**From:** C Deibele (*deibele@fnal.gov*)

**Date:** Fri Sep 01 2000 - 05:34:54 PDT

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Hi Everyone,

Yesterday we discussed so many different parts of skin effect and copper

losses. After reading so many good points brought up by everyone,

namely the

effect of the perimeter of the trace, as well as the characteristic

impedance of

the trace, I had really wanted to study the effects of increasing the

trace

width and the copper losses associated with increasing the trace width.

As a background, remember that

I(z)=Const/Zo exp^{-i Beta z}

where Beta is the wavenumber...

Const is a constant

z is a position on the z axis

and Zo is the characteristic impedance.

so dI/I = -dZo/Zo (for a given point in z)

==>Increasing Zo decreases the current flowing on the conductor.

==>decreasing the current on the conductor decreases the copper losses

on the

conductor.

==>Increasing Zo means the trace width gets *smaller*

So, the question I ask is: Which effect wins?

There is no simple answer to this question because the characteristic

impedance

is not a simple linear function of geometry. Take for instance coax,

which is

one of the easiest geometries/topologies to study:

Zo=60 ln(b/a)

If we restrict our attention to geometries where the frequency is high

enough

such that the skin depth, delta, is much greater than a

i.e. delta >>a, then quite simply,

dZo/Zo=-da(r)/[a(r) ln (b/a(r))]

so, inserting this relationship into the equation of I

dI/I=-dZo/Zo=da(r)/[a(r) ln (b/a(r))]

the denominator clearly shows that there is not a nice linear

relationship. Now stripline is also nonlinear....So, this says that

there is probably NEVER a nice and easy rule to decide whether making a

trace wider, or thicker will reduce copper losses.

so, the change in the characteristic impedance is not something simple

to

describe -- namely there are regimes where a simple change in radius may

be

easily described by the inverse nature, other regimes where the

logarithmic

properties are dominated...and a regime where the mix is the important

property.

So, I attempted to study one simple property. I used Ensemble v. 7 and

simulated a simple stripline geometry. copper upper conductor, copper

lower

conductor, copper trace, and perfect vacuum in the middle (We were

restricting

our discussions to copper loss...*NOT* dielectric loss)

So, the results were fairly convincing, for a stackup as follows

copper ground

50 mil vacuum

copper trace

50 mil vacuum

copper ground

All the traces were simulated were 1800 mils long

The reflection data was nearly identical (on the order of 0.005%) ...so

I won't

post it....

trace at 140 mils http://www.geocities.com/simulations00/140.jpg

trace at 130 mils http://www.geocities.com/simulations00/130.jpg

and examine the following:

trace at 14 mils http://www.geocities.com/simulations00/14.jpg

trace at 13 mils http://www.geocities.com/simulations00/13.jpg

These two sets of results clearly show that there is a discrepancy based

on the distributed nature of the transmission line circuit.

So, the first set shows that a wider trace has less loss.

the second set shows that the skinnier trace has less loss.

I do not want to get into any argument about semantics about the English

of the problem. This is to say, skin effect, or skin depth.

Craig Deibele

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