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Oscillators are critical components in electrical and electronic engineering and other engineering and sciences. Oscillators are classified as free-running oscillators and injected oscillators. This chapter describes the background necessary for the analysis and design of injected oscillators. When an oscillator is injected by an external periodic signal mentioned as an injection signal, it is called an injected oscillator. Consequently, two phenomena occur in the injected oscillators: (I) pulling phenomena and (II) locking phenomena. For locking phenomena, the oscillation frequency of the injection signal must be near free-running oscillation frequency or its sub-/super-harmonics. Due to these phenomena are nonlinear phenomena, it is tough to achieve the exact equation or closed-form equation of them. Therefore, researchers are scrutinizing them by different analytical and numerical methods for accomplishing an exact inside view of their performances. In this chapter, injected oscillators are investigated in two main subjects: first, analytical methods on locking and pulling phenomena are reviewed, and second, applications of injected oscillators are reviewed such as injection-locked frequency dividers at the latter. Furthermore, methods of enhancing the locking range are introduced.
Department of Electrical and Electronic Engineering, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran
*Address all correspondence to: alirezahazeri@iauksh.ac.ir
1. Introduction
One of the most important blocks in the electronic systems is undoubtedly the oscillator block. The oscillator converts a direct current (DC) generated by power supply into an alternating current (AC) signal. The design of this block involves many trade-offs between phase noise, oscillation frequency range, power consumption, layout size, etc. Oscillators are employed in many applications such as phase-locked loops (PLL), frequency dividers/multipliers, clock recovery, frequency synthesizers, etc. Nowadays, the demand for compact and portable systems has increased. Hence, the oscillators, amplifiers, mixers, power amplifiers, etc. may be integrated on a chip. For example, FM radio, Bluetooth, GPRS, Wi-Fi, NFC, and GPS are integrated into modern mobile systems. According to the application, the oscillators are classified as the free-running oscillators and injected oscillators. Free-running oscillators have been extremely studied in many electronic engineering books and articles so far [1, 2, 3, 4, 5, 6]. In the free-running oscillators, there is no external signal injected into the oscillator. In the injected oscillators, an external periodic signal mentioned as an injection signal is injected to the oscillator, which may be deliberately applied by the designers to make an injected oscillator, or any injection signals are accidentally injected to the oscillator which may be generated from other blocks such as power amplifiers and other oscillators. By injected oscillators, designers can implement many high-performance blocks such as quadrature oscillators and frequency dividers/multipliers, frequency synthesizers without a frequency-locked loop, which are useful in the fast frequency locking-loop systems. Thus, the cost of fabrication is cheaper than a frequency-locked loop. There are many devices operating with different center frequencies. As a result, several oscillators and other devices are placed together for implementing a system called system on chip (SOC). However, when the oscillators are integrated with other devices on the chip, various signals with different center frequencies may leak through the substrate, parasitic elements, or packaging and be injected into the oscillator. Hence, the performance of the oscillators is changed which can be suitable or not dependent on their functionality. When an oscillator is injected with an injection signal, the pulling and locking phenomena occur. Thus, pulling and locking phenomena are important parameters for designers.
The injected locked oscillators designed by engineers are classified into three classes. While the oscillation frequency of the injection signal is near to the free-running oscillator, the first-harmonic injection locking takes place. When the oscillation frequency of the injection signal is near to the sub-/super-harmonic of the oscillation frequency of the free-running oscillator, frequency dividers/multipliers are realized. These phenomena occur since the nonlinear performance of the injected oscillators. So, the size of the layout and complexity of designing frequency dividers/multipliers are lower than the frequency-locked loop because they do not employ many blocks in the frequency-locked loop such as filters, charge pomp, and frequency detector [7]. Consequently, power consumption is reduced. Furthermore, they are fast and may be applied to the high-speed or high clock data recovery and fast-locking systems. At last, the phase noise of the injected oscillator is different from the free-running oscillator and is dependent on the phase noise of the injection signal. Therefore, while the injection signal is generated by an oscillator which owns excellent phase noise, the injected oscillator will have a better phase noise [7, 8]. This chapter tries to disclose all subtleties and challenges encountered during the design of injected oscillators.
The presented chapter aims to investigate the injected oscillators. A summary of the injected oscillator specifications regarding locking and pulling phenomena, previously significantly published papers about the first harmonic injection oscillator, frequency dividers and enhancing locking range are presented. Section 2 covers with introducing pulling and locking phenomena in the first-harmonic injection locking oscillator. Then, a free-running oscillator is implemented for exploring the pulling and locking phenomena by various injection signals. Section 3 will be dedicated to the pulling and locking formula, and beat frequency equation is discussed for injected oscillators for nonharmonic (LC) and harmonic (ring) oscillators. Section 4 will treat the implementation of injected locked frequency dividers and increase the locking range. First, a block of the injection-locked frequency dividers/multiplier is displayed. In a literature overview, two classes of realization will be recognized: the conventional LC-injection-locked frequency dividers where the injection signal is applied to the oscillator by the tail current source or a transistor connected to the output nodes. Furthermore, the previously important published papers are reviewed. Finally, some structures and techniques employed in order to extend the locking range of frequency dividers are exhibited from previously significant published papers. Section 5 will conclude with the main contributions of the presented chapter.
When an oscillator is injected with an external signal, two phenomena occur for the oscillator. Once the strength of the external signal and frequency difference between the external signal and free-running oscillator are not suitable for locking, the oscillator is perturbed, and the output signal is modulated called pulling phenomenon. In the pulling case, the output spectrum of the injected oscillators has some spurious tones along with the effective oscillation frequency of the external signal. While the strength of the external signal and frequency difference between the external signal and free-running oscillator are suitable, the oscillator is locked, and the frequency of the output signal is locked to the first, sub- or super-harmonic of the oscillation frequency of the external signal called locking phenomenon. The range of the oscillation frequency of the external signal causes a locking phenomenon called the locking range. In Figures 1 and 2, these phenomena are simply presented in the time domain and frequency domain.
Figure 1.
(a) Three-stage differential ring oscillator, (b) delay cell, (c) output voltage at the time domain, (d) output voltage at the frequency domain.
Figure 2.
(a) Injected delay cell, (b) locking phenomenon in frequency domain (Finj = 260 MHz and Iinj = Iosc/20), (c) locking phenomenon in time domain (Finj = 260 MHz and Iinj = Iosc/20), (d) pulling phenomenon in frequency domain (Finj = 262.5 MHz and Iinj = Iosc/20), (e) pulling phenomenon in time domain (Finj = 262.5 MHz and Iinj = Iosc/20), (f) pulling phenomenon in frequency domain (Finj = 265 MHz and Iinj = Iosc/20), (g) pulling phenomenon in the time domain (Finj = 265 MHz and Iinj = Iosc/20).
At first, a free-running oscillator is chosen. Next, the oscillation frequency, the output voltage oscillation amplitude, and the current oscillation amplitude are obtained. Second, the given oscillator is injected with the external signal, which is modeled by a current source and in parallel with the output nodes. Third, the results of the injected oscillator are revealed under locking and pulling phenomena.
The free-running oscillator, implemented by a three-stage differential ring oscillator, is depicted in Figure 1a along with the delay cell in Figure 1b. In the time and frequency domain, the output voltage of the oscillator is presented in Figure 1c and d. According to Figure 1b and c, there is no frequency or phase modulation on the output voltage. It has a center frequency (257 MHz). In addition, the output voltage amplitude is constant (0.481 V), and the oscillation current (Iosc) is equal to 0.18 mA. In fact, there is not any amplitude modulation (AM).
Figure 2a shows the given three-stage differential ring oscillator under an injection signal, which its current (Iinj) and oscillation frequency (Finj) are equal to Iosc/20 and 260 MHz, respectively. Assuming locking conditions are covered, the oscillation frequency of the injected oscillator is the same as the injected signal as shown in Figure 2b and c. Figure 2b and c displays the output voltage of the injected oscillator under locking phenomenon in the time and frequency domain, respectively. Figure 2b and c illustrates the output voltage of the injected oscillator locked in Finj. It is clear that there is not any modulation on the output voltage at the time and frequency domain. Then, the oscillation frequency of the injected oscillator is equal to the oscillation frequency of the injected signal.
Figure 2d–g illustrates the output voltage of the injected oscillator in the pulling phenomenon when Iinj = Iosc/20 and Finj is equal to 262.5 and 265 MHz. It is obvious that there are modulations on the output voltage at the time and frequency domain. The injected oscillator produces amplitude and frequency (or phase) modulation in the output voltage. According to Figure 2d and f, the center frequency of the oscillator is pulled to the frequency of the external signal. Furthermore, the beat frequency, instance variation of oscillation frequency, is reduced when Finj becomes near locking range as portrayed in Figure 2.
3. Review of the previously significantly published papers
The pulling and locking phenomena have been investigated in previously published papers [2, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. According to the output waveform, oscillators may be categorized as nonharmonic oscillators, for example, LC oscillators and harmonic oscillators such as ring oscillators. For the nonharmonic oscillators, the output waveform has a center frequency near the resonance frequency of the LC tank. Consequently, the output waveform is almost sinusoidal. For harmonic oscillators, since the output waveform is not sinusoidal, the higher harmonics effect on the output waveform. In fact, nonharmonic oscillators have an LC tank block operated similarly to a band-pass filter, and harmonic oscillators have a low-pass filter. The pulling and locking equations are different in these classes. Most of the methods can be used for a class. The locking phenomena have been detected in pendulum clocks in 1629–1695. At first, Adler explained pulling and locking phenomena for LC oscillators for a weak injection signal [9, 54]. Then, case studies that are more special have been reported such as [8, 10, 11, 12, 13, 14, 15, 16, 17]. In [10], Adler’s equation was extended for applications that the strength of the injection signal is not weak. In [8], by vector diagram of the instantaneous currents, the locking range (LR) equation was improved. The same locking range equation was obtained by another method in [16] and partly in [24, 29]. In [22] using nonlinear feedback analysis, injection locking was investigated. A cross-coupled oscillator under the injection signal is depicted in Figure 3. By writing differential equations on the output nodes of the oscillator and solving them in nonlinear, vector diagram or empirical methods, the locking range of the cross-coupled oscillator is achieved.
Figure 3.
Cross-coupled LC oscillator.
In Table 1, previously significantly published locking ranges are offered, which Q is the quality factor of the LC tank circuit. When Iinj/Iosc≪1, it is clear that the locking range is proportional to Iinj and inversely proportional Q and Iosc.
Previously significant published locking ranges for LC oscillators.
The pulling phenomena for LC oscillators were completely investigated in [8, 25, 27, 28, 33, 39, 41, 47, 48, 49]. When Iinj/Iosc≪1, the beat frequency mathematical formula (ωb) for LC oscillators is as follows:
ωbLC=Ω2−K2,K=ω0Iinj/2QIosc,Ω=ωosc−ωinjE1
According to Eq. (1), ωb is increased by increasing Ω or Iosc when other parameters are constant. Eq. (1) was improved for strong injection in [41, 47]. The multi-injection signals with different total phases have been reported for the LC oscillators in [24, 29, 56].
Nonharmonic injected oscillators, for instance, ring oscillators and relaxation oscillators, have been studied in several papers [21, 23, 26, 30, 46, 50, 52]. In [21], by using approach of the LC oscillators, a three-stage single-ended ring oscillator was studied. In [23, 30], a locking range equation was obtained at the time domain analysis. In [26], by using current phase diagrams, a locking range was calculated. The locking range of [26] was improved in [50] for a larger injection level. In [52], various locking range equations for ring oscillators were introduced. In addition, comprehensive and exact analyses of the ring oscillators were presented to both pulling case and locking case considering higher-order harmonics. Moreover, the multi-injection signals with the same frequency and different initial phase (δ) have been achieved for the ring oscillators. Like locking range for LC oscillators, the locking range for rings oscillators are calculated by solving nonlinear differential equations in the output nodes and proportional to Iinj/Iosc and inversely proportional to the number of stages (N) [52]. In Table 2, previously significantly published locking ranges are exhibited. Moreover, the beat frequency mathematical formula for ring oscillators is expressed as below [5]:
Previously significant published locking ranges for ring oscillators.
Ai is the amplitude of the ith output voltage harmonic and N is the number of stages.
ωbring=Ω2−k2,k=Iinj/NA1CL,Ω=ωosc−ωinjE2
Furthermore, the relaxation oscillators with injection signals were presented in [46]. In addition, a perturbation projection vector (PPV) was employed for analyzing both harmonic and nonharmonic oscillators [18, 19, 20]. However, using PPV is complicated, and most of the time it needs preprocessing by simulators such as Cadence.
At first, fractional frequency generators utilizing regenerative modulation have been introduced by [61]. A block diagram of the injection frequency divider/multiplier is displayed in Figure 4. By subharmonic (ωinj≈ω0/M) or super-harmonic (ωinj≈Mω0) injection signals, frequency dividers and multipliers were reported in many papers such as [13, 15, 17], where, M is a positive number. Frequency dividers were explored in [36, 38, 40, 43, 44, 45, 49, 60]. Studies on frequency multipliers/dividers are generally in two sections. The first one is to obtain an approximation equation of the locking and pulling phenomena [36, 38, 40, 43, 44, 45, 49]. The second one is increasing the locking range. In the frequency multipliers/dividers, injection signals may be injected from tail node or output node called direct injection as demonstrated in Figure 5. In [13, 15, 17], the general models of the frequency multiplier/dividers have been proposed. The transistor is modeled as a nonlinear block (NB). By using a summer and nonlinear block, a conceptual model was introduced [13]. Nevertheless, once an oscillator behaves similar mixers for the injection signal, for example, when the injection signal is parallel with the tail current source, this model is not correct. In order to achieve a general conceptual model, the summer block is replaced with a multiplier block [17]. However, this model is dependent on SPICE parameters and preprocessing. By phase-domain macromodel, injection-locked frequency dividers were analyzed. Nonetheless, the phase-domain macromodel requests preprocessing and time-consuming. For ÷ 2, an injection signal has been paralleled to the tail current source of the cross-coupled oscillator, and then, the injection signal is modeled as an equivalent injection signal at output nodes whose oscillation frequency and amplitude are ωinj/2 and 2Iinj/π, respectively [8]. Therefore, the locking range was accomplished similar to the first-harmonic injection locking. This locking range has been acquired by a slightly different analysis in [62]. The asymptotic analysis, or slowly varying amplitude, averaging method, and phase analysis have been utilized to analyze the injected oscillators which make frequency dividers [43, 44, 45]. In [45], the locking range has been obtained when the injection signal is applied to the tail current source. Moreover, by the numerical bifurcation analysis using continuation software such as AUTO, they have been analyzed [36, 40]. In [60], an exact analysis for the locking range in injection-locked frequency dividers has been proposed by phasor diagrams and differential equations.
Figure 4.
Block diagram of a frequency divider/multiplier.
Figure 5.
Conventional injection locking frequency divider/multipliers, (a) injection in the tail, (b) injection in the drain.
Due to the small locking range of injected LC oscillators, various techniques have been realized to enhance the locking range. Passive and active structures are explored for improving the injection efficiency such as combining inductors in series or parallel with the injection mixer to enhancing its transconductance, body biasing, transformer feedback, dual-resonance RLC resonators, dual injection for increasing the voltage and current injection paths, tapped resonators, switched resonators, harmonic suppression, and distributed injection to distribute the injection signals; in other words, the injection component is divided to several smaller components; input-power-matching and inductive input-matching network is located to the gate of the NMOS switch to heighten the injection power [63, 64, 65, 66, 67, 68, 69, 70, 71, 72]. Figure 6 discloses a quadrature LC oscillator employed in the injection signal.
Some basic concepts and definitions have been presented in this chapter. First, pulling and locking phenomena have been introduced which both contain for injection oscillators. Next, previously significantly published papers have been explored. Furthermore, locking range and beat frequency formula have been studied for both first-harmonic injected LC and ring oscillators. Finally, previously significantly published papers about injected locked frequency dividers have been reviewed. Moreover, some previously important published papers about increasing the locking range of the injected locked frequency dividers have been introduced.
The author would like to thank Kermanshah Branch, Islamic Azad University, Kermanshah, Iran, and my wife Dr. Z. Ebrahimipour for any supports.
References
1.Hong B, Hajimiri A. A phasor-based analysis of sinusoidal injection locking in LC and ring oscillators. IEEE Transactions on Circuits and Systems I: Regular Papers. 2018;66(1):355-368
2.Farahabadi PM, Miar-Naimi H, Ebrahimzadeh A. Closed-form analytical equations for amplitude and frequency of high-frequency CMOS ring oscillators. IEEE Transactions on Circuits and Systems I: Regular Papers. 2009;56(12):2669-2677
3.Leung B. VLSI for Wireless Communication. Springer Science & Business Media; 2011
4.Razavi B. Microelectronics. Upper Saddel River, NJ, USA: Prentice Hall; 2011
5.Hazeri AR, Miar-Naimi H. Generalized analytical equations for injected ring oscillator with RC-load. IEEE Transactions on Circuits and Systems I: Regular Papers. 2018;65(1):223-234
6.Hazeri AR, Miar-Naimi H. Novel closed-form equation for oscillation frequency range of differential ring oscillator. Analog Integrated Circuits and Signal Processing. 2018;96(1):117-123
7.Yuan F. Injection-Locking in Mixed-Mode Signal Processing. Springer; 2020
8.Razavi B. A study of injection locking and pulling in oscillators. IEEE Journal of Solid-State Circuits. 2004;39(9):1415-1424
9.Adler R. A study of locking phenomena in oscillators. Proceedings of the IRE. 1946;34(6):351-357
10.Paciorek L. Injection locking of oscillators. Proceedings of the IEEE. 1965;53(11):1723-1727
11.Stover HL. Theoretical explanation for the output spectra of unlocked driven oscillators. Proceedings of the IEEE. 1966;54(2):310-311
12.Campbell C. Beat-frequency spectra in a driven unlocked multimode SAW comb oscillator. In: IEEE 1987 Ultrasonics Symposium. IEEE; 1987. pp. 69-72
13.Rategh HR, Lee TH. Superharmonic injection-locked frequency dividers. IEEE Journal of Solid-State Circuits. 1999;34(6):813-821
14.Banai A, Farzaneh F. Locked and unlocked behaviour of mutually coupled microwave oscillators. IEE Proceedings - Microwaves Antennas and Propagation. 2000;147(1):13-18
15.Betancourt-Zamora R, Verma S, Lee T. 1-GHz and 2.8-GHz injection-locked ring oscillator prescalers. In: IEEE Symposium on VLSI Circuits, Digest of Technical Papers. 2001. pp. 47-50
16.Kuo J, Shih E. A 60-GHz 0.13 um CMOS divide-by-three frequency divider. IEEE Transactions on Microwave Theory and Techniques. 2003;51(5):1554-1559
17.Verma S, Rategh HR, Lee TH. A unified model for injection-locked frequency dividers. IEEE Journal of Solid-State Circuits. 2003;38(6):1015-1027
18.Lai X, Roychowdhury J. Capturing oscillator injection locking via nonlinear phase-domain macromodels. IEEE Transactions on Microwave Theory and Techniques. 2004;52(9):2251-2261
19.Lai X, Roychowdhury J. Automated oscillator macromodelling techniques for capturing amplitude variations and injection locking. In: Proceedings of the 2004 IEEE/ACM International Conference on Computer-Aided Design. IEEE Computer Society; 2004. pp. 687-694
20.Lai X, Roychowdhury J. Analytical equations for predicting injection locking in LC and ring oscillators. In: Proceedings of the IEEE of Custom Integrated Circuits Conference. IEEE; 2005. pp. 461-464
21.Mesgarzadeh B, Alvandpour A. A study of injection locking in ring oscillators. In: IEEE International Symposium on Circuits and Systems, ISCAS. IEEE; 2005. pp. 5465-5468
22.Wan Y, Lai X, Roychowdhury J. Understanding injection locking in negative-resistance LC oscillators intuitively using nonlinear feedback analysis. In: Proceedings of the IEEE of Custom Integrated Circuits Conference. IEEE; 2005. pp. 729-732
23.Gangasani GR, Kinget PR. A time-domain model for predicting the injection locking bandwidth of nonharmonic oscillators. IEEE Transactions on Circuits and Systems II: Express Briefs. 2006;53(10):1035-1038
24.Mirzaei A, Heidari ME, Abidi AA. Analysis of oscillators locked by large injection signals: Generalized Adler’s equation and geometrical interpretation. In: Custom Integrated Circuits Conference. CICC’06. IEEE; 2006. pp. 737-740
25.Razavi B. Mutual injection pulling between oscillators. In: Custom Integrated Circuits Conference. CICC’06. IEEE; 2006. pp. 675-678
26.Chien J-C, Lu L-H. Analysis and design of wideband injection-locked ring oscillators with multiple-input injection. IEEE Journal of Solid-State Circuits. 2007;42(9):1906-1915
27.Heidari ME, Abidi A. Behavioral models of frequency pulling in oscillators. In: IEEE International of Behavioral Modeling and Simulation Workshop. BMAS. IEEE; 2007. pp. 100-104
28.Maffezzoni P, Codecasa L, D’Amore D, Santomauro M. Closed-form expression of frequency pulling in unlocked-driven nonlinear oscillators. In: 18th European Conference on Circuit Theory and Design. ECCTD. IEEE; 2007. pp. 914-917
29.Mirzaei A, Heidari ME, Bagheri R, Chehrazi S, Abidi AA. The quadrature LC oscillator: A complete portrait based on injection locking. IEEE Journal of Solid-State Circuits. 2007;42(9):1916-1932
30.Gangasani GR, Kinget PR. Time-domain model for injection locking in nonharmonic oscillators. IEEE Transactions on Circuits and Systems I: Regular Papers. 2008;55(6):1648-1658
31.Bhansali P, Roychowdhury J. Gen-Adler: The generalized Adler’s equation for injection locking analysis in oscillators. In: Proceedings of the 2009 Asia and South Pacific Design Automation Conference. IEEE Press; 2009. pp. 522-527
32.Harutyunyan D, Rommes J, Ter Maten J, Schilders W. Simulation of mutually coupled oscillators using nonlinear phase macromodels. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. 2009;28(10):1456-1466
33.Maffezzoni P, D’Amore D. Evaluating pulling effects in oscillators due to small-signal injection. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. 2009;28(1):22-31
34.Shekhar S et al. Strong injection locking in low-$Q$LC oscillators: Modeling and application in a forwarded-clock I/O receiver. IEEE Transactions on Circuits and Systems I: Regular Papers. 2009;56(8):1818-1829
35.Dal Toso S. Analysis and Design of Injection-Locked Building Blocks for RF Frequncy Generation in Ultra-Scaled CMOS Technologies. PhD [Thesis]. 2010. Available from: http://paduaresearch.cab.unipd.it/3151/
36.Daneshgar S, De Feo O, Kennedy MP. Observations concerning the locking range in a complementary differential LC injection-locked frequency divider—Part I: Qualitative analysis. IEEE Transactions on Circuits and Systems I: Regular Papers. 2010;57(1):179-188
37.Maffezzoni P. Synchronization analysis of two weakly coupled oscillators through a PPV macromodel. IEEE Transactions on Circuits and Systems I: Regular Papers. 2010;57(3):654-663
38.Maffezzoni P, D’Amore D, Daneshgar S, Kennedy MP. Analysis and design of injection-locked frequency dividers by means of a phase-domain macromodel. IEEE Transactions on Circuits and Systems I: Regular Papers. 2010;57(11):2956-2966
39.Mirzaei A, Abidi AA. The spectrum of a noisy free-running oscillator explained by random frequency pulling. IEEE Transactions on Circuits and Systems I: Regular Papers. 2010;57(3):642-653
40.Daneshgar S, De Feo O, Kennedy MP. Observations concerning the locking range in a complementary differential LC injection-locked frequency divider—Part II: Design methodology. IEEE Transactions on Circuits and Systems I: Regular Papers. 2011;58(4):765-776
41.Ali I, Banerjee A, Mukherjee A, Biswas B. Study of injection locking with amplitude perturbation and its effect on pulling of oscillator. IEEE Transactions on Circuits and Systems I: Regular Papers. 2012;59(1):137-147
42.Yeh Y-L, Chang H-Y. Design and analysis of a W-band divide-by-three injection-locked frequency divider using second harmonic enhancement technique. IEEE Transactions on Microwave Theory and Techniques. 2012;60(6):1617-1625
43.Buonomo A, Schiavo AL. A deep investigation of the synchronization mechanisms in LC-CMOS frequency dividers. IEEE Transactions on Circuits and Systems I: Regular Papers. 2013;60(11):2857-2866
44.Buonomo A, Schiavo AL. Analytical approach to the study of injection-locked frequency dividers. IEEE Transactions on Circuits and Systems I: Regular Papers. 2013;60(1):51-62
45.Buonomo A, Schiavo AL, Awan MA, Asghar MS, Kennedy MP. A CMOS injection-locked frequency divider optimized for divide-by-two and divide-by-three operation. IEEE Transactions on Circuits and Systems I: Regular Papers. 2013;60(12):3126-3135
46.Yuan F, Zhou Y. Frequency-domain study of lock range of non-harmonic oscillators with multiple multi-tone injections. IEEE Transactions on Circuits and Systems I: Regular Papers. 2013;60(6):1395-1406
47.Ali I, Biswas B, Ray S. Improved closed form large injection perturbation analytical model on the output spectrum of unlocked driven oscillator—Part I: Phase perturbation. IEEE Transactions on Circuits and Systems I: Regular Papers. 2014;61(1):106-119
48.Mirzaei A, Darabi H. Mutual pulling between two oscillators. IEEE Journal of Solid-State Circuits. 2014;49(2):360-372
49.Buonomo A, Lo Schiavo A. Evaluating the spectrum of periodic pulling in subharmonic resonant LC circuits. International Journal of Circuit Theory and Applications. 2015;43(12):1899-1913
50.Tofangdarzade A, Jalali A. An efficient method to analyze lock range in ring oscillators with multiple injections. IEEE Transactions on Circuits and Systems II: Express Briefs. 2015;62(11):1013-1017
51.Ghonoodi H, Miar-Naimi H, Gholami M. Analysis of frequency and amplitude in CMOS differential ring oscillators. Integration, the VLSI Journal. 2016;52:253-259
52.Hazeri AR, Miar-Naimi H. Generalized analytical equations for injected ring oscillator with RC-load. In: IEEE Transactions on Circuits and Systems I: Regular Papers. 2018;65(1):223-234
53.Imani A, Hashemi H. Distributed injection-locked frequency dividers. IEEE Journal of Solid-State Circuits. 2017;52(8):2083-2093
54.Adler R. A study of locking phenomena in oscillators. Proceedings of the IEEE. 1973;61(10):1380-1385
55.Cheng J-H, Tsai J-H, Huang T-W. Design of a 90.9% locking range injection-locked frequency divider with device ratio optimization in 90-nm CMOS. IEEE Transactions on Microwave Theory and Techniques. 2017;65(1):187-197
56.Tofangdarzade A, Tofangdarzade A, Saniei N. Strong injection locking and pulling in LC multiphase oscillators with multiple injection signals. In: IEEE Transactions on Circuits and Systems II: Express Briefs. 2018;66(8):1336-1340
57.Hong B, Hajimiri A. A Phasor-based analysis of sinusoidal injection locking in LC and ring oscillators. IEEE Transactions on Circuits and Systems I: Regular Papers. 2018;66(1):355-368
58.Hong B, Hajimiri A. A general theory of injection locking and pulling in electrical oscillators—Part I: Time-synchronous modeling and injection waveform design. IEEE Journal of Solid-State Circuits. 2019;54(8):2109-2121
59.Hong B, Hajimiri A. A general theory of injection locking and pulling in electrical oscillators—Part II: Amplitude modulation in LC oscillators, transient behavior, and frequency division. IEEE Journal of Solid-State Circuits. 2019;54(8):2122-2139
60.Mohammadjany A, Hazeri AR, Miar-Naimi H. Exact analyses for locking range in injection-locked frequency dividers. Integration. 2018;63:93-100
61.Miller R. Fractional-frequency generators utilizing regenerative modulation. Proceedings of the IRE. 1939;27(7):446-457
62.Tang-Nian L, Shuen-Yin B, Chen Y. A 60-GHz 0.13 μm CMOS divide by-three frequency divider. IEEE Transactions on Microwave Theory and Techniques. 2008;56(11):2409-2415
63.Jang SL, Chang YT, Hsue CW, Juang MH. Wide-locking range divide-by-4 injection-locked frequency divider using injection MOSFET DC-biased above threshold region. International Journal of Circuit Theory and Applications. 2016;44(5):968-976
64.Jang SL, Lin GY. Wide-locking range single-injection divide-by-3 injection-locked frequency divider. Microwave and Optical Technology Letters. 2015;57(12):2720-2723
65.Jang S-L, Chang C-W. A 90 nm CMOS LC-tank divide-by-3 injection-locked frequency divider with record locking range. IEEE Microwave and Wireless Components Letters. 2010;20(4):229-231
66.Jang S-L, Cheng W-C, Hsue C-W. Wide-locking range divide-by-3 injection-locked frequency divider using sixth-order RLC resonator. IEEE Transactions on Very Large Scale Integration (VLSI) Systems. 2016;24(7):2598-2602
67.Lee SH, Jang SL, Lee CF, Juang MH. Wide locking range divide-by-4 injection locked frequency dividers. Microwave and Optical Technology Letters. 2007;49(7):1533-1536
68.Jang SL, Han JC, Lee CF, Huang JF. A small die area and wide locking range CMOS frequency divider. Microwave and Optical Technology Letters. 2008;50(2):541-544
69.Jang SL, Liu CC. Wide-locking range divide-by-4 injection-locked frequency dividers. Microwave and Optical Technology Letters. 2008;50(12):3229-3232
70.Chen CZ, Hsu WL, Lin YS. A 58-GHz wide-locking range CMOS direct injection-locked frequency divider using input-power-matching technique. Microwave and Optical Technology Letters. 2009;51(3):685-689
71.Jang SL, Chang CW, Yang SM. Low power wide-locking range CMOS quadrature injection-locked frequency divider. Microwave and Optical Technology Letters. 2009;51(10):2420-2423
72.Jang SL, Huang JF, Lin FB. Wide-locking range LC-tank divide-by-4 injection-locked frequency divider using transformer feedback. International Journal of RF and Microwave Computer-Aided Engineering. 2015;25(7):557-562
Written By
Ali Reza Hazeri
Submitted: 28 October 2019Reviewed: 10 February 2020Published: 13 March 2020
Open access peer-reviewed chapter
