# [SI-LIST] non-negative off diagonal capacitive matrix elements

• From: "Eric Bogatin" <eric@xxxxxxxxxxxx>
• To: "SI list" <si-list@xxxxxxxxxxxxx>
• Date: Sun, 2 Sep 2001 09:25:30 -0500
```Ray, and others-

I wanted to shed some insight on the negative off diagonal capacitance
matrix elements question that might quiet some of the confusion on the
SI list. This is a confusing issue in the industry because there are
really two different capacitance matrices, each with a different
definition. By force of habit, we often don't explicitly identify
which one we are referring to.

The short explanation is that in the SPICE capacitance matrix, all
elements are always positive. The diagonal elements are the
capacitance to ground and the off diagonals represent the coupling
between conductors.

The other matrix is the Maxwell capacitance matrix. The definition of
the Maxwell C matrix elements is different from the SPICE C matrix. In
the Maxwell C matrix, all off diagonal elements are always negative
and the diagonal elements represent the "loaded capacitance" or "total
capacitance". The off diagonal elements of each matrix are numerically
equal. If you take one row of the Maxwell C matrix and add up all the
elements, it will be equal to the diagonal element in the
corresponding row of the SPICE C matrix.

The quick way to tell if you've got a C matrix from a field solver
result is to look at the off diagonal elements. If they are negative,
it came from a field solver. I have tried in vane, to get field solver
companies to label their capacitance matrices as Maxwell Capacitance
matrices, to help avoid the confusion and emphasize the fact that
there really are two different matrices, each with a slightly
different definition. So far, only Ansoft has done this.

The reason you sometimes see positive values of the off diagonal
elements in the Maxwell C matrix is numerical error- especially far
off the diagonal where you are looking at incredibly weak couplings.

There are a few application notes on the www.gigatest.com  web site if
you want to see the details on paper. This topic is covered in our
class, GTL262, creating interconnect models from calculations.

That's the short answer. If anyone wants the longer answer of why the
off diagonal elements of the Maxwell C matrix are negative, and why
not interested, see ya. --eric

In the Maxwell Capacitance matrix, the capacitor elements are defined
based on the definition that is used to extract the matrix elements
from a collection of conductors using a static 2D or 3D field solver.

The definition of the Maxwell capacitor matrix elements is: Ckm =
Qk/Vm, in the following situation:

1. take the collection of all conductors and any associated
dielectrics.

2. connect each and every one of then to ground with a conducting wire

3. take one conductor, the m'th one, and disconnect it from ground,
and place a 1 volt potential on it, wrt ground.

4. This m'th conductor has a voltage on it, wrt all the other
conductors, since they are all at ground, and tied there. There will
be field lines between this conductor and every other conductor,
especially the place that is defined as ground (this could be a
reference conductor, like a plane, or the boundary of space- i.e.,
infinity).

5. By making conductor m have 1 volt, we dumped some positive charge
on it. Imagine we walk along the surface of conductor m and measure
how much charge we had to dump on it to get the 1 volt potential
difference, given the proximity of all the other grounded conductors.
We count the total charge on the m'th conductor, Qm. The diagonal
element of the Maxwell capacitance element, Cmm, is Qm/Vm. Since Vm is
1 volt, Cmm is numerically equal to Qm.

6. Keeping conductor m with 1 volt and everyone else connected to
ground, look at a nearby conductor, k. Since we dumped some positive
charge on conductor m to get it to 1 volt, it will attract some
negative charge on conductor k. The excess charge on conductor k will
be negative, since it was induced to flow onto k from ground by the
proximity of the positive charges on conductor m. The charge on every
other conductor will thus be negative.

7. In the definition of the Maxwell Capacitance matrix elements, the
capacitor matrix element, Ckm, will always be negative because the
induced charge on every other conductor will be negative and Ckm =
Qk/Vm.

8. Of course, the process of "walking over the surface of conductor k
and counting the total charges there" is precisely what the field
solver engine does. It sets the boundary conditions based on the
distribution of conductors and dielectrics and solves for all the
electric fields using LaPlace's equation. Then it uses Gauss' law to
calculate the total charge on each conductor. The direct output of
every field solver is this special capacitance matrix. It is not the
same as the SPICE capacitance matrix.

In the Maxwell matrix, the diagonal elements are the capacitance of
each conductor to ground, when every other conductor is grounded. This
is often called the "loaded capacitance" or the total capacitance. In
the SPICE capacitance matrix, each diagonal element is the capacitance
between each conductor to ground, with every other capacitor "guarded"
to the same potential as the isolated conductor. In this way, the only
current that will flow to ground is through the singled out capacitor
element.

In the SPICE matrix, the off diagonal elements are always positive-
after all, the capacitance of an ideal capacitor is always positive.
It is related to the current that would flow between two conductors
when all other conductors (and ground) are guarded to the potential of
one of the conductors. Since the Maxwell and the SPICE off diagonal
capacitors both represent the amount of coupling between the
conductors, their magnitudes are the same, it's just their signs that
are different.

To avoid confusion, we should all get in the habit of referring
explicitly to the Maxwell C matrix and the SPICE C matrix.

--eric

**************************************
Eric Bogatin
CTO, Giga Test Labs
v: 913-393-1305
f: 913-393-1306
e: eric@xxxxxxxxxxxx
26235 W. 110th Terr.  Olathe, KS  66061
corporate office:
408-524-2700
134 S. Wolfe Rd Sunnyvale, CA 94086
web: www.gigatest.com
**************************************

From: si-list-bounce@xxxxxxxxxxxxx
[mailto:si-list-bounce@xxxxxxxxxxxxx]On Behalf Of Ray Anderson
Sent: Friday, August 31, 2001 10:08 AM
To: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] non-negative off diagonal capacitive matrix
elements
??

I've just extracted the RLGC matrices for 5 coupled
striplines using Apsimtech RLGC.

As expected, the diagonal elements of the capacitive matrix are
positive capacitance values. All the off-diagonal elements are
negative EXCEPT one. Is this correct. It has been a while since I've
off diag element were all negative. Anyone know for sure ??

The element in column 1 row 5 is the one I question:

3.1600e-15
-4.9360e-16   3.2540e-15
-2.0080e-17  -4.8990e-16   3.2540e-15
-1.3380e-18  -1.9820e-17  -4.8990e-16   3.2540e-15
3.1280e-20  -1.3380e-18  -2.0080e-17  -4.9360e-16   3.1600e-15

-Ray

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