fundamentals of pwm dc-to-dc power conversion dynamic performance of pwm dc-to-dc converters
TRANSCRIPT
![Page 1: Fundamentals of PWM Dc-to-Dc Power Conversion Dynamic Performance of PWM Dc-to-Dc Converters](https://reader035.vdocuments.mx/reader035/viewer/2022081506/56649f1c5503460f94c32bc6/html5/thumbnails/1.jpg)
Fun
dam
enta
ls o
f PW
M D
c-to
-Dc
Pow
er C
onve
rsio
n
Dynamic Performance ofPWM Dc-to-Dc Converters
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2
Performance of PWM Dc-to-Dc Converters
Power
stage
Control
Load
Dc-
to-D
c C
on
vert
er P
erf
orm
anc
e
● Static performance:
● Dynamics performance
Stability Frequency-domain response:
Time-domain response:
![Page 3: Fundamentals of PWM Dc-to-Dc Power Conversion Dynamic Performance of PWM Dc-to-Dc Converters](https://reader035.vdocuments.mx/reader035/viewer/2022081506/56649f1c5503460f94c32bc6/html5/thumbnails/3.jpg)
3
Buck Converter ExampleS
tab
ility
PWM rampV
ref 4.0VV
20 s0V
3.8V
470 F
0.0540 H
16V
0.1
1
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4
Stability of Buck ConverterS
tab
ility 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0
2
4
6
8
2
3
4
5
6
i L(t
) [A
]
Time [ms]
v O(t
) [V
]
Unstable operation
Stable operation
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5
Loop GainS
tab
ility
v ( s )s
i ( s )o
d ( s )
v ( s )o
conv ( s )Fm - Fv(s)
Gvd(s)
Zp(s)
Gvs(s)
Tm
mT ( s )
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6
Stability of Buck ConverterS
tab
ility
0.1 1 10
-180
-150
-120
-90
-60
-30
0 -40
-20
0
20
40
Pha
se [
deg]
Frequency [kHz]
Mag
nitu
de [
dB]
Stable compensation
Stable compensation
-3 -2 -1 0 1 2 3
3
2
1
-1
-2
-3
Stable design
Unit circle
Re
Im
Bode plot of loop gain Polar plot of loop gain
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7
Input-to-Output Frequency Response:
Power
stage
Control
Sv
Input
Ov
Output
● Input-to-output transfer function:
: Laplace transformation of ac component of
: Laplace transformation of ac component of
(s)ov ( )ov t
(s)sv ( )sv t
Fre
que
ncy-
Do
ma
in P
erfo
rma
nce
Cri
teri
a
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Audio-SusceptibilityT
ime
-Do
ma
in P
erfo
rma
nce
Cri
teri
a
v ( s )s
i ( s )o
d ( s )
v ( s )o
conv ( s )Fm - Fv(s)
Gvd(s)
Zp(s)
Gvs(s)
Tm
0.01 0.1 1 10 100-70
-60
-50
-40
-30
-20
Mag
nitu
de [
dB]
Frequency [kHz]
Audio-susceptibility
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9
Input-to-Output Frequency Response
-60
-40
-20
0
20
Ma
gn
itud
e[d
B]
Buck converter
with 0.25D Sv
ˆsv
20 Vov
Sv
20
5
ov
Fre
que
ncy-
Do
ma
in P
erfo
rma
nce
Cri
teri
a
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10
Load Current-to-Output Transfer Function:
Power
stage
Control
oi
Loadcurrent
oV
● Load current-to-output transfer function: :
: Laplace transformation of ac component of
: Laplace transformation of ac component of
(s)ov ( )ov t
(s)oi ( )oi t
Fre
que
ncy-
Do
ma
in P
erfo
rma
nce
Cri
teri
a
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11
Output ImpedanceT
ime
-Do
ma
in P
erfo
rma
nce
Cri
teri
a
v ( s )s
i ( s )o
d ( s )
v ( s )o
cv ( s )Fm - Fv(s)
Gvd(s)
Zp(s)
Gvs(s)
Tm
0.01 0.1 1 10 100
-80
-60
-40
-20
0
Mag
nitu
de [
dB]
Frequency [kHz]
Output impedance
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12
Load Current-to-Output Frequency Response
-60
-40
-20
0
20
Buck converter
with 0.5D 10 V
5
ov
1 ˆoi
Tim
e-D
om
ain
Per
form
an
ce C
rite
ria
Output impedance
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13
Step Load Response
● Transient response of the output voltage due to step change in the input voltage
Tim
e-D
om
ain
Per
form
an
ce C
rite
ria
Power
stage
Control
10 A
5 A
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14
Output Impedance and Step Load ResponseT
ime
-Do
ma
in P
erfo
rma
nce
Cri
teri
a
0.0 0.5 1.0 1.5 2.03.6
4.0
4.4
v O(t
) [V
]
Time [ms]
0.01 0.1 1 10 100
-80
-60
-40
-20
0
Mag
nitu
de [
dB]
Frequency [kHz]
Output impedance Step load response
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Step Input ResponseT
ime
-Do
ma
in P
erfo
rma
nce
Cri
teri
a
● Transient response of the output voltage due to step change in load current
Power
stage
Control
24 V
20 V
Control
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16
Audio-Susceptibility and Step Input ResponseT
ime
-Do
ma
in P
erfo
rma
nce
Cri
teri
a
0 1 2 3 4 53.5
4.0
4.5
Time [ms]
v O(t
) [V
]
0.01 0.1 1 10 100-70
-60
-50
-40
-30
-20
Mag
nitu
de [
dB]
Frequency [kHz]
Audio-susceptibility Step input response
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17
Definition of Stability
● Stability of transfer function T(s) or stability of an LTI system having T(s) as its transfer function
2 10 1 1 1
20 1 2
2 2 21 2 1 1 1
( )T(s)=
( )
( ) ( )( ) 2 ( ) ( )( )
( ) 0 :
Roots of (s)=0 :
nn
nn
b b s b s b s N s
D sa a s a s a s
D s s a s a s s
D s
D
● T(s) or LTI system is stable if and only if all the roots of the characteristic equation are located in the left-half plane(LHP) of the s-plane
Sta
bili
ty D
efin
itio
n
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18
Nyquist (Stability) Criterion
● Nyquist(Stability) Criterion: Graphical method to determine the number of RHP roots in 1+Tm(s)=0
N Z P
Im[T( )]j
mRe[T ( )]j( 1,0)
Nyq
uist
Cri
teri
on
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19
Application Example
● 2 1
0 1 1 12
0 1 2
T(s)=
nn
nn
b b s b s b s
a a s a s a s
● Characteristic equation
20 1 2
22
0 1
0
1+ 0
nn
nn
a a s a s a s
a s a s
a a s
Nyq
uist
Cri
teri
on
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20
Stability Analysis of PWM Converters
( )Sv s
( )Oi s
( )d s
( )vsG s
( )pZ s
( )vdG s
mF vF
( )Ov s
( )
( )
O
S
v s
v s
Nyq
uist
Cri
teri
on
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21
Nyquist Analysis on 1+Tm(s)=0Im[T(j )]
mRe[T (j )]( 1,0)
m
m
: number of encirclements of (-1,0) point
: number of RHP roots in 1+T (s) =0
: number of RHP poles in T (s)
N Z P
N
Z
P
Nyq
uist
Cri
teri
on
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Absolute Stability
Im
Re
Im
Re( 1,0) ( 1,0)
Im
Re( 1,0)
Im
Re
( 1,0)
Stable Unstable
Nyq
uist
Cri
teri
on
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23
Stability Analysis Using Bode Plot
mT mTmT
mTmT
mT
0dB0dB 0dB
180 180 180
Nyq
uist
Cri
teri
on
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24
Marginally Stable Buck ConverterM
arg
ina
lly S
tab
le B
uck
Co
nve
rte
r
0.1 1 10
-180
-135
-90
-45
0 -40
-20
0
20
40
Phas
e [d
eg]
Frequency [kHz]
Mag
nitu
de [
dB]
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Bode plot of loop gainPolar plot of loop gain
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25
Marginally Stable Buck ConverterM
arg
ina
lly S
tab
le B
uck
Co
nve
rte
r
0.0 0.5 1.0 1.5 2.0 2.5
3.0
3.5
4.0
4.5
5.0
Time [ms]
Vol
tage
[V
]
3
3
2 2 2 10 1 180 1
1 2 2 10 0
m c m c m
m
T ( j ) T ( j f ) T ( j )
T ( j )
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26
Conditionally Stable SystemM
arg
ina
lly S
tab
le B
uck
Co
nve
rte
r Re
Iv
Ov
Region A
Im
I oRegion A: Large slope of v -v curve Largegain Stable
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27
Effect of Gain and Phase Delay
m( )
T (s)( )( )
mK
s
Re
lativ
e S
tab
ility
Im
Re
Larger gain
More phase delay
Im
Re
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28
Gain Margin
● Gain margin : the amount of gain increase that can be added to before the system becomes unstable,
mT
mT
Re
Im
(-1,0)6dB
mT
0dB
Sta
bili
ty M
arg
ins
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29
Phase Margin
mT
Re
Im
mT
0dB
( 1,0)
PM
● Phase margin : the amount of phase delay that can be added to before the system becomes unstable,
mT
Sta
bili
ty M
arg
ins
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30
Gain Margin and Phase Margin
( 1,0)
( K,0) Re
Im
-180
| |mT
mT
0 dB
Sta
bili
ty M
arg
ins
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31
Stability Margins and Closed-Loop Performance
mPolar plot of T (s) mLocation of roots in 1+T (s) 0
mIm[T ]
mRe[T ]
j
Smallstabilitymargins
Nearness toimaginary axis
Sta
bili
ty M
arg
ins
and
Clo
sed
-Loo
p P
erf
orm
an
ce
● Proximity to (-1,0) point
Small stability margin
Nearness of poles to imaginary axis:
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32
Buck Converter ExampleS
tab
ility
Ma
rgin
s an
d C
lose
d-L
oop
Pe
rfo
rma
nce
0.1 1 10 100
0.1 1 10 100
0.1 1 10 100
-180
-135
-90
-45
0
-40
-20
0
20
40
Pha
se [
deg]
Frequency [kHz]
Mag
nitu
de [
dB]
0.1 1 10 100
0.1 1 10 100
°PM = 60
°0
°0
°60
-2 -1 1 20
1
2
-1
-2
°PM = 60°45
°30
°15
°0
Bode plot of loop gain Polar plot of loop gain
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33
Buck Converter ExampleS
tab
ility
Ma
rgin
s an
d C
lose
d-L
oop
Pe
rfo
rma
nce
0.1 1 10 100
0.1 1 10 100
0.1 1 10 100
-180
-135
-90
-45
0
-40
-20
0
20
40
Phas
e [d
eg]
Frequency [kHz]
Mag
nitu
de [
dB]
0.1 1 10 100
0.1 1 10 100
°PM = 60
°0
°0
°60
°PM = 60
0
45
30
15
Bode plot of loop gain
0.1 1 10 100-80
-60
-40
-20
0
20
Mag
nitu
de [
dB]
Frequency [kHz]
°PM = 60
°45°35
°15
°0
Output impedance
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34
Buck Converter Example: Step Load Response
2.0 2.5 3.0 3.5
3.8
4.0
4.2
3.8
4.0
4.2
3.8
4.0
4.2
3.8
4.0
4.2
3.8
4.0
4.2
v(t)
[V
]
Time [ms]
v(t)
[V
]
v(
t) [
V]
v(t)
[V
]
v(
t)[V
]
Sta
bili
ty M
arg
ins
and
Clo
sed
-Loo
p P
erf
orm
an
ce