Magnetic Circuits Notes - Electrical Engineering (EE) PDF Download

 Page 1


 
Chapter 1  
Magnetic Circuits 
 
1.1 Introduction 
Practically all transformers and electric machinery use magnetic 
material for shaping and directing the magnetic fields which act as 
the medium for transferring and converting energy. Thus it is 
important to analyze and describe magnetic field quantities for 
understanding these devices. Magnetic materials play a big role in 
determining the properties of a piece of electromagnetic equipment 
or the electric machine and affect its size and efficiency. 
 
In electrical machines, ferromagnetic materials may form the 
magnetic circuits only (as in transformers) or by ferromagnetic 
materials in conjunction with an air medium (as in rotating 
machines). In most electrical machines, except permanent magnet 
machines, the magnetic field (or flux) is produced by passing an 
electrical current through coils wound on ferromagnetic materials. 
 
This chapter will develop some basic tools for the analysis of 
magnetic field systems and will provide a brief introduction to the 
Page 2


 
Chapter 1  
Magnetic Circuits 
 
1.1 Introduction 
Practically all transformers and electric machinery use magnetic 
material for shaping and directing the magnetic fields which act as 
the medium for transferring and converting energy. Thus it is 
important to analyze and describe magnetic field quantities for 
understanding these devices. Magnetic materials play a big role in 
determining the properties of a piece of electromagnetic equipment 
or the electric machine and affect its size and efficiency. 
 
In electrical machines, ferromagnetic materials may form the 
magnetic circuits only (as in transformers) or by ferromagnetic 
materials in conjunction with an air medium (as in rotating 
machines). In most electrical machines, except permanent magnet 
machines, the magnetic field (or flux) is produced by passing an 
electrical current through coils wound on ferromagnetic materials. 
 
This chapter will develop some basic tools for the analysis of 
magnetic field systems and will provide a brief introduction to the 
     2  Chapter One       
properties of practical magnetic materials. These results will then be 
applied to the analysis of transformers and rotating machines. So a 
carfull study for this chapter is recommended to fully understand 
the next chapters. 
 
1.2 Magnetic Field Intensity, H And Flux 
Density, B 
When a conductor carries current a magnetic field is produced 
around it, as shown in Fig.1.1. The direction of flux lines or 
magnetic field intensity H (A/m) can be determined by what is 
known as the thumb rule. 
thumb rule  
“If the conductor is held with the right hand with the thumb 
indicating the direction of current in the conductor then, the 
fingertips will indicate the direction of magnetic field intensity ”. 
 Fig.1.1 can explains Thumb rule  
 
 
 
 
 
 
 
 
I
H
Page 3


 
Chapter 1  
Magnetic Circuits 
 
1.1 Introduction 
Practically all transformers and electric machinery use magnetic 
material for shaping and directing the magnetic fields which act as 
the medium for transferring and converting energy. Thus it is 
important to analyze and describe magnetic field quantities for 
understanding these devices. Magnetic materials play a big role in 
determining the properties of a piece of electromagnetic equipment 
or the electric machine and affect its size and efficiency. 
 
In electrical machines, ferromagnetic materials may form the 
magnetic circuits only (as in transformers) or by ferromagnetic 
materials in conjunction with an air medium (as in rotating 
machines). In most electrical machines, except permanent magnet 
machines, the magnetic field (or flux) is produced by passing an 
electrical current through coils wound on ferromagnetic materials. 
 
This chapter will develop some basic tools for the analysis of 
magnetic field systems and will provide a brief introduction to the 
     2  Chapter One       
properties of practical magnetic materials. These results will then be 
applied to the analysis of transformers and rotating machines. So a 
carfull study for this chapter is recommended to fully understand 
the next chapters. 
 
1.2 Magnetic Field Intensity, H And Flux 
Density, B 
When a conductor carries current a magnetic field is produced 
around it, as shown in Fig.1.1. The direction of flux lines or 
magnetic field intensity H (A/m) can be determined by what is 
known as the thumb rule. 
thumb rule  
“If the conductor is held with the right hand with the thumb 
indicating the direction of current in the conductor then, the 
fingertips will indicate the direction of magnetic field intensity ”. 
 Fig.1.1 can explains Thumb rule  
 
 
 
 
 
 
 
 
I
H
Magnetic Circuit       3  
Fig.1.1 Field around an infinitely long, straight conductor carrying a 
current. 
Ampere’s law: 
The magnetic field intensity H around a closed contour C is 
equal to the total current passing through any surface S linking that 
contour which is known as Ampere’s law as shown in equation 
(1.1) 
?
?
= i dl H.       (1.1) 
where H is the magnetic field intensity at a point on the contour 
and dl is the incremental length at that point. 
 
Suppose that the field strength at point C distant r meters from 
the center of the conductor is H. Then it means that if a unit N-pole 
is placed at C, it will experience a force of H Newton. The direction 
of this force would be tangential to the circular line of force passing 
through C. If the unit N-pole is moved once round the conductor 
against this force, then work done, this work can be obtained from 
the following releation: 
 
r H ce dis Force Work * 2 * tan * p = =   (1.2) 
 
The relationship between the magnetic field intensity H and the 
magnetic flux density B is a property of the material in which the 
Page 4


 
Chapter 1  
Magnetic Circuits 
 
1.1 Introduction 
Practically all transformers and electric machinery use magnetic 
material for shaping and directing the magnetic fields which act as 
the medium for transferring and converting energy. Thus it is 
important to analyze and describe magnetic field quantities for 
understanding these devices. Magnetic materials play a big role in 
determining the properties of a piece of electromagnetic equipment 
or the electric machine and affect its size and efficiency. 
 
In electrical machines, ferromagnetic materials may form the 
magnetic circuits only (as in transformers) or by ferromagnetic 
materials in conjunction with an air medium (as in rotating 
machines). In most electrical machines, except permanent magnet 
machines, the magnetic field (or flux) is produced by passing an 
electrical current through coils wound on ferromagnetic materials. 
 
This chapter will develop some basic tools for the analysis of 
magnetic field systems and will provide a brief introduction to the 
     2  Chapter One       
properties of practical magnetic materials. These results will then be 
applied to the analysis of transformers and rotating machines. So a 
carfull study for this chapter is recommended to fully understand 
the next chapters. 
 
1.2 Magnetic Field Intensity, H And Flux 
Density, B 
When a conductor carries current a magnetic field is produced 
around it, as shown in Fig.1.1. The direction of flux lines or 
magnetic field intensity H (A/m) can be determined by what is 
known as the thumb rule. 
thumb rule  
“If the conductor is held with the right hand with the thumb 
indicating the direction of current in the conductor then, the 
fingertips will indicate the direction of magnetic field intensity ”. 
 Fig.1.1 can explains Thumb rule  
 
 
 
 
 
 
 
 
I
H
Magnetic Circuit       3  
Fig.1.1 Field around an infinitely long, straight conductor carrying a 
current. 
Ampere’s law: 
The magnetic field intensity H around a closed contour C is 
equal to the total current passing through any surface S linking that 
contour which is known as Ampere’s law as shown in equation 
(1.1) 
?
?
= i dl H.       (1.1) 
where H is the magnetic field intensity at a point on the contour 
and dl is the incremental length at that point. 
 
Suppose that the field strength at point C distant r meters from 
the center of the conductor is H. Then it means that if a unit N-pole 
is placed at C, it will experience a force of H Newton. The direction 
of this force would be tangential to the circular line of force passing 
through C. If the unit N-pole is moved once round the conductor 
against this force, then work done, this work can be obtained from 
the following releation: 
 
r H ce dis Force Work * 2 * tan * p = =   (1.2) 
 
The relationship between the magnetic field intensity H and the 
magnetic flux density B is a property of the material in which the 
     4  Chapter One       
field exists which is known as the permeability of the material; 
Thus,  
uH B =        (1.3) 
where u is the permeability. 
In SI units B is in webers per square meter, known as tesla (T), 
and  
In SI units the permeability of free space, Vacume or 
nonmagnetic materials is 
7
0
10 * 4
-
= p u . The permeability of 
ferromagnetic material can be expressed in terms of  its value 
relative to that of free space, or 
r
u u u *
0
= . Where 
r
u is known as 
relative permeability of the material. Typical values of 
r
u range 
from 2000 to 80,000 for feromagnetic materials used in 
transformers and rotating machines. For the present we assume 
that 
r
u is a known constant for specific material, although it 
actually varies appreciably with the magnitude of the magnetic 
flux density. 
 
Fig.1.2 shows a simple magnetic circuit having a ring-shaped 
magnetic core, called toroid, and a coil that extends around it. When 
current i flows through the coil of N turns, magnetic flux is mostly 
confined in the core material. The flux outside the toroid, called 
leakage flux, is so small that for all practical purposes it can be 
neglected. Consider a path at main radius r. The magnetic intensity 
Page 5


 
Chapter 1  
Magnetic Circuits 
 
1.1 Introduction 
Practically all transformers and electric machinery use magnetic 
material for shaping and directing the magnetic fields which act as 
the medium for transferring and converting energy. Thus it is 
important to analyze and describe magnetic field quantities for 
understanding these devices. Magnetic materials play a big role in 
determining the properties of a piece of electromagnetic equipment 
or the electric machine and affect its size and efficiency. 
 
In electrical machines, ferromagnetic materials may form the 
magnetic circuits only (as in transformers) or by ferromagnetic 
materials in conjunction with an air medium (as in rotating 
machines). In most electrical machines, except permanent magnet 
machines, the magnetic field (or flux) is produced by passing an 
electrical current through coils wound on ferromagnetic materials. 
 
This chapter will develop some basic tools for the analysis of 
magnetic field systems and will provide a brief introduction to the 
     2  Chapter One       
properties of practical magnetic materials. These results will then be 
applied to the analysis of transformers and rotating machines. So a 
carfull study for this chapter is recommended to fully understand 
the next chapters. 
 
1.2 Magnetic Field Intensity, H And Flux 
Density, B 
When a conductor carries current a magnetic field is produced 
around it, as shown in Fig.1.1. The direction of flux lines or 
magnetic field intensity H (A/m) can be determined by what is 
known as the thumb rule. 
thumb rule  
“If the conductor is held with the right hand with the thumb 
indicating the direction of current in the conductor then, the 
fingertips will indicate the direction of magnetic field intensity ”. 
 Fig.1.1 can explains Thumb rule  
 
 
 
 
 
 
 
 
I
H
Magnetic Circuit       3  
Fig.1.1 Field around an infinitely long, straight conductor carrying a 
current. 
Ampere’s law: 
The magnetic field intensity H around a closed contour C is 
equal to the total current passing through any surface S linking that 
contour which is known as Ampere’s law as shown in equation 
(1.1) 
?
?
= i dl H.       (1.1) 
where H is the magnetic field intensity at a point on the contour 
and dl is the incremental length at that point. 
 
Suppose that the field strength at point C distant r meters from 
the center of the conductor is H. Then it means that if a unit N-pole 
is placed at C, it will experience a force of H Newton. The direction 
of this force would be tangential to the circular line of force passing 
through C. If the unit N-pole is moved once round the conductor 
against this force, then work done, this work can be obtained from 
the following releation: 
 
r H ce dis Force Work * 2 * tan * p = =   (1.2) 
 
The relationship between the magnetic field intensity H and the 
magnetic flux density B is a property of the material in which the 
     4  Chapter One       
field exists which is known as the permeability of the material; 
Thus,  
uH B =        (1.3) 
where u is the permeability. 
In SI units B is in webers per square meter, known as tesla (T), 
and  
In SI units the permeability of free space, Vacume or 
nonmagnetic materials is 
7
0
10 * 4
-
= p u . The permeability of 
ferromagnetic material can be expressed in terms of  its value 
relative to that of free space, or 
r
u u u *
0
= . Where 
r
u is known as 
relative permeability of the material. Typical values of 
r
u range 
from 2000 to 80,000 for feromagnetic materials used in 
transformers and rotating machines. For the present we assume 
that 
r
u is a known constant for specific material, although it 
actually varies appreciably with the magnitude of the magnetic 
flux density. 
 
Fig.1.2 shows a simple magnetic circuit having a ring-shaped 
magnetic core, called toroid, and a coil that extends around it. When 
current i flows through the coil of N turns, magnetic flux is mostly 
confined in the core material. The flux outside the toroid, called 
leakage flux, is so small that for all practical purposes it can be 
neglected. Consider a path at main radius r. The magnetic intensity 
Magnetic Circuit       5  
on this path is H and, from Ampere's circuit law, the following 
relation can be obtained: 
?
= Ni dl H.       (1.4) 
Then Ni r H Hl = = p 2 *     (1.5) 
Where 
?
?
?
?
?
?
+
=
2
*
2
1 OD ID
r 
Where as shown in Fig.1.2 ID and OD are inner and outer 
diameter of the core of the triode. 
The quantity Ni is called the magnetomotive force (mmf), and its 
unit is Ampere-turn (At). 
i
l
N
H = At/m       (1.6) 
From Eqs. (1.3) And (1.6) 
l
uNi
B = Tesla       (1.7)