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Advanced Simulation
Techniques
Part 6:Stability Analysis
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Advanced Simulation Techniques
Table of Contents
Open Loop Transfer Function
The Cross Over Frequency
Stability Criteria
Various RC Networks
Type 1, 2 and 3 Amplifier
K Factor Technique
Example of a Stabilized Buck Converter
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Advanced Simulation Techniques
Buck Converter Open Loop Transfer Function
Gpwm
Error amplifier
H(s)
D1
SW1
+1
-2 D
3-
+
V1
{Vin}
IN 1 2L1
{L}
C1{C}
R1{R}
Out
Gate_Drv
Gate_Drv
{R_Comp}
C3
{C_Comp}
0
10k
0
+
-
OUT
{Vref}
H( s)
GPWM
Er r or Ampl i f i er
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Advanced Simulation Techniques
Buck Converter Open Loop Transfer Function
+=
+=
)0(1
)0(1
)0()0(1
)0()0(1
T
T
HG
HG
V
V
ref
out
1/ +Vref
VoutG(o) H(o)-
Error amplifier Power stage
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Advanced Simulation Techniques
Small Signal Representation
wmH(s)G(s)GpT(s)=Duty_Cycle
G( s)
IN2
V2
{Vin}
1 2L2
{L}
C2{C}
R2{R}
Out2
Duty_Cycle
0.5
a c
d p
SW2
Duty_Cycle
Gpwm
{R_Comp}
C5
{C_Comp}
0
10k
0
+
-
OUT
{Vref}
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Advanced Simulation Techniques
Voltage Injection
{R_Comp}
C5
{C_Comp}
0
10k
0
+
-
OUT
{Vref}
{Gpwm}
V2
{Vin}
1 2L2
{L}
C2{C}
R2{R}
Duty_Cycle
a c
d p
SW2Out3
Verr
Duty_Cycle
V61Vac
0.5Vdc
Vinjector
Vinjector
10k
)V
V(20.logT(s)
inj
err10=
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Advanced Simulation Techniques
Voltage Injection
Gain-40db/decade
0
-180
H(s) T(s)
+
+180
0
-20db/decade
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Advanced Simulation Techniques
Method#1
injectorV
errorV
db{R_Comp}
C6
{C_Comp}
0
10k
0
+
-
OUT
{Vref}
{Gpwm}
V3
{Vin}
1 2L3
{L}
C3{C}
R3{R}
Duty_Cycle
a c
d p
SW3Out3
Duty_Cycle
Verr
V81Vac
0.5Vdc
Vinjector
Vinjector
10k
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Advanced Simulation Techniques
Method#2
in
out
V
Vdb
Verr
1Vac
Vin
Ac=1
+Duty_cycle
{R_Comp}
C7
{C_Comp}
0
10k
0
+
-
OUT
{Vref}
{Gpwm}
V4
{Vin}
1 2L4
{L}
C4{C}
R4{R}
Duty_Cycle
Out3a c
d p
SW4
10k
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Advanced Simulation Techniques
The Cross Over Frequency: Example
outCf
1
out
ESRC
fout
C
outI
pV
c
c
2
2
Voltage Undershoot
With RHP ZeroSet fc
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Advanced Simulation Techniques
Stability Criteria
0
01
+
Vout
Verr
GH
High DC Gain by G(s) Reduce the Static Error Reduce output
Impedance Improve the Input line
Rejection
Phase Margin=45-80
Gain Margin=10-15db Phase Margn0
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Advanced Simulation Techniques
Stability Criteria
Dc Gain0 Gain
Cross over frequency
0
0 Phase180
When Phase=0 Gain0
Gain=0 Phase0
0
-180
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Advanced Simulation Techniques
Bode Plot for RC Network
R
c
Vout
Vin
0o
o
w
w1tan
ws1
1log20
V
V10
in
out
=
+= )(.
0
-20db/Decade
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Advanced Simulation Techniques
Bode Plot for CR Network
C
R
VinVout
o
o
ws
ws
Vin
Vout
+=1
-
+20db/Decade
+90o
0o
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Advanced Simulation Technique
Right Half Plane Zero
Occurs in Indirect Energy Transfer Topologies
Flyback Boost
Buck Boost
Position Changes with the Duty Cycle
Select a Cross Over Frequency to be of 1/3rd of RHPZ
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Advanced Simulation Techniques
Right Half Plane Zero
)ww(120.log
)w
s(1T(s)
o10
o
=
=
0o
-90o
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Advanced Simulation Techniques
When to Use Type 1 Amplifier
Power Stage Phase Shift is Small
Roll of the Gain far away from the ResonantFrequency
Brings the Largest Overshoot in the Presenceof Sudden Load
Widely Used in PFC Application
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Advanced Simulation TechniquesType 1 Amplifier, No Phase Boost Just dcGain
C
+
-
OUT
U6
OPAMPR
Vin
VVref
2.5
V1Vac
0VdcR
+Vout
11CsR1G(s) =
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Advanced Simulation Techniques
Type 1 Amplifier, The K Factor
0
G
fc
1
db
K=1Log(f)
1c
20
Gf
GRf2
1C
10Gc
=
=
Type 1 Derivation
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Advanced Simulation Techniques
When to Use Type 2 Amplifier
Power Stage Phase Shift Lags Down of -90
Phase Boost of ESR Must be Canceled
Use for Current Mode (CCM) and VoltageMode(DCM)
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Advanced Simulation Techniques
Type 2 Amplifier Can Boost the Phase
+
+
+=
=
+++
+=
10
1
0
1
1
1
)(
pp
z
211P
12z
21
1
2211
12
ss
s
sG
)C(CR
1
CR1
)CC
CCsR)(1C(CsR
CsR1G(s)
2
C
C
R
+
-
OUT
U6
OPAMPR
Vin
VVref
2.5
V1Vac
0VdcR
+Vout
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Advanced Simulation Techniques
Type 2 Amplifier Can Boost the Phase
+
=
21
212
11
CC
CCR
p
But if C2
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Advanced Simulation Techniques
Type 2 Amplifier, The K Factor
0
G
k
k
Log(f)1
0
Type 1 Reference
Pole 1Zero 1
1
=
+
+==
=+
00
0
0
arctanarctan
1
1
arg))(arg(
arctan)arg(
pz
p
z
ff
ff
ff
ff
boostfT
a
bjba
Let us assume we place a zero at frequencyf/k and a pole at a frequency kf
Type 2 Derivation
k
fc kfccf
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Advanced Simulation Techniques
Type 2 Amplifier, The K Factor
+=
+=
+=
=
+
=
452
tan
452
)arctan(
90)arctan(2
)arctan(90)arctan(
90
1
arctan)arctan(
1arctan)arctan(
boostk
boostk
k
kkBoost
xx
kkBoost
Type 2 Derivation
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Advanced Simulation Techniques
Type 2 Amplifier, The K Factor
Type 2 Derivation
12
2
21
12
2
)1(
2
190
18090
Cf
kR
kCC
GkRfC
PSPMBoost
PMBoostPS
c
c
=
=
=
=
+=+
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Advanced Simulation Technique
When to Use Type 3 Amplifier
Power Stage Phase Shift Lags Down to-180 CCM Voltage mode Buck Boost Type
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Advanced Simulation Technique
Type 3 Amplifier Associated with anIntegrator
22p
33p
11p
31z
12z
PPP
ZZ
CR1
CR
1
CR1
CR
1
CR
1
)s)(1s(1s
)s)(1s(1
G(s)
2
1
0
2
1
210
21
=
=
=
=
=
++
++
C
C
R
+
-
OUT
U6
OPAMP
R
R Vin
VVref
2.5
V1Vac
0VdcR
+
C
Vout
AssumingC2
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Advanced Simulation Techniques
Type 3 Amplifier, The K Factor
0
GDoubleZero
k
k
1
1Log(f)
+1
Type 1 reference
Double Pole
2
454
tan
1arctan)arctan(2
1arctan)arctan(
+=
=
=
boostk
kkBoost
kkBoost
Type 3 Derivation
k
fc cf
k
fc
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Advanced Simulation Techniques
Type 3 Amplifier, The K Factor
Type 3 Derivation
3
3
13
12
21
12
2
1
1
2
)1(
2
1
RkfC
k
R
R
Cf
kR
kCC
GRfC
c
c
c
=
=
=
=
=
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Advanced Simulation Techniques
Stabilizing a Voltage-Mode Buck Converterwith k Factor
Generate an Open-Loop Bode Plot
Select a crossover Frequency and a Phase Margin
Read the Bode Plot at the Crossover Frequency
Select the Amplifier type
Apply Formulas
Sweep the Open-Loop Gain with the Above Values
Vary the output Capacitor(s) ESRs Step Load the Output
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Advanced Simulation Techniques
Stability Analysis: Demonstration
Demonstration#24: Buck Converter Compensated