multiquadrant converters

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NPTEL – Electrical Engineering – Introduction to Hybrid and Electric Vehicles

Joint initiative of IITs and IISc – Funded by MHRD Page 24 of 55

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




Joint initiative of IITs and IISc – Funded by MHRD 5 of 55

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





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)





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





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





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





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





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





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





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)





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





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





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





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





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.





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





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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