# Two-input NAND gate sizing (GP)

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    % Boyd, Kim, Patil, and Horowitz, "Digital circuit optimization
    % via geometric programming"
    % Written for CVX by Almir Mutapcic 02/08/06
    % (a figure is generated)
    %
    % This is an example taken directly from the paper:
    %
    %   Digital circuit optimization via geometrical programming
    %   by Boyd, Kim, Patil, and Horowitz
    %   Operations Research 53(6): 899-932, 2005.
    %
    % Solves the problem of choosing device widths w_i for the given
    % NAND2 gate in order to achive minimum Elmore delay for different
    % gate transitions, subject to limits on the device widths,
    % gate area, power, and so on. The problem is a GP:
    %
    %   minimize   D = max( D_1, ..., D_k )  for k transitions
    %       s.t.   w_min <= w <= w_max
    %              A <= Amax, etc.
    %
    % where variables are widths w.
    %
    % This code is specific to the NAND2 gate shown in figure 19
    % (page 926) of the paper. All the constraints and the objective
    % are hard-coded for this particular circuit.

    %********************************************************************
    % problem data and hard-coded GP specs (evaluate all transitions)
    %********************************************************************
    N = 4;       % number of devices
    Cload = 12;  % load capacitance
    Vdd = 1.5;   % voltage

    % device specs
    NMOS = struct('R',0.4831, 'Cdb',0.6, 'Csb',0.6, 'Cgb',1, 'Cgs',1);
    PMOS = struct('R',2*0.4831, 'Cdb',0.6, 'Csb',0.6, 'Cgb',1, 'Cgs',1);

    % maximum area and power specification
    Amax = 24;
    wmin = 1;

    % varying parameters for the tradeoff curve
    Npoints = 25;
    Amax = linspace(5,45,Npoints);
    Dopt = [];

    disp('Generating the optimal tradeoff curve...')
    for k = 1:Npoints
        fprintf(1,'  Amax = %5.2f:', Amax(k));
        cvx_begin gp quiet
            % device width variables
            variable w(N)

            % device specs
            device(1:2) = PMOS; device(3:4) = NMOS;

            for num = 1:N
                device(num).R   = device(num).R/w(num);
                device(num).Cdb = device(num).Cdb*w(num);
                device(num).Csb = device(num).Csb*w(num);
                device(num).Cgb = device(num).Cgb*w(num);
                device(num).Cgs = device(num).Cgs*w(num);
            end

            % capacitances
            C1 = sum([device(1:3).Cdb]) + Cload;
            C2 = device(3).Csb + device(4).Cdb;

            % input capacitances
            Cin_A = sum([ device([2 3]).Cgb ]) + sum([ device([2 3]).Cgs ]);
            Cin_B = sum([ device([1 4]).Cgb ]) + sum([ device([1 4]).Cgs ]);

            % resistances
            R = [device.R]';

            % area definition
            area = sum(w);

            % delays and dissipated energies for all six possible transitions
            % transition 1 is A: 1->1, B: 1->0, Z: 0->1
            D1 = R(1)*(C1 + C2);
            E1 = (C1 + C2)*Vdd^2/2;
            % transition 2 is A: 1->0, B: 1->1, Z: 0->1
            D2 = R(2)*C1;
            E2 = C1*Vdd^2/2;
            % transition 3 is A: 1->0, B: 1->0, Z: 0->1
            % D3 = C1*R(1)*R(2)/(R(1) + R(2)); % not a posynomial
            E3 = C1*Vdd^2/2;
            % transition 4 is A: 1->1, B: 0->1, Z: 1->0
            D4 = C1*R(3) + R(4)*(C1 + C2);
            E4 = (C1 + C2)*Vdd^2/2;
            % transition 5 is A: 0->1, B: 1->1, Z: 1->0
            D5 = C1*(R(3) + R(4));
            E5 = (C1 + C2)*Vdd^2/2;
            % transition 6 is A: 0->1, B: 0->1, Z: 1->0
            D6 = C1*R(3) + R(4)*(C1 + C2);
            E6 = (C1 + C2)*Vdd^2/2;

            % objective is the worst-case delay
            minimize( max( [D1 D2 D4] ) )
            subject to
                area <= Amax(k);
                w >= wmin;
        cvx_end
        % display and store computed values
        fprintf(1,' delay = %3.2f\n',cvx_optval);
        Dopt = [Dopt cvx_optval];
    end

    % plot the tradeoff curve
    plot(Dopt,Amax);
    xlabel('Dmin'); ylabel('Amax');
    disp('Optimal tradeoff curve plotted.')

    Generating the optimal tradeoff curve...
      Amax =  5.00: delay = 11.56
      Amax =  6.67: delay = 9.23
      Amax =  8.33: delay = 7.84
      Amax = 10.00: delay = 6.90
      Amax = 11.67: delay = 6.23
      Amax = 13.33: delay = 5.73
      Amax = 15.00: delay = 5.34
      Amax = 16.67: delay = 5.03
      Amax = 18.33: delay = 4.77
      Amax = 20.00: delay = 4.55
      Amax = 21.67: delay = 4.37
      Amax = 23.33: delay = 4.22
      Amax = 25.00: delay = 4.08
      Amax = 26.67: delay = 3.96
      Amax = 28.33: delay = 3.86
      Amax = 30.00: delay = 3.76
      Amax = 31.67: delay = 3.68
      Amax = 33.33: delay = 3.60
      Amax = 35.00: delay = 3.54
      Amax = 36.67: delay = 3.47
      Amax = 38.33: delay = 3.42
      Amax = 40.00: delay = 3.36
      Amax = 41.67: delay = 3.32
      Amax = 43.33: delay = 3.27
      Amax = 45.00: delay = 3.23
    Optimal tradeoff curve plotted.