2. The basic tuiTeiiL-pragiammed conliDller is unstable when D > 0.5, reganlless of (he ctmverler topology. The controllercan be stabilized by aJJilion of an artificial ramp having slope >n. When lil > 0.5wiji Ihen the controller is stable for all duty cycles.

3. The behavior of current-programmed converters can be modeled in a simple and intuitive manner by the first-order approiiraation (/.(O);; УО-The averaged terminal waveforms ofthe switch network can then be modeled simply by a current source of value (., in conjunction with a power sink or power source element. Perturbation anJ linearization uf these elements leads to the small-signal model. Alternatively, the small-signal converter equations derived in Chapter 7 tan be adapted lo cover the current programmed mtsle, using the simple approximation i{t) - ,.()-

4. The simple model predicts that one pole is eliminated from the conveiter line-to-outpul and control-to-out-pul transfer functions, Current programming does not alter the Iransfer function zeroes, The dc gains become k>ad-depenJent.

5. The more accurate model of Section 12.3 conecdy accounts for the difference between the average inductor current (/i (f))7jand the control input ijf). This model preditls the nonzero line-to-outpul transfer function G,(i ) t>f the buck tonverler. The current-programmed controller behavior is modeled by a block diagram, which is appended to the small-signal converter models derived in Chapter 7. Analysis of the resuUing multiloop feedback system then leads to the relevant transfer funttions.

6. The more accurate model predicts that the induclor pole occurs al the crossover frequencyof Ihe effective current feedback loop gain T[s). The freqtiency typically occurs in the vicinity of the converter switching frequency The more accurate raoJel also predicts lhat the line-to-output transfer function Gj,(i) of die buck converter is nulled when m = O.Sntj.

7. Current prograraraeJ converters operating in the discontinuous conduction mode are modeled in Section

12.4. The averageJ transistor waveforms can be modeled by a power sink, while the averaged diode waveforms are modeled by a power source. The power is controlled by j(f). Perturbation and bnearization of these averaged mt>dels, as usual, leads to smalt-signal equivalent circuits.

References

[1] C. Deiscu, Simple Switching Control Method Changes Power Converter into a Current Sotirce, IEEE

Power Elecmmics Specialists Cotiference, 1978 Record, pp. 300-306.

]2] A. Capel, G. FereajiITE, D. OSullivajiI, and A. Weikbero, Apphcation of Ihe Injected Current Model for the Dynamic Analysis of Switching Regulators wilh the New Concept of LC Modulator, IEEE Power Electronics Specialists Conference, 1978 Record, pp. 135-147.

[3] S. Hsu, A. Brown, L. Rensink, and R. D. Middlebrook, Modeling and Analysis uf Switching Dc-tu-

I>c Converters in Constant-Frequency Current Programmed Mode, IEEE Power Electronics Specialists Conference, 1979 Record, pp. 284-301.

[4] F. C. Lee and R. A. Carter, Investigations iif Stability and Dynamic Performances of Switching Regulators Employing Current-Injected Control, IEEE Power Electronics Specialists Conference, 1981 Record, pp. 3-16.

[5] R. D. Middlebrook, Topics in Multiple-Loop Regulators and Current-Mode Programming, IEEE Power Elearonics Specialists Conference. 1985 Record, pp. 716-732.

[6] R, D. Middlebrook, Modeling Current Programmed Buck and Boost Regulattus, IEEE Transactions on Power Electronics. Vol. 4, No. 1, January 1989, pp. 36-52.

П] G. Veflcnese, C. Bruzos, und K. MAiiABiR, Averaged and Sampled-Dala Models for Current Mode Control: A Reexaminiilion, IEEE Power Electronici Specialise Conference, 1989 Record, pp. 484-491.

[8] D. M. MiTCiiELL, Dc-Dc Switching Reguiatcr Analysis. Mew York: McGraw-Hill, 1988, Chapter 6.

[9] A. KiSLOVSKi, R. Redl, and N. Sokal, Dynamic Analysis of Swiiching-Mode DC/DC Conveners, New

York: VanNoslrandReinhold, 1994.

[10] A. Brown and R. D. Middlebrook, Sampled-Data Modeling of Swilching Regulators, IEEE Power Electronics Specialists Conference, 1981 Record, pp. 716-732.

[11] R. Ridley, A New Continuous-Time Model for Current-Mode Control, IEEE Transactions on Power Electronic.:;, Vol. 6, Mo. 2, April 1991, pp. 271-2SO

[12] F. D. Так and R. D. Middlebrook, Unified Modeling and Measurement of Current-Programmed Converters, IEEE Power Electronics Specialists Conference, 1993 Record, pp. 380-387.

[13] R. TvMER.SKi, Sampled-Data Modeling of Switched Circuits, Revisited, IEEE Power Electronics Specialists Conference, 1993 Record, pp. 395-401.

[14] W. Tang, F. C. Lee, R. B. Ridley and I. Cohen, Charge Control: Modehng, Analysis and Design, IEEE Power Electronics Specialists Conference, 1992 Record, pp. 503-51 1.

[15] K. Sledlev and S. (5uK, One-Cycle Contrul of Switching Converters, IEEE Power Elearonics Specialists Conference. 1991 Record, pp. 888-896.

Problems

12J. A nonideal btick converter tjperates in the continuotis conduction mode, wilh the values = 10 V,/2 =

100 kHz, L = 4;iH, C= T.i др, and = t),23 Й. The desired full-load output is 5 V al 20 A. The power stage contains the lollovving loss elements; MOSFET on-resislance Я, = 0.1 £i, Schottky diode forward vollage drop Vjr, = O..) V, inductor winding resistance = 0.03 Q.

(a) Steady-state analysis: determine the converter steady-state dtily cycle D, the inductor currenl ripple slopes and tll, and the dimensionless parameter К - 2URT. Determine the small-signal equations for this converter, for duty cycle control.

A current-programmed controller is now implemented for this converter. An artificial ramp is used, having a fixed slope - 0.5M, where Л/ is (he steady-stale slope /iFj obtained wilh an output of 5 V at20 A.

(c) Over what range of D is the currenl programmed controller stable? Is h stable at rated output? Note that the nonidealities affect the stiibihty boundary.

(d) Determine the control-to-output transfer function G.fs), using the simple approximation (/.C))r, {-(0. Give analytical exp ressions for the corner frequency and dc gain. Sketch the Bode plot of Gj(.v).

12J Use lhe averaged switch modeling approach lo model lhe CCVI boost converter with currenl-pro-

grammed contrtd:

(a) Define the swiich netwtirk terminal quantities as in Fig. 7.46(a). Wilh the assumption that Oj.W)]- ~ i,.-(0, determine expressions for the average values of the swiich network terminal waveforms, and hence derive the equivalent circuit of Fig. 12.18(a).

(b) Perturb anJ linearize yuur mcxiel of pari (a), [ooblain Che equivalent tircuil of Fig, 12.22.

(c) Sulve your moJel of pari (b), to derive expressions for the control-to-oulput transfer function Cfijfi) and the line-to-output transferfunclion tivU) Express your results in standard normal-izeJ form, and give analytical expressions for the corner frequencies iind dc gains.

12.3 Use the averaged switch modeling approach to moJel the CCM Ctik converter with current-programrned control. A converler is diagrammed in Fig. 2.20.

(a) It isdesireJ lo model the switch iielwork wilh an f. current sotuxe and a depenJent power source or sink, using the approach of Section 12.2.2. How should the switch neiwork terminal voltages and currents be Jefined?

(b) Sketch the switch network terminal voltage and current waveforms. With the assumption that ([[(0)7;. ~ .Wtwhere if and ij are the inductorcurrenls defined in Fig. 2.20), Jeter-mine expressions lor the average values of the switch network terminiil waveforms, and hence derive an equivalent circuit similar to the equivalent circuits of Fig. 12.18.

(c) Perturb and linearize your model of pan (b), to obtain a small signal equivalent circuit similar to the model of Fig. 12.19. El is not necessary to solve your model.

12.4 The full-bridge converter of Fig. 6.19(a) operateswith = 320V, and supplies 10(DW to a 42 V resistive load. Losses can be neglected, the duty cycle is 0.7, and the switching period redefined in Fig. 6.20 is 10 ftsec. f. = 50 ДHand C= 100 др. A current-programmed controller is employed, whose waveforms are referred lo the secondary side ofthe transformer. In the following calculalicns, you may neglect the transformer magnetizing current.

(a) What is the minimum artificial ramp slope Jfl lhal will stabilize the controller at the given oper-adng point? Express your result in terms of liij.

(b) An artificial ramp having the slope = Mij is employed. Sketch the Bode plot of the current hK>p gai]i 7J), and Itibel numerical values ofthe cor]ier frequencies and dc gains. Il is not necessary to re-derive the analytical expression for 7j.. Delermine the crossover frequencyy..

(c) ¥atm~ Hij, sketch the Bode plots cflhe control-lo-outpLl transferfunclion Cf.fi) and line-lo-output transfer function G ,(.v). and label numerical values of the corner frequencies and dc gains, It is not necessary to re-derive analytical expressions for these Iransfer functions,

12i In a CCM current-programmed buck converter, it is desireJ to minimize the line-to-output Iransfer func-

tion G ] via the choice 1)1 - U.SWj. However, because of component tolerances, the value of inductance Lean vary by ±10% from its nominal value of 100 fiH. Hence, r is fixed in valuewhile ffi, varies, and - O.iii is obtaineJ only al the nominal value ofL The switching frequency is 100 kHz, the output voltage is 15 V, the load current varies over the range 2 to 4 A, anJ the input voltage varies over the range 22 to 32 V. You may neglect losses. Delermine the worst-case (maximum) value of the line-to-()Ut-put dc gain jijfW)-

12.6 The noniJeal flyback converter of Fig. 7.18 employs currenl-programmeJ control, wilh artificial ramp having slope 01, MOSFET exhibits On-resistance R. All current prtjgrammeJ controller waveforms are referred to the transformer primary side.

(a) Derive a block Jiagram which moJels the current-programmed controller, of form similar to Fig. 12.24, Give analytical expressiuns for the gains in your block Jiagram.

(b) Combine your result ofpart (a) with the converter small-signal nioJel. Derive a new expression forthecontrol-to-oulpultransferfunclion CfJj).

12.7 A buck ctmverler operates wilh current-programmed control. The elemeni values are:

V=120V D = 0.6

Д=10П /,= 100 kHz

L = S50;iH C=100fiF