pll_synth_3nd_order.m

function [cap res] = pll_synth_3rd_order(ipump, vco_sensitivity, fout, fcomp, bandwidth, pm)

% Synthesizes PLL loop components for a 3rd order system using the
% topology shown below.  
%
% Input - ipump is the charge pump current in Amperes
%       - vco_sensitivity is the VCO sensitivity in Hertz/Volt
%       - fout is the output frequency in Hertz
%       - fcomp is the comparison frequency in Hertz
%       - bandwidth is the open loop bandwidth in Hertz
%       - pm is the phase margin in degrees
% Output - cap is the capacitors of the loop in Farads [C1, C2, C3, 0]
%        - res is the resistors of the  loop in Ohms [ R2, R3, 0]   
%
%
% The method used here is derived from that presented in Dean Banerjee's
% Book "PLL Performance, Simulation, and Design" 4th Ed available at
% National Semiconductors site www.national.com.  Most of the work is
% derived from Chapter 22 pp.178-185.
%
% No closed form solution exists for 3rd
% order systems, therefore some simplification of the model has been made
% to derive the results.  Due to the simplifications the results will
% also not always be 100% accurate.  It is recommended that the
% results are verified using the pll_simulation.m script, which does not
% use a simplified model.
% Loop Topology
%
%        +  _____                                   ______
% fcomp -->|Phase|-----------------R3---------------| VCO |---->fout
%          |Det. |     |      |         |           |     |  |
%           -----      |      |         |            -----   |
%            ^ -      C1     R2        C3                    |
%            |         |      |         |                    |
%            |        GND    C2        GND                   |
%            |                |                              |
%            |               GND         _______             |
%            ----------------------------| 1/N |--------------
%                                        -------    
%
%   Author: Ben Gilbert
%   Homepage: http://nicta.com.au/people/gilbertb
%   Email: ben.gilbert (wibble) nicta.com.au
%   (c) 2009 by National ICT Australia (NICTA)
%   %


%% Design Parameters
% The following values will give sensible results but not necessarily
% optimal
T31 = 0.6; % T3/T1 ratio of time constants
gamma = 1.0; % optimization factor p172 and pp220-227 of Banerjee

% Conversion of parameters to more convenient units
Kpd = ipump/2/pi; % phase detector gain
Kvco = vco_sensitivity*2*pi; % vco gain
omega = 2*pi* bandwidth; % open loop bandwidth in radians/sec

%% Synthesis
%% Find T1 using numerical methods
t1 = (sec(pm*pi/180)-tan(pm*pi/180))/(omega*(1+T31)); % approximate value for T1

% T1 is the root of this equation
num1 = @(t) atan(gamma./(omega.*t*(1+T31)))-atan(omega.*t)-atan(omega.*t*T31)- pm*pi/180;
T1 = fzero(num1,t1);% find root numerically

%% Find T2 and T3
T3 = T1*T31;
T2 = gamma/omega^2/(T1+T3);

%% Solving for Loop Components
A0 = Kpd * Kvco * fcomp/fout / omega^2 * sqrt((1+omega^2*T2^2)/(1+omega^2*T1^2)/(1+omega^2*T3^2));
A1 = A0*(T1+T3);
A2 = A0*T1*T3;

C1 = A2/T2^2*(1+sqrt(1+T2/A2*(T2*A0-A1)));
C3 = (-T2^2*C1^2+T2*A1*C1-A2*A0)/(T2^2*C1-A2);
C2 = A0-C1-C3;
R2 = T2/C2;
R3 = A2/C1/C3/T2;

cap = [C1 C2 C3 0];
res = [R2 R3 0];