closed-loop control of dc drives with controlled rectifier by mr.m.kaliamoorthy department of...
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Closed-loop Control of DC Drives with Controlled RectifierByMr.M.KaliamoorthyDepartment of Electrical & Electronics EngineeringPSNA College of Engineering and Technology
Solid State Drives 1
OutlineClosed Loop Control of DC DrivesClosed-loop Control with Controlled Rectifier –
Two-quadrantTransfer Functions of SubsystemsDesign of Controllers
Closed-loop Control with Field Weakening – Two-quadrant
Closed-loop Control with Controlled Rectifier – Four-quadrant
ReferencesSolid State Drives 2
Closed Loop Control of DC Drives
• Closed loop control is when the firing angle 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
Solid State Drives 3
Controller Plant
Sensor
+
Referencesignal
Outputsignal
Closed Loop Control of DC Drives
Feedback loops may be provided to satisfy one or more of the following:ProtectionEnhancement of response – fast response with small
overshootImprove steady-state accuracy
Variables to be controlled in drives:Torque – achieved by controlling currentSpeedPosition
Solid State Drives 4
Closed Loop Control of DC Drives• Cascade 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
– Torque loop is fastest, speed loop – slower and position loop - slowest
Solid State Drives 5
Closed Loop Control of DC Drives• Cascade control structure:
– Inner Torque (Current) Control Loop:• Current control loop is used to control torque via armature
current (ia) and maintains current within a safe limit
• Accelerates and decelerates the drive at maximum permissible current and torque during transient operations
Solid State Drives 6
Torque (Current)
Control Loop
Closed Loop Control of DC Drives• Cascade control structure
– Speed Control Loop:• Ensures that the actual speed is always equal to reference speed *• Provides fast response to changes in *, TL and supply voltage (i.e. any
transients are overcome within the shortest feasible time) without exceeding motor and converter capability
Solid State Drives 7
Speed Control
Loop
Closed Loop Control with Controlled Rectifiers – Two-quadrant
• Two-quadrant Three-phase Controlled Rectifier DC Motor Drives
Solid State Drives 8
Current Control Loop
Speed Control Loop
Closed Loop Control with Controlled Rectifiers – Two-quadrant
• Actual motor speed m measured using the tachogenerator (Tach) is filtered to produce feedback signal mr
• The reference speed r* is compared to mr to obtain a speed error signal
• The speed (PI) controller processes the speed error and produces the torque command Te*
• Te* is limited by the limiter to keep within the safe current limits and the armature current command ia* is produced
• ia* is compared to actual current ia to obtain a current error signal
• The current (PI) controller processes the error to alter the control signal vc
• vc modifies the firing angle to be sent to the converter to obtained the motor armature voltage for the desired motor operation speed
Solid State Drives 9
Closed Loop Control with Controlled Rectifiers – Two-quadrant
• Design of speed and current controller (gain and time constants) is crucial in meeting the dynamic specifications of the drive system
• Controller design procedure:1. Obtain the transfer function of all drive subsystems
a) DC Motor & Loadb) Current feedback loop sensorc) Speed feedback loop sensor
2. Design current (torque) control loop first3. Then design the speed control loop
Solid State Drives 10
Transfer Function of Subsystems – DC Motor and Load
• Assume load is proportional to speed
• DC motor has inner loop due to induced emf magnetic coupling, which is not physically seen
• This creates complexity in current control loop design
Solid State Drives 11
mLL BT
Transfer Function of Subsystems – DC Motor and Load
• Need to split the DC motor transfer function between m and Va
(1)
• where(2)
(3)
• This is achieved through redrawing of the DC motor and load block diagram.
Solid State Drives 12
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
Back
Transfer Function of Subsystems – DC Motor and Load
• In (2),- mechanical motor time constant: (4)
- motor and load friction coefficient: (5)• In (3),
(6)
(7)
Note: J = motor inertia, B1 = motor friction coefficient, BL = load friction coefficient
Solid State Drives 13
tm B
JT
Lt BBB 1
a
b
a
tat
a
at
a
a
JL
K
JL
BR
J
B
L
R
J
B
L
R
TT
22
21 4
1
2
11,
1
tab
t
BRK
BK
21
Back
Transfer Function of Subsystems – Three-phase Converter
• Need to obtain linear relationship between control signal vc
and delay angle (i.e. using ‘cosine wave crossing’ method)(8)
where vc = control signal (output of current controller)
Vcm = maximum value of the control voltage
• Thus, dc output voltage of the three-phase converter
(9)
Solid State Drives 14
cm
c
V
v1cos
crccm
m
cm
cmmdc vKv
V
V
V
vVVV
L,L
L,L L,L
3
coscos3
cos3 1
Transfer Function of Subsystems – Three-phase Converter
Gain of the converter
(10)
where V = rms line-to-line voltage of 3-phase supplyConverter also has a delay
(11) where fs = supply voltage frequencyHence, the converter transfer function
(12)
Solid State Drives 15
rr
sT
K
1sG r
cmcmcm
mr V
V
V
V
V
VK 35.1
233
L,L
ssr ffT
1
12
11
360
60
2
1
Back
Transfer Function of Subsystems – Current and Speed Feedback
Current FeedbackTransfer function:No filtering is required in most casesIf filtering is required, a low pass-filter can be included (time
constant < 1ms).Speed Feedback
Transfer function:(13)
where K = gain, T = time constantMost high performance systems use dc tacho generator and low-
pass filterFilter time constant < 10 ms
Solid State Drives 16
sT
K
1sGω
cH
Design of Controllers – Block Diagram of Motor Drive
Control loop design starts from inner (fastest) loop to outer(slowest) loop
Only have to solve for one controller at a timeNot all drive applications require speed control (outer loop)Performance of outer loop depends on inner loop
Solid State Drives 17
Speed Control Loop
Current Control Loop
Design of Controllers– Current Controller
PI type current controller: (14)Open loop gain function:
(15)
From the open loop gain, the system is of 4th order (due to 4 poles of system)
Solid State Drives 18
c
cc
sT
sTK
1sGc
r
mc
c
crc
sTsTsTs
sTsT
T
HKKK
111
11sGH
21
1ol
DC Motor & LoadConverterController
Design of Controllers– Current Controller
• If designing without computers, simplification is needed.• Simplification 1: Tm is in order of 1 second. Hence,
(16)Hence, the open loop gain function becomes:
i.e. system zero cancels the controller pole at origin.Solid State Drives 19
mm sTsT 1
c
mcrc
r
c
r
mc
c
crc
r
mc
c
crc
T
THKKKK
sTsTsT
sTK
sTsTsTs
sTsT
T
HKKK
sTsTsTs
sTsT
T
HKKK
1
21ol
21
1
21
1ol
where111
1sGH
111
1
111
11sGH
(17)
Design of Controllers– Current Controller
• Relationship between the denominator time constants in (17):
• Simplification 2: Make controller time constant equal to T2
(18) Hence, the open loop gain function becomes:
i.e. controller zero cancels one of the system poles.
Solid State Drives 20
12 TTTr
2TTc
c
mcrc
r
r
r
c
T
THKKKK
sTsT
K
sTsTsT
sTK
sTsTsT
sTK
1
1ol
21
2
21ol
where11
sGH
111
1
111
1sGH
Design of Controllers– Current Controller
• After simplification, the final open loop gain function:(19)
where (20)
• The system is now of 2nd order.• From the closed loop transfer function: , the closed loop characteristic equation is:
or when expanded becomes: (21) Solid State Drives 21
rsTsT
K
11sGH
1ol
c
mcrc
T
THKKKK 1
KsTsT r 11 1
sGH1
sGHsG
ol
olcl
rr
rr TT
K
TT
TTssTT
11
121
1
Design of Controllers– Current Controller
• Design the controller by comparing system characteristic equation (eq. 21) with the standard 2nd order system equation:
• Hence,
• So, for good dynamic performance =0.707 – Hence equating the damping ratio to 0.707 in (23) we get
Solid State Drives 22
22 2 nnss
(23) 1
21
1
1
r
r
r
TTK
TTTT
(22) 1
1
2
rn TT
K
23
1
2
707.0
1
1
1
r
r
r
TTK
TTTT
Squaring the equation on both sides
r1
2
1
1
r1
2
1
1
2
1
1
1
TT1K
x 21
TT1K
x 2 x 20.5
12
5.0
r
r
r
r
r
r
r
TTTT
TTTT
TTK
TTTT
rTT
rTTK
rTTX
rTT
rTT
rTT
rTT
rTT
K
12
21
12
1
2
1
11K
1
2
2
1
1
1
24
rTT
rTTK
rTTX
rTT
rTT
rTT
rTT
rTT
K
12
21
12
1
2
1
11K
1
2
2
1
1
1
An approximation K >> 1 & rTT 1 Which leads to
rr T
T
TT
TK
221
1
21
Equating above expression with (20) we get the gain of current controller
rc
mcrc
T
T
T
THKKK
211
mcrr
cc THKKT
TTK
1
1 1
2Back
Design of Controllers– Current loop 1st order approximation • To design the speed loop, the 2nd order model of current loop
must be replaced with an approximate 1st order model• Why?• To reduce the order of the overall speed loop gain function
Solid State Drives 25
2nd order current loop
model
Design of Controllers– Current loop 1st order approximation • Approximated by adding Tr to T1
• Hence, current model transfer function is given by:
(24)
Solid State Drives 26
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
Full derivation available here.
1st order approximation of current loop
Design of Controllers– Current Controller
• After simplification, the final open loop gain function:
Solid State Drives 27
rrr TTsTTs
K
sTsT
K
12
11ol 111
sGH
c
mrc
T
TKKKK 1
rTTsTs
K
12
3ol 1
sGH
31 TTT r Since
and since rTT 1 3
ol 1sGH
sT
K
Therefore
Where
Design of Controllers– Current loop 1st order approximation where (26)
(27)
(28)
• 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.
Solid State Drives 28
c
mcrcfi T
THKKKK 1
fii K
TT
13
fic
fii KH
KK
1
1
Design of Controllers– Speed Controller
• PI type speed controller: (29)
• Assume there is unity speed feedback: (30)
Solid State Drives 29
s
sss sT
sTK
1sG
11
sGω
sT
H
DC Motor & Load
1st order approximation of current
loop
Design of Controllers– Speed Controller
Open loop gain function:
(31)
From the loop gain, the system is of 3rd order.If designing without computers, simplification is needed.
Solid State Drives 30
1
mi
s
st
isB
sTsTs
sT
TB
KKK
11
1sGH
DC Motor & Load
1st order approximation of current
loop
Design of Controllers– Speed Controller
• Relationship between the denominator time constants in (31):(32)
• Hence, design the speed controller such that:(33)
The open loop gain function becomes:
i.e. controller zero cancels one of the system poles.Solid State Drives 31
mi TT
ms TT
st
isB
i
mi
m
st
isB
mi
s
st
isB
TB
KKKK
sTs
K
sTsTs
sT
TB
KKK
sTsTs
sT
TB
KKK
where
1sGH
11
1
11
1sGH
Design of Controllers– Speed Controller
• After simplification, loop gain function:(34)
where (35)
• The controller is now of 2nd order.• From the closed loop transfer function: ,
the closed loop characteristic equation is:
or when expanded becomes: (36)Solid State Drives 32
isTs
K
1sGH
st
isB
TB
KKKK
KsTs i 1
sGH1
sGHsGcl
iii T
K
TssT 12
Design of Controllers– Speed Controller
• Design the controller by comparing system characteristic equation with the standard equation:
• Hence:(37)
(38)
• So, for a given value of :– use (37) to calculate n
– Then use (38) to calculate the controller gain KS
Solid State Drives 33
22 2 nnss
n2
2n
Closed Loop Control with Field Weakening – Two-quadrant
Motor operation above base speed requires field weakening
Field weakening obtained by varying field winding voltage using controlled rectifier in:single-phase orthree-phase
Field current has no ripple – due to large LfConverter time lag negligible compared to field time
constant Consists of two additional control loops on field circuit:
Field current control loop (inner)Induced emf control loop (outer)
Solid State Drives 34
Closed Loop Control with Field Weakening – Two-quadrant
Solid State Drives 35
Field weakening
Closed Loop Control with Field Weakening – Two-quadrant
Solid State Drives 36dt
diLiRVe aaaaa
Induced emf controller
(PI-type with limiter)
Field weakening
Field current
controller(PI-type)
Field current reference
Estimated machine -induced emf
Induced emf reference
Closed Loop Control with Field Weakening – Two-quadrant
• The estimated machine-induced emf is obtained from:
(the estimated emf is machine-parameter sensitive and must be adaptive)• The reference induced emf e* is compared to e to obtain the induced emf
error signal (for speed above base speed, e* kept constant at rated emf value so that 1/)
• The induced emf (PI) controller processes the error and produces the field current reference if*
• if* is limited by the limiter to keep within the safe field current limits• if* is compared to actual field current if to obtain a current error signal• The field current (PI) controller processes the error to alter the control
signal vcf (similar to armature current ia control loop)• vcf modifies the firing angle f to be sent to the converter to obtained the
motor field voltage for the desired motor field fluxSolid State Drives 37
dt
diLiRVe aaaaa
Closed Loop Control with Controlled Rectifiers – Four-quadrant
• Four-quadrant Three-phase Controlled Rectifier DC Motor Drives
Solid State Drives 38
Closed Loop Control with Controlled Rectifiers – Four-quadrant
• Control very similar to the two-quadrant dc motor drive.• Each converter must be energized depending on quadrant of operation:
– Converter 1 – for forward direction / rotation– Converter 2 – for reverse direction / rotation
• Changeover between Converters 1 & 2 handled by monitoring– Speed– Current-command– Zero-crossing current signals
• ‘Selector’ block determines which converter has to operate by assigning pulse-control signals
• Speed and current loops shared by both converters• Converters switched only when current in outgoing converter is zero (i.e.
does not allow circulating current. One converter is on at a time.)Solid State Drives 39
Inputs to ‘Selector’ block
References
• Krishnan, R., Electric Motor Drives: Modeling, Analysis and Control, Prentice-Hall, New Jersey, 2001.
• Rashid, M.H, Power Electronics: Circuit, Devices and Applictions, 3rd ed., Pearson, New-Jersey, 2004.
• Nik Idris, N. R., Short Course Notes on Electrical Drives, UNITEN/UTM, 2008.
Solid State Drives 40
DC Motor and Load Transfer Function - Decoupling of Induced EMF Loop
• Step 1:
• Step 2:
Solid State Drives 41
DC Motor and Load Transfer Function - Decoupling of Induced EMF Loop
• Step 3:
• Step 4:
Solid State Drives 42
Back
Cosine-wave Crossing Control for Controlled Rectifiers
Solid State Drives 43
Vm
Vcmvc
0 2 3 4
Input voltageto rectifier
Cosine wave compared with control voltage vc
Results of comparison trigger SCRs
Output voltageof rectifier
Vcmcos() = vc
cm
c
V
v1cos
Cosine voltage
Back
Design of Controllers– Current loop 1st order approximation
Solid State Drives 44
i
cc
c
c
m
c
m
sT
K
K
Ts
KH
K
KsT
H
K
sTK
sTH
K
sTT
THKKKsTT
TKKK
i
fi
fi
fi
fi
fi
fi
fi
crc
rc
1
11
11
1
11
1
11
11
1
11
sI
sI
33
3
3
3
1
3
1
*a
a
Back
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