U. Glaeser

© 2002 by CRC Press LLC
TABLE 10.2 Equations for Second-Order PLL Impulse, Step, and Ramp Time
Domain Responses
Define:
c1 = −ζ ⋅ ω N + ω N ⋅ (ζ 2 − 1) 0.5
c2 = −ζ ⋅ ω N − ω N ⋅ (ζ 2 − 1) 0.5
T1 = C ⋅ R = 2 ⋅ ζ�ωN 2
Note: c1 ⋅ c2 = ω N
Impulse Response (Input is δ(t)):
ζ > 1:
h(t) = (N ⋅ ω N�(2 ⋅ (ζ 2 − 1) 0.5 )) ⋅
((1 + T1⋅ c1) ⋅ e (c1 ⋅ t) − (1 + T1 ⋅ c2) e (c2 ⋅ t) ) ⋅ u(t)
ζ = 1:
h(t) = N ⋅ ω N ⋅ e (−ωN ⋅ t)
⋅ (2 − ωN ⋅ t) ⋅ u(t)
0 < ζ < 1:
h(t) = (N ⋅ ω N�(1 − ζ 2 ) 0.5 ) ⋅
e (−ζ ⋅ ωN ⋅ t)
⋅ cos (ωN ⋅ (1 − ζ 2 ) 0.5 ⋅ t − φ) ⋅ u(t)
where:
φ = tan −1 ((1 − 2 ⋅ ζ 2 )�(2 ⋅ ζ ⋅ (1 − ζ 2 ) 0.5 ))
Step Response (Input is u(t)):
ζ > 1:
s(t) = N ⋅ (1 + (ω N�(2 ⋅ (ζ 2 − 1) 0.5 )) ⋅
((1�c 1 + T 1) ⋅ e (c1 ⋅ t) − (1/c 2 + T 1) ⋅ e (c2 ⋅ t) )) ⋅ u(t)
ζ = 1:
s(t) = N ⋅ (1 + e (−ω N ⋅ t) ⋅ (ωN ⋅ t − 1)) ⋅ u(t)
0 < ζ < 1:
s(t) = N ⋅ (1 − (1/(1 − ζ 2 ) 0.5 ) ⋅
e (−ζ ⋅ ω N ⋅ t) ⋅ cos (ω N ⋅ (1 − ζ 2 ) 0.5 ⋅ t + φ′)) ⋅ u(t)
where:
φ′ = sin −1 (ζ)
Ramp Response (input is t ⋅ u(t)):
r′(t) = r(t) − N ⋅ t ⋅ u(t) = PO(t) − N ⋅ PI(t) ζ > 1:
r(t) = N ⋅ (t − (1/(2 ⋅ ω N ⋅ (ζ 2 − 1) 0.5 )) ⋅
(e (c1 ⋅ t) − e (c2 ⋅ t) ) ⋅ u(t)
r′(t) = −(N/(2 ⋅ ω N ⋅ (ζ 2 − 1) 0.5 )) ⋅
(e (c1 ⋅ t) − e (c2 ⋅ t) ) ⋅ u(t)
ζ = 1:
r(t) = N ⋅ t ⋅ (1 − e (−ωN ⋅ t)
) ⋅ u(t)
r′(t) = − N ⋅ t ⋅ e (−ωN ⋅ t)
⋅ u(t)
0 < ζ < 1:
r(t) = N ⋅ (t − (1/(ω N ⋅ (1 − ζ 2 ) 0.5 )) ⋅ e (−ζ ⋅ ωN ⋅ t) ⋅
sin (ω N ⋅ (1 − ζ 2 ) 0.5 ⋅ t)) ⋅ u(t)
r′(t) = −(N/(ω N ⋅ (1 − ζ 2 ) 0.5 )) ⋅ e (−ζ ⋅ ω N ⋅ t)
⋅
sin (ω N ⋅ (1 − ζ 2 ) 0.5 ⋅ t) ⋅ u(t)
Slow Step Response (d(t) = (r(t) − r(t − dt))�dt):
d′(t) = d(t) − N ⋅ (t ⋅ u(t) − (t − dt) ⋅ u(t − dt)))
= r′(t) − r′(t − dt)
= PO(t) − N ⋅ PI(t) 0 < ζ < 1:
d′(t) = −(N�(dt ⋅ ω N ⋅ (1 − ζ 2 ) 0.5 )) ⋅ e (−ζ ⋅ ωN ⋅ t)
⋅
(sin (ω N ⋅ (1 − ζ 2 ) 0.5 ⋅ t) ⋅ u(t) − e (ζ ⋅ ωN ⋅ dt)
⋅
sin (ω N ⋅ (1 − ζ 2 ) 0.5 ⋅ (t − dt)) ⋅ u(t − dt))