Fundamentals of Engineering Electromagnetics - Chapter 7 pptx

The characteristics of these modes dependupon the cross-sectional dimensions of the conducting cylinder, the type of dielectricmaterial inside the waveguide, and the frequency of operati

Dielectric waveguides guide electromagnetic waves by using the reflections thatoccur at interfaces between dissimilar dielectric materials They can be constructed foruse at microwave frequencies, but are most commonly used at optical frequencies, wherethey can offer extremely low loss propagation The most common dielectric waveguidesare optical fibers, which are discussed elsewhere in this handbook (Chapter 14: OpticalCommunications).

Resonators are either metal or dielectric enclosures that exhibit sharp resonances atfrequencies that can be controlled by choosing the size and material construction of theresonator They are electromagnetic analogs of lumped resonant circuits and are typicallyused at microwave frequencies and above Resonators can be constructed using alarge variety of shaped enclosures, but simple shapes are usually chosen so that theirresonant frequencies can be easily predicted and controlled Typical shapes are rectangularand circular cylinders

infinite number of distinct electromagnetic field configurations that can exist inside it.Each of these configurations is called a mode The characteristics of these modes dependupon the cross-sectional dimensions of the conducting cylinder, the type of dielectricmaterial inside the waveguide, and the frequency of operation.

When waveguide properties are uniform along the z axis, the phasors representingthe forward-propagating (i.e., þ z) time-harmonic modes vary with the longitudinal

is called the propagation constant of the mode and is, in general, complex valued:

ð7:1Þwhere j ¼ ffiffiffiffiffiffiffi

Waveguide modes are typically classed according to the nature of the electric and

are called the longitudinal components From Maxwell’s equations, it follows that the

related to the longitudinal components by the relations [1]

expressions for the transverse fields can be derived in other coordinate systems, butregardless of the coordinate system, the transverse fields are completely determined by thespatial derivatives of longitudinal field components across the cross section of thewaveguide.

Several types of modes are possible in waveguides

vector is always perpendicular (i.e., transverse) to the waveguide axis Thesemodes are always possible in metal waveguides with homogeneous dielectrics

field vector is perpendicular to the waveguide axis Like TE modes, they arealways possible in metal waveguides with homogeneous dielectrics

characteristics of the transverse fields are controlled more by Ezthan Hz Thesemodes usually occur in dielectric waveguides and metal waveguides withinhomogeneous dielectrics

characteristics of the transverse fields are controlled more by Hzthan Ez Like

EH modes, these modes usually occur in dielectric waveguides and in metalwaveguides with inhomogeneous dielectrics

These modes can exist only when more than one conductor with a complete dccircuit path is present in the waveguide, such as the inner and outer conductors

of a coaxial cable These modes are not considered to be waveguide modes.Both transmission lines and waveguides are capable of guiding electromagneticsignal energy over long distances, but waveguide modes behave quite differently withchanges in frequency than do transmission-line modes The most important difference isthat waveguide modes can typically transport energy only at frequencies above distinctcutoff frequencies, whereas transmission line modes can transport energy at frequencies allthe way down to dc For this reason, the term transmission line is reserved for structurescapable of supporting TEM modes, whereas the term waveguide is typically reserved forstructures that can only support waveguide modes

t ¼@2=@x2þ@2=@y2 When more than one dielectric is present, Ez and Hz

must satisfy Eqs (7.9) and (7.10) in each region for the appropriate value of k in eachregion

Modal solutions are obtained by first finding general solutions to Eqs (7.9)and (7.10) and then applying boundary conditions that are appropriate for the particular

where p is the direction perpendicular to the waveguide wall At dielectric–dielectricinterfaces, the E- and H-field components tangent to the interfaces must be continuous.Solutions exist for only certain values of h, called modal eigenvalues For metal waveguideswith homogeneous dielectrics, each mode has a single modal eigenvalue, whose value isindependent of frequency Waveguides with multiple dielectrics, on the other hand, havedifferent modal eigenvalues in each dielectric region and are functions of frequency, but

is determined by its modal eigenvalue, the frequency of operation, and the dielectricproperties From Eqs (7.1) and (7.6), it follows that

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h2k2

p

ð7:11Þwhere h is the modal eigenvalue associated with the dielectric wave number k When

a waveguide has no material or radiation (i.e., leakage) loss, the modal eigenvalues are

2

> h2,
¼ 0 and

and decay exponentially with distance Fields of this type are called evanescent fields The

at frequencies above its cutoff frequency is a propagating mode Conversely, a modeoperated below its cutoff frequency is an evanescent mode

The dominant mode of a waveguide is the one with the lowest cutoff frequency.Although higher order modes are often useful for a variety of specialized uses of wave-guides, signal distortion is usually minimized when a waveguide is operated in thefrequency range where only the dominant mode is propagating This range of frequencies

is called the dominant range of the waveguide

Metal waveguides are the most commonly used waveguides at RF and microwave cies Like coaxial transmission lines, they confine fields within a conducting shell, whichreduces cross talk with other circuits In addition, metal waveguides usually exhibit lowerlosses than coaxial transmission lines of the same size Although they can be constructedusing more than one dielectric, most metal waveguides are simply metal pipes filled with

frequen-a homogeneous dielectric—usufrequen-ally frequen-air In the remfrequen-ainder of this chfrequen-apter, the term metfrequen-al

Metal waveguides have the simplest electrical characteristics of all waveguide types,since their modal eigenvalues are functions only of the cross-sectional shape of the metalcylinder and are independent of frequency For this case, the amplitude and phaseconstants of any allowed mode can be written in the form:

The distance over which the phase of a propagating mode in a waveguide advances by 2

is called the guide wavelength lg For metal waveguides,  is given by Eq (7.13), so lgforany mode can be expressed as

lg¼2

lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ð fc=f Þ2

where l ¼ ð f ffiffiffiffiffiffi

"

p

why it is usually undesirable to operate a waveguide mode near modal cutoff frequencies

Although waveguide modes are not plane waves, the ratio of their transverse electric andmagnetic field magnitudes are constant throughout the cross sections of the metalwaveguides, just as for plane waves This ratio is called the modal wave impedance and hasthe following values for TE and TM modes [1]:

HT ¼j!"¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 fð c=fÞ2

is the intrinsic impedance of the dielectric In the limit as

cutoff and the transverse magnetic fields are dominant in TM modes near cutoff

Unlike transmission-line modes, where  is a linear function frequency,  is not

for waveguide modes For metal waveguides, it is found from Eqs (7.13), (7.18), and(7.19) that

q

ð7:21Þ

where uTEM¼1= ffiffiffiffiffiffip"is the velocity of a plane wave in the dielectric

modes appear more and more like TEM modes at high frequencies But near cutoff,

which states that energy and matter cannot travel faster than the vacuum speed of light c.But this result is not a violation of Einstein’s theory since neither information nor energy

is conveyed by the phase of a steady-state waveform Rather, the energy and informationare transported at the group velocity, which is always less than or equal to c

7.4.4 Dispersion

Unlike the modes on transmission lines, which exhibit differential propagation delays (i.e.,dispersion) only when the materials are lossy or frequency dependent, waveguide modesare always dispersive, even when the dielectric is lossless and walls are perfectlyconducting The pulse spread per meter t experienced by a modulated pulse is equal tothe difference between the arrival times of the lowest and highest frequency portions of thepulse Since the envelope delay per meter for each narrow-band components of a pulse isequal to the inverse of the group velocity at that frequency, we find that the pulsespreading t for the entire pulse is given by

encountered within the pulse bandwidth, respectively Using Eq (7.21), the pulsespreading in metal waveguides can be written as

uTEM

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 fð c=fminÞ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 fð c=fmaxÞ2q

0B

1

bandwidth From this expression, it is apparent that pulse broadening is most pronounced

The pulse spreading specified by Eq (7.23) is the result of waveguide dispersion,which is produced solely by the confinement of a wave by a guiding structure and hasnothing to do with any frequency-dependent parameters of the waveguide materials Otherdispersive effects in waveguides are material dispersion and modal dispersion Materialdispersion is the result of frequency-dependent characteristics of the materials used in thewaveguide, usually the dielectric Typically, material dispersion causes higher frequencies

to propagate more slowly than lower frequencies This is often termed normal dispersion.Waveguide dispersion, on the other hand, causes the opposite effect and is often termedanomalous dispersion

Modal dispersion is the spreading that occurs when the signal energy is carried bymore than one waveguide mode Since each mode has a distinct group velocity, the effects

of modal dispersion can be very severe However, unlike waveguide dispersion, modaldispersion can be eliminated simply by insuring that a waveguide is operated only in itsdominant frequency range

There are two mechanisms that cause losses in metal waveguides: dielectric losses andmetal losses In both cases, these losses cause the amplitudes of the propagating modes to

Typically, the attenuation constant is considered as the sum of two components:

losses alone, respectively In most cases, dielectric losses are negligible compared to metallosses, in which case

c

Often, it is useful to specify the attenuation constant of a mode in terms of its decibelloss per meter length, rather than in Nepers per meter The conversion formula betweenthe two unit conventions is

Both unit systems are useful, but it should be noted that
must be specified in Np/m when

generalizing the dielectric wave number k to include the effect of the dielectric conductivity

 For a lossy dielectric, the wave number is given by k2¼!2"ð1 þ =j!"Þ, where  is theconductivity of the dielectric, so the attenuation constant
ddue to dielectric losses alone isgiven by

where Re signifies ‘‘the real part of ’’ and h is the modal eigenvalue

The effect of metal loss is that the tangential electric fields at the conductorboundary are no longer zero This means that the modal fields exist both in the dielectricand the metal walls Exact solutions for this case are much more complicated than thelossless case Fortunately, a perturbational approach can be used when wall conductivitiesare high, as is usually the case For this case, the modal field distributions over the crosssection of the waveguide are disturbed only slightly; so a perturbational approach can beused to estimate the metal losses except at frequencies very close to the modal cutofffrequency [2]

This perturbational approach starts by noting that the power transmitted by awaveguide mode decays as

the transmitted power P is integral of the average Poynting vector over the cross section S

of the waveguide [1]:

ðS

Similarly, the power loss per meter can be estimated by noting that the wall currentsare controlled by the tangential H field at the conducting walls When conductivities arehigh, the wall currents can be treated as if they flow uniformly within a skin depth of thesurface The resulting expression can be expressed as [1]

2Rs

þCH

As long as the metal losses are small and the operation frequency is not too close tocuttoff, the modal fields for the perfectly conducting case can be used in the above integral

waveguide modes are presented later in this chapter

A rectangular waveguide is shown in Fig 7.2, consisting of a rectangular metal cylinder

of width a and height b, filled with a homogenous dielectric with permeability and

The modal eigenvalues, propagation constants, and cutoff frequencies are

a

b 2r

ð7:33Þ

as both are not zero

h2 mn

where the values of hmn mn, and fc mnare the same as for the TEmnmodes [Eqs (7.31)–(7.33)]

frequency of

cutoff frequencies of the lowest order rectangular waveguide modes (referenced to the

Figure 7.3 Field configuration for the TE10 (dominant) mode of a rectangular waveguide.(Adapted from Ref 2 with permission.)

showsTable 7.1

cutoff frequency of the dominant mode) when a/b ¼ 2.1 The modal field patterns ofseveral lower order modes are shown in Fig 7.4.

The attenuation constants that result from metal losses alone can be obtained bysubstituting the modal fields into Eqs (7.27)–(7.29) The resulting expressions are [3]

Table 7.1 Cutoff Frequencies of the

Lowest Order Rectangular Waveguide

Modes for a/b ¼ 2.1

Frequencies are Referenced to the Cutoff

Frequency of the Dominant Mode.

Figure 7.4 Field configurations for the TE11, TM11, and the TE21 modes in rectangularwaveguides (Adapted from Ref 2 with permission.)

7.6 CIRCULAR WAVEGUIDES

metal cylinder with inside radius a, filled with a homogenous dielectric The axis of thewaveguide is aligned with the z axis of a circular-cylindrical coordinate system, where and  are the radial and azimuthal coordinates, respectively If the walls are perfectly

nm

Here, the values p0

the modal eigenvalues are given by

Table 7.2shows the cutoff frequencies of the lowest order modes for circular waveguides,

The attenuation constants that result from metal losses alone can be obtained bysubstituting the modal fields into Eqs (7.27)–(7.29) The resulting expressions are [3]

p0 nm

frequencies, making them useful for transporting microwave energy over large distances

When coupling electromagnetic energy into a waveguide, it is important to ensure thatthe desired mode is excited and that reflections back to the source are minimized, and

Table 7.2 Cutoff Frequencies of

the Lowest Order Circular

Frequencies are Referenced to the Cutoff

Frequency of the Dominant Mode.

Figure 7.8 Field configurations of the TM01, TE01, and TE21 modes in a circular waveguide.(Adapted from Ref 2 with permission.)

Figure 7.9shows the metal attenuation constants for several circular waveguide modes,

that undesired higher order modes are not excited Similar concerns must be consideredwhen coupling energy from a waveguide to a transmission line or circuit element This

is achieved by using launching (or coupling) structures that allow strong coupling betweenthe desired modes on both structures

rectangular waveguide from a coaxial transmission line This structure provides goodcoupling between the TEM (transmission line) mode on the coaxial line and the

Figure 7.9 The attenuation constant of several lower order modes due to metal losses in circularwaveguides with diameter d, plotted against normalized wavelength (Adapted from Baden Fuller,A.J Microwaves, 2nd Ed.; Oxford: Pergamon Press Ltd., 1979, with permission.)

7.10

Figure

Table 7. 1 Cutoff Frequencies of the

Lowest Order Rectangular Waveguide

Modes... TEmnmodes [Eqs (7. 31)– (7. 33)]

frequency of

cutoff frequencies of the lowest order rectangular waveguide modes (referenced to the

Figure 7. 3 Field configuration for... class="page_container" data-page="12">

7. 6 CIRCULAR WAVEGUIDES

metal cylinder with inside radius a, filled with a homogenous dielectric The axis of thewaveguide is aligned with the z axis of a circular-cylindrical