Transcript EEEB283 Electrical Machines & Drives

Closed-loop Control of DC Drives with
Controlled Rectifier
By
Mr.M.Kaliamoorthy
Department of Electrical & Electronics Engineering
PSNA College of Engineering and Technology
Solid State Drives
1
Outline
 Closed Loop Control of DC Drives
 Closed-loop Control with Controlled Rectifier –
Two-quadrant
 Transfer Functions of Subsystems
 Design 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
Reference
signal
+
Plant
Controller

Output
signal
Sensor
Solid State Drives
3
Closed Loop Control of DC Drives
 Feedback loops may be provided to satisfy one or
more of the following:
 Protection
 Enhancement of response – fast response with small
overshoot
 Improve steady-state accuracy
 Variables to be controlled in drives:
 Torque – achieved by controlling current
 Speed
 Position
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
Torque
current and torque during transient operations
(Current)
Control Loop
Solid State Drives
6
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
Speed
Control
Loop
Solid State Drives
7
Closed Loop Control with Controlled
Rectifiers – Two-quadrant
• Two-quadrant Three-phase Controlled
Rectifier
DC Control
Motor Drives
Speed
Current
Control Loop
Loop
Solid State Drives
8
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 & Load
b) Current feedback loop sensor
c) Speed feedback loop sensor
2. Design current (torque) control loop first
3. Then design the speed control loop
Solid State Drives
10
Transfer Function of Subsystems –
DC Motor and Load
• Assume load is proportional to speed
TL  BLm
• 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
Transfer Function of Subsystems –
DC Motor and Load
•
Need to split the DC motor transfer function between m and Va
ωm s  ωm s  I a s 


Va s  Ia s  Va s 
•
where
(1)
ωm s 
Kb

Ia s  Bt 1  sTm 
(2)
1  sTm 
Ia s 
 K1
1  sT1 1  sT2 
Va s 
•
(3)
This is achieved through redrawing of the DC motor and load block diagram.
Back
Solid State Drives
12
Transfer Function of Subsystems –
DC Motor and Load
• In (2),
- mechanical motor time constant:
J
Tm 
Bt
(4)
- motor and load friction coefficient: Bt  B1  BL
• In (3),
B
K1  2 t
K b  Ra Bt
(5)
(6)
2
2
1 1
1  Ra Bt 
1  Ra Bt   Ra Bt Kb 

    
 ,      


T1 T2
2  La J 
4  La J   JLa JLa 
Note: J = motor inertia, B1 = motor friction coefficient,
BL = load friction coefficient
Solid State Drives
(7)
Back
13
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)
1  vc 
  cos  
 Vcm 
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)
 1 vc  3 VLL, m
 
Vdc  VLL, m cos   VLL, m cos  cos
vc  K r vc


Vcm   Vcm

3
3
Solid State Drives
14
Transfer Function of Subsystems –
Three-phase Converter
 Gain of the converter
3 VLL, m 3 2V
V
Kr 

 1.35
 Vcm
 Vcm
Vcm
(10)
where V = rms line-to-line voltage of 3-phase supply
 Converter also has a delay
1 60 1
1 1
Tr  
  
2 360 f s 12 f s
(11)
where fs = supply voltage frequency
 Hence, the converter transfer function
(12)
Kr
G r s  
1  sTr 
Solid State Drives
Back
15
Transfer Function of Subsystems –
Current and Speed Feedback
 Current Feedback
 Transfer function: H c
 No filtering is required in most cases
 If filtering is required, a low pass-filter can be included (time
constant < 1ms).
 Speed Feedback
 Transfer function:
K
(13)
G ω s  
1  sT 
where K = gain, T = time constant
 Most high performance systems use dc tacho generator and lowpass filter
 Filter time constant < 10 ms
Solid State Drives
16
Design of Controllers –
Block Diagram of Motor Drive
Current
Control Loop
Speed Control
Loop
 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
Solid State Drives
17
Design of Controllers–
Current Controller
DC Motor
Controller
Converter
& Load
K 1  sTc 
 PI type current controller: G c s   c
(14)
sTc
 Open loop gain function:
 K1K c K r H c 
(15)
1  sTc 1  sTm 
GH ol s   

T
c

 s1  sT1 1  sT2 1  sTr 
 From the open loop gain, the system is of 4th order (due to 4 poles
of system)
Solid State Drives
18
Design of Controllers–
Current Controller
• If designing without computers, simplification is needed.
• Simplification 1: Tm is in order of 1 second. Hence,
1  sTm   sTm
(16)
Hence, the open loop gain function becomes:
 K1 K c K r H c 
1  sTc 1  sTm 
GH ol s   

Tc

 s1  sT1 1  sT2 1  sTr 
 K1 K c K r H c 

1  sTc sTm 


T
c

 s1  sT1 1  sT2 1  sTr 

1  sTc 
GH ol s   K
1  sT1 1  sT2 1  sTr 
K1 K c K r H cTm
(17)
where K 
Tc
i.e. system zero cancels the controller pole at origin.
Solid State Drives
19
Design of Controllers–
Current Controller
• Relationship between the denominator time constants in (17):
Tr  T2  T1
• Simplification 2: Make controller time constant equal to T2
Tc  T2
(18)
Hence, the open loop gain function becomes:

1  sTc 
GH ol s   K
1  sT1 1  sT2 1  sTr 

1  sT2 
K
1  sT1 1  sT2 1  sTr 
GH ol s  
KK K HT
K
where K  1 c r c m
1  sT1 1  sTr 
Tc
i.e. controller zero cancels one of the system poles.
Solid State Drives
20
Design of Controllers–
Current Controller
• After simplification, the final open loop gain function:
GH ol s  
where
K
K
1  sT1 1  sTr 
(19)
K1K c K r H cTm
Tc
(20)
• The system is now of 2nd order.
GH ol s 
• From the closed loop transfer function: G cl s  
,
1  GH ol s 
the closed loop characteristic equation is:
1  sT1 1  sTr   K
or when expanded becomes:
 2  T1  Tr
T1Tr s  s
 T1Tr

Solid State Drives
 K  1
 

T
T
1 r 

(21)
21
Design of Controllers–
Current Controller
• Design the controller by comparing system characteristic
equation (eq. 21) with the standard 2nd order system
equation:
s 2  2 s   2
n
• Hence,
K 1
n 
T1Tr
2
(22)
n
 T1  Tr 


T1Tr 

 
K 1
2
T1Tr
(23)
• So, for good dynamic performance =0.707
– Hence equating the damping ratio to 0.707 in (23) we get
Solid State Drives
22
 T1  Tr 


T1Tr 

0.707 
K 1
2
T1Tr
Squaring the equation on both sides
2
  T1  Tr  
 
 
  T1Tr  
0.5  
  0.5 
 2 K 1 


T
T
1
r


 T1  Tr

T1Tr

K 1 
2
T1Tr



2
2
 T1  Tr 
 T1  Tr 




 T1Tr   1   T1Tr 
K 1
K 1
2x2x
2x
T1Tr
T1Tr
2
 T1  Tr
 K  1  
 TT
 1r



2
X
T1Tr
2
T1  Tr 
2
 K  1 
2T1Tr
23
 T1  Tr

T1Tr

K 1 
2



2
 T1  Tr
 K  1  
 TT
 1r



2
X
T1Tr
T1  Tr 
2
 K  1 
2
2T1Tr
T1Tr
An approximation K >> 1 &
T1  Tr
Which leads to
2
1
T
T1
K

2T1Tr 2Tr
Equating above expression with (20) we get the gain of current
controller
K1K c K r H cTm T1

Tc
2Tr
T1Tc
Kc 
2Tr


1


 K1 K r H cTm 
Back
24
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
2nd order
current loop
model
Solid State Drives
25
Design of Controllers–
Current loop 1st order approximation
• Approximated by adding Tr to T1  T3  T1  Tr
• Hence, current model transfer function is given by:
K c K r K 1Tm
1

Tc
1  sT3 
Ia s 
Ki


*
K
K
K
H
T
1

Ia s 
1  sTi 
c r 1
c m
1
1  sT3 
Tc
Solid State Drives
1st order
approximation
of current loop
(24)
Full derivation
available here.
26
Design of Controllers– Current Controller
• After simplification, the final open loop gain function:
K
K
GH ol s  

1  sT1 1  sTr  1  sT1  Tr   s 2T1Tr
K
GH ol s  
1  sT3   s 2T1Tr
and since T1  Tr
Where
Since
Therefore GH ol s  
Solid State Drives
K
K1 K c K rTm
Tc
T1  Tr  T3
K
1  sT3
27
Design of Controllers–
Current loop 1st order approximation
where
Ti 
Ki 
K fi 
T3
1  K fi
K fi
(26)
1
H c 1  K fi 
(27)
K1K c K r H cTm
Tc
(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
Design of Controllers–
DC Motor
Speed Controller
& Load
1st order
approximation
of current
loop
• PI type speed controller: G s   K s 1  sTs 
s
(29)
sTs
• Assume there is unity speed feedback:
H
G ω s  
1
1  sT 
Solid State Drives
(30)
29
Design of Controllers–
DC Motor
Speed Controller
& Load
1
 Open loop gain function:
 K B K s Ki 

1  sTs 
GHs   

B
T
 t s  s1  sTi 1  sTm 
1st order
approximation
of current
loop
(31)
 From the loop gain, the system is of 3rd order.
 If designing without computers, simplification is needed.
Solid State Drives
30
Design of Controllers–
Speed Controller
• Relationship between the denominator time constants in (31):
Ti  Tm
(32)
• Hence, design the speed controller such that:
Ts  Tm
(33)
The open loop gain function becomes:
1  sTs 
 BtTs  s1  sTi 1  sTm 
 K B K s Ki 

1  sTm 


B
T
t s

 s1  sTi 1  sTm 


K B K s Ki


GH s  

K
K B K s Ki
GH s  
where K 
s1  sTi 
BtTs
i.e. controller zero cancels one of the system poles.
Solid State Drives
31
Design of Controllers–
Speed Controller
• After simplification, loop gain function:
GHs  
where
K 
K
s 1  sTi 
K B K s Ki
BtTs
(34)
(35)
• The controller is now of 2nd order.
GH s 
• From the closed loop transfer function: G cl s  
,
1  GH s 
the closed loop characteristic equation is:
s1  sTi   K
or when expanded becomes:
 2  1  K 
Ti s  s  

 Ti  Ti 

Solid State Drives
(36)
32
Design of Controllers–
Speed Controller
• Design the controller by comparing system characteristic
equation with the standard equation:
s 2  2 n s  n2
• Hence:
2n 
(37)
n 
2
(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
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 or
 three-phase
 Field current has no ripple – due to large Lf
 Converter 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
Field weakening
Solid State Drives
35
Closed Loop Control with Field
Weakening – Two-quadrant
Field weakening
Field
current
controller
(PI-type)
Estimated machine induced emf
dia
e  Va  Ra ia  La
dt
Induced emf
reference
Solid State Drives
Induced emf
controller
(PI-type with
limiter)
Field current
reference
36
Closed Loop Control with Field
Weakening – Two-quadrant
• The estimated machine-induced emf is obtained from:
dia
e  Va  Ra ia  La
dt
•
•
•
•
•
•
(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 flux
Solid State Drives
37
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
Inputs to
– Current-command
‘Selector’ block
– 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
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:
Back
Solid State Drives
42
Cosine-wave Crossing Control for
Controlled Rectifiers
Input voltage
to rectifier
Vm
0

2
3
4
Cosine voltage
Vcm
vc

 vc 

 Vcm 
  cos 1 
Results of
comparison
trigger SCRs
Output voltage
of rectifier
Cosine wave compared with
control voltage vc
Vcmcos() = vc

Back
Solid State Drives
43
Design of Controllers–
Current loop 1st order approximation
Ia s 

*
Ia s 

K c K r K 1Tm
1
K fi
1
H c 1  sT3 

1
1  K fi
1  sT3 
1  sT3 
1
1  sT3 
Tc
K c K r K 1 H cTm
1
Tc
K fi
Hc
1  sT   K
3

fi
K fi
1
H c 1  K fi 
 T3 

1  s

1

K
fi

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Solid State Drives
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220
x10  7.09
310 .5
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