Fields and Waves I

1
Fields and Waves I

• Lecture 16
• Faradays Law
• 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.
3
Overview

• Review
• Amperes Law
• Magnetic Flux
• Magnetic Vector Potential
• Faradays Law
• EMF
• Induced Voltage/Current
• Moving Magnet or Loop
• Inductance
• Self Inductance
• Mutual Inductance
• Quiz review

4
Amperes Law
Maxwells Equations
Amperes Law
Integral form
Magnetostatics
Electrostatics
5
B-Fields
wraps around

Direction of
http//encarta.msn.com/media_701504656_761566543_-
1_1/Right-Hand_Rule.html
multiple wires or segments - use superposition
6
Magnetic Flux Magnetic Vector Potential
Magnetic Vector potential definition
Flux definition
Alternative way to find FLUX
Magnetic FLUX
7
Example Field Due To Several Wires
What is the direction of B and A at the 4
indicated points?
z-direction is into the page
8
Field Due To Long Straight Wire
First, determine the magnetic field due to a long
straight wire carrying a current I. (See Example
5-5 of Ulaby)
Line for Amperes Law
I
http//www.ee.surrey.ac.uk/Workshop/advice/coils/t
erms.html
9
Long Straight Wire
The magnetic vector potential can be determined
from first principles or from the magnetic field.
We will do the latter.
Specifying the zero reference will determine this
constant
From the curl expression
Note that the vector potential is always in the
direction of the current
10
Magnetic Vector Potential Direction
All currents are in the z-direction and, thus,
the vector potential will also be in the
z-direction. Its sign is arbitrary, since we are
free to select the reference potential point
anywhere. That is, we could chose all potentials
to be positive or all to be negative.
11
Magnetic Field Direction
1
2
At point 1 (point 3 has to opposite sign)
3
4
12
Magnetic Field Direction
1
2
At point 2
3
4
13
Faradays Law and dynamic fields

• Faradays Law

14
Faradays Law
In electrostatics, we used
Faradays Law comes from Maxwells equation

• inductors
• transformers
• motors
• generators
• noise

Applications
15
Faradays Law - concept of EMF
Time varying flux through a coil
is the electromotive force
The emf is similar to a VOLTAGE
Orientation issues
16
Faradays Law various types of EMF
What does the flux derivative means ?
The emf may come from

• A dynamic field and a stationary loop
• A moving loop in a static field
• Both moving loop and dynamic field

17
Faradays Law dynamic field experiment
Assume that we hook up the experiment as shown
where the 1 micro Henry inductor is connected
across the output of the function generator and
monitor the output of the generator using one
scope channel. Then, place a coil facing the
inductor and connect it to the other scope
channel. An induced voltage is observed at the
ends of the pickup coil (in phase with the
generator)
18
Transformers Faradays law with dynamic fields
http//www.transformerfactory.com/e1-model-small-p
ower-transformer-1va-70a.html

• A huge range in sizes

http//www.meppi.com/Products/Transformers/Power/P
ages/Core-formTransformers.aspx
http//en.ferilex.eu/transformers.html
19
Faradays Law moving loop experiment
A loop falls through the magnetic field between
two pole faces at a constant velocity, u0.
A current is flowing in the loop as it pass
through the magnets
20
Generators Faradays Law
Hoover Dam
http//isu.indstate.edu/jspeer/conservation/
http//www.wenzelontheweb.de/Hoover20Dam.htm
21
Faradays Law and dynamic fields

• Dynamic fields

22
Example 1

• For the solenoid, inside and 0
  outside
• n is the number of turns per unit length
• a is the radius of the solenoid and the coil

23
Example 1

• Circuit analysis. What are the current I and
  voltage V through the inductor?
• What is the flux, ? ?B ? ds, through the loop?
  Do this analytically and then obtain a numerical
  value for n 1560 and solenoid radius a 2.5
  mm. Pay attention to the signs/direction of dl
  and ds.
• c. What is the emf induced around the loop?
  Again do an analytical calculation, but then plug
  in the numbers from above.
• 1) At t0, does a scope read V1 - V2 gt or lt 0?
• 2) If the clip leads were connected through a
  low impedance, which way would current flow at
  t0?
• d. Sketch emf and ? vs time. What is the flux
  when the emf is largest?

24
Example 1
25
Example 1
26
Faradays Law and Lenzs law
Previous result t0, flux decreasing, I as
shown
Lenzs law The current in the loop is always
in such a direction as to oppose the change of
magnetic flux that produced it.
l
1
2
Low impedance Output
Ulaby
High impedance Output
27
Faradays Law and dynamic fields

• Moving conductors

28
Faradays Law for moving loop
0
29
Faradays Law for moving loop
30
Faradays Law for moving loop
Approximate flux derivative as
Dt
A general form
Alternate expression
31
Example 2
A loop falls through the magnetic field between
two pole faces at a constant velocity, u0.
Assume that the magnetic field is B0 between the
pole faces and that the fringe fields are 0.
Plot the flux through the loop, ? ?B ? ds, as a
function of time. Calculate the emf around the
loop for all times by derivation of the
flux. Calculate the emf around the loop for all
times indicated using the uxB. If the loop is
connected across a low impedance output, will the
current be in the clockwise direction, 0, or in
the counter-clockwise direction?
32
Example 2
33
Example 2
34
Faradays Law and dynamic fields

• Inductances

35
Inductances

• self inductance - e.g. inductors

Two types of Inductances

• mutual inductance - e.g. transformers

Self Inductance
http//www.gaussbusters.com/ppm93.html
36
Inductor
Geometric parameters for a solenoidal inductor
http//www3.telus.net/chemelec/Calculators/Helical
-Coil-Calc.htm
37
Example 3 Solenoid Inductance

• Consider a solenoid with N turns, length l , and
  radius a . Assume the current is sinusoidal with
  a frequency f and ignore fringing effects.
• What emf, ? E ? dl is induced around the solenoid
  (include all turns)?
• The "voltage" across an inductor is the emf (with
  care taken about signs). Find the solenoid
  inductance by substituting the absolute value of
  the emf in part b. for the voltage in V L
  dI/dt.
• What is the flux linkage through all N turns?
• Calculate L Flux/I and compare with your answer
  to part c.

38
Example 3
39
Self inductance
Two ways to calculate the inductance

• Calculate the emf then use L dI/dt.
• Calculate the total flux linkage and use L
  Total Flux / I

or
Things to remember
The flux linkage,

• only if all loops intersect same flux

40
Self inductance

• x N, because

• x N, because

thus
L depends on materials (through m) and geometry
(like C)
41
Mutual inductance
Current through Coil 1 induces e.m.f. in Coil 2
Mutual Inductance
42
Mutual inductance
where,
Also,
And,