Fields and Waves I

Fields and Waves I

Lecture 16Faraday’s Law

K. A. ConnorElectrical, Computer, and Systems Engineering Department

Rensselaer Polytechnic Institute, Troy, NY

Y. MaréchalPower Engineering Department

Institut National Polytechnique de Grenoble, France

April 19, 2023 Fields and Waves I 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 Ulaby’s textbook.


Overview

Review• Ampere’s Law• Magnetic Flux• Magnetic Vector Potential

Faraday’s Law• EMF• Induced Voltage/Current• Moving Magnet or Loop

Inductance• Self Inductance• Mutual Inductance

Quiz review


Ampere’s Law

Maxwell’s Equations:

jH

0B Integral form

netIsdjldH

0sdB

HB 0

Ampere’s Law

jH

0B

Magnetostatics Electrostatics

0E

D


Direction of B

B

wraps around j

I &

Use right-hand rule• thumb along

• fingers are in

j

I &

B

multiple wires or segments - use superposition

http://encarta.msn.com/media_701504656_761566543_-1_1/Right-Hand_Rule.html

B-Fields


Magnetic Flux & Magnetic Vector Potential

ldAsdAsdB

Alternative way to find

FLUX

BA

Magnetic FLUX

Magnetic Vector potential definition:

Flux definition:


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


Field Due To Long Straight Wire

H dl I enclosed

B rI

o

2

First, determine the magnetic field due to a long straight wire carrying a current I. (See Example 5-5 of Ulaby)

I

Line for Ampere’s Law

BI

ro

2

http://www.ee.surrey.ac.uk/Workshop/advice/coils/terms.html


Long Straight Wire

BI

r

A

ro z

2

The magnetic vector potential can be determined from first principles or from the magnetic field. We will do the latter.

AI

r constzo 2ln

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


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.


Magnetic Field Direction1

3

2

4

At point 1 (point 3 has to opposite sign):

B

I

da

I

daC E N TE R

oy

oy

2

2

B

I

d da a

I

da aLE F T

ox y

ox y

2

1

2

1

2 42 2

B

I

d da a

I

da aR IG H T

ox y

ox y

2

1

2

1

2 42 2

B

I

daTO TA L

oy

2


Magnetic Field Direction1

3

2

4

At point 2:

B

I

d da a

I

da aC E N TE R

ox y

ox y

2

2

1

2

1

2 22 2

B

I

d da a

I

da aLE F T

ox y

ox y

2 4

2

5

1

5 102

2 2

B

I

daR IG H T

oy

2

B TO TA L ?

Faraday’s Law and dynamic fields

Faraday’s Law


Faraday’s Law comes from Maxwell’s equation:

t

BE

sdB

dt

dldE

In electrostatics, we used: 0ldE

Applications:

• inductors• transformers• motors• generators• noise

Faraday’s Law


Faraday’s Law - concept of EMF

sdBdt

d

dt

dldEVemf

is the electromotive force

Time varying flux through a coil

ld

sd

The emf is similar to a VOLTAGE

sdBdt

dldE

Use right hand rule for sd

and ld

Orientation issues :

ld

sd


Faraday’s Law – various types of EMF

dt

sdBsd

t

BsdB

dt

d

dt

dVemf

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

What does the flux derivative means ?


dl

12

coil

Faraday’s 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)

1 H

function generator

~

50

10 cos(t) V1 MHz IIIII

B

solenoid


Transformers : Faraday’s law with dynamic fields

A huge range in sizeshttp://www.meppi.com/Products/Transformers/Power/Pages/Core-formTransformers.aspx

http://www.transformerfactory.com/e1-model-small-power-transformer-1va-70a.html

http://en.ferilex.eu/transformers.html


Faraday’s Law: moving loop experiment

A loop falls through the magnetic field between two pole faces at a constant velocity, u0.

B

side view of loop

magnets

SIDE VIEWFRONT VIEW

magnet face

loopu

B

t = t1FRONT VIEW

uB

t = t2

u

B

FRONT VIEWt = t3

A current is flowing in the loop as it pass through the magnets


Generators : Faraday’s Law Hoover Dam

http://isu.indstate.edu/jspeer/conservation/ http://www.wenzelontheweb.de/Hoover%20Dam.htm


Dynamic fields


dl

12

coil

Example 1

1 H

function generator

~

50

10 cos(t) V1 MHz IIIII

B

solenoid

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

zanIB ˆ0


Example 1

a. Circuit analysis. What are the current I and voltage V through the inductor?

b. 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 t=0+, does a scope read V1 - V2 > or < 0?2) If the clip leads were connected through a low impedance, which way would current flow at t=0+?

d. Sketch emf and vs time. What is the flux when the emf is largest?


Example 1

April 19, 2023 Fields and Waves I 26High impedance Output

Faraday’s Law and Lenz’s law

Low impedance Output

l

12

IIIIIB

Previous result : t=0+, flux decreasing, I as shown

Lenz’s law : “The current in the loop is always in such a direction as to oppose the change of magnetic flux that produced it.”

Ulaby


Moving conductors


Previous example had: 0t

B

In moving loop example: 0t

B

,but sd

changes with time

)(tsd

is a function of t

dt

sdBsd

t

BsdB

dt

dldE

0

Faraday’s Law for moving loop

dt

sdB

dt

dldEVemf


sI B

B-field into page

sI

sI

sI

u

sliding bar

At time t2 = t1 + t

sI z h

sd

B

ld

u

At time t1

sI sliding bar



ldBulduB

)()(

Approximate flux derivative as:

uhBt

zhB

t

sdB

t

u

sdt

BldBusdB

dt

dldE

Alternate expression

dt

sdB

A general form:

sI z h


t


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.

B

side view of loop

magnets

SIDE VIEWFRONT VIEW

magnet face

loopu

B

t = t1FRONT VIEW

uB

t = t2

u

B

FRONT VIEWt = t3

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?


Example 2


Inductances


Two types of Inductances:• self inductance - e.g. inductors

• mutual inductance - e.g. transformers

Self Inductance:

• coil of wire with 1I , creates B

• wire loop intersects sdB

• this creates sdBdt

dfme

...

Inductances

http://www.gaussbusters.com/ppm93.html


Inductor

Geometric parameters for a solenoidal inductor

http://www3.telus.net/chemelec/Calculators/Helical-Coil-Calc.htm


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.

a. What emf, E dl is induced around the solenoid (include all turns)?

b. 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.

c. What is the flux linkage through all N turns?

d. Calculate L = Flux/I and compare with your answer to part c.


Example 3


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

Things to remember :

or

The flux linkage, N• only if all loops intersect same flux• not true for finite solenoid and will need:


Self inductance

IB

I ,I

L is independent of I

L depends on materials (through ) and geometry (like C)

2NL , because• x N, because NB

• x N, because N

Note: To calculate L, don’t need Faraday’s Law just need:I

L

thus


Coil 1 Coil 2

1I

1B1B

Mutual Inductance: Current through Coil 1 induces e.m.f. in Coil 2

Mutual inductance


1

2121 I

L

2121 sdB

Mutual Inductance

Also,dt

dILemf 1

212

And, 2112 LL

where,

Mutual inductance

Coil 1 Coil 2

1I

1B1B

Fields and Waves I

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