[SI-LIST] Re: what is the conductivity of a dielectric?

  • From: Rob Hinz <rob@xxxxxxxxxx>
  • To: RayCaliendo@xxxxxxxxxx
  • Date: Fri, 12 Apr 2002 00:15:45 -0700

Hi Roy,

Actually that is incorrect. Conductivity, as related to conduction current, 
is a useful and appropriate concept for any real material, even 
dielectrics. Now conductivity may be practically zero, for good PCB 
substrate material. Because of this, the conduction current in a good 
dielectric is also very small, but the equations remain valid.

Also, displacement current has nothing to do with moving charge in a 
dielectric. It has to do with time varying electric fields producing 
magnetic fields and conservation of charge. It is a little hard to explain 
so I will state what it is and then walk you through Maxwell's postulation 
of the effect.

 From Maxwell's equations:

curl(H) = J + d(D)/dt

The displacement current is the d(D)/dt part. Thus:

curl(H) = J + Jd

Here is how we/Maxwell arrive at that. For static fields, the field 
equations are:

1) div(D) = rho (charge density)
2) div(B) = 0 (no magnetic charges)
3) curl(E) = 0
4) curl(H) = J (electric current density)

Now conservation of charge demands that:

5) div(J) = -d(rho)/dt

What this equation is saying is that the current flowing out of a small 
volume is equal to the negative of the time rate of change of the charge 
within the volume. If we apply the divergence operation to equation 4 about 
we get:

6) div(J) = div(curl(H)) = 0            (The divergence of the curl of any 
vector is zero)

This result is clearly at odds with eqn. 5, and violates the conservation 
of charge. Maxwell recognized that eqn. 4 was not complete and proposed an 
extension to the static field equations as follows:

7) curl(H) = J + d(D)/dt

Now when the divergence operation is applied to eqn. 7, we get:

8a) div(J) + div(d(D)/dt) = 0
8b) div(J) = -div(d(D)/dt) = -d(div(D))/dt = -d(rho)/dt

And charge is conserved, as 8b is clearly the same as eqn. 5.

It is the additional component that Maxwell added to the statics version of 
eqn. 7 that is called displacement current. It is so named because is 
arises from the displacement vector "D." The added term contributes to the 
curl of the magnetic field in the same way as an actual conduction current 
density "J" does. But, displacement current can be non-zero even in a 
vacuum, where there is no charge at all.

Regarding the units, as another writer indicated, the loss tangent is 
unit-less. There are several different forms for the loss tangent depending 
on what approximations you want to make and how you want to express things. 
The formulation I used is very general. If you look at the numerator of the 
loss tangent as I stated it:

tan(delta) = (we''+sigma)/(we')   [or rearranged:  tan(delta) = (e'' + 
sigma/w)/e']

we'' is really indistinguishable from sigma mathematically. They add 
directly. Physically, however, the we'' term arises from the work done to 
move bound charges. This is how water heats in a microwave oven. These 
charges move a very small distance within the material creating a dipole 
moment in the material (Pe) that reduces the field strength within the 
material (think about Q=CV, if capacitance goes up with increasing er, 
voltage comes down, for fixed charge). If in that process, any work is 
done, heat is generated and the we'' term becomes non-zero. Some references 
do not make this distinction and lump the we'' term with sigma. There is no 
real problem with this. It gives the same answer, but it can lead to 
confusion because the loss tangent is defined a little differently in this 
case, as follows:

e = e' -j(sigma)/(w) = e' - je''

and

tan(delta) = e''/e'

With regard to what Howard Johnson proposes for loss tangent, I suspect he 
is trying to make things a little easier than I have. Capacitance, 
capacitive reactance, and resistance are pretty ill-defined (or undefined) 
terms relating to a general dielectric material. I would not attempt to 
make a circuit based analogy in this way. Circuit theory is merely a 
simplification of Maxwell's equations for low frequencies. I would have to 
see Howard's treatment of loss tangent using R's and Xc's to be certain but 
I have never seen it defined other than I have described, and I have 
checked it across multiple EM references. That doesn't mean it hasn't been 
done, for good or ill.

Hopefully this is at least sort of clear...!

Regards,

Rob Hinz
Principal Engineer
SiQual Corporation
rob@xxxxxxxxxx
phone (503)885-1231
fax   (503)885-0550
http://www.siqual.com
At 04:24 PM 4/10/2002 -0700, RayCaliendo@xxxxxxxxxx wrote:


>         Rob et. al.,
>
>         I believe the word 'conductivity' (sigma) should be used  for a
>conductor, while the movement of charge in a dielectric is the 'Displacement
>current' (D = eE), which, if I understand it correctly, behaves "like" a
>conduction current.  Also, It looks to me that the units of some of the
>equations' here don't seem to balance. What have I missed?  I found some
>other explanations for loss tangent :
>                 - Tan(delta) = er'' / er'
>                 - Howard Johnson article "Dielectric Loss Tangents"
>                         Theta = Im(Capacitance) / Re (Capacitance)
>                 - Tan(delta) = Resistance / Reactance (parallel equivalent
>circuit)
>
>         Regards,
>
>         Ray Caliendo
>         Solectron Corp
>         (408)956-6294
>
> > ----------
> > From:         Rob Hinz[SMTP:rob@xxxxxxxxxx]
> > Reply To:     rob@xxxxxxxxxx
> > Sent:         Tuesday, April 09, 2002 2:29 PM
> > To:   Patrick_Carrier@xxxxxxxx
> > Cc:   si-list@xxxxxxxxxxxxx
> > Subject:      [SI-LIST] Re: what is the conductivity of a dielectric?
> >
> >
> >
> > 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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Rob Hinz
Principal Engineer
SiQual Corporation
rob@xxxxxxxxxx
phone (503)885-1231
fax   (503)885-0550
http://www.siqual.com

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