Download Linear Motors

Lesson 27:
Linear Motors
1
Learning Objectives
• Explain the difference between permanent magnets and
electromagnets.
• Identify lines of magnetic flux in a permanent magnet,
straight line current carrying conductor, and current-carrying
coil.
• Define flux density, magnetic field intensity, and magnetic
flux.
• Understand the direction of force on a current-carrying
conductor in a magnetic field (Lorentz Force Law).
• Analyze the Lorentz Force Law in a DC linear motor.
• Understand the effect of a changing magnetic field upon a
current-carrying closed path conductor
(Faraday/Lenz/Electromotive Force).
2
Magnets
• All magnets have two poles, north and south.
• Opposite poles attract, similar poles repel.
• Lines of magnetic flux flow from the north pole to the south
pole.
• Magnetic field is strongest close to the magnet.
3
Magnet Types
• Permanent Magnets – Constantly Magnetized (iron,
nickel,..).
• Electromagnets – Exist When Electric Current Is Flowing.
Large Electromagnet
Neodymium Rare Earth
Permanent Magnet
4
Current Carrying Wire
• Flowing current produces
magnetic field.
5
Right-Hand Rule
Point right thumb in direction of
current flow…
Fingers indicate direction of lines of
magnetic flux…
6
Current Carrying Coil
• Current creates
magnetic field.
• Closely spaced wires
create lines of magnetic
flux that reinforce each
other and create a larger
magnetic field.
7
Magnetic Flux (Φ)
• Number of lines between north and south poles.
• Unit of Weber (Wb) (volts-seconds).
8
Magnetic Flux Density (B)
• Magnitude of magnetic field
• Unit of Tesla (T), or Weber (Wb) per square meter.
V sec
N
Tesla 

2
Am
m
9
Magnetic Flux (Φ)
of a Current Carrying Wire.
• Same number of lines leaves the pole of the magnet
and re-enter the south pole.
• Lines are denser close to the magnet, especially near
the poles.
• The direction of the lines depends on the direction of
the current through the coil.
• Changing current direction changes the poles of the
magnet.
• Higher currents produces more lines of flux.
10
Lorentz Force Law
•
•
Suppose a wire is placed in a magnetic field, as shown below. If we now
force a current to flow through the wire, the magnetic field created by the
current carrying wire will interact with the existing magnetic field to exert a
force on the wire.
The magnitude and the direction of this force is given by the Lorentz Force
Law…
11
Lorentz Force Law
• Lorentz Force Law states that a magnetic field created by a
current carrying wire interacts with an existing magnetic field to
exert a developed force (Fd) on the wire.
Fd  IL B
• Force is proportional to current (I), length of wire (L),
magnitude of magnetic field (B), and the angle between vectors
L and B. The angular dependence can be shown as the cross
product:
L  B  L  B sin(angle between L and B)
• Therefore, the force is maximum when angle between current
and magnetic field is 90 degrees:
F  ILB
12
Faraday Experiment #2:
Motional EMF
• Moving a conductor through a magnetic field induces
a voltage in the conductor.
• The magnitude of the voltage is proportional to the
velocity of the conductor.
13
Linear Motors
• A linear motor is a machine that converts electrical
energy into mechanical energy.
• A linear motor consists of a current source, moveable
wire, and a magnetic field – hence all the magnet,
flux, and force theory in the previous slides…
.
ume.gatech.edu/mechatronics_course/Motors_F09
14
Into the screen.
Out of the screen.
Faraday’s Law
•
•
In the image below, applying a current to the conductor (wire) sitting in the
magnetic field, causes the wire’s own magnetic field to interact with the
existing magnetic field, inducing the Lorentz Force (Fd) that will cause the
wire to move.
The movement of the conductor in the magnetic field induces a voltage
across the wire, given by Faraday’s Law:


Einduced  u x B L
•
The polarity of this induced voltage opposes the current from the current
source.
− Note that in Faraday’s Law, u us the velocity.
− Note also that the geometry (where the velocity is perpendicular to the magnetic field)
means that the magnitude of the induced voltage is given by:
Einduced  uBL
15
Linear Motor Startup
• Initially, applied voltage (VB) is zero.
− Current is zero thus Force on wire is zero.
• Wire is initially at rest.
− Induced voltage across wire is zero.
16
Linear Motor Acceleration
• Now the voltage source is turned on -> large current begins to
flow through wire that has a value of:
VB
I
Rrail
− Note that the current is limited by Rrail, which is the resistance of the
movable wire.
• Initial current results in Lorentz force being applied to the bar:
F  ILB
• Bar begins to move and accelerate…
17
Linear Motor Operation
• Voltage is induced in bar as it picks up speed:


Einduced  u x B L
• KVL equation for circuit becomes:
VB – IRrail – Einduced = 0
VB  Einduced
I
Rrail
18
Linear Motor Operation
• As speed
Einduced
current
VB  Einduced
I
R
http://www.parkermotion.com/video/Braas_Trilogy_T3E_Video.MPG
19
Linear Motor at Steady State
• If there are no frictional forces (or other loads) on the
wire, eventually Einduced will match VB:
VB  Einduced
I
0
R
• Zero current will flow through the wire, which means
Lorentz force will be zero:
Fload
I
BL
F d  I LxB  Fl oad
• The machine will maintain a constant speed. This is
called STEADY STATE.
20
Linear Motor Operation
w/ Frictional Forces
• If frictional forces (or other loads) exist, these can be
treated as a force (Fload) that opposes the Lorentz
force.
• When the Lorentz force equals Fload, a steady state
condition will be reached, and the bar will maintain a
constant speed.
21
Linear Motor Operation
• The steady state current can be determined:
F d  I LxB  Fload
Fload
I
BL
• This means the steady state speed (u) can be calculated:
VDC  Einduced
I
R

Eind VDC  IRrail
u

BL
BL

Einduced  u x B L
22
Example Problem 1
A 100V linear motor operates with a magnetic field of 0.5 Tesla, and a
mechanical loading of 1.0N. The effective length of the bar is 0.1m, and the
rail resistance is 0.02 Ω.
Find the current flowing through the motor and the velocity of the bar when
steady-state conditions are achieved.
Current at steady-state:
To find velocity (u):
Fd  Fload  1.0 N
Einduced  VDC  IRrail
Fd  Fload
Einduced  100V  (20 A)(0.02)  99.6V
Fd  BLI
Einduced  BLu
1.0 N  (0.5T )(0.1m)( I )
I  20 A
Einduced
99.6V
u

 1992 m/s
BL
(0.5T )(0.1m)
23
Power Balance
Ploss  I 2 Rrail
Pout  Fload u
PIN  VDC I
 Eind I
Eind  BLu
24
Example Problem 2
•
Design a 10 kW (output power) roller coaster that reaches 100 km/hr. Maximum
B-field is 3 T. We have a 450 V DC source available. The system desired
efficiency is 95%.
Find the required rail resistance, source current, and bar length.
•
Pout  Fload u  Eind I
and
Pin  VDC I
10kW  Eind I

P
Pout
10kW
 Pin  out 
 10.526kW
Pin

0.95
Pin
10.526W

 23.39 A
VDC
450V
Pout
10kW
Now find E ind : Eind 

 427.5V
I
23.39 A
Now find I: I 
Now find R rail : E ind  VDC  IRrail  Rrail 
u
100000m
1hr
1min
*
*
 27.7m / s
1hr
60 min 60sec
u
25
VDC  Eind
450V  427.5V

 0.962
I
23.39 A
Eind
E
427.5V
 L  ind 
 5.12m
BL
Bu (3T )(27.78m / s )
Bottom Line
• F = ILB
• Force (Newtons) = Current (amps) *Length (meters) * Bfield (Tesla)
• E = uBL
• Induced voltage = velocity of bar (meters/sec) *B*L
• Take givens, draw what you know on the single loop model.
• Use, P = VI, V=IR, KVL around loop … that’s all you need to
solve these types of problems.
• Two important laws to remember (and not confuse):
− Faraday’s Law: Movement of conductor (wire) in magnetic field
induces a voltage.
− Lorentz Force Law: a force will be exerted on a current carrying
conductor when it is placed in a magnetic field.
26
QUESTIONS?
27