UNIT-5

C

IRCULAR WAVEGUIDES & RESONATORS

CONTENTS

  1. Bessel functions – Solution of field equations in cylindrical co-ordinates
  2. TM andTE waves in circular guides
  3. Wave impedances and characteristic impedance
  4. Dominant mode in circular waveguide
  5. Excitation of modes – Microwavecavities
  6. Rectangular cavity resonators, circular cavity resonator
  7. Semicircularcavity resonator
  8. Q factor of a cavity resonator for TE101 mode.

LEARNING OBJECTIVES

  1. Identify the modes of operation in circular waveguides.
  2. Explain the basic input/output methods used in circular waveguides.
  3. Describe the basic theory of operation, construction, and applications of cavity resonators.
  4. Explain the propagation of energy in waveguides in terms of electromagnetic field theory.
  5. Calculation of Q factor in TE101 mode

THEORETICAL BACKGROUND

In 1897 Lord Rayleigh performed the first theoretical analysis of a wave in a circular waveguide, but by the time World War II came around, the men in charge of the war effort did not think to look at Lord Rayleigh's manuscripts for guidance. The legend of its rediscovery is this: The US military (Air Force, I think?) was transmitting radio waves using glass dielectric waveguides (and using it much like we now use optical fiber), but they needed something a little more bendable and fluid than glass. So, thinking about the most fluid thing around, they decided to try water as a dielectric medium. They designed an experiment, built their water waveguide, and behold, the water was a great dielectric waveguide. Then, being good scientists, they wanted to re-test their results the next morning before writing up the official report.

They tested their waveguide again and found that it was even better than the night before! Then one of the technicians found a puddle of water underneath their waveguide. This puzzled the men, so they opened up the pipe that had contained their water waveguide and found that it was empty. All of the water had leaked out in the night. After some thinking, they figured out that the water had nothing to do with the propagation of the wave. They realized that the reason the wave was confined in their dielectric (air/water) was because of the iron walls of the pipe. Thus the first practical metallic waveguide was a round air-filled pipe. It was not until some time later that it was found that Lord Rayleigh had predicted the behavior (but not necessarily the application) of metallic waveguides.

BESSEL FUNCTIONS

One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Bessel functions. They are solutions to a very important differential equation, the Bessel equation:

The solutions to this equation are in the form of infinite series which are called Bessel funtions of the first kind. The expression for the sum is

Values for the Bessel functions can be found in most collections of mathematical tables. Bessel functions are encountered in physical situations where there is cylindrical symmetry. This occurs in problems involving electric fields, vibrations, heat conduction, optical diffraction and others.

Here are some of the basic properties of Bessel functions:

where Z is any Bessel function. Figures 5.1 and 5.2 show Bessel functions of the first and second kinds of orders 0, 1, 2, 3.

Figure 5.1 : Bessel functions of the first kind.

Figure 5.2 : Bessel functions of the second kind.

CIRCULAR WAVEGUIDE

Figure 2.5: A circular waveguide of radius a.

For a circular waveguide of radius a (Fig. 2.5), we can perform the same sequence of steps in cylindricalcoordinates as we did in rectangular coordinates to find the transverse field components in terms of thelongitudinal (i.e. Ez, Hz) components. In cylindrical coordinates, the transverse field is

Using this in Maxwell’s equations (where the curl is applied in cylindrical coordinates) leads to

TE MODES

We don’t need to prove that the wave travels as e§j¯z again since the differentiation in z for the Laplacianis the same in cylindrical coordinates as it is in rectangular coordinates (@2=@z2). However, the ½ and Áderivatives of the Laplacian are different than the x and y derivatives. The wave equation for Hz is

Using the separation of variables approach, we let Hz(½; Á; z) = and obtain

Multiplying by a common factor leads to

Because the terms in this equation sum to a constant, yet each depends only on a single coordinate, eachterm must be constant:

This is known as Bessel’s Differential Equation.

Now, we could use the Method of Frobenius to solve this equation, but we would just be repeating a wellknownsolution. The series you obtain from such a solution has very special properties (a lot like sine andcosine: you may recall that sin(x) and cos(x) are really just shorthand for power series that have specialproperties).

The solution is

where Jº(x) is the Bessel function of the first kind of order º and Nº(x) is the Bessel function of the secondkind of order º.

1. First, let’s examine kÁ.

where ` is an integer

2. ItturnsoutthatNν(kcρ) →−∞asρ→0. Clearly,ρ=0isinthedomainofthewaveguide.Physically,however,wecan’t haveinfinitefieldintensityatthis point.ThisleadsustoconcludethatD0 =0.Wenowhave

Hz(ρ,φ,z)=[Asin(νφ)+Bcos(νφ)]Jν(kcρ)e−jβz

3. TherelativevaluesofAandB havetodowiththeabsolutecoordinateframeweusetodefine the waveguide.Forexample,letA=Fcos(νφ0)andB=−Fsin(νφ0)(youcanfindavalueofFandφ0 tomakethiswork).Then

Asin(νφ)+Bcos(νφ)=Fsin[ν(φ−φ0)]

4. Thevalueofφ0 thatmakesthisworkcanbethoughtofasthe coordinatereferenceformeasuringφ.So,wereallyareleftwithfindingF,whichissimplythemodeamplitudeandisthereforedeterminedbytheexcitation.

Cutofffrequency(β=0):Sincek=kc =2πfc,νn/catthemodecutofffrequency,

The expressionsforwavelengthand phasevelocityderived fortherectangularwaveguideapplyhereaswell.However,youmustusetheproper valueforthecutofffrequencyintheseexpressions.

TM MODES

ThederivationisthesameexceptthatwearesolvingforEz.Wecanthereforewrite

Ez(ρ,φ,z)=[Asin(νφ)+Bcos(νφ)]Jν(kcρ)e−jβz

OurboundaryconditioninthiscaseisEz(a,φ,z)=0orJν(kca)=0.Thisleadsto

where pºn is the nth zero of Jº(x).

In this case, we have

It becomes clear the the TE11 mode is the dominant overall mode of the waveguide.

DOMINANT MODE IN CIRCULAR WAVEGUIDE

In the transverse magnetic (TM) mode, the entire magnetic field is in the transverse plane and has no portion parallel to the length axis.

Since there are several TE and TM modes, subscripts are used to complete the description of the field pattern. In rectangular waveguides, the first subscript indicates the number of half-wave patterns in the "a" dimension, and the second subscript indicates the number of half-wave patterns in the "b" dimension.

The dominant mode for rectangular waveguides is shown in figure 1-36. It is designated as the TE mode because the E fields are perpendicular to the "a" walls. The first subscript is 1 since there is only one half-wave pattern across the "a" dimension. There are no E-field patterns across the "b" dimension, so the second subscript is 0. The complete mode description of the dominant mode in rectangular waveguides is TE1,0. Subsequent descriptions of waveguide operation in this text will assume the dominant (TE1,0) mode unless otherwise noted.

A similar system is used to identify the modes of circular waveguides. The general classification of TE and TM is true for both circular and rectangular waveguides. In circular waveguides the subscripts have a different meaning. The first subscript indicates the number of full-wave patterns around the circumference of the waveguide. The second subscript indicates the number of half-wave patterns across the diameter.

In the circular waveguide in figure 1-37, the E field is perpendicular to the length of the waveguide with no E lines parallel to the direction of propagation. Thus, it must be classified as operating in the TE mode. If you follow the E line pattern in a counterclockwise direction starting at the top, the E lines go from zero, through maximum positive (tail of arrows), back to zero, through maximum negative (head of arrows), and then back to zero again. This is one full wave, so the first subscript is 1. Along the diameter, the E lines go from zero through maximum and back to zero, making a half-wave variation. The second subscript, therefore, is also 1. TE1,1is the complete mode description of the dominant mode in circular waveguides. Several modes are possible in both circular and rectangular waveguides. Figure 1-38 illustrates several different modes that can be used to verify the mode numbering system.

MICROWAVE CAVITY

A microwave cavity is a closed metal structure that confines electromagnetic fields in the microwave region of the spectrum. Such cavities act as resonant circuits with extremely low loss at their frequency of operation. Their Q factor may reach several hundred thousand compared to a few hundred for resonant circuits made with inductors and capacitors at the same frequency.

Resonant cavities are usually made from closed (or short-circuited) sections of waveguide or high-permittivitydielectric material (see dielectric resonator). Electric and magnetic energy is stored in the cavity and only losses are due to finite conductivity of cavity walls and dielectric losses of material filling the cavity. Every cavity has numerous resonant frequencies that correspond to electromagnetic field modes satisfying necessary boundary conditions on the walls of the cavity. Because of these boundary conditions that must be satisfied at resonance (tangential electric fields must be zero at cavity walls), it follows that cavity length must be an integer multiple of half-wavelength at resonance [1]. Hence, a resonant cavity can be thought of as a waveguide equivalent of short circuited half-wavelength transmission line resonator[2]. Q factor of a resonant cavity can be calculated using cavity perturbation theory and expressions for stored electric and magnetic energy.

The electromagnetic fields in the cavity are excited via external coupling. External power source is usually coupled to the cavity by a small aperture, a small wire probe or a loop [3]. External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis[4].

Rectangular Microwave Cavity

Resonant frequency of a rectangular microwave cavity for any TEmnl or TMmnl resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. This frequency is given by

where kmnl is the wavenumber, c is the speed of light in vacuum, and μr and εr are relative permeability and permittivity respectively.

Q factor of rectangular microwave cavity can be decomposed in two parts. First part is Qc, the Q factor of the cavity with lossy walls filled with lossless dielectric, and the second part is Qd, the Q factor of the cavity with perfectly conducting wall filled with lossy dielectric. Total Q factor of the cavity can be found as [6]:

Qc and Qd are calculated as follows:

where k is the wavenumber, η is the intrinsic impedance of the dielectric, Rs is the surface resistivity of the cavity walls, μr and εr are relative permeability and permittivity respectively and tanδ is the loss tangentof the dielectric.

Case 1 TMmnp mode:Hz=0, neither m nor n =0, p can be 0.

,

where

Case 2TEmnp mode:Ez=0, p≠0. Either m or n =0, but not both.

Both have the same resonant frequency (degenerate modes):

Note:TE101 mode is the dominant mode of the rectangular resonator in case of abd.

CIRCULAR CAVITY RESONATORS

For an air-filled circular cylindrical cavity resonator of radius a and length d. The resonant frequencies are

, where Jm(Xmn)=0

, where J’m(X’mn)=0

In case of 2d>2ad, the dominant mode of the circular cylindrical cavity isTM010mode:

,

But the tangential component of the electric fields and normal components of the magnetic

fields suffer 180 o phase shift and the normal components of the electric fields and tangential

components of the magnetic fields suffer no phase shifts on reflection at the surface of the

perfect conductor resulting in

And hence the fields in the standing wave become

When the both ends are closed, the distance between the ends must correspond to an integer

no. of half(guide) wave lengths.

Resonant frequency:

leading to the resonating frequency of

For TM mode n0,1,2,... ,m1,2,3..... , q0,1,2,....

  • For TE mode n0,1,2,... ,m1,2,3..... , q1,2,3....
  • n … indicates the periodicity in the direction
  • m ….indicates the number of the zeros of the field in the radial direction.
  • q …the number of half-waves in the axial direction

1 | Page Prepared by V.NAVANEETHAKRISHNAN AP/ECE