1. Chapter 12– 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. 2

3. 12.2 – Magnetic Field Flux and Flux Density
In the SI system of units, magnetic flux is measured in webers (Wb) and is represented using the symbol .
The number of flux lines per unit area is called flux density (B). Flux density is measured in teslas (T).
Its magnitude is determined by the following equation: 8

4. Magnetic Fields 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

5. 12.3 – 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 12

6. 12.4 – 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 . 13

7. 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

8. 12.5 – 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

9. 12.6 – Hysteresis
Hysteresis – The lagging effect between the flux density of a material and the magnetizing force applied.
The curve of the flux density (B) versus the magnetic force (H) is of particular interest to engineers. 16

10. Hysteresis The entire curve (shaded) 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

11. 12.7 – 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.
F = 0 18

12. 12.8 – Flux 
The sum of the fluxes entering a junction is equal to the sum of the fluxes leaving a junction.
a= b + c
or b + c = a
both of which are equivalent 19

13. 12.9 – 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

14. Effects of air gaps on a magnetic circuit
The flux density of the air gap is given by
where g = core
Ag = Acore
Assuming the permeability of air is equal to that of free space, the magnetizing force of the air gap is determined by
And the mmf drop across the air gap is equal to Hg Lg 21

15. 12.11 – 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

16. 12.12 – 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

17. 12.13 – Applications Speaker and microphones Hall effect sensor
Magnetic reed switch
Magnetic resonance imaging 24