Unformatted text preview: frequency signal components
in the phase diﬀerence.
PLLs 272 Analog ICs; JiehTsorng Wu PhaseLocked Loops (PLLs)
Applications:
• Automatic frequency control.
• Frequency and phase demodulation.
• Data and clock recovery.
• Frequency synthesis.
References:
• Roland E. Best, “PhaseLocked Loops,”, 2nd Edition, McGrawHill, Inc., 1993.
• Dan H. Wolaver, “PhaseLocked Loop Circuit Design,” PrenticeHall, Inc., 1991.
• Floyd M. Gardner, “Phaselock Techniques,” 2nd Edition, John Wiley & Sons, 1979.
PLLs 273 Analog ICs; JiehTsorng 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 phasedetector gain factor, and Ko is the VCO gain
factor.
PLLs 274 Analog ICs; JiehTsorng 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 closedloop transfer function.
PLLs 275 Analog ICs; JiehTsorng Wu SecondOrder PLL — Active LagLead 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 276 Analog ICs; JiehTsorng Wu SecondOrder PLL — Passive LagLead 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 277 ωn
ζ=
2KoKd H (s) = ω2
n
s2 + 2ζ ωns + ω2
n Analog ICs; JiehTsorng Wu HighGain SecondOrder 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 highfrequency components at ωi , 2ωi ,
. . . , existing in the phase detector’s output.
• The PD output’s highfrequency components can show up as spurious tones in the
frequency spectrum of the PLL’s output. PLLs 278 Analog ICs; JiehTsorng Wu HighGain SecondOrder 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 279 Analog ICs; JiehTsorng Wu Step Response of a TwoPole System
Consider the following twopole 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 2710 if 4ζ 2 1 Analog ICs; JiehTsorng Wu Step Response of a TwoPole 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 2711 Analog ICs; JiehTsorng 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 2712 Analog ICs; JiehTsorng Wu Phase Jitter
Assume that
n2 = 1212
·n...
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 Winter '09
 Choma
 Integrated Circuit, Transistor, The Land, Bipolar junction transistor, VDS, Analog ICs, JiehTsorng Wu

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