Advanced Signal Integrity for High-Speed Digital Designs

APPENDIX E

CAUSAL RELATIONSHIP BETWEEN SKIN EFFECT RESISTANCE AND INTERNAL INDUCTANCE FOR ROUGH CONDUCTORS

To derive a causal relationship between the skin effect resistance calculated with rough copper and the internal inductance, we start with the equation for the surface impedance, which was derived in equation (5.29): (5-29)

For a function to be causal, it must be zero for negative time, as described in Section 8.2.1.†. To be causal, the function must obey the relationship.

(8-17)

From Section 8.2.1, equation (8-20) shows that the function H(ω) [which is the Fourier transform of h(t)] can be written in terms of only the even part:

(8-20)

where_e (ω) is the Hilbert transform of the even part of the function.

The similarity of (8-20) and (5-29) allows us to conclude that H_e = R_ac and —_e = ωL_internal. So, quite simply, the internal inductance for a causal relationship is

(E-1)

where the hat ...