2nd order-typ2-II PLL loop filter analysis

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% design a loop filter for a typeII PLL % the folrmulas are based on this analysis: http://sss-mag.com/pdf/pllfil.pdf % % Icp_\___________________________Vctrl % / | | % \ | % / R1 ohms | % \ | % / | % | _|_ % _|_ ___ C2 F % ___ C1 F | % | | % _|_ _|_ % \ / \ / % % function [R1,C1,C2]=pll_typeII_get_loop_filter (Icp,kvco,pm,bw,N) % Icp: charge pump current [A] % kvco: VCO gain [Hz/V] % pm: phase margin [deg] % bw: 3dB closed loop bandwidth [Hz] % N: feeback divider ratio % Z(s): impedance of loop filter % Z(s) = (s+1/R1/C1)/(C2*s*(s+(C1+C2)/C1/C2/R1)) % z1=1/R1/C1 % p1=(C1+C2)/C1/C2/R1 % Z(s) = (s+z1)/C2/s/(s+p1) % A(s): open loop transfer function % ---------- |\ % | phi_in |--------|+\ __________ ________ ____________ % ---------- | \________|Icp/2/pi|__________| Z(s) |_______|2*pi*kvco/s|______________phi_out % | / --------- ------- ------------ | % ----|-/ | % | |/ ----- | % |___________________________________________________| %N |_______| % ----- % % A(s) = Icp*kvco*Z(s)/s/N = Icp*kvco/C2/N/s^2*(s+z1)/(s+p1) % phase[A(s=jw)]=-180+arctan(w/z1)-arctan(w/p1) % PM: phase margin % PM = arctan(w/z1)-arctan(w/p1) = arctan[(p1-z1)*w/(w^2+p1*z1)] % design loop filter to maximize PM: d[(p1-z1)*w/(w^2+p1*z1)]/dw=0 --> wp=sqrt(p1*z1) % by PM definition we have: |A(s=jwp)|=1 % closed loop transfer function: T(s) % (phi_in-phi_out/N)*[A(s)*N] = phi_out --> T(s) = phi_out/phi_in = N*A(s)/(1+A(s)) % 3dB bandwidth, w0=2*pi*bw, is achived when |T(s=jw0)| = N/2; this can be achived for A(s=jw0) = j % we know that at wp, |A(s=jwp)|=1 and by maximizing PM at wp, we can assume that w0 is approximately equal to wp % using trigonometry equality of sec(tetha)=sqrt(1+tan^2(tetha)): z1=wp*(sec(PM)-tan(PM)) wp = 2*pi*bw; tetha = pm*pi/180; z1 = wp*(sec(tetha)-tan(tetha)); p1 = wp^2/z1; C2= Icp*kvco/N/wp^2*sqrt(wp^2+z1^2)/sqrt(wp^2+p1^2); % using |A(s=jwp)|=1 C1=(p1/z1-1)*C2; R1=1/C1/z1;