# Midterm Exam One-week take-home exam, to cover material of HW#1-6

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Midterm Exam One-week take-home exam, to cover material of HW#1-6
Averaged switch modeling Simulation Extra element theorem Input filter design N-extra element theorem Feedback theorem DCM modeling On-campus: handed out this Friday March 6, due at beginning of class of Friday March 13 Off-campus: start exam at time of your choosing, after you at least have completed HW#6.

Sampling v*(t) v(t) T Sampler v(t) t v*(t) t Unit impulse (Dirac)

Unit impulse Properties t Laplace transform unit step d(t) area = 1

Sampling in frequency domain

Sampling in frequency domain: derivation

Sampling in frequency domain: derivation

Aliasing

Zero-order hold H v*(t) vo(t) v*(t) t vo(t) t Zero-order hold nT
T = sampling period 1/T = sampling frequency

Zero-order hold: time domain
vo(t) Zero-order hold

Zero-order hold: frequency domain
u(t) vo(t) Zero-order hold

Sampled-data system example: frequency domain
v*(t) H v(t) vo(t) T Sampler Zero-order hold Consider only low-frequency signals: System “transfer function” =

Zero-order hold: frequency responses

Zero-order hold: frequency responses
fs = 1 MHz MATLAB file: zohfr.m

Zero-order hold: 1st-order approximation

Zero-order hold: frequency responses
fs = 1 MHz MATLAB file: zohfr.m

How does any of this apply to converter modeling?

PWM is a small-signal sampler!
PWM sampling occurs at tp (i.e. at dTs, periodically, in each switching period)

General sampled-data model
Sampled-data model valid at all frequencies Equivalent hold describes the converter small-signal response to the sampled duty-cycle perturbations [Billy Lau, PESC 1986] State-space averaging or averaged-switch models are low-frequency continuous-time approximations to this sampled-data model

Application to DCM high-frequency modeling
iL c dTs d2Ts Ts

Application to DCM high-frequency modeling
iL c dTs d2Ts Ts

DCM inductor current high-frequency response
High-frequency pole due to the inductor current dynamics in DCM, see (11.77) in Section 11.3

Conclusions PWM is a small-signal sampler
Switching converter is a sampled-data system Duty-cycle perturbations act as a string of impulses Converter response to the duty-cycle perturbations can be modeled as an equivalent hold Averaged small-signal models are low-frequency approximations to the equivalent hold In DCM, at high frequencies, the inductor-current dynamic response is described by an equivalent hold that behaves as zero-order hold of length D2Ts Approximate continuous-time model based on the DCM sampled-data model correlates with the analysis of Section 11.3: the same high-frequency pole at fs/(pD2) is obtained Next: current-mode control (Chapter 12)

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