Chapter 14: Control of D.C. Drives
I
CLOSED-LOOP CONTROL OF
D.C. DRIVES
Motor drives are used in a very wide power range, from a few watts to many
thousands of kilowatts in applications ranging from very precise, high performance
position controlled drives in robotics to variable speed drives for adjusting flow
rates in pumps. In all drives, where the speed and position are controlled, a
power electronic converter is needed as an interface between the input power
and the motor. In many applications, open-loop operation of d.c. motors may not
be satisfactory because the speed changes if the firing angle is kept constant and
the torque applied to the d.c. motor is increased. However, if the drive requires
constant-speed operation the firing angle has to change to maintain a constant
speed. This can be achieved in a closed-loop control system. The closed-loop
control system has the advantages of improved accuracy, fast dynamic response,
and reduced effects of load disturbances and system nonlinearities.
When the drive requirements include rapid acceleration and deceleration, closedloop control is necessary. The system can be made to operate at constant torque
or constant horse power over a certain speed range. In practice, most industrial
drive system operates on closed-loop feedback system.
Figure 14.1 shows the basic block diagram of a closed-loop control system.
If the motor speed decreases due to application of additional load torque, the
speed error en increases, which increases the control signal Ec. This, in turn,
changes the firing angle of the converter and thus increases the motor armature
voltage, Ea. An increase in the motor voltage develops more torque to restore the
speed of the drive system. The system thus passes through a transient period
until the developed torque matches the applied torque.
Fig. 14.1
Basic block diagram of a closed-loop speed control system
The response of a closed-loop system can be studied using transfer function
techniques. In the following sections, a systematic development of the transfer
function of various blocks as well as the whole system is presented.
1
Open-loop Transfer Function
Consider the circuit arrangement of a converter fed separately excited d.c. motor
drive, as shown in Fig. 14.2. Separate excitation in a separately excited d.c.
Power-14 OLC.p65
110
9/27/07, 2:07 PM
Control of D.C. Drives
motor makes the speed control of the motor relatively easy. The motor speed is
adjusted by setting reference or control voltage, er. Assuming a linear power
converter of gain k c, the armature voltage of the motor is
ea = k c ◊ er
(14.1)
ia
a.c. power
supply
Ra
+
+
er
Converter
of
gain ,
Kc
–
If
+
ea
La
–
–
ef
Lf ◊ Rf
+
eb
M
–
W
Td
TL
B
Fig. 14.2 Converter fed separately excited d.c. motor drive
Assuming that the motor field current If and the back-emf constant k a remain
constant during any transient disturbances, the system equations are
e a = R a ia + L a
dia
+ eb
dt
where
eb = Ka f w
\
eb = K a K f If w, But f = K f If
(14.2)
= K a If w
(14.3)
Substituting Eq. (14.3) in Eq. (14.2) yields
dia
+ K a If w
(14.4)
dt
The developed torque balance equation is
dw
+ Bw + TL
(14.5)
T d = K t I f ia = J
dt
The transient behaviour may be analyzed by changing the system equation into
Laplace transforms with zero initial conditions.
Transforming Eqs (14.1) through (14.2) yields
ea = R a ia + L a
Power-14 OLC.p65
Ea(s) = K c er(s)
(14.6)
Ea(s) = R a Ia(s) + s La Ia(s) + Ka If w (s)
(14.7)
111
9/27/07, 2:07 PM
Power Electronics
Td(s) = Kt If Ia (s) = sJw (s) + Bw (s) + T L(s)
(14.8)
From Eq. (14.7), the armature current is
Ia(s) =
Ea (s ) - K a ◊ I f ◊ w (s )
sLa + Ra
=
Ea (s ) - K a ◊ I f ◊ w (s )
Ra (st a + 1)
(14.9)
where ta = La/R a is known as the time constant of motor armature circuit. From
Eq. (14.8), the motor speed is
w(s) =
Td ( s ) - TL ( s )
sJ + B
(14.10)
=
Td ( s ) - TL ( s )
B( st m + 1)
(14.11)
where tm = J/B is known as the mechanical time constant of the motor. Equations
(14.6), (14.9) and (14.11) can be used to draw the open-loop block diagram as
shown in Fig. 14.3. Two possible disturbances are control voltage, er, and load
torque, T L . The steady-state responses can be determined by combining the
individual response due to er and T L .
Fig. 14.3 Open-loop block diagram of separately excited d.c. motor drive
From Fig. 14.3, an expression can be obtained for the change in speed due to
disturbances in control voltage er(s) and load torque TL (s). The response due to
a step change in the reference voltage is obtained by setting T L to zero from
Fig. 14.3, we can have the speed response due to reference voltage as
Kc ◊ Ka ◊ I f ( Ra ◊ B)
w ( s)
= 2
Er (s )
s (t a ◊ t m ) + s (t a + t m ) + 1 + ( Ka ◊ I f )2 / Ra B
(14.12)
Now, the response due to a change in load torque, T L , can be obtained by setting
Er to zero. The block diagram for a step-change in load torque disturbance is
shown in Fig. 14.4.
(1/ B) (st a + 1)
w (s )
= 2
TL (s )
s (t a ◊ t m ) + s(t a + t m ) + 1 + ( Ka ◊ I f )2 / Ra B
Power-14 OLC.p65
112
9/27/07, 2:07 PM
(14.13)
!
Control of D.C. Drives
Using the final value theorem, the steady-state relationship of a change in speed,
Dw, due to a step change in control voltage, D Er, and a step change in load
torque, DT L , can be found from Eqs (14.12) and (14.13), respectively, by
substituting s = 0.
Kc ◊K a I f
\
Dw =
and
Dw = –
Ra B + (K a I f )2
- Ra
(14.14)
DEr
Ra B + (K a I f )2
(14.15)
DTL
Fig. 14.4 Block diagram for torque disturbance input
As discussed in previous sections that the d.c. series motors are used
extensively in traction applications where the steady-state speed is determined by
the friction and gradient forces. The motor can be operated at a constant torque
(or current) up to the base speed by adjusting the armature voltage, which
corresponds to the maximum armature voltage. Figure 14.5 shows the chopperfed d.c. series motor drive.
Fig. 14.5 Chopper-controlled d.c. series motor drive
The armature voltage is related to the control voltage by a linear gain of the
chopper, kc. Assuming that the back emf constant k a does not change with the
armature current and remains constant, the system equation are:
Power-14 OLC.p65
113
9/27/07, 2:07 PM
"
Power Electronics
e a = k c ◊ er
(14.16)
eb = k a ◊ i a ◊ w
(14.17)
e a = R m ia + L m
Td = K t ◊ i2a
dia
+ eb
dt
(14.18)
(14.19)
dw
+ Bw + T L
(14.20)
dt
The application of a transfer function techniques would no longer be valid
since Eq. (14.19) contains a product of variable-type nonlinearities. However,
these equations can be linearized by considering a small perturbation at the operating
point. Let us define the system parameters around the operating point as
Td = J
eb = Eb0 + Deb,
ea = Ea0 + Dea
i a = Ia0 + Dia,
Td = T d0 + DT d
w = w0 + Dw ,
er = Er0 + Der
TL = TL0 + DT L
Recognizing that Dia Dw and (Dia)2 are very small, tending to zero, Eqs (14.16)
to (14.20) can be linearized to
Dea = KcDer
Deb = K a (Ia0 Dw + w0 Dia)
Dea = R m Dia + L m
d( Dia )
+ Deb.
dt
Dtd = 2K a Ia0 Dia.
d( Dw )
+ B ◊ Dw + DT L
dt
Transforming these equations into Laplace transform form, yields
DTd = J
DEa(s) = Kc DEr(s)
(14.21)
DEb(s) = Ka[Ia0 Dw (s) + w0 DIa(s)]
(14.22)
DEa(s) = R m Dia(s) + sLm DIa(s) + DEb(s)
(14.23)
DTd(s) = 2 K c Ia0 DIa(s)
(14.24)
DTd(s) = sJ Dw(s) + BDs(s) + DT L (s)
(14.25)
Equations (14.21) to (14.25) are sufficient to establish the block diagram of a
d.c. series motor drive, as shown in Fig. 14.6. From Fig. 14.6, it becomes clear
that any change in either control voltage or load torque will result in a speed
change. The block diagram for a change in reference voltage is shown in
Fig. 14.7(a) and that for a change in load torque is shown in Fig. 14.7(b).
Power-14 OLC.p65
114
9/27/07, 2:07 PM
#
Control of D.C. Drives
2
Closed-Loop Transfer Function
To change the open-loop arrangement in Fig. 14.2 into a closed-loop system, a
speed sensor is attached to the motor-shaft. The output of the sensor, which is
proportional to the speed, is amplified by a factor of k s and is compared with the
reference voltage er to form the error voltage, eN. The complete block diagram
for a closed-loop control of a separately excited d.c. motor is shown in Fig. 14.8.
Fig. 14.6
Open-loop block diagram of a chopper-fed d.c. series drive
Fig. 14.7
Power-14 OLC.p65
115
Block diagram for reference voltage load torque
9/27/07, 2:07 PM
$
Power Electronics
Fig. 14.8 Block diagram for closed-loop control of separately
excited d.c. motor
The closed-loop step response due to a change in reference voltage can be
obtained from Fig. 14.4 with TL = 0. The transfer function becomes,
Kc Ka I f / Ra ◊B
w ( s)
= 2
Er (s )
s (t a t m ) + s (t a + t m ) +1 + [(K a ◊I f )2 + K s K c K a ◊I f ] Ra B
(14.26)
The response due to a change in the load torque T L can also be obtained from
Fig. 14.8 by setting Er to zero. The transfer function becomes,
(1/ B)(st a + 1)
w (s )
= - 2
TL (s )
s (t at m ) + s(t a + t m ) + 1 + [(K a ◊I f )2 + K s K c ◊I f ]/ Ra B
(14.27)
Using the final value theorem, the steady-state change in speed, Dw, due to a
step change in control voltage, DEr, and a step change in load torque, DTL , can
be obtained from Eqs (14.26) and (14.27), respectively, by substituting s = 0.
Dw =
and
Kc K a I f
Ra B + (K a I f )2 + K s K c K a I f
Dw = -
(14.28)
DEr
Ra
DI L
Ra B + ( Ka I f )2 + Ks Kc Ka I f
(14.29)
Figure 14.8 uses a speed feedback only. In practice, the motor is required to
operate at a desired speed but it has to meet the load torque which depends on the
armature current. When the motor is operating at a particular speed and, if a load
is applied suddenly, the speed will fall and the motor will take time to come up to
the desired speed. A speed feedback with an inner current loop, as shown in
Fig. 14.9, provides faster response to any disturbances in speed command, load
torque and supply voltage.
Power-14 OLC.p65
116
9/27/07, 2:07 PM
%
Control of D.C. Drives
Fig. 14.9 Closed-loop speed control with inner current loop
and field weakening
The purpose of the current loop is to cope with a sudden torque demand
under transient operations such as starting, braking, speed reversal, etc. The
output of the speed controller, ec, is applied to the current limiter which sets the
current reference, Iar, for the current loop. The armature current Ia is sensed by
a current sensor, filtered normally by an active filter to remove ripple, and compared
with the current reference Iar. The current error is processed through a current
controller whose output ec adjusts the firing angle of the converter and brings the
motor speed to the desired value.
Any positive speed error caused by either an increase in the speed command
or an increase in the load torque, produces a higher current reference, Iar. The
motor accelerates due to an increase in Iar, to correct the speed error and finally
settles at a new Iar, which makes the motor torque equal to the load torque and
the speed error close to zero. For any large positive speed error, the current
limiter saturates and the current reference Iar is limited to a value Iam and the
drive current is not allowed to exceed the maximum permissible value. The speed
error is corrected at the maximum permissible armature current until the speed
error becomes small and the current limiter comes out of saturation. Now, the
speed error is corrected with Ia less than the permissible value Iam.
Power-14 OLC.p65
117
9/27/07, 2:07 PM
&
Power Electronics
The speed control from zero to base-speed is normally done at the maximum
field by armature voltage control, and control above the base-speed should be
done by field weakening at the rated armature voltage. In the field control loop,
the back emf Eb (= Ea – Ia Ra) is compared with a reference voltage Eb(ref), which
is chosen to be between 0.85 to 0.95 of the rated armature voltage. The higher
value is used for motors with a low armature circuit resistance. For speeds
below base speed, the field controller saturates due to a large value of error ef,
thereby applying the maximum field voltage and current.
When the speed is closed to the base speed, Ea is almost near to the rated
value and the field controller comes out of saturation. For a speed command
above the base speed, the speed error causes a higher value of Ea. The motor
accelerates, the back emf Eb increases, and the field error ef decreases. The field
current then decreases and the motor speed continues to increase until the motor
speed reaches the desired speed. Thus, the speed control above the base speed is
obtained by the field weakening while the armature terminal voltage is maintained
at near the rated value.
In the field weakening region, the drive responds very slowly due to the large
field time constant. A fully-controlled rectifier is normally used in the field, because
it has the ability to reverse the voltage, thereby reducing the field current much
faster than a half-controlled rectifier.
SOLVED EXAMPLES
Example 14.1 A 60 HP, 230 V, 1750 rpm, separately excited d.c. motor is
controlled by a converter as shown in the block diagram of Fig. 14.8. The machine back
emf constant is Ka = 0.9 V/A rad/s and the field current is maintained constant at 1.2 A.
The armature resistance is R a = 0.1 W and the viscous friction constant is b = 0.25 N-m/
rad/s. The amplification of the speed sensor is K s = 90 mV/rad/s and the gain of the
power control is K c = 100. Determine the following:
(a) The rated torque of the motor.
(b) The reference voltage Er to drive the motor at the rated speed.
(c) The speed at which the motor develops the rated torque if the reference
voltage is kept unchanged.
(d) The motor speed if the load torque is increased by 10% of the rated value.
(e) The motor speed, if the reference voltage is reduced 10%.
(f) The motor speed, if the load torque is increased by 10% of the rated value and
the reference voltage is reduced by 10%.
(g) The speed regulation for a reference voltage of Er = 2.2 V if there was no
feedback as in an open-loop control.
(h) The speed regulation with a closed-loop control.
Solution: wr =
Power-14 OLC.p65
118
1750 ¥ 2p
= 183.26 rad/s.
60
9/27/07, 2:07 PM
'
Control of D.C. Drives
(a) The rated torque, TL =
60 ¥ 746
= 244.24 N-m.
183.26
(b) Since Ea = Kc Er, for open control Eq. (14.14) gives,
KaI f
0.90 ¥ 1.2
w
w
=
=
= 0.903
=
2
0.1 ¥ 0.25 + (0.90 ¥ 1.2)2
Ea
Kc Er Ra B + ( Ka I f )
Now, at rated speed,
Ea =
w
183.26
=
= 202.95 V
0.903 0.903
And, feedback voltage,
E b = Ks w = 90 ¥ 10–3 ¥ 183.26 = 16.49 V
With closed-loop control, (Er – Eb)K c = Ea
or (Er – 16.45) ¥ 100 = 202.95 \ Er = 18.48 V
(c)
For Er = 18.48 V, and DT L = 244.54 N-m, Eq. (14.29) gives
Dw = -
0.1 ¥ 244.24
= – 2.24 rad/s.
(0.1 ¥ 0.25) + (0.9 ¥ 1.2)2 + 90 ¥ 10 - 3 ¥ 100 ¥ 0.9 ¥ 1.2
The speed at rated torque,
w = 183.26 – 2.24 = 181.02 rad/s = 1728.61 rpm.
(d) DT L = 1.1 ¥ 244.24 268.66 N-m. From Eq. (14.29)
Dw = \
0.1 ¥ 268.66
= – 2.46 rad/s.
(0.1 ¥ 0.25) + (0.9 ¥ 1.2)2 + 90 ¥ 10 - 3 ¥ 100 ¥ 0.9 ¥ 1.2
Motor speed, w = 183.26 – 2.46 = 180.8 rad/s = 1726.51 rpm.
(e)
Now,
DE r = – 0.1 ¥ 18.48 = – 1.848 V.
From Eq. (14.28), change in speed
Dw = -
100 ¥ 0.9 ¥ 1.2 ¥ 1.848
= 18.29 rad/s.
(0.1 ¥ 0.25) + (0.9 ¥ 1.2) 2 + 90 ¥ 10 - 3 ¥ 100 ¥ 0.9 ¥ 1.2
Motor speed is
w = 183.26 – 18.29 = 164.97 rad/s = 1575.35 rpm.
(f)
Now, the motor speed can be obtained by using superposition:
w = 183.26 – 2.46 – 18.29 = 162.51 rad/s = 1551.86 rpm.
(g)
DEr = 2.2 V
Equation (14.14) gives, Dw =
100 ¥ 0.9 ¥ 1.2 ¥ 2.2
= 199.43 rad/s = 1904.3 rpm.
0.1 ¥ 0.25 + (0.9 ¥ 1.2) 2
and the no-load speed is w = 1904.38 rpm.
For full-load, DT L = 244.24 N-m.
Power-14 OLC.p65
119
9/27/07, 2:07 PM
Power Electronics
Now, Eq. (14.14) gives Dw =
0.1 ¥ 244.24
= – 20.5 rad/s.
0.1 ¥ 0.25 + (0.9 ¥ 1.2) 2
and the full-load speed,
w = 183.26 – 20.5 = 162.76 rad/s = 1554.24 rpm.
The speed regulation with open-loop control is
=
(h)
1750 - 1524.24
= 14.81%
1524.24
Using the speed from part (c), the speed regulation with closed-loop control is
1750 - 1728.61
= 1.24%
1728.61
From parts (g) and (b), it becomes clear that with closed-loop control, the speed
regulation is reduced by a factor of 11.944.
Power-14 OLC.p65
120
9/27/07, 2:07 PM