Chapter 8: DC-DC Converters

8.1: Introduction

The topics covered in this chapter are as follows:

1. Classification of converters.
2. Principle of Step Down Operation.
3. Buck Converter with R-L-E Load.
4. Buck Converter with R Load and Filter.

8.2: Classification of Converters

The converter topologies are classified as:

1. Buck Converter: In figure 8.1a
   
   Figure 8.1a: General configuration of buck converter., a buck converter is shown. The buck converter is step down converter and produces a lower average output voltage than the DC input voltage.
2. Boost converter: In figure 8.1b
   
   Figure 8.1b:General Configuration Boost Converter., a boost converter is shown. The output voltage is always greater than the input voltage.
3. Buck-Boost converter: In figure 8.1c
   
   Figure 8.1c: General Configuration Buck-Boost Converter., a buck-boost converter is shown. The output voltage can be either higher or lower than the input voltage.

Figure 8.1a: General Configuration Buck Converter.

Figure 8.1b: General Configuration Boost Converter.

Figure 8.1c: General Configuration Buck-Boost Converter.

8.3: Principle of step down operation

The principle of step down operation of DC-DC converter is explained using the circuit shown in figure 8.2a

Figure 8.2a: Step down operation.. When the switch S₁ is closed for time duration T₁, the input voltage V_in appears across the load. For the time duration T₂ switch S₁ remains open and the voltage across the load is zero. The waveforms of the output voltage across the load are shown in figure 8.2b

Figure 8.2b: Voltage across the load resistance.

Figure 8.2a: Step down operation.

Figure 8.2b: Voltage across the load resistance.

The average output voltage is given by

(8.1)

The average load current is given by

(8.2)

where

T is the chopping period

is the duty cycle

f is the chopping frequency

The rms value of the output voltage is given by

(8.3)

In case the converter is assumed to be lossless, the input power to the converter will be equal to the output power. Hence, the input power (P_in) is given by

(8.4)

The effective resistance seen by the source is (using equation 8.2
)

(8.5)

The duty cycle D can be varied from 0 to 1 by varying T₁, T or f. Thus, the output voltage V_oavg can be varied from 0 to V_in by controlling D and eventually the power flow can be controlled.

8.4: The Buck Converter with R-L-E Load

A buck converter with R-L-E load is shown in figure 8.3

Figure 8.3: Schematics of a buck converter feeding a R-L-E load..

Figure 8.3: Schematics of a buck converter feeding a R-L-E load.

The buck converter is a voltage step down and current step up converter. The two modes in steady state operations are:

Mode 1 Operation

In this mode the switch S₁ is turned on and the diode D₁ is reversed biased, the current flows through the load. The time domain circuit is shown in figure 8.4a

Figure 8.4a: Time domain circuit of buck converter in mode 1.. The load current, in s domain, for mode 1 can be found from

(8.6)

Where

I₀₁ is the initial value of the current and I₀₁=I₁

Figure 8.4a: Time domain circuit of buck converter in mode 1.

From equation 8.6
, the current i₁(s) is given by

(8.7)

In time domain the solution of equation 8.7
is given by

(8.8)

The mode 1 is valid for the time duration 0 ≤ t ≤ T₁ Þ 0 ≤ t ≤ DT . At the end of this mode, the load current becomes

(8.9)

Mode 2 Operation

In this mode the switch S₁ is turned off and the diode D₁ is forward biased. The time domain circuit is shown in figure 8.4b

Figure 8.4b: Time domain circuit of buck converter in mode 2.. The load current, in domain, can be found from

(8.10)

Where

I₀₂ is the initial value of the load current

The current at the end of mode 1 is equal to the current at the beginning of mode 2. Hence, from equation 8.9
T₀₂ is obtained as

(8.11)

Figure 8.4b: Time domain circuit of buck converter in mode 2.

Hence, the load current in time domain is obtained from equation 8.10
as

(8.12)

The equation 8.12
is valid for the time duration DT ≤ t ≤ T

Determination of I₁ and I₂

At the end of mode 2 the load current becomes

(8.13)

At the end of mode 2, the converter enters mode 1 again. Hence, the initial value of current in mode 1 is

(8.14)

From equation 8.8
and equation 8.12
the following relation between I₁ and I₂ is obtained as

(8.15)

(8.16)

Solving equation 8.15
and equation 8.16
for I₁ and I₂ gives

(8.17)

(8.18)

Where

(8.19)

where f is the chopping frequency.

The load current waveform for a buck converter with R-L-E load is shown in figure 8.5

Figure 8.5: Current through R-L-E load..

Figure 8.5: Current through R - L - E load.

The operation of the buck converter with R-L-E load is shown in Animation 8.1

\

Animation 8.1: Buck converter for continouse conduction.

Current Ripple

The peak to peak current ripple is given by

(8.20)

To determine the current the maximum current ripple (∆I_max), the equation 8.20
is differentiated w.r.t D. This give

(8.21)

The duty ratio (D_max) at which maximum current ripple occurs is given by setting equation 8.21
equal to zero, i.e.

(8.22)

Substituting the value of D_max=0.5 in equation 8.20
gives the maximum current ripple (∆I_max) as

(8.23)

For the condition 4fL >> R,

(8.24)

Hence the maximum current ripple is given by

(8.25)

From equation 8.20
the normalized peak to peak ripple current (∆I) is given by

(8.26)

In figure 8.6

figure 8.6: Normalized current ripple versus duty ratio. the duty ratio versus normalized current ripple (∆I_normalized) curve is shown by

Figure 8.6: Normalized current ripple versus duty ratio.

From figure 8.6

figure 8.6: Normalized current ripple versus duty ratio. the following conclusions can be drawn

1. The ripple current reduces to zero as D → 0 and D → 1 .
2. The maximum ripple current occurs at D = 0 .

From figure 8.7

figure 8.7: Current ripple versus duty ratio for a fixed switching frequency., the ripple current (∆I) versus duty ratio for different values of time constant () is shown. The time constant is varied by changing the values inductance (L)

Figure 8.7: Current ripple versus duty ratio for a fixed switching frequency.

From figure 8.7

Figure 8.7: General configuration of buck converter., it can be seen that, as time constant increases (keeping the resistance constant and increasing the inductance), the magnitude of the current ripple increases. In figure 8.8

Figure 8.8: General configuration of buck converter., the current ripple versus duty ratio for different switching frequency (keeping the resistance and inductance constant) is shown. From figure 8.8

Figure 8.8: General configuration of buck converter., it can be seen that as the switching frequency increases, the current ripple decreases. Hence, from equation 8.20
, figure 8.6

Figure 8.6: Normalized current ripple versus duty ratio., figure 8.7

Figure 8.7: General configuration of buck converter. and figure 8.8

Figure 8.8: Current ripple versus duty ratio for varying switching frequency. the following conclusions can be drawn:

1. The current ripple is independent of load voltage E.
2. The ripple current reduces to zero as D → 0 and D → 1 .
3. The maximum ripple current occurs at D = 0 .
4. As time constant increases (keeping the resistance constant and increasing the inductance), the magnitude of the current ripple increases.
5. As the switching frequency increases (keeping the resistance and inductance constant), the current ripple decreases.

Figure 8.8: Current ripple versus duty ratio for a varying switching frequency.

Average and r.m.s values of input and diode currents

In mode 1 operation, the switch S₁ is on and the current passes through the switch (figure 8.4a

Figure 8.4a: Time domain circuit of buck converter in mode 1.) and the instantaneous current is given by equation 8.8
. In mode 2, the switch is off and the load current flows through the diode D₁ (figure 8.4b

Figure 8.4b: Time domain circuit of buck converter in mode 2.) and the instantaneous current is given by equation 8.12
. Hence, the average load current is given by

(8.27)

Substituting the values of i₁(t) and i₂(t) from equation 8.8
and equation 8.12
into equation 8.27
and solving it gives

(8.28)

In mode 1 the current flows through the switch, hence, the average switch current can be obtained by integrating the instantaneous load current given by equation 8.8
over a time period T, that is

(8.29)

The currents I₁ and I₂ are given by equation 8.17
and equation 8.18
. In mode 2 the diode conducts and the instantaneous current is given by equation 8.12
. The average diode current can be obtained by integrating equation 8.12
over a time period, that is

(8.30)

Fourier series analysis of the output voltage and load current for R-L-E load is given in Box 8.1.

The analysis of a buck converter is explained in example 8.1.

Discontinuous Load Current

For a buck converter with R-L-E load it is possible that the load current can reach zero during the period when the switch S₁ is off (T₁≤ t ≥ T), at a time T_x. The load current thus becomes discontinuous. In figure 8.9

Figure 8.9: Current waveform for buck converter with R-L-E load in discontinous conduction. the discontinuous load current is shown.

Figure 8.9: Current waveform for buck converter with R-L-E load in discontinous conduction.

From figure 8.9

Figure 8.9: General configuration of buck converter. it can be seen that the current through the load becomes zero before the mode 2 ends. The instantaneous current for mode 1 (when the switch S₁) is on is given by equation 8.8
. However, in mode 1 the current starts from zero (that is I₁=0) and the instantaneous current can be written as

(8.31)

In mode 2 the instantaneous load current is given by equation 8.12
. However, unlike the continuous conduction scenario, in discontinuous conduction the current exists only up to time instant T_x. Hence, the current in discontinuous conduction mode can be written as,

(8.32)

To determine I₂ we use the following condition

(8.33)

From equation 8.33
, I₂ is obtained as

(8.34)

The time instant at which the load current becomes zero (T_x) can be obtained by substituting t =T_x in equation 8.32
, that is

(8.35)

Solving equation 8.35
for T_x gives

(8.36)

The load voltage for discontinuous conduction is given by

(8.37)

The load voltage is shown in figure 8.10

Figure 8.10: Voltage across the load in a buck converter with R-L-E load in case of discontinuous conduction..

Figure 8.10: Voltage across the load in a buck converter with R - L - E load in case of discontinuous conduction.

The average value of voltage across the load is given by

(8.38)

The r.m.s value of voltage across the load is given by

(8.39)

The current ripple in case of discontinuous mode of conduction is given by

(8.40)

In case of discontinuous mode of conduction I₁ = 0 and substituting for I₂ from equation 8.34
into equation 8.40
gives

(8.41)

The average load current is given by

(8.42)

Substituting I₁ = 0 and I₂ from equation 8.34
into equation 8.41
gives

(8.43)

Substituting for V_out from equation 8.38
into equation 8.43
gives

(8.44)

In mode 1 the switch (S₁) conducts, hence the average switch current is given by

(8.45)

In mode 2 the diode conducts for an interval T₁≤ t ≤ T_x , hence the average diode current is given by

(8.46)

The operation of buck converter in discontinuous mode is shown in Animation 8.2. The Fourier analysis of the load voltage and current is given in Box 8.2.

Animation 8.2: Buck converter with discontinuous load current.

8.5: The Buck Converter with R Load and Filter

The output voltage and current of the converter contain harmonics due to the switching action. In order to remove the harmonics LC filters are used. The circuit diagram of the buck converter with LC filter is shown in figure 8.11

Figure 8.11: Buck converter with L-C filter.. There are two modes of operation as explained in the previous section.

The voltage drop across the inductor in mode 1 is

(8.47)

where i_L is the current through the inductor L_f

i_switch is the current through the switch

The switching frequency of the converter is very high and hence, i_L changes linearly. Thus, equation 8.47
can be written as

(8.48)

where T₁ is the duration for which the switch S remains on

T is the switching time period

Figure 8.11: Buck converter with L-C filter.

Hence, the current ripple ∆i_L is given by

(8.49)

When the switch S is turned off, the current through the filter inductor decreases and the current through the switch S is zero. The voltage equation is

(8.50)

where i_diode is the current through the diode D

Due to high switching frequency, the equation 8.50
can be written as

(8.51)

where T₂ is the duration in which switch S remains off the diode D conducts. Neglecting the very small current in the capacitor C_f, it can be seen that (i_load = i_switch) for time duration in which switch S conducts and (i_load = i_diode) for the time duration in which the diode D conducts.

The current ripple obtained from equation 8.30
is

(8.52)

The voltage and current waveforms are shown in figure 8.12

Figure 8.12: The voltage and current through the inductor in a buck conveter with L-C filter..

Figure 8.12: The voltage and current through the inductor in a buck conveter with L-C filter.

From equation 8.49
and equation 8.52
the following relation is obtained for the current ripple

(8.53)

Hence, from equation 8.53
the relation between input and output voltage is obtained as

(8.54)

If the converter is assumed to be lossless, then

(8.55)

The switching period T can be expressed as

(8.56)

From equation 8.56
the current ripple is given by

(8.57)

Substituting the value of V_o from equation 8.54
into equation 8.57
gives

(8.58)

Using the Kirchhoff’s current law, the inductor current is expressed as (see figure 8.11

Figure 8.11: Buck converter with L-C filter.)

(8.59)

If the ripple in load current (i_load) is assumed to be small and negligible, then

(8.60)

The average capacitor current, which flows into it for (T₁/2 + T₂/2 = T/2) is

(8.61)

The capacitor voltage is given by

(8.62)

The peak-to-peak ripple voltage of the capacitor is

(8.63)

Substituting the value of ∆I_L from equation 8.52
into equation 8.2
gives

(8.64)

Boundary between Continuous and Discontinuous Conduction

The inductor (i_L ) and the voltage drop across the inductor (e_L ) are shown in figure 8.13

Figure 8.13: The inductor voltage and current waveforms for discontinuous operation..

Figure 8.13: The inductor voltage and current waveforms for discontinuous operation.

Being at the boundary between the continuous and the discontinuous mode, the inductor current i_L goes to zero at the end of the off period. At this boundary, the average inductor current is (B referes to the boundary)

(8.65)

Hence, during an operating condition, if the average output current (I_L ) becomes less than I_LB, then I_L will become discontinuous.

Discontinuous Conduction Mode with Constant Input Voltage V_in

In applications such as speed control of DC motors, the input voltage (V_in) remains constant and the output voltage (V_out) is controlled by varying the duty ratio D. Since V_out = DV_in, the average inductor current at the edge of continuous conduction mode is obtained from equation 8.65
as

(8.66)

In figure 8.14

Figure 8.14: General configuration of buck converter. the plot of I_LB as a function of D, keeping all other parameters constant, is shown. The output current required for a continuous conduction mode is maximum at D = 0.5 and by substituting this value of duty ratio in equation 8.66
the maximum current (I_{LB, max}) is obtained as

Figure 8.14: Current versus duty ratio keeping input voltage constant.

(8.67)

From equation 8.66
and equation 8.67
, the relation between I_LB and I_LB,max is obtained as

(8.68)

To understand the ratio of output voltage to input voltage (V_out/V_in) in the discontinuous mode, it is assumed that initially the converter is operating at the edge of the continuous conduction (figure 8.13

Figure 8.13: General configuration of buck converter.), for given values of T, L, V_d and D. Keeping these parameters constant, if the load power is decreased (i.e., the load resistance is increased), then the average inductor current will decrease. As is shown in figure 8.15

Figure 8.15: Load current and voltage in a buck converter with L-C fileter in discontinuous operation is buck converter., this dictates a higher value of V_o than before and results in a discontinuous inductor current

Figure 8.15: Load current and voltage in a buck converter with L-C filter in discontinuous operation is buck converter.

Figure 8.16: Buck converter characteristics for constant input current.

In the time interval ∆₂T the current in the inductor L_f is zero and the power to the load resistance is supplied by the filter capacitor alone. The inductor voltage e_L during this time interval is zero. The integral of the inductor voltage over one time period is zero and in this case is given by

(8.69)

In the interval 0 ≤ t ≤ ∆₁T_s (figure 8.15

Figure 8.15: Load current and voltage in a buck converter with L-C fileter in discontinuous operation is buck converter.) the current ripple in L_f is

(8.70)

From figure 8.15

Figure 8.15: General configuration of buck converter. it can be seen that

(since the current falls)

(8.71)

(8.72)

Substituting the values of ∆i_L and e_L from equation 8.71
and equation 8.72
into equation 8.70
gives

(8.73)

(8.74)

Hence,

(8.75)

From equation 8.68
and equation 8.74
the ratio V_out / V_in is obtained as

(8.76)

In figure 8.16

Figure 8.16: Buck converter characteristics for constant input current. the step down characteristics in continuous and discontinuous modes of operation is shown. In this figure the voltage ratio (V_o / V_in) is plotted as a function of I_o / I_{LB, max} for various duty ratios using equation 8.55
and equation 8.76
. The boundary between the continuous and the discontinuous mode, shown by dashed line in figure 8.16

Figure 8.16: Buck converter characteristics for constant input current., is obtained using equation 8.54
and equation 8.70
.

Discontinuous-Conduction Mode with Constant V_o

In some applications such as regulated DC power supplies, V_in may vary but V_o is kept constant by adjusting the duty ratio. From equation 8.66
the average inductor current at the boundary of continuous conduction is obtained as

(8.77)

From equation 8.77
it can be seen that, for a given value of V_o the maximum value of I_LB occurs at D = 0 and is given by

(8.78)

From equation 8.77
and equation 8.78
the relation between I_LB and I_{LB, max} is

(8.79)

From equation 8.74
, the output current is obtained as

(8.80)

Solving the equation 8.80
for ∆₁ and substituting its value in equation 8.69
gives

(8.81)

8.6: Conclusion:

In this chapter the analysis of DC-DC buck converter was presented. The Fourier series analysis of the output voltage and load current for continuous and discontinuous mode of operation was also presented in this chapter.

The key points of this chapter are:

1. When the DC-DC converter feed a resistive load, the current is discontinuous.
2. An inductive load can make the load current continuous.
3. The critical value of inductance that is required for continuous current depends on the load to e.m.f. ratio.
4. The load current ripple becomes maximum for a duty ratio of 0.5.

Chapter 8 Video Lecture 1

Chapter 8 Video Lecture 2

Chapter 8 Video Lecture 3

Chapter 8 Video Lecture 4