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Conduction Modes of a Peak Limiting Current Mode Controlled Buck - - PowerPoint PPT Presentation
Conduction Modes of a Peak Limiting Current Mode Controlled Buck - - PowerPoint PPT Presentation
Conduction Modes of a Peak Limiting Current Mode Controlled Buck Converter Predrag Pejovi, Marija Glii Introduction peak limiting current mode control . . . known since 1978, C. W. Deisch, Simple switching control method changes
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what is in the paper?
◮ nonlinear dynamics methods applied to analyze a peak
limiting current mode controlled buck converter . . .
◮ which required an iterated map model . . . ◮ and resulted in an infinite number of the discontinuous
conduction modes!
◮ regions where the modes occur identified ◮ clarification of notions of stability:
◮ limit cycle stability ◮ open loop (averaged model) stability
◮ just a homework assignment in nonlinear dynamics . . . ◮ but haven’t been done before!
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what is not in the paper?
◮ this paper does not contain an algorithm that would earn
you money . . .
◮ but would help you understand some phenomena you might
- bserve in some 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! ◮ so, I would like to share that with you!
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the circuit . . . constant current load!
+ − S D L iS vIN iC C iL iD iOUT vOUT + − vX The constant current load model affects the open loop (averaged model) stability! And some people (RWE) prefer a resistor load . . .
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and the control . . .
iL Im d TS TS t The discontinuous conduction mode (DCM) . . . Actually, period-1 DCM! . . . where it all started! We just wanted to study open loop instability of the averaged model for VOUT > 1
2 VIN, and landed
in nonlinear dynamics! We observed period-n DCM!
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and the control . . .
iL Im d TS TS t The continuous conduction mode (CCM) . . . Actually, period-1 CCM! . . . which is known to have limit cycle stability issues . . . and what is the averaged model then?
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questions?
◮ limit cycle stability?
- 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
◮ open loop (averaged model) stability?
- 1. something quite different!
- 2. depends on the load!
- 3. analysis requires averaged circuit model
- 4. averaged circuit models require periodicity
- 5. well, at least in some sense . . .
- 6. the DCM might expose an open loop (averaged model)
instability!!!
- 7. which is a result of the previous paper
◮ periodicity?
- 1. DCMs have periodic limit cycle
- 2. CCM might have a periodic limit cycle
- 3. but also, there are aperiodic attractors for the CCM
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clarifications needed?
1 2 3 4 5 6 7 8 9 10 11 12 vOUT [V] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 iL, iOUT [A]
Im = 0.5 A
We expected this, since we assumed period-1 stable limit cycle
- peration . . .
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clarifications needed?
1 2 3 4 5 6 7 8 9 10 11 12 vOUT [V] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 iL, iOUT [A]
Im = 0.5 A
But, we obtained this, since in the CCM for D > 1
2 the limit
cycle is unstable, and we reach stable period-n stable limit cycle in the DCM; nothing to say about open loop (averaged model) stability!
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just a closer look . . .
10.0 10.5 11.0 11.5 12.0 vOUT [V] 0.0 0.1 0.2 0.3 0.4 0.5 iL [A]
For the thin lines we have a closed-form solution . . . And we know what is going on there . . . twin-peaks (yellow) and triangular (red) DCM waveforms . . . infinity of DCMs . . .
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steady state waveform of iL, . . . “twin peaks”
5 10 15 20 t [µs] −0.1 0.0 0.1 0.2 0.3 0.4 0.5 iL [A]
vOUT = 10.123190 V; Im = 0.5 A; IOUT = 0.2 A
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actually happens . . .
5 10 15 20 25 30 35 40 45 50 t [µs] −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 iL [A]
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Reduction to a Switching Cell Model
+ − + − S D L iS vIN iL iD vOUT vX used to draw iL, to compute iL . . . vIN, vOUT assumed constant over TS . . .
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circuit equations . . .
L d iL dt = vL vL = VIN − VOUT , S − on, D − off −VOUT , S − off, D − on 0, S − off, D − off
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methods applied . . .
◮ numerical simulation of iterated maps ◮ Python, PyLab, lists . . . ◮ a way to generalize results and conclusions? ◮ normalization! ◮ I’m becoming boring!
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normalization . . .
Vbase = VIN: m v VIN MIN = 1 M VOUT VIN Ibase = VIN/ (fS L): j fS L VIN i Tbase = TS: τ t TS
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voilà!
d jL dτ = mL mL = 1 − M, S − on, D − off −M, S − off, D − on 0, S − off, D − off
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Discrete Time Model of the Switching Cell
+ − + − S D L iS vIN iL iD vOUT vX All we need is jL as a function of Jm and M!
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and when we get it . . . averaged circuit model!
iC C iL iOUT vOUT + − C d vOUT d t = iL − iOUT Decoupled!
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iterated map, case 1, no switching jL Jm 1 τ jL(0) jL(1)
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iterated map, case 2, switch turn-off jL Jm 1 τ jL(0) jL(1) τ1
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iterated map, case 3, switch turn-off, diode turn-off jL Jm 1 τ jL(0) jL(1) = 0 τ1 τ1 + τ2
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iterated map model . . .
◮ 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?
◮ specify M and Jm ◮ start from jL(0) = 0, at least for the CCM ◮ iterate till jL(k) = 0; we got the periodicity! ◮ sum all the charges, Q = k
n=1 qk
◮ jL = jOUT = Q/k ◮ all the rest is the matter of presentation . . .
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A Glimpse on the Period-1 Model
◮ assumed period-1 operation, regardless the limit cycle
stability
◮ DCM occurs for Jm < M (1 − M) ◮ CCM occurs for Jm > M (1 − M) ◮ in DCM jOUT = J2
m
2 M (1−M) ◮ in CCM jOUT = Jm − 1 2 M (1 − M) ◮ open loop (averaged model) instability for d jOUT dM
> 0
◮ in DCM d jOUT dM
=
J2
m(2M−1)
2(M−1)2M2 ◮ in CCM d jOUT dM
= M − 1
2 ◮ in both cases open loop (averaged model) instability for
M > 1
2 ◮ elementary?
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Conduction Modes
◮ effects caused by the period-1 limit cycle instability for
supposed-to-be CCM
◮ where the phenomena occur?
- 1. M > 1
2 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!
◮ 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”
◮ . . . and this is period-n DCM!
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chart of modes
1/2 1 M 1/4 1/2 1 Jm period-1 CCM period-1 DCM period-n DCM period-n CCM
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DCM-1
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DCM-2
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DCM-3
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DCM-4
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DCM-5
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DCM-6
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DCM-7
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DCM-8
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DCM-9
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DCM-10
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dependence of the period number on M and Jm
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Output Current and Stability
◮ jOUT = jL ◮ interested in jOUT (M, Jm) ◮ open loop (averaged model) stability for d jOUT d M
< 0
◮ comparison to period-1 model
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jOUT(M, Jm), period-1 assumed
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jOUT(M, Jm), actual
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jOUT(M, Jm)
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jOUT(M, Jm) in 3D, period-1
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jOUT(M, Jm) in 3D
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stability, different than 50 : 50
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Conclusions 1
◮ analysis of a PLCMC buck converter ◮ reduction to a switching cell ◮ discrete-time model ◮ normalized discrete time model ◮ limit cycle instability (CCM) causes all the problems . . . ◮ where the period-1 limit cycle is stable everything is as
predicted by the averaged model obtained assuming the period-1 mode
◮ unstable limit cycle does not hold for long, does not repeat,
not suitable for average modelling
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Conclusions 2
◮ the converter might end up in period-n discontinuous
conduction mode, with stable limit cycle, an infinite number of such modes
◮ region where period-n DCM occurs identified ◮ regions for DCM-1, DCM-2, DCM-3, DCM-4, . . .
identified, some of them numerically
◮ complex behavior observed
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Conclusions 3
◮ out of the DCM region, the converter might reach some
sort of period-n CCM, where n might be ∞, not so easy to model in that case
◮ for averaged models it is nice to have periodic behavior . . . ◮ but long-term statistics seem to converge . . . (?)
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