# Wire sizing and topology design

Jump to: Source code Text output Plots Library index

    % Section 5.3,  L. Vandenberghe, S. Boyd, and A. El Gamal
    % "Optimizing dominant time constant in RC circuits"
    % Original by Lieven Vandenberghe
    % Adapted for CVX by Joelle Skaf - 11/25/05
    % Modified by Michael Grant - 3/8/06
    %
    % We size the wires for an interconnect circuit with four nodes. The
    % topology of the circuit is more complex; the wires don't even form a tree
    % (refer to Figure 13 in the paper).
    % The problem can be formulated with the following SDP:
    %               minimize        sum(xi*li)
    %                   s.t.        0 <= xi <= wmax
    %                               Tmax*G(x) - C(x) >= 0
    % Please refer to the paper (section 2) to find what G(x) and C(x) are.

    %
    % Circuit parameters
    %

    n      = 4;    % number of nodes
    m      = 6;    % number of branches
    G      = 0.1;  % resistor between node 1 and 0
    Co     = 10;   % load capacitance
    wmax   = 10.0; % maximum width
    % alpha: conductance per segment
    % 2 * beta: capacitance per segment
    alpha  = [ 1.0, 1.0, 1.0, 1.0, 1.0, 1.0 ];
    beta   = [ 10,  10,  100, 1,   1,   1   ];

    %
    % Build capacitance and conductance matrices
    %

    CC = zeros(n,n,m+1);
    GG = zeros(n,n,m+1);
    % constant terms
    CC(3,3,1) = Co;
    GG(1,1,1) = G;
    % branch 1
    CC(1,1,2) = + beta(1);
    CC(2,2,2) = + beta(1);
    GG(1,1,2) = + alpha(1);
    GG(2,1,2) = - alpha(1);
    GG(1,2,2) = - alpha(1);
    GG(2,2,2) = + alpha(1);
    % branch 2
    CC(2,2,3) = + beta(2);
    CC(3,3,3) = + beta(2);
    GG(2,2,3) = + alpha(2);
    GG(3,2,3) = - alpha(2);
    GG(2,3,3) = - alpha(2);
    GG(3,3,3) = + alpha(2);
    % branch 3
    CC(1,1,4) = + beta(3);
    CC(3,3,4) = + beta(3);
    GG(1,1,4) = + alpha(3);
    GG(3,1,4) = - alpha(3);
    GG(1,3,4) = - alpha(3);
    GG(3,3,4) = + alpha(3);
    % branch 4
    CC(1,1,5) = + beta(4);
    CC(4,4,5) = + beta(4);
    GG(1,1,5) = + alpha(4);
    GG(4,1,5) = - alpha(4);
    GG(1,4,5) = - alpha(4);
    GG(4,4,5) = + alpha(4);
    % branch 5
    CC(2,2,6) = + beta(5);
    CC(4,4,6) = + beta(5);
    GG(2,2,6) = + alpha(5);
    GG(2,4,6) = - alpha(5);
    GG(4,2,6) = - alpha(5);
    GG(4,4,6) = + alpha(5);
    % branch 6
    CC(3,3,7) = + beta(6);
    CC(4,4,7) = + beta(6);
    GG(3,3,7) = + alpha(6);
    GG(4,3,7) = - alpha(6);
    GG(3,4,7) = - alpha(6);
    GG(4,4,7) = + alpha(6);

    % Reshape for easy Matlab use
    CC = reshape(CC,n*n,m+1);
    GG = reshape(GG,n*n,m+1);

    %
    % Compute points the tradeoff curve, and the three sample points
    %

    npts    = 50;
    delays  = linspace( 180, 800, npts );
    xdelays = [ 200, 400, 600 ];
    xnpts   = length(xdelays);
    areas   = zeros(1,npts);
    sizes   = zeros(6,xnpts);
    for i = 1 : npts  + xnpts,

        if i > npts,
            xi = i - npts;
            delay = xdelays(xi);
            disp( sprintf( 'Particular solution %d of %d (Tmax = %g)', xi, xnpts, delay ) );
        else,
            delay = delays(i);
            disp( sprintf( 'Point %d of %d on the tradeoff curve (Tmax = %g)', i, npts, delay ) );
        end

        %
        % Construct and solve the convex model
        %

        cvx_begin sdp quiet
            variable x(6)
            variable G(n,n) symmetric
            variable C(n,n) symmetric
            minimize( sum( x ) )
            subject to
                G == reshape( GG * [ 1 ; x ], n, n );
                C == reshape( CC * [ 1 ; x ], n, n );
                delay * G - C >= 0;
                0 <= x <= wmax;
        cvx_end

        if i <= npts,
            areas(i) = cvx_optval;
        else,
            xareas(xi) = cvx_optval;
            sizes(:,xi) = x;

            %
            % Plot the step response
            %

            figure(xi+1);
            A = -inv(C)*G;
            B = -A*ones(n,1);
            T = linspace(0,1000,1000);
            Y = simple_step(A,B,T(2),length(T));
            hold off; plot(T,Y([1,3,4],:),'-');  hold on;

            % compute threshold delay, elmore delay, dominant time constant
            tthres=T(min(find(Y(3,:)>0.5)));
            tdom=max(eig(inv(G)*C));
            telm=max(sum((inv(G)*C)'));
            plot(tdom*[1;1], [0;1], '--', telm*[1;1], [0;1],'--', ...
                 tthres*[1;1], [0;1], '--');
            text(tdom,0,'d');
            text(telm,0,'e');
            text(tthres,0,'t');
            title(sprintf('Step response for solution (%d), Tmax=%g', xi, delay ));

        end

    end

    %
    % Plot the tradeoff curve
    %

    figure(1)
    ind = isfinite(areas);
    plot(areas(ind), delays(ind));
    xlabel('Area');
    ylabel('Tdom');
    title('Area-delay tradeoff curve');
    hold on
    for k = 1 : xnpts,
        text( xareas(k), xdelays(k), sprintf( '(%d)', k ) );
    end

    %
    % Display sizes for the three solutions
    %

    disp(['Three specific solutions:']);
    sizes

    Point 1 of 50 on the tradeoff curve (Tmax = 180)
    Point 2 of 50 on the tradeoff curve (Tmax = 192.653)
    Point 3 of 50 on the tradeoff curve (Tmax = 205.306)
    Point 4 of 50 on the tradeoff curve (Tmax = 217.959)
    Point 5 of 50 on the tradeoff curve (Tmax = 230.612)
    Point 6 of 50 on the tradeoff curve (Tmax = 243.265)
    Point 7 of 50 on the tradeoff curve (Tmax = 255.918)
    Point 8 of 50 on the tradeoff curve (Tmax = 268.571)
    Point 9 of 50 on the tradeoff curve (Tmax = 281.224)
    Point 10 of 50 on the tradeoff curve (Tmax = 293.878)
    Point 11 of 50 on the tradeoff curve (Tmax = 306.531)
    Point 12 of 50 on the tradeoff curve (Tmax = 319.184)
    Point 13 of 50 on the tradeoff curve (Tmax = 331.837)
    Point 14 of 50 on the tradeoff curve (Tmax = 344.49)
    Point 15 of 50 on the tradeoff curve (Tmax = 357.143)
    Point 16 of 50 on the tradeoff curve (Tmax = 369.796)
    Point 17 of 50 on the tradeoff curve (Tmax = 382.449)
    Point 18 of 50 on the tradeoff curve (Tmax = 395.102)
    Point 19 of 50 on the tradeoff curve (Tmax = 407.755)
    Point 20 of 50 on the tradeoff curve (Tmax = 420.408)
    Point 21 of 50 on the tradeoff curve (Tmax = 433.061)
    Point 22 of 50 on the tradeoff curve (Tmax = 445.714)
    Point 23 of 50 on the tradeoff curve (Tmax = 458.367)
    Point 24 of 50 on the tradeoff curve (Tmax = 471.02)
    Point 25 of 50 on the tradeoff curve (Tmax = 483.673)
    Point 26 of 50 on the tradeoff curve (Tmax = 496.327)
    Point 27 of 50 on the tradeoff curve (Tmax = 508.98)
    Point 28 of 50 on the tradeoff curve (Tmax = 521.633)
    Point 29 of 50 on the tradeoff curve (Tmax = 534.286)
    Point 30 of 50 on the tradeoff curve (Tmax = 546.939)
    Point 31 of 50 on the tradeoff curve (Tmax = 559.592)
    Point 32 of 50 on the tradeoff curve (Tmax = 572.245)
    Point 33 of 50 on the tradeoff curve (Tmax = 584.898)
    Point 34 of 50 on the tradeoff curve (Tmax = 597.551)
    Point 35 of 50 on the tradeoff curve (Tmax = 610.204)
    Point 36 of 50 on the tradeoff curve (Tmax = 622.857)
    Point 37 of 50 on the tradeoff curve (Tmax = 635.51)
    Point 38 of 50 on the tradeoff curve (Tmax = 648.163)
    Point 39 of 50 on the tradeoff curve (Tmax = 660.816)
    Point 40 of 50 on the tradeoff curve (Tmax = 673.469)
    Point 41 of 50 on the tradeoff curve (Tmax = 686.122)
    Point 42 of 50 on the tradeoff curve (Tmax = 698.776)
    Point 43 of 50 on the tradeoff curve (Tmax = 711.429)
    Point 44 of 50 on the tradeoff curve (Tmax = 724.082)
    Point 45 of 50 on the tradeoff curve (Tmax = 736.735)
    Point 46 of 50 on the tradeoff curve (Tmax = 749.388)
    Point 47 of 50 on the tradeoff curve (Tmax = 762.041)
    Point 48 of 50 on the tradeoff curve (Tmax = 774.694)
    Point 49 of 50 on the tradeoff curve (Tmax = 787.347)
    Point 50 of 50 on the tradeoff curve (Tmax = 800)
    Particular solution 1 of 3 (Tmax = 200)
    Particular solution 2 of 3 (Tmax = 400)
    Particular solution 3 of 3 (Tmax = 600)
    Three specific solutions:

    sizes =

        0.0000    0.0000    0.0000
        0.0000    0.0000    0.0000
       -0.0000    0.0381    0.0273
        0.2303    0.0369    0.0000
        0.0000   -0.0000    0.0000
        0.2156    0.0361    0.0000