Oscillators are integral parts of many electronic devices. Majority of fundamental signal processing operations such as frequency translation, synchronization, and timing, require stable clocks derived from an electrical oscillator. An oscillator is generally susceptible to any external periodic signal whose frequency is close to the oscillation frequency or its integer harmonics. Without the loss of the generality, here we assume that the frequency of the injection signal is close to that of the oscillator. The external periodic signal injected into an autonomous oscillator can speed up or slow down the oscillation and may even make the oscillator lock to itself, if the two injection and free-running frequencies are adequately close [1-2].
The external signal injected into the oscillator can be either intentional or spurious. If intentional, the target is usually to lock the oscillator to a predefined frequency, which is commonly utilized in injection-locked dividers. However, in many occasions the injection signal is an unwelcomed spurious tone picked up through some unwanted parasitic paths due to the limited isolation between the source of the injection signal and the oscillator. In such cases, the injection signal typically doesn't have enough strength to lock the oscillator, but still pulls it causing the oscillation frequency to be displaced from the center. The pulled oscillator’s spectrum is no longer a clean single tone. It contains other frequency components that can be in the form of sidebands at different frequency offsets [1-4]. The pulled oscillator exhibits an interesting behavior, which is the subject of this post.
Consider a simplified model of an LC oscillator shown in Fig. 1(a), with a free-running frequency . The negative resistance –R compensates the loss of the LC tank modeled with a shunt resistor R. In practice the oscillator is designed differentially and the negative resistance is typically a regenerative differential pair. The negative resistance is usually non-linear and the oscillation amplitude is regulated by this non-linearity in a way to ensure the effective negative resistance is exactly R. The free-running frequency is approximately equal to . Now, as shown in Fig. 1(b), a sinusoidal current whose frequency is relatively close to the free-running is injected into the LC tank. The injection is assumed to be weak, meaning that the amplitude of the injection current is much smaller than the signal current , where is the fundamental component of the current flowing into the LC tank. The ratio of is called the injection strength. Due to the presence of the nonlinearity in the negative resistance, the injection signal perturbs the oscillation’s amplitude as well as its phase. The fluctuations in the amplitude are typically stripped away by following hard-limiting buffers. However, phase fluctuations can never be corrected and would last perpetually. Under the condition of weak injection, i.e. , the dynamics of the oscillator phase are described by the following first-order non-linear differential equation, widely known as Adler’s equation [1-4]:
where is the instantaneous phase of the oscillator and is the quality factor of the LC tank at . According to the Adler’s equation, the phase fluctuations are decoupled from those of the amplitude. This is not exactly true and in reality the phase fluctuations are coupled to variations of the amplitude albeit weakly. To understand impacts of the injection signal on the oscillation spectrum, Adler’s equation needs to be solved. The solution exists in the literature , which is described as follows. The following key parameter is defined:
is called the pulling strength, and the rational of this definition will be apparent shortly. Note that no matter how weak the injection strength is, the pulling strength can assume arbitrarily large values if the injection frequency is sufficiently close to the free-running.
When > 1, the oscillator is locked to the injection signal, meaning that the oscillation frequency would become . What is more interesting is the case when < 1 or the oscillator is unable to lock. The injection signal pulls the oscillator frequency toward itself and the oscillator resists to this frequency displacement. As a result of this action and reaction, infinite number of decaying sidebands appear in the opposite side of the injection signal (Fig. 2). However, at the side where the injection signal is located, theoretically no sidebands besides the one at the injection frequency is generated. The impact of the injection signal whose strength is not large enough to lock the oscillator can be summarized as follows :
- The injection signal displaces the oscillator frequency toward itself by .
- Decaying sideband are spaced from each other by . This quantity is called the beat frequency denoted by .
- Spectrum of the pulled oscillator is heavily asymmetric and the sidebands decay with a slope of dB/ (log-scale vertical axis and linear-scale the horizontal axis).
If the injection strength is not week, meaning that and are comparable, the modified differential equation describing the dynamic of the perturbed oscillator phase would be:
which is called the generalized Adler’s equation . A strong injection seldom happens in spurious oscillator pulling cases. However, (3) proves to be very useful in analyzing coupled oscillators such as a quadrature oscillator, where the two oscillator cores mutually injection pull each other.
 R. Adler, "A study of locking phenomena in oscillators," Proceedings of the IEEE , vol.61, no.10, pp.1380-1385, Oct. 1973.
 B. Razavi, "A study of injection locking and pulling in oscillators," IEEE Journal of Solid-State Circuits, vol.39, no.9, pp.1415-1424, Sept. 2004.
 T.J. Buchanan,"The Frequency Spectrum of a Pulled Oscillator," Proceedings of the IRE, vol.40, no.8, pp.958-961, Aug. 1952.
 A. Mirzaei, A.A. Abidi, "The Spectrum of a Noisy Free-Running Oscillator Explained by Random Frequency Pulling," IEEE Transactions on Circuits and Systems I: Regular Papers, vol.57, no.3, pp.642-653, March 2010.
 A. Mirzaei, M.E. Heidari, and A.A. Abidi, "Analysis of Oscillators Locked by Large Injection Signals: Generalized Adler's Equation and Geometrical Interpretation," Custom Integrated Circuits Conference, pp.737-740, Sept. 2006.
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