Waveguides - ieeetsu

www.ece.rutgers.edu/∼orfanidi/ewa 261
262 Electromagnetic Waves & Antennas – S. J. Orfanidis – June 21, 2004
∫
P T = P z dS ,
S
where P z = 1 2 Re(E × H∗ )·ẑ (8.2.1)
It is easily verified that only the transverse components of the fields contribute to
the power flow, that is, P z can be written in the form:
Fig. 8.1.1
Cylindrical coordinates.
(
1 ∂
ρ ∂E )
z
+ 1 ∂ 2 E z
ρ ∂ρ ∂ρ ρ 2 ∂φ + 2 k2 cE z = 0
(
1 ∂
ρ ∂H )
z
+ 1 ∂ 2 H z
ρ ∂ρ ∂ρ ρ 2 ∂φ + 2 k2 c H z = 0
(8.1.23)
P z = 1 2 Re(E T × H ∗ T )·ẑ (8.2.2)
For waveguides with conducting walls, the transmission losses are due primarily to
ohmic losses in (a) the conductors and (b) the dielectric medium filling the space between
the conductors and in which the fields propagate. In dielectric waveguides the losses
are due to absorption and scattering by imperfections.
The transmission losses can be quantified by replacing the propagation wavenumber
β by its complex-valued version β c = β − jα, where α is the attenuation constant. The
z-dependence of all the field components is replaced by:
Noting that ẑ × ˆρ = ˆφ and ẑ × ˆφ =−ˆρ, we obtain:
ẑ ×∇ T H z = ˆφ(∂ ρ H z )−ˆρ 1 ρ (∂ φH z )
The decomposition of a transverse vector is E T = ˆρE ρ + ˆφE φ . The cylindrical
coordinates version of (8.1.16) are:
E ρ =− jβ ( 1
∂ρ
k 2 E z − η TE
c
ρ ∂ )
φH z
E φ =− jβ ( 1
k 2 c ρ ∂ ) ,
φE z + η TE ∂ ρ H z
H ρ =− jβ (
∂ρ
kc
2 H z + 1
η TM ρ ∂ )
φE z
H φ =− jβ ( 1
kc
2 ρ ∂ φH z − 1 )
∂ ρ E z
η TM
(8.1.24)
For either coordinate system, the equations for H T may be obtained from those of
E T by a so-called duality transformation, that is, making the substitutions:
E → H , H →−E , ɛ → µ, µ→ ɛ (duality transformation) (8.1.25)
These imply that η → η −1 and η TE → η −1
TM. Duality is discussed in greater detail in
Sec. 16.2.
8.2 Power Transfer and Attenuation
With the field solutions at hand, one can determine the amount of power transmitted
along the guide, as well as the transmission losses. The total power carried by the fields
along the guide direction is obtained by integrating the z-component of the Poynting
vector over the cross-sectional area of the guide:
e −jβz → e −jβcz = e −(α+jβ)z = e −αz e −jβz (8.2.3)
The quantity α is the sum of the attenuation constants arising from the various loss
mechanisms. For example, if α d and α c are the attenuations due to the ohmic losses in
the dielectric and in the conducting walls, then
α = α d + α c (8.2.4)
The ohmic losses in the dielectric can be characterized either by its loss tangent, say
tan δ, or by its conductivity σ d —the two being related by σ d = ωɛ tan δ. The effective
dielectric constant of the medium is then ɛ(ω)= ɛ − jσ d /ω = ɛ(1 − j tan δ). The
corresponding complex-valued wavenumber β c is obtained by the replacement:
β =
√
ω 2 µɛ − k 2 c → β c =
√
ω 2 µɛ(ω)−k 2 c
For weakly conducting dielectrics, we may make the approximation:
√
β c =
ω 2 µɛ ( 1 − j σ d
ωɛ
) √
√
− k
2
c = β 2 − jωµσ d = β 1 − j ωµσ d
≃ β − j 1 β 2
2 σ ωµ
d
β
Recalling the definition η TE = ωµ/β, we obtain for the attenuation constant:
α d = 1 2 σ dη TE = 1 2
ω 2
βc 2 tan δ =
ω tan δ
√1
(dielectric losses) (8.2.5)
2c − ω 2 c/ω 2
which is similar to Eq. (2.7.2), but with the replacement η d → η TE .
The conductor losses are more complicated to calculate. In practice, the following
approximate procedure is adequate. First, the fields are determined on the assumption
that the conductors are perfect.