Study of Injection Locking With Amplitude Perturbation ... - IEEE Xplore

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 1, JANUARY 2012

137

Study of Injection Locking With Amplitude Perturbation and Its Effect on Pulling of Oscillator Ikbal Ali, Abhijit Banerjee, Arindum Mukherjee, and B. N. Biswas

Abstract—Injection locking characteristics of oscillators are studied both qualitatively and analytically and the closed-form expressions of frequency-pulling and spectrum of the unlocked driven oscillator is estimated with negligible amplitude perturbation. Modifications in spectrum, lock range, and pulling of the oscillator are shown under significant amplitude perturbation. Also an approximate expression of amplitude perturbation is presented. Numerical simulations and experimental results describing the oscillator’s pulling process under significant amplitude perturbation are given. Index Terms—Amplitude perturbation, ideal lock range, injection locking, modified lock range, nonlinearity, pulling.

I

I. INTRODUCTION

NJECTING A very minute periodic signal of frequency in the close vicinity of the free running oscillator completely changes the oscillator dynamics. Interesting pulling or locking phenomena are observed, describing the way every oscillatory system in nature responds to the external stimuli. Although rigorously studied over the past few decades by Adler [1], Kurokawa [2], and others [3]–[8], is still in its infancy from practical point of view. Injection of a small external signal close to the oscillator frequency perturbs both the phase and the amplitude of the oscillator and ultimately it locks to the injection signal frequency. Only the phase perturbation under small amplitude perturbation (weak injection) has been extensively reported in the literature since Adler published his classic paper in 1946. Understanding the potential application of the phenomena in the communication link, recent study on it under low level of injection has been published during last few years [7]–[13], [19]. It is well known that amplitude perturbation is a characteristic feature of injection locking and it becomes significant and can no longer be avoided under strong injection. Study of injection locking under strong injection can also be found in literature [15]–[18], but very little study of the amplitude perturbation and its effect on the locking characteristics of the oscillator with the injection signal can be found in the literature. Recent study of injection locking of oscillator by Razavi [7] also carefully avoided the amplitude perturbation Manuscript received October 09, 2010; revised March 26, 2011; accepted June 15, 2011. Date of publication December 20, 2011; date of current version January 11, 2012 This paper was recommended by Associate Editor I. M. Filanovsky. I. Ali is with the Department of Electronics and Communication Engineering, Academy of Technology, Adisaptagram, Hooghly 712 121, India (e-mail: ([email protected]). A. Banerjee and A. Mukherjee are with with the Department of Applied Electronics and Instrumentation Engineering, Academy of Technology, Adisaptagram, Hooghly 712 121, India (e-mail: ([email protected]; [email protected]). B. N. Biswas is with the Sir J. C. Bose School of Engineering, SKFG Institutions Mankundu, Hooghly 712 139, India e-mail: (baidyanathbiswas@gmail. com). Digital Object Identifier 10.1109/TCSI.2011.2161361

Fig. 1. Equivalent circuit diagram of a negative differential conductance oscil. lator with injection signal

issue of injection locking of oscillator. Most recently Lai and Roychowdhury [11] presented a oscillator macromodeling technique to capture the amplitude perturbation. But the technique cannot tell us the basic structure of the amplitude perturbation signal and the modifications of the oscillator locking properties compared to small amplitude perturbation. We have captured and studied the modifications of the locking characteristics of the oscillator under noticeable amplitude perturbation both analytically and experimentally in this paper. II. ANALYTICAL APPROACH The nonlinearity is the heart of injection locking. Each and every oscillator in nature has inherent nonlinearity, making injection locking a natural phenomena. Deep understanding and more insight of the phenomena will make us to use it more advantageously and simultaneously to reduce its effect whenever undesirable. Surprisingly, it is the unlocked driven state rather than the locked state of the oscillator that will give us the clear picture of the locking dynamics within the oscillator. In this paper we present a comprehensive analysis to knit together the various loose strands in the literature and to give other new aspects of injection locking, so far not reported in the literature. Our approach is truly analytical based on the negative differential conductance oscillator. The analytical equivalent circuit is shown in Fig. 1. The analytical approach that is presented here differs from the three terminal oscillators [14], but the final governing equations are of the same form. The output of such an oscillator and the synchronizing signal are respectively taken as (1) (2) where represents the equivalent injection signal voltage applied to the negative differential conductance device (see Appendix A, if the device is a transistor). As a result of the interaction of the two signals, the instantaneous phase and amplitude of the oscillator change, which can be assumed to be of the form

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(3)

138

The instantaneous frequency


can be expressed as

(4) The diode dynamic characteristic

is expressed as (5)

The fundamental component of the diode current is expressed as (6) That is, the equivalent diode conductance can be given as (7) Therefore, the system equation can be written as

(8) is the conductance of the equivalent oscillator’s where tank circuit and is given by

Fig. 2. Steady state phase variation of the oscillator with normalized detuning.

where , is the unperturbed frequency difference between the oscillator and the injection signal or the detuning term. Here (12) and (13) are two coupled nonlinear differential equations, indicating that not only the phase but also the amplitude of the oscillator are perturbed or modulated by the external signal. In other words the locking dynamics is governed by these two equations, from the beginning, i.e., at (when the injection ) through unlocked driven state and to locked state. Unfortunately, the exact solution of is not possible considering both the modulations simultaneously. Though an approximate expressions can be obtained to understand the effect of amplitude perturbation on the locking , when the injection mechanism. At the beginning, i.e., at , the oscillator oscillates with the signal is absent, i.e., . Hence from unperturbed steady amplitude, i.e., , i.e., the unperturbed (12), and being the normalized amplitude of the oscillator is amplitude. A. Phase Modulation Let us assume that the degree of amplitude modulation is negligible (i.e., under weak injection) and it can be ignored, so that and normalized amthe oscillator maintain its amplitude . Then the external signal will only perturb the plitude phase of the oscillator. In that case (13) will become

(9)

(14)

Assuming is close to and substituting from (4). Using (3), (7), and (9) into (8) and equating the real and imaginary parts of (8), one finds

. The aforesaid equation is where the familiar Adler’s equation [1]. The denotes the maximum possible synchronizing range without amplitude modulation being the instantaneous phase of the local oscillator and difference between the oscillator and the impressed signal. Phase dynamics of the system (14) signifies that the steady state is achievable if (as ) without amplitude moduand the stationary phase angle . If lation can be written as: and steady state is achieved, , otherwise for in steady state . For . The steady state phase variation without amplitude modulation is shown in Fig. 2. It is interesting to note that for , the detuning is positive and in the the first case second case is negative. Now for the case where there is no amplitude modulation one can write (according to , the phase difference Adler [1]), for

(10) (11) (12)

(13)

(15)

ALI et al.: STUDY OF INJECTION LOCKING WITH AMPLITUDE PERTURBATION

assuming , where constant. Putting

and

is an integration

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Combining (22), (23), and (24), one can write

(16) (17) and differentiating (15) one finds

(25) (18)

From (18) it can easily be shown that

(19)

Thus the unlocked driven oscillator contains a carrier component at injection signal frequency and the side band components which are separated from each other by the beat-frequency . In order to get a complete picture of the locking mechanism we should analyze (25) under different injection conditions: 1) Fast Beat: Let us first examine the above result for the , the lock range. In this case obviously X detuning is large so that one can write:

where (20) The average value of the oscillator frequency averaged over the time interval , in the presence of the synchronizing signal is thus given by , where is the beat frequency

and the beat frequency becomes . Thus the beat frequency is high and the sideband components are wide apart. We can approximately write (25) as

(21) . A similar closed form expression has where been derived in [8] and [10] for the frequency shift [the term within the parenthesis of (21)] following PPV phase-domain , the average oscilmacromodel. We note here that if lator frequency will be higher than the free running frequency and vice versa. Also if the detuning term , the lock range, then average frequency of the oscillator will be close to , i.e., the perturbation of the external signal is small. Similarly, as we gradually decrease the detuning while fixed, the oscillator will be more and more pulled keeping toward the frequency of the impressed signal and ultimately the center frequency of the oscillator becomes equal to that of the external signal for . In that case oscillator loses its identity and its frequency of oscillation becomes entrained by the external signal. We say that the oscillator is locked by the injected signal. Now we are in a position to find out the spectral components of the unlocked driven oscillator without amplitude modulation. Writing the oscillator output as

(26) where . We see from (26) that the spectrum presents a symmetric shape with two equal amplitude sidebands when the perturbation is small as shown in Fig. 3(a). Sideband components are of much smaller strength than the central one. We now reduce fixed then we can approxithe detuning , keeping mately write: and . Again (25) can be written as

(22) (27)

Using (14) one can easily show that

(23) Also from (15), (16), and (17), one gets

(24)

The spectrum contains a carrier component at frequency and two nonsymmetric side bands: one at frequency corresponds to impressed signal and the , other one of smaller amplitude at frequency shown in Fig. 3(b). 2) Quasi-Lock: If we gradually decrease the detuning maintaining fixed, the amplitude at the driver component will increase faster compared to the other components and the asymmetry will increase more and more. Now the condition is reached, when the amplitude

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nonlinear differential equations. We will try to find an approxthat will able to give us the modificaimate expression of tion of the locking characteristics of the oscillator. We first see , considthe effect on the oscillator steady state amplitude , leading ering amplitude modulation. We assume that to be positive. Thus (12) to steady state phase difference and (13) in the steady state modifies into (29) (30)

Fig. 3. Oscillator spectrum components for different decreasing detuning

.

of the carrier and the component at driver becomes same and equal to [see frequency and amplitude Fig. 3(c)]. The value of under such circumstances can be found in the literature [8]. The slight mismatch is due to the amplitude approximations in [8, eq. (32)]. If the detuning is further reduced the spectrum component at driver frequency becomes the dominant one and the oscillator response becomes quasi-locked. Thus we can say that the condition separates the fast beat state and quasi-locked state of the oscillator qualitatively. We further reduce the detuning to such extent that it is close to , i.e., leads to . Therefore from (25) one finds

(28) Equation (28) shows that there are no spectral components above , but a large number of spectral components with . The progressively decreasing strength is present below spectral distribution is quite asymmetric [see Fig. 3(d)]. 3) Locked: Further reducing detuning, we obtain the condi. Under this condition the beat frequency tion and the amplitude of the harmonics are zero. Actually as we gradually decrease the detuning, the beat frequency decreases, signifying that all the harmonic components shift toward (with continuous decrease of amplitude) the driving component and ultimately all harmonic components of the driven oscillator merge together at the frequency of the external driver. We say that the oscillator is locked to the injected signal [see Fig. 3(e)]. B. Amplitude Modulation The complete picture of the locking dynamics will not be achieved if we ignore the amplitude perturbation of the oscillator by the impressed signal. In doing so we have to extract by solving both (12) and (13) simultaexact expression of neously. But it is almost impossible to solve these two coupled

where . Equation (29) shows that steady state equivalent normalized locked amplitude of the oscillator under injection, i.e., , where is the steady state equivalent locked amplitude of the oscillator considering amplitude modulation. Similarly (30) shows that the lock range of the oscillator modifies to . Hence we can say that the amplitude modulation reduces the lock compared to range of the oscillator by a factor that of without amplitude modulation, i.e., ideal lock range In conclusion we can say that significant amplitude perturbation of the oscillator reduces the synchronization range of the oscillator, so far discussed as and we should define modified to account amplitude lock range as perturbation instead of , as the synchronization range of the oscillator. Thus (30) can be written as (31) Equation (31) shows that , in locked state, which becomes unrealistic if we do not replace by , . We conclude that amplitude perturbation rewhere duces the lock range and thus we also have to reduce detuning, to lock the oscillator with the impressed signal. In other i.e., words we should decrease injection frequency to where . Hence (32) represents the locking condition of the oscillator considering amplitude modulation (32) or ) to the Further if there is no injection (i.e., or . We can oscillator, (29) shows that now write the amplitude perturbation and phase perturbation equation as

(33) (34) where , modified normalized amplitude of the oscillator including amplitude perturbation. Now we are in a position to show that the amplitude modulation reduces the pull-in of the oscillator and it will not lock to the injection any more. In doing so we consider the signal at frequency , where is amplitude perturbation as the amplitude perturbation and . From (34) one can show that (35)


where

141

and

(36) where . The solution of (35) can be written as

(37)

Fig. 4. Variation of normalized amplitude of harmonics with normalized detuning .

being the integration constant. Substituting (36) into (37) one can easily show (42) (38)

where

where and

Now we can write from (34)

(39) where

can be approximately written as

(40) From (38), (39), and (40) one can easily write where

. If we consider , then . Thus the amplitude modulation or it forces to reduce forces to reduce the detuning to to . In other word, the oscillator will no longer lock frequency of the impressed signal at the same injection to so level. Hence (34) should be modified by replacing to as to represent the locked state of the oscillator precisely (41) It is not also difficult to understand that we should modify the beat frequency term with in all perturbed oscillator response expressions to capture the effect of amplitude perturbation. Thus the amplitude modulated signal can be rewritten from (38) as

Though (42) is an approximate expression, still it reveals the basic structure of the amplitude modulated signal, consisting of a dc together with the beat frequency component and its har, monics with decreasing amplitudes. If the detuning the amplitude of the harmonic components are very small, i.e., the amplitude perturbation is negligible. Continuous decrease keeping constant increases the degree of of the detuning the amplitude perturbation of the oscillator with significant injection and ultimately it becomes maximum when the oscillator and very close to is in its quasi-locked state the locked state. The strength of the th order harmonic component present in the amplitude modulated signal can be written as (43)

The variation of strength of the harmonic components versus is shown in Fig. 4. The fundathe normalized detuning is dominant over other harmonics mental component present and hence we can say that the amplitude perturbation of the oscillator will be maximum when this fundamental compo. From (43) one can easily nent achieves its peak value at

142


Fig. 5. Variation of normalized detuning at maximum amplitude modulation for different values of .

show that this maxima governed by

for the fundamental component is

Fig. 6. Spectrum components of Oscillator output and amplitude modulated signal for different decreasing detuning with significant amplitude perturbation.

. One can approximately show that (44) The numerical solution of (44) with different values of is shown in Fig. 5. It reveals that amplitude modulation becomes maximum in the quasi-locked state and very close to the locked state of the oscillator. Equation (43) also reveals that the amplitude perturbation vanishes (i.e., amplitude of the components becomes zero) as soon as the oscillator achieves the . Simultaneously, gradual decrease locked state for keeping constant reduces the beat freof the detuning continuously, i.e., shifting the amquency plitude harmonics toward zero frequency component and finally merges to the dc component in the locked state as can be seen from Fig. 6. The phase modulated signal at least should be modified considering amplitude perturbation by replacing to and to as

(46) where

(45) It is obvious from (42) and (45) that amplitude of the th harmonic component of the amplitude modulated signal is reduced compared to the phase by a factor modulated signal and also the higher harmonic components are reduced more. Hence the degree of nonlinearity present in the amplitude modulation is less than that of phase modulation. Thus one should capture the information efficiently from the envelope of the ILO rather than from its phase if the injected signal contains information in its phase. Considerable amplitude perturbation modifies the spectral components of the oscillator and significant modifications can be captured if we write the amplitude perturbation as (42) and the output of the oscillator as

and . Hence the oscillator output contains progressively decreasing spectral components below the driver frequency separated by the beat frequency (originating mainly from phase perturbation) together with very weak spectral composeparated by (orignents of decreasing strength above inating mainly from amplitude perturbation). Thus the single sided spectrum (opposite to the driver) of the unlocked driven oscillator does not exactly unfold the mystery of the driven oscillator. The weak components on the same side of the driver were also experimentally reported by Stover [4]. These weak components rise to its peak value when the amplitude perturbation gets maximum, deep into the quasi-locked state and ultimately vanishes when the oscillator becomes locked with the impressed signal as shown in Fig. 6. III. SIMULATION RESULTS A numerical simulation technique is used to capture the phase and amplitude perturbation and injection locking of a Colpitts oscillator in a prototype electrical simulator developed in the


Fig. 7. Colpitts oscillator.

Fig. 8. Variation of lock range with increasing perturbation signal strength in simulation. “ ”—Full Simulation and “ ”—Analytical.

frame of TINA circuit simulator. In this example we have analyzed injection pulling in the Colpitts oscillator shown in Fig. 7. The Colpitts oscillator circuit chosen (Fig. 7) in this example can also be represented by the equivalent circuit of analytical model as shown in Fig. 1, presented in Appendix B [see (56)]. The nonlinearity of the active core of the transistor is also similar to that of the chosen expression () of analytical model (see (50) of V, Appendix A). The parameters are F, pF, pF, nH, with a series resistance of 11 . The bipolar transistor is a SPICE-BJT model of type and is described through the conventional Ebers-Moll Spice model represented in the right-hand side of Fig. 7. The model parameters are pF, pF, . With this model, the free running frequency of the Colpitts oscillator is found to be 75.04 MHz with peak to peak amplitude 2.44 V as shown in Fig. 9(a) and (f). An external current signal is injected at the base node of the BJT to analyze the injection perturbation effect of the oscillator. In this injection locking analysis we plot (Fig. 8) the relationship between injection amplitude and the lock range using full simulation. It is interesting to note that lock range maintains a linear relationbelow a threshold value of ship with the injection amplitude mA and it deviates from the linear relationinjection ship above the threshold. Significant amplitude perturbation is responsible for this nonlinearity and the synchronization range to the of the oscillator deviates from the ideal lock range . We have measured modified lock range and the quality factor of the tank circuit in the simulation experiment and hence calculated the lock range from analytical expression (Table III). It seems that analytical results closely match with the full simulation value except in the very high injection level, where it slightly deviates from the

143

TABLE I FULL SIMULATION AND ANALYTICAL DERIVATION NEGLIGIBLE AMPLITUDE PERTURBATON

FOR

TABLE II FULL SIMULATION AND ANALYTICAL DERIVATION SIGNIFICANT AMPLITUDE PERTURBATON

FOR

COMPARISON

OF

COMPARISON

OF

full simulation value (Fig. 8 and Table III). This discrepancy can be improved if we take the more accurate phase perturbation equation for large injection level [15, eq. (7)]) instead of [15, eq. (13) and the lock range expression as (9)]. We also have measured the locked amplitude of the oscillator for different injection level (Table III) and it is interesting to see that locked amplitude gradually decreases from the unperturbed amplitude, signifying that lock range should increase with the injection signal amplitude. This phenomena can also be found in literature [16, Fig. 5], [18, Fig. 4]. Pulling effect of the oscillator is studied for both negligible amplitude perturbation at injection amplitude 0.55 mA and large amplitude perturbation at injection amplitude 2.4 mA for many different values of driving frequency, shown in Figs. 10 and 9 respectively. Both the time domain and the frequency domain representation of the output reveal the locking mechanism of the oscillator with the synchronization signal. For both cases the lock range is found to be 0.402 MHz and 2.56 MHz respectively. We have chosen the external signal frequency to be lower than the oscillator frequency. Gradual increase of the external signal frequency towards oscillator frequency (maintaining injection signal amplitude constant), perturbs both the phase and amplitude of the oscillator and the oscillator frequency is pulled towards the injection signal frequency (Figs. 9 and 10). Initially the detuning and beat frequency is high and the oscillator is in the fast beat conditions [Fig. 9(b), (g), and Fig. 10(a), (f), (b), (g)]. The oscillator just enters into for (small the quasi-locked state [large perturperturbation in Fig. 10(c)) and for bation in Fig. 9(c)]. Further reducing the detuning makes the oscillator to enter deep into the quasi-locked state [Fig. 9(d), (i) and Fig. 10(d), (i)]. Ultimately the oscillator just locks to the injection signal and its frequency is entrained by the external signal [Fig. 9(e), (j) and Fig. 10(e), (j)]. It can also be seen that the repetition frequency of the envelope of the unlocked driven oscillator gradually decreases as the detuning (beat frequency) is reduced. Tables I and II compare full simulation variation of the detuning versus frequency shift (beat frequency) with the analytical derivation for both negligible and significant amplitude perturbation simultaneously. The results closely match for the small perturbation case, but it deviates for large perturbation due to approximation in the analytical model. Our phase modulation (13) does not accurately account the nature

144


Fig. 9. Output and spectra of the 75.04 MHz Colpitts oscillator under different perturbation frequencies in simulation (large amplitude perturbation). TABLE III COMPARISON OF LOCK RANGE AT DIFFERENT INJECTION SIGNAL IN FULL SIMULATION EXPERIMENT WITH ANALYTICAL DERIVATION

Fig. 10. Output and spectra of the 75.04 MHz Colpitts oscillator under different perturbation frequencies in simulation (small amplitude perturbation).

components on the same side of the synchronizing signal, originating from large amplitude perturbation can be clearly seen in Fig. 9. They are separated by the beat frequency comdecreases they grow up, and ponents and as the detuning [Fig. 9(d)] in the achieves maximum value for quasi-locked state, close to the locked state. IV. EXPERIMENTAL RESULTS

of phase perturbation under significant injection. Instead [15, eq. (7)] will give us more accurate beat frequency expression rather than for significant injection level, that is applicable for both low and high level injection. The weak

An implementation and study of injection pulling of Colpitts oscillator of Fig. 7, has been experimentally demonstrated. The oscillator is designed to achieve a free-running frequency of 5.31253 MHz and peak to peak amplitude of 342 mV. The deV, signed parameters are F, pF, pF, , with a series resistance of 337 . The bipolar transistor is of model no. . The external signal is injected into the base from a signal generator through a series combination of a resistor and and F. The perturbed oscila capacitor of value lator response is measured with the help of a DSO of model name, GDS-1022 of GW-INSTEK and stored as excel format in a PC, interfaced with the DSO. The time-domain and the frequency-domain oscillator response is generated with the help of MATLAB language. The FFT is done over 4000 data points and no window function is used to truncate the data. We have


145

Fig. 11. Variation of lock range with increasing perturbation signal strength in experiment. “ ”—Experimental and “ ”—Analytical. TABLE IV COMPARISON OF LOCK RANGE AT DIFFERENT INJECTION SIGNAL EXPERIMENT WITH ANALYTICAL DERIVATION

IN

Fig. 12. Output and spectra of the 5.31235 MHz Colpitts oscillator under different perturbation frequencies in experiment, fast beat condition.

first measured the lock range of the oscillator under different injection signal strength and also the locked amplitude of the oscillator as presented in Table IV. It is interesting to see that locked amplitude of the oscillator gradually increases from the unperturbed amplitude with the injection signal strength, signifying that lock range of the oscillator should not follow the linear ideal lock range relation; instead it varies rather nonlinearly with the injection signal amplitude, as in Fig. 11. However this nonlinearity becomes significant if the injection amplitude exceeds 3.363 A and the oscillator response contains large amplitude perturbation in its unlocked driven state. Otherwise the amplitude perturbation appears to be negligible and the lock range nearly traces the ideal lock range curve, as evident from Fig. 11. The quality factor of the tank circuit is measured and it is found out to be . and the lock from analytical expression is then calculated for difrange ferent injection signal strength and is presented in Table IV. It is interesting to see that, calculated lock range closely matches with the experimental lock range (Fig. 11 and Table IV). The oscillator response is then measured under large injection signal amplitude by choosing an amplitude of 7.607 A of the injection signal and with different detuning frequencies. The lock range is found to be 0.3735 MHz. Fig. 12 shows the various stages of the fast-beat and Fig. 13 shows different stages of quasi-locked

and locked states of unlocked driven oscillator both in time domain as well as in frequency domain. We have chosen the frequency of the injection signal lower than that of oscillator free running frequency. Now gradual increase of the injection signal frequency towards oscillator free running frequency pulls the oscillator frequency towards external signal, like 5.31 MHz oscillator frequency with 4.34 MHz injection signal in Fig. 12(b) to 5.228 MHz oscillator frequency with 4.77 MHz injection signal in Fig. 12(e) in fast-beat condition. As the detuning is reduced spectral components of the unlocked driven oscillator grow up above the injection signal frequency mainly from phase perturbation, as well as weak spectral components below the injection signal frequency (4 MHz in (c), 4.248 MHz in (d) and 4.305 MHz in (e) of Fig. 12) mainly from amplitude perturbation. Fig. 12(e) describes the boundary between fast-beat and quasi-locked states where the driver component (4.772 MHz) strength (0.994 V) becomes equal to that of the oscillator carrier component (5.228 MHz, 1 V). Further reducing the detuning causes the oscillator dominant component to be entrained by the external signal [Fig. 13(a) to (d)]. Finally the oscillator is entrained completely by the external signal in the locked state of the oscillator [Fig. 13(e)]. The repetition frequency of the envelope of the unlocked driven oscillator decreases continuously as the detuning and hence the beat frequency is reduced and ultimately vanishes in locked state of the oscillator [Fig. 12(f) to (j) and Fig. 13(f) to (j)]. Significant amplitude perturbation and the corresponding spectral components below the perturbation

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Fig. 14. Simplified transistor circuit to calculate large signal transconductance gain of it.

V. CONCLUSION It should be understood from the discussion so far that lock range is not a linear function of the injection signal amplitude (according to Adler [1]), rather obeys a nonlinear relationship in general. Small injection signal cannot exploit the amplitude nonlinearity to become significant and the steady amplitude of oscillator thus remains constant for certain range of injection amplitude, making the lock range to follow a linear relationship. However, nonlinearity grows with the injection signal forcing the lock range curve to increase or decrease nonlinearly as the steady amplitude of the oscillator decreases or increases depending on the pattern of the nonlinearity present. Though approximate, our large perturbation model still successfully captures the modifications of the injection locking dynamics under significant injection perturbation. APPENDIX A NONLINEARITY PRESENT IN TRANSISTOR ACTIVE CORE

Fig. 13. Output and spectra of the 5.31235 MHz Colpitts oscillator under different perturbation frequencies in experiment, quasi-locked, and locked states.

The large signal transconductance gain of the transistor with the nonlinearity present as a function of the input signal can be explicitly derived considering the transistor model as shown in at the base node generates Fig. 14. The input signal signal voltage across the base emitter junction. Thus the total base emitter voltage can be given as (47)

COMPARISON

OF

TABLE V EXPERIMENTAL BEAT FREQUENCY WITH ANALYTICAL DERIVATION

where is the constant voltage across the base emitter junction. The total collector current flowing through the transistor can be written as

(48) and is the modified Bessel’s function being the reverse saturaof first kind and of order n, being the thermal equivalent tion current of BE junction, voltage, and being the transistor current gain. The large signal transconductance gain of the transistor is defined as where

signal is observed. These components grow up to a maximum value and again die out with the amplitude modulation. Injecting 4.82 MHz signal produces spectral components [Fig. 13(a)] separated from each other by 0.387 MHz, 0.375 MHz, 0.38 MHz, and 0.375 MHz respectively above the driver component and 0.373 MHz below the driver component, i.e., the spectral components originating from amplitude perturbation are separated . The experimental from each other by the beat frequency beat frequency at different detuning frequencies are compared with the analytical calculation, presented in Table V. The discrepancy among them is due to the approximate beat frequency expression (for low level injection) is used for strong injection.

(49)


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Fig. 15. Hybrid-pi equivalent model of Colpitts oscillator (Fig. 7).

(56)

where is the fundamental component of the collector and is the current, small signal transconductance gain of the transistor. Now and as , one can easily show Thus that (50) APPENDIX B EQUIVALENT CIRCUIT MODEL OF COLPITTS OSCILLATOR The hybrid-pi equivalent model of Colpitts oscillator (Fig. 7.) and can be conis shown in Fig. 15. Now as , one can write the node equation at the sidered as part of collector node as

(51) and

where node as

being the load. Also at the base (52) , in (52) one can write

Now writing

(53) Substituting , the resonant frequency of the tank circuit of Colpitts oscillator operating at freinto (53) quency

(54) As

and

one can write (55)

Now (51) can be written as

Substituting

from (55)

Thus it is not difficult to understand from (56) that Colpitts oscillator can easily be modelled as the equivalent circuit shown in Fig. 1 without injection signal applied. ACKNOWLEDGMENT The author I. Ali would like to thank Israj Ali for his help and support in preparation of the final copy of the paper. REFERENCES [1] R. Adler, “A study of locking phenomena in oscillators,” Proc. IEEE, vol. 61, pp. 1380–1385, Oct. 1973. [2] K. Kurokawa, “Injection locking of microwave solid-state oscillators,” Proc. IEEE, vol. 61, pp. 1386–1410, Oct. 1973. [3] L. J. Paciorek, “Injection locking of oscillators,” Proc. IEEE, vol. 53, pp. 1723–1727, Nov. 1965. [4] H. L. Stover, “Theoretical explanation of the output spectra of unlocked driven oscillators,” Proc. IEEE, vol. 54, pp. 310–311, Feb. 1966. [5] M. Armand, “On the output spectrum of unlocked driven oscillators,” Proc. IEEE, vol. 59, pp. 798–799, May 1969. [6] B. N. Biswas, “On the output spectrum of unlocked driven oscillators,” Proc. IEEE, vol. 44, pp. 833–834, May 1970. [7] B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuits, vol. 39, no. 9, pp. 1415–1424, Sept. 2004. [8] P. Maffezzoni and D. D’ Amore, “Evaluating pulling effects in oscillators due to small-signal injection,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 28, no. 1, pp. 22–31, Jan. 2009. [9] X. Lai and J. Roychowdhury, “Capturing oscillator injection locking via nonlinear phase-domain macromodels,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2251–2261, Sep. 2004. [10] P. Maffezzoni, L. Codecasa, D. D’Amore, and M. Santomauro, “Closed-form expression of frequency pulling in unlocked-driven nonlinear oscillators,” Proc. 18th Eur. Conf. Circuit Theory Design, pp. 914–917, Aug. 2007. [11] X. Lai and J. Roychowdhury, “Automated oscillator macromodelling techniques for capturing amplitude variations and injection locking,” in Proc. IEEE ICCAD, Nov. 2004, pp. 687–694. [12] X. Lai, Y. Wan, and J. Roychowdhury, “Understanding injection locking in negative-resistance lc oscillators intuitively using nonlinear feedback analysis,” in Proc. IEEE Custom Integr. Circuits Conf., Sep. 2005, pp. 729–732. [13] P. Bhansali and J. Roychowdhury, “Gen-Adler: The generalized Adler’s equation for injection locking analysis in oscillators,” in Proc. IEEE Design Autom. Conf., Jan. 2009, pp. 522–527. [14] B. N. Biswas, Phase Lock Theories and Applications. New Delhi, India: Oxford and IBH, 1988. [15] S. Shekhar, M. Mansuri, F. O’Mahony, G. Balamurugan, J. E. Jaussi, J. Kennedy, D. J. Allstot, R. Mooney, and B. Casper, “Strong injection locking of low-Q LC oscillators: Modeling and application in a forwarded-clock I/O receiver,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 56, no. 8, pp. 1818–1829, Aug. 2009. [16] E. Monaco, M. Pozzoni, F. Svelto, and A. Mazzanti, “Injection-locked CMOS frequency doublers for -wave and mm-wave applications,” IEEE J. Solid-State Circuits, vol. 45, no. 8, pp. 1565–1574, Aug. 2010. [17] M. Mansuri, F. O’Mahony, G. Balamurugan, J. Jaussi, J. Kennedy, S. Shekhar, R. Mooney, and B. Casper, “Strong injection locking of low-Q LC oscillators,” in Proc. IEEE Custom Integr. Circuits Conf., Sep. 2008, pp. 699–702. [18] T. Kun-Hung, W. Jia-Hao, and L. Shen-Iuan, “Frequency dividers with enhanced locking range,” in Proc. IEEE Radio Freq. Integr. Circuits Symp., Apr. 2008, pp. 661–664. [19] F. C. Plessas, A. Papalambrou, and G. Kalivas, “A 5-GHz subharmonic injection-locked oscillator and self-oscillating mixer,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 7, pp. 633–637, Jul. 2008. Authors’ photographs and biographies not available at the time of publication.