A Study of Injection Pulling and Locking in Oscillators Behzad Razavi Electrical Engineering Department University of Califomia, Lo s Angeles Abstract This paper presents an an alysis that confers new insights into injection pulling and locking of oscillators and the re- duction of phase noiseun der locked condition. A graphical interpretation of Adler’s equation pred icts the behavior of injection-pulled oscillators in time and frequency dom ains. An identity derived from the phase and envelope equations expresses the required oscillator nonlinearity across the lock range. I. INTRODUCTION The phenomenon of injection locking was observed as early as the 17th century, when Christian Huygens. confined to bed by illness, noticed that the pendulums of two clocks on the wall moved in unison if the clocks were hung close to each other [l]. Attributing the coupling to mechanical vibrations transmitted through the wall, Huygens was able to explain the locking between the two cl0cks.l Injection pulling and locking can occur in any oscillatory system, includinglasers, electrical oscillators, and mechanical and biological machines. For exam ple, humans left in isolated bunkers reveal a free-running sleep-wake period of about 25 hours [2], ut, when brought back to nature, they are injection- locked to the Earth’s cycle. This paper deals with the study of injection pulling and locking in oscillators, presenting new insights that prove use- fu l in circuit and system design. Section I I describes examples wherein injection pulling becomes critical. Following qualita- tive observations i n Section m, Section IV provides a detailed analysis leading t o A dler’s equation [3] and employs a new graphical interpretation to predict the time- and frequency- domain behavior of pulled oscillators. Sections V and VI deal with the effect of oscillat or nonlinearity and phase noise, respectively. 1 1 . MOTIVATION Analog and m ixed-signal systems containing oscillators must often deal with the problem of injection pulling. A few exam- ples demonstrate the difficulty. Consider the broadband transceiver show n in Fig. 1 . Here, a phase-locked loop including VCOl provides a retiming clock ‘Huygens s known for inventing the pendu lum clock, disco vering th e naNre of the nng i m u d a m, and numerous other accomplishments. ........ : . ................................................. Retimer Fig. 1. Injection pulling in a broadbandmrceiver, for thetransm itteddata, which is subsequently amplifiedby the driver to deliver large currents or voltages to a low-impedance load, e.g., a laser or a 50-C2 line. The receive path incorporates VCOz in a clock and data recovery loop. In practice, VCO l is phase-locked to a local crystal oscillator, and VC Ol to incom- ingdata. As aresult, the two oscillatorsmay operate at slightly different frequencies, suffering from injection pulling due to substrate coupling. Similarly, the high-swing broadband data at theoutputof theTXd river may contain substantial energy in the vicinit y of the oscillation frequencies of VCOl and VC0 2, thus pulling both. Another example of pulling arises in RF transceivers if the power amplifier (PA) output spectrum lies close to the fre- quency of an oscillator (Fig. 2). The large swings produced Fig. 2. Injection pulling in an RF transceiver. by the PA couple to the oscillator through the substrate or the package, leading to considerable pulling. While injection pulling typically proves undesirable, injec- tion locking can be exploited as a useful design technique. For example, a n oscillator running at W O can be locked to a signal at 2W o to petform frequency division [4 , 5 , 61. Simi- larly, two identical oscillators operating at WO can be locked 1 3-4- 0-7803-7842-3/03/$17.00 0 2003 IEEE IEEE 2003 CUSTOM INTEGRATED CIRCUITS CONFERENCE 305
Fig. 3(d) operates with a unity loop gain, gmlRp= 1, then:
In other words, as winj approaches w0. the injected signal
circulates around the loop with a larger am plitude, "hogging"
a greater fraction of the availab le power. We then expect thatthe component at W O begins to lose energy to that at winj,eventually vanishing if the latter is sufficiently close. This
corresponds to injection locking.Le t us now compute the injection level and frequency.such
that the amplified input reaches an amplitude equal to that
of the free-running oscillator. For the circuit of Fig. 3(d),
the oscillation amplitude is approximately equa l to I o 8 c , p R p ,
where Iorc,penotes the peak excursion in the transistor draincur rent . Equ atin g V,,, fro m (3) to Iorc,pRp, e obtain,
(4 )
As we will see later, once win, enters the range given by
this equation, the circuit locks to the inpu t. In other words,
injection-lockingoccurs if the oscillator am plifies the input somuch as to raise its level to that of t he free-running circuit.
IV. FIRST-ORDERNALYSIS
With the foregoing observations in mind, w e can now con-sider a more general LC oscillator under injection.
A. Assumptions
In order to arrive at a mathematically-tractable formulation,
we make the following assumptions: (1) the injection level is
much less than the free-running oscillation amplitude; (2) theinput frequency is close to W O,.e., 1wi-j - O I <W O / & ;3)
the input is an unmodulated sinusoid.
For subsequent derivations, we need an expression for the
phase shift introduced by a tank in the vicinity of resonance.The circuit of Fig. 5 exhibits a phase shift of
Since w$ - w2 = Zwo(w0 - w ), L w l R p = 1/Q, and $712-tan-' I = tan-'(z-'). we have
t a n s -w ). ( 6 )WO
If the input current in Fig. 5 contains phase modulation, i.e.,
I,, = Iocos[wt+$(t)], thent heph ases hiftca n beobtained by
replacing w in Eq. (6 )with the instanta neous input frequency,
w + d $ / d t :
2Q d$t a n s = -(WO -w - -).WO dt (7)
Valid for narrowband phase modulation (slowly-varying $),
this approximation holds well for typ ical injection phenomena.
)Even for large-signal oscillation,we can a ~ ~ u m ehe "average" value of
y, is equal lo Rp'.
Fig. 5 . Phase shift in a nnk around resonance.
B . Oscillator und er Injection
The objective of our analysis is to determine the effect of
injection on the phase and envelope of the oscillator output.
In this section, we deal with the phase response - the moreimportant aspect.
Consider the feedback oscillatory system show n in Fig. 6,
where the injection is modeled as an additive input. The
t ' I
Fig. 6. Oscillatory syrtemunderinjection.
output is represented by a phase-modulated signal having a
carrier frequency of w;,j (rather than WO ). n other words,
the output is assumed to track the input except for a (possibly
time-varying) phase difference. This representation is justified
Equating this resu lt to V,,,,, cos(w ;,t + 8) . we obtain
$ + [ - winj - )]$ = 8, (14)at high frequencies. For example, quadrature oscillators may
WO dt face more severe trade-offs due to this trend.
w e also note from (10) thatThird, injection locking to a frequency winj # WO mandates
overation away from the tank resonance, where the Q begins-
V A C , ,+ V ~ , c , p l i n j , p c o s ~ dB* -dt VAC,, + 2 V 0 8 , , p K n j , p ~ ~ s ~ +ij,, dt
to degrade. This is in stark contrast to phase-locking, in whichcase theoscillation frequen cy can be varied (e.g.; by a varactor)while maintaining the tank at resonance.
Fourth, Eq . (19) confirms the trend predicted by Fig. 4,
indicating that injection locking is accompanied by a static
-
d8(15)-&>
and phase error:
(21)WO - Wi" j
0 = sin-'
K,j,,sinB (16) W L
which, as depicted in Fig. 7. eaches f r / 2 a t the edges of thelock range. Asmentioned in Section 11, at 8 = f a / 2 , t he zero
tan(8 - *) = V,,,,, + K,j,,cos8
4 7h . p si.n@, (17)VO,,,,
It followsfrom (14), (15), and (17) that
Originally derived by Adler [31 using a somewhat different
approach, this equation serve s as a versatile ind pow erful ex-
pression for the behavior of oscillators u nder injection.
C. Injection Locking 2 j
For the oscillator to lock to t he input, the phase difference,8, must remain constant with time. Adler's equation therefore
requires that:
Since I sin 81 5 1, the condition f or lock emerges as:
which is th e same as that expressed by Eq. (4). We denote[WO/(ZQ)](K,~ , , /V, ,~ ,, ) y WL with the understanding thatthe overall lock rq g e is in fact +WL around WO?
This study confirms the hypothesis illustrated in Fig. 3:
injection locking is simply a shift in the oscillation frequencyin response to the add itional phase shif t that arises from addin gan external signal to the feedba ck signal.
Several important conclusions can be drawn from theseequations. First, low-Q oscillators such as resistively-loaded
4We all wL the "one-sided ack range.
Fig. 7.Phase shift in an njection-lockedoscillator.
crossing s of the input fail to synch ronize the oscillator.
Varying with process and temperature to some extent, 0
may prove problematic if the phase relationship between the
input and the outpu t is critical in an app lication. For example,if an injection-locked oscillator serves a s a frequency dividerin a tree multiplexer or demultiplexer environment, then thevariation of the phase becomes undesirable.
D. Injection Pulling
If the injected signal frequency falls out of, but not veryfar from, the lock range, then the oscillator is "pulled." Thisbehavior can be studied by solving Adler's equation with the
assumption IWO win j l > W L = [~0/(2Q)](li~j,~/V~~~,
N o t e f r o m ( l8 ) t h a td 8 l d t re ac hes a m a x i m u m o f w o - ~ ; , ~+W L , asmallvalueco mpare d towo. Similarly, higherderivatives
of 8 are also small. That is. 8 indeed varies slowly.Adler's equation can be rewritten as:
Noting that sin8 = 2tan(9/2)/ [1 + tan2(8/2)]. making achange of variable tan(8/2) = U, and carrying out the in-tegration, we ani ve at:
+ W b tan-, (23)W L
2 WO -wi.j WO - i " j 2
where wb =JW.his paper introduces a
graphical interpretation of this equation that confers a great
deal of insight into the phenomenon of injection pulling.
Case I: Quasi-Lock Le t us first examine the above result
for an input frequency just below the lock range, i.e., win , <W O - W L but (w0 - u in , ) /w~ m 1. Under this condition,
W b is relatively small, and the right hand side of Q . ( 2 3 ) is
dominated by the first term (zz 1) so long as tan(Wbt/2) is
less than one, approaching a large magn itude only for a shortduration [Fig. 8(a)]. Notin g that the cycle repeats with a
period equal to Wb we plot 8 as shown in Fig. 8(h). The key
W b ttan- =
tan!
y1~
+1 .. ... ...,.. .......
-no b
(a)
of both WO -wjnj and WL (and henc e the injection level). (2 )Since the oscillator is almost injection-locked to the input for
a large fraction of the period, we expect the spectrum to con-tain significant energy at wnj. ( 3 ) Writing the instantaneous
frequency of the output as d(winjt + 8)/dt = wi,j + d8/dtand redrawing Fig. 8(b) with the mo dulo -2n transitions at
the end of each period removed F ig . 9(a)], we obtain the
:...
+n..i ..............................
wl"l+% K'"n_nOnnl ..;.... ......... ........
Quasi-Lock Phase Slip
(4
Fig. 8 . Phase variation of gn njection-pulled oscillator.
observation here is that 8 is near 90' most of the time - as ifthe oscillator were injection-locked to the inp ut at the edge ofthe lock range. At the end of each period and the beginnin g of
the next period, 9 undergoes a rapid 360' change and retumsto the quai -lock condition [Fig. 8(c)l.
We now study the spectrum of the pulled oscillator. Thespectrum has been analytically derived using different tech-niques [9, IO]. but additional insight can be gained if the re-
sults in Fig. 8 are utilized a he starting point. The following
observa tions can be made. (1)The perio dic variation of 9 at arate of W b implies that the outpu t beats with th e input, exhihit-
ing sidebands with a spacing of Wb. Note that W b is a function
result depicted in Fig. 9 (b). T he interestin g point here is that,fo r wjn j below the lock range, the instantaneous frequency of
the oscillator goes only above win, , exhibiting a peak valueof WO + WL as obtained from Q. (18). That is, the output
spectrum contains mostly sidebands above w+.
We now invoke a useful observation that the shape of the
spectrum is given by the probabilityd ensity function (PDF)of the instantaneous frequency [ I l l . Th e PD F is qualitatively
plotted in Fig. 9(c), revealing that most of the energy is con-fined to the range [win, W O+ W L ] and leading to the actualspectrum in Fig. 9(d). The m agnitude of the sidebands drops
Fig. 12. One-ponrepresentation of an oscillatorunder injection.
( 2 5 ) and separating the real and im aginary parts, we have,
1
LI+-V,,, = ~ i , j I i " ~ , ~ine (26)
d0 dv.,. d20
dt dtC l ( W i " j + -)- + c13vn".d0
dt(G I - G m ) ( ~ i n j --)!&e = ~ i , j I i , , j , ~ c 0 ~ 8 .27 )
To simplify these equations, we assume: (1) the envelopevaries slowly and by a small amount: (2 ) the magnitude ofthe envelope can be approximated as the tank peak current
produced by the - G , circuit, Io,,,p, ultiplied by the tank
resistance, G-I = Q L t w o ; (3 ) w:nj - w i rz h o ( w o - w;. , j);(4) winj WO where applicable; (5) the phase and its deriva-tives vary slowly. Equations (26)an d (27) thus reduce to
The first isAdler's e quation whereas the second expresses the
behavior for the envelope.To develop more insight, let us study these results within the
lock range, i.e., if d8fdt = dV, , , /d t = 0. Writingsin'B +cos28 = 1 gives the follow ing useful identity,
( w a ~ ~ ) Z +G 1 - G m V e , , u , p" j ,P
Fo r w i n j = WO ,
(31)i lLj,p
V,*,,p
Iinj,p
RpIosc,p
G, = G I - -
(32)
that is, thecircuitresponds by weakening he -G, circuit (i.e.,allowing more saturation) because the injection adds energy
to the oscillator. On the other hand, for Iwo - winj = W L ,
we have G, = GI, ecognizing that the -G, circuit must besufficiently strong under this c ondition. Figure 13 illustrates
the behavior of G, across the lock range.
= GI--.
Fig. 13. BehaviorofG, across the lack range
VI. PHASENOISE
The phase noise of oscillators can be substantially reduced
by injection locking to a low-noise source. From a time-
domain perspective. the "synchronizing" effect of injection
manifests itself as correction of the oscillator zero crossings
in every period, thereby lowering the acc umulation of jitter.
This viewpoint also reveals that (1) the reduction of phase
noise depends on the injection level, and (2) the reduction
reaches a maximum for win , = WO [Fig. 14(a)l (where thezero crossings of Iinj greatly impact those of IoaJ nd a
minimum for wi,j = W O W L [Fig. 14(b)] (where the zerocrossings of Iinjcoincide with the zero-slope points on
We present a new analysis of phase noise under injection
locking using the the one-port model of Fig. 12 an d Eq.
(32). As depicted in Fig. 15, the noise of the tank and the
running noise shaping function of Eq. (3) to Rp Iorc,p/Iinj,p:%I = 0 0
(33)c , pR - R -
2 Q b n - W O / 1”J .P
p - P I . . I
WO 4 ” j , ,I W * - W o I = - . - .
29 Io, , , ,
obtaining
(34)
+(a) t
Thus, the free-running and locked phase noise profiles meet at
the edges of th e lock range.
I t is interesting to note that the d erivations in [121 and [131conclude that thenear-carrier phase noise is nearly equal to that
of the synch ronizing input and relatively independent of th e
free-running oscillatorphase n oise. By contrast. ourderivation
reveals a significant contribution by the oscillator itself within
th e lock range and its dependence on the injection level. Thishas been indeed verified by measurements on a 1-GHz CMOSLC oscillator.
As illustrated in Fig. 14(b). if the input frequency deviatesfrom WO , the resulting phase noise reduction becomes less
pronounced. This can also be seen from Eq. (30) because
GI - G, drops to zero as the input frequency approacheswo*W L . General equations for this case and the case of noisy
input are given in [12, 131.
O y - O O f O L
1-c ‘v\/Lltnl
*i
Fig. 14. Effect of injectionlocking on itter (a) in l h e middle and (b) at lhe
edge of Ih e lockmge.
-I.)
+m:‘1
Fig. 15. Model forrtudyingphasenoise. REFERENCES
-G, cell can be represented as a current source I,,. Withno injection input, the average v alue of -G, cancels GI, nd
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noise frequency approaches WO!
Now suppose a finite injection with n o ph ase noise is ap-plied at the center of the lock range, w j n j = WO . Then,
Eq. (32) predicts that the overall tank admittance rises to
G I-
G , = I j n j , p / ( R ~ I o s c . p ) .n other words, the tankimpedance seen by I, at W O falls from infinity (with no in-
jection) to R p l o s c , p / I , n j , pnder injection locking. As th efrequency of I , deviates from WO, R pI o s e , p / I ~ nj , pontinues
to dominate the tank impedance a p to the frequency offset at
which the phase noise approaches that of the free-running os-cillator (Fig. 16 ). To determine this point. we equ ate the free-
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