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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.

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

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

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**Sampling in frequency domain**

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**Sampling in frequency domain: derivation**

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**Sampling in frequency domain: derivation**

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Aliasing

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**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

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**Zero-order hold: time domain**

vo(t) Zero-order hold

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**Zero-order hold: frequency domain**

u(t) vo(t) Zero-order hold

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**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” =

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**Zero-order hold: frequency responses**

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**Zero-order hold: frequency responses**

fs = 1 MHz MATLAB file: zohfr.m

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**Zero-order hold: 1st-order approximation**

1st-order Pade approximation

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**Zero-order hold: frequency responses**

fs = 1 MHz MATLAB file: zohfr.m

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**How does any of this apply to converter modeling?**

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**PWM is a small-signal sampler!**

PWM sampling occurs at tp (i.e. at dTs, periodically, in each switching period)

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**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

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**Application to DCM high-frequency modeling**

iL c dTs d2Ts Ts

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**Application to DCM high-frequency modeling**

iL c dTs d2Ts Ts

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**DCM inductor current high-frequency response**

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

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**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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