Microwave Magnetics 5.ppt - Sharifee.sharif.edu/~mwmagnetics/Microwave Magnetics_5.pdf ·...

Microwave Magnetics

Graduate CourseElectrical Engineering (Communications)2nd Semester, 1394-1395Sharif University of Technology

Waveguides and resonators with magnetic media 2

General information Contents of lecture 5:

• Waveguides with transversely magnetized media Modes in a rectangular waveguide Field displacement effect Partially filled waveguides

Thin magnetic slab at the guide edge Thin magnetic slab at the guide centre General case


(v) Waveguides with transversely magnetized media Let us turn to waveguides filled with a gyromagnetic

medium. We restrict ourselves to the case where the magnetization is perpendicular to the direction of propagation

Again magnetization is chosen in the z-direction. But now propagation takes place along y

zx

y( , ) exp( ) ( , ) exp( )x z j y x z j y e e h h

0 0,M H


(v) Waveguides with transversely magnetized media Let us return to our main equations ( )

zx

y

2 2 20 02 2 0akx z z

2z z zz

h h eh

2 2 20 02 2 0akx z z

2

z z zze e he

These are complicated equations and exact solution is impossible in general. But results are simple if we focus on modes in which fields do not change along z

1


(v) Waveguides with transversely magnetized media Then:

zx

y 2 2

02 0kx

2z z

h h

2 202 0kx

2

z ze e

We have separate TE and TM modes. But note that propagation takes place along y not z

These are not the only modes. Generally, modes are of hybrid type. Usually, however, if the height is small, these modes have the lowest cutoff frequency.

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


(v) Waveguides with transversely magnetized media Other field components follow from Maxwell equations:

0/a

a

jj j jd dx

z x

z y

e he h

0/j jd dx

xz

yz

eheh

0 jx z z x yh h e

0 jy z yz x

hh e 0 aj jy z yz x y

ee h h

0 aj jx z z x x ye e h h


(v) Waveguides with transversely magnetized media 1st case:

1

0

2 20

1/

1 ( ) /

a

a

a

a a

j jj j d dx

j jj j d dx

zx

zy

z

z

eheh

ee

0 e

0, 0 z zh e

2 202 0kx

2

z ze e

z

xy

e

These are TE waves


(v) Waveguides with transversely magnetized media 2nd case:

0 h

0, 0 z zh e

2 202 0kx

2

z zh h

0

1/

jj d dx

x z

y z

e he h

zx

yh

There is no solution satisfying the boundary conditions on perfectly conducting walls!


(v) Waveguides with transversely magnetized media 1st case: 0, 0 z zh e

z

xy

e0 0( , ) sin exp( )n nn xx y A j ya

n0ze

a2

20 0n

nk a

For lossless magnetic materials propagation occurs when

22 222 20 0 0 2 2 0M Hn nk a a

2 ( )H H M

1n


(v) Waveguides with transversely magnetized media

For each n, there are two branches and a propagation gap

22 2 22 2

M H n ca

Necessary

condition:

0n

Note also that n0 depends on the static magnetic field through H. This can be used to build phase shifters.


(vi) Field displacement effect Magnetic field distribution:

00 0

2 20

0

sin cosexp( )( ) sin cos

n an n

an a

n x n n xa a aA j y

n x n n xj ja a a

n0xn0y

hh

02 2

0

exp( )( ) /

ana a

j jj yj j d dx

n0x zn0y z

h eh e

Electric field symmetric or anti-symmetric in the x-direction; but no such property for the magnetic field


(vi) Field displacement effect For example:

0 0 002 20

exp( ) ( )cos( )n n

a

jA j y n x xa

n0yh

220 0n a

na

00 arctan n aaax n n

0x z

x


(vi) Field displacement effect Note that the magnetic field profile changes when the

sign of the propagation constant changes. Waves traveling in opposite directions have different magnetic field profiles

0 00 0n n

x x

00 arctan n aaax n n

0x 0x

0 0n 0 0n


(vi) Field displacement effect To understand the physical origin of field displacement

consider an empty rectangular waveguide. The TEn0modes are now given by

0 0( , ) sin exp( )n nn xx y A j ya

n0ze 2

20 0n

nk a

00 0

0

sinexp( )cos

nn n

n xaA j y

n n xj a a

n0xn0y

hh

At each point, the magnetic field is elliptically polarized.


(vi) Field displacement effect

y

x

0x x a

But: direction of rotation is different near the left and right edges. For n=1, for a wave moving along +y, magnetic field is right handed near the left edge, and left handed near the right edge.

When included, magnetic medium will affect the edges differently. This is because different polarizations see different permeability

Behavior reversed for propagation in –y direction


(vii) Partially filled waveguide Consider now a waveguide, partly

filled by a vertically magnetized magnetic medium

We again restrict ourselves to modes which do not change in z-direction

Also, as before, we only consider TE modes, where the electric field only has a z-component, and magnetic field lies in the x-y plane (TM modes not changing in z-direction are not allowed)

zx

y

e0 0,M H

g g d

d


(vii) Partially filled waveguide For simplicity, consider the case with

the magnetic slab touching the wall

x g

d

0 x g

2 202 0kx

2

z ze e

2 202 0kx

2z z

e e

g x a

a

Solution:0 x g g x a

,0sin xA k xze sin ( )xB k a x ze

2 2 20 ,0xk k 2 2 20 xk k

0x


(vii) Partially filled waveguide Matching the electric field:

We get another equation by considering the tangential magnetic field at x = g

,0sin sinx xk g A k d B

0

1 dj dx zy

eh 0 x g

g x a 2 20

1( ) a

a

dj dx

zy z

eh e

g a d

da


(vii) Partially filled waveguide

a ,0 ,0cos cos sinx x x x xk k g A k k d k d B

Resulting equations:

,0

,0 ,0

sin sin 0cos cos sinx x

x x x x x

k g k d Ak k g k k d k d B


(vii) Partially filled waveguide Setting determinant equal to zero:

,0 ,0cot cot 0x x x xk k d k k g a 2 2 2

,0 0xk k 2 2 20xk k 2

a

The limit d0 yields results for an empty waveguide Propagation constants in +y and –y directions differ There might be solutions for negative One might have:

0k


(vii) Partially filled waveguide

,0 ,0cot cot 0x x x xk k d k k g a 2 2 2

,0 0xk k 2 2 20xk k 2

a

Example of a thin magnetic plate: 1xk d

g

d

,0 ,01 cot 0x xk k gd


(vii) Partially filled waveguide: thin magnetic slab at the guide edge

Graphic solution: plot the left and right functions, first assume

2 2 2 20 0

1cot a ggg k g k d

0k0k

2 2 2 20 0cotg k g k

1 ,a ggd

1 ,a M Hg gd

0


(vii) Partially filled waveguide: thin magnetic slab at the guide edge Solution depends on the sign of : waves moving in

+y and –y directions do not have the same propagation constant!

0k0k10

10 20

20


(vii) Partially filled waveguide: thin magnetic slab at the guide edge For a thin layer, the solution (propagating modes) are

close to those of an empty waveguide Are solutions smaller than ?? If the magnetic layer

is thin yes. Remember that also in an empty waveguide

This not necessarily true for a thick layer. For instance, for a completely filled guide:

In this case may not be smaller than Also for intermediate values of thickness no general

statement may be made.

0k

220 0/k n a k

220 /k n a

0k


(vii) Partially filled waveguide: thin magnetic slab at the guide edge But, what about the field profile?

0 x g g x a

,0sin xA k xze sin ( )xB k a x ze

gd

a

x ,0,0

sin sin for sinx

xx

k gB a xA k g g x aA k d d ze

,01,0sin ( 1) 1

xnx

k dk g d

a 1n 0

0 0


(vii) Partially filled waveguide: numerical results Numerical results for a waveguide filled with a magnetic

slab (using the full equation). • Saturation magnetization Ms = 1.6x105 A/m (0.2 T)• Total internal dc field H0 = 0.53x105 A/m

6 GHz2 2sM M

0 2 GHz2 2

H H

2 22

( )( )H M

H H M

1

f4GHz 8GHz


(vii) Partially filled waveguide: numerical results Waveguide width: 2 cm, frequency: 10 GHz,

d2 cma

0.429

0

0

0/ k

(mm)d

1n


(vii) Partially filled waveguide: numerical results Electric field profile (d=6mm):

ze

(m)x

d2 cma

0 0


(vii) Partially filled waveguide: numerical results Magnetic field profile (d=6mm):

0j yh

(m)x

d2 cma

0 0


(vii) Partially filled waveguide: numerical results Forward direction: magnetic field stronger

inside the magnetic slab Waves moving backwards: magnetic field

pushed out of the slab

y

x

0x x a

2GHz 8GHz1

This difference is again related to the relative dominance of the two circular polarizations

Forward direction: left-hand dominant near slab, backward: right-hand dominant near slab


(vii) Partially filled waveguide: thin magnetic slab at the guide edge So far we assumed a positive . What if ? Remember: if the waveguide is completely filled, there

exists a propagation gap in this region But in a partially filled waveguide, propagation is

possible To see this let us return to our approximate equation

and its graphic solution

0


(vii) Partially filled waveguide: thin magnetic slab at the guide edge Graphic solution:

0k0k

2 2 2 20 0cotg k g k 1 a gg

d

0

10 10

20 20

We again have solutions: propagation not generally forbidden


(vii) Partially filled waveguide: thin magnetic slab at the guide edge Again we have dependence of propagation constant on

direction of propagation Note that for a thin layer we may have two solution types:

solutions with < k0 (similar to that of an empty waveguide) and those with > k0

Solutions with < k0 are close to the conventional waveguide modes (except for dependence on direction)

Solutions with > k0 are different. These are the surface wave modes (for a thin magnetic layer).


(vii) Partially filled waveguide: thin magnetic slab at the guide edge First situation

0 x g g x a

,0sin xA k xze ,0sin x

a xA k g dze

,01,0sin ( 1) 1

xnx

k dk g d

1n 0

0

0

2 2 20 ,0 0 0xk k k

Field has sinusoidal behavior outside the magnetic plate

x


(vii) Partially filled waveguide: thin magnetic slab at the guide edge 2nd situation

0 x g g x a

,0sinh xjA q xze x ,0sinh x

a xjA q g dze

2 2 20 ,0 0 0xk k k

Field has exponential behavior outside the magnetic plate

2 2,0 0xq k


(vii) Partially filled waveguide: numerical results Numerical results for a waveguide filled with a magnetic

slab with (using the full equation)• Saturation magnetization Ms = 4x105 A/m (0.5 T)• Total internal dc field H0 = 0.53x105 A/m • = 1

15 GHz2 2sM M

0 2 GHz2 2

H H

2 22

( )( )H M

H H M

1

f5.8GHz 17GHz


(vii) Partially filled waveguide: numerical results Numerical results at 10 GHz 1st type solutions

d2 cma

0

0 0/ k

(mm)d

2.86 1n


(vii) Partially filled waveguide: numerical results Electric field profile (d=1.2mm):

ze

(m)x

2 cma 0

0


(vii) Partially filled waveguide: numerical results Magnetic field profile (d=1.2mm):

0j yh

(m)x

0 0


(vii) Partially filled waveguide: numerical results Magnetic field profile (d=1.2mm):

0 xh

(m)x

0

0


(vii) Partially filled waveguide: numerical results 2nd type solutions (surface waves)

0 0/ k

(mm)d

10 GHzf 2 cma

8 GHzf

0

0 0/ k

(mm)d


(vii) Partially filled waveguide: numerical results Field profile (10GHz, 0.5mm)

- j ze

(m)x (m)x

0 yh2 cma

The field is confined to the vicinity of the magnetic film, which is typical of surface waves


(vii) Partially filled waveguide: thin magnetic slab at the guide edge Note that the above conclusions are not all valid for

arbitrarily thick magnetic layers. In general: propagation constant depends on direction of propagation. Also surface wave modes may exist in general at the ferromagnetic-air or ferromagnetic-dielectric interfaces

Note that sometimes a mode becomes uni-directional: it propagates in one direction only. Like for the surface waves of the previous example

For a thick layer this phenomenon may not be limited to surface waves. Ordinary modes may also become uni-directional.


(vii) Partially filled waveguide: general case We study another example which

is symmetric It will not produce a dependence

of on direction, but is useful to study field distribution

First, however, we consider the general case to derive equations for propagation constant

g g

d

a

2a dg


(vii) Partially filled waveguide: general case First consider the general case:

g g d

d0 , x gg d x a

22 0xkx

2

z ze e

2,02 0xkx

2

z ze e

g x g d

a

Solution:0 x g

g d x a g x g d

,0sin xA k xze sin ( ) cos ( )x xB k x g C k x g ze

,0sin ( )xD k a x ze

2 2 2,0 0xk k

2 2 20xk k


(vii) Partially filled waveguide: general case Matching the electric field:

g

d

a

Tangential magnetic field:

,0sin xk g A C ,0sin cos sinx x xk d B k d C k l D

l a g d

0

1 dj dx zy

eh 0 , x g g d x a g x g d 2 2

0

0

1( )

1 a

a

dj dx

dj dx

zy z

z z

eh ee e

z

x


(vii) Partially filled waveguide: general case ,0 ,0cosx x xk k g A k B C

,0 ,0

cos sin sin cos cos

x x x

x x x x x

k k d k d Bk k d k d C k k l D

,0sin xk g A C ,0sin cos sinx x xk d B k d C k l D


(vii) Partially filled waveguide: symmetric configuration

,0 ,0cosx x xk k g A k B C

,0 ,0

cos sin sin cos cos

x x x

x x x x x

k k d k d Bk k d k d C k k g D

,0sin xk g A C ,0sin cos sinx x xk d B k d C k g D

g

da

l g

0cot cotx

x x x x

k Bk k d k k d C

,0 ,0cotx xk k g


(vii) Partially filled waveguide: symmetric configuration Setting the determinant equal to zero:

2 22 2,0 ,0 ,0 ,02 cot cot cot 0xx x x x x x

kk k k d k g k k g

Now, no dependence of propagation constant on propagation direction (due to symmetry)

Again, let us consider a case which is easy to analyze Let

gda1xk d


(vii) Partially filled waveguide: symmetric configuration, thin slab

2 2 20xk k

gda

2 2,0 2 2,0 ,0 ,02 cot cot 0x xx x x

k kk g k k gd

1xk d

2 2,0 ,0 2

1 1 1cotx x xk k g kd d

a

2 2 2,0 0xk k



2 2 2 20 0

2 2 20

cot 1 1 /g k g k

g k dd

0k0k

2 2 2 20 0cotg k g k

0 2 2 201 1 /g k dd



2 2 2 20 0

2 2 20

cot 1 1 /g k g k

g k dd

0k0k

2 2 2 20 0cotg k g k

0


(vii) Partially filled waveguide: symmetric configuration, thin slab For a thin magnetic slab at the guide centre, most

solutions resemble those of an empty waveguide:

2 2 2 2 2 20 0 0cot 1g k g k g k n n

2 2 2 2 2 20 0 0

1cot 0 02g k g k g k n n 2 2

2 20 0

2 1 2 , 2 2n nk kg g

But, if , there may be two extra solutions with

||> k0 which again correspond to surface waves0


(vii) Partially filled symmetric waveguide: numerical results Numerical results:

• Saturation magnetization Ms = 1.6x105 A/m and Ms = 4x105 A/m • Total internal dc field H0 = 0.53x105 A/m • Frequency=10 GHz, = 1

6 and 15 GHz2 2sM M

0 2 GHz2 2

H H

2 22

( )( )H M

H H M

1

f


(vii) Partially filled waveguide: numerical results Waveguide width: 2 cm, frequency: 10 GHz,

0.429

0/ k

1n

(mm)d

Conventional moded

(mm)d

2.86



ze

(m)x

0 0

(m)x

0

0

0.429

0j yh



ze

(m)x (m)x

0 0

2.86

0j yh

0

0


(vii) Partially filled waveguide: numerical results Surface waves (f=8 GHz):

(m)x

0 0 7.5

0/ kd

(mm)d

ze

2mmd


(vii) Partially filled waveguide: general case General case: setting the determinant to zero:

2 2,0 ,0 ,0 ,0 ,0

2,0 ,0 ,0 ,0

cot tan tan tan tantan tan 0

xx x x x x x x

x x x x

kk k k d k g k l k g k lk k k g k l

a 2 2 2,0 0xk k

2 2 20xk k

g

d

a

l2a


(vii) Numerical results: general case

a

l

• Saturation magnetization Ms = 1.6x105 A/m (fM = 6 GHz)• Total internal dc field H0 = 0.53x105 A/m (fH = 2 GHz)• Frequency=10 GHz, = 1, d = 2 mm, a = 2 cm

0.429

0/ k

(mm)l

0

0

d

Microwave Magnetics 5.ppt - Sharifee.sharif.edu/~mwmagnetics/Microwave Magnetics_5.pdf ·...

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