Basic Circuit Operation

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 would be 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.

When the switch is closed, the elements carrying current are shown in red colour in Fig. 6, whereas the diode is in gray, indicting that it is in the off state. In Fig. 6(a), the capacitor is getting charged, whereas it is discharging in Fig. 6(b).

FIG. 6(a):

FIG. 6(b): Fig. 6.: Buck Converter : First State

The equations that govern the operation of the circuit in the first state are shown below.

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 Figure 7 illustrates the second state.

FIG. 7(a):

FIG. 7(b): Fig. 7.: Buck Converter : Second State

The equations that govern the operation of the circuit in the second state are shown below.

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.8 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. 8: Fig. 8.: Buck Converter : Third State

The equation that governs the operation of the circuit in the third state is shown below.

When the circuit receives a periodic signal, the response of the circuit also becomes periodic. Here it is assumed that the source voltage remains constant with no ripple, and the frequency of operation is kept fixed with a fixed duty cycle. If the RC time constant due to the load resistor and the filter capacitor is very large compared to the cycle period of the switching frequency, the output voltage is more or less constant, with no noticeable ripple. When both the input voltage and the output voltage are constant, the current through the inductor rises linearly when the switch is ON and it falls linearly when the switch is OFF. Under this condition, the current through the capacitor also varies linearly when it is getting charged or discharged.

The responses obtained for a particular set of parameters are displayed in Fig. 9. The values of parameters used are:

    Source voltage = 100 V dc,
    Switching frequency = 20 kHz,
    L = 500 mH,
    C = 500 mF,
    R = 10 W, and
    duty cycle = 0.5.

The value of 1 in a voltage plot in Fig. 9 corresponds to 100 V and the value of 1 in a current plot corresponds to 10 A. Figure 9 displays the responses over one cycle.

Fig.9: Periodic Response with a dc voltage Input

An expression for the average output voltage can be obtained as follows. It is assumed that there is continuous conduction in the inductor. Given that the cycle period is T, the ON-period is DT, and the source voltage is E,

In eqn. (6), D stands for the duty cycle. The same expression for output voltage can be obtained in another way. When the responses in the circuit are periodic, the inductor current is the same at the beginning and end of a cycle. That is,

Equation (7) can be expressed as follows:

When the switch is ON, v_L(t) = E - V_o,avg and when the diode is conducting,

v_L(t) = - V_o,avg. Therefore

On evaluation,

The change in inductor current when the switch is On can be determined as follows. Let the change in inductor current be DI, as shown in Fig. 10. In this figure, the change in output voltage has been exaggerated for sake of clarity. When the switch is ON, the voltage across the inductor can be expressed as:

When the output voltage remains steady at V_o,avg, the inductor current linearly during the ON period of the switch. Then

During the ON-period, the inductor current rises from (V_o,avg/R - DI/2) to (V_o,avg/R + DI/2). That is,

The capacitor current i_C is expressed as follows:

Now an assumption is made to find out the change in output voltage. It is assumed that the capacitor gets charged for half of the cycle period and gets discharged during the other half , as shown in Fig. 11. Since the current through the capacitor varies linearly, the average charging current is half of its peak value of the triangular waveform. The peak value of its triangular waveform is shown to be DI/2. Hence

If a periodic signal has zero dc value over its cycle period, its average is defined based only on its positive part and hence the average capacitor current is obtained as shown above. For a capacitor

Based on the average charging current and half of the cycle period as the charging period, we get the change in output voltage DV as:

Using equation (12), the above equation can be expressed as:

Assuming that the ripple in output voltage is sinusoidal, the rms value of ripple content in output voltage is:

Note that the variation in output voltage is not shown to be sinusoidal in Fig. 10.. Even though the variation appears to be triangular, equation (20) gives a better approximation of the rms value of ripple content in output voltage.

Given that source voltage = 100 V dc, switching frequency = 20 kHz, L = 500 mH, C = 500 mF, R = 10 W, and duty cycle = 0.5, the results obtained are:

V_o,avg = 50 V,

DI = 2.5 A,

DV = 31.25 mV, and

V_rms,ripple = 11.05 mV.

In order that the capacitor current and the inductor current vary linearly, it is necessary that the RC time constant should be relatively large, equal to about four or five times the cycle period. When the RC time constant is much smaller than the cycle period, the responses obtained are not linear. To illustrate, Fig. 12 displays another set of responses. The only change is that a 1 mF capacitor is used in place of the 500 mF capacitor.

When the RC time constant is small, the output voltage contains noticeable ripple. In addition, the ripple in output voltage appears to be sinusoidal, justifying the equation used for finding out the rms value of the ripple content in output voltage. Another aspect can be noticed in Fig. 12. It is that the ripple in output voltage goes through its negative half-cycle when the switch is ON. When this circuit is to be controlled by negative feedback, the feedback at the ripple frequency would become positive and closed-loop control effected without taking this aspect into account would produce a larger ripple in the output.

Fig.12: Periodic Response with a dc voltage Input

Transient Response

Fig. 13 Transient Response

Let us assume that the output voltage is zero and that there is no current through the inductor at start. If the source voltage is connected suddenly and the switch is turned ON and OFF at a fixed frequency with a preset duty cycle, the transient response of the circuit lasts for several cycles before it settles down to periodic response. The inrush current through the inductor is quite high, several times the maximum current that can flow under settled conditions. The transient response obtained over the first 10 ms with source voltage = 100 V dc, switching frequency = 20 kHz, L = 500 mH, C = 500 mF, R = 10 W, and duty cycle = 0.5, is shown in Fig. 13 . Even after 200 cycles, the response is still in the transient state.

Effect of ripple in input voltage

Usually the input to an SMPS happens to be an unregulated dc voltage provided by a rectifier-filter circuit. Such a filter contains significant ripple content at double the line frequency. The first applet in this page illustrates the response obtained when the input voltage contains ripple. In this program, the ripple frequency, the peak-to-peak ripple and the source voltage can be set. If the source voltage = 100 V, peak-to-peak ripple voltage = 20 V, and the ripple frequency = 100 Hz, then the input voltage falls from 110 V to 90 V in the first 7.5 ms and rises from 90 V to 110 V in the remaining 25% of the input cycle period. The input voltage falls linearly in the initially for 75% of the cycle period from (E + V_rip,pk-to-pk/2) to (E - V_rip,pk-to-pk/2) and then rises linearly during the remainder of the input cycle period.

When the input voltage varies cyclically, the response of the circuit is periodic over its input cycle period and it is not periodic not over the period corresponding the switching frequency. In addition, there is significant overshoot in the inductor current when the input voltage is rising linearly. It can also be seen that the output voltage has nearly the same waveform as the input voltage, which is only to be expected. Since the duty cycle is kept fixed, the output voltage would tend to rise as the input voltage rises. The response obtained with a peak-to-peak ripple voltage of 20 V at 100 Hz is shown in Fig. 14. It becomes clear from the response that closed-loop control is necessary to maintain the output voltage when the input voltage has some ripple content. The closed-loop control circuit has to be designed with care, since the duty cycle has to be continually varied to maintain the output voltage at its set value.

Fig. 14: Periodic Response over One Input Cycle Period

Effect of Step Change in Load

When there is a step change in load, the circuit goes through transient response before it settles back to periodic response. Here the circuit is allowed to be a settled state with a load resistance of 10 W. Then the load resistance is changed to 5 W and the transient response that is obtained is presented in Fig. 15. In Fig. 15, '1' on the axis for corresponds to 10 A.

When the load resistor was 10 W, the load current would have been 5 A given that the duty cycle is 0.5 and the input voltage is 100 V. It can be seen that after the step change in load resistor, the output voltage dips first and then recovers. If the change is in the other direction from 5 W to 10 W, the output voltage rises first before falling back, as shown in Fig. 16. For this figure, '1' on the axis for corresponds to 20 A.

Effective closed-loop control would reduce the transients. The under-damped nature of inductor current response can be improved.

Fig. 15: Effect of Step Load Change: Increase in Load

 
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