FDTD measurement of reflection coefficient

Let assume a cylindrical waveguide with an arbitary scatterer inside. I would like to calculate the reflection coefficient of the structure without any previous pre-simulation. I have (as it's known already) total field values in time domain. How can from these values extract the reflection coefficient? I'm trying using the wave impendance using the relation

but the results are not correct. I have to take the real part of the relation or the magnitude? And if I take the magnitude, the magnitude of the whole relation or each value"s magnitude?

Thanks in advance for your help

If I am understanding your configuration, things are a bit more complicated. The equation you included is for a single interface. Your "arbitrary scatterer" probably has at least two interfaces. Even more complicated is if it does not completely fill the waveguide or has surfaces not perpendicular to the direction of the waveguide. Anyway, you can probably model whatever it is as just a slab of some thickness. You can search the internet for things like Fabry-Perot, or scattering from dielectric slabs. I put some information on this topic here:

http://emlab.utep.edu/ee5390em21.htm

See Lecture 1

Remember that a wave in a waveguide is not exactly like a plane wave in free space. So, the answer you get from a simple slab analysis may different slightly from your waveguide problem. Still, they should be reasonably close if the waveguide is single mode. If it is multimode, I don't think there is any closed form model you could use for an arbitrary scatterer.

Hope this helps!

Thanks for the reply.

I'm trying to calculate PML reflection error in a waveguide, when PML is lies perpendicular to the propagation axis. But I don't want to use the method that uses a reference signal, because with this you can find the relative reflection error. I think the method with the wave impedance is more accurate, assuming to have plane wave. If I have an axial component, how can I measure the reflection coefficient? Any ideas will be appreciated

Regards

The PML should have a graded profile making your simple equation for reflection invalid. I suggest running a simple little simulation with just a PML and calculate reflection from it. There are some papers that provide theoretical reflection for plane waves, but you have a guided mode, which is a bit different.

Thank you for the answer, I suppose the fact that the waves are guided makes the method inappropiate. One more question, in reflection coefficient should I take the magnitude of the ratio

or the ratio of the magnitudes,

. I know that these expressions are not equivalent, but under some conditions they can be.

Magnitude of ratio, not ratio of magnitudes...then square that quantity.

Why square the magnitude? To take the reflection coefficient of Power? But in order to square the magnitude and go from voltages to power calculations, power waves according to Kurokawa have to be supposed

Any wave carries power. A wave "has" some amplitude in order to "carry" some amount of power. Looking at it one or another does not mean there is a different wave. The formulation you gave is for the amplitude reflection coefficient...

r = (Z2 - Z1)/(Z2 + Z1)

This tells you the relative amplitude of the reflected wave. Sometimes knowing what fraction of power that was reflected is more meaningful. This is called reflectance.

R = abs(r)^2