# Combined wire sizing and spacing

Jump to: Source code Text output Plots Library index

    % Section 5.5,  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 - 11/27/05
    % Modified by Michael Grant - 3/8/06
    %
    % The problem is to determine the optimal sizes of interconnect wires and
    % the optimal distances between them. We will consider an example with 3
    % wires, each consisting of 5 segments (see paper, fig.21). The variables
    % are the widths wij , and the distances s1 and s2 between the wires.
    % The difference with the models used in other scripts is that we include a
    % parasitic capacitance between the wires.
    % The objective is to minimize the total width s1+s2.
    % The problem can be formulated with the following SDP:
    %               mimimize    s1 + s2
    %                    s.t.   Tmax*G(w11,..,w35)-C(w11,..,w35,t11,..,t23) >=0
    %                           1/t1j <= s1 - w1j - 0.5*w2j     , j = 1,..,5
    %                           1/t2j <= s2 - w3j - 0.5*w2j     , j = 1,..,5
    %                           0 <= tij <= 1         , i = 1,2 , j = 1,..,5
    %                           t1 >=0, t2 >= 0, s1 >=smin, s2>=smin
    %                           0 <= wij <= wmax
    % the 2nd and 3rd constraints are nonlinear convex constraints that can be
    % cast as 3 x 3-LMIs (Please refer to the paper for more details).

    %
    % Circuit parameters
    %

    n = 6;           % number of nodes per wire
    N = 3*n;         % total number of nodes
    m = n-1;         % number of segments per wire
    alpha = 1;       % conductance per segment is is alpha*size
    beta = 0.5;      % capacitance per segment is twice beta*size
    gamma = 2;       % coupling capacitance is twice gamma*distance
    G0 = 100;        % source conductance
    C0 = [10,20,30]; % loads of first, second, third wires
    wmin = 0.1;      % minimum width
    wmax = 2.0;      % maximum width
    smin = 1.0;      % minimum distance between wires
    smax = 50;       % upper bound on s1 and s2  (meant to be inactive)

    %
    % Construct the capacitance and conductance matrices
    %   C(x) = C0 + w11 * C1 + w21 * C2 + ...
    %   G(x) = G0 + w11 * G1 + w21 * G2 + ...
    % and we assemble the coefficient matrices together as follows:
    %   CC = [ C0(:) C1(:) C2(:) ... ]
    %   GG = [ G0(:) G1(:) G2(:) ... ]
    %

    CC = zeros(N,N,5*m+1);
    GG = zeros(N,N,3*m+1);
    for w = 0 : 2,
        % Constant terms
        CC(w*n+n,w*n+n,1) = C0(w+1);
        GG(w*n+1,w*n+1,1) = G0;
        for i = 1 : m,
            % capacitances to ground
            CC(w*n+[i,i+1],w*n+[i,i+1],w*m+i+1) = beta*[1,0;0,1];
            if w < 2,
                % coupling capacitors
                CC(w*n+[i,  n+i  ],w*n+[i,  n+i  ],(w+3)*m+i+1) = gamma*[1,-1;-1,1];
                CC(w*n+[i+1,n+i+1],w*n+[i+1,n+i+1],(w+3)*m+i+1) = gamma*[1,-1;-1,1];
            end
            % segment conductances
            GG(w*n+[i,i+1],w*n+[i,i+1],w*m+i+1) = alpha*[1,-1;-1,1];
        end
    end
    % Reshape for Matlab use
    CC = reshape(CC,N*N,5*m+1);
    GG = reshape(GG,N*N,3*m+1);

    %
    % Compute points the tradeoff curve and the two desired points
    %

    npts    = 50;
    delays  = linspace( 85, 200, npts );
    xdelays = [ 130, 90 ];
    xnpts   = length(xdelays);
    areas   = zeros(1,npts);
    xareas  = zeros(1,xnpts);
    for j = 1 : npts + xnpts,

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

        %
        % Construct and solve the convex model
        %

        cvx_begin sdp quiet
            variables w(m,3) t(m,2) s(1,2)
            variable G(N,N) symmetric
            variable C(N,N) symmetric
            minimize( sum(s) )
            subject to
                G == reshape( GG * [ 1 ; w(:) ], N, N );
                C == reshape( CC * [ 1 ; w(:) ; t(:) ], N, N );
                delay * G - C >= 0;
                wmin <= w(:) <= wmax;
                t( : ) <= 1 / smin;
                s( : ) <= smax;
                inv_pos( t(:,1) ) <= s(1) - w(:,1) - 0.5 * w(:,2);
                inv_pos( t(:,2) ) <= s(2) - w(:,3) - 0.5 * w(:,2);
        cvx_end
        ss = cvx_optval;

        if j <= npts,
            areas(j) = ss;
        else,
            xareas(xj) = ss;

            %
            % Draw the wires
            %

            figure(4*xj-2);
            m2 = 2 * m;
            x1 = reshape( [ 1 : m ; 1 : m ], 1, m2 );
            x2 = x1( 1, end : -1 : 1 );
            y  = [ ss*ones(m2,1), s(2) + 0.5*w(x1,2), zeros(m2,1) ; ...
                   ss-w(x2,1),    s(2) - 0.5*w(x2,2), w(x2,3)     ; ...
                   ss,            s(2) + 0.5*w(1,2),  0           ];
            x1 = reshape( [ 0 : m - 1 ; 1 : m ], m2, 1 );
            x2 = x1( end : -1 : 1, 1 );
            x  = [ x1 ; x2 ; 0 ];
            hold off;
            fill( x, y, 0.9 * ones(size(y)) );
            hold on
            plot( x, y, '-' );
            axis( [-0.1, m+0.1,-0.1, ss+0.1]);
            colormap(gray);
            caxis([-1,1])
            title(sprintf('Solution (%d), Tmax = %g',xj,delay));

            %
            % Build the state space models and plot step responses
            %

            A = -inv(C)*G;
            T = linspace(0,2*delay,1000);
            B = -A * kron( eye(3), ones(n,1) );
            for inp = 1 : 3,
                figure(4*xj-2+inp);
                Y1 = simple_step(A,B(:,inp),T(2),length(T));
                hold off;
                plot(T,Y1([n,2*n,3*n],:),'-');
                hold on;
                text(T(1000),Y1(  n,1000),'v1');
                text(T(1000),Y1(2*n,1000),'v2');
                text(T(1000),Y1(3*n,1000),'v3');
                axis([0 2*delay -0.1 1.1]);
                % show dominant time constant
                plot(delay*[1;1], [-0.1;1.1], '--');
                title(sprintf('Solution (%d), Tmax = %g, step applied to wire %d',xj,delay,inp));
            end

        end

    end

    %
    % Plot the tradeoff curve
    %

    figure(1);
    ind = isfinite(areas);
    plot(areas(ind), delays(ind));
    xlabel('total width s_1 + s_2');
    ylabel('dominant time constant');
    title('Width-delay tradeoff curve')
    hold on;
    for k = 1 : xnpts,
        text( xareas(k), xdelays(k), sprintf( '(%d)', k ) );
    end

    Point 1 of 50 on the tradeoff curve (Tmax = 85)
    Point 2 of 50 on the tradeoff curve (Tmax = 87.3469)
    Point 3 of 50 on the tradeoff curve (Tmax = 89.6939)
    Point 4 of 50 on the tradeoff curve (Tmax = 92.0408)
    Point 5 of 50 on the tradeoff curve (Tmax = 94.3878)
    Point 6 of 50 on the tradeoff curve (Tmax = 96.7347)
    Point 7 of 50 on the tradeoff curve (Tmax = 99.0816)
    Point 8 of 50 on the tradeoff curve (Tmax = 101.429)
    Point 9 of 50 on the tradeoff curve (Tmax = 103.776)
    Point 10 of 50 on the tradeoff curve (Tmax = 106.122)
    Point 11 of 50 on the tradeoff curve (Tmax = 108.469)
    Point 12 of 50 on the tradeoff curve (Tmax = 110.816)
    Point 13 of 50 on the tradeoff curve (Tmax = 113.163)
    Point 14 of 50 on the tradeoff curve (Tmax = 115.51)
    Point 15 of 50 on the tradeoff curve (Tmax = 117.857)
    Point 16 of 50 on the tradeoff curve (Tmax = 120.204)
    Point 17 of 50 on the tradeoff curve (Tmax = 122.551)
    Point 18 of 50 on the tradeoff curve (Tmax = 124.898)
    Point 19 of 50 on the tradeoff curve (Tmax = 127.245)
    Point 20 of 50 on the tradeoff curve (Tmax = 129.592)
    Point 21 of 50 on the tradeoff curve (Tmax = 131.939)
    Point 22 of 50 on the tradeoff curve (Tmax = 134.286)
    Point 23 of 50 on the tradeoff curve (Tmax = 136.633)
    Point 24 of 50 on the tradeoff curve (Tmax = 138.98)
    Point 25 of 50 on the tradeoff curve (Tmax = 141.327)
    Point 26 of 50 on the tradeoff curve (Tmax = 143.673)
    Point 27 of 50 on the tradeoff curve (Tmax = 146.02)
    Point 28 of 50 on the tradeoff curve (Tmax = 148.367)
    Point 29 of 50 on the tradeoff curve (Tmax = 150.714)
    Point 30 of 50 on the tradeoff curve (Tmax = 153.061)
    Point 31 of 50 on the tradeoff curve (Tmax = 155.408)
    Point 32 of 50 on the tradeoff curve (Tmax = 157.755)
    Point 33 of 50 on the tradeoff curve (Tmax = 160.102)
    Point 34 of 50 on the tradeoff curve (Tmax = 162.449)
    Point 35 of 50 on the tradeoff curve (Tmax = 164.796)
    Point 36 of 50 on the tradeoff curve (Tmax = 167.143)
    Point 37 of 50 on the tradeoff curve (Tmax = 169.49)
    Point 38 of 50 on the tradeoff curve (Tmax = 171.837)
    Point 39 of 50 on the tradeoff curve (Tmax = 174.184)
    Point 40 of 50 on the tradeoff curve (Tmax = 176.531)
    Point 41 of 50 on the tradeoff curve (Tmax = 178.878)
    Point 42 of 50 on the tradeoff curve (Tmax = 181.224)
    Point 43 of 50 on the tradeoff curve (Tmax = 183.571)
    Point 44 of 50 on the tradeoff curve (Tmax = 185.918)
    Point 45 of 50 on the tradeoff curve (Tmax = 188.265)
    Point 46 of 50 on the tradeoff curve (Tmax = 190.612)
    Point 47 of 50 on the tradeoff curve (Tmax = 192.959)
    Point 48 of 50 on the tradeoff curve (Tmax = 195.306)
    Point 49 of 50 on the tradeoff curve (Tmax = 197.653)
    Point 50 of 50 on the tradeoff curve (Tmax = 200)
    Particular solution 1 of 2 (Tmax = 130)
    Particular solution 2 of 2 (Tmax = 90)