Chapter 6 Magnetic Circuits
SUJIT K. PATEL
Topics to be Discussed
Magneto Magnetomot motive ive Force Force (MMF). (MMF). Magnetic Field Strength (H ). ). Magnetic Permeability. Reluctance (R ). Analogy between Electric and Magnetic Circuits. Composite Magnetic Circuit. Magnetic Leakage and Fringing. Air Gaps in Magnetic Circuits.
Introduction
Unlike electric field lines, the lines of magnetic flux form closed loops . A magnetic circuit is a closed path followed by lines of magnetic flux. A copper wire, because of its high conductivity, confines the electric current within itself. Similarly, a ferromagnetic material (such as iron or steel), due to its high permeability , confines magnetic flux within itself.
Magnetomotive Force (MMF)
The electric current is due to the existence of an electromotive force (emf). By analogy, we may say that in a magnetic circuit, the magnetic flux is due to the existence of a magnetomotive force (mmf ). mmf is caused by a current flowing through one or more turns. The value of the mmf is proportional to the current and the number of turns. It is expressed in ampere turns (At). But for the purpose of dimensional analysis, it is expressed in amperes.
Magnetic Field Strength (H )
The mmf per metre length of the magnetic circuit is termed as the magnetic field strength , magnetic field intensity , or magnetizing force . Its units are ampere-turns per metre (At/m) . Its value is independent of the medium .
H
F
IN
l
l
Magnetic Permeability (μ )
If the core of the toroid is vacuum or air, the magnetic flux density B in the core bears a definite ratio to the magnetic field strength H. This ratio is called permeability of free space . Thus, for vacuum or air, B H
0
4
10
7
Tm/A
The flux produced by the given mmf is greatly increased, if iron replaces the air in the core. As a result, the flux density B also increases many times.
In general, we can write B = μ H.
μ is called the permeability of the material.
Normally, we write μ = μ r μ 0.
μ r is called relative permeability (just a number).
Reluctance ( ) and Permeance ( )
The current in an electric circuit is limited by the presence of resistance of the electric circuit. Similarly, the flux Φ in a magnetic circuit is limited by the presence of the reluctance of the magnetic circuit, R
1 l A
1 r
l 0
A
The reciprocal of reluctance is known as permeance (G ).
Magnetic Circuit Theory
For a toroid, mmf, F = NI ampere-turns.
Because of this mmf, a magnetic field of strength H is set up throughout the length l.
Therefore, F = Hl
If, B is the flux density, total flux is given as Φ = B
Φ
F
A
Dividing, we get BA Hl
B A H l
A l
A r
0
l
Φ
F l /(
r
Click
)
0 A
Comparing this with We get R
1 r
l 0
A
I
E R
Analogy between Electric and Magnetic Circuits
Example 1 Calculate the magnetomotive force (mmf) required to produce a flux of 0.015 Wb across an air gap of 2.5 mm long, having an effective area of 200 cm2. Solution :
B
Φ
A H
B 0
F
Hl
0.015 200 10 0.75 4π 10
4
-7
597 000
0.75 T 597 000 A/m
2.5 10
3
1492 At
Composite Magnetic Circuit Case 1 : R 1
l1 A1
1
R 2
l2 A2
2
Total Reluctance , R
R 1
R 2
l1
l2
1 A1
2 A2
Total flux,
mmf of coil total reluctance NI
F R
l1
l2
1 A1
2 A2
Case 2 : (with air gap) Total reluctance, R
l1 1 A
1
l2 0 A
l1
A
0
1 A
0
( 1 / l1 r
0
) l2
l2
Since the relative permeability μr (= μ1 / μ0) of steel is very large (of the order of thousand), the major contribution in the total reluctance R is by the air-gap, though its length l2 may be quite small (say, a few millimetres).
Magnetic Leakage and Fringing
The flux lines, such as a, b and c, leak through the core. This is called leakage flux , since it does not contribute to the useful flux passing through the metallic ring. We define leakage factor as the ratio of total flux through the exciting winding to the useful flux. The value of the leakage factor for electrical machines is about 1.15 to 1.25.
There is another effect called fringing . The useful flux passing across the air-gap tends to bulge outward. This happens because the magnetic flux lines tend to repel each other in Its effect is to cause a slight increase (say, about 10 %) in the cross-sectional area at the air gap.
Kirchhoff’s Laws
Kirchhoff’s Flux Law (KFL) : The total magnetic flux towards a junction is equal to the total magnetic flux away from that junction. Kirchhoff’s Magnetomotive Force Law (KML) In a closed magnetic circuit , the algebraic sum of the product of the magnetic field strength and the length of each part of the circuit is equal to the resultant magnetomotive force.
Steps to solve a problem on magnetic circuit
Air Gaps in Magnetic Circuits
Two purposes :
1.
2.
To permit part of a magnetic circuit to move, for example, in relays and in electrical machines. To make the magnetization characteristic of the circuit more linear.
Review
Magnetomotive Force (MMF). Magnetic Field Strength (H ). Magnetic Permeability. Reluctance (R ). Analogy between Electric and Magnetic Circuits. Composite Magnetic Circuit. Magnetic Leakage and Fringing. Air Gaps in Magnetic Circuits.
