Linearized PLL The linear model of the analog PLL is depicted in Figure 6.22.

(a) Derive the closed-loop transfer function o (f )/i(f ).

(b) If F(s)¼1/(s¼

ffiffiffi2 p

), using the Laplace transform and the final value theorem, find an expression for the steady-state phase error.

Hints: The Laplace transform of a general signal x(t) is defined as XðsÞ

Z1

1 xðtÞðeÞstdt:

Σ Kd LPF F1( f ) = F ( f )

VCO j2πf Kv F2( f ) =

θi (t )

θo (t )

+ θe (t )

–

v2 (t )

Figure 6.22 Linearized PLL.

Under the constraints that x(t)¼0, t

Transcribed Image Text:

ww R F() Figure 6.23 Low-pass loop filter. 8 (t) + Be(t) LPF V(1) Kd F(f)=F() 8(t) 8 (1) + VCO Ky F(f)= j2f Figure 6.24 Phase noise model for VCO.