1. Lesson 27: Linear Motors

2. 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).

3. 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.

4. Magnet Types
Permanent Magnets – Constantly Magnetized (iron, nickel,..).
Electromagnets – Exist When Electric Current Is Flowing.
Neodymium Rare Earth
Permanent Magnet
Large Electromagnet

5. Current Carrying Wire
Flowing current produces magnetic field.

6. Right-Hand Rule Point right thumb in direction of current flow…
Fingers indicate direction of lines of magnetic flux…

7. 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.

8. Magnetic Flux (Φ) Number of lines between north and south poles.
Unit of Weber (Wb) (volts-seconds).

9. Magnetic Flux Density (B)
Magnitude of magnetic field
Unit of Tesla (T), or Weber (Wb) per square meter.
Nikola Tesla, inventor and electrical engineer. Born in Austria in 1856 to Serbian parents. He studied Electrical Engineering at the Austrian Polytechnic in 1875, however did not complete his 2/C year. He immigrated to the US in 1884, where he was hired by Thomas Edison as an Electrical Engineer at Edison’s Machine Works. He resigned following disagreement about a “promised” bonus for redesigning Edison’s DC generators. He developed the polyphase power systems while an employee of Westinghouse Electric, resulting in the design and eventual use of the modern AC power system.

10. 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.

11. 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…
Hendrick Lorentz was a Dutch physicist, Educated at the University of Leiden in Physics and Mathematics. Awarded Nobel prize in 1902 for his studies in electromagnetism and charged particles, which resulted in the formulation of the Lorentz-Lorenz formula, the Lorentz force, the Lorentzian distribution, and the Lorentz transformation. His research was used heavily by Einstein in the development of the Theory of Special Relativity (originally called the Lorentz-Einstein theory).

12. 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.
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:
Therefore, the force is maximum when angle between current and magnetic field is 90 degrees:
Hendrick Lorentz was a Dutch physicist, Educated at the University of Leiden in Physics and Mathematics. Awarded Nobel prize in 1902 for his studies in electromagnetism and charged particles, which resulted in the formulation of the Lorentz-Lorenz formula, the Lorentz force, the Lorentzian distribution, and the Lorentz transformation. His research was used heavily by Einstein in the development of the Theory of Special Relativity (originally called the Lorentz-Einstein theory).

13. 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.

14. 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… .
Into the screen.
ume.gatech.edu/mechatronics_course/Motors_F09
Out of the screen.

15. 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:
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:
Michael Faraday was an English chemist and physicist, Experimented with induced voltages between conductors and early electric motors.
L is the length and direction of a moveable wire
V is the velocity of the wire

16. 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.

17. Linear Motor Acceleration
Now the voltage source is turned on -> large current begins to flow through wire that has a value of:
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:
Bar begins to move and accelerate…

18. Linear Motor Operation
Voltage is induced in bar as it picks up speed:
KVL equation for circuit becomes:
VB – IRrail – Einduced = 0

19. Linear Motor Operation
As speed Einduced current

20. Linear Motor at Steady State
If there are no frictional forces (or other loads) on the wire, eventually Einduced will match VB:
Zero current will flow through the wire, which means Lorentz force will be zero:
The machine will maintain a constant speed. This is called STEADY STATE.

21. 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.

22. Linear Motor Operation
The steady state current can be determined:
This means the steady state speed (u) can be calculated:

23. 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):

24. Power Balance

25. 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.

26. 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.

27. QUESTIONS?