ELET 4143 Electrical Machines
and Controls
Electromagnetism
Spring 2008
Energy in an Inductor
A coils stores in its magnetic field when it carries a
current I
The energy is given by:
W=½ LI2
W = Energy stored in the coil (Joules)
L = Inductance of the coil (Henry)
I = Current (Ampres)
Energy in an Capacitor
A capacitor stores energy in its electric field
whenever a voltage E appears across its terminals
The energy is given by:
W=½ CE2
W = Energy stored in the capacitor (Joules)
C = Capacitance of the capacitor (Farads)
E = Voltage (Volts)
Magnetism concepts
ƒ Magnetic Field, B (Tesla)
ƒ Magnetic Flux, φ (Weber)
ƒ Magnetomotive Force, U (Ampere-turn)
ƒ Magnetic Field Strength, H (Ampere/meter)
ƒ Magnetic constant, μο (Henry/meter)
(or magnetic permeability)
Magnetic Field, B, and Magnetic Flux, φ
Magnetic flux, φ, is the product of the average magnetic
field, B, times the perpendicular area, A, that it penetrates.
Units:
φ Weber (Wb)
B Tesla (T)
A square meters (m2)
Magnetomotive Force
• Magnetomotive force (mmf) is any physical cause
that produces magnetic flux.
• Magnetomotive Force (mmf) is the strength of a
magnetic field in a coil of wire.
• The mmf depends on how much current flows in the
turns of coil.
• The more current, the stronger the magnetic field.
• The more turns of wire, the more concentrated the lines
of force.
mmf = U = N I
(ampere-turns)
N = number of turns
I =current
To determine the direction of the north pole:
• Using your right hand, take your four fingers and curl
them around in the direction of positive charge current
around the coils.
• Stick your thumb out and that should be the direction of
the B field inside the solenoid and also the direction of the
north pole.
Magnetic Field Strength, H
Whenever a magnetic field exists in a body or component, it is
due to the presence of a magnetic field strength H, given by:
H=U/l
H = Magnetic field strength ( ampere/meter)
U = Magnetomotive force acting on the component (ampere-turn)
l = Length of the component (meters)
The resulting magnetic field is given by:
B = φ/A
B = Magnetic Field (Tesla)
φ = Magnetic Flux (Weber)
A = Cross section area of the component ( m2)
Relationship between the magnetic field, B, and
the Magnetic Field Strength, H
B = μ o μr H
B = Magnetic field
H = Magnetic Field Strength (A/m)
μ0 = Magnetic constant or permeability of vacuum
(Henry/meter)
μr = Relative permeability of the magnetic material
μ0 = 4 π x 10 – 7 (H/m) = 1 / 800 000
The value of μr is not constant and varies with the
magnetic field in the material.
Therefore the relationship between B-H is not linear
and the equation B = μo μr H is not practical.
Instead B-H saturation curves are used.
Magnetization B-H Curves
• Consider a toroidal solenoid wound on a non-magnetic core.
• If the magnetic flux density (B) is measured on the center
line of the torioid, the relationship between B and H is the
straight line OA.
• If now the space is filled with a ferromagnetic material, the
B-H curve is OBCDE
The difference in the magnetic flux, B, between the
saturation curve and the air line, OA, at any magnetising
force, is due to the contribution of the magnetic material.
This flux is known as the intrinsic flux (relative permeability)
and gives a measure of magnetic properties.
μr ≈ 800 000 B/H
The magnetization curves can be determined using a
galvanometer or a fluxmeter.
Example
An air cored toroidal coil has 1500 turns and carries a
current of 0.2 A. The length of the coil is 300 mm and its
cross-sectional area is 800 mm2. Determine:
a) The magnetic field strength, H
b) The magnetic field, B
c) The total flux within the coil, φ
a)
H = N I / l = (1500 x0.2) / 0.3 = 1000 A T/m
b)
B = μoμr H = 4 π x 10 – 7 x 1000 = 1.25 x 10 – 3 T
c)
φ = B x A = 1.25 x 10 – 3 x 800 x 10 – 6 = 1.005 x 10– 6
=
Wb
Given a B-H saturation curve, the relative
permeability, μr, can be calculated:
μr ≈ 800 000 B/H
Faraday’s Law of Electromagnetic Induction
ƒ This law describes the fundamental relationship
between the voltage and the flux in a circuit
ƒ If a flux in a turn of coil varies as a function of time,
then a voltage is induced between tis terminals.
ƒ The value of the induced voltage is proportional to the
rate of change of the flux.
Generated Voltage from Changing φ
E=N (Δφ/Δt)
E = Induced voltage (volts)
N = Number of turns in the coil (turns)
Δφ = Change of flux inside the coil (Wb)
Δt =Time interval during which the flux changes (s)
Homework. Study the following laws
ƒ Faraday’s Law
ƒ Lenz’s Law