Microwave Magnetics 5.ppt - Sharifee.sharif.edu/~mwmagnetics/Microwave Magnetics_5.pdf ·...
Transcript of 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
Waveguides and resonators with magnetic media 3
(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
Waveguides and resonators with magnetic media 4
(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
Waveguides and resonators with magnetic media 5
(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
Waveguides and resonators with magnetic media 6
(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
Waveguides and resonators with magnetic media 7
(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
Waveguides and resonators with magnetic media 8
(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!
Waveguides and resonators with magnetic media 9
(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
Waveguides and resonators with magnetic media 10
(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.
Waveguides and resonators with magnetic media 11
(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
Waveguides and resonators with magnetic media 12
(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
Waveguides and resonators with magnetic media 13
(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
Waveguides and resonators with magnetic media 14
(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.
Waveguides and resonators with magnetic media 15
(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
Waveguides and resonators with magnetic media 16
(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
Waveguides and resonators with magnetic media 17
(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
Waveguides and resonators with magnetic media 18
(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
Waveguides and resonators with magnetic media 19
(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
Waveguides and resonators with magnetic media 20
(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
Waveguides and resonators with magnetic media 21
(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
Waveguides and resonators with magnetic media 22
(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
Waveguides and resonators with magnetic media 23
(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
Waveguides and resonators with magnetic media 24
(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
Waveguides and resonators with magnetic media 25
(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
Waveguides and resonators with magnetic media 26
(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
Waveguides and resonators with magnetic media 27
(vii) Partially filled waveguide: numerical results Waveguide width: 2 cm, frequency: 10 GHz,
d2 cma
0.429
0
0
0/ k
(mm)d
1n
Waveguides and resonators with magnetic media 28
(vii) Partially filled waveguide: numerical results Electric field profile (d=6mm):
ze
(m)x
d2 cma
0 0
Waveguides and resonators with magnetic media 29
(vii) Partially filled waveguide: numerical results Magnetic field profile (d=6mm):
0j yh
(m)x
d2 cma
0 0
Waveguides and resonators with magnetic media 30
(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
Waveguides and resonators with magnetic media 31
(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
Waveguides and resonators with magnetic media 32
(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
Waveguides and resonators with magnetic media 33
(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).
Waveguides and resonators with magnetic media 34
(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
Waveguides and resonators with magnetic media 35
(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
Waveguides and resonators with magnetic media 36
(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
Waveguides and resonators with magnetic media 37
(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
Waveguides and resonators with magnetic media 38
(vii) Partially filled waveguide: numerical results Electric field profile (d=1.2mm):
ze
(m)x
2 cma 0
0
Waveguides and resonators with magnetic media 39
(vii) Partially filled waveguide: numerical results Magnetic field profile (d=1.2mm):
0j yh
(m)x
0 0
Waveguides and resonators with magnetic media 40
(vii) Partially filled waveguide: numerical results Magnetic field profile (d=1.2mm):
0 xh
(m)x
0
0
Waveguides and resonators with magnetic media 41
(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
Waveguides and resonators with magnetic media 42
(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
Waveguides and resonators with magnetic media 43
(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.
Waveguides and resonators with magnetic media 44
(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
Waveguides and resonators with magnetic media 45
(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
Waveguides and resonators with magnetic media 46
(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
Waveguides and resonators with magnetic media 47
(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
Waveguides and resonators with magnetic media 48
(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
Waveguides and resonators with magnetic media 49
(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
Waveguides and resonators with magnetic media 50
(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
Waveguides and resonators with magnetic media 51
(vii) Partially filled waveguide: symmetric configuration, thin slab
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
Waveguides and resonators with magnetic media 52
(vii) Partially filled waveguide: symmetric configuration, thin slab
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
Waveguides and resonators with magnetic media 53
(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
Waveguides and resonators with magnetic media 54
(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
Waveguides and resonators with magnetic media 55
(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
Waveguides and resonators with magnetic media 56
(vii) Partially filled waveguide: numerical results Electric field profile (d=9mm):
ze
(m)x
0 0
(m)x
0
0
0.429
0j yh
Waveguides and resonators with magnetic media 57
(vii) Partially filled waveguide: numerical results Electric field profile (d=9mm):
ze
(m)x (m)x
0 0
2.86
0j yh
0
0
Waveguides and resonators with magnetic media 58
(vii) Partially filled waveguide: numerical results Surface waves (f=8 GHz):
(m)x
0 0 7.5
0/ kd
(mm)d
ze
2mmd
Waveguides and resonators with magnetic media 59
(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
Waveguides and resonators with magnetic media 60
(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