Magnetic Circuits - B-H Relationship (Magnetization Curve)

Summary

In 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

00:00

🧲 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.

05:04

🔌 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).

10:06

📈 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.

15:07

🔌 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.

Keywords

💡BH Relationship

The BH relationship, as discussed in the video, refers to the direct proportionality between the magnetic field intensity (H) and the magnetic field density (B) in a magnetic circuit. This relationship is fundamental in magnetic circuit analysis and design. The video explains that B equals to mu times H, where mu is the permeability of the medium. This relationship is crucial for understanding how magnetic fields are generated and controlled in various applications, such as transformers and electric machines.

💡Magnetic Field Intensity (H)

Magnetic field intensity, denoted as H, is a measure of the force exerted by an electric current that produces a magnetic field. It is defined as the number of ampere-turns per unit length of the magnetic circuit. In the context of the video, H is used to describe the strength of the magnetic field produced by a current in a coil. The script mentions that increasing the magnetizing current (I) will linearly increase the magnetic field intensity, which in turn increases the magnetic field density (B).

💡Magnetic Field Density (B)

Magnetic field density, represented by B, is a measure of the strength of the magnetic field and is expressed in webers per square meter. The video explains that B is directly proportional to the magnetic field intensity (H) and the permeability of the medium (mu). It is a key parameter in magnetic circuit design as it indicates the amount of magnetic field that can be produced for a given amount of current excitation.

💡Permeability (mu)

Permeability (mu) is a property of a material that describes its ability to support the formation of a magnetic field. It is defined as the ratio of the magnetic field density (B) to the magnetic field intensity (H). The video discusses how permeability is a characteristic of the medium and is inversely proportional to the reluctance of the material to the magnetic field. The permeability of free space (mu naught) is a constant value, while the relative permeability (mu r) varies with different materials.

💡Reluctance

Reluctance is a term used to describe the opposition that a material offers to the formation of a magnetic field. It is analogous to electrical resistance in a circuit. The video mentions that materials with high permeability have low reluctance, meaning they are more conducive to the flow of the magnetic field. Reluctance is inversely proportional to permeability, as explained in the context of the BH relationship.

💡Magnetization Curve

The magnetization curve is a graphical representation of the relationship between the magnetic field intensity (H) and the magnetic field density (B) for a magnetic material. The video describes how this curve initially shows a linear relationship at low values of H, indicating that B increases linearly with H. However, at higher values of H, the curve becomes non-linear, indicating the onset of magnetic saturation where the material can no longer support an increase in B with further increases in H.

💡Saturation

Saturation in the context of magnetic materials refers to the point at which the material can no longer increase its magnetic field density (B) significantly, even with an increase in magnetic field intensity (H). The video explains that at high values of H, the change in B becomes non-linear, indicating that the material has reached its saturation point. This is an important consideration in magnetic circuit design to ensure efficient operation and avoid excessive current requirements.

💡Magnetizing Current (I)

The magnetizing current is the current that flows through the windings of a magnetic circuit to establish the magnetic field. The video emphasizes that this current is essential for creating the magnetic field in devices like transformers and electric machines. It is usually a portion of the total current drawn by the machine, and its primary role is to excite the magnetic field, which is then used to perform work or transfer energy.

💡Demagnetization

Demagnetization is the process of reducing or eliminating the residual magnetism in a magnetic material. The video describes how, when the magnetizing current is reduced to zero, the aligned dipoles in the material return to a random distribution, effectively demagnetizing the material. This process is important for understanding how magnetic materials can be reset after use in devices such as motors and transformers.

💡Electrical Machines

Electrical machines, such as motors and generators, are devices that convert electrical energy into mechanical energy or vice versa. The video discusses how these machines rely on the principles of magnetic circuits, including the BH relationship, to function effectively. The design of the magnetizing curve is crucial for these machines to ensure they operate within their linear region for efficiency and to minimize losses due to saturation.

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