Current Programmed Control(i.e. Peak Current-Mode Control)

(1) Fi i h S l d D t M d li(1) Finish Sampled-Data Modeling(2) More Accurate Averaged Models

ECEN 5807

ECEN5807

Lecture 26

Announcements

For on-campus students, HW 8 due in class on Friday, March 18 Grace period for off campus students expires March 18. Grace period for off-campus students expires Friday, March 25, 5pm (Mountain)

Midterm exam: grace period for off-campus students expires Friday, March 18

March 21-25 is Spring break, no classes

ECEN5807 2

Discrete-time dynamics with compensation ramp: ][ˆ][ˆ nini Lc

Tndmmnini ][ˆ)(]1[ˆ][ˆ sLL Tndmmnini ][)(]1[][ 21

saLc Tndmmnini ][ˆ)(]1[ˆ][ˆ1

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saLc )( 1

Complete sampled-data “transfer function”

Control-to-inductor current small-signal response:

sT

sTL

Tesi s

1

11

)(ˆ)(ˆ

ssT

c sTesi s1)(

22

1mm

mma

a

2

1'

mm

DDmm aa

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Example

CPM buck converter:Vg = 10V, L = 5 H, C = 75 F, D = 0.5, V = 5 V, g , , , , ,I = 20 A, R = V/I = 0.25 , fs = 100 kHz

Inductor current slopes: (V V)/L 1 A/m1 = (Vg – V)/L = 1 A/s

m2 = V/L = 1 A/s

mm22

1

2

1

1

'

1 m

mm

mDmm

mmmm

a

a

a

a

a

a

s

sT

sTc

L

sTe

esisi s

s

1

11

)(ˆ)(ˆ

22

1mmD

D = 0.5: CPM controller is stable for any compensation ramp, ma/m2 > 0

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y p p, a 2

Control-to-inductor current responses for several compensation ramps (ma/m2 is a parameter)

10

20iL/ic magnitude and phase responses ma/m2 = 0.1

ma/m2 = 0.5m /m = 1

-20

-10

0

mag

nitu

de [

db] ma/m2= 1

ma/m2 = 5

MATLAB fil CPMf

102

103

104

105

-40

-30 MATLAB file: CPMfr.m

-50

0

eg] 1

0.50.1

-150

-100

phas

e [d

e

5

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102

103

104

105

frequency [Hz] fs = 100 kHz

First-order approximation

)/(1

ssT

s

First order Pade approximation applied to get control-to-inductor

)/(1

)(

s

ssT

se s

applied to get control to inductor

current “transfer function” as a rational function of s

hf

s

sT

sTc

L

sssTe

esisi s

s

1

1

)/(111

11 1

1)(ˆ)(ˆ

hfs )/(1

Control-to-inductor current response behaves approximately as a single-pole transfer function with a high-frequency pole at

s

a

shf

fmDD

ff221

111

ECEN5807

mDD

2

221

Control-to-inductor current responses for several compensation ramps (ma/m2 = 0.1, 0.5, 1, 5)

10

20iL/ic magnitude and phase responses

-20

-10

0

mag

nitu

de [

db]

102

103

104

105

-40

-30

1st-order transfer-function approximation

-50

0

eg]

-150

-100

phas

e [d

e

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102

103

104

105

frequency [Hz]

Second-order approximation

2

2/)2/(21

sssT

ss

s

Second order Pade approximation applied to get control-to-inductor

2

2/)2/(21

ss

sssT

sse s

pp gcurrent “transfer function” as a rational function of s

211

11 1

1)(ˆ)(ˆ

s

sT

sTc

L

sssTe

esisi s

s

2/)2/(12

ss

1212 Control-to-inductor current response

2

221

12112

mmDD

Qa

behaves approximately as a second-order transfer function with corner frequency fs/2 and Q-factor given by

ECEN5807

Control-to-inductor current responses for several compensation ramps (ma/m2 = 0.1, 0.5, 1, 5)

10

20]

iL/ic magnitude and phase responses

30

-20

-10

0

mag

nitu

de [

db]

102

103

104

105

-40

-30

2nd-order transfer-function approximation

-50

0

deg]

-150

-100

phas

e [d

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102

103

104

105

frequency [Hz]

Conclusions

• In CPM converters, high-frequency inductor dynamics depend strongly on the compensation (“artificial”) ramp slope ma

• Without compensation ramp (ma = 0), CPM controller is unstable for D > 0.5, resulting in period-doubling or other sub-harmonic (or even chaotic) oscillations

• For ma = 0.5m2, CPM controller is stable for all D

• Relatively large compensation ramp (ma > 0.5m2) is a practical choice not just to ensure stability of the CPM controller but also tochoice not just to ensure stability of the CPM controller, but also to reduce sensitivity to noise

• For relatively large values of ma, high-frequency inductor current dynamics can be well approximated by a single high-frequency pole

• Second-order approximation is very accurate for any ma

N t t d d l i l di hi h f

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• Next: more accurate averaged model, including high-frequency dynamics

More Accurate Averaged CPM ModelgTextbook Section 12.3 with updates

ECEN 5807

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ECEN 5807

More Accurate Averaged Model

Objectives• Effects of inductor current ripple and compensation (“artificial”)

ramp

• Modeling of all transfer functions of interest: control-to-output, line-to-output, input and output impedances

• More accurate model at high-frequencies to enable wide-bandwidth designs

ApproachApproach• Large-signal averaged CPM controller model: relationship

between control input ic, average inductor current iL, averaged lt d d t l dvoltages, and duty-cycle d

• Small-signal averaged CPM controller model

• Complete CPM controlled converter models incorporating well-

ECEN5807

gknown small-signal models for duty-cycle control

Averaged CPM Controller Modelic(t)

ma

m (t)m2(t)

i (t) m1(t)

dT

iL(t)

d’TdTs d Ts

Goal: find average inductor current iL as a function of Goal: find average inductor current iL as a function of

Control input icDuty-cycle d

ECEN5807

Slopes m1, m2, ma

Averaged CPM Controlled Model

ic(t)

maip

m (t)m2(t)

i (t)

i1 i2

m1(t)

dT

iL(t)

d’TdTs d Ts

sacp dTmii p

ssacsp dTmdTmidTmii 111 21

21

11?Li

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ssacsp TdmdTmiTdmii '21'

21

222

?issac dTmdTmii 11 2

1

?Lissac TdmdTmii '

21

22

ic(t)

imaip

i1 i2

m1(t)m2(t)

iL(t)

i2

dT d’T

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dTs d Ts

Averaged CPM Controlled Modelic(t)

ma

ip

issac dTmdTmii 11 2

1

iL

m1(t)m2(t)

iL(t)

i1 i22

ssac TdmdTmii '21

22

dTs d’Ts

Model iL Comment

i1 Correct in steady-state (DC)1i

2i

21 'iddi

1

2

3

Correct in steady state (DC)

Correct in steady-state (DC)

iL = average over the entire period: textbook

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21

21' diid 4

iL = average over the entire period: textbook

iL = average at the sampling instant

Large-signal averaged CPM models

All models give the same results at DCAll models give very similar results at low-frequenciesAll models include inductor current dynamicsAll models include inductor current dynamicsModel (3) (average over the entire period) is the textbook approach

(Section 12.3)

sssacL TdmTdmdTmiiddii 222121 '

22'

Textbook(Eq. 12.59)

• Model (4) (average current at the sampling instant) results in high-frequency dynamics consistent with the sampled-data model

ssacL TddmmdTmiiddii '2

' 2121

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Small-signal averaged CPM model

Model (3) (textbook)

sssacL TdmTdmdTmiiddii 222121 '

22'

22

2

2

1

2

21 ˆ2'ˆ

2ˆ'ˆˆ mTDmTDdTDMDMMii ss

sacL

vFvFiiFd vggLcm ˆˆˆˆˆ

1

sam TM

F 1

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Model (3) (textbook)

ECEN5807

Small-signal averaged CPM model

Model (4)

ssacL TddmmdTmidiidi '2

' 2121

2

)ˆˆ(2'ˆ)21(

2ˆˆ

2121 mmTDDdTDMMMii s

sacL

vFvFiiFd vggLcm ˆˆˆˆˆ

11

sa

m TDMMMF 1

)21(2

121

11

sa

m TMMMF 1

2

121

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Small-signal averaged CPM model

Model vFvFiiFd vggLcm ˆˆˆˆˆ

Model (4)

sa

m TMMMF 1

2

121

Converter

B k

Fg Fv

TDD s' 0Buck

Boost

Ls

20

0LTDD s

2'

Buck-boost

L2

LTDD s

2'

LTDD s

2'

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Model (4) version of Table 12.2