A buck converter is used in place of a linear regulator. The reasons for using a buck converter or a step-down converter are as follows.
Now let us find out what a buck converter is. First the aim of building a buck converter is stated. Then the circuit to perform the task is developed in a progressive manner.
The aim is presented by means of a sketch, presented in Fig. 1.
As shown in the diagram in Fig. 1, the task of a buck converter is produce an output voltage that can be varied from zero to the source voltage. Ideally the conversion efficiency should be 100 %. It is also desirable that the output voltage should be without any ripple content.
In practice, it is difficult to achieve a conversion efficiency of even 85 % . The output voltage does contain some ripple. Because switching of power at a high frequency, the buck converter does generate sufficient amount of electro-magnetic interference, abbreviated as EMI.
The buck converter is introduced in this page using the evolutionary approach. Let us first consider the circuit in Fig. 2, containing a single pole double-throw switch. For the circuit in Fig. 2, the output voltage equals the input voltage when the switch is in position A and it is zero when the switch is in position B. By varying the duration for which the switch is in position A and B, it can be seen that the average output voltage can be varied, but the output voltage is not pure dc. The output voltage contains an average voltage with a square-voltage superimposed on it, as shown in Fig. 3. Usually the desired outcome is a dc voltage without any noticeable ripple content and the circuit in Fig. 2 needs to be modified.
Let us find out the dc content and the ac content of output waveform shown in Fig. 2. We can express the output waveform as the sum of two waveforms, as shown below in Fig. 3.
In Fig. 3, the cycle period of switching is T seconds. In each period, the switch is at position A for DT seconds, and it is at position for (1-D)T seconds. In such as case, D is known as the duty cycle. On seeing the waveforms in Fig. 3, the aspects that are to be made clear are the following:
The dc content is obtained as shown in Fig. 4.
An expression for the dc content in output voltage can be obtained from the plot in Fig. 4.
Equation (1) states what the shaded area per cycle is. Knowing the area and the corresponding time period, we can obtain the average or the dc content of output voltage, as shown by equation (2). Mathematically, it is preferable to calculate the average of output voltage as shown by equation (3). A graphical representation leads to better understanding, whereas the mathematical representation leads to easier calculation. Equation (4) states the conditions used by equation (3) to obtain the average of output voltage.
How do we calculate the ac content? The difference between the actual output and the average of output is the ac content. The plot in Fig. 5 expresses the ac content.
We can find the rms value of the ac content, as follows. Let us call it as Vrms,ripple.
The plot of ratio of rms value of the ac content to input voltage as a function of duty cycle can be obtained and it is displayed in Fig. 6. In Fig. 6, the ratio of rms value of the ac content to input voltage is called RC(D).
Fig. 6: Ripple Content as a function of duty cycle
It is seen from the plot in Fig. 6 that the rms value of ripple content is highest when the duty cycle is 0.5.
We can find out when the rms value of ripple content is highest as shown above. Equation (7) is obtained from equation (6).
It is possible to get the rms value of ac content in another way. Let us find the rms value of output voltage waveform shown in Fig. 4.
The rms value of output voltage is expressed by equation (10), Given that the integral of the ripple content over a cycle period is zero, we can express the rms value of ac content or ripple content as shown by equations (11) and (12). It can be seen that equations (6) and (7) are the same.
The ratio of of rms value of the ac content to maximum average output voltage can be called the ripple factor. The maximum average output voltage equals the source voltage when the duty cycle is zero. The ripple factor is defined differently from how it is in some of the texts. In some of the texts, the ripple factor is defined as the ratio of of rms value of the ac content to average output voltage, where the average output voltage is a function of duty cycle. But here the the ripple factor is defined as the ratio of of rms value of the ac content to average maximum output voltage.
The ripple factor, however it is defined, is a measure of how good the output waveform is. For output voltage of a SMPS or a rectifier circuit, the ripple factor is the goodness factor or the figure of merit, expressing how closely the output voltage corresponds to a ripple-free dc voltage. When the ripple factor of a voltage signal is zero, then that voltage signal is pure dc with no ripple. The higher the ripple factor is, the higher is the ripple content in the voltage signal.
The question now is how to improve the performance of the circuit shown in Fig.2. The improved circuit is shown next.
We can add an inductor in series with the load resistor, and we get the circuit in Fig. 3. An inductor reduces ripple in current passing through it and the output voltage would contain less ripple content since the current through the load resistor is the same as that of the inductor. When the switch is in position A, the current through the inductor increases and the energy stored in the inductor increases. When the switch is in position B, the inductor acts as a source of energy and maintains the current through the load resistor. During this period, the energy stored in the inductor decreases and its current falls. It is important to note that there is continuous conduction through the load for this circuit. If the time constant due to the inductor and load resistor is relatively large compared with the period for which the switch is in position A or B, then the rise and fall of current through inductor is more or less linear, as shown in Fig. 3.
For the circuit in Fig.3, we can form the following equations. These equations are formed with simulation in mind.
When simulation using a numerical method is resorted to, it is necessary to define the input signal, as seen by the circuit over a cycle. In Fig. 7, this input voltage is marked as vS, and this voltage can be defined by equation (13). When the switch is in position A, vS = E. When the switch is in position B, vS = 0. The sum of voltage drop across the resistor and the voltage drop across the inductor equals vS, as indicated by equation (14). The direction of loop current, i(t) is marked in Fig. 7. From equation (14), we get equation (15) by algebraic manipulation. The change in inductor current for each step in numerical simulation is expressed by equation (16). It is preferable to replace Dt by (wDt) by multiplying both the numerator and the denominator by w where w = 2 pf = (2 p)/T. In equation (16), (wDt) is the step size in radians. It is preferable to specify the step size in degrees. In equation (16), hh is the step size in degrees. The advantage of equation (16) is that it normalizes the circuit. We can assign a value of 1 to the ratio of E to R. If the actual value of E is 100 volts, and the value of R is 5 W, the scale factor for obtaining the actual current from the normalized current is E/R, which is 20. The circuit in Fig. 7 can be simulated just by specifying the ratio of (wL)/R, and the duty cycle D.
An applet that simulates the circuit in Fig. 7 is presented below. The typical value ratio of (wL)/R is about 20, but it is preferable to use a much lower value in order to see change in the output voltage over a cycle. You can change the ratio and the duty cycle, and then click on the REFRESH button to see their effect on the output voltage.
The matlab script for simulation is presented below.
% Program to simulate the circuit in Fig. 7 % Aim: To obtain the waveshape of output voltage and % to get the average, rms and the ripple content in output
disp('You may key-in a value and then press Enter when prompted');
disp('If you press Enter key without keying-in any value, then');
disp('the default value is assigned to the variable.');
disp('Default value for wL/R is 3 radians, & it is 0.5 for duty cycle.');
dfVal = 3; % Default Value for wL/R
kk1 = input('Ratio (wL)/R in radians: >');
kk2 = size(kk1);
if (kk2(1,1) == 0) tau = dfVal; else tau = kk1; end;
dfVal = 0.5; % Default Value for dutyC
kk1 = input('Duty Cycle between 0 & 1 is: >');
kk2 = size(kk1);
if (kk2(1,1) == 0) dutyC = dfVal; else dutyC = kk1; end;
% Define values to get simulation started
hh = pi/180.0; % step size 1 degree delI = 0.0; % increment in load current cur = 0.0; % Inductor Current vOut = 0.0; % output voltage across 1 ohm load resistor nKnt = 360; % One Cycle corresponds to 360 degrees iFin = 0.0; % Inductor current at the end of a cycle % Some arbitrary value for Inductor current at the start of a cycle iStart = 0.5;
dVal = 0.0; % temporary variable avgOutV = 0.0; % Average Output Voltage rmsOutV = 0.0; % RMS Output Voltage pkpkRipple= 0.0; % Peak-to-peak Ripple in Output Voltage rmsRipple = 0.0; % RMS value of Ripple in Output Voltage lowOutV = 0.0; % Lowest Output Voltage in a periodic cycle highOutV = 0.0; % Highest Output Voltage in a periodic cycle
m = 0; % Loop Variable for FOR loop
dutyDeg = dutyC*360.0; % Duty Cycle in degrees
% Looping till the response is periodic.
% When the difference between iFin and iStart is less than 0.005
% the response is almost periodic
while (abs(iFin - iStart) > 0.005)
% Inductor current starts with 0 value for the first cycle,
% assigned last value of previous cycle for the next cycle
iStart = iFin;
dVal = 0.0;
for m =1:nKnt;
if (dVal < dutyDeg) delI = ((1.0 - cur)/tau)*hh;
else delI = ((0.0 - cur)/tau)*hh; end;
cur = cur + delI;
vOut = cur;
dVal = dVal + 1.0;
end;
iFin = cur;
end;
% Periodic Response obtained and plotted
bCycle = 1; lowOutV = vOut; dVal = 0.0;
for m =1:(nKnt + 1);
if (dVal < dutyDeg) delI = ((1.0 - cur)/tau)*hh;
else
delI = ((0.0 - cur)/tau)*hh;
if (bCycle == 1)
highOutV = vOut;
bCycle = 0;
end;
end;
Deg(m) = m -1;
VoltOut(m) = vOut;
if (m <= nKnt)
cur = cur + delI;
vOut = cur;
avgOutV = avgOutV + vOut;
rmsOutV = rmsOutV + vOut*vOut;
end;
dVal = dVal + 1;
end;
plot(Deg, VoltOut)
title('Output Voltage')
xlabel('Degrees')
ylabel('Volts')
axis([0 360 0 1])
grid
avgOutV = avgOutV/360.0; rmsOutV = sqrt(rmsOutV/360.0); pkpkRipple = highOutV - lowOutV; rmsRipple = sqrt( rmsOutV*rmsOutV - avgOutV*avgOutV);
avgOutV rmsOutV pkpkRipple rmsRipple
The plot obtained from the Matlab program is presented below.
The values output by the Matlab program are presented below. The plot shown above and the results presented below are for the default values of 3 for (wL)/R and 0.5 for duty cycle.
avgOutV =0.4998 rmsOutV = 0.5201 pkpkRipple =0.4820 rmsRipple = 0.1439
It can be seen that the values obtained from the Matlab simulation and the applet are the same for the default values for (wL)/R and duty cycle.
Since the performance of the circuit in Fig. 7 is still poor, further modification is needed. We can add a capacitor across the output, as shown next.
The circuit in Fig. 8 has a filter capacitor, connected across the load resistor. A capacitor reduces the ripple content in voltage across it, whereas an inductor smoothes the current passing through it. The combination of capacitor and inductor works well. One works on the
voltage, and the other on current. Operation at high frequency leads to relatively small values
of L and C. The combined action of LC filter reduces the ripple in output to a very low level. The operation of the circuit in Fig. 8 is explained with the help of diagrams, shown in Figs 9 and 10.
The explanation presented here is based on the following assumptions.
When the components used are ideal, it means that there is no loss associated with any of them. When the circuit is in a periodic state, the responses contain terms related to the poles of the circuit. This aspect is explained later. If the inductance is sufficiently large, the rise and fall in inductor current is linear. When the switch is at position A, the inductor current rises linearly and it falls linearly when the switch is at position B. When the response of the circuit is periodic, the value of inductor current at the start of each cycle is the same. The plots in Fig. 9 explain how the inductor current varies. It is also assumed that the voltage across the capacitor is steady. The ripple content of capacitor voltage is ignored. It is a valid assumption when the ripple content is low, which is usually the case.
There is some ripple in output voltage, because of the charging and discharging of the capacitor over a cycle. We know the inductor current waveform. We can assume that the output voltage remains relatively steady and hence the current through the load resistor is a steady-value, equaling the average of inductor current. It can be seen from the circuit in Fig.8 that the inductor current is the sum of the load resistor current and the capacitor current. Hence the capacitor current is obtained by subtracting the load resistor current from the inductor current. The waveform of capacitor current is shown in Fig. 10 and it is triangular in waveshape. For about half-of the cycle, the capacitor current is positive. When the capacitor current is positive, the capacitor gets charged and its voltage rises, as shown in Fig. 11. For about half-of the cycle, the capacitor current is negative. When the capacitor current is negative, the capacitor gets discharged and its voltage falls, as shown in Fig. 10. The ripple in capacitor voltage, shown in Fig. 10, is exaggerated, to make the changes appear more prominent.
In practice, the double throw single-pole switch is replaced by a MOSFET and a diode. When the switch in the circuit shown in Fig. 8 is in position A, the current through the inductor and it decreases when the switch is in position B. It is possible to have a power semiconductor switch to correspond to the switch in position A. When switch is in position B, the inductor current free-wheels through it and hence a diode can be used for free-wheeling operation. Then only the power semiconductor switch needs to be controlled, and in practice, a pulse-width modulating IC is used for controlling the power semiconductor switch. The circuit that results is shown in Fig. 11.
The circuit with an n-channel MOSFET and a diode is shown in Fig. 11. This circuit has an n-channel MOSFET as the power semiconductor switch.
Generally any basic switched power supply consists of five standard components:
Control by pulse-width modulation, usually effected by an IC, is necessary for regulating the output. The transistor switch is the heart of the switched supply and it controls the power supplied to the load. Power MOSFETs are more suited than BJTs at power outputs of the order of 50 W. Transistors chosen for use in switching power supplies must have fast switching times and should be able to withstand the voltage spikes produced by the inductor.
An inductor is used in a filter to reduce the ripple in current. This reduction occurs because current through the inductor cannot change suddenly. When the current through an inductor tends to fall, the inductor tends to maintain the current by acting as a source. Inductors used in switched supplies are usually wound on toroidal cores, often made of ferrite or powdered iron core with distributed air-gap to minimize core losses at high frequencies.
A capacitor is used in a filter to reduce ripple in voltage. Since switched power regulators are usually used in high current, high-performance power supplies, the capacitor should be chosen for minimum loss. Loss in a capacitor occurs because of its internal series resistance and inductance. Capacitors for switched regulators are chosen on the basis of effective series resistance (ESR). Solid tantalum capacitors are good in this respect. Some other types of capacitors are also suitable. For very high performance power supplies, sometimes it is necessary to parallel capacitors to get a low enough effective series resistance.
The diode used in a switched regulator is usually referred to as the free-wheeling diode or sometimes as a catch diode. The purpose of this diode is not to rectify, but to direct current flow in the circuit and to ensure that there is always a path for the current to flow into the inductor. It is also necessary that this diode should be able to turn off relatively fast. Diodes known as the fast recovery diodes are used in these applications.
A switch-mode power supply needs a minimum load. When the current through the load resistor is sufficiently high, then the inductor carries current always. Analysis of the circuit is easier, if the inductor current is continuous.
The buck converter can be controlled in several ways. The control techniques can be classified in quite a few ways. The most general classification is presented below.
Traditionally, switch-mode power supplies have been controlled using analogue signals. Of late, there is a tendency towards digital control.
Based on the type of feedback used, a buck converter can be controlled in a few ways.
There is yet another classification, which is presented below.
In order to describe the behaviour of the circuit, it is easiest to start with analog control. The circuit can voltage feedback or voltage-mode control, and the circuit can be operated at fixed frequency with a variable duty cycle. Operation at fixed frequency with a variable duty cycle is best suited when the current through the inductor is continuous.
To achieve regulation of output voltage by varying the duty cycle of the switch and keeping the frequency of operation constant, the technique of pulse-width modulation control is used. Duty cycle refers to the ratio of the period for which the power semiconductor is kept ON to the cycle period. Usually control by pulse width modulation is the preferred method since constant frequency operation leads to optimization of LC filter and the ripple content in output voltage can be controlled within the set limits. On the other hand, if the load on the converter is below a certain level, voltage regulation of output becomes a problem and in such a case, control by frequency modulation is to be preferred. The basic operation of the circuit in Fig. 11 is now explained.
The operation of the buck converter is explained first. This circuit can operate in any of the three states as explained below. The first state corresponds to the case when the switch is ON. In this state, the current through the inductor rises, as the source voltage IS greater than the output voltage, whereas the capacitor current may be in either direction, depending on the inductor current and the load current. When the inductor current rises, the energy stored in it increases. During this state, the inductor acquires energy.
An applet is presented below to illustrate how the circuit operates. It is assumed that the frequency is held constant and the duty cycle can be varied. The components used are ideal, and the MOSFET is represented by a switch.
The circuit is shown on the top left part of the applet window. You can vary the duty cycle, and the relative values of components R, L and C. It is also possible to speed up or slow down the simulation. The waveforms on the right are shown for a cycle.The waveform with the title Ramp shows how the PWM controller operates. A saw-tooth signal is generated for every cycle. The value of the saw-tooth signal is compared with a control voltage shown in blue colour. The level of control voltage varies proportionately with the duty cycle. If the duty cycle increases, the level of control voltage goes up, and it goes down when the duty cycle falls. During a cycle, the saw-tooth voltage rises from zero to its full value. The saw-As long as the control voltage level is higher than the voltage of the saw-tooth signal, the switch is ON. When the control voltage level is less than the voltage of the saw-tooth signal, the switch is OFF. The waveforms of inductor current, the output voltage and the ripple in output voltage are also displayed. It can be seen that the ripple voltage waveform is sinusoidal in waveshape.
When the switch is closed, the elements carrying current are shown in red colour in Fig. 12(a), whereas the diode is in gray, indicting that it is in the off state. In Fig. 12(a), the capacitor is getting charged, whereas it is discharging in Fig. 12(b).
Fig. 12(a): Switch closed, Capacitor getting charged
Fig. 12(b): Switch closed, Capacitor getting discharged
The equations that govern the operation of the circuit in the first state are shown below. Let vS be the source voltage, iL be the inductor current, iC be the capacitor current and vO be the output voltage.
The voltage across the inductor is the difference between the source voltage and the output voltage. From the inductor voltage, the rate of change of inductor current is expressed by equation (17). Note that the inductor current qualifies to be a state variable, since it reflects energy stored in a system. The capacitor current is the difference of the inductor current and the load resistor current. From the capacitor current, the rate of change of capacitor voltage is expressed by equation (18). Note that the capacitor voltage qualifies to be a state variable, since it reflects energy stored in a system.
The second state relates to the condition when the switch is off and the diode is ON. In this state, the inductor current free-wheels through the diode and the inductor supplies energy to the RC network at the output. The energy stored in the inductor falls in this state. In this state, the inductor discharges its energy and the capacitor current may be in either direction, depending on the inductor current and the load current. Figures 13(a) and (13(b) illustrate the second state.
Fig. 13(a): Switch open, Capacitor getting charged
Fig. 13(b): Switch open, Capacitor getting discharged
The equations that govern the operation of the circuit in the second state are shown below.
When the diode is ideal, the voltage drop across it is zero, given that the diode is in conduction. Hence we get equation (19). Equation (20) reflects the rate of change of capacitor voltage.
When the switch is open, the inductor discharges its energy. When it has discharged all its energy, its current falls to zero and tends to reverse, but the diode blocks conduction in the reverse direction. In the third state, both the diode and the switch are OFF and Fig.14 illustrates the third state. During this state, the capacitor discharges its energy and the inductor is at rest, with no energy stored in it. The inductor does not acquire energy or discharge energy in this state.
Fig. 14: Switch open, Zero Inductor Current
The equation that governs the operation of the circuit in the third state is shown below.
The applet displayed for illustration of the basic circuit operation has been developed using equations (17) to (21).
The program code written in C can be accessed by clicking on the link following this line.
Download the program code in C.
Download the header file for C code
Even though the file extension is cpp, it is only a program in C language. It is ANSI C compliant. You can edit the program to vary the values of components and the duty cycle. The program creates a file with extension CSV. This file can be opened using a spreadsheet program such as EXCEL and the plots can be created using the data.
The same program has been coded as a Matlab, which can be download from the link presented below.
Download the Matlab Code
The plots obtained are presented below. The values used to produce these waveforms are:
E = 100 V, R = 10 W , L = 500 mH, C = 500 mF, frequency = 20 kHz, and D = 0.5.
Fig. 15: Plots obtained from the Matlab Program
It is necessary to illustrate how the PWM takes place. This aspect is explained next.
A circuit is presented to explain the operation of pulse-width modulation is presented in Fig. 16. Typically an integrated circuit for pulse-width modulation contains the error amplifier, the saw-tooth generator and the comparator. In steady-state, the reference voltage and the voltage fedback to the error amplifier are equal. In this circuit, the circuit is shown to have unity feedback, but in practice a fraction of output may be fed back. The output of the error amplifier is such that the voltage fed back equals the reference voltage. The saw-tooth generator produces a saw-tooth waveform at a frequency, determined by the values of R and C. As the values of R and C increase, the period of saw-tooth signal increases, leading to a lower frequency. In a cycle, the comparator output is high, when the error amplifier output voltage is higher than the saw-tooth signal and hence the error amplifier output voltage determines the duty cycle. If the voltage fed back is less than the reference voltage, the output voltage of the error amplifier increases, the duty cycle increases and the voltage across the load resistor increases. If the voltage fed back is higher than the reference voltage, the output voltage of the error amplifier decreases, the duty cycle decreases and the voltage across the load resistor falls. Ultimately the duty cycle gets adjusted such that the the reference voltage and the voltage fedback to the error amplifier are equal.
A practical PWM IC is more sophisticated than that described here. It has to be more sophisticated in order to ensure reliable operation. The block diagram of a practical PWM is presented later in the next page.
The applet presented above to illustrate the operation of the basic circuit also illustrates what pulse-width modulation is.
This page has presented the basic aspects of a buck converter. The next page presents the analysis of this circuits and the closed loop operation of this circuit. In order to explain how the closed-loop control can be brought about, it is shown how his circuit can be modeled. The next page also contains more information on PWM operation and variable frequency control of a buck converter.