Arabian Journal for Science and Engineering

, Volume 38, Issue 12, pp 3383–3397

# Analytical Solution for Steady and Transient States of Buck DC–DC Converter in CCM

Research Article - Electrical Engineering

## Abstract

In this paper, a new method is proposed for mathematical modeling of buck dc–dc converter in continuous conduction mode. In this method, the differential equations of inductor current and capacitor voltage are firstly obtained. Then, using Laplace and Z transforms, the differential equations of inductor current and output voltage of converter are solved to obtain the relations of them. In the proposed method, the Laplace transform is used to determine the general equations of inductor current and output voltage and the Z-transform is used to determine their initial conditions. The transient and steady states responses of converter are analyzed using the obtained mathematical model. In addition, the effect of converter components on output voltage and inductor current ripples is investigated. The validity of proposed method is reconfirmed by experimental and simulation results using PSCAD/EMTDC software.

### Keywords

Buck dc–dc converter Modeling Laplace transform Z-transform

## Preview

Unable to display preview. Download preview PDF.

### References

1. 1.
Camara M.B., Gualous H., Gustin F., Berthon A., Dakyo B.: DC–DC converter design for super capacitor and battery power management in hybrid vehicle applications polynominal control strategy. IEEE Trans. Ind. Electron. 57(2), 587–597 (2010)
2. 2.
Kimball J.W., Krein P.T.: Singular perturbation theory for dc–dc converters and application to PFC converters. IEEE Trans. Power Electron. 23(6), 1–12 (2008)
3. 3.
Rivetta C.H., Emadi A., Widliamson G.A., Jayabalan R., Fahimi B.: Analysis and control of a buck dc–dc converter operating with constant power load in sea and undersea vehicles. IEEE Trans. Ind. Appl. 42(2), 559–572 (2006)
4. 4.
Vinnikov D., Laugis J.: An improved high-voltage IGBT-based half-bridge dc–dc converter for railway applications. COMPEL: Int. J. Comput. Math. Electr. Electron. Eng. 30(1), 280–299 (2011)
5. 5.
Villava M.G., Siqueira T.G., Ruppert E.: Voltage regulation of photovoltaic arrays small signal analysis and control design. IET Trans. Power Electron. 3(6), 869–880 (2010)
6. 6.
Maksimovic D., Stankovic A.M., Thottuvelil V.J., Verghese G.C.: Modeling and simulation of power electronic converters. Proc. IEEE. 89(6), 898–912 (2001)
7. 7.
Tasi-Fu W., Yu-Kai C.: Modeling PWM dc–dc converters out of basic converter units. IEEE Trans. Power Electron. 13(5), 870–881 (1998)
8. 8.
Rim, C.T.; Joung, G.B.; Cho, G.H.: A state-space modeling of non-ideal dc–dc converters. In: Power electronics specialists conference 19th annual IEEE, pp 943–970 (1988)Google Scholar
9. 9.
Hamar, J.; Nagy, I.; Funata, H.; Nishida, Y.; Ohsaki, H.; Masada, E.: Discrete-time modeling tools for dc–dc converters. In: International power electronics conference, pp 1088–1093 (2010)Google Scholar
10. 10.
Sun J., Mitchell D.M., Greuel M.F., Krein PT., Bass R.M.: Averaged modeling of PWM converters operating in discontinuos conduction mode. IEEE Trans. Power Electron. 16(4), 482–492 (2001)
11. 11.
Chetty P.R.K.: Current injected equivalent circuit approach modeling and analysis of current programmed switching dc–dc converters DCM. IEEE Trans. Ind. Appl. 18(3), 295–299 (1982)
12. 12.
Charles W.G.: Simultaneous numerical solution of differential-algebraic equations. IEEE Trans. Circuit Theory 18(1), 89–95 (1971)
13. 13.
Opal A., Vlach J.: Consistent initial conditions of linear switched networks. IEEE Trans. Circuits Syst. 37, 364–372 (1990)
14. 14.
Kelkar S.S., Lee F.C.Y.: A fast time domain digital simulation technique for power converters application to a buck converter with feed forward compensation. IEEE Trans. Power Electron. 1(1), 21–31 (1986)
15. 15.
Luo F.L., Ye H.: Small signal analysis of energy factor and mathematical modeling for power dc–dc converters. IEEE Trans. Power Electron. 22(1), 69–79 (2007)
16. 16.
Czarkowski D., Kazimierczuk M.K.: Energy-conservation approach to modeling PWM dc–dc converters. IEEE Trans. Aerosp. Electron. Syst. 29(7), 1059–1063 (1993)
17. 17.
Bryant B., Kazimierczuk M.K.: Voltage-loop power stage transfer function with mosfet delay for boost PWM converter operating in CCM. IEEE Trans. Ind. Electron. 57(1), 347–353 (2007)
18. 18.
Cuk, S.; Middlebrook, R.D.: A general unified approach to modeling switching converter power stage. In: Proceeding of the power electronics specialists conference, pp 18–34 (1976)Google Scholar
19. 19.
Biolek D.: Modeling of periodically switched networks by mixed S–Z description. IEEE Trans. CAS-I, 44(8), 750–758 (1997)
20. 20.
Zarudi M., Shenkman A.: Analysis of common switching converters using Z-transform. IEEE Trans. Circuits Syst. I: Fund. Theory Appl. 40(9), 602–605 (1993)
21. 21.
Biolek, D.; Biolkova, V.; Dobes, J.: Modeling of switched DC–DC converters by mixed s–z description. In: IEEE international symposium on circuits and systems, (2006)Google Scholar
22. 22.
Babaei E., Seyed Mahmoodieh M.E., Mashinchi Mahery H.: Operational modes and output voltage ripple analysis and design considerations of buck-boost dc–dc converters. IEEE Trans. Ind. Electron. 59(1), 381–391 (2012)