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the current at the limit between continuous and discontinuous mode is I o lim = V i T 2 L D ( 1 − D ) = I o lim 2 | I o | D ( 1 − D ) {\displaystyle \scriptstyle I_{o_{\text{lim}}}={\frac {V_{i}\,T}{2L}}D\left(1-D\right)={\frac {I_{o_{\text{lim}}}}{2\left|I_{o}\right|}}D\left(1-D\right)} . When it is off, the diode is forward biased (we consider the continuous mode operation), therefore V S = V i − V o {\displaystyle \scriptstyle V_{S}=V_{i}-V_{o}} . From this equation, it can be seen that the output voltage of the converter varies linearly with the duty cycle for a given input voltage. However, parasitic resistances exist in all circuits, due to the resistivity of the materials they are made from.
The output current delivered to the load ( I o {\displaystyle I_{\text{o}}} ) is constant, as we consider that the output capacitor is large enough to maintain a constant voltage across its terminals during a commutation cycle. A mechanical analogy for a buck converter would be to pedal a bicycle in single, strong bursts (Force ~ Voltage), and let the bicycle roll in between (inertia ~ inductor). 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. As told at the beginning of this section, the converter operates in discontinuous mode when low current is drawn by the load, and in continuous mode at higher load current levels. As the duty cycle D {\displaystyle D} is equal to the ratio between t on {\displaystyle t_{\text{on}}} and the period T {\displaystyle T} , it cannot be more than 1.The two circuit configurations of a buck converter: on-state, when the switch is closed; and off-state, when the switch is open (arrows indicate current according to the direction conventional current model).
As I o lim {\displaystyle \scriptstyle I_{o_{\text{lim}}}} is the current at the limit between continuous and discontinuous modes of operations, it satisfies the expressions of both modes. This section may be written in a style that is too abstract to be readily understandable by general audiences. A buck converter or step-down converter is a DC-to-DC converter which decreases voltage, while increasing current, from its input ( supply) to its output ( load). 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).However, this drawback is of no consequence if the power supply is isolated from the load circuit (if, for example, the supply is a battery) because the supply and diode polarity can simply be reversed. By integrating Id t (= d Q ; as I = d Q/d t, C = Q/ V so d V = d Q/ C) under the output current waveform through writing output ripple voltage as d V = Id t/ C we integrate the area above the axis to get the peak-to-peak ripple voltage as: Δ V = Δ I T/8 C (where Δ I is the peak-to-peak ripple current and T is the time period of ripple.
The voltage across the inductor is V L = − V o {\displaystyle V_{\text{L}}=-V_{\text{o}}} (neglecting diode drop). Use the higher-than-needed voltage of the source to quickly induce a current into an inductor ("on" in fig. With V L {\displaystyle V_{\text{L}}} equal to V i − V o {\displaystyle V_{\text{i}}-V_{\text{o}}} during the on-state and to − V o {\displaystyle -V_{\text{o}}} during the off-state.the normalized current, defined by | I o | = L T V i I o {\displaystyle \left|I_{\text{o}}\right|={\frac {L}{TV_{\text{i}}}}I_{\text{o}}} . the normalized voltage, defined by | V o | = V o V i {\displaystyle \scriptstyle \left|V_{o}\right|={\frac {V_{o}}{V_{i}}}} . The simplified analysis above, does not account for non-idealities of the circuit components nor does it account for the required control circuitry. Capacitor selection is normally determined based on cost, physical size and non-idealities of various capacitor types.
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