Microwave Magnetics 10 - ee.sharif.edu

Microwave Magnetics

Graduate Course

Electrical Engineering (Communications)

2nd Semester, 1389-1390

Sharif University of Technology

Magnetostatic waves and oscillations 2

General information

� Contents of lecture 10:

• Magnetostatic waves and oscillations

� Introduction

� Magnetostatic waves

� Magnetic potential

� Magnetostatic waves in metallized plates

� Normally magnetized plates

� Transversely magnetized plates

� Magnetostatic waves in tangentially magnetized free plates

� Surface waves

� Excitation of magnetostatic waves

� Magnetostatic wave devices

(i) Introduction

� Let us start our discussion with an example: an

unbounded magnetic film placed on top of a ground plane

� The film is magnetized parallel to its plane in the z-

direction

0 0,M Hx

(i) Introduction

� We look for solutions which represent waves

propagating in the y-direction, we assume them to be

uniform along the magnetization (z-direction)

( ) exp( ) ( ) exp( )x j y x j y� �� e e h h

(i) Introduction

� To analyze the structure we resort to equations we had

used before (inside the magnetic material)

� �2

2 20 02 2

0akx z z

� ��

� � � � ��

2z z z

h h eh�

� � ��

� �2

2 20 02 2

0akx z z

� ��

��

� ��

��

z z zz

e e he

(i) Introduction

� Two decoupled sets result.

� 1st set (inside magnetic material):

� �2

��

j j d dx

� � �

��

� � � � ��

��

� �

0� � �� x y ze e h

(i) Introduction

� 2nd set (inside magnetic material): x

0� � �� x y zh h e

� �2

��

��z

j d dx

��

� ��

��

� 2nd set not affected by magnetic properties since �|| = 1 (ac

magnetic field parallel to static magnetization).

(i) Introduction

� 1st set: Region I: 0 < x < d

� �2

��

� �( ) sin xx A k x��ze � � � �

cos sinx x x

Ak k x k x

��

� ��

a��

(i) Introduction

� 1st set: Region II (air): x >d

� �2

��

� �,0( ) exp xx B jk x� ��ze � �,0

k Bjk x

��

(i) Introduction

� Resulting propagation equation:

� � ,0cotx x xk k d jk� ��

� �2 2 2 2 2 20 0 0cot ak d k j k

��

� The left hand side is always real. Since

the right hand side should be real �0k� �

2 2 2 20 0k j k� �� Why the minus sign?

(i) Introduction

� There are two classes of solutions. The 1st class satisfies

2 2 20 0k k� �� Condition: �� > 0

� These solutions are surface waves (why?) Like ordinary

(TE) surface waves on grounded dielectrics, they have a

frequency cutoff which depends on the film thickness,

dielectric constant, etc.

� The magnetic material, however, makes � dependent on

the direction of propagation.

(i) Introduction

� Numerical example:

��

-1 (mm )�

(GHz)f

� In any case 0k� ��

0� �

� Note also that the phase velocities are comparable to

that of light (e.g. at 50 GHz: vp = 2�f / � ~ 2.85 x 108 m/s)

0� �

(i) Introduction

� However, there is a second branch of solutions in which

2 20k� ��

� These are also surface waves (why?). They only exist for

particular frequencies, but the range mainly depends on

magnetic properties

� �2 2 2 2 2 20 0 0coth ak d k k

��

(i) Introduction

-1 (mm )�

��

H Mf f�

ff �

� These �’s are large compared to to k0 !

� Wavelengths (2�/�) are short compared to

electromagnetic wavelength 2�/k0.

f�0� �

0� �

(i) Introduction

� These waves are called slow waves

� In thinner films (here the thickness was 1mm) the velocity

and wavelength can be even 2-3 orders of magnitude

smaller than electromagnetic velocities!

� Note also that the phase velocities can be much smaller

than that of light: for example

• left moving wave at 7 GHz: vp = 2�f / � ~ 0.665 x 108 m/s

• right moving wave at 4.7 GHz: vp = 2�f / � ~ 0.45 x 108 m/s

(i) Introduction

� These slow waves can satisfy

� Dispersion equation may have been approximated by

2 20k� � ��

� �2 2 2 2 2 20 0 0coth ak d k k

��

� �coth ad�

� � � � ��

2 20k� �

(i) Introduction

� Comparison with the exact result (lines: exact, points:

approximate) show the accuracy of this approximation,

in particular for short wavelength’s (large �’s)

-1 (mm )�

H Mf f�

ff �

f�0� �

0� �

(ii) Magnetostatic waves

� So, with magnetic materials, it is possible to have slow

waves with short wavelength’s

0pv c� 00

f f� ��

� What is more: to obtain these solutions we may

assume that the velocity of light is infinite

0 0pk v c� ��

� This approximation means that we neglect all the

propagation effects!

� This means that, when dealing with slow wave

solutions, in Maxwell equations we are allowed to

neglect the displacement current term everywhere:

� �

��

� �

��

h e j h

� This leads to the magnetostatic approximation:

� � 0�� h j h�

� In electromagnetic problems we are accustomed to the

idea that wave propagation requires the displacement

current.

� Clearly here we have waves which do not need the

electric field for propagation!

� These are called magnetostatic waves. They are, in fact,

waves of the magnetization propagating inside the

medium.

� To see the physical origin of these waves recall the

linearized Landau-Lifshitz equation

( , )( , ) ( , )

d tt t

dt� ��

m rm r H M h r

( , )tM r0M

( , )tm r

� � � �0x

dmM h H m

dt� ��

� � � �0y

dmM h H m

dt� ��

M� H�

( , )tM r0M

( , )tm r

� Differentiation with respect to time:

yxH x M H M x

dhd mm h

dt dt� � � ��

y xH y M H M y

d m dhm h

dt dt� � � ��

� The field h is the ac magnetic field which contains the

externally applied field, and the “demagnetization”

field generated by the magnetization itself.

� Let us assume that no external ac field is applied.

� Besides, since we are adopting the magnetostatic

approximation anyway, we use the approximation

0( , ) ( , ) ( ) ( , )M

t t G t dV� � � �� h r h r r r m r

� If we neglect the effect of the boundary of the volume V

(in reality this is wrong but this is just a qualitative

argument)

� �0( , ) ( ) ( , )V

t G t dV� � � � �� h r r r m r

� The demagnetization field is related to the change of the

magnetization in space (its second derivative)

� It can lead to wavelike behavior. For instance, imagine

for some reason we can neglect my and hy

yxH x M H M x H M x

dhd mm h h

dt dt� � � � � ��

( , )( ) x

m th G dV

��

� This is a wavelike equation (not a wave equation

because of the integration involved)

� Hence: change of magnetization in space induces

demagnetization fields which interact with the motion

(rotation) of the magnetization

� This leads to wavelike phenomena

( , )( ) 0x x

H M H x

d m m tG dV m

dt x� � �

��

� Remarks:

• Whatever the mechanism, everything ‘is’ already covered by

the full Maxwell equations. The magnetostatic approach

does not bring about new phenomena such as slow waves:

these are already in the Maxwell equations.

• Magnetostatic approach only leads to a simplified formalism

which allows us to study a certain class of solutions.

• Yet, magnetostatic approach cannot cover all possible

solutions. Had we neglected the displacement current in our

example, we could not have found the conventional surface

waves with 2 2 20 0k k� ��

• Short wavelength magnetostatic waves can be used to

design compact devices at microwave frequencies (MSW

devices)

• Magnetostatic approach allows us to perform an

approximate, but simple analysis of these components.

• Magnetostatic waves are sometimes called spin waves, but

that is not completely accurate. They are, actually, the long

wavelength limit of the spin waves. A more accurate terms

is: non-exchange spin waves.

(iii) Magnetic potential

� The magnetostatic equations are often solved in

materials where the electric (conduction) currents are

negligible. Therefore

� �0 0�� h h�

� Let us introduce

�� h Magnetic potential

� Then the first equation is automatically satisfied

(iii) Magnetic potential

� 2nd equation:

� � 0�� h�

2 2 20

� � ��

� �a does not appear in this equation! But it affects the

problem through boundary conditions.

� ��

Walker equation

(iv) Magnetostatic waves in metallized plates

� Consider an unbounded magnetic plate metallized on

both sides

� We consider two cases:

• Normally magnetized plate

• Tangentially magnetized plate

0 0,M H 0 0,M H

(iv) Normally magnetized metallized plates

� 1st case: normally magnetized plate

� We had seen similar systems before (transversely

magnetized waveguides, microstrips on normally

magnetized magnetic substrates)

� But now we will not restrict ourselves to solutions

uniform along z-direction

0 0,M H

� It suffices to consider waves along one direction only

(because of the rotational symmetry of the system)

� Solution written as

( , ) ( ) exp( )yy z f z jk y� � �

� Boundary conditions on perfect metallized surfaces

( 0) ( ) 0z z d� � � �z zb b

�� b h� � z

��

� Solution:

( ) cosn z

f z Ad

��

� Such solutions only exist when �< 0:

� ��

1n � 2n �H� � ��

��

� ��

� Dispersion curves

� ��

2 22 2

n My H H

d n k d

��

� �

� ��

-1 (mm )yk

0.1 mm

��

(GHz)f

1n �

2n �

3n �

1(3.5 GHz) 0.157mm!n� � �

� Overall solution:

� �( , ) cos exp nn y

n zy z A jk y

��

� �

� �( , ) cos expn ny n y

n zy z jk A jk y

��

� �( , ) sin exp nn y

n n zy z A jk y

� ��

� These waves have sinusoidal behavior inside the

sample. They are called volume waves.

� Remarks about volume waves:

• Dispersion curves are independent of the propagation direction

because of the rotational symmetry of the problem. The same

result is found for waves propagating in the x-direction or along

any other direction.

• All modes can propagate between �H and �� . There is no

size-dependent cutoff frequency as in a classical

electromagnetic waveguide.

• The mode n=0 is not a solution. It results in a zero magnetic

field (constant �).

• The waves have a group velocity parallel to the phase velocity:

they are forward waves.

� The electric field does not enter the magnetostatic

equations in the first place, but can be perturbatively

found after the magnetic field h is solved

��

�� e h e�

� For simplicity assume there are no charges

(conduction or external)

0 0j�� e h e�

� We expect

dz�� y

dz�� x

�� x ze h

dz� � �z

� �( ) exp nyz jk y� ��e e

� �( , ) cos expn ny n y

n zy z jk A jk y

��

� �( , ) sin exp nn y

n n zy z A jk y

� ��

� �0( , ) sin exp nn yn

n n zy z A jk y

� ��

� �xe

� We shall not compute the other components here!

� Note: the electric field satisfies the boundary conditions.

(v) Tangentially magnetized metallized plates

� Let us now turn our attention to a tangentially

magnetized, metallized plate

� Consider waves propagating with a wave vector

y0 0,M H

� � � �0, , 0, sin , cosy z k kk k k � �� k

� We look for wave solutions of the type:

0 0,M H

� �( , , ) ( ) exp y zx y z f x jk y jk z� � � �

� Walker’s equation leads to

� �2

( )( ) 0y z

d f xk k f x

dx� ��

� Boundary conditions lead to:

0 0,M H

0 0 and aj x x dx y

� ��

� � � ��

0 0 and a y

dfk f x x d

dx� ��

� General solution:

� � � �( ) sin cosx xf x A k x B k x� �

��

� Boundary conditions �

0x y ak A k B� ��

� � � �sin 0y a x xk A k B k d� ��

� Setting determinant to zero

� 1st solution:

� � � �2 2 2 2 sin 0x y a xk k k d� ��

� � 2 2 2 2sin 0 or 0x x y ak d k k� ��

kn nk k

� ��

222 2 cos

sin kk

� ��

��

� ��

� Result:

� Regardless of the value of n, in order to have

propagation: 2

2 cossin 0k

��

� We should have µ < 0 like in the previous case;

otherwise this condition cannot be satisfied

� Even then, propagation only occurs for certain angles

2 1tan k� �

� �

Propagation region

� This limitation does not depend on thickness

� The width of the propagation cone at each frequency:

1tan H

� ��

� � ��

��

H� ��

� Propagation cone becomes very narrow near �H , but

covers the whole plane near ��

H� � ��

� Within the propagation cone the dispersion relation is:

222 2 cos

sin kn k

� ��

��

� ��

1 /k n

� ��

��

� The dispersion curve depends on the propagation angle

� For every angle, the limit k � 0 corresponds to the same

frequency ��

-1 (mm )k

� But, with increasing k,

frequency decreases

finally reaching

� �2sin

H H M k

� � � �

��

1n �2n �

3n �

� The higher n, the later this limit is reached

� Propagation along static magnetization (z)

-1 (mm )k

1n �2n �

3n �

� �0k� �

0 0,M Hk

-1 (mm )k

� Propagation nearly perpendicular to z � �/ 2k� ��

0 0,M Hk

� The dispersion curves become nearly flat

� Magnetic potential:

( ) cos sin

cos sin sin

k dn x n xf x B

k dn x n xB

��

� �

� ��

� �( , , ) ( ) expn y zx y z f x jk y jk z� � � �

( , , )x y z�� h

� Note that for all angles the dispersion curves have a

negative slope: the group velocity is opposite to phase

velocity

� These waves are called backward volume waves

� The are called volume waves since fields have a

sinusoidal behavior inside the magnetic material

� How does the wave front of these waves look like if we

have an isotropic source?

� But, there also exist a second class of solutions:

2 2 2 2 2 20x y a z yk k k k� � ��

� ky and kz are real numbers (related to propagation).

Therefore this solution exists only if

� �2 2

2 20 0M H

� � ��

� ��

� Propagation

angle: � �

1tan k

� ��

� � � ��

��

� �

� Propagation angle is a function of frequency:

f� H Mf f�

� Note that this is a single mode (there is no mode

number n like in the previous case)

� Besides, apart from the propagation angle, nothing can

be said about the relation between the frequency and

the magnitude of the wave number k.

� This means the dispersion relation is flat: at any allowed

frequency, every value of k is allowed� flat dispersion

� Of course, this is an artifact of the magnetostatic

approximation. But it can be said that in the true case

the dispersion is “almost” flat.

� Also, note that2

2 2 2 2 2 20 ax y a x yk k k k

��

� � � � � � ��

� Since ky is real, kx should be imaginary

� The corresponding magnetic potential and field should

have an exponential behavior inside the magnetic

material

� � � �( ) cos siny ax x

kf x B k x k x

��

� ��

� �

a yx x x

kk jq q

��

� � � �( ) exp xf x B q x� �

� This type of solution is called a magnetostatic surface

wave because the field drops exponentially inside the

magnetic material

� It is concentrated near the top or bottom surface

depending on the sign of qx

� Summarizing the results:

• Between �H and �� we have volume waves. There are different

modes. At each frequency, the modes can propagate within a

certain range of angles with respect to the static magnetization.

All modes represent backward waves: their group velocity is

opposite to their phase velocity.

• Between �� and �H + �M we have a surface wave. To each

frequency their corresponds a specific propagation angle. But

the dispersion curve is flat. The field is concentrated at the top or

bottom interface (with the metal) depending on the propagation

direction.

� Remember, for normally

magnetized metallized

plates:

� Forward volume waves

between �H and ��

� Independent of the

propagation angle

0 0,M H

-1 (mm )yk

(GHz)f

1n �

2n �

3n �

� Tangentially magnetized

metallized plates:0 0,M H

-1 (mm )k

1n �2n �

3n �

H Mf f�� Angle-dependent, back-

ward volume waves

between �H and �� .

� Flat-dispersion surface

wave between �� and �H

+ �M . Frequency

depends on angle.

(vi) Tangentially magnetized free plate

� We now turn to the case of a tangentially magnetized

free plate (no metallization)

� This case is more difficult to solve, but it is important

due to its rich physics, and application in MSW devices

0 0,M Hk�

� Walker’s equation

0 0,M H

2 2 20

� � ��

� �( , , ) ( ) exp y zx y z f x jk y jk z� � � �

� Inside the magnetic material (0<x<d)

� �2

( )( ) 0y z

d f xk k f x

dx� ��

� Outside the magnetic material (x<0, x>d) we have �=1:

� �2

( )( ) 0y z

d f xk k f x

dx� � �

� Inside the magnetic material (0<x<d)

� � � �( ) sin cosx xf x A k x B k x� �

��

� Outside the magnetic material (x<0, x>d):

� �exp ( ) ( )

exp( ) 0

C k x d x df x

D kx x

� � � ��

� �2

( )( ) 0y z

d f xk k f x

dx� � �

2 2y zk k k� �

� Boundary conditions at magnetic material/air interface:

: continuousy z xh ,h ,b

0 0,M H

: continuousy

��

0 ajx y

� ��

� Inside magnetic plate:

� Outside magnetic plate:

��

� System

to solve:

��

� � � �sin cos exp( )x x y zA k x B k x jk y jk z� � � � ��

0 ajx y

� ��

� �exp ( ) exp( )y zC k x d jk y jk z� � � � � �

��

� �exp exp( )y zD kx jk y jk z� � � �

� Matching:

D B�

x y akD k A k B� ��

� � � �sin cosx xC A k d B k d� �

� � � ��

x y a x

kC k A k B k d

k B k A k d

� �

� � �

� �

� Dispersion equation:

� �2 2 2 2 2

x y ax x

k k kk k d

� �

� ��

� � 2 21 1cot

2x x z yk k d k kk

��

� ��

� �

� Note that: 2

2 cossin k

x kk k�

��

� � �

� Hence, we distinguish two situations:

Volume waves: 2

: real

10, tan

� ��

� � �

Surface waves: 2

: imaginary

��

� �

22 cos

sin 0kk

��

22 cos

sin 0kk

��

� Again, volume waves are

allowed within certain angles

1tan k� �

��

� � � �2 21 1

cot cos sin2k k k

� � � ��

� ��

� �

� For any angle, the dispersion equation of volume waves is

� � 2 2sin cos /k k kg � � � ��

� Example: volume waves for

0k� �k

-1 (mm )k

1n �

2n �

3n �� Again different modes

appear which are all

backward waves

0k� �

1 1cot

� �

� � � ��

� For surface waves we have

� �2

2 cossin 0k

k kg�

� ��

� � �

� � � �2 21 1

coth cos sin2k k k

� � � ��

� ��

� �

� Left hand side positive� there are solutions only if

2 21 1cos sin 0

2 k k� � ��

� ��

� �

We can also take the minus sign, it does not matter

� We should have2 2 2

cossin 0k

� ��

� � � �

� ��

2 22 2

1 1cos sin 0

��

� � ��

� � � ��

� �� 2 2 21 1sin

2 2H M H M k H� � � � � ��

� �2 2sinH H M k� � � ��

� If � � ��

� � � �� 2 21 1sin

2 2H H M H M H M k H� � � � � � � � ��

�� Impossible (Why?)

� Only possibility: � � ��

� But then: ��

2sin Hk

��

� ��

�Necessary condition

� But these conditions are not enough, we should also have

� �2 21 1cos sin

2 k k kg� � � ��

� ��

� �2 2 2 2

2 2 2 2

� ��

� � ��

� �

� � � �� 22 2 2 2 2 2� � � ��

� These relations lead to an upper frequency limit dependent

of propagation angle

� �� 2 2sin

2 sinH M H M k H

� � � � � ��

� �

� � ��

-1 (mm )k

2H Mf f�

��

� The opposite is also true: like volume waves, surface

waves on a free plate are only allowed within a certain

range of angles

Propagation region

sin kM H

� � ��

� ��

��

� Summarizing:

• Free magnetized plate allows

backward volume modes in

the frequency range between

�H and �� . Between �� and

�H + �M/2 a surface waves

mode is allowed.

• But the actual dispersion

relation and frequency range

of propagation depends on the

propagation angle in both

cases. -1 (mm )k

1n �2n �

3n �

2H Mf f�

Volume modes

Surface mode

(vii) Surface waves

� A particularly important case is that of surface waves

propagating perpendicular to the direction of static

magnetization

� We analyze this case in more detail

0 0,M H

��

(vii) Surface waves

� The dispersion equation becomes

� � 1 1coth

2kd �

��

� ��

� �

2y yk k k� �

� ��

12 1exp(2 )

2 1 1a

kd� ��

� � � � ��

� ��

� � � �

� ��

� �

� ��

(vii) Surface waves

� These waves propagate in the range

��

� Magnetic potential inside the magnetic layer:

� � � �sin cos exp( )x x yA k x B k x jk y� � � ��

2x y yk k jk� � �

2y yk k k� �

(vii) Surface waves

� From the matching equation it follows that

y y ay a a

k kk k sAj

B k jk

��

� � � �sinh cosh exp( )ay y y

sB k x k x jk y

��

� � ��

(vii) Surface waves

� We rewrite this as

� � � � � �1 sinh cosh exp( )a y

Bs kx kx jk y� � �

��

� � � �exp( ) exp ( ) expyjk y U k x d U kx� � ��

� �21aU s� �� 2

1aU s� ��

(vii) Surface waves

� Plotting the magnetic potential, it is found that waves

moving in the +y direction (s=1) concentrate near the

bottom surface of the magnetic plate

� Those moving in the –y direction (s=-1) concentrate

near the top surface of the plate

S =1 S = -1

(vii) Surface waves

� What about a half infinite magnetic plate?

� What is its dispersion equation?

� What about the field profile?

(vii) Surface waves

� Free plate is not the only structure whose surface waves

have been studied (in view of possible application)

� Various structures have been considered containing

metallic ground planes

� Solid lines: waves

moving in –y direction

(near top surface)

� Dashed lines: waves

moving in +y direction

(near bottom surface)

(viii) Excitation of MSW’s in magnetic films

� Open or half-open (grounded) magnetic plates form the

basis of MSW devices. But how are these waves excited?

� By conventional current carrying lines (transducers) on the

surface or close to the surface of the magnetic film

Microstriptransducer

Meandertransducer

Lattice transducer

Magnetic film

(viii) Excitation of MSW’s in magnetic films

� Roughly speaking, the amplitude of the excited MSW with

a (tangential) wave vector k is proportional to Fourier

transform of current density at that wave vector:

Microstriptransducer

� �( ) expV

j dV�� J r k r

(ix) MSW devices

� MSW devices: based on

excitation and reception

of MSW’s in a finite or

infinite magnetic film

� Mostly based on

excitation of surface

waves, but volume

waves used as well

(ix) MSW devices

� These devices benefit from the following properties of the

MSW’s:

• Broad frequency range

( )H H H M� � � � � �� Volume waves

H M� � � �� Surface waves (grounded plates)

By applying very high dc magnetic fields (up to 2 Tesla by using

permanent magnets) or using materials with a high saturation

magnetization the range of 1-50GHz may be covered

(ix) MSW devices

• Properties of MSW devices can be tuned by changing the applied

magnetic field

• Wavelength’s are short, for instance for surface MSW’s propagating

perpendicular to the magnetization in a tangentially magnetized

free plate wavelength is proportional to the film thickness. Using

thin films leads to very short wavelengths

� ��

� �

� ��

124 / ln

� ��

� �

� ��

(ix) MSW devices

• It is possible to change the dispersion properties of MSW’s by the

choice of the wave type and by changing the layer thickness,

adding ground planes, etc.

(ix) MSW devices

• The losses are comparatively law (if single-crystalline high quality

films are used)

• Transducers (for exciting MSW’s) are easy to design

� For these reasons MSW devices were investigated in the

1970’s and early 1980’s:

• Delay lines (phase shifters)

• Filters

• Resonators

(ix) MSW devices

� Delay lines: consist of a transmitting transducer and a

receiving transducer

� The resulting time delay (phase shift) can be large because

MSW’s are slow (propagation constants are large)

(ix) MSW devices

� These devices can be tuned by changing the dc magnetic

bias. They can also be reciprocal or non-reciprocal

� The dispersion characteristics can be engineered to

realize true wideband delay lines (small dispersion over a

wide frequency band) or to have other properties

(ix) MSW devices

� MSW filters:

• Wide band filters can be built by using the natural propagation

ranges of MSW’s between transducers

• Narrow-band filters built by engineering the transducers. For

instance note that the amplitude of the wave is proportional to

� �( ) expV

j dV�� J r k r

(ix) MSW devices

� For a lattice transducer with N elements each carrying a

current density

( ) ( , , )N

J r J x x y z�

� � ��

� � � � � �1

( ) exp ( ) exp expN

j dV j dV jk ns�

� ��

� �� J r k r J r k r

(ix) MSW devices

� � � ��

1 expexp

sin / 2 exp 1 / 2

sin / 2

jk sNjk ns

k sNjk s N

��

� Therefore, one can select

just certain values of wave

number (thus certain

frequencies) for excitation

� ��

sin / 2

sin / 2x

(ix) MSW devices

� MSW resonators: utilize the formation of standing

MSW’s in a ‘finite’ magnetic sample excited by a

transducer

� The standing wave is formed by the reflection of the

MSW off the edges of the finite film

� Since wavelength is short, resonators are small

Standing wave

(ix) MSW devices

� For a more detailed overview see

W.S. Ishak, “Magnetostatic wave technology: a review”, Proceedings of

the IEEE, Vol. 78, Issue 2, 1988.

��

Microwave Magnetics 10 - ee.sharif.edu

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