An efficient linear-time variant simulation technique of ...

An efficient linear-time variant simulation technique of oscillator phase sensitivity function Abstract—This paper presents a fast and accurate simulation technique to evaluate the impulse sensitivity function (ISF) of an oscillator. The proposed method, based on the linear-time variant (LTV) analysis of oscillators, computes the impulse phase response by means of periodic steady-state (PSS) and periodic transfer function (PXF) simulations available in commercial simulators (Spectre, Eldo, etc.). This technique overwhelms the classical simulation method based on transient analysis and injection of charge pulses along the oscillator period in terms of both speed and precision. The good accuracy of the proposed method has been verified in two oscillator topologies, namely a Van der Pol and a ring oscillator.

I. I NTRODUCTION Phase noise of VCOs (Voltage Controlled Oscillators) and PLLs (Phase Locked Loops) is still a major issue in the design of wireless communication systems limiting the maximum achievable data transfer rate [1]. Since bit-rates keep increasing with time, this results in more stringent requirements of the transceiver and a deep insight of phase noise generation mechanism is necessary to perform an optimum design. In the framework of linear time variant (LTV) model, which is the most suitable for circuits with a periodically varying operating point such as oscillators, the impulse sensitivity function (ISF) introduced by Hajimiri and Lee [2] provides a very useful tool for the analysis of the oscillator system dynamics under small signal excitations, such as MOSFET current noise, and thus for evaluating the oscillator phase noise. The above mentioned sensitivity function is also adopted to analyze injection locking mechanism in oscillators [3]. This phenomenon is observed when an oscillator is perturbed by a weak external signal whose frequency is close enough but not equal to that of the unperturbed oscillator and has been exploited to realize quadrature oscillators or oscillators with finer phase separation [4] and low-power frequency dividers [5]. At the present time, the ISF cannot be directly computed by commercially available circuit simulators, thus requiring ad hoc simulations. The traditional technique, proposed in [2] and adopted by RF designers, is based on the definition of impulse phase response of an oscillator. By means of timedomain transient simulations, current pulses are injected in an oscillator node and the phase response of the system is evaluated. This technique basically trades accuracy off against implementation simplicity and simulation speed. In this paper, a frequency-domain simulation method for the ISF is presented, based on a small-signal time-varying analysis, namely the periodic transfer function (PXF) simulation available in Cadence environment [6]. It overwhelms the transient analysisbased method in terms of both accuracy and speed.

This paper is organized as follows. Section II briefly reviews some basic concepts of Hajimiri’s theory of the impulse sensitivity function. Section III describes the proposed method to simulate the ISF by means of periodic small signal analysis. Section IV shows the comparison between the PXF and transient analysis-based methods in terms of implementation simplicity and simulation speed in a Van der Pol and a ring oscillator, highlighting the advantages of the proposed technique. Finally, conclusions are drawn in Section V. II. I MPULSE S ENSITIVITY F UNCTION Consider a generic oscillator  voltage can be  whose output expressed as V (t)=A(t) cos ω0 t + φ0 (t) , being A(t) and φ0 (t) the amplitude and the phase of the oscillator perturbed by electronic noise and ω0 = 2π T0 the oscillation frequency. A time-domain analysis can be performed considering that (i) a linear model is valid under the hypothesis of small perturbation (always valid when considering oscillator noise) and (ii) both amplitude and phase of the output voltage are perturbed. Thus, the oscillator acts like a multiple-input multiple-output (MIMO) system, where the inputs are the noise sources while the outputs are the oscillation amplitude A(t) and phase φ0 (t). It is thus possible to define two different impulse responses for each noise source, hA (t, τ ) and hφ0 (t, τ ). The impulse response associated to the oscillation amplitude hA (t, τ ) is of little interest since it rapidly fades asymptotically because of the amplitude limitation mechanism due to the transconductor non-linearity. The impulse response of the phase is instead of greater interest. It results in a step function u(t), as visible in Fig. 1, meaning that the phase shift arisen from a single charge pulse is not recovered since the oscillator has no time reference. The height of the step function is highly dependent on the time instant τ of the input impulse. Thus hφ0 (t, τ ) can be written as hφ0 (t, τ ) = hφ (τ ) u(t − τ ), where hφ (τ ) is a T0 -periodic function. The perturbed phase of the oscillator output voltage caused by a noisy current in (t) applied between two generic nodes can be expressed as: Z t Z +∞ hφ0 (t, τ ) in (τ ) dτ = hφ (τ ) in (τ ) dτ . φ(t) = −∞

−∞

(1) In [2] the impulse sensitivity function is defined as a dimensionless function, Γ(τ ), given by hφ (τ ) · qmax , qmax being the maximum charge displacement across the capacitor between the nodes. In this paper, we will instead refer to the impulse sensitivity function as hφ (τ ). In both cases, being the ISF closely related to the phase impulse response

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hφ0 (t, τ ), its definition suggests a method for its simulation, reported in [2]. Perturbing the oscillator with a charge pulse ) in (t) = ∆q · δ(t − τ ), hφ (τ ) is given by ∆φ(τ ∆q , where ∆φ is the asymptotical output phase shift. The required steps for computing hφ (τ ) are the following: 1) Run a simulation of the unperturbed oscillator and annotate a time instant tref when the output voltage crosses an arbitrarily chosen value Vref , once the initial transient has faded. Referring to Fig. 1, Vref has been chosen equal to 0. 2) Apply a short current pulse of area ∆q at instant τ at the nodes where the current noise source is applied. If the injected charge produces a phase shift much smaller than 2π, the output voltage will cross Vref at the instant t1 close to tref (see Fig. 1). 3) The phase shift ∆φ is obtained as ∆φ = ω0 (tref − t1 ), resulting in hφ (τ ) = ∆φ ∆q . 4) Sweep the injection time instant τ over one oscillation period computing for each instant the phase shift. The more accuracy is needed, the more transient simulations must be run. Despite its simplicity, this procedure is time-consuming. In addition, since the injected charge has to be small, the results may be affected by numerical noise. In [2] an analytical method to compute the above mentioned phase shift and thus the phase sensitivity function is proposed; a current impulse applied to the i-th node of the oscillator causes an initial change ∆Vi =∆q/Ci of the node voltage and an excess phase equal to: ∆q v˙i ∆φi = ω0 , (2) − C i |→ v˙i |2 being ∆q the excess charge on the node capacitor Ci , ω0 the oscillation frequency, v˙i the time derivative of the i-th − v˙i |2 the norm of the first derivative of the node voltage and |→ waveform vector. Equation (2) refers to oscillators where state variables are node voltages, while the reader can find in [7]

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how (2) can be applied in the case of an LC oscillator whose state variables are capacitor voltage and inductor current. It should be noted that (2) is not correct in general [7], [8], since it assumes that noise perpendicular to state-space trajectory of the oscillator does not generate any phase noise. In particular, (2) leads to incorrect determination of the ISF each time amplitude to phase noise conversion takes place in the oscillator, as verified in Section IV by an appropriate simulation. III. T HE PROPOSED METHOD The method proposed in this work derives the oscillator impulse phase response in the frequency domain by means of a PXF (periodic transfer function) analysis available in commercial RF circuit simulators. Let us consider the output voltage of the unperturbed oscillator at the fundamental frequency as V (t) = A0 cos [ω0 t + φ0 ] and the ISF to be evaluated: +∞

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In the case of the hφ (t) DC component, ϕ0 can be either 0 or π. These equations pose the basis of the numerical methodology to derive the ISF function by means of a PXF V PM simulation, which computes the magnitudes kin and the phases Φ+ k of the periodic small-signal transfer function from a single current tone at frequency kω0 − ωm to the output voltage at frequency ω0 +ωm . The index k is an integer number automatically swept between 0 and K, where K can be limited to 5-6 for sinusoidal oscillators. In the above mentioned equations A0 and φ0 are the magnitude and the initial phase of the first harmonic of the output voltage, respectively, and can be estimated through a single PSS (Periodic Steady State) simulation. Finally, note that the PXF simulation is a linear analysis; the circuit is linearized around the time-varying operating point, thus avoiding the problem of large injected signal occurring when performing transient analysis. Fig. 2 shows the spectra of the injected current tone in (t), hφ (t), output phase φ (t) and modulated output voltage V (t).

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IV. S IMULATION RESULTS The computation of the ISF is greatly simplified using a simulation environment that supports scripting, such as Cadence Open Command Environment for Analysis (OCEAN) [9]. However, while with the proposed technique the results can be achieved in a completely automatic way, the transient analysis-based method requires a manual calibration before being performed. In particular, the amount of charge injected into the circuit with each single pulse must be properly chosen in order to be sufficiently small to assure a linear relationship between charge and phase and sufficiently large to make the simulator numerical error negligible. As a practical case, the ISF has been simulated with both the Hajimiri’s and the proposed method for two oscillator topologies, namely a Van der Pol and a ring oscillator shown in Fig. 3 and Fig. 4, respectively. Both oscillators are designed in a 0.35-µm CMOS technology and run at 1-GHz frequency. In the Van der Pol oscillator the ISF is simulated for a current noise source connected between one of the output and ground, as shown in Fig. 3. Fig. 5 shows the comparison between the results obtained with the PXF and transient analysis-based methods, the latter being performed with three different values of the ratio between the injected charge ∆q and the maximum charge stored in the tank capacitor C, qmax . In particular, a perfect matching between the results of the two methods is achieved for a ∆q/qmax ratio equal to 10−2 . As this ratio is raised by two order of magnitude, the resulting ISF is distorted, while, as the ratio is lowered by three orders of magnitude, the resulting waveform is affected by numerical error. In order to verify that phase sensitivity function resulting from (2) can lead to an incorrect result, the oscillator in Fig. 3 has been simulated adopting as tank capacitors two accumulation PMOS varactors with the same capacitance value (22 pF). Thus, due to the non linear C-V characteristic of the varactors, the VCO is affected by an amplitude to

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phase modulation conversion mechanism [10]. Fig. 6 shows the phase sensitivity evaluated via the PXF-based method and by means of (2). The latter estimation confirms the incorrect prediction of the closed-form formula reported in [2]. Finally, the ring oscillator of Fig. 4 has been considered as case study. The ISF has been evaluated for the tail current noise source and for a noise source across the differential output, namely Iinj,tail and Iinj,core . The results, plotted in Fig. 7, suggest a perfect matching between the ISF waveforms obtained with the two above mentioned methods. Table 1 reports a comparison between the ISF simulation times for the two methods in both oscillators. The simulations have been performed in the Cadence Ocean environment with a 3-GHz Pentium Xeon featuring a 4-Gbyte main memory. A total of 50 time instants over one oscillation period have been simulated by means of transient analyses. Each analysis has been run for 200 oscillation periods, injecting the current

Topology Van der Pol Ring

TABLE I ISF SIMULATION TIME Simulation Time (TRAN) Simulation Time (PXF) 6m 41s 5s 7m 40s 6s

Fig. 7. Simulated ISFs of the ring oscillator in Fig. 4 using the proposed and transient analysis-based simulation method.

pulse in the middle of the simulation time, with the following accuracy parameters: reltol=10−6 and errpreset=conservative [6]. Regarding the PXF-based method, the PSS stabilization time (tstab) has been set equal to 100 periods (with the same accuracy parameters) and 20 harmonic terms have been computed. The computation time with the transient analysisbased method is approximately 80 times larger than adopting the proposed simulation technique. V. C ONCLUSIONS In this work a fast and accurate simulation method based on PXF analysis for the impulse phase response in oscillators has been presented. The proposed technique overwhelms the traditional simulation method based on transient analysis in terms of both computation time and accuracy. In particular, the method is helpful every time the magnitude and phase of each ISF component is of interest. R EFERENCES [1] B. Razavi, RF Microelectronics. Prentice Hall, 1998. [2] A. Hajimiri and T. Lee, “A general theory of phase noise in oscillators,” IEEE J. Solid State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998. [3] P. Maffezzoni, “Analysis of oscillator injection locking through phasedomain impulse-response,” IEEE Trans. Circuits Syst. I: Reg. Paper, vol. 55, no. 5, pp. 1297–1305, June 2008. [4] P. Andreani, A. Bonfanti, L. Romano’, and C. Samori, “Analysis and design of a 1.8-GHz CMOS LC quadrature VCO,” IEEE J. Solid-State Circuit, vol. 37, no. 12, pp. 1737–1747, Dec. 2002. [5] A. Bonfanti, A. Tedesco, C. Samori, and A. Lacaita, “A 15-GHz broadband divided by 2 frequency divider in 0.13-µ m CMOS for quadrature generation,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 9, pp. 724–726, Sept. 2005. [6] Affirma RF Simulator User Guide, Cadence Product Documentation, version 4.4.6, Apr. 2001. [7] P. Andreani and X. Wang, “On the phase-noise and phase-error performances of multiphase LC CMOS VCO,” IEEE J. Solid State Circuits, vol. 39, no. 11, pp. 1883–1893, Nov. 2004. [8] G. Coram, “A simple 2-D oscillator to determine the correct decomposition of perturbations into amplitude and phase noise,” IEEE Trans. Circuits Syst. I, vol. 48, pp. 896–898, July 2001. [9] OCEAN Reference, Cadence Product Documentation, version 4.4.5, Dec. 1999. [10] A. Bonfanti, S. Levantino, C. Samori, and A. L. Lacaita, “A varactor configuration minimizing the amplitude-to-phase noise conversion in VCOs,” IEEE Trans. Circuits Syst. I: Reg. Paper, vol. 53, no. 3, pp. 481–488, March 2011.