Chapter 12
APPLICATIONS OF SOFT MAGNETS.
Soft magnets are needed to enhance and guide the ‡ux denstity produced by currents in
electric circuits. Electrical sheet steel is used in electromagnetic machinery - transformers,
motors and generators. Insulating ferrites are mainly used for high-frequency inductors and
antennae, and for microwave devices.
12.1
Introduction
A good soft magnetic material should have minimal hysteresis and a high permeability. Permeability is usually quoted for the internal …eld, because soft magnets are
normally used in a toroidal geometry where demagnetizing e¤ects are negligible. In
some range of internal …eld, the B(H) response is linear B = H or
B=
0 rH
(12.1)
where the relative permeability r is a pure number: The initial permeability i near
the origin of the hysteresis loop, is smaller thanthe slope B/H for slightly larger …elds,
where it attains its maximum value max , as shown in Fig 12.1.
Fig 12.1 Hysteresis in a soft magnetic material. B(H) and J(H) are indistinguishable in small …elds.
2
Applications of soft magnets.
Values of max can reach 106 in the softest materials. Hence B is can be
hugely enhanced, up to a limit set by the spontaneous induction Bs Js ; compared
to the free space induction 0 H that produces it, where H is the external applied
…eld: Permeability and loop shape can be modi…ed by annealing, especially in a weak
external …eld. The relation between susceptibility and relative permeability follows
from B = 0 (H + M ); it is r = 1 + .
Soft materials are used for static and alternating current applications. Static
and low-frequency ac applications are ‡ux guidance and concentration in magnetic
circuits, including cores for transformers and inductors. In the microwave frequency
range applications involve modifying and guiding the propagation of electromagnetic
waves. Metals are generally used up to about 50 kHz, but insulating ferrites are
needed in the radio frequency and microwave ranges to avoid eddy-current losses.
Generally the spontaneous induction Bs
0 Ms is lower in the materials
usable at higher frequency, and they operate at a smaller fraction of saturation.
Hysteresis increases with frequency, and the maximum permeability falls to a hundred
or less for ferrites in the megahertz range. The main frequency ranges and typical
materials and applications are given in Table 12.1
Table 12.1. Soft magnetic materials and applications.
static
low-frequency
audio-frequency
radio-frequency
microwave
frequency
materials
< 1 Hz
1 Hz - 1 kHz
100 Hz - 100 kHz
0.1 - 1000 MHz
> 1 GHz
Soft iron, Fe-Co (permendur),i
ditto
permalloy foils, magnetic glasses, sendust powder, Mn-Zn ferrit
Mn-Zn ferrite, Ni-Zn ferrite
YIG, Li ferrite
Whenever metals are used in any ac application, ‡ux penetration is opposed
by the induced eddy currents, which limit the depth of penetration of the …eld. The
skin depth s is de…ned as the depth where B falls to 1/e of its value at the surface.
s
=
p
( =
r 0f )
(12.2)
Here is the resistivity and f is the ac frequency. For electrical steel Fe94 Si6 at 50
Hz ( = 0.5
m, r = 2 104 ) the value of s is 0.36 mm. At 500 kHz, s is 3.6 m.
Cores made from soft magnetic metals are often assembled from a stack of insulated
laminations, where the lamination thickness is chosen to be close to s so that the
applied …eld can penetrate each one.
12.1.1 Losses
The energy losses are critical in any ac application. Traditionally three main sources
are identi…ed:
hysteresis losses Ph
eddy-current losses Ped
anomalous losses Pa
Introduction
3
The total losses per cubic metre are therefore
Ptot = Phy + Ped + Pa
(12.3)
Hysteresis losses are related
to irreversibility of the static B(H) loop. The
R
energy loss per cycle is Eh = loop HdB:(see Eq. 2.70). At frequency f the losses Phy
are f Eh :
Fig 12.2 Hysteresis loss per cycle is the area of the B(H) loop
Eddy current losses are inevitable when a conducting ferromagnet is subject
to an alternating …eld. The induced currents dissipate their energy as heat. In a
sheet of thickness t and resistivity cycled to a maximum induction Bmax the losses
Ped vary as f 2 :
Ped = ( tf Bmax )2 =6
(12.4)
These losses can be minimized by using thin laminations or a highly resistive material.
For example, electrical steel Fe94 Si6 (3 wt% Si -Fe) is usually made in laminations of
0.5
m. and density 7650 kg m 3 . For Bmax = 1
thickness 300 m. It has
tesla, it follows from (12.4) that at 50 Hz Ped 0.1 W kg 1 .
Anomalous losses are the losses left over after Phy and Ped are taken into
account. They are comparable in magnitude to Ped and are associated with extra
eddy current losses due to domain wall motion, non-uniform magnetization and sample inhomogeneity. Essentially, they re‡ect the increase of hysteresis with frequency,
so the separation of static hysteresis losses and anomalous losses is rather arti…cial.
Anomalous losses are reduced by a structure of many parallel domain walls, which reduces the distance the walls must move during the magnetization process. High-grade
electrical steels are laser-scribed to ensure a structure with narrow stripe domains.
The distinction between static and anomalous losses is arti…cial. The hysteresis loop itself depends on measuring frequency, and coercivity increases with f . A
modern physical approach is due to Bertotti, who aascribes the anomalous losses to
the movement of elementary magnetic objects, identi…ed as domain walls, or groups
of correlated walls. N e¤ective …eld Han = Pan =(dJ/dt) is introduced, and the anomalous losses are found to depend on the domain wall mobility (Eq 7.21) and f 1=2 :
Fig 12.3. Total loss per cycle showing the three contributions.
Fig 12.4. Total losses per kg for permalloy at di¤erent frequencies. Thickness is 350
m.
4
Applications of soft magnets.
Fig 12.5 Total losses in transformer cores over the 20th century
Losses at high frequencies are best represented in terms of a complex permeability. The magnetization process involves magnetization rotation rather than
domain-wall motion, and the losses are in‡uences by the ferromagnetic resonances.
If the applied …eld is Hei!t ;the induced ‡ux density generally lags by a phase angle ; it is Bei(!t ) :The complex permeability = (B=H)ei can be expressed as
(B=H)(cos + i sin ), or
=
i
(12.5)
where = (B=H)cos and = (B=H)sin :Losses are proportional to the response
in quadrature with the driving …eld.
12.2
Static Applications
Electromagnets consist of …eld coils to generate the …eld, an iron yoke to guide the
‡ux and pole pieces to concentrate the ‡ux in the airgap. Flux guidance and concentration in electromagnets requires material with the highest polarization and very
little remanence. Usually pure soft iron or Fe65 Co35 is used. For best results, the
pole pieces are tapered at an angle of 55 . Electromagnetic relays and solenoid valves
are miniature electromagnets where an iron core is magnetized and exerts a force on
another iron member. The force per unit area if the gap ‡ux density is Bg is Bg2 /2 0
(Eq.11.13).
Fig 12.5 An electromagnet.
Magnetic shielding of weak dc or low-frequency ac …elds (e.g. the Earth’s …eld)
requires material to provide a low reluctance ‡ux path around the shielded volume.
The shielding ratio R is the ratio of the …eld outside to the …eld inside. Values of R
100 are achieved in low …elds. The shielding material is chosen so that its polarization
is unsaturated by the ‡ux collected. Generally it is more e¤ective to use several thin
shields rather than one thick one. Dc shields are often made of permalloy Ni80 Fe20
or supermalloy Ni77 Fe14 Cu5 Mo4 , which has no anisotropy or magnetostriction, and is
therfore immune to shock and strain. Properties of some soft magnetic materials are
listed in Table 12.2.
Fig 12.6. Magnetic shielding. The shielding ratio R is H /Hin
Table 12.2. Properties. soft materials for static applications.
material
name
Fe
Fe49 Co49 V2
Ni50 Fe50
Ni77 Fe14 Cu5 Mo4
a-Fe40 Ni38 Mo4 B18
Soft iron
Supermendur
Hipernik
Supermalloy
Metglas 2628SC
Finemet
i
300
1000
6000
80000
50000
max
5000
50000
40000
300000
400000
Js (T) Hc (A/m)
2.15
2.40
1.60
0.80
0.88
70
40
8
1
0.5
Low-frequency applications.
12.3
5
Low-frequency applications.
Transformers, motors and generators include soft iron cores to generate and guide
the ‡ux. Eddy-current loses (12.2) are reduced by using thin laminations of material
with a high resistivity. E¢ ciencies of well-designed transformers are > 99%; they
are probably the most e¢ cient energy converters ever made. Core losses represent
about a quarter of the total, the remainder being in the windings. Global energy
consumption is of order 15 1012 kW h, and the core losses in transformers nonetheless
cost several billion euros per year.
Isotropic and grain-oriented Fe94 Si6 (
0.5
m) is produced in huge quantities, in sheets about 300 m thick, for low-frequency electrical applications. Losses
are about ten times lower than for soft iron, which continues to be used for consumer applications where there is no cost premium for e¢ ciency. Silicon increases
the resistivity, and reduces the anisotropy and magnetostriction of the iron. The 6
at% Si composition is su¢ ciently ductile to be rolled into thin sheets. Grain-oriented
material with the Goss texture has the lowest losses. {110} planes are parallel to the
sheets and a [100] easy axis is parallel to the roll direction.
Melt-spun amorphous alloy ribbons can be produced with thickness of about 50
m; resistivity is 1.5
m. Cobalt-rich compositions exhibit zero magnetostriction,
and huge permeability. These materials can be wound into cores for use up to about
100 kHz, and applied, for example, in switched mode power supplies.
Powder of iron or the zero-anisotropy, zero magnetostriction alloy ‘Sendust’
(Fe85 Si10 Si5 ) can be insulated and used at higher-frequencies in cores. Permeability
is limited to 10 - 100 because of the demagnetizing …elds. Long telephone lines are
loaded with powder-core inductors (loading coils) to balance their capacitance.
Fig. 12.7 Types of cores. a) stacked laminaations, b) tape-wound core, c) powder
core (sectioned to show internal structure) and d) ferrite C-core.
Magnetic ampli…ers use square loop cores. The principle is shown in Fig 12.8.
When the dc current is zero, the load current is very small because the voltage drop
across the right hand winding proportional to d /dt almost cancels the source signal.
As the current from the dc source saturates the core, the change in ‡ux in the core
(a-b) becomes negligible and the current in the load rises.
Fig 12.8 A magnetic ampli…er.
Table 12.3. Properties of soft materials for low-frequency applications
Material
Js (T)
Mild steel
Si-iron
Grain-oriented Si-iron
2.15
2.12
2.00
r
at 1.5 T Hc (A/m) Wtot (W/kg)
500
1000
20000
80
40
8
12
4
1.2
(
0.15
0.60
0.50
m)
6
12.4
Applications of soft magnets.
High frequency applications.
Ferrites are most suitable for rf and microwave applications on account of their high
intrinsic resistivity. Saturation polarisation is only 0.2 - 0.5 T, much less than for
metals. Conduction is usually due to traces of Fe2+ ions in the structure. Mn - Zn
ferrites are used up to about 1 MHz, and Ni - Zn ferrites from 1 - 300 MHz. The
latter have lower polarization, but higher resistivity.
Ferrite cores are used in chokes, inductors and high-frequency transformers for
switched-mode power supplies. They are also used for broad-band ampli…ers and pulse
transformers where the Fourier components of the pulse extend over a wide frequenct
range. Losses at 100 kHz are about 50 W/kg. Another common application of ferrites
is as antenna rods in am radios.
Miniature inductors can be electoplated directly onto a chip with a copper coil
and a thin permalloy core.
Fig 12.9 Frequency response of some soft magnetic materials.
The frequency response of the permeability of high-frequency ferrites is ‡at
up to the ferromagnetic resonance frequency, where it falls o¤. The higher the cut-o¤
frequency, the lower the permeability.
The equation of motion in the absence of damping is
dM =dt =
0 (M
(12.6)
H)
where
is the gyromagnetic ratio of 28 GHz/T.
A damping term of the Landau-Lifschitz form M M B/M 2 or the
Gilbert form M (@M /@t)/Ms can be added to the right-hand side. The ferromagnetic resonance can occur in in the externally-applied …eld H, or in the demagnetizing
…eld -N M . In general, the ferromagnetic resonance frequency is
! 20 =
2
[B0 + (Ny
Nz ) 0 M ][B0 + (Nx
Nz ) 0 M ]
(12.7)
where the external …eld B0 = 0 H is applied in the z-direction.
Microwave ferrites are used in the frequency range from 300 MHz to 50 GHz.
At these frequencies, it is necessary to consider the electromagnetic wave rather than
the current in a circuit. The magnetic component is a waveguide. Microwave devices
exploit the nonreciprocal interaction of the electromagnetic …eld with the ferromgnetic
medium, specially in the vicinty of the ferromagnetic resonanc frequency. Generally,
if the external …eld is applied along zand an ac …eld h is applied in the xy plane, the
resulting ‡ux density b is given by the tensor permeability
2 3
bx
4 by 5
bz
2
= 4i
0
i
0
3
0
05
0
2 3
hx
4hy 5
hz
(12.8)
High frequency applications.
7
The o¤-diagonal terms here produce the nonreciprocal e¤ects.
The e¤ective permeability for right and left polarizations are + =
and
= + . The …rst shows a resonance at the ferromagnetic resonance frequency,
the second does not.
Fig 12.10 Absorption and transmission for left and right-polarized radiation.
Microwave ferrite devices are often made of YIG, which has a particularly
sharp microwave resonance. For resonance at 10 GHz, a YIG sphere can have Q =
! 0 / ! 105 . A tunable narrow-band …lter is made by winding two orthogonal coils
on the sphere. The signal from one coil will only be detected in the other at resonance,
which can be tuned by applying an external …eld. Other devices exploit the di¤erence
between + and
in various ways. Resonant isolators use the resonance at B0 to
absorb signals re‡ected back along a waveguide. Phase shifters exploit di¤erences
between + and
above and below the resonance peak. Three port circulators
transmit signals from one port to the next one, while strongly absorbing other paths.
Faraday rotators rotate the plane of polarization of the microwaves. A YIG sphere
acts as a resonant …lter (Fig 12.11).
Fig 12.11 A resonant …lter. The two coils are orthogonal, and the device transmits
the signal in a narrow frequency range around the ferromagnetic resonance frequency
Fig 12.12 A wire loop antenna, and an equivalent ferrite rod with much smaller
cross section.
Exercises.
1. Deduce (12.2). Estimate how thin a soft magnetic …lm should be if it is to
operate at 1 GHz.
2. By considering the …eld at the apex of a cone of fully-saturated magnetic
material produced by the surface magnetic pole density
J en show that the …eld is
p
1
maximum when the half-angle of the cone is tan
2.
3. Deduce (12.5) from (12.4) by writing (12.4) in components, and looking for
solutions with time-dependence exp (-i!t).
Further reading. Hysteresis, G. Bertotti, Academic Press
Magnétisme et matériaux magnétiques pour l’électrotechnique, P. Brissonneau,
Editions Hermès, Paris 1997