The capacitance of integrated-circuit elements
4
ation ---- NOTES The Capacitance of Integrated-Circuit Elements he purpose of this note is to demon- strate that it is possible to modify a technique that was originally devel- oped for calculations involving the scat- tering of electromagnetic waves from diverse-shaped objects and use it in calcu- lations involving integrated-circuit ele- ments. The technique is the “method of moments” [l], which was previously kept in the domain of researchers and graduate students in electromagnetic theory. It is now entering into undergraduate educa- tion (e.g, [Z-51). We develop the technique using no more knowledge than that gained in an introductory courses in electrostatics and in matrix manipulation. PROCEDURE With reference to Fig. 1, we subdivide the area a x a of the circuit element into N x N = 2 x 2 subareas, each of which has an area AA. The charge in a particular subarea AQ is assumed to be concentrated at its cen- ter, and it has a value (1) where ps is the charge density that exists on the sheet. The potential at the center of another subarea is found from (2) In Fig. 1, there will be N x N = 4 terms that will contribute to this potential I/. 24 %- ~ Karl E. Lonngren and Er- Wei Bai Three of the terms will have rj # ri and will create no difficulty in the calculation. However, the remaining term will have a singularity. To remove this singularity, we use a standard technique that is de- scribed in several textbooks [2-51. The technique can be briefly described as fol- lows: we approximate the square subarea with a circular disk that has the same area bl as the square area with a uniform change distribution. The potential at the center of the disk, owing to its own uniform change distribu- tion, is given by (3) Hence, the potentials at all N x N = 4 sub- areas in Fig. 1 can be calculated. The calculation outlined above is straightforward, but it has made the tacit assumption that the charge density ps is known a priori. However, this is the un- known. In the plane, it has a specified value, say Vi. Herein, we develop such a program. The major steps are: + Divide a square area into N x N sub- areas and identify each subarea. + Calculate the matrix elements that relate the potential in one subarea to a charge in a different subarea. + Set the voltage V(i, j) = 1 if the sub- area contains a conducting surface or set V(i, j) = 0 if the subarea does not contain a conducting surface. This is the step that defines the par- ticular circuit element of interest. It is the only step that requires modifi- cation for different elements. 4 Calculate and plot the charge distri- bution. Since the voltage of the element is set equal to 1, the calculated charge distribu- tion is also proportional to the self-capacitance of the element. The MATLAB “dot-m” file that effects this cal- culation is available from the authors. EXA~P&ES In Fig. 2, the computed charge distribu- tion for a square [Fig. 2(a)] and a circular [Fig. Z(b)] element are shown. The poten- tial of either element was constant as shown in the top figures. We chose N = 24 and absorbed the factor ~/~KEO into the charge distribution. The charge density in the middle of the metal surface as shown in the bottom figures is almost constant. At the edges of the metal surface, the X 1. Aplanedividedinto NxN=ZxZsubareas. CIRCUITS 8 DEVICES MAY 1999
Transcript of The capacitance of integrated-circuit elements
ation ----
NOTES The Capacitance of Integrated-Circuit Elements
he purpose of this note is to demon- strate that it is possible to modify a technique that was originally devel-
oped for calculations involving the scat- tering of electromagnetic waves from diverse-shaped objects and use it in calcu- lations involving integrated-circuit ele- ments. The technique is the “method of moments” [l], which was previously kept in the domain of researchers and graduate students in electromagnetic theory. It is now entering into undergraduate educa- tion (e.g, [Z-51). We develop the technique using no more knowledge than that gained in an introductory courses in electrostatics and in matrix manipulation.
PROCEDURE With reference to Fig. 1, we subdivide the area a x a of the circuit element into N x N = 2 x 2 subareas, each of which has an area AA. The charge in a particular subarea AQ is assumed to be concentrated at its cen- ter, and it has a value
(1) where ps is the charge density that exists on the sheet. The potential at the center of another subarea is found from
(2) In Fig. 1, there will be N x N = 4 terms
that will contribute to this potential I/.
24 %- ~
Karl E. Lonngren and Er- Wei Bai
Three of the terms will have rj # ri and will create no difficulty in the calculation.
However, the remaining term will have a singularity. To remove this singularity, we use a standard technique that is de- scribed in several textbooks [2-51. The technique can be briefly described as fol- lows: we approximate the square subarea with a circular disk that has the same area b l as the square area with a uniform change distribution.
The potential at the center of the disk, owing to its own uniform change distribu- tion, is given by
(3) Hence, the potentials at all N x N = 4 sub- areas in Fig. 1 can be calculated.
The calculation outlined above is straightforward, but it has made the tacit assumption that the charge density ps is known a priori. However, this is the un- known. In the plane, it has a specified value, say Vi. Herein, we develop such a program. The major steps are:
+ Divide a square area into N x N sub- areas and identify each subarea.
+ Calculate the matrix elements that relate the potential in one subarea to a charge in a different subarea.
+ Set the voltage V(i, j ) = 1 if the sub- area contains a conducting surface
or set V(i, j ) = 0 if the subarea does not contain a conducting surface. This is the step that defines the par- ticular circuit element of interest. It is the only step that requires modifi- cation for different elements.
4 Calculate and plot the charge distri- bution.
Since the voltage of the element is set equal to 1, the calculated charge distribu- t i o n is a lso propor t iona l t o t h e self-capacitance of the element. The MATLAB “dot-m” file that effects this cal- culation is available from the authors.
E X A ~ P & E S In Fig. 2, the computed charge distribu- tion for a square [Fig. 2(a)] and a circular [Fig. Z(b)] element are shown. The poten- tial of either element was constant as shown in the top figures. We chose N = 24 and absorbed the factor ~ / ~ K E O into the charge distribution. The charge density in the middle of the metal surface as shown in the bottom figures is almost constant. At the edges of the metal surface, the
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charge density drastically changes due to the loss of symmetry at those locations.
In Figure 3, the computed charge dis- tribution for an inhomogeneous ele- ment [Fig. 3(a)] and the intersection of two elements [Fig. 3(b)] are shown. The potential of either element was constant as shown in the top figures. We absorbed the factor 1/4n&o into the charge distri- bution. The charge density in the middle of the metal surface as shown in the bot- tom figures is almost constant. At the edges of the metal surface, the charge density drastically changes due to the loss of symmetry at those locations. In order to increase the clarity of the last
picture, the value of N = 48 was chosen for the calculation.
C@)WlSlONS The “method ofkoments, ”which was origi- nally developed and well used in electromag- netic theory, can be applied in integrated-circuit technology. As transit times from one element to another decrease, a lumped circuit model is no longer valid and one may have to calculate the capacitance for a given potential. The method outlined here is useful to this end.
Karl E. Lonngren and Er- Wei Bai (e-mail: [email protected]) are with the Depart-
ment of Electrical and Computer Engi- neering at the University of Iowa in Iowa City, Iowa, USA. 52240
FERENGES [ I ] R.F. Harrington, Field Computation by Mo-
ment Methods. New York MacMillan, 1968. 121 C.R. Paul and S.A. Nasar, Introduction to Elec-
tromagnetic Fields, New York McGraw Hill, 1982, pp. 473-480.
[3] N.N. Rao, Elements of Engineering Electromagnetics. Englewood Cliffs, NJ: Prentice Hall, 1991, pp. 243-248,365-367.
[4] M.F. Iskander, Electromagnetic Fields & Waves. Englewood Cliffs, NJ: Prentice-Hall, 1992, pp. 313-322.
[5] K.E. Lonngren, Electromagnetics with MATLAB. Cambridge, U K Cambridge Interna- tional Science Press, 1997, pp. 211-226. CDW
fa meteorite drifts slowly int I sphere, at an altitude of ab0 avitational field, how fastwill it be movingwhen it reaches the earth’s atmo-
ANSWER TO LAST ISSUE’S BRAINBUSTER e, it must win three of the next six games to win the series. The number ofways to do that is the
number of combinations of six things taken at a time, or (3 =20.
The opponents, team B, mustwin four of the next six games in order towin the series. The number ofways to do that is (3 = 15.
Of the 35 outcomes, 20 produce a win for team A, so its probability of winning the series is
If team A wins the second game, its probability of winning the series is
And if it wins the third game,
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