The step-down dc-dc converter, commonly known as a buck converter, is shown in Fig.1.a. It consists of dc input voltage source $V_{S},$ controlled switch $S,$ diode $D,$ filter inductor $L,$ filter capacitor $C,$ and load resistance $R .$ Typical waveforms in the convertor are shown in Fig.1.b under the assumption the the inductor current is always positive. The state of the converter in which the inductor current is never zero for any period of time is called the continuous conduction mode (CCM). It can be seen from the circuit that when the switch S is commanded to the on state, the diode D is reverse-biased. When the switch S is off, the diode conducts to support an uninterrupted current in the inductor.

The relationship among the input voltage, output voltage, and the switch duty ratio D can be derived, for instance, from the inductor voltage $v_{L}$ waveform. According to Faraday's law, the inductor volt-second product over a period of steady-state operation is zero. For the buck converter

$$\left(V_{S}-V_{O}\right) D T=-V_{O}(1-D) T-----(1)$$

Hence, the dc voltage transfer function, defined as the ratio of the output voltage to the input voltage, is

$$M_{V} \equiv \frac{V_{O}}{V_{S}}=D-----(2)$$

It can be seen from Eq. 2 that the output voltage is always smaller that the input voltage.

The dc-dc converters can operate in two distinct modes with respect to the inductor current $i_{L} .$ Figure 1.b depicts the CCM in which the inductor current is always greater than zero. When the average value of the output current is low (high $R )$ and/or the switching frequency f is low, the converter may enter the discontinuous conduction mode (DCM).

In the DCM, the inductor current is zero during a portion of the switching period. The CCM is preferred for high efficiency and good utilization of semiconductor switches and passive components. The DCM may be used in applications with special control requirements because the dynamic order of the converter is reduced (the energy stored in the inductor is zero at the beginning and at the end of each switching period). It is uncommon to mix these two operating modes because of different control algorithms. For the buck converter, the value of the filter inductance that determines the boundary between $\mathrm{CCM}$ and DCM is given by

$$L_{b}=\frac{(1-D) R}{2 f}-----(3)$$

For typical values of $D=0.5, R=10 \Omega,$ and $f=100 \mathrm{kHz}$ , the boundary is $L_{b}=25 \mu \mathrm{H} .$ For $L \gt L_{b}$ , the converter operates in the CCM.

The filter inductor current $i_{L}$ in the CCM consists of a dc component $I_{O}$ with a superimposed triangular ac component. Almost all of this ac component flows through the filter capacitor as a current $i_{c}$ . Current $i_{c}$ causes a small voltage ripple across the dc output voltage $V_{O} .$ To limit the peak-to- peak value of the ripple voltage below a certain value $V_{r},$ the filter capacitance C must be greater than

$$C_{\min }=\frac{(1-D) V_{O}}{8 V_{r} L f^{2}}-----(4)$$

At $D=0.5, V_{r} / V_{O}=1 \%, L=25 \mu \mathrm{H},$ and $f=100 \mathrm{kHz}$ , the minimum capacitance is $C_{\min }=25 \mu \mathrm{F}$ .

Equations 4 and Equation 5 are the key design equations for the buck converter. The input and output dc voltages , and the range of load resistances R are usually determined by preliminary specifications. The designer needs to determine values of passive components $L,$ and $C,$ and of the switching frequency $f.$ The value of the filter inductor L is calculated from the CCM/DCM condition using Eq. 3.

The value of the filter capacitor C is obtained from the voltage ripple condition Eq. 4 . For the compactness and low conduction losses of a converter, it is desirable to use small passive components. Equations 4 and 5 show that it can be accomplished by using a however, by the type of semiconductor switches used and by switching losses. It should also be noted that values of L and C may be altered, by the effects of parasitic components in the converter, especially by the equivalent series resistance of the capacitor.