Magnetic Circuits - B-H Relationship (Magnetization Curve)
Summary
TLDRIn this lecture, the BH relationship in magnetic circuit analysis is explored, detailing how magnetic field intensity (H) generates magnetic field density (B). The permeability (μ) of a material, which influences its magnetic field resistance, is discussed, with μ being the product of μ₀ (free space permeability) and μr (relative permeability). The lecture covers the linear and non-linear behaviors of B with respect to H, using the magnetization curve to illustrate the magnetic material's response to varying H. Practical applications, such as motor design and the importance of operating within the linear region of the magnetization curve for efficiency, are highlighted.
Takeaways
- 🧲 The BH relationship is fundamental in magnetic circuit analysis, where B (magnetic field density) is directly proportional to H (magnetic field intensity) through the permeability µ.
- 🌐 Permeability µ is the influence of a material on the magnetic field and is inversely proportional to the material's reluctance to magnetic field flow.
- 🔗 The permeability µ can be expressed as µ = µ₀ * µ_r, where µ₀ is the permeability of free space, and µ_r is the relative permeability of the medium.
- 🌌 µ₀, the permeability of free space, is a constant value of 4π × 10^-7 H/m.
- 🛠 In electrical machines, a high µ_r value implies that a small electric current can produce a large magnetic field density, which is crucial for efficient operation.
- 🔋 The magnetic field intensity H can also be referred to as the magnetic field excitation, which is the driving force behind establishing a magnetic field.
- 🔗 Ampere's Law is applied in magnetic circuits to relate the total ampere turns to the magnetic field intensity and permeability.
- 📉 The BH relationship for non-magnetic materials like air, aluminum, and copper is linear with a slope equal to the permeability of free space µ₀.
- 📈 For magnetic materials, the BH relationship is initially linear but becomes non-linear at higher magnetic field intensities due to saturation effects.
- 🔧 The shape of the magnetization curve explains the behavior of magnetic materials under varying magnetic field intensities, showing linearity at low fields and saturation at high fields.
- ⚙️ In the design of electrical machines, it's important to operate within the linear region of the magnetization curve to ensure efficiency and avoid excessive current draw.
Q & A
What is the BH relationship in magnetic circuits?
-The BH relationship refers to the relationship where the magnetic field intensity H produces a magnetic field density B in any medium. It is expressed as B equals to μ times H, where μ is the permeability of the medium.
What is the unit of magnetic field density B?
-The unit of magnetic field density B is weber per meter squared (Wb/m²).
How is permeability (μ) related to the reluctance of a material to the magnetic field?
-Permeability (μ) is inversely proportional to the reluctance of the material to the magnetic field. High permeability means low reluctance, indicating the material offers less resistance to the magnetic field.
What is the permeability of free space (μ₀) and its unit?
-The permeability of free space (μ₀) is defined as 4 times pi times 10 to the power of -7 henry per meter (H/m).
What is relative permeability (μ_r) and what does its value imply for a material?
-Relative permeability (μ_r) is a dimensionless quantity that represents the ability of a material to support the formation of a magnetic field compared to the ability of free space. A high μ_r value implies that a small electric current can produce a large amount of magnetic field density in the material.
How does the BH relationship change for non-magnetic materials?
-For non-magnetic materials like air, aluminum, plastic, wood, and copper, the relative permeability (μ_r) is unity, so the permeability (μ) equals μ₀, and the magnetic field density B equals μ₀ times H.
What is the significance of the magnetizing current (i) in the BH relationship?
-The magnetizing current (i) is the variable used to produce the BH relationship and is used to excite and establish the magnetic field density. It is a key factor in determining the magnetic field intensity H and, consequently, the magnetic field density B.
What is the shape of the magnetization curve for magnetic materials, and what does it indicate?
-The magnetization curve for magnetic materials shows an almost linear increase in magnetic field density B at low magnetic field intensity H, but becomes non-linear at higher values of H, indicating the onset of saturation where the material's permeability decreases.
What happens to the magnetic dipoles in a material when the magnetizing current is increased?
-As the magnetizing current is increased, the magnetic field intensity also increases, causing more dipoles to align with the magnetic field, which in turn increases the magnetic field density B.
Why is it important for designers to keep the operating points within the linear region of the magnetization curve?
-Designers aim to keep the operating points within the linear region of the magnetization curve to avoid the non-linear or saturation region, which requires too much current to achieve only a small increase in magnetic field density, leading to inefficiency and unnecessary losses.
How does an AC source affect the magnetization curve in a magnetic circuit?
-When an AC source is applied to a magnetic circuit, the full cycle of the magnetization curve, including both positive and negative sides, is repeated multiple times per second depending on the system frequency, such as 60 times per second for a 60 Hz system.
Outlines
🧲 Magnetic Field Intensity and Density Relationship
This paragraph introduces the BH relationship in magnetic circuit analysis, which describes how magnetic field intensity (H) generates magnetic field density (B). The relationship is given by B = μH, where μ (permeability) is the product of μ₀ (permeability of free space) and μ_r (relative permeability of the medium). The permeability is inversely proportional to the reluctance of the material, indicating how much a material resists the magnetic field. The paragraph also discusses how selecting materials with high permeability can result in a greater magnetic field density for the same amount of magnetic field excitation. The permeability of free space (μ₀) is a constant, and the relative permeability (μ_r) varies with different materials, significantly affecting the magnetic field density.
🔌 Derivation of BH Relationship and Magnetic Circuit Analysis
The paragraph delves into the derivation of the BH relationship for a magnetic circuit comprising a winding and core. It applies Ampere's Law to establish the relationship between the total ampere turns, the air gap, and the magnetic material. The assumption is made that the ampere turns in the air gap are significantly higher than those in the magnetic material, simplifying the law. The magnetic field intensity (H) is calculated as the number of turns (N) times the current (I) divided by the length of the air gap (Lg). The magnetic field density (B) is then expressed as μ₀μ_rN/L, where μ₀ is the permeability of free space and μ_r is the relative permeability of the material. The paragraph explains how the BH relationship can be graphically represented, showing that an increase in the magnetizing current (I) leads to a linear increase in magnetic field density (B), with the slope determined by the permeability of the air (μ₀).
📈 Magnetization Curve and Saturation in Magnetic Materials
This section explains the shape of the magnetization curve, which describes the relationship between magnetic field intensity (H) and magnetic field density (B) in magnetic materials. At low excitation levels, the curve is almost linear, indicating that the permeability is high and the reluctance to the magnetic field is low. However, as the magnetic field intensity increases, the change in magnetic field density becomes non-linear, indicating the onset of saturation. The paragraph also discusses the behavior of magnetic materials under varying current conditions, explaining how increasing current aligns more dipoles with the magnetic field, leading to higher magnetic field density. Conversely, reducing the current demagnetizes the material, causing the dipoles to return to a random distribution.
🔌 Design Considerations for Rotating Electrical Machines
The final paragraph discusses the practical implications of the BH relationship and magnetization curve in the design of rotating electrical machines and transformers. It emphasizes the importance of operating within the linear region of the magnetization curve to maintain high efficiency and avoid unnecessary losses. The total current drawn by the motor is divided into the magnetizing current, which establishes the magnetic field, and the load current, which is converted into mechanical energy. The magnetizing current typically constitutes 5 to 20 percent of the machine's total current. The paragraph also notes that in AC systems, the magnetization curve is cycled multiple times per second, depending on the system's frequency.
Mindmap
Keywords
💡BH Relationship
💡Magnetic Field Intensity (H)
💡Magnetic Field Density (B)
💡Permeability (mu)
💡Reluctance
💡Magnetization Curve
💡Saturation
💡Magnetizing Current (I)
💡Demagnetization
💡Electrical Machines
Highlights
Introduction to the BH relationship in magnetic circuit analysis, linking magnetic field intensity (H) to magnetic flux density (B).
BH relationship states that magnetic field intensity (H) produces magnetic flux density (B), defined by B = μ * H, where μ is the permeability of the medium.
Permeability (μ) is the property of a material that affects the magnetic field. It is inversely proportional to the reluctance of the material.
μ₀, the permeability of free space, is defined as 4π × 10⁻⁷ Henry per meter.
Relative permeability (μr) varies depending on the material; it is 1 for free space and ranges from 2000 to 6000 for materials used in electrical machines.
High relative permeability (μr) means that a small electric current can generate a large magnetic flux density in materials like iron or steel.
In non-magnetic materials such as air, aluminum, plastic, wood, and copper, μr equals 1, meaning μ = μ₀ and B = μ₀ * H.
Using Ampere’s law, the relationship between magnetizing current (I) and magnetic flux density (B) is derived for both air gaps and magnetic materials.
The magnetic field density (B) increases linearly with increasing magnetizing current (I) in non-magnetic materials.
In magnetic materials (e.g., iron, cobalt, steel), the relationship becomes non-linear at higher magnetic field intensities due to saturation effects.
The BH curve, or magnetization curve, shows linear behavior at low H values but becomes non-linear as H increases due to saturation of the magnetic material.
Magnetic materials show saturation, where further increases in H result in only small increases in B, due to alignment of magnetic domains reaching a limit.
Demagnetization occurs when magnetizing current (I) is reduced, causing the alignment of magnetic domains to revert to random orientation.
When designing electric machines, keeping the operating points within the linear region of the BH curve avoids unnecessary energy loss and inefficiency.
The magnetizing current (I) is typically 5–20% of a machine’s total current, depending on the machine type, used to establish the magnetic field for energy conversion.
Transcripts
welcome back to the energy conversion
lectures
in this lecture i will review and
discuss another important relationship
in magnetic circuit analysis and design
this relationship is the bh relationship
the bh relationship stated that
the magnetic field intensity h
produces a magnetic field density b
everywhere it exists
or at any medium it exists
these two magnetic field variables
are related to each other
as follows
b equals to mu times h
and the unit is weber
over meter square
mu is equal to mu naught times mu r
basically
if we choose a proper magnetic material
represented by mu
you could have more magnetic field
density b
for the same amount
of magnetic field excitation h
as mentioned in previous lectures
the magnetic field intensity h
can also be called magnetic field
excitation
mu is defined as a characteristics of
the medium
and it is called permeability of the
medium
permeability can also be defined
as the influence of the material on the
magnetic field
basically
if the material has high permeability
means that the material has low
reluctance to the magnetic field
in other words
the permeability mu
is inversely proportional to the
reductance of the material
for example
if permeability is high
the reluctance is low
you can think about the reluctance
as a resistance to the flow of magnetic
field
mu naught is defined as a permeability
of the free space and it is equal to
4 times pi times 10 to the power -7
and the unit is henry over meter
mu r
is defined as a relative permeability of
the medium
for instance
mu r of the free space is equal to unity
and mu r of the material used in
electrical machines is varies in the
range between 2000 to 6000
the large value of mu r
implies that a small electric current
can produce a large amount
of magnetic field density in the machine
let's study and see how the ph
relationship behaves
in case of different materials
let's starts
with non-magnetic material
in case of non-magnetic material such as
air aluminum plastic wood and copper
the value of mu r is unity
and therefore
mu will be equal to mu naught
and the magnetic field density b
will be equal to mu naught times h
and the unit is weber over meter square
now let's derive the bh relationship
for a magnetic circle that consists of
winding and core as shown
first let's apply ampere's law
to this magnetic circuit as follows
basically this equation shows that the
total ampere turns an i
is equals to the air gap and paired
turns sglg
plus the magnetic material ampere turns
at c lc
for more details
please review ampere's law
lecture
now let's assume that impaired turns of
the air gap hglg
is much higher than the impaired turns
of the magnetic material at clc
this is a valid assumption and we will
learn more about it later during the air
gap lecture
based on this assumption
sclc can be ignored
and the ampere's law
for the magnetic circuit can be
simplified
as follows
by arranging this equation
the magnetic field intensity hg
will be equal to ni over l g
and the unit is ampere turns over meter
by substituting equation two into
equation 1
the magnetic field density b
will be equal to
mu n i over lg
and this would equal to
mu naught and i
over lg and the unit is weber over meter
square
if we assume that the quantities lg
n
and mu
are constant and already pre-designed
and selected
then the only variable used to produce
the ph relationship is the electric
excitation current i
the current i
is called the magnetizing current
basically
this current is used to excite
and establish the magnetic field density
by using equations 2 and 3
the bh relationship
can be drawn as follows
it is very clear that if the magnetic
field intensity hg
is increased by increasing the
magnetizing current i
the magnetic field density
will be increased linearly
the slope or the angle of this linear
relationship is equal to the
permeability of the air mu naught
let's move on to the magnetic material
case
in case of a magnetic material
such as iron cobalt
nickel steel and ferrite
the value of the relative permeability
mu r
of the magnetic materials varies from
several hundreds to several thousands
therefore
the magnetic field density b
will be equal
to mu naught times mu r times h
and the unit is weber over meter square
now
let's derive the bh relationship for a
magnetic circuit that consists of
winding and core as shown
first let's apply ampere's law
to this magnetic circuit as follows
by arranging this equation
the magnetic field intensity h
will be equal to
n i over l
and the unit is ampere turns
over meter
now by substituting equation 5 into
equation 4
the magnetic field density
b
will be equal to mu naught mu r n i over
l
and the unit is weber over meter square
based on equation 5 and equation 6
the bh relationship can be drawn as
shown
it is very clear
that if the magnetic field intensity h
is increased by increasing the
magnetizing current i the magnetic field
density
b
is increased as well
this ph curve is called the
magnetization curve
as you can see the magnetic field
density b
increases almost linearly in the region
of low value of magnetic field intensity
h
however
at higher value of magnetic field
intensity h
the change of the magnetic field density
b
is non-linear
in other words
the magnetic material shows the effect
of saturation at high magnetic field
intensity h
let's explain in details
why the magnetization curve has this
shape
first let's take a small piece of the
core and zoom it in
when there is no current
the magnetic material has random
arrangement of dipoles or domains
once the magnetizing current i
is slowly increasing in a small value
the magnetic field intensity h
will also increase in a small value
based on this
low value of excitation
part of the dipoles will be aligned with
the magnetic field
this will increase the magnetic field
density b
as you can see
the magnetic field density b
increases almost linearly in the region
of low values of magnetic field
intensity h
in the region of low excitation such as
h1 and h2
the magnetic material permeability mu
has high value
and therefore
the reluctance to the magnetic field is
low
now
as the current and therefore the
magnetic field intensity h
start to increase higher and higher
only a few of the dipoles align with the
magnetic field
and the magnetic material starts showing
higher reluctance
or we can say the permeability becomes
smaller
now if we start to reduce the
magnetizing current i
the magnetic field density b
will reduce accordingly
and eventually the dipole becomes random
distribution when the current is zero
this process called
demagnetization of magnetic material
in other words
the process of forcing the aligned
dipoles
of the magnetic material to be unaligned
is called
demagnetization
if we apply negative current the
magnetic material dipoles will be
aligned in opposite direction until it
reaches the saturation
but in opposite direction
basically
the negative side
is just equivalent to the concept of
what i explained
in the positive side but in opposite
direction
you probably got bored from all the
equations and the mathematical formulas
therefore let's talk about information
that you need to always remember
assume we have a motor controlling a
mechanical load
as shown
the motor converts
the electrical energy into rotating
mechanical energy
only after storing some energy as a
magnetic field
this stored energy
is represented by the shaded area of the
magnetization
curve
the magnetizing current i
is used to establish the magnetic field
now
we need to focus on this point
the total current drawn by the motor is
divided into two parts
the magnetizing current i
and the load current
the magnetizing current is required to
establish the magnetic field
and the load current will be converted
to mechanical energy to support the load
the magnetizing current is usually
between 5 to 20 percent of the machine
total current
depending on the type and size of the
machine
designing the magnetizing curve
is an important part during the design
of any rotating electrical machine or
transformer
usually any designer trying to keep
the operating points
within the linear region of the
magnetization curve or a little bit
further to avoid any unnecessary loss
and keep high efficiency design
in other words
we need to avoid the non-linear or the
saturation region of the magnetization
curve because it requires too much
current
to achieve only a small amount of
magnetic field density
another important note to mention here
is that if we apply
an ac source to a magnetic circuit
the full cycle
or the positive and negative sides of
the magnetization curve
will be repeated 60 times per second in
case of 60 hertz system
and 50 times per second
in case of 50 hertz system
let's conclude this lecture at this
point and we'll continue in the next
lecture
thanks for listening i am essan and nabi
and it was a pleasure sharing this
lecture with you thank you
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