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
References
Solid 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
sVsI
sIsω
sVsω
a
a
a
m
a
m
mt
b
sTBK
1sI
sω
a
m
21
1a
a
111
sVsI
sTsTsTK 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
JLK
JLBR
JB
LR
JB
LR
TT
22
21 41
211,1
tab
t
BRKBK
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
Vv1cos
crccm
m
cm
cmmdc vKv
VV
VvVVV
L,L
L,L L,L
3coscos3cos3 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
sTK
1
sG r
cmcmcm
mr V
VVV
VV
K 35.1233
L,L
ssr ffT 1
1211
36060
21
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
sTK
1
sGω
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
sTsTK
1sGc
r
mc
c
crc
sTsTsTssTsT
THKKK
111
11sGH21
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
TTHKKKK
sTsTsTsTK
sTsTsTssTsT
THKKK
sTsTsTssTsT
THKKK
1
21ol
21
1
21
1ol
where111
1sGH
1111
11111sGH
(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
TTHKKKK
sTsTK
sTsTsTsTK
sTsTsTsTK
1
1ol
21
2
21ol
where11
sGH
1111
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
rsTsTK
11sGH
1ol
c
mcrc
TTHKKKK 1
KsTsT r 11 1
sGH1
sGHsGol
olcl
rr
rr TT
KTTTTssTT
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) 12
1
1
1
r
r
r
TTKTTTT
(22) 1
1
2
rn TT
K
23
12
707.0
1
1
1
r
r
r
TTKTTTT
Squaring the equation on both sides
r1
2
1
1
r1
2
1
1
2
1
1
1
TT1K x 2
1
TT1K x 2 x 2
0.5 12
5.0
r
r
r
r
r
r
r
TTTT
TTTT
TTKTTTT
rTT
rTTKrTTX
rTT
rTT
rTT
rTT
rTT
K12
211
2
1
2
1
11K
1
2
2
1
1
1
24
rTT
rTTKrTTX
rTT
rTT
rTT
rTT
rTT
K12
211
2
1
2
1
11K
1
2
2
1
1
1
An approximation K >> 1 & rTT 1 Which leads to
rr TT
TTTK
221
1
21
Equating above expression with (20) we get the gain of current controller
rc
mcrc
TT
TTHKKK
211
mcrr
cc THKKT
TTK1
1 12
Back
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
sTK
sTTTHKKK
sTTTKKK
i
crc
rc
11
11
11
sIsI
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 TTsTTsK
sTsTK
12
11ol 111
sGH
c
mrc
TTKKKK 1
rTTsTsK
12
3ol 1
sGH
31 TTT r Since
and since rTT 1 3
ol 1sGH
sTK
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
1
3
fic
fii KHK
K
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
sTH
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
sTsTssT
TBKKK
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
TBKKKK
sTsK
sTsTssT
TBKKK
sTsTssT
TBKKK
where
1sGH
111
111sGH
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
isTsK
1
sGH
st
isB
TBKKKK
KsTs i 1
sGH1
sGHsGcl
iii T
KT
ssT 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 36dtdiLiRVe a
aaaa 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
dtdiLiRVe a
aaaa
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
Vv1cos
Cosine voltage
Back
Design of Controllers– Current loop 1st order approximation
Solid State Drives 44
i
cc
c
c
m
c
m
sTK
KT
s
KHK
KsTHK
sTK
sTHK
sTTTHKKK
sTTTKKK
i
fi
fi
fi
fi
fi
fi
fi
crc
rc
11
1
11
1
111
11
111
11
sIsI
33
3
3
3
1
3
1
*a
a
Back
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