Electrical and information technology Projects in microwave theory 2009 Write a short report on the project that includes a short abstract, an introduction, a theory section, a section on the results and conclusions. The deadline for the report is December 7. The lecture on Thursday December 3, 8-10 is devoted for the assignments and projects. You can then get help from Anders. Those of you who like to get a grade 4 or 5 need to give an oral presentation of the project. The plan is to have oral presentations during December 8-11. You then sign up for a two-hour session where you present your project and listen to the other presentations. Some of the projects require that you discuss with Anders before you start the project.
1 Bandstop SIW-filter Construct a bandstop filter by using a substrate integrated waveguide with a trap. The vias are circular rods, that for convenience are assumed to be perfectly conducting. The radius of the vias is a = 1 mm. The distance between the centers of two vias should be 3 mm. The substrate has a relative permittivity of approximately ε = 4. Try to make an SIW that is a bandstop filter such that the frequency f = 30 GHz is stopped. You may vary ε in order to tune the filter. What is the smallest S 12 you can achieve? Try also to estimate the bandwidth of your filter. Is it possible to change the bandwidth? Try some different types of structures. Help: Before you do the bandstop filter for the SIW it is wise to do it for a hollow waveguide first. You may try a waveguide filter similar to the one in figure 1. It consists of a resonator that is attached to the waveguide. The resonance frequency is stopped. There is a hole between the resonator and the waveguide. The S21 parameter (the transmission coefficient) is seen in figure 2. To obtain the S-parameter as a function of frequency you do the following: 1. Select Port as boundary condition for the entrance (port 1) and exit (port 2) of the waveguide. At the entrance choose Incident wave at this port. For both ports click on port and choose Analytic and mode number 1 for specification of mode. 2. Go to Solver Parameters dialog box and select the Parametric solver (in the Swedish version this is Lösarinställningar Parametrisk under Lös). Set Parameter name to freq and Parameter values to linspace(28e9,32e9,50) where 28e9 is the start frequency, 32e9 the stop frequency and 50 the number of frequency steps. 3. Open the Application Scalar Variables dialog box from the Physics menu and clear the Synchronize equivalent variables check box. Set the frequency variable nu rfw to freq. This will make the frequency sweep from 28 GHz to 30 GHz. 4. Solve the problem. 5. On the Point tab in Domain Plot Parameters dialog box select Point 1 and select Electromagnetic Waves (rfw) S-parameter db (S21). Click Apply to plot S21 on a db scale. 2 Silica on silicon waveguides: leakage in bends In photonics it is possible to make integrated circuits for light directly on silicon. Instead of using microstrips, as in ordinary integrated circuits, one uses silica waveguides embedded in silicon. The silica waveguides have rectangular cross section. The waves are confined to the waveguide, just like it is for an optical fiber. On a circuit
Figure 1: Waveguide with bandstop filter 0-4 S 21 db -8-12 -16 27.5 28 28.5 29 29.5 30 frequency/ghz Figure 2: S21
it is often necessary to bend waveguides in order to send signals from one point to another. The drawback is that there is a leakage of power at the bends. Use a rectangular silica on silicon waveguide and try different radius of curvature (measured in wavelengths) for bends and different widths of the waveguide to get relations between the power leakage and the power transported in the waveguide. In order to get simple calculations in COMSOL you can consider that the waveguide is between two perfectly conducting horizontal plates. In that case you can use RF 2d in-plane waves in COMSOL. Use refractive index of 3.5 for silicon and 1.45 for silica. The vacuum wavelength is 1.55 µm. You are only interested in propagation of TE waves. At the entrance and exit of the waveguide you should use port as boundary condition. You may specify the port with β equal to the wavenumber in silicon. In the COMSOL help menu there is an example of propagation of light in a silica waveguide. You may take a look at that example. The example consider birefringence of waves. Due to mechanical stresses in the material the silica becomes anisotropic which means that the permittivity depends on the direction of the electric field. In this project you should not take anisotropic effects into account. 3 Determination of transmission line parameters for microstrips Analyze how to achieve the line parameters R, L, G, and C, as well as the characteristic impedance Z 0, by COMSOL for a microstrip line on a substrate above a ground plane. The substrate should have a low but non-zero conductivity and the micro strip and ground plane have large but finite conductivity. Try to find realistic values for the substrate and the microstrip. Consider micro strips with thickness much larger than the skin depth. You may also discuss how to find the line parameters when the thickness is in the order of the skin depth. In the book there are two different methods described how to determine the line parameters (see pages 67 70 and 142 143). Use both methods and try to improve the results obtained in the book. 4 Microwave oven Examine a microwave oven. In particular you may examine and describe the magnetron, the waveguide from the magnetron to the oven, the oven as a resonance cavity, and the microwave filter in the door. Here are some things you may calculate by using COMSOL. It is enough if you pick one or two of these. 1. The magnetron is in fact a resonance cavity with a resonance frequency of 2.45 GHz. Try to make a 2-D copy of the geometry and calculate the resonance frequency in COMSOL. There are different resonances in the magnetron. At
http://www.radartutorial.eu/08.transmitters/tx08.en.html you find information on the one that is used in most magnetrons. 2. Simulate the quarter wave filter in the door of the oven. 3. Simulate the electromagnetic field in the oven. Here you can get help from the help desk of COMSOL. They have the microwave oven as an example. Anders has a magnetron and a dismantled microwave oven that you can look at. Only one group can pick this project. 5 Limitations of transmission line theory 0 z The figure shows two transmission lines with different characteristic impedances that are connected at z = 0. The width of the microstrips is much larger than the distances between the ground plane and the strip lines and hence the structure can then be approximated by a 2-d structure. The distance between the ground plane and the microstrip to the left is d 1 = 1.6 mm and the corresponding distance for the microstrip to the right is d 2 = 2.4 mm. The substrate has a permittivity ε r = 2. The width of the micro strips are such that the characteristic impedance is 50 Ω for the microstrip to the left and Z 0 = 75 Ω for the microstrip to the right. a) Use the mode matching method, described in Section 6.11 in the book, to determine the reflection and transmission coefficients when the TEM mode, traveling in the positive z direction, impinges on the interface between the two transmission lines. Depict the variation of the reflection and transmission coefficients in the frequency interval 0 Hz to 60 GHz in a graph. Compare with the values of reflection and transmission coefficients you get from transmission line theory. At what frequency does transmission line theory start to give wrong results? b) Do the same calculations in COMSOL. Use 2-D, In plane waves and TMwaves. The reflection and transmission coefficients are obtained by using ports as boundary condition at the entrance and exit. Follow the instructions in project 2 but use the user defined port values rather than the analytic. Specify the wavenumber in the port as β =wavenumber in waveguide. c) Use Comsol and transmission line theory to check the S-parameters for some other structures.
6 Determination of the Q value of an axially symmetric cavity At electron accelerators, e.g. Maxlab, one uses microwave cavities to accelerate the electrons. The cavities are axially symmetric and the mode that is used corresponds to the TM 010 mode of a circular cylindrical cavity. The reason is that the electron beam passes along the axis of symmetry of the cavity and hence the electric field needs to be very strong at the axis of symmetry. One of the main problems with the cavities used in accelerators is the losses due to currents in the walls. It is important that the Q-value of the cavity is very large otherwise the heating of the cavity becomes a problem. Try to design a microwave resonator for the frequency 500 MHz with a large Q value. You may use the axially symmetric version of COMSOL to calculate the electric and magnetic fields in the resonator. Below it is described how the 3-D problem is reduced to a 2-D problem for the axially symmetric resonator. Since the new version of COMSOL can handle axially symmetric structures it is not necessary to use the theory. If we use non-relativistic formulas the energy that is added to the electron when it passes through the cavity is W e = qed where q is the charge of the electron, E is the electric field on the axis of symmetry and d is the distance the electron travels. Determine W e /W L where W L is the Ohmic energy loss during one cycle and try to make the quotient large. Theory Introduce cylindrical coordinates (ρ, φ, z). The cavity mode that is of interest is the φ independent TM-mode with lowest resonance frequency. For that mode the magnetic field has only a φ-component, i.e., The corresponding electric field reads H(ρ, z) = H(ρ, z)ˆφ E(ρ, z) = E ρ (ρ, z)ˆρ + E z (ρ, z)ẑ A necessary condition for the solution is that E z (ρ, z) is maximal on the z axis. In the case of a general φ independent geometry, it is preferable to study the equation for H(ρ, z) (why is it not good to study E z?). The equation reads 1 z) ρ H(ρ, ρ ρ ρ 1 ρ 2H(ρ, z) + k2 H(ρ, z) = 0 (6.1) Notice the second term which is due to the cylindrical coordinate system, cf Appendix B. The boundary condition for H(ρ, z) is obtained from the condition that the tangential component of the electric field is zero at the surface, ˆn E(ρ, z) = 0, where ˆn is the normal unit vector. From Amperes law it is seen that ˆn ( H(ρ, z)) = iωǫ 0ˆn E(ρ, z) = 0, ons
Thus H(ρ, z) ˆn (ˆρ z Since ˆn = n ρˆρ + n z ẑ it follows that ẑ 1 (ρh(ρ, z)) = 0 ρ ρ 1 n ρ ρ ρ (ρh(ρ, z)) + n H(ρ, z) z z Unfortunately this is not a Neumann boundary condition and hence not suitable for COMSOL. In order to obtain a Neumann boundary condition we introduce Then Ψ(ρ, z) = ρh(ρ, z) ˆn Ψ(ρ, z) = 0 The eigenvalue problem for Ψ(ρ, z) follows from equation (6.1). ( ) 2 ρ + 2 Ψ(ρ, z) 1 Ψ(ρ, z) + k 2 Ψ(ρ, z) = 0 in V 2 z 2 ρ ρ ˆn Ψ(ρ, z) = 0 on S = 0 The Q value Once the eigenvalue problem has been solved for the resonator with perfectly conducting walls one may calculate the Q value for the resonator with walls that have finite conductivity. Since rms of stored EM-energy in cavity Q = ω dissipated energy during one time period rms of stored magnetic energy in cavity = 2ω dissipated energy during one time period the Q-value can be obtained. The final result is Q = ω µ 0 R s S H(ρ, z) 2 ρds C H(ρ, z) 2 ρdl where S is the area of the cross section in the xz-plane, C is the curve around S, and R s is the surface resistance. The following parts are included in the project. 1. Check the analysis above. 2. Check numerically that the resonance frequency and the Q value agree with the analytic results for a circular cylindric cavity. 3. Calculate the resonance frequencies and the Q values for cavities that you find interesting.
x y 45 o z 7 Horn antenna Examine the electric field in a horn antenna with opening angle of 45, cf figure. The horn is fed by the fundamental mode in a rectangular waveguide with dimension a b, where a = 5 cm and b = 2.5 cm. Notice that the distance between the horizontal plates is equal to b. The frequency is 4 GHz. First determine the different TE-modes that the TE 10 mode can generate in the horn by solving Helmholtz equation in cylindrical coordinates. Try to plot the different modes in the horn. You don t have to take the feeding waveguide into account and you may consider a horn with infinite extension in the radial direction. The solutions you look for have to satisfy the radiation condition that the wave is an outward traveling wave for large value of kρ. You might need some help from Anders here. Next solve the same problem in COMSOL for a horn with length R = 30 cm. Use in-plane waves. Now take into account the feeding waveguide as well as the finite length of the horn. At the aperture of the horn you need an absorbing boundary condition. It is possible to animate the wave in COMSOL. You can do that by using animate under the menu Postprocessing. 8 Photonic band-gaps It is hard to guide light unless you are using an optical fiber or some other dielectric waveguide. During the last 10 years there have been efforts to create waveguides for light that propagates in air. This is done by using channels in a periodic structure of dielectric rods. The periodic structure should then have a band-gap for the wavelength used by the light. A band gap is a frequency interval where the light cannot propagate. Try to find some more information on band-gap structures on internet. In the help-desk of COMSOL you may find a simulation of a photonic band gap structure. Go to RF and Users guide in the help desk. Pick Optics and photonics models. There you find photonic crystals. Instead of using gallium-arsenide you may use a material with constant index of refraction, say n = 5. First analyze the cut-off frequency and bandwidth of the photonic band gap structure. This is done by determining the leakage of power from the waveguide. Integrate the power transported through the surfaces that are not the input port or the output port. If this leakage is negligible compared to the power transported in the waveguide then you are in the frequency band of the waveguide. If the frequency is too small
then there is no propagating mode in the waveguide and you are below the cut-off frequency. Try to make a waveguide that splits up the wave in two parts that travel in opposite direction. You may also try to make a bandstop filter, cf project Bandstop SIW filter. It is possible to animate the wave in COMSOL. You can do that by using animate under the menu Postprocessing.