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## PowerPoint Slideshow about ' (Tan)' - ulric-padilla

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Presentation Transcript

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

Discrete-time dynamics:

Difference equation:

Z-transform:

Discrete-time (z-domain) control-to-inductor current transfer function:

- Pole at z = a
- Stability condition: pole inside the unit circle, |a| < 1
- Frequency response (note that z-1 corresponds to a delay of Ts in time domain):

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:

Complete sampled-data “transfer function”

Control-to-inductor current small-signal response:

Example

- CPM buck converter:
Vg = 10V, L = 5 mH, C = 75 mF, D = 0.5, V = 5 V,

I = 20 A, R = V/I = 0.25 W, fs = 100 kHz

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

m2 = V/L = 1 A/ms

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

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

ma/m2=0.1

ma/m2=0.5

ma/m2=1

ma/m2=5

MATLAB file: CPMfr.m

0.1

0.5

1

5

First-order approximation

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)

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

1st-order transfer-function approximation

Second-order approximation

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

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

2nd-order transfer-function approximation

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

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

Example

- CPM buck converter:
Vg = 10V, L = 5 mH, C = 75 mF, D = 0.5, V = 5 V,

I = 20 A, R = V/I = 0.25 W, fs = 100 kHz

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

m2 = V/L = 1 A/ms

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

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

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

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

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