an RSFQ ring oscillator. The experimental results are compared with results from a stochastic circuit simu- lator. We determined the value of jitter to be 1.52.
Short-Term Frequency Stability of RSFQ Ring Oscillators Cesar A. Mancini and Mark F. Bocko
Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627
Abstract| The temporal stability of the clock sig-
nal has a profound e ect on the performance of synchronous RSFQ digital systems. Short-term clock
uctuations, or clock jitter, can severely degrade system performance due to the hazard of timing constraint violations. Successful large-scale RSFQ digital systems will require highly stable multi-Gigahertz onchip clock sources. To meet this need, methods for characterizing and measuring the short-term stability of such sources are required. In this paper we identify the relevant gure of merit to characterize and compare various clocks: the cycle-to-cycle standard deviation of the clock periods. We have developed experimental techniques for the measurement of this gure of merit and applied it to the characterization of an RSFQ ring oscillator. The experimental results are compared with results from a stochastic circuit simulator. We determined the value of jitter to be 1.52% at 10GHz. I. Introduction
Synchronous digital systems use a global clock signal to de ne a time reference for the movement of data and for the change of state in the system. The clock timing parameters of this clock signal, such as its period, are often used as one of the performance metrics of the system. The clock signal is not immutable and is subject to timing parameter variations which are the superposition of two e ects: one is the result of the fabrication process parameter deviations that a ect the circuits of the clock source; the second is the random uctuations induced by electrical noise in the clock circuit. Random uctuations occurring on the time scale of the clock period are referred to as short-term uctuations; long-term uctuations are de ned to occur over times of several clock cycles. The impact of long and short-term timing uctuations on system performance depends on the type of system. Long-term clock stability is critical in A/D converters and communication blocks such as clock-recovery circuits. In digital computing systems, long-term uctuations are usually not important and the short-term uctuations lower the maximum frequency at which the system can reliably operate. Depending on the timing strategy used it can also prevent correct operation at any frequency. The word jitter will be used in the context of this manuscript as a reference to short term clock period uctuations and Manuscript received September 15, 1998. This work was supported in part by the University Research Initiative at the University of Rochester, sponsored by the Army Research Oce under grant DAAL03-92-G-0012
SW
Bias DC voltage JTL
External Clk DC/SFQ
v Clk
CB
JTL
S
JTL
Fig. 1. Block diagram for the ring oscillator. The \CB" block denotes a con uence bu er, the \JTL" block a Josephson transmission line, the \S" block a splitter. The DC/SFQ produces an SFQ pulse on the rising edge of the input current.
the term cycle-to-cycle jitter refers speci cally to timing
uctuations in two consecutive clock cycles. There exist several choices for on-chip clock sources in RSFQ circuits among them a single over-biased Josephson junction, a ring oscillator and a long Josephson junction. The rst option is the simplest of all; a single Josephson junction whose bias current is higher than its critical current. In this regime the junction produces a continuous stream of SFQ pulses. If the junction is underdamped, a monotonic increase in bias current will produce a monotonic increase in frequency, thus the stability of this frequency depends on the stability of the bias current. Thermal noise from the shunt resistor typically present in such applications induces a modulation e ect on the frequency. This in turn induces short-term uctuations in the period of oscillation of the clock. A ring oscillator is essentially a circular active transmission line (Figure 1). A con uence bu er cell allows the entry of an SFQ pulse, which then circulates inde nitely. A splitter cell produces an output pulse every time the pulse inside the ring completes one round trip. The length of the ring and the bias current determine the time the SFQ pulse takes to complete one round trip, and therefore the frequency of the output pulses. The DC/SFQ converter produces an SFQ pulse in the rising edge of a input current pulse. The switch SW allows one to open and close the ring using an external control line. The oscillator is started by closing the switch SW and then applying a current pulse to the DC/SFQ converter. The voltage monitor line supplies a voltage V proportional to the frequency of the circulating SFQ pulse according to the Josephson frequency fJ = 2eV=h. A long Josephson junction is characterized by having its length longer than the Josephson penetration depth J , while having its width smaller than J . For certain combination of the junction's critical current density, subgap resistance, and external stimuli such as bias current and external magnetic eld, an unshunted long junction is capable of holding one or more uxons bouncing back and forth from the ends of the long junction. A drawback of this oscillator is that the length of the junction sets the resonant frequency, which is then xed once the junction
is fabricated. This type of oscillator is also known as a resonant soliton oscillator (RSO). There has been a lot of e ort in the past in the characterization of long-term frequency uctuations of oscillators [1],[2]. These e orts were basically concentrated in the area of time and frequency standards for metrology applications. As a result, a set of de nitions were developed in order to characterize and compare such standards in the time and frequency domains [3]. Short-term timing uctuations have not been studied to a similar extent [4]. The key reason is that until recently, the e ect of short-term uctuations on system performance has been insigni cant. But as clock frequencies increase, the effects of jitter will begin to have a signi cant impact on the degradation of system performance. In fact, this has been observed in communication circuits, such as clock and data recovery circuits based on phase-locked loops (PLL) [5]. Superconducting oscillators have been studied for the past twenty years. The main e ort has been on the construction of millimeter and sub-millimeter wavelength sources for on-chip local oscillators in integrated receivers [6]. The gure of merit used for the characterization of these sources has invariably been the -3dB linewidth as measured by a spectrum analyzer. Most recently, and following the trend, an RSFQ ring oscillator [7], [8] and a long Josephson junction connected to an RSFQ frequency divider [9], have also been characterized using this gure of merit. Unfortunately it is not possible to obtain a quantitative estimate of jitter by just using the -3dB linewidth, and previous attempts to do so have been misinterpretations of the measured data. Nevertheless, it does provide a qualitative measure of the \goodness" of the oscillator, that is, an oscillator with a small linewidth is \better" than one with a large linewidth at a given frequency. Oscillators used in analog applications are conventionally characterized by their phase noise at a given o set from the carrier. It will be shown later that this gure of merit provides a way to obtain a quantitative measure of jitter; they are in fact equivalent. In this contribution we adopt the use of this standard for the characterization of superconducting oscillators as well.
SFQ/DC
SFQ/DC
SFQ/DC
SFQ/DC
Consider a sinusoidal oscillator whose amplitude and phase are subject to some small uctuations respectively: v(t) = (A + ) sin(!0t + ): (1) Where =A 1 and 1 radians. These restrictions are called the small amplitude modulation and the small angle modulation conditions respectively. Expanding and keeping only rst order terms we get: v(t) = (A + ) sin !0t + A( ) cos !0t (2) The amplitude uctuations modulate a carrier in phase with the original signal. On the other hand the phase uctuations modulate the amplitude of a carrier which is in quadrature (90 degrees out of phase) with the original signal. Therefore, for the small angle modulation condition, phase modulation is equivalent to amplitude modulation of a source in quadrature with the clock oscillator. In general, the phase and the amplitude uctuations are the result of "noise" in the oscillator system. This unwanted e ect of "noise" can can be categorized into two broad groups: a) Deterministic signals such as electromagnetic or direct coupling of power line (60Hz) and broadcast (TV, radio, etc.) signals. b) Random signals which are the product of random processes such as thermal, shot and icker noise. The characterization of deterministic noise in the frequency domain can be obtained by taking the Fourier transform of equation (2) and noting that and can be approximated as a linear combination of harmonic signals. The end result is a spectrum with well de ned discrete lines. By examining the magnitude of the Fourier transform of equation (2), the result will be a superposition of the spectrum of and that of . Therefore by just looking at the magnitude of the Fourier transform it is not possible to separate the contribution of each component unless there is some previous certainty of the nature of them. One of the weaknesses of previous studies has been the inability to properly discriminate between the phase noise and the amplitude noise component. Random noise requires a more elaborate treatment. The phase and amplitude uctuations can be represented by random processes, which in the general case are completely speci ed in terms of the joint probability density II. Modeling of the Jitter Process function for all times [10]. This can be quite dicult in Jitter can be analyzed in both the time domain and the most cases and it is necessary to make some assumptions frequency domain. Figures of merit for the evaluation of in order to simplify the analysis. For example, if these ranjitter can be developed in both domains and then related dom processes are independent and are known to be Gausto one another. sian distributed, the process is completely speci ed by the mean and the autocorrelation function. If a random process is known to be wide-sense stationary (WSS), the A. Frequency domain analysis Wiener-Khinchin theorem states that the Fourier transFrequency domain analysis is appropriate when the form of the autocorrelation function RNN yields the power measurements are carried out by a spectrum analyzer. spectral density (PSD) SN of that process. The power spectral density of the phase uctuations is usually called Ring the phase noise spectrum. S T S T S T Oscillator From the physical standpoint, these phase uctuations can be related to current or voltage uctuations in the circuit which could also directly modulate the frequency Ring instead of the phase of the oscillator. Both e ects can Bias also be present simultaneously. In the case of uctuations in frequency due to a noise current, the oscillator can be modeled as a current-controlled oscillator (CCO) whose Fig. 2. Block diagram of the ring oscillator circuit proposed for for input is fed by a current noise source. Using phase as the high-speed test. output variable and current as the input variable, we can
≈
We now consider a time domain statistical model used in [11] to describe jitter in the period of an oscillator. We can consider each period of a clock signal as a discrete event that indicates when the phase has advanced by a speci c amount (2). Fig. 3 shows the output of a clock with jitter. Each period of the clock is a time interval that is the result of an accumulation of phase equal to 2. The interval of time for the nth period is represented by T [n]. A clock with jitter implies that the periods are not uniform and T [n] describes a discrete time domain random process where: T [n] = T0 + x[n]; (5) where T0 is the average period and x[n] is a zero-mean discrete-time random process that represents the deviation of the period from the average. Since x[n] is de ned as a zero-mean random variable, this implies that T0 exists. When there is a frequency drift, T0 will not exist if it is de ned as the mean over all time. Nevertheless, in practice all time records are nite and therefore it is always possible to de ne an average T0 .
B. Time domain analysis
where KI is the CCO current-to-frequency conversion constant in units of Arads s . Converting to the frequency domain this equation becomes: 2 S (!) = jHCCO (!)j2 SI (!) = K!2I SI (!): (4) Notice that this power spectral density diverges at the origin. This is a direct consequence of the integral in (3) extending to ,1. Therefore the variance of this process is in nite. The phase can wander with no nite bound. This process is also known as a \random walk" process. However, in real systems the lower limit of the integral is always nite; it is limited by the observation time, or the time that the spectrum analyzer takes to do the measurement. The former discussion implies that under this model the concept of a -3dB linewidth cannot be de ned, since there is not a well de ned peak for the spectrum of the uctuations. In addition to this, the value of the PSD for small shifts from the carrier is related to the value of the autocorrelation function for long periods of time. Therefore the concept of linewidth can be more closely connected with long-term rather than short-term uctuations.
,1
The circuit depicted in Fig. 2 was used for the circuit simulations and experiments. It consists of a ring oscillator (RO), followed by a chain of three T ip- ops (TFF). Each TFF divides the frequency by half. Therefore a chain of N ip- ops divides the frequency by 2N . The SFQ/DC converter provides a conversion to voltage levels that can be measured with conventional room temperature electronics; it toggles between 0 and 200V for each incoming SFQ pulse, dividing by half the frequency output. The RO consist of 25 JTL stages in addition to a con uence bu er and a splitter. The bias current can be adjusted to get a frequency of oscillation between 5 and 12GHz. The circuit parameters of these cells are optimized for maximum yield [12] according to parametervariation data supplied by the foundry service. The circuit fabrication was done by Hypres Inc. using the standard 1kA=cm2 Niobium-based superconducting process [13]. A stochastic simulator of SFQ circuits [14] was used to obtain a theoretical measure of jitter for each SFQ/DC
III. Simulations and Experiments
where Rxx [i; j ] is the autocorrelation function of x[n]. If the x[n] is wide-sense stationary (WSS) then this last result can be simpli ed by writing Rxx [i; j ] as Rxx [i , j ]. If the condition x[n] T0 is satis ed then: Z (8) x[n] = K! I I ( )d 0 T0 and the x[n] process is therefore a WSS process. For white Gaussian noise with a spectral density of In2 =2 we obtain: 2 T02 In2 K j j I Rxx[ ] = 2!2 1 , 2T (9) 0 0 and therefore (10) T = KpI T0 In 2! 0 The result obtained in (10) is the cycle-to-cycle jitter. Substituting into (7) 2 2 2 T2 M = M KI2T!02In (11) 0 This result will be used later to analyze the results from simulations and experiments.
i=n,M+1 j=n,M+1
The measure of jitter at a given delay is given by the sum of the jitter at each clock period. Suppose that we are looking at the jitter of the previous M periods starting at clock period n; we can now de ne a new random process T[1] T[2] T[n] T[2-M] t which is the sum of the previous M periods: To + x[1] To + x[2] To + x[n] n X T [ n ] = T [i] (6) M Fig. 3. Description of the random process of jitter in a clock source. i=n,M+1 T [n] is the nth clock period. TM [n] is the sum of the T [n] for the previous M periods starting at period n. This new random process describes the jitter process for M periods of the clock. The mean of this random express the phase of the oscillator as the integral in time process is MT0 and its variance is given by: of the current: n n Zt X 2 = X Rxx [i; j ] (7) TM (t) = KI I ( )d ; (3) TM[1]
≈
TABLE I SIMULATED AND MEASURED CYCLE-TO-CYCLE JITTER
−44.0
Division Simulated Measured Factor Jitter(%) Jitter(%) 1 0.58 1.52a 2 0.46 1.20 4 0.33 0.93 8 0.21 0.55 16 0.14 0.28 a Extrapolated value from least-squares t
Power (dBm)
−54.0
−64.0
−74.0
Slope -2.09 -2.08 -2.08 -2.08
division factor, yields a slope of -0.49 and -0.30 respectively. This implies that the ratio of standard deviations 1.40 and 1.23 respectively. From (11), the assumption Fig. 4. Direct spectrum measurement of the fundamental frequency is of p white noise added to the clock period yields a ratio of of the divide-by-two ring oscillator output. 2. The simulation result is closer to this analytical result than the experimental one. This might indicate the converter output. In these simulations each resistor is of an additional noise component that introduces shunted with a current noise source with the power spec- presence some amount of correlation in the uctuation of consecutral density of Johnson noise at 4.2K. A MATLAB script tive clock periods. Additional simulations are needed to was used to process the output of the transient simula- con rm this hypothesis. tions and compute the average period and the standard deviation from an ensemble of 50 iterations of 10 clock References cycles each. The step size for each transient simulation is 0.01ps. The results are presented on column 2 of Table I. \Characterization of phase and frequency stabiliExperiments were carried out using a cryogenic probe [1] J.tiesRutman, in precision frequency sources: fteen years of progress," with high-bandwidth coaxial cable connections between Proc. IEEE, vol. 66, pp. 1048{1075, Sept 1978. the chip holder and room temperature. The output of [2] L. S. Cutler and C. L. Searle, \Some aspects of the theory the SFQ/DC converter is ampli ed by 60dB and then fed and measurement of frequency uctuations in frequency stanto a HP8569B spectrum analyzer. The resolution banddards," Proc. IEEE, vol. 54, pp. 136{154, Feb 1966. width of the instrument is set to the minimum possible [3] J. A. Barnes et al., \Characterization of frequency stabilsetting that renders a stable frequency peak. The specity," IEEE Transactions on Instrumentation and Measurement, vol. IM-20, pp. 105{120, May 1971. trum data is transfered to a computer for further analysis through a IEEE-488 connection. A typical spectrum of [4] E. J. Baghdady, R. N. Lincoln, and D. B. Nelin, \Short-term frequency stability: Characterization, theory, and measurethe fundamental frequency is shown in Figure 4. Another ment," Proc. IEEE, vol. 53, pp. 704{722, July 1965. MATLAB script was used to take the data points from [5] A. W. Bushwald, K. W. Martin, A. Oki, and K. W. Kobayashi, one side of the spectrum and plot them on a log-log scale, \A 6GHz integrated phase-locked loop using AlGaAs/GaAs then a least-squared linear interpolation is done on the heterojunction bipolar transistors," IEEE Journal of Solid data points. The slope of this line re ects the exponent State Circuits, vol. 27, pp. 1752{1762, December 1992. of the frequency dependence. Using equation (11) we ob- [6] P. L. Richards and Q. Hu, \Superconductive components for tain the cycle-to-cycle jitter. The results can be seen in infrared and millimeter-wave receivers," Proc. IEEE, vol. 77, column 3 of Table I. The division factor includes the e ect pp. 1233{1246, August 1989. of the SFQ/DC. The value of jitter for the ring oscillator [7] J.-C. Lin and V. Semenov, \Timing circuits for RSFQ digital systems," IEEE Trans. Appl. Supercond., vol. 5, pp. 3472{ was determined from a linear t of the measured jitter 3477, June 1995. versus the division factor. [8] V. Kaplunenko, V. Borzenets, and N. B. Dubash, \Supercon−84.0 4.989
4.994
4.999 5.004 Frequency (GHz)
5.009
IV. Analysis and Discussion
The experimental results in column 4 of Table I indicate an inverse squared frequency dependence. This corroborates the validity of the model presented earlier and summarized in (4). The values for jitter are consistently higher than the simulation results by an average factor of 2.4. This excess noise can be modeled as an equivalent noise temperature Tn for the ring oscillator and assuming that Tn is proportional to the square of this factor we obtain Tn 25K . A more precise value of Tn can be determined by setting the temperature as a free parameter in the simulation runs. From the data presented in Table I one can also obtain a relationship for the ratio of the standard deviation of the period between two consecutive divider outputs. A least-squares linear t on a log-log scale applied to both the simulations and the experimental results versus the
[9] [10] [11] [12] [13] [14]
ducting single ux quantum 20 Gb/s clock recovery circuit," Appl. Phys. Lett., vol. 71, pp. 128{130, July 1997. Y. Zhang, V. Borzenets, V. K. Kaplunenko, and N. B. Dubash, \Underdamped long josephson junction coupled to overdamped single- ux-quantum circuits," Appl. Phys. Lett., vol. 71, pp. 1863{1865, Sep 1997. A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 1991. J. McNeill, Jitter in Ring Oscillators. PhD thesis, Boston University, Boston, MA, 1994. Q. Herr and M. Feldman, \Multiparameter optimization of RSFQ circuits using the method of inscribed hyperspheres," IEEE Trans. Appl. Supercond., vol. 5, pp. 3327{3340, June 1995. Hypres, Inc., Elmsford, NY 10523, HYPRES Niobium Process Flow and Design Rules, 1993. J. Satchell, \Stochastic simulation of SFQ logic," IEEE Trans. Appl. Supercond., vol. 7, pp. 3315{3318, June 1997.
