Lesson 14: Transfer Functions of Dc Motors
10/28/2015
lesson14et438a.pptx
LESSON 14: TRANSFER
FUNCTIONS OF DC MOTORS
1
ET 438a Automatic Control Systems Technology
lesson14et438a.pptx
LEARNING OBJECTIVES
After this presentation you will be able to: Write the transfer function for an armature
controlled dc motor. Write a transfer function for a dc motor that
relates input voltage to shaft position. Represent a mechanical load using a mathematical
model. Explain how negative feedback affects dc motor
performance.
2
1
10/28/2015
lesson14et438a.pptx
STEADY-STATE OPERATION OF SEPARATELY EXCITED DC MOTORS
Consider steady-state model
ia = armature current
eb= back emf
ea= armature terminal voltage
wm
wm = motor speed (rad/sec)
T = motor torque
Tf = static friction torque Ra = armature resistance La = armature inductance Jm = rotational inertia Bm = viscous friction
Review the steady-state relationships
Of machine
3
lesson14et438a.pptx
STEADY-STATE OPERATION OF SEPARATELY EXCITED DC MOTORS
Relationships of Separately Excited Dc Motor
Torque-Current Curve
Back EMF Curve
T KT=DT/Dia
eb KT=DT/Dia
DT
Dia ia
-Tf
wm
wm=wnl ? (Dwm/DT)T
Speed-Torque Curve wnl
Dwm
DT
T
DT Dia
wm
4
2
10/28/2015
lesson14et438a.pptx
STEADY-STATE MOTOR EQUATIONS
Developed Torque
T = KT ia Tf N - m
T = motor torque KT = torque constant Tf = motor friction torque ia = armature current
Back EMF
eb = Ke wm V
wm= shaft speed (rad/s) eb = back emf Ke = back emf constant
KVL in Armature Circuit
ea = ia Ra eb V
ea= armature voltage eb = back emf Ra = armature resistance
Developed Power
P = wm T W
P = shaft power
5
lesson14et438a.pptx
STEADY-STATE MOTOR EQUATIONS
Combining the previous equations gives:
wm
=
KT
ea
(T Tf KT Ke
)Ra
(1)
wm
=
ea
ia Ra Ke
(2)
If the load torque is zero (T=0) then the above equation (1) gives the no-load speed
wnl
=
KT
ea (Tf ) KT Ke
Ra
6
3
10/28/2015
STEADY-STATE MOTOR OPERATION
Example 14-1: An armature-controlled dc motor has the following ratings: Tf=0.012 N-m, Ra=1.2 ohms, KT=0.06 N-m/A, Ke=0.06 V-s/rad. It has a maximum speed of 500 rad/s with a maximum current of 2 A. Find: a) maximum output torque, b) maximum mechanical output power, c) maximum armature voltage, d) no-load speed at maximum armature voltage.
7
lesson14et438a.pptx
lesson14et438a.pptx
EXAMPLE 14-1 SOLUTION (1)
Define given variables
a) Tmax occurs at Imax so....
b) Find Pmax
Answer
8
Answer
4
EXAMPLE 14-1 SOLUTION (2)
c) Find maximum back emf
Answer d) Find no-load motor speed
At no-load, T=0. Load torque is zero.
T=0
lesson14et438a.pptx
10/28/2015
9
lesson14et438a.pptx
TRANSFER FUNCTION OF ARMATURECONTROLLED DC MOTOR
Write all variables as time functions
Write electrical equations and mechanical equations. Use the electromechanical relationships to couple the two equations.
Ra + ia(t) ea(t)
La
+ T(t)
eb(t)
Jm Bm
Consider ea(t) and eb(t) as inputs and ia(t) as output. Write KVL around armature
ea (t)
=
Ra
ia
(t
)
L
dia (t dt
)
eb (t)
Mechanical Dynamics
T(t)
=J
m
dwm (t) dt
Bm
wm
(
t
)
10
5
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