Closed-loop Control of DC Drives with ChopperByDr. Ungku Anisa Ungku AmirulddinDepartment of Electrical Power EngineeringCollege of Engineering
Dr. Ungku Anisa, July 2008 1EEEB443 - Control & Drives
OutlineClosed Loop Control of DC Drives with ChoppersCurrent Control for DC Drives with Choppers
Pulse-Width-Modulation (PWM) ControllerHysteresis-Current ControllerComparison between PWM and Hysteresis Controller
Transfer Function of PWM-Controlled ChopperTwo-quadrantFour-quadrant
Design of ControllersReferences
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 2
Closed Loop Control of DC DrivesClosed 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
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 3
Controller Plant
Sensor
+
Referencesignal
Outputsignal
Closed Loop Control of DC DrivesFeedback loops may be provided to satisfy one or more
of the following:ProtectionEnhancement of speed responseImprove steady-state accuracy
Variables to be controlled in drives:Torque – achieved by controlling currentSpeedPosition
Dr. Ungku Anisa, July 2008 4EEEB443 - Control & Drives
Closed Loop Control of DC DrivesCascade control structure
Flexible – outer loops can be added/removed depending on control requirements.
Control variable of inner loop (eg: speed, torque) can be limited by limiting its reference value
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 5
For DC Drive, this can be:•Controlled rectifier or•DC-DC converter
Closed Loop Control for DC Drives with ChoppersOuter speed loop very similar to that in the controlled
rectifier dc driveInner current control loop – different
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 6
Current Control LoopSpeed Control
Loop
Current Control for DC Drives with ChoppersCurrent control loop is used to control torque via
armature current (ia)Output of current controller determines duty cycle (i.e.
switching) of DC-DC converterCurrent 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 responseHence, affects speed loop bandwidth.
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 7
Current Control for Chopper Drives – PWM ControllerIn two quadrant chopper, upper and
lower switches are complementaryOnly ONE control signal requiredCurrent 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
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 8
ierr
Vdc
Pulse WidthModulator
(PWM)
vcia
*
PI+
q
T1
T2
D1
+
Va
-
D2
ia
+
Vdc
ia
vtri
01
qvc > Vtri
vc < Vtri
T1 ‘on’, Va = Vdc
T2 ‘on’, Va = 0
Current Control for Chopper Drives – PWM Controller
In the PWM circuit:vc is compared with a triangular
waveformif vc > Vtri ‘on’ signal is produced (q
= 1)if vc < Vtri ‘off’ signal is produced (q
= 0)
(1)
Chopper switching frequency is fixed by triangular waveform frequency regardless of operating conditions
Bandwith of current loop controller is limited by frequency of VtriDr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 9
01
qvc > Vtri
vc < Vtri
Ttri
ton
0
1
vc
0
Vdc
q
va
q = 1 T1 ‘on’, va = Vdc
q = 0 T2 ‘on’, va = 0
vc > Vtri vc < Vtri
In the PWM circuit:Average value of q over a cycle
determines duty cycle of chopper:
Average armature voltage:
Current Control for Chopper Drives – PWM Controller
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 10
Ttri
ton
0
1
vc
0
Vdc
Va
tri
onTt
ttri T
tdtq
T
tri
T1 , 1
dc
T
aa VdtvT
V 1
0
q
va
va switches between Vdc and 0
average armature voltage Va depends on duty cycle (i.e. how long T1 is on)
Current Control for Chopper Drives – PWM ControllerPWM controls chopper duty cycle once in every
cycleFrequency of Va fixed by frequency of Vtri
Hence, chopper is a variable voltage source with average current control
Instantaneous current control is not exercisedCurrent can exceed maximum armature current
between two consecutive switching
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 11
Current Control for Chopper Drives – Hysteresis ControllerInstantaneous current controlCurrent controlled within a narrow
band of excursion from the desired value ia
*
Hysteresis window determines allowable deviation of ia
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 12
Vdc
ia* q
T1
T2
D1
+
Va
-
D2
ia
+
Vdc
ierr
ia
ia*
q
ia
ia
ia
01
qia ia
* - ia
ia ia* + ia
HysteresisController
Current Control for Chopper Drives – Hysteresis ControllerActual current ia compared with reference
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
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 13
ia*
q
ia
ia
ia
q = 1 T1 ‘on’, va = Vdc
ia ia* - ia
q = 0 T2 ‘on’, va = 0
ia ia* + ia
01
qia ia
* - ia
ia ia* + ia
Current Control for Chopper Drives – Qualitative Comparison
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 14
Characteristics Hysteresis Controller PWM Controller
Switching frequency
Varying Fixed (follows sawtooth waveform
frequency i.e. carrier frequency)
Switching losses High (due to varying
switching frequency)
Low
Speed of response
Fastest(due to instantanous
change in current)
Fast
Ripple current Adjustable(depends on hysteresis
window ia )
Fixed
Filter size Depends on ia SmallPreferred method !
Closed Loop Control for DC Drives with ChoppersController design procedure:
1. Obtain the transfer function of all drive subsystemsa) DC Motor & Loadb) Current feedback loop sensorc) Speed feedback loop sensor
2. Design torque (current) control loop first 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 chopperii. design PI controller using the same procedure as in closed
loop control using controlled rectifier Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 15
Exactly the same as before (i.e. transfer functions obtained in closed loop control using controlled rectifier)
Closed Loop Control for DC Drives with ChoppersController design procedure (continued):
3. Then design the speed control loopi. Obtain 1st order model of the designed current controllerii. 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 ChopperPWM current controller is preferred over Hysteresis
ControllerBefore we can design the PI controller, need to obtain linear
relationship between control input vc and average armature voltage Va for PWM method
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 17
Pulse WidthModulator
(PWM)
vcia
*
PI+
q
ia
vtri
Chopper DC motor
Va ia
Need transfer function for PWM-controlled chopper
Transfer Function of PWM-Controlled Two-quadrant ChopperNeed to obtain linear relationship between control input vc
and average armature voltage Va for PWM method
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 18
Vtri
-Vtri vc
0q
01
qvc > Vtri
vc < Vtri
vc
-Vtri
Case 1:tric Vv
T1 off all the timei.e. ton, T1 = 0
0 1 T1 ,
tri
onTt
ttri T
tdtq
T
tri
Transfer Function of PWM-Controlled Two-quadrant Chopper
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 19
Vtri
-Vtri vc
cycle a 1/2for ,0
cycle a 1/2for ,1 q
01
qvc > Vtri
vc < Vtri
vc
-Vtri
0.5
Case 2:0 cv
T1 on ½ cyclei.e. ton, T1 = 0.5Ttri
5.0 1 T1 ,
tri
onTt
ttri T
tdtq
T
tri
Transfer Function of PWM-Controlled Two-quadrant Chopper
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 20
Vtri
-Vtri vc
1 q
01
qvc > Vtri
vc < Vtri
vc
-Vtri
0.5
1
Vtri
Case 3:
tric Vv
T1 on all the timei.e. ton, T1 = Ttri
1 1 T1 ,
tri
onTt
ttri T
tdtq
T
tri
Relationship between and vc :
(2)
For the two-quadrant chopper: (3)
Hence, considering only the term due to vc, the two–quadrant chopper gain is:
(4)
Transfer Function of PWM-Controlled Two-quadrant Chopper
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 21
ctri
vV2
15.0
0.5
vc
-Vtri+Vtri
ctri
dcdcdca v
V
VVVV
25.0
tri
dc
c
a
V
V
v
VK
2r
1
Transfer Function of PWM-Controlled Four-quadrant Chopper
Recap Chopper operation:Positive current:
Va = Vdc when T1 and T2 onVa = 0 when current
freewheels through T2 and D4
+ Va -T1
D1
T2D2
D3
D4
T3
T4
+
Vdc
-
Dr. Ungku Anisa, July 2008 22EEEB443 - Control & Drives
Transfer Function of PWM-Controlled Four-quadrant Chopper
Recap Chopper operation:Positive current:
Va = Vdc when T1 and T2 onVa = 0 when current
freewheels through T2 and D4Negative current:
Va = -Vdc when T3 and T4 onVa = 0 when current
freewheels through T4 and D2Output voltage can swing
between:Vdc and -Vdc Vdc and 0
+ Va -T1
D1
T2D2
D3
D4
T3
T4
+
Vdc
-
Dr. Ungku Anisa, July 2008 23EEEB443 - Control & Drives
Transfer Function of PWM-Controlled Four-quadrant ChopperNeed to obtain linear relationship between control input vc
and average armature voltage Va for PWM methodFour 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 switchingUnipolar switching
Switching schemedetermines output voltage swing betweenVdc and -Vdc or Vdc and 0.
+ Va -T1 D1
T2D2
D3
D4
T3
T4
Leg A Leg B
+
Vdc
−
Dr. Ungku Anisa, July 2008 24EEEB443 - Control & Drives
EEEB443 - Control & Drives
Transfer Function of PWM-Controlled Four-quadrant Chopper (Bipolar Switching)Bipolar Switching PWMLeg A and Leg B obtain switching signals from the same control signal vc
Switching of Leg A and Leg B are always complementary
+ Va -T1 D1
T2D2
D3
D4
T3
T4
+
Vdc
−vc
vtri
q
q
Leg A
Leg B
Dr. Ungku Anisa, July 2008 25
01
q vc > Vtri
vc < Vtri
q = 1,q =0 T1 on, T2 on Va= Vdc
q = 0, q =1 T4 on, T3 on Va= -Vdc
Transfer Function of PWM-Controlled Four-quadrant Chopper (Bipolar Switching)Bipolar Switching PWM
+ Va -T1 D1
T2D2
D3
D4
T3
T4
+
Vdc
−vc
vtri
q
q
Leg A
Leg BVa+ Va
-
Va-
Vdc
0
Va+
Vdc
0
2vtri vc
Va
Vdc
-Vdc
Va jumps between +Vdc and –Vdc Bipolar Switching PWMDr. Ungku Anisa, July 2008 26EEEB443 - Control & Drives
Va = Va+- Va
-
Transfer Function of PWM-Controlled Four-quadrant Chopper (Bipolar Switching)Bipolar Switching PWM
Va-
Vdc
0
qVdc
0
2vtri vc
Va
Vdc
-Vdc
Va jumps between +Vdc and –Vdc
Bipolar Switching PWMDr. Ungku Anisa, July 2008 27EEEB443 - Control & Drives
Va = Va+- Va
-
qVdc
0
2vtri vc
Va+
Vdc
0dca VV A
T
t T1 on,A
0
1q
vc > Vtri
vc < Vtri
dcdca VVV AB 1 T
t T3 on,B
dca VV B
)1( qq
Transfer Function PWM-Controlled Four-quadrant Chopper (Bipolar Switching)Each leg is a two-quadrant chopper. Output of Leg A (average):
(5)where
(6)Output of Leg B (average):
(7)where
(8)Hence, average voltage across the motor:
(9)Subt. (6) into (9)
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 28
Vdc
Va-
Vdc
0
Va
Vdc
-Vdc
2vtrivc
dca VV A
dcdca VVV AB 1
dcaaa VVVV 12 A
ctri
dca vV
VV
Bipolar Switching PWM
Va+
0
Vdc
AtriofftrionB TtTt 1T1 ,T3 ,
ctritri
on vVT
t
2
15.0T1 ,
A
Transfer Function PWM-Controlled Four-quadrant Chopper (Unipolar Switching)
Unipolar Switching PWMLeg B switching signals obtained from the inverse of control signal for
Leg A
Leg A
Leg B
+ Va -T1 D1
T2D2
D3
D4
T3
T4
+
Vdc
−
vc
vtri
qa
-vc
vtri
qb
Dr. Ungku Anisa, July 2008 29EEEB443 - Control & Drives
0
1qa
vc > Vtri
vc < Vtri
0
1qb
-vc > Vtri
-vc < Vtri
Transfer Function PWM-Controlled Four-quadrant Chopper (Unipolar Switching)
Unipolar Switching PWMLeg A
Leg B
+
Vdc
−
vc
vtri
qa
-vc
vtri
qb
+ Va -T1 D1
T2D2
D3
D4
T3
T4
2Vtri
vc
-vc
Va+ Va
-
Va jumps between +Vdc and 0 Unipolar Switching PWM
Va+
Vdc
0
Va-
Vdc
0
Va
Vdc
0
Dr. Ungku Anisa, July 2008 30EEEB443 - Control & Drives
Va = Va+- Va
-
Transfer Function PWM-Controlled Four-quadrant Chopper (Unipolar Switching)Unipolar Switching PWM
2Vtri
vc
-vc
Va jumps between +Vdc and 0 Unipolar Switching PWM
Va
Vdc
0
Dr. Ungku Anisa, July 2008 31EEEB443 - Control & Drives
2Vtri
vc
-vc
qa
Vdc
0
Va+
Vdc
0
qb
Vdc
0
Va-
Vdc
0dca VV A
dca VV B
Va = Va+- Va
-
T
t T1 on,A
T
t T3 on,B
0
1qa
vc > Vtri
vc < Vtri
0
1qb
-vc > Vtri
-vc < Vtri
Transfer Function PWM-Controlled Four-quadrant Chopper (Unipolar Switching)
2Vtri
vc
-vc
Va+
Vdc
0
Va-
Vdc
0
Va
Vdc
0
Each leg is a two-quadrant chopper. Output of Leg A (average):
(10)where
(11)Output of Leg B (average):
(12) where
(13)Hence, average voltage across motor armature:
(14)
dca VV A
dca VV B
dcdcaaa VVVVV 12 ABA
Same as Bipolar Switching Scheme!
Unipolar Switching PWM
Dr. Ungku Anisa, July 2008 32EEEB443 - Control & Drives
ctritri
on vVT
t
2
15.0T1 ,
A
AT3 ,
B 12
15.0
c
tritri
on vVT
t
PWM-Controlled Four-quadrant ChopperComparison between Bipolar & Unipolar SwitchingBipolar Switching PWM Unipolar Switching PWM
2Vtri
vc
-vc
Va+
Vdc
0
Va-
Vdc
0
Va
Vdc
0
Dr. Ungku Anisa, July 2008 33EEEB443 - Control & Drives
Va-
Vdc
0
2vtri vc
Va+
Vdc
0
Va
Vdc
-Vdc
ctri
dca vV
VV
• Output voltage swings from Vdc and –Vdc
• Output voltage frequency equal to frequency of triangle voltage (ftri)
• Output voltage swings from Vdc and 0• Output voltage frequency equal to 2 times frequency of triangle voltage (ftri)
PWM-Controlled Four-quadrant ChopperComparison between Bipolar & Unipolar Switching
Current ripple =
For same ftri and Vdc, unipolar scheme gives:better output voltage waveform (less ripple)lower current ripplebetter frequency response
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 34
Characteristics Bipolar Switching Unipolar Switching
Output voltage swing Vdc and -Vdc Vdc and 0
Output voltage frequency
ftri = frequency of Vtri 2ftri
tageoutput volripple f
Vi dc
Gain of the PWM-controlled chopper:
Two -quadrant: (15)
Four–quadrant: (16)
where Vdc = dc link voltage Vtri = maximum control voltage
(i.e. peak of the triangular waveform)Chopper also has a delay:
(17) where fc = carrier (triangular) waveform frequency
Transfer Function PWM-Controlled Chopper: Two and Four Quadrant
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 35
tri
dc
c
a
V
V
v
VK r
tri
dc
c
a
V
V
v
VK
2r
cr fT
2
1
PWM-controlled Chopper: (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:
Current Feedback: Speed feedback:
Transfer Function of Subsystems
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 36
rr
sT
K
1sG r
sV
sI
sI
sω
sV
sω
a
a
a
m
a
m
mt
b
sTB
K
1sI
sω
a
m
21
1a
a
11
1
sV
sI
sTsT
sTK m
cH
sT
K
1sGω
Design of Controllers – Block Diagram of Motor Drive
Assume that we are using PWM controlled chopperControl 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
Speed Control Loop
Current Control Loop
PI type current controller: (19)Loop gain function:
(20)
Design procedure - same as for current controller in closed-loop control using controlled rectifiers
Design of Controllers– Current Controller
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 38
c
cc
sT
sTK
1sGc
r
mc
c
crc
sTsTsTs
sTsT
T
HKKK
111
11sHG
21
1i
DC Motor & Load
PWM-controlled Chopper
Approximated by adding Tr to T1
(21)
Design of Controllers– Current loop 1st order approximation
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 39
i
c
m
c
m
sT
K
sTT
THKKKsTT
TKKK
i
crc
rc
11
11
11
sI
sI
3
1
3
1
*a
a
rTTT 13
Design of Controllers– Current loop 1st order approximation
where (22)
(23)
(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
c
mcrcfi T
THKKKK 1
fii K
TT
13
fic
fii KH
KK
1
1
PI type current controller: (25)
Assume there is unity speed feedback: (26)
Design of Controllers– Speed Controller
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 41
s
sss sT
sTK
1sG
11
sGω
sT
H
DC Motor & Load
1st order approximation of current
loop
1
Loop gain function: (27)
Design procedure - same as for speed controller in closed-loop control using controlled rectifiers
Design of Controllers– Speed Controller
Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 42
mi
s
st
isB
sTsTs
sT
TB
KKK
11
1sGH
ReferencesKrishnan, 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 43EEEB443 - Control & Drives
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