MAGNETIC CIRCUITS

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

A

l

A

lR

At/Wbor rels,A

l

tyconductivi

1

Ampere-turns per weber

Ohm’s Law for Magnetic Circuits

opposition

causeeffect

• 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, .

Ohm’s Law for Magnetic Circuits

• Substituting:

Ohm’s Law for Magnetic Circuits

• The magnetomotive force 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 AtNI

• An increase in the number of turns of the current through the wire, results in an increased “pressure” on the system to establish the flux lines through the core.

Ohm’s Law for Magnetic Circuits

Magnetizing Force• The magnetomotive force per unit

length is called the magnetizing force (H).

• Magnetizing force is independent of the type of core material.

At/ml

H

Magnetizing Force• Magnetizing force, H is determined

solely by the number of turns, N the current, I and the length of the core, l:

At/ml

NIH

Magnetizing Force• The flux density and the magnetizing

force are related by the equation:

HB

Hysteresis

• Hysteresis – The lagging effect between the flux density, B of a material and the magnetizing force, H applied.

• The curve of the flux density, (B) versus the magnetizing force, (H) is of particular interest to engineers.

Hysteresis

Series magnetic circuit used to define the hysteresis curve.

Hysteresis Curve

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

Hysteresis

• If the entire cycle is repeated, the curve obtained for the same core will be determined by the maximum H applied.

Hysteresis

Normal magnetization curve for three ferromagnetic materials.

Hysteresis

Expanded view for the low magnetizing force region.

Ampere’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.

0

IRV

or Hl

Ampere’s Circuital Law• As an example:

0 cacabcbcabab lHlHlHNI

0

cacabcbcabab lHlHlHNI

Hl

Steel

Cobalt

Iron

Flux • The sum of the fluxes entering a

junction is equal to the sum of the fluxes leaving a junction

Flux acba juction at

bacb junction at or

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

Series Magnetic Circuits : Determining NI• 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

Series Magnetic Circuits : Determining NI

Example 1:

(a) Determine the secondary current I2 for the transformer if the resultant clockwise flux in the core is 1.5 x 10-5 Wb.

Series Magnetic Circuits : Determining NI

Example 1 – solution

Tx

x

AB 1.0

1015.0

105.13

5

At/m30abcdH

Tx

x

AB 1.0

1015.0

105.13

5

Series Magnetic Circuits : Determining NIExample 1: – solution (cont’d)

Based on B- H curve: mAtH abcd /30

Using ampere circuital law :

A 84.330

)16.0(30260

2

2

112

1122

2211

I

xx

N

lHINI

lHININ

lHININ

abcdabcd

abcdabcd

abcdabcd

abcdabcd2211 lHININ

Series Magnetic Circuits : Determining NIExample 2 – solution

3108.30470 efabefablH

Calculate Hl for each section;

At 34.21

3101271600 bcdebcdelH

At 2.203

1 Inch=0.0254m 1 Inch2=6.451x10-

4m2

Series Magnetic Circuits : Determining NIExample 2 – solution (cont’d)

The magnetic circuit equivalent

The electric circuit analogy

At 54.2242.20334.21 bcdebcdeefabefab lHlHNI

A 4.4950

54.224

N

NII

At/Wbor rels,,ReluctanceA

l

At,force iveMagnetomot NI

Summary:

]Tesla[,densityFlux A

B

IRV

At/ml

NI

lH