Electrical Machines I
WEEK 2: MAGNETIC CIRCUITS
Sources of Magnetic Fields
“Magnetic fields are an essential element in the conversion of mechanical energy to electrical energy and vice versa”
Permanent Magnets
These include iron, nickel, cobalt, some alloys of rare
earth metals. Example: Alnico family, Samarium
Cobalt family, Boron family
Electro-Magnets and Current carrying conductors
A current-carrying wire produces a magnetic field in the area
around it
The magnetic field lines around a long wire which carries an
electric current form concentric circles around the wire. The
direction of the magnetic field is perpendicular to the wire
and is in the direction the fingers of your right hand would
curl if you wrapped them around the wire with your thumb
in the direction of the current
Sources of Magnetic Fields: Notes
 The polarity of the mmf from a coil of wire can be determined from a
modification of the right-hand rule: If the fingers of the right hand curl
in the direction of the current flow in a coil of wire, then the thumb will
point in the direction of the positive mmf .
 The magnetic circuit concept assumes that all flux is confined within a
magnetic core. Unfortunately, this is not quite true. The permeability of
a ferromagnetic core is 2000 to 6000 times that of air, but a small
fraction of the flux escapes from the core into the surrounding lowpermeability air. This flux outside the core is called leakage flux
 In many applications, magnetic flux must cross one or more air gaps.
As the magnetic lines of force cross the air gap, they spread out
because the Individual lines repel each other. This spreading out is
called Fringing
Faraday law of induced voltage from a time-changing magnetic field
Faraday's law 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”
𝑑ф
e= average emf (V)
𝑒 = −𝑁
𝑑𝑡
N= number of turns
ф = flux passing through the turn
t= time
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 would produce current that would cause a flux opposing the original flux change.
Instead of the moving magnet, if a time varying current is passed through a coil, a timechanging flux is produced which when transferred to another coil, induces a voltage across its
terminals.
Analogy between Magnetic and Electrical Circuits

N: number of turns (T)
i= current (A)
H= magnetic field intensity (AT/m)
l= MEAN length of the core (m)
A= CROSS sectional area of core (m2)
F= Magneto motive force, (AT)
i
Toroidal core
N
Ferromagnetic core: Iron, steel
THERFORE, for the
ferromagnetic core shown
H : some people call it
the magnetizing force
𝐻. 𝑙 = 𝑁 𝑖
𝐹 = 𝑁𝑖
Ferromagnetic core C core
EI core
𝐻=
1 turn coil
l
Side view
𝑁𝑖
𝐻=
𝑙
Ampere’s Law:
ර 𝐻. 𝑑𝑙 = 𝐼
N turn coil
𝐹
𝑙
The magnetic field
intensity H is in a sense
a measure of the
“effort” that a current
is putting into the
establishment of the
magnetic field.
Quantity
Electrical Circuit
Magnetic Circuit
Driving force
V (volt) EMF
F (NI) MMF
Produce
i (A)
Ф (weber)
Limited by
R (Ω)
ℜ (AT/weber)

I
V
R
F
F =MMF is analogous to Electromotive force (EMF) =E
 = Flux is analogous to i = Current
 = Reluctance is analogous to R = Resistance

Since magnetic and
electrical circuits have
similar characteristics,
then we can apply the
traditional circuits laws
to magnetic circuits
Important Relations: Part ‘1’
𝑉 = 𝑖𝑅
OHM’s law
𝑙
𝑅=𝜌
𝐴
𝐹 = 𝐻𝑙 = 𝑁 𝑖
= 𝜑ℜ
R 
Resistance depends on length,
cross sectional area of cable AND
the material from which the
resistance is made
1 l
1 l

 A  r 0 A
Relative permeability is
a way to compare the
“magnetisability” of
materials
𝜇0 = permeability of free space = 4𝜋 10−7 H/m
𝜇𝑟 = relative permeability of material compared to free space
𝜇
𝜇𝑟 =
𝜇0
Steel relative permeability could reach
up to 6000!!! AND is used in machines.
Electrical ccts
Magnetic ccts
Important Relations: Part ‘2’
𝑖
𝐽=
𝐴
Current density
(A/m2)
𝜑
𝐵=
𝐴
Flux density
(wb/m2)= Tesla
Series Magnetic ccts
Kirchoff voltage law
Kirchoff voltage law
Series resistances law
Series resistances law
Kirchoff Current law
Kirchoff Current law
Parallel resistances law
Parallel resistances law
Parallel Magnetic ccts
Electrical ccts
Magnetic ccts
Important Relations: Part ‘3’
saturation
i
B
1
R
  0 r
V
Electrical ccts
Current and voltage have a
“linear relationship”, the
slope of which determines
the resistance of the
electrical circuit
knee
B
Linear
H
H
Magnetization curve
(linear) (Ideal)
Magnetization curve
(actual) (non-Ideal)
Assume that A= constant
l= constant
N= constant
𝐵=
𝜑
𝐴
𝑁𝑖
𝐻=
𝑙
B is proportional to ф
H is proportional to i
Electrical ccts
Magnetic ccts
i
ф
Magnetization Characteristics
(BH curve)
WHAT DOES THAT
REALLY MEAN??!!!!!!
Check the graph to the
right. Silicon steel sheets
have higher slope than
cast iron. This means that
for the same amount of
magnetic force H, silicon
steel will produce more
magnetizing flux density B
and thus more flux ф
B2
B2 > B1
This could be very useful
if selecting cores used in
motor and transformer
applications
B1
Magnetization Characteristics (BH curve) : USES
It has been concluded
that “turning the atoms”
will require ENERGY!! This
energy must be taken
from the source, which
will lead to LOSSES!!
Magnetization Characteristics: LOSSES
1- Hystresis Losses:
•
HYSTRESIS LOSSES= The energy required to accomplish orientation of domains
during each cycle of the applied ac current to the core
The area enclosed in the hystresis loop formed by applying an AC
current to the core is directly proportional to the energy lost in a
given ac cycle. The smaller the applied MMF on the core, the smaller
the area of the resulting hystresis loop and so the smaller the
resulting losses
we can't eliminate the loss but we can reduce it to some extent by using appropriate
cores for each type of application as mentioned in the previous slide materials
with thin hysteresis have minimum hystresis losses
Losses cause
heating of core
and may cause
fatigue to
material
Magnetization Characteristics: LOSSES
2- Eddy Current:
•
EDDY CURRENT LOSSES= Induced currents in the core will cause
current to circulate in the core causing heat to the magnetic core
D
e  -N
Dt
Faraday law 1 states that if a flux passes through a turn of a coil of a wire, a voltage will
be induced in the wire that is directly proportional to the rate of change of flux with
respect to time. This “time changing flux” induces voltage WITHIN a ferromagnetic core in
just the same manner as it does in a wire wrapped around the core !!!! They act exactly like
when current passes through a resistance and causes heat losses and they depend on the
resistivity of material in which the current swirls and the size of the swirl.
we reduce eddy currents by making the core of thin laminations OR use high resistivity
material. Thin laminations will cause current swirl to be reduced, thus lower emf induced
and lower current will circulate.
Questions
• What are the sources of magnetic field?
• Demonstrate the analogy between electrical and magnetic circuits
• Explain the theory of hystresis curve
• Explain the types of losses occurring in magnetic cores and how can you
reduce them