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];