Advanced engineering electromagnetics - Hanyang

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Advanced engineering electromagnetics

Chapter 8.5~8.7.3 : Partially filled waveguide –Transverse Electric modes

Sangeun Jang

Applied Electromagnetic Technology Laboratory

Department of Electronics and Computer EngineeringHanyang University, Seoul, Korea

jangse1006@naver.com

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Outline

❑ Partially filled waveguide

❑ Transverse resonance method

❑ Dielectric waveguide

1. Dielectric slab waveguide

2. Transverse magnetic modes

3. Transverse electric modes

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Partially filled waveguide(LSEy or TEy)

▪ For each region, the TEy field components are those of (8-95) and the corresponding potential functions

are

▪ For the configuration of Fig.8-15a, there are two sets of fields: one for the dielectric region (0≤x≤a, 0≤y

≤h, z), designated by superscript d, and the other for the free-space region (0≤x≤a, h≤y≤b, z),

designated by superscript 0.

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Partially filled waveguide(LSEy or TEy)

for the dielectric region, and

for the free-space region. In both sets of fields, βz is the same, since for propagation along the interface both

sets of fields must be common.

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Partially filled waveguide(LSEy or TEy)

▪ For this waveguide configuration, the appropriate independent boundary conditions are

▪ Another set of dependent boundary conditions is

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Partially filled waveguide(LSEy or TEy)

▪ By using (8-95) and (8-103), we can write that

▪ Application of boundary condition (8-104d) leads to

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Partially filled waveguide(LSEy or TEy)

▪ Application of (8-104e) leads to

▪ Thus, (8-103) reduces to

with

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Partially filled waveguide(LSEy or TEy)

▪ Use of (8-95) and (8-102) gives

▪ Application of boundary condition (8-104a) leads to

▪ Application of (8-104b) leads to

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Partially filled waveguide(LSEy or TEy)

▪ Thus, (8-102) reduces to

with

▪ Application of boundary condition (8-104c) and use of (8-108) and (8-111) leads to

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Partially filled waveguide(LSEy or TEy)

▪ By using (8-107) and (8-110), the z component of the H field from (8-95) can be written as

▪ Application of the boundary condition of (8-104f) reduces, with βxd = βx0, to

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Partially filled waveguide(LSEy or TEy)

▪ Division of (8-114) by (8-112) leads to

▪ For m =0, the modes will be denoted as TE0n. For these modes, (8-115a) and (8-115b) reduce to

▪ Cutoff occurs when βz =0. Thus, at cutoff (8-116a) and (8-116b) reduce to

which can be used to find βy0 and βyd at cutoff (actually slightly above), once the cutoff frequency has been

determined.

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Partially filled waveguide(LSEy or TEy)

▪ By using (8-117a) and (8-117b), we can write (8-115) as

or

which can be used to find the cutoff frequencies of the TEy0n modes in a partially filled waveguide.

▪ For a rectangular waveguide filled completely either with free space or with a dielectric material with εd,

μd, the cutoff frequency of the TEy01 mode is given, respectively, according to (8-100b), by

▪ Thus, the cutoff frequency of the TEy01 mode of a partially filled waveguide is greater than (8-119b) and

smaller than (8-119a), that is,

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Partially filled waveguide(LSMy or TMy)

▪ The TMy field components are those of (8-101), where the corresponding vector potentials in the

dielectric and free space regions for the waves traveling in the +z direction are given, respectively, by

▪ Application of the boundary conditions (8-104a) through (8-105f) shows that the following relations follow:

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Partially filled waveguide(LSMy or TMy)

▪ Application of the boundary conditions (8-104a) through (8-105f) shows that the following relations follow:

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Partially filled waveguide(LSMy or TMy)

▪ For m =1, the modes will be denoted as TMy1n. For these modes, (8-127a) and (8-127b) reduce to

▪ Cutoff occurs when βz =0. Thus, at cutoff, (8-128a) and (8-128b) reduce to

which can be used to find βy0 and βyd slightly above cutoff, once the cutoff frequency has been determined.

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Partially filled waveguide(LSMy or TMy)

▪ By using (8-129a) and (8-129b), we can write (8-127) as

or

which can be used to find the cutoff frequencies of the TMy1n modes in a partially filled waveguide.

▪ For a rectangular waveguide filled completely either with free space (μ0, ε0) or with a dielectric material (εd,

μd), the cutoff frequency of the hybrid TMy10 mode is given, respectively, according to (8-101d) by

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Partially filled waveguide(LSMy or TMy)

▪ The cutoff frequency of the TMy10 mode of a partially filled waveguide (part free space and part dielectric)

is greater than (8-131b) and smaller than (8-131a), that is

or

▪ When the dielectric properties of the dielectric material inserted into the waveguide are such that εd ≅ ε0

and μd ≅ μ0, the wave constants of the TMmn modes of the partially filled waveguide are approximately

equal to the corresponding wave constants of the TMymn modes of the totally filled waveguide.

▪ Therefore, (8-127) can be approximated by

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Partially filled waveguide(LSMy or TMy)

▪ At cutoff, βz =0. Therefore, using (8-129a) and (8-129b), we can write (8-134) as

which reduces to

where

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Transverse resonance method

▪ The transverse resonance method is a technique that can be used to find the propagation constant of

many practical composite waveguide structures, as well as many traveling wave antenna systems.

▪ By using this method, the cross section of the waveguide or traveling wave antenna structure is

represented as a transmission line system.

▪ The fields of such a structure must satisfy the transverse wave equation, and the resonances of this

transverse network will yield expressions for the propagation constants of the waveguide or antenna

structure.

▪ Whereas the formulations of this method are much simpler when applied to finding the propagation

constants, they do not contain the details for finding other parameters of interest.

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Transverse resonance method

▪ Although the method will not yield all the details of the analysis of Sections 8.5.1 and 8.5.2, it will lead to

the same characteristic equations 8-115 and 8-127.

▪ The objective here is to analyze the waveguide geometry of Figure 8-15a using the transverse resonance

method.

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Transverse resonance method

▪ One dielectric-filled (0 ≤ y ≤ h) with characteristic impedance 𝑍𝑐𝑑 and wave number 𝛽𝑡𝑑.

▪ The problem will be modeled as a two-dimensional structure represented by two transmission lines.

▪ The other air-filled (h ≤ y ≤ b) with characteristic impedance 𝑍𝑐0 and wave number 𝛽𝑡0, as shown in

Figure 8-18.

▪ Each line is considered shorted at its load, that is, 𝑍𝐿 = 0 at y = 0 and y = b.

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Transverse resonance method

▪ It was shown in Section 3.4.1A that the solution to the scalar wave equation for any of the electric field

components, for example, that for 𝐸𝑥 of (3-22) as given by (3-23), takes the general form of

where

▪ The scalar function ψ represents any of the electric or magnetic field components. For waves traveling in

the z direction, the variations of h(z) are represented by exponentials of the form 𝑒±𝑗𝛽𝑧𝑧.

where

where

transverse wave equation

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Transverse resonance method

▪ Each of the electric and magnetic field components in the dielectric- and air-filled sections of the two-

dimensional structure of Figure 8-18 must satisfy the transverse wave equation 8-137 with corresponding

transverse wave numbers of 𝛽𝑡𝑑 and 𝛽𝑡0, where

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Transverse resonance method

▪ The input impedance of a line is defined as the ratio of the electric/magnetic field components (or

voltage/current), then at any point along the transverse direction of the waveguide structure, the input

impedance of the transmission line network looking in the positive y direction is equal in magnitude but

opposite in phase to that looking in the negative y direction.

▪ This follows from the boundary conditions that require continuous tangential components of the electric

(E) and magnetic (H) fields at any point on a plane orthogonal to the transverse structure of the

waveguide.

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Transverse resonance method

▪ For the transmission line model of Figure 8-18b, the input impedance at the interface looking in the +y

direction of the air-filled portion toward the shorted load is given, according to the impedance transfer

equation 5-66d, as

▪ In a similar manner, the input impedance at the interface looking in the −y direction of the dielectric-filled

portion toward the shorted load is given by

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Transverse resonance method

▪ Since these two impedances must be equal in magnitude but of opposite signs, then

▪ The preceding equation is applicable for both TE and TM modes. It will be applied in the next two

sections to examine the TEy and TMy modes of the partially filled waveguide of Figure 8-18a.

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TEy or LSEy or Hy

▪ The characteristic equation 8-141 will now be applied to examine the TEy modes of the partially filled

waveguide of Figure 8-18a.

▪ It was shown in Section 8.2.1 that the wave impedance of the TE𝑚𝑛𝑧 modes is given by (8-19)

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TEy or LSEy or Hy

▪ Allow the characteristic impedances for the TEy modes of the dielectric- and air-filled sections of the

waveguide, represented by the two-section transmission line of Figure 8-18b, to be of the same form as

(8-142), or

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TEy or LSEy or Hy

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TMy or LSMy or Ey

▪ According to (8-29a), the wave impedance of TM𝑚𝑛𝑧 modes is given by

▪ Allow the characteristic impedances for the TMy modes of the dielectric- and air-filled sections of the

waveguide to be of the same form as (8-145), or

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TMy or LSMy or Ey

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Dielectric waveguide

▪ Transmission lines are used to contain the energy associated with a wave within a given space and guide

it in a given direction.

▪ Typically, many people associate these types of transmission lines with either coaxial and twin lead lines

or metal pipes (usually referred to as waveguides) with part or all of their structure being metal.

▪ However, dielectric slabs and rods, with or without any associated metal, can also be used to guide

waves and serve as transmission lines.

▪ Usually these are referred to as dielectric waveguides, and the field modes that they can support are

known as surface wave modes.

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Dielectric slab waveguide

▪ We also assume that the waves are traveling in the ±z directions, and the structure is infinite in that

direction, as illustrated in Figure 8-19b.

▪ To simplify the analysis of the structure, we reduce the problem to a two-dimensional one (its width in the

x direction is infinite) so that𝜕

𝜕𝑥= 0.

▪ The objective in a dielectric slab waveguide, or any type of waveguide, is to contain the energy within the

structure and direct it toward a given direction.

▪ For the dielectric slab waveguide this is accomplished by having the wave bounce back and forth

between its upper and lower interfaces at an incidence angle greater than the critical angle.

▪ When this is accomplished, the refracted fields outside the dielectric form evanescent (decaying) waves

and all the real energy is reflected and contained within the structure.

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Dielectric slab waveguide

▪ The characteristics of this line can be analyzed by treating the structure as a boundary-value problem

whose modal solution is obtained by solving the wave equation and enforcing the boundary conditions.

▪ The other approach is to examine the characteristics of the line using ray-tracing (geometrical optics)

techniques.

▪ This approach is simpler and sheds more physical insight onto the propagation characteristics of the line

but does not provide the details of the more cumbersome modal solution.

▪ Both methods will be examined here. We will begin with the modal solution approach.

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TMz modes

▪ The TMz mode fields that can exist within and outside the dielectric slab of Figure 8-19 must satisfy (8-

24), where 𝐴𝑧 is the potential function representing the fields either within or outside the dielectric slab.

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TMz modes

▪ For the fields within the dielectric slab, the potential function Az takes the following form:

where

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TMz modes

▪ For the slab to function as a waveguide, the fields outside the dielectric slab must be of evanescent form.

Therefore, the potential function 𝐴𝑧 takes the following form:

where

where

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TMz modes

▪ Since the fields within and outside the slab have been separated into even and odd modes, we can

examine them separately and then apply superposition.

▪ For each mode (even or odd), a number of dependent and independent boundary conditions must be

satisfied.

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TMz modes

A. TMz (Even)

▪ By using (8-24) along with the appropriate potential function of (8-148) through (8-150c), we can write the

field components as follows.

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TMz modes

▪ Applying the boundary condition (8-151a) and using (8-148c) and (8-149c) yields

▪ In a similar manner, enforcing (8-151b) and using (8-148c) and (8-149c) yields

▪ Comparison of (8-155a) and (8-155b) makes it apparent that

▪ Thus, (8-155a) and (8-155b) are the same and both can be represented by

▪ Follow a similar procedure by applying (8-151c) and (8-151d) and using (8-155c). Then we arrive at

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TMz modes

▪ Division of (8-157) by (8-156) allows us to write that

where according to (8-148c) and (8-149c)

▪ From the free space looking down the slab we can define an impedance, which, by using (8-153a)

through (8-153f) and (8-158b), can be written as

which is inductive, and it indicates that TM mode surface waves are supported by inductive surfaces.

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TMz modes

B. TMz (Odd)

▪ By following a procedure similar to that used for the TMz (even), utilizing the odd mode TMz potential

functions (8-148) through (8-150c), it can be shown that the expression corresponding to (8-158) is

where (8-158a) and (8-158b) also apply for the TMz odd modes.

C. Summary of TMz (Even) and TMz (Odd) modes

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TMz modes

▪ It is apparent from (8-160c) and (8-160d) that if 𝛽𝑧 is real, then

▪ For the dielectric slab to perform as a lossless transmission line, 𝛽𝑦𝑑 , 𝛼𝑦0 , and 𝛽𝑧 must all be real.

▪ The lowest frequency for which unattenuated propagation occurs is called the cutoff frequency.

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TMz modes

▪ For the dielectric slab this occurs when 𝛽𝑧 = 𝛽0. Thus, at cutoff, 𝛽𝑧 = 𝛽0, and (8-158a) and (8-158b)

reduce to

▪ Through the use of (8-165a) and (8-165b), the nonlinear transcendental equations 8-160a and 8-160b

are satisfied, respectively, when the following equations hold.

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TMz modes

D. Graphical solution for TM𝑚𝑧 (Even) and TM𝑚

𝑧 (Odd) modes

▪ Equations 8-160a through 8-160d can be solved graphically for the characteristics of the TMz even and

odd modes.

▪ This is accomplished by referring to Figure 8-21 where the abscissa represents 𝛽𝑧ℎ and the ordinate,

𝛼𝑦0ℎ.

▪ The procedure can best be illustrated by considering a specific value of 𝜀0/𝜀𝑑.

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TMz modes

▪ Let us assume that𝜀0

𝜀𝑑= 1/2.56.

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TMz modes

▪ The next step is to solve graphically (8-160c) and (8-160d). By combining (8-160c) and (8-160d), we can

write that

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TMz modes

▪ By multiplying both sides by ℎ2, we can write (8-167) as

where

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TEz modes

▪ By following a procedure similar to that for the TMz modes,

▪ Therefore, TE surface waves are capacitive and are supported by capacitive surfaces, whether they are

dielectric slabs, dielectric covered ground planes, or corrugated surfaces.