7 Chopper Controlled DC Drives-closed Loop

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Closed-loop Control of DC Drives with
Chopper
By
Dr. Ungku Anisa Ungku Amirulddin
Department of Electrical Power Engineering
College of Engineering

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

1

Outline
 Closed Loop Control of DC Drives with Choppers
 Current Control for DC Drives with Choppers
 Pulse-Width-Modulation (PWM) Controller
 Hysteresis-Current Controller
 Comparison between PWM and Hysteresis Controller
 Transfer Function of PWM-Controlled Chopper
 Two-quadrant
 Four-quadrant
 Design of Controllers
 References
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

2

Closed Loop Control of DC Drives
 Closed loop control is when the duty cycle is varied
automatically by a controller to achieve a reference
speed or torque
 This requires the use of sensors to feed back the
actual motor speed and torque to be compared with
the reference values
Reference
signal

+

Plant

Controller


Output
signal

Sensor
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

3

Closed Loop Control of DC Drives
 Feedback loops may be provided to satisfy one or more of

the following:
 Protection
 Enhancement of speed response
 Improve steady-state accuracy

 Variables to be controlled in drives:
 Torque – achieved by controlling current
 Speed
 Position
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

4

Closed Loop Control of DC Drives
For DC Drive,
this can be:
 Flexible – outer loops can be added/removed depending on control •Controlled
rectifier or
requirements.
•DC-DC
 Control variable of inner loop (eg: speed, torque) can be limited by
converter

 Cascade control structure

limiting its reference value

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

5

Closed Loop Control for DC
Drives with Choppers
 Outer speed loop very similar to that in the controlled

rectifier dc drive
 Inner current control loop – different
Current
Control Loop

Speed Control
Loop

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

6

Current Control for DC Drives
with Choppers

 Current control loop is used to control torque via

armature current (ia)

 Output of current controller determines duty cycle (i.e.

switching) of DC-DC converter

 Current controller can be either:
 Pulse-Width-Modulation (PWM) Controller
 contain PI controllers, i.e. linear
 fixed switching frequency
 Hysteresis (bang-bang) controller
 on-off controllers, i.e. non-linear
 varying switching frequency

 Selection of controller affects current control loop

transient response
 Hence, affects speed loop bandwidth.

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

7

Current Control for Chopper
Drives – PWM Controller
 In two quadrant chopper, upper and

lower switches are complementary
 Only ONE control signal required
 Current error is passed to PI controller
to produce control voltage vc
 vc is then passed to a PWM circuit to
produce the switching signal q.
 q = 1  T1 ‘on’, T2 ‘off’  Va = Vdc
 q = 0  T1 ‘off’, T2 ‘on’  Va = 0

1
q
0

vtri
i a* +

ierr


PI

vc

T1 ‘on’, Va = Vdc

vc < Vtri

T2 ‘on’, Va = 0

+
T1

Vdc

D1

ia
Vdc

+
T2


Pulse Width
Modulator
(PWM)

vc > Vtri

D2

Va
-

q

ia
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

8

Current Control for Chopper
Drives – PWM Controller
In the PWM circuit:
 vc is compared with a triangular
waveform
 if vc > Vtri  ‘on’ signal is produced
(q = 1)
 if vc < Vtri  ‘off’ signal is produced
(q = 0)

1
q
0

vc > Vtri
Ttri

vc

vc > Vtri

vc < Vtri

1

(1)

q
0

 Chopper switching frequency is fixed

by triangular waveform frequency
regardless of operating conditions
 Bandwith of current loop controller
is limited by frequency of Vtri
Dr. Ungku Anisa, July 2008

vc < Vtri

EEEB443 - Control & Drives

ton

Vdc

va
0

q=1
T1 ‘on’, va = Vdc

q=0
T2 ‘on’, va = 0

9

Current Control for Chopper Drives
– PWM Controller
Ttri

In the PWM circuit:
 Average value of q over a cycle

determines duty cycle  of
chopper:

1

Ttri

t Ttri

 q dt 
t

ton, T1
Ttri

vc

T


0

 Average armature voltage:

1
Va   va dt  Vdc
T 0

1

q

ton

va

Vdc

Va
0

va switches between Vdc and 0

average armature voltage Va depends on duty cycle (i.e. how long T1 is on)
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

10

Current Control for Chopper Drives
– PWM Controller
 PWM controls chopper duty cycle  once in every
cycle
 Frequency of Va fixed by frequency of Vtri
 Hence, chopper is a variable voltage source with
average current control
 Instantaneous current control is not exercised
 Current can exceed maximum armature current
between two consecutive switching
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

11

Current Control for Chopper Drives
– Hysteresis Controller
1 ia  ia* - ia
q
* + i
i

i
0
a
a
a


 Instantaneous current control
 Current controlled within a narrow

band of excursion from the desired
value ia*
 Hysteresis window determines
allowable deviation of ia

+
T1

ia

Vdc

Hysteresis
Controller 
i a*

ia

+
T2

ia*
ia
ia

ierr

Vdc

D1

D2

Va
-

q

q

ia
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

12

Current Control for Chopper
Drives – Hysteresis Controller
 Actual current ia compared with reference

ia  ia* + ia

current ia* to obtain error signal ierr
 If ia  ia* + ia  q = 0, T2 ‘on’ and Va = 0
 If ia  ia* - ia  q = 1, T1 ‘on’ and Va = Vdc

1 ia  ia* - ia
q
* + i
i

i
0
a
a
a

 Value of ia can be externally

set or made to be a fraction of ia
 Chopper switching frequency is
not fixed
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

ia*
ia
ia

ia

ia  ia* - ia

q=1
T1 ‘on’, va = Vdc

q

q=0
T2 ‘on’, va = 0
13

Current Control for Chopper Drives
– Qualitative Comparison
Characteristics

Switching
frequency

Hysteresis Controller

PWM Controller

Varying

Fixed
(follows sawtooth waveform
frequency i.e. carrier frequency)

Switching losses

High

Low

(due to varying
switching frequency)

Speed of
response
Ripple current

Fastest

Fast

(due to instantanous
change in current)

Adjustable

Fixed

(depends on hysteresis
window ia )

Filter size
Dr. Ungku Anisa, July 2008

Depends on ia
EEEB443 - Control & Drives

Small

Preferred method !

14

Closed Loop Control for DC Drives
with Choppers
 Controller design procedure:
Obtain the transfer function of all drive subsystems

1.
a)

b)
c)

DC Motor & Load
Current feedback loop sensor
Speed feedback loop sensor

Exactly the same as before
(i.e. transfer functions obtained
in closed loop control using
controlled rectifier)

Design torque (current) control loop first

2.


Two options to choose from:
A.
Hysteresis Controller – to design just choose value of ia
B.
PWM Controller (contains PI controller)
i.
determine transfer function of PWM-controlled chopper
ii.
design PI controller using the same procedure as in closed
loop control using controlled rectifier

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

15

Closed Loop Control for DC Drives
with Choppers
 Controller design procedure (continued):
Then design the speed control loop

3.
i.

ii.

Obtain 1st order model of the designed current controller
Design the speed PI controller using the same procedure as
in closed loop control using controlled rectifier

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

16

Transfer Function of PWM-Controlled
Chopper
 PWM current controller is preferred over Hysteresis

Controller
 Before we can design the PI controller, need to obtain linear
relationship between control input vc and average armature
voltage Va for PWM method
Need transfer function
for PWM-controlled
chopper
vtri
i a* +


PI

vc

Pulse Width
Modulator
(PWM)

q

Chopper

Va

DC
motor

ia

ia

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

17

Transfer Function of PWM-Controlled
Two-quadrant Chopper
 Need to obtain linear relationship between control input vc

and average armature voltage Va for PWM method
Case 1:

1
q
0

vc   Vtri
Vtri

vc > Vtri
vc < Vtri


vc

-Vtri

T1 off all the time
i.e. ton, T1 = 0

1

Ttri
Dr. Ungku Anisa, July 2008

q0
t Ttri

 q dt 
t

ton, T1
Ttri

0

EEEB443 - Control & Drives

-Vtri

vc
18

Transfer Function of PWM-Controlled
Two-quadrant Chopper
Case 2:
vc  0
Vtri

vc

-Vtri

T1 on ½ cycle
i.e.
ton, T1 = 0.5Ttri

1
q
0

1

Ttri

Dr. Ungku Anisa, July 2008

 q dt 
t

ton, T1
Ttri

vc < Vtri


1, for 1/2 a cycle
q
0, for 1/2 a cycle
t Ttri

vc > Vtri

0.5

 0.5

EEEB443 - Control & Drives

-Vtri

vc
19

Transfer Function of PWM-Controlled
Two-quadrant Chopper
Case 3:
vc  Vtri

1
q
0

Vtri

vc

-Vtri

T1 on all the time
i.e. ton, T1 = Ttri

1

Ttri

 q dt 
t

1

ton, T1
Ttri

0.5

1
-Vtri

Dr. Ungku Anisa, July 2008

vc < Vtri


q 1
t Ttri

vc > Vtri

EEEB443 - Control & Drives

Vtri

vc
20

Transfer Function of PWM-Controlled
Two-quadrant Chopper
 Relationship between  and vc :
1
  0.5 
vc
2Vtri



(2)

0.5

 For the two-quadrant chopper:
Vdc
Va  Vdc  0.5Vdc 
vc
2Vtri

1

vc

+Vtri

-Vtri

(3)

 Hence, considering only the term due to vc, the two–quadrant

chopper gain is:
Va Vdc
Kr  
vc 2Vtri
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

(4)

21

Transfer Function of PWM-Controlled
Four-quadrant Chopper
 Recap Chopper operation:
 Positive current:
 Va = Vdc when T1 and T2 on
 Va = 0 when current
freewheels through T2 and D4

+
T1

D1

D3

+ Va -

T3

Vdc
T4

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

D4

D2

T2

22

Transfer Function of PWM-Controlled
Four-quadrant Chopper
 Recap Chopper operation:
 Positive current:
 Va = Vdc when T1 and T2 on
 Va = 0 when current
freewheels through T2 and D4
 Negative current:
 Va = -Vdc when T3 and T4 on +
 Va = 0 when current
freewheels through T4 and D2
Vdc
 Output voltage can swing

between:
 Vdc and -Vdc
 Vdc and 0
Dr. Ungku Anisa, July 2008

T1

EEEB443 - Control & Drives

D3

+ Va -

T4

-

D1

D4

D2

T3

T2

23

Transfer Function of PWM-Controlled
Four-quadrant Chopper
 Need to obtain linear relationship between control input vc

and average armature voltage Va for PWM method
 Four quadrant chopper has two legs, so it requires two
switching signals (one for each leg)
 Depending on relationship between the two switching signals,
4-quadrant chopper has two switching schemes:
 Bipolar switching
+
D1
D3
T1
 Unipolar switching
T3
+ Va Vdc
 Switching scheme
determines output
T4
T2
D2
D4

voltage swing between
Vdc and -Vdc or Vdc and 0.
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

Leg A

Leg B

24

Transfer Function of PWM-Controlled
Four-quadrant Chopper (Bipolar Switching)
Bipolar Switching PWM
 Leg A and Leg B obtain switching signals from the same control signal vc
 Switching of Leg A and Leg B are always complementary
q

1 vc > Vtri
q
0 vc < Vtri

Leg A

 q = 1,q =0

+
T1
vtri

D3

+ Va -

T3

 T1 on, T2 on
 Va= Vdc

T2

 q = 0, q =1
 T4 on, T3 on
 Va= -Vdc

Vdc
T4


vc
q
Dr. Ungku Anisa, July 2008

D1

EEEB443 - Control & Drives

D4

D2

Leg B
25

Transfer Function of PWM-Controlled
Four-quadrant Chopper (Bipolar Switching)
 Bipolar Switching PWM

Va = Va+- Va-

Leg A

q

2vtri

vc

+
T1

D1

D3

+ Va -

T3

Vdc

vtri

Vdc

Va+

0
Vdc

T4



D2

D4

T2

Va-

0
Vdc

vc

Va

q

Va+

Va-

-Vdc

Leg B

Va jumps between +Vdc and –Vdc  Bipolar Switching PWM
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

26

Transfer Function of PWM-Controlled
Four-quadrant Chopper (Bipolar Switching)
 Bipolar Switching PWM
2vtri

1 vc > Vtri
q
0 vc < Vtri

Va   AVdc
ton, T1
A 
T

2vtri

vc

q

Va+

vc

Vdc

Vdc

0

0

Vdc

Vdc
Va0

0

Va = Va+- VaVdc

Va

q

q  (1  q)

Va   BVdc
ton, T3
B 
T

Va   BVdc  1   A Vdc

-Vdc

Va jumps between +Vdc and –Vdc
 Bipolar Switching PWM
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

27

Transfer Function PWM-Controlled
Four-quadrant Chopper (Bipolar Switching)
 Each leg is a two-quadrant chopper.

Bipolar
Switching PWM

 Output of Leg A (average):

Va   AVdc

(5)

 ton, T1 
1


A  
 0.5 
vc

2Vtri
 Ttri 
 Output of Leg B (average):
where

(6)

V   BVdc  1   A Vdc

a

2vtri

Vdc

(7)

Va+

 B  ton, T3 Ttri   toff , T1 Ttri   1   A  (8)

Va-

where

 Subt. (6) into (9)
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

(9)

V
Va  dc vc
Vtri

Vdc
0
Vdc

 Hence, average voltage across the motor:

Va  Va  Va  2 A  1Vdc

vc

0
Vdc

Va
-Vdc

28

Transfer Function PWM-Controlled
Four-quadrant Chopper (Unipolar Switching)
Unipolar Switching PWM
 Leg B switching signals obtained from the inverse of control signal for

Leg A
1 vc > Vtri
qa  
0 vc < Vtri

qa

Leg A

+

vtri
vc

T1

D1

D3

+ Va -

T3

Vdc
vtri

T4



-vc

1 -vc > Vtri
qb 
0 -vc < Vtri
Dr. Ungku Anisa, July 2008

qb

EEEB443 - Control & Drives

D4

D2

T2

Leg B
29

Transfer Function PWM-Controlled
Four-quadrant Chopper (Unipolar Switching)
Unipolar Switching PWM
qa

vc
2Vtri

Leg A

-vc

+

vtri

vc

T1

D1

D3

+ Va -

T3

Vdc

Va+

0

Vdc
vtri

Vdc

T4
-vc



Va-

D2

D4

0

T2

Vdc
Va

qb

0

Va+

Va-

Leg B

Va = Va+- Va-

Va jumps between +Vdc and 0  Unipolar Switching PWM
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

30

Transfer Function PWM-Controlled
Four-quadrant Chopper (Unipolar Switching)
Unipolar Switching PWM
vc

vc
2Vtri

-vc

1 vc > Vtri
qa  
q
0 vc < Vtri a

V   AVdc
ton, T1
A 
T

a

2Vtri

-vc
Vdc
qb q 
b
0

Vdc
0
Vdc

Va+

Vdc
Va0

0

Va = Va+- VaVdc

1 -vc > Vtri
 -v < V
0 c tri

Va   BVdc
ton, T3
B 
T

Va
0

Va jumps between +Vdc and 0 
Unipolar Switching PWM
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

31

Transfer Function PWM-Controlled
Four-quadrant Chopper (Unipolar Switching)
 Each leg is a two-quadrant chopper.

Unipolar
Switching PWM

 Output of Leg A (average):

Va   AVdc

(10)
vc

 ton, T1 
1
  0.5 
 A  
vc
2Vtri
 Ttri 
 Output of Leg B (average):
Va   BVdc
where

2Vtri

-vc

(11)
(12)

Vdc

Va+

0

where  ton, T3 
1


 vc   1   A  (13)V
B  
 0.5 

2Vtri
 Ttri 
 Hence, average voltage across motor armature:
(14) V
V  V   V      V  2  1V

Vdc

a

a

a

a

Dr. Ungku Anisa, July 2008

A

B

dc

EEEB443 - Control & Drives

A

dc

0
Vdc

a

Same as Bipolar Switching Scheme!

0
32

PWM-Controlled Four-quadrant Chopper
Comparison between Bipolar & Unipolar Switching
Bipolar Switching PWM
2vtri

Unipolar Switching PWM
vc

vc
2Vtri

Va+

Vdc

-vc

Va+

0

0
Vdc

Vdc
Va-

Va-

0
Vdc

Va
-Vdc

V
Va  dc vc
Vtri

• Output voltage swings from Vdc and –Vdc
• Output voltage frequency equal to
frequency of triangle voltage (ftri)
Dr. Ungku Anisa, July 2008

Vdc

EEEB443 - Control & Drives

0
Vdc

Va
0

• Output voltage swings from Vdc and 0
• Output voltage frequency equal to
2 times frequency of triangle voltage
(ftri)
33

PWM-Controlled Four-quadrant Chopper
Comparison between Bipolar & Unipolar Switching
Characteristics

Bipolar Switching

Unipolar Switching

Vdc and -Vdc

Vdc and 0

ftri = frequency of Vtri

2ftri

Output voltage swing
Output voltage
frequency

 Current ripple = i

ripple 

Vdc
f output voltage

 For same ftri and Vdc, unipolar scheme gives:
 better output voltage waveform (less ripple)
 lower current ripple
 better frequency response
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

34

Transfer Function PWM-Controlled
Chopper: Two and Four Quadrant
 Gain of the PWM-controlled chopper:


Va Vdc
Two -quadrant: K r  v  2V
c
tri

 Four–quadrant:

Kr 

Va Vdc

vc Vtri

(15)
(16)

where Vdc = dc link voltage
Vtri = maximum control voltage
(i.e. peak of the triangular waveform)

 Chopper also has a delay:

Tr 

1
2 fc

(17)

where fc = carrier (triangular) waveform frequency
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

35

Transfer Function of Subsystems
 PWM-controlled Chopper: G s  
r

Kr
1  sTr 

(18)

Note: Kr and Tr as given in equations (15) – (17) above.
 Other subsystem transfer functions are as observed in ‘Closed-loop
Control of DC Drives with Controlled Rectifier’.
 DC Motor and Load:
ωm s  ωm s  Ia s 


Va s  Ia s  Va s 

ωm s 
Kb

Ia s  Bt 1  sTm 


Ia s 
1  sTm 
 K1
1  sT1 1  sT2 
Va s 

 Current Feedback: H c
 Speed feedback: G s  
ω
Dr. Ungku Anisa, July 2008

K
1  sT 

EEEB443 - Control & Drives

36

Design of Controllers –
Block Diagram of Motor Drive
Current
Control Loop

Speed Control
Loop

Assume that we are using PWM controlled chopper
 Control loop design starts from inner (fastest) loop to
outer(slowest) loop





Only have to solve for one controller at a time
Not all drive applications require speed control (outer loop)
Performance of outer loop depends on inner loop

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

37

Design of Controllers–
Current Controller
PWM-controlled
Chopper

 PI type current controller:
 Loop gain function:

G c s  

DC Motor
& Load

K c 1  sTc 
sTc

 K1Kc Kr H c 

1  sTc 1  sTm 
G Hi s   

T
c

 s1  sT1 1  sT2 1  sTr 

(19)
(20)

 Design procedure - same as for current controller in closed-

loop control using controlled rectifiers

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

38

Design of Controllers–
Current loop 1st order approximation
 Approximated by adding Tr to T1  T3  T1  Tr

Ia s 

*
Ia s 
Dr. Ungku Anisa, July 2008

K c K r K 1Tm
1

1
1  sT3 
Tc
K c K r K 1 H cTm
1
Tc

EEEB443 - Control & Drives

1  sT 

Ki

1  sTi 

(21)

3

39

Design of Controllers–
Current loop 1st order approximation
where

T3
Ti 
1  K fi
Ki 

K fi 

(22)

K fi

1
H c 1  K fi 

(23)

K1K c K r H cTm
Tc

(24)

 1st order approximation of current loop used in speed loop

design.
 If more accurate speed controller design is required, values of
Ki and Ti should be obtained experimentally.
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

40

Design of Controllers–
Speed Controller

 PI type current controller:

DC Motor
& Load

K s 1  sTs 
G s s  
sTs

1st order
approximation
of current
loop

(25)

 Assume there is unity speed feedback:
H
G ω s  
1
1  sT 
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

(26)

41

Design of Controllers–
Speed Controller

1

 Loop gain function:
 K B K s Ki 

1  sTs 
GHs   

 BtTs  s1  sTi 1  sTm 

(27)

 Design procedure - same as for speed controller in

closed-loop control using controlled rectifiers
Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

42

References
 Krishnan, R., Electric Motor Drives: Modeling, Analysis and

Control, Prentice-Hall, New Jersey, 2001.
 Mohan, Underland, Robbins, Power Electronics: Converters,
Applications and Design, 2nd ed., John Wiley & Sons, USA,
1995.
 Nik Idris, N. R., Short Course Notes on Electrical Drives,
UNITEN/UTM, 2008.

Dr. Ungku Anisa, July 2008

EEEB443 - Control & Drives

43

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