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Three-dimensional Integrated Circuit Design by Eby G. Friedman, Vasilis F. Pavlidis

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Appendix B. Formal Proof of Optimum Single Via Placement

In this appendix, the Lemma used to determine the optimum via location for an interplane interconnect that includes only one via for different values of r21, c12, and l1 is stated and proved.

Lemma 1: If f(x) = Ax2+Bx+C and

(a) for xmax

[x0, x1], x1 > x0 >0,

(i) if

, f(x0) < f(x1), (ii) if

, f(x1) < f(x0),

(b) for xmax < x0, f(x0) > f(x1),

(c) for xmax > x1, f(x0) < f(x1).

Proof: (a) (i) f(x) ...

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