1. Chapter 11 – Magnetic Circuits
Introductory Circuit Analysis
Robert L. Boylestad 1

2. Introduction
Magnetism is an integral part of almost every electrical device used today in industry, research, or the home
Generators, motors, transformers, circuit breakers, televisions, computers, tape recorders and telephones all employ magnetic effects to perform a variety of important tasks
Chinese sailors used compasses, as early as the second century A.D., that relied on a permanent magnet
In 1820, the Danish physicist Hans Christian Oersted discovered that a needle of a compass would deflect if brought near a current-carrying conductor 2

3. Magnetic Fields
In the region surrounding a permanent magnet there exists a magnetic field, which can be represented by magnetic flux lines similar to electric flux lines
Magnetic flux lines differ from electric flux lines in that they don’t have an origin or termination point
Magnetic flux lines radiate from the north pole to the south pole through the magnetic bar 3

4. Magnetic Fields
Continuous magnetic flux lines will strive to occupy as small an area as possible
The strength of a magnetic field in a given region is directly related to the density of flux lines in that region
If unlike poles of two permanent magnets are brought together the magnets will attract, and the flux distribution will be as shown below 4

5. Magnetic Fields
If like poles are brought together, the magnets will repel, and the flux distribution will be as shown
If a nonmagnetic material, such as glass or copper, is placed in the flux paths surrounding a permanent magnet, there will be an almost unnoticeable change in the flux distribution 5

6. Magnetic Fields
If a magnetic material, such as soft iron, is placed in the flux path, the flux lines will pass through the soft iron rather than the surrounding air because the flux lines pass with greater ease through magnetic materials than through air
This principle is put to use in the shielding of sensitive electrical elements and instruments that can be affected by stray magnetic fields 6

7. Magnetic Fields
The direction of the magnetic flux lines can be found by placing the thumb of the right hand in the direction of conventional current flow and noting the direction of the fingers (commonly called the right hand rule) 7

8. Flux Density
In the SI system of units, magnetic flux is measured in webers and has the symbol 
The number of flex lines per unit area is called the flux density, is denoted by B, and is measured in teslas
Its magnitude is determined by the following equation: 8

9. Permeability
If cores of different materials with the same physical dimensions are used in the electromagnet, the strength of the magnet will vary in accordance with the core used
The variation in strength is due to the number of flux lines passing through the core
Magnetic material is material in which flux lines can readily be created and is said to have high permeability
Permeability ()is a measure of the ease with which magnetic flux lines can be established in the material 9

10. Permeability Permeability of free space 0 (vacuum) is
Materials that have permeability slightly less than that of free space are said to be diamagnetic and those with permeability slightly greater than that of free space are said to be paramagnetic
0 = 4 x 10 -7 Wb
A • m 10

11. Permeability
Magnetic materials, such as iron, nickel, steel and alloys of these materials, have permeability hundreds and even thousands of times that of free space and are referred to as ferromagnetic
The ratio of the permeability of a material to that of free space is called relative permeability 11

12. Reluctance
The resistance of a material to the flow of charge (current) is determined for electric circuits by the equation
The reluctance of a material to the setting up of magnetic flux lines in a material is determined by the following equation l
R =  A 12

13. 11.6 - Ohm’s Law For Magnetic Circuits
For magnetic circuits, the effect is the flux 
The cause is the magnetomotive force (mmf) F, which is the external force (or “pressure”) required to set up the magnetic flux lines within the magnetic material
The opposition to the setting up of the flux  is the reluctance R
Effect =
cause
opposition 13

14. Ohm’s Law For Magnetic Circuits
Substituting
The magnetomotive force F is proportional to the product of the number of turns around the core (in which the flux is to be established) and the current through the turns of wire 14

15. Magnetizing Force
The magnetomotive force per unit length is called the magnetizing force (H)
Magnetizing force is independent of the type of core material
Magnetizing force is determined solely by the number of turns, the current and the length of the core 15

16. Hysteresis
Hysteresis curve is named from the Greek hysterein, meaning “to lag behind”
To an engineer it is important to know the curve of the flux density of B versus the magnetic force H of a material 16

17. Hysteresis
The entire curve represented by bcdefb is called the hysteresis curve
The flux density B lagged behind the magnetizing force H during the entire plotting of the curve. When H was zero at c, B was not zero but had only begun to decline. Long after H had passed through zero and had equaled to –Hd did the flux density B finally become equal to zero 17

18. 11.9 - Ampère’s Circuital Law
Ampère’s circuital law: The algebraic sum of the rises and drops of the mmf around a closed loop of a magnetic circuit is equal to zero; that is, the sum of the rises in mmf equals the sum drops in mmf around a closed loop 18

19. The Flux 
The sum of the fluxes entering a junction is equal to the sum of the fluxes leaving a junction.
a= b + c
b + c = a
both of which are equivalent 19

20. 11.11 - Series Magnetic Circuits: Determining NI
Two types of problems
 is given, and the impressed mmf NI must be computed – design of motors, generators and transformers
NI is given, and the flux  of the magnetic circuit must be found – design of magnetic amplifiers
Table method
A table is prepared listing in the extreme left-hand column the various sections of the magnetic circuit. The columns on the right are reserved for the quantities to be found for each section 20

21. 11.12 - Air Gaps Effects of air gaps on a magnetic circuit
The flux density of the air gap is given by
Where:
g = core
Ag = Acore
The permeability of air is taken to be equal to that of free space. The magnetizing force of the air gap is then determined by
And the mmf drop across the air gap is equal to Hg Ig 21

22. 11.13 - Series-Parallel Magnetic Circuits
Close analogies between electric and magnetic circuits will eventually lead to series-parallel magnetic circuits similar in many respects to electric circuits encountered previously (in Chapter 7) 22

23. Determining 
When determining magnetic circuits with more than one section, there is no set order of steps that will lead to an exact solution for every problem on the first attempt
Find the impressed mmf for a calculated guess of the flux  and then compare this with the specified value of mmf
Make adjustments to the guess to bring it closer to the actual value
For most applications, a value within 5% of the actual  or specified NI is acceptable 23

24. 11.15 - Applications Recording systems Speaker and microphones
Computer hard disks
Hall effect sensor
Magnetic reed switch
Magnetic resonance imaging 24