% Two-input NAND gate sizing (GP)
% 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 = [];

% set the quiet flag (no solver reporting)
cvxq = cvx_quiet(true);

disp('Generating the optimal tradeoff curve...')

for k = 1:Npoints
  cvx_begin gp
    % 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,'  Amax = %5.2f   delay = %3.2f\n',Amax(k),cvx_optval);
  Dopt = [Dopt cvx_optval];
end

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

% restore initial solver reporting state
cvx_quiet(cvxq);