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794 IEEE JOURNAL OF SOLID-STATECIRCUITS,VOL. SC-18,NO. 6,DECEMBER 1983 Noise in Relaxation Oscillators ASAD A. ABIDI AND ROBERT G, MEYER, FELLOW, IEEE Abstract—Thetiming jitter in relaxation oscillators is analyzed. This ture, however, on either measurements of jitter in relaxa- jitter is described by a single normalized equation whose solution allows tion oscillators, or on an analysis of how noise voltages and prediction of noise in practical oscillators. The theory is confirmed by currents in the components of the oscillator randomly measurements on practical oscillators and is used to develop a prototype low noise oscillator with a measured jitter of 1.5 ppm rms. modulate the periodic waveform. Such an analysis was perhaps discouraged by the nonlinear fashion in which the oscillator operates, as shall become evident in Section III I. INTRODUCTION below. Despite this state of affairs, circuits designers have suc- OW noise in the output of an oscillator is important in cessfully used methods based on qualitative reasoning to many applications. The noise produced in the active L reduce the jitter in relaxation oscillators. The purpose of and passive components of the oscillator circuit adds ran- this study has been to develop an explicit background for dom perturbations to the amplitude and phase of the these methods. By analyzing the switching of such oscilla- oscillatory waveform at its output. These perturbations tors in the presence of noise, circuit methods are developed then set a limit on the sensitivity of such systems as to reduce the timing jitter. receivers, detectors, and data transmission links whose The results of this study have been verified experimen- performance relies on the precise periodicity of an oscilla- tally, and a prototype low jitter oscillator was built with tion. jitter less than 2 parts per million, nearly an order of A number of papers [1]–[6] were published over the past magnitude better than most commonly available circuits. several decades in which theories were developed for the prediction of noise in high-Q LC and crystal oscillators which are widely used in high-frequency receivers. Noise in II. OSCILLATOR TOPOLOGIES AND DEFINITION OF these circuits is filtered into a narrow bandwidth by the JITTER high-Q frequency-selective elements. This fact allows a relatively simple analysis of the noise in the oscillation, the One of the most popular relaxation oscillator circuits is results of which show that the signal to noise ratio of the the emitter-coupled multivibrator [7] with a floating timing oscillation varies inversely with Q. capacitor shown in Fig. 1, which uses bipolar transistors as Relaxation oscillators are an important class of oscilla- the active devices. Transistors Q1 and Q2 alternately switch tors used in applications such as voltage-controllable on and off, and the timing capacitor C is charged and frequency and waveform generation. In contrast to LC discharged via current sources 1. Transistors Q3 and Q4 oscillators, they require only one energy storage element, are level shifting emitter followers, and diodes 1)1 and D2 and rely on the nonlinear characteristics of the circuit define the voltage swings at the collectors of Q1 and Q2. A rather than on a frequency-selective element to define an triangle wave is obtained across the capacitor and square oscillatory waveform. These circuits have recently become waves at the collectors of Q1 and Q2. common because they are easy to fabricate as monolithic This circuit is sometimes modified for greater stability integrated circuits. against temperature drifts, and other types of active de- Due to their broad-band nature, these oscillators often vices are used, but in essence it is always equivalent to Fig. suffer from large random fluctuations in the period of their 1. The oscillator operates by sensing the capacitor voltage output waveforms, termed the timing jitter, or simply, jitter and reversing the current through it when this voltage in the oscillator. In an application such as an FM demodu- exceeds a predetermined threshold, lator, the relaxation oscillator in a phase-locked loop will Another common relaxation oscillator is shown in Fig. be limited in its dynamic range, and hence sensitivity, due 2(a), which uses a grounded timing capacitor. The charging to this jitter. There are no systematic studies in the litera- current is reversed by the Schmitt trigger output, whose two input thresholds determine the peak-to-peak amplitude of the triangle wave across the capacitor. The block dia- Manuscript recewed November 16, 1982; rewsed April 14, 1983. This gram of this circuit is shown in Fig. 2(b). As the ground in research work was supported by the U,S. Army Research OffIce under Grant DAAG29-80-K-0067. an oscillator is defined only with respect to a load, the A A Abldi is with the Bell Laboratories, Murray Hill, NJ 07974. circuit of Fig. 1 is also represented by the block diagram of R, G. Meyer N with the Electronics Research Laboratory, University of California, Berkeley, CA 94720. Fig. 2(b). 0018-9200/83 /1200-0794$01 .00 01983 IEEE ABIDIANDMEYER:NOISEIN RELAXATIONOSCILLATORS 795 Vcc D, D2 ! I I ● t QI tl 1~ ‘3 Fig. 4. Typical waveform of the current m a switching dewce m a relaxation oscillator. 1 1 I 1 VEE b Regenerollon Bias Re~t::on Fig. 1. Cross-coupled relaxation oscillator with floating timing capaci- —. Current ‘– { tor. — —— ——— .—— ——— ——— Regenerotlon Level +V I T I / * I !, ?~ t3 t Fig. 4 Typicaf waveform of the current m a switching device in a relaxation oscillator. ‘%EP-+ Noise Uncertolnty 1+ Bias b 0* Current ~ _\ (a) — ——. ————— Regenerotlon Level . t + tl tz t3 voltage 0 Flg 5 Actual waveform (including noise) of the current in a switching Sensor device in a relaxation oscdlator. + = A’‘ Actuol Waveform output 4 (b) I 4 I+ Voltoge Fig. 2. (a) Relaxation oscillator with grounded timing capacitor. (b) tpw Topological equivalent of oscillator. I I The voltage sensor in such oscillators is a bistable circuit. I * t When the capacitor voltage crosses one of the two trigger / points at its input, the sensor changes state. The sensor has “d Ideol a vanishingly small gain, while the capacitor voltage is Tronsltlon Points between these trigger points; but as a trigger point is Fig. 6. Output voltage waveform of a relaxation oscillator showing the approached, the operating points of the active devices in effect of noise. the sensor change in such a way that it becomes an amplifier of varying gain. The small-signal gain of the circuit is determined by an internal positive feedback loop, Fig. 6. The timing jitter may be defined in terms of the and becomes unfoundedly large at the trigger point, caus- mean p~and standard deviation ul of the pulsewidth as ing the sensor to switch regeneratively and change the direction of the capacitor current. jitter = u,/pI (1) In contrast to the linear voltage waveform on the capaci- tor (Fig. 3), the currents in the active devices of the sensor which is usually expressed in parts per million. circuit are quite nonlinear because of this regeneration. The To determine the noise in FM demodulation, it is more slope of these currents increases as the trigger point is desirable to know the frequency spectrum of an oscillation approached, as shown in (Fig. 4), so that noise in the with jitter. However, the problem is more clearly stated, circuit is amplified and randomly modulates the time at and solved, in the time domain, and the spectrum of the which the circuit switches. Thus, the noisy current of Fig. 5 jitter should, in principle, be obtainable from a Fourier produces the randomly pulsewidth modulated waveform of transformation. 796 IEEEJOURNALOF SOLID-STATECIRCUITS,VOL.SC-18,NO.6, DECEMBER1983 +v~ III. THE PROCESS OF JITTER PRODUCTION I I R R The switching of a floating capacitor oscillator in the presence of noise is now analyzed by examining the circuit 21.-1, + I. I II of Fig. 7 as it approaches regeneration. The device and al I parasitic capacitance are assumed to be negligibly small. VI V2 Q2 ~3 V4 Suppose Q2 conducts a small current 11, while Ql carries c ~ 10 + 10 the larger current 210 – ll. The current flowing through the ~ timing capacitor produces a negative-going ramp at V4, -v~ causing an increase in VBE(Qz ) and thus in the current Fig. 7 Generalized equivalentwcircuit of a relaxation oscillator for noise through Qz. A single stationary noise source In is assumed anatysis. to be present in the circuit, as shown in Fig. 7. The equations describing the circuit are (6) can then be integrated from t~ to tz as follows: 210 – II VBE,= Vrloge ~ (2) ‘R 21v~I +~–2R dI1 s J(1A 01 1 ) v~~, = vTloge : (3) s dVC =~o–~l (5) so that dt C where VT = kT/q and Is is the reverse leakage current at VK=$(t2– t4)–~:’;dt –R{In(t2)– In(tA)} (10) the base of the transistor. Substituting (2) and (3) into (4), and applying (5), the where VK, the left-hand side of (9), is a constant which differential equation describing the circuit is depends only on the choice of initial current 1A, l., and, from (8), on VT and R. The infkence of the noise current on the instant of switching is now evident from (10): random fluctuations in the value of In(rz) must induce corresponding fluctuations This may be rewritten as in t2 so that the sum of the terms on the right-hand side of (10) remains equal to the constant VK. More precisely, if t,4= O and t2= T (the half-period of the oscillation), then F(Z,, (T), T)~f$T-&!dt-R {L(T) -L@)} = constant (11) where the right-hand side consists of two terms, one due to and thus the autonomous dynamics of the circuit and one due to noise.

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