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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
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:
3
Buck Converter ExampleS
tab
ility
PWM rampV
ref 4.0VV
20 s0V
3.8V
470 F
0.0540 H
16V
0.1
1
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
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 )
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
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
8
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
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
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
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
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
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
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
15
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
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
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
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
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
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
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
22
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
23
Stability Analysis Using Bode Plot
mT mTmT
mTmT
mT
0dB0dB 0dB
180 180 180
Nyq
uist
Cri
teri
on
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
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 )
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
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
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
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
30
Gain Margin and Phase Margin
( 1,0)
( K,0) Re
Im
-180
| |mT
mT
0 dB
Sta
bili
ty M
arg
ins
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:
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
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
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
