FreeLists / si-list / [SI-LIST] Re: non-negative off diagonal capacitive matrix elements

[John Howard <jhoward@xxxxxxxxxxx>]

Good Explanation!  Thanks Eric.
John

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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