phy 712 electrodynamics 10-10:50 am mwf olin 107 plan for lecture 19:
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PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 19: Start reading Chap. 8 in Jackson. Examples of waveguides TEM, TE, TM modes Resonant cavities. Reminder : Topic choices for “computational project” – suggestions available upon request -- due Monday March 24 th ?. - PowerPoint PPT PresentationTRANSCRIPT
PHY 712 Spring 2014 -- Lecture 19 1
PHY 712 Electrodynamics10-10:50 AM MWF Olin 107
Plan for Lecture 19:Start reading Chap. 8 in Jackson.
A. Examples of waveguidesB. TEM, TE, TM modesC. Resonant cavities
03/17/2013
PHY 712 Spring 2014 -- Lecture 19 203/17/2013
PHY 712 Spring 2014 -- Lecture 19 3
Reminder: Topic choices for “computational project” – suggestions available upon request-- due Monday March 24th?
03/17/2013
PHY 712 Spring 2014 -- Lecture 19 4
Fields near the surface on an ideal conductor
22
2
Suppose for an isotropic medium: Maxwell's equations in terms of and :
0 0
0 ,
Plan
b
b
b
t t
t t
D E J EH E
E HH EE H E
F F E H
0
e wave form for :
ˆ, where i i tR It e n in
c k r
E
E r E k k
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PHY 712 Spring 2014 -- Lecture 19 5
Fields near the surface on an ideal conductor -- continued
12
1For
112
112
:systemour For
2/12
2/12
IR
b
bI
b
bR
nc
nc
nc
nc
ˆ ˆ/ /0,
1ˆ ˆ, , ,
i i tt e e
n it t tc
k r k rE r E
H r k E r k E r03/17/2013
PHY 712 Spring 2014 -- Lecture 19 6
Fields near the surface on an ideal conductor -- continued
icinnc
nc
nc
IR
IR
11 limit, In this
12
1For
00
ˆ ˆ/ /0,
, , ,
1ˆ ˆ, , ,
1ˆ ˆ, , , ,
i i tt e e
it t t
n it t tc
n it t t tc
k r k rE r E
D r E r E r
H r k E r k E r
B r H r k E r k E r03/17/2013
PHY 712 Spring 2014 -- Lecture 19 7
Fields near the surface on an ideal conductor -- continued
ˆ ˆ/ /0,
, , ,
1ˆ ˆ, , ,
1ˆ ˆ, , , ,
i i tt e e
it t t
n it t tc
n it t t tc
k r k rE r E
D r E r E r
H r k E r k E r
B r H r k E r k E r
ˆ ˆ/ /0
Note that we can express the results in terms of the field:
,
1 ˆ, ,2
i i tt e e
it t
k r k r
H
H r H
E r k H r
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PHY 712 Spring 2014 -- Lecture 19 8
Boundary values for ideal conductorAt the boundary of an ideal conductor, the E and H fields decay in the direction normal to the interface.
tit
eet tii
,ˆ2
1,
,
:conductor theInside/ˆ
0/ˆ
rHkrE
HrH rkrk
k̂H0
n̂
Ideal conductor boundary condit
0 0
ions:
S S n E n H
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PHY 712 Spring 2014 -- Lecture 19 9
Waveguide terminology• TEM: transverse electric and magnetic (both E and H
fields are perpendicular to wave propagation direction)• TM: transverse magnetic (H field is perpendicular to
wave propagation direction)• TE: transverse electric (E field is perpendicular to wave
propagation direction)
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PHY 712 Spring 2014 -- Lecture 19 10
Wave guides
Coaxial cable TEM modes
Top view:
a
b
z
0 0
for :medium inside equations sMaxwell'
BEBEBE
εiωi
ba
(following problem 8.2 in Jackson’s text)
Inside medium, assumed to be real
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PHY 712 Spring 2014 -- Lecture 19 11
Electromagnetic waves in a coaxial cable -- continued
ˆ cos ˆ sinˆ
ˆ sinˆ cosˆ
ˆ
ˆ
for solution Example
0
0
yxφyxρ
φB
ρE
tiikz
tiikz
eaB
eaE
baTop view:
a
b
zBES ˆ22
1
: ),(with medium cablethin vector wiPoynting22
0*
aBavg
00
:Find
BE
k
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PHY 712 Spring 2014 -- Lecture 19 12
Electromagnetic waves in a coaxial cable -- continuedTop view:
a
b
abaB
ddavg
b
a
lnˆ
:material cablein power averaged Time22
02
0
zS
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PHY 712 Spring 2014 -- Lecture 19 13
Analysis of rectangular waveguide
Boundary conditions at surface of waveguide: Etangential=0, Bnormal=0
z x
y
Cross section view b a
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PHY 712 Spring 2014 -- Lecture 19 14
Analysis of rectangular waveguide
z x
y
tiikz
zyx
tiikzzyx
eyxEyxEyxE
eyxByxByxB
zyxE
zyxB
ˆ,ˆ,ˆ,
ˆ,ˆ,ˆ,
2222
bn
amk
03/17/2013
PHY 712 Spring 2014 -- Lecture 19 15
Maxwell’s equations within the pipe:
0.
0.
yxz
yxz
BB ikBx y
EE ikEx y
.
.
.
zy x
zx y
y xz
E ikE i By
EikE i Bx
E Ei B
x y
.
.
.
zy x
zx y
y xz
B ikB i Ey
BikB i Ex
B Bi E
x y
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PHY 712 Spring 2014 -- Lecture 19 16
Solution of Maxwell’s equations within the pipe:
2 22 2
2 2
Combining Faraday's Law and Ampere's Law, we find that each fieldcomponent must satisfy a two-dimensional Helmholz equation:
( , ) 0.xkx y
E x y
0
2 22 2 2
For the rectangular wave guide discussed in Section 8.4 of yourtext a solution for a TE mode can have
( , ) 0 and ( , ) cos cos ,
with
S
:
z z
mn
m x n yE x y B x y Ba b
m nk ka b
ome of the other field components are:
and .x y y xk kB E B E
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PHY 712 Spring 2014 -- Lecture 19 17
TE modes for retangular wave guide continued:
0
02 2 2 2
02 2 2 2
( , ) 0 and ( , ) cos cos ,
cos sin ,
sin
z z
zx y
zy x
m x n yE x y B x y Ba b
Bi i n m x n yE B Bk k y b a bm n
a b
Bi i mE B Bk k x am n
a b
cos .m x n ya b
tangential
normal
0 because: ( ,0) ( , ) 0
and
Check boundary conditions:
(0, ) ( , ) 0
0
.x x
y y
E x E x b
E y E a y
E
B
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PHY 712 Spring 2014 -- Lecture 19 18
Solution for m=n=1
byn
axmamByxiE
byn
axmbnByxiE
byn
axmByxB
bn
amy
bn
amx
z
cossin/,
sincos/,
coscos,
220
220
0
yxBz ,
yxiEx , yxiE y ,
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PHY 712 Spring 2014 -- Lecture 19 19
Solution for m=n=12 2
2 2 2mn
m nk ka b
k
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PHY 712 Spring 2014 -- Lecture 19 20
dpk
kzyxBkzyxBzyxBkzyxEkzyxEzyxE
ezyxEzyxEzyxE
ezyxBzyxBzyxB
iii
iii
tizyx
tizyx
cos,or sin,,, cos,or sin,,, :generalIn
ˆ,,ˆ,,ˆ,,
ˆ,,ˆ,,ˆ,,
zyxE
zyxB
20
Resonant cavity
z x
y
dzbyax
000
03/17/2013
PHY 712 Spring 2014 -- Lecture 19 2103/17/2013
Resonant cavity
z x
y
dzbyax
000
2222
222
22
1dp
bn
am
bn
am
dpk