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Advanced Signal Integrity For High-Speed Digital Designs by HOWARD L. HECK, STEPHEN H. HALL

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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): c05-eq029.jpg(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)c08-eq017.jpg

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)c08-eq020.jpg

wherebapp05-1.jpge (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 He = Rac and bapp05-1.jpge = ωLinternal. So, quite simply, the internal inductance for a causal relationship is

(E-1)c19-eq001.jpg

where the hat ...

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