Buck converter

A buck converter or step-down converter is a DC-to-DC converter which decreases voltage, while increasing current, from its input to its output. It is a class of switched-mode power supply. Switching converters provide much greater power efficiency as DC-to-DC converters than linear regulators, which are simpler circuits that dissipate power as heat, but do not step up output current. The efficiency of buck converters can be very high, often over 90%, making them useful for tasks such as converting a computer's main supply voltage, which is usually 12V, down to lower voltages needed by USB, DRAM and the CPU, which are usually 5, 3.3 or 1.8V.
Buck converters typically contain at least two semiconductors and at least an inductor as energy storage element, usually in combination with a capacitor. To reduce voltage ripple, filters made of capacitors are normally added to such a converter's output and input. Its name derives from the inductor that “bucks” or opposes the supply voltage.
Buck converters typically operate with a switching frequency range from 100 kHz to a few MHz. A higher switching frequency allows for use of smaller inductors and capacitors, but also increases lost efficiency to more frequent transistor switching.

Theory

The basic concept of a buck converter is:

Use the higher-than-needed voltage of the source to quickly induce a current into an inductor.
Disconnect the source and use the inertia of the current in the inductor to provide more current than the source delivers. To close the circuit with the source disconnected, a second switch, usually a diode, is needed.

During on-state, the source may need to momentarily provide more current than its rating for constant load allows, but the on-time is too short for the source to take damage. During off-state, no current is drawn from the source, and the components can cool down. The average current draw over both states needs to be below the source specification.
To even out voltage spikes from the switching between on-state and off-state, a capacitor is used on the output side.
A mechanical analogy for a buck converter would be to pedal a bicycle in single, strong bursts, and let the bicycle roll in between.
The basic operation of the buck converter has the current in an inductor controlled by two switches. In a physical implementation, these switches are realized by a transistor and a diode, or two transistors.

Idealised case

The conceptual model of the buck converter is best understood in terms of the relation between current and voltage of the inductor. Beginning with the switch open, the current in the circuit is zero. When the switch is first closed, the current will begin to increase, and the inductor will produce an opposing voltage across its terminals in response to the changing current. This voltage drop counteracts the voltage of the source and therefore reduces the net voltage across the load. Over time, the rate of change of current decreases, and the voltage across the inductor also then decreases, increasing the voltage at the load. During this time, the inductor stores energy in the form of a magnetic field.
If the switch is opened while the current is still changing, then there will always be a voltage drop across the inductor, so the net voltage at the load will always be less than the input voltage source. When the switch is opened again, the voltage source will be removed from the circuit, and the current will decrease. The decreasing current will produce a voltage drop across the inductor, and now the inductor becomes a current source. The stored energy in the inductor's magnetic field supports the current flow through the load. This current, flowing while the input voltage source is disconnected, when appended to the current flowing during on-state, totals to current greater than the average input current.
The "increase" in average current makes up for the reduction in voltage, and ideally preserves the power provided to the load. During the off-state, the inductor is discharging its stored energy into the rest of the circuit. If the switch is closed again before the inductor fully discharges, the voltage at the load will always be greater than zero.

Continuous mode

Buck converters operate in continuous mode if the current through the inductor never falls to zero during the commutation cycle. In this mode, the operating principle is described by the plots in figure 4:

When the switch pictured above is closed, the voltage across the inductor is. The current through the inductor rises linearly. As the diode is reverse-biased by the voltage source, no current flows through it;
When the switch is opened, the diode is forward biased. The voltage across the inductor is . Current decreases.

The energy stored in inductor L is
Therefore, it can be seen that the energy stored in L increases during on-time as increases and then decreases during the off-state. L is used to transfer energy from the input to the output of the converter.
The rate of change of can be calculated from:
With equal to during the on-state and to during the off-state. Therefore, the increase in current during the on-state is given by:
where is a scalar called the duty cycle with a value between 0 and 1.
Conversely, the decrease in current during the off-state is given by:
Assuming that the converter operates in the steady state, the energy stored in each component at the end of a commutation cycle T is equal to that at the beginning of the cycle. That means that the current is the same at and at .
So, from the above equations it can be written as:
The above integrations can be done graphically. In figure 4, is proportional to the area of the yellow surface, and to the area of the orange surface, as these surfaces are defined by the inductor voltage. As these surfaces are simple rectangles, their areas can be found easily: for the yellow rectangle and for the orange one. For steady state operation, these areas must be equal.
As can be seen in figure 4, and.
This yields:
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. As the duty cycle is equal to the ratio between and the period, it cannot be more than 1. Therefore,. This is why this converter is referred to as step-down converter.
So, for example, stepping 12 V down to 3 V would require a duty cycle of 25%, in this theoretically ideal circuit.

Discontinuous mode

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. 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. 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. This means that the average value of the inductor voltage 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 I_L can be sorted out geometrically as follows:
The inductor current is zero at the beginning and rises during t_on up to I_Lmax. That means that I_{L_max} is equal to:
Substituting the value of I_Lmax 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 and the duty cycle D, but also of the inductor value, the commutation period and the output current.

From discontinuous to continuous mode (and vice versa)

The converter operates in discontinuous mode when low current is drawn by the load, and in continuous mode at higher load current levels. The limit between discontinuous and continuous modes is reached when the inductor current falls to zero exactly at the end of the commutation cycle. Using the notations of figure 5, this corresponds to :
Therefore, the output current at the limit between discontinuous and continuous modes is :
Substituting I_Lmax by its value:
On the limit between the two modes, the output voltage obeys both the expressions given respectively in the continuous and the discontinuous sections. In particular, the former is
So I_olim can be written as:
Let's now introduce two more notations:

the normalized voltage, defined by. It is zero when, and 1 when ;
the normalized current, defined by. The term is equal to the maximum increase of the inductor current during a cycle; i.e., the increase of the inductor current with a duty cycle D=1. So, in steady state operation of the converter, this means that equals 0 for no output current, and 1 for the maximum current the converter can deliver.

Using these notations, we have:

in continuous mode:
:
in discontinuous mode:
:

the current at the limit between continuous and discontinuous mode is:
Therefore, the locus of the limit between continuous and discontinuous modes is given by:
These expressions have been plotted in figure 6. From this, it can be deduced that in continuous mode, the output voltage does only depend on the duty cycle, whereas it is far more complex in the discontinuous mode. This is important from a control point of view.
On the circuit level, the detection of the boundary between CCM and DCM are usually provided by an inductor current sensing, requiring high accuracy and fast detectors as: