where the pole frequency Д is given by

To simplify the expression for the pole frequency /j, we use the steady-state relationship that foiltiws from Eq. (11.12):

--f (.1.75)

Also, recall that the steatly-state equivalent resistaitce R(D) can be written as

= (11.76)

whereis the switching frequency. Upoit substitutioit of Eqs. (11.65), (11.75) and (11.76) into Eq. (11.74) we get:

This is an expression for the frequency of the high-frequency pole that is caused by the inductor dynamics ofthe DCM buck-boost converter. It can be shown that Eq. (11.77) is a general result for the high-frequency pole, valid for all basic converters operating in DCM. Since 0< D< I, Eq. (11.77) implies that the high-frequency pole is always greater than approximately one tfurd of the switching frequency.

Table 11.4 summarizes the expressions forthe high-frequency pole tOj and the RHP zero caused by the iitductor dynamics in coittroi-to-output transfer functions Gj-s) of basic DCM coitverters [6]. The high-frequency pole and the RHP zero occur at frequencies close to or exceeding the switching frequency/,. This is why, in practice, the high-frequency inductra: dynamics can usually be neglected.

Table 11,4 High-frequency pole and RHP zero of the DCM converter eonlral-to-output transfer function G/s)

11Л SUMMARY OF KEY POINTS

1. In the discontintious condtiction mode, the average transistor voltage and cunent are proportional, and hence obey Ohms law. An averaged equivalent circuit can be obtained by replacing the transistor with an effective resistor R(d). The average diode voltage and current obey a power source characteristic, with power equal to the power effectively dissipated by R. In the averaged equivaleni circuit, the diode is replaced with a dependent power source.

2. The two-port lossless network consisting of an effective resistor and power source, which results from averaging the transistor and diode waveforms of DCM converters, is called a loss-free resistor. This network models the basic power-processing functions of DCM converters, much in the same way lhat the ideal dc uansformer models the basic functions of CCM converters,

3. The large-signal averaged model can be solved under equilibrium conditions to determine the quiescent values of the converter currents and voltages. Average power arguments can often be used.

4. A small-signal ac model for the DCM swiich network can be derived by perturbing and linearizing the loss-free resistor network. The result has the form of a two-port v-parameter model. The model describes the small-signal variations in the transistor and diode currents, as functions of variations in the duty cycle and in the transistor and diode ac voltage variations.

5. To simplify the ac analysis of the DCM buck and boost converters, it is convenient to define two other forms of the small-signal switch model, corresponding to the switch networks of Figs, 11,16(a) and 11.16(b). These models are also y-parameter two-port models, but have different parameter values.

6. The inductor dynamics of the DCM buck, boost, and buck-boost converters occur at high frequency, above orjusl below the switching frequency. Hence, in most ca.ses the high frequency inductor dynamics can be ignored. In the small-signal ac model, the indtictance L is set to zero, and the remaining model is solved relatively easily for the low-frequency converter dynamics. The DCM buck, boost, and buck-boost converters exhibit transfer functions containing essentially a single low-frequency dominant pole.

7. To obtain a more accurate model of the itiductor dynamics in DCM, it is necessary to write the equations of the averaged inductor waveforms in a way that does not assume that the average inductor voltage is zero.

References

[I] V. VORPERtAN, R. Tymerski, and F. C. Lee, Equivalent Circuit Models for Resonant and PWM

Switches, IEEE Tmnsactian.. on Power Electronkx, Vol. 4, No. 2, pp. 205-214, April 1989.

[2] V. VORPERIAN, Simplified Analysis of PWM Converters Using die Model of die PWM Switch, parts I and ii, IEEE TrmtMictioiis on Aerospace am! Elearoiiic Systems. Vol. 26, No. 3, May 1990, pp. 490-505.

[3] D. MAKiiMOVid and S. Cuk, A Unified Analysis of PWM Converters in Discontinuous Modes, IEEE Tromacmm on Power Eiecfroiiics. Vol. 6, No. 3, pp. 476-490, luly 1991.

[4] i. Sun, D. M. MrrcHEU., M. Greuel, P. T. Krein, and R. M. Bass, Averaged Modelhng of PWM Converters in Discontinuous Conduction Mode: a Reexamination, /£££ Power Electronics Specialist. Conference. 1998 Record, pp. 615-622, lune 1998.

[5] S. Ben-Yaakov and D. Adar, Average Models as Tools for Studying Dynamics of Switch Mode DC-DC Converters, IEEE Power Electronics Specialists Conference, 1994 Record, pp. 1369-1376, Jane 1994.

[6] J. Sun, D. M. Mitchell, M. Greuel, P. T. Krein, and R. M. Bass, Average Models tor PWM Converters in Discontinuous Conduction Mode, Proceedms\s of the I99S hiternational High Frequency Power Covnsion Conference ЩРРСЧЩ, pp. 61-72, Novcinlicr 1УУ8.

[7] A. WlTLLSKi and R. Erickson, Extension of State-Space Averaging to Resonant Switclies -and Beyond, !EEE Transactions on Power Electronics, Vol. 5. No. 1. pp. yS-lt)9, Jantiary 1990.

[8] S. Freeland and R. D. MlDDLEBRtraK, A Unified Analysis of Converters with Resonant Switches, IEEE Power Electronics Specialists Conference, 19S7 Record, pp. 20-30, June 1987.

[9] S. .Singer, Realization of Loss-Free Resistive Elements, IEEE Trairiactions on Circuits wai Systejiis,

Vol. CA.S-.K), No. 12, January 1990.

[10] S. Singer and R.W. Erickson, Power-Source Element and Its Properties, lEE Proceedings-Circuits Devices aiul Systems, Vol. 141, No. 3, pp. 220-226, June 1994.

[11] S. Singer and R. Erickson, Canonical Modeling of Power Processing Circuits Based on the POPl Concept, IEEE Transactions on Power Electioiiics, Vol. 7, No. 1, Jannary 1992.

[12] S. Cuk and R. D. Middlebrook. A General Unified Approach to Modeling Switching Dc-to-Dc Converters in Discontinuotis Conduction Mode, IEEE Power Electronics Specialists Conference, 1977 Record, pp. 36-57.

[13] S. <itJK, Modeling, Analysis, and Design of Switching Conveners. Ph.D. Thesis, Califoritia litstitutc of Technology, November 1976.

Problems

11.1 Averaged iwlLch modeling of a flyback converter. The converter of Fig. 11.23 operates in the discontinuous conduction mode. The two-winding inductor has a V.n turns ratio and negligible leakage inductance, and can be modeled as an ideal transformer in parallel with primary-side magnetizing inductance L.

(a) Sketch the transiitor and diode voltage and ctirrent wavelormsi, and derive expresion:: for their average values.

(b) Sketch an averaged model for the converter lhat includes a los::-lree resistor network, and give an expression Ifx R(d).

(t) Solve your model lo determine ihe voltage ralio V/V in the discontinuous conduction mode.

(d) Over whai range ol load current / is your answer ol part (c) valid? Express the DCM boundary in the form I < f.JP, V, ti).

(e) Derive an expression for ihe small-signal contml-to-oulpul Iransler lunclion Gco. You may neglect inductor dynamics.

11.2 Averaged switch modeling of a nonisolated Watkins-Johnstin converter, The ctinverier iii Fig. 11.24 Operates in the discontinuous conduction mode. The two-winding induclor has a 1:1 turns ratio and negligible leakage inductance, and can be modeled as an ideal transformer in parallel with magnetizing inductance L

(a) Sketch the transistor and diode vohage and current waveforms, and derive expressions for their average values.

(b) Sketch an averaged model for the converter lhat includes ti loss-lree resisttjr network, tind give an expression for PJ.d).