page_title: Buck converter
section_title: Discontinuous mode
hierarchical_section_title: Buck converter / Concept / Discontinuous mode
caption_attribution_description: Cyril BUTTAY Waveforms of current and voltage in a buck converter operating in discontinuous mode. Although I made this figure using Inkscape, this is a png version because of the presence of one greek letter. I'll use the svg version as soon as I know how to deal with greek letters...
context_page_description: A buck converter is a DC-to-DC power converter which steps down voltage from its input to its output. It is a class of switched-mode power supply typically containing at least two semiconductors and at least one energy storage element, a capacitor, inductor, or the two in combination. To reduce voltage ripple, filters made of capacitors are normally added to such a converter's output and input.
  Switching converters provide much greater power efficiency as DC-to-DC converters than linear regulators, which are simpler circuits that lower voltages by dissipating power as heat, but do not step up output current.
  Buck converters can be highly efficient, making them useful for tasks such as converting a computer's main supply voltage down to lower voltages needed by USB, DRAM and the CPU.
context_section_description: In some cases, the amount of energy required by the load is too small. In this case, the current through the inductor falls to zero during part of the period. The only difference in the principle described above is that the inductor is completely discharged at the end of the commutation cycle (see figure 5). This has, however, some effect on the previous equations.
  The inductor current falling below zero results in the discharging of the output capacitor during each cycle and therefore higher switching losses. A different control technique known as Pulse-frequency modulation can be used to minimize these losses.
  We still consider that the converter operates in steady state. Therefore, the energy in the inductor is the same at the beginning and at the end of the cycle (in the case of discontinuous mode, it is zero). This means that the average value of the inductor voltage (VL) is zero; i.e., that the area of the yellow and orange rectangles in figure 5 are the same. This yields:
  So the value of δ is:
  The output current delivered to the load () is constant, as we consider that the output capacitor is large enough to maintain a constant voltage across its terminals during a commutation cycle. This implies that the current flowing through the capacitor has a zero average value. Therefore, we have :
  Where  is the average value of the inductor current. As can be seen in figure 5, the inductor current waveform has a triangular shape. Therefore, the average value of IL can be sorted out geometrically as follows:
  The inductor current is zero at the beginning and rises during tₒₙ up to ILmax. That means that ILmax is equal to:
  Substituting the value of ILmax in the previous equation leads to:
  And substituting δ by the expression given above yields:
  This expression can be rewritten as:
  It can be seen that the output voltage of a buck converter operating in discontinuous mode is much more complicated than its counterpart of the continuous mode. Furthermore, the output voltage is now a function not only of the input voltage (Vᵢ) and the duty cycle D, but also of the inductor value (L), the commutation period (T) and the output current (Iₒ).
caption: Fig. 5: Evolution of the voltages and currents with time in an ideal buck converter operating in discontinuous mode.
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