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Texas Instruments Analysis and Design of Input Filter for DC-DC Circuit Application notes
Application Report
SNVA801 – November 2017
Analysis and Design of Input Filter for DC-DC Circuit
Charles Zhang
ABSTRACT
When designing DC-DC circuits, it is common to add the input filter circuit before the power stage. An LC
filter is one of most frequently used input filter circuit. However, if a design is not optimal, input filter circuit
may cause large output noise instead of suppressing the noise, and may even cause loop stability
problems. This application report analyzes the influence of the input filter on the DC-DC control loop
transfer function, and the influence of a closed loop on the input filter, explains why input filter causes
unexpected problem, and suggests how to eliminate the side effect of the input filter.
1
2
3
4
5
6
Contents
Function of Input Filter Circuit .............................................................................................. 2
Loop Gain of Buck With Input LC Filter ................................................................................... 2
2.1
Open Loop Small Signal Model ................................................................................... 2
2.2
Closed Loop of the Buck Converter .............................................................................. 4
2.3
Loop Gain With Input LC Filter .................................................................................... 5
Closed-Loop Input Impedance ............................................................................................. 7
3.1
Closed-Loop Input Impedance Calculation ...................................................................... 7
3.2
Influence of the Input Impedance on Input Filter ............................................................... 8
Designing the Input Filter Circuit ........................................................................................... 9
4.1
Stability Criteria With Input Filter.................................................................................. 9
4.2
Damping the Input Filter .......................................................................................... 10
Conclusion .................................................................................................................. 11
References .................................................................................................................. 11
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Analysis and Design of Input Filter for DC-DC Circuit
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1
Function of Input Filter Circuit
1
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Function of Input Filter Circuit
Input filters are widely used in power design. They have two main purposes: one is to suppress the noise
and surge from the front stage power supply, another is to decrease the interference signal at switching
frequency and its harmonic frequency to go back to the power supply and interfere other devices which
uses the power supply. The input filter design is very important to pass the electromagnetic compatibility
(EMC) test.
Simple LC passive filters are among the common filters for DC-DC converters. They can attenuate the
high frequency noise from the power supply and can also suppress the switching noise to go back to the
power supply.
If LC filter is not well damped, the frequency response will peak near resonant frequency, which means
the LC actually is amplifying the noise signal. Apart from the amplification, sometimes, LC filter design
also has effect on the control loop of the DC-DC converter, some poor LC filter will decrease the phase
margin and degrade the transient response performance, some even cause stability problem. This is
discussed in Section 2 based on a buck converter.
2
Loop Gain of Buck With Input LC Filter
2.1
Open Loop Small Signal Model
Figure 1 is a traditional synchronous buck converter. During the on and off status of one switching cycle,
FET shows non-linear characteristic. According to the average model, the small signal of input current and
switching node voltage can be calculated as Equation 1, and the open loop small signal model of the buck
can be expressed as Figure 2.
I LdÖ
iÖin
uÖCP
I in
A Q1
DiÖL
U in dÖ
C
DuÖin
I L RL
(1)
L
IO
C
U in +±
D
Q2
IÖin A
UO
RC
Rload
±+
C IÖL RL
1: D U in dÖ
I L dÖ
UÖ in +
±
L
UÖ O
C
RC
Rload
IÖO
P
P
GND
GND
Figure 1. Buck Circuit Topology
Figure 2. Buck Circuit Small Signal Model
It is normal that input voltage and output current may change, which influences the output voltage and
inductor current. When adopting closed loop control, duty cycle may change to maintain the output
voltage, at the same time, the inductor current will change. So we can build the open loop control block
diagram as Figure 3.
2
Analysis and Design of Input Filter for DC-DC Circuit
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Loop Gain of Buck With Input LC Filter
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Gii
Giu
iÖL
Gid
iÖo
Zout
uÖi
dÖ
Guu
uÖo
Gud
Figure 3. Open-Loop Buck Controller Block Diagram
For simplicity, ignore the inductor DCR and output capacitor ESR when calculated the transfer function.
Then the six transfer functions in Figure 3 are calculated as Equation 2 to Equation 7. All the six transfer
functions will be used in following sections, especially at Section 3.1.
uÖo ( s )
Ui
|uÖ ( s ),iÖ ( s )
2
i
o
Ö
LCs RL s 1
d ( s)
uÖo ( s )
D
|dÖ ( s ),iÖ ( s )
Guu ( s )
2
o
uÖi ( s )
LCs RL s 1
uÖo ( s )
Ls
Z out ( s )
|dÖ ( s ),uÖ ( s )
2
i
LCs RL s 1
iÖo ( s )
Uin
iÖ ( s )
RCs )
R (1
Gid ( s ) L
|uÖ ( s ),iÖ ( s )
2
LCs RL s 1
dÖ ( s ) i o
D
iÖ ( s )
RCs )
R (1
|dÖ ( s ),iÖ ( s )
Giu ( s ) L
2
o
uÖi ( s )
LCs RL s 1
iÖ ( s)
1
Gii ( s) L |dÖ ( s ),uÖ ( s )
2
i
Öio ( s)
LCs RL s 1
Gud ( s )
(2)
(3)
(4)
(5)
(6)
(7)
Input impedance is a key parameter in this paper, which is not obvious to drawn in Figure 3; nevertheless,
open loop input impedance can be defined and calculated as Equation 8 using a similar method.
Zin ( s)
uÖi ( s )
uÖi ( s)
|
|
Öiin ( s ) dÖ ( s ),iÖo ( s ) iÖL ( s) dÖ ( s ),iÖo ( s )
D
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R LCs 2 RL s 1
D 2 (1 RCs )
(8)
Analysis and Design of Input Filter for DC-DC Circuit
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3
Loop Gain of Buck With Input LC Filter
2.2
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Closed Loop of the Buck Converter
For a buck converter, it is normally desired that the output voltage is constant. Feedback topology is used
to close the control loop, it automatically changes the duty cycle to decrease or even eliminate the
influence from the input voltage or output current. A typical closed loop of VMC buck converter is as
Figure 4. H is the feedback resistor network, Gc represents the error amplifier, and Gm is for the modulator
gain.
Gii
Giu
iÖL
Gid
iÖo
Zout
uÖi
uÖref
uÖerr
uÖc
Gc
Gm
dÖ
Guu
uÖo
Gud
H
Figure 4. VMC Closed Loop Buck Controller Block Diagram
By adopting closed loop topology, loop gain is defined as the product of the gains in feedforward path and
feedback path in the closed loop as Equation 9. Take the VMC plus type III compensator as example, the
frequency response of Tloop can be plotted as Figure 5.
Tloop(s) = HGcGmGud
(9)
Figure 5. Loop Gain Frequency Response
4
Analysis and Design of Input Filter for DC-DC Circuit
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Loop Gain of Buck With Input LC Filter
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Under the effect of the closed loop, the transfer function from the perturbation to the output voltage is
multiplied by a factor 1 /(1 + T(s)). The closed loop audio susceptibility and output impedance can be
expressed as Equation 10 and Equation 11. And the open loop and closed loop frequency response can
be drawn as Figure 6 and Figure 7, it can be seen from the picture, low frequency perturbation can be well
suppressed by closing the control loop.
Guu _ CL ( s )
Zout _ CL ( s)
Guu ( s )
1 Tloop ( s )
D
(1 Tloop ( s )) ˜ ( LCs 2
L
R
Zout ( s )
1 Tloop ( s)
Ls
(1 Tloop ( s)) ˜ ( LCs 2
L
R
Figure 6. Closed Loop Audio Susceptibility
2.3
s 1)
(10)
s 1)
(11)
Figure 7. Closed Loop Output Impedance
Loop Gain With Input LC Filter
Even though closed loop suppresses the audio susceptibility a lot at low frequency, but it doesn’t change
too much for the middle and high frequency. The designer should also consider the interference from the
DC-DC back to the power bus. Thus, a LC filter is often used for DC-DC circuit to pass EMI test.
When adopting LC filter to solve an EMI problem, another problem may arise, when the load transient
performance changes, audio stability becomes worse at some frequency, some even cause the stability
problem. These problems can be explained as that input filter circuit modifies the transfer function of the
buck converter, including the control to output transfer function Gud, which then changes the loop gain,
phase margin, etc.
According to Middlebrook’s extra element theorem, adding the filter circuit to the buck converter adds a
correction factor to the original transfer function.
UÖ in +
±
DCDC
Converter
Power
stage
Input
filter
Rload
dÖ
Z filter
Z in
Controller
Figure 8. Buck Control Block Diagram With LC Filter
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Analysis and Design of Input Filter for DC-DC Circuit
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Loop Gain of Buck With Input LC Filter
Gud _ f ( s )
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Gud ( s ) |Z filter ( s )
0
·
§ Z filter ( s )
¨1
Z N ( s ) ¹¸
©
·
§ Z filter ( s )
¨1
Z D ( s ) ¹¸
©
where
•
Z filter ( s )
Gud ( s ) |Z filter ( s )
sL f
0
is the original transfer function before addition of input filter.
§ 1
RDCR || ¨
¨ sC f
©
(12)
·
RESR ¸
¸
¹ is the output impedance of the LC filter, looking from DC-DC
side.
Z D ( s)
Z in ( s ) |dÖ ( s )
Z N ( s)
Z in ( s ) |uÖo ( s )o0
0
R LCs 2 RL s 1
D2
1 RCs is the converter input impedance, with dÖ ( s )set to zero.
R
D 2 is the converter input impedance, with the output nulled to zero.
Using a poorly designed LC filter as an example, after addition of correct factor, the new loop gain transfer
function can be calculated from Equation 13. The frequency response can be calculated as Figure 9,
simulated as Figure 10 using Simplis, the two waveforms show good consistency of the LC resonant
frequency.
Tloop(s) = HGcGmGud_f
(13)
Figure 9. Mathcad Calculation Loop Gain
Figure 10. Simplis Simulation Loop Gain
Comparing the new frequency response waveform with original one, it can be easily found that the LC
filter brings a dip to the original frequency response, the magnitude-frequency curve drops to close to 0 db
in this case, and the phase-frequency curve also drops near that point, which makes the loop gain
performance worse.
6
Analysis and Design of Input Filter for DC-DC Circuit
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Closed-Loop Input Impedance
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3
Closed-Loop Input Impedance
3.1
Closed-Loop Input Impedance Calculation
The open loop input impedance is defined as Equation 8, after applying the closed loop, the closed loop
changes the input impedance, because when input voltage change, duty cycle changes to maintain the
output voltage, which also brings additional change to the input current.
To calculate the closed loop input impedance, combined with Equation 1, the block diagram can be draw
as Figure 11. By changing the feedback and feed-forward position, a simplified block diagram can be
drawn as in Figure 12. By comparing Equation 2 through Equation 8, it can be calculated that
Guu
Gud
D
, DGiu
UI
1
, GuuGid
Zin
Gud
DZin
(14)
combined with Figure 12, a single block from input voltage to input current can be expressed as Figure 13.
iÖin
IL
D
Giu
iÖL
Gid
uÖi
Guu
dÖ
uÖc
Gc
Gm
uÖo
Gud
H
Figure 11. Closed Loop Input Impedance Calculation Block Diagram
1 HGcGmGud
Giu
HGcGmGuu
uÖi
uÖo
Guu
1 HGcGmGud
HGcGm
dÖ
iÖL
Gid
D
iÖin
IL
Figure 12. Simplified Closed Loop Input Impedance Calculation Block Diagram
uÖi
1
(1 Tloop ) Zin
T
D
u loop u I L
U I 1 Tloop
iÖin
Figure 13. Single Block From Input Voltage to Input Current
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Closed-Loop Input Impedance
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So the closed loop input impedance can be calculated as Equation 15:
Zin _ CL ( s )
uÖi _ CL ( s )
iÖin _ CL ( s )
Zin
(1 Tloop )
DI LTloop Zin
1
UI
(15)
The frequency response of closed loop input impedance is plotted as Figure 14 in comparison with open
loop input impedance. It can be seen that closed loop impedance and open loop impedance shows similar
characteristic when frequency goes high; during middle frequency, the closed loop makes the impedance
amplitude larger, which means that same input voltage change will brings smaller input current variation.
At low frequency, they show same amplitude characteristic, but closed loop makes the phase
characteristic changes, the phase at low frequency is close to –180°, it functions as negative incremental
input impedance. For example, the output voltage and current are fixed, if input voltage increases, the
input current decreases, it looks like negative resistor.
Figure 14. Open-Loop and Closed-Loop Input Impedance
3.2
Influence of the Input Impedance on Input Filter
For a DC-DC circuit, if the input filter is utilized, this negative incremental input impedance also has some
influence on the input filter circuit. From Figure 14 it can also be seen, when loop is open, the input
impedance is very low at middle frequency, and the low impedance brings more damping to the LC filter
circuit, but when adopting closed loop, the impedance become negative, this is decreasing the damping to
the LC filter. Define the transfer function from input voltage to the output of the LC filter as Gfv(s). Use
Mathcad to draw the frequency response of Gfv(s) as Figure 15. It can be seen, when loop is open, the
peak value of the amplitude is very small, much less than 20 dB. But when adopting the closed loop, the
peak value is close to 20 dB, which means very small damping of the LC circuit.
Figure 15. Input Filter Frequency Response With Closed Loop
8
Analysis and Design of Input Filter for DC-DC Circuit
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Designing the Input Filter Circuit
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When the damping ratio is too small, normally there will be large amplitude oscillation in time-domain
waveform. Figure 16 is the test result based on the TPS40040EVM by adding a 3.5-µH inductor as filter
circuit. Taking the de-rating of the inductor and MLCC into consideration, a Simplis simulation result is
plotted as Figure 17, which shows good consistence with the EVM test result.
Figure 16. Load Transient Test on EVM
4
Designing the Input Filter Circuit
4.1
Stability Criteria With Input Filter
Figure 17. Load Transient Test of Simplis Simulation
From above load transient test, it can be seen that when the load is heavy, the oscillation is much larger
than light load, because larger load current brings smaller negative incremental input impedance, which
means much smaller damping ratio, or even brings stability problem. Capacitor ESR, Inductor DCR and
power source output impedance RO are also very critical for stability, because they also bring damping to
the LC filter. To analyze the stability of a buck circuit with an LC filter, use a pure resistor –Rin = Zin(s)|| s = 0
as input impedance, then equivalent circuit is Figure 18.
L
RDCR
Uf
C
RO
Rin
U in +
RESR
±
Figure 18. Equivalent Circuit for Stability Analysis
RESR is normally much smaller than –Rin, so the paralleled impedance of the capacitor, and the input
impedance can be expressed as Equation 16 and Gfv(s) can be expressed as Equation 17.
Rin / / 1
G fv ( s)
sC
RESR |
uÖ f ( s)
uÖin ( s)
2
Rin 1 sCRESR
1 sCRESR
s LCRin
(16)
Rin 1 sCRESR
s ª¬ RinC RO RDCR RESR L º¼ ( Rin
RO
RDCR )
(17)
For stability, there should be no positive pole for Gfv(S) , then
LCRin ! 0
RinC RO
RDCR
Rin
RDCR ! 0
RO
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RESR
L!0
(18)
Analysis and Design of Input Filter for DC-DC Circuit
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Designing the Input Filter Circuit
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So the stability requirement is as Equation 19 and Equation 20, and the damping ratio is as Equation 21.
Rin !
C RO
Rin ! RO
L
RDCR
RESR
(19)
RDCR
(20)
ª¬ RinC RO RDCR RESR L º¼
2 LCRin u ( Rin RO RDCR )
]
(21)
Normally it is not very hard to meet Equation 19 and Equation 20. But in some cases, even though meets
Equation 19 and Equation 20, if Rin is too small, according to Equation 21, the damping ratio of the system
is also small, which brings large oscillation. Normally this happens with the condition of large output
current and large duty cycle, and the input cap is using MLCC which has smaller ESR. That’s why in
Figure 9 and Figure 10, when load is heavy, the oscillation amplitude is large.
4.2
Damping the Input Filter
If oscillation happens after adding a LC filter circuit, normally decreasing the inductor value or increasing
the capacitor value may increase the damping ratio and remove the oscillation, but this will also change
the target LC filter characteristic. Sometimes replacing MLCC with electrolytic capacitor may also solve the
problem. But the large ESR of the electrolytic capacitor will also bring large input voltage ripple when the
MOSFET is switching. Another way is to add an additional damping circuit after the LC filter circuit like
Figure 19. The resistor is to damp the filter, and because we only want to damp the middle frequency
amplitude peak value, there is no need to use a pure resistor as the damping circuit. Rather, a series
capacitor Cd can be used to avoid large power dissipation on the resistor. A capacitor much larger than C
is required for the LC filter to detect Rd at the middle frequency, Cd ≥ 4C may be an acceptable value, and
there is no need to use good-quality capacitor like C to saving cost.
L
RDCR
C
RO
Uf
L
Cd
DCDC
Converter
U in +
RESR
±
Rd
C
RO
Uf
-R'in
Rd
U in +
±
Figure 19. LC Filter With Damping Circuit
RDCR
Rin
RESR
Figure 20. Equivalent Circuit at Middle Frequency
By applying the damping circuit, the approximate circuit at middle frequency is as Figure 20, the
equivalent input impedance is shown in Equation 22: by adding the damping circuit, the absolute value of
the equivalent input impedance value will increase, which increases the damping ratio of the LC filter.
Rin'
( Rin ) ˜ Rd
Rin Rd
§
·
1
Rin ¨
¸
© 1 Rd Rin ¹
(22)
By adding additional damping circuit, EVM test result and the Simplis simulation result are as Figure 21
and Figure 22, which shows good consistency.
10
Analysis and Design of Input Filter for DC-DC Circuit
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Conclusion
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Figure 21. Load Transient Test on EVM
5
Figure 22. Load Transient Test of Simplis Simulation
Conclusion
An LC filter is widely used to optimize the EMC during DC-DC design, but improper design of the LC filter
may cause the stability problem. This paper analyzes the influence of the input filter on the closed-loop
gain, also explains how the closed loop brings the negative incremental input impedance and influence the
LC filter. At the end, this paper describes stability criteria for LC filter design and also suggests some
methods to solve oscillation problem when adopting LC filter.
6
References
1. R.D. Middlebrook, “Input filter considerations, in design and application of switching regulators” 1976
IEEE
2. Texas Instruments, Input Filter Design for Switching Power Supplies application note
3. Texas Instruments, Low Pin Count, Low VIN (2.5V to 5.5V Synchronous Buck DC-to-DC Controller
TPS40040/1 data sheet
4. Texas Instruments, AN-2162 Simple Success With Conducted EMI From DCDC Converters application
report
5. R. Ahmadi, D. Paschedag, and M. Ferdowsi, “Closed-loop input and output impedances of DC-DC
switching converters operating in voltage and current mode control,” Proc. IECON ’10, 2010, pp.
2311–2316.
6. R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 2nd ed. New York: Kluwer,
2001.
7. Marian K. Kazimierczuk, Pulse Width Modulated DC-DC Power Converters, 2nd ed. Ohio: Wiley, 2016.
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