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Nonlinear systems, such as switching DC-DC boost or buck converters, haverich dynamics. A simple one-dimensional discrete-time model is used to analyzethe boost or buck converter in discontinuous conduction mode. Seven differentcontrol schemes (open-loop power stage, voltage mode control, current modecontrol, constant power load, constant current load, constant-on-time control,and boundary conduction mode) are analyzed systematically. The linearizeddynamics is obtained simply by taking partial derivatives with respect todynamic variables. In the discrete-time model, there is only a single pole andno zero. The single closed-loop pole is a linear combination of three terms:the open-loop pole, a term due to the control scheme, and a term due to thenon-resistive load. Even with a single pole, the phase response of thediscrete-time model can go beyond -90 degrees as in the two-pole averagemodels. In the boost converter with a resistive load under current modecontrol, adding the compensating ramp has no effect on the pole location.Increasing the ramp slope decreases the DC gain of control-to-output transferfunction and increases the audio-susceptibility. Similar analysis is applied tothe buck converter with a non-resistive load or variable switching frequency.The derived dynamics agrees closely with the exact switching model and the pastresearch results.

Discrete-Time Poles and Dynamics of Discontinuous Mode Boost and Buck Converters Under Various Control Schemes