Motivation
The circular PEC waveguide is the simplest waveguide, due to its rotational symmetry. It consists of a circular domain enclosed in a PEC wall. To find analytical solutions, polar coordinates are applied in the transverse plane. The separation of variables leads to a Bessel equation for the radial dependence and to a harmonic differential equation for the angular dependence. This is exactly what one has when one derives 2D multipole and Bessel expansions.
Wave types
As in the parallel plate waveguide, one has E and H waves in the circular PEC waveguide. The transverse field of these modes can be derived from the longitudinal field, i.e., from Ez for E or TM waves and from Hz for H or TE waves.
Degenerated modes
It is obvious that the field of each mode found can be rotated around the z axis by an arbitrary angle. The resulting field will automatically fulfill all boundary conditions. The propagation constant of all of these derived modes will the same, i.e., the modes are degenerate. The angular dependence of all modes has the form harm(m*phi+phi0), where m is an integer number, phi the angle, phi0 a constant, and harm a harmonic function. All derived modes can be written as a superposition of two orthogonal modes, one with the angular dependence sin(m*phi) and one with the angular dependence cos(m*phi) for the longitudinal component of the field.
Modes
For a given angular dependence, one can find several solutions to the eigenvalue problem. These solutions are numbered according to the cutoff frequency. Hmn denotes an H wave. The first index m indicates its angular dependence and varies from 0 to infinity, the second index n affects the radial dependence and the cutoff frequency. This index starts at 1. For example, H01 is the first H0n mode, i.e., the H0n mode with the lowest cutoff frequency. Note that one has two orthogonal Hnm modes, one with the angular dependence cos(m*phi) and one with sin(m*phi). To distinguish these degenerate modes, one can use the notations Hcmn and Hsmn. Note that no Hs0n modes exist because sin(0*phi) is zero. Similar statements hold for Emn modes.
Expansions
The field of all Hmn and Emn modes is entirely defined by a single 2D Bessel expansion that may be selected in the Expansion dialog. Set only one parameter for the expansion and set the minimum order equal to the first index m.
Symmetry
When you want to compute a mode with the angular dependence sin(m*phi), you must set the symmetry numbers in the Project dialog appropriately. For example, Esmn modes have an Ez field that is anti-symmetric with respect to the XZ plane. Moreover, the field is symmetric with respect to the YZ plane when m is an even number.
Boundary
Since the MMP model has only one expansion with a single parameter, the MMP matrix has only one column and the corresponding parameter can be set equal to one, i.e., no matrix equation needs to be solved at all. The definition of the boundary is required to obtain a transcendent function for computing the eigenvalue, i.e., the propagation constant. The natural boundary is a circle. For finding the MMP solution, it is sufficient to have a single matching point on this boundary. If you set the matching point definition parameters in the MMP dialog in such a way that only one matching point is generated, this matching point is probably on the x axis. On the x axis some of the boundary conditions are automatically fulfilled because of the symmetry of the corresponding expansions. Therefore, you may obtain a trivial matrix with zero elements only. Such a matrix would evidently cause numerical problems. To avoid that, you can either use more than two matching points, which would increase the computation time, or you can set appropriate symmetry numbers in the Project dialog and define a quarter of the circle only. Note that such an arc is an open C-polygon with three corners.
Minimum search
Although the MMP matrix has a single column, you can search for the minimum of the function as for all eigenvalue problems, i.e., E2/Amp2. The search interval for the normalized propagation is the interval 0…1. The mode with the lowest index n has the highest propagation constant. Therefore, you best set the search interval 1…0 and search downward. The n-th minimum that is found will correspond to the Emn or Hmn mode respectively, provided that the search grid is sufficiently fine. When the search grid is too coarse, the search routine might overlook some modes. To avoid numerical problems, set the eigenvalue search Corner 1, for example, to 0.99999+i*0 and the Corner 2 to 0.00001+i*0. This is done in the Project dialog.
Amplitude definition
It has been mentioned that the definition of the amplitude is important for the eigenvalue search. Since the circular PEC waveguide is very easy, you can omit a sophisticated definition of the amplitude and set the Amplitude definition to 1 in the MMP dialog. This should not cause any problems.
Validation
When MMP has found a minimum, verify that the MMP errors are sufficiently small. Let MMP write the search data to a function file and let MaX-1 draw the contents of this file. Analyze the function plot. Are the minims low and sharp? Do you expect that there are additional minims that might not have been detected by the search routine? Note that such minims are often close to the borders of the search interval.
Find all modes
Increase the Search Evl. number in the Project dialog to find higher modes.
Modify the minimum order of the Bessel expansion in the Expansion dialog to find modes with another angular dependence. When you do that, keep in mind that the symmetry numbers in the Project dialog must be adapted.
Frequency dependence, cutoff frequencies
To obtain the frequency dependence of the propagation constants, you can proceed as in the previous sections. Read and adapt the directive file PARW130.DIR to find the cutoff frequencies of higher order modes and to trace them while increasing the frequency.
Evanescent modes
Search for evanescent modes by selecting a search interval along the imaginary axis of the gamma plane. What does the transverse field pattern of these modes look like?
CIRW1?0.PRO
On the CD-ROM you will find some simple project files for computing some modes of the circular PEC waveguide. Adapt these files to find more modes, and compute the frequency dependence, and cutoff frequencies.
