2. 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
3. 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
+
Reference
signal
Output
signal
4. 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
5. 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
6. 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
7. 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
8. 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
9. 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
10. 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
11. 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
m
L
L B
T
12. 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
s
V
s
I
s
I
s
ω
s
V
s
ω
a
a
a
m
a
m
m
t
b
sT
B
K
1
s
I
s
ω
a
m
2
1
1
a
a
1
1
1
s
V
s
I
sT
sT
sT
K m
Back
13. 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
t
m
B
J
T
L
t B
B
B
1
a
b
a
t
a
t
a
a
t
a
a
JL
K
JL
B
R
J
B
L
R
J
B
L
R
T
T
2
2
2
1 4
1
2
1
1
,
1
t
a
b
t
B
R
K
B
K
2
1
Back
14. 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
v
1
cos
c
r
c
cm
m
cm
c
m
m
dc v
K
v
V
V
V
v
V
V
V
L,
L
L,
L
L,
L
3
cos
cos
3
cos
3 1
15. Transfer Function of Subsystems –
Three-phase Converter
Gain of the converter
(10)
where V = rms line-to-line voltage of 3-phase supply
Converter also has a delay
(11)
where fs = supply voltage frequency
Hence, the converter transfer function
(12)
Solid State Drives 15
r
r
sT
K
1
s
Gr
cm
cm
cm
m
r
V
V
V
V
V
V
K 35
.
1
2
3
3
L,
L
s
s
r
f
f
T
1
12
1
1
360
60
2
1
Back
16. Transfer Function of Subsystems –
Current and Speed Feedback
Current Feedback
Transfer function:
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:
(13)
where K = gain, T = time constant
Most high performance systems use dc tacho generator and low-
pass filter
Filter time constant < 10 ms
Solid State Drives 16
sT
K
1
s
Gω
c
H
17. 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 time
Not 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
18. 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
c
c
sT
sT
K
1
s
Gc
r
m
c
c
c
r
c
sT
sT
sT
s
sT
sT
T
H
K
K
K
1
1
1
1
1
s
GH
2
1
1
ol
DC Motor
& Load
Converter
Controller
19. 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
m
m sT
sT
1
c
m
c
r
c
r
c
r
m
c
c
c
r
c
r
m
c
c
c
r
c
T
T
H
K
K
K
K
sT
sT
sT
sT
K
sT
sT
sT
s
sT
sT
T
H
K
K
K
sT
sT
sT
s
sT
sT
T
H
K
K
K
1
2
1
ol
2
1
1
2
1
1
ol
where
1
1
1
1
s
GH
1
1
1
1
1
1
1
1
1
s
GH
(17)
20. 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
1
2 T
T
Tr
2
T
Tc
c
m
c
r
c
r
r
r
c
T
T
H
K
K
K
K
sT
sT
K
sT
sT
sT
sT
K
sT
sT
sT
sT
K
1
1
ol
2
1
2
2
1
ol
where
1
1
s
GH
1
1
1
1
1
1
1
1
s
GH
21. 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
r
sT
sT
K
1
1
s
GH
1
ol
c
m
c
r
c
T
T
H
K
K
K
K 1
K
sT
sT r
1
1 1
s
GH
1
s
GH
s
G
ol
ol
cl
r
r
r
r
T
T
K
T
T
T
T
s
s
T
T
1
1
1
2
1
1
22. 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
2
2
2 n
ns
s
(23)
1
2
1
1
1
r
r
r
T
T
K
T
T
T
T
(22)
1
1
2
r
n
T
T
K
23. 23
1
2
707
.
0
1
1
1
r
r
r
T
T
K
T
T
T
T
Squaring the equation on both sides
r
1
2
1
1
r
1
2
1
1
2
1
1
1
T
T
1
K
x
2
1
T
T
1
K
x
2
x
2
0.5
1
2
5
.
0
r
r
r
r
r
r
r
T
T
T
T
T
T
T
T
T
T
K
T
T
T
T
r
T
T
r
T
T
K
r
T
T
X
r
T
T
r
T
T
r
T
T
r
T
T
r
T
T
K
1
2
2
1
1
2
1
2
1
1
1
K
1
2
2
1
1
1
25. 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
26. 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
sT
T
T
H
K
K
K
sT
T
T
K
K
K
i
c
r
c
r
c
1
1
1
1
1
1
s
I
s
I
3
1
3
1
*
a
a
r
T
T
T
1
3
Full derivation a
vailable here.
1st
order
approximation
of current loop
27. Design of Controllers– Current Controller
• After simplification, the final open loop gain function:
Solid State Drives 27
r
r
r T
T
s
T
T
s
K
sT
sT
K
1
2
1
1
ol
1
1
1
s
GH
c
m
r
c
T
T
K
K
K
K 1
r
T
T
s
T
s
K
1
2
3
ol
1
s
GH
3
1 T
T
T r
Since
and since r
T
T
1
3
ol
1
s
GH
sT
K
Therefore
Where
28. 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
m
c
r
c
fi
T
T
H
K
K
K
K 1
fi
i
K
T
T
1
3
fi
c
fi
i
K
H
K
K
1
1
29. Design of Controllers–
Speed Controller
• PI type speed controller:
(29)
• Assume there is unity speed feedback:
(30)
Solid State Drives 29
s
s
s
s
sT
sT
K
1
s
G
1
1
s
Gω
sT
H
DC Motor
& Load
1st
order
approximatio
n of current
loop
30. 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
m
i
s
s
t
i
s
B
sT
sT
s
sT
T
B
K
K
K
1
1
1
s
GH
DC Motor
& Load
1st
order
approximatio
n of current
loop
31. 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
m
i T
T
m
s T
T
s
t
i
s
B
i
m
i
m
s
t
i
s
B
m
i
s
s
t
i
s
B
T
B
K
K
K
K
sT
s
K
sT
sT
s
sT
T
B
K
K
K
sT
sT
s
sT
T
B
K
K
K
where
1
s
GH
1
1
1
1
1
1
s
GH
32. 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
i
sT
s
K
1
s
GH
s
t
i
s
B
T
B
K
K
K
K
K
sT
s i
1
s
GH
1
s
GH
s
Gcl
i
i
i
T
K
T
s
s
T
1
2
33. 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
2
2
2 n
ns
s
n
2
2
n
34. 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
35. Closed Loop Control with Field
Weakening – Two-quadrant
Solid State Drives 35
Field weakening
36. Closed Loop Control with Field
Weakening – Two-quadrant
Solid State Drives 36
dt
di
L
i
R
V
e a
a
a
a
a
Induced emf
controller
(PI-type with
limiter)
Field weakening
Field
current
controller
(PI-type)
Field current
reference
Estimated machine -
induced emf
Induced emf
reference
37. 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 flux
Solid State Drives 37
dt
di
L
i
R
V
e a
a
a
a
a
38. Closed Loop Control with Controlled
Rectifiers – Four-quadrant
• Four-quadrant Three-phase Controlled
Rectifier DC Motor Drives
Solid State Drives 38
39. 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
40. 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
41. DC Motor and Load Transfer Function -
Decoupling of Induced EMF Loop
• Step 1:
• Step 2:
Solid State Drives 41
42. DC Motor and Load Transfer Function -
Decoupling of Induced EMF Loop
• Step 3:
• Step 4:
Solid State Drives 42
Back
43. Cosine-wave Crossing Control for
Controlled Rectifiers
Solid State Drives 43
Vm
Vcm
vc
0 2 3 4
Input voltage
to rectifier
Cosine wave compared with
control voltage vc
Results of
comparison
trigger SCRs
Output voltage
of rectifier
Vcmcos() = vc
cm
c
V
v
1
cos
Cosine voltage
Back
44. Design of Controllers–
Current loop 1st
order approximation
Solid State Drives 44
i
c
c
c
c
m
c
m
sT
K
K
T
s
K
H
K
K
sT
H
K
sT
K
sT
H
K
sT
T
T
H
K
K
K
sT
T
T
K
K
K
i
fi
fi
fi
fi
fi
fi
fi
c
r
c
r
c
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
s
I
s
I
3
3
3
3
3
1
3
1
*
a
a
Back
