3. Microwave Engineering
1. Waveguide
Transmission line
Voltage applied between conductors(E: vertically
between the conductors)
Interior fields: TEM (Transverse ElectroMagnetic)
wave (wave vector indicates the direction of wave
propagation as well as the direction of power flow)
3 EM Wave Lab
5. Microwave Engineering
1. Waveguide
Waveguide modes
TE (Transverse Electric) mode
- E parallel to the transverse plane of the waveguide
TM (Transverse Magnetic) mode
- H is within the transverse plane of the waveguide
5 EM Wave Lab
7. Microwave Engineering
2. Parallel-plate
Wavenumbers
β m = k 2 − km
2
Lossless and nonmagnetic medium
ω ε r ωn
k = ω µ o ε = ω µ oε o ε r = =
c c
where n is a refrective index
7 EM Wave Lab
10. Microwave Engineering
2. Parallel-plate
Cutoff frequency
k mπ mπc mλ
k m = k cos θ m → θ m = cos −1 m = cos −1 = cos −1 = cos −1
k kd ωnd 2nd
2 2
mπ mπc
β m = k − km = k 1 −
2 2
= k 1−
kd ωnd
mπc
cutoff frequency for mode, m : ω cm =
nd
2
nω ω cm
βm = 1−
c ω
If ω > ω cm , real - valued phase constant β m and the mode will propagate.
If ω < ω cm , β m is imaginary and the mode does not propagate.
10 EM Wave Lab
12. Microwave Engineering
2. Parallel-plate
TE mode representation
E ys = E0 e − jk u •r − E0 e − jk d •r
k u = k ma x + β ma z and k d = −k ma x + β ma z , r = xa x + za z
E ys = E0 (e − jk m x − e jk m x )e − jβ m z = 2 jE0 sin ( k m x ) e − jβ m z = E0 sin ( k m x ) e − jβ m z
'
E y ( z , t ) = Re( E ys e jωt ) = E0 sin ( k m x ) cos(ωt − β m z )
'
(TE mode above cutoff)
The TE mode field is the interference pattern resulting from
the superposition of the upward and downward plane waves.
12 EM Wave Lab
13. Microwave Engineering
2. Parallel-plate
TE mode representation
2 2
nω cm ω 2πn λcm
If ω < ω cm , α m = jβ m =
1− =
1−
c ω cm λcm λ
E ys = E0 sin ( k m x ) e −α m z and E y ( z , t ) = E0 sin ( k m x ) e −α m z cos(ωt )
' '
(TE mode below cutoff )
ω cm λ
cos θ m = =
ω λcm
At cutoff (ω = ω cm ), θ m = 0 and the plane waves are just reflecting back and forth;
they are making no forward progress down the guide.
As ω is increased beyond cutoff ( or λ is decreased), the wave angle increases,
approaching 90 o as ω approaches infinity.
13 EM Wave Lab
14. Microwave Engineering
2. Parallel-plate
Phase and group velocity
nω
β m = k sin θ m = sin θ m
c
ω c
Phase velocity v pm = β = n sin θ
m m
This may exceed the speed of light in the medium
: not violate the principle of a special relativity.
2
dω c ω cm c
Group velocity : vgm = = 1− = sin θ m
dβ m n ω n
14 EM Wave Lab
15. Microwave Engineering
2. Parallel-plate
Field analysis
∇ 2 E s = −k 2 E s where k = nω / c
TE modes (only a y component of E)
∂2 ∂2 ∂2 ∂2
E ys + 2 E ys + 2 E ys + k 2 E ys = 0 ( 2 E ys = 0, z - variation : e − jβ m z )
∂x 2 ∂y ∂z ∂y
E ys = E 0 f m ( x)e − jβ m z
2
d f m ( x) d 2 f m ( x)
+ (k 2 − β m ) f m ( x) = 0
2
k2 − β m = km ,
2 2
+ k m f m ( x) = 0
2
dx 2 dx 2
f m ( x) = cos(k m x) + sin( k m x),
mπx
BC : E y must be zero at x = 0 and x = d . → f m ( x) = sin
d
15 EM Wave Lab
16. Microwave Engineering
2. Parallel-plate
Characteristics of TE mode
At cutoff, the plane wave angle of incidence in the guide is zero.
The wave simply bounces up and down between the conducting walls.
Net round - trip phase shift is 2mπ .
mπ 2nπ mλcm
β m = 0 and k m = k = 2nπ / λcm ⇒ At cutoff, = ⇒d =
d λcm 2n
mπx 2nπx mπx − jβ m z
At cutoff, E ys = E 0 sin = E ys = E 0 sin
λ E ys = E 0 sin e
d cm d
One dimensional resonant cavity
16 EM Wave Lab
17. Microwave Engineering
2. Parallel-plate
Field representations
x and z components of H s for a TE mode
∇ × E s = − jωµH s
Only a y component of E s ,
∂E ys ∂E ys
∇ × Es = az − a x = k m E 0 cos(k m x)e − jβ m z a z + jβ m E 0 sin( k m x)e − jβ m z a x
∂x ∂z
βm − jβ m z km
H xs =− E 0 sin( k m x)e , H zs = j E 0 cos(k m x)e − jβ m z
ωµ ωµ
17 EM Wave Lab
18. Microwave Engineering
2. Parallel-plate
Intrinsic impedance
| H s |= H s • H * = H xs H xs xs + H zs H zs
s
* *
| H s |=
ωµ
( k m + β m ) (sin (k m x) + cos (k m x)) = ωµ = η
E0 2 2 1/ 2 2 2 kE0 E 0
( k 2
m + β m = k 2 and sin 2 A + cos 2 A = 1)
2
18 EM Wave Lab
19. Microwave Engineering
3. Waveguide
Rectangular waveguide
WR (Waveguide Rectangular) series
- EIA (Electronic Industry Association) designation
WR-62 1.58 cmx0.79 cm
- Size:
- Recommended range: 12.4-18.0 GHz
- Cutoff: 9.486 GHz
a = 62 inch/100 = 62 × 2.54 cm/100
b ≈ a/2
19 EM Wave Lab
21. Microwave Engineering
3. Waveguide
TE and TM modes
Hz and Ez fields: TE and TM modes
Non-TEM modes: Hz = Ez = 0
Concept of a dominant mode: TE10 mode
Boundary condition enforcements
: PEC (Perfect Electric Conductor)
H z ( x, y ) = cos(am x) cos(bn y )e − jβ mn z : TEmn mode
E z ( x, y ) = sin( am x) sin(bn y )e − jβ mn z : TM mn mode
mπ nπ
where am = , bn = , and β mn = k 2 − am − bn 2 2
a b
21 EM Wave Lab
22. Microwave Engineering
3. Waveguide
Dominant mode: TE 10 mode
H z ( x, y ) = cos(a1 x) cos(b0 y )e − jβ10 z
= cos(πx / a)e − jβ10 z
where m = 1 and n = 0.
π
β10 = k − 2
a
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24. Microwave Engineering
4. Tx line
Microstrip line
Quasi-TEM line
Easy fabrication: etching
Substrate
Characteristic impedance
24 EM Wave Lab
25. Microwave Engineering
4. Tx line
Substrate
Relative permittivity
Thickness of a substrate: mil (inch/1000)
Thickness of a metal: oz (almost 1.4 mils)
Loss: loss tangent
Temperature
Power amplifier module
25 EM Wave Lab
26. Microwave Engineering
4. Tx line
Etching
Film
Photoresist (PR)
Toluene
Ultraviolet
Iron chloride
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