Chapter 6
Dynamic analysis
of switching converters
Avg switch
a
c
p
L1
L1
C1
Vdc
a) averaged buck converter.
R1
Avg switch
c
p
a
C1
Vdc
b) averaged boost converter.
Step Response
From: U(1)
0.35
0.3
Overview
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2






Switching converter analysis using classical control
techniques
Averaged switching converter models
Review of negative feedback using classical-control
techniques
Feedback compensation
State-space representation of switching converters
Input EMI filters
Power switching converters
3
3.5
4
4.5
-3
Continuous-Time Linear Models

2.5
Time (sec.)
Dynamic analysis of switching converters
2
x 10
Step Response
From: U(1)
0.35
0.3
Overview
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2

Continuous-time and discrete-time domains

Continuous-time state-space model

Discrete-time model of the switching
converter

Design of a discrete control system with
complete state feedback
Power switching converters
3
3.5
4
4.5
-3
Discrete-time models

2.5
Time (sec.)
Dynamic analysis of switching converters
3
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
Dynamic analysis
0.05
0
0
0.5
1

Dynamic or small-signal analysis of the switching
converter enables designers to predict the dynamic
performance of the switching converter to reduce
prototyping cost and design cycle time

Dynamic analysis can be either numerical or
analytical
Power switching converters
Dynamic analysis of switching converters
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
4
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
Dynamic analysis
0.05
0
0
0.5
1

Switching converters are non-linear time-variant
circuits

Nevertheless, it is possible to derive a continuous
time-invariant linear model to represent a switching
converter

Continuous-time models are easier to handle, but
not very accurate

Since a switching converter is a sampled system, a
discrete model gives a higher level of accuracy
Power switching converters
Dynamic analysis of switching converters
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
5
x 10
Step Response
From: U(1)
0.35
0.3
Linear model of a switching
converter
Vref
Vo 
PWM
Switch
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Vo
LPF
Load
(a)
Z
Vo
k
Load
ZL
(b)
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
ZL
k Vref
Zo  Z L
Vref
2.5
Time (sec.)
6
x 10
Step Response
From: U(1)
0.35
0.3
PWM modulator model
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Sensitivity of the duty cycle with respect to vref
Vp
^
D
Vref
v ref
Vp
^
D
d
D
1
v ref  v ref
Vref
Vp
Power switching converters
d
T
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Voltage-mode control
Vref
2.5
Time (sec.)
7
x 10
Step Response
From: U(1)
0.35
0.3
PWM modulator model
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Current- mode control
diL VL I P  I1


dt
L
D
D
( I P  I1 ) L
VL
d ^ d ^ d ^
d
iL 
vc 
Ip
iL
vc
I p
^
Variation of the duty cycle due to a perturbation in the inductor current
r
Vd  Vc
L
^
d T
^
i L( 0)
^
^
i L  r ( d T )
r
^
L
d 
iL
(Vd  Vc ) T
r
^
^
i L( 0)
^
d
i L

L
(Vd  Vc ) T
Power switching converters
^
(D+ d) T
Dynamic analysis of switching converters
4.5
-3
Time (sec.)
D T
8
x 10
Step Response
From: U(1)
0.35
0.3
PWM modulator model
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
Current- mode control
Variation of the duty cycle due to a perturbation in the output voltage
r1  (Vd  Vc ) L
r1  I DT
^
r '  [Vd  (Vc  v c )] L
r '  I ( D  d ) T
(Vd  Vc ) L
r1
I DT


r ' I ( D  d^ ) T [V  (V  v^ )] L
c
d
c
^


^
v
c

d  D
^
V  V  v 
c 
c
 d
^
^ 
D 
d  D 
d  vc 



V

V
V

V

v
c 
c 
 d
 d
Power switching converters
^
^
d T
DT
Ip
r1
I
iL
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Time (sec.)
r’
D T
^
(D d ) T
9
x 10
Step Response
From: U(1)
0.35
0.3
PWM modulator model
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
Current- mode control
Variation of the duty cycle due to a perturbation on the peak current
^
Ip
dT 
r
^
r
^
d T
Vd  Vc
L
^
Ip L
d
T Vd  Vc
^
^
d
1 L

^
 I p T Vd  Vc
Power switching converters
Ip+Îp
Îp
Ip
r
iL
D T
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Time (sec.)
^
(D + d ) T
10
x 10
Step Response
From: U(1)
0.35
0.3
Averaged switching converter
models
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Three-terminal averaged-switch model
v1 
Vap
D
v1
d
i1
ic
c
(common)
1
D
i1  I c d
p
(passive)
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Averaged-switch model for voltage-mode control
a
(active)
2.5
Time (sec.)
11
x 10
Step Response
From: U(1)
0.35
0.3
Averaged switching converter
models
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
L1
L1
C1
Vdc
a) averaged buck converter.
Power switching converters
R1
Avg switch
c
p
a
C1
Vdc
R1
b) averaged boost converter.
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Examples of switching converters with an averaged switch
Avg switch
a
c
p
2.5
Time (sec.)
12
x 10
Step Response
From: U(1)
0.35
0.3
Averaged switching converter
models
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Ia
,
Vac
a
Ia
d,
D
I
I 2  2 p v ac ,
Vac
p
I1  2
I3  2
go 
Ip
D
gi
I2
I1
I3
go
d,
Ip
c
Vcp
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Small-signal averaged-switch model for the discontinuous mode
gi 
2.5
Time (sec.)
13
x 10
Step Response
From: U(1)
0.35
0.3
Averaged switching converter
models
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
^
Vo
Qz 
n 
POWER
STAGE
MODEL
^
Vg
^
IL
2


Ts
Fm 
Power switching converters
^
d
 DTs Ri
L
 D
1   ,
2

(1  D) 2 Ts Ri
kr 
,
2L
and
kf 
Fm
^
Von
+
K’f
1
.
( S n  Sc )Ts
Dynamic analysis of switching converters
+
^
Voff
K’r
+
H e (s)
+
3
3.5
4
4.5
-3
Small-signal model for current-mode control
s
s2
H e ( s)  1 
 2
 nQz  n
2.5
Time (sec.)
Ri
^
Vc
14
x 10
Step Response
From: U(1)
0.35
0.3
Output filter model
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Vo (s) =
1
sC o

1 
sL o +  R o //
sC o 

1
sLo
V s(s) .
2

o
H(s) = 2
,
2
+
2

s
+
s
o o
Power switching converters
+
+
1
sCo
Vs (s)
V o(s)
C o Lo
=
.
s
1
(s)
2
Vs
+
s +
C o R o L oC o
Ro
-
=
Vo (s)
-
1
2R C o
Lo
3
3.5
4
4.5
-3
Output filter of a switching converter
R o //
2.5
Time (sec.)
o=
Dynamic analysis of switching converters
1
.
L oC o
15
x 10
Step Response
From: U(1)
0.35
0.3
Output filter model
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
30
Ro
>
Lo
Co
Magnitude Response (dB)
10
0
-10
Ro =
-20
1 Lo
2 Co
-30
Ro
2 2
2









20 log G ( )  10 log  1      4  
 
    o  
 o  



Power switching converters
<
Lo
Co
-40
-50
0.01
3
3.5
4
4.5
-3
Magnitude response of the output filter for several values of the
output resistance Ro
20
2.5
Time (sec.)
0.1
Dynamic analysis of switching converters
1
10
100
f / fo
16
x 10
Step Response
From: U(1)
0.35
0.3
Output filter model
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
0
-20
Ro =
Phase Response (degree)



 2




o
( )   tan 1 
2 



 1

   
  o 
-60
1 Lo
2 Co
-80
Ro
-100
<
Ro
>
Lo
Co
Lo
Co
-120
-140
-160
-180
0.01
0.1
1
10
100
f / fo
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Phase response of the output filter for several values of the output
resistance Ro
-40
2.5
Time (sec.)
17
x 10
Step Response
From: U(1)
0.35
0.3
Output filter model
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
Output filter with a capacitor Resr
s+
1
H(s) =
R o R esr
C o R esr
.
+
C
L
R
R
R
(
+
)
o
o o esr
L o R o R esr s 2 + o
s+
L oC o( R o + R esr )
L oC o( R o + R esr )
f ESR 
1
2 R e sr Co
sLo
+
+
Vs (s)
1
sCo
Ro
Vo (s)
Resr
-
Power switching converters
3.5
4
4.5
-3
Time (sec.)
Dynamic analysis of switching converters
-
18
x 10
Step Response
From: U(1)
0.35
0.3
Output filter model
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
2 2
2









20 log G ( )  10 log(1   2 2 )  10 log  1      4  

    o  
  o  




20
0
Ro
Magnitude Response (dB)
Magnitude response of
an output filter with a
capacitor having a Resr
for several values of
the output resistance
Ro
-20
Ro =
>
Lo
Co
1 Lo
2 Co
-40
Ro
-60
<
Lo
Co
-80
-100
0.001
0.01
0.1
1
10
fo
Power switching converters
3.5
4
4.5
-3
Time (sec.)
Dynamic analysis of switching converters
f / fo
100
1000
fesr
19
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
Output filter model
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
Phase response of an output filter with a capacitor having a Resr
for several values of the output resistance Ro
0
-20
Ro
>
Lo
Co
Ro
<
Lo
Co
-40
Phase Response (degree)


f
 2

 f 
fo 
1 
-1
=
tan
 LC
 .
2  tan 

 f esr 
 1  f  
 f  
  o 
Ro
o 
LoCo ( Ro  R e sr )
-60
Ro =
1 Lo
2 Co
-80
-100
-120
 Lo

 Co R e sr  o
 Ro
 2
 
  Co R e sr
-140
-160
-180
0.001
0.01
0.1
1
100
10
fo
1000
fesr
f / fo
Power switching converters
4
4.5
-3
Time (sec.)
Dynamic analysis of switching converters
20
x 10
Step Response
From: U(1)
0.35
0.3
Example 6.4

To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
3
3.5
4
4.5
-3
The boost converter shown in Figure 2.10 has the
following parameters: Vin = 10 V, Vo = 20 V, fs = 1
kHz, L = 10 mH, C = 100 µF and RL = 20 Ω. The
reference voltage is 5 V. The converter operates in
the continuous-conduction mode under the voltagemode. Using (a) the averaged-switch model,
calculate the output-to-control transfer function, and
(b) Matlab to draw the Bode plot of the transfer
function found in (a) .
Power switching converters
2.5
Time (sec.)
Dynamic analysis of switching converters
21
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
Example 6.4
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
(a)
The nominal duty cycle can be calculated as
x 10
V0
V
V
1

 1 D  d  D  1 d
Vd 1  D
V0
V0
for the given input and output voltages, we have D=0.5.
Small-signal model of the boost converter
L
V
I C  (1  D)I 0  (1  D) 0
R0
10mH
v0

d


1  D 2 s 2

 ωM
1  D

LC
LC
1  D 2
s
ξ
Power switching converters
L
R 1  D 
2
Vap 
d
D

 1

L
1
C 2R 1  D 
i1-i2
v o
passive
D
i1
Vap D - 1  sLI C
+ v2 -
Ic
Vap  V0

i2
common
1
+ v1 -
R
Icd
C
20
100uF
active
0
Dynamic analysis of switching converters
22
Step Response
From: U(1)
0.35
0.3
Example 6.4
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
3
3.5
4
4.5
-3

d
1


vc VP
V
VP VC
 VP  ref  10 V

D
1
D
Power switching converters
2.5
Time (sec.)
Dynamic analysis of switching converters
23
x 10
Step Response
From: U(1)
0.35
0.3
Example 6.4
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
Bode plot of the small-signal transfer function of the boost converter
Bode Diagram
Magnitude (dB)
50
0
-50
-100
0
Phase (deg)
-45
-90
-135
-180
-225
-270 10 1
Power switching converters
3
3.5
4
4.5
-3
Time (sec.)
10
0
10
1
Frequency (rad/sec)
10
2
Dynamic analysis of switching converters
10
3
24
x 10
Step Response
From: U(1)
0.35
Power switching converters
Dynamic analysis of switching converters
To: Y(1)
0.25
Amplitude
Small-signal models of switching
converters
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
25
x 10
Step Response
From: U(1)
0.35
Power switching converters
Dynamic analysis of switching converters
To: Y(1)
0.25
Amplitude
Small-signal models of switching
converters
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
26
x 10
Step Response
From: U(1)
0.35
Power switching converters
Dynamic analysis of switching converters
To: Y(1)
0.25
Amplitude
Small-signal models of switching
converters
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
27
x 10
Step Response
From: U(1)
0.35
Power switching converters
Dynamic analysis of switching converters
To: Y(1)
0.25
Amplitude
Small-signal models of switching
converters
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
28
x 10
Step Response
From: U(1)
0.35
Power switching converters
Dynamic analysis of switching converters
To: Y(1)
0.25
Amplitude
Small-signal models of switching
converters
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
29
x 10
Step Response
From: U(1)
0.35
Power switching converters
Dynamic analysis of switching converters
To: Y(1)
0.25
Amplitude
Small-signal models of switching
converters
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
30
x 10
Step Response
From: U(1)
0.35
Power switching converters
Dynamic analysis of switching converters
To: Y(1)
0.25
Amplitude
Small-signal models of switching
converters
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
31
x 10
Step Response
From: U(1)
0.35
Power switching converters
Dynamic analysis of switching converters
To: Y(1)
0.25
Amplitude
Small-signal models of switching
converters
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
32
x 10
Step Response
From: U(1)
0.35
0.3
Review of negative feedback
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Vo
A (s)
+
ß (s)
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Block diagram representation for a closed-loop system
Vref
2.5
Time (sec.)
33
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
Review of negative feedback
0.05
0
0
0.5

Closed-loop gain
Vo
A

Vref 1   A

Loop gain
TL   A
For TL>>1
Vo
1

Vref 
Stability analysis

  A 1
 A  1 or 

 phase(  )  phase( A)  180


Power switching converters
Dynamic analysis of switching converters
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
34
x 10
Step Response
From: U(1)
0.35
0.3
Relative stability
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
20
0
-50°
-20
-100°
-40
-150°
Phase margin = 40°
Phase Angle
Magnitude (dB)
Gain margin = 8dB
-180°
-60
-200°
-80
-250°
Power switching converters
1.0
f1
3
3.5
4
4.5
-3
Definitions of gain and phase margins
0.2
2.5
Time (sec.)
f'
Dynamic analysis of switching converters
10
20
35
x 10
Step Response
From: U(1)
0.35
0.3
Relative stability
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Magnitude
0°
-20dB /decade
-45°
-40dB /decade
Phase

-90°
Gain
margin
-60dB /decade
-135°
g
fg
0
f p1
fp2
p
fp
f p3
Phase
margin
-180°
-225°
-270°
Power switching converters
3
3.5
4
4.5
-3
Loop gain of a system with three poles
20 log |TL( j)|
2.5
Time (sec.)
Dynamic analysis of switching converters
36
x 10
Step Response
From: U(1)
0.35
Closed-loop switching
converter
Vref
Compensation
PWM
Switch
ß
Power switching converters
(s)
Dynamic analysis of switching converters
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
LPF
Vo
Load
37
x 10
Step Response
From: U(1)
0.35
0.3
Feedback network
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
L1
Qs
+
C2
Dfw
Vs
RL
Va
Z2
_
Base Drive
Circuitry
R2 Va
.
.
-
Rc
Error
Amplifier
Sawtooth
signal
Vcc
CLOCK
Vref
+
.
R2
R1  R2
_
-
. +
Comparator
R2
Z1
Ve
PWM out
Power switching converters
Vsp =
R1 + R2
-
Va '  Vo
R1
Qst
Cc
(a)
Dynamic analysis of switching converters
38
x 10
Step Response
From: U(1)
0.35
0.3
Error amplifier compensation
networks
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
PI Compensation network
1 
 1 
+
R

 2

sC 2  
sC 1 

H(s) =
.
1
1 

+
R1  R 2 +

sC
sC 2 

1
f p=
C2
1
2 R 2C 2
1
2 R 2C 1
C1
R2
 f1
 lag = tan   .
 f p
-1
R1
Vfb
-
8
 f 1
-1
=
 lead tan  
 f z
f z=
1 + sR 2C 1
.
sR 1( C 1 + C 2 + sR 2C 1C 2 )
H(s) =
+
The total phase lag
Power switching converters
 f1
 f 1
-1
+
 .
 tan 
 f z
 f p
 = 270 o - tan -1 
Dynamic analysis of switching converters
4
4.5
-3
Time (sec.)
.
Verror
Vref
39
x 10
Step Response
From: U(1)
0.35
0.3
Error amplifier compensation
networks
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
70
Magnitude response (dB)
50
30
10
0
-10
0.01
0.1
1
10
fz = 854.3
100
1000
10000
fp = 21,832
f (kHz)
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Frequency response of the PI compensation network
-30
0.001
2.5
Time (sec.)
40
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Error amplifier compensation
networks
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
-20
Phase response (degrees)
-30
-40
-50
-60
-70
-80
0.01
0.1
1
10
100
1000
10000
f (kHz)
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Phase response of the PI compensation network
-90
0.001
2.5
Time (sec.)
41
x 10
Step Response
From: U(1)
0.35
0.3
Error amplifier compensation
networks
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
PID Compensation network
H(j ) =
f p1 =
(1+ j R 2C 1 )
(1+ j( R 1 + R 3 )C 3 )
.
- 2 R 2C 1C 2 + j ( C 1 + C 2 ) R1 + j R1 R 3C 3
1
C2
( C1+ C 2 )
f p2 =
2 R 2C 1C 2
2 R 3C 3
R3
f z1 =
1
2 R 2C 1
f z2 =
1
2 ( R 1 + R 3 )C 3
C3
R1
-
8
Vref
R2
K 1=
R1
C1
R2
Vfb
Verror
.
+
R 2( R 1 + R 3 )
=
K2
R1R 3
Power switching converters
Dynamic analysis of switching converters
4.5
-3
Time (sec.)
42
x 10
Step Response
From: U(1)
0.35
0.3
Error amplifier compensation
networks
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Magnitude responce (dB)
K2
+ 20dB / decade
K1
- 20dB / decade
Part of interest
f
f z1
Power switching converters
f z2
f p1
f p2
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Magnitude response of the PID compensation network
- 20dB / decade
2.5
Time (sec.)
43
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Error amplifier compensation
networks
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
20
Magnitude response (dB)
15
10
5
0
-5
-10
0.1
1
10
fz1
fz2
fp1
100
1000
10000
fp2
f (kHz)
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Magnitude response of the PID compensation network
-15
0.01
2.5
Time (sec.)
44
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Error amplifier compensation
networks
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
60
40
Phase response (degrees)
20
0
-20
-40
-60
-80
0.01
0.1
1
10
fz1
fz2
fp1
100
1000
10000
fp2
f (kHz)
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Phase response of the PID compensation network
-100
0.001
2.5
Time (sec.)
45
x 10
Step Response
From: U(1)
0.35
0.3
Error amplifier compensation
networks
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
x 10
Asymptotic approximated magnitude response of the PID
compensation network
2 R 2C 1
=
1
2 ( R1 + R 3 )C 3
 f1 
-1
=
2
 zd
tan 
 R 2C 1 = ( R 1 + R 3 )C 3 .
 f zd 
1
( C 1 + C 2 ) R3
C 1C 2
=
.
C3=
2 R 3C 3 2 R 2C 1C 2 R 2
C1 + C 2
 f1 
 .
 pd = 2 tan 
f
 pd 
-1
Magnitude responce (dB)
1
K2
+ 20dB / decade
- 20dB / decade
-1
Power switching converters
- 20dB / decade
Region of interest
 f1 
 f1 
-1
 = 270 - 2 tan 
 .
 + 2 tan 
f
f
 zd 
 pd 
o
K1
Dynamic analysis of switching converters
f zd
f
f pd
46
Step Response
From: U(1)
0.35
0.3
Compensation in a buck converter
with output capacitor ESR
average output voltage: 5 V
input voltage: 12 V
load resistance RL = 5 Ω
Power switching converters
To: Y(1)
Amplitude
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
Lo
Qs
100uH
Co
100uF
R1
Dfw
R3
100K
RL
Resr
0.5
R2
R4
100K
Z2
Comparator
Z1
Ve
+
Currente
Driver
Vref
+
-
Error
Amplifier
Sawtooth
Oscillator
Pulse-Width Modulator
Dynamic analysis of switching converters
4
4.5
-3
Time (sec.)
Output Filter
Vs
12V
Design the compensation
to shape the closed-loop
magnitude response of the
switching converter to
achieve a -20 dB/decade
roll-off rate at the unitygain crossover frequency
with a sufficient phase
margin for stability
0.25
47
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Compensation in a buck converter
with output capacitor ESR
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
20 [ log10 ( 2.5 / 5) ] = - 6 dB
20 [ log10 ( V s / V p ) ]
fo 
=
5
(100 x106 )(100 x106 )(5  0.5)
 1.517 kHz
2
1
= 3.18 kHz
2 (0.5)100x10 -6

f1, is chosen to be one-fifth of the switching
frequency
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
R4
R3 + R4
Ro
LoCo ( Ro  R e sr )

2
2.5
Time (sec.)
48
x 10
Step Response
From: U(1)
0.35
0.3
Compensation in a buck converter
with output capacitor ESR
Magnitude Response (dB)
Magnitude response of the buck converter
40
open-loop (ABCD)
- 20dB / decade
closed-loop (JKLMNO)
J
20
K
L
error amplifier EFGH
E
F
0
To: Y(1)
Amplitude
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
Compensation
network
G
- 20dB / decade
C
Open loop
gain GH(s)
-20
- 20dB / decade
H
N
D
-40
-60
Power switching converters
O
0.1
1
Dynamic analysis of switching converters
10
f (kHz)
4.5
-3
f1
B
4
x 10
Time (sec.)
- 40dB / decade
M
A
0.25
100
1000
49
Step Response
From: U(1)
0.35

 2
1 
 LC = tan 
 1 
 
 
f 
 f 
fo 
-1
  70.1 .
2  tan 
f
f  
 esr 
 
fo  
 5 
 5 
-1
 = 64.9 .
tan   - tan 
 f z
 f p
-1
Power switching converters
Dynamic analysis of switching converters
To: Y(1)
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
C2=
2 R 2 f
= 774 pF
z
1
2 R 2 f
3
3.5
4
4.5
-3
-1  1 
f"
tan
tan  "  = 64.9 .
f 
1
2.5
Time (sec.)
-1
C1=
 ea = 315  70.1 = 244.9 .
0.25
Amplitude
Compensation in a buck converter
with output capacitor ESR
0.3
= 38 pF .
p
50
x 10
Step Response
From: U(1)
0.35
 f 
 f 
2 tan -1  1  - 2 tan -1  1  = 270 o - 135 o = 135 o .
f 
 f zd 
 pd 

 1 
2  tan -1 f" - tan -1  "   = 135 o .
 f 

C3=
1
2 f
pd
= 0.16  F .
To: Y(1)
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Dynamic analysis of switching converters
2.5
3
3.5
4
4.5
-3
Time (sec.)
( R 1 + R 3 )C 3
=
= 0.13  F .
C1
R2
R3
C 1C 3
R 2 = 5.29 nF .
C2=
[ C 1 - C 3 R3 ]
R2
phase(delay )  360* tdelay * f1
R3
Power switching converters
0.25
Amplitude
Compensation in a buck converter
with no output capacitor ESR
0.3
51
x 10
Step Response
From: U(1)
0.35
Magnitude response (dB)
Magnitude response of the buck converter
50
open-loop ABC
closed-loop HIJKL
error amplifier DEFG 30 H
+ 20dB / decade
0
To: Y(1)
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
A
E
J
f1
B
G
- 20dB / decade
-10
K
- 40dB / decade
- 40dB / decade
-30
-50
C
0.1
1
10
L
100
1000
f (kHz)
Power switching converters
Dynamic analysis of switching converters
4.5
-3
- 20dB / decade
I
4
x 10
Time (sec.)
F
D
10
0.25
Amplitude
Compensation in a buck converter
with no output capacitor ESR
0.3
52
Step Response
From: U(1)
0.35
Linear model of a voltage
regulator including external
perturbances
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
GVref 
vo
vref

io 0
vo
v ref
v DC  0
io 0
Switching
v^ DC
GVDC 
vo
vDC

io 0
vo
v DC
VDC
io 0
output impedance
Gio 
Vo
I o
v ref  0

v DC  0
îo
v ref  0
v ref  0
v ref  0
Io
converter
audio susceptibility
vo
io
Power switching converters
Vref
^
vref
 Zo
v DC  0
Dynamic analysis of switching converters
4.5
-3
vo  GVref vref  GVDC v DC  Gio i o
v DC  0
4
x 10
Time (sec.)
53
Step Response
From: U(1)
0.35
0.3
Output impedance and stability
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
Z of 
0
Output impedance
Z oo
1  A
0.5
1
1.5
2
Vo
k
1
1
1


Zof Zo Z L
Load
ZL
Zoo
 1
1 
Zo  


Z
Z
L 
 of
1
 1
1 
Zo  


Z
Z
L 
 of
1
A
1 Z of
(a)
Z
Vref
Vo
k
Z oo
ß
Z of
Z L Z of  Z o
(s)
ZL
Zo
Zof
(b)
Power switching converters
3
3.5
4
4.5
-3
Z
Vref
2.5
Time (sec.)
Dynamic analysis of switching converters
54
x 10
Step Response
From: U(1)
0.35
0.3
State-space representation of
switching converters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
x1
u1  L x1  x2
L
+

dx
x1 = 1
dt


x1 = C x 2 +

x2 =
u1
x2
R
dx 2
dt

x1 = -
x2 u1
+
L L

-x
x
x2 = 1 + 2 .
C RC
Power switching converters
C
x2
R
-

x= A x+ B u
1
0
1

 x 1
L
x =   A, = 
 , u = [ u1 ] , B =  L  .
 
 x2
1 - 1 
 0
 C
RC 
Dynamic analysis of switching converters
4.5
-3
Review of Linear System Analysis
A simple second-order low-pass circuit

4
x 10
Time (sec.)
55
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
State-space representation of
switching converters
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
1
1
(s +
)
X 1(s) = L
RC
U 1(s) ( s 2 + s + 1 )
RC LC
X (s) = (s I - A )-1 B U(s)
1
1

s
+
1
 RC
L  

 L U(s)
1
s

  0
 X 1(s) 
  
C
.
 (s) =
s
1
2
X 2 
+
s +
RC LC
Power switching converters
1
X 2(s) =
LC
.
U 1(s) ( s 2 + s + 1 )
RC LC
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Review of Linear System Analysis
A simple second-order low-pass circuit
s X (s) = A X (s) + B U(s)
2.5
Time (sec.)
56
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
State-Space Averaging
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)

approximates the switching converter as a
continuous linear system

requires that the effective output filter corner
frequency to be much smaller than the
switching frequency
Power switching converters
Dynamic analysis of switching converters
57
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
State-Space Averaging
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
Procedures for state-space averaging



Step 1: Identify switched models over a switching cycle. Draw
the linear switched circuit model for each state of the switching
converter (e.g., currents through inductors and voltages across
capacitors).
Step 2: Identify state variables of the switching converter. Write
state equations for each switched circuit model using Kirchoff's
voltage and current laws.
Step 3: Perform state-space averaging using the duty cycle as a
weighting factor and combine state equations into a single
averaged state equation. The state-space averaged equation is

x = [ A1d + A2 (1- d)] x + [ B1 d + B 2 (1- d)] u .
Power switching converters
Dynamic analysis of switching converters
58
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
State-Space Averaging
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)

Step 4: Perturb the averaged state equation
to yield steady-state (DC) and dynamic (AC)
terms and eliminate the product of any AC
terms.

Step 5: Draw the linearized equivalent circuit
model.

Step 6: Perform hybrid modeling using a DC
transformer, if desired.
Power switching converters
Dynamic analysis of switching converters
59
x 10
Step Response
From: U(1)
0.35
Qs
x1
To: Y(1)
0.25
Amplitude
State-Space Averaged Model for an
Ideal Buck Converter
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Dfw
L
C
x2
R
-
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
+
u1
2.5
Time (sec.)
60
x 10
Step Response
From: U(1)
0.35

u1  L x1  x2

x1 = C x 2 +
x2
R
0 = L x1+ x2
x1 = C x 2 +
To: Y(1)
0.1
0.05
0
0
0.5
1
1.5
2
x2
R
C
u1
R
x2
(a) dT interval
L
1
+
1
0
  
L   x 1 0 
 x1  = 
   +   [ u1 ]

1   x 2  0 
x   1
 2
 C
RC 

C
R
x2
(b) (1-d)T interval
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
-
x
2.5
Time (sec.)
+
1
0
  
  x1  1 
x
L
1
 = 
   +  L  [ u1 ]

1   x2  
x   1
 2
 0
 C

RC 

0.2
0.15
L
x1


0.25
Amplitude
State-Space Averaged Model for an
Ideal Buck Converter
0.3
61
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
State-Space Averaged Model for an
Ideal Buck Converter
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
1
1
0
0


L
L
A= 
 d+
 (1- d)
1
1
1
1




RC 
RC 
 C
 C
1
0

L
A= 
 .
1 - 1 
 C
RC 
1
d 
0


B =  L  d +   (1- d) =  L  .
 
 
0 
0
 
 0
1
0
  
  x1  d 
x
L
1
 L  [ u1 ] .
 = 
+



1   x2  
 x   1
 2
 0
 C
RC 
Power switching converters

Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Time (sec.)
62
x 10
Step Response
From: U(1)
0.35

x1 = -
x2 d
+ u1
L L
x1
To: Y(1)
0.25
Amplitude
A nonlinear continuous equivalent
circuit of the ideal buck converter
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
+
1
1
x 2 = x1 x2 .
C
RC
u1 d
C
R
x2

u1d = L x1 + x2

x1 = C x 2 +
-
x2
.
R
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
L

2.5
Time (sec.)
63
x 10
Step Response
From: U(1)
0.35
0.3
A linear equivalent circuit of the
ideal buck converter
^
du 10
 
To: Y(1)
Amplitude
0.1
0.05
0
0
0.5
1
1.5
2
2.5

x1
C
Du1
R
x2


d x2 1
1

x10  x1 
x20  x 2
dt
C
RC


-

0=
1
1
x1    x20  D u10    x 2  D u1  d u10
L
L

x2 =
1
1
x
xˆ
( x 10 - 20 ) + ( xˆ 1 - 2 ) .
C
R
C
R
Power switching converters
3.5

1
(- x 20 + Du 10 )
L
ˆ 1 1
dx
= ( x 2  Du1  du10 )
dt L
0=
1
x
( x 10 - 20 )
C
R
Dynamic analysis of switching converters
x 20
=D .
u 10
ˆ 1
dx
ˆ
x 2 = D uˆ 1 + d u 10 - L
.
dt
x 10 =
4
4.5
-3
d x1
1
1
  x20  x 2 
D  d  u10  u1 
dt
L
L

3
Time (sec.)
+
d = D + dˆ .

0.2
0.15
+
-
x1 = x10 + x1 , x2 = x20 + x 2 ,
u1 = u10 + u1 ,
L
0.25
x 20
.
R
64
x 10
Step Response
From: U(1)
0.35
^
du 10
xˆ 1 = C
To: Y(1)
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
x1
C
Du1
R
x2
-


x2 = Du 1 + dˆ u 10 - L x1
x20 + x 2 = - L x1 + D(u10 +u1 ) + dˆ u10 .

x10 + x1 = C x 2 +
x20 + x 2
R
Power switching converters

x1 = C x 2 +
x2
.
R
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
+
d x 2 xˆ 2
+
.
dt
R
2.5
Time (sec.)
+
-
d x2 1
xˆ
= ( xˆ 1 - 2 )
dt
C
R
L
0.25
Amplitude
A linear equivalent circuit of the
ideal buck converter
0.3
65
x 10
Step Response
From: U(1)
0.35
^
_ u10
d
D
L
__
2
D
u1
x 2 x2 D 2
2
D x1   D C  
D D R
Power switching converters
To: Y(1)
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
+
-
D2 C
x
2
__
D
__
R
2
D
-
Dynamic analysis of switching converters
4
4.5
-3
Dx 1
+
3.5
x 10
Time (sec.)
+
-
dˆ
x2 L 
u1 + u10 = + 2 x1 D
D
D D

0.25
Amplitude
A source-reflected linearized
equivalent circuit of the ideal buck
converter
0.3
66
Step Response
From: U(1)
0.35
0.3
^
du 10
To: Y(1)
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
x1
C
x2
R
-
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
+
u1
2.5
Time (sec.)
+
-
1:D
L
Amplitude
A linearized equivalent circuit of
the ideal buck converter using a DC
transformer
0.25
67
x 10
Step Response
From: U(1)
0.35
0.3
State-space averaged model for the
discontinuous-mode buck converter
A  A1 d1  A2 d 2  A3 1  d1  d 2 
L
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
-3
x1
+
B  B1 d1  B 2 d 2  B3 1  d1  d 2 
u1
C
x2
R

x1 = 0
(a) d1 T interval

x
C x2 + 2 = 0 .
R
0 
0

A3  
0 - 1 
RC 

0 
B3 =   .
0 
Power switching converters
L
x1
L
+
C
x2
+
R
C
(b) d 2 T interval
Dynamic analysis of switching converters
4.5
x 10
Time (sec.)
x2
(c) (1 - d1 - d 2 ) T interval
68
R
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
State-space averaged model for the
discontinuous-mode buck converter
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
d1
B=  L  .
 
 0
0 ( d 1+ d 2 )

  
  x 1  d 1 
x
L
1
 = 
   +  L  u1

1   x2  
x   d 1+ d 2
 2
 0
 C

RC 
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
0 ( d 1+ d 2 )



L
A= 

+
1
d
d
1
2


 C
RC 
Power switching converters
2.5
Time (sec.)
69
x 10

x1 = -
( d1+ d 2 )
d1
x2 + u1
L
L
u1 d1   d1  d2  x2

x2 =
From: U(1)
0.3
To: Y(1)
0.25
Amplitude
A nonlinear continuous equivalent
circuit for the discontinuous-mode
buck converter
Step Response
0.35
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
3
+
d1 u 1
(d 1 +d2 ) x 2 +
-
(d 1 +d2 ) x 1
C
R
x2
-
x1 (0)  x1 (T )  0
Dynamic analysis of switching converters
3.5
4
4.5
-3
x1
( d1+ d 2 )
x2
.
x1 C
RC
Power switching converters
2.5
Time (sec.)
70
x 10
Step Response
From: U(1)
0.3
To: Y(1)
0.25
Amplitude
A nonlinear continuous equivalent
circuit for the discontinuous-mode
buck converter
0.35
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
u1 d1   d1  d2  x2
( u 10 + uˆ 1 )( D1 + dˆ 1 ) = ( D1 + dˆ 1 + D 2 + dˆ 2 )( x 20 + xˆ 2 ) ,

d x 2 ( D 1 + dˆ 1 + D 2 + dˆ 2 )
( x + xˆ 2 )
=
( x 10 + xˆ 1 ) - 20
,
dt
C
RC
x2 =
( d1+ d 2 )
x2
.
x1 C
RC
x1 = ( u 1 - x 2 )
d1
2Lf s
Power switching converters
x10 + xˆ 1 = ( u 10 + uˆ 1 - x 20 - xˆ 2 )
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Time (sec.)
( D1 + dˆ 1 )
.
2Lf s
71
x 10
Step Response
From: U(1)
0.35
A linearized equivalent circuit
for the discontinuous-mode
buck converter
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
u^ 1
r1
g1 x^ 2
g2 ^u1
r2
^
j 2d1
C
x2
R
-
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
+
^
j 1 d1
2.5
Time (sec.)
72
x 10
Step Response
From: U(1)
0.35
State-Space Averaged Model for a
Buck Converter with a Capacitor
ESR
Qs
x1
To: Y(1)
Amplitude
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
-
Dfw
Dynamic analysis of switching converters
R
2.5
3
3.5
4
4.5
-3
Time (sec.)
+
+
x2
Resr
Power switching converters
0.25
L
C
u1
0.3
y2
-
73
x 10
Step Response
From: U(1)




x1
x + C x2
x1 = C x 2 + 2 R esr
R
u1 = L x1  x2 + R e sr C x 2
 - R esrR
   L( R + R)
esr
 x1  = 

R
x  
 2 
 C( R esr + R)

1
L( R esr + R)   x 1   
   + L [ u1 ] .
-1
  x2  
 0
C( R esr + R) 
-R
To: Y(1)
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
x 10
Time (sec.)
L
C

0.3
Amplitude
Switched models for the buck
converter with a Resr
0.35
-
u1
+
+
x2
R
Resr
y2
-
(a) dT interval



0 = L x1+ x2 + R esrC x 2 .
 -R R e sr
   L( R + R)
e sr
 x1  = 

R
x  
 2 
 C( R e sr + R)

x1 = C x 2 +

x2 + R esrC x 2
R
x1

L( R e sr + R)   x 1 0 
   +   [ u1 ] .
-1
  x 2  0 
C( R e sr + R) 
-R
L
C
+
+
x2
-
R
Resr
-
(b) (1-d) T interval
Power switching converters
Dynamic analysis of switching converters
y2
74
Step Response
From: U(1)
0.35
 - R esrR
 L(
R esr + R)
A = A1 d  A2 (1  d )  A1  A2 = 
R

 C(
 R esr + R)
To: Y(1)
0.25
Amplitude
Switched models for the buck
converter with a Resr
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
 - R esrR
   L( R + R)
esr
 x1  = 
R
 x  
 2 
 C( R esr + R)
Power switching converters

L( R esr + R) 
 .
-1

C( R esr + R) 
-R
-R

d 
L( R esr + R)   x 1  
   + L [ u1 ]
-1
  x2  
 0
C( R esr + R) 
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
d 
B = B1 d  B2 (1  d ) =  L  .
 
 0

2.5
Time (sec.)
75
x 10
Step Response
From: U(1)
0.35

x1 =
- R esrR
R
d
x1 x2 + u1
L( R esr + R)
L( R esr + R)
L

x2 =
R
C( R esr + R)

d u 1 = L x1+
x1 -
x2
C( R esr + R)
x1
u1 d
R
R esrR
x1 +
x2
( R esr + R)
( R esr + R)
To: Y(1)
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
+
x2
-
R
Resr

Power switching converters
y2 =
4
4.5
x 10
y2
-
R x2
R esrR x1 + R x 2 = (
R esr //R)x 1 +
( R esr + R) ( R esr + R)
( R esr + R)
Dynamic analysis of switching converters
3.5
+

( x 2 + R esrC x 2 )
.
x1 = C x 2 +
R
3
-3
d u 1 = L x1 + y 2

2.5
Time (sec.)
L
C
.
0.25
Amplitude
A nonlinear continuous
equivalent circuit for the buck
converter with a Resr
0.3
76
A linearized continuous
equivalent circuit for the buck
converter with a Resr
ˆ u 10 + uˆ 1 )
d x1 - R esrR( x 10 + xˆ 1 ) R( x 20 + xˆ 2 ) (D+ d)(
=
+
dt
L( R esr + R)
L( R esr + R)
L
L
From: U(1)
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Du1
0=
R
C( R esr + R)
x10 -
x 20
.
C( R esr + R)
Power switching converters
4
4.5
x 10
+
+
x2
-
R
d x1
- R esrR
R
D uˆ 1
=
xˆ 1 xˆ 2 +
dt
L( R esr + R)
L( R esr + R)
L
dˆ u 10
+
L
d x2
R
xˆ 2
=
.
xˆ 1 dt
C( R esr + R)
C( R esr + R)
Dynamic analysis of switching converters
y2
-
The AC terms are
- R esrR
R
Du 10
x10 x 20 +
L( R esr + R)
L( R esr + R)
L
3.5
-3
Resr
0=
3
x1
C
The DC terms are
2.5
Time (sec.)
+
-
d x 2 R( x 10 + xˆ 1 ) ( x 20 + xˆ 2 )
=
.
dt C( R esr + R) C( R esr + R)
^
du10
Step Response
0.35
77
A linearized equivalent circuit
using DC transformer with a
turns-ratio of D
^
du 1
To: Y(1)
Amplitude
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Resr

Du 1 + dˆ u 10 = L x1+ ( R esr //R)x 1 +
R
y2
-

R x2
= L x1+ y 2
( R esr + R)
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
R
1
x1
x2 .
R esr + R
R esr + R
Power switching converters
2.5
Time (sec.)
+
+
x2
-
u1
C x2 =
0.25
x1
C

From: U(1)
0.3
+
-
1:D
L
Step Response
0.35
78
x 10
Step Response
From: U(1)
0.35
L
x1
To: Y(1)
0.25
Amplitude
State-Space Averaged Model for
an Ideal Boost Converter
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Dfw
Qs
C
x2
R
u2
-
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
+
u1
2.5
Time (sec.)
79
x 10
Step Response
From: U(1)
0.35
State-Space Averaged Model
for an Ideal Boost Converter

u 1 = L x1

u 2 = C x2 +
1
0
   0
 x 1  L

 x1  = 0
1   +
 x    x2  0
 2 
RC 


u 1 = L x1 + x 2
x2
R
L
0
  u 1
.

1  u 2 
C 
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
x1
+
C
u1
x2
R
u2
R
u2
-

(a) dT interval
x2
x1 + u 2 = C x2 +
R
L
x1
+
1
0
1
- 
  
L  x 1  L
 x1  = 
+

1   x 2   0
 x   1
 2
 C

RC 
o
  u 1
.

1  u 2 
C 
u1
C
x2
(b) (1-d) T interval
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Time (sec.)
80
x 10
Step Response
From: U(1)
0.35
1
0
0
0

L

A = A1 d  A2 (1  d ) = 0
d
+

 (1- d)
 - 1 
1
1


RC 

RC 
 C
0


A= 
 (1- d)
 C
1
L
B = B1 d  B2 (1  d ) = 
0

1
L
B= 
0

0

  
 x1  = 
 x   (1- d)
 2
 C
Power switching converters
0
1

L
 d+
1
0

C 
-(1- d) 
1

 x 1  L
L
+

1   x 2   0

RC 
0

 (1- d)
1
C 
0
  u 1

1  u 2 
C 
Dynamic analysis of switching converters
To: Y(1)
0.25
Amplitude
State-Space Averaged Model for
an Ideal Boost Converter
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
-(1- d) 
L 
 .
1 
RC 
0

 .
1
C 
-(1- d)x 2   u 1 

  
  L
L
 x1  = 
+  .
 x   (1- d)x 1 x 2   u 2 
 2
RC   C 
 C
81
x 10
Nonlinear continuous
equivalent circuit of the ideal
boost converter
-(1- d) x 2 1
x1 =
+ u1
L
L
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
u1
___
1- d
C
___
1- d
3
3.5
x2
2
___
u
1- d
R (1- d)
C 
u2
x2
+ x1 =
x2 +
.
(1- d)
(1- d)
R(1- d)
Dynamic analysis of switching converters
4
4.5
-3
-
(1- d)x 1 x 2 u 2
+
C
RC C
Power switching converters
2.5
Time (sec.)
+
L
___
1- d
L 
u1
=
x1 + x 2
(1- d) (1- d)
x2 =
From: U(1)
0.3
x1


Step Response
0.35
82
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Linearized equivalent circuit of
the ideal boost converter
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
ˆ
d x 2 (1- D - d)
1
1
=
( x 10 + xˆ 1 ) ( x 20 + xˆ 2 ) + uˆ 2 ,
dt
C
RC
C
d x1 -(1- D)
dˆ
1
1 D
1
=
x20  u10
xˆ 2 + x 20 + uˆ 
dt
L
L
L
L
L
d x 2 (1- D)
dˆ
1
1
(1- D)
1
=
xˆ 1 - x 10 xˆ 2 + uˆ 2 +
x 10 x 20 .
dt
C
C
RC
C
C
RC
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
ˆ
d x1 -(1- D - d)
1
=
( x 20 + xˆ 2 ) + ( u 10 + uˆ 1 )
dt
L
L
Power switching converters
2.5
Time (sec.)
83
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Linearized equivalent circuit of
the ideal boost converter
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
0=
(1- D)
u 10
x 20 +
L
L
(1- D)
1
x 10 x 20 .
C
RC
1
x 20
=
u 10 (1- D)
x 20
x 10 = R
(1- D)
x 10 =
u 10
,
2
R(1- D )
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
DC solutions
0= -
2.5
Time (sec.)
84
x 10
Step Response
From: U(1)
0.3
To: Y(1)
0.25
Amplitude
Linearized equivalent circuit
of the ideal boost converter
0.35
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
d x1
= - (1- D)xˆ 2 + dˆ x 20 + uˆ 1
dt
C
d x2
xˆ
= (1- D)xˆ 1 - x 10dˆ - 2 + uˆ 2 .
dt
R
small-signal averaged state-space equation

 0
x
 (1  D)
 L

Power switching converters

(1  D) 
 x20 
1
 L 
L 
d L
x

1 
  x10 

0
 C 
RC 

0   u1 
 
 u 2 
1  
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
AC solutions
L
2.5
Time (sec.)
85
x 10
Step Response
From: U(1)
0.3
To: Y(1)
0.25
Amplitude
Linearized equivalent circuit of
the ideal boost converter
0.35
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
+
-
+
L
___
1-D
u1
___
1-D
C
___
1-D
x2
R (1-D)
^
x10 d
____
1-D
u2
___
1-D
-
L 
1
x 20dˆ
x1 = - x 2 +
+
u1
(1- D)
(1- D) (1- D)
C 
1
1
x10 dˆ
x 2 = x1 x2 +
u2 .
(1- D)
(1- D) R(1- D)
(1- D)
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
x1
^
x20 d
____
1-D
2.5
Time (sec.)
86
x 10
Step Response
From: U(1)
0.35
0.3
Source-reflected linearized
equivalent circuit for the ideal
boost converter
^
x 20 d
L
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
+
-
+
u1
x2 (1-D)
R (1-D)
2
^
x10 d
____
1-D
u2
___
1-D

L x1 = - x 2(1- D) + x 20dˆ + u 1 .

x2 (1- D)
C
x 10dˆ
u2
[
x
(1D)]
=
+
,
2
x
1
2
2
(1- D) R(1- D ) (1- D)
(1- D )
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
x1
C
____
2
(1-D)
2.5
Time (sec.)
87
x 10
Step Response
From: U(1)
0.35
Load-reflected linearized
circuit for the ideal boost
converter
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
u1
___
1-D
+
L
____
2
(1-D)
x1 (1-D)
C
x2
R
x 10 d^
u2
-

C x 2 = x 1(1- D) -
x2
- x 10dˆ + u 2 .
R

L
u1
x 20dˆ
x (1- D) = - x 2 +
+
.
2 1
(1- D) (1- D)
(1- D )
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
+
-
^
x20 d
____
1-D
2.5
Time (sec.)
88
x 10
Step Response
From: U(1)
0.35
0.3
DC transformer equivalent
circuit for the ideal boost
converter
x1
^
x20 d
+
-
u1
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
+
x2
C
R
x 10 d^
u2
-
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
L
(1-D) : 1
2.5
Time (sec.)
89
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Switching Converter Transfer
Functions
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
ˆ
s X(s) = A0 X(s) + B0U(s) + Ed(s)
x  A0 x  B 0 u  E d
A0 = [ A1D + A2(1- D)]
ˆ
(sI - A0 )X(s) = B0 U(s) + E d(s)
B0 = [ B1D + B 2(1- D)]
E = ( A1 - A2 )x0 + ( B1 - B 2 )u 0 .
Power switching converters
-1
-1
ˆ .
X (s) = (s I - A0 ) B 0U(s) + (s I - A0 ) E d(s)
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Source-to-State Transfer Functions

2.5
Time (sec.)
90
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Switching Converter Transfer
Functions
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
Source-to-State Transfer Functions
linearized control law
ˆ = T (s)X (s) + QT (s)U(s)
d(s)
F
T
-1
X (s)= (sI - A0 ) ( B 0U(s)+ E[ F T(s)X (s)+ Q (s)U(s)])
X (s)
T
= [ sI - A0 - E F T (s) ] -1( B 0 + E Q (s))
U(s)
T
X (s) = [ sI - A0 - E F T (s) ] -1[ B 0 + E Q (s)]U(s) .
Power switching converters
Dynamic analysis of switching converters
3.5
4
4.5
-3
Time (sec.)
91
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Switching Converter Transfer
Functions
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
1
0
- 
D
 u 10 
  
 x1  
x
L
1
 = 
   + L [ u 1 ] +  L  dˆ .

 
1   x2  
x   1
0
 2
 
 0
RC 
 C
d(s) =
Power switching converters
 u 10 
E=  L  .
 
 0
V e(s) [1 + H(s)]V R(s) - H(s) X 2(s)
=
VP
VP


F (s) = 0

T
D
 
B0 =  L 
 0

-H(s) 

VP 
ˆ = - H(s) Xˆ 2(s) .
d(s)
VP
Q (s) = 0 0  .
T
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
BUCK CONVERTER
1
0

L
=

A0 
1 - 1 
 C
RC 
2.5
Time (sec.)
92
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Switching Converter Transfer
Functions
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
1 u 10H(s) 
+

L
LV P 
1 
s+
RC 
-1
D
 L  [ U 1(s)]
 
 0
1

u 10H(s) + 1 )
s
+
-(
D
 RC
L  
LV
P

 L [ U 1(s)]
1
s  0

  
 X 1(s) 
C
.
 (s) =
s
1
H(s)
u
10
2
X
2


+
+
s +
RC LC LCV P
Power switching converters
D
1
[s +
]
X 1(s) =
L
RC
U 1(s) S 2 + s + 1 + u 10H(s)
RC LC LCV P
D
X 2(s) =
LC
.
s
1
H(s)
u
(s)
10
2
U1
+
+
S +
RC LC LCV P
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
BUCK CONVERTER
 s
 X 1(s) 
 (s) = 
 X 2  - 1
 C
2.5
Time (sec.)
93
x 10
Step Response
From: U(1)
0.35
0.3
Switching Converter Transfer
Functions
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
x 10
BOOST CONVERTER
0

  
 x1  = 
 x   (1- D)
 2
 C
-(1- D) 
1
L   x 1  L
+

1   x 2   0

RC 
0
 x 20 
  u 1  L 
+

 dˆ .
1  u 2   x 10 
 C 
C 
s
(1- D) x 20H(s) 

+
L
 X 1(s) 
LV P 

=
 (s) -(1- D)
1
x 10H(s) 
X 2  
(s
+
)
 C
RC
CV P 

Power switching converters
-1
 1 o
L
 U 1(s)


 .
o
1
(s)
U

 2 

C 
0


A0 = 
 (1- D)
 C
-(1- D) 
L 

1 
RC 
1
L
B0 = 
0

0


1
C 
s
(1- D) x 20H(s) 

+
L
 X 1(s) 
LV P 

=
 (s) -(1- D)
1
x 10H(s) 
X 2  
(s
+
)
 C
RC
CV P 

Dynamic analysis of switching converters
-1
 x 20 
 L 
E=
 .
- x 10 
 C 
 1 o
L
 U 1(s)


 .
o
1
(s)
U

 2 

C 
94
Step Response
From: U(1)
0.35
0.3
Switching Converter Transfer
Functions
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
x 10
Time (sec.)
BOOST CONVERTER
1
1 x 10H(s)
[s +
]
L
RC CV P
X 1(s) =
.
x
10H(s)R
U 1(s)
s[1]
(1- D )2 x 20H(S)(1- D)
V
2
P
+
+
s +
RC
LC
LCV P
-(1- D)
H(s)
[1+ u 10
]
2
LC
X 1(s) =
V P(1- D )
.
2
s
H(s)
(1D
H(s)
)
U 2(s)
u
u
2
[1- 10
]+
[1+ 10
]
s +
2
2
RC
LC
V P(1- D )
V P(1- D )
Power switching converters
1- D
X 2(s) =
LC
.
s
(1- D )2
U 1(s)
u
u
10H(s)
10H(s)
2
[1]+
[1+
]
s +
2
2
RC
LC
V P(1- D )
V P(1- D )
X 2(s) =
U 2(s)
s
C
2
s
u 10H(s) ] + (1- D ) [1+ u 10H(s) ]
+
[1s
2
2
RC
LC
V P(1- D )
V P(1- D )
2
Dynamic analysis of switching converters
95
.
Step Response
From: U(1)
0.35
0.3
Complete state feedback
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
This technique allows us to calculate the
gains of the feedback vector required to place
the closed-loop poles at a desired location

All the states of the converter are sensed and
multiplied by a feedback gain
Dynamic analysis of switching converters
3
3.5
4
4.5
-3

Power switching converters
2.5
Time (sec.)
96
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Design of a control system with
complete state feedback
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
x  A x  B u
u   F  x

x  (A  B F) x
closed-loop poles
det[ s  I  A  B F ]  0
The closed-loop poles can be arbitrarily placed by choosing the
elements of F
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3

control strategy
2.5
Time (sec.)
97
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Design of a control system with
complete state feedback
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
One way of choosing the closed-loop poles is to select an ith
order low-pass Bessel filter for the transfer function, where i is
the order of the system that is being designed
Feedback gains
F  PLACE ( A B P)
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Pole selection
Power switching converters
2.5
Time (sec.)
98
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Design of a control system with
complete state feedback
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2

A buck converter designed to operate in the
continuous conduction mode has the following
parameters: R = 4 Ω, L = 1.330 mH, C = 94 µF, Vs =
42 V, and Va = 12 V.

Calculate (a) the open-loop poles, (b) the feedback
gains to locate the closed loop poles at P = 1000 * {0.3298 + 0.10i -0.3298 - 0.10i}, (c) the closed loop
system matrix ACL.
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Example
Power switching converters
2.5
Time (sec.)
99
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Design of a control system with
complete state feedback
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
1 
 0
0 
L 
A2  
; B2   
1
1 
0 
RC 
 C
A  A1 D  A2  (1  D);
B  B1 D  B 2  (1  D)
1 
 0
D 
L 

A
; B   L
1
1 
 0 
RC 
 C
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Solution
1 
 0
1 
L 

A1 
; B1   L 
1
1 
 0 
RC 
 C
2.5
Time (sec.)
100
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Design of a control system with
complete state feedback
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
polesOL = 1000 * { -1.3298 + 2.4961i, -1.3298 - 2.4961i}

^
^
^
1 ^
x1  ( x 2  D u  d U )
L

^
1 ^ 1 ^
x 2  ( x1  x 2 )
C
R
1   ^   D  ^ U  ^
 0
L   x1   

x
 L u Ld
^
 
1
1     
x
0
RC   2   
 C
0

^
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
polesOL = eig(A)
Power switching converters
2.5
Time (sec.)
101
x 10
Step Response
From: U(1)
0.35
0.3
Design of a control system with
complete state feedback
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
Power switching converters
0.5
1
1.5
Dynamic analysis of switching converters
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
3
3.5
4
4.5
-3
Step response of the linearized buck converter
sysOL=ss(A,B,C,0)
step(sysOL)
2.5
Time (sec.)
x 10
102
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Design of a control system with
complete state feedback
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
design the control strategy

^
^
^
U 
E L
 
0
^
x  A x E d
u0
D ^
d
v ref
Vref
^
for voltage-mode control
^
^
If we apply complete state feedback v ref   F x

^
D
x  A x  E (
F x)
Vref
^
Power switching converters
^

^
^
D
x  (A  E
F) x
Vref
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Time (sec.)
D
ACL  A  E
F
Vref
103
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Design of a control system with
complete state feedback
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
D
, P)
Vref
Then, F = {-2.6600 -0.3202}.
ACL  A  E
D
F
Vref
0.2000 -0.0511
ACL  1e4 

1.0638 -0.2660 
check the locations of the closed loop poles
eig(ACL); which gives
ans = 1e+2 * [ -3.2980 + 1.0000i -3.2980 - 1.0000i]
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
we calculate the feedback gains as
P=1000 *[-0.3298 + 0.10i -0.3298 - 0.10i]'
F  place( A, E
2.5
Time (sec.)
104
x 10
Step Response
From: U(1)
0.35
PSpice schematic
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
L1
0.05
out
pwm
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
-
S
+
S1
+
-
Time (sec.)
V1 = 42
V2 = 44
TD = 5m
TR = 1n
TF = 1n
PW = 1
PER = 1
V2
1.33mH
IC = 0
VON = 1.0V
VOFF = 0.0V
ROFF = 1e6
RON = 0.05
R1
C1
D1
Dbreak
4
94uF
IC = 0
E1 GAIN = 1
+
if ( V(%IN1)>V(%IN2),0,1)
0
pwm_out
PWM modulator
x2
+
-
E
control
PARAMETERS:
0
loop = 0
saw
x1_
12
V1 = 0
V2 = 10
TD = 0
TR = 99.9u
TF = 1n
PW = 1n
PER = 0.1m
V4
{loop}
-0.3202
0
3
-2.6600
Vref
x1
I(L1)
x2_
3
Power switching converters
Dynamic analysis of switching converters
105
x 10
Transient response of the
open-loop and closed-loop
converters
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Open loop
13V
12V
11V
Closed loop
10V
9V
8V
7V
6V
5V
4V
3V
2V
1V
0V
0s
0.5ms
V(OUT)
1.5ms
2.5ms
3.5ms
4.5ms
5.5ms
6.5ms
7.5ms
8.5ms
9.5ms
Time
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
15V
14V
2.5
Time (sec.)
106
x 10
Step Response
From: U(1)
0.3
To: Y(1)
0.25
Amplitude
Expanded view of the transient
at 5 ms
0.35
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Open loop
12.000V
Closed loop
5.00ms
V(OUT)
5.50ms
6.00ms
6.50ms
7.00ms
7.50ms
8.00ms
8.50ms
Time
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
12.602V
11.104V
4.61ms
2.5
Time (sec.)
107
x 10
Step Response
From: U(1)
0.35
0.3
Input EMI filters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
An input EMI filter placed between the power source
and the switching converter is often required to
preserve the integrity of the power source

The major purpose of the input EMI filter is to
prevent the input current waveform of the switching
converter from interfering with the power source

As such, the major role of the input EMI filter is to
optimize the mismatch between the power source
and switching converter impedances
Dynamic analysis of switching converters
3
3.5
4
4.5
-3

Power switching converters
2.5
Time (sec.)
108
x 10
Step Response
From: U(1)
0.35
0.3
Input EMI filters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
LI
Lo
Re
1:D
u1
CI
Input EMI Filter
Power switching converters
3
3.5
4
4.5
-3
Circuit model of a buck converter with an input EMI filter
Rs
2.5
Time (sec.)
Co
RL
Buck Converter
Dynamic analysis of switching converters
109
x 10
Step Response
From: U(1)
0.35
0.3
Input EMI filters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2

The stability of a closed-loop switching
converter with an input EMI filter can be
found by comparing the output impedance of
the input EMI filter to the input impedance of
the switching converter

The closed-loop switching converter exhibits
a negative input impedance
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Stability Considerations
Power switching converters
2.5
Time (sec.)
110
x 10
Step Response
From: U(1)
0.35
0.3
Input EMI filters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
Output impedance of the EMI filter
Z EMI =
R s + j L I
.
2
(1 -  C I L I ) + j R sC I
0.05
0
0
0.5
1
1.5
2
2.5
3
2
RL
R LC o
+
]
+
j

[
] ),
Z in = _ 2 ( [
R
L
e
o
2
2
1+(  R LC o )
1+(  R LC o )
D
Zin
At the resonant frequency
RL + RE
D2
Re
Z in = - 2 .
D
Z in = 2  f L
2
D
Above the resonant frequency
Z in = -
j L
D
2
Re
.
D2
f
1
2  RL Co
Power switching converters
Dynamic analysis of switching converters
4
4.5
-3
Input impedance versus frequency for a buck converter
1
3.5
x 10
Time (sec.)
1
2  Lo Co
111
Step Response
From: U(1)
0.35
0.3
Input EMI filters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
The maximum output impedance of the input EMI filter, ZEMI,max,
must be less than the magnitude of the input impedance of the
switching converter to avoid instability
Z in  Z EMI,max .


The switching converter negative input impedance in
combination with the input EMI filter can under certain conditions
constitute a negative resistance oscillator
Zin // Z EMI
To ensure stability, however, the poles of
should lie in the left-hand plane
Power switching converters
3
3.5
4
4.5
-3
Stability Considerations

2.5
Time (sec.)
Dynamic analysis of switching converters
112
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
Input EMI filters
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2

A resistance in series with the input EMI filter inductor can be
added to improve stability

However, it is undesirable to increase the series resistance of the
input EMI filter to improve stability since it increases conduction
losses
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Stability Considerations
Power switching converters
2.5
Time (sec.)
113
x 10
Step Response
From: U(1)
0.35
0.3
Input EMI filters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Rd
LI
u1
Power switching converters
3
3.5
4
4.5
-3
Input EMI filter with LR reactive damping
Ld
2.5
Time (sec.)
CI
Dynamic analysis of switching converters
-RL
114
x 10
Step Response
From: U(1)
0.35
0.3
Input EMI filters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
3
3.5
4
4.5
-3
Input EMI filter with RC reactive damping
LI
Rd
CI
u1
-RL
Cd
Power switching converters
2.5
Time (sec.)
Dynamic analysis of switching converters
115
x 10
Step Response
From: U(1)
0.35
0.3
Input EMI filters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
It should be noted that high core losses in the input EMI filter
inductor is desirable to dissipate the energy at the EMI frequency
so as to prevent it from being reflected back to the power source
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Stability Considerations

2.5
Time (sec.)
116
x 10
Step Response
From: U(1)
0.35
0.3
Input EMI filters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
Rd
Ld
u1
Power switching converters
3
3.5
4
4.5
-3
A fourth-order input EMI filter with LR reactive damping
LI 2
LlI
2.5
Time (sec.)
CI1
Dynamic analysis of switching converters
CI2
-RL
117
x 10
Step Response
From: U(1)
0.35
0.3
Input EMI filters
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
Input impedance, Zin(f), of the buck converter and output
impedance, ZEMI(f), of the input EMI filter
Z (  ) =
1
D
2
2
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
x 10
2
2



RL
R
LC o
2
+
+


,
R
L
e

 o
2
2
1+(

1+(

)
)
C
C
R
R
L
L
o
o




50
| Z in (f) |
30
+(  L I )2
R
.
Z EMI (  ) =
2
2
2
1-  C I L I  +(  R sC I )
2
s
Magnitude (dB)
10
0
-10
| Z EMI (f) |
-30
-50
-70
0.01
Power switching converters
0.1
Dynamic analysis of switching converters
1
10
Frequency (Hz)
100
1000
118
10000
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
Part 2
Discrete-time models
Power switching converters
Dynamic analysis of switching converters
119
x 10
Step Response
From: U(1)
0.35
continuous-time system
To: Y(1)
0.25
Amplitude
Continuous-time and discrete-time
domains
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
3
3.5
4
x  A  x(t )  B  u (t )
The solution for the differential equation
x(t )  e  x(to )   e A(t  )  B  u ( )  d
At
to
e At  I  At  A2  t 2 2 
t
A( t  )
At
1


e

B

u
(

)

d


e

I

A
 B u



to
t
A( t  )
1
e

B

u
(

)

d


I

A

t

I

A
 B u



to
x(t )  e At  x(to )  t  B  u (to )
Dynamic analysis of switching converters
4.5
-3
t
Power switching converters
2.5
Time (sec.)
120
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Continuous-time and discrete-time
domains
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
to  (n  D)  Ts
t  (n  1  D)  Ts
x[(n  1  D)  Ts ]  e ATs  x[(n  D)  Ts ]  Ts  B  u[(n  D)  Ts ]
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
the discrete-time expression
Power switching converters
2.5
Time (sec.)
121
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
Continuous-time state-space model
0.05
0
0
0.5
1
1.5
2
RL
L

C
R
Vd
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Equivalent circuit during ton: A1
Ron
2.5
Time (sec.)
x  A1 x  B1 u
122
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
Continuous-time state-space model
0.05
0
0
0.5
1
1.5
2
Ron
Power switching converters
L
C
R
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Equivalent circuit during toff: A2
RL
2.5
Time (sec.)

x  A2 x  B2 u
123
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
Continuous-time state-space model
0.05
0
0
0.5
1
1.5
2
1 if nTs  t  (n  d n )Ts
d (t )  
0 if (n  d n )Ts  t  (n  1)Ts
d '(t )  1  d (t )
nonlinear model

x   d (t ) A1  d '(t ) A2  x   d (t ) B1  d '(t ) B2  u
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
switching functions
Power switching converters
2.5
Time (sec.)
124
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
Continuous-time state-space model
0.05
0
0
0.5
1
1.5
2
dn  D  d n
d d d
d '  1 d
x  xx
1 if nTs  t  (n  D)Ts
d (t )  
0 if (n  D)Ts  t  (n  1)Ts
sgn(d n  D) if t   (n  D)Ts , (n  d n )Ts 

d (t )  

0 otherwise
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
small-signal model
u  Vs  v s
2.5
Time (sec.)
125
x 10
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
Continuous-time state-space model
0.05
0
0
0.5
1
1.5
2

 

x  d A1  d ' A2 x  d B1  d ' B2 Vs
perturbation in the state vector

x   d  A1  d '  A2  xˆ   d B1  d ' B2  v s    A1  A2  x   B1  B2  Vs  dˆ
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
steady-state equation

2.5
Time (sec.)
126
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Discrete-time model of the switching
converter
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2

x  A2 x  B2 v s  K d n Ts  t   n  D  Ts 
 A1  A2  x  n  D  Ts    B1  B2 Vs
x  n  1 Ts   e A2 D 'Ts e A2 D 'Ts x  n  D  Ts   e A2 D 'Ts K Ts d n 
 n 1Ts


 n  D Ts
e
A2  n 1Ts  
B2 v s d
 n  1 Ts ,  n  1  D  Ts 

x  A1 x
x  n  1  D  Ts   e A1 DTs e A2 D 'Ts x  n  D  Ts   e A1DTs e A2 D 'Ts K Ts d n
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
 n  D  Ts ,  n  1 Ts 
K
2.5
Time (sec.)
127
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Design of a discrete control
system with complete state
feedback
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
u (n)   F  x(n)
x(n  1)  ( A  F  B)  x(n)
det[ z  I  A  F  B ]  0
The closed-loop poles can be arbitrarily placed by choosing the
elements of F
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
x(n  1)  A  x(n)  B  u (n)
Power switching converters
2.5
Time (sec.)
128
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Design of a discrete control
system with complete state
feedback
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
Pole selection
One way of choosing the closed-loop poles is to design a lowpass Bessel filter of the same order
The step response of a Bessel filter has no overshoot, thus it is
suitable for a voltage regulator
The desired filter can then be selected for a step response that
meets a specified settling time
Feedback gains
Power switching converters
L  PLACE ( d d  P)
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Time (sec.)
129
x 10
Design of a discrete control
system with complete state
feedback
Step Response
From: U(1)
0.35
0.3
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Voltage mode control
^
x  n  1 Ts   e
d
^
D
v ref
Vref
A1 DTs
e
A2 D 'Ts
^
x  nTs   e
^
A1 DTs
e
A2 D 'Ts
^
K Ts d n
x[(n  1) Ts ]   x[nTs]   d n
  e A1 DTs e A2 D 'Ts

   K Ts
 K  ( B  B )V
1
2
s

x  n  1 Ts    x  nTs   
Power switching converters
v ref   F x  nTs 

D 
x  n  1 Ts     
F  x  nTs 
Vref 

^
 CL
D
v ref
Vref
Dynamic analysis of switching converters
4.5
-3
Time (sec.)

D 
   
F
Vref 

x  n  1 Ts   CL x nTs 
130
x 10
Step Response
From: U(1)
0.35
0.3
Extended-state model for a
tracking regulator
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2

( a,a,L2)

D/A
(VM, )
-L1
A/D
c
c
 iL [n] 
0
 
 


xd [n]   vc [n]   d  
 ; d   0  L   L1

c

 
a
 a
 xa [n]
ya  L2 xa
y cx
Power switching converters
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Digital tracking system with full-state feedback
Vref
2.5
Time (sec.)
L2 
131
x 10
Step Response
From: U(1)
0.35
0.3
Current mode control
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
Sensitivities of the duty cycle
^
^
dn 
d n ^ d n ^ d n ^
x1 
x2 
Ip
x1
x2
I p
x1  iL
x2  vC
d n ^ d n ^ d n ^
dn 
iL 
vc 
Ip
iL
vc
I p
dn  
^
L
iL
(Vd  Vc ) Ts
D ^
dn 
vc
Vd  Vc
^
^
L
dn 
Ip
(Vd  Vc ) Ts
^
^
1 
^
dn
d
d
; 2  n ; 3  n ; and   [1 2 ]
x1
x2
I p
d n  1
^ 
^
iL
2   ^   3 I p
 
^
v c 
dF
^
^
^
d n  d CM  d F
^
d CM
Power switching converters
Dynamic analysis of switching converters
132
x 10
Step Response
From: U(1)
0.35
0.3
Current mode control
To: Y(1)
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3
Time (sec.)
With complete state feedback
^
^
CM     
^
x[(n  1) Ts ]   x[nTs]   d n


Iˆ p   F  xˆ  nTs 
^
^
^
^
x[(n  1) Ts ]   x[nTs ]   (d CM  d F )
^
^
^
x  n  1 Ts   CM  3  F  x  nTs 
^
x[(n  1) Ts ]   x[nTs]    x[nTs]   3 I p
^
^
^
x[(n  1) Ts ]   CM x[nTs ]   3 I p
Power switching converters
CL  CM  3  F 
x  n  1 Ts   CL x  nTs 
Dynamic analysis of switching converters
133
x 10
Step Response
From: U(1)
0.35
To: Y(1)
0.25
Amplitude
Extended-state model for a tracking
regulator
0.3
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2

(a, a, L2)

D/A
-L1
(current-mode,3)
c
A/D
c
 iL [n] 
xd [n]   vc [n] 
 xa [n]
Power switching converters

d   CM
 a c
0
 
,


L   L1
d



a 
0
Dynamic analysis of switching converters
3
3.5
4
4.5
-3
Digital tracking system with full-state feedback
Vref
2.5
Time (sec.)
L2 
134
x 10