1. Application of Bifurcation Theory To Current Mode Controlled (CMC) DC-DC Converters

2. :Introduction Switching-mode DC-DC regulators are in general highly nonlinear systems. Switching-mode DC-DC regulators are in general highly nonlinear systems. The complete dynamic behavior of switching regulators still has to be further understood and improved. The complete dynamic behavior of switching regulators still has to be further understood and improved. The control of dc-dc converters usually takes on two approaches, namely voltage feedback control and current-programmed control. The control of dc-dc converters usually takes on two approaches, namely voltage feedback control and current-programmed control.

3. Recently, DC-DC switching regulators were observed to behave in a chaotic manner. Recently, DC-DC switching regulators were observed to behave in a chaotic manner. To explore these interesting phenomena, one needs discrete models. To explore these interesting phenomena, one needs discrete models. The mapping in closed form without approximations is derived. The mapping in closed form without approximations is derived.

4. Mathematical Modeling In this paper, we will deal with a parallel-input / parallel-output two- module synchronized current- programmed boost DC-DC converter. In this paper, we will deal with a parallel-input / parallel-output two- module synchronized current- programmed boost DC-DC converter. The current passing through the load is the sum of the currents passing through the two inductors as shown in Fig. 2.1. The current passing through the load is the sum of the currents passing through the two inductors as shown in Fig. 2.1.

5. Figure 2.1 Circuit diagram of parallel - input / parallel-output two-module current- programmed boost converter.

6. Figure 2.2 Sketch of currents and voltages waveforms appearing in the circuit.

7. Derivation of the Iterative Map for Parallel-Input / Parallel-Output Two-Module Current-Programmed Boost Converter. Derivation of the Iterative Map for Parallel-Input / Parallel-Output Two-Module Current-Programmed Boost Converter. In this section we derive a difference equation for the system which takes the form: In this section we derive a difference equation for the system which takes the form:

8. By definition:, and., and. So the nonlinear mapping of currents: Where:

9. By definition:, and., and. So the nonlinear mapping of voltages:

10. Periodic Solutions and Bifurcation Analysis In this section, we use the modern nonlinear theory, such as, bifurcation theory and chaos theory, to analyze the two- module parallel input / parallel output boost DC-DC converter using peak current-control. In this section, we use the modern nonlinear theory, such as, bifurcation theory and chaos theory, to analyze the two- module parallel input / parallel output boost DC-DC converter using peak current-control. Bifurcation theory is introduced into nonlinear dynamics by a French man named Poincare. Bifurcation theory is introduced into nonlinear dynamics by a French man named Poincare.

11. Numerical Analysis. The mapping is a function that relates the voltage and current vector The mapping is a function that relates the voltage and current vector sampled at one instant, to the vector sampled at one instant, to the vector at a previous instant. at a previous instant. The instants in question are the arrival of a triggering clock pulse. The instants in question are the arrival of a triggering clock pulse. For obtaining the bifurcation diagrams, we start by specifying an initial condition and a given. For obtaining the bifurcation diagrams, we start by specifying an initial condition and a given.

12. The iterations are continued for 750 times. The iterations are continued for 750 times. The first 500 iterations are discarded and the last 250 are plotted taking The first 500 iterations are discarded and the last 250 are plotted taking as the bifurcation parameter ( was swept from 0.5 to 5.5 ). as the bifurcation parameter ( was swept from 0.5 to 5.5 ). Figs. 3.1 and 3.2 show the bifurcation diagrams for the proposed system, where the y-axis represents the module inductor current. Figs. 3.1 and 3.2 show the bifurcation diagrams for the proposed system, where the y-axis represents the module inductor current.

13. Figure 3.1 Bifurcation diagram for the proposed system.

14. Figure 3.2 Bifurcation diagrams of and for the proposed system.

15. Figure 3.3 Bifurcation diagram for single boost converter.

16. Figure 3.4 Fundamental periodic operations at.

17. Figure 3.5 subharmonic operations at.

18. Figure 3.6 subharmonic operations at.

19. Figure 3.7 Chaotic operations at.

20. To compare different regulator systems with different compensation networks, the control (design) parameter should be independent from the compensator design. To compare different regulator systems with different compensation networks, the control (design) parameter should be independent from the compensator design. A good choice would be either the input voltage or the load resistance. A good choice would be either the input voltage or the load resistance. Hence we repeated the calculations for the same feedback system with the input voltage as the control parameter. Hence we repeated the calculations for the same feedback system with the input voltage as the control parameter.

21. Figure 3.8 Bifurcation diagram for the regulator system with as the control parameter.

22. Figure 3.9 Bifurcation diagrams of and for the regulator system with as the control parameter.

23. Figure 3.10 Bifurcation diagram for single boost converter with as the control parameter.

24. Figure 3.11 Fundamental periodic operations at.

25. Figure 3.12 subharmonic operations at.

26. Figure 3.13 subharmonic operations at.

27. Figure 3.14 Chaotic operations at.

28. The nonlinear mapping that describes the boost converter under current-mode control in continuous conduction mode has been derived. The nonlinear mapping that describes the boost converter under current-mode control in continuous conduction mode has been derived. It is unusual to find a switching regulator circuit for which the (four-dimensional) mapping is available in closed form without approximations. It is unusual to find a switching regulator circuit for which the (four-dimensional) mapping is available in closed form without approximations.Conclusions

29. When taking as the bifurcation parameter, bifurcation diagram of a single boost converter is generated and compared with that of the proposed converter. When taking as the bifurcation parameter, bifurcation diagram of a single boost converter is generated and compared with that of the proposed converter. The bifurcation point for this converter is found to be less than that of a single boost converter that uses the same component values. The bifurcation point for this converter is found to be less than that of a single boost converter that uses the same component values.

30. When taking as the bifurcation parameter for the proposed system, the period-doubling route to chaos is from right to left, as opposed to the one obtained when as the control parameter. When taking as the bifurcation parameter for the proposed system, the period-doubling route to chaos is from right to left, as opposed to the one obtained when as the control parameter. The bifurcation point for this converter is found to be greater than that of a single boost converter that uses the same components values. The bifurcation point for this converter is found to be greater than that of a single boost converter that uses the same components values.

31. The end