Introduction
Conduction Modes of a Peak Limiting
Current Mode Controlled Buck Converter
Predrag Pejović, Marija Glišić
what is in the paper?
nonlinear dynamics analysis of a peak limiting current
mode controlled buck converter . . .
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which required an iterated map model . . .
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and resulted in an infinite number of the discontinuous
conduction modes!
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peak limiting current mode control . . .
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known since 1978, C. W. Deisch, “Simple switching control
method changes power converter into a current source,”
PESC’78 [2]
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revisited many times, e.g. in 2001 [6] and 2011 (!) [7]
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still something to say?
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CCM, DCM, stability, D > 0.5, chaos, . . .
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artificial ramp . . .
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purpose of the paper to clarify the issues . . .
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continuation of our Ee 2013 paper, “Stability Issues in Peak
Limiting Current Mode Controlled Buck Converter” . . .
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initial plan turned out to be not ambitious enough . . .
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since there is an infinity of DCMs!
what is not in the paper?
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region where the modes occur identified
clarification of notions of stability:
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this paper does not contain an algorithm that would earn
you money . . .
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limit cycle stability
open loop stability
closed loop stability
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just a homework assignment in nonlinear dynamics . . .
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but never done before!
the circuit . . . constant current load!
but would help you understand some phenomena you might
observe in the circuits you build . . .
or at least helped me understand what happened in some of
my designs . . .
and helped me understand phenomena I would rather avoid!
and the control . . .
iL
iS
S
vX iL
L
Im
iC
iD
vIN
+
−
D
C
+
iOU T
vOU T
−
0
d TS
TS
t
The discontinuous conduction mode (DCM) . . .
The constant current load model affects open loop stability!
Actually, period-1 DCM!
And some people (RWE) prefer a resistor load . . .
Where it all started! We just wanted to study open loop
instability of the averaged model, and landed in nonlinear
dynamics! We observed period-n DCM!
and the control . . .
questions?
iL
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1. limit cycles are stable in the DCM
2. there are unstable limit cycles in the CCM, well known . . .
3. both are results of small perturbation analysis
Im
I
0
limit cycle stability?
d TS TS
1.
2.
3.
4.
5.
6.
7.
t
The continuous conduction mode (CCM) . . .
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Actually, period-1 CCM!
Which is known to have limit cycle stability issues . . . And what
is the averaged model then?
open loop stability?
something quite different!
depends on the load!
analysis requires averaged circuit model
averaged circuit models require periodicity
well, at least in some sense . . .
the DCM might expose open loop instability!!!
which is a result of the previous paper
periodicity?
1. DCM, periodic limit cycle
2. CCM might have a periodic limit cycle
3. but also, there are aperiodic attractors for the CCM
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
clarifications needed?
Im = 0.5 A
iL , iOU T [A]
iL , iOU T [A]
clarifications needed?
0
1
2
3
4
5 6 7
vOU T [V]
8
9
10 11 12
We expected this, since we assumed period-1 stable limit cycle
operation . . .
0
1
2
3
4
5 6 7
vOU T [V]
8
9
10 11 12
But, we obtained this, since in the CCM for D > 21 the limit
cycle is unstable, and we reach stable period-n stable limit cycle
in the DCM; nothing to say about open loop stability!
vOU T = 10.123190 V;
0.5
0.4
Im = 0.5 A;
IOU T = 0.2 A
10
t [µs]
15
0.5
0.3
0.4
0.2
iL [A]
iL [A]
Im = 0.5 A
steady state waveform of iL , . . . “twin peaks”
just a closer look . . .
0.1
0.0
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
10.0
10.5
11.0
vOU T [V]
11.5
12.0
0.3
0.2
0.1
0.0
For the thin lines we have a closed-form solution . . .
−0.1
0
5
And we know what is going on there . . . twin-peaks (yellow)
and triangular (red) DCM waveforms . . . infinity of DCMs . . .
Reduction to a Switching Cell Model
actually happens . . .
1.2
1.0
iS
S
vX iL
0.8
iD
0.6
iL [A]
L
vIN
0.4
+
−
D
+ v
− OU T
0.2
0.0
−0.2
−0.4
used to draw iL , to compute iL . . .
0
5
10
15
20
25 30
t [µs]
35
40
circuit equations . . .
45
50
vIN , vOU T assumed constant over TS . . .
methods applied . . .
L
d iL
= vL
dt

 VIN − VOU T , S − on, D − off
−VOU T ,
S − off, D − on
vL =

0,
S − off, D − off
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numerical simulation of iterated maps
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Python, PyLab, lists . . .
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a way to generalize results and conclusions?
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normalization!
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I’m becoming boring!
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normalization . . .
voilà!
Vbase = VIN :
m,
v
VIN
d jL
= mL
dτ
MIN = 1
M,
VOU T
VIN
j,
fS L
i
VIN

 1 − M, S − on, D − off
−M,
S − off, D − on
mL =

0,
S − off, D − off
Ibase = VIN / (fS L):
Tbase = TS :
τ,
t
TS
Discrete Time Model of the Switching Cell
iS
S
vX iL
and when we get it . . . averaged circuit model!
iC
L
iL
iD
vIN
+
−
+
iOU T
C
vOU T
−
+ v
− OU T
D
C
d vOU T
= iL − iOU T
dt
All we need is jL as a function of Jm and M !
Decoupled!
iterated map, case 1, no switching
iterated map, case 2, switch turn-off
jL
jL
Jm
Jm
jL (1)
jL (1)
jL (0)
jL (0)
0
1
0
τ
iterated map, case 3, switch turn-off, diode turn-off
jL
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Jm
I
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0
τ1
1
τ1 + τ2
τ
τ
essentially jL (n) as a function of jL (n − 1), M , and Jm
auxiliary, compute the charge qn carried over each period
and store it
equations are in the paper . . .
simulation? numerical solution?
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jL (1) = 0
1
iterated map model . . .
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jL (0)
τ1
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specify M and Jm
start from jL (0) = 0, at least for the CCM
iterate till jL (k) = 0; we got
P the periodicity!
sum all the charges, Q = n = 1k qk
jL = jOU T = Q/k
all the rest is the matter of presentation . . .
A Glimpse on the Period-1 Model
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Conduction Modes
assumed period-1 operation, regardless the limit cycle
stability
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DCM occurs for Jm < M (1 − M )
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CCM occurs for Jm > M (1 − M )
in DCM jOU T =
effects caused by the period-1 limit cycle instability for
supposed-to-be CCM
where the phenomena occur?
1. M > 12 to ensure the period-1 limit cycle instability
2. Jm > M (1 − M ) to put period-1 mode in the continuous
conduction
3. Jm < M to allow the inductor discharge over one period;
this is new!
2
Jm
2 M (1−M )
in CCM jOU T = Jm − 21 M (1 − M )
OU T
open loop instability for d jdM
>0
2 (2M −1)
Jm
d jOU T
in DCM dM = 2(M −1)2 M 2
OU T
= M − 12
in CCM d jdM
in both cases open loop instability for M >
1
2
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what happens? start from zero, instability, eventually
return to zero; once returned, “it will start again, it
won’t be any different, will be exactly the same”
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. . . and this is period-n DCM!
elementary?
chart of modes
DCM-1
period-n CCM
period-n DCM
Jm
period-1 CCM
1
1/2
1/4
period-1 DCM
0
0
1/2
M
1
DCM-2
DCM-3
DCM-4
DCM-5
DCM-6
DCM-7
DCM-8
DCM-9
DCM-10
dependence of the period number on M and Jm
Output Current and Stability
jOU T (M, Jm ), period-1 assumed
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jOU T = jL
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interested in jOU T (M, Jm )
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open loop stability for
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comparison to period-1 model
d jOU T
dM
<0
jOU T (M, Jm ), actual
jOU T (M, Jm )
jOU T (M, Jm ) in 3D
jOU T (M, Jm ) in 3D, period-1
stability, different than 50 : 50
Conclusions 1
Conclusions 2
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control model of the switching cell “derived”, jOU T (M, Jm )
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. . . applying simulation of the normalized discrete-time
model
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open loop stability analyzed
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instability of the period-1 limit cycle in CCM modifies the
model . . .
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resulting in a complex behavior . . .
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where small-signal models are of limited value
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conclusion: avoid period-n modes
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which we knew before
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but I understand it better now!
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analysis of a PLCMC buck converter
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reduction to a switching cell
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discrete-time model
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normalized discrete time model
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limit cycle instability (CCM) causes all the problems . . .
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limit cycle instability is different than the open loop
instability
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the converter might operate in period-n discontinuous
conduction mode, stable limit cycle
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region where period-n DCM occurs identified
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or it can stay in some sort of period-n CCM
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for averaged models it is nice to have periodic behavior . . .