The frequency acquisition aid provided by the pfd is

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: frequency signal components in the phase diﬀerence. PLLs 27-2 Analog ICs; Jieh-Tsorng Wu Phase-Locked Loops (PLLs) Applications: • Automatic frequency control. • Frequency and phase demodulation. • Data and clock recovery. • Frequency synthesis. References: • Roland E. Best, “Phase-Locked Loops,”, 2nd Edition, McGraw-Hill, Inc., 1993. • Dan H. Wolaver, “Phase-Locked Loop Circuit Design,” Prentice-Hall, Inc., 1991. • Floyd M. Gardner, “Phaselock Techniques,” 2nd Edition, John Wiley & Sons, 1979. PLLs 27-3 Analog ICs; Jieh-Tsorng Wu Basic Model Vd = Kd (θi − θo) θi ωo = ωoo + Ko × Vc Vd PD F(s) Phase Detector Vc Filter Ko/s θo VFO When the PLL is locked, Vd (s) = Kd · [θi (s) − θo(s)] = Kd θe(s) θe = θi − θo Vc(s) = F (s) · Vd (s) ωod t = ωoot + KoVcd t = ωoot + θo ⇒ Ko θo(s) = Vc(s) · s • θe is the phase error, Kd is the phase-detector gain factor, and Ko is the VCO gain factor. PLLs 27-4 Analog ICs; Jieh-Tsorng Wu Basic Model System equations are Vd = Kd · (θi − θo) = Kd · θe Vc = F (s) · Vd θo = Vc · Ko s The transfer functions are θo θi = KoKd F (s) s + KoKd F (s) = H (s) θe s = = 1 − H (s) θi s + KoKd F (s) sKd F (s) Vc s = · H (s) = θi s + KoKd F (s) Ko ⇒ ∆ωo Vc H (s) = = Ko · ∆ωi ∆ωi ∆ωi = ωi − ωoo ∆ωo = ωo − ωoo • H (s) is the closed-loop transfer function. PLLs 27-5 Analog ICs; Jieh-Tsorng Wu Second-Order PLL — Active Lag-Lead Filter R2 C R1 Vo Vi sτ2 + 1 F (s ) = − sτ1 τ1 = R1 C τ2 = R2 C 2 H (s) = 2ζ ωns + ωn s2 + 2ζ ωns + ω2 n ωn = KoKd τ1 ωn · τ2 ζ= 2 • ωn is the pole frequency of the loop. • ζ is the damping factor. Qp = 1/(2ζ ) is the pole quality factor. PLLs 27-6 Analog ICs; Jieh-Tsorng Wu Second-Order PLL — Passive Lag-Lead Filter R1 Vo Vi F (s) = R2 τ1 = (R1 + R2)C C 2 H (s) = τ2 = R2C 2 s 2ζ ωn − ωn/(KoKd ) + ωn s2 + 2ζ ωns + sτ2 + 1 sτ1 + 1 ωn = ω2 n KoKd τ1 ωn 1 τ2 + ζ= 2 KoKd • If R2 = 0, then 1 1 τ1 = = R1C ωLF PLLs ωn = KoKd ωLF 27-7 ωn ζ= 2KoKd H (s) = ω2 n s2 + 2ζ ωns + ω2 n Analog ICs; Jieh-Tsorng Wu High-Gain Second-Order PLL Frequency Response If KoKd τ2 1 in the passive ﬁlter, then 2 Hpassive(s) ≈ Hactive(s) = 2ζ ωns + ωn s2 + 2ζ ωns + ω2 n And the −3 dB bandwidth of H (s) is 1/2 ω−3dB = ωn 2ζ 2 + 1 + (2ζ 2 + 1)2 + 1 • Usually choose ωn < ωi /10 to remove the high-frequency components at ωi , 2ωi , . . . , existing in the phase detector’s output. • The PD output’s high-frequency components can show up as spurious tones in the frequency spectrum of the PLL’s output. PLLs 27-8 Analog ICs; Jieh-Tsorng Wu High-Gain Second-Order PLL Frequency Response 10 ζ = 5.0 5 | H ( j ω) | (dB) ζ = 2. 0 0 -5 -10 ζ = 0.3 ζ = 0.5 -15 ζ = 0.707 -20 0.1 1 10 Frequency (ω/ωn ) PLLs 27-9 Analog ICs; Jieh-Tsorng Wu Step Response of a Two-Pole System Consider the following two-pole transfer function 2 H (s) = ωn Poles = s1,2 = −ζ ± s2 + 2ζ ωns + ω2 n ζ 2 − 1 ωn • If ζ > 1, the system is overdamped, and both poles are real. Step Response = 1 − k1 = ζ − 1 −k1ωnt 1 −k2ωnt e −e k2 2 ζ 2 − 1 k1 1 ζ2 − 1 k2 = ζ + ζ2 − 1 • If ζ = 1, the system is critically damped, and both poles are at −ωn . Step Response = 1 − (1 + ωnt )e−ωnt ≈ 1 − e−ωnt/(2ζ ) PLLs 27-10 if 4ζ 2 1 Analog ICs; Jieh-Tsorng Wu Step Response of a Two-Pole System • If ζ < 1, the system is underdamped. Step Response ζ ωn Step Response = 1 − · sin ωd t + cos ωd t e−ζ ωnt ωd √ 2 % Overshoot = 100e−π/ 1/ζ −1 ωd = 1 − ζ 2 · ωn Overshoot 1 Error Band t √ • For PLL, choose ζ > 1/ 2 = 0.707 to avoid excessive ringing. PLLs 27-11 Analog ICs; Jieh-Tsorng Wu Phase Jitter Probability Density V nt N θn Vs θn nc pdf = 2 2 −θ /(2σ ) 1 √ en n 2πσn v (t ) = s(t ) + n(t ) = Vs sin(2πfot ) + n(t ) n(t ) = nc(t ) sin(2πfot ) + nt (t ) cos(2πfot ) The phase jitter is nt (t ) ≈ θn(t ) = tan Vs Vs + nc(t ) nt (t ) PLLs 27-12 Analog ICs; Jieh-Tsorng Wu Phase Jitter Assume that n2 = 1212 ·n...
View Full Document

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern