ON Semiconductor
The “PWM Switch” in mode
transitioning SPICE models
PCIM Germany 2005
Christophe Basso - Application Manager
The “PWM Switch” in mode transitioning SPICE models
Agenda
‰ Why average simulations?
‰ What techniques already exist?
‰ The PWM Switch concept
‰ The voltage-mode case
‰ The current-mode case
‰ Checking averaged model’s validity
‰ Conclusion
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2
The “PWM Switch” in mode transitioning SPICE models
Why average simulations?
‰ Unveil open-loop ac response for stabilization purposes
‰ Helps to assess impact of stray elements variations on stability
‰ Can predict transient response with large-signal models
‰ Simulation time is quick as frequency component fades away
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3
The “PWM Switch” in mode transitioning SPICE models
What techniques already exist?
‰ State-Space Averaging (SSA)
‰ Introduced by Slobodan Ćuk in the 80’
‰ Long and painful process
‰ Fails to predict sub-harmonic oscillations
x1
L
x1
L
Vout
Vout
on
off
dx1
1
1
= − x2 + u
dt
L
L
u1
x2
C
R
dx 2
1
1
=
x1 −
x2
dt
Cout
Rload .Cout
x2
C
R
dx1
1
= − x2
dt
L
dx 2
1
1
=
x1 −
x2
dt
Cout
Rload .Cout
Pfffff!
Apply smoothing process
Linearize
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/
The “PWM Switch” in mode transitioning SPICE models
What techniques already exist?
‰ The GSIM concept
‰ Introduced by Sam Ben-Yaakov in the 90’
‰ Easy to derive but not fully invariant (dual inductors converters?)
‰ Fully auto-toggling mode models
‰ Fails to predict sub-harmonic oscillations
Ib
Ic
on
b
Lf
off
Fsw
Ia
c
a
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5
V(a,c)
Vin
Lf
on
a
V(a,b)
Fsw
V(a,b)
V(a,c)
c
off
b
C
R
The “PWM Switch” in mode transitioning SPICE models
What techniques already exist?
‰ The CoPEC model
‰ Introduced by the Colorado Power Electronic Center in the 90’
‰ Easy to derive and fully invariant
‰ Fully auto-toggling mode models
‰ Fails to predict sub-harmonic oscillations
i1(t)
i1(t)
v2(t)
v1(t)
Q1
v2(t)
v1(t)
D1
Q1
D1
i12(t)
CCM
i1(t)
d1(t)
i1(t)
v2(t)
v1(t)
v1(t)
i12(t)
Re(d1)
I2
v2(t)
i12(t)
D':D
I2=(Re x i1^2) / v2
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November 2001
i12(t)
DCM
d(t)
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The “PWM Switch” in mode transitioning SPICE models
What techniques already exist?
‰ The Ridley models
‰ Introduced by Raymond Ridley from VPEC in the 90’
‰ Use z-transform method
‰ No auto-toggling mode models
‰ Can only work in ac
‰ Can predict sub-harmonic oscillations in CCM
AC model
Vin
Vout
0
2
D Gnd
Vg
AC = 0
0.458
4
Duty
Ctrl
0
3
V2
AC = 1
Vout
RS = 20m
FS = 50k
VOUT = 5
RL = 3
VIN = 11
X1
RI = 0.33
L = 37.5u
Resr
100m 0
1
0
5
Cout
220uF
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7
Rload
3
The “PWM Switch” in mode transitioning SPICE models
What techniques already exist?
‰ The PWM Switch
‰ Introduced by Vatché Vorpérian in the mid-80’
‰ Easy to derive and fully invariant
‰ No auto-toggling mode models
‰ Can predict sub-harmonic oscillations in CCM
‰ DCM model was never published!
a
c
L
d
a
Ia(t)
Ic(t)
d
d'
c
PWM switch
Vin
d'
Vap(t)
p
Vcp(t)
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p
C
R
Vout
The “PWM Switch” in mode transitioning SPICE models
The PWM Switch concept
L
Linear network
a
Vin
PWM switch
d
d'
p
c
on
off
Linear network
What do
you plead?
diode + transistor = guilty for non-linearity!
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November 2001
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9
C
R
Vout
The “PWM Switch” in mode transitioning SPICE models
The PWM Switch concept
5
ib
Rc
10k
Rb_upper
1Meg
Vin
7
Beta.Ib
e
Vg
Vin
4
Q1
Vout
8
h11
Vout
1
ic
c
b
Req
Rb_upper//Rb_lower
3
Rc
10k
ie
2
Rb_lower
100k
Remember the bipolars
Ebers-Moll model…
Re
150
Re
150
Ce
1nF
Ve
Replace Q1 by its small-signal model
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November 2001
Ce
1nF
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The “PWM Switch” in mode transitioning SPICE models
An invariant association
R
p
d'
a
Vin
C
Boost
PWM switch
d
p
c
Vin
c
Buckboost
d'
d
a
PWM switch
L
Vout
C
R
Vout
L
a
a
Ia(t)
Ic(t)
d
c
Vcp(t)
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November 2001
Buck
d'
PWM switch
Vin
p
L
d
d'
Vap(t)
c
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p
C
R
Vout
The “PWM Switch” in mode transitioning SPICE models
Observe waveforms and average them
a
Ia(t)
Ic(t)
d
L
c
d'
Vout
Vin
I a (t )
Tsw
= Ia =
1
Tsw
Tsw
∫
0
I a (t )dt =d I c (t )
Vap(t)
Tsw
C
Vcp(t)
p
= dI c
Vcp (t )
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12
Tsw
= Vcp =
R
1
Tsw
Tsw
∫V
cp
0
(t )dt =d Vap (t )
Tsw
= dVap
The “PWM Switch” in mode transitioning SPICE models
PWM Switch model in CCM: a 1:D transformer!
a
c
Ia(t)
a
Ic(t)
Vap
p
p
Large-signal (non-linear) model
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November 2001
d
1
V=d.V(a,p)
I=d.Ic
p
Ic(t)
Ia(t)
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c
Vcp
The “PWM Switch” in mode transitioning SPICE models
Use it immediately, SPICE linearizes it for you!
10.0
L
c
p
Vout
d
100uH
10.0
a
Vin
10
0.400
16.7
Vbias
0.4
AC = 1
C1
100u
R1
10
Always verify the dc operating point!
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The “PWM Switch” in mode transitioning SPICE models
The original CCM/DCM PWM Switch models
d
1-d
c
a
vap (t )
CCM: common « passive »
vcp (t )
p
d1
d2
a
Looks like
auto-toggling
is impossible…
p
DCM: common « common »
d3
vcp (t )
vac (t )
c
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15
The “PWM Switch” in mode transitioning SPICE models
Deriving the DCM PWM Switch in common « passive »
Ia
a
Ipeak
Ia(t)
L
Ic(t)
d1
d2
c
d3
Vout
Vin
t
Vap
Ic
Vap(t)
t
Ipeak
Vcp
t
Vap
Ia =
I peak d1
Ic =
I peak d1
Vcp
t
d1Tsw
d2Tsw
d3Tsw
p
Ic =
2
2
+
I peak d 2
2
=
R
I peak ( d1 + d 2 )
2
(d + d2 )
2 I a ( d1 + d 2 )
= Ia 1
2
d1
d1
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November 2001
C
Vcp(t)
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The “PWM Switch” in mode transitioning SPICE models
An auto-toggling version: clamp the equation!
a
Vap
Ic(t)
Ia(t)
N
1
p
Clamp d2:
d2 CCM = 1- d1
d2 DCM = 1- d1- d3
c
Vcp
N=d1/(d1+d2)
2 I c L − Vac d1 ²Tsw 2 LFsw I c
d2 =
=
− d1
Vac d1Tsw
d1 Vac
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17
d2 < d2 CCM
model is in DCM!
Model input
The “PWM Switch” in mode transitioning SPICE models
In voltage mode, add the PWM modulator gain
X4
PWMCCMVM
a
0.500
10.0
17
V4
10
GAIN
3
5.00
c
L1
75u
4
d
PWM switch VM
XPWM
GAIN
K = 0.5
K PWM =
p
5.0
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18
1
Vpeak
The “PWM Switch” in mode transitioning SPICE models
Testing the auto-toggling model
c
R5
20m
L1
75u
4.99
a
4
vout
4.99
17
L1
75u
IC =
X2
PSW1
Vout
R4
20m
1
0.499
10.0
5
V4
10
GAIN
3
d
PWM switch VM
Resr
70m
p
X10
PWMVM2
L = 75u
Fs = 100k
XPWM
GAIN
K = 0.5
Xstep
PSW1
vout
Vout
4
Resr
70m
7
10
4.99
2
Cout
100u
IC = 0
V4
10
19
16
D2
N = 0.01
Vstep
Cout
100u
IC =
20
Xstep
PSW1
Vstep
vout
vout
C2
{C2}
R3
{R3}
C1
{C1}
R2
{R2}
Rupper
{Rupper}
14
0.329
X8
COMPAR
3
C3
{C3}
2.50
Vsaw
Tran Generators = PULSE
V2
2.5
Rlower
10k
Averaged model
8
Verr
X1
AMPSIMP
VHIGH = 1.9
Cycle-by-cycle
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November 2001
18
13
13
2.50
X2
AMPSIMP
VHIGH = 1.9
Rupper
{Rupper}
19
8
Verr
R3
{R3}
C1
{C1}
R2
{R2}
5.00
7
6
6
+
-
C2
{C2}
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V2
2.5
Rlower
10k
C3
{C3}
The “PWM Switch” in mode transitioning SPICE models
Comparing results with a stepload…
5.40
Cycle-by-cycle
5.20
,
5.00
4.80
Averaged model
Output voltage
4.60
Averaged model
1.10
Error voltage
I can’t believe
this result…
Cycle-by-cycle
900m
700m
500m
300m
9.75m
11.2m
12.7m
time in seconds
14.2m
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20
15.7m
The “PWM Switch” in mode transitioning SPICE models
Current-mode PWM switch
‰ Same approach as before:
9 observe and average waveforms
9 get the equivalent representation
9 perturb for small-signal analysis
CCM DCM
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The “PWM Switch” in mode transitioning SPICE models
CCM operation, current expression
3
1
2
I c (t ) Ri = Vc (t ) − d (t )Tsw Se −
1
Ic =
2
S f d '(t )Tsw
2
Vc
T S
T
− d sw e − Vcp (1 − d ) sw
Ri
Ri
2L
3
a
c
Ia(t)
Ic(t)
I=Vc/Ri
I=d.Ic
p
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22
p
I=Iu
Cs
The “PWM Switch” in mode transitioning SPICE models
DCM operation, current expression
Vc − d1Tsw × Se
Ri
V −dT S
I c = c 1 sw e − α d 2Tsw S f
Ri
I peak d1 I peak d 2 I peak ( d1 + d 2 )
Ic =
+
=
2
2
2
I peak =
Iµ =
d1 × Tsw × Se
V (c, p ) ⎛ d1 + d 2 ⎞
+ d 2 × Tsw ×
× ⎜1 −
⎟
2 ⎠
Ri
L
⎝
a
c
Ia(t)
Ic(t)
I=Vc/Ri
I=(d1/(d1+d2)).Ic
p
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November 2001
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23
p
I=Iu
The “PWM Switch” in mode transitioning SPICE models
The PWM Switch,
Switch the final encapsulation
duty_cycle
a
c
R4
20m
L1
75u
2
duty-cycle
1
L1
75u
IC = 250m
Vout
4
1
R4
1
vout
Vout
4
Resr
70m
12
d
PWM switch CM
V4
25
X2
PSW1
vout
Resr
70m
p
X1
PWMDCMCM
V4
25
16
D1
1n5818
Cout
220uF
IC = 4.6
16
20
V2
Cout
220uF
10
X5
PSW1
V2
C2
470p
vout
X5
PSW1
2
S
V1
Q
Q
C1
10n
R1
20k
R7
10k
12
10
R
13
L4
1p
C1x
100p
6
7
C3
1p
Y4
X4
AMPSIMP
V6
2.5
R8
10k
-
X3
COMPAR
8
+
9
14
6
R3
470
B1
Voltage
17
V(4,vout)
C2
470p
11
V7
AC = 1
vout
C1
10n
R1
20k
R7
10k
24
18
19
Verr
A buck: averaged model vs cycle-by-cycle
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24
X4
AMPSIMP
V6
2.5
R8
10k
The “PWM Switch” in mode transitioning SPICE models
Good matching between both models
Error voltage
Output voltage
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25
It can’t
be, he is
cheating!
The “PWM Switch” in mode transitioning SPICE models
Testing the ac response
0.649
vc
a
duty-cycle
0.408
dc
X1x
XFMR
-127RATIO = -0.1
1
Vin
126
0
4
Vout
D1
MBR140P
L1
out1 2.2uH
12.2
12.2
out2
6
12.7
3
R4
100m
126
L3
1.8m
PWMCM
X1
L = 1.8m
Fs = 66k
Ri = 1.5
12.2
15
C1
470uF
out1
R7
8k
LoL
1kH
0.649
0.649
14
9
C5
470uF
out2
R15
1.5k
10.6
VFB
16
5
CoL
1kF
0
Cf
100nF
9.88
12
10
Vstim
AC = 1
X3
TL431
Rupp
3.9k
2.50
13
Rlow
1k
A dcm current-mode flyback
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26
Rload
14
12.2
7
V3
4.8
11
R17
300m
R5
100m
12.2
4.80
November 2001
Rs
10m12.2
c
p
2
PWM switch CM
18
C2
10uF
The “PWM Switch” in mode transitioning SPICE models
Ac simulation results of the flyback converter
Phase
Gain
BW = 600Hz
0
Y = 20dB/div
10
100
Y = 45°/div
1K
10K
100K
Simulation
Measurement
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The “PWM Switch” in mode transitioning SPICE models
If the load increases…
plot1
vdbfb in db(volts)
60.0
gain
Sub-harmonic oscillations!
40.0
20.0
0
-20.0
220
CCM operation
1
phase
Plot2
ph_vfb in degrees
2
180
140
100
60.0
10
100
1k
frequency in hertz
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November 2001
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28
10k
100k
The “PWM Switch” in mode transitioning SPICE models
Testing on a multi-output forward
L2
{L2}12.2
12.2
4
R2x
70m
12.0
5
Vout2
9
X1
XFMR
RATIO = N2
12.0
Rload2
6
2
Coupled
inductances
Cout2
100u
X7
XFMR
RATIO = N1/N2
0.356
160
a
10
0.861
1
parameters
Rsense=0.35
Vout=28
L1=130u
L2=130u
N1=0.5
N2=0.215
duty-cycle
c
PWM switch CM
L1
{L1}28.3
28.3
56.6
8
11
R9
70m
vout
28.0
Vout1
3
X6
XFMR
RATIO = N1
p
X5
PWMCM2
L = L1/N1^2+L2/N2^2
Fs = 200k
Ri = Rsense
Se = 0
Resr1
245m
28.0
Rload1
7
6
Cout1
48u
vout
C2
{C2}
Rupper=(Vout-2.5)/250u
fc=5k
pm=50
Gfc=8.84
pfc=-66
G=10^(-Gfc/20)
boost=pm-(pfc)-90
pi=3.14159
K=tan((boost/2+45)*pi/180)
C2=1/(2*pi*fc*G*k*Rupper)
C1=C2*(K^2-1)
R2=k/(2*pi*fc*C1)
dc
vc
V6
160
C1
{C1}
R2
{R2}
Rupper
{Rupper}
20
2.50
0.861
13
LoL
1kH
19
0.861
CoL
1kF
0
2.50
Verrx
18
X4
AMPSIMP
V3
2.5
Rlower
10k
7
V1
AC = 1
A multi-output forward
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November 2001
12 V
Resr2
100m
Christophe Basso – PCIM 2005
29
28 V
The “PWM Switch” in mode transitioning SPICE models
Output voltage bang on the 28 V output…
Plot1
vout, vout1 in volts
28.8
28 V
28.4
28.0
3
1
27.6
Inductances are
coupled…
27.2
12.4
12.5
12.2
12.3
12.0
vout2#a in volts
Plot2
vout2 in volts
Averaged
Cycle-by-cycle
12 V
12.1
2
4
11.8
11.9
11.6
11.7
3.93m
4.83m
5.74m
time in seconds
6.64m
A forward converter
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November 2001
7.55m
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The “PWM Switch” in mode transitioning SPICE models
Output voltage bang on the 28 V output…
plot1
vout, vout1 in volts
28.8
28 V
28.4
28.0
4
1
27.6
Inductances are
un-coupled…
27.2
Averaged
Cycle-by-cycle
Plot2
vout2, vout2#a in volts
12.8
12 V
12.4
12.0
3
2
11.6
11.2
3.83m
4.75m
5.68m
time in seconds
6.61m
7.54m
A forward converter
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31
The “PWM Switch” in mode transitioning SPICE models
Instability in the buck DCM current-mode
Gain
Vin = 7.5 V
Rload = 50 Ω
80.0
180
> 20 dB increase
-90.0
vdbout, vdbout#1, vdbout#2 in db(volts)
Plot1
vphout, vphout#1, vphout#2 in degrees
0
Gain
Vin = 10 V
Rload = 50 Ω
0
3
5
1
Phase
Vin = 10 V
Rload = 50 Ω
2
4
6
Phase
Vin = 7.5 V
Rload = 50 Ω
-40.0
Phase jumps
to –180°
-180
Gain
Vin = 7 V
Rload = 100 Ω
40.0
90.0
The DCM buck
shows instability
as M > 0.66
without ramp
Vin = 7V M = 0.71
Vin = 7.5V M = 0.66
Vin = 10V M = 0.5
Phase
Vin = 7 V
Rload = 100 Ω
-80.0
1
10
100
1k
frequency in hertz
10k
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32
100k
1Meg
The “PWM Switch” in mode transitioning SPICE models
Instability in the buck DCM current-mode
40.0
180
-20.0
-40.0
Adding 0.086 x Soff
cures the problem
Gain
Vin = 7 V
Rload = 100 Ω
Sa = 12.28 kV/s
90.0
vphout, vphout#2 in degrees
Plot1
vdbout, vdbout#2 in db(volts)
20.0
0
Gain
Vin = 7 V
Rload = 100 Ω
No ramp
2
4
0
Phase
Vin = 7 V
Rload = 100 Ω
Sa = 12.28 kV/s
-90.0
3
1
Phase
Vin = 7 V
Rload = 100 Ω
No ramp
-180
1
10
100
1k
frequency in hertz
10k
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November 2001
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33
100k
1Meg
The “PWM Switch” in mode transitioning SPICE models
The conclusion
‰ The CM PWM Switch DCM was derived
‰ Two auto-toggling models developed
‰ Good matching of average vs reality
‰ Models also exist in BCM (PFC simulations)
‰ Exist in both IsSpice and PSpice
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34