Eigen Mode Solver & Complex Frequency (HFSS)

mode solver with loss

Hi all,

This is probably a simple question, but the answer is not coming to me right away so I thought I'd post here.

Recently, I've begun using the Eigen Mode solver in HFSS. At first I was viewing coupling between resonators, then moved onto loading a resonator with loss.

When loss was introduced frequencies became complex.

My question is simple: why is the frequency complex with loss? I assume the reason is fundamental and not specific to HFSS (i.e. pure math).

Thanks!

David

This can be explained with basic circuit theory. Let say you have an RLC circuit. The resistor models the loss, the capacitor and inductor is assumed lossless. The eigenfrequencies will in general be complex. But if we remove the resistor the resonances will be purely real. So losses induce the imaginary part. You can see this by determining the poles of the transfer function of an RLC circuit.

Thanks for the reply Jone.

This was the first thing that I thought of (going back to the days of school and looking at tank circuits). What confuses me here (with HFSS) is that the frequencies are real for lossless resonators then become complex upon the introduction of loss.

From circuit theory (s-plane) an LC tank (no loss) has an imaginary frequency (in conjugate pairs) and when loss is introduced the frequency takes on a real part as well.

What do you make of that? Are the imaginary and real parts simply swapped when viewing the "Eigen-Mode Data" in HFSS or is there still something I don't know/understand?

David

Dear Felis Silvestris,

In eigenmode solutions, when you take the conductors as pec and dielectrics as perfect, there will be no losses and frequency will be real only. It is obvious.
Now consider the case of lossy dielectrics and imperfect conductors. So there must be some losses in the structure when wave exists. In your model, you have specified that the boundary conditions are Perfect E or Perfect H or Master Slave. It means that in propagation of wave [e^j(wt-bz) * e^(-az)], you have told the simulator the bz part (phase) and you are asking for w(frequency). But you haven't told about az part (losses). Simulator calculates it and incorporates it into imaginary part of frequency.
Thats why in results, eigenmode frequency comes with imaginary part such that when you put it in wave propagation, losses part is obtained. In this way, your Q factor is determined.

Best wishes,

Amitesh Kumar
:lol:

If the resonance frequency is complex, the imaginary part describes the decay. This is how Q is defined: Q = 2*Im(f)/Re(f).
Physically, it means that somehow, we load the structure with energy and then leave it free. When the conditions are right, the energy begins oscillating. If there are losses, the oscillation decays. The imaginary part will characterize this decay.