# Sizing of clock meshes

Jump to: Source code Text output Plots Library index

    % Section 4, L. Vandenberghe, S. Boyd, and A. El Gamal
    % "Optimal Wire and Transistor Sizing for Circuits with Non-Tree Topology"
    % Original by Lieven Vanderberghe
    % Adapted to CVX by Argyris Zymnis - 12/04/05
    % Modified by Michael Grant - 3/8/06
    %
    % We consider the problem of sizing a clock mesh, so as to minimize the
    % total dissipated power under a constraint on the dominant time constant.
    % The numbers of nodes in the mesh is N per row or column (thus n=(N+1)^2
    % in total). We divide the wire into m segments of width xi, i = 1,...,m
    % which is constrained as 0 <= xi <= Wmax. We use a pi-model of each wire
    % segment, with capacitance beta_i*xi and conductance alpha_i*xi.
    % Defining C(x) = C0+x1*C1+x2*C2+...+xm*Cm we have that the dissipated
    % power is equal to ones(1,n)*C(x)*ones(n,1). Thus to minimize the
    % dissipated power subject to a constraint in the widths and a constraint
    % in the dominant time constant, we solve the SDP
    %               minimize        ones(1,m)*C(x)*ones(m,1)
    %                   s.t.        Tmax*G(x) - C(x) >= 0
    %                               0 <= xi <= Wmax

    %
    % Circuit parameters
    %

    dim=4;           % grid is dimxdim (assume dim is even)
    n=(dim+1)^2;     % number of nodes
    m=2*dim*(dim+1); % number of wires
                     % 1...dim(dim+1) are horizontal segments
                     % (numbered rowwise);
                     % dim(dim+1)+1 ... 2*dim(dim+1) are vertical
                     % (numbered columnwise)
    beta = 0.5;      % capacitance per segment is twice beta times xi
    alpha = 1;       % conductance per segment is alpha times xi
    G0 = 1;          % source conductance
    C0 = [ 10     2     7     5     3;
            8     3     9     5     5;
            1     8     4     9     3;
            7     3     6     8     2;
            5     2     1     9    10 ];
    wmax = 1;       % upper bound on x

    %
    % Build capacitance and conductance matrices
    %

    CC = zeros(dim+1,dim+1,dim+1,dim+1,m+1);
    GG = zeros(dim+1,dim+1,dim+1,dim+1,m+1);

    % constant term
    CC(:,:,:,:,1) = reshape( diag(C0(:)), dim+1, dim+1, dim+1, dim+1 );
    zo13 = reshape( [1,0;0,1],   2, 1, 2, 1 );
    zo24 = reshape( zo13,        1, 2, 1, 2 );
    pn13 = reshape( [1,-1;-1,1], 2, 1, 2, 1 );
    pn24 = reshape( pn13,        1, 2, 1, 2 );
    for i = 1 : dim+1,
        % source conductance
        % first driver in the middle of row 1
        GG(dim/2+1,i,dim/2+1,i,1) = G0;
        for j = 1 : dim,
            % horizontal segments
            node = 1 + j + ( i - 1 ) * dim;
            CC([j,j+1],i,[j,j+1],i,node) = beta * zo13;
            GG([j,j+1],i,[j,j+1],i,node) = alpha * pn13;
            % vertical segments
            node = node + dim * ( dim + 1 );
            CC(i,[j,j+1],i,[j,j+1],node) = beta * zo24;
            GG(i,[j,j+1],i,[j,j+1],node) = alpha * pn24;
        end
    end
    % reshape for ease of use in Matlab
    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( 50, 150, npts );
    xdelays = [ 50, 100 ];
    xnpts   = length( xdelays );
    areas   = zeros(1,npts);
    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(m)
            variable G(n,n) symmetric
            variable C(n,n) symmetric
            dual variables Y1 Y2 Y3 Y4 Y5
            minimize( sum( C(:) ) )
            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) = sum(x);
        else,
            xareas(xi) = sum(x);

            %
            % Display sizes
            %

            disp( sprintf( 'Solution %d:', xi ) );
            disp( 'Vertical segments:' );
            reshape( x(1:dim*(dim+1),1), dim, dim+1 )
            disp( 'Horizontal segments:' );
            reshape( x(dim*(dim+1)+1:end), dim, dim+1 )

            %
            % Determine the step responses
            %

            figure(xi+1);
            A = -inv(C)*G;
            B = -A*ones(n,1);
            T = linspace(0,500,2000);
            Y = simple_step(A,B,T(2),length(T));
            indmax = 0;
            indmin = Inf;
            for j = 1 : size(Y,1),
               inds = min(find(Y(j,:) >= 0.5));
               if ( inds > indmax )
                  indmax = inds;
                  jmax = j;
               end;
               if ( inds < indmin )
                  indmin = inds;
                  jmin = j;
               end;
            end;
            tthres = T(indmax);
            GinvC  = full( G \ C );
            tdom   = max(eig(GinvC));
            elmore = max(sum(GinvC'));
            hold off; plot(T,Y(jmax,:),'-',T,Y(jmin,:));  hold on;
            plot( tdom   * [1;1], [0;1], '--', ...
                  elmore * [1;1], [0;1], '--', ...
                  tthres * [1;1], [0;1], '--');
            axis([0 500 0 1])
            text(tdom,1,'d');
            text(elmore,1,'e');
            text(tthres,1,'t');
            text( T(600), Y(jmax,600), sprintf( 'v%d', jmax ) );
            text( T(600), Y(jmin,600), sprintf( 'v%d', jmin ) );
            title( sprintf( 'Solution %d (Tmax=%g), fastest and slowest step responses', 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

    Point 1 of 50 on the tradeoff curve (Tmax = 50)
    Point 2 of 50 on the tradeoff curve (Tmax = 52.0408)
    Point 3 of 50 on the tradeoff curve (Tmax = 54.0816)
    Point 4 of 50 on the tradeoff curve (Tmax = 56.1224)
    Point 5 of 50 on the tradeoff curve (Tmax = 58.1633)
    Point 6 of 50 on the tradeoff curve (Tmax = 60.2041)
    Point 7 of 50 on the tradeoff curve (Tmax = 62.2449)
    Point 8 of 50 on the tradeoff curve (Tmax = 64.2857)
    Point 9 of 50 on the tradeoff curve (Tmax = 66.3265)
    Point 10 of 50 on the tradeoff curve (Tmax = 68.3673)
    Point 11 of 50 on the tradeoff curve (Tmax = 70.4082)
    Point 12 of 50 on the tradeoff curve (Tmax = 72.449)
    Point 13 of 50 on the tradeoff curve (Tmax = 74.4898)
    Point 14 of 50 on the tradeoff curve (Tmax = 76.5306)
    Point 15 of 50 on the tradeoff curve (Tmax = 78.5714)
    Point 16 of 50 on the tradeoff curve (Tmax = 80.6122)
    Point 17 of 50 on the tradeoff curve (Tmax = 82.6531)
    Point 18 of 50 on the tradeoff curve (Tmax = 84.6939)
    Point 19 of 50 on the tradeoff curve (Tmax = 86.7347)
    Point 20 of 50 on the tradeoff curve (Tmax = 88.7755)
    Point 21 of 50 on the tradeoff curve (Tmax = 90.8163)
    Point 22 of 50 on the tradeoff curve (Tmax = 92.8571)
    Point 23 of 50 on the tradeoff curve (Tmax = 94.898)
    Point 24 of 50 on the tradeoff curve (Tmax = 96.9388)
    Point 25 of 50 on the tradeoff curve (Tmax = 98.9796)
    Point 26 of 50 on the tradeoff curve (Tmax = 101.02)
    Point 27 of 50 on the tradeoff curve (Tmax = 103.061)
    Point 28 of 50 on the tradeoff curve (Tmax = 105.102)
    Point 29 of 50 on the tradeoff curve (Tmax = 107.143)
    Point 30 of 50 on the tradeoff curve (Tmax = 109.184)
    Point 31 of 50 on the tradeoff curve (Tmax = 111.224)
    Point 32 of 50 on the tradeoff curve (Tmax = 113.265)
    Point 33 of 50 on the tradeoff curve (Tmax = 115.306)
    Point 34 of 50 on the tradeoff curve (Tmax = 117.347)
    Point 35 of 50 on the tradeoff curve (Tmax = 119.388)
    Point 36 of 50 on the tradeoff curve (Tmax = 121.429)
    Point 37 of 50 on the tradeoff curve (Tmax = 123.469)
    Point 38 of 50 on the tradeoff curve (Tmax = 125.51)
    Point 39 of 50 on the tradeoff curve (Tmax = 127.551)
    Point 40 of 50 on the tradeoff curve (Tmax = 129.592)
    Point 41 of 50 on the tradeoff curve (Tmax = 131.633)
    Point 42 of 50 on the tradeoff curve (Tmax = 133.673)
    Point 43 of 50 on the tradeoff curve (Tmax = 135.714)
    Point 44 of 50 on the tradeoff curve (Tmax = 137.755)
    Point 45 of 50 on the tradeoff curve (Tmax = 139.796)
    Point 46 of 50 on the tradeoff curve (Tmax = 141.837)
    Point 47 of 50 on the tradeoff curve (Tmax = 143.878)
    Point 48 of 50 on the tradeoff curve (Tmax = 145.918)
    Point 49 of 50 on the tradeoff curve (Tmax = 147.959)
    Point 50 of 50 on the tradeoff curve (Tmax = 150)
    Particular solution 1 of 2 (Tmax = 50)
    Solution 1:
    Vertical segments:

    ans =

        0.6536    0.4385    0.5236    0.4709    0.2363
        1.0000    0.8536    1.0000    0.9360    0.5700
        0.9232    0.2956    0.8004    1.0000    1.0000
        0.4130    0.1355    0.2668    0.6705    0.8895

    Horizontal segments:

    ans =

        0.1957    0.1415   -0.0000   -0.0000   -0.0000
        0.0710    0.0636   -0.0000   -0.0000   -0.0000
       -0.0000   -0.0000   -0.0000    0.0950    0.1579
        0.0000    0.0000    0.0000    0.0837    0.0541

    Particular solution 2 of 2 (Tmax = 100)
    Solution 2:
    Vertical segments:

    ans =

        0.2688    0.0437    0.1712    0.1338    0.0736
        0.4135    0.0802    0.3064    0.2224    0.1485
        0.2576    0.0802    0.1120    0.3835    0.2816
        0.1344    0.0437    0.0245    0.2408    0.2453

    Horizontal segments:

    ans =

       1.0e-06 *

        0.0454   -0.1821   -0.2023   -0.1763    0.0936
        0.0608   -0.1976   -0.1993   -0.1900    0.0740
        0.0813   -0.1935   -0.1955   -0.1908    0.0737
        0.0991   -0.1851   -0.2120   -0.2055    0.0602