[SI-LIST] Re: non-negative off diagonal capacitive matrix elements
- From: Steve Corey <steve@xxxxxxxxxxxxxx>
- To: SI list <si-list@xxxxxxxxxxxxx>
- Date: Sun, 02 Sep 2001 19:59:14 -0700
Hi Eric -- my two cents on the same subject --
Interestingly, when SPICE reads in a netlist of discrete capacitors and
loads the capacitance information into its internal matrix for solution,
the internal matrix it forms for computation is exactly what you have
described as the "Maxwell capacitance matrix" (except, or course, values
are not per-unit-length). That matrix is usually the most concise
representation of the capacitance information for computation and
mathematical manipulation, so it's no wonder the EM types have chosen to
use it rather than use the arguably more visually intuitive "SPICE
capacitance matrix".
The SPICE capacitance matrix is more visually intuitive to many circuit
designers since it allows them to easily interpret the capacitance matrix
as a group of discrete capacitors, whose values are the matrix entries,
connected between circuit nodes. However, it doesn't lend itself well to
the algebraic formulations which are quite common in EM and transmission
line theory, and may not be as intuitive or general to hardcore EM types
who would rather think in terms of fields than discrete circuit elements.
-- Steve
-------------------------------------------
Steven D. Corey, Ph.D.
Time Domain Analysis Systems, Inc.
"The Interconnect Modeling Company."
http://www.tdasystems.com
email: steve@xxxxxxxxxxxxxx
phone: (503) 246-2272
fax: (503) 246-2282
-------------------------------------------
Eric Bogatin wrote:
> 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
> the diagonal elements are the loaded capacitance, read on. If you're
> 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.
>
> Check out our web site for more info. www.gigatest.com
>
> --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
> thought about this, but I was under the impression the capacitive
> 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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