Texas Instruments | Optimizing Transient Response of Internally Compensated DC-DC Converters (Rev. B) | Application notes | Texas Instruments Optimizing Transient Response of Internally Compensated DC-DC Converters (Rev. B) Application notes

Texas Instruments Optimizing Transient Response of Internally Compensated DC-DC Converters (Rev. B) Application notes
Application Report
SLVA289B – January 2008 – Revised November 2017
Optimizing Transient Response of Internally Compensated
dc-dc Converters With Feedforward Capacitor
Brian Butterfield .................................................................................................. PMP - Portable Power
ABSTRACT
This application report describes how to choose the feedforward capacitor value (Cff) of internally
compensated dc-dc power supplies to achieve optimum transient response. The described procedure in
this application report provides guidance in optimizing transient response by increasing converter
bandwidth while retaining acceptable phase margin. This document is intended for all power supply
designers who want to optimize the transient response of a working, internally compensated dc-dc
converter.
1
2
3
4
Contents
Introduction ................................................................................................................... 2
Feedback Network With and Without the Feedforward Capacitor .................................................... 2
Conclusion .................................................................................................................. 11
References .................................................................................................................. 11
List of Figures
1
Feedback Network Consisting of Two Bias Resistors Used to Set Output Voltage ................................ 2
2
Standard Feedback Divider Transfer Function
3
4
5
6
7
8
9
10
11
12
13
14
15
.......................................................................... 2
Feedback Network With Addition of Feedforward Capacitor .......................................................... 3
Standard Feedback Divider With Feedfoward Capacitor Transfer Function ......................................... 3
Internally Compensated Converter Without Feedforward Capacitor .................................................. 4
Tip and Barrel Measurement Technique ................................................................................. 5
Voltage Transient in Response to Load Transient Without Feedforward Capacitor ................................ 5
Step Response vs Loop Phase Margin ................................................................................... 6
Loop Gain and Phase Plot of TPS61081 Circuit Without Feedforward Capacitor .................................. 6
Voltage Transient in Response to a Load Transient With 82-pF Feedforward Capacitor.......................... 8
Loop Gain and Phase Plot of TPS61081 Circuit With 82-pF Feedforward Capacitor .............................. 8
Voltage Transient in Response to a 0 to 100% Load Transient With 1000-pF Feedforward Capacitor ......... 9
Loop Gain and Phase Plot of TPS61081 Circuit With 1000-pF Feedforward Capacitor ........................... 9
Voltage Transient in Response to Load Transient With 33-pF Feedforward Capacitor .......................... 10
Loop Gain and Phase Plot of TPS61081 Circuit With 33-pF Feedforward Capacitor ............................. 10
List of Tables
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1
Introduction
1
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Introduction
Internally compensated dc-dc converters allow designers to save time in the design and in debug
processes by minimizing the number of external components that they must select. This simplification
inherently narrows a designer's ability to optimize the transient response of the converter. However, with
some internally compensated converters, the use of a feedforward capacitor in the feedback network is
recommended, but only general guidance is provided for choosing this capacitor value to improve
transient response. With measured transient or loop characteristics of a working dc-dc converter, a
feedforward capacitor value can be chosen such that the converter bandwidth is significantly improved
while still maintaining adequate phase margin. Furthermore, with a better understanding of the
feedforward capacitor, the designer can go one step further to optimize either higher bandwidth or greater
phase margin to meet specific performance requirements.
2
Feedback Network With and Without the Feedforward Capacitor
Without a feedforward capacitor, the feedback network of an internally compensated dc-dc converter
consists of two feedback resistors used to set the output voltage of the converter, as shown in Figure 1.
Figure 2 shows the corresponding gain and phase plot.
Converter
VOUT
R1
470 kW
VFB
R2
180 kW
Figure 1. Feedback Network Consisting of Two Bias Resistors Used to Set Output Voltage
5
0
Gc(f)
-5
fc(f)
DC gain = 20log [R2/(R1 + R2)]
-10
-15
1-10
-3
0.01
0.1
1
10
f/kHz
3
100 1-10 1-10
4
Figure 2. Standard Feedback Divider Transfer Function
Figure 3 shows the addition of the feedforward capacitor, C1 (Cff), in the feedback network and Figure 4
shows the corresponding gain and phase plot. With the addition of the feedforward capacitor network, the
converter can more effectively respond to high-frequency disturbances on the output voltage rail. The
bode plots in Figure 2 and Figure 4 show that the responses of each feedback network are identical at
lower frequencies. At mid-to-higher frequencies, disturbances on the output rail are attenuated less as the
impendence path through C1 decreases and effectively provides a boost in gain and phase. In a working
power supply, the increased gain and phase correlates to the converter responding faster to transient
loads because the voltage deviation, sensed at the feedback node, is attenuated less at higher
frequencies. The converter reacts by adjusting the duty cycle to more quickly correct the output voltage
deviation.
2
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With Feedforward Capacitor
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Converter
VOUT
C1
10 pF
R1
470 kW
VFB
R2
180 kW
Figure 3. Feedback Network With Addition of Feedforward Capacitor
Although Cff introduces a gain boost after its zero frequency, loop phase boost is at a maximum between
the zero and pole frequencies; see the following Equation 1 and Equation 2. Increasing the value of Cff
shifts the zero and pole in Equation 1 to lower frequencies, and decreasing the value Cff shifts the zero
and pole to higher frequencies. The gain at dc is set by R1 and R2. The following equations calculate the
pole, zero, and the dc gain of the feedback network as is shown in Figure 4.
1
fz +
2p R1 Cff
(1)
Equation 1 calculates the zero frequency based on the feedforward capacitor value and the top bias
resistor, R1. fz is shown on the plot in Figure 4.
fp =
(
1
1 + 1
2p x Cff R2 R1
)
(2)
Equation 2 calculates the pole frequency based on the feedforward capacitor value and both top and
bottom bias resistors, R1 and R2. fp is shown in on the plot in Figure 4.
The transfer function is plotted as:
40
30
20
Gc(f)
10
fc(f)
fp
0
-10
-20
1-10
fz
-3
0.01
0.1
1
10
f/kHz
3
100 1-10 1-10
4
Figure 4. Standard Feedback Divider With Feedfoward Capacitor Transfer Function
To optimize transient response, a Cff value is chosen such that the gain and phase boost of the feedback
increases the bandwidth of the converter, while still maintaining an acceptable phase margin. In general,
larger values of Cff provide greater bandwidth improvements. However, if Cff is too large, the feedforward
capacitor causes the loop gain to crossover too high in frequency and the Cff phase boost contribution is
insufficient, resulting in unacceptable phase margin or instability. Recommended limitations of Cff is
discussed later in this document.
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Feedback Network With and Without the Feedforward Capacitor
2.1
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Feedforward Capacitor Value Optimization Process
The following process outlines a step-by-step procedure for optimizing the feedforward capacitor.
1. Determine the crossover frequency of an internally compensated dc-dc converter with an unpopulated
feedforward capacitor (f_nocff). In certain circumstance, this can be calculated, but for this application
report, this optimization procedure is based on measured converter characteristics. You can determine
the crossover frequency (converter bandwidth) with transient analysis or by using a network analyzer.
Both methods are shown.
2. Once the crossover frequency is known, a few equations allow calculation of a feedforward capacitor
value which prompts a good compromise between bandwidth improvement and acceptable phase
margin. Improvement in transient and loop response is shown with transient analysis and frequency
analysis to confirm the design.
3. If the designer chooses to optimize for higher bandwidth or increased phase margin (more damping),
guidance is provided.
2.2
Determining the Crossover Frequency
The TPS61081 is used in this example to determine the crossover frequency. This example can be
applied to other internally compensated dc-dc converters which recommend external feedforward
capacitors in the feedback network.
After using the data sheet guidelines to choose all appropriate external components, remove the
feedforward capacitor, and measure the converters crossover frequency by using transient analysis or a
network analyzer. Note that to determine the crossover frequency, f_nocff, the feedforward capacitor must
be left open as shown in Figure 5.
Open
Figure 5. Internally Compensated Converter Without Feedforward Capacitor
Figure 6 shows the tip and barrel measurement method set up for transient analysis. A transient load is
connected to the output of the power supply circuit, while a current probe measures the transient load
current, and a tip and barrel voltage probe measures the voltage deviation during transient load conditions
on the output.
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With Feedforward Capacitor
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Figure 6. Tip and Barrel Measurement Technique
The tip and barrel measurement technique in Figure 6 is used to minimize coupling magnetic fields and
obtain a more accurate voltage waveform during transient load transitions. TP14 is connected to the
measured signal whereas TP15 is connected to ground. TP14 and TP15 are not shown in Figure 5. If the
power supply does not include the appropriate test points, the test points can be strategically placed using
bus wire. It is recommended that the bus wire test points be tacked onto the converter output capacitor
closest to the load. Figure 7 shows the TPS61081 transient response as measured with the tip and barrel
technique. The plots are taken using the TPS61081EVM-147 with Vin = 5 V, Vout = 12 V, and a load
transient from 0 mA to 160 mA.
Approximate
Crossover Frequency
Figure 7. Voltage Transient in Response to Load Transient Without Feedforward Capacitor
About 0.9 V of output voltage deviation from the dc voltage set point is observed. The voltage waveform in
Figure 7 provides insight to the converter crossover frequency as described in Evaluation and
Performance Optimization of Fully Integrated DC/DC Converters (Topic 7 of the 2006 Portable Power
Design Seminar). The frequency of the voltage deviation waveform in response to a load transient is
related to the crossover frequency of the converter. Using the oscilloscope's cursors, the crossover
frequency is approximated. The frequency of the transient ripple in this example is approximately 15 kHz.
Note that the voltage deviation begins to correct 30 μs after the transient occurs. As the crossover
frequency of the converter is increased, it is confirmed that the converter response is improved as the
voltage deviation begins to correct in less time, resulting in less voltage deviation.
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2
j = 15°
j = 30°
j = 45°
j = 60°
1.5
1
Crossover
Frequency
0.5
0
0
5
10
wt
15
20
Figure 8. Step Response vs Loop Phase Margin
Using Figure 8 from the 2006 Portable Power Design Seminar topic paper Evaluation and Performance
Optimization of Fully Integrated DC/DC Converters, the phase margin of the loop can be adequately
approximated. Comparing the two plots, the TPS61081 measured the transient response most resembling
the number of oscillations of the blue trace with just slightly more oscillation. This means that the
measured loop has just slightly less than 30° of phase margin.
2.3
Determining the Crossover Frequency Using Frequency Analysis
Because frequency analysis equipment is costly, using such equipment is not always an option. However,
when such equipment is available, the crossover frequency can be quickly measured. This method is
more accurate than the transient analysis approximation and should be used when possible. Figure 9
shows the frequency analysis of the control loop for the example circuit in Figure 5.
1 Side Bar:
15.73 k Frequency (Hz)
0
Gain (dB)
27.87 Phase (deg)
-1.78 Slope (20 dB/decade)
Figure 9. Loop Gain and Phase Plot of TPS61081 Circuit Without Feedforward Capacitor
Once the loop gain and phase plot is obtained with a network analyzer, the crossover frequency is quickly
noted. The phase margin is 28° which confirms the transient analysis approximation of being just less than
30°. The crossover frequency also is measured at 16 kHz which is close to the 15-kHz approximation.
6
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With Feedforward Capacitor
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2.4
Calculating the Feedforward Capacitor for Optimum Loop Response
With the crossover frequency with no Cff identified (f_nocff), the value of Cff can be calculated for
optimum transient response by choosing Cff such that the zero and pole frequency geometrically straddle
f_nocff.
f_nocff = 16 kHz
Converter crossover frequency with no Cff
R1 = 442 kΩ
R2 = 49.9 kΩ
R1 and R2 are the feedback bias resistors used to set the output voltage of the converter as shown in
Figure 5.
Equation 3 calculates the geometric mean of the feedback network's zero and pole frequencies. The
geometric mean frequency equation is used to calculate the frequency where the phase boost from the
zero and pole is at a maximum. However, because Cff is currently unknown, equations fz and fp are left in
variable form.
F geometric_mean +
Ǹǒfz
fp
Ǔ
(3)
Equation 4 sets the geometric mean frequency equal to the converter crossover frequency with no Cff.
f _noCff +
Ǹǒfz
fp
Ǔ
(4)
Setting the geometric mean frequency equal to the converter crossover frequency with no Cff positions the
maximum phase boost of Cff at f_nocff. However, because Cff introduces a boost in phase and in gain,
the new crossover frequency occurs at a frequency greater than the geometric mean frequency.
Therefore, the new converter crossover frequency does not occur at the maximum phase boost frequency
due to Cff, but crosses over at a higher frequency facilitating a faster converter response time, while still
benefitting from additional phase boost. The following plots confirm that the converter response time does
indeed improve, and as a result, less transient voltage deviation is observed.
Substituting Equation 1 and Equation 2 into Equation 4 results in Equation 5, which is now a function of
R1, R2, and Cff.
f _noCff +
Ǹǒ
2p
1
R1
Cff
Ǔƪ2p 1 Cff ǒR21 ) R11 Ǔƫ
(5)
Solving for Cff results in a feedforward capacitor value for optimum transient response, Cff_op.
Cff_op +
2p
Ǹ
1
f _nocff
Cff_op + 7.066
1
R1
ǒR11 ) R21 Ǔ
10*11
(6)
Where f_nocff = 16 kHz, R1 = 442 kHz, and R2 = 49.9 kHz. Rounding the calculated Cff value up to the
next nearest standard capacitor value, rounds to 82 pF.
Cff_std = 82 pF
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Improvement
Figure 10 shows the improved transient response with the addition of the 82-pF Cff capacitor. The
converter responds in 14 μs with Cff = 82 pF compared to 30 μs without Cff. The maximum transient
voltage deviation is 377 mV with Cff compared to 900 mV without Cff.
Figure 10. Voltage Transient in Response to a Load Transient With 82-pF Feedforward Capacitor
7 Side Bar:
47.89 k Frequency (Hz)
0
Gain (dB)
54.91 Phase (deg)
-0.976 Slope (20 dB/decade)
Figure 11. Loop Gain and Phase Plot of TPS61081 Circuit With 82-pF Feedforward Capacitor
Figure 11 shows that the network analyzer also confirms the improved bandwidth with adequate phase
margin. For this example, the addition of the Cff capacitor increased the bandwidth by a factor of 3, from
16 kHz to 48 kHz, and increased the phase margin to an acceptable 55°.
For most applications, this is an optimum placement of the feedforward capacitor response. Increasing the
feedforward capacitance value pushes both the zero and pole frequencies closer to the origin which
increases the crossover frequency but can result in lower overall phase margin. This corresponds to a
faster loop at the expense of lower phase margin. Decreasing the Cff value results in the opposite result
until a certain point where the feedforward capacitor gain and phase boost contribution diminishes and
approaches the response of having no Cff. Having too small a Cff value injects a zero and pole at
frequencies too high and effectively too late in loop response, resulting in little or no performance
improvement
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2.6
Optimizing Toward a Faster Loop At the Expense of Less Phase Margin
To reduce transient ripple even more, the feedforward capacitor value can be increased to push the
crossover to higher frequencies. Although this can decrease the voltage deviation even more and speed
up loop response, more ringing is observed because less phase boost is added from the feedforward
capacitor at the new crossover frequency. Larger Cff values provide less phase boost because increasing
Cff causes the converter to cross over at higher frequencies while the maximum phase boost moves to
lower frequencies. It is recommended to keep the feedforward capacitor value smaller than the value
which corresponds to 30° of phase margin, so that a phase margin ≥30° is the minimum phase margin
target. This corresponding Cff value limit must be determined empirically, if required. It is generally not
recommended to increase Cff significantly greater than the calculated optimized Cff.
Figure 12 and Figure 13 show the same converter using a 1000-pF feedforward capacitor which is much
larger than the initially optimized capacitor value in an attempt to speed up the loop response. It is seen
that the converter begins to correct the deviation from dc faster and results in less voltage deviation, at
258 mV. Using the network analyzer, the resulting crossover frequency is improved from 48 kHz to 73 kHz
at the expense of lower phase margin, now at 22°, which is lower than that generally recommended. In
this transient response, the voltage deviation begins to correct in 9 μs as opposed to the optimized 14-μs
response.
Figure 12. Voltage Transient in Response to a 0 to 100% Load Transient With 1000-pF Feedforward
Capacitor
21 Side Bar:
73.15 k Frequency (Hz)
0
Gain (dB)
21.97 Phase (deg)
-1.37 Slope (20 dB/decade)
Figure 13. Loop Gain and Phase Plot of TPS61081 Circuit With 1000-pF Feedforward Capacitor
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Optimizing Toward Greater Phase Margin for Less Ringing
Conversely, if more phase margin is desired, a smaller feedforward capacitor allows the loop to crossover
at a lower frequency and position the maximum phase boost from the feedforward capacitor closer to the
new crossover frequency, with the tradeoff of lower bandwidth. As the Cff capacitor value is reduced, the
bandwidth of the converter approaches the bandwidth of the converter without the feedforward capacitor.
Figure 14 and Figure 15 show the transient and loop response of a converter with the 82-pF Cff replaced
with a 33-pF Cff. With only 33 pF, the converters response time has increased to 22 μs, resulting in a
larger transient voltage deviation of 613 mV. This, however, is still better than the 30-μs response, 900-mV
voltage deviation of the converter without the feedforward capacitor. Also, note that a 0-A to full-load
transient is very aggressive testing and was used to show more clearly the optimization throughout this
application report.
Figure 14. Voltage Transient in Response to Load Transient With 33-pF Feedforward Capacitor
15 Side Bar:
22.98 k Frequency (Hz)
0
Gain (dB)
75.31 Phase (deg)
-1.01 Slope (20 dB/decade)
Figure 15. Loop Gain and Phase Plot of TPS61081 Circuit With 33-pF Feedforward Capacitor
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3
Conclusion
The feedforward capacitor used in the feedback network improves the performance of internally
compensated dc-dc converters. The respective data sheet describes how to generally size the
feedforward capacitor for improved loop response. However, with measured response characteristics of a
working design, the feedforward capacitor can be sized to facilitate an improved transient response. The
calculated optimal Cff value can be increased or decreased to optimize the converters transient response
for minimum voltage deviation or higher phase margin.
4
References
1. Evaluation and Performance Optimization of Fully Integrated DC/DC Converters (Topic 7 of 2006
Portable Power Design Seminar)
2. Using the TPS40040EVM-001: A 12-V Input, 1.8-V Output, 10-A Synchronous Buck Converter User's
Guide (SLUU266)
3. TPS6108xEVM-147 User's Guide (SLVU144)
4. TPS61080/81, High Voltage DC/DC Boost Converter With 0.5-A/1.3-A Integrated Switch data sheet
(SLVS644)
SPace
Revision History
NOTE: Page numbers for previous revisions may differ from page numbers in the current version.
Changes from A Revision (May 2015) to B Revision ...................................................................................................... Page
•
Changed Figure 8 ......................................................................................................................... 6
Revision History
Changes from Original (January 2008) to A Revision .................................................................................................... Page
•
Changed From: [R2/(R1xR2)] To: [R2/R1 + R2)] in Figure 2........................................................................ 2
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