STABILITY ISSUES IN PEAK LIMITING CURRENT MODE CONTROLLED BUCK - - PowerPoint PPT Presentation

STABILITY ISSUES IN PEAK LIMITING CURRENT MODE CONTROLLED BUCK CONVERTER

Marija Glišić, Predrag Pejović

Introduction

◮ peak limiting current mode control . . . ◮ known since 1978, C. W. Deisch, “Simple switching control

method changes power converter into a current source,” PESC’78 [2]

◮ revisited many times, e.g. in 2001 [6] and 2011 (!) [7] ◮ still something to say? ◮ CCM, DCM, stability, D > 0.5, chaos, . . . ◮ artificial ramp . . . ◮ purpose of the paper to clarify the issues . . . ◮ and this presentation contains more than the paper does!

the circuit . . . constant current load!

+ − S D L iS vIN iC C iL iD iOUT vOUT + − vX

the waveform, . . . DCM assumed!

iL Im d TS TS t iL = fS L 2 I2

m

vIN vOUT 1 vIN − vOUT

decoupling, switching cell . . . also assumed, implicitly!

+ − + − S D L iS vIN iL iD vOUT vX used to draw iL, to compute iL . . . vOUT assumed constant over TS . . . implicitly!

decoupling, averaged model . . .

+ − + − iS vIN iC C iL iD iOUT vOUT + − vL

decoupling, averaged model simplified . . .

iC C iL iOUT vOUT + − C d vOUT d t = iL − iOUT and iL is given three slides above . . . and our story begins here . . .

Averaging

C d vOUT d t = fS L 2 I2

m

vIN vOUT 1 vIN − vOUT − iOUT

• verline notation consistent?

d vOUT d t = 0 ⇒ fixed points two fixed points . . . (overline notation dropped) vOUT 1, 2 = vIN 2 ±

• v2

IN

4 − fS L I2

m vIN

2 iOUT and it is not a good practice to have two when you need only

• ne . . .

an example . . .

vIN = 12 V, fS = 100 kHz, L = 36 µH, Im = 0.5 A, iOUT = 0.2 A, C = 200 µF iL = 0.45 A 12 V vOUT 1 12 V − vOUT iL = iOUT = 0.2 A fixed points: vOUT =    3 V 9 V

fixed points . . .

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

detour: normalization

mX vX vIN jY fS L iY vIN τ t TS = fS t result: L d iL d t = vL ⇒ d jL d τ = mL special: vIN ⇒ 1, vOUT ⇒ M, Im ⇒ Jm, iOUT ⇒ jOUT

fixed points, normalized

Vbase = 12 V, Ibase = 10 3 A Jm = 0.15, jOUT = 0.06 M1, 2 = 1 2 ±

• 1

4 − J2

m

2 jOUT fixed ponts: M =    1/4 3/4

fixed points . . . normalized!

1/4 1/2 3/4 1 M 3/100 6/100 9/100 12/100 15/100 jL, jOUT

Jm = 15/100

Linearization

C d vOUT d t = fS L 2 I2

m

vIN vOUT 1 vIN − vOUT s C vOUT = gIN vIN + gOUT vOUT + αm Im − iOUT gIN = ∂ iL ∂ vIN = − fS L I2

M

2 (VIN − VOUT )2 gOUT = ∂ iL ∂ vOUT = fS L I2

M VIN (2 VOUT − VIN)

2 V 2

OUT (VIN − VOUT )2

αm = ∂ iL ∂ Im = fS L IM VIN VOUT (VIN − VOUT )

transfer functions . . .

• vOUT = HIN

vIN + Hm Im − HOUT iOUT HIN = gIN s C − gOUT Hm = αm s C − gOUT HOUT = 1 s C − gOUT stability: gOUT < 0, ∂ iL ∂ vOUT < 0 . . . previous slide: 2 VOUT − VIN < 0, VOUT < VIN/2

fixed points, once again . . . what’s going on?

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

Discrete Time Model

+ − + − S D L iS vIN iL iD vOUT vX

◮ vOUT assumed constant over TS ◮ want to know mapping iL(0) → iL(TS)

. . . knowing Im, vIN, vOUT , fS, L . . . . . . or just Jm and M? (5 → 2)

◮ iL(n) 1 TS

n TS

(n−1) TS iL(t) dt is an auxiliary (but important!)

result

◮ normalization is useful here!!!

normalization, three cases . . .

◮ S-on, D-off:

d jL dt = 1 − M

◮ S-off, D-on:

d jL dt = −M

◮ S-off, D-off:

d jL dt = 0 But only two parameters, Jm and M! Look for jL(n − 1) → jL(n) and jL(n)!

three cases, again . . . case 1, no switching interval jL Jm 1 τ jL(0) jL(1)

three cases, again . . . case 2, continuous conduction interval jL Jm 1 τ jL(0) jL(1) τ1

three cases, again . . . case 3, discontinuous conduction interval jL Jm 1 τ jL(0) jL(1) = 0 τ1 τ1 + τ2

analytical . . .

jL(1) (jL(0), M, Jm) =                                                      1 − M + jL(0), if jL(0) < Jm + M − 1 1 1 − M Jm − M − M 1 − M jL(0), if Jm + M − 1 < jL(0) and jL(0) < Jm M + M − 1 0, if Jm M + M − 1 < jL(0) . . . similar for jOUT . . .

Basins of Attraction

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

trajectory of vOUT . . .

10 20 30 40 50 t [ms] 2 4 6 8 10 12 vOUT [V]

vOUT(0) = 9.1 V vOUT(0) = 8.9 V

trajectory of iL . . .

10 20 30 40 50 t [ms] 0.10 0.15 0.20 0.25 iL [A]

vOUT(0) = 9.1 V vOUT(0) = 8.9 V

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

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]

Limit Cycles

◮ the problem begins when the converter would enter CCM

for D > 1/2

◮ supposed limit cycle is unstable! ◮ but the converter operates in a stable limit cycle, regardless

• ur assumptions . . .

◮ . . . it happened to be period-2 DCM . . . ◮ . . . and here the mess starts . . .

supposed . . .

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

actual . . .

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

and we have it analytical . . . in the paper! (boring)

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]

iterate over Jm . . . not in the paper!

iterate over Jm . . . not in the paper!

important: operating mode chart . . . not in the paper!

1/2 1 M 1/4 1/2 1 Jm period-1 CCM period-1 DCM period-n DCM period-n CCM

iterate over Jm . . . not in the paper!

iterate over Jm . . . not in the paper!

DCM, period number, n = 1, not in the paper!

DCM, period number, n = 2, not in the paper!

DCM, period number, n = 3, not in the paper!

DCM, period number, n = 4, not in the paper!

DCM, period number, n = 5, not in the paper!

DCM, period number, n = 6, not in the paper!

DCM, period number, n = 7, not in the paper!

DCM, period number, n = 8, not in the paper!

DCM, period number, n = 9, not in the paper!

DCM, period number, n = 10, not in the paper!

period number, in color, . . . not in the paper!

stability . . . not in the paper!

Conclusions, 1

◮ buck converter analyzed, PLCMC applied ◮ decoupling (in reversed order):

• 1. “averaged” model, linearized later on . . .
• 2. “discrete time” model

◮ stability:

• 1. stability of the averaged model
• 2. limit cycle stability (stability of the discrete time model)

◮ limit cycle instability:

• 1. occurs in would-be CCM for D > 1/2
• 2. results in sensitive small-signal parameters
• 3. affects averaged model stability!

◮ analytical techniques, models, normalization . . .

Conclusions, 2

◮ in the paper, case study for Jm = 0.15 ◮ analytical techniques developed, discrete time model ◮ detailed study of the discrete time model ◮ identification of modes ◮ pretty good analytical description . . . ◮ analysis of stability

Conclusions, 3

◮ in this presentation, generalized over Jm, the remaining

degree of freedom, along with M, completeness achieved

◮ important:

• 1. occurrence of period-n modes when assumed period-1 CCM

has unstable limit cycle, for D > 1/2

• 2. both period-n CCM and period-n DCM exist

◮ charts:

• 1. chart of modes
• 2. chart of periodicity (chart of n)
• 3. chart of stability

Conclusion

avoid period-n modes!