PB 151408

technical r[ot&

jioulder laboratories

920.49

DYNAMIC MEASUREMENTS OF THE

MAGNETOELASTIC PROPERTIES OF

FERRITES

VIRGIL E. BOTTOM

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

technical *ylote

49

DYNAMIC MEASUREMENTS OF THE

MAGNETOELASTIC PROPERTIES OF FERRITES

by

Virgil E. Bottom

This work was partially supported by

Department of the Navy

Bureau of Ships

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TABLE OF CONTENTS

DYNAMIC MEASUREMENTS OF THE

MAGNETOELASTIC PROPERTIES OF FERRITES

by

Virgil E. Bottom

Page

ABSTRACT 1 TABLE OF SYMBOLS ii Rationalized M. K. S. System V

CONVERSION TABLE vi

LIST OF FIGURES viii

1. INTRODUCTION 1

2. DEFINITIONS OF THE ELASTIC, MAGNETIC AND

MAGNETOELASTIC COEFFICIENTS 4

3. DERIVATION OF THE EQUIVALENT CIRCUIT OF THE

VIBRATING RING DRIVEN BY MAGNETOSTRICTION 8

4. THE EQUIVALENT ELECTRICAL CIRCUIT 17

4. 1 Determination of Circuit Parameters (High Q) 19

4. 2 Determination of Circuit Parameters (Low Q) 22

5. EVALUATION OF THE PHYSICAL COEFFICIENTS OF

THE FERRITE 25

5. 1 Young's Modulus 25

5. 2 Magnetostriction Coefficients 26

5. 3 Permeability 26

5.4 Q of the Ferrite 27

5. 5 The Magnetoelastic Coupling Coefficient (mecc) 29

5. 6 The Strain and Displacement at the Resonant

Frequency 33

TABLE OF CONTENTS (Cont. ) Page

EXPERIMENTAL PROCEDURE 34

6. 1 Dimensions 34

6. 2 Density 35

6. 3 Inductance 35

6. 4 Measurement of Series and Parallel Impedances

and Frequencies 36

6. 5 The Magnetoelastic Coupling Coefficient 38

6. 6 The Parameters of the Equivalent Circuit 38

6. 7 The Quality Factor 39

6.8 Young's Modulus 39

6. 9 The Magnetostriction Coefficient 40

6. 10 The Permeability 41

6.11 The Biasing Field 41

REFERENCES 43

-1-

DYNAMIC MEASUREMENTS OF THE

MAGNETOELASTIC PROPERTIES OF FERRITES

by

Virgil E. Bottom

ABSTRACT

The relations between the mechanical and magnetic properties

of a ferrite are derived for the small signal or reversible condition

using thermodynamic principles. The equation of motion of a ferrite

ring driven in its fundamental mode is set up and solved leading to the

equivalent electrical circuit of the magnetostriction resonator. From

mechanical measurements of the density and dimensions of the ring

and electrical and frequency measurements of the resonator, the

elastic modulus, permeability, magnetostriction coefficients and loss

factors in the ferrite can be determined. Apparatus is proposed for

performing the above measurements.

-II- TABLE OF SYMBOLS

Symbol Definition Units

d =

d> =

A

A

c

a

B

C = Cm

8e_

8H

8B

H

( BB )

=

0" ( 8. )

e

H

Jk

Km |jl L = L

Area of cross section

of ring

Area of air space between

ring and toroid

Mean radius of ring

Flux density in air = u H

oFlux density or magnetic

induction

Motional capacitance of

equivalent network

Magnetostriction strain

coefficient

Magnetostriction strain

coefficient

Instantaneous voltage

meter

meter

meter

weber / meter

weber/

meter

farad

weber

/

newton

ampere meter/

newton

volt

Frequency at resonance cycles

/

(no damping) second

Frequency at parallel cycles /

resonance (reactance zero) second

Frequency at series cycles /

resonance (reactance zero) second

Magnetic field intensity ampere /meter

Maximum instantaneous ampere

current

nTH" (none)

Magnetoelastic coupling (none)

m

coefficient

Relative permeability

Motional inductance of

equivalent circuit

(none)

henry

-111-

TABLE OF SYMBOLS

(Cont. )

Symbol Definition Units

L Inductance with ferrite henry

c clamped

N Number of turns on toroid

q • ( *i Elastic (Young's) modulus newton/

at constant field meter

q' <n Elastic modulus at constant newton/

induction meter

Q Quality factor (none)

r Damping coefficient

7

kg /meter

second

R = Rm

tW

Z

m

Motional resistance of ohms

equivalent network

Resistance of circuit with ohms

ferrite clamped

Time second

Work or energy

Impedance of equivalent

circuit at parallel resonance

Impedance of equivalent ohms

circuit at series resonance

Motional impedance of ohms

equivalent circuit

Impedance of circuit with ohms

ferrite clamped

joule = newton meter

ohms

Strain (none)

Maximum strain ampli tude (none)

Magnetic flux weber — - paw + joor

J newton/

meter

-IVTABLE OF SYMBOLS

(Cont. )

Symbol Definition Units

(fri (s-i

/ 8o_\ / 8H\

!

m

H'

y. =

(8).

8B

8H

<r

9

P

xo

COo

COa

CO s

Aco = ZirAf

Magnetostriction stress

coefficient

Magnetostriction stress

coefficient

newton/

ampere meter

newton /

weber

Permeability of space (air) weber/

(12.57 X 10 ) ampere meter

Static permeabilityMotional permeability

Permeability at constant

stress

Permeability at constant

strain

Complex permeability of

ferrite

weber /

ampere meter

weber/

ampere meter

weber/

ampere meter

weber/

ampere meter

weber/

ampere meter

newton /mete r

'

radian

kg /meter

Stress

Angle

Density

Instantaneous displacement meter

Displacement amplitude meter

Angular frequency of radian/sec

resonance

Angular frequency at parallel radian/sec

resonance (zero reactance)

Angular frequency at series radian/sec

resonance

(co - co ) difference between radian/sec

s a

series and parallel resonance

frequencies

-VRationalized M. K. S. System

All quantities in the above table are expressed in the rational'

ized M* K. S. system. In this system

B = u H + xH

o

where x is called the susceptibility and xH is the intensity of magnet!

zation or magnetic moment per unit volume.

B = (p. + X )H

o

B = \x H

\i = a + X o

iL_ = i + JL

o o

K = — is called the relative permeability, m u, o

-viCONVERSION TABLE

Each of the four magnetostriction coefficients, X, X', d and d'

may be expressed in terms of the other and the appropriate modulus

of elasticity and permeability. From the definitions of the quantities

on page 5 we have

* < {&). ( mb-x

*

» < *V VS-XO' • «<

d = 8e_

8H (H-K&X-v

d

/ _ 9H \ / 9H \ / 8e

»° 4 v^yB v^yB

q'

from which the following conversion table can be constructed with the

aid of the relation *— = — ( e q. 32, page 31).

|j/ q

X = X'u. d = -

q

X = dq d = d'}j/

X = d'qV = d'qp/ d = X'-^ = X'-^-

q q'

jj. q'

X' = d'q' /- 1

»'

-viiX/= d 5 = d ^ d'= -A.

jx M-' q'p. qn'

Use of the above table permits comparison of data expressed in

different ways so long as the definitions used in obtaining the data are

consistent.

The older data on ferrites are usually expressed in c. g. s. units

and later data in m. k. s. units. The conversion table is, of course,

applicable to either system. To convert from c. g. s. units to m. k. s.

units the following conversion factors are required.

5

Force 1 newton = 10 dynes

Mass 1 kilogram = 10 grams

4

Induction 1 weber /square meter = 10 gauss

-3

Field 1 ampere /meter = 4ir X 10 oersteds

2 4 2

Area 1 meter = 10 cm

-VlllLIST OF FIGURES

Fig. No. Title Page

1 Magnetostriction Ring Resonator 8

2 Displacement in Ring Resonator 9

3 Forces on Element of Vibrating Ring 9

4 Motional Part of Equivalent Circuit 16

5 Complete Equivalent Circuit 17

6 Resonance Curve of the Magnetostriction

Resonator 24

7 Bridge and Biasing Circuit 36

8 Magnetostriction Resonator Apparatus 37

-1-

DYNAMIC MEASUREMENTS OF THE

MAGNETOELASTIC PROPERTIES OF FERRITES

by

Virgil E. Bottom

1. INTRODUCTION

In common with other ferromagnetic materials, the ferrites

exhibit the property of magnetostriction or changes of dimensions upon

magnetization. The simplest form of the phenomenon is the Joule

effect or longitudinal magnetostriction discovered in 1847 by J. P. Joule*

Although magnetostriction in general involves other forms of stresses

and strains, the term is commonly taken to mean the Joule effect and

it is in this sense that the term is used in this paper. The fractional

change in length or strain in ferrites at saturation magnetization is

-6 10 reported to range from about -0. 5 to + 40 X 10 (Popper , 1953)

depending upon the composition and method of preparation.

Numerous static methods have been devised for measuring the

magnetostriction strain, These include optical and mechanical levers,

strain gages, interferometer and capacitance techniques and combinations of these methods.

For various reasons, dynamic measurements are desired and

a number of experimenters have devised methods of measuring the

strain, the magnetostriction coefficients, the elastic coefficients and

permeability by observations made on vibrating rods and rings.

As pointed out by Rowland, many years ago, magnetic measurements are much more simply made using rings than rods due to the

absence of demagnetization due to the poles at the ends of the rod.

The theory of the vibrating ring driven by magnetostriction is also

much simpler than that of the driven rod because the strain is uniform
-2-

in the ring whereas it varies along the length of the rod in a complex

manner.

One difficulty which is encountered in the use of the vibrating

ring is the absence of nodal points for supporting it. The ring vibrates,

in the fundamental mode, in such a way that the displacement of every

point in the ring is radial leaving no nodal points for support. However the losses are not very great if the ring is supported on thin

radially stretched fibres. Another difficulty with the use of ring

oscillators is that of winding accurately dimensioned toroidal coils on

the specimen. One solution to this problem is to construct demountable half toroids which can be assembled over the specimen.

The theory derived in this paper is based upon the assumption

that the ferrite is isotropic. The magnetic and mechanical properties

are treated as scalars instead of tensors as will be required when single

crystalline ferrites are to be investigated. The extension of the theory

to include the effects of tensor properties can readily be made.

Much confusion exists in the literature concerning the relationships between the magnetic and mechanical properties. In a number

of cases, authors have not been careful to define quantities completely

by specifying the conditions under which the measurements were made.

Analytical expressions relating the various magnetic and mechanical

coefficients exist, in general, only for the small signal or reversible

case, under which condition the relationships can be derived by the

use of the methods of thermodynamics. There are many ways of

setting up the relationships, each being equally logical. The choice,

therefore, is determined by practical consideration and in particular

by the ease with which the conditions can be realized experimentally.

The coefficients obtained by a dynamic method are the adiabatic

coefficients whereas those obtained by static methods are the isothermal coefficients. Generally the difference is less than the error of

measurement and this will probably be the case with ferrites.
-3-

Equivalent circuits have been used to represent electromechanical transducers for many years. One of the first investigations of the

1 theory of equivalent circuits was made by Butte rworth in 1915 in connection with his studies on the vibration galvanometer* Butterworth

showed quite generally that any mechanical vibrating system with one

degree of freedom driven by the interaction between a current and a magnetic field behaves as if it were an electrical circuit consisting of an

inductance, a resistance and a capacitance in parallel. He also showed

that a mechanical system excited by the interaction between charged

bodies and an electric field is equivalent to an electrical circuit consisting of the same three elements in series. The latter equivalent circuit

has been shown to represent the piezoelectric resonator with very high

precision and has been of great value in the development of filter and

oscillator circuits employing quartz crystal units.

The first paper on the magnetostriction resonator was published

2

by Pierce in 1928. In his paper Pierce derived the equations for the

electrical impedance of the magnetostriction resonator but he did not

interpret the equations in terms of an equivalent circuit. In 1931

5

Butterworth and Smith published a paper in which they used a series network of a capacitance, an inductance and a resistance to represent the

motional impedance of the magnetostriction resonator. It can readily be

shown that the equivalent network used by Butterworth and Smith is a limited equivalent network, applicable at only one frequency whereas the

absolute equivalent circuit of a transducer is applicable at all frequencies.

3, 10

4 8 9 11 12 A number of authors " have proposed limited equivalent circuits for the magnetostriction resonator and at least two authors

have referred to the absolute equivalent network but the literature on the

subject does not appear to provide a rigorous derivation of the absolute

equivalent network. It is the purpose of this paper to derive the absolute

equivalent circuit of the magnetostriction resonator and to relate the constants of the equivalent circuit to the physical properties and the dimensions of the resonator. The equivalent circuit is essential to the design
-4-

of filters and oscillators -which employ ferrite elements as resonators

and in determining the values of the coefficients of the physical properties

of fer rites by dynamic methods.

2. DEFINITIONS OF THE ELASTIC,

MAGNETIC AND MAGNETOELAS TIC COEFFICIENTS

The relationships between the magnetic and elastic properties

of an isotropic ferrite may be expressed in a number of different ways

depending upon whether the induction or the field is expressed in terms

of the stress or the strain and vice versa. Since the elastic modulus

depends upon the state of magnetization in a ferrite, it is necessary

to define two values of the modulus, i. e. , the modulus at constant

field and the modulus at constant induction. In the same way the permeability depends upon the elastic state so that it becomes necessary

to distinguish between the permeability at constant stress and the permeability at constant strain.

The magnetostriction coefficient may be defined in four ways

depending upon whether the stress or the strain is related to the field

or the induction. In addition there are four inverse coefficients relating

the stress and the strain to the field and the induction. These eight

coefficients are not independent and by the use of thermodynamic

relations the number may be reduced to four independent coefficients.

In order, therefore, to completely describe the magnetoelastic

relations in a ferrite it is necessary to define the following quantities:

e = the strain. (Change in length per unit length)

v = the stress. (Force per unit area)

H = the magnetic field.

B = the magnetic induction.

In each of these definitions it is assumed that the change in the

quantity is small, corresponding to the reversible or small signal condition. It is further understood that the quantity is defined for a par-
-5-

ticular magnetoelastic state of the ferrite. For example, the stress

is defined for a particular value of the strain and the field or of the

strain and the induction. It should be noted that none of the quantities

is a constant and therefore the word "coefficient" is used instead of the

word "constant".

The following coefficients relating the magnetoelastic properties

may now be defined.

q

8a

8e Young's modulus at constant field.

H

q' =

00" 8c B

Young's modulus at constant

induction.

\i = 8B

8H Permeability at constant strain.

H' =

X -

\' = 8B

8B

8H

or

(Sir) -(fV

8H

8e

Permeability at constant stress.

Magnetostriction stress coeffi'

cients.

t -

8e_

8H

8e_

8B

or

<J

da

8H

toH

B

Magnetostriction strain coefficients.

The equalities in the last four definitions follow from the ther

modynamic relations

dU = BdH + o-de

dU = HdB + o- de
-6-

dU = BdH + ed<r

dU = HdB + edoTo prove the first of the four equalities we use the first ther

modynamic equation to write

and 7: I = cr 8H

>c V^/ H

Taking a second derivative

8

2 U /8B\ , a

2

u

H v / e

»cr

Therefore

9B

9e ;__ v du

/ H v / e

The remaining equalities may be proved in the same way using

the other three thermodynamic relations.

The stress in a ferrite may be expressed as a function of the

strain and the magnetic field intensity cr = (r(e, H) or as a function of

the strain and the induction <r = cr (e, B). Similarly we have

£ = £((T , H) and £ = £(cr,B) .

The induction and the field may also be expressed as functions

of the stress and strain leading to the following relationships:

B = B (cr , H)

B = B (£, H)

H = H (cr , B)

H 3 H (e , B)

From these relationships and the definitions above we obtain the four
-7-

direct relations

fa) dor = qde + XdH

(b) da- = q'de + X'dB

(c) ds = - da- + d dH

q

(d) de = —, do- + d'dB

q'

and from the inverse relations

(e) dB = ddo- + jx'dH

(f) dB = Xde + ^dH

(g) dH = d'dcr + — dB

(h) dH = X'de + - dB

VIn all the relationships it is understood that the changes are

reversible, i.e. , infinite simally small and that the values are for the

particular magnetoelastic state specified.

It is, of course, never possible to achieve perfect conditions

of constant stress, strain, field or induction. However, the conditions

may be approximated experimentally in the case of a vibrating ferrite

ring in the following ways!

Constant stress . The condition of constant stress is approximated by

mounting the ring so that it is free of all external constraints. The

ring may be supported on three points or on fine threads in vacuum.

Unfortunately, in the case of the vibrating ring, no nodal points are

available for support.

Constant strain. The condition of constant strain implies that the ring

is not free to move. In theory the ring could be clamped in such a way

that no displacement is possible. In practice it is sufficient to drive
-8-

the ring at a high but non-resonant frequency. For small driving forces,

the displacement and the strain are negligible at a non-resonant frequency.

Constant field. The condition of constant field implies that the magnetic

field due to a coil surrounding the ring is unaffected by changes in the

induction in the ring. Changes in the elastic state of the ferrite produce

changes in the induction which, in general, produce currents in the

coil, thereby changing the field. However, if the impedance of the

circuit is very high, the induced current is small and the field is approximately constant.

Constant induction. The condition of constant induction implies that the

induction in the ferrite is not affected by changes in the elastic state.

This condition can be approximated by surrounding the ferrite with a

shorted coil of very low resistance. Then any changes in the induction

set up currents in the coil which are in the direction to oppose the

changes of induction. If the resistance of the circuit is zero, no change

of induction is possible.

3. DERIVATION OF THE EQUIVALENT CIRCUIT

OF THE VIBRATING RING DRIVEN BY MAGNETOSTRICTION

We consider a uniform, homogeneous polycrystalline, isotropic

ferrite ring of mean radius a with a uniform toroidal winding having

N turns. The area of the cross section of

the ring is A and the radial dimension of

the cross section is small compared with

a, the mean radius of the ring.

Consider an element of the ring

a d 0, fig. 2, having a mass pAad0 where p

is the density of the ferrite. An alternating

Fig. 1 Magnetostriction Ring Resonator
-9-

current in the coil causes the ring to expand radially by an amount \\i.

2

8 \h The kinetic reaction of the element is pAadG

j- . As the ring expands

the length of the arc is increased from ad0 to (a + i(j)d0 and the strain

dtf _ ^d_9 _ ^

io ad 9 a

(i;

The stress normal to the end of the' element required to produce

the strain e is, from eq. a, page 7 S

«r = qe + XH = q^+ \H

a

The tangential force on the end of the element is g-A and the radial component of the

resultant of the tangential forces on the ends

of the element is (fig* 3)

Fig. 2

Displacement in Ring

Resonator

F = a- A dG

r

= (q^ + XH) A dt

a

Fig. 3

Forces on Element of

Vibrating Ring

If we assume that H is produced by a sinus

oidal current 1 = 1 exp jcot then

o

NI

o

2ira

exp jo>t

We further assume the existence of a damping or frictional force proportional to the

velocity having the form rAdG —- =
-10-

The equation of motion thus becomes

2 . ~ . VNI

11 + r *± + 3

9t

Each term in (2) has the dimensions of force /area, i. e. , of

stress. The damping coefficient r has the dimensions of mass/area

per unit time. The assumption that the damping is proportional to the

velocity is, of course, arbitrary, and other assumptions might be used.

It is found, however, for a large variety of damped vibrating systems,

that the assumption leads to results which can be verified experimentally. The justification for the present assumption must rest upon the

consistency of the results obtained. Fortunately, most of the results

do not depend at all upon the nature of the damping force so long as it

is small, a condition which is readily satisfied in the case of many

mechanical vibrating systems.

To solve the equation of motion in the absence of a driving force

we set I =0 and assume

o

i(j = i|j exp joot

which gives for the resonance frequency

r 1 r ; (0 = j ± \J 4pq - r o 2pa 2pa

and for the displacement

qj = l|j exp f )»exp —i- v4pq-r t (3)

\_ 2pa J

2pa
-11-

which is, of course, a damped harmonic motion. The frequency in the

absence of damping is

w = - /5 or f = _L /I e ( 3a ) o a ^ p o 2ira *J p

Just as in any other mechanical system, the actual resonance

frequency is slightly reduced by frictional damping. From (3a) it follows that the velocity of sound in the ring is /— and the period of the

vibration is therefore the time required for a sound wave to travel

around the ring, a distance 2ua. Since the value of q depends upon the

state of magnetization and upon whether B or H is fixed, the velocity of

sound and the period are not constant except under fixed conditions.

We are interested, however, in the ring driven in the steady

state condition. As a solution we try

ip = i|/ exp jo)t

ip = jwip and ip = -co ip

where the dots signify time derivatives,, Substituting into (2) we obtain

2 q , o ip -paco i|i + J«ri|, + -i|i = -^- ^

from which
-12

where

— - paw + ja>r (4a)

a

and the solution of (2) is

XNI

^

= 2^ eXp jWt * ^ 4b)

The motion of the ring is such that all points in the ring experience sinusoidal, radial displacement with the same phase and (if the

ring is thin) the same amplitude. The strain and the induction are uniform at all points in the ring. It is for this reason that the vibrating

ring is to be preferred over vibrating rods in which both the strain and

the induction vary along the length of the rod. The problem is further

complicated, in the case of the rod, by the demagnetizing effect produced by the poles formed at the ends of the rod.

If r = corresponding to the ideal case of zero damping, the

amplitude becomes infinite if

q 2 - - paw

a

or when

1 /q w = o = —

o avp

which is the same as the undamped resonance frequency.

The strain, e s which for the vibrating ring is — , is given by
-13-

— Io exp jot . (5)

2Tra v

From eq. f, page 7, the reversible induction in the ferrite is given

by

B = Xe + [iH (6)

where e and H are the small signal strain and field respectively. Substituting the values of e and H into (6) gives for the induction in the ferrite

b =( -^|L + ^L )Iexpjut . 2 2-rra / o

2ira -y

If the area of the cross section A, of the ring, is equal to that

of the toroidal coil, the flux through the coil is BA. It is not practical,

however, to make the coil fit so closely to the ferrite ring as to make

the areas sensibly equal without introducing damping in the ring. It is

therefore preferable to make the area of the cross section of the coil

larger than that of the ring.

If A is the area of the cross section of the ring and A is the o area of the air space between the ferrite ring and the toroid, then the

flux through the toroid is

= B A + BA

o o

where B is the induction in the air and B the induction in the ferrite. o The changing flux associated with the vibration of the ferrite and the al-
-14-

ternating current in the coil results in a back emf in the coil, the mag'

nitude of which is given by

e = N-

= NA _JEL + NA C o dt dt

But

dB ,m

« „ o dH

B = u H so —-

= jjl —

o o dt o dt

and

dB % de . dH

= Xe+ uH so — = X— +

dt dt ndt

Therefore

e = NA u. ^-+ NAX^J + NAu^- . (7) o r

o dt dt n dt

_,,..,. „de , dH

Substituting for — and we have

r. n^a/ a

o \, . nVa 1 T

. . Ial

I \ / 2ira v J

which may be written as
-15

jW

[ ^1

+ ^

]

2

" N

2A

I exp jcot (8a)

2ira o

where

A . A ^=-%+[i=(-^+K ) fi (8b] r l A

ro V A m / "o

where K = — is the relative permeability of the fernte.

m u,ro

If A is made zero by fitting the coil closely to the ferrite ring

° / dB \

then p = p, the permeability of the ferrite at constant strain, I 7—7-

J •

Eq. (8a) shows that the effect of magnetostriction is to increase the

2

permeability and to make it complex since \ is necessarily positive

and v is in general complex.

The term in the bracket of (8) has the dimensions of an impedance. The first term is the inductive reactance of the coil assuming

the ferrite to be non-vibrating or clamped. The second term is due to

the motion of the ferrite ring excited through the magnetostriction

effect. We may therefore write (8) as

e = f Z + Z II exp jwt L c m J o

where

Z = jX

c c

2 2

N Ap. N Ap

X = u —5 or L = — . (9! c 2ira c Zira
16-

For certain purposes it may be desirable to include the resistance R of the coil and to write

c

Z = R + jooL

c c c

for the impedance of the coil with the ferrite clamped. The resistance

R is principally due to the ohmic resistance (copper loss) in the windc ing of the toroid since eddy current losses should be negligible in most

ferrites, although residual losses may still be present.

The motional impedance Z is from (8) and (4a) m

m

= jw2 2 , X N A/2Traq

2 2

1 _ P a g j. J cora (10)

Z has the same form as that of the impedance of the electrical circuit m

of fig. 4 consisting of an inductance, capacitance and a resistance in

parallel the impedance of which is

AB = jw

m

jooL

(10a)

- to L C + m

mm Rm

l m

n

m 'm

6

B

Hence the equivalent motional inductance is

2 2

X N A

m 2irqa (11)

Fig. 4

Motional Part of

Equivalent Circuit

The equivalent motional capacitance is
-17-

m

2irpa

2 2

X N A

(12)

and the equivalent motional resistance is

R

2 2

X N A

m „ 2

2-rrra

(13)

m

The equivalent circuit of the resonator as seen at the terminal of the

resonator is that of the circuit of fig. 5.

It is of interest to note that R , m

the motional resistance of the equivalent

circuit is very large if the damping coefficient r is small. This somewhat unusual

relationship is due to the presence of R

Lm -T Cm in parallel with C and L . r mm m

Fig. 5

Complete Equivalent Circuit

4. THE EQUIVALENT ELECTRICAL CIRCUIT

The impedance of the equivalent circuit of fig. 5 is

jwL Z = R + jcoL +

c c

I . w

2 lC + *£R

(14)

where for convenience the subscripts have been omitted from the terms

representing the motional parameters. After writing (14) over a common denominator and multiplying the numerator and denominator by

the complex conjugate of the latter , we obtain
-18-

z =

JJVI — < 14a )

(1 - to LC) + —

R^

where the real part of the numerator is

M = R +w (-TT+ ^" ~ 2LCR )+ o> L C R (14b)

\ R

R

2 c 7

and the imaginary part of the numerator is

L

2 L

co 5

(L

2 L_C2

) - co 3

( L

2

C + 2LLC ~ L ) / 2 ^ L . LCI 2LL C r^- ) + (o V R

2 /

The frequencies at which the reactance is zero, which are very

close to the resonant frequencies, are found by setting M' = and solving for oo. If the Q of the circuit is high enough the term involving R

may be neglected. (The low Q case is treated later). Neglecting the

term in R in (14c) we have

co4L

2 L C2

- oo 2

(L

2

C + 2LL C) + (L + L ) = (14d)

c c c

from which, by use of the quadratic formula

co2= —

1 2 1, —

and co = ——- + LC LC L C

c

The first frequency is identified with the mechanical resonance

of the ring and may be termed, from the nature of the equivalent circuit, the parallel resonant frequency. The second and slightly higher

frequency occurs at the frequency at which the motional reactance is

capacitive and numerically equal to the inductive reactance of the coil

L . This frequency may be termed the series resonant frequency. We

c
-19-

may therefore write

2 _ 1 a LC and f = — iff^ (15a)

a 2ir \/ LC

and

2 1 1

CO + -^-r and f = — /~ + t^: (15b)

3 J_i(_/ LJLi C\~» s 2w \/ LC L C

C C

NOTE: Some authors use the opposite designation for the two resonant

frequencies.

4. 1 Determination of Circuit Parameters (High Q)

From (15a) and (15b) we have

2 2 1 a L C

c

1 (w + to ) (to - tO ) =

s as a L C

c

But since to and to are very close together and (to - o> )

' = Ato«co , we

s a s a s may write

2(0 Ato = L C

c
-20

and

C = , A

. = — (16) 2coAcoL 2

c 8tt f AfL

c

where L is the inductance of the coil measured at a non-resonant fre

c quency and Af is the difference between the two frequencies of zero

reactance f and f . s a

The value of L is also readily determined since

1 LC

2 f L

c *X (17 »

where k is called the magnetoelastic coupling coefficient.

The Q of the ferrite may also be determined from the parameters of the equivalent circuit. However, the Q of the circuit of

coL

fig. 4 is not —— as is the usual case with resonant circuits in which

R

the resistance is in series with the coil but is —=- . This may be shown

coL as follows: The impedance Z of fig. 4 is

Z

2

w = a

1 8ir

2

f AfL

c 2~

w C

a

2 2

4tt f

AB . 2 T _ , . coL

1 - o) LC + j —

Rationalizing the denominator gives

2,2 , . 3.2

co L . / coL co L C \

AB RD J ^"D" D J
-21-

2 2 w

2 L2

where D = (1 - w LC) + =— . The above equation represents the

R i impedance of the series circuit Z , , = R' + j (uL' —) which has

coLi a Q given by . In terms of the parameters of the original equiva- R

lent circuit

q = JL = RooC . (18)

COL

This somewhat unusual form of the expression for Q is a consequence

of the relationship between R = R and r of (13). m

In terms of measureable quantities

R R

coL 4tt L Af v ' c

where Af = f -f . s a

In order to find the resistance at the terminals of the toroid at

2 1 the parallel resonant frequency f we substitute oo = —— into the real

a a i-iC

part of (14a) and obtain

Z = R + R . (19)

a c

In the same way the impedance at the series resonant fre2 1 1 quency is found by substituting to = —— + -—— into the real part of (14a).

s LC Li C

c The result, after a considerable amount of algebra, is found to be

z = R + R = R + 5_— ~ • (20]

s c „ 2 c , , , 2 ^ . 2 R 1 + ( k Q)

1 +

2 2

CO Lic
-22-

Taking the difference between (19) and (20) we obtain

R +u L 1 +

c V ,2 rt k Q

Unfortunately both terms in the denominator of (21) are often of the

same order of magnitude so that the determination of the value of R

involves the solution of a cubic equation. The solution can most readily

be obtained by numerical methods.

4. 2 Determination of Circuit Parameters (Low Q)

Thus far we have considered the damping to have negligible

effect upon the frequencies of parallel and series resonance. If, however, as is the case in many ferrite materials, the damping is large

and the Q is small it is necessary to further examine the solution for

the frequencies of resonance. If the term involving R is reinserted

into (14d) the resulting equation has real roots only if the discriminant

is equal to or greater than zero. This leads to the condition that for

2

to to be real

Q > 2

L

c 2

2

or that

2 Qk > 2
-23-

2 Physically this result means that if k Q is less than two, the

motional part of the equivalent circuit is unable to develop a capacitive

reactance numerically equal to the inductive reactance of L . 2

C

Assuming that k Q > 2, equation (14c), including the term in

2

R, may be set equal to zero and solved for co . To a second approximation it is found that

co2

= ± ( 1 + a LCI ' ,22 k Q

and

2 _ _1_ _1_ /. _1_ W

s " L C

+ LC ( " 2 2 c V k Q

Hence the effect of damping is to cause the parallel resonant

frequency to be increased and the series resonant frequency to be

decreased thereby bringing the two frequencies closer together. If 2

k Q is considerably greater than two and if Q is large the effect of

damping on the difference between the two frequencies is small compared with the difference between the two and equations (16) through (21)

are good approximations. In cases where both k and Q are small these

equations may not yield satisfactory results.

2

If k Q is less than two, no frequencies exist (except f = 0) for

which the reactance of (14a) is zero. Consequently the impedances at

the maximum and minimum points are complex. Fair approximations

2

to the values of the complex impedances may be obtained provided k Q

is not much less than two and the damping is not too great by substituting (15a) and (15b) into (14a). The results are found to be

Z = R I R + j(o L

a c c
-24-

and Z = R +

s c

R + jwL

2 2

1 + (k Q)

These equations may be compared with (19) and (20) which are

derived on the assumption that damping has negligible effect upon the

resonant frequencies.

Fig. 6 illustrates qualitatively the effect of damping on the shape

of the resonance curve of the magnetostriction resonator. The effect of

damping is to reduce the value of Q which from (18) results in decrease

ing the value of R. Therefore, the antiresonant impedance is lower in

low Q resonators. Similarly the effect of damping is to slightly increase

the series resonance impedance. As shown before, damping decreases

the interval between the series and parallel resonant frequencies.

|Z|

HIGH Q

f FREQUENCY

Fig. 6 Resonance Curve of the Magnetostriction Resonator
-25-

5. EVALUATION OF THE PHYSICAL COEFFICIENTS

OF THE FERRITE

The values of the elastic modulus, the magnetostriction stress

and strain coefficients, the permeability, Q, and the magnetoelastical

coupling coefficient are all related to the values of the parameters of

the equivalent circuit which can be determined by electrical and frequency measurements. In addition, measurements of the density of

the ferrite and of the dimensions of the ring and toroid are required.

It must be kept in mind that the values of all the parameters are dependent upon the state of magnetization of the ring.

5. 1 Young's Modulus

The value of L can be determined by the use of a bridge or

c Q-meter at a frequency near but not too near the resonance frequency.

The resonance frequency f which, for this purpose may be taken as

either f or f , must be measured. Then from (3a)

a s

f - /*

o 2ira n/ p

from which the elastic (Young's) modulus

q = (§§) = 4.

2 a¥p (22)

H
-26-

5. 2 Magnetostriction Coefficients

From (11) the motional inductance

2 2

X N A

-Li — Xj — m 2irqa

and from (18a)

Lm

= 2L

c £fo

so that

X

4it q a L Af

2 c N

2 Af

o

which may be written in a different form by the use of (22) as

m

^irpaL f Af

= TT V a^^ (23 >

5. 3 Permeability

The effective permeability of the core (ferrite and surrounding

air space) of the toroid in the absence of magnetostrictive strain may

be determined from a measurement of L, , made at a nonre sonant frec quency. From (9)
-27-

2tt a L, P, = T-^ • (24)

N A

Using (8b) and (24) the static permeability

>'(&)

2tt a L A

c o

TVT2A A °

E N A

u . (25)

Eq. (25) gives the permeability of the ferrite in the absence of magneto^

striction strain. This is the value of the permeability which is applicable to the ferrite when clamped or at a nonresonant frequency. At

or near a resonant frequency the permeability becomes complex.

At the frequency of parallel resonance f , v = jrw and the cona a

tribution to the permeability due to the magnetostriction effect is from

(8a)

2 2 m X .X

V- = — = " J va roo aa

which using (13) may be written

,2 aR m X . m

Y

f N A

a

from which the complex permeability of the ferrite at the frequency of

mechanical resonance may be written as
-28-

s j_ . m f _c ^> \ . _ m . (26)

\i* = \i + j\i

2tt aL A X aR

c ° \ m

2 . '"X^o )

"

J , 2 N A / A f N A

a

5.4 Q of the Ferrite

The Q of the vibrator is equal to the Q of the circuit of fig. 4

representing the motional impedance of the vibrating ring. Unfortunately

the Q includes not only the internal losses in the ferrite but also the

losses due to sound radiated in the air and frictional forces at the points

of support. The latter may be minimized by placing the vibrating ring

in vacuum and by careful design of the supporting mechanism but the Q

of the ferrite must always be higher than the measured value.

We have shown that the Q of the equivalent circuit of fig. 4 is

R

Q = — . Using (11) and (1 3) we have

CO Li a m

m to ra r r a

The damping coefficient r is therefore given by

n/ qp r = Q

It is now quite easy to show that the damping has negligible effect upon the frequency of mechanical resonance if Q is large. From

(3) we have that the frequency, including damping effects is
-29-

I-4pq - r w s ^

a 2pa

Substituting for r from (27) we obtain

a a ^ P V 4Q2 /

So that with a high degree of approximation

x) O) W O) ( 1 - 8Q

Hence if Q > 100, co differs from go by less than 0. 002%.

a o

5. 5 The Magnetoelastic Coupling Coefficient (mecc)

Interesting relationships between the elastic moduli at constant

field and induction and between the permeabilities at constant stress

and strain are obtained in terms of the magnetoelastic coupling coefficient k. The magnetoelastic coupling coefficient is defined as the

square root of the ratio of energy stored mechanically to that stored in

the magnetic field. The coefficient k must not be confused with r, the

damping coefficient which is related to the energy dissipated; k relates

only to conservative effects.

The magnetic energy stored in a magnetic field (per unit volume)

is

W = IpH'

2
-30-

If the field is of the type H = H exp jurt then

W = ^ M-H

2

2 o

is the maximum instantaneous energy stored during the cycle. If the

ferrite is clamped, (constant strain)

W = 4 M<H

2

2 o

but if it is free to vibrate (constant stress)

W = \ ix'H

2

2 o

The elastic energy stored at constant stress is greater than that stored

at constant strain since some work is done against elastic forces in

the first case. From the definition of k we then have

2 _ W - W _ p.' - p. (28)

from which

^7 = 1 - k

2

. (29)

It is, of course, also possible to drive the ring mechanically by

applying a sinusoidal stress of the form <r = <r exp jcot. The maximum

instantaneous energy stored (per unit volume) at constant magnetic

field is
-31-

».!'•

2 q

But if the induction is held constant by winding the ring with a shorted

coil of negligible resistance

2 q'

W < W because the contribution of the magnetostriction effect to the

strain is eliminated. Therefore, from the definition of k

S - ^ - ^

(30.

from which

^r = 1 - k

2

. (31)

q

We thus have the interesting relation that

i = ± = 1 - k

2

. (32)

q' h-'

The mecc can also be related to the parameters of the equivalent circuit. The power into the circuit is given by

I 2

co L

c
-32-

while the power into the resonator is

I2

u) Lm

From the definition of k we have

n 2 m

c

which agrees with (17)

.2

Lm Af

k = IT =

Z T

Therefore

q=q'(l-2f I < 33 )

and

V. = »>( 1 -2^ ) (34)

where f may be taken as either f or f , the difference being ordinarily

o as

negligibly small.
-33-

5. 6 The Strain and Displacement at the Resonant Frequency

From (4), (4a) and (4b) the displacement in a ring vibrating in a simple radial mode is

XN T

ib = I exp jcot T

2iraY o

r

At the resonant frequency f , v = jco r

The strain e in the ring is — so that at the frequency f

d. 3.

z = ^ = - j r I exp jot . (35)

a _ 2 o

2ira w r a

Eq. (35) shows the strain amplitude to be

e =

X]

?

I (36) o 2 o

2"ira cor

or using (27) and (13)

R I XNQ _ mo /,_» e = I = . (37) o 2 , o co\NA

2ira con/ qp

The radial displacement amplitude

\ tvt ^ a R I

i XNQ m o Mo\

ib = & a =

, I = ——— . (^>o; o o 2Traco\/ qp o coXNA
-34-

6. EXPERIMENTAL PROCEDURE

One advantage of the dynamic method is that a number of coefficients can be evaluated from a relatively small number of measurements. In order to evaluate the magnetoelastic coupling coefficient k,

the reversible complex permeability p.*, the magnetostrictive coefficient X, Young's modulus q, and the elastic quality factor Q of the material, the following mechanical measurements are required:

A = the area of the cross section of the ring.

A = the area of the air space between the toroid and the

o

ring.

N = the number of turns on the toroid. a = the me an radius of the ring.

p = the density of the ferrite.

In addition, the following measurements must be made electrically.

L = the inductance of the coil at a non -re sonant frequency,

c

f = the antiresonant frequency.

f = the series resonant frequency.

Z = the resistance at the antiresonant frequency.

Z = the resistance at the series resonant frequency.

6. 1 Dimensions

It is desirable that the thickness of the ring in the radial direction be made as thin as possible in order that the mean radius may be

more precisely defined. The mean radius may be taken to a first approximation to be one -fourth the sum of the inside and outside diameters.

The area of the cross section should be determined with an accuracy of

1% or better. To permit a measurement of this accuracy, the radii

and thickness of the ring must be uniform to better than 0. 5%.
-35-

6. 2 Density

The density of the ferrite may be determined by taking the ratio

of the mass to the volume computed from the relationships p = — and

y-hDt- D>

where D and D are the outside and inside diameters respectively and

h is the thickness of the ring. Alternatively the density may be determined by immersing the ring in a fluid of density p and using the relationship

W

a

P = P

i W - W„

a SL

where W is the weight in air (vacuum) and W is the weight immersed

a x in the liquid.

6. 3 Inductance

The inductance L. is measured at a non-resonant frequency

c close but not too close to the resonant frequency. It is desirable to

measure L. at several frequencies to insure that the measured value

c

is not being influenced by mechanical resonance in the ring. JL may

be measured by means of a Q-meter or better by means of a suitable

radio frequency bridge. Fig. 7 shows the basic circuit diagram of a

radio frequency bridge with the magnetostriction apparatus and the

biasing circuit.
-36-

Fig. 7. Bridge and Biasing Circuit

The circuit of the bridge permits the exciting winding

of the resonator to be used also for supplying the biasing field since

one terminal of the coil is grounded and the other is in series with

capacitor C which isolates it from the rest of the bridge for direct

P

currents. It is therefore possible to pass a direct current through the

coil without affecting the bridge. It is necessary, however, to provide

an isolating inductance L. having an impedance which is high compared

with that of the resonator.

6.4 Measurement of Series and Parallel

Impedances and Frequencie

Fig. 8 shows the apparatus used for measuring the series and

parallel resonant frequencies and impedances.
-37-

COUNTER

OSCILLATOR

AND

AMPLIFIER

V\A/—

±r

L^^v-I

Fig. 8. Magnetostriction Resonator Apparatus

The oscillator must be adjustable and stable to one cycle per

second or better. The amplifier must be capable of developing from

1 to 10 volts across R -which is small compared with the series resonant resistance of the resonator. R is small compared with R . Typical values for R and R are 50 ohms and 1 ohm respectively but

the actual values are determined by the series resonant impedance of

the resonator. The current through the resonator is measured by the

vacuum tube voltmeter and the frequency by the frequency counter.

The values of Zasand Z are measured by the substitution

method using a calibrated variable carbon resistor or a decade resistance box of suitable frequency range.

The frequency of parallel resonance is, by definition, the frequency at which the reactive component of the motional impedance of the

equivalent network is zero. This frequency differs only very slightly

from the frequency of mechanical resonance in the absence of damping

and from the frequency of maximum impedance. If the Q is high, the

differences between the three frequencies are very small and may be

neglected. To a very high degree of approximation, even when the Q

is not high, the frequency of series resonance is equal to the frequency

of minimum impedance. It is therefore usually adequate to consider
-38-

Af to be the difference between the frequencies of maximum and minimum impedance.

6. 5 The Magnetoelastic Coupling Coefficient

The magnetoelastic coupling coefficient (mecc) k is given by

2 Af k = 2=y

where Af is the difference between the frequencies of series and parallel resonance and approximately the difference between the frequencies

of maximum and minimum impedance. To a sufficient degree of accuracy f may be taken as either frequency.

6. 6 The Parameters of the Equivalent Circuit

The inductance in the motional part of the equivalent circuit is

given by

L = k

2 L m c

The motional capacitance is

cm

8*

2 fAfL

c
-39-

The motional resistance R is found from the relationship m

R

3

m

l Zal-l Zsl = -2 22

R + w L m c

6.7 The Quality Factor

The mechanical Q or elastic quality factor is determined from

the relation

R

Qm 4ir L Af

c

The measured value of Q will always be somewhat lower than the m

true Q of the ferrite because of the effects of air damping and fric- m

tional losses at the support points. These may be minimized by careful design of the mounting system and by placing the ring in vacuum.

Rings with chips or cracks may be expected to exhibit abnormally low

Q's because of frictional losses at these points.

6. 8 Young's Modulus

Young's modulus is determined from the relationship for the

resonant frequency giving

q = 4t fa p
-40-

where a is the mean radius of the ring. The above relationship is derived on the assumption that the radial thickness of the ring is negligible compared with the mean radius of the ring. If the ring is not thin,

it may be necessary to consider the effects of lateral inertia. It can

be shown that the error made in the determination of q as a result of

neglecting the radial thickness of the ring is of the order of

2

T

loir a

where t is the radial thickness and a the mean radius of the ring. For

a small ring having an inside radius of 0. 6 cm and an outside radius

of 1. cm the error introduced into the determination of q by neglecting the radial thickness of the ring is about 0.2%.

6. 9 The Magnetostriction Coefficient

The magnetostriction stress coefficient is given by

N A f

The other magnetostriction stress and strain coefficients may be cal'

culated using the conversion table on page vi.
-41-

6. 10 The Pe rmeability

The permeability at the resonant frequency is complex. The

real part is the ordinary permeability of the non-vibrating ring. The

imaginary part is due to the motion of the ring. The complex permeability is given by

u* = (

Ztt aL A V aR

c o Nk

. m

. T2 A " A" ^o ,

) "

J 2 N A / f N A

6. 11 The Biasing Field

In order to obtain a complete picture of the behaviour of a ferrite it is necessary to measure the permeability, coupling coefficient,

magnetostriction coefficient, Q, etc. , under different magnetic and

elastic conditions. In particular it is desirable to determine the values

of the various parameters at different values of the magnetic field or

induction from zero to saturation. For this purpose it is necessary to

provide a biasing field of known intensity. The biasing field may be

provided in various ways; by use of a second winding over the exciting

winding, by a heavy conductor along the axis of the ring or by passing

a polarizing current through the exciting winding. In every case it is

necessary to suppress the currents induced in the conductor by the

exciting current and by the change of induction resulting from the mechanical motion of the ferrite if constant field conditions are desired.

It is necessary, therefore, to include in the biasing circuit an inductive

reactance which offers high impedance at the resonant frequency of the

ringo

On the other hand, if conditions of constant induction are desireds the biasing circuit must include substantially zero impedance at the
-42-

resonant frequency. This requires that the direct current supply be

shunted with a capacitor having an extremely low impedance at the

resonant frequency. Neither of the two conditions is completely realizable but the condition of constant field appears to be more readily

approximated.

The magnetic field around a long straight conductor at a distance

a from the conductor is

H = * 2tra

where I is the current in the conductor. For a toroid of N turns on a

core of mean radius a

H - NI

2ira

For either method of furnishing the biasing field the number of ampere

turns required to produce induction B is

aB 7

NI = —-10

m
-43-

7. REFERENCES

1. S. Butterworth, Proc. Phy. Soc. , 27, 410-424(1915).

2. G. W. Pierce, Proc. Amer. Acad. Sci. , 63, 1 (1928); Proc. IRE,

J7, 42-88 (1929).

3. J. H. Vincent, Proc. Phy. Soc, 41, 476-485 (1929).

4. F. D. Smith, Proc. Phy. Soc, 42, 181-191 (1930).

5. S. Butterworth and F. D. Smith, Proc Phy. Soc, 43, 166(1931).

6. J. H. Vincent, Proc Phy. Soc, 43, 157-165(1931).

7. L.. F. Bates, "Modern Magnetism", 418-421, (Cambridge University Press, 1939).

8. F. J. Bech, J. S. Kouvelites and L. W. McKeehan, Phys. Rev. , 84, 957-963 (1951).

9. C. M. van der Burgt, Phillips Res. Rep., 8, 91-132 (1953).

10. Richards and Lynch, "Soft Magnetic Materials for Telecommunications", chapter by P. Popper, p. 323, (Interscience, 1956).

11. C. M. van der Burgt, J. Acoust. Soc. Am., 28, 1020-1032 (1956).

12. C. M. van der Burgt, Phillips Tech. Rev., 18, 285-298(1956-57).

U.S. DEPARTMENT OF COMMERCE

Frederick XI. Mueller, Secretary NATIONAL BUREAU OF STANDARDS

A. V. Aslin, Director THE NATIONAL BUREAU OF STANDARDS

Tin- scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder. Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

In general, each section carries out specialized research, development, and engineering in the field indicated by

its title. A brief description of the activities, and of the resultant publications, appears on the inside of the front

cover. WASHINGTON, D.C.

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Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion. Metal

Physics.

Mineral Products. Engineering Ceramics. Glass. Refractories. Enameled Metals. Constitution and Microstructure.

Building Technology. Structural Engineering. Fire Protection. Air Conditioning, Heating, and Refrigeration. Floor, Roof, and Wall Coverings. Codes and Safety Standards. Heat Transfer. Concreting Materials.

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics. Data Processing Systems. SEAC Engineering Group. Components and Techniques. Digital Circuitry.

Digital Systems. Analog Systems. Application Engineering.

• Office of Basic Instrumentation. • Office of Weights and Measures. BOULDER, COLORADO

Cryogenic Engineering. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Gas Liquefaction. Radio Propagation Physics. Upper Atmosphere Research. Ionospheric Research. Regular Propagation

Services. Sun-Earth Relationships. VI1F Research. Radio Warning Services. Airglow and Aurora. Radio

Astronomy and Arctic Propagation.

Radio Propagation Engineering. Data Reduction Instrumentation. Modulation Research. Radio Noise. Tropospheric Measurements. Tropospheric Analysis. Propagation Obstacles Engineering. Radio-Meteorology. Lower Atmosphere Physics.

Radio Standards. High Frequency Electrical Standards. Radio Broadcast Service. High Frequencj Im- pedance Standards. Electronic Calibration Center. Microwave Physics. Microwave Circuit Standards.

Radio Communication and Systems. Low Frequency and Verj Low Frequency Research. High Fre- quency and Very High Frequency Research. Ultra High Frequency ami Super I ligh Frequencj Research. Modulation Research. Antenna Research. Navigation Systems. Systems Analysis. Field Operations.

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