7. (Tan)

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

9. Discrete-time dynamics: Z-transform: Discrete-time (z-domain) control-to- inductor current transfer function: Difference equation: Pole at z =  Stability condition: pole inside the unit circle, |  | < 1 Frequency response (note that z  1 corresponds to a delay of T s in time domain):

10. Equivalent hold: ic[n]ic[n] m1m1 m2m2 i c + i c iL[n]iL[n] d[n]Tsd[n]Ts i L [n-1] ma(t)ma(t) iL(t)iL(t) iL[n]iL[n] TsTs

11. Equivalent hold The response from the samples i L [n] of the inductor current to the inductor current perturbation i L (t) is a pulse of amplitude i L [n] and length T s Hence, in frequency domain, the equivalent hold has the transfer function previously derived for the zero- order hold:

12. Complete sampled-data “transfer function” Control-to-inductor current small-signal response:

13. Example CPM buck converter: V g = 10V, L = 5  H, C = 75  F, D = 0.5, V = 5 V, I = 20 A, R = V/I = 0.25 , f s = 100 kHz Inductor current slopes: m 1 = (V g – V)/L = 1 A/  s m 2 = V/L = 1 A/  s D = 0.5: CPM controller is stable for any compensation ramp, m a /m 2 > 0

14. Control-to-inductor current responses for several compensation ramps (m a /m 2 is a parameter) m a /m 2 =0.1 m a /m 2 =0.5 m a /m 2 =1 m a /m 2 =5 5 1 0.5 0.1 MATLAB file: CPMfr.m

15. 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)

16. Control-to-inductor current responses for several compensation ramps (m a /m 2 = 0.1, 0.5, 1, 5) 1 st -order transfer-function approximation

17. Second-order approximation Control-to-inductor current response behaves approximately as a second- order transfer function with corner frequency f s /2 and Q-factor given by

18. Control-to-inductor current responses for several compensation ramps (m a /m 2 = 0.1, 0.5, 1, 5) 2 nd -order transfer-function approximation

19. 2 nd -order approximation in the small-signal averaged model

22. DC gain of line-to-output G vg-cpm (based on model (4))

23. Example CPM buck converter: V g = 10V, L = 5  H, C = 75  F, D = 0.5, V = 5 V, I = 20 A, R = V/I = 0.25 , f s = 100 kHz Inductor current slopes: m 1 = (V g – V)/L = 1 A/  s m 2 = V/L = 1 A/  s D = 0.5: CPM controller is stable for any compensation ramp, m a /m 2 > 0 Select: m a /m 2 = M a /M 2 = 1, M a = 1 A/  s

24. Example (cont.) Duty-cycle control Peak current-mode control (CPM)

25. 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