Introduction to Electrical machines-chapter 1 magnetics

1. Chapter One
Magnetics
 Introduction
 Magnetic circuits
 Magnetic Materials and Their Properties
 Magnetically Induced Emf and Force
 Ac Operation of Magnetic Circuits
 Hysteresis and eddy current losses
By :Yimam A.(MSc)

2. Introduction
 An electrical machine is a device which converts electrical power (voltages
and currents) into mechanical power(torque and rotational speed), and/or
vice versa.
 A motor describes a machine which converts electrical power to mechanical
power; a generator (or alternator) converts mechanical power to electrical
power.
 Almost all practical motors and generators convert energy from one form
to another through the action of a magnetic field.
 Transformers are usually studied together with generators and motors
because they operate on the same principle, the difference is just in the
action of a magnetic field to accomplish the change in voltage level.2

3. Principle of Electromagnet
 The principles of magnetism play an important role in the operation of an
electric machine.
 The basic idea behind an electromagnet is a magnetic field around the
conductor can be produced when current flows through a conductor. In
other word, the magnetic field only exists when electric current is flowing
 By using this simple principle, you can create all sorts of things, including
motors, solenoids, read/write heads for hard disks and tape drives,
speakers, and so on.
3

4. Magnetic Field
 magnetic field encircle their current
source.
 field is perpendicular to the wire and
that the field's direction depends on
which direction the current is
flowing in the wire.
 A circular magnetic field develops
around the wire follows right-hand
rules.
4

5. Properties of Magnetic Lines of Force
 Magnetic lines of force are
directed from north to south
outside a magnet.
 Magnetic lines of force are
continuous.
 Magnetic lines of force in the same
direction tend to repel each other.
 Magnetic lines of force tend to be
as short as possible.
 Magnetic lines of force occupy
three-dimensional space extending
(theoretically) to infinity.
 Magnetic lines of force enter or
leave a magnetic surface at right
angles.
 Magnetic lines of force cannot
cross each other.
5

6. Cont.…
 magnetic fields are the fundamental mechanism by which energy is
converted from one form to another in motors, generators, and
transformers. Four basic principles describe how magnetic fields are used in
these devices:
1. A current-carrying wire produces a magnetic field in the area around it.
2. A time-changing magnetic field induces a voltage in a coil of wire if it
passes through that coil. (This is the basis of transformer action.)
3. A current-carrying wire in the presence of a magnetic field has a force
induced on it. (This is the basis of motor action.)
4. A moving wire in the presence of a magnetic field has a voltage induced in
it.(This is the basis of generator action.) 6

7. Example of Electromagnet
 An electromagnet can be made by
winding the conductor into a coil and
applying a DC voltage.
 The lines of flux, formed by current
flow through the conductor, combine
to produce a larger and stronger
magnetic field.
 The center of the coil is known as
the core. In this simple electromagnet
the core is air.
7

8. Cont.…
 Iron is a better conductor of flux
than air. The air core of an
electromagnet can be replaced by a
piece of soft iron.
 When a piece of iron is placed in
the center of the coil more lines of
flux can flow and the magnetic
field is strengthened.
8

9. Cont.…
 Because the magnetic field around a
wire is circular and perpendicular to
the wire, an easy way to amplify the
wire's magnetic field is to coil
the wire.
 The strength of the magnetic field in
the DC electromagnet can be
increased by increasing the
number of turns in the coil.
 The greater the number of turns the
stronger the magnetic field will be. 9

10. Basics of Magnetic Circuits
1. Magnetic flux(ϕ):
 The magnetic lines of force produced by a magnet is called magnetic flux.
 It is denoted by ϕ and its unit is Weber.
 1 weber = 108 lines of force
2. Flux density(B)
 The total number of lines of force per square metre of the cross-
sectional area of the magnetic core is called flux density.
 Its SI unit is Tesla (weber per metre square).
B= ϕ/A Wb/m2 or Tesla
Where ϕ -total flux in webers A - area of the core in square metres
B - flux density in weber/metre square.
10

11. Cont.…
3 . Magneto-Motive Force
 The amount of flux density setup in the core is dependent upon five
factors - the current, number of turns, material of the magnetic core,
length of core and the cross-sectional area of the core.
 More current and the more turns of wire we use, the greater will be the
magnetizing effect.
 This ability of a coil to produce magnetic flux is called the magneto
motive force.
mmf = NI ampere - turns
Where mmf is the magneto motive force in ampere turns
N is the number of turns.
11

12. Cont.…
4. Magnetic field Intensity(H)
 The magnetic field intensity is the mmf per unit length along the path of
the flux.
 Is also known as magnetic flux intensity and is represented by the letter
H. Its unit is ampere turns per meter.
H= mmf/ Length
H = NI/l AT/m
Where H is magnetic field intensity
N is the number of turns
l is average path length of the magnetic flux 12

13. Cont.…
5. Magnetic Flux Linkage(𝝀):
 The product of magnetic coupling to a conductor, or the flux thru a single
turn times the number of turns in coils.
𝜆 = 𝑛∅
 Which also relates to define inductance as
𝜆 = 𝐿𝑖
Where 𝑣 =
𝑑
𝑑𝑡
𝜆 and 𝑣 =
𝑑
𝑑𝑡
𝐿𝑖, L is inductance
13

14. Cont.…
6. Reluctance [S] or
 It is the opposition of a magnetic circuit to setting up of a magnetic flux in
it.
𝑓𝑙𝑢𝑥 = ∅ = 𝐵𝐴; 𝐹 = 𝑚𝑚𝑓 = 𝐻𝑙; 𝐵 = 𝜇𝐻
∅
𝐹
=
𝐵𝐴
𝐻𝑙
=
𝜇 𝑜 𝜇 𝑟
𝑙
; ℎ𝑒𝑛𝑐𝑒 ∅ =
𝜇 𝑜 𝜇 𝑟 𝐴
𝑙
F
∅ =
𝐹
𝑙
𝜇 𝑜 𝜇 𝑟 𝐴
=
𝐹
𝑆
; 𝑆 =
𝐹
∅
𝑤ℎ𝑒𝑟𝑒 𝑆 =
𝑙
𝜇 𝑜 𝜇 𝑟 𝐴
Where, S – reluctance of the magnetic circuit
l - length of the magnetic path in meters
μo- permeability of free space µr - relative permeability
14

15. Cont.…
7. Permeability [μ]
 A property of a magnetic material which indicates the ability of
magnetic circuit to carry electromagnetic flux.
 Ratio of flux density to the magnetizing force, μ = B / H
 Unit: henry / meter
 Permeability of free space or air or non magnetic material
𝜇 𝑜 = 4𝜋 × 10−7 Τ𝐻 𝑚
Relative permeability [𝜇 𝑟]:
𝜇 𝑟 =
𝜇
𝜇 𝑜
15

16. Cont.…
8. Residual Magnetism
 It is the magnetism which remains in a material when the effective
magnetizing force has been reduced to zero.
9. Magnetic Saturation
 The limit beyond which the strength of a magnet cannot be increased is
called magnetic saturation.
16

17. Cont.…
10. End Rule
 According to this rule the current direction when looked from one end of
the coil is in clock wise direction then that end is South Pole. If the current
direction is in anti clock wise direction then that end is North Pole.
11. Lenz’s Law
 When an emf is induced in a circuit electromagnetically the current set up
always opposes the motion or change in current which produces it.
17

18. Cont.…
12. Electro magnetic induction
 Electromagnetic induction means the electricity induced by the magnetic
field.
Faraday's Laws of Electro Magnetic Induction
 There are two laws of Faraday's laws of electromagnetic induction.
They are,
1) First Law
2) Second Law
18

19. Cont.…
First Law
 Whenever a conductor cuts the magnetic flux lines an emf is induced in
the conductor.
Second Law
 The magnitude of the induced emf is equal to the rate of change of flux-
linkages
𝑣 = −𝑁
𝑑∅
𝑑𝑡
Where V is induced voltage N is number of turns in coil
𝑑∅ is change of flux in coil 𝑑𝑡 is time interval
19

20. Magnetic Materials
 Ferro Magnetic Materials: these materials are strongly attracted by a
magnet. example: iron, steel, nickel, cobalt, some metallic alloys. The
relative permeability of these materials is very high.
 Para Magnetic Materials: these materials are attracted by a magnet but
not very strongly. example: aluminum, tin, platinum, magnesium,
manganese etc. The relative permeability of these materials is slightly more
than one.
 Dia Magnetic Materials: these materials are not at all attracted by any
magnet. The relative permeability of these materials is less than one.
example: zinc, mercury, lead, sulfur, copper, silver etc.
20

21. Magnetic Circuit
 The complete closed path followed by any group of magnetic lines of flux
is referred to as magnetic circuit.
Equivalent electrical circuit
21

22. Analogy with Electric circuits
Similarities
Electric circuit
o Emf (volt)
o Current(ampere)
o Resistance(ohm)
o Current density(A/𝑚2)
o Conductivity
Difference
 Current actually flows
 Circuit may be open or closed
Magnetic circuit
o m.m.f (AT)
o Flux(weber)
o Reluctance(A/Wb)
o flux density(T or Wb/𝑚2)
o Permeability
 flux is created, but does not flow
 Circuit is always closed 22

23. Cont.…
23
Electric circuit magnetic circuit

24. Cont.…
 The equivalent reluctance of a
number of reluctances in series is
just the sum of the individual
reluctances:
 Similarly, reluctances in parallel
combine according to the equation
Important formulas
24

25. Leakage Flux and Fringing
 Leakage Flux : the magnetic flux
which does not follow the
particularly intended path in a
magnetic circuit.
 When a current is passed through a
solenoid, magnetic flux is produced
by it.
25

26. Cont.…
 Most of the flux is set up in the core of the solenoid and passes through
the particular path that is through the air gap and is utilised in the magnetic
circuit. This flux is known as Useful flux ∅ 𝒖
 Practically it is not possible that all the flux in the circuit follows a
particularly intended path and sets up in the magnetic core and thus some
of the flux also sets up around the coil or surrounds the core of the coil,
and is not utilised for any work in the magnetic circuit. This type of flux
which is not used for any work is called Leakage Flux and is denoted by
∅𝒍.
 The total flux Φ produced by the solenoid in the magnetic circuit is the
sum of the leakage flux and the useful flux. 26

27. Cont.…
Leakage coefficient
 The ratio of the total flux produced
to the useful flux set up in the air
gap of the magnetic circuit is called
leakage coefficient or leakage
factor. It is denoted by (λ).
λ=
𝑇𝑜𝑡𝑎𝑙 𝑓𝑙𝑢𝑥(𝑓𝑙𝑢𝑥 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑟𝑜𝑛 𝑝𝑎𝑡ℎ)
𝑢𝑠𝑒𝑓𝑢𝑙 𝑓𝑙𝑢𝑥(𝑓𝑙𝑢𝑥 𝑖𝑛 𝑡ℎ𝑒 𝑎𝑖𝑟𝑔𝑎𝑝)
Fringing
 The useful flux when sets up in the
air gap, it tends to bulge outward at
(b and b’) as shown in figure,
because of this bulging the
effective area of the air gap
increases and the flux density of
the air gap decreases. This effect is
known as Fringing and the longer
the air gap the greater is the
fringing. 27

28. Series magnetic circuits
 Magnetic circuit composed of
various materials of different
permeabilities.
 When composite magnetic circuit
parts are connected one after the
other the circuit is called series
magnetic circuit.
 Consider a circular ring made up of
different materials of lengths
𝑙1, 𝑙2 𝑎𝑛𝑑 𝑙3and with cross sectional
areas 𝑎1, 𝑎2 𝑎𝑛𝑑 𝑎3 with absolute
permeabilities 𝜇1, 𝜇2 and 𝜇3.
28

29. Cont.…
Equivalent electric circuit
29

30. Series magnetic circuit with air gap
 Consider a ring having mean
length of iron part as 𝑙𝑖
Where 𝑆𝑖=reluctance of iron path
𝑆 𝑔=reluctance of air gap
𝑆𝑖 =
𝑙 𝑖
𝜇𝑎 𝑖
𝑆 𝑔 =
𝑙 𝑔
𝜇 𝑜 𝑎 𝑔
𝑆 𝑇 =
𝑙𝑖
𝜇𝑎𝑖
+
𝑙 𝑔
𝜇 𝑜 𝑎𝑖
∅ =
𝑚. 𝑚. 𝑓
𝑟𝑒𝑙𝑢𝑐𝑡𝑎𝑛𝑐𝑒
=
𝑁𝐼
𝑆 𝑇
Total 𝑚. 𝑚. 𝑓 = 𝑁𝐼 AT
Total reluctance 𝑆 𝑇 = 𝑆𝑖 + 𝑆 𝑔
30

31. Parallel magnetic circuit
 A magnetic circuit which has more
than one path for the flux is
known as a parallel magnetic
circuit.
 At point A the total flux ∅ divides
into two parts ∅1 𝑎𝑛𝑑 ∅2.
∅ = ∅1 + ∅2
 The fluxes ∅1 𝑎𝑛𝑑 ∅2 have their
paths completed through ABCD
and AFED respectively
Magnetic core Equivalent electrical circuit 31

32. Cont.…
 Total 𝑚. 𝑚. 𝑓 = 𝑁𝐼 AT
𝑓𝑙𝑢𝑥 =
𝑚. 𝑚. 𝑓
𝑟𝑒𝑙𝑢𝑐𝑡𝑎𝑛𝑐𝑒
𝑚. 𝑚. 𝑓 = ∅ × 𝑆
 For path ABCDA
𝑁𝐼 = ∅1 𝑆1 + ∅𝑆 𝐶
 For path AFEDA
𝑁𝐼 = ∅2 𝑆2 + ∅𝑆 𝐶
Where
S1 =
l1
μa1
, S2 =
l2
μa2
and Sc =
lc
μac
For parallel circuit
Total m.m.f = m.m.f required by central limb
+ m.m.f required by any one of outer limbs.
𝑁𝐼 = (𝑁𝐼) 𝐴𝐷+(𝑁𝐼) 𝐴𝐵𝐶𝐷 𝑜𝑟 (𝑁𝐼) 𝐴𝐹𝐸𝐷
𝑁𝐼 = ∅𝑆𝑐 + [∅1 𝑆1 𝑜𝑟 ∅2 𝑆2]
32

33. Parallel magnetic circuits with air gap
 Consider a parallel circuit with air
gap in the central limb
 The analysis of this circuit is exactly
similar to the parallel circuit.
 The only change is the analysis of
central limb. The central limb is
series combination of iron path and
air gap.
 The central limb is made up of
Path GD=iron path=𝑙 𝑐
Path GA=air gap=𝑙 𝑔
33

34. Cont.…
∅ = ∅1 + ∅2
 The reluctance of central limb is
𝑆𝑐 = 𝑆𝑖 + 𝑆 𝑔 =
𝑙 𝑐
μ𝑎 𝑐
+
𝑙 𝑔
μ 𝑜 𝑎 𝑐
 m.m.f of central limb is
(𝑚. 𝑚. 𝑓) 𝐴𝐷= (𝑚. 𝑚. 𝑓) 𝐺𝐷+(𝑚. 𝑚. 𝑓) 𝐺𝐴
 The total m.m.f can be expressed as
(𝑁𝐼) 𝑡𝑜𝑡𝑎𝑙= (𝑁𝐼) 𝐺𝐷+(𝑁𝐼) 𝐺𝐴+ 𝑁𝐼 𝐴𝐵𝐶𝐷 𝑜𝑟 (𝑁𝐼) 𝐴𝐹𝐸𝐷
Examples:?
Equivalent electrical circuit
34

35. Magnetic Behavior of Ferromagnetic Materials
 To illustrate the behavior of magnetic permeability in a ferromagnetic
material, apply a direct current to the core, starting with 0 A and slowly
working up to the maximum permissible current.
 At first, a small increase in the magnetomotive force produces a huge
increase in the resulting flux. After a certain point, though, further
increases in the magnetomotive force produce relatively smaller increases in
the flux. Finally, an increase in the magnetomotive force produces almost
no change at all.
 The graph between the flux density(B) and the magnetic field intensity(H)
for the magnetic material is called its magnetization curve or B-H curve.
 It is also called a saturation curve. 35

36. Cont.…
36
Experimental set up to obtain B-H curve
Knee
unsaturation
saturation

37. Cont.…
 Magnetic Saturation is The limit beyond which magnetic flux density in a
magnetic area does not increase sharply further with increase of mmf.
 Residual magnetism is the amount of magnetization left behind after
removing the external magnetic field from the circuit. In another word the
value of the flux density retained by the magnetic material is called Residual
Magnetism and the power of retaining this magnetism is called retentivity
of the material. or
 Residual flux density is the certain value of magnetic flux per unit area
that remains in the magnetic material without presence of magnetizing
force (i.e. H = 0).
37

38. Cont.…
38
The magnetization curve expressed
in terms of flux density and
magnetic field intensity.
Magnetization curve of
different magnetic materials

39. Cont.…
 The region of this figure in which the curve flattens out is called the
saturation region, and the core is said to be saturated.
 In contrast, the region where the flux changes very rapidly is called the
unsaturated region of the curve, and the core is said to be unsaturated.
 The transition region between the unsaturated region and the saturated
region is sometimes called the knee of the curve.
 The value of relative permeability mainly depends on the value of flux
density. But for the non-magnetic materials like plastic, rubber, etc. and for
the magnetic circuit having an air gap, its value is constant, denoted by (µ0).
Its value is 4πx10-7H/m and commonly known as absolute permeability or
permeability of free space. 39

40. Magnetic Hysteresis
 1: When supply current I = 0, so no existence of flux density (B) and
magnetizing force (H). The corresponding point is ‘O’ in the graph below.
 2: When current is increased from zero value to a certain value, magnetizing
force (H) and flux density (B) both are set up and increased following the
path o – a.
 3: For a certain value of current, flux density (B) becomes maximum (Bmax).
The point indicates the magnetic saturation or maximum flux density of
this core material. All element of core material get aligned perfectly. Hence
Hmax is marked on H axis. So no change of value of B with further
increment of H occurs beyond point ‘a’.
40

41. Cont.…
41
Hysteresis Loop
Experimental set up to obtain Hysteresis Loop

42. Cont.…
 4: When the value of current is decreased from its value of magnetic flux
saturation, H is decreased along with decrement of B not following the
previous path rather following the curve a – b.
 5: The point ‘b’ indicates H = 0 for I = 0 with a certain value of B. This
lagging of B behind H is called hysteresis. The point ‘b’ explains that after
removing of magnetizing force (H), magnetism property with little value
remains in this magnetic material and it is known as residual magnetism
(Br). Here o – b is the value of residual flux density due to retentivity of the
material.
42

43. Cont.…
 6: If the direction of the current I is reversed, the direction of H also gets
reversed. The increment of H in reverse direction following path b – c
decreases the value of residual magnetism (Br) that gets zero at point ‘c’
with certain negative value of H. This negative value of H is called coercive
force (Hc).
 7: H is increased more in negative direction further; B gets reverses
following path c – d. At point ‘d’, again magnetic saturation takes place but
in opposite direction with respect to previous case. At point ‘d’, B and H
get maximum values in reverse direction, i.e. (-Bm and -Hm).
43

44. Cont.…
 8: If we decrease the value of H in this direction, again B decreases
following the path de. At point ‘e’, H gets zero valued but B is with finite
value. The point ‘e’ stands for residual magnetism (-Br) of the magnetic
core material in opposite direction with respect to previous case.
 9: If the direction of H again reversed by reversing the current I, then
residual magnetism or residual flux density (-Br) again decreases and gets
zero at point ‘f’ following the path e – f. Again further increment of H, the
value of B increases from zero to its maximum value or saturation level at
point a following path f – a.
 The path a – b – c – d – e – f – a forms hysteresis loop.
[NB: The shape and the size of the hysteresis loop depend on the nature of
the material chosen]
44

45. Cont.…
 Hysteresis: The phenomenon of flux density(B) lagging behind the magnetizing force
(H) in a magnetic material is known as Magnetic Hysteresis.
 Coercive force is defined as the negative value of magnetizing force (-H) that reduces
residual flux density of a material to zero.
 Retentivity:It is defined as the degree to which a magnetic material gains its magnetism
after magnetizing force (H) is reduced to zero.
 The hysteresis loss in an iron core is the energy required to accomplish the
reorientation of domains during each cycle of the alternating current applied to the
core.
 The area enclosed in the hysteresis loop formed by applying an alternating current to
the core is directly proportional to the energy lost in a given ac cycle.
45

46. Hysteresis Loss
 The work done by the magnetizing force against the internal friction of the
molecules of the magnet, produces heat. This energy which is wasted in the
form of heat due to hysteresis is called hysteresis loss.
Where, Ph – hysteresis loss in watts
Ƞ – hysteresis or Steinmetz’s constant in J/m3,
Bmax – maximum value of the flux density in the magnetic material in
wb/m2
𝑓 – number of cycles of magnetization made per second
𝑣- volume of the magnetic material (part in which magnetic
reversal occur) in m3
46

47. Cont.…
Soft magnetic material
 The soft magnetic material has a narrow magnetic
hysteresis loop which has a small amount of
dissipated energy. They are made up of material
like iron, silicon steel, etc.
 It is used in the devices that require alternating
magnetic field.
 It has low coercivity
 Low magnetization
 Low retentivity
47

48. Cont.…
Hard magnetic material
 The Hard magnetic material has a
wider hysteresis loop and results in a
large amount of energy dissipation
and the demagnetization process is
more difficult to achieve.
 It has high retentivity
 High coercivity
 High saturation
48

49. Importance of Hysteresis Loop
 Smaller hysteresis loop area symbolizes less hysteresis loss.
 Hysteresis loop provides the value of retentivity and coercivity of a
material. Thus the way to choose perfect material to make permanent
magnet, core of machines becomes easier.
 From B-H graph, residual magnetism can be determined and thus choosing
of material for electromagnets is easy.
 Magnetic material having a wider hysteresis loop is used in the devices like magnetic
tape, hard disk, credit cards, audio recordings as its memory isn’t easily erased.
 Magnetic materials having a narrow hysteresis loop are used as electromagnets, solenoid,
transformers and relays which require minimum energy dissipation.
49

50. Eddy Current Loss
 When an alternating magnetic field is applied to a magnetic material an emf
is induced in the material itself.
 Since the magnetic material is a conducting material, these EMFs circulates
currents within the body of the material. These circulating currents are
called Eddy Currents. They will occur when the conductor experiences a
changing magnetic field.
 As these currents are not responsible for doing any useful work, and it
produces a loss (𝐼2
𝑅 loss) in the magnetic material known as an Eddy
Current Loss. Similar to hysteresis loss, eddy current loss also increases the
temperature of the magnetic material.
50

51. Cont.…
 The hysteresis and the eddy
current losses in a magnetic
material are also known by the
name iron losses or core losses or
magnetic losses.
 When the changing flux links with
the core itself, it induces emf in
the core which in turns sets up the
circulating current called Eddy
Current and these current in return
produces a loss called eddy current
loss or (𝐼2
𝑅) loss.
where I is the value of the current and
R is the resistance of the eddy current path. 51

52. Cont.…
 If the core is made up of solid iron of larger cross-sectional area, the
magnitude of I will be very large and hence losses will be high. To reduce
the eddy current loss mainly there are two methods.
 By reducing the magnitude of the eddy current.
 The magnitude of the current can be reduced by splitting the solid core into thin
sheets called laminations, in the plane parallel to the magnetic field. Each lamination
is insulated from each other by a thin layer of coating of varnish or oxide film. By
laminating the core, the area of each section is reduced and hence the induced emf
also reduces. As the area through which the current is passed is smaller, the
resistance of eddy current path increases.
 The eddy current loss is also reduced by using a magnetic material having
the higher value of resistivity like silicon steel. 52

53. Cont.…
 It is difficult to determine the eddy current loss from the resistance and
current values, but by the experiments, the eddy current power loss in a
magnetic material is given by the equation
Where, Pe=eddy current loss in watts
Ke=coefficient of eddy current.
Bm= maximum value of flux density in ΤWb m2
𝑡 =thickness of lamination in meters
𝑓 =frequency of reversal of magnetic field in Hz
𝑣 =volume of magnetic material in 𝑚3. 53

54. Faraday's law- induced voltage from a time-changing magnetic field
 From the various ways in which an existing magnetic field can affect its
surroundings, the first major effect is Faraday's law.
 It states that if a flux passes through a turn of a coil of wire, a voltage will
be induced in the turn of wire that is directly proportional to the rate of
change in the flux with respect to time.
Where 𝑒𝑖𝑛𝑑 is the voltage induced in the turn of the coil and
∅ is the flux passing through the turn.
 If a coil has N turns and if the same flux passes through all of them, then the
voltage induced across the whole coil is given by
54

55. Cont.…
Where 𝑒𝑖𝑛𝑑 = voltage induced in the coil
N = number of turns of wire in coil
∅ =flux passing through coil
 The minus sign in the equations is an expression of Lenz's law.
 Lenz's law states that the direction of the voltage buildup in the coil is such
that if the coil ends were short circuited, it would produce current that
would cause a flux opposing the original flux change. Since the induced
voltage opposes the change that causes it, a minus sign is included.
55

56. Cont.…
 The above equation assumes that
exactly the same flux is present in
each turn of the coil. Unfortunately,
the flux leaking out of the core into
the surrounding air prevents this
from being true.
 The magnitude of the voltage in the
𝑖 𝑡ℎ
turn of the coil is always given
by
 If there are N turns in the coil of
wire, the total voltage on the coil is
56
Where
𝜆 =Flux linkage

57. Production of induced force on a wire
 A second major effect of a magnetic field
on its surroundings is that it induces a force
on a current-carrying wire within the field.
 The force induced on the conductor is
given by
Where i = magnitude of current in wire
𝐿 = length of wire, with direction of I
defined to be in the direction
of current flow
B = magnetic flux density vector
57
Fleming's Right-hand Rule

58. Cont.…
 The direction of the force is given
by the right-hand rule: If the index
finger of the right hand points in
the direction of the vector I and
the middle finger points in the
direction of the flux density vector
B, then the thumb points in the
direction of the resultant force on
the wire.
 The magnitude of the force is
given by
 where 𝜃 is the angle between the
wire and the flux density vector.
58current-carrying wire in the
presence of a magnetic field

59. Induced voltage on a conductor moving in a magnetic field
 If a wire with the proper orientation moves through a magnetic field, a
voltage is induced in it. The voltage induced in the wire is given by
Where 𝑣= velocity of the wire
B = magnetic flux density vector
𝑙 = length of conductor in the magnetic field
59

60. Cont.…
 Vector 𝑙 points along the direction
of the wire toward the end making
the smallest angle with respect to
the vector 𝑣 × 𝐵.
 The voltage in the wire will be built
up so that the positive end is in the
direction of the vector 𝑣 × 𝐵.
60

61. 61