2.6 SUMMARY OF KEY POINTS

1. The dc compoiiem ot ii converter waveform is given by its average value, or die integral over one switching period, divided by ihe switching period. Soluiion of a dc-dc converiei lo find iis dc, or steady-state, voltageR and currents therefore involves averagiiig the waveforms.

2. The linear- (or small-) ripple approximation greatly simplifies the analysis, tn a well-designed converter, the switching ripples in the inductor currents and capacilor voltages are small compared to Ihe respective dc componeiits, and can be neglected.

3. The piincipte of inductor volt-second balance allows determination of the dc voltage components in any switching converter. In steady state, the average voltage applied to an inductor must be zero.

4. The priticiple of capacitor charge balance allows deiertuination of the dc componeiits of the iiiducior currents in a swilching converter, tn steady stale, Ihe average currenl applied to a capacilor must be zero.

5. By kiiowledge of the slopes of the inductor current and capacitor voltage waveforms, the ac switching ripple magnitudes may be computed. Inductance and capacitance values can then be chosen to obtain desired ripple magnitudes.

ft. In converlers coiitaining tmiltiple-pole filters,contiiinous (nonpulsating) voltages and currents are applied 10 one or more of the inductors or capacitors. Computation of the ac switching ripple in these elements can be done using capacitor chaige and/or iiiditctor fliix-linkage arguments, without use of the small-ripple approximaiioti.

7. Converters capable of increasing (boost), decreasing (buck), and inverting the vohage polarity (buck-boost and Clik) have been described. Conveiter circuits are explored more fully in the problems and in a later chapter.

REFERENCES

llj S. Ci;k, Basics of Switched-Mode Power Conversioii: Topologies, Magnetics, and Control, in Advances

in SwiKhei!-Mi)de Power Conversiim, Vol. Z, pp. 27У-31(), Irvine, CA; Teslaco, 1W81.

12! N. Mohan, T. UlSDEUkKD, and W. Robbin.s, Posver Elearonics: Converters, Applications, and Design.

2 edir. New York: John Wiley & Sonn, 1995.

[i\ J. K.ASSAKIAN, М. SciiLECltr, and G. Veroese, Principles of Power Etectrtmics, Reading, MA: Addisoti-

Wesley, 1991.

(4] R. Severns and G. E. Bloom, Madern Dc-to-dc Switch Mode Power Converter Circuits. New York: Van

Nostrand Reiiihold, I9S5.

15] D. Hart, Introduction to Power Electronics. New York: Prentice Hall, 1997.

[6] M. RAsmn, Power Electronic.:: Circuits. Devices, and Applications, 2 edit., New York: Prentice Hall,

1993.

17] P. Krein, Elements ofPower Electronics, New York: Oxford University Press, 199s.

IS] K. Kit Sum, Switch Mode Power Conversion-Ba.iif Theoiy and Design. New York: Marcel Dekker,

19S4.

Problems

Problems 2.1

Analysis and design of a buck-boost converter: A buck-boosL converter is illLstraLed in Fig. 2.28(a), and apractical implementation using a iransisiorand diode is shovn in Fig, 2,28(b).

Fig. 2,2S Buck-boost convenei of Problem 2.1: (a) idtal convener circuit, (b) implemctitatioti using MOSFET and diode.

-O I 0-[-

(b) (c)

Find tlie dependence of ttie equilibrium output voltage Vand inductor current / on tlie duty ratio

A input voltage V and load resistance R. Уш may assume that ihe inductor current ripple and

capacitor voltage npple are small.

Plot your results of part (a) over the range 0 < Л £ 1,

DC design: for the specifications

V = 30V U = -20V

R = 4L1 y; = 40 kHz

(i) Find D and /

(ii) Calculate the value ofL that will make the peak inductor current ripple Де equal to ten per-

cem of the average inducior current /. fiiij Choose С such that the peak output voltage ripple Atis 0,1 V.

Sketch the transistor drain current vaveform for your design of pari (c). Include the effects of inductor current ripple. What is the peak value of f . Also sketch ij(f) for the case when L is decreased such that Ai is 50% off. What happens to the peak value of fj- in this case? Sketch the diode current waveform ( (г) for the two cases of pan (d).

In a certain application, an unregulated dc input voltage can vary between IS and 36 V. ll is desired to produce a regulated output of 28 V to supply a 2 A load. Hence, a converter is needed that is capable of both increasing and decreasing ihe voltage. Since the input and output voltages are boih positive, converters that invert the voltage polarity (such as the basic buck-boost converter) are not suited for this application.

One convener thai is capable of performing ihe required function is the nonisolated SEPIC (single-ended primary inductance convener) shown m Fig. 229. This convener has a conversion raho M(fl) that can both buck and boost the voltage, but the voltage polarity is not invened. In the normal converter operaiing mode, the Iransistorconducls during ihe first subinterval [0 < t < DTJ, smd the diode conducts during the secondsubinterval {DT < j < 7 ). You may assume that all elements are ideal, (a) Derive expressions for the dc components of each capacitor voltage and inductor current, as

functionsof the duly cycle D, the input voltage V , and the load resistance R.

Fig. 2.29 sepic of Problems 2.2 and 2.3,

(b) A control circuit aatomaticitlly adjusts the conveiler dttty cycle D, to maintain a constant output voltage of 1= 2K V. Jhe iiiput toltage slowly varies over ttie raiige LS V < 36 V. The load current is constant and equal to 2 A. Over wtiat range will the duty cycle D vary? Over what range will ttie input inductor current dc component vary?

For the SEPIC of Problem 2.2,

Derive expressions for each inductor curiCiit ripple and capacitor voltage ripple. Expiess ttiese quantities as functions of the switching period V,; the component val ties l L, C, C; iheduiy cycle D: the inpttt voltage V; and the load resistance R.

(b) Sketch the waveforms of the transistor voltage 1,и(г} and transistor current r ((}, and give expi-essions for tiieir peak values.

lA The switches in the convener of Fig. 2.30 operate synchronously: each is in position ] for 0 < t < DT,

and in position 2 for D/,<t<1\. Derive an expression for the voltage conversion ratio M(D) = V/V. Sketch M{D] vs. O.

ТПЛР

+ V

Fig. 2.30 H-htidge converter of Ptobleiiis 2.4 and 2.6.

2,5 Tlie switches in the converter of Fig. 2.31 operate synchronously: each is in position 1 for (i < t < D7,

and in position 2 for Dl\<t <1\. Derive an expression for the voltage conversion ratio М(1У) = VIV. Sketch M(D) vs, D,

-*-nm-

+ 1 -

Fig. 2.31 Cuircnt-fed bridge converter of Problems 2.7, and 2.K,

2.6 For the converter of Fig. 2.30, derive expiessions for the inductor curreiit ripple Aij aiid the capacilor voltage ripple Av..

2.7 For the coiiveiler of Fig. 2.31, derive an analytical expression for the dc component of the inductor ciir-