1
DC Motor Controller Design
1. INTRODUCTION
The DC motors have been dominating the field of adjustable speed drives over a century. They are still the most common
choice if a controlled electric drive operating over a wide speed range is specified. This is due to their excellent operational
properties and control characteristics; the only disadvantage is the mechanical commutator, which restricts power and speed;
increases the inertia and the axial length and requires frequent maintenance.
The electric circuit of the armature and of the rotor are shown in the following figure:
Figure 1: DC Motor equivalent circuit
The equivalent circuit of separately excited DC machine can be represented in schematic form as shown in Fig.1. The pertinent
dynamic equations for the DC motor are as follows.
di (t )
R a i a (t ) L a a E b (t ) V a (t ) (1)
dt
E b (t ) C1 (t ) (2)
dw(t )
J M d (t ) M L (t ) (3)
dt
M d (t ) C 2 i a (t ) (4)
d (t )
(t ) ( 5)
dt
It is assumed that the field flux f is constant and the dynamics associated with the field circuit is slow as compared to the
armature circuit. From (2) and (4), it can be seen that
2
E a (t )i a (t ) (C1 / C 2 ) M d (t ) (t ) ( 6)
Therefore C1 C 2 = C . The armature voltage Va and field voltage V f are assumed to be independently controllable.
The machine model can be built from the dynamic equations as described above; and is shown in Fig.2.
Figure 2: The equivalent model of DC motor
The input output relationship can be determined for the above model in terms of transfer functions. Here the input to the
system are ML(s) and Va(S) while the outputs are Ia(s) and w(s.
Ia(s) (1 / R a ) sTem
(7)
Va(s) 1 sTem (1 sTa )
(s) (1 / C )
(8)
Va(s) 1 sT em (1 sT a )
Ia(s) (1 / C ) (9)
M L (s) 1 sT em (1 sT a )
(s) ( R /(C ) 2 )(1 sTa )
a ( 10 )
M L (s) 1 sTem (1 sTa )
The current loop can be represented as a simple lag with a time constant Ta=La/Ra (called electrical time constant of motor).
Similarly the mechanical time constant is defined as Tem JR a /(C ) 2 . The equivalent model may be presented as shown in Fig.
3.
Figure 3: Alternate representation in terms of armature constant and mechanical constant
3
2. POWER CONVERTER
Figure 4: Power converter
The converter used to control the DC voltage across the armature is as shown in Fig.4. Uni-polar switching is used for
switching Legs A and B. The control signal Vc is compared with the triangular carrier to produce PWM for phase A while -Vc is
compared with triangular carrier to produce PWM pulses for Leg B. The advantage of this method over bi-polar switching is that
harmonics generated at carrier frequency by Leg A and Leg B cancels each other and the first harmonics produced are around
twice the switching frequency.
2.THE CONTROL STRUCTURE
Figure 5: The DC motor speed control scheme
4
The speed and current controllers are PI controllers (with limits on the integrator and well as the proportional gain) as shown in
Fig.5. The current and speed feedbacks are taken through filters shown as first order delay blocks. The converter is also modeled
as a gain KA with a delay TA. The design of the PI controller constants is done in two steps. First the faster inner loop is designed.
The approximate model for the overall inner loop is used to design the outer speed loop.
A. Current Loop Design
Ia(s) (1 / R a ) sTem Ia(s) (1 / R a )
The can be approximated to as the mechanical time constant is much greater than
Va(s) 1 sTem (1 sTa ) Va(s) 1 sTa
electrical time constant. Thus the overall current loop is as shown in Fig.5.
Figure 6: Current control loop
The pole-zero cancellation technique is used to determine the gain and time constants for PI controller. The transfer function
for the current control loop is
(1 / Ra ) K c (1 sTc ) K A
Ia(s) 1 sTa sTc 1 sT A
( 11 )
Ia (s) 1 (1 / Ra ) K c (1 sTc ) K A
* K2
1 sTa sTc 1 sT A 1 sT2
The zero of the system is selected in such a way that it cancels the pole with largest time constant in the system. The power
converter time constant depends upon the type of converter is used and switching frequency. When the sine triangle comparison
method is used to switch the devices, the time constant of converter is much smaller than the motor electrical time constant. So
the Tc is selected equal to Ta.
1 Kc K A
Ia(s) R a sTa 1 sT A
Therefore ( 12 )
Ia (s) 1 1 K c K A
* K2
R a sTa 1 sT A 1 sT2
Kc K A
(1 sT2 )
Ia(s) Ra
( 13 )
Ia* (s) Kc K AK2
sTa (1 sT2 )(1 sT A )
Ra
(1 sT2 )(1 sT A ) is approximated as 1 s (T2 T A ) . Let = (T2 T A ) . Therefore, 1 s (T2 T A ) 1 s
KcK A
(1 sT2 )
Ia(s) R a Ta
( 14 )
Ia* (s) s 2 s K c K A K 2
R a Ta
1 R a Ta
The denominator is compared with s 2 2 n n 2 . With , we get K c . Thus the current controller
2 2 K A K 2
5
Ra Ta
constants are Tc Ta and K c
2 K A K 2
Ia(s) 1 1 (1 sT2 )
( 15 )
* 2
K2 2 s 1
Ia (s) s
2 2
B. Speed controller design
Ia(s) 1 1
Eqn. (15) can be approximated to first order system, by neglecting the second order terms. Thus the
Ia (s) * K 2 1 2 s
overall speed control loop is represented as shown in Fig.7.
Figure 7: Speed controller loop
The time constant and gain for PI controller is Tw and Kw. The loop has one pole at zero, which causes continuous integration.
For such system, the method of pole-zero cancellation does not work. Instead the method of symmetric optimum is used to
place the PI controller zero to obtain sufficient phase margin. The open loop gain is
(1 / K 2 ) R a K w (1 sT w ) 1 K1
G (s) H (s ) ( 16 )
1 s 2 C sT w sTem 1 sT1
K w K1 Ra (1 sTw )
( 17 )
(C ) K 2 T wTem s (1 s 2 ) (1 sT )
2
1
K w K1 Ra (1 sT w )
( 18 )
(C ) K 2 T wTem s 2 (1 s ))
where (2 T1 )
Thus the open loop transfer function has two poles at zero and one pole at 1 / and one zero at 1 / Tw . The value of Tw is
selected such that we get sufficient phase margin at the gain crossover frequency. Therefore T w 4
The crossover frequency is 1 / T w and the gain at crossover frequency is selected equal to one. This results into
K 2 (C)Tem
Kw ( 19 )
2Ra K 1
6
Typical Simulation Results
