Noise Analysis of Phase-Locked Loops Amit Mehrotra Department of Electrical and Computer Engineering, Univerity of Illinois at Urbana-Champaign Abstract tem of equations was presented. However, the PLL is a phase feedback system and special techniques are required for solving the associated sys- This work addresses the problem of noise analysis of phase locked tem of equations. In this work, a system of stochastic differential equa- loops (PLLs). The problem is formulated as a stochastic differential equa- tions governing the behaviour of the PLL VCO phase are developed. The tion and is solved in presence of circuit white noise sources yielding the PLL is assumed to be locked to a reference periodic signal which is as- spectrum of the PLL output. Specifically, the effect of loop filter charac- sumed to have Brownian motion phase deviation. It is shown in Section 3 teristics, phase-frequency detector and phase noise of the open loop volt- that the PLL output phase, in locked condition, is a sum of two stochas- age controlled oscillator (VCO) on the PLL output spectrum is quantified. tic processes: the Brownian motion phase deviation of the reference sig- These results are derived using a full nonlinear analysis of the VCO in the nal and one component of an appropriate multi-dimensional Ornstein- feedback loop and cannot be predicted using traditional linear analyses or Uhlenbeck process. Similar to [2], it is shown that the PLL output is the phase noise analysis of open loop oscillators. The computed spectrum asymptotically wide-sense stationary. Using the statistics of the phase de- matches well with measured results, specifically, the shape of the output viation process, a general expression for the power spectral density (PSD) spectrum matches very well with measured PLL output spectra reported of the PLL output is obtained. This expression is used to derive the PSD in the literature for different kinds of loop filters and phase detectors. The of PLL output for some specific loop filter configurations in Section 4. PLL output spectrum computation only requires the phase noise of the From the output spectrum, it can be observed that that the PLL output VCO, loop filter and phase detector noise, phase detector gain and loop PSD closely follows the reference signal spectrum for very small offset filter transfer function and does not require the transient simulation of the frequencies and follows the open loop VCO output spectrum for high off- entire PLL which can be very expensive. The noise analysis technique is set frequencies. This fact has been experimentally observed and widely illustrated with some examples. reported in the literature. Finally, experimental results on an example cir- cuit are presented in Section 5. 1 Introduction Phase and delay locked loops (PLL and DLL) are extensively used in 2 Previous Works microprocessors and digital signal processors for clock generation and as Noise analysis of PLLs is probably the least understood topics in RF frequency synthesizers in RF communication systems for clock extraction noise analysis. Some existing works present an intuitive explanation of and generation of a low phase noise local oscillator signal from an on- how various noise sources affect the overall noise of PLL [3, 4, 5, 6, 7]. chip voltage controlled oscillator (VCO) which might have a higher open- Techniques borrowed from linear noise analysis are used to predict the loop noise performance. The basic block diagram of a PLL is shown in output phase noise spectrum and the relative importance of VCO phase Figure 1. The phase of a local VCO signal is compared with the phase of a noise, reference signal phase noise and noise in the phase detector and (hopefully) low noise reference signal and the difference of the two phases loop filter as a function of the loop bandwidth [5, 7, 8, 9]. Output char- is low-pass filtered and applied to the controlling node of the VCO. If the acteristics of sampling PLLs has also been analyzed in presence of white input signal frequency is within the VCO tuning range, the VCO output noise at the PLL input [10, 11]. However, the PLL circuit noise and refer- is also “locked” to the same frequency as the input signal and the phase ence signal phase noise have not been considered. In all these approaches, difference between the two signals is very small. In RF communication a precise mathematical characterization of the noisy PLL output and quan- systems, frequency synthesizer noise directly degrades the overall noise tification of how much each noise sources contribute to the PLL output performance of the system. Similarly, timing jitter in phase-locked loops noise is not present. Moreover, it has been shown [1, 2] that noise genera- of high performance processors degrades the timing margins of the overall tion in an oscillator is inherently a nonlinear phenomenon and linear noise design. Hence accurate prediction of PLL noise performance is critical for analysis techniques for circuits containing VCOs are not rigorously justi- the design of these systems. fied. Also the VCO output itself is a stochastic process and VCO phase Noise generation mechanisms for PLLs and DLLs are very different. noise cannot be viewed as an additive noise source. Similarly, translation In a DLL, the voltage noise from each of the delay stages accumulates of random phase deviations in the VCO phase to the VCO output PSD is for one period of the input reference signal and then the output phase a nonlinear phenomenon and the use of linear analysis based techniques is aligned with the input signal phase. On the other hand, in a PLL, a for this purpose is also not justified. Behavioural level noise analysis tech- VCO is present in the feedback loop and the difference in the phase of the niques for PLLs have also been reported [12, 13] which can also include reference and the VCO signal is filtered and used as the control signal of power supply noise [14, 15]. However, numerical integration involved in the VCO. Hence the difference of the phase noise of the reference signal such methods can be expensive. Also the oscillator phase noise models and the VCO output acts as one of the noise sources for the VCO. This used in these works [14] is not rigorously justified. The VCO modelled paper addresses the problem of noise analysis of phase-locked loops. in these works is a ring oscillator and it is not clear how to extend this The starting point of this work is [1, 2] where noise analysis of open approach to PLLs with other kinds of VCOs such as one based on LC loop oscillators based on a novel perturbation analysis of oscillatory sys- tank. Vin PD LPF VCO Vout 3 PLL Noise Analysis Let the input reference signal of the PLL in Figure 1 be periodic with ω 2π period T, i.e., of angular frequency 0 = T . Since this reference is also Figure 1: PLL block diagram generated by a real oscillator, it also has Brownian motion phase error αin(t) [1]. I.e., the reference signal is of the form xin(t + αin(t)) where and the phase detector. Note that the coefficient matrices G, E and F are αin(t) = pcinBin. Bin(t) is a one-dimensional Brownian motion process. independent of time. This assumes that the reference signal frequency Note that throughout this discussion, “phase” has units of time. Phase in is not drifting with time and the VCO remains locked to the reference. radians can be recovered by multiplying α(t) by the appropriate angular This implies that the variations around the VCO control voltage are very frequency ω. The advantage of using this formulation is that the process small. This also follows from the assumption that in locked state γ(t) has of zero-delay frequency division of either the reference signal or the VCO bounded variance. If the filter transfer function is stable, bounded variance output does not affect the analysis presented below (except for adding of γ(t) also implied bounded variance of β(t). more noise). PLL noise analysis now proceeds as follows: Noise analysis of the open loop VCO yields that [1, 2] the phase devia- tion of the VCO output is governed by the following stochastic differential 1. (2), (3) and (4) are solved using stochastic differential equation tech- β( ) γ( ) equation niques and an expression of t is obtained. t and other compo- nents of x are not required for the output spectrum calculation and dα need not be computed separately. open loop vco = vT (t + α (t))ξ (t) (1) dt open loop vco p 2. Since αvco(t) is a stochastic process, PLL VCO output xs(t + p α ( )) β( ) where ξp(t) R is a vector of p uncorrelated VCO white noise sources vco t is also a stochastic process. Using the expression of t 2p and v( ) R = v1( ) v2( ) ::: vp( ) is a periodic function which obtained in step 1, the following autocorrelation can be computed · 2 · · · depends on the noise source intensities and the response of the linearized τ E α τ α τ Rxs;xs (t; ) = [xs(t + vco(t))xs(t + + vco(t + ))] oscillator circuit [1]. It was shown that asymptotically αopen loop vco(t) becomes a Brownian motion process whose variance increases linearly 3. It can be shown that the asymptotically R (t;τ) is independent with time at a rate c, i.e., α = pcB(t), where c is the time xs;xs open loop vco of t, i.e., the PLL VCO output is a wide-sense stationary stochastic average of the inner product of the vector v( ).