ECEN5807

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

More Accurate Models

ECEN 5807Lecture 25

ECEN5807

More Accurate CPM Models: Outline

Sampled-data modeling of inductor dynamics in current programmed mode

• Instability of the current loop and the need for compensation (“artificial”) ramp (Textbook Section 12.1)

• Improved modeling of high-frequency dynamics to enable design of wide-bandwidth voltage loops

More accurate averaged model• Large-signal and small-signal averaged modulator model• Accurate averaged small-signal models, including high-

frequency dynamics• Accurate modeling of line-to-output responses

Discussion of results for basic convertersCPM model for simulationsDesign examples

ECEN5807

High-frequency small-signal inductor-current dynamics

Assume that voltage perturbations are negligibly small at high frequencies: the slopes m1 and m2 can be considered constant

Apply sampled-data modeling:

)(ˆ][ˆ][ˆˆ)(ˆ * tininiiti LLccc

)(ˆ)(ˆ)(ˆ)(ˆ1)(ˆ sizizijksiT

si LLck

scs

c

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Discrete-time dynamics ][ˆ][ˆ nini Lc

sLL Tndmmnini ][ˆ)(]1[ˆ][ˆ21

sLc Tndmnini ][ˆ]1[ˆ][ˆ1

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Discrete-time dynamics ][ˆ][ˆ nini Lc

sLL Tndmmnini ][ˆ)(]1[ˆ][ˆ21

sLc Tndmnini ][ˆ]1[ˆ][ˆ1

][ˆ)1(]1[ˆ ][ˆ]1[ˆ][ˆ1

21

1

2 nininim

mmnimmni cLcLL

'

1

2

1

2

DD

MM

mm

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Instability for D > 0.5

][ˆ)1(]1[ˆ ][ˆ ninini cLL

'

1

2

1

2

DD

MM

mm

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Small-signal perturbation with compensation ramp

ic[n]

m1(t) m2(t)

ic + ic

iL[n]

d[n]TsiL[n-1]

ma(t)

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

sLL Tndmmnini ][ˆ)(]1[ˆ][ˆ21

saLc Tndmmnini ][ˆ)(]1[ˆ][ˆ1

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sLL Tndmmnini ][ˆ)(]1[ˆ][ˆ21

][ˆ)1(]1[ˆ ][ˆ]1[ˆ][ˆ1

21

1

2 nininimmmmni

mmmmni cLc

aL

a

aL

2

2

1

2

'

1

mm

DD

mm

mmmm

a

a

a

a

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

saLc Tndmmnini ][ˆ)(]1[ˆ][ˆ1

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][ˆ)1(]1[ˆ ][ˆ ninini cLL

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][ˆ)1(]1[ˆ ][ˆ ninini cLL

Discrete-time dynamics: )(ˆ)(ˆ zizi Lc

Z-transform: )(ˆ)1()(ˆ )(ˆ 1 zizzizi cLL

1 11

)(ˆ)(ˆ

zzi

zi

c

L

Discrete-time (z-domain) control-

to-inductor current transfer function:

ss TjsT ee

11

11

Difference equation:

• Pole at z = • Stability condition: pole inside the unit circle, || < 1

• Frequency response (note that z1 corresponds to a delay of Ts in time domain):

ECEN5807

Equivalent hold:

ic[n]

m1(t) m2(t)

ic + ic

iL[n]

d[n]TsiL[n-1]

ma(t)

)(ˆ)(ˆ ),(ˆ][ˆ sizitini LLLL

iL(t) iL[n]

Ts

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Equivalent hold

The response from the samples iL[n] of the inductor current to the inductor current perturbation iL(t) is a pulse of amplitude iL[n] and length Ts

Hence, in frequency domain, the equivalent hold has the transfer function previously derived for the zero-order hold:

se ssT1

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Complete sampled-data “transfer function”

s

sT

sTc

L

sTe

esisi s

s

1

11

)(ˆ)(ˆ

2

2

1

2

'

1

mm

DD

mm

mmmm

a

a

a

a

Control-to-inductor current small-signal response:

ECEN5807

Example

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

Inductor current slopes:m1 = (Vg – V)/L = 1 A/sm2 = V/L = 1 A/s

2

2

2

2

1

2

1

1

'

1

mmmm

mm

DD

mm

mmmm

a

a

a

a

a

a

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

s

sT

sTc

L

sTe

esisi s

s

1

11

)(ˆ)(ˆ

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

102

103

104

105

-40

-30

-20

-10

0

10

20

mag

nitu

de [

db]

iL/ic magnitude and phase responses

102

103

104

105

-150

-100

-50

0

frequency [Hz]

phas

e [d

eg]

ma/m2 = 0.1

ma/m2 = 0.5ma/m2= 1

ma/m2 = 5

5

10.5

0.1

MATLAB file: CPMfr.m

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First-order approximation

hfs

s

sT

sTc

L

sssTe

esisi s

s

1

1

)/(111

11 1

1)(ˆ)(ˆ

)/(1

)/(1

s

ssT

s

s

e s

s

a

shf

f

mmDD

ff

2

221

111

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

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

102

103

104

105

-40

-30

-20

-10

0

10

20

mag

nitu

de [

db]

iL/ic magnitude and phase responses

102

103

104

105

-150

-100

-50

0

frequency [Hz]

phas

e [d

eg]

1st-order transfer-function approximation

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Second-order approximation

2

2/)2/(11

21

11 1

1)(ˆ)(ˆ

ss

s

sT

sTc

L

sssTe

esisi s

s

2

2

2/)2/(21

2/)2/(21

ss

sssT

ss

ss

e s

2

221

12112

mmDD

Qa

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

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102

103

104

105

-40

-30

-20

-10

0

10

20m

agni

tude

[db

]iL/ic magnitude and phase responses

102

103

104

105

-150

-100

-50

0

frequency [Hz]

phas

e [d

eg]

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

2nd-order transfer-function approximation

ECEN5807

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 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

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