Fields and Waves I

1
Fields and Waves I

• Lecture 18
• Magnetic Force and Energy
• K. A. Connor
• Electrical, Computer, and Systems Engineering
  Department
• Rensselaer Polytechnic Institute, Troy, NY
• Y. Maréchal
• Power Engineering Department
• Institut National Polytechnique de Grenoble,
  France

2
These Slides Were Prepared by Prof. Kenneth A.
Connor Using Original Materials Written Mostly by
the Following

• Kenneth A. Connor ECSE Department, Rensselaer
  Polytechnic Institute, Troy, NY
• J. Darryl Michael GE Global Research Center,
  Niskayuna, NY
• Thomas P. Crowley National Institute of
  Standards and Technology, Boulder, CO
• Sheppard J. Salon ECSE Department, Rensselaer
  Polytechnic Institute, Troy, NY
• Lale Ergene ITU Informatics Institute,
  Istanbul, Turkey
• Jeffrey Braunstein Chung-Ang University, Seoul,
  Korea

Materials from other sources are referenced where
they are used. Those listed as Ulaby are figures
from Ulabys textbook.
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Overview

• Review
• Magnetic Materials
• Magnetic Circuits
• Energy and Force
• Energy
• Force
• Applications
• DC motors
• Induction heating

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Magnetic Force and Energy

• Magnetic circuits and Reluctances

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Boundary Conditions
Arguing from analogy with Electric Fields
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Introduction
MAGNETIC CIRCUITS used to analyze relays,
switches, speakers...
In a simple experiment
R
http//www.cedrat.com/
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Magnetic Flux - TOROID
Flux is a constant

• Flux stays in toroid - so area is a constant

B and H are constant along the path
L is much higher that with air
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Magnetic Flux - TOROID with GAP
Introduce an air gap to toroid
Apply boundary conditions across gap
Can get very large H in gap
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Magnetic Circuits - Analogy to E-circuits

• enables us to draw analogy to electric circuits

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Magnetic Circuits Electric Circuits
Electric Circuits
Magnetic Circuits
Magneto motive force
V or e.m.f
NI or m.m.f
I
Flux
High m - low reluctance path
Magnetic circuits model
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Example 4
a. Evaluate ? H ? dl around the dashed line in
the figure on the left below. Then, determine
H and B in the iron core. Make reasonable
approximations. b. What is the inductance,
L? c. For the figure on the left, what are the
reluctance and magnetomotive force? Draw a
magnetic circuit equivalent and show how to solve
for the inductance using the circuit. d. Analyze
the situation on the right using magnetic
circuits. Determine the flux through the iron
core. What is the inductance? What is H in the
core and in the gap? e. Calculate numerical
values for L, Hgap and Hcore when N 1000,
I  1 A, w 5 cm, g 1 cm, l 20 cm, and
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Example 4
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Example 4 Continued
Lwa19H
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Flux distribution in a relay
High reluctance, low flux density
Low reluctance, High flux density
http//www.cedrat.com/
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Magnetic Force and Energy

• Energy

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Energy
Power in inductor
Can we obtain energy in terms of B and H fields ?
Flux linkage
Also,
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Magnetic Energy
Energy
Energy Density
(per unit volume)
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Example 1 Coaxial Cable
One of the three standard configurations
Detailed Solution for Coax
Amperes Law Contours
Amperes Law
Left Hand Side
a
Right Hand Side
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Example 1 Coaxial Cable
Assume the outer conductor is very thin
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Example 1 Coaxial Cable
The energy in the magnetic field can be divided
into two terms
r
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Example 1 Coaxial Cable
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Example 1 Coaxial Cable
Ulaby
To compute the inductance per unit length, we
need to determine the magnetic flux through the
area S between the conductors
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Example 1 Coaxial Cable
The flux through the surface S
Note that the flux is linked only once since
there is only one turn. Thus, the inductance is
given by
or
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Problem
Using the flux
External inductance
???
Using the energy
Total inductance
Additional term
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Example 1 Coaxial Cable
Note that this analysis does not incorporate the
flux inside the center conductor so it does not
give us the total inductance. However, figuring
out the flux linking this current is difficult.
Thus we leave this to our method based on energy.
Ulaby
External Inductance What we have determined is
called the external inductance, since it is
inductance due to the magnetic field external to
the current-carrying wires. Internal Inductance
What we have neglected is the contribution to the
inductance from the field inside the wires.
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Magnetic Force and Energy

• Force

http//www.sumitomokenki.co.jp/english/special/lm_
sh330lc-3lm.html
http//library.thinkquest.org/28032/cgi-bin/frames
.cgi?srclessons0206
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Protection relay
High sensitivity relay
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Protection relay
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Protection relay
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Force first approach from energy
First approach - F does work and changes energy
with the energy stored being
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Example Simple Relay
Consider a simple electromagnetic relay
consisting of a solenoid and a moveable arm.
In the region of the gap, the normal component of
the magnetic field will be continuous.
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Alternative Calculation Magnetic Pressure
The magnetic field intensity H is very different
in the gap and the core, since B is the same.
The magnetic energy density is also very
different.
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Alternative Calculation Magnetic Pressure
The difference in the magnetic field energy
density on the two sides produces a pressure
difference.
Pressure
Energy Density
Force
High Pressure
Low Pressure
Net Force
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Alternative Calculation Magnetic Pressure
Since the pressure is so much higher on the gap
side than in the core we only need to evaluate
the pressure on the gap side to figure out the
force. S is the area of the gap and core.
To figure out the force, we first need to find
the magnetic field, which we can do using the
magnetic circuits technique.
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Alternative Calculation Magnetic Pressure
To analyze this configuration, we will use the
idealized version at the right. Assume that each
leg has a length lo and the area of each leg is
S. The gap length is . The reluctances are
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Alternative Calculation Magnetic Pressure
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Forces on current wires why Beakmans Motor
turns ?

• A simple DC motor with brushes made with a
  battery, two paperclips, a rubber band and about
  1 meter of enameled wire.

http//fly.hiwaay.net/palmer/motor.html
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Force on currents
First approach - similar to that for individual
particles
For one particle
For many particles
For a wire in a magnetic field.
http//www.ac.wwu.edu/vawter/PhysicsNet/Topics/Ma
gneticField/MFOnWire.html
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FORCE
The force on a current loop in a magnetic field
can result in rotational torque if the loop has a
fixed axis as shown. This is the basic
configuration for the Beakmans motor.
Figure reference placeholder
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DC motor the commutator
http//teamster.usc.edu/fixture/Robotics/Course.h
tm
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Example Rail Gun
If a sliding contact is placed across a two wire
transmission line carrying a large current, a
very large force can result on the contact.
Assume that all the wires (including the slider)
have a radius a and that the transmission line
wires are separated by a distance d.
This material is discussed more extensively in
Unit 9 of the class notes.
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Example Rail Gun
The external inductance of a two wire line of
length l is given by (one of many forms
where we have used the fact that typically d gtgt
a. The force on the sliding conductor will be
If d/a 5 and I 105 A, F 100 Newtons
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Magnetic Force and Energy

• Motors and other applications

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DC Motors

• The stator is the stationary outside part of a
  motor. The rotor is the inner part which rotates.
  
• In the motor animations, red represents a magnet
  or winding with a north polarization, while green
  represents a magnet or winding with a south
  polarization. Opposite, red and green, polarities
  attract.

http//www.freescale.com/files/microcontrollers/do
c/train_ref_material/MOTORDCTUT.html
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DC Motors

• Just as the rotor reaches alignment, the brushes
  move across the commutator contacts and energize
  the next winding. In the animation the commutator
  contacts are brown and the brushes are dark grey.
  A yellow spark shows when the brushes switch to
  the next winding.

http//www.freescale.com/files/microcontrollers/do
c/train_ref_material/MOTORDCTUT.html
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DC Motor Applications

• Automobiles
• Windshield Wipers
• Door locks
• Window lifts
• Antenna retractor
• Seat adjust
• Mirror adjust
• Anti-lock Braking System

• Elsewhere
• Cordless hand drill
• Electric lawnmower
• Fans
• Toys
• Electric toothbrush
• Servo Motor

http//stuffo.howstuffworks.com/rc-toy3.htm
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DC Motor
http//www.cedrat.com/
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Brushless DC Motors

• A brushless DC motor has a rotor with permanent
  magnets and a stator with windings. It is
  essentially a dc motor turned inside out. The
  control electronics replace the function of the
  commutator and energize the proper winding.

http//www.freescale.com/files/microcontrollers/do
c/train_ref_material/MOTORDCTUT.html
49
Full Stepper Motor

• This animation demonstrates the principle for a
  stepper motor using full step commutation. The
  rotor of a permanent magnet stepper motor
  consists of permanent magnets and the stator has
  two pairs of windings. Just as the rotor aligns
  with one of the stator poles, the second phase is
  energized. The two phases alternate on and off
  and also reverse polarity. There are four steps.
  One phase lags the other phase by one step. This
  is equivalent to one forth of an electrical cycle
  or 90.

http//www.freescale.com/files/microcontrollers/do
c/train_ref_material/MOTORDCTUT.html
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Half Stepper Motor
Full stepper
Half stepper

• This animation shows the stepping pattern for a
  half-step stepper motor. The commutation sequence
  for a half-step stepper motor has eight steps
  instead of four. The main difference is that the
  second phase is turned on before the first phase
  is turned off. Thus, sometimes both phases are
  energized at the same time.
• A half-step motor has twice the resolution of a
  full step motor. It is very popular for this
  reason.

http//www.freescale.com/files/microcontrollers/do
c/train_ref_material/MOTORDCTUT.html
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Real DC brushless Motors

• This stepper motor is very simplified.
• The rotor of a real stepper motor usually has
  many poles (hundred of them).
• The stator poles of a real stepper motor also has
  many teeth.

Demo
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Some Interesting Inductors

• Induction Heating

http//rsc.ambrell.com/videostill_brzstlstl.jpg
http//zzkechuang.ecvv.com/products/1474115.html
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Some Interesting Inductors

• Induction Heating in Aerospace
• http//www.ameritherm.com/videoindex.html

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Numerical simulation of Induction heating
Induction heating of a gear box gearing