Lecture # 11Magnetism and Inductance

Magnetism

• Magnetism is a phenomenon which includes forces exerted by magnets on other magnets• It has its origin in electric currents and the fundamental magnetic

moments of elementary particles

Magnetic Fields

• Magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials• The magnetic field at any given point is specified by both a direction and a

magnitude (or strength); as such it is a vector field• The term is used for two distinct but closely related fields denoted by the

symbols B and H• B refers to magnetic flux density, and H to magnetic field strength• A permanent magnet has a magnetic field surrounding it• Magnetic field is consists of lines of forces which radiate from north pole

(N) to south pole (S) and back to north pole through the magnetic material

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

Inductance

Inductance (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).

Flux in Inductors• The relation between the flux in inductor and the

current through the inductor is given below.

Li

i

φ Linear

Nonlinear

I-V Relation of Inductors

• An inductor consists of a coil of conducting wire.

dt

diL

dt

dv

+

-

v

i

L

Figure 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.

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

According to Eq. a discontinuous change in the current through an inductor requires an infinite voltage, which is not physically possible. Thus, an inductor opposes an abrupt change in the current through it. For example, the current through an inductor may take the form shown in Fig.(a), whereas the inductor current cannot take the form shown in Fig. (b) in real-life situations due to the discontinuities. However, the voltage across an inductor can change abruptly.

• Like the ideal capacitor, the ideal inductor does not dissipate energy. The energy stored in it can be retrieved at a later time. The inductor takes power from the circuit when storing energy and delivers power to the circuit when returning previously stored energy.• The inductor can be used to generate a high voltage, for example,

used as an igniting element.

Example 1

The 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

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,• But Niiiii ...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 3

Find 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

Example 4Solution:

.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

Example 4

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

Table

Applications of Capacitors and Inductors

Circuit 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.

Capacitors and inductors possess the following three special properties that make them very useful in electric circuits:

1. The capacity to store energy makes them useful as temporary voltage or current sources. Thus, they can be used for generating a large amount of current or voltage for a short period of time.

2. Capacitors oppose any abrupt change in voltage, while inductors oppose any abrupt change in current. This property makes inductors useful for spark or arc suppression and for converting pulsating dc voltage into relatively smooth dc voltage.3. Capacitors and inductors are frequency sensitive. This property makes them useful for frequency discrimination.

The first two properties are put to use in dc circuits, while the third one is taken advantage of in ac circuits.

Practice Problem

Practice Problem• Consider the circuit in Fig

(a). Under dc conditions, find:

(a) i, vC, and iL. (b) the energy stored in

the capacitor and inductor.