Transcript Introduction to Converter Sampled-Data Modeling ECEN 5807 Dragan Maksimović 1

Introduction to
Converter Sampled-Data Modeling
ECEN 5807 Dragan Maksimović
ECEN5807 Intro to Converter Sampled-Data Modeling
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Objectives
• Better understanding of converter small-signal
dynamics, especially at high frequencies
• Applications
– DCM high-frequency modeling
– Current mode control
– Digital control
ECEN5807 Intro to Converter Sampled-Data Modeling
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Example: A/D and D/A conversion
v(t)
A/D
v*(t)
Analog-todigital converter
D/A
vo(t)
Digital-toanalog converter
v(t)
t
v*(t)
t
vo(t)
t
nT
ECEN5807 Intro to Converter Sampled-Data Modeling
(n+1)T (n+2)T
T = sampling period
1/T = sampling frequency
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Modeling objectives
• Relationships: v to v* to vo
– Time domain: v(t) to v*(t) to vo(t)
– Frequency domain: v(s) to v*(s) to vo(s)
v(t)
t
v*(t)
t
vo(t)
t
nT
ECEN5807 Intro to Converter Sampled-Data Modeling
(n+1)T (n+2)T
T = sampling period
1/T = sampling frequency
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Model
v(t)
A/D
v*(t)
Analog-todigital converter
v(t)
ECEN5807 Intro to Converter Sampled-Data Modeling
D/A
vo(t)
Digital-toanalog converter
v*(t)
T
H
Sampler
Zero-order hold
vo(t)
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Sampling
v(t)
v*(t)
T
Sampler
v(t)
t
v*(t)
t

v * (t )  v(t )  (t  nT )

Unit impulse (Dirac)
ECEN5807 Intro to Converter Sampled-Data Modeling
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Unit impulse
(t)
s (t )   (t )
area = 1
s(t)
t
Dt

Properties
  (t )dt  1
Laplace transform



 v(t ) (t  t )dt  v(t )
s

s
 st

(
t
)
e
dt  1


t
  ( )d  h(t )

unit step
ECEN5807 Intro to Converter Sampled-Data Modeling
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Sampling in frequency domain

v * (t )  v(t )  (t  nT )


v( s )   v(t )e  st dt


v * ( s )   v * (t )e  st dt

1 
v * ( s )   v( s  jks )
T k  
ECEN5807 Intro to Converter Sampled-Data Modeling
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Sampling in frequency domain: derivation

v * (t )  v(t )  (t  nT )


v * ( s )   v * (t )e  st dt


  (t  nT ) 


jk s t
C
e
 k
k  
s 
2
 2f s
T
1
1
 n
  jkst
Ck 

(
t

nT
)
e
dt




T T/ 2  n
T

T /2
ECEN5807 Intro to Converter Sampled-Data Modeling
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Sampling in frequency domain: derivation
ECEN5807 Intro to Converter Sampled-Data Modeling
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Aliasing
ECEN5807 Intro to Converter Sampled-Data Modeling
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Zero-order hold
v*(t)
H
vo(t)
Zero-order hold
v*(t)
t
vo(t)
t
nT
ECEN5807 Intro to Converter Sampled-Data Modeling
(n+1)T (n+2)T
T = sampling period
1/T = sampling frequency
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Zero-order hold: time domain
(t)
H
vo(t)
Zero-order hold
t
vo (t )    ( )d
t T
ECEN5807 Intro to Converter Sampled-Data Modeling
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Zero-order hold: frequency domain
u(t)
H
vo(t)
Zero-order hold
t
vo (t ) 
 u( )d
t T
1  e  sT
H
s
ECEN5807 Intro to Converter Sampled-Data Modeling
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Sampled-data system example: frequency domain
v*(t)
v(t)
T
Sampler
H
vo(t)
Zero-order hold
1  e  sT
H
s
1 
v * ( s )   v( s  jks )
T k  
1  e  sT
1  e  sT
vo ( s) 
v * ( s) 
s
sT

 v(s  jk )
k  
s
1  e  sT
Consider only low-frequency signals: vo ( s ) 
v( s )
sT
vo 1  e  sT

System “transfer function” =
v
sT
ECEN5807 Intro to Converter Sampled-Data Modeling
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Zero-order hold: frequency responses
jT / 2
 jT / 2
1  e  jT
e

e
1
sin( T / 2)  jT / 2
 e  jT / 2

e
 sinc (T / 2)e  jT / 2
jT
2j
T / 2
T / 2
ECEN5807 Intro to Converter Sampled-Data Modeling
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Zero-order hold: frequency responses
fs = 1 MHz
Zero-Order Hold magnitude and phase responses
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magnitude [db]
0
-20
1  e  sT
H /T 
sT
-40
-60
MATLAB file: zohfr.m
-80
-100
2
10
3
10
4
5
10
10
6
10
7
10
phase [deg]
0
-50
-100
-150
2
10
3
10
ECEN5807 Intro to Converter Sampled-Data Modeling
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5
10
10
frequency [Hz]
6
10
7
10
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Zero-order hold: 1st-order approximation
1
e
 sT

1
 sT
1 e
sT
ECEN5807 Intro to Converter Sampled-Data Modeling

s
p
1st-order Pade approximation
s
p
p 
1
1
s
p
fp 
2
T
f
1
 s
T 
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Zero-order hold: frequency responses
fs = 1 MHz
Zero-Order Hold magnitude and phase responses
20
magnitude [db]
0
-20
-40
-60
MATLAB file: zohfr.m
-80
-100
2
10
3
10
4
5
10
10
6
10
7
10
phase [deg]
0
-50
-100
-150
2
10
3
10
ECEN5807 Intro to Converter Sampled-Data Modeling
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5
10
10
frequency [Hz]
6
10
7
10
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How does any of this apply to converter modeling?
Vg d
L
i
+
vg
+
–
+
–
Di
C
D vg
Id
v
–
d
1
VM
ECEN5807 Intro to Converter Sampled-Data Modeling
R
_
u
Gc
vref
+
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PWM is a small-signal sampler!
u  uˆ
u
d̂Ts
c
ĉ
dˆTs t  t p 
tp
PWM sampling occurs at tp (i.e. at dTs, periodically, in each switching period)
ECEN5807 Intro to Converter Sampled-Data Modeling
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General sampled-data model
Equivalent hold
Gh(s)
d Ts(t nTs), d = u
v
_
u
Ts
Gc(s)
vref
+
• Sampled-data model valid at all frequencies
• Equivalent hold describes the converter small-signal response to the
sampled duty-cycle perturbations [Billy Lau, PESC 1986]
• State-space averaging or averaged-switch models are low-frequency
continuous-time approximations to this sampled-data model
ECEN5807 Intro to Converter Sampled-Data Modeling
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Application to DCM high-frequency modeling
iL
c
dTs
ECEN5807 Intro to Converter Sampled-Data Modeling
d2Ts
Ts
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Application to DCM high-frequency modeling
iL
c
dTs
ECEN5807 Intro to Converter Sampled-Data Modeling
d2Ts
Ts
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DCM inductor current high-frequency response
 sD2Ts
 sD2Ts
V

V
1

e
V

V
1

e
1
iˆL ( s)  1 2 Ts
dˆ * ( s)  1 2 Ts
L
s
L
s
Ts

 dˆ (s  jk )
k  
s
 sD2Ts
V

V
1

e
1
2
iˆL ( s) 
D2Ts
dˆ ( s)
L
D2Ts s
iˆL ( s) V1  V2

D2Ts
ˆ
L
d (s)
f
f2  s
D2
ECEN5807 Intro to Converter Sampled-Data Modeling
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1
s
2
2 
2
D2Ts
High-frequency pole due to the
inductor current dynamics in
DCM, see (11.77) in Section 11.3
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Conclusions
•
•
•
•
•
•
•
•
PWM is a small-signal sampler
Switching converter is a sampled-data system
Duty-cycle perturbations act as a string of impulses
Converter response to the duty-cycle perturbations can be modeled as an
equivalent hold
Averaged small-signal models are low-frequency approximations to the
equivalent hold
In DCM, at high frequencies, the inductor-current dynamic response is
described by an equivalent hold that behaves as zero-order hold of length D2Ts
Approximate continuous-time model based on the DCM sampled-data model
correlates with the analysis of Section 11.3: the same high-frequency pole at
fs/(D2) is obtained
Next: current-mode control (Chapter 12)
ECEN5807 Intro to Converter Sampled-Data Modeling
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