a guide. For intermediate power levels, interpolate.
Table 1. Maximum Permissible Inductor DCR Vs Output Power
6
(1)
6.1
Po, W at 1% THD+N
1.0
1.5
2.0
DCR, mΩ maximum
300
180
70
Boost Circuit Capacitor
The output capacitor is typically smaller and less expensive than the inductor, but it is just as important to
the performance of a boost circuit. It is desirable to minimize its size and cost and therefore its
capacitance. However, lower capacitance can compromise circuit stability and EMC (1). Although it is
tempting to use the smallest SMD ceramic capacitors, these components are subject to capacitance
losses that can compromise stability and EMC even more than low nominal capacitance.
EMC is compliance with regulatory requirements in standardized tests. Electromagnetic interference (EMI) is audible interference above
a specification level caused by electromagnetic fields. Because the design objective in this document is for EMC, EMI is not considered.
Capacitor Value
Output capacitor ripple voltage varies inversely with capacitance as well as directly with the output current
and duty cycle of switch SI. It is greatest at minimum input voltage, where switch duty cycle is greatest.
The following graph (Figure 9) compares capacitor voltage during one cycle with a TPA2013D1 boost
converter with two different capacitors, an input of 3 Vdc, a nonsaturating 4.7-µH inductor, a load of 10 Ω,
and an output target of 5.5 Vdc.
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40
30
22 mF:
9.5 mV rms
Output Ripple - mV
20
47 mF:
4.2 mV rms
10
0
-10
-20
-30
-40
0
1
2
3
t - Time - ms
5
4
Figure 9. Boost Converter Ripple Voltage Vs Output Capacitance
The ripple voltage waveform is nominally triangular because of the relatively constant currents into and out
of the capacitor. However, at low levels like those shown in Figure 9, it is complicated by switching
artifacts. The capacitors have nominal values of 47 µF and 22 µF and produce ripple voltages of 4.2
mVrms and 9.5 mVrms. The smaller value produces a larger ripple voltage and is thus more likely to
cause difficulty in achieving EMC.
The following equation can be used to try to predict rms ripple voltage.
Capacitor RMS ripple voltage: ΔVC RMS ≈ 0.3 × Io × Tc × [(Vo – VI) / Vo] / Co
(RMS value is √ { (1/T) × ∫ (At/T)2 dt, t = 0 to T }, or √ { (1/T) × A2t3/T2/3, t = 0 to T }, or A/√3, where A
is peak value. So compared to peak-to-peak value, ΔVC, RMS is 0.5/1.723 ≈ 0.3.)
Calculations with this equation predict ripple values of approximately 2.7 mVrms and 5.7 mVrms for these
two capacitors, but the measured quantities are 4.2 mVrms and 9.5 mVrms. These differences are not
caused by switching artifacts, as might be expected. The artifacts obscure the underlying triangle
waveform but do not add significantly to its rms value. The differences are caused by a loss of
capacitance in the high-K capacitors used in these measurements, a typical problem with parts like these.
(K is dielectric constant, the relationship between electric field strength in a material and the voltage
across it. In high-K materials this “constant” is only a nominal or approximate quantity, because the
dielectric constant varies with applied voltage. This will be discussed below.)
High-K capacitors, made of material with high dielectric coefficients like X7R or Y5V, can be smaller than
capacitors made of low-K materials like COG. However, high-K materials are less stable than low-K
materials. The temperature dependence of various materials, shown in Table 2, is generally well known.
Table 2. Tolerance and Temperature Coefficients for Different Capacitor Materials
Material:
COG/NP0
X7R
X5R
Y5V
Tolerance:
±5%
±10%
±10%
+80/–20%
Temperature
Coefficient:
±30 ppm
±15%
±15%
+22/–82%
Range,°C:
–55/+125°C
–55/+125°C
–55/+85°C
–30/+85°C
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A capacitor made of X7R material can lose 15% of its initial capacitance at 25°C at any point in the range
–55°C to 125°C. Y5V is worse: it can lose 22% over a more narrow temperature range. However, this is
not the worst effect encountered in high-K capacitors. Sensitivity to applied voltage is a much greater
problem.
In addition to temperature, high-K ceramic capacitors are relatively sensitive to applied voltage, both ac
and dc, as well as frequency. Because sensitivity to applied dc voltage is usually a much greater problem
than any of the others, the following discussion concentrates on dc voltage sensitivity. Figure 10 and
Figure 11 show relative capacitance, capacitance as a percentage of initial capacitance at 0 Vdc voltage
versus applied dc voltage as a percentage of rated dc voltage. Figure 10 shows the range of variation that
has been observed for X5R and Y5V, with a purple band for the range of variation for X5R and a red band
for the range of variation for Y5V. Figure 11 includes only the worst examples.
X5R and Y5V Capacitance Ranges vs. DCV
120
Relative Capacitance - %
100
80
X5R
Y5V
60
40
20
0
0
5
10
25
50
DC Voltage Re. Voltage Rating - %
75
100
Figure 10. Relative Capacitance Ranges Vs Applied DC Voltage Measured for X5R and Y5V
100
Relative Capacitance - %
80
X5R
60
40
Y5V
20
0
0
20
60
40
DCV re.Voltage Rating - %
80
100
Figure 11. Worst Relative Capacitance Vs Applied DC Voltage Measured for X5R and Y5V
It is possible for an X5R capacitor to lose 60% or more of its initial capacitance at its rated dc voltage. For
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Y5V parts, the loss is typically more like 80%. In the graph of ripple voltage measurements (Figure 9), the
two capacitors were both X5R, 10 V. A comparison of measured rms ripple voltage to calculations shows
that the effective values of the two capacitors were more like 30 µF than 47 µF and more like 13 µF than
22 µF. In each case, the effective value is a little more than 60% of nominal. Boost circuit output voltage of
5.5 Vdc is 55% of the 10-V rating, so this percentage lies in the range for X5R shown in Figure 11.
Results are typically worse with lower-voltage and higher-K capacitors. The following graph (Figure 12) of
capacitor ripple voltages repeats the previous one with a third trace added, for a 22 µF, 6.3-V X5R
capacitor. The 6.3-V capacitor produces ripple voltage of 16 mVrms. It can be concluded that its effective
value is less than 8 µF, about 36% of its nominal value at 0 Vdc. This percentage also lies in the range for
X5R shown in Figure 11.
40
30
22 mF 6.3 V:
16 mV rms
22 mF 10 V:
9.5 mV rms
Output Ripple - mV
20
10
0
-10
47 mF 10 V:
4.2 mVms
-20
-30
-40
0
1
2
3
t - Time - mS
4
5
Figure 12. Boost Converter Ripple Voltage Vs Output Capacitor Voltage Rating
While dc voltage sensitivity in the output capacitor can cause increased output ripple voltage and therefore
problems with EMC, another issue is potentially worse. As mentioned previously, the response of the
feedback loop depends partly on the filter formed by the inductor and output capacitor. If either of these
loses a significant part of its expected value, the result can be instability in the boost circuit output.
The graphs shown in Figure 13 compare boost circuit output and ripple voltages against audio output
voltages with input voltage 3.6 Vdc, output voltage 5.5 Vdc, load 8 Ω, and two different capacitors. The
Figure 13a graph uses the 47-µF, 10-V X5R part from previous measurements, and the Figure 13b graph
uses the 22-µF, 6.3-V X5R part. The audio output is set just below the onset of clipping, with THD+N less
than 1%. The measurement with the 47-µF, 10-V part shows low boost circuit output ripple and no
instability. The measurement with the 22-µF, 6.3-V part, however, shows instability around 60 kHz and
slightly higher boost circuit output ripple as a result of the instability.
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13
0.30
6
0.30
4
0.25
4
0.25
2
0.20
2
0.20
0
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0
0.15
-2
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-2
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-4
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-4
0.05
-6
0
-6
0
-8
-0.05
-8
-0.05
-0.10
-10
-10
CO = 47 mF, 10 V
-12
0
0.2
-0.15
0.4
0.6
0.8
1.0
VOUT and VCC - V
6
Ripple Voltage - V
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Ripple Voltage - V
VOUT and VCC - V
Boost Circuit Capacitor
-0.10
CO = 22 mF, 6.3 V
-0.15
-12
0
0.2
0.4
0.6
t - Time - ms
t - Time - ms
a)
b)
0.8
1
Figure 13. Boost Converter Stability Vs Output Capacitance and Voltage Rating
Audio output power in each case is 1.6 W, so no significant change occurs with the lower value, lower
voltage capacitor. However, the instability drives the boost circuit output voltage to peaks that are a
significant fraction of a volt over the expected range. The increased voltage produces some risk to
long-term reliability of the boost converter. In a circuit with greater instability, the increase in risk is also
greater. (A circuit using L2 from the previous discussion and the 22-µF, 6.3-V X5R or a Y5V capacitor is
considerably less stable.)
Note: the 22-µF, 6.3-V X5R capacitor is not a defective component. It can be used in other applications
with success. However, it is inappropriate to use this capacitor with the TPA2013D1 because of its dc
sensitivity and because the TPA2013D1 boost circuit output voltage is a large portion of its rated voltage.
Thus, both nominal value and voltage sensitivity must be considered in choosing a capacitor. The
following rule applies.
Capacitance and effective capacitance used with TPA2013D1 must never be less than 12 µF.
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7
Boost Circuit Stability
The preceding discussion of instability leads to a consideration of the boost circuit feedback loop
response. This response is extremely complex, and a full analysis is therefore beyond the scope of this
application report. (1) Instead, the elements of the response and how passive components affect its
closed-loop stability are discussed. Figure 14 shows the closed-loop circuit schematic (a) and the
equivalent circuit (b).
• Rf and Rg are the feedback and ground shunt resistors. They cause a fixed low-frequency loss.
• Calculate Rg from Rf: Rg = Rf × 0.5 / (VCC – 0.5), where the value 0.5 is an internal feedback reference.
• Rz and Cz are internal to TPA2013D1. With Rf and Rg they produce a zero and a high-frequency pole.
The zero is used to offset phase lag of the LC filter and restore phase margin near unity loop
response.
• Ai is a fixed gain block that provides gain of 4 with phase inversion.
• Rp and Cp are internal to TPA2013D1. They introduce a pole at 180 kHz to eliminate noise.
• Ae is a gain block that includes PWM action and voltage boost. Ae = Vi / D,2, where D,2 = (Vi / Vo)2.
• Re is the effective total of switch resistance Rs of Si and So and DCR of Li. Re = (Rs + DCR) / D,2.
• Le is effective inductance. Le = Li / D,2.
• Co is output capacitance and Ro is load resistance.
• An element that is not revealed in the schematic is a right-half-plane zero, a characteristic of all boost
circuit feedback loops. This zero occurs at approximately frhp = Rload × D,2 / Li.
Rg
Rf
Rg
R z 10 kW
C z 20 pF
Rp 180 kW
A i -4
Cp 5 pF
Rf
Rz 10 kW
Cz 20 pF
Rp 180 kW
=>
A i -4
Cp 5 pF
TPA2013D1
Li
PWM
Vi
Le
Re
Vo
Control
Ae
Co
(a)
R load
Co
(b)
Figure 14. Boost Converter Loop Response Equivalent Circuits
The sections that follow examine the various elements of TPA2013D1 boost circuit response and their
combined response. The first section examines the response of gain block Ae and the LC filter plus the
RHP zero for a circuit using an appropriate Li and Co (see Figure 15). The next section examines the
feedback circuit, zero circuit Rz and Cz, fixed gain block Ai, and pole circuit Rf and Rg together as a fairly
fixed response. This shows that feedback resistances Rf and Rg must be set to fairly constant values.
Finally, the results of the two sections are combined to permit examining gain/phase margin .
Consider a circuit in which Li is a nonsaturating 4.7-µH inductor with DCR of approximately 30 mΩ and Co
is constant 15 µF. The boost circuit output voltage Vo and current Io are 5.5 Vdc and 1 A, so the effective
Rload is 5.5 Ω. Set input voltage Vi to 3 Vdc and 4.2 Vdc. Note the effect on the response of the combined
PWM gain block Ae and the LC filter and the response of the RHP zero.
(1)
Full details of boost converter operation can be found in the TI application report Understanding Boost Power Stages in Switchmode
Power Supplies (SLVA061).
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30
180
20
135
10
90
0
45
Phase - deg
Magnitude - dB
Boost Circuit Stability
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-50
1
1.6
2.5
4
6.3
10
15.8 25.1 39.8 63.1
f - Frequency - kHz
100 158.5 251.2 398.1 631
Vi = 3 Vdc: A e = 10.1 and L e = 15.8 mH
= gain block + LC filter response magnitude;
= right half plane zero response magnitude;
Vi = 4.2 Vdc: A e = 7.2 and L e = 8.1 mH
= gain block + LC filter response magnitude;
= right half plane zero response magnitude;
= phase
= phase
= phase
= phase
Figure 15. Boost Converter Effective Output Filter Response With Nonsaturating 4.7-µH Inductor
Output voltage of TPA2013D1 boost circuit is typically around 5 Vdc, and the reference voltage for the
circuit is 0.5 Vdc. The feedback circuit divides the output voltage by approximately 10 at the feedback
input, introducing a loss of approximately 20 dB. The balance of gain in the feedback circuit is Ai, with
magnitude 4, or 12 dB. Therefore, net gain of the feedback circuit, independent of the effect of the zero
circuit and the pole circuit, is approximately –8 dB. Adding this to the preceding responses shows that the
unity magnitude loop response occurs in the range of approximately 25 to 50 kHz, where the LC filter
response is approximately +10 dB. Figure 15 indicates that phase margin from the LC filter is small at this
point, around 20 degrees. Therefore, it is clear that the margin needs to be improved to ensure stability.
This is done in the feedback circuit.
Note that, in this fairly normal case, the phase lag introduced by the RHP zero is negligible compared to
phase lag of the LC filter. This is not always the case.
The next graph (Figure 16) shows the response of the combination of the full feedback circuit, including
the zero circuit, fixed gain block Ai, and pole circuit. In this case Rf is set to 499 kΩ, so Rg is 49.9 kΩ. This
places the zero produced by Rz and Cz at 15.6 kHz and the pole that follows at 123 kHz. Note the addition
of the 180-degree inversion in Ai to phase.
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Magnitude - dB
30
360
20
315
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0
225
-10
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135
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0
1
1.6
2.5
4
6.3
10
15.8 25.1 39.8 63.1
f - Frequency - kHz
= response magnitude;
100 158.5 251.2 398.1 631
= phase
Figure 16. Boost Converter Fixed Gain and Loop Compensation Response
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The zero circuit adds a small increase to response magnitude in the range of approximately 25 to 40 kHz.
The responses shown in Figure 15 and Figure 16 are combined in the graph of total response (Figure 17),
which shows good phase margin.
-90
1
1.6
2.5
4
6.3
10
15.8 25.1 39.8 63.1
f - Frequency - kHz
100 158.5 251.2 398.1 631
Vi = 3 Vdc:
= total loop response magnitude;
= phase
= total loop response magnitude;
= phase
Vi = 4.2 Vdc:
Figure 17. Boost Converter Overall Loop Response With Nonsaturating 4.7-µH Inductor
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Boost Circuit Stability
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For the circuit with Li = 4.7 µH, DCR = 30 mΩ, Co = 15 µF, Rf = 499 kΩ, Vo = 5.5 Vdc, Io = 1 A, phase and
gain margins predicted by simulation are as follows. This circuit has a small risk of instability. See
Figure 17.
• Vi = 3 Vdc
– Phase margin ~= 56 degrees at 32 kHz
– Gain margin ~= 22 dB at 180 kHz
• Vi = 4.2 Vdc
– Phase margin ~= 56 degrees at 40 kHz
– Gain margin ~= 23 dB at 220 kHz
30
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However, margins deteriorate with a saturating inductor, and a voltage-sensitive output capacitor is
changed. Presume the inductor is replaced with one that saturates to 1 µH at the given output, and the
output capacitor is changed to a Y5V part with capacitance that falls to 5 µF. At the same time, the
feedback circuit remains the same as before. A graph of response of the gain plus LC filter block and RHP
zero follow (Figure 18).
-180
-50
1
1.6
2.5
4
6.3
10
15.8 25.1 39.8 63.1
f - Frequency - kHz
100 158.5 251.2 398.1 631
Vi = 3 Vdc: A e = 10.1 and L e = 3.4 mH
= gain block + LC filter response magnitude;
= right half plane zero response magnitude;
Vi = 4.2 Vdc: A e = 7.2 and L e = 1.7 mH
= gain block + LC filter response magnitude;
= right half plane zero response magnitude;
= phase
= phase
= phase
= phase
Figure 18. Boost Converter Effective Output Filter Response With Saturating 4.7-µH Inductor
The 10-dB-gain response frequency of the gain block plus LC filter has increased dramatically. The next
graph (Figure 19) shows that the unity-gain frequency has increased even more, because the rolloff of the
gain block plus LC filter have moved well above the point where the feedback circuit zero begins
increasing feedback response gain. The total response is shown in Figure 19. The picture there is not
good.
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Magnitude - dB
30
270
20
225
10
180
0
135
-10
90
-20
45
-30
0
-40
-45
Phase - deg
Boost Circuit Stability
www.ti.com
-90
-50
1
1.6
2.5
4
6.3
10
15.8 25.1 39.8 63.1
f - Frequency - kHz
100 158.5 251.2 398.1 631
Vi = 3 Vdc:
= total loop response magnitude;
= phase
= total loop response magnitude;
= phase
Vi = 4.2 Vdc:
Figure 19. Boost Converter Overall Loop Response With Saturating 4.7-µH Inductor
For the circuit with Li = 1 µH, DCR = 30 mΩ, Co = 5 µF, Rf = 499 kΩ, Vo = 5.5 Vdc, Io = 1 A, phase and
gain margins have been reduced dramatically, because of the increase in LC filter rolloff frequency. This
circuit is marginally stable. See Figure 19.
• Vi = 3 Vdc
– Phase margin ~= 28 degrees at 180 kHz
– Gain margin ~= 11 dB at 350 kHz
• Vi = 4.2 Vdc
– Phase margin ~= 22 degrees at 220 kHz
– Gain margin ~= 12 dB at 450 kHz
In the simulation model, a 10-pF capacitor has been added across Rg to represent possible stray
capacitance there. However, actual parasitic effects are impossible to predict completely. In an actual
implementation, parasitic elements can make the situation worse than shown here. In any case, the
margins in the revised circuit are not large enough to prevent significant ringing and overshoot at the loop
unity-gain frequency. Therefore, the circuit with a saturating inductor and voltage-sensitive output
capacitor is unacceptable.
It is difficult to improve margins by varying Rf. In the first case presented, with Li= 4.7 µH, Co = 15 µF, and
input voltage fixed at 3 Vdc, varying Rf, by factors of 1/2 and 2 produces the following graph (Figure 20).
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30
360
20
315
10
270
0
225
-10
180
-20
135
-30
90
-40
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-50
Phase - deg
Magnitude - dB
Boost Circuit Stability
0
1
1.6
2.5
4
6.3
10
15.8 25.1 39.8 63.1
f - Frequency - kHz
100 158.5 251.2 398.1 631
= Rf = 249 kW, feedback circuit response magnitude;
= Rf = 1 MW, feedback circuit response magnitude;
= phase
= phase
Figure 20. Boost Converter Loop Compensation Response Vs Feedback Resistance and Zero Frequency
With Rf = 250 kΩ, the zero and following pole occur at twice the previous frequencies. The phase benefit
of the zero is not great around the unity-gain frequency of the circuit. With Rf = 1 MΩ, the zero and
following pole occur at half the previous frequencies. So, where the gain block plus LC filter response
starts to roll off, the phase benefit of the zero is fading but its gain contribution is high.
The effect can be seen in the following graph (Figure 21) in which Rf = 249 kΩ versus 1 MΩ.
20
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Conclusions
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135
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-90
-50
1
1.6
2.5
4
6.3
10
15.8 25.1 39.8 63.1
f - Frequency - kHz
= Rf = 249k, total loop response magnitude;
= Rf = 1.0M, total loop response magnitude;
100 158.5 251.2 398.1 631
= phase.
= phase.
Figure 21. Boost Converter Overall Loop Response Vs Feedback Resistance and Zero Frequency
For the circuit with Li = 4.7 µH, DCR = 30 mΩ, Co = 15 µF, Vo = 5.5 Vdc, Io = 1 A, phase and gain margins
predicted by simulation are as follows. In each case, some phase margin has been lost.
• Rf = 249 kΩ:
– Phase margin ≈ 47 degrees at 25 kHz
– Gain margin ≈ 27 dB at 220 kHz
• Rf = 1 MΩ:
– Phase margin ≈ 43 degrees at 45 kHz
– Gain margin ≈ 15 dB at 130 kHz
This indicates that a value near 499 kΩ is optimal for Rf.
8
Conclusions
Conclusions that can be drawn from this application report follow.
• Inductor:
– Output power is nearly constant for high inductor values but falls at low values. Use Li = 3.3 to 6.8
µH for TPA2013D1 in most applications.
– Limit peak-to-peak ripple current ΔIL to 40% of average IL or less to avoid losing output power.
– Remember that input current Ii is larger than output current Io (Ii = Io × Vo / VI). Ensure that the
inductor is rated for input current Ii, not output current Io.
– Ensure that inductor Li retains at least 70% of its nominal value at the peak input current Ii and
maximum temperature for a given application.
– Ensure that inductance of Li is always more than 2.2 µH, even in saturation and at high
temperatures. Otherwise, long-term reliability may be reduced by repeated overcurrent.
– Minimize inductor DCR to avoid losing output power. Use Table 1 as a guide.
• Output Capacitor:
– Ensure that effective capacitance is >12 µF for 1-W applications and >25 µF for 2-W applications,
even with full dc voltages and at high temperatures. Otherwise, long-term reliability may be reduced
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21
Conclusions
•
22
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by overvoltage from instability.
– If working capacitance cannot be determined, do the following.
• Use materials with temperature coefficients at least as good as X5R. Do not use materials like
Y5V or Z5U.
• Use capacitors with voltage ratings at least twice the maximum application voltage. For
TPA2013D1, this means at least 10 Vdc.
• Use capacitors with values 2x calculated values. This plus voltage rating gives the right
capacitance.
Feedback Resistance:
– Set feedback resistor Rf = 499 kΩ.
– Set ground resistor Rg = Rf × 0.5 / (VCC – 0.5).
– This optimizes phase and gain margin.
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