How to explain L drop in sub-zero-GHz freqency?

I observed many cases in papers, in our RF measurement and also in full-wave simulation that the "inductance vs freqeuncy" usually drops in sub-zero-GHz range, i.e. several mega hertz range

For our case with a micromachined 3D solenoid inductor, the drop is quite obvious. And I guess similar drop occurs in planar spiral or symmetrical indutors.

The inductance is extracted using expression:
-1/(2*pi*freq*imag(Y11)))

How to explain this phemonenon?
I guess measurement errors in sub-zero-GHz might has some effect. But how about the results from full wave simulation as in HFSS?
Other research findings indicate that internal inductance of a conductor drops with increase in frequency.
Or the method of extraction will introduce some effect.

Anyone can help me with that?

Hi, wizardz:

This is due to the field and current distribution in the cross-section of a TLN changes very much at frequency close to DC, especially for lossy structure. It is similar phenomenon to the skin depth effect. At very low frequency, the current is spreaded into more cross-section area and the L and R change at very low frequency. You can read the Appendix BB of IE3D User's Manual for a more complete explanation. Regards.

Thank you jian for your kindness.

I thus have another question.
Many article still use analytical model as "Greenhouse method" or formula in Grover's book. But these inductance calculations are frequency independent(I think these are low frequency case). Does it mean they are the results near DC?
Or shall we really use the value at certain frequency of interest extracted from -1/(2*pi*freq*imag(Y11))) when designing circuit with inductors?

In my case, the drop from DC to GHz is quite large(30%). I guess this greater drop than widely used planar inductors is due to the thick metal I use in the inductor.

Hi, wizardz:

If the formula yields L independent of frequency, it normally should be from static solver.

Normally, conductivity in the metal and the surrounding materials may change the current and field distribution of a structure much when frequency is changing. This change may create signficant change in R and L values of a TLN. It normally does not change the C value much. For most microstrip line, you may not see the L and R change very much. It is because a regular microstrip line does not have much loss. However, if you try to reduce the width and thickness of the strip, you will see the L value changes wildly at low frequency.

Regards.