[SI-LIST] Re: conductor losses

Hello,
your doubt is justified, the simple formula indeed violates causality and 
as a result will also produce incorrect simulation results.

The problem is that it neglects the change in internal inductance. Well 
above skin effect onset (i.e. in the region where the skin depth is small 
against the conductore diameter or thickness), you can approximate this 
with the classic formula

R_AC = (1 + j) x k_skin x sqrt(f) = k_skin x sqrt(s/PI)

where j is the imaginary unit (sqrt(-1)), s is the Laplace parameter 2 x 
PI x f x j, k_skin is yout fit parameter, and of course PI=3.14...

In essence, this corresponds to a resistive part that increases with 
sqrt(f), in series with an internal inductance L_int that decreases with 
sqrt(f) (so the inductive reactance 2*PI*j*f*L increases with sqrt(f)).

As said above, this formula breaks down at low frequencies, it would 
predict zero resistance (in even worse, infinite internal inductance) 
there. In his last book Howard Johnson gives a simple approximation 
formula to extend this formula towards DC (with R_DC being the DC 
resistance), although I have to admit I did not yet check how well this 
fares in terms of causality:

R_total = sqrt(R_DC^2 + R_AC^2)

Hope that helps!

Wolfgang






"Saoer Sinaga" <saoer.sinaga@xxxxxxxxx> 
Sent by: si-list-bounce@xxxxxxxxxxxxx
06/27/2008 06:04 AM

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Subject
[SI-LIST] conductor losses






Hi all,
Some approximate the resistance of a conductor with the following formula 
R(
f ) = R_DC + R_AC*sqrt( f ). But I read some where, R_AC*sqrt( f ) 
violates
the causality and passivity.

I also found a formula to approximate the conductor loss of a plane R( f ) 
=
R_DC + sqrt( j * 2 pi * f * mu / sigma).
Question: Can I also use the same formula for e.g. via ?

many thanks,
Saoer Sinaga
NXP


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