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
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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
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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
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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
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
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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
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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
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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
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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
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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
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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
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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
q0
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
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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
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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 1Vdc
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
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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
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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 1V
• 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
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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
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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
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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
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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
s1 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
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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
GHs
BtTs s1 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
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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.