% LC oscillator design.
% (a tradeoff figure between the power and phase noise is generated)
%
% This is an example taken directly from the paper:
%
%   Geometric Programming and its Applications to EDA Problems
%   (DATE Tutorial 2005) by Boyd, Kim, and Mohan.
%   (see pages 102-113)
%
% Designs an LC oscillator consisting of a loop inductor, varactors
% for fine tuning, binary weighted switching capacitors for coarse
% tuning, cross coupled NMOS transistors, and tail current source.
% The optimal LC oscillator design iwith minimum power consumption,
% and limits on phase noise, area, etc... can be formulated as a GP:
%
%   minimize   P
%       s.t.   N <= Nmax, A <= Amax, l >= lmin, etc.
%
% where optimization variables are loop inductor dimensions D,W,
% size of varactor Vc, size of switching caps Csw, width and length
% of transistors Wnmos, Lnmos, bias current Ibias, etc.
%
% Original code by S. Mohan.
% Almir Mutapcic 02/03/06

%********************************************************************
% problem data
%********************************************************************
Vdd   = 1.2;         % voltage
CL    = 0.2e-12;     % load capcitance
F     = 5e9;         % operating frequency in Hz
omega = 2*pi*F;      % operating freq. in radians

PNSpec = 0.7e-11;    % phase noise specs
FOff   = 6e5;        % offset frequency for phase noise calculation
LoopGainSpec = 2.0;  % loop gain spec
Vbias  = 0.2;        % non ideality of current mirror

% tuning specs
T         = 0.1;     % +/- tuning range as a normalized value
CvarRatio = 3;       % maximum to minimum value of CVar
CswBits   = 3;
CswSegs   = 2^(CswBits);
CvarCswLSBOverlap = 2;

%********************************************************************
% optimization of LC oscillator
%********************************************************************
% optimization variables
gpvar D;        % diameter of loop inductor
gpvar W;        % width of loop inductor
gpvar SRF;      % self resonance frequency
gpvar l;        % length of CMOS transistor
gpvar w;        % width of CMOS transistor
gpvar I;        % maximum current through CMOS transistor
gpvar VOsc;     % differential voltage amplitude
gpvar CT;       % total capacitance of oscillator
gpvar Csw;      % maximum switching capacitance
gpvar Cvar;     % minimum variable capacitance
gpvar IBias;    % bias current
gpvar CMaxFreq; % capacitor max frequency

%********************************************************************
% loop inductor definitions and constraints
%********************************************************************
SRFSpec  = 3*F;
omegaSRF = 2*pi*SRF;

% inductance
L = 2.1e-06*D^(1.28)*(W)^(-0.25)*(F)^(-0.01);
% series resistance
R = 0.1*D/W+3e-6*D*W^(-0.84)*F^(0.5)+5e-9*D*W^(-0.76)*F^(0.75)+0.02*D*W*F;
% effective capacitance
C = 1e-11*D+5e-6*D*W;

% area, tank conductance, and inverse quality factor
Area = (D+W)^2;
G    = R/(omega*L)^2;
invQ = R/(omega*L);

loop_constr = [ Area <= 0.25e-6;
W <= 30e-6;
5e-6 <= W;
10*W <= D;
D <= 100*W;
SRFSpec <= SRF;
omegaSRF^2*L*C <= 1 ];

%********************************************************************
% transistor definitions and constraints
%********************************************************************
GM  = 6e-3*(w/l)^0.6*(I/2)^(0.4);
GD  = 4e-10*(w/l)^0.4*(I/2)^(0.6)*1/l;
Vgs = 0.34+1e-8/l+800*(I*l/(2*w))^0.7;
Cgs = 1e-2*w*l;
Cgd = 1e-9*w;
Cdb = 1e-9*w;

transistor_constr = [ 2e-6 <= w;
0.13e-6 <= l;
l <= 1e-6 ];

%********************************************************************
% overall LC oscillator definitions and constraints
%********************************************************************
% power
power   = Vdd*IBias;
invVOsc = (G+GD)/IBias;

% phase noise
kT4  = 4*1.38e-23*300;
kT4G = 2*kT4;
LoopCurrentNoise = kT4*G;
TransistorCurrentNoise = 0.5*kT4G*GM;
PN = 1/(160*(FOff*VOsc*CT)^2)*(LoopCurrentNoise+TransistorCurrentNoise);

% capacitance
Cfix = C+0.5*(CL+Cgs+Cdb+4*Cgd); % fixed capacitance
CDiffMaxFreq = Cfix+0.5*Cvar;

invLoopGain=(G+0.5*GD)/(0.5*GM);

% collect all of the LC oscillator constraints
constr = [ PN <= PNSpec;
loop_constr;
transistor_constr;
omega^2*L*CT == 1;
omega^2*(1+T)^2*L*CMaxFreq == 1;
4*T/((1-T^2)^2)*CT <= Csw*(1+CvarCswLSBOverlap/CswSegs);
Csw*CvarCswLSBOverlap/CswSegs <= 0.5*Cvar*(CvarRatio-1);
CDiffMaxFreq <= CMaxFreq;
VOsc+2*Vbias <= 2*Vdd;
VOsc*invVOsc <= 1;
invLoopGain*LoopGainSpec <= 1; % loop gain spec
Vbias+Vgs+IBias/2*R/2 <= Vdd;  % bias constraint spec
I == IBias ];


%********************************************************************
% solve the problem and generate the tradeoff curve
%********************************************************************
% set the quiet flag (no solver reporting)
global QUIET; QUIET = 1;

powers = [];
for T = 0.15 % can also vary tuning T = 0.05:0.05:0.2
  for PNSpec=0.7e-12:0.2e-12:1e-11
    constr(1) = PN <= PNSpec;
    [min_power solution status] = gpsolve(power,constr);
    powers = [powers min_power];
  end
end

% enable solver reporting again
global QUIET; QUIET = 0;

% plot the tradeoff curve
PNSpec = [0.7e-12:0.2e-12:1e-11];
plot(10*log10(PNSpec),powers/1e-3);
xlabel('Phase Noise (dBc/Hz)')
ylabel('Power (mW)')

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