Electromagnetism Lecture#06 MUHAMMAD MATEEN YAQOOB THE UNIVERSITY OF LAHORE SARGODHA CAMPUS.

Electromagnetism

Lecture#06MUHAMMAD MATEEN YAQOOB

THE UNIVERSITY OF LAHORE

SARGODHA CAMPUS

MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Faraday’s Law The electric fields and magnetic fields considered up to now have been produced by stationary charges and moving charges respectively.

Imposing an electric field on a conductor gives rise to a current which in turn generates a magnetic field.

In 1831, Michael Faraday discovered that, by varying magnetic field with time, an electric field could be generated. The phenomenon is known as electromagnetic induction.

Faraday’s experiment demonstrates that an electric current is induced in the loop by changing the magnetic field.

The coil behaves as if it were connected to a source.


Consider a uniform magnetic field passing through a surface S

The magnetic flux through the surface is given by

Faraday’s law of induction may be stated as:

The induced emf ε in a coil is proportional to the negative of the rate of change of magnetic flux

For a coil that consists of N loops


Lenz’s Law The direction of the induced current is determined by Lenz’s law

To illustrate how Lenz’s law works, let’s consider a conducting loop placed in a magnetic field. We follow the procedure below:



Ampere’s Law We have seen that moving charges or currents are the source of magnetism. This can be readily demonstrated by placing compass needles near a wire. As shown in Figure, all compass needles point in the same direction in the absence of current. However, when I is non zero, the needles will be deflected along the tangential direction of the circular path.






Field inside and outside a Current-carrying wire

Consider a long straight wire of radius R carrying a current I of uniform current density, as shown in Figure. Find the magnetic field everywhere.


Solution


Magnetic flux (Φ) The group of force lines going from north pole to south pole of a magnet is called magnetic flux

Number of lines of force in a magnetic field determines the value of flux

Unit of magnetic flux is Weber (Wb)

One weber is 108 lines

It is a huge unit; so in most of applications micro-weber (µWb) is used


Magnetic flux density (B) It is the amount of flux per unit area perpendicular to the magnetic field

Its symbol is B and its unit is Tesla (T)

One tesla equals one weber per square meter (Wb/m2)

B = Φ / A


Inductor An inductor is a passive element designed to store energy in its magnetic field.

Inductors find numerous applications in electronic and power systems. They are used in power supplies, transformers, radios, TVs, radars and electric motors.

Any conductor of electric current has inductive properties and may be regarded as an inductor. But in order to enhance the inductive effect, a practical inductor is usually formed into a cylindrical coil with many turns of conducting wire.

Inductor An inductor is made of a coil of conducting wire

Inductors are formed with wire tightly wrapped around a solid central core

lAN

L2



InductanceInductance (or electric inductance) is a measure of the amount of magnetic flux produced for a given electric current.

The inductance has the following relationship:

L= Φ/i

where L is the inductance in henrys, i is the current in amperes, Φ is the magnetic flux in webers


If current is allowed to pass through an inductor, it is found that the voltage across the inductor is directly proportional to the time rate of change of the current. Using the passive sign convention,

where L is the constant of proportionality called the inductance of the inductor. The unit of inductance is the henry (H), named in honor of the American inventor Joseph Henry (1797–1878).

I-V Relation of Inductors An inductor consists of a coil of conducting wire.

dt

diL

dt

dv

+

-

v

i

LFigure shows this relationship graphically for an inductor whose inductance is independent of current. Such an inductor is known as a linear inductor. For a nonlinear inductor, the plot of Eq. will not be a straight line because its inductance varies with current.



Flux in Inductors The relation between the flux in inductor and the current through the inductor is given below.

Li

i

φ Linear

Nonlinear


t

to

otidttv

Li )()(

1

tdttv

Li )(

1

memory. hasinductor The

vdtL

di1

+

-

vL

dt

diL

dt

dv

where i(t0) is the total current for −∞ < t < t0 and i(−∞) = 0. The idea of making i(−∞) = 0 is practical and reasonable, because there must be a time in the past when there was no current in the inductor.

The inductor is designed to store energy in its magnetic field

The energy stored in an inductor

idt

diLviP

t tidt

dtdiLpdtw

)(

)(

22 )(21

)(21ti

iLitLidiiL ,0)( i

)(2

1)( 2 tLitw

+

-

vL


Important properties of inductor

When the current through an inductor is a constant, then the voltage across the inductor is zero, same as a short circuit.

An inductor acts like a short circuit to dc.

The current through an inductor cannot change instantaneously.

dt

diL

dt

dv



Example 1The current through a 0.1-H inductor is i(t) = 10te-5t A. Find the voltage across the inductor and the energy stored in it.

Solution:

V)51()5()10(1.0 5555 teetetedtd

v tttt

J5100)1.0(21

21 1021022 tt etetLiw

,H1.0andSince LdtdiLv

isstoredenergyThe


Assignment # 2 Date of submission: On the day of mid term paper

Assignment # 2 Consider the circuit in Fig (a). Under dc conditions, find:

(a) i, vC, and iL.

(b) the energy stored in the capacitor and inductor.


Example 2 Find the current through a 5-H inductor if the voltage across it is

Also find the energy stored within 0 < t < 5s. Assume i(0)=0.

Solution:

0,00,30)(

2

ttttv

.H5and L)()(1

Since0

0 t

ttidttv

Li

A23

6 33

tt

tdtti

0

2 03051


Example 2

5

0

65 kJ25.156

05

66060t

dttpdtw

thenisstoredenergytheand,60powerThe 5tvip

before.obtainedas Same

usingstoredenergytheobtaincanweely,Alternativ

)0(21

)5(21

)0()5( 2 LiLiww

kJ25.1560)52)(5(21 23


Inductors in Series

Neq LLLLL ...321


Series Inductor Applying KVL to the loop,

Substituting vk = Lk di/dt results in

Nvvvvv ...321

dtdi

Ldtdi

Ldtdi

Ldtdi

Lv N ...321

dtdi

LLLL N )...( 321

dtdi

Ldtdi

L eq

N

KK

1

Neq LLLLL ...321


Inductors in Parallel

Neq LLLL

1111

21


Parallel Inductors Using KCL,

ButNiiiii ...321

t

t kk

k otivdt

Li )(

10

t

t

t

t sk

tivdtL

tivdtL

i0 0

)(1

)(1

02

01 t

t NN

tivdtL 0

)(1

... 0

)(...)()(1

...11

0020121

0

tititivdtLLL N

t

tN

t

teq

N

kk

t

t

N

k k

tivdtL

tivdtL 00

)(1

)(1

01

01

Neq LLLL

1111

21


Example 3Find the equivalent inductance of the circuit shown in Fig.


Example 3 Solution: 10H12H,,H20:Series

H6427427

: Parallel

H18864 eqL

H42


Example 4 For the circuit in Fig,

If

find :

.mA)2(4)( 10teti ,mA 1)0(2 i

)0( (a)1i );(and),(),((b) 21 tvtvtv )(and)((c) 21 titi


Solution.mA4)12(4)0(mA)2(4)()(a 10 ieti t

mA5)1(4)0()0()0( 21 iii

H53212||42 eqL

mV200mV)10)(1)(4(5)( 1010 tteq eedtdi

Ltv

mV120)()()( 1012

tetvtvtv

mV80mV)10)(4(22)( 10101

tt eedtdi

tv

isinductanceequivalentThe)(b


t t t dteidtvti0 0

10121 mA5

4120

)0(41

)(

mA38533mA50

3 101010 ttt eet

e

t ttdteidtvti

0

1020 22 mA1

12120

)0(121

)(

mA11mA10

101010 ttt eet

e

)()()(thatNote 21 tititi

t

idttvL

i0

)0()(1

)(c


Applications of Capacitors and InductorsCircuit elements such as resistors and capacitors are commercially available in either discrete form or integrated-circuit (IC) form. Unlike capacitors and resistors, inductors with appreciable inductance are difficult to produce on IC substrates. Therefore, inductors (coils) usually come in discrete form and tend to be more bulky and expensive. For this reason, inductors are not as versatile as capacitors and resistors, and they are more limited in applications. However, there are several applications in which inductors have no practical substitute. They are routinely used in relays, delays, sensing devices, pick-up heads, telephone circuits, radio and TV receivers, power supplies, electric motors, microphones, and loudspeakers, to mention a few.

Introduction

The term alternating indicates only that the waveform alternates between two prescribed levels in a set time sequence.


Sinusoidal AC Voltage Characteristics and Definitions

Generation An ac generator (or alternator) powered by water power, gas, or nuclear fusion is the primary component in the energy-conversion process.

The energy source turns a rotor (constructed of alternating magnetic poles) inside a set of windings housed in the stator (the stationary part of the dynamo) and will induce voltage across the windings of the stator.



Generation

Wind power and solar power energy are receiving increased interest from various districts of the world.

The turning propellers of the wind-power station are connected directly to the shaft of an ac generator.

Light energy in the form of photons can be absorbed by solar cells. Solar cells produce dc, which can be electronically converted to ac with an inverter.

A function generator, as used in the lab, can generate and control alternating waveforms.



Definitions Waveform: The path traced by a quantity, such as voltage, plotted as a function of some variable such as time, position, degree, radius, temperature and so on. Instantaneous value: The magnitude of a waveform at any instant of time; denoted by the lowercase letters (e1, e2).Peak amplitude: The maximum value of the waveform as measured from its average (or mean) value, denoted by the uppercase letters Em (source of voltage) and Vm (voltage drop across a load).



DefinitionsPeak value: The maximum instantaneous value of a function as measured from zero-volt level.

Peak-to-peak value: Denoted by Ep-p or Vp-p, the full voltage between positive and negative peaks of the waveform, that is, the sum of the magnitude of the positive and negative peaks.Periodic waveform: A waveform that continually repeats itself after the same time interval.


DefinitionsPeriod (T): The time interval between successive repetitions of a periodic waveform (the period T1 = T2 = T3), as long as successive similar points of the periodic waveform are used in determining T Cycle: The portion of a waveform contained in one period of timeFrequency: (Hertz) the number of cycles that occur in 1 s

Hz) (hertz, 1

Tf



AmplitudePEAK AMPLITUDE PEAK-TO-PEAK AMPLITUDE

ppp

ppp

II

VV

2

2


Instantaneous Value Instantaneous value or amplitude is the magnitude of the sinusoid at a point in time.

VssradVtvmst

VssradVtvst

tsradVtv

94.2)]01.0)(/377sin[(5)( 10

0)]0)(/377sin[(5)( 0

])/377sin[(5)(


Average Value The average value of a sinusoid signal is the integral of the sine wave over one full cycle. This is always equal to zero.

If the average of an ac signal is not zero, then there is a dc component known as a DC offset.


Root Mean Square (RMS) Most equipment that measure the amplitude of a sinusoidal signal displays the results as a root mean square value. This is signified by the unit Vac or VRMS. RMS voltage and current are used to calculate the average power associated with

the voltage or current signal in one cycle.

T

RMS dttvT

V0

2)(1

RVP

VVV

RMSAve

ppRMS

2

707.02

2


Current (I)

Electrical current is the rate of flow of charges

where: I = current in amperes (A)

Q = charge in coulombs (C)

t = time in seconds (s) the rate of flow of charge.

Random motion of free electrons in a material.

Electrons flow from negative to positive when a voltage is applied across a conductive or semiconductive material.


Definition of Current One ampere (1 A) is the amount of current that exists when a number of electrons having a total charge of one coulomb (1 C) move through a given cross-sectional area in one second (1 s).


Example


Resistance (R) Resistance is the opposition to current.

Definition of resistance One ohm (1 Ω) of resistance exists if there is one ampere (1 A) of current in a material when one volt (1 V) is applied across the material.


Conductance (G) The reciprocal of resistance is conductance, symbolized by G. It is a measure of the ease with which current is established.

The formula is

Unit is siemens.


Types of Resistor

Fixed Resistor


Carbon-composition resistor This resistor is made with a mixture of finely ground carbon, insulating filler, and a resin binder. The ratio of carbon to insulating filler sets the resistance value.


Resistor Color Code


Resistor 4-band color code


Example What is the resistance and tolerance of each of the four-band resistors?

5.1 kW ± 5%

820 kW ± 5%

47 W ± 10%

1.0 W ± 5%

Tolerance= 0.255KΩ4.845------------5.355


Variable Resistor Variable resistors include the potentiometer and rheostat. A potentiometer can be connected as a rheostat

13

2

Resistiveelement

Wiper

Shaft

The center terminal is connected to the wiper

R

Variable(potentiometer)

R

Variable(rheostat)

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Electromagnetism Lecture#06 MUHAMMAD MATEEN YAQOOB THE UNIVERSITY OF LAHORE SARGODHA CAMPUS.

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