MESB374 System Modeling and Analysis Electro-mechanical Systems.

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 ?