Problems

rent, [, as a funcLion of Д V, and R. Sltelch your resulL -vs. D.

2JS For tfie converter of Fig. 2.31, derive expressions for Itie inductor current ripple Ai iind the ciiputitor

voltage ripple Av,

2.9 To reduce the switching harmonics pi-esent in the input cuiTent of a certain buck converter, an input filter

consisting of inductor i[ and capacitor is iidJed as shown in Fig. 2.32, Such filters are commonly used to meet regulations limiting conducted electromagnetic interference (emi). Fcrthis problem, you may iissume thiit all inductance anJ capacitance values are sufficiently large, such that all ripple magnitudes are sniiill.

Fig. 2,32 Addition of L-C input filter to buck converter, РгоЫе m 2.У.

(a) Sketch the triinsistor current waveform i,{()

(b) Derive analytical expressions for the dc components of the ciipacitor voltages iind inductor currents.

(c) Derive analytical expressions for the peak ripple magnitudes of the input filter inductor current and capacitor vohage.

fd) Given the following vjlues;

Input voliage

Output voltage

Switching frequency

Load resistance

V, = 48V V= ЗЙ V /,= too kHz R = (,il

Select values for i and such that fi ) ihe peak voltage ripple on Cj, Avj, is two percent of the dc component Vi- tf) input peak current ripple Ai is 20mA,

Extra credit problem: Derive exact analytical expressions for (i) the dc component of the output voltage, and (fi ) the peak-lo-peak inductor current ripple, uf the ideal buck-boost converter operating in steady stale. Do nol make the small-ripple approximation.

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Steady-State Equivalent Circuit Modeling, Losses, and Efficiency

Let us now consider the basic functions performed by a switching converter, and attempt to represent these fuitctions by a simple equivalent circuit. The tlesigner of a converter power stage must calculate the network voltages and ttirrents, and specify the power components accordingly. Losses and efficiency are of prime importance. The use of equivalent circuits is a physical ami intuitive approach which allows the well-known techniques of circuit analysis to be eniployetl. As notetl in the previous chapter, it is desirable to ignore the small but complicated switching ripple, and model only the important dc components of the wavefonns.

The dc transformer is used to model the ideal functions performed by a dc-dc converter [1-4]. This simple model correctly represents the relationships between the dc voltages and currents of the converter. The model can be refined by including losses, such as semiconductor forward voltage drops and on-resistances, inductor core and copper losses, etc. The resulting mtxiel can be directly solved, to find the vohages, currents, losses, imd efficiency in the actual nonideal converter.

3.1 THE DC TRANSFORMER MODEL

As illustrated in Fig. 3.1, any switching converter contains three ports: a power input, a power output, and a control input. The inptit power is prtK;essed as specified by the control input, and then is output to the load. Ideally, these functions are performed with 100% efficiency, and hence

Я =f (3.1)

Vl=VI (3.2)