Motors

Lecture 11.4

Hans Christian Oersted (1777 – 1851)

Ref: http://chem.ch.huji.ac.il/~eugeniik/history/oersted.htm

1822

In 1820 he showed that a current produces a magnetic field.

X

André-Marie Ampère (1775 – 1836)French mathematics professor who only a week after learning of Oersted’s discoveries in Sept. 1820 demonstrated that parallel wires carrying currents attract and repel each other.

attract

repel

A moving charge of 1 coulomb per second is a current of 1 ampere (amp).

What do Maxwell’s Eqs. Predict?

B = magnetic flux density (magnetic induction)

magnetic permeability

B( )q v F B

0 B

B H

Magnetic field lines must be closed loops

Force on moving charge qLorentz force

Vector MultiplicationScalar (Dot) Product

A

B

C A B

cosC A B

Vector (Cross) Product

A

B

C A B

sinC A B

Direction of C given by "Right-hand rule" (into the screen)

Force on current in a magnetic field( )q v F BForce on moving charge q -- Lorentz force

Current density, j, is the amount of charge passing per unit area per unit time. N = number of charges, q, per unit volume moving with mean velocity, v.

v t

Sj

j Nqv

( )N V q v F B

( ) V F j B

L

dQ Nq Sv ti j S

dt t

V S L

( ) L F i B

i BForce per unit length on a wire is

Rotating Machine

B

i

i

X

Force in

Force out

+-

Rotating Machine

B

i

i

X

Force in

Force out

+-

Rotating Machine

B

i

i

X

Force in

Force out

+-

brushes

commutator

What do Maxwell’s Eqs. Predict?

t

B

ECorresponds to Faraday’s law of electromagnetic induction

A changing magnetic flux B density induces a curl of E

S S

d da dat

B

E s E n n

flux through SS

d dat t

E s B n

B

i

iX

Force in

Force out

+-

X

r

B

2 cosw r

flux 2 cosA rl B Β

area 2 cosA lw l r

l

d

dt

emf flux 2 cos d rlt t

E s B

emf 2 sin 2 sinabe rl rlt

B B

a

b

Back emf

B

i

i

X

Force in

Force out

+-

2 sinabe rl Ba

b

Back emf

abe

2 sinabe rl B

Back emf

abe

B

i

i

X

Force in

Force out

+-

brushes

commutator

Armature with four coil loops

XXX

X

abe

N

S

aE

aR

tV

aI

Motor Circuit

t a a aV E I R

a aE K

Power and Torque

d a a dP E I

/a t a aI V E R

d a aK I

/a t a aI V K R

2

at ad

a a

KV K

R R

Armature Current, Speed, Power and Torque

d a a dP E I

d a aK I

2

at ad

a a

KV K

R R

1a d

a

IK

2t a

da a

V R

K K

2

2t a

d d da a

V RP

K K

FF-130 Series Motors

1a d

a

IK

2t a

da a

V R

K K

2

2t a

d d da a

V RP

K K

2t a

da a

V R

K K

2 0t ad

a a

V R

K K

ω=0 6.4 mN-mt ad

a

V K

R

2t a

da a

V R

K K

0 7100 rpmd

t

a

V

K

0d

1a d

a

IK

6.4 mN-mt ad

a

V K

R

At 0

ω 0t

aa

VI

R

4.84.0

1.21t

aa

VR

I

2

2t a

d d da a

V RP

K K

2

20d t a

dd a a

dP V R

d K K

Maximum Power

max 2t a

da

V K

R

0t a

da

V K

R

2

max 4t

da

VP

R

4.80.00646 volt-sec

7100 rpm 7100(2 / 60)t

a

VK

ω=0

4.8 0.006467.6 mN-m 6.4 mN-m

4t a

da

V K

R

0 7100 rpmd

t

a

V

K

ω 0t

aa

VI

R

2 2

max

4.81.4 W 1.18 W

4 4 4t

da

VP

R

4.84.0

1.21t

aa

VR

I

aE

aR

tV

aI

Generator Circuit

t a a aV E I R

a aE K

0.00646 volt-secaK

0.00646 7100 2 / 60 4.8 VaE

4.8 0.15 4 4.2 VtV