Lesson 30: Linear Motors
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 currentcarrying 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).
Linear motors

Machines which convert electrical energy into
mechanical energy.
Magnets
Permanent Magnets – Constantly
Magnetized (iron, nickel,..)
 Electromagnets – Exist When Electric
Current Is Flowing

neodymium rare earth magnet
Large electromagnet
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
Current Carrying Wire

Flowing current produces
magnetic field
Right-Hand Rule
Point right thumb in direction of
current flow
Finger indicate direction of lines of
magnetic flux
Current Carrying Coil


Current creates
magnetic field
Closely spaced wires
create lines of
magnetic flux that
reinforce each other
and create larger
magnetic field
Magnetic Flux (Φ)


Number of lines between north and south poles
Unit of Weber (Wb) (volts-seconds)
Magnetic Flux Density (B)


Magnitude of magnetic field
Unit of Tesla (T), or Weber (Wb) per square
meter
V sec
N
Tesla 

2
m
Am
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
Lorentz Force Law

Magnetic field created by a current carrying
wire interacts with an existing magnetic field to
exert a developed force (Fd) on the wire
F d  I LxB
Lorentz Force Law


Force is proportional to the current, length of
wire, magnitude of magnetic field, and angle
between vectors
Force is maximum when angle between current
and magnetic field is 90 degrees
F d  I LxB
Linear Motor


Linear motor consists of current source,
moveable wire, and a magnetic field
Magnetic field is going into screen
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.
Faraday’s Law

Movement of conductor (wire) in magnetic field induces a
voltage


Einduced  u x B L

Induced voltage opposes current from current source
VIDEO

http://m.youtube.com/watch?v=yGWHdum
VCSI

http://m.youtube.com/watch?v=FnmetchLOk
Linear Motor Startup

Initially, applied voltage (VDC) is zero
 Force

on wire is zero
Wire is initially at rest
 Induced
voltage across wire is zero
Linear Motor Acceleration

Voltage source is turned on ->large current begins to flow
through wire
VDC
I 
Rrail

Initial current results in Lorentz force being applied to the
bar
F d  I LxB

Bar begins to move and accelerate
Linear Motor Operation

Voltage is induced in bar as it picks up speed


Einduced  u x B L

KVL equation for circuit becomes
VDC – IRrail – Einduced = 0
VDC  Einduced
I
Rrail
Linear Motor Operation

As speed
Einduced
I
VDC  Einduced
R
current
Linear Motor at Steady State

If there are no frictional forces (or other loads) on
the wire, eventually Einduced will match VDC
VDC  Einduced
I
0
R

Zero current will flow through the wire, which
means Lorentz force will be zero
F d  I LxB

The machine will maintain a constant speed.
This is called STEADY STATE.
Linear Motor Operation


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.
Linear Motor Operation

The steady state current can be determined:
F d  I LxB  Fload

Fload
I
BL
This means the steady state speed can be
calculated:
VDC  Einduced
I
R


Einduced  u x B L
Eind VDC  IRrail
u

BL
BL
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.
Power Balance
Ploss  I 2 Rrail
PIN  VDC I
Pout  Fload u
 Eind I
Example
#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.
Desired efficiency is 95%.
Find the required rail resistance, source current, and
bar length.
Bottom Line

F = ILB

Force (Newtons) = Current (amps) *Length (meters) * B-field (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.