[SI-LIST] Re: what is the conductivity of a dielectric?
- From: Rob Hinz <rob@xxxxxxxxxx>
- To: Patrick_Carrier@xxxxxxxx
- Date: Tue, 09 Apr 2002 14:29:16 -0700
Patrick,
The definition of loss tangent, tan(delta) is:
tan(delta) = (we'' + cond)/(we')
Where:
w = 2*pi*freq
e' = eo*er (dielectric constant real part) This is the one we are used to
seeing...
e'' = imaginary (and therefore loss generating) part of the dielectric constant
cond = electrical conductivity of the material.
Thus, in general, the dielectric constant is expressed as a complex number as:
e = e'-je''
Now to your question, if you assume that the dielectric is otherwise
lossless, that is, e''=0, then conductivity is:
cond = tan(delta)*2*pi*freq*eo*er.
So I would agree with the equation you propose except that it is missing a
key term eo=8.854e-12. The should correct the scale problem you are noting...
cond = .02*2*pi*100e6*8.854e-12*4 = 4.5e-4 S/m
On background, the loss tangent equation is easily understood from first
principles. If you recall the relationship between Electric flux (D) and
Electric field (E) in free space:
D = eo*E;
the addition of a material to the space causes a polarization of the
molecules of that material resulting in additional electric flux that can
be represented as a polarization vector as:
D = eo*E + Pe (the same can be said of the magnetic field, for that, Pm
is used)
Pe is consequence of the applied E field and for linear materials,
(generally true for the material we use in SI work), Pe = eo*Xe*E. Xe is
the relative electric susceptibility of the material. In general, it may be
complex resulting in the following:
D = eo*E + Pe = eo*(1+Xe)*E = eo*er*E = e*E
e = eo*(1+Xe) = e'-je''
The complex part accounts for damping effects on the polarizing dipole
vibrations. Like a finite Q tank circuit or a spring and dash pot, this
loss is generally in the form of heat. You might ask why -je'' and not
+je''? This is because choosing +je'' would violate the conservation of
energy by allowing the dielectric to add energy to the system.
Finally the equation for loss tangent can be arrived at using Maxwell's
equations for time harmonic fields. I should point out that this is a
sticky issue for those of us doing SI analysis in the time domain and wish
to use the concept of loss tangent for that analysis. The assumption of
constant loss tangent, brings with it all sorts of complex and probably
non-causal time domain behavior. So BE CAREFUL!
curl(H) = jwD + J (J is electric current density, J = cond *E)
curl(H) = jweE + cond*E
curl(H) = jwe'E + (we'' + cond)*E
curl(H) = jw(e'-je''-j(cond/w))*E
As you can see here the e' term is the lossless part and j(e''+cond/w) is
the "lossy" part and if we think of the lossless part, e', as being on the
real axis and the "lossy" part (e'' + cond/w) as being on the imaginary
axis and we take the ratio of imaginary and real parts to get a "tangent"
that gives us a loss perfomance metric:
tan(delta) = (we''+cond)/(we')
for a SINGLE frequency!
I hope this helps your understanding.
Rob Hinz
Principal Engineer
SiQual Corporation
rob@xxxxxxxxxx
phone (503)885-1231
fax (503)885-0550
http://www.siqual.com
At 01:33 PM 4/9/2002 -0500, Patrick_Carrier@xxxxxxxx wrote:
>Transmission line gurus and people who love dielectrics--
>
>I am trying to figure out the conductivity of a dielectric.
>I have an equation that gives me:
>tanD = 1/(2*pi*Freq*Er*rd) where rd is the resistivity of the dielectric
>I assume that 1/rd is the conductivity of the dielectric. Is that an
>erroneous assumption?
>That gives me the equation:
>conductivity of dielectric = 2*pi*Freq*Er*tanD
>
>This second equation makes sense to me in that increasing your frequency
>increases the dielectric conductivity, causing more "leakage" of your
>transmitted energy. However, using this equation, that would indicate that
>the conductivity of a dielectric with Er=4 and tanD=0.02 would have a
>conductivity approaching that of copper at 100MHz. Now that does not make
>sense.
>
>Is there a such thing as non-frequency-dependent conductivity of a
>dielectric? How would I obtain such a number?
>Is there something else I am missing?
>
>Any guidance would be greatly appreciated. Thanks.
>--Pat
>
>
>
>
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Rob Hinz
Senior Electromagnetics Specialist
SiQual Corporation
rob@xxxxxxxxxx
phone (503)885-1231
fax (503)885-0550
http://www.siqual.com
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