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(Tan). 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

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


Discrete time dynamics
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
Equivalent hold:

ic + ic

-ma(t)

ic[n]

iL[n]

iL[n-1]

d[n]Ts

-m2

m1

iL(t)

iL[n]

Ts


Equivalent hold1
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
Complete sampled-data “transfer function”

Control-to-inductor current small-signal response:


Example
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 m a m 2 is a parameter
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
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 m a m 2 0 1 0 5 1 5
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
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 m a m 2 0 1 0 5 1 51
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 g vg cpm based on model 4
DC gain of line-to-output Gvg-cpm(based on model (4))


Example1
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


Example cont
Example (cont.)

Duty-cycle control

Peak current-mode control (CPM)


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