Finding the coupling coefficient between coupled cavity resonators using electromagnetic simulations with Microwave CST
In this example we will explore how to use the eigenmode solver in Microwave CST to find the coupling coefficient between two microwave box resonators coupled together via a circular aperture. Its useful to begin by examining the resonant modes of a single, isolated box resonator. The cavity to be analyzed is shown below.
The dimensions of the cavity are: 22.86 mm × 10.16 mm × 18.371 mm, as indicated below in the component definition box.
We now analyze the box using the eigenmode solver, which will determine the resonant frequencies (eigen-frequencies) and field distribution (E and H fields) of each eigenmode. But before that, here is a little thought on eigenmodes you might find helpful.
Whenever I get hazy on what is meant by an electromagnetic eigenmode, I think of harmonics on a guitar string. A guitar is a very tangible object to me (actually a bass guitar to be specific) that doesnt take much effort or mathematics on my part to imagine. A guitar string is pinned or fixed on two ends of the guitar but otherwise free to move. These end positions are where the amplitude of the string are fixed and thus define the boundary conditions. This is analogous to the conducting walls of an electromagnetic box resonator where the tangential component of the electric field is fixed at zero. When you pluck a guitar string and let it vibrate on its own, you hear a distribution of sine waves or harmonics supported by the string. The particular spectrum of harmonics depends on the length, tightness, and physical properties of the guitar strings. In the case of a box resonator, the electromagnetic modes refer to the fields that slosh around the metal box as governed by Maxwells Equations. These electromagnetic modes can be excited by a source such as an antenna, and the spectrum of possible harmonics (eigen-frequencies) depends on the size and material properties of the conducting box itself, as well as the properties of the air inside the box.
Anyways, back to the matter at hand. To run the eigenmode solver, we first need to set the frequency range of the solver. To do this, click on Frequency under the Solve dropdown menu:
Next, we set the frequency range to be between 10 GHz and 17 GHz:
Now click on Eigenmode Solver under the Solve dropdown menu and set the number of modes to 4 in the corresponding dialogue box. Also, make sure you select the option to Store all result data in cache as shown below:
Click on the Start button to run the eigenmode solver. After a minute or so the solver will be done. You will see a box telling you the eigen-frequencies of each mode, as shown below:
To see the E or H field distribution of any of these four modes, look under the 2D/3D results folder under the left window pane, as illustrated below:
Lets look a little closer at the results. Here are the eigen-frequencies associated with each mode:
Full E- and H-field distribution plots (for the x-, y-, and z-components) for modes 1 and 2 are shown below.
Sometimes plotting the field distributions for all three x-, y-, and z-coordinates can be a little overwhelming, i.e. it gives more information than we really need. Instead it is sometime useful to simply view the absolute value of the field. Below we show the absolute value of the E-field for modes 1 through 4.
Absolute value of E-field:
So much for the single box resonator. Now we connect two box resonators together by a thin circular aperture and run the eigensolver for the entire structure as illustrated in the screenshots below:
Running the eigenmode solver for the six modes yields the following results:
How are we to use this information to find the coupling coefficient k? Luckily there is an easy equation we can use to solve for k:
where fodd and feven are the even and odd mode frequencies, respectively. I discussed this equation in the tutorial Positive coupling, negative coupling, and all that. Note that this equation works only in the case of two identical coupled resonators with a definite plane of symmetry (synchronously tuned resonators) as opposed to two resonators of different shapes coupled together (asynchronously tuned resonators). Such an assumption is inherent in the grammatical use of the terms even and odd.
To actually identify what mode is even and what mode is odd we need to look at the field distributions. Below we show the electric fields for modes 1 and 2 (y-component).
Mode 1 (y-component of E):
Mode 2 (y-component of E):
Note that these two modes look identical except for one crucial difference. In mode 1, both peaks are in phase and exhibits even symmetry (both bumps are colored red), whereas in mode 2 the peaks are out of phase and exhibits odd symmetry (one bump is colored red while the other is blue). We therefore, by visual inspection, identify mode 1 as the even mode and mode 2 as the odd mode as illustrated below:
From this we find the coupling coefficient k1 is equal to:
This is the first coupling coefficient, which describes the behavior of the box only in a narrow bandwidth near the first resonance. To find the second coupling coefficient, we look at the electric field distributions for modes 3 and 4 (y-component):
Mode 3 (y-component of E):
Mode 4 (y-component of E)
As before, these two modes look identical except for one crucial difference. In mode 3, the field distribution is even symmetrical, whereas in mode 4 the field distribution is odd symmetrical. We can therefore identify mode 3 as the even mode and mode 4 as the odd mode. From this we find the coupling coefficient k2 to be:
We can keep playing this game to find the coupling coefficient for the higher order modes.
Alternative method
An alternative method (which is quicker from a computational level) for identifying the even and odd modes is to split the structure right down the center as shown below (i.e. delete one of the boxes and split the coupling iris down the center):
The circular iris butts up against the plane of symmetry. To find the odd modes, we set this circular boundary as a PEC wall. To find the even modes, we set this circular boundary as a PMC wall. Running the eigenmode solver with a PEC wall and a PMC wall at the plane of symmetric yields the following results:
PEC wall (odd modes):
PMC wall (even modes):
These results yield the same even and mode frequencies we found by visually looking at the field distributions for each mode.
Dual mode resonator
The dual mode resonator is a single physical cavity that supports two eigenmodes with identical eigen-frequencies. A classic example of a dual mode resonator is a cylindrical cavity as illustrated below:
Running the eigensolver for this structure yields:
Note that modes 1 and 2 have the same eigenfrequency (16.83 GHz). The electric field for these two modes are plotted below. We can see from these plots that the field vectors for the two modes are physically perpendicular (orthogonal) to each other.
Mode 1
E normal component:
Mode 2
E normal component:
Now we break the symmetry of the structure with a little cylindrical indentation as shown below. Such a perturbation splits the eigenfrequencies of modes 1 and 2 from the original 16.83 GHz to 16.5 GHz and 16.87 GHz, as found by the eigensolver tool.
Plots of the electric fields for modes 1 and 2 are shown below:
Mode 1:
Mode 2:
To find the coupling coefficient k we recall our old faithful equation:
Unfortunately, for the case of the coupled dual mode resonator, it is not as clear (at least to me) what mode corresponds to the odd mode and what mode corresponded to the even mode. This means that while we have the magnitude of the coupling coefficient k, the sign of k for the dual mode cylindrical cavity remains inherently ambiguous. In practice, this poses no problem as long as we are consistent in defining which field orientation corresponds to which type of mode (even or odd). The sign of a coupling coefficient in a circuit only ever matters relative to another coupling coefficient in the same circuit. This is similar to how only the voltage difference matters in a circuit, not the absolute voltage at a given node.