Unit-1
Chapter 1.1 Magnetic Fields and Magnetic Circuits
LECTURE TOPICS:
1.1.1 Review of magnetic circuits - MMF, flux, reluctance, inductance
1.1.2 Review of Ampere Law and Biot Savart Law
1.1.3 Visualization of magnetic fields produced by a bar magnet and a current
carrying coil - through air and through a combination of iron and air
1.1.4 Influence of highly permeable materials on the magnetic flux lines
1.1 .1 Lecture topic
Review of magnetic circuits - MMF, flux, reluctance, inductance:
Web link: http://ocw.nthu.edu.tw/ocw/upload/124/news/Saadany_Magnetic%20Circuits.pdf
Video link: https://www.youtube.com/watch?v=q-nTad8SYOw
Electric machines and transformers have electric circuits and magnetic circuits
interlinked through the medium of magnetic flux. Electric current flow through the
electric circuits, which are made up of windings. On the other hand, magnetic fluxes
flow through the magnetic circuits, which are made up of iron cores. The interaction
between the currents and the fluxes is the basic of the electromechanical energy
conversion process that takes place in generators and motors. However, in
transformers it is more feasible to think about the process in terms of an energy
transfer. In transformers, the energy transfer is normally associated with change in
voltage and current levels. Thus, magnetic circuits play an essential role.
The magnetic flux is produced due to the flow of a current in a wire (electric magnet).
The direction of the produced magnetic flux is determined by “the right-hand rule”
as shown in Fig. 1.
Fig. 1 The Right-hand rule for magnetic flux
The unit for the flux Φ is (weber) and the magnetic flux density B is given as:
Φ
𝐴
B=
Wb/m2 (Tesla). The magneto-motive force mmf is the ability of a coil to produce
magnetic flux. The mmf unit is Amp-turn and is given by: mmf = NI (AT). The
magnetic flux intensity H is the mmf per unit length along the path of the flux and is
given by: 𝐻 =
𝑚𝑚𝑓
𝑙
AT/m, where l is the mean or average path length of the
magnetic flux in meters.
The relation between the mmf and the flux is governed by the system reluctance ℜ,
such that mmf = ℜ φ, where the reluctance is given by ℜ =
𝜌𝑙
𝐴
, where
l = The average length of the magnetic core (m)
A = The cross section area (m2 )
μ = The permeability of the material (AT/m2 )
The permeability of the material is given by
μ = μ0μr
where: μ0 is the permeability of air and μr is the relative permeability.
From the above relationships, we can conclude that:
𝑙
Φ 𝑚𝑚𝑓/𝑅 (𝐻𝑙)/(𝜇𝐴)
𝐵= =
=
= 𝜇𝐻
𝐴
𝐴
𝐴
The relation B-H is known as the magnetization characteristics of the material and
is broken to three different regions: Linear, knee and saturation as shown in Fig. 2.
Fig. 2 Magnetization Curve
2. Analogy between magnetic and electric circuits
3. Magnetic Circuit Analysis:
In order to analyze any magnetic circuit, two steps are mandatory as illustrated by
Fig. 3:
• Step #1: Find the electric equivalent circuit that represents the magnetic circuit.
• Step #2: Analyze the electric circuit to solve for the magnetic circuit quantities.
Fig. 3 Magnetic circuit analysis
Inductances:
The inductance (in Henry) is given by:
𝐿=
𝜆 𝑁∅
=
𝐼
𝐼
Since: 𝐹 = 𝑁𝐼 = ∅𝑅,
and ℜ =
𝜌𝑙
𝐴
Therefore,
𝜆 𝑁∅ 𝑁 2 𝑙𝜇𝐴 𝑁 2
𝐿= =
=
=
𝐼
𝐼
𝐼𝑙
ℜ
𝐿11 =
𝜆11
= 𝑠𝑒𝑙𝑓 𝑖𝑛𝑑𝑢𝑐𝑡𝑎𝑛𝑐𝑒
𝐼1
𝐿21 =
𝜆21
= 𝑚𝑢𝑡𝑢𝑎𝑙 𝑖𝑛𝑑𝑢𝑐𝑡𝑎𝑛𝑐𝑒
𝐼1
where: 𝜆11 is the total flux linking coil 1 due to 𝐼1 when 𝐼2 and 𝜆21 is the total flux
linking coil 2 due to 𝐼1 when 𝐼2 =0
𝐿11
𝑁1 2
=
,
𝑅
𝐿22
𝑁2 2
=
,
𝑅
𝐿21 =
𝑁1 𝑁2
= 𝐿12
𝑅
1.1.2 Lecture Topic
Review of Ampere Law and Biot Savart Law:
Web link:
https://ciet.nic.in/moocspdf/Physics03/leph_10403_eContent2020.pdf
https://ciet.nic.in/moocspdf/Physics03/leph_10402_eContent2020.pdf
Video link: https://www.youtube.com/watch?v=JHNloU9Rfow
https://www.youtube.com/watch?v=1kydon2HxQA
You will recall that electric fields and magnetic fields might seem different, but
they're actually part of one larger force called the electromagnetic force. Charges
that aren't moving produce electric fields. But when those charges do move, they
instead create magnetic fields. Charges moving in an electric wire also produce
magnetic fields. If we move a compass near to an electric wire, the compass needle
changes direction or deflects.
The Biot-Savart Law is a mathematical expression that describes the magnetic field
created by a current carrying wire, and allows you to calculate its strength at
various points.
To derive this law, we first take this equation for electric field. This is the full
version, where we use µ0/4π instead of the electrostatic constant k. Since we're
looking at a wire, we replace the charge q with Idl, which is the current in the wire,
multiplied by a length element in the wire. Basically, it's treating this little chunk of
the wire as our charge. And we also replace the electric field E with a magnetic field
element dB because a moving charge produces a magnetic field, not an electric field.
Last of all, we have to realize that a current has a direction (unlike a charge). So, we
need to make sure the direction of the current affects our result. We do that by adding
sine of the angle between the current and the radius. That way, if the wire is curvy,
we'll take that into account. And that's it - that's the Biot-Savart law.
Fig. 3 Biot-Savart law
The magnetic field dB due to this element is to be determined at a point P which is
at a distance r from it. Let θ be the angle between dl and the displacement vector r.
According to Biot-Savart’s law, the magnitude of the magnetic field dB is
proportional to the current I, the element length |dl|, and inversely proportional to
the square of the distance r.
Its direction is perpendicular to the plane containing dl and r.
Thus, in vector notation:
𝑑𝐵 =
𝑑𝐵 =
𝐼𝑑𝑙 × 𝑟
𝑟3
𝜇0 𝐼𝑑𝑙 × 𝑟
4𝜋 𝑟 3
Direction of the field is given by Right hand grip rule.
AMPERE'S CIRCUITAL LAW:
This law, given by Ampere, provides us with an alternative way of calculating the
magnetic field due to a given current distribution.
This law, is, in a way, similar to the Gauss’s law in electrostatics, which again
provides us with an alternative way of calculating the electric field due to a given
charge distribution.
Ampere’s circuital law can be written as:
The line integral of the magnetic field around some closed loop is equal to μ0
times the algebraic sum of the currents which pass through the loop.
So, let us attempt to understand what is meant by:
 line integral  closed loop  algebraic sum of currents
Fig. 4 Magnetic field around a current carrying conductor
The figure shows the magnetic field around a current carrying conductor. From our
previous knowledge we take the conventional direction of current (from +ve to –ve),
the red concentric circles represent the magnetic field in a plane perpendicular to the
wire.
From Biot Savart’s Law, B is inversely proportional to the square of the distance
from the source to the point of interest.
The source of the field is a vector given by Idl or the magnetic field is produced by
a vector source Idl.
The magnetic field is perpendicular to the plane containing the displacement vector
r and the current element Idl.
Amperes’ circuital law makes the calculation of B easier in many cases:
Consider a long straight current carrying wire encircled by magnetic field lines and
imagine travelling around a closed path that also encircles the wire.
Ampere’s law relates the magnetic field along the path to the electric current
enclosed by this path.
Let us travel along the path taking steps of length ∆l and let B be the component of
the magnetic field parallel to these steps.
According to ampere’s law over the entire closed loop, which we have taken as a
circular loop.
∑ 𝐵. Δ𝐼 = 𝜇𝑜 𝐼
1.1.3 Lecture Topic
Visualization of magnetic fields produced by a bar magnet and a current
carrying coil - through air and through a combination of iron and air
Web link: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html#c1
Video link: https://www.youtube.com/watch?v=nfSJ62mzKyY
Bar Magnet
The lines of magnetic field from a bar magnet form closed lines. By convention, the
field direction is taken to be outward from the North pole and in to the South pole of
the magnet. Permanent magnets can be made from ferromagnetic materials.
As can be visualized with the magnetic field lines, the magnetic field is strongest
inside the magnetic material. The strongest external magnetic fields are near the
poles. A magnetic north pole will attract the south pole of another magnet, and repel
a north pole.
As can be visualized with the magnetic field lines, the magnetic field is strongest
inside the magnetic material. The strongest external magnetic fields are near the
poles. A magnetic north pole will attract the south pole of another magnet, and
repel a north pole.
Fig. 6 Magnetic field of a bar magnet
The magnetic field lines of a bar magnet can be traced out with the use of a compass.
The needle of a compass is itself a permanent magnet and the north indicator of the
compass is a magnetic north pole. The north pole of a magnet will tend to line up
with the magnetic field, so a suspended compass needle will rotate until it lines up
with the magnetic field. Unlike magnetic poles attract, so the north indicator of the
compass will point toward the south pole of a magnet. In response to the Earth's
magnetic field, the compass will point toward the geographic North Pole of the Earth
because it is in fact a magnetic south pole. The magnetic field lines of the Earth enter
the Earth near the geographic North Pole.
Fig. 7 Magnetic field using compass
Electric and Magnetic Sources
The electric field of a point charge is radially outward from a positive charge.
Electric sources are inherently "monopole" or point charge sources.
The magnetic field of a bar magnet.
Magnetic sources are inherently dipole sources - you can't isolate North or South
"monopoles".
Bar Magnet and Solenoid
The magnetic field produced by electric current in a solenoid coil is similar to that
of a bar magnet.
The magnetic field lines can be thought of as a map representing the magnetic
influence of the source object in the space surrounding it. The properties of the
magnetic field lines are can be summarized by:
1. The direction of the magnetic field is tangent to the magnetic field line at any
point in space.
2. The strength of the magnetic field is visualized by the closeness of the lines
to each other. It is proportional to the number of lines per unit area
perpendicular to the lines. a commonly used phrase is "magnetic flux density".
3. Magnetic field lines never cross. The magnetic field at any point is unique.
4. Magnetic field lines are continuous, forming closed loops without beginning
or end.
Iron Core Solenoid
An iron core has the effect of multiplying greatly the magnetic field of a solenoid
compared to the air core solenoid on the left.
Electromagnet:
Electromagnets are usually in the form of iron core solenoids.
The ferromagnetic property of the iron core causes the internal magnetic domains of
the iron to line up with the smaller driving magnetic field produced by the current in
the solenoid. The effect is the multiplication of the magnetic field by factors of tens
to even thousands. The solenoid field relationship is
𝐵 = 𝑘𝜇0 𝑛𝐼
𝑤ℎ𝑒𝑟𝑒 𝜇 = 𝑘𝜇0
and k is the relative permeability of the iron, shows the magnifying effect of the iron
core.
1.1.4 Lecture Topic
Influence of highly permeable materials on the magnetic flux lines
Web link: http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/
Video link: https://study.com/academy/lesson/what-is-magnetic-permeability-definitionexamples.html
Permeability
Quantity name
permeability
alias absolute permeability
Quantity symbol
μ
Unit name
henrys per metre
Unit symbols
H m-1
Base units
kg m s-2 A-2
Duality with the Electric World
Quantity
Unit
Formula
Permeability henrys per metre μ = L/d
Permittivity farads per metre ε = C/d
Although, magnetic permeability is related in physical terms most closely to electric
permittivity, it is probably easier to think of permeability as representing
'conductivity for magnetic flux'; just as those materials with high electrical
conductivity let electric current through easily so materials with high permeabilities
allow magnetic flux through more easily than others. Materials with high
permeabilities include iron and the other ferromagnetic materials. Most plastics,
wood, non-ferrous metals, air and other fluids have permeabilities very much
lower: μ0.
Just as electrical conductivity is defined as the ratio of the current density to the
electric field strength, so the magnetic permeability, μ, of a particular material is
defined as the ratio of flux density to magnetic field strength μ=B/H
This information is most easily obtained from the magnetization curve. Figure shows
the permeability (in black) derived from the magnetization curve (in color) using
above equation. Note carefully that permeability so defined is not the same as the
slope of a tangent to the B-H curve except at the peak (around 80 A m-1 in this case).
The latter is called differential permeability, μ′ = dB/dH.
In ferromagnetic materials the hysteresis phenomenon means that if the field
strength is increasing then the flux density is less than when the field strength is
decreasing. This means that the permeability must also be lower during 'charge up'
than it is during 'relaxation', even for the same value of H. In the extreme case of a
permanent magnet the permeability within it will be negative. There is an analogy
here with electric cells, since they may be said to have 'negative resistance'.
If you use a core with a high value of permeability then fewer turns will be required
to produce a coil with a given value of inductance. For a given core B is proportional
to flux and H is proportional to the current so that inductance is also proportional to
μ: the ratio of B to H.
Unlike electrical conductivity, permeability is often a highly non-linear quantity.
Most coil design formula, however, pretend that μ is a linear quantity. If you were
working at a peak value of H of 100 A m-1, for example, then you might take an
average value for μ of about 0.006 H m-1. This is all very approximate, but you must
accept inaccuracy if you insist on treating a non-linear quantity as though it was
actually linear.
This form of permeability, where μ is written without a subscript, is known
in SI parlance as absolute permeability. It is seldom quoted in engineering texts.
Instead a variant is used called relative permeability described next.
μ = μ0 × μr
Relative permeability
Quantity name
Relative permeability
Quantity symbol
μr
Unit symbols
dimensionless
Relative permeability is a very frequently used parameter. It is a variation upon
'straight' or absolute permeability, μ, but is more useful to you because it makes
clearer how the presence of a particular material affects the relationship between flux
density and field strength. The term 'relative' arises because this permeability is
defined in relation to the permeability of a vacuum, μ0
μr = μ / μ0
For example, if you use a material for which μr = 3 then you know that the flux
density will be three times as great as it would be if we just applied the same field
strength to a vacuum. This is simply a more user-friendly way of saying that
μ
-6
-1
= 3.77×10 H m . Note that because μr is a dimensionless ratio that there are no
units associated with it.
Many authors simply say "permeability" and leave you to infer that they mean
relative permeability. In the CGS system of units these are one and the same thing
really. If a figure greater than 1.0 is quoted then you can be almost certain it is μ r.
Material
μ/(H m-1) μr
Application
Ferrite U 60
1.00E-05 8
UHF chokes
Ferrite M33
9.42E-04 750
Resonant circuit RM cores
Nickel (99% pure)
7.54E-04 600
-
Ferrite N41
3.77E-03 3000
Power circuits
Iron (99.8% pure)
6.28E-03 5000
-
Ferrite T38
1.26E-02 10000
Broadband transformers
Silicon GO steel
5.03E-02 40000
Dynamos, mains transformers
superalloy
1.26
1000000 Recording heads
Approximate maximum permeabilities
Note that, unlike μ0, μr is not constant and changes with flux density. Also, if the
temperature is increased from, say, 20 to 80 centigrade then a typical ferrite can
suffer a 25% drop in permeability. This is a big problem in high-Q tuned circuits.
Another factor, with steel cores especially, is the microstructure, in particular grain
orientation. Silicon steel sheet is often made by cold rolling to orient the grains along
the laminations (rather than allowing them to lie randomly) giving increased μ. We
call such material anisotropic.
Before you pull any value of μ from a data sheet ask yourself if it is appropriate for
your material under the actual conditions under which you use it. Finally, if you do
not know the permeability of your core then build a simple circuit to measure it.