multiquadrant converters
TRANSCRIPT
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Lecture 11: Multi Quadrant DC-DC Converters I
Multi Quadrant DC-DC Converters I
Introduction
The topics covered in this chapter are as follows:
Converter classification
Two Quadrant Converters
Converter Classification
DC-DC converters in an EV may be classified into unidirectional and bidirectional
converters. Unidirectional converters are used to supply power to various onboard loads
such as sensors, controls, entertainment and safety equipments. Bidirectional DC-DC
converters are used where regenerative braking is required. During regenerative braking
the power flows back to the voltage bus to recharge the batteries.
The buck, boost and the buck-boost converters discussed so far allow power to flow
from the supply to load and hence are unidirectional converters. Depending on the
directions of current and voltage flows, dc converters can be classified into five types:
First quadrant converter
Second quadrant converter
First and second quadrant converter
Third and fourth quadrant converter
Four quadrant converter
Among the above five converters, the first and second quadrant converrters are
unidirectional where as the first and second , third and fourth and four quadrant
converters are bidi rectional converters . In Figure 1 the relation between the load or
output voltage out V and load or output current out I for the five types of converters is
shown.
First Quadrant Second Quadrant First and Second Quadrant
out V
out I i
v
out V
out I
v
i
out V
out I
v
i
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Third and Fourth Quadrant Four Quadrant
Figure 1: Possible converter operation quadrants.
Second Quadrant Converter
The second quadrant chopper gets its name from the fact that the flow of current is from
the load to the source, the voltage remaining positive throughout the range of operation.
Such a reversal of power can take place only if the load is active, i.e., the load is capable
of providing continuous power output. In Figure 2 the general configuration of the
second quadrant converter consisting of a emf source in the load side is shown. The emf
sour ce can be a separately exci ted dc motor with a back emf of E and armature resistace
and inductance of R and L respectively.
Figure 2: Second Quadrant DC-DC Converter Figure 3: Current and voltage waveform
The load current flows out of the load. The load voltage is positive but the load current is
negative as shown in Figure 2. This is a single quadrant converter but operates in the
second quadrant. In Figure 2 it can be seen that switch 4S is turned on, the voltage E
drives current through inductor L and the output voltage is zero. The instantaneous
output current and output voltage are shown in Figure 3. The system equation when theswitch 4S is on (mode 1 ) is given by
0 oo
di L Ri E
dt (1)
out V
out I
v
iout I
out V
out I
v
iout I
out V
E
R L D
inV
oV 4S
o I
1 I
2 I
oi
t
DT T 1 D T t
inV
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With initial condition 1( 0)oi t I , gives
11 for 0
R Rt t
L L
o
E i I e e t DT
R
(2)
At time t DT the output current is given by reaches a value of 2 I , i.e., 2( )oi t DT I .
When the switch 4S is turned off (mode 2 ), a magnitude of the energy stored in the
inductor L is returned to the input voltagein
V via the diode 1 D and the output currento
I
falls. Redefining the time origin 0t , the load current is described as
out in out
diV L Ri E
dt (3)
At the beginning of mode 2 the initial value of the current is same as the final value of
current at the end of mode 1 . Hence, the initial condition at the beginning of mode 2 is 2 I .
With this initial condition, the solution of equation 3 is
21 for
R Rt t
L Lino
V E i I e e DT t T
R
(4)
At the end of mode 2 the load current becomes
2 2 3( (1 ) )i t T D T I (5)
However, at the end of mode 2 , the converter enters mode 1 again. Hence, the initial
value of current in mode 1 is 3 1 I I .
From equation 2 and equation 4 the values of1 I and
2 I is obtained as
1
1
2
1
1
1
D z
in
z
Dz z
in
z
V e E I
R e R
V e e E I
R e R
where
TR z
L
(6)
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Two Quadrant Converters
This converter is a combination of the first and second quadrant converters. Two such
converters are discussed here:
operating in first and second quadrant
operating in first and fourth quadrant
The following assumptions are made for ease of analysis:
The input voltage is greater than the load voltageinV E
The positive direction of the current is taken to be the direction from source to
load.
F ir st and Second Quadrant Converter
In Fugure 4a the configuration of a two quadrant converter providing operation in first
and second quadrants is shown.
Figure 4: First and Second Quadrant Converter
The converter works in fi rst quadrant when2S is off , diode
2 D is not conducting and1S is
on. If the switch1S is off,
2S is on and diode1 D is not forward biased, then the converter
operates in second quadrant . There are four possible modes of operation of this
converter. These four possibilities are:
i. The minimum curr ent 1 0 I and minimum 1 I and maximum 2 I currents
are positive: In this mode, only the switch1S and the diode
1 D operate. When1S is
switched on at time 0t (Figure 5a), current flows from the source to the motor
and the inductor L gains energy. At time1
t T 1S is turned off but the current
continuous to flow in the same direction and finds a closed path through the load,
L R
E
2 D
1 D
1S
2S
inV
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the freewheeling diode1 D (Figure 5b). Hence, the instantaneous output current
oi
is positive throughout and hence the average output currento
I is also positive.
Therefore, the converter operates in first quadrant . The waveforms in this
condition are shown in Figure 5c.
Figure 5a: The switch1S is on Figure 5b: The diode
1 D is conducts
Figure 5c: The current waveform
ii. The minimum cur rent1 0 I , maximum cur rent
20 I and average load cur rent
o I is positive: In this case the instantaneous load current
oi can be positive or
negative but its profile is such that the average load currento
I is positive. In order
to analyse the operation of the converter it is assumed that the converter is in
steady state. The1S is turned on at 0t , the instantaneous load current is negative
0oi and 2 D conducts it (Figure 6a). The drop across the 2 D reverse biases 1S
thus preventing conduction. The input voltagein
V is greater than the load voltage
E , hence, odi
dt is positive. When 0
oi , the switch 1S starts conduction and
continuous to do so till 1T (Figure 6b). At time 1T the switch 1S is turned off and
1S L R
E inV
oi
1 D
L R
E
oi
1T T t
1S 1 D 1S 1 D
t
1 I
2 I
Gate signal of 1S
o I
o I
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the switch 2S is turned on. At this instant the switch 2S cannot conduct because
the current is in positive direction. Since the source is isolated, 1 D freewheels the
inductive current (Figure 6c). The slope odi
dt being negative,
oi becomes zero
after some time and1 D stops conduction. When
oi becomes negative,
2S starts
conduction (Figure 6d). This condition remains till time T at which instant1S is
turned on again. The quantities1T and T are such that the average load current
o I is positive. The presence of the
2S and2 D facilitate continuous flow of current
irrespective of its direction. The current waveforms for this mode of operation are
shown in Figure 6e.
Figure 6a: Load current – ve and2 D conducts Figure 6b: Load current +ve and
1S conducts
Figure 6c: Load current +ve and1 D conducts Figure 6d: Load current -ve and
2S conducts
2
D
L R
E inV
oi
1
S L R
E inV
oi
1 D
L R
E
oi
2S
L R
E
oi
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Figure 6e: Current waveforms
iii. The minimum cur rent1 0 I , maximum cur rent
2 0 I and average load cur rent
o I is negative: The sequence of events for this case is same as case i i except that
1T and T are such that the average load currento
I is negative. Hence, the converter
operates in second quadrant. The current waveforms are shown in Figure 7.
iv. 2 0 I : In this case the instantaneous load current is always negative. Hence, the
average load current is also negative and the converter operates in the second
quadrant. The diode2 D conducts till time
1T , odi
dt being positive. The current rises
from1 I to 2 I at
1T . The switch2S starts conduction at
1T and this conduction
continuous tillT , from which moment onwards the sequence repeats. The
waveforms are shown in Figure 8.
1T T t
1S 1 D
t
1 I
2 I
Gate signal of1S
o I
Gate signal of 2S
2 D 2S
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Figure 7: Current waveforms for case iii
Figure 8: Current waveforms for case iv
The following can be observed from the four cases discussed above:
a. For the cases i and iv , during the conduction of2 D , 0
oi but the load voltage
0 E and hence, the load power is negative. This can be interpreted as that the
kinetic energy of the motor gets converted into electrical energy and fed back to
the source, thereby implying that the motor operates in regenerative braking
mode.
b. The switches1S and
2S can conduct only when their respective triggering signals
are present and the instantaneous current through them is positive.
1T T
t
2 D1 D
t
1 I
2 I
Gate signal of1S
o I
Gate signal of 2S
1S 2S
1T T t
2 D
t
1 I
2 I
o I
Gate signal of 2S
2S
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c. The average current through the load is given by
1in
o
T V E
T I
R
(7)
This current is either positve or negative, respectively, depending on whether
1in
T V E
T
or 1
in
T V E
T
.
Suggested Reading:
[1] M. H. Rashid, Power Electronics: Circuits, Devices and Applications, 3rd
edition,
Pearson, 2004
[2] V. R. Moorthi, Power Electronics: Devices, Circuits and Industrial Applications, Oxford University Press, 2007
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Lecture 12: Multi Quadrant DC-DC Converters II
Multi Quadrant DC-DC Converters II
Introduction
The following topics are covered in this lecture:
First and Second Quadrant Converter
Four Quadrant Chopper
First and Fourth Quadrant Converter
In Figure 1a the configuration of two quandrant converter capable of operating in first
and fourth quadrants is shown. Both the switches 1S and 2S are turned on for a duration
0t to 1t T (Figure 1b) and off for a duration 1t T to t T . The instantaneous
output voltage appearing across the load out v is:
1
1
0
out in
out in
v V t T
v V T t T
(1)
When the switches 1S and 2S are turned off , the current throught the inductor L continues
to flow in the same direction, making the diodes 1 D and 2 D conduct thus feeding the load
energy back to the dc source (Figure 1c). The average load voltageout
V is obtained as
1
1
1
0
1
1 ( )
T T
inout in in off
T
off
V V V dt V dt T T T T
where
T T T
(2)
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From equation 2 it can be seen that for 1 off T T ,out
V is positive and the current flows
from the DC source to load. Both the average load voltageout
V and load currento
I being
positive, the operation of the converter is in fi rst quadrant (Figure 1d). When 1 off T T ,
out V is negative but
o I is positive and the converter operates in fourth quadrant (Figure
1e).
Figure 1a: First and Fourth quadrant converter Figure 1b: When switches are on (First quadrant operation)
Figure 1c: When freewheeling diodes operate (Fourth quadrant operation)
L R E
2 D
1 D
1S
2S
inV a b
L R E
inV
a b
out V
oi
L R E
inV
a b
out V
oi
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Figure 1d: Waveforms when1 off T T
Figure 1e: Waveforms when1 off T T
1T T
t
1 2, D D
t
1 I
2 I
Gate signal of1S
o I
Gate signal of 2S
1 2,S S
t
inV
inV
t
out V
1T T
t
1 2, D D
t
1 I
2 I
Gate signal of1S
o I
Gate signal of 2S
1 2,S S
t
inV
inV
t
out V
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Four Quadrant Converters
A four quadrant converter is shown in Figure 2. The circuit is operated as a two
quadrant converter to obtain:
a. Sequence 1: First and second quadrant operation
b. Sequence 2: Third and fourth quadrant operation
Figure 2: Four quadrant converter
Sequence 1 Operation
In this mode 4S is kept permanently on . The switches 1S and 2S are controlled as per the
following four steps:
Mode 1: If 1S and 4S are turned on , the input voltage inV is applied across the load
and current flows in the positve direction from a to b Figure 3a. The
instantaneous output voltage across the load is out inv V .
Mode 2: When 1S is turned off at time 1t T , the current due to the stored
212 o Li energy of the inductor L drives through 2
D and 4S as shown in Figure
3b. The switch 2S is turned on at 1t T but it doesnot conduct because it is shorted
by 2 D .
Mode 3: The switch 2S conducts when the cureent reverses its direction (Figure
3c).
L R E
2 D
1 D1S 3 D
inV
2S
3S
4S 4 D
a b
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Mode 4: Finally when 2S is turned of f at t T , current flows in the negative
direction (Figure 3d). The converter operates in the fourth quadrant and the
power flows from load to source.
Figure 3a: Mode 1 operation of sequence 1 Figure 3b: Mode 2 operation of sequence 1
Figure 3c: Mode 3 operation of sequence 1 Figure 3d: Mode 4 operation of sequence 1
The wavforms for sequence 1 are shown in Figure 4.
Figure 4: Waveforms for sequence 1
L R E
inV
a b
out V
oi
1S
4S
oi
2 D4S
L R E
oi
4S
L R E
2S o
i
L R E
inV
1T T
t
1 4,S S
t
1 I
2 I
Gate signal of1S
Gate signal of 2S
1 4, D D
t
inV
t
out V
Gate signal of 4S
2 4, D S 4 2, D S 1 4, D D
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Sequence 2Operation
In this sequence, the converter operates in third and fourth quadrant and the switch 3S is
permanently kept on . The switches 1S and 2S are controlled as per the following four
steps:
Mode 1: 2S is turned on at 0t but starts conduction only when the current
changes sign. The diodes 2 D and 3 D conduct (Figure 5a) till the current changes
itrs sign. The instantaneous output voltage across the load is out inv V .
Mode 2: When2S is turned off at 1t T , the inductor continuous to drive the
current in the reverse direction through 3S and 1 D (Figure 5b).
Mode 3:The switch 1
S is turned
on at 1t T
but does not conduct because the
current flows in the negative direction and 1 D and 3S conduct. Once the current
changes the sign 1S and 3 D conduct 1 D (Figure 5c).
Mode 4: When 1S is turned off at t T , out inV V but positive current flows,
hence, 2 D and 3 D conduct 1 D (Figure 5d).
The waveforms for this sequence are shown in Figure 6.
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Figure 5a: Mode 1 operation of sequence 2 Figure 5b: Mode 2 operation of sequence 2
Figure 5c: Mode 3 operation of sequence 2 Figure 5d: Mode 4 operation of sequence 2
Figure 6: Waveforms for sequence 2
L R E
inV
a b
out V
oi
L R E
inV
a b
out V
oi
L R E
inV
a b
oi
L R E
inV
a b
oi
1T T t
2 3,S S
t 1 I
2 I
Gate signal of1S
Gate signal of 2S
2 3, D D
t
inV t
out V
Gate signal of 3S
1 3, D S
3 1, D S 2 3, D D
t
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Suggested Reading:
[1] M. H. Rashid, Power Electronics: Circuits, Devices and Applications, 3rd
edition,
Pearson, 2004
[2] V. R. Moorthi, Power Electronics: Devices, Circuits and Industrial Applications,
Oxford University Press, 2007