Difference of resonant frequency between theory and simulation

Hi all members! I'm a newbie in antenna design. I have a question hope every one help.

I'm designing a rectangular patch antenna. I used formulas in book Antenna Theory: Analysis and Design of Balanis. But when i simulate by using CST transient solver, the resonant frequency is quite different from the theory. Theory is 3.5 GHz and result of simulation is 3.36 GHz. I'm use FR-4 lossy for substrate. So i want to ask everyone why have a difference between theory and simulation in CST?
Thank you very much!

As a simulator executes each frame, it does so by stepping forward a small amount of time (a time-step).

The larger the time-step, the more 'coarse' the simulator results are. The more simulation departs from reality.

There is always some error in the simulation, because it is impractical to make the time-steps infinitely small.

The difference between 3.5 and 3.36 is only 4.17 percent.
See if it makes much difference when you change the time step.

Also as a secondary influence, change ohmic resistances in components.

Thank you for your help! But could you show me how can i change the time step in CST studio? I usually use default setting in transient solver!

Check this out:
https://www.edaboard.com/thread318946.html
https://www.edaboard.com/thread318299.html

Also you can try some other simulator, results can be different! At your frequency trial version limitations will be acceptable. Some simulators are better with antennas.

The best way i found to check if simulator is ok:

1. Find PCB of antenna with known substrate and center frequency
2. Measure antenna sizes using ruler
3. Put substrate parameters and antenna size to simulator

Then you will see if it is ok, and what simulator settings need to be tweaked.

If you have good equipment you can make test PCBs, measure their S-parameters and see if simulator results are ok.

Check FR4 data for simulation. If selected from menu in simulator it can be different from what you used for calculation.

Sorry, I have no experience with CST.

You may find related discussions, if you click the similar threads listed at the bottom of this page.

Or use the Search. Here is one thread which turned up.

https://www.edaboard.com/thread114189.html

I have tried to increase the line per wavelength. The S-parameter is deeper at resonant frequency but it is not change resonant frequency. I was thinked because dielectric of substrate change with frequency but seem this is not main cause :(
@Terminator3: Thank you so much. But my condition doesn't allow me do that. And my purpose is explain why it happen in CST :)
@Borber: I use data in CST to calculate.

I am not sure which equations/theory you have used, but all patch antenna equations I have seen are only approximate.

@volker@muehlhaus
I attached my CST file. In this have all my calculation dimension. Could you check or recalculate for help me? Thank you!
My purpose is design an Rectangular patch antenna use coaxial probe feed. F0 is 3.5. height of substrate is 1.6. substrate is FR4 lossy, epsilon is 4.3

Sorry, I am using other EM tools and don't have CST. What I was trying to say: don't expect that patch antenna design equations are accurate. They are based on approximations.

Thank you! I just a few worried because i don't know what is tolerance. May be 4% is ok with FR4 :D