As discussed, it is imperative that the eigenvalue extracted by ANSYS be correct since it is used by ANSYS to expand the fields. This in itself provides good reason for confirming the accuracy of the extracted eigenvalue. There is another reason as well. The eigenvalue is actually the resonant frequency of the cavity, which, as we have demonstrated, is related to the cutoff wavelength and hence the cutoff frequency of the mn^th mode. Therefore the extracted eigenvalue is of significant practical importance. That is, the main purpose of using ridged waveguide is to achieve a broad operating bandwidth. We showed that this is obtained by lowering the cutoff frequency of the TE₁₀ mode through the use of ridges. Determining whether or not a ridged waveguide will meet the bandwidth needs of a specific application requires knowledge of the cutoff frequency. Hence the cutoff frequency of ridged waveguide is an extremely important parameter in its own right. It follows then that the corresponding eigenvalue is also of great importance.

We are fortunate that the behavior of the ridged cavity is completely characterized by two parameters. These two parameters are the resonant frequency and field distribution within the cavity, where the latter is dependent upon the former. We may use either of these parameters to evaluate the accuracy of the FEA model. The resonant frequency has the advantage that it is a single number. Moreover the resonant frequency is related to the cutoff frequency which has already been computed for the ridged guide of interest, in equation (40).

In contrast the field distribution is not easily specified (i.e. requires several data points), nor is it published in quantitative form. For these reasons the TE₁₀ cutoff frequency will be used to evaluate the accuracy of the FEA model.

Each cavity possesses an unlimited number of resonant frequencies, or, eigenvalues. Moreover, each resonant frequency has associated with it a cutoff frequency. Therefore, if we are to use the cutoff frequency to assess the accuracy of the FEA model, we must be sure to use the correct cutoff frequency. Fortunately we are interested in the cutoff frequency of the TE₁₀ mode which is related to the resonant frequency of the TE₁₀₁ cavity mode. It is more fortunate still that the TE₁₀₁ cavity mode is the primary cavity mode. Recall that we have defined the primary cavity mode as that mode with the lowest resonant frequency for a cavity with dimensions of b < a < L. This makes it very easy to find the resonant frequency of the TE₁₀₁ mode (i.e. f_r101). We find f_r101 by choosing the start frequency of the modal analysis to be extremely low. We may request any number of modes to be extracted, but it is the results of the first load step that will be of interest to us (i.e. the first eigenvalue extracted). Actually, we need only request one mode extraction and this will yield f_r101.

So far we have settled on an assessment parameter, and we have described how the assessment parameter is found. All that remains now is to use the assessment parameter to determine the accuracy of the FEA model. A convergence study is not absolutely necessary to do this. After all, a target value has been presented for the cutoff frequency in equation (40). Convergence will however be demonstrated. In fact, it follows naturally from trying to push the limits of the model through mesh refinement. We shall begin with a baseline model and compare the results to the target values. Additional models will be generated, and analyzed, until the predicted results are within 0.5% of the target values, or until the limitations of the software have been reached, whichever comes first.

In light of the previous statement, it is necessary to define the "limitations of the software". As discussed, the University version of ANSYS 5.4 allows a maximum of 8,000 nodes. One might argue that it is possible to successively tighten the mesh and still remain below 8,000 nodes provided that the length of the cavity is successively decreased. Indeed this statement is true. However, the resulting model would be impractical for a ridged cavity of arbitrary cross section. The reason is that the TE₁₀₁ cavity mode is only the primary cavity mode for a cavity with dimensions of b < a < L. Therefore, once the cavity width exceeds the cavity length we have no easy way of knowing which extracted eigenvalue corresponds to f_r101. Surely we could view the field distribution associated with each eigenvalue until the TE₁₀₁ mode is found, but this would be very time consuming.

The ridged cavity of interest is not of arbitrary cross section. Rather, the dimensions are well known and so too is the cutoff wavelength of the TE₁₀ mode. It is given by equation (39). Hence it is possible to compute f_r101 for a cavity of arbitrary length L using equation (31). Therefore, we could reduce the cavity length to a value lower than that of the cavity width and still identify f_r101 amongst the many other eigenvalues extracted by ANSYS. However, if ANSYS is used to analyze other ridged cavities, the user will not know the value of f_r101 a priori. In keeping with that spirit we shall define the limitation of the software as follows: The software is limited by the tightest possible mesh that can be achieved for a cavity with a length to width ratio approaching the value 1 from the right.

Five ridged cavity models were constructed in ANSYS. Before providing a complete description of each model, let us consider the similarities and differences between the models. All five of the models bear the same cross sectional dimensions, element type, element order, and material properties. The models differ only in length and/or in the number of elements and nodes. Table 3 below highlights the differences between the models.

An isometric view of a typical model is shown below in Figure 19. A front view of the model is shown in Figure 20. An electric wall boundary condition is applied to all surfaces of the model with the exception of the surface at x=0.00635 m. The default boundary condition at x=0.00635 m is that of a magnetic wall as desired.

Model 1 was constructed with the following dimensions, elements, and material properties:

Dimensions: a=0.006350 m s=0.001397 m L=0.073 m

Element Type: HF120

Element Order: 1^st order element

Element Edge Size: 0.002 m

Number of Elements: 370

Number of Nodes 2,376

Material Properties: m =1; e =1 (i.e. air)

Model Size: ¼ (Upper Left Quadrant)

The cutoff frequency of a ridged waveguide is a parameter, which in itself is dependent only upon the cross sectional dimensions of the guide. Therefore, our model must be capable of yielding the same cutoff frequency regardless of the cavity length, provided that the element edge size is unaltered. This behavior is not to be confused with that of the resonant frequency, which must change with changes in cavity length. To confirm this behavior, Model 2 was created with the same element edge size as Model 1. However, the cavity length of Model 2 is ½ that of Model 1.

Model 2 was constructed with the following dimensions, elements, and material properties:

Dimensions: a=0.006350 m s=0.001397 m L=0.0365 m

Element Type: HF120

Element Order: 1^st order element

Element Edge Size: 0.002 m

Number of Elements: 190

Number of Nodes 1,242

Material Properties: m =1; e =1 (i.e. air)

Model Size: ¼ (Upper Left Quadrant)

Clearly Model 2 is far from approaching the limit of 8,000 nodes. As such it is possible to refine Model 2 by reducing the element edge size. To that end, Model 3 was created with the same cavity length as Model 2 and an element edge size of 0.001 meters.

Model 3 was constructed with the following dimensions, elements, and material properties:

Dimensions: a=0.006350 m s=0.001397 m L=0.0365 m

Element Type: HF120

Element Order: 1^st order element

Element Edge Size: 0.001 m

Number of Elements: 1,480

Number of Nodes 7,697

Material Properties: m =1; e =1 (i.e. air)

Model Size: ¼ (Upper Left Quadrant)

Further mesh refinement of Model 3 is not possible since Model 3 has approached the maximum allowable number of nodes. Hence further refinement will require a reduction in cavity length. In keeping with the approach used thus far, we shall validate the new "reduced length" model before refining the mesh. To that end Model 4 was created with the same element edge size as Model 3. However, the cavity length of Model 4 is ½ that of Model 3.

Model 4 was constructed with the following dimensions, elements, and material properties:

Dimensions: a=0.006350 m s=0.001397 m L=0.01825 m

Element Type: HF120

Element Order: 1^st order element

Element Edge Size: 0.001 m

Number of Elements: 760

Number of Nodes 4,025

Material Properties: m =1; e =1 (i.e. air)

Model Size: ¼ (Upper Left Quadrant)

Clearly Model 4 has not reached the limit of 8,000 nodes. As such it is possible to refine Model 4 by reducing the element edge size. To that end, Model 5 was created with the same cavity length as Model 4 and an element edge size of 0.00073 meters.

Model 5 was constructed with the following dimensions, elements, and material properties:

Dimensions: a=0.006350 m s=0.001397 m L=0.01825 m

Element Type: HF120

Element Order: 1^st order element

Element Edge Size: 0.00073 m

Number of Elements: 1,500

Number of Nodes 7,540

Material Properties: m =1; e =1 (i.e. air)

Model Size: ¼ (Upper Left Quadrant)

Model 5 has approached the limit of 8,000 nodes. Hence further mesh refinement would require an additional reduction in cavity length. However, the cavity length of Model 5 is already approaching the cavity width. To be exact, the length to width ratio of Model 5 is 1.437. Of course we must honor our previous definition regarding the "limitations of the software". Therefore, we can only reduce the length of the cavity by as little as 0.00555 meters. We will soon see that this reduction in cavity length will most likely yield only a marginal improvement in the accuracy of the model. To that end, Model 5 is the last of the ridged cavity models to be considered.

Extractor: Block Lanczos

Excitation: None

Frequency Range: 0.1 GHz to 12.0 GHz

Number of Modes Extracted: 1

Number of Modes Expanded: 1

A modal analysis was performed on each model from 0.1 GHz to 12.0 GHz. Only one mode extraction was requested for each model. The start frequency of the modal analysis is extremely low. Moreover, the length to width ratio is greater than 1 for all Models 1-5. Therefore, the eigenvalue extracted by ANSYS will be the resonant frequency of the TE₁₀₁ cavity mode for all Models 1-5 as expected.

The results of Model 1 are summarized below in Table 4:

The results of Model 1 are quite good! Especially since this is our baseline model. Both Models 1 & 2 share the same element edge size. Thus we should expect the percent error of Model 2 to be nearly identical to that of Model 1.

The results of Model 2 are summarized below in Table 5:

Indeed the percent error of Model 2 is nearly identical to that of Model 1 as expected. This validates the performance of the ½ length model. Both Models 2 & 3 share the same cavity length. However, Model 3 boasts a reduced element edge size. Hence we should expect the percent error of Model 3 to be much improved over that of Model 2.

The results of Model 3 are summarized below in Table 6:

As expected the percent error of Model 3 is much improved over that of Model 2. This shows that there is indeed room for improvement. Both Models 3 & 4 share the same element edge size. Thus we should expect the percent error of Model 4 to be nearly identical to that of Model 3.

The results of Model 4 are summarized below in Table 7:

Indeed the percent error of Model 4 is nearly identical to that of Model 3 as expected. This validates the performance of the ¼ length model. Both Models 4 & 5 share the same cavity length. However, Model 5 boasts a reduced element edge size. Hence we should expect the percent error of Model 5 to be much improved over that of Model 4.

The results of Model 5 are summarized below in Table 8:

The percent error of Model 5 is improved over that of Model 4 as expected. However, the improvement is marginal. The results of Models 1-5 are summarized below in Table 9.

Table 9 shows a delta of 0.137% between Models 4 & 5. This improvement is dwarfed in comparison to the delta of 1.463% evident between Models 2 & 3. Thus the FEM has converged quite rapidly. While there is always room for improvement, it is clear that we are reaching the limits of the software as defined earlier. That is, a reduction in the cavity length of Model 5 by 0.00555 meters will not yield an appreciable improvement in the accuracy of the model. Moreover, the percent error of Model 5 is fairly close to our original goal of 0.5%. Thus Model 5 is the model that we shall use to examine the field distribution.

We will examine the field distribution throughout the entire cavity of Model 5. However, we are particularly concerned with the electric field distribution in the z=L/2 plane (i.e. at z=0.009125 m). For it is the field distribution at this location that corresponds to the electric field distribution within a rectangular waveguide of identical cross section. Contour plots have been obtained for each component of the electric and magnetic fields. The results are presented below in Figures 21-26. Each of these figures has four parts. Each part simply shows a different view of the same field component. For example, Figure 21 shows four different views of the y-directed electric field component (E_y). Each view of Figure 21 is designated as follows:

Parts c & d of each figure were obtained by first moving the working plane to z=L/2. The plot style was then changed to "capped-hidden" with the cutting plane set equal to the working plane.

Figures 21-29 are surrounded by brief descriptions of the field behavior within the cavity. These descriptions compare the results of the ridged cavity to those expected for a rectangular cavity. Where appropriate, the field distribution predicted by ANSYS will be justified for the ridged cavity on the grounds of physical intuition.

Comparison of Equation (35) with Figure 21 shows that the y directed electric field distribution in a ridged cavity is similar, but not identical, to that in a rectangular cavity. Recall that in a rectangular cavity, E_y is sinusoidal in x and z, but constant in y. To highlight the similarities between ridged and rectangular cavities we note that E_y for the ridged cavity is somewhat sinusoidal in x and z. However, the sinusoidal behavior in x is more prominent in the region between the ridges. As for the differences, perhaps the most obvious is that for the ridged cavity, E_y is not constant in y, with the exception of the region between the ridges.

The x directed electric field component shown in Figure 22 is somewhat sinusoidal in z with an irregular distribution in x and y. Unfortunately we can not compare the behavior of this field component (i.e. E_x) in the ridged cavity to that in a rectangular cavity. For E_x is nonexistent in a rectangular cavity as dictated by equation (38). That is, the entire electric field in a rectangular cavity is purely y-polarized. However, for the ridged cavity the electric field has both x and y components. The y-polarized field component is commonly referred to as the co-polarized field component and the x-polarized field component is referred to as the cross-polarized field component. The cross-polarized field component is of serious concern to the antenna designer. Generally it is undesirable since it degrades the rejection purity of a transmitting/receiving system.

The existence of a cross-polarized component indeed agrees with physical intuition. That is, the very presence of the metal ridge demands that the electric field be always and everywhere perpendicular to its surface. Thus if we were to plot the total electric field in vector form (i.e. the vector sum of E_x, E_y, & E_z) we would expect the resultant electric field vector to be perpendicular to the ridge at every point along the surface of the ridge. Such a vector plot has been obtained from ANSYS at z=L/2. This vector plot appears in Figure 29. The z-component of electric field (E_z) is shown below in Figure 23. The plots in Figures 21-26 are rather small resulting in an illegible colorbar. However, the values of E_z are on the order of .001x10^-6 everywhere inside the cavity. In contrast, the maximum value of E_y is 2607. Therefore, if we normalize the values of E_z to the maximum value of E_y we see that E_z is essentially zero everywhere inside the cavity. Of course this is mandatory for any TE_z mode as dictated by equations (9) & (34), and evidenced by equation (38). Therefore, the vector sum is essentially a combination of the E_y and E_x field components shown in Figures 21(d) and 22(d) respectively. As such these two figures have been enlarged. The enlarged contour plots appear in Figures 27 and 28 respectively.

Comparison of Equation (36) with Figure 24 shows that the x directed magnetic field distribution in a ridged cavity is similar, but not identical, to that in a rectangular cavity. Recall that in a rectangular cavity, H_x is sinusoidal in x and co-sinusoidal in z, but constant in y. To highlight the similarities between ridged and rectangular cavities we note that H_x for the ridged cavity is somewhat sinusoidal in x and somewhat co-sinusoidal in z. However, the sinusoidal behavior in x is more prominent in the region between the ridges. As for the differences, perhaps the most obvious is that for the ridged cavity, H_x is not constant in y, with the exception of the region between the ridges.

The y directed magnetic field component shown in Figure 25 is somewhat co-sinusoidal in z with an irregular distribution in x and y. Unfortunately we can not compare the behavior of this field component (i.e. H_y) in the ridged cavity to that in a rectangular cavity. For H_y is nonexistent in a rectangular cavity as dictated by equation (38). We have already described how the electric field vector has both x and y components. It stands to reason then that the magnetic field vector must also have x and y components. That is, the total electric and magnetic field vectors are always and everywhere orthogonal.

Comparison of Equation (37) with Figure 26 shows that the z directed magnetic field distribution in a ridged cavity is similar, but not identical, to that in a rectangular cavity. Recall that in a rectangular cavity, H_z is sinusoidal in z and co-sinusoidal in x, but constant in y. To highlight the similarities between ridged and rectangular cavities we note that H_z for the ridged cavity is somewhat sinusoidal in z and somewhat co-sinusoidal in x. However, the co-sinusoidal behavior in x is more prominent in the region between the ridges. As for the differences, perhaps the most obvious is that for the ridged cavity, H_z is not constant in y, with the exception of the region between the ridges.

We discussed how the vector plot of Figure 29 is essentially the vector sum of E_y and E_x shown in Figures 27 and 28 respectively. Moreover we argued that the total electric field must me perpendicular to the ridge at all points along its surface in order to satisfy the boundary conditions. Inspection of Figure 29 reveals that the boundary conditions have indeed been satisfied.

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