It looks to me as if you have plotted a dispersion curve, with frequency plotted against phase per period. It looks like the phase is not in degrees, but in radians.

I would smooth the curves. You might first average them over lots of periods. The spikes almost certainly shouldn't be there so need to be removed by some form of processing, except the ones at the periodic boundary which are probably correct.

The highest point in each smoothed curve is the cutoff frequency.

I think the real part is the frequency (radians per second?), and the imaginary part will be to do with loss (nepers per second?)

Malcolm White thanks Malcolm for your response. the phase (x-axis) is in degrees. to add to my explanation that I have simulated one unit-cell in HFSS, just for test to understand how this type of simulation works. And the structure is a simple transmission line with a 10 pF capacitance at the middle.

1- I am not sure if I have understood what you mean by "You might first average them over lots of periods". HFSS simulates one unit cell and generates that result. where to include lots of periods?

2- Same for this "except the ones at the periodic boundary". How to know which spike belongs to the boundary?

3- Regarding your question, I do not know what quantity does re(mode(1)) represents. this actually is my question, too.

thanks again for your help:)

Yousef Vahabzadeh I would concur with Malcolm White that this dispersion response seems to be taken over a larger range of angles than 180 degrees. Please show us the setup of your boundary conditions and analysis. Also, what is the length of the transmission line, and its propagation constant?

Stuart Barth thanks, Stuart. I added a photo of the structure here, too. The length of the transmission line is 15 mm and 10 mm, in the y and x directions, respectively. In the attached image, the purple boundaries are set periodic (master-slave boundary). The phase between these two boundaries is varied between 0 to 180 degrees. the analysis mode is eigenmode simulation. I have chosen 5modes in the analysis setup and put a minimum frequency of 1 GHz. We cannot set the propagation constant in this simulation, so it is not known.

Yousef Vahabzadeh thanks, could you also show us the settings used for the boundaries and analysis setup, along with your variable definitions.

If the host is a microstrip transmission line on FR-4, we can expect its propagation constant to be around 60 rad/m at 2 GHz -- or 52 degrees over the 15mm length of the line. A 10pF capacitor shouldn't be adding too much to this (the value of the capacitor is relatively large), so I would expect to see something in this range. However, your plot shows something closer to about 1 degree. The ratio here is very close to that of degrees to radians (180/pi~=57.3). Therefore I strongly suspect that (as usual) Malcolm White is correct: you have plotted radians rather than degrees.

In this case, I would suspect that your phase variable used in the master-slave boundaries (beta_d) is defined in terms of radians (or unitless), instead of specifically being defined in degrees.

The period of the variation in your plot is 2 pi. That means that the scale is almost certainly in radians, and each hump and half the spikes each side of it make up one period of the dispersion plot. You need to average over many periods.

Dispersion curves are usually plots of angular frequency against phase change per space period (omega/beta), and are periodic in the phase domain too, so only one period needs to be plotted, or only half a period if it is symmetrical, as yours seems to be. Based on the repeating shape and the period of 2 pi, It would be exceedingly unusual if the horizontal axis was in degrees. I suggest you reconsider your maths or read the help and find out why it is in radians when you think it is degrees.

Stuart Barth

Thank you.

I am wrong more often than I would like.

Stuart Barth attached is the setting for the boundary conditions, the analysis set up, and variable definitions. Malcolm White was right the x-axis is unitless, thus it is in radian, as you suggested, too. I should have set the unit for that, too.

The issue that I have is how to interpret the results. I have attached both the real and imaginary for all 4modes for beta_d=[0-10]. Since im(mode) is very small compared to the re(mode), we can conclude that all of these 4modes are propagating(?).

1- Do I need to consider more modes in the analysis setup until I get the first evanescent mode to make sure the siulation is correct? how many modes should be included in this simulation?

2- what do those big falls in the plots mean? For example, in the re(mode(2)), we have a big fall around beta_d=6. It is not a computational issue for sure. Then, why does it happen?

3- How to interpret the result to find the cut-off frequency of a mode?

I appreciate you so much for your time and help.

The slope of the plot (dw/dk) is the group velocity of the signal (the speed and direction the energy travels along the structure), and the w/k is the phase velocity (the speed and direction the pattern travels along the structure).

You need to smooth your plot somehow to get rid of the jagged shape, but the big dips are probably real and correspond to the real behaviour that the group velocity is high in these regions, as it often is. The narrower peaks indicate that there may be a region where the group velocity reverses and the phase velocity becomes faster than light (which it does in many structures. A peak at zero phase indicates a signal with infinite wavelength and zero net movement of power. Structures that do this have been demonstrated as antennas, as a whole surface can have equal phase when fed from any point.

The vertical axis is frequency. The minimum frequency is 1 GHz and the bottom of the axis is 1e9. To get the group and phase velocities you will need to multiply the ratios by 2 pi.

It looks to me as if the slopes are too steep for the group velocity to be less than the speed of light, which it has to be, so it may be that the solutions are jumping between branches in the regions around 0 and 2 pi. Or there may be another solution I haven't thought of. I think a slope going up 2 main divisions while along 1 is about lightspeed on this plot.