1.1 Introduction

The injection-locking or synchronization of an electronic oscillator by an external periodic signal hereafter referred to as the injection signal whose frequency is in the proximity of the natural or free-running frequency, more specifically the frequency of the first harmonic, of the oscillator under injection shifts the frequency of the oscillator from its natural frequency to the frequency of the injection signal without a frequency-locked loop, which is more costly in terms of both silicon area and power consumption. First-harmonic injection-locking achieves frequency synchronization by utilizing the frequency dependence of the amplitude and phase of the output of the oscillators under injection, whereas superharmonic/subharmonic injection-locking achieves frequency-locking by utilizing the internal nonlinear mechanism of the oscillator under injection that functions as a frequency mixer to generate the desired frequency component and the internal frequency selection mechanism of the oscillator to select the desired frequency component generated by the nonlinear mechanism of the oscillator. In both cases, Barkhausen criteria for oscillation must be satisfied. The absence of an explicit frequency detector, a charge pump, and a loop filter that typically exist in a FLL not only greatly shortens the frequency-locking process, but also significantly lowers power consumption. Injection-locking is capable of achieving rapid frequency-locking pivotal to a broad range of applications. Not only the frequency of the oscillator under injection is shifted to that of the injection signal, the phase noise of the oscillator under injection will also become comparable to that of the injection signal once the oscillator is locked to the injection signal whose phase noise performance is superior. This unique characteristic of injection-locked oscillators allows low-cost oscillators whose phase noise is inferior typically to behave as oscillators with superior phase noise performance once the oscillators are locked to a clean signal.

This chapter provides an overview of the injection-locking of oscillators and its applications in mixed-mode signal processing. The classification of oscillators is provided in Sect. 1.2. Section 1.3 briefly browses through the development of the synchronization of oscillators prior to 1946 in which the ground-breaking work of Adler on the injection-locking of harmonic oscillators was published. Section 1.4 presents Adler’s first-harmonic method for analysis of harmonic oscillators in weak injection. It is followed with the presentation of the first-harmonic method for analysis of harmonic oscillators in strong injection in Sect. 1.5. Frequency regenerative injection specifically tailored for frequency multiplication and frequency division is explored in Sect. 1.6. Section 1.7 studies the first-harmonic balance method capable of analyzing harmonic oscillators in first-harmonic, superharmonic, and subharmonic injections. The progressive multiphase injection of ring oscillators with multiple injections is examined in Sect. 1.8. Further development of this method for multiphase harmonic oscillators and injection-locked ring oscillator frequency dividers is also referenced. Section 1.9 briefly presents a Volterra series-based approach to investigate the impact of the degree of the nonlinearity of an oscillator under injection on the lock range of the oscillator. Section 1.10 presents some key applications of the injection-locking of oscillators. The chapter is summarized in Sect. 1.11.

1.2 Classification of Oscillators

Oscillators can be loosely classified into harmonic oscillators and nonharmonic oscillators. The former have a sinusoidal output in the time domain and a unitone spectrum in the frequency domain, whereas the latter have a non-sinusoidal output in the time domain and a multitone spectrum in the frequency domain . The typical examples of harmonic oscillators are LC (inductor-capacitor) oscillators with either a spiral inductor/transformer resonator or an active inductor/transformer resonator. The representative nonharmonic oscillators include ring oscillators and relaxation oscillators.

Architecturally a harmonic oscillator is made of two distinct blocks, namely (a) a nonlinear transconductor that functions as an amplitude limiter needed to sustain oscillation and a frequency mixer where various frequency components are generated from the injection signal and the output of the oscillator when the oscillator is under the injection of an external signal and (b) a linear frequency discriminator that functions as a high-Q band-pass filter to select the desired output frequency . Figure 1.1a shows the simplified block diagram of a harmonic oscillator under the injection of unitone external signal v inj. Similarly, a nonharmonic oscillator is composed of a nonlinear transconductor that functions as both an amplitude limiter and a frequency mixer when the oscillator is under injection of an external signal and a linear RC block that functions as a low-pass filter, as shown in Fig. 1.1b.

Fig. 1.1
figure 1

Block diagram of oscillators under injection: (a) harmonic oscillators. (b) Nonharmonic oscillators

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Harmonic oscillators distinct themselves from nonharmonic oscillators with a small frequency tuning range and low phase noise, accredited mainly to the higher quality factor of the frequency discriminator of the oscillators. The superior phase noise performance of harmonic oscillators allows them to be used in wireless communication systems where a stringent constraint on the phase noise of oscillators exists. Nonharmonic oscillators, on the other hand, are advantageous over their harmonic counterparts in silicon area, power consumption, and frequency tuning range. They are widely used in digital systems as clock generators.

1.3 Injection-Locking of Oscillator: Before 1946

The earliest observation of the synchronized oscillation of two oscillating systems is perhaps the synchronized pendulums of two pendulum clocks by C. Huygens (1629–1695), a Dutch physicist and the inventor of the pendulum clock, in 1665 . Huygens noticed that no matter how the pendulums on these clocks began, within about half an hour, they ended up swinging in exactly the opposite directions from each other. Perhaps the earliest reported work on the synchronization of oscillators are those by J. Vincent in 1919 [84] and E. Appleton in 1922 [2] . One of the first theoretical analyses of the nonlinear characteristics of oscillators was provided by van der Pol in 1934 [81] . An early application of modulating the frequency of an electronic oscillator by an external signal is the frequency-modulation receiver by G. Beers in 1944 where the frequency of the oscillator of the receiver is modulated in accordance with the frequency of an external modulating signal [6] . An in-depth study of forced oscillation in oscillator circuits was conducted by D. Tucker in 1945 [79, 80] . In Tucker’s approach, an oscillator was partitioned into two functionally distinct blocks, specifically a frequency discriminating block that functions as a band-pass filter and a maintaining block whose transfer voltage ratio is only dependent of the amplitude of the input and independent of the frequency of the input of the block. Using the phasor representation of the injection signal, the behavior of forced oscillators was analyzed.

1.4 Adler: Weak First-Harmonic Injection

In 1946, R. Adler, an American physicist (1913–2007), published his widely cited theoretical work on locking phenomena in oscillators [1] . Adler’s approach was developed for harmonic oscillators under the injection of a weak continuous-wave injection signal. The following three constraints were imposed:

  1. (a)

    The frequency of the injection signal is in the proximity of the natural also known as free-running frequency of the oscillator under injection.

  2. (b)

    The strength of the injection signal is significantly smaller as compared with the output of the oscillator, i.e., weak injection.

  3. (c)

    The feedback time from the output of the oscillator to the node at which the output voltage of the oscillator is subtracted from the injection signal is negligible as compared with beat period 1∕(ω inj − ω o) where ω inj and ω o are the frequency of the injection signal and the natural frequency of the oscillator under injection, respectively.

Although the derivation of Adler’s theoretical results was rigorous and correct, no experimental validation of the theoretical results was provided in the paper.

Given the significance of Adler’s work on the injection-locking of oscillators and its profound impact on the investigation of the injection-locking of oscillators, a brief representation of Adler’s approach on the injection-locking of oscillators is clearly warranted and well justified. In what follows we use the simple LC oscillator shown in Fig. 1.2 to briefly present Adler’s approach on analysis of the injection-locking of oscillators. As lock range is of our primary interest, we will focus on the lock range of the oscillator.

Fig. 1.2
figure 2

Injection-locking in harmonic oscillators

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Consider the simplified schematic of the LC oscillator shown in Fig. 1.2. It is a tuned common-source amplifier with the addition of an inverting amplifier of unity gain. Since the RLC network functions as a pure resistor at its resonant frequency \(\omega _o=1 / \sqrt {LC}\), the common-source tuned amplifier only provides a phase shift of − 180∘ at ω o. An additional − 180∘ phase shift needed for oscillation is provided by the unity-gain inverting amplifier. When the injection signal of frequency ω inj is injected into the oscillator with ω inj in the close proximity of ω o, the frequency of the oscillator will start to shift from its natural frequency ω o to ω inj. Let the instantaneous frequency of the oscillator during the locking process be ω. The instantaneous variation of the frequency also known as the beat frequency of the oscillator is given by ω − ω o . Further let θ be the phase between the injection signal I inj and the output of the oscillator I o. Since when I inj is absent, I T and I o will be in phase, I inj is therefore the cause of both frequency shift Δω and phase angle ϕ between I T and I o. Referring to Fig. 1.2, the law of sine gives

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{I_{inj}}{\sin \phi}=\frac{I_o}{\sin \theta} \end{array} \end{aligned} $$
(1.1)

from which we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \sin \phi=\frac{I_{inj}}{I_o}\sin \theta. \end{array} \end{aligned} $$
(1.2)

If I inj ≪ I o, i.e., weak injection, ϕ will be small and

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sin \phi \approx \phi \end{array} \end{aligned} $$
(1.3)

will hold. Equation (1.2) can be simplified to

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \phi \approx\frac{I_{inj}}{I_o}\sin \theta. \end{array} \end{aligned} $$
(1.4)

As mentioned earlier, the injection signal causes the frequency of the oscillator to deviate from its natural frequency ω o and gives rise to a phase angle between I T and I o. Since the frequency of the injection signal is in the proximity of ω o, the rate of the variation of ϕ with respect to frequency

$$\displaystyle \begin{aligned} \begin{array}{rcl} A=\frac{d\phi}{d\omega} \end{array} \end{aligned} $$
(1.5)

can be considered to be constant. We therefore have

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \phi &\displaystyle \approx&\displaystyle A(\omega-\omega_o) \\ &\displaystyle =&\displaystyle A\left(\varDelta \omega+\varDelta \omega_L\right). \end{array} \end{aligned} $$
(1.6)

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta \omega=\omega-\omega_{inj} \end{array} \end{aligned} $$
(1.7)

is the instantaneous beat frequency and

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta \omega_L=\omega_{inj}-\omega_o \end{array} \end{aligned} $$
(1.8)

is the undisturbed beat frequency. Equating (1.4) and (1.6) and noting that

$$\displaystyle \begin{aligned} \begin{array}{rcl} \omega=\frac{d\theta}{dt}+\omega_{inj}, \end{array} \end{aligned} $$
(1.9)

we arrive at

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{d\theta}{dt}+\varDelta \omega_L=\frac{1}{A}\frac{I_{inj}}{I_o}\sin \theta. \end{array} \end{aligned} $$
(1.10)

The impedance of the parallel RLC network is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} Z(s)=\frac{1}{C}\frac{s}{s^2+s\frac{1}{RC}+\frac{1}{LC}}. \end{array} \end{aligned} $$
(1.11)

It can be shown that the phase of Z(jω), denoted by \(\angle Z(j\omega )\), is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \angle Z(j\omega)=-\tan^{-1}\left(\frac{R}{\omega L}\frac{\omega_o^2-\omega^2}{\omega_o^2}\right). \end{array} \end{aligned} $$
(1.12)

When the injection signal is absent,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} I_o=g_mZ_T(j\omega_o)I_T, \end{array} \end{aligned} $$
(1.13)

where g m is the transconductance of the transistor. Note Z(jω o) = R. I o and I T are in phase in this case. When the injection current I inj is present,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} I_o=g_mZ_T(j\omega_{inj})I_T. \end{array} \end{aligned} $$
(1.14)

I o and I T in this case are not in phase but rather have a phase angle \(\angle Z(j\omega _{inj})\), which is ϕ. We therefore have

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \phi=-\tan^{-1}\left(\frac{R}{\omega L}\frac{\omega_o^2-\omega^2}{\omega_o^2}\right). \end{array} \end{aligned} $$
(1.15)

Since ω is in the vicinity of ω o,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \omega_o^2-\omega^2 &\displaystyle =&\displaystyle (\omega_o+\omega)(\omega_o-\omega) \\ &\displaystyle \approx&\displaystyle 2\omega_o(\omega_o-\omega). \end{array} \end{aligned} $$
(1.16)

Further noting that the quality factor of the RLC network is given byFootnote 1

$$\displaystyle \begin{aligned} \begin{array}{rcl} Q=\frac{\omega L}{R} \end{array} \end{aligned} $$
(1.17)

and making use of the identity of trigonometric functions

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tan^{-1}\left(x^{-1}\right)=\frac{\pi}{2}-\tan^{-1}x, \end{array} \end{aligned} $$
(1.18)

we arrive at

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \tan \phi \approx -\frac{2Q}{\omega_o}(\omega_o-\omega). \end{array} \end{aligned} $$
(1.19)

For weak injection, ϕ is small,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tan \phi \approx \phi \end{array} \end{aligned} $$
(1.20)

holds. Equation (1.19) can be approximated as

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \phi \approx -\frac{2Q}{\omega_o}(\omega_o-\omega). \end{array} \end{aligned} $$
(1.21)

It follows that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} A=\frac{d\phi}{d\omega}=\frac{2Q}{\omega_o}. \end{array} \end{aligned} $$
(1.22)

Equation (1.10) becomes

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{d\theta}{dt}+\varDelta \omega_L=\frac{\omega_o}{2Q}\frac{I_{inj}}{I_o}\sin \theta. \end{array} \end{aligned} $$
(1.23)

When the frequency of the oscillator is the same as that of the injection signal, i.e., the oscillator is locked to the injection signal, their phase difference will be stationary. As a result,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{d\theta}{dt}=0. \end{array} \end{aligned} $$
(1.24)

Equation (1.23) in this case becomes

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \omega_L=\frac{\omega_o}{2Q}\frac{I_{inj}}{I_o}\sin \theta. \end{array} \end{aligned} $$
(1.25)

Δω L is maximized when

$$\displaystyle \begin{aligned} \begin{array}{rcl} \theta=\frac{\pi}{2}. \end{array} \end{aligned} $$
(1.26)

The maximum lock range is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \omega_{L,max}=\frac{\omega_o}{2Q}\frac{I_{inj}}{I_o}. \end{array} \end{aligned} $$
(1.27)

Δω L,max is termed the maximum lock range or simply the lock range of the oscillator. Clearly the lock range of the oscillator is inversely proportional to the quality factor of the oscillator and directly proportional to the relative injection strength I inj∕I o of the oscillator.

Adler’s theoretical treatment of the injection-locking of oscillators was validated experimentally by Huntoon and Weiss [31] . The theoretical results obtained by Adler assumed that the injection signal was a continuous wave. Fraser extended Adler’s theory on the injection-locking of oscillators to oscillators under the injection of a periodically interrupted wave [21]. Mackey demonstrated experimentally that Adler’s theory of the injection-locking of oscillators also accurately describes the locking phenomena of microwave X-band reflex klystron oscillators [49] .

1.5 Paciorek: General First-Harmonic Injection

Alder’s formula on the lock range of injection-locked harmonic oscillators is valid only if the strength of the injection signal is significantly smaller as compared with the output of the oscillator, i.e., weak injection. The other two constraints upon which Adler’s formula was derived are usually not of a concern. This is because the frequency of the external injection signal is always chosen to be within the lock range of the oscillator under injection for the given injection strength. For applications where the strength of the injection signal is not small, is Adler’s formula of lock range still valid? To answer this question, in this section we briefly present Paciorek’s formula on the lock range of injection-locked oscillators that removes the weak injection constraint imposed on Adler’s formula. Paciorek’s formula was developed nearly 20 years after the inception of Adler’s formula [61] .

We use the same LC oscillator shown in Fig. 1.2 to present Paciorek’s general formula of the lock range of injection-locked oscillators. The law of sine gives

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{I_o}{\sin (\theta-\phi)}=\frac{I_{inj}}{\sin \phi}, \end{array} \end{aligned} $$
(1.28)

from which we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \tan \phi=\frac{I_{inj}\sin \theta}{I_o+I_{inj}\cos \theta}. \end{array} \end{aligned} $$
(1.29)

Equation (1.15) gives the phase of the RLC network and is repeated here for convenience

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \tan \phi=-\frac{R}{\omega L}\frac{\omega_o^2-\omega^2}{\omega_o^2}. \end{array} \end{aligned} $$
(1.30)

Since ω is in the vicinity of ω o,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \omega_o^2-\omega^2 \approx 2\omega_o(\omega_o-\omega). \end{array} \end{aligned} $$
(1.31)

Equation (1.30) can be simplified to

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \tan \phi \approx \frac{2Q(\omega-\omega_o)}{\omega_o}. \end{array} \end{aligned} $$
(1.32)

Equating (1.29) and (1.32) and noting

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \omega-\omega_o=(\omega-\omega_{inj})+(\omega_{inj}-\omega_o)=\varDelta \omega+\varDelta \omega_L \end{array} \end{aligned} $$
(1.33)

yield

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{d\theta}{dt}+\varDelta \omega_L=\frac{\omega_o}{2Q}\frac{I_{inj}}{I_o}\frac{\sin \theta}{1+\frac{I_{inj}}{I_o}\cos \theta}. \end{array} \end{aligned} $$
(1.34)

Let us comment on the preceding results prior to further development:

  1. (a)

    It is seen through the steps of the derivation of (1.34) that no any constraint was imposed. Equation (1.34) is therefore valid for both weak injection and strong injection. Note that in the derivation of Adler’s formula, I inj ≪ I o was imposed.

  2. (b)

    Equation (1.34) is similar to (1.23) except the denominator and can be simplified to (1.23) if \(\frac {I_{inj}}{I_o}\cos \theta \ll 1\), i.e., weak injection.

  3. (c)

    When the oscillator is locked to the injection signal, \(\frac {d\theta }{dt}=0\) will hold. We therefore arrive from (1.34)

    $$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \omega_L=\frac{\omega_o}{2Q}\frac{I_{inj}}{I_o}\frac{\sin \theta}{1+\frac{I_{inj}}{I_o}\cos \theta}. \end{array} \end{aligned} $$
    (1.35)
  4. (d)

    The optimal θ ∗ at which the lock range is maximized can be obtained by letting

    $$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{d\left(\varDelta \omega_L\right)}{d\theta}=0. \end{array} \end{aligned} $$
    (1.36)

    The result is given by

    $$\displaystyle \begin{aligned} \begin{array}{rcl} {} \cos \theta^*=-\frac{I_{inj}}{I_o}. \end{array} \end{aligned} $$
    (1.37)

    Substituting (1.37) into (1.35) yields the maximum lock range

    $$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \omega_{L,max}=\frac{\omega_o}{2Q}\frac{I_{inj}}{I_o}\frac{1}{\sqrt{1-\left(\frac{I_{inj}}{I_o}\right)^2}}. \end{array} \end{aligned} $$
    (1.38)

    If I inj ≪ I o, Eq. (1.38) is simplified to

    $$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \omega_{L,max} \approx \frac{\omega_o}{2Q}\frac{I_{inj}}{I_o}. \end{array} \end{aligned} $$
    (1.39)

    Equations (1.39) and (1.39) show that Paciorek’s maximum lock range formula given in (1.38) is simplified to Adler’s maximum lock range formula given in (1.27) in the case of weak injection.

Paciorek’s formula of the lock range of harmonic oscillators was also derived by Mirzaei et al. with a geometrical interpretation so as to provide a graphical interpretation of the locking process of harmonic oscillators [54, 56].

1.6 Miller: Frequency Regenerative Injection

In 1939, R. Miller proposed a frequency regenerative injection-locking scheme that can be used for injection-locked frequency multiplication/division [53] . Figure 1.3 shows the block diagram of an injection-locked oscillator that uses Miller’s frequency regenerative injection-locking scheme. It consists of two nonlinear blocks f 1 and f 2 that are used to generate various frequency components, a multiplier that multiplies the output of the nonlinear block f 1 and that of the nonlinear block f 2, and a linear frequency discriminator that functions as a high-Q band-pass filter. The nonlinear blocks can be characterized with their Taylor series expansion

$$\displaystyle \begin{aligned} v_1=f_1(v_{in}) \approx \sum_{m=1}a_mv_{in}^m, \end{aligned} $$
(1.40a)
$$\displaystyle \begin{aligned} v_2=f_2(v_o) \approx \sum_{n=1}b_nv_o^n. \end{aligned} $$
(1.40b)

Let

$$\displaystyle \begin{aligned} v_{in}=V_{in}\cos{}(\omega_{in}t+\phi), \end{aligned} $$
(1.41a)

and

$$\displaystyle \begin{aligned} v_o=V_o\cos{}(\omega_o t), \end{aligned} $$
(1.41b)

we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} y=\left[\sum_{m=1}a_mV^m_{in}\cos^m(\omega_{in}t+\phi)\right] \left[\sum_{n=1}b_nV^n_o\cos^n(\omega_ot)\right]. \end{array} \end{aligned} $$
(1.42)
Fig. 1.3
figure 3

Block diagram of an injection-locked oscillator using Miller’s frequency regenerative injection-locking scheme

Full size image

The mixer will generate various frequency components. Only those that are at ω o will pass through the band-pass filter and reach the output of the oscillator. The oscillator can therefore perform either frequency multiplication if ω o is a multiple of ω in or frequency division if ω o is a fraction of ω in, depending upon the configuration of the nonlinear blocks. A further development of Miller’s approach was given by Verma et al. where the two nonlinear blocks in Fig. 1.3 were combined into one so as to account for any interaction between them [83].

1.7 Schmideg: Superharmonic and Subharmonic Injection

The approaches of Adler and Paciorek for analysis of injection-locked harmonic oscillators are similar except that Paciorek’s approach removes the weak inversion constraint imposed on Adler’s formula. As a result, Paciorek’s formula is valid for both weak and strong injection. Both Adler and Paciorek’s approaches can only be used to analyze the first-harmonic injection of harmonic oscillators, i.e., the frequency of the injection signal needs to be located close to the natural frequency of the oscillator under injection. They cannot be used to analyze both superharmonic injection where the frequency of the injection signal is a superharmonic of the natural frequency of the oscillator or subharmonic injection where the frequency of the injection signal is a subharmonic of the natural frequency of the oscillator. In 1971, Schmideg proposed a first-harmonic balance method capable of analyzing injection-locked harmonic oscillators under the injection of either a superharmonic or a subharmonic signal [68]. Schmideg’s approach forms the basis for analysis of injection-locked frequency dividers [63, 83]. In this section, we briefly present Schmideg’s approach.

As mentioned earlier that a harmonic oscillator can be partitioned into a nonlinear block that performs frequency mixing and a high-Q band-pass filter that performs frequency discrimination, as shown in Fig. 1.4. Let the output of the oscillator in the lock state be a single tone of frequency ω in the vicinity of the passband center frequency ω o of the bandpass filter, i.e.,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} v_o=V_o \cos{}(\omega t)=\frac{V_o}{2}\left(e^{j\omega t}+e^{-j\omega t}\right).\vspace{-2pt} \end{array} \end{aligned} $$
(1.43)

Further let the injection signal be a single tone whose frequency is the kth harmonic of the output of the oscillator and the phase of the injection signal be ϕ.

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} v_{inj}=V_{inj} \cos{}(k\omega t+\phi)=\frac{V_{inj}}{2}\left(e^{jk\omega t}e^{j\phi}+e^{-jk\omega t}e^{-j\phi}\right).\vspace{-2pt} \end{array} \end{aligned} $$
(1.44)

The nonlinear block is depicted by the Taylor series expansion of the governing equation of the nonlinearity

$$\displaystyle \begin{aligned} y=a_1x+a_2x^2+a_3x^3+\cdots. \end{aligned} $$
(1.45)
Fig. 1.4
figure 4

Schmideg’s model of injection-locked harmonic oscillators

Full size image

The output of the nonlinear block of the oscillator is obtained by substituting (1.43) and (1.44) into (1.45).

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} y&\displaystyle =&\displaystyle a_1\left[\frac{V_{inj}}{2}\left(e^{jk\omega t}e^{j\phi}+e^{-jk\omega t}e^{-j\phi}\right)+\frac{V_o}{2}\left(e^{j\omega t}+e^{-j\omega t}\right)\right] \\ &\displaystyle &\displaystyle +\,a_2\left[\frac{V_{inj}}{2}\left(e^{jk\omega t}e^{j\phi}+e^{-jk\omega t}e^{-j\phi}\right)+\frac{V_o}{2}\left(e^{j\omega t}+e^{-j\omega t}\right)\right]^2 \\ &\displaystyle &\displaystyle +\,a_3\left[\frac{V_{inj}}{2}\left(e^{jk\omega t}e^{j\phi}+e^{-jk\omega t}e^{-j\phi}\right)+\frac{V_o}{2}\left(e^{j\omega t}+e^{-j\omega t}\right)\right]^3 \\ &\displaystyle &\displaystyle +\ldots. \end{array} \end{aligned} $$
(1.46)

Among the frequency components in (1.46), only those that fall into the passband of the band-pass filter pass through the filter and reach the output of the oscillator while all other frequency components are suppressed by the band pass filter. The transfer function of the band-pass filter is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} H_{BP}(j\omega)=\frac{H_o}{1+j2Q\left(\frac{\omega-\omega_o}{\omega_o}\right)}, \end{array} \end{aligned} $$
(1.47)

where H o is the gain of the filter at the passband center frequency ω o and Q is the quality factor of the filter.

For the sake of simplicity, let us consider the simplest case where k = 1 (first-harmonic injection) and y = a 1 x. Focusing on frequency components associated with e jωt, we have the following phasor relation

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{V_o}{2} \approx -\frac{H_o}{1+j2Q\left(\frac{\omega-\omega_o}{\omega_o}\right)}\frac{a_1}{2}\left(V_{inj}e^{j\phi_{inj}}+ V_o\right). \end{array} \end{aligned} $$
(1.48)

Separating the real and imaginary parts of (1.48) yields

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} 1+a_1H_o+a_1H_o\frac{V_{inj}}{V_o}{\cos}\phi_{inj}=0, \end{array} \end{aligned} $$
(1.49)

and

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \omega_L=\omega-\omega_o=\frac{\omega_o}{2Q}\frac{V_{inj}}{V_o}(a_1H_o)\sin \phi_{inj}. \end{array} \end{aligned} $$
(1.50)

Equation (1.50) shows that Δω L will be maximized when

$$\displaystyle \begin{aligned} \begin{array}{rcl} \phi_{inj}^*=\frac{\pi}{2}. \end{array} \end{aligned} $$
(1.51)

The maximum lock range is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \omega_{L,max}=\frac{\omega_o}{2Q}\frac{V_{inj}}{V_o}(a_1H_o). \end{array} \end{aligned} $$
(1.52)

At \(\phi _{inj}^*=\frac {\pi }{2}\), Eq. (1.49) becomes

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} 1+a_1H_o=0. \end{array} \end{aligned} $$
(1.53)

Equation (1.53) is Barkhausen criteria. Substituting (1.53) into (1.52) yields

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \omega_{L,max}=-\frac{\omega_o}{2Q}\frac{V_{inj}}{V_o}. \end{array} \end{aligned} $$
(1.54)

It is seen that (1.54) is similar to (1.39) except in (1.39) the injection signal and the output of the oscillator are currents whereas in (1.54) the injection signal and the output of the oscillator are voltages.

It is evident that Schmideg’s approach was derived using an approach that is more general and completely different from the approaches of Adler and Paciorek. It is capable of handling both a single injection signal and multiple injection signals. In addition, it is capable of handling both unitone injections and multitone injections as (1.44) can be readily modified to accommodate them [57]. Moreover, it is capable of handling superharmonic injection and subharmonic injection [110]. Furthermore, it is capable of handling an arbitrary degree of nonlinearity by properly truncating the Taylor series expansion of the nonlinear characteristic equation of the nonlinearity.

1.8 Progressive Multiphase Injection

For most harmonic oscillators, the number of injection signals is typically limited to two as these oscillators normally use a pair of cross-coupled common-source tuned amplifiers to generate required − 360∘ phase shift and synthesize a negative resistor, which is needed to compensate for the loss of the resonator so as to sustain oscillation. Unlike harmonic oscillators, a ring oscillator typically has more than two stages with the number of stages often set by the number of phases needed for a particular application. The presence of the multiple delay stages in ring oscillators makes us wonder whether only one external signal should be injected into a particular node of the oscillator or the number of external injection signals should be the same as that of the delay stages of the oscillator so as to maximize the lock range. If multiple injections indeed outperform single injection in terms of lock range, is there an optimal injection sequence or equivalently an optimal injection phase of each injection signal that yields the maximum lock range? To answer these questions, in this section we follow the approach of A. Mirzaei et al. [55].

Consider the ring oscillator shown in Fig. 1.5. The load capacitor of each stage is made of the output capacitance of the driving stage, the input capacitance of the driven stage, and the capacitance of the interconnect connecting the driving and driven stages. The load resistance comes from the output resistance of the driving stage only as the input resistance of the driven stage is infinite. Let us first assume that there is only one current i inj1 injected into node 1 of the oscillator and all other nodes have no injection. Assume that the output currents of all transconductors are the same. Let the per-stage delay of the oscillator be τ without injection. Note that τ is the average propagation delay of the delay stage. The free-running period of the oscillator is given by T o = Nτ where N is the number of the stages of the oscillator. Let the injection signal be a square wave of period T inj. Since C 1 is charged by both i o1 and i inj1, v 1 will rise faster as compared with the voltage of other nodes. As a result, the per-stage delay of stage 1 will be τ + Δτ 1 where Δτ 1 < 0 is the variation of the delay of stage 1 caused by i inj1. For the sake of simplicity, we assume that the injection current impacts the rising and falling edges equivalently.Footnote 2 Since in the lock stage, the period of the oscillator will be the same as that of the injection signal, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} (N-1)\tau+\left(\tau+\varDelta \tau_1\right)=T_{inj}\vspace{-3pt} \end{array} \end{aligned} $$
(1.55)

or

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \tau_1=T_{inj}-T_o.\vspace{-3pt} \end{array} \end{aligned} $$
(1.56)

The lock range is therefore given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \omega&\displaystyle =&\displaystyle \omega_{inj}-\omega_o \\ &\displaystyle =&\displaystyle \frac{2\pi \left(T_o-T_{inj}\right)}{T_oT_{inj}}\\ &\displaystyle =&\displaystyle -\left(\frac{\varDelta \tau_1}{T_{inj}}\right)\omega_o. \end{array} \end{aligned} $$
(1.57)

It becomes evident in (1.57) that the lock range is directly proportional to injection-induced delay variation of stage 1, which is proportional to i inj1, specifically the duration in which C 1 is charged by i inj1. Clearly one can increase the lock range by injecting i inj2 into node 2 when C 2 is charged by i o2, i inj3 into node 3 when C 3 is charged by i o3, …, and i injN into node N when C N is charged by i oN. This is known as progressive multiphase injection as the injection time of kth injection current i inj,k needs to be properly set such that the current injected into each node speeds up the charging process of the load capacitor of the node.

Fig. 1.5
figure 5

Progressive multiphase injection in ring oscillators

Full size image

If each stage has one injection current and the injection phase of the injection current is set in a preceding progressive fashion, the load capacitor of each stage will be charged by both the output current of the transconductor and the injection current. We continue our early assumption that the injection current of each delay stage of the oscillator impacts the rising and falling edges of the output voltage of the stage equivalently. Since in the lock stage the period of the oscillator will be the same as that of the injection signal, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \left(\tau+\varDelta \tau_1\right)+\left(\tau+\varDelta \tau_2\right)+\cdots+\left(\tau+\varDelta \tau_N\right)=T_{inj}. \end{array} \end{aligned} $$
(1.58)

As a result,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} T_{inj}-T_o=\varDelta \tau_1+\varDelta \tau_2+\cdots+\varDelta \tau_N. \end{array} \end{aligned} $$
(1.59)

The lock range is therefore given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varDelta \omega&\displaystyle =&\displaystyle \omega_{inj}-\omega_o \\ &\displaystyle =&\displaystyle \frac{2\pi \left(T_o-T_{inj}\right)}{T_oT_{inj}}\\ &\displaystyle =&\displaystyle -\left(\frac{\varDelta \tau_1+\varDelta \tau_2+\cdots+\varDelta \tau_N}{T_{inj}}\right)\omega_o. \end{array} \end{aligned} $$
(1.60)

Equation (1.60) shows that N-stage progressive multiphase injection increases the lock range by N times if the change of the per-stage delay induced by the injection signals is identical. Since for a given ring oscillator the number of the delay stages of the oscillator is known, one can properly assign the injection phase of each injection signal according to the delay of each stage of the oscillator so that each injection signal is progressively injected into the oscillator to achieve the maximum lock range.

The ring oscillator with progressive multiphase injection can be analyzed by assuming that the injection current at kth node is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} i_{inj,k}(t)=I_{inj}e^{j(\omega_{inj}t+\phi_{inj,k})}, \end{array} \end{aligned} $$
(1.61)

where the injection phase is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \phi_{inj,k}=\omega_{inj}t-\frac{k-1}{N}\pi. \end{array} \end{aligned} $$
(1.62)

If the injection current is a square wave, only the first harmonic of the square wave whose frequency is ω inj is considered as all other harmonics are assumed to be sufficiently suppressed by the RC network at the output of the delay stages of the oscillator. There are two currents flowing into the output node of each transconductor representing the delay stage: One is the output current of the transconductor and the other is the injection current. Since both the amplitude and phase of the voltage of the node vary with time, the output voltage of the kth transconductor can be written as

$$\displaystyle \begin{aligned} \begin{array}{rcl} v_k=A(t)e^{j\phi(t)}, \end{array} \end{aligned} $$
(1.63)

where A(t) and ϕ(t) are the amplitude and phase of the voltage of the node, respectively. Writing KCL at the output node of each transconductor in the time domain yields a total of N differential equations of the phase of the voltage of each node of the oscillator. They are termed generalized Adler’s equations of ring oscillators . No closed-form solutions, however, were given in [55].

In [75, 76], Tofangdarzade et al. further developed the progressive multiphase injection approach developed by Mirzaei et al. and derived the closed-form solution of the generalized Adler’s equations of ring oscillators and multiphase LC oscillators in the lock state and validated the obtained closed-form solutions using simulation results. Readers are referred to the cited references for the details of this approach .

Progressive multiphase injection was also developed for LC oscillators that can be treated as two-stage ring oscillators with each stage composed of a nonlinear transconductor and a RLC network. Using a negative feedback system approach, Yuan and Zhou showed that the lock range of the oscillator with a single injection signal is smaller as compared with that with dual injection signals. Also the lock range of the oscillator with dual injection signal depends upon the injection phases of the injection signals. The lock range is maximized once the difference between the injection phase of the injection signals is 180∘ [105, 112].

Mirzaei’s progressive multiphase injection was originally developed for first-harmonic injection, i.e., the frequency of the progressively phased injection signals is in the vicinity of the frequency of the first harmonic of ring oscillators under injection. A. Musa et al. further developed the progressive mixing technique to injection-locked ring oscillator frequency dividers so as to obtain a large lock range [58] . Readers are referred to the cited references for the details of this approach.

1.9 Effective Injection Signaling

It has been well recognized that the nonlinear characteristics of oscillators, both harmonic oscillators and nonharmonic oscillators, are the root cause of injection-locking phenomena in oscillators. It has also been widely observed that the severer the degree of the nonlinearity of the oscillator under injection, the larger the lock range of the oscillator. In order to seek the theoretical explanation of this fundamental characteristic of oscillators, Yuan and Zhou utilized a Volterra series approach to quantify the impact of the degree of the nonlinearity of oscillators on their lock range [106, 107, 113]. In this approach, the nonlinearity of the oscillator under injection is depicted by its truncated Taylor series expansion first. The variables of the oscillator such as nodal voltages and branch currents are then represented by their truncated Volterra series expansion, thereby converting the nonlinear differential equations depicting the oscillator to a set of linear differential equations characterizing a set of linear circuits called Volterra circuits. Each of the Volterra circuits represents a harmonic oscillator and the frequencies of these harmonic oscillators are harmonically related. The input of high-order Volterra circuits is obtained from the response of lower-order Volterra circuits. The response of the oscillator is obtained by summing up that of the Volterra circuits. By solving Volterra circuits sequentially starting from the first-order Volterra circuit, the response of the oscillator can be obtained. Since the frequency of the first-order Volterra circuit is the same as that of the first harmonic of the oscillator, one only needs to find the lock range of the first-order Volterra circuit in order to obtain the lock range of the oscillator. The total or effective injection signals of the first-order Volterra circuit consist of two distinct parts: One from the external injection signals and the other contributed by higher-order Volterra circuits. It is the contribution of higher-order Volterra circuits that strengthens the effective injection signal of the first-order Volterra circuit subsequently a larger lock range. The severer the degree of the nonlinearity of the oscillator, the more the higher-order Volterra circuits contribute to the effective injection signal of the first-order Volterra circuit, the larger the lock range of the oscillator. The details of this approach will be provided in the later chapters.

1.10 Applications of Injection-Locking of Oscillators

The injection-locking of oscillators has found a broad range of applications in mixed-mode signal processing. In this section, we briefly browse through some of the representative applications of the injection-locking of oscillators. This, by no means, is an attempt to include all key applications of the injection-locking of oscillators. As the design of systems utilizing the injection-locking of oscillators is not the focus of this monograph, no details on the operation of these systems are provided. Readers are referred to the cited references for the detailed information of the principle and operation of these systems.

1.10.1 Frequency Division

Perhaps the most representative application of the injection-locking of oscillators is the pre-scalar of frequency synthesizers in both RF communication systems and digital systems. In an integer-N frequency synthesizer, a frequency divider with a large division ratio is needed in the feedback path to generate a low-frequency signal from the output of the high-frequency oscillator of the synthesizer so that the frequency of the low-frequency signal can be compared with that of a low-frequency reference, often from a crystal oscillator, such that the frequency of the oscillator can be stabilized. A large frequency division ratio is often obtained using a multistage approach in which multiple frequency dividers are cascaded. The first stage is most difficult to design and also consumes the most power. This is because not only the first-stage must operate at the same frequency as that of the oscillator of the synthesizer, it must also cope with the large variation of the frequency of its input arising from the impact of PVT (process, voltage, and temperature) uncertainty on the frequency of the oscillator of the synthesizer. It becomes increasingly difficult to design frequency dividers operating at high frequencies using conventional digital logic such as current-mode logic (CML) not because of their speed limitation but also due to their excessive dynamic power consumption. Attributive to the intrinsic characteristics of injection-locked oscillators including high frequency accuracy, fast locking process, low power consumption, and low phase noise once locked, injection-locked frequency dividers are attractive in realization of the first stage of the divide-by-N block of frequency synthesizers known as the pre-scalar . Both injection-locked harmonic oscillator frequency dividers [8, 11, 14, 19, 33, 34, 74, 94, 99] and injection-locked nonharmonic oscillator frequency dividers are widely popular [18, 30, 47, 48, 98].

1.10.2 Parallel Links

High-bandwidth short-distance parallel data links are critical for multi-core processing and networking applications. Forwarded-clock (FC) also known as source synchronous architecture where the sampling clock of the receiver of a parallel data link is transmitted directly from the transmitter and multiple data channels share the same clock channel so as to minimize routing cost is widely favored due to the inherent jitter correlation between data and clock as both clock and data are generated by the same transmitter and the fact that the performance of data links is dominated by the relative jitter between clock and data rather than the absolute jitter of either clock or data [41] . Jitter correlation between clock and data is only beneficial over a certain frequency range called jitter transfer bandwidth beyond which tracking high-frequency jitter is detrimental .

As lossy channels amplify data jitter at high frequencies [5], filtering high-frequency data jitter beyond the jitter transfer bandwidth in which jitter correlation between data and clock exists is important . As delay mismatch between data and clock channels can be several unit intervals (UIs), which reduces jitter correlation between clock and data [70], minimizing channel mismatch via deskewing is critical. Channel mismatch induced clock and data skews also mandate that deskewing be performed at the receiver such that data can be sampled at the center of data eyes .

To lower power consumption and minimize interference from the clock channel, it is desirable to transmit a low-frequency clock over the clock channel and generate a high-frequency clock from the transmitted low-frequency clock at the receiver using a frequency multiplier. Frequency multiplication can be performed using a phase-locked loop (PLL) or a delay-locked loop (DLL). PLL frequency multiplication removes high-frequency data jitter, accredited to its finite loop bandwidth. It, however, also removes desired correlated jitter between data and clock that resides in jitter transfer bandwidth [9, 27] . The loop bandwidth, on the other hand, must be sufficiently large in order to allow PLLs to track data. This also helps retain the desirable correlated jitter. The loop bandwidth of PLLs, however, is limited to less than 1/10 of the frequency of the input due to loop stability constraints [46, 59] . On top of that, PLL-frequency multiplication suffers from phase accumulation. As a result, PLL frequency multiplication is not particularly attractive at high data rates. DLL frequency multiplication uses a multiplying delay-locked loop (MDLL) to generate a high-frequency clock from a low-frequency clock [40] . It is generally preferred over their PLL counterparts as phase accumulation only exists within one period of the input . This differs from PLL frequency multiplication where phase accumulation takes place over multiple periods of the input until it becomes sufficiently large such that it can be detected by the phase detector and subsequently corrected by the feedback loop. DLL frequency multiplication, however, suffers from jitter amplification at high frequencies, which undesirably increases high-frequency data jitter beyond jitter transfer bandwidth.

Zhang et al. proposed an injection-locking clock distribution scheme in which the frequency of local oscillators responsible for local clocking is synchronized to the global clock via injection-locking [108]. With superharmonic injection, the frequency of the global clock can be lowered, thereby lowering both power consumption and interference from global clock trees. As the injection signal need not to be large, the number of buffers needed to relay global clocks can also be lowered. This is echoed with both power consumption reduction and clock jitter improvement.

O’Mahony et al. proposed a forwarded-clock parallel I/O architecture with an injection-locked oscillator to generate required deskewing clocks at the receiver [60, 70] . The injection-locking signal is the clock transmitted by the clock channel. The injection-locked oscillator behaves as a low-pass filter that filters out unwanted high-frequency jitter beyond the desired jitter transfer bandwidth and duty-cycle errors present in the received clock. Since the inception of this source synchronous parallel data link architecture, numerous studies have been reported and the effectiveness of this architecture has been validated by measurement results [27–29].

1.10.3 Frequency Calibration

The system clock of a PWM such as an embedded temperature sensor or a radio-frequency identification (RFID) tag that controls the operation of both the RF and baseband blocks of the PWM can be sent directly by its base station either via its carrier or the envelope of the carrier when amplitude shift key (ASK) modulation scheme is used [36] . This approach suffers from a number of drawbacks: (a) High power consumption as the comparator in the clock recovery block must operate at the carrier frequency, which is significantly higher as compared with the frequency of the system clock of the PWM typically. The need for a chain of frequency dividers to generate the system clock whose frequency is typically much lower than that of the carrier also greatly increases power consumption. (b) A large modulation index is preferred from a data and clock recovery point of view . Because the amplitude of the received RF signal is usually small, especially in logic-0 states when the modulation index of the ASK-modulated RF signal is large, a large timing error will exist in the recovered clock. (c) The availability of the system clock in down-link operation only greatly restrains the flexibility of the system.

In order to avoid the drawbacks of generating the system clock of a PMW directly from its carrier, it is desirable to generate the system clock of the PWM locally using either a ring oscillator or a relaxation oscillator. The frequency of the local oscillator, however, is subject to the impact of PVT uncertainty and needs to be calibrated with reference to a clock of a known frequency prior to the intended operations of the PWM. The reference clock with respect to which the frequency of the PWM is to be calibrated is sent from the base station rather than generated locally using a timing oscillator whose frequency is independent of the effect of PVT uncertainty. This is because it is difficult to generate a precision band-gap reference voltage or current in the presence of a fluctuating supply voltage and varying temperature with the constraint of a limited amount of power consumption allowed.

Frequency calibration can be performed using a frequency-locked loop. FLL-based frequency calibration generally suffers from high power consumption, a low frequency accuracy, and a long calibration time, making it less attractive for PWMs. Frequency calibration using injection-locking with the frequency reference against which the frequency of the local oscillator is calibrated as the injection signal sent from the base station is an effective way to lower power consumption, ameliorate low frequency accuracy, and shorten calibration time while ensuring a good frequency calibration accuracy [3, 4, 37, 73].

1.10.4 Phase-Locked Loops

Low phase noise phase-locked loops (PLLs) are important building blocks for a broad range of applications including data links and frequency synthesizers. The phase noise of PLLs is dominated by either the phase noise of its input or the phase noise of its oscillator, application-dependent. For example, for the transmitter PLL of a serial data link, since the input of the PLL in this case is often a crystal oscillator whose phase noise is low, the phase noise of the PLL is dominated by that of its oscillator whereas for the receiver PLL of the serial data link, the phase noise of the PLL is dictated by the jitter of its input, which is the data received at the far end of the channel, specifically, the equalized data. One can manipulate the phase noise of a PLL by adjusting the loop bandwidth of the PLL [43]. For example, for the transmitter PLL, the loop bandwidth should be maximized so as to minimize the contribution of the phase noise of the oscillator to the overall phase noise of the PLL whereas for the received PLL, the loop bandwidth should be minimized in order to minimize the contribution of the jitter of the received data to the overall phase noise of the PLL. The deployment of these effective design techniques, however, is confronted by other constraints such as stability and lock time imposed on the PLL. For example, for the received PLL, the loop bandwidth of the PLL must also be maximized so as to allow the PLL to lock to incoming data in typically less than half of the unit interval (UI) or symbol time even though minimizing loop bandwidth is preferred from a phase noise point of view .

The spectrum of the phase noise of a PLL typically consists of three distinct sections: A low frequency offset section in which the phase noise of the PLL follows the profile of that of the input but is larger, mainly dominated by flicker noise, a high frequency offset section near and beyond the loop bandwidth where the phase noise of the PLL is nearly the same as that of the oscillator, and an intermediate frequency offset section up to the loop bandwidth where the phase noise of the PLL is nearly constant [16, 17, 24, 39, 42, 62, 72, 90, 109]. Lowering the phase noise of the PLL in both the low frequency offset and intermediate frequency offset sections, i.e., within the loop bandwidth of the PLL, is critical .

As to be shown in later chapters that injection-locking is capable of lowering the phase noise of an oscillator once the oscillator is locked to a clean reference signal, i.e., its phase noise is significantly lower than that of the free-running oscillator that is to be injection-locked. This unique characteristic of injection-locking has attracted a lot of attention recently as an effective means to lower the phase noise of PLLs without sacrificing the loop bandwidth of the PLLs. For the PLL of frequency synthesizers, as a clean injection-signal is provided by a crystal oscillator while the frequency of the oscillator of the PLL for wireless and wireline communications is in the range of multi-GHz, the subharmonic injection of the oscillator of the PLLs is generally performed [43]. Injection-locking should always be accompanied with the PLL to ensure lock as the lock range of oscillators is much smaller as compared that of the PLL such that injection-locking alone cannot guarantees lock. The injection-locking signal is typically a train of narrow pulses generated from the reference signal of the PLL using a pulse generator. As to be shown in the later chapters that injection will be most effective if injection-locking signal is injected at the threshold-crossing of the output of the oscillator under injection. The effectiveness of injection-locking critically depends upon the time at which the injection-locking pulse is applied to the oscillator. Timing mismatch between the time instant at which the injection signal is injected into the oscillator and the time instant at which the output of the oscillator crosses the threshold gives rise to reference spurs [72].

1.11 Chapter Summary

A general classification of oscillators was provided. We showed that oscillators can be loosely classified into harmonic oscillators and nonharmonic oscillators. The brief history of the injection-locking of oscillators was presented with an emphasis on the analysis rather than design of injection-locked oscillators. The first-harmonic method for analysis of harmonic oscillators in weak inversion developed by Adler was studied in detail. It was followed with the presentation of the first-harmonic method for analysis of harmonic oscillators in both weak and strong developed by Paciorek. Frequency regenerative injection method developed by Miller and further enhanced by other researchers for the realization of frequency multiplication and frequency division was explored. Schmideg’s first-harmonic balance method capable of analyzing harmonic oscillators in first-harmonic, superharmonic, or subharmonic injections was studied. The progressive multiphase injection of ring oscillators was examined. We showed that the lock range of a ring oscillator with properly phased N injections of equal strength is N times that of the oscillator with only one injection. The further development of progressive multiphase injection for multiphase harmonic oscillators and injection-locked ring oscillator frequency dividers was referenced. The effective injection signaling arising from the nonlinearity of oscillators under injection and obtained by analyzing the Volterra circuits of the oscillators was described. Finally, the representative applications of the injection-locking of oscillator were briefly explored.