(Tan) High-frequency pole (from the Tan averaged model (4))

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Transcript of (Tan) High-frequency pole (from the Tan averaged model (4))

Page 1: (Tan) High-frequency pole (from the Tan averaged model (4))
Page 2: (Tan) High-frequency pole (from the Tan averaged model (4))
Page 3: (Tan) High-frequency pole (from the Tan averaged model (4))
Page 4: (Tan) High-frequency pole (from the Tan averaged model (4))
Page 5: (Tan) High-frequency pole (from the Tan averaged model (4))
Page 6: (Tan) High-frequency pole (from the Tan averaged model (4))
Page 7: (Tan) High-frequency pole (from the Tan averaged model (4))

(Tan)

Page 8: (Tan) High-frequency pole (from the Tan averaged model (4))

High-frequency pole(from the Tan averaged model (4))

Page 9: (Tan) High-frequency pole (from the Tan averaged model (4))

][ˆ)1(]1[ˆ ][ˆ ninini cLL αα −+−=

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

Z-transform: )(ˆ)1()(ˆ )(ˆ 1 zizzizi cLL αα −+= −

1 1

1

)(ˆ)(ˆ

−−−

=zzi

zi

c

L

ααDiscrete-time (z-domain) control-to-

inductor current transfer function:

ss TjsT ee ωαα

αα

−− −−

→−−

11

11

Difference equation:

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

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

Page 10: (Tan) High-frequency pole (from the Tan averaged model (4))

Equivalent hold:

ic[n]

m1m2

ic + ic

iL[n]

d[n]TsiL[n-1]

ma(t)

)(ˆ)(ˆ ),(ˆ][ˆ sizitini LLLL →→

iL(t) iL[n]

Ts

Page 11: (Tan) High-frequency pole (from the Tan averaged model (4))

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:

s

e ssT−−1

Page 12: (Tan) High-frequency pole (from the Tan averaged model (4))

Complete sampled-data “transfer function”

s

sT

sTc

L

sT

e

esi

si s

s

−−−

=1

11

)(ˆ)(ˆ

αα

2

2

1

2

'

1

mm

DD

mm

mm

mm

a

a

a

a

+

−−=

+−

−=α

Control-to-inductor current small-signal response:

Page 13: (Tan) High-frequency pole (from the Tan averaged model (4))

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/s

m2 = V/L = 1 A/s

2

2

2

2

1

2

1

1

'

1

mmmm

mm

DD

mm

mm

mm

a

a

a

a

a

a

+

−−=

+

−−=

+−

−=α

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

s

sT

sTc

L

sT

e

esi

si s

s

−−−

=1

11

)(ˆ)(ˆ

αα

Page 14: (Tan) High-frequency pole (from the Tan averaged model (4))

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

magnitude [db] iL/ic magnitude and phase responses

102 103 104 105

-150

-100

-50

0

frequency [Hz]

phase [deg]

ma/m2=0.1

ma/m2=0.5

ma/m2=1

ma/m2=5

5

10.5

0.1

MATLAB file: CPMfr.m

Page 15: (Tan) High-frequency pole (from the Tan averaged model (4))

First-order approximation

hfs

s

sT

sTc

L

sssT

e

esi

si s

s

ωπωααα

α

+=

−++

≈−

−−

=−

1

1

)/(111

11 1

1)(ˆ)(ˆ

)/(1

)/(1

πω

πω

s

ssT

s

s

e s

+

−≈−

ππαα s

a

shf

f

mm

DD

ff

2

221

1

1

1

+−=

+

−=

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

Same prediction as HF pole in basic model (4) (Tan)

Page 16: (Tan) High-frequency pole (from the Tan averaged model (4))

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

magnitude [db] iL/ic magnitude and phase responses

102 103 104 105

-150

-100

-50

0

frequency [Hz]

phase [deg]

1st-order transfer-function approximation

Page 17: (Tan) High-frequency pole (from the Tan averaged model (4))

Second-order approximation

2

2/)2/(11

21

11

1

1

)(ˆ)(ˆ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛

−++

≈−

−−

=−

ss

s

sT

sTc

L

sssTe

esisi s

s

ωωααπα

α

2

2

2/)2/(21

2/)2/(21

⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

≈−

ss

sssT

ss

ss

e s

ωωπ

ωωπ

2

221

12

1

12

m

mDD

Qa+−

=+−

=πα

απ

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

Page 18: (Tan) High-frequency pole (from the Tan averaged model (4))

102 103 104 105-40

-30

-20

-10

0

10

20

magnitude [db] iL/ic magnitude and phase responses

102 103 104 105

-150

-100

-50

0

frequency [Hz]

phase [deg]

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

2nd-order transfer-function approximation

Page 19: (Tan) High-frequency pole (from the Tan averaged model (4))

2nd-order approximation in the small-signal averaged model

Page 20: (Tan) High-frequency pole (from the Tan averaged model (4))
Page 21: (Tan) High-frequency pole (from the Tan averaged model (4))
Page 22: (Tan) High-frequency pole (from the Tan averaged model (4))

DC gain of line-to-output Gvg-cpm(based on model (4))

Page 23: (Tan) High-frequency pole (from the Tan averaged model (4))

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/s

m2 = V/L = 1 A/sD = 0.5: CPM controller is stable for any compensation ramp, ma/m2 > 0

Select: ma/m2 = Ma/M2 = 1, Ma = 1 A/s

A/V 25.02

'==

LTDD

F sg

1/A 1.01

2

1

21

=−+

=s

a

m TMMMF

Page 24: (Tan) High-frequency pole (from the Tan averaged model (4))

Example (cont.)

kHz 2.81

2

1==

LCfo π

1==LC

RQ

47.047.01

1==

+

+= Q

LVRCFRVF

QQgm

gm

ckHz 3.1851 ==+= ogm

oc fRVF

ff

kHz 4.81 =≈ ccp fQf

kHz 39/2 =≈= cchfp Qfff

Duty-cycle control

Peak current-mode control (CPM)

Page 25: (Tan) High-frequency pole (from the Tan averaged model (4))

Compare to first-order approximation of the high-frequency sampled-data control-to-current model

hfs

s

sT

sTc

L

sssT

e

esi

si s

s

ωπωααα

α

+=

−++

≈−

−−

=−

1

1

)/(111

11 1

1)(ˆ)(ˆ

)/(1

)/(1

πω

πω

s

ssT

s

s

e s

+

−≈−

kHz 32221

1

1

1

2

==+−

=+−

=πππα

α ss

a

shf

ff

mm

DD

ff

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