MESB374 System Modeling and Analysis
Electro-mechanical Systems
Electro-Mechanical Systems
• DC Motors– Principles of Operation
– Modeling ( EOM)
• Block Diagram Representations– A Convenient Graphic Representations of Interconnections
among various Subsystems of a complex system described by Equations in s - Domain
– Block Diagram Representation of DC Motors
• Example
DC Motors• Terminology
– Rotor : the rotating part of the motor.
– Stator : the stationary part of the motor.
– Field System : the part of the motor that provides the magnetic flux.
– Armature : the part of the motor which carries current that interacts with the magnetic flux to produce torque.
– Brushes : the part of the electrical circuit through which the current is supplied to the armature.
– Commutator : the part of the rotor that is in contact with the brushes.
Motors are actuation devices (actuators) Motors are actuation devices (actuators) that generate that generate torquetorque as actuation. as actuation.
i
DC Motors - Principles of Operation• Torque Generation
d f i dLa
B
Perpendicular
a af i L i L
B B
Coil 2 2 aFR i L R B
dL
B
d f
i
B
B
Force will act on a conductor in a magnetic field with current flowing through the conductor:
Total torque generated:
Integrate over the entire length:
Magnetic fieldConductorCurrent
Needs three elements:
DC Motors - Principles of OperationLet N be the number of coils in the motor. The total torque generated from the N coils is:
For a given motor, (N, B, L, R) are fixed. We can define
as the Torque Constant of the motor.
The torque generated by a DC motor is proportional to the armature current ia :
For a DC motor, it is desirable to have a large KT . However, size and other physical limitations often limits the achievable KT .
(2 )
(2 )
T
m a
F
a
K
N i L R
N L R i
B
B
K N L RT 2 B Nm / A
m T aK i
Large KT :
– Large (N, L, R).
(N, L, R) is limited by the size and weight of the motor.
– Large B:
Need to understand the methods of generating flux ...
• Back-EMF Generation
Electromotive force (EMF) is generated in a conductor moving in a magnetic field:
Integrate over the entire length L:
Since the N armature coils are in series, the total EMF is:
Define the Back-EMF Constant Kb :
The Back-EMF generated due to the rotation of the motor armature is opposing the applied voltage and is proportional to the angular speed of the motor:
Note: KT = Kb is true only if SI unit is used !
DC Motors - Principles of Operation
( )emfde v dL B
( )emfe v L B
2 ( ) 2b
emfKv
E N R L NR L B B
K N R Lb 2 B V / (rad / sec)
E Kemf b
v B
B
v
DC Motors - ModelingSchematic Element Laws:
iA
+ eLa
LA
+ eRa
RA+
ei(t)_
+Eemf_
JA
B
m
L
FBD:
Interconnection Laws:
Mechanical Subsystem:
Electrical Subsystem
JA
L
fm
1 2
4
3
41
0Ra
La
AA A A emf i
e ee
diR i L E e
dt
f
A m LJ B
12 23 34 41 0e e e e
emf bE K
m T AK i
DC Motors - ModelingDerive I/O Model:
41
0R emfa
La
AA A A b i
e E ee
diR i L K e
dt
L J
K
L B
K
R J
K
R B
KK e t
L R
KA A
T
A
T
A A
T
A
Tb i
A L A L
T
( )
FHG
IKJ FHG
IKJ
b g
L J
K
L B
K
R J
K
R B
KK e t
L R
KA A
T
A
T
A A
T
A
Tb i
A L A L
T
( )
FHG
IKJ FHG
IKJ
b g
I/O Model from ei(t) andLto angular speed :
I/O Model from ei(t) andLto angular position :
fm
A T A LJ K i B
Eliminate iA 1A A L
T
i J BK
1 1
A A
A A L A A L b iT T
i i
dR J B L J B K e
K dt K
DC Motors - Modeling
Q: Let the load torque L = T, what is the steady state speed of the motor for a constant input voltage V ?
1 1( ) ( ) ( )E i T Ls G E s G T s
s s
2
2
( ) ( )
( )
E
T
Ti
A A A A A A b T
G
A AL
A A A A A A b T
G
Ks E s
L J s L B R J s R B K K
L s Rs
L J s L B R J s R B K K
Transfer Functions:
Q: Let the load torque be zero (No Load), what is the steady state speed (No-Load Speed) of the motor for a constant input voltage V ?
0
( ) ( )
( ) lim
i L
T Ass E E
sA b T A b T
E s T s
V T K Rt s G s G s V T
s s R B K K R B K K
( ) ( )E is G E s 0
( )
( ) lim
i
Tss Es
A b TE s
V Kt sG s V
s R B K K
• Signal Addition/Subtraction
Ex: Draw the block diagram for the following DE:
Block Diagram Representation• Differential Equation Transfer
Function (System & Signals)
Ex: Draw the block diagram for the following DE:
1
Js
Y s G s U s( ) ( ) ( )
G(s)U(s) Y(s)
InputSignal
OutputSignal
J
Y s U s U s( ) ( ) ( ) 1 2
J B
U1(s) Y(s)
U2(s)
+
T(s)
InputSignal
OutputSignal
s
f
J B
1
Js
T(s) s
1
Js BT(s) s
B-
• Multiple Inputs
Ex: Draw the block diagram for
Block Diagram Representation• Transfer Function in Series
• Transfer Function in Parallel
Y s G s Y s Y s G s U s
Y s G s G s U s
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 1 1 1
2 1
,
b gU(s) Y(s)
InputSignal
OutputSignal
G2 (s)G1 (s)Y1(s)
X s G s U s X s G s U s
Y s X s X s
G s G s U s
1 1 2 2
1 2
1 2
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
,
b g
U(s) Y(s)
InputSignal
OutputSignalG2 (s)
G1 (s) X1(s)
X2(s)
Y s G s U s Y s G s U s
Y s Y s Y s
G s U s G s U s
1 1 1 2 2 2
1 2
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
,
U1(s)
U2(s)
InputSignals
G2 (s)
G1 (s)
Y(s)
OutputSignal
Y1(s)
Y2(s)
Ld
dti Ri K ei
Ei(s)
K
I(s)1
Ls R
s
-
Block Diagram Representation of DC MotorsSchematic
Governing Equations:
EOM in s-Domain:
iA
+ eLa
LA
+ eRa
RA+
ei(t)_
+Eemf_
JA
B
m
L
( ) ( )
( )
A A A A emf i
A m L
m T A
emf b
dL i R i E e t
dt
J B
K i
E K
Electrical
Mechanical
EM Coupling
1A i emf
A A
I s E s E sL s R
1m L
A
s T s T sJ s B
m T A
emf b
T s K I s
E s K s
Block Diagram Representation of DC Motors
TK AI s s
Q: Now that we generated a block diagram of a voltage driven DC Motor, can we derive the transfer function of this motor from its block diagram ? ( This is the same as asking you to reduce the multi-block diagram to a simpler form just relating inputs e i(t) and L to the output, either or )
Ei(s) 1
A AL s R s
-
bK
1
AJ s B1
s
s
mT s
Electrical System Mechanical SystemEM Coupling
LT s
-
emfE s
Block Diagram ReductionFrom Block Diagram to Transfer Function• Label each signal and block
• Write down the relationships between signals
emfE s
1JA s + BKT
1LA s + RA
Kb
Ei(s) AI s mT s
LT s
s E s
T s
1
1
A
m L
m T A
AA A
i emf
emf b
s T sJ s B
T s T s T s
T s K I s
I s E sL s R
E s E s E s
E s K s
6 equations
6 unknowns
, , ,
, , and
m
A emf
s T s T s
I s E s E s
• Solve for the output signal in terms of the input signals
• Substitute the transfer functions’ label with the actual formula and simplify
Block Diagram Reduction
1
1
A
m L
m T A
AA A
i emf
emf b
s T sJ s B
T s T s T s
T s K I s
I s E sL s R
E s E s E s
E s K s
( )( ) ( )
( )( ) ( )
( )sK
L J s BL R J s R B K KE s
L s R
L J s BL R J s R B K KsT
A A A A A A b Ti
A A
A A A A A A b TL
2 2
( )( ( ) ( ))
( )( ( ) ( ))
( )sK
s L J s BL R J s R B K KE s
L s R
s L J s BL R J s R B K KsT
A A A A A A b Ti
A A
A A A A A A b TL
2 2
1
1 1
1
1
m
A
emf
T A LA
T s
T s
T LA A A
I s
Ti b L
A A A AE s
E s
T T bi L
A A A A A A
s K I s T sJ s B
K E s T sJ s B L s R
KE s K s T s
J s B L s R J s B
K K KE s T s
J s B L s R J s B J s B L
A
ss R
Example(A) Given the following specification of a DC
motor and assume there is no load, find its transfer function from input voltage to motor angular speed
LA = 2 mH
RA = 10
KT = 0.06 Nm/A
JA = 5 10-6 Kg m2
B = 3 10-6 Nm/(rad/sec)
(B)Find the poles of the transfer function.
(C) Plot the Bode diagram of the transfer function
2
3 6 2 6 3 6 6
8 2 5 3
6
2 5
( ) ( )
0.06
2 10 5 10 3 10 2 10 10 5 10 10 3 10 0.06 0.06
0.06
10 5 10 3.63 10
6 10
5000 3.63 10
T
A A A A A A b T
KG
L J s BL R J s R B K K
s s
s s
s s
2 55000 3.63 10 0s s
1
2
73.68
4926
p
p
6
2 5
6
6
6 10
5000 3.63 10
6 10
73.68 4926
6 10 1 11 173.68 4926 1 1
73.68 49261 1
16.531 1
1 173.68 4926
G ss s
s s
s s
s s
ExampleP
ha
se (
de
g)
M
ag
nitu
de
(d
B)
100
101
102
103
104
105
Frequency (rad/sec)
-80
-60
-40
-20
0
20
-180
-135
-90
-45
0
Q: If we are only interested in the system response up to 400 rad/sec, can we simplify our model ?How would you simplify the model ?