Figure 3 shows a typical second-order PLL circuit that can be used as the starting point for building a frequency synthesizer. It uses a phase detector, loop filter, oscillator, and feedback divider.10-12 Parameter Kφ is the gain of the phase detector, F(s) is the transfer characteristic of the loop filter, K0 is the VCO sensitivity, and N is the feedback divider ratio. Considering the PLL a control system, the closed-loop transfer function (transfer function of the input phase noise to the output signal) can be described by Eq. 1:

B(s) = φ0(s)/φi(s) = N[(2ξωns + ω2n)/(s2 + 2ξωns + ω2n)]     (1)

with F(s), the natural pulsation (ωn), and the damping factor of the control system (ξ) described by Eqs. 2, 3, and 4, respectively:

F(s) = (1 + τ2s)/(τ1s)     (2)

ωn = [KφK0/(Nτ2)]0.5     (3)

ξ = ωnτ2/2     (4)

By analyzing Eq. 1, as the input phase-noise frequency offset decreases towards zero, the transfer function approaches N. Essentially, the input phase noise is amplified by the feedback divider ratio, increasing by a factor of 20logN (in dBc). The unity gain frequency (f0) is independent on the damping factor and is given by Eq. 513:

f0 = [(2)0.5/2π]ρn     (5)

The VCO noise transfers to the output of the PLL by a second-order highpass relationship given by Eq. 6:

R(s)  = s2/(s2 + 2ξωns + ω2n     (6)

For low offset frequencies, the VCO noise is attenuated with a 40-dB/decade slope, while for far offset frequencies, the VCO noise passing to the output is unaffected by the loop.

The magnitudes of the B(s) and K(s) noise transfer functions are plotted in Fig. 4. For a damping factor of 0.707, shown by the blue trace on Fig. 4, the peaking has a value of 2.09 dB. If the PLL is underdamped (ξ < 0.707), peaking can be large, making the system unstable. Peaking can be reduced by increasing the damping factor, but this takes the B(s) and R(s) trace decrease less sharply beyond the natural frequency.

Noise models can be categorized as linear time invariant (LTIV), linear time variant (LTV), and nonlinear time variant (NLTV) models, in order of increasing complexity. Leeson’s model15 is based on LTIV oscillator properties, such as resonator Q, feedback gain, output power, and noise figure. Additional models include an LTV model16 and an NLTV configuration using a perturbation model based on numerical techniques.17-20

Phase noise has been analyzed by means of a number of different models, with both time- and frequency-domain techniques applied.21-26 (The relative strengths and weaknesses of the three phase-noise models are compared in Table 2, included in the online version of this article). When comparing noise models for harmonic (LC-resonator-type) and nonharmonic oscillator circuits (RC-oscillator-type) oscillator circuits, a designer must choose the noise model since none of the models provide closed-form solutions for phase noise.