Split waveguide and a waveguide acting as an antenna

Transcription

1 Department of Physics Olexii Iukhymenko Computerlab 4 Split waveguide and a waveguide acting as an antenna Introduction: Split waveguide: Picture 1 This is a picture of a metallic waveguide. An electromagnetic wave is entering from the left and is split into two braches. A problem is unwanted reflections and therefore a small cylindrical dielectric object is inserted into the waveguide (the small circle in the middle of the picture). We will try to find the best choice of dielectric constant that will minimize reflections

2 Antenna: Picture 2 This is a picture of a waveguide acting as an antenna. An electromagnetic wave is entering from the left and then the waveguide opens up and we get the impression of a wave freely moving into open space. Antenna and obstacle Picture 3 In this picture we can observe the effect of a metallic object inserted in the wave.

3 Interference Picture 4 Two waves enter from the left in the two waveguides. We can see an interference pattern. Interference using the split waveguide Picture 5

4 2D symmetry We consider in this lab fields with the symmetry 0 z and J z i.e. Actually we consider an important modification of this by allowing for a finite geometry in the z-direction. While the above may seem to make necessary structures very large in the z-direction this is actually not at all necessary. The trick is to consider regions 0 < z< b where the boundaries z and z = b are perfect conductors. The only interface condition on these boundaries is that E which is directly satisfied with the above symmetry. The structures in the pictures in the Introduction may be assumed to be confined in this way in the z-direction by 0 < z< b. TE m0 modes in a metallic waveguide Consider first a rectangular waveguide (see lecture notes or Griffiths Introduction to electrodynamics). The TEmn -modes are determined by 0 z ( x, y) = Bcos ( mπx/ a) cos ( nπ y/ b) where B just determines the amplitude and phase of the wave. The z-component the magnetic field is i( kz ωt ) Bz ( x, y, z, t) = Re ( x, y) e We will however only consider the TEm0 modes, then ( x) = Bcos ( mπ x/ a) and the z-component of the magnetic field is independent of the y-coordinate i( kz ωt ) Bz ( xzt,, ) = Re ( x) e The other components are obtained from

5 We also have 0 y 2 2 ( ω / c) = k y i = ω k x i ( ω / c) ( ) 2 2 ω / c k ωm 20 k x = where ωm0 = cmπ / a We note that b, the size of the waveguide in the y-direction, does not enter at all and any value of b give the same TEm0 -modes. Also the TEm0 -modes are independent of y. We will use this independence of the y-coordinate. However, we like to have z-independence instead as in the the Comsol figure above We then make the following permutation in the expressions for the TEm0 x y, y z, z x Thus i( kx ωt ) Bx ( xyt,, ) = Re ( y) e where ( y) = Bcos ( mπ y/ a) The other components 0 y 2 2 i = ω k y ( ω / c) -modes above i 0 y = k 2 2 ( ω / c) k y Note how these fields agree with the figure above. A side view of the rectangular waveguide is (we have 0 < z< b) y y=a x

6 The waveguide in the z-direction is defined by 0 < z < b. However the TEm0 -modes is independent of b and of the z-coordinate (inside the waveguide). We have a system with only x- and y-dependence but no z-dependence. The Radio Frequency module has the application mode Out-of-plane vector, TE waves for this situation. Above we have x, y -dependence. harmonic x-dependence but CMPH now allow us to have arbitrary ( ) We may now consider the Split Waveguide example below.

7 Wave Propagation in Split Waveguide (from lab developed by Comsol) Model background A metal waveguide for microwaves is designed to split the incoming radiation into two branches in order to feed a group antenna. A sudden change in waveguide cross-section introduces a change in the characteristic impedance and give rise to unwanted reflections. Inserting a dielectric cylindrical post of suitable permittivity in the branching region can reduce this effect. This model takes you through all the steps from defining the geometry to finding the optimal permittivity of the dielectric cylinder in Comsol Multiphysics, RF Module. Figure showing the part of the waveguide which includes the branching. The pattern of the electromagnetic field is also seen.

8 Select the application mode 2D/RF Module/Electromagnetic Waves, Frequency Domain (emw)/frequency Domain. In Parameter define a constant eps_match to be 1 Set the frequency to be 20 GHz (Study/Step1:Frequency Domain/ Study Settings) Create the geometry below: The small circle has radius and (as seen in the figure) center with coordinates (0.05,0.005). In the figure there are 8 circular arcs of 90 or less each. These are associated with centers at coordinates (0.04,-0.02), (0.04,0.03), (0.08,-0.02) and (0.08,0.03). You may create this geometry in several different ways, you may for example construct it by using circles and rectangles and use of the Boolean operations on the Geometry. If you don`t want no keep boundaries between the united objects turn off Keep interior boundaries option. An alternative is to draw straight lines (with the tool for this) and circular arcs of 90 using Quadratic Bezier Polygon. You will still need to use the Boolean operations. The two arcs in the figure which are less than 90 degrees (these two arcs meet in a point a little to the right of the small circle) are obtained from the union between two domains each involving full 90 degree arcs in their boundaries. The next two pictures indicate one way to construct the wanted geometry

9 You may easily obtain reversed copy of the object using Transforms/ Mirror (turn on keep input objects option) in the Geometry instruments.

10 Locate Electromagnetic Waves, Frequency Domain (emw)/ Components. Select Out-ofplane vector from the list. In the Electromagnetic Waves, Frequency Domain (emw) (on Physics menu) we take ε r = eps_match on the small circle (we use this name for a parametric sweep). All other ε r, and µ r are 1. Conductivity σ is 0 in whole the domain. Add ports to the model as shown on the snapshot above. In the Port settings window, locate the Port Properties section. From the Type of port list, choose Rectangular for all three ports. From the Wave excitation at this port list, choose On for Port 1 only. It would be better if you take Extremely fine meshing. Compute the problem. Hopefully you will see something like picture 1 (plot E z component). How to minimize refections We like to make a parametric study of this power outflow at port 2.Add parametric sweep for the relative permittivity eps_match take values from 1 up to 5. Compute the problem. We may calculate the power that leaves through port 2 (and by symmetry the same power must leave through port 3). To do this you use from Results/Derived values menu the Line Integration where Power outflow, time average is predefined with name emw.npoav. You may produce a plot instantly after the integration by clicking Graph Plot button above the table. Observe a maximum at eps_match=3.7, this corresponds to minimal reflection.

11 The antenna examples (Picture 2-5) The wave (or waves) in these examples comes through the port (ports) on the left. The outflow is through the port on the right. In order to get a good picture you may need create messing better then Extremely fine option. In this case use User-controled mesh/calibrate for Custom and set proper element size options manually. How to report the results of this laboration: The report is a word document that you send to me by as an attachment (oleksii.iukhymenko@physics.umu.se). Always fill in the Subject in the mail. For example write eldyn_lab4. Otherwise your mail may easily be treated as Spam The word document includes the curve of P_out as a function of permittivity as seen below in this instruction The word document includes Picture 1 Picture 2-5 refers to the antenna case and show the wave (Picture 2), the effect of an obstacle (Picture 3) and interference (Picture 4 and 5). Send me similar or different pictures illustrating these phenomena. Feel free to find your own examples.