text stringlengths 1 1k ⌀ | source stringclasses 12
values |
|---|---|
n1 sin ψi = n2 sin ψt (5.214)
When both media are non-magnetic, Equation 5.212
simplifies to
ψt = arcsin
(√
ǫr1
ǫr2
sin ψi
)
(5.215)
When ǫr2 >
ǫr1, we observe that ψt <ψi. In other
words, the transmitted wave travels in a direction that
is closer to the surface normal than the angle of
incidence. This scenario is demon... | Electromagnetics_Vol2.pdf |
82 CHAPTER 5. W A VE REFLECTION AND TRANSMISSION
c⃝ Z. S´andor CC BY -SA 3.0
Figure 5.15: Angles of reflection and refraction for a
light wave incident from air onto glass.
As expected, the angle of reflection ψr is
observ
ed to be equal to ψi. The angle of
refraction ψt is observed to be 35◦. What is the
relative permit... | Electromagnetics_Vol2.pdf |
the surface normal. Figure 5.16 shows an example
from common experience.
In non-magnetic media, when ǫr1 < ǫr2, ψt <
ψi (refraction
toward the surface normal). When
ǫr1 > ǫr2, ψt > ψi (refraction away from the
surface normal).
c⃝ G. Saini CC BY -SA 4.0
Figure 5.16: Refraction accounts for the apparent dis-
placement of... | Electromagnetics_Vol2.pdf |
all we can say is that when ǫr2 <ǫr1, ψt is able to
reach π/2 radians, which corresponds to propagation
parallel to the boundary . Beyond that threshold, we
must account for the unique physical considerations
associated with total internal reflection.
W e conclude this section with a description of the
common waveguidin... | Electromagnetics_Vol2.pdf |
frequency . Thus, each frequency is refracted by a
different amount. Conversely , a prism comprised of a
material whose permittivity exhibits negligible
variation with frequency will not separate incident | Electromagnetics_Vol2.pdf |
5.9. TE REFLECTION IN NON-MAGNETIC MEDIA 83
c⃝ D-K uru CC BY -SA 3.0
Figure 5.17: A typical triangular prism.
white light into its constituent colors since each color
will be refracted by the same amount.
Additional Reading:
• “Prism” on Wikipedia.
• “Refraction” on Wikipedia.
• “Refractive index” on Wikipedia.
• “Snel... | Electromagnetics_Vol2.pdf |
respectively . Many materials of practical interest are
non-magnetic; that is, they have permeability that is
not significantly different from the permeability of
free space. In this section, we consider the behavior
of the reflection coefficient for this class of materials.
T o begin, recall the general form of Snell’s l... | Electromagnetics_Vol2.pdf |
84 CHAPTER 5. W A VE REFLECTION AND TRANSMISSION
Since permittivity ǫcan be expressed as ǫ0 times the
relative permittivity ǫr, we may reduce further to:
β1
β2
=
√ ǫr1
ǫr2
(5.220)
No
w Equation 5.218 reduces to:
sin ψt =
√ ǫr1
ǫr2
sin ψi (5.221)
Ne
xt, note that for any value ψ, one may write cosine
in terms of sine as... | Electromagnetics_Vol2.pdf |
Finally
, by substituting Equation 5.223, we obtain:
Γ TE =
cos ψi −
√
ǫr2/ǫr1 − s in 2 ψi
cos ψi +
√
ǫr2/ǫr1 − s in 2 ψi
(5.228)
This
expression has the advantage that it is now
entirely in terms of ψi, with no need to first calculate
ψt.
Using Equation 5.228, we can see how different
combinations of material affect th... | Electromagnetics_Vol2.pdf |
ǫr2/ǫr1 − sin2 ψi to be negative, which makes Γ TE
complex-valued. This results in total internal
reflection, and is addressed elsewhere in another
section. When ǫr1 <ǫr2 (e.g., wave traveling in air
toward glass), we see that ǫr2/ǫr1 − sin2 ψi is always
positive, so Γ TE is always real-valued.
Let us continue with the ... | Electromagnetics_Vol2.pdf |
5.10. TM REFLECTION IN NON-MAGNETIC MEDIA 85
5.10 TM Reflection in
Non-magnetic Media
[m0172]
Figure 5.21 shows a TM uniform plane wave incident
on
the planar boundary between two semi-infinite
material regions. In this case, the reflection coefficient
is given by:
Γ TM = −η1 cos ψi + η2 cos ψt
+η1 cos ψi + η2 cos ψt (5.22... | Electromagnetics_Vol2.pdf |
In
non-magnetic media, the permeabilities µ1 and µ2
are assumed equal to µ0. Thus:
β1
β2
= ω√µ1ǫ1
ω√µ2ǫ2
=
√ ǫ1
ǫ2
(5.231)
c⃝ C. W ang CC BY -SA 4.0
Figure 5.21: A transverse magnetic uniform plane
wave obliquely incident on the planar boundary be-
tween two semi-infinite material regions.
Since permittivity ǫcan be exp... | Electromagnetics_Vol2.pdf |
Γ TM = −
(
η0/√ǫr1
)
cos ψi +
(
η0/√ǫr2
)
cos ψt
+
(
η0/√ǫr1
)
cos ψi +
(
η0/√ǫr2
)
cos ψt
(5.238)
Multiplying
numerator and denominator by √ǫr2/η0,
we
obtain:
Γ TM = −
√
ǫr2/ǫr1 cos ψi + cos ψt
+
√
ǫr2/ǫr1 cos ψi + cos ψt (5.239)
Substituting Equation 5.235, we obtain:
Γ TM =
−
√
ǫr2/ǫr1 cos ψi +
√
1 − (ǫr1/ǫr2) sin2 ... | Electromagnetics_Vol2.pdf |
86 CHAPTER 5. W A VE REFLECTION AND TRANSMISSION
-1
-0.5
0
0.5
1
0 10 20 30 40 50 60 70 80 90
2
10
100
Reflection Coefficient
angle of incidence [deg]
Figure 5.22: The reflection coefficient Γ TM as a func-
tion of angle of incidence ψi for various media combi-
nations, parameterized as ǫr2/ǫr1.
Using Equati... | Electromagnetics_Vol2.pdf |
Γ TM is always real-valued.
Let us continue with the ǫr1 <ǫr2 condition.
Figure 5.22 shows Γ TM plotted for various
combinations of media over all possible angles of
incidence from 0 (normal incidence) to π/2 (grazing
incidence). W e observe:
In non-magnetic media with ǫr1 < ǫr2, Γ T M is
real-valued and increases from... | Electromagnetics_Vol2.pdf |
would obtain for a perfect conductor in Region 2.
Also note that when ǫr1 <ǫr2, Γ TM changes sign
from negative to positive as angle of incidence
increases from 0 to π/2. This behavior is quite
different from that of the TE component, which is
always negative for ǫr1 <ǫr2. The angle of incidence
at which Γ TM = 0 is re... | Electromagnetics_Vol2.pdf |
Applying the principle of superposition, we may
consider these components separately . The TE
component of the incident wave will scatter as
reflected and transmitted waves which are also TE.
However, Γ TM = 0 when ψi = ψi
B, so the TM | Electromagnetics_Vol2.pdf |
5.10. TM REFLECTION IN NON-MAGNETIC MEDIA 87
component of the transmitted wave will be TM, but
the TM component of the reflected wave will be zero.
Thus, the total (TE+ TM) reflected wave will be
purely TE, regardless of the TM component of the
incident wave. This principle can be exploited to
suppress the TM component o... | Electromagnetics_Vol2.pdf |
B = R− sin2 ψi
B (5.244)
Now employing a trigonometric identity on the left
side of the equation, we obtain:
R2 (
1 − sin2 ψi
B
)
= R− sin2 ψi
B (5.245)
R2 − R2 sin2 ψi
B = R− sin2 ψi
B (5.246)
(
1 − R2)
sin2 ψi
B = R− R2 (5.247)
and finally
sin ψi
B =
√
R− R2
1 − R2 (5.248)
Although
this equation gets the job done, it ... | Electromagnetics_Vol2.pdf |
ster’s angle for non-magnetic media.
Thus, we have found
tan ψi
B =
√ ǫr2
ǫr1
(5.253)
Example 5.10. Polarizing angle for an
air-to-glass interface.
A plane wave is incident from air onto the planar
boundary with a glass region. The glass exhibits
relative permittivity of 2.1. The incident wave
contains both TE and TM c... | Electromagnetics_Vol2.pdf |
88 CHAPTER 5. W A VE REFLECTION AND TRANSMISSION
5.11 T otal Internal Reflection
[m0169]
T otal internal reflection refers to a particular
condition resulting in the complete reflection of a
wave at the boundary between two media, with no
power transmitted into the second region. One way to
achieve complete reflection with... | Electromagnetics_Vol2.pdf |
ψr = ψi (5.255)
i.e., angle of reflection equals angle of incidence.
Also, from Snell’s law:
√
µr1ǫr1 sin ψi = √µr2ǫr2 sin ψt (5.256)
where
“r” in the subscripts indicates the relative
(unitless) quantities. The associated formula for ψt
c⃝ C. W ang CC BY -SA 4.0 (modified)
Figure 5.25: A uniform plane wave obliquely inc... | Electromagnetics_Vol2.pdf |
increased? When calculating ψt using
Equation 5.257, one finds that the argument of the
arcsine function becomes greater than 1. Since the
possible values of the sine function are between −1
and +1, the arcsine function is undefined. Clearly our
analysis is inadequate in this situation.
T o make sense of this, let us beg... | Electromagnetics_Vol2.pdf |
TE component and non-magnetic materials is
Γ TE =
cos ψi −
√
ǫr2/ǫr1 − s in 2 ψi
cos ψi +
√
ǫr2/ǫr1 − s in 2 ψi
(5.261)
From Equation 5.260, we see that
sin2 ψi
c = ǫr2
ǫr1
(5.262) | Electromagnetics_Vol2.pdf |
5.11. TOT AL INTERNAL REFLECTION 89
So, when ψi >ψi
c, we see that
ǫr2/ǫr1 − sin2 ψi <0 (ψ i >ψi
c) (5.263)
and therefore√
ǫr2/ǫr1 − s in 2 ψi = jB (ψi >ψi
c) (5.264)
where Bis a positive real-valued number. Now we
may write Equation 5.261 as follows:
Γ TE = A− jB
A+ jB ( ψi >ψi
c) (5.265)
where A≜ cos ψi is also a pos... | Electromagnetics_Vol2.pdf |
component, the identical conclusion is obtained for
TM component as well. This is left as an exercise for
the student.
Example 5.11. T otal internal reflection in glass.
Figure 5.15 (Section 5.8) shows a demonstration
of refraction of a beam of light incident from air
onto a planar boundary with glass. Analysis of
that ... | Electromagnetics_Vol2.pdf |
Figure 5.26: T otal internal reflection of a light wave
incident on a planar boundary between glass and air.
c⃝ Ti mwether CC BY -SA 3.0
Figure 5.27: Laser light in a dielectric rod exhibiting
the Goos-H ¨anchen effect.
The angle of incidence in Figure 5.26 is seen to
be
about ∼= 50◦, which is greater than the critical
... | Electromagnetics_Vol2.pdf |
90 CHAPTER 5. W A VE REFLECTION AND TRANSMISSION
The presence of an imaginary component in the
reflection coefficient is odd for two reasons. First, we
are not accustomed to seeing a complex-valued
reflection coefficient emerge when the wave
impedances of the associated media are real-valued.
Second, the total reflection of... | Electromagnetics_Vol2.pdf |
• “Goos-H ¨anchen effect” on Wikipedia.
• “Snell’s law” on Wikipedia.
• “T otal internal reflection” on Wikipedia.
5.12 Evanescent W aves
[m0170]
Consider the situation shown in Figure 5.28: A
uniform
plane wave obliquely incident on the planar
boundary between two semi-infinite material regions,
and total internal reflec... | Electromagnetics_Vol2.pdf |
the boundary if no power is transmitted across the
boundary? There must be a field on the opposite side
of the boundary , but – somehow – it must have zero
power. T o make sense of this, let us attempt to find a
solution for the transmitted field.
c⃝ C. W ang CC BY -SA 4.0
Figure 5.28: A uniform plane wave obliquely inci-... | Electromagnetics_Vol2.pdf |
5.12. EV ANESCENT W A VES 91
W e begin by postulating a complex-valued angle of
transmission ψtc. Although the concept of a
complex-valued angle may seem counterintuitive,
there is mathematical support for this concept. For
example, consider the well-known trigonometric
identities:
sin θ= 1
j2
(
ejθ − e−j
θ)
(5.268)
co... | Electromagnetics_Vol2.pdf |
real part of ψtc remains fixed (parallel to the
boundary) and that an imaginary component jψ′′
emerges to satisfy the boundary conditions.
For clarity , let us assign the variable ψ′ to represent
the real part of ψtc in Equations 5.270–5.272. Then
we may refer to all three cases using a single
expression as follows:
ψtc... | Electromagnetics_Vol2.pdf |
sin jψ′′ = 1
j2
(
ej(jψ′
′) − e−j(jψ′′))
(5.279)
= 1
j2
(
e−ψ′′
− e+ψ′
′ )
(5.280)
= j1
2
(
e+ψ′′
− e−ψ′
′ )
(5.281)
= jsinh ψ′′ (5.282)
In other words, the sine of jψ′′ is jtimes hyperbolic
sine (“sinh ”) of ψ′′. Now note that sinh of a
real-valued argument is real-valued, so sin jψ′′ is
imaginary-valued.
Using these ... | Electromagnetics_Vol2.pdf |
When total internal reflection is in effect, ψi >ψi
c, so
ψ′ = π/2. In this case, Equations 5.283 and 5.284
yield
sin ψtc = cosh ψ′′ (5.287)
cos ψtc = −jsinh ψ′′ (5.288)
Let us now consider what this means for the field in
Region 2. According to the formalism adopted in
previous sections, the propagation of wave | Electromagnetics_Vol2.pdf |
92 CHAPTER 5. W A VE REFLECTION AND TRANSMISSION
components in this region is described by the factor
e−jkt·r where
kt = β2 ˆkt
= β2
(ˆx sin ψtc + ˆz cos ψtc)
(5.289)
and
r = ˆxx+ ˆyy+ ˆzz (5.290)
so
kt · r =
(
β2 sin ψtc)
x+
(
β2 cos ψtc)
z
= ( β2 cosh ψ′′) x+ (−jβ2 sinh ψ′′) z (5.291)
Therefore, the wave in Region 2 ... | Electromagnetics_Vol2.pdf |
unlike the transmitted wave in the ψi <ψi
c case (also
a uniform plane wave). The transmitted wave that we
have derived in the ψi >ψi
c case gives the impression
of being somehow attached to the boundary , and so
may be described as a surface wave. However, in this
case we have a particular kind of surface wave,
known ... | Electromagnetics_Vol2.pdf |
propagation and attenuation constants for the
evanescent wave. It suffices to say that the magnitude
of the evanescent field becomes negligible beyond a
few wavelengths of the boundary .
Finally , we return to the strangest characteristic of
this field: It acts like a wave, but conveys no power.
This field exists solely to... | Electromagnetics_Vol2.pdf |
and continues to propagate to infinity , even after light
is no longer incident on the boundary . In contrast, the
evanescent wave vanishes at the moment laser light
ceases to illuminate the boundary . In other words, the
evanescent field does not continue to propagate along
the boundary to infinity . The reason for this ... | Electromagnetics_Vol2.pdf |
5.12. EV ANESCENT W A VES 93
Image Credits
Fig. 5.1: c⃝ Sevenchw (C. W ang),
https://commons.wikimedia.org/wiki/File:Upw incident on planar boundary .svg,
CC
BY -SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.2: c⃝ Sevenchw (C. W ang), https://commons.wikimedia.org/wiki/File:Upw incident on a slab.svg... | Electromagnetics_Vol2.pdf |
CC
BY -SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.6: c⃝ Sevenchw (C. W ang),
https://commons.wikimedia.org/wiki/File:Plane wave in another rotation coord.svg,
CC
BY -SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.7: c⃝ Sevenchw (C. W ang),
https://commons.wikimedia.org/wiki/File:... | Electromagnetics_Vol2.pdf |
CC
BY -SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.11: c⃝ Sevenchw (C. W ang),
https://commons.wikimedia.org/wiki/File:TE-polarized upw incident from air to glass.svg,
CC
BY -SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.12: c⃝ Sevenchw (C. W ang),
https://commons.wikimedia.org/w... | Electromagnetics_Vol2.pdf |
94 CHAPTER 5. W A VE REFLECTION AND TRANSMISSION
Fig. 5.15: c⃝ Z. S ´andor, https://commons.wikimedia.org/wiki/File:F%C3%A9nyt%C3%B6r%C3%A9s.jpg,
CC BY -SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/).
Fig. 5.16: c⃝ G. Saini, https://kids.kiddle.co/Image:Refractionn.jpg,
CC BY -SA 4.0 (https://creativecommons.... | Electromagnetics_Vol2.pdf |
Fig. 5.21: c⃝ Sevenchw (C. W ang),
https://commons.wikimedia.org/wiki/File:TM-polarized upw incident on planar boundary .svg,
CC
BY -SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/).
Fig. 5.23: c⃝ Sevenchw (C. W ang),
https://commons.wikimedia.org/wiki/File:Reflection of plane wave incidence angle equals polariz... | Electromagnetics_Vol2.pdf |
https://en.wikipedia.org/wiki/File:T eljes f%C3%A9nyvisszaver%C5%91d%C3%A9s.jpg,
CC
BY -SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/).
Fig. 5.27: c⃝ Timwether, https://en.wikipedia.org/wiki/File:Laser in fibre.jpg,
CC
BY -SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/).
Fig. 5.28: c⃝ Sevenchw (C. W a... | Electromagnetics_Vol2.pdf |
Chapter 6
W a
veguides
6.1 Phase and Group V elocity
[m0176]
Phase velocity is the speed at which a point of
constant
phase travels as the wave propagates. 1 For a
sinusoidally-varying wave, this speed is easy to
quantify . T o see this, consider the wave:
Acos (ωt− βz+ ψ) (6.1)
where ω= 2πf is angular frequency , zis ... | Electromagnetics_Vol2.pdf |
1 Formally , “velocity” is a vector which indicates both the di-
rection and rate of motion. It is common practice to use the terms
“phase velocity” and “group velocity” even though we are actually
referring merely to rate of motion. The direction is, of course, in the
direction of propagation.
Central to the concept o... | Electromagnetics_Vol2.pdf |
phase modulation). In other words, information can
be transmitted only by making the wave non-uniform
in some respect. Furthermore, some materials and
structures can cause changes in ψor other
combinations of parameters which vary with position
or time. Examples include dispersion and propagation
within waveguides. Reg... | Electromagnetics_Vol2.pdf |
Group velocity, vg, is the ratio of the apparent
change in frequency ωto the associated change in
the phase propagation constant β; i.e., ∆ω/∆β .
Electromagnetics V ol. 2. c⃝ 2020 S.W . Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics- vol- 2 | Electromagnetics_Vol2.pdf |
96 CHAPTER 6. W A VEGUIDES
Letting ∆β become vanishingly small, we obtain
vg ≜ ∂ω
∂β (6.5)
Note
the similarity to the definition of phase velocity
in Equation 6.3. Group velocity can be interpreted as
the speed at which a disturbance in the wave
propagates. Information may be conveyed as
meaningful disturbances relative... | Electromagnetics_Vol2.pdf |
and dispersion (frequency-dependent constitutive
parameters) are examples in which vg is not
necessarily equal to vp. Here’s an example involving
dispersion:
Example 6.1. Phase and group velocity for a
material exhibiting square-law dispersion.
A broad class of non-magnetic dispersive media
exhibit relative permittivit... | Electromagnetics_Vol2.pdf |
(6.12)
No
w simplifying using Equation 6.8:
vg =
(
2 β
ω
)−1
(6.13)
= 1
2
ω
β (6.14)
= 1
2vp (6.15)
Thus,
we see that in this case the group velocity
is always half the phase velocity .
Another commonly-encountered example for which
vg is not necessarily equal to vp is the propagation of
guided waves; e.g., waves withi... | Electromagnetics_Vol2.pdf |
6.2. P ARALLEL PLA TE W A VEGUIDE: INTRODUCTION 97
• “Phase velocity” on Wikipedia.
• “Speed of light” on Wikipedia.
6.2 Parallel Plate W aveguide:
Introduction
[m0173]
A parallel plate waveguide is a device for guiding the
propag
ation of waves between two
perfectly-conducting plates. Our primary interest in
this stru... | Electromagnetics_Vol2.pdf |
exhibiting real-valued permeability µand real-valued
permittivity ǫ.
Let us limit our attention to a region within the
waveguide which is free of sources. Expressed in
phasor form, the electric field intensity is governed by
the wave equation
∇2 ˜E + β2 ˜E = 0 (6.16)
where
β = ω√
µǫ (6.17)
Equation
6.16 is a partial dif... | Electromagnetics_Vol2.pdf |
98 CHAPTER 6. W A VEGUIDES
imposed by the perfectly-conducting plates, is
sufficient to determine a unique solution. This is most
easily done in Cartesian coordinates, as we shall now
demonstrate. First we express ˜E in Cartesian
coordinates:
˜E = ˆx ˜Ex + ˆy ˜Ey + ˆz ˜Ez (6.18)
This facilitates the decomposition of Equ... | Electromagnetics_Vol2.pdf |
∂y2
˜Ez + ∂2
∂z2
˜Ez = −β2 ˜Ez (6.25)
Let
us restrict our attention to scenarios that can be
completely described in two dimensions; namely x
and z, and for which there is no variation in y. This is
not necessarily required, however, it is representative
of a broad class of relevant problems, and allows
Equations 6.23–... | Electromagnetics_Vol2.pdf |
z= d; namely , that the tangent component of ˜E is
zero at these surfaces.
At this point, the problem has been reduced to a
routine exercise in the solution of partial differential
equations. However, a somewhat more useful
approach is to first decompose the total electric field
into transverse electric (TE) and transver... | Electromagnetics_Vol2.pdf |
components.
The TE and TM solutions are presented in
Sections 6.3 (Electric component of the TE solution),
6.4 (Magnetic component of the TE solution), and 6.5
(Electric component of the TM solution). The
magnetic component of the TM solution can be
determined via a straightforward variation of the
preceding three case... | Electromagnetics_Vol2.pdf |
6.3. P ARALLEL PLA TE W A VEGUIDE: TE CASE, ELECTRIC FIELD 99
6.3 Parallel Plate W aveguide: TE
Case, Electric Field
[m0174]
In Section 6.2, the parallel plate waveguide was
introduced.
At the end of that section, we described
the decomposition of the problem into its TE and TM
components. In this section, we find the e... | Electromagnetics_Vol2.pdf |
and kx and kz are real-valued constants. W e have
assigned variable names to these constants with
advance knowledge of their physical interpretation;
however, at this moment they remain simply unknown
constants whose values must be determined by
enforcement of boundary conditions.
c⃝ C. W ang CC BY -SA 4.0
Figure 6.2: ... | Electromagnetics_Vol2.pdf |
be no wave components propagating in the −ˆz
direction. In this case, C = D= 0 and Equation 6.30
simplifies to:
˜Ey = e−jkz z[
Ae−jkxx + Be+jkxx]
(6.31)
Before proceeding, let’s make sure that Equation 6.31
is actually a solution to Equation 6.29. As we shall
see in a moment, performing this check will reveal
some addit... | Electromagnetics_Vol2.pdf |
100 CHAPTER 6. W A VEGUIDES
This confirms that kx and kz are in fact the
components of the propagation vector
k ≜ βˆk = ˆxkx + ˆyky + ˆzkz (6.39)
where ˆk is the unit vector pointing in the direction of
propagation, and ky = 0 in this particular problem.
The solution has now been reduced to finding the
constants A, B, an... | Electromagnetics_Vol2.pdf |
This expression is simplified using a trigonometric
identity:
1
2j
[
e+jkxx − e−j
kxx]
= sin kxx (6.42)
Let us now make the definition Ey0 ≜ j2B. Then:
˜Ey = Ey0e−jkz zsin kxx (6.43)
Now applying the boundary condition at x= a:
Ey0e−jkz zsin kxa= 0 (6.44)
The factor e−jkz z cannot be zero, and Ey0 = 0 yields
only trivial... | Electromagnetics_Vol2.pdf |
given by Equation 6.43 and Equation 6.46 with
m= 1,2,.... Each solution associated with a
particular value of mis referred to as a mode, which
(via Equation 6.46) has a particular value of kx. The
value of kz for mode mis obtained using
Equation 6.38 as follows:
kz =
√
β2 − k2x
=
√
β2 −
(mπ
a
)2
(6.47)
Since kz is
spec... | Electromagnetics_Vol2.pdf |
component in order to propagate; therefore, these
modes do not propagate and may be ignored.
Let us now summarize the solution. For the scenario
depicted in Figure 6.2, the electric field component of
the TE solution is given by:
ˆy ˜Ey = ˆy
∞∑
m=1
˜E(m)
y
(6.52) | Electromagnetics_Vol2.pdf |
6.3. P ARALLEL PLA TE W A VEGUIDE: TE CASE, ELECTRIC FIELD 101
where
˜E(m)
y ≜
{
0, f <f(m)
c
E(m)
y0 e−jk(m)
z zsin k(m)
x x, f ≥ f(m)
c
(6.53)
where menumerates
modes (m = 1,2,...),
k(m)
z ≜
√
β2 −
[
k(m)
x
] 2
(6.54)
and
k(m)
x ≜ mπ
/a (6.55)
Finally
, E(m)
y0 is a complex-valued constant that
depends on sources or ... | Electromagnetics_Vol2.pdf |
f
>1/2a√µǫ. Also k( 1)
x = π/a, so
k(1)
z =
√
β2 −
(π
a
)2
(6.56)
Subsequently
,
˜E(1)
y = E(1)
y0 e−jk(1)
z zsin πx
a (6.57)
Note
that this mode has the form of a plane wave. The
plane wave propagates in the +ˆz direction with phase
propagation constant k(1)
z . Also, we observe that the
apparent plane wave is non-uni... | Electromagnetics_Vol2.pdf |
Right: m= 2.
f >1/a√µǫ. This frequency is higher than f( 1)
c , so
the m= 1 mode can exist at any frequency at which
the m= 2 mode exists. Also k(2)
x = 2π/a, so
k(2)
z =
√
β2 −
(2π
a
)2
(6.58)
Subsequently
,
˜E(2)
y = E(2)
y0 e−jk(2)
z zsin 2πx
a (6.59)
In
this case, the apparent plane wave propagates in the
+ˆz direc... | Electromagnetics_Vol2.pdf |
increasing integer number of sinusoidal half-periods
in magnitude.
Example 6.2. Single-mode TE propagation in a
parallel plate waveguide.
Consider an air-filled parallel plate waveguide
consisting of plates separated by 1 cm.
Determine the frequency range for which one | Electromagnetics_Vol2.pdf |
102 CHAPTER 6. W A VEGUIDES
(and only one) propagating TE mode is assured.
Solution. Single-mode
TE propagation is
assured by limiting frequency f to greater than
the cutoff frequency for m= 1, but lower than
the cutoff frequency for m= 2. (Any frequency
higher than the cutoff frequency for m= 2
allows at least 2 modes... | Electromagnetics_Vol2.pdf |
1/√µ0ǫ0 = c. However, the phase velocity indicated
by
Equation 6.62 is greater than 1/√µǫ; e.g., faster
than
light would travel in the same material
(presuming it were transparent). At first glance, this
may seem to be impossible. However, recall that
information travels at the group velocity vg, and not
necessarily the... | Electromagnetics_Vol2.pdf |
dispersion, and sometimes specifically as mode
dispersion or modal dispersion.
6.4 Parallel Plate W aveguide: TE
Case, Magnetic Field
[m0175]
In Section 6.2, the parallel plate waveguide was
introduced.
In Section 6.3, we determined the TE
component of the electric field. In this section, we
determine the TE component of... | Electromagnetics_Vol2.pdf |
of ˜E are zero. The two remaining terms are
−ˆx∂˜Ey/∂z and +ˆz∂˜Ey/∂x. Thus:
˜H = j
ωµ
(
−ˆx∂˜Ey
∂z + ˆz∂˜Ey
∂x
)
(6.66)
Recall
that ˜Ey is the sum of modes, as indicated in
Equation 6.63. Since differentiation (i.e., ∂/∂z and
∂/∂x) is a linear operator, we may evaluate
Equation 6.66 for modes one at a time, and then s... | Electromagnetics_Vol2.pdf |
6.4. P ARALLEL PLA TE W A VEGUIDE: TE CASE, MAGNETIC FIELD 103
and
∂˜E(m)
y
∂x = ∂
∂x E(m)
y0 e−jk(m)
z zs in k(m)
x x
=
(
E(m)
y0 e−jk(m)
z zcos k(m)
x x
)(
+k(m)
x
)
(6.68)
W e may now assemble a solution for the magnetic
field as follows:
ˆx ˜Hx + ˆz ˜Hz = ˆx
∞∑
m=1
˜H(m)
x
+ ˆz
∞∑
m=1
˜H(m)
z (6.69)
where
˜H(m)
x = ... | Electromagnetics_Vol2.pdf |
surface at x= 0, we see
˜H(m)
x (x= 0) = 0
˜H(m)
z (x= 0) = +j k(m)
x
ωµ E(m)
y0 e−jk(m)
z z (6.72)
Similarly
, on the PEC surface at x= a, we see
˜H(m)
x (x= a) = 0
˜H(m)
z (x= a) = −jk(m)
x
ωµ E(m)
y0 e−jk(m)
z z (6.73)
Thus,
we see the magnetic field vector at the PEC
surfaces is non-zero and parallel to the PEC surf... | Electromagnetics_Vol2.pdf |
on the inner and outer conductors, or the
electromagnetic fields between the conductors. The
parallel plate waveguide is only slightly more
complicated because the field in a properly-designed
coaxial cable is a single transverse electromagnetic
(TEM) mode, whereas the fields in a parallel plate
waveguide are combinations... | Electromagnetics_Vol2.pdf |
104 CHAPTER 6. W A VEGUIDES
6.5 Parallel Plate W aveguide:
TM Case, Electric Field
[m0177]
In Section 6.2, the parallel plate waveguide shown in
Figure
6.4 was introduced. At the end of that section,
we decomposed the problem into its TE and TM
components. In this section, we find the TM
component of the fields in the wa... | Electromagnetics_Vol2.pdf |
+ e+jkz z[
Ce−jkxx + De+jkxx]
(6.75)
where A, B, C, and Dare complex-valued constants;
and kx and kz are real-valued constants. W e have
assigned variable names to these constants with
advance knowledge of their physical interpretation;
however, at this moment they remain simply unknown
constants whose values must be d... | Electromagnetics_Vol2.pdf |
on the right (z > 0) side of Figure 6.4, then there can
be no wave components propagating in the −ˆz
direction. In this case, C = D= 0 and Equation 6.75
simplifies to:
˜Hy = e−jkz z[
Ae−jkxx + Be+jkxx]
(6.76)
Before proceeding, let’s make sure that Equation 6.76
is actually a solution to Equation 6.74. As in the TE
case... | Electromagnetics_Vol2.pdf |
6.5. P ARALLEL PLA TE W A VEGUIDE: TM CASE, ELECTRIC FIELD 105
This is precisely the same constraint identified in the
TE case, and confirms that kx and ky are in fact the
components of the propagation vector
k ≜ βˆk = ˆxkx + ˆyky + ˆzkz (6.83)
where ˆk is the unit vector pointing in the direction of
propagation, and ky ... | Electromagnetics_Vol2.pdf |
−ˆx∂˜Hy/∂z and +ˆz∂˜Hy/∂x. Thus:
˜E = 1
jωǫ
(
−ˆx∂˜Hy
∂z + ˆz∂˜Hy
∂x
)
(6.86)
W
e may further develop this expression using
Equations 6.77 and 6.79. W e find the ˆx component of
˜E is:
˜Ex = kz
ωǫ e−jkz z[
Ae−j
kxx + Be+jkxx]
(6.87)
and the ˆz component of ˜E is:
˜Ez = kx
ωǫ e−jkz z[
−Ae−j
kxx + Be+jkxx]
(6.88)
The solu... | Electromagnetics_Vol2.pdf |
this is unnecessarily restrictive. Instead, we require
A= Band we may rewrite Equation 6.88 as follows:
˜Ez = Bkx
ωǫ e−jkz z[
e+j
kxx − e−jkxx]
(6.90)
This expression is simplified using a trigonometric
identity:
sin kxa= 1
2j
[
e+jkxa − e−j
kxa]
(6.91)
Thus:
˜Ez = j2Bkx
ωǫ e−jkz zs in kxx (6.92)
Now following up with ˜... | Electromagnetics_Vol2.pdf |
106 CHAPTER 6. W A VEGUIDES
where mis an integer. Note that this is precisely the
same relationship that we identified in the TE case.
There is an important difference, however. In the TE
case, m= 0 was not of interest because this yields
kx = 0, and the associated field turned out to be
identically zero. In the present ... | Electromagnetics_Vol2.pdf |
Continuing: The value of kz for mode mis obtained
using Equation 6.82 as follows:
kz =
√
β2 − k2x
=
√
β2 −
(mπ
a
)2
(6.99)
Since kz is
specified to be real-valued, we require:
β2 −
(mπ
a
)2
>0 (6.100)
This
constrains β; specifically:
β >mπ
a (6.101)
Recall
that β = ω√µǫand ω= 2πf where f is
frequency . Solving for f, we ... | Electromagnetics_Vol2.pdf |
which m= 0 is not available.
Let us now summarize the solution. With respect to
Figure 6.4, we find that the electric field component
of the TM field is given by:
˜E =
∞∑
m=0
[
ˆx ˜E(m)
x + ˆz ˜E(m)
z
]
(6.104)
where
˜E(m)
x ≜
{
0
, f <f(m)
c
E(m)
x0 e−jk(m)
z zcos k(m)
x x, f ≥ f(m)
c
(6.105)
and
˜E(m)
z ≜
{
0
, f <f(m)
... | Electromagnetics_Vol2.pdf |
left of the region of interest, with no additional
sources or boundary conditions to the right of the
region of interest.
The m= 0 mode, commonly referred to as the
“TM 0” mode, is of particular importance in the
analysis of microstrip transmission line, and is
addressed in Section 6.6. | Electromagnetics_Vol2.pdf |
6.6. P ARALLEL PLA TE W A VEGUIDE: THE TM 0 MODE 107
6.6 Parallel Plate W aveguide:
The TM 0 Mode
[m0220]
In Section 6.2, the parallel plate waveguide (also
sho
wn in Figure 6.5) was introduced. At the end of
that section we decomposed the problem into its
constituent TE and TM fields. In Section 6.5, we
determined the ... | Electromagnetics_Vol2.pdf |
TE case. W e also noted that the cutoff frequency for
this mode is zero, so it may exist at any frequency ,
and within any non-zero plate separation a. For this
mode k(0)
x = 0, k(0)
z = β, and we find
˜E = ˆxE(0)
x0 e−jβz (TM0 mode) (6.109)
Remarkably , we find that this mode has the form of a
uniform plane wave which p... | Electromagnetics_Vol2.pdf |
(6.111)
= ˆy E(0)
x0
η e−jβ
z (TM0 mode) (6.112)
Example 6.3. Guided waves in a printed circuit
board (PCB).
A very common form of PCB consists of a
1.575 mm-thick slab of low-loss dielectric
having relative permittivity ≈ 4.5 sandwiched
between two copper planes. Characterize the
electromagnetic field in a long strip o... | Electromagnetics_Vol2.pdf |
of TE and TM modes. The active (non-zero)
modes depend on the source (a mode must be
“stimulated” by the source in order to propagate)
and modal cutoff frequencies. The cutoff
frequency for mode mis
f(m)
c = m
2a√µǫ (6.113)
In
this case, a= 1.575 mm, µ≈ µ0, and
ǫ≈ 4.5ǫ0. Therefore:
f(m)
c ≈ (44.9 GHz) m (6.114)
Since t... | Electromagnetics_Vol2.pdf |
108 CHAPTER 6. W A VEGUIDES
the only mode that can propagate inside the PCB
is
TM0. Therefore, the field deep inside the PCB
may be interpreted as a single plane wave having
the TM 0 structure shown in Figure 6.5,
propagating away from the source end of the
PCB. The phase velocity is simply that of the
apparent plane wa... | Electromagnetics_Vol2.pdf |
Within a straight waveguide, waves can travel either
“forward” or “backward. ” The principle of
superposition allows one to consider these two
unidirectional cases separately , and then to simply
sum the results.
In this section, the equations that relate the various
components of a unidirectional wave are derived. The... | Electromagnetics_Vol2.pdf |
as follows:
˜E = ˆx ˜Ex + ˆy ˜Ey + ˆz ˜Ez (6.119)
˜H = ˆx ˜Hx + ˆy ˜Hy + ˆz ˜Hz (6.120)
Now applying the equation for curl in Cartesian
coordinates (Equation B.16 in Appendix B.2), we | Electromagnetics_Vol2.pdf |
6.7. GENERAL RELA TIONSHIPS FOR UNIDIRECTIONAL W A VES 109
find:
˜Ex = 1
jωǫ
(
∂˜Hz
∂y − ∂˜Hy
∂z
)
(6.121)
˜Ey = 1
jωǫ
(
∂˜Hx
∂z − ∂˜Hz
∂x
)
(6.122)
˜Ez = 1
jωǫ
(
∂˜Hy
∂x − ∂˜Hx
∂y
)
(6.123)
W
ithout loss of generality , we may assume that the
single direction in which the wave is traveling is in
the +ˆz direction. If t... | Electromagnetics_Vol2.pdf |
to zreduce to algebraic operations; i.e.:
∂˜Hx
∂z = −jkz˜hx( x,y)e−jkz z = −jkz ˜Hx (6.127)
∂˜Hy
∂z = −jkz˜hy( x,y)e−jkz z = −jkz ˜Hy (6.128)
∂˜Hz
∂z = −jkz˜hz( x,y)e−jkz z = −jkz ˜Hz (6.129)
W e now substitute Equations 6.124–6.126 into
Equations 6.121–6.123 and then use
Equations 6.127–6.129 to eliminate partial deri... | Electromagnetics_Vol2.pdf |
(specifically , a +ˆz-traveling) wave. With just a little
bit of algebraic manipulation of these equations, it is
possible to obtain expressions for the ˆx and ˆy
components of ˜E and ˜H which depend only on the ˆz
components of ˜E and ˜H. Here they are: 3
˜Ex = −j
k2ρ
(
+kz
∂˜Ez
∂x + ωµ∂˜Hz
∂y
)
(6.136)
˜Ey = +j
k2ρ
(
... | Electromagnetics_Vol2.pdf |
propagation in the ˆz direction in the waveguide, kρ
must be associated with variation in fields in
directions perpendicular to ˆz. In the cylindrical
coordinate system, this is the ˆρdirection, hence the
subscript “ρ. ” W e shall see later that kρ plays a special
3 Students are encouraged to derive these for themselves... | Electromagnetics_Vol2.pdf |
110 CHAPTER 6. W A VEGUIDES
role in determining the structure of fields within the
waveguide, and this provides additional motivation to
identify this quantity explicitly in the field equations.
Summarizing: If you know the wave is unidirectional,
then knowledge of the components of ˜E and ˜H in the
direction of propagat... | Electromagnetics_Vol2.pdf |
– either ˜Ez or ˜Hz must be non-zero and kρ will be
non-zero. In this case, Equations 6.136–6.139 are
both usable and useful since they allow determination
of all field components given just the z components.
In fact, we may further exploit this simplicity by
taking one additional step: Decomposition of the
unidirection... | Electromagnetics_Vol2.pdf |
and higher. The fields in a rectangular waveguide
consist of a number of propagating modes which
depends on the electrical dimensions of the
waveguide. These modes are broadly classified as
either transverse magnetic (TM) or transverse electric
(TE). In this section, we consider the TM modes.
Figure 6.6 shows the geometr... | Electromagnetics_Vol2.pdf |
6.8. RECT ANGULAR W A VEGUIDE: TM MODES 111
Equation 6.141 is a partial differential equation. This
equation, combined with boundary conditions
imposed by the perfectly-conducting plates, is
sufficient to determine a unique solution. This
solution is most easily determined in Cartesian
coordinates, as we shall now demon... | Electromagnetics_Vol2.pdf |
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