**3. Review of the previously significantly published papers**

The pulling and locking phenomena have been investigated in previously published papers [2, 8–60]. 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

**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.*

**91**

*Review of Injected Oscillators*

*DOI: http://dx.doi.org/10.5772/intechopen.91687*

*Review of Injected Oscillators DOI: http://dx.doi.org/10.5772/intechopen.91687*

**Figure 1.**

**90**

*voltage at the frequency domain.*

*Modulation in Electronics and Telecommunications*

*(a) Three-stage differential ring oscillator, (b) delay cell, (c) output voltage at the time domain, (d) output*

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, 10–17]. 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

*(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).*

**Figure 2.**

*Review of Injected Oscillators*

*DOI: http://dx.doi.org/10.5772/intechopen.91687*

**93**

of the oscillator and solving them in nonlinear, vector diagram or empirical methods, the locking range of the cross-coupled oscillator is achieved.

**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).*

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, 10–17]. 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.

**92**

*Modulation in Electronics and Telecommunications*

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*.

The pulling phenomena for LC oscillators were completely investigated in [8, 25, 27, 28, 33, 39, 41, 47–49]. When *Iinj=Iosc* ≪ 1, the beat frequency mathematical formula (*ωb*) for LC oscillators is as follows:

$$\left|o\_{b}\right|\_{LC} = \sqrt{\Omega^{2} - K^{2}}, K = \left(o\_{0}I\_{\text{inj}}\right) / (2\Omega I\_{\text{occ}}), \Omega = \left|o\_{\text{occ}} - o\_{\text{inj}}\right|\tag{1}$$

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

th *output voltage harmonic and* N *is the number of stages.*

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 [18–20]. However, using PPV is complicated, and most of the time it needs preprocessing by simulators such as

At first, fractional frequency generators utilizing regenerative modulation have

, *k* ¼ *Iinj=*ð Þ *NA*1*CL* , Ω ¼ *ωosc* � *ωinj*

*NCL* P<sup>∞</sup> *i*¼1 ð Þ �<sup>1</sup> *<sup>i</sup>*þ<sup>1</sup>

*N*

*NIosc* sin 2ð Þ *π=N*

*N* <sup>1</sup><sup>þ</sup> tan <sup>2</sup> ð Þ *<sup>π</sup>=<sup>N</sup>* tan ð Þ *<sup>π</sup>=<sup>N</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>� *Iinj Iosc* � �<sup>2</sup> q

<sup>1</sup> <sup>þ</sup> tan <sup>2</sup> ð Þ *<sup>π</sup>=<sup>N</sup>* tan ð Þ *π=N*

ð Þ 2*i*�1 *A*2*i*�<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>� *<sup>I</sup> Iosc* ð Þ<sup>2</sup> <sup>q</sup> *Iinj Iosc*

� tan ð Þ *<sup>π</sup>=<sup>N</sup> Iinj Iosc Iinj Itrans:*

�

� �

� (2)

for ring oscillators is expressed as below [5]:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>Ω</sup><sup>2</sup> � *<sup>k</sup>*<sup>2</sup> <sup>p</sup>

**Refs. Locking range** [5] *Iinj*

[23, 30] <sup>2</sup>*ωoscIinj*

[50] *<sup>ω</sup>osc*

[26] *<sup>ω</sup>osc*

*Previously significant published locking ranges for ring oscillators.*

**4. Review of the frequency multipliers/dividers**

been introduced by [61]. A block diagram of the injection frequency

divider/multiplier is displayed in **Figure 4**. By subharmonic (*ωinj* ≈ *ω*0*=M*) or super-harmonic (*ωinj* ≈ *Mω*0) 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, 43–45, 49, 60]. Studies on

*ωb*j *ring* ¼

Ai *is the amplitude of the* i

*Review of Injected Oscillators*

*DOI: http://dx.doi.org/10.5772/intechopen.91687*

**Table 2.**

Cadence.

**Figure 4.**

**95**

*Block diagram of a frequency divider/multiplier.*

According to Eq. (1), *ω<sup>b</sup>* is increased by increasing Ω or *Iosc* when other parameters are constant. Eq. (1) was improved for strong injection in [41, 47]. The multiinjection 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


**Table 1.** *Previously significant published locking ranges for LC oscillators.*


#### **Table 2.**

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

The pulling phenomena for LC oscillators were completely investigated in [8, 25,

According to Eq. (1), *ω<sup>b</sup>* is increased by increasing Ω or *Iosc* when other parameters are constant. Eq. (1) was improved for strong injection in [41, 47]. The multiinjection signals with different total phases have been reported for the LC oscillators

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

� �*<sup>=</sup>* <sup>2</sup>*QIosc* ð Þ, <sup>Ω</sup> <sup>¼</sup> *<sup>ω</sup>osc* � *<sup>ω</sup>inj*

�

� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � *Iinj=Iosc* � �<sup>2</sup> � � q

� (1)

locking range is proportional to *Iinj* and inversely proportional Q and *Iosc*.

formula (*ωb*) for LC oscillators is as follows:

*Modulation in Electronics and Telecommunications*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>Ω</sup><sup>2</sup> � *<sup>K</sup>*<sup>2</sup> <sup>p</sup>

*ωb*j*LC* ¼

in [24, 29, 56].

**Table 1.**

**94**

**Figure 3.**

*Cross-coupled LC oscillator.*

27, 28, 33, 39, 41, 47–49]. When *Iinj=Iosc* ≪ 1, the beat frequency mathematical

,*K* ¼ *ω*0*Iinj*

**Refs. Locking range** [9, 44, 54] *ωoscIinj=* 2*QIosc* ð Þ [8, 24, 47] *<sup>ω</sup>oscIinj<sup>=</sup>* <sup>2</sup>*QIosc*

*Previously significant published locking ranges for LC oscillators.*

*Previously significant published locking ranges for ring oscillators.*

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]:

$$|o o\_b|\_{\text{ring}} = \sqrt{\Omega^2 - k^2}, k = I\_{\text{inj}} / (\text{NA}\_1 \text{C}\_L), \Omega = \left| o\_{o \text{sc}} - o\_{\text{inj}} \right| \tag{2}$$

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 [18–20]. However, using PPV is complicated, and most of the time it needs preprocessing by simulators such as Cadence.

## **4. Review of the frequency multipliers/dividers**

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* ≈ *ω*0*=M*) or super-harmonic (*ωinj* ≈ *Mω*0) 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, 43–45, 49, 60]. Studies on

**Figure 4.** *Block diagram of a frequency divider/multiplier.*

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, 43–45, 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 timeconsuming. 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 2*Iinj=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 [43–45]. 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 pro-

Due to the small locking range of injected LC oscillators, various techniques have

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.

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 [63–72]. **Figure 6** discloses a quadrature LC oscillator employed in the injection signal.

posed by phasor diagrams and differential equations.

**5. Conclusions**

**97**

**Figure 6.**

*The quadrature LC oscillator [29].*

*Review of Injected Oscillators*

*DOI: http://dx.doi.org/10.5772/intechopen.91687*

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, 43–45, 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 timeconsuming. 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 2*Iinj=π*, respectively [8]. Therefore, the locking range was

*Modulation in Electronics and Telecommunications*

*Conventional injection locking frequency divider/multipliers, (a) injection in the tail, (b) injection in the drain.*

**Figure 5.**

**96**

**Figure 6.** *The quadrature LC oscillator [29].*

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 [43–45]. 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.

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 [63–72]. **Figure 6** discloses a quadrature LC oscillator employed in the injection signal.

## **5. Conclusions**

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.
