# Texas Instruments | How to Calculate the Load Pole and ESR Zero When Using Hybrid Output Capacitors (Rev. A) | Application notes | Texas Instruments How to Calculate the Load Pole and ESR Zero When Using Hybrid Output Capacitors (Rev. A) Application notes

```Application Report
SLVAE26A – September 2018 – Revised April 2019
How to Calculate the Load Pole and ESR Zero When
Using Hybrid Output Capacitors
Hao Zhang, Jason Wang
ABSTRACT
Multi-layer ceramic (MLCC), aluminum electrolytic, tantalum, and polymer are the capacitor types most
widely used in DC/DC switching regulator circuits. In practical applications, a power designer can use a
hybrid capacitor network, formed by combining different capacitor types, in an effort to achieve low ESR
and high capacitance. This can be a very effective method of reducing output ripple and improving load
transient performance. This application report provides a method to analyze how the hybrid capacitor
network affects the loop. The Section 1 section introduces the key characteristics of each of the different
types of capacitors. The Section 2 section discusses peak current mode power stage small signal
modeling with a hybrid output capacitor network. Finally, the Section 3 section verifies the analysis with
bench test results on TPS65400EVM.
1
2
3
4
5
Contents
Introduction ................................................................................................................... 2
Current Mode Power Stage Small Signal Modeling ..................................................................... 3
Bench Verification ........................................................................................................... 7
Summary .................................................................................................................... 12
References .................................................................................................................. 12
List of Figures
1
2
3
4
5
6
7
8
9
10
11
.................................................................... 3
Current Mode Buck Converter Circuit With Hybrid Output Capacitor Network ...................................... 4
Simplified Power Stage Bode Plot of Single Output Capacitor Configuration ....................................... 6
Simplified Power Stage Bode Plot of Hybrid Output Capacitor Network Configuration ............................ 7
Simplified Schematic of All MLCC Output Capacitor Configuration ................................................... 8
Bode Plot of All MLCC Output Capacitor Configuration (VOUT = 3.3 V, IOUT = 1.5 A) ................................ 8
Simplified Schematic of Hybrid Output Capacitor Configuration ...................................................... 9
Bode Plot of Hybrid Output Capacitor Configuration .................................................................. 10
Simplified Schematic of Single Polymer Output Capacitor Configuration .......................................... 10
Bode Plot of Single Polymer Output Capacitor Configuration (VOUT = 3.3 V, IOUT = 1.5 A) ....................... 11
Simplified Current Mode Functional Block Diagram
Gain Frequency Response Comparison Among Hybrid, MLCC, and Polymer Output Capacitor
Configuration ............................................................................................................... 11
List of Tables
1
Comparison of Different Types of Capacitors ............................................................................ 2
2
Basic Configurations of TPS65400EVM .................................................................................. 7
3
Parameters of Capacitors Used in Output Capacitor Network......................................................... 9
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1
Introduction
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Introduction
Output capacitors are critical components for a switch-mode power supply. To select output capacitors for
a DC/DC switching regulator application, the basic parameters you must pay attention to include
capacitance, equivalent series resistance (ESR), rated voltage, and size. Table 1 compares the four types
of capacitors.
Table 1. Comparison of Different Types of Capacitors
CAPACITANCE
ESR
RATED VOLTAGE
SIZE
Multi-layer Ceramic
Capacitor
Low
Low
Medium
Small
Aluminum Electrolytic
Capacitor
High
High
Various
Large
Tantalum Capacitor
Medium
Medium
Medium
Medium
Polymer Capacitor
High
Medium
Medium
Medium
Generally speaking, MLCC provides very low ESR, which is critical to compress the resistive output ripple.
Aluminum electrolytic capacitors provide a large amount of capacitance, but have the highest ESR among
the four capacitor types. Tantalum and polymer capacitors have medium-range capacitance values, ESR,
and rated voltage. By using a hybrid capacitor network, designers can take advantage of the benefits of
each capacitor type. In applications where small ripple, overshoot, and undershoot are required, hybrid
output capacitor networks are very common.
Loop stability is another important topic for DC/DC switching regulator circuit design. In DC/DC converter
small signal modeling, the capacitance and ESR values of the output capacitor have a direct effect on the
poles and zeros in the open loop transfer function. With the presence of a hybrid output capacitor network,
new poles and zeros are introduced into the loop by the network itself. This application report discusses
how hybrid output capacitors influence the loop, and then verifies the analysis using the TPS65400EVM.
2
How to Calculate the Load Pole and ESR Zero When Using Hybrid Output
Capacitors
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Current Mode Power Stage Small Signal Modeling
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2
Current Mode Power Stage Small Signal Modeling
Figure 1 shows the simplified functional block diagram of a peak current mode DC/DC circuit.
+
Current
Sense
±
VIN
Power stage
dÖ
vÖout
Control
Logic
L
ESR
COUT
RFBT
vÖFB
RFBB
EA
+
Compensation
Network
±
vÖ COMP
VREF
Control stage
Figure 1. Simplified Current Mode Functional Block Diagram
To analyze and judge the loop stability by open loop transfer function and Bode plot, the loop is split into
two components:
Gopen (s )
Gdv (s ) ˜ Hvd (s )
where
•
Gdv(s) are the transfer function from control to the output ( dÖ to vÖ out ), including the PWM modulator
and power stage.
•
Hvd(s) is the transfer function from the output to control ( vÖ out to dÖ ), including the feedback path,
compensation network.
(1)
Poles and zeros in power stage have a direct effect on the loop transfer function. To understand how the
hybrid output capacitor network affects the loop, calculate the poles and zeros in the power stage. The
calculation can vary in the different control mode, as they have different control to output transfer function
Gdv(s). This application report shows how to do the calculation based on a current mode DC/DC converter
circuit. Impedance of hybrid output capacitor network is discussed first. Based on that, the effect of the
network on current mode DC/DC circuit is analyzed.
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Current Mode Power Stage Small Signal Modeling
2.1
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Impedance of Hybrid Output Capacitor Network
Figure 1 shows that the output capacitor in the functional block diagram is simplified as a single capacitor.
Impedance of a single capacitor with a specific ESR is:
1 s ˜ ESR ˜ C
sC
Z cap (s )
(2)
When the output capacitor is substituted by an output capacitor network, the situation can be different.
Figure 2 shows two capacitors with same or different capacitance and ESR that are put in parallel.
+
Current
Sense
±
VIN
Hybrid Output
Capacitor Netwrok
Vout
Control
Logic
L
ESR1
ESR2
RFBT
Cout1
Cout2
RFBB
GND
±
EA
+
Compensation
Network
VREF
Figure 2. Current Mode Buck Converter Circuit With Hybrid Output Capacitor Network
Impedance of the hybrid capacitor network Zcap(s) can be expressed as:
Z cap (s )
(R1
1
) //(R 2
sC1
1
)
sC 2
where
•
•
R1 is the ESR of C1
R2 is the ESR of C2
(3)
If you expand Equation 3, you have:
Z cap ( s )
s(C 1
(1 sR 1C 1 ) ˜ (1 sR 2 C 2 )
ª
C 1C 2 º
C 2 ) ˜ «1 s ˜ (R 1 R 2 ) ˜
C 1 C 2 »¼
¬
(4)
Equation 4 indicates when the capacitance and ESR are different. There is an initial pole, a parallel pole,
and two ESR zeros in the impedance expression.
A specific situation is where the two capacitors are same, C1 = C2 = C, R1 = R2 = R. Equation 4 can be
further simplified as:
4
How to Calculate the Load Pole and ESR Zero When Using Hybrid Output
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Zcap (s)
1 sRC
s ˜ 2C
(5)
In this situation, two of the same output capacitors can be equalized to one single capacitor with two times
capacitance and half ESR.
2.2
Pole and Zero Calculation in Current Mode Power Stage
The distribution of poles and zeros in the loop transfer function depends on both power stage
characteristic and control mode. As discussed in the Current-Mode Modeling for Peak, Valley and
Emulated Control Methods Application Report, the transfer function of peak current mode buck converter
power stage is:
vÖ out
vÖ c
1
1
Ri
Ro
1
Km
˜
(1
s
Z ESR
s
) ˜ (1
Z0
s
)
ZL
where
•
•
•
•
•
•
Km
VIN
Vramp
Km is the modulator voltage gain, which is given by
Ri is the product of current sense gain and current sense resistance
ω0 is the dominant pole
ωL is the inductor pole
ωESR is the ESR zero of the output capacitor
(6)
The dominant pole locates at the frequency where the impedance of output capacitor equals to the loading
resistance:
1
ZESR =
ESR u COUT
(7)
The inductor pole locates at the frequency where the impedance of the inductor equals to the current-loop
gain:
ZL
K m ˜ Ri
L
(8)
In most cases, Km × Ri » 1, so ω0 can be simplified as:
Z0
1
Cout ˜ Ro
(9)
Also, as Km × Ri » 1, ωL is usually much higher than the bandwidth, which means the pole has limited
effect on the loop stability and can be ignored in the following analysis.
The ESR zero of output capacitor locates at:
1
ZESR =
ESR u COUT
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(10)
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Current Mode Power Stage Small Signal Modeling
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Equation 6 is further simplified as:
vÖ out
vÖ c
Ro
˜
Ri
1
s
Z ESR
s
1
Z0
(11)
Figure 3 shows the simplified Bode plot.
Gain
Z0
ESR zero
Z ESR
Frequency
Figure 3. Simplified Power Stage Bode Plot of Single Output Capacitor Configuration
When the output capacitor is substituted by a hybrid capacitor network, modify Equation 11 as:
vÖ out
vÖ c
Ro
˜
Ri
(1
s
)( 1
Z ESR 1
s
(1
)( 1
Z0
s
Z ESR
s
)
Zp
)
2
where
•
Z0
1
Ro ˜ (C1 C2 )
Zp
•
•
ωp is the parallel pole introduced by the parallel capacitors, given by
ωESR1 and ωESR2 are the ESR zeros of the capacitors, given by
ZESR1
6
1
ZESR 2
ESR1 ˜ C1
1
ESR 2 ˜ C2
How to Calculate the Load Pole and ESR Zero When Using Hybrid Output
Capacitors
1
(R 1 R 2 ) ˜
C1C 2
C1 C 2
(12)
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Bench Verification
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Figure 4 shows the simplified Bode plot.
Gain
Z0
ESR zero
ZESR1
Zp
Parallel pole
ESR zero
ZESR2
Frequency
Figure 4. Simplified Power Stage Bode Plot of Hybrid Output Capacitor Network Configuration
If you compare Equation 11 and Equation 12, you have:
● The dominant pole ω0 is determined by loading resistance and the sum of all capacitance in the
power stage.
● ESR zero of the electrolytic capacitor remains at the same frequency.
● An additional pole ωp is introduced by the two different capacitors in output capacitor network.
Take these poles and zeros into consideration when designing the loop compensation with a hybrid output
capacitor network. These results are verified on a TPS65400EVM in Section 3.
3
Bench Verification
3.1
TPS65400 Introduction
The TPS65400 is a 4.5-V to 18-V, synchronous quad buck converter with PMBus/I2C interface. The device
works in peak current mode with an external loop compensation network. To support high output
capacitance in this test, compensation zero formed by Rcomp and Ccomp is placed at 0.6 kHz. The rolling
pole formed by Rcomp and Croll locates at 141.6 kHz. The test is performed on the first channel of the
TPS65400EVM. Table 2 lists some of the basic configurations. See the TPS65400 EVM User's Guide for
other detailed descriptions and configurations.
Table 2. Basic Configurations of TPS65400EVM
3.2
VIN
12 V
VOUT
3.3 V
Switching Frequency
500 kHz
1.5 A
0.33 kHz
Compensation Zero
0.6 kHz
Rolling pole
141.6 kHz
Bench Test
To verify the calculation in the previous section, a Bode plot is done for the output capacitor network
formed by all MLCC, hybrid, and single polymer capacitors.
3.2.1
All MLCC
Figure 5 shows the output capacitor network is formed all by MLCC. The part number of the capacitor is
GRM31CR61A476KE15L. Taking the DC bias effect into consideration, the total effective capacitance is
about 224.8 µF. Figure 6 shows the Bode plot of all MLCC configuration as the output capacitor.
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All MLCC
12V
VIN
Vout=3.3V
Iout=1.5A
SW
L
2.2uH
RFBT
TPS65400
Cout1
Cout2
47uF*8
COMP
RFBB
FB
Rcomp
51.1k
Croll
22pF
GND
Ccomp
4.7nF
Figure 5. Simplified Schematic of All MLCC Output Capacitor Configuration
Figure 6. Bode Plot of All MLCC Output Capacitor Configuration
(VOUT = 3.3 V, IOUT = 1.5 A)
According to the characteristic curve, the capacitor ESR is about 3 mΩ. ESR zero of single capacitor
locates at about 1.95 MHz, which is much higher than the bandwidth. As in the previous analysis, putting
the capacitors in parallel has no effect on the frequency of ESR zero. That is to say, besides the error
amplifier pole, the loading pole and the compensation zero, there is no any other pole nor zero within the
bandwidth. The gain curve keeps going down with a –20 dB/dec slope after the loading pole and
compensation zero. The bandwidth is about 22.2 kHz, and the phase margin is 62.9 degrees.
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3.2.2
Hybrid Output Capacitor Network
Figure 7 shows the hybrid output capacitor network is formed by one polymer capacitor and four MLCCs.
Total effective output capacitance is about 222 µF. Table 3 lists the parameters of the components.
Hybrid Output
Capacitor
12V
VIN
Vout=3.3V
Iout=1.5A
SW
L
2.2uH
22uF*4
TPS65400
COMP
Cout1
Cout4
Cout5
RFBT
FB
RFBB
Rcomp
51.1k
Croll
22pF
150uF/
50mŸ
GND
Ccomp
4.7nF
Figure 7. Simplified Schematic of Hybrid Output Capacitor Configuration
Table 3. Parameters of Capacitors Used in Output Capacitor Network
PART NUMBER
CAPACITANCE/µF
ESR /mΩ
COUNT
16TQC150MYF
150
50
1
GRM31CR71A226KE
15L
22
3
4
To analyze this hybrid network, the same four MLCC can be equalized to one single MLCC by Equation 5.
Then, according to the Equation 12, in the hybrid output capacitor network, ESR zero of the polymer
capacitor locates at Equation 13.
fESR 1
1
2S ˜ ESR 1 ˜ C1
21 .2kHz
(13)
The pole formed by putting different capacitors in parallel locates at Equation 14.
fp
2 S ˜ (ESR
1
1
ESR 2 ) ˜ ( C 1 // C 2 )
61 . 8kHz
(14)
Figure 8 shows the Bode plot. The existence of the polymer capacitor ESR zero fESR1 and the parallel pole
fp extends the bandwidth to 43 kHz, with a phase margin of 79.7 degrees. Judging from the gain curve, the
zero and pole can be identified at the calculated frequency above. At a higher frequency, the parallel pole
together with the rolling pole formed by Rcomp and Croll keeps the gain curve rapidly going down.
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Figure 8. Bode Plot of Hybrid Output Capacitor Configuration
3.2.3
Single Polymer Capacitor
Figure 9 shows the output capacitance consists of only one polymer capacitor with high ESR value.
Single Polymer
Output Capacitor
12V
VIN
Vout=3.3V
Iout=1.5A
SW
L
2.2uH
RFBT
TPS65400
COMP
Cout
FB
RFBB
Rcomp
51.1k
Croll
22pF
220uF/
35mŸ
GND
Ccomp
4.7nF
Figure 9. Simplified Schematic of Single Polymer Output Capacitor Configuration
The part number of the polymer capacitor is 6TPE220MAZB. It has a capacitance of 220 µF, with 35 mΩ
ESR. ESR zero locates at Equation 15.
fESR
10
1
2 S ˜ ESR ˜ C
20 . 7 kHz
(15)
How to Calculate the Load Pole and ESR Zero When Using Hybrid Output
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To make the comparison more convincing, the total effective output capacitance is set to be almost the
same with the previous test. Figure 10 shows the Bode plot.
Figure 10. Bode Plot of Single Polymer Output Capacitor Configuration
(VOUT = 3.3 V, IOUT = 1.5 A)
Because of the low frequency ESR zero, the gain curve is going down very slowly until the rolling pole
(141 kHz) takes effect. The bandwidth is extended to 164 kHz, which has exceeded 1/5 switching
frequency. Phase margin is about 35.6 degree. Obviously, the compensation network designed for other
configurations is not doing well in this situation.
3.2.4
Comparison
To show how the hybrid configuration affect the loop, the Bode plot of three configurations are drawn
together. Figure 11 shows the gain-frequency response curves.
60
Hybrid
MLCC
Polymer
40
Gain (dB)
20
0
-20
-40
-60
-80
102
103
104
Frequency (Hz)
105
106
Plot
Figure 11. Gain Frequency Response Comparison Among Hybrid, MLCC, and
Polymer Output Capacitor Configuration
Effective output capacitance is almost the same among the three configurations. However, MLCC
configuration has the smallest bandwidth because there is no ESR zero introduced by the high ESR
capacitor. For the hybrid configuration consisting of MLCC and polymer capacitors, the ESR zero locates
at the same frequency with the standalone polymer capacitor configuration. As calculated in Equation 14,
the parallel pole introduced by hybrid configuration locates at 61.8 kHz. It is the parallel pole which leads
to the significant difference between the hybrid configuration and the standalone polymer capacitor.
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3.3
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Conclusion
From the analysis and bench verification, three conclusions can be drawn.
• The frequency of loading pole depends on the total effective capacitance of the output capacitor
network.
• In the parallel output capacitor network, ESR zero of each capacitor is still taking effect independently.
Also, by putting two output capacitors in parallel, a parallel pole is introduced into the loop.
• The effect brought by the ESR zero and parallel pole depends on how close they are on the frequency
axis. In most cases, due to the DC bias degrading of MLCC, solution size, and cost, the total effective
capacitance of MLCC is usually much smaller than that of electrolytic or other capacitor with higher
capacitance and ESR. See the expression of ωp,
Zp
1
(R1 R 2 ) ˜ C1 // C2 .
It can be identified that the bigger difference between C1 and C2, the further the parallel pole and the ESR
zero are, and the less cancellation between ESR zero and parallel pole. This means its Bode plot curve
looks more like single output capacitor with high ESR. Take this into consideration when doing the loop
compensation design.
4
Summary
Hybrid output capacitor network with high capacitance is widely used to support applications with strict
limitation on ripple, overshoot and undershoot. This application report analyzed the poles and zeros in the
power stage when using a hybrid output capacitor network theoretically. Furthermore, the analysis is
verified on TPS65400EVM. The conclusion can be a guidance for the compensation design of DC/DC
circuit with hybrid output capacitor network.
5
References
•
•
•
12
Texas Instruments, TPS65400 4.5- to 18-V Input Flexible Power Management Unit with PMBus/I2C
Interface Data Sheet
Texas Instruments, TPS65400EVM 4.5- to 18-V Input Flexible Power Management Unit with
PMBus/I2C Interface Evaluation Module User's Guide
Texas Instruments, Current-Mode Modeling for Peak, Valley and Emulated Control Methods
Application Report
How to Calculate the Load Pole and ESR Zero When Using Hybrid Output
Capacitors
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Revision History
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Revision History
NOTE: Page numbers for previous revisions may differ from page numbers in the current version.
Changes from Original (September 2018) to A Revision ............................................................................................... Page
•
Edited the application report for clarity.
................................................................................................
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