Mag Ckt Ex Holey Notes 1

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Magnetic Circuits and Examples EE201

Denard Lynch of 13 Sep 3, 2009

Review of Magnetism

Originates at the molecular level:

• electrons orbiting a nucleus induce a magnetic “moment”

• each “orbit induces its own separate moment

• depending on particular molecule, can have a net magneticmoment (i.e. ferromagnetic)

This implies and illustrates a fundamental relationship between

electricity and magnetism

Two important and related facts:

• a moving charge generates a magnetic field, and although not obvious from the above…

• relative motion between a magnetic field and a charge exerts a force on the charge(s)

(and on that which contains them; i.e. a conductor)

So what is this “magnetic field”?

• a force field• visualized as “lines of magnetic flux” that appear to follow rules:

o form closed loops from one “pole” to another

o directional (by convention, from north to south pole of magnet)

o repel each other *

o follow the path of least resistance*

o *these last two rules conflict with one another!

• Remember… the field consists of continuous lines, but are really a visual aid

• Magnetic flux can also be visualized as flowing, similar to water or electric current

(symbol:Φ)

• We are often interested in its density (symbol B)

Depending on the material, the magnetic moments from each set of orbiting electrons in amolecule can reinforce or cancel each other to give the molecule a net magnetic moment.

Various molecular arrangements lead to three types of materials:

o Paramagnetic – only very slightly receptive to a magnetic field (e.g. air, glass, wood,

paper, plastics etc.)

o Diamagnetic (anti-magnetic) – form opposite dipoles in response to an imposed

magnetic field (bismuth, pyrolytic graphite), but only a weak response

o Ferromagnetic – net magnetism at the molecular level; get together into polarized“domains”. They are normally oriented randomly, but may be aligned temporarily or

permanently. (E.g. iron, cobalt etc.)

We are interested in this last category because it is relatively easy* to establish magnetic flux,

Φ, in these materials in response to a magnetic field.

• this “relative ease” is somewhat temperature dependent. Each material has a “Currietemperature”, above which they behave like paramagnetic materials.





Why is this (ferromagnetic) ability of interest to us?

So we can design “magnetic circuits” to route flux in certain ways and to certain places; thesame way we route electric current or hydraulic fluid. We route this to certain places for one of

two reasons:

1) The force it can provide

a.

Speakers, sound reproduction b. Motors, generators

c. Door bells, electromagnets2) its magnetizing effect

a. HDDs, magnetic tapes

b. Transformers, inductors

How can we create a magnetic field, or a magnet?

Back to moving charges…

Moving charges, like a current flowing in a conductor, create a magnetic field. The induced“lines of flux” are formed in concentric circles in a plane perpendicular to the direction of

current flow.

The direction of the flux, Φ, can be determined using a “right-hand-rule” (RHR)

Note: diagrammatic convention: into page/board = ⊗; out of page/board =

Looking at a current carrying conductor from one end, the density of the magnetic field, B, is

given by:

B =

µ 0 I

2π r , outside the conductor, and B =

µ 0 Ir

2π R2

inside the conductor.

B

r R

R

Charge-carrying conductor

⊗





If we route current through a cylindrical coil of wire, we will end up with a fairly uniformmagnetic field inside the coil. The coil, in effect, becomes a magnet (electromagnet), which is a

source of magnetomotive force (mmf), symbol: ℑ.

A source of magnetomotive force (mmf), which could be an electromagnet or a permanentmagnet, will cause an amount of magnetic flux, Φ, to ‘flow’. The amount of flux that floes

depends on the magnetic ‘resistance’ of the circuit or path. Again, we can use a right-hand-rule(RHR) to determine the north pole of the source of mmf. (For an electromagnet consisting of acoil of wire, if you wrap the fingers of your right hand around the coil in the direction of current

flow, your thumb points in the direction of the flux flow, and the north pole.)

The ‘resistance’ of the path is proportional to the length, l , inversely proportional to the area, A,

and is know as the reluctance, ℜ. Similar to electrical resistance where, the magnetic

reluctance, ℜ =

l

µ A, where µ is the permeability of the material and is the product of the

relative permability, µr , and the permability of free space, µ0 (µ=µr µ0).

Also similar to the electric circuit case, there is a fundamental relationship between the driving

force, ℑ, the reluctance of the path, ℜ, and the amount of flux, Φ, that flows in the circuit. This

relationship is know as Ohm’s Law for Magnetic Circuits: ℑ=Φℜ, which will be a fundamental

too for analysis of magnetic circuits.

Ω’s Law for Magnetic Circuits: ℑ = Φℜ, where:

ℑ = magnetomotive force (mmf) = (number of turns)(current in Amperes) = NI

Φ = total magnetic flux in Webers, Wb

ℜ =

l

µ A= reluctance in Amp-turns per Weber (A-t/Wb) or rels, and

l = average path length in meters A = cross-sectional area in m

2

µ = µr µ0, and µr is unit-less and specific to the material,

and µ0 = 4π X 10-7

Wb/At-m.

µr ~ 1 for paramagnetic materials,

and >>1 for ferromagnetic materials

Substituting ℑ=NI and ℜ =

l

µ Ainto Ω’s Law for Magnetic Circuits gives us:

Φ =

µ A

l NI , with a slight rearrangement…

Φ

A= µ

NI

l , now defining two new quantities: B =

Φ

Aand H =

NI

l , where:





B is the flux density in Webers per square meter (Wb/m2) or Teslas, (T)

(1T=10,000Gauss)H is the magnetizing force or magnetic field intensity in Amp-turns/m (A-t/m),

and substituting these expressions for B and H in the last expression, we have:

B = µ H

Take the expression for flux density: B =

Φ

A, or Φ = BA. Recalling Ω’s Law and that the mmf,

ℑ = NI = Φℜ, we can do another manipulation:

NI = Φ( ) ℜ( ) = BA( )l

µ A

⎛

⎝ ⎜⎞

⎠ ⎟ =

B

µ l = Hl

or NI = Hl , which is a variation of Kirchhoff’s Voltage Law for electric circuits, which

can also be stated as: NI = Hl ∑∑ , or the sum of the mmf rises around any loop in a magnetic

circuit must equal the sum of the magnetizing force drops. This relationship will prove very

useful in analyzing magnetic circuits!

Let us re-examine the relationship between the flux density, B (which is proportional to Φ) and

the magnetizing force, H (proportional to NI ) given as B= µ H . µ is the slope of the relationship

between B and H and is typically not linear for magnetic materials. It is usually given ingraphical form in a “B-H curve”.

Hysteresis

Hysteresis is a property of magnetic material that causes some residual magnetism to remain in

a material after it has been exposed to an external source of mmf. Once a ferromagneticmaterial is subjected to a magnetic field, the domains align. If the magnetic field is removed,

most of the domains will return to (approximately) their original orientation. Dependent on thetype of material, some domains may stay oriented in the induced direction, giving the sample

some residual magnetism of its own. The amount of (reverse) field strength needed to return

the magnetic moment of the sample back to zero is called the coercive force.

Magnetic Circuit Problems

Magnetic circuit problems faced by a designer are generally one of two types:

1. the NI (mmf) is known, and we need to find the flux, Φ or B, or

2. the desired flux, Φ, or flux density, B, is known and we want to find the required NI.

Useful Circuit Comparisons:

Fluids Electricity Magnetics

Pressure

(pump)

Restriction

(valve)

Flow (fluid) Flow (current)

Restriction

(resistance)Pressure

(voltage)

Pressure

(mmf)

Flow (flux)

Restriction

(reluctance)

S

N





The parallel between magnetic circuits and (particularly) electric circuits is very useful for

analyzing and designing magnetic circuits!!

Channeling Flux: Gaps and Fringing; Laminations and Magnetic Force of Attraction

Fringing:

Recall that two opposing properties of magnetic ‘lines’ of flux is that they will “avoid” each

other by spreading out as much as possible, but that they also take the path of least resistance,so they will “get together” to follow a lower reluctance path. This means that when they have

been channeled through a ferromagnetic material (easy path) and then come to a paramagnetic

(e.g. air) gap, they will start to “spread out” to avoid each other again over the length of the gap,which will affect the density, B. To estimate the effective area of the path in an air gap:

•

add the length of the gap, g, to the length and width of a rectangular or square cross-section:

A Eff − gap

= l + g ( ) w+ g ( )

• add the length of the gap, g, tothe radius of a circular (or

elliptical) cross-section:

A Eff − gap

= π r + g ( )2

Laminations:

Magnetic cores are often laminated to reduce heating and losses from eddy currents (thosecurrents induced in a conductor by a varying magnetic field). For laminated cores, the effectivearea is the nominal area multiplied by the Stacking Factor (S.F.).

StackingFactor = A

Eff

AOverall

For a rectangular cross-section, for example, A Eff. = (S.F.)(length X width)

Note: the Stacking Factor is sometimes given

as a percentage: e.g. S.F. = .95 or 95%.

Overlapping Areas:Where two legs of a magnetic circuit cometogether, the areas may not be the same or

oriented to exactly match. In these cases, the

flux will almost entirely pass through theoverlapping area. Although the lines of flux will have to make some path adjustment when

transitioning from one piece to the other, this happens quickly enough so an assumption that it

happens immediately at the transition is a reasonable one.

l

w

effective area

physical area

l+g

w+g

r+g

r

2cmX4cm

Effective overlap:2cmX2cmX0.95

S.F.=0.95





Magnetic Force of Attraction:

In any magnetic circuit, there is a ‘magnetic force of attraction’. This attractive force exists

anywhere in the circuit, but is most useful where there is a physical break or separation between

one piece and another. This makes it possible to lift or hold ferromagnetic loads with a magnet

(or electromagnet). The magnetic force of attraction is given by:

F Mag − Attr

=

B gap

2 A gap

2µ 0

.

The designation ‘gap’ in this formula should really be interpreted as ‘interface’, as this forceexists at any interface (or other point in the circuit), regardless of whether an actual gap exists

or not. Also note, it depends only on the flux density, B, and the area of the contact , not on the

length of any ‘gap’. The gap length, however, may indirectly influence the force by increasingthe reluctance of the magnetic path and thus reducing the total flux.

Remember, when dealing with overlap situations, use the smallest common area as the ‘Agap_ ’.

SUMMARYElectric Magnetic

E = IR ℑ = Φℜ

R =ρ l A

ℜ = l µ A

Φ = BA

KVL

Σ V loop = 0 Σℑ loop = 0

Σ V loop = Σ IR Σ NI loop = Σ Hl

KCL

Σ I node = 0 ΣΦ node = 0

Σ I in = Σ I out ΣΦ in = ΣΦ out

B – H Curves

Where R typically stays reasonably constant with varying current, ℜ (reluctance) does not

usually stay constant with varying levels of flux!

B = µ H

µ = B

H

Flow (current)

Restriction

(resistance)Pressure

(voltage)Pressure

(mmf)

Flow (flux)

Restriction

(reluctance)

S

N





Some Quick Examples:

Fringing:

Overlap/Laminations:

Hysteresis:





Ex. 1

a) What magnetomotive force, NI isneeded to provide a flux of 0.35

mWb in this cast steel core?

b) What is the direction of the flux in the

core if the current flows into the top

terminal as shown?

a)

leg A(m2) l(m) Φ(Wb) B(T) H(A-t/m) Hl(A-t)

Cast S.

Sheet S.





Ex. 2

N = 1000t; I = .25A

Find the total flux, Φ, produced by the coil if

the core is made of:a) sheet steel

b) cast steel

c) cast iron

Leg A(m2) l(m) Φ(Wb) B(T) H(A-t/m) Hl(A-t)

a) Sheet S.

b) Cast S.

c) Cast Iron

10cm

10cm

2cm2cm

I

N





Ex: 3

a) The core in the figure

is made of Cast Steel. If

the coil is 1000 turns and

the current is 1.35A,determine the flux in the

core.

b) Given the same

conditions as in a),

determine the flux if a2mm air gap is cut in the

core as shown. Consider

fringing at the gap.

**Considering Fringing**


a)

b)

cast steel

gap

cast steel

gap

2mm gap – part L= .5m

2cmX2cm

I

N





Ex: 4

N = 500t; I = 0.6A

*Neglecting Fringing*

a)Find the flux in the air gap, and

b) the total flux

produced by theoil.


L-2 .0003 .1 .000465 1.55(sat!) 2440 244

L-3 .0003 .099 .000085 .283 195 19

L-gap .0003 .001 .000085 .283 225,000 Guess 75%225

L-1(SS) .0009 .4 .000550 .611 75 30

=274

274/300=91%, too low!

L-2

L-3

L-gap

L-1(SS)

So…

a) the flux in the gap is 85µWb, and

b) the total flux produced by the coil is .544mWb

I

N

L1 = 0.4m

L2 = 0.1m

1cmX3cm3cmX3cm

Sheet steel

Cast steel

1mm gap





Eg. 5: Consider the magnetic circuit shown

in the figure.

N1 = N2 = 200t

I1 = 8A

Stacking Factor for the [laminated] cast steelsection is 0.93. The sheet steel sections (with

the gap) are solid.

Consider fringing effects at the air gap.

Find the current, I2, required to establish a flux density, B, of 1.1T in the air gap.

First draw an Electric Equivalent…

Note: You can combine the cast segments into one reluctance

and also the 2 sheet steel segments into 1 ℜ because:i) same material

ii) same physical characteristics (dimensions)iii) same flux

Second, make a table, then put in what you know (bold), and finally calculate the rest.

leg A(m2) Φ(Wb) B(T) H(A-t/m) l(m) Hl(A-t)

N1

N2

laminatedCastSteel

SheetSteel

I1

I2

2mm gap

3cm

3cm

all x-sections

15cm

15cm





Eg. 6: The electromagnet (EM) shown at right is madeof solid sheet steel. The coil has 200 turns of 12ga

insulated copper wire.

If the EM is brought to within 1mm of a 20kg cast steel bar of the same cross-section, find the current required

to just start lifting the bar.

Consider the effects of fringing at the air gaps.

Draw Elec. Equiv.

Now use this to do the rest of the mag ckt calculations:

leg A(m2) Φ(Wb) B(T) H(A-t/m) l(m) Hl(A-t)

sheet steel

EM

gap

cast steel

bar

2cm

2cm

all x-sections

20 kg

1mm gap

N

I

l=24cm

8cm

Mag Ckt Ex Holey Notes 1

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