RECTANGULAR WAVEGUIDE ANALYSIS
A three dimensional model of rectangular waveguide was constructed in ANSYS with the following dimensions, elements, and material properties:
Dimensions: a=0.03 m; b=0.01 m; L=0.048 m
Element Type: HF120
Element Order: 2^{nd} order element
Element Edge Size: 0.0032 m
Number of Elements: 600
Number of Nodes 3,209
Material Properties: m =1; e =1 (i.e. air)
Model Size: Full
An isometric view of the model is shown below in Figure 5. Using equation (18), we find the guide wavelength to be l _{g} = 0.048 m at 8GHz for a rectangular waveguide of these dimensions. Hence the waveguide model is one guide wavelength long at 8 GHz. A similar harmonic analysis of this very structure using the HF119 element is presented in the ANSYS verification manual. An electric wall boundary condition was applied to all surfaces of the model with the exception of the surfaces at z=0 and z=0.048m. I have chosen the surface at z=0 to correspond to the input port of the waveguide. Likewise I have chosen the surface at z=0.048 m to correspond to the output port of the waveguide. The load requirements for the input and output surfaces are dependent upon the analysis type. Thus the input/output load requirements will be discussed separately for each analysis type.
Figure 5: Isometric View of a Rectangular Waveguide Model in ANSYS
Solution Method: Full
Equation Solver: Frontal
Excitation: TE_{10} waveguide mode
Frequency: 8 GHz
A full harmonic analysis requires the specification of an excitation condition at the input and output ports of the waveguide. In particular it is necessary to specify an input excitation condition at z=0 and an output excitation at z=0.048. The first step in applying the port excitation is to define a local coordinate system at each port. For example, if the input port is defined at z=0 and the global coordinate system is as shown in Figure 5, the origin of the local coordinate system must be located at x=a/2, y=0, and z=0. Moreover the local x axis must be co-directed with the global x axis (i.e. x_{L} = x_{G}) and likewise for the local/global y & z axes (i.e. y_{L} = y_{G} and z_{L} = z_{G}).
The location and orientation of the local coordinate system is slightly different for the output port. For the global coordinate system shown in Figure 5 and an output port defined at z=0.048, the origin of the local coordinate system must be located at x=a/2, y=0, and z=0.048. Moreover the local y axis must be co-directed with the global y axis (i.e. y_{L} = y_{G}). However, local/global x & z axes must be anti-directed (i.e. x_{L} = -x_{G} and z_{L} = -z_{G}).
Once the local coordinate systems have been defined it is possible to specify the port excitations. In specifying an input excitation, the ANSYS user must choose one of two allowable forms. The choices are that of a coaxial input excitation or a TE_{10} waveguide mode excitation. The TE_{10} waveguide mode excitation is of course the proper choice for the analysis at hand. In addition to selecting an excitation form, the user must specify the input dimensions of the waveguide relative to the local coordinate system. Finally, it is necessary to specify an input amplitude and phase term for the incident wave.
The excitation specification process for the output port is identical to that for the input port with one exception. That is, in specifying the output excitation, the amplitude & phase fields of the "Define Port Options" menu must be left blank. Leaving these fields blank identifies the output port as a "match terminated" port. The characteristics of a matched output port are such that all power incident upon the output port from the input port will be absorbed by the "matched termination". In other words, all power presented to the output port will be absorbed and none of it will be reflected back into the waveguide structure.
A full-harmonic analysis was performed at 8 GHz using the full frontal solver. All components of the electric and magnetic fields were evaluated using the general postprocessor. Figure 6 below is a contour plot of the y component of electric field E_{y}.
Figure 6: Contour Plot of E_{y} (Isometric View)
Indeed Figure 6 agrees with our discussion of the field behavior of the TE_{10} mode in rectangular waveguide. In particular the field strength is sinusoidal in x and constant in y as described in equation (19). Moreover the field has gone through one full cycle in z as it should for a waveguide that is one guide wavelength long. A contour plot of the x component of magnetic field (H_{x}) is presented in Figure 7 below. It too is sinusoidal in x, constant in y, and completes one full cycle in z as dictated by equation (20). Moreover, H_{x} is spatially co-phased with E_{y} as discussed. The z component of magnetic field (H_{z}), shown in Figure 8, is co-sinusoidal in x and constant in y as described by equation (21). It too completes one full cycle in z, however it is displaced from E_{y} and H_{x} by l _{g}/4. This is due to a 90° phase differential in time and is not due to differences in the functional dependence of z between the field components. This was discussed earlier and can be verified upon inspection of equations (19), (20), and (21).
Figure 7: Contour Plot of H_{x} (Isometric View)
Figure 8: Contour Plot of H_{z} (Isometric View)
Vector plots of the electric and magnetic fields were also obtained from the general postprocessor. Figure 9 below shows a vector plot of the total electric field in the waveguide. More specifically, Figure 9 is a top view that illustrates the electric field distribution in the x-z plane. The field distribution in the y-z plane is nicely depicted in Figure 10. Figure 10 is a vector plot of the total electric field at x=a/2.
Figure 9: Vector Plot of E_{total} (Top View)
Figure 10: Vector Plot of E_{total} (Section View at x=a/2)
A mental merging of Figures 7 & 8 reveals that the magnetic field forms loops within the waveguide. This behavior is easily visualized through the use of a magnetic field vector plot as shown in Figure 11 below.
Figure 11: Vector Plot of H_{total} (Isometric View)
Extractor: Block Lanczos
Excitation: None
Frequency Range: 4.9 GHz to 10.1 GHz
Number of Modes Extracted: 2
Number of Modes Expanded: 2
The excitation requirements for a high frequency modal analysis are described in Chapter 10 of the ANSYS Electromagnetic Field Analysis Guide. In particular, page 10-26 of this guide states that the "Input excitation is ignored, and the input port is treated as an open circuit condition." Page 10-12 of the guide states that "For an open circuit condition, do not specify any boundary condition or excitation to the port region." With regard to output excitation, page 10-26 states that "Infinite surfaces are treated as a magnetic wall boundary condition", which is the default boundary condition of ANSYS. Hence the TE_{10} waveguide mode excitations (previously applied to the model) were deleted for the modal analysis.
A modal analysis was performed on the revised waveguide model from 4.9 GHz to 10.1 GHz. Five modes were requested, however, only two modes were found in this frequency range. We will demonstrate that the two modes found were the TE_{10} and TE_{20} modes as expected.
Results of mode set number one (TE_{10})
The first eigenvalue extracted by ANSYS was 5 GHz. We will later show that this is precisely the cutoff frequency of the TE_{10} mode in this waveguide. A contour plot of the y component of electric field (E_{y}) is shown below in Figure 12. From Figure 12 it is clear the electric field distribution is sinusoidal in x and constant in y as described by equation (19). However, the electric field distribution is shown to be constant in z as well. This is inconsistent with the results of the harmonic analysis (Figures 6, 9, &10) and equation (19). A constant z behavior was also noted in the magnetic field contour plots (not shown). Clearly the modal analysis does not accurately represent the z dependent nature of the electric field in a rectangular waveguide. However, the results of the modal analysis will suffice since the focus of this project is on the cross sectional field distribution within waveguides of constant cross section.
Figure 12: Contour Plot of E_{y} (Isometric View)
A vector plot of the y component of electric field (E_{y}) is shown below in Figure 13. This plot depicts the sinusoidal nature of the y-directed electric field as a function of x. As discussed, the "1" in TE_{10} infers a one (1) half sine variation in x. Likewise the "0" in TE_{10} infers a zero (0) sine variation in y (i.e. constant in y). To quantify the results, the nodal solution was queried at several values of x. The y-directed electric field values are listed below in Table 1 for several values of x.
TABLE 1
Distance x (m) |
E_{y} (V/m) |
0.015 |
804.123 |
0.018 |
764.767 |
0.021 |
650.550 |
0.024 |
472.652 |
0.027 |
248.488 |
0.030 |
0.65340 |
Figure 13: Vector Plot of E_{total} (Front View Rotated 180° )
Results of mode set number two (TE_{20})
The second eigenvalue extracted by ANSYS was 10 GHz. We will later show that this is precisely the TE_{20} cutoff frequency of the waveguide. A contour plot of the y component of electric field (E_{y}) is shown below in Figure 14. From Figure 14 it is clear the electric field distribution is sinusoidal in x and constant in y as described by equation (22). However, the electric field distribution is shown to be constant in z as well. A constant z behavior was also noted in the magnetic field contour plots (not shown). This constant z dependence is inconsistent with equations (22), (23), and (24). Once again the modal analysis does not accurately represent the z dependent nature of the electric field in a rectangular waveguide. As argued earlier, the results of the modal analysis, though limited, are indeed sufficient for the purpose of this project.
A vector plot of the y component of electric field (E_{y}) is shown below in Figure 15. This plot depicts the sinusoidal nature of the y-directed electric field as a function of x. As discussed, the "2" in TE_{20} infers two (2) half sine variations in x. Likewise the "0" in TE_{20} infers a zero (0) sine variation in y (i.e. constant in y). To quantify the results, the nodal solution was queried at several values of x. The y-directed electric field values are listed in Table 2 below for several values of x.
TABLE 2
Distance x (m) |
E_{y} (V/m) |
0.015 |
0.195e-9 |
0.018 |
504.177 |
0.021 |
815.775 |
0.024 |
815.775 |
0.027 |
504.177 |
0.030 |
5.38900 |
Figure 14: Contour Plot of E_{y} (Isometric View)
Figure 15: Vector Plot of E_{total} (Front View)
Comparison of FEM Results with Textbook Solutions
Harmonic analysis (TE_{10} mode)
As discussed above, the results of the harmonic analysis visually agree with the field equations given by (19), (20), and (21). However, I did not query the results of the harmonic analysis. Instead I choose to quantify the results of the modal analysis since it is the analysis type that must be used for ridged geometries.
Modal analysis (TE_{10} mode)
Before we consider the field distribution, we shall examine the first eigenvalue extracted by ANSYS. It is extremely important that ANSYS extract the proper eigenvalue since the field distribution is expanded in terms of the extracted value. The first eigenvalue extracted by ANSYS was 5 GHz. This is actually the TE_{10} mode cutoff frequency for this particular rectangular waveguide. We can confirm this result by recalling our expression for l _{c10}. We said that l _{c10} = 2a, where a=0.03 m is the width of the waveguide. The cutoff frequency is given by equation (16). Substituting our expression for l _{c10} into equation (16) yields a cutoff frequency of exactly 5 GHz. This is precisely the value extracted by ANSYS!
The cross sectional field distribution predicted by ANSYS is easily compared to equation (19) by means of a graph. It is possible to normalize the values of E_{y} listed in Table 1 to the maximum value at x=0.015 m. Indeed this has been done and the normalized values have been plotted below in Figure 16 for several values of x. The sinusoidal dependence of equation (19) is also plotted (with ‘+’ tic marks) on the same graph for comparison purposes. It is clear from this discussion, and Figure 16, that the ANSYS model yields results that are in complete agreement with textbook solutions for the TE_{10} mode.
Figure 16: Plot of Normalized E_{y} and Equation (19) versus x
Modal analysis (TE_{20} mode)
The second eigenvalue extracted by ANSYS was 10 GHz. This is actually the TE_{20} mode cutoff frequency for this particular rectangular waveguide. We can confirm this result by recalling our expression for l _{c20}. We said that l _{c20} = a, where a=0.03 m is the width of the waveguide. The cutoff frequency is given by equation (16). Substituting our expression for l _{c20} into equation (16) yields a cutoff frequency of exactly 10 GHz. This is precisely the value extracted by ANSYS!
The cross sectional field distribution predicted by ANSYS is easily compared to equation (22) by means of a graph. It is possible to normalize the values of E_{y} listed in Table 2 to the maximum value at x=0.0225 m. Indeed this has been done and the normalized values have been plotted below in Figure 17 for several values of x. The sinusoidal dependence of equation (22) is also plotted (with ‘+’ tic marks) on the same graph for comparison purposes. It is clear from this discussion, and Figure 17, that the ANSYS model yields results that are in complete agreement with textbook solutions for the TE_{20} mode.
Figure 17: Plot of Normalized E_{y} and Equation (22) versus x