The Use of Ferrites At Microwave Frequencies

109 Pages·1964·3.441 MB·English

by  L. Thourel (Auth.)

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1
L. THOUREL
The Use of Ferrites
at
Microwave Frequencies
TRANSLATED BY
J. B. ARTHUR
PERGAMON PRESS
OXFORD · LONDON · EDINBURGH
NEW YORK · PARIS · FRANKFURT
SOCIÉTÉ FRANCAISE
DE DOCUMENTATION ÉLECTRONIQUE
PARIS
1964
PERGAMON PRESS LTD.
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Distributed in the Western Hemisphere by
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Copyright © 1964
PERGAMON PRESS LTD. AND SOFRADEL (FRANCE)
Library of Congress Catalog Card Number 63-13500
A translation of the original volume "Emploi des
Ferrites en Hyperfréquence," Société Frangaise de
Documentation Électronique, Paris 1962
Set in Monotype Times 11 on 12 pt.
and printed in Great Britain by
Charles Birchall and Sons Limited, Liverpool and London.
INTRODUCTION
THE past few years have witnessed the appearance of more and more
appUcations of soUd state physics in the field of electronics. The
transistor is certainly the best known and most spectacular, but there
are others, more recent still, which are no less important and which
permit the solution of problems hitherto thought to be insoluble (the non-
reciprocal isolator, for example). We shall mention, in particular,
parametric amplifiers, which use a variety of junction diodes, and
masers, whose operation is based on gyromagnetic resonance effects.
In ferrites, which are ferromagnetic dielectrics, there are two
effects which can be exploited, namely variable permeability and non
reciprocal phenomena. The first appUcations of ferrites were in the low
or video frequency domain: magnetic cores, memories, for example.
But, in parallel with these techniques, applications in the domain of
decimetre and centimetre waves have been developed, applications
which are today an industrial reality and which have made possible
devices which were unfeasible without the use of ferrites. We shall quote
in particular: isolators, which allow energy to pass in one direction
and absorb it in the reverse direction; circulators, used in radio beam
and radar equipment; electronic tuning filters; waveguide modulation
devices; ultra-high speed switches; electronic scanning devices for radar
beams. There are other possible applications, such as in detection,
frequency multiplication and even parametric amplification; experi­
mental units have been set up in the laboratory, but they have so far
had no practical application since there already exist simpler devices
which have the same function.
It is essential nowadays that the microwave engineer should keep
up to date with these new techniques. L'École Nationale Supérieure
d'Aéronautique has introduced several lectures on ferrites into its
curricula and it was thought to be useful to condense them into this
book. Essentially, it is an attempt to clarify ideas on the relevant
phenomena while at the same time familiarizing the reader with the
apparatus described. Mathematical explanations are reduced to the
strict minimum and only the results of calculations are indicated. There
is a bibliography at the end of the book referring the reader to special­
ized articles or books in which he can find fuller explanations than are
here possible.
L. THOUREL
vii
CHAPTER I
A REVIEW OF THE THEORY
OF MAGNETISM
I.l. DEFINITIONS
A permanent magnet (or an electric current) creates a certain magnetic
field around itself. An ideal magnetic mass m placed in this field is
subjected to a force F proportional to m:
F = mB. [I.l]
By definition, Β is the magnetic induction of the medium surrounding
the mass m. This induction depends on the nature of the medium; it is
convenient to define a quantity, independent of the latter, called the
magnitude of the magnetic field and related to the induction by the
expression:
Β = /^H, [1.2]
where μ is the permeability of the medium.
For a given medium, the value of μ depends on the system of units
chosen. If His expressed in oersteds and Β in gauss, = 1 for a vacuum
and, for practical puφoses, for air. In these conditions, the value of μ
to be taken in formula [1.2] is the relative permeability of the medium
with respect to a vacuum.
When a small magnet consisting of a thin rod of length / with masses
+ m and — m at its extremities is placed in a magnetic field, it is acted on
by a couple which is given by the relation:
C = Μ Λ B. [1.3]
In this formula Μ is the magnetic moment of the magnet. It is equal in
magnitude to the product of m and /.
In a magnetized body, each molecule can be considered as a small
magnet: an element of volume dv will therefore have a certain magnetic
moment dM which will be the sum of the moments of all the molecules
in dv. dM is proportional to dv and the magnetic state of the body can
be represented by the magnetic moment per unit volume or the magnetic
intensity which we designate by the letter Μ (in some works, this
quantity is denoted by the letter J).
2 THE USE OF FERRITES AT MICROWAVE FREQUENCIES
The induction inside a magnetized body is given by the formula:
Β = Η + 4 π M, [1.4]
or Β = Η (1 + 4 TT χ), [1.5]
Μ
with χ = -. [1.6]
The coefficient χ is called the magnetic susceptibility of the body.
1.2. THE BOHR MAGNETON
The whole modern theory of magnetism is based entirely on the law
of the equivalence of currents and magnets established by Ampere.
This theory states, in fact, that the origin of magnetism is the rotation
of electrons, partly round the orbits which they describe around the
atomic nucleus and partly round themselves.
Mo
FIG. I.l.
An electron of charge — e rotating on a circular orbit (Fig. I.l) at a
frequency of/rotations per second is equivalent to a circular current:
/ = - ? / ·.
According to Ampere's law, this current produces a magnetic mo­
ment given by:
MQ = μο S /,
where /^o is the permeability of the medium and S is the area enclosed
by the orbit. Thus:
M,= -μ,Sef. [1.7]
However, the electron is also rotating on its own axis (spin); this
fact gives rise to a second magnetic moment or spin moment. Finally,
the magnetic moment of the electron is the vector sum of the orbital
moment and the spin moment. The latter moment, called the Bohr
magneton, is the smallest quantity of magnetic moment that can exist.
Its magnitude is:
MB = 9^27 X 10-21 Mx-cm. [1.8]
1.3. DIAMAGNETISM AND PARAMAGNETISM
For any material the magnetic moment of an atom is equal to the
sum of the moments of its electrons. In a large number of cases, this
A REVIEW OF THE THEORY OF MAGNETISM 3
moment is zero due to the symmetry of the atom which contains as
many electrons rotating in one sense as in the other sense. Just as in
electromagnetism, where a flux gives rise to an induced current tending
to oppose any change, a substance which is acted on by a magnetic field
tends to protect itself by the creation of internal charges. If the moments
of the atoms are zero, then the induction is smaller than the applied field
and the magnetic susceptibility must be negative. Such a substance is
said to be diamagnetic.
The magnetic susceptibility is very small; here are three values:
Water: X = - 0-91 X 10-^
Copper: χ = - 1-39 χ 10"^
Bismuth: X = - 16-7 χ 10"^
If the magnetic moment of the atom is not zero, the body is said to
be paramagnetic: substances whose atoms have an odd number of
electrons are paramagnetic. This is the case for atoms which have an
incomplete internal shell as well as for free ions. Metals are often
paramagnetic because of the presence of conduction electrons, as are
also some bodies which have an even number of electrons.
The susceptibility is now positive but it is still very small (of the order
of 10~^ to 10~*). It varies inversely with absolute temperature:
X = C/T,
C being Curie's constant.
1.4. FERROMAGNETISM
A substance is said to be ferromagnetic when it possesses a spon­
taneous magnetic moment in the absence of any applied magnetic
field. The spontaneous magnetic moment per unit volume is called the
saturation intensity M^. From [1.4], the saturation induction is thus:
Β, = 4π Μ, [1.9]
An explanation of this phenomenon was first given by Weiss who
postulated the existence of a force tending to align all the magnetic
moments of the atoms in the material in one direction. Thus, a ferro­
magnetic substance is a paramagnetic substance in which there exists
a field, which Weiss called the molecular or exchange field, which aligns
the magnetic moments of all the paramagnetic atoms into one direction.
Heisenberg later showed that the molecular field was a coupling of
electrical origin which existed only if the distances between the atoms
in the material lay between certain well-defined limits. The energy of
this coupling is called the exchange energy.
4 THE USE OF FERRITES AT MICROWAVE FREQUENCIES
It can be shown that, for two neighbouring atoms, whose spin
moments equal to S and at an angle φ to each other, the exchange
energy is given by:
w = J 8^φ\ [1.10]
where J is a coefficient called the exchange integral.
However, the exchange energy which tends to align the moments is
more or less counterbalanced by the thermal energy which, by random
agitation of the atoms, tends to destroy the alignment. This energy is
proportional to the product of Boltzmann's constant {k = 138 χ lO'^»
erg/X) and the absolute temperature.
FIG. 1.2.
At low temperatures the exchange energy is preponderant and a large
number of the moments are aUgned. The saturation intensity M, is,
thus, relatively important. As the temperature rises, the increase in the
thermal energy gradually destroys the alignment and the saturation
intensity decreases. When the two energies are equal, the saturation
intensity becomes statistically zero. The temperature at which this
occurs is called the Curie point or temperature.
Above the Curie point, the substance is no longer ferromagnetic and
becomes paramagnetic. Below it, the moment M, varies as shown in
Fig. 1.2.
The value of the molecular field can be deduced from a knowledge
of the Curie temperature T^: in the case of iron it turns out to be of the
order of 5 milUon oersteds.
Since the exchange energy is important only if the distances between
atoms lie between very narrow limits, the number of ferromagnetic
materials is very small: in fact, only iron, cobalt and nickel exhibit this
property at normal temperatures (soUd oxygen is ferromagnetic).
However, a ferromagnetic material is not necessarily magnetized to
saturation. The reason is that in the absence of an external magnetic
A REVIEW OF THE THEORY OF MAGNETISM 5
field, the substance is divided into small regions, called Weiss domains,
in which the moments are aligned spontaneously and which are therefore
saturated. But the direction of magnetization varies from one domain
to another so that the overall magnetization can be quite small or even
zero. Figure 1.3 shows schematically the distribution of Weiss domains
FIG. 1.3.
in a ferromagnetic material and the direction of magnetization for each
one. This microscopic domain structure has been verified photo­
graphically. It is assumed that each domain is separated from its
neighbour by a thin region, called a Bloch wall, in which the magnetiza­
tion changes direction.
In the presence of an external field, two things can happen: an increase
in the size of the domains in which the magnetization lies in the direc­
tion of the appHed field, or a rotation of the magnetization directions
in the various domains tending to align them with the field. The first
of these two effects seems to exist for relatively weak fields and the
second for fields approaching saturation. When the latter condition is
reached (Fig. 1.4), all the domains are aligned with the applied field.
Η applied
FIG. L4.
1. Zone where the phenomena are reversible.
2. Zone where magnetization is due to an increase in the size of certain domains.
3. Zone where magnetization is due to rotation of the magnetization'of the domains.
6 THE USE OF FERRITES AT MICROWAVE FREQUENCIES
If the appHed field is reduced or removed after a specimen of ferro­
magnetic metal has been magnetized to saturation, the domains do not
revert to their original positions. A certain amount of magnetization,
or remanent induction B,, is left. If it is large, we have a permanent
magnet; if it is small, the material is said to be soft. The coercive field is
the field which must be applied to reduce the remanent induction to
zero.
1.5. FERROMAGNETIC CRYSTALS. ENERGY OF ANISOTROPY
In a crystal lattice the atoms are arranged in a regular manner and
the planes of the electron orbits lie in fixed directions. In the absence
of any external field, the distribution of the domains is such that the
magnetization in the crystal is zero. Figure 1.5 shows a possible domain
distribution for a cubic crystal.
1
Ho
\ ^
t Vi
/ —\
(a) (b) (c)
FiG. 1.5.
The orbital moment having a well-defined direction, the action of an
external field will depend upon the angle between this direction and that
of the applied field. Magnetization is due either to an increase in the
size of one domain (Fig. I.5b) or to a rotation of the magnetization
(Fig. I.5c). In either case, the energy required for magnetization will be
small if the orbital and spin moments are oriented so that they align
easily with the direction of the field. Otherwise, it will be large. Thus, as a
result of the anisotropic nature of the medium, there will be directions
of easy magnetization and directions of difficult magnetization in a
crystal.
The extra magnetic energy required to magnetize the crystals to
saturation in the latter case is called the energy of anisotropy or
magnetocrystalline energy.
Finally, it appears that certain fields attract the elementary moments
into the directions of easy magnetization. These fields are called
anisotropy fields. It can be shown that if the magnetization is near an
"easy" axis, the anisotropy field is: