# DC-DC a boost converter is shown. TheDC-DC a boost converter is shown. The

DC-DC CONVERTERS

Classification
of DC- DC Converters

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The converter
topologies are classified as:

Ø
Buck Converter: In Figure 1a a buck converter is
shown. The buck converter is step down converter and produces a lower average output
voltage than the dc input voltage.

Ø
Boost
converter: In Figure 1b a
boost converter is shown. The output voltage is always greater than the input
voltage.

Ø
Buck-Boost converter: In Figure 1c a buck-boost converter
is shown. The output voltage can be either higher or lower than the input
voltage.

Figure 1a: General
Configuration Buck Converter       Figure
1b: General Configuration Boost Converter

Figure 1c: General Configuration Buck-Boost Converter

Principle of Step Down Operation

The principle of step
down operation of DC-DC converter is explained using the circuit shown in Figure 2a. When the switch
S1 is
closed for time duration T1, the input
voltage Vin appears across the load. For the
time duration T2 is
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
2b.

Figure 2a: Step down operation                                     Figure 2b: Voltage across the

The average output voltage is given by

current is given by

Where

is the chopping period

D = T1/ T2

F is the chopping frequency

The rms value of the output voltage is
given by

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) is given by

The effective resistance seen by the
source is

The duty cycle can be varied from 0 to 1
by varying T1 , T or f . Thus, the output voltage V0avg
can be varied from 0 to Vin by controlling D and eventually the
power flow can be controlled.

The Buck Converter with RLE 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 Figure3. The load current, in s-domain, for mode 1 can
be found from

Where

I0 is the initial value of
the current and I0 =I1.

Figure
3: Time domain circuit of buck converter in mode 1

From equation 6, the current I1(s)
is given by

In time domain the solution of equation 7 is given by

The mode1 is valid for the time duration
0? t ?T1
? 0? t ?DT. At the end
of this mode, the load current becomes

i1(t
= T1 = DT) = I2                                                                                                                                    (9)

Mode 2 Operation

Figure 4: Time domain circuit
of buck converter in mode 2

In this mode the switch 1 S is
turned off and the diode 1
D is forward biased. The time domain circuit is shown
in Figure4. The load current, in
s domain, can be found from

Where

I02 is the initial value of load
current.

The current at
the end of mode1
is equal to the current at the beginning of mode 2.

Hence, from equation9  I02 is
obtained as

I02 = I2                                                                                                                                                                              (11)

Hence, the load current is time domain is obtained
from equation 10 as

Determination of I1 and I2

At the end of mode 2 the load current becomes

i2 = (t = T2 =
(1-D)T) = I3                                                                                                                                       (13)

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

I01 = I3 = I1                                                                                                                                                                    (14)

From equation 8 and equation 12
the following relation between and is obtained as

Solving equation 15 and equation
16 for I1 and I2 gives 1

Where

Where f is the chopping frequency.

Current Ripple

The
peak to peak current ripple is given by

In case fL >>R , a?0 . Hence, for the limit a?0 equation 20 becomes

To determine the maximum current ripple
(?Imax ), the equation 20a is
differentiated w.r.t. D. The value of ?Imax is given by

For the condition , 4fL>> R

Hence, the maximum current ripple is
given by

If equation 20b is used to determine the maximum current ripple, the
same result is obtained.

Continuous and Discontinuous Conduction
Modes

In case of large off time, particularly
at low switching frequencies, the load current may be discontinuous, i.e. i2(t
= T2 = (1-D)T)  will be zero.
The necessary condition to ensure continuous conduction is given by

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 5.
There are two modes of operation as explained in the previous section.

The voltage drop across the inductor in mode 1 is

Where iL is the current through the inductor Lf

isw is the current through
the switch

The switching frequency of the converter is very
high and hence, iL changes linearly.

Thus, equation 25 can be written as

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

T is the switching time
period

Figure 5: Buck converter with resistive load and
filter           Figure 6: Voltage and
current waveform

Hence, the current ripple ?IL
is given by

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

Where iD is the
current through the diode D

Due to high switching
frequency, the equation 28 can
be written as

Where Toff is the
duration in which switch S remains off the diode D conducts Neglecting the very
small current in the capacitor Cf, it can be seen that

i0 = iSW for
time duration in which switch conducts and

i0 = iD
for the time duration in which the diode D conducts

The current ripple obtained
from equation 29 is

The voltage and current
waveforms are shown in Figure 6.

From equation 27 and equation
30 the following relation is obtained for the current ripple

Hence, from equation 31 the relation between input
and output voltage is obtained as

If the converter is assumed
to be lossless, then

The switching period T can
be expressed as

From equation 34 the current ripple is given by

Substituting the value of V0
from equation 32 into equation 35 gives

Using the Kirchhoff’s
current law, the inductor current iL is expressed as

If the ripple in load
current (i0 ) is assumed to be small and negligible, then

The incremental voltage ?VC
across the capacitor (Cf ) is associated with incremental charge ?Q
by the relation

The area of each of the
isoceles triangles representing ?Q in Figure
6 is given by

Combining equation 39 and equation 40 gives

Substituting the value of ?IL
from equation 31 into equation 41 gives

MODELING AND DESIGN OF BASIC DC-DC CONVERTERS:

BUCK CONVERTER:

Figure7 circuit diagram of
buck converter

Figure8 input and output
waveforms of buck converter

Boost converter:

Figure9 circuit diagram of
boost converter

Figure10 input and output
waveforms of boost converter

Buck Boost converter:

Figure11 circuit diagram of
buck boost converter

Figure12 input and output
waveforms of buck boost converter