Informatica
Parallel Plate Waveguide
w
a
x
0
z
y
a
Assume uniform waves along the y-direction Assume no fringing effects
⇒ w >> a
∂ ⇒ ( ∂y
)=0
Propagation along the z-direction
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Electromagnetic Fields
Maxwell’s equations
∇ × E = − jω µ H ⇓ ˆ ˆ ˆ ix i y iz ∂ ∂ ∂ det ∂ x ∂ y ∂ z Ex E y Ez ∂ ∂ E z − E y = − jωµH x ∂y ∂z ⇒ (1)
∂ ∂ E x − E z = − jωµH y (2) ∂z ∂x ∂ ∂ E y − E x = − jω µH z ∂x ∂y (3)
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Electromagnetic Fields
∇ × H = jω ε E ⇓ ˆ ˆ ˆ ix iy iz ∂ ∂ ∂ det ∂ x ∂ y ∂ z H x H y H z ∂ ∂ H z − H y = jωεE x ∂y ∂z ⇒ (4)
∂ ∂ H x − H z = jωεE y (5) ∂z ∂x ∂ ∂ H y − H x = jωεE z ∂y ∂x(6)
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Electromagnetic Fields
From (1) & (2) & (5)
∂ (1) ∂z
∂ ⇒ E y = jωµ H x ∂z ∂ z2 ∂ E = − jω µ H z 2 y ∂x ∂x ∂2
∂2
∂ (3) ⇒ ∂x ∂2 ∂2
∂ ∂ Ey + E y = jωµ H x − H z = −ω2µ ε E y ∂x ∂z ∂ z2 ∂ x2 From (5) ⇓ jωε E y
Wave equation for Transverse Electric (TE) modes
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From (4) & (6) & (2)
∂ (4) ∂z
⇒
∂ H y = − jωε E x ∂z ∂ z2 ∂ H = jωε E z 2 y ∂x ∂x ∂2
∂2
∂ (6) ⇒ ∂x ∂2 ∂2
∂ ∂ Hy + H y = − jωε E x − E z = −ω2µ ε H y ∂x ∂z ∂ z2 ∂ x2 From (2) ⇓ − jωµ H y
Wave equation for Transverse Magnetic (TM) modes
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Electromagnetic Fields
Transverse Electric (TE)modes
E H
β
θ θ
H
×
E
θ θ
β
H
E
Boundary Conditions
Ey = 0
x = 0 x = a
This solution satisfies the boundary conditions:
E y = Eo sin ( β x x ) e
− jβ z z
Eo − jβ x x − jβ z = j e − e jβ x x e z 2
x z
135
(
)
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Electromagnetic Fields
We have
β2 =
4π2 λ2
= β 2 + β 2 = ω 2µ ε x z
andfrom boundary conditions at the conductor plates
x = 0) E y = 0 x = a) sin ( β x a ) = 0 ⇒ β x a = m π m = 1, 2, 3… mπ β x = β cos θ = a mλ mπ β z = β sin θ = ω µ ε − = ω µ ε 1 − 2a a
2 2
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2 1 / 2
136
Electromagnetic Fields
For each possible index m we have a mode of propagation. Modes are labeled TE10, TE20 , TE30 , …. The first index gives the periodicity (number of half sinusoidal oscillations) between the plates, along the x-direction. The second index is zero to indicate uniform solution along the y-direction. Note that the solution m = 0 (or mode TE00) is not acceptable, because it would require a field configuration with uniform electric field tangent to the metal plates. This is anunphysical boundary condition, which is possible only for the case of trivial solution of zero field everywhere.
E H
β
TE00 ⇒ m = 0 ⇒ βx = 0 & β = βz Unphysical !!!
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Electromagnetic Fields
A mode can propagate only if the frequency is sufficiently high, so that βz > 0. We have the cut-off condition when
mπ 2 π 2a β = βx = = ⇒ λc = aλc m
2 mλ 2 2 mπ ⇒ β z = ω2 µ ε − = ω µ ε 1 − =0 2a a vp m fc = Cut - off frequency for mode m = λ c 2 a µε
Exactly at cut-off the wave would bounce between the plates, without propagation along the wave guide axis.
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1
Electromagnetic Fields
When the frequency is below the cut-off value
2a f < fc ⇒λ > λ c = m mπ 2 βz = ± ω µ ε − a
2
2 mλ 2 = ±ω µ ε 1 − 2a >1
1
1
mλ 2 2 = ± j ω µε − 1 2a ⇒ β z = − jα ⇒ e− j( − jα ) z = e−αz
The mode attenuates entering the guide as an evanescent wave.
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Electromagnetic Fields
Transverse Magnetic (TM) modes
E
β...
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