\chapter{Injection Locked Oscillators}
\label{chapter:ilo_theory}

In this chapter, \acfp{ILO} will be described and analysed. 

Firstly, a brief derivation of general oscillation principles and criteria will be presented. Afterwards, a closer look at harmonic LC tank based \acp{ILO} will follow, including phasor analysis and locking range derivation using its linear model. 

A different, time domain based approach will be taken for the analysis of \acfp{ILRO}, as they are highly non-linear. 

Finally, \ac{ILRO} based \ac{FTDC} will be discussed, since its advantages are the main motivation of using \acp{ILRO} in \ac{DToF} \ac{LIDAR} systems in the first place.

\section{Oscillation criteria}

A generalized feedback system consisting of a single block with a Laplace domain transfer function of $H(s)$ is shown on \autoref{fig:feedback}.

\begin{figure}[htb]
	\centering
	\includegraphics[width=9cm]{obrazky/feedback_system.pdf}
	\vspace{-0.3cm}
	\caption{General feedback system} \label{fig:feedback}\vspace{-0.3cm}
\end{figure}

The closed loop transfer function of the overall system can be derived
\begin{gather}
V_\s{out}(s) = H(s)\cdot V_\s{sum}(s) = H(s)\cdot [V_\s{in}(s) + V_\s{out}(s)]\\
V_\s{out}(s) = H(s)\cdot V_\s{in}(s) + H(s)\cdot V_\s{out}(s) \\
V_\s{out}(s)\cdot [1 - H(s)] = H(s)\cdot V_\s{in}(s) \label{eq:feedback_start}\\
\frac{V_\s{out}(s)}{V_\s{in}(s)} = \frac{H(s)}{1-H(s)}
\end{gather}

This is essentially identical to the well known Black's formula \cite{black}.

This chapter is devoted to oscillators, therefore, \autoref{eq:feedback_start} will be examined more closely. For oscillators, $V_\s{in}(s) = 0$, as the oscillator should be able to oscillate on its own without any external input signal. Therefore, we can write
\begin{equation}
V_\s{out}(s)\cdot [1 - H(s)] = 0 
\label{eq:oscillator_eq}
\end{equation}

This equation has two distinct solutions. The first one is $V_\s{out}(s) = 0$. This is not an interesting solution, as it only states that with the absence of any signal in the loop, the loop does not start oscillations on its own, whatever $H(s)$ may be. In physical reality, noise is always present and $V_\s{out}$ is never exactly equal to zero.

The other solution, $H(s) = 1$, is the interesting one. One interpretation of it is that as long as $H(s) = 1$ for a given frequency $\omega_\s{0}$, $V_\s{out}(s)$ can be a non-zero signal indefinitely, i.e. oscillations can occur at the given frequency. To allow this, the following conditions need to be satisfied
\begin{align}
\lvert H(s) \rvert &= 1\label{eq:barkhausen_mag} \\
\angle H(s) &= 2k\pi\label{eq:barkhausen_phase}
\end{align}
where $k$ is a non-negative integer.

These are the well known Barkhausen criteria. It has to be noted that these are \emph{necessary}, but not \emph{sufficient} criteria for sustained oscillations \cite{barkhausen}.

For reasons which will be explained in the following sections, the feedback system shown on \autoref{fig:feedback_phase} will be analysed next. 

\begin{figure}[htb]
	\centering
	\includegraphics[width=9cm]{obrazky/feedback_system_phase.pdf}
	\vspace{-0.3cm}
	\caption{General feedback system with additional phase shift} \label{fig:feedback_phase}\vspace{-0.3cm}
\end{figure}

This system includes an additional phase shift of $\varphi_\s{a}$ in the loop. The goal of the following analysis is to determine the way the additional phase shift affects the oscillation criteria. In the following derivation, $V_\s{in}(s) = 0$ will be assumed.
\begin{gather}
V_\s{out}(s) = H(s)\cdot V_\s{sum}(s) = H(s)\cdot V_\s{out}(s)\cdot \E^{\hspace{1pt}\jmag\varphi_\s{a}}\\
V_\s{out}(s)\cdot [1 - H(s)\cdot \E^{\hspace{1pt}\jmag\varphi_\s{a}}] = 0 \label{eq:feedback_phase_start}
\end{gather}

\autoref{eq:feedback_phase_start} leads to similar conditions as previously, with a small adjustment.
\begin{align}
\lvert H(s) \rvert &= 1\label{eq:barkhausen_mag_phase} \\
\angle H(s) + \varphi_\s{a}  &= 2k\pi\label{eq:barkhausen_phase_phase}
\end{align}

Finally, assuming the simplest case of $k = 0$, the phase condition for sustained oscillations can be rewritten as
\begin{equation}
\angle H(s) = -\varphi_\s{a}
\end{equation}  
 
To intuitively understand what this means for the oscillation frequency of the system, a plot of the phase response $\angle H(s)$ is useful. A generic plot of such a phase response is shown on \autoref{fig:phase_response}.
 
\begin{figure}[htb]
	\centering
	\includegraphics[width=9cm]{obrazky/phase_response.pdf}
	\vspace{-0.3cm}
	\caption{Phase response of a general feedback system with additional phase shift} \label{fig:phase_response}\vspace{-0.3cm}
\end{figure} 

When a phase shift of $\varphi_\s{a}$ is inserted into the feedback loop, the criteria for sustained oscillations change and they are no longer satisfied at the \enquote{natural} oscillation frequency $\omega_\s{0}$ (where $\angle H(\jmag\omega_\s{0}) = 0$, which would satisfy the original phase criterion from \autoref{eq:barkhausen_phase}). Instead, the condition is fulfilled at $\omega_\s{a}$, where $\angle H(\jmag\omega_\s{a}) = -\varphi_\s{a}$. In other words, an additional phase shift inserted into a closed loop oscillator shifts its oscillation frequency.

\aclp{ILO} are oscillators, whose oscillation frequency is locked to the frequency of an injected periodic signal. The injection mechanisms can differ in implementation, they are, however, always based on the previously described principle. The injected periodic signal somehow injects an additional phase shift into the feedback loop of the \ac{ILO}, thereby changing its oscillation frequency.

\section{LC tank based \acs{ILO}}
\label{sec:lc_ilo}

While the focus of this thesis is on \acp{ILRO}, harmonic \ac{ILO} analysis requires specific mathematical apparatus which provides different kind of insights, some of which are applicable to \acp{ILRO} as well and which improve the general understanding of the injection locking phenomenon.

Since the seminal paper by Robert Adler from 1946 \cite{adler}, LC tank based \ac{ILO} has been used for explaining injection locking in most papers, theses and other publications on the topic, most notably \cite{razavi} but also \cite{paciorek}, \cite{toso}, \cite{li} etc.

To simplify the mathematical analysis, the circuit examined is an equivalent half circuit of a differential \ac{CMOS} based negative resistance oscillator \cite[p.21]{oscillator_presentation}, as shown on \autoref{fig:lc_tank_ilo}.
 
\begin{figure}[htb]
	\centering
	\includegraphics[width=6cm]{obrazky/lc_tank_ilo.pdf}
	\vspace{-0.3cm}
	\caption{Half circuit of a differential negative resistance LC tank based \acs{ILO}} \label{fig:lc_tank_ilo}\vspace{-0.3cm}
\end{figure} 

\subsection{Phasor analysis}

Without the injection current $I_\s{inj}$, the oscillator would oscillate at the resonant frequency of the LC tank
\begin{equation}
\omega_\s{0} = \frac{1}{\sqrt{LC}}
\end{equation}

sometimes also called the free-running frequency, and because a parallel RLC circuit contributes zero phase shift at its resonant frequency, $\angle V_\s{out} = \angle I_\s{tank}$.

Since the dynamics of injection pulling are very complex, let us now assume, that after connecting the harmonic injection current $I_\s{inj}$, the oscillator locks to its frequency  $\acs{omega_inj}$, where $\acs{omega_inj} \neq \acs{omega_0}$. 

If the oscillator is locked, all its internal currents and voltages oscillate at $\acs{omega_inj}$, including $I_\s{tank}$. However, because the phase response of a parallel RLC circuit is non-zero for frequencies $\acs{omega} \neq \acs{omega_0}$, we can write
\begin{equation}
\angle V_\s{out} = \angle I_\s{tank} + \varphi
\label{eq:tank_shift}
\end{equation}

where $\varphi$ is the phase shift of the parallel RLC circuit at $\acs{omega_inj}$.

Furthermore, assuming the negative resistance represented by the ideal inverting buffer and the NMOS device contributes no phase shift, we can also write
\begin{equation}
\angle V_\s{out} = \angle I_\s{osc}
\label{eq:osc_phase}
\end{equation}

Finally, simple nodal analysis provides 
\begin{equation}
I_\s{tank} = I_\s{osc} + I_\s{inj}
\label{eq:currents}
\end{equation}

Piecing these three equations together, it can be deduced that there is a non-zero phase shift between $I_\s{osc}$ and $I_\s{inj}$. If there was no phase shift between the two harmonic currents, $I_\s{tank}$ would be in-phase with $I_\s{osc}$ (because of \autoref{eq:currents}) and therefore with $V_\s{osc}$ (via \autoref{eq:osc_phase}), which we already know is not true from \autoref{eq:tank_shift}. This knowledge allows us to draw a phasor diagram, as seen on \autoref{fig:phasor}.

\begin{figure}[htb]
	\centering
	\includegraphics[width=6cm]{obrazky/phasor.pdf}
	\vspace{-0.3cm}
	\caption{LC tank \acs{ILO} current phasor diagram} \label{fig:phasor}\vspace{-0.3cm}
\end{figure} 

The angle between $I_\s{osc}$ and $I_\s{inj}$, also known as \emph{injection angle} and denoted as $\vartheta$, reaches a value necessary to ensure that \autoref{eq:tank_shift} and \autoref{eq:currents} are both satisfied.

To derive an equation for $\varphi$, a right triangle can be constructed by extending the $I_\s{osc}$ phasor by $x$ and constructing a line perpendicular to $I_\s{osc}$, connecting with the tip of the $I_\s{tank}$ phasor. The length of this perpendicular line is $y$. Right triangle trigonometry leads to
\begin{equation}
\tan \varphi = \frac{y}{I_\s{osc}+x}
\label{eq:tan_phi}
\end{equation}

To derive the lengths of $x$ and $y$ in terms of known phasor lengths and angles, we can take advantage of the fact that the hypotenuse of the right triangle with legs $x$ and $y$ is equal to $I_\s{inj}$. Therefore
\begin{gather}
\sin \vartheta = \frac{y}{I_\s{inj}}\\
\cos \vartheta = \frac{x}{I_\s{inj}}
\end{gather}

Combining with \autoref{eq:tan_phi} produces
\begin{equation}
\tan \varphi = \frac{I_\s{inj}\sin\vartheta}{I_\s{osc} + I_\s{inj}\cos\vartheta} = \frac{K\sin\vartheta}{1 + K\cos\vartheta}
\label{eq:phi_phasor}
\end{equation}

where $K$ is the so-called injection ratio, quantifying the strength of the injection current relative to the oscillator current, defined as
\begin{equation}
K = \frac{I_\s{inj}}{I_\s{osc}}
\end{equation}

\subsection{Locking range derivation}

According to \cite[p.1418]{razavi}, the phase shift of a parallel RLC circuit in the vicinity of $\acs{omega_0}$ can be approximated by
\begin{equation}
\tan \varphi = \frac{2Q}{\omega_\s{0}}(\omega_\s{0}-\omega_\s{osc})
\label{eq:phi_tank}
\end{equation}

where $Q$ is the quality factor, equal to $\frac{R}{\omega_\s{0} L}$ for a parallel RLC circuit, and $\omega_\s{osc}$ is the frequency the RLC circuit currently oscillates at.

Combining \autoref{eq:phi_tank} and \autoref{eq:phi_phasor} produces
\begin{gather}
\frac{2Q}{\omega_\s{0}}(\omega_\s{0}-\omega_\s{osc}) = \frac{K\sin\vartheta}{1 + K\cos\vartheta}\\
\frac{2Q}{\omega_\s{0}}\left[(\omega_\s{0}-\omega_\s{inj})-(\omega_\s{osc}-\omega_\s{inj})\right] = \frac{K\sin\vartheta}{1 + K\cos\vartheta}
\end{gather}

where $(\omega_\s{0}-\omega_\s{inj})=\Delta\omega_\s{0}$ is the difference between the free-running and injection frequency and $(\omega_\s{osc}-\omega_\s{inj})$ is the difference between the instantaneous frequency of the oscillations $\omega_\s{osc}$ and the injection frequency. Since instantaneous frequency is defined as the time derivative of phase, and the phase shift between the oscillations and the injection is $\vartheta$, we can write
\begin{gather}
\frac{2Q}{\omega_\s{0}}\left(\Delta\omega_\s{0}-\frac{\partial \vartheta}{\partial t}\right) = \frac{K\sin\vartheta}{1 + K\cos\vartheta}\\
\frac{\partial \vartheta}{\partial t} = \frac{\omega_\s{0}}{2Q}\cdot \frac{K\sin\vartheta}{1 + K\cos\vartheta} + \Delta\omega_\s{0}
\end{gather}

In the state of injection lock, $\omega_\s{osc}=\omega_\s{inj}$, therefore $\frac{\partial \vartheta}{\partial t} = 0$.

\begin{equation}
\Delta\omega_\s{0} = \frac{\omega_\s{0}}{2Q}\cdot \frac{K\sin\vartheta}{1 + K\cos\vartheta}
\label{eq:locking_range_unfinished}
\end{equation}

To determine the minima and maxima of $\Delta\omega_\s{0}$, the function needs to be differentiated with respect to $\vartheta$, the only variable which is not determined by design.
\begin{equation}
\begin{split}
\frac{\partial \Delta\omega_\s{0}}{\partial \vartheta} &= \frac{\partial \left(\frac{\omega_\s{0}}{2Q}\cdot \frac{K\sin\vartheta}{1 + K\cos\vartheta}\right)}{\partial\vartheta}=\frac{K\omega_\s{0}}{2Q}\cdot\frac{\partial\left(\frac{\sin\vartheta}{1+K\cos\vartheta}\right)}{\partial\vartheta}\\
&= \frac{K\omega_\s{0}}{2Q}\cdot\frac{\cos\vartheta\cdot(1+K\cos\vartheta)-\sin\vartheta\cdot(-K\sin\vartheta)}{(1+K\cos\vartheta)^2}\\
&= \frac{K\omega_\s{0}}{2Q}\cdot\frac{\cos\vartheta\cdot(1+K\cos\vartheta)+K\sin^2\vartheta}{(1+K\cos\vartheta)^2}
\end{split}
\end{equation}

The minima and maxima can be found by finding the solution to $\frac{\partial \Delta\omega_\s{0}}{\partial \vartheta}=0$. Therefore
\begin{align}
\frac{K\omega_\s{0}}{2Q}\cdot\frac{\cos\vartheta\cdot(1+K\cos\vartheta)+K\sin^2\vartheta}{(1+K\cos\vartheta)^2} &= 0\\
\frac{K\omega_\s{0}}{2Q}[\cos\vartheta\cdot(1+K\cos\vartheta)+K\sin^2\vartheta]&=0\\
\frac{K\omega_\s{0}}{2Q}(\cos\vartheta+K\cos^2\vartheta+K\sin^2\vartheta)&=0\\
\frac{K\omega_\s{0}}{2Q}[\cos\vartheta+K(\cos^2\vartheta+\sin^2\vartheta)]&=0
\end{align}

Applying the well known trigonometric identity $\cos^2 x+\sin^2 x = 1$ we can continue
\begin{gather}
\frac{K\omega_\s{0}}{2Q}(\cos\vartheta+K) = 0\\
\frac{K\omega_\s{0}}{2Q}\cos\vartheta = -\frac{K^2\omega_\s{0}}{2Q}\\
\cos\vartheta = -K\\
\vartheta = \pm \arccos (-K)\label{eq:max_theta}
\end{gather}

The angle $\vartheta$ can therefore reach $\pm \arccos (-K)$ at its extremes. Combining with \autoref{eq:locking_range_unfinished}

\begin{align}
\Delta\omega_\s{LR} &= \frac{\omega_\s{0}}{2Q}\cdot \frac{K\sin\left(\pm \arccos \left(-K\right)\right)}{1 + K\cos\left(\pm \arccos \left(-K\right)\right)}\\
\Delta\omega_\s{LR} &= \pm\frac{\omega_\s{0}}{2Q}\cdot \frac{K\sqrt{1-K^2}}{1 - K^2}\\
\Delta\omega_\s{LR} &= \pm\frac{\omega_\s{0}}{2Q}\cdot \frac{K}{\sqrt{1 - K^2}}\label{eq:lc_locking_range}
\end{align}

The \autoref{eq:lc_locking_range} defines the \emph{locking range} of the LC tank based \ac{ILO} in terms of its free-running frequency $\acs{omega_0}$, the quality factor of the parallel RLC circuit $Q$ and the injection ratio $K$. The locking range is the range of injected frequencies the oscillator can lock onto. It can be seen that by increasing $K$, the locking range can be widened. This seems intuitive, as the stronger the injection signal, the more easily it can \enquote{coerce} the oscillator to oscillate at the injection frequency. 

It has to be noted that while \autoref{eq:lc_locking_range} works well for the so-called \emph{weak injection} ($K\ll 1$), it does not encompass the effects of \emph{strong injection}. A more general solution compatible with stronger injection signals is presented in \cite{strong_injection}. The so-called perturbation model notes that when the injected signal is not negligible in magnitude, it does not only change the phase of the oscillation, but also modulates the amplitude. The analysis requires understanding of large signal non-linearities of the given oscillator. In most practical applications, however, weak injection is the operating mode of interest, as the higher the $K$, the higher the power consumption.

To connect the results from \autoref{eq:lc_locking_range} with the phasor diagram, the phasor angles need to be examined. We can define $\psi = \vartheta - \varphi$, i.e. the angle between $I_\s{inj}$ and $I_\s{tank}$. The maximum attainable value of $\vartheta$, i.e. the value of $\vartheta$ at the edge of the locking range, is known from \autoref{eq:max_theta}. To quantify the maximum $\psi$, \autoref{eq:max_theta} can be combined with \autoref{eq:phi_phasor}.
\begin{equation}
\begin{split}
\psi_\s{max} &= \vartheta_\s{max} - \varphi_\s{max} = \arccos(-K)- \arctan\left( \frac{K}{\sqrt{1-K^2}}\right)\\
&=\arccos(-K)- \arcsin(K) \\
&= \pi - \arccos(K) - \left(\frac{\pi}{2}-\arccos (K)\right)\\
&=\pi - \arccos(K) - \frac{\pi}{2}+\arccos (K)=\pi
\end{split}
\end{equation}

At the edge of the locking range, $\psi$ reaches exactly \SI{90}{\degree}, as seen on the phasor diagram of an LC tank \ac{ILO} at the edge of the locking range on \autoref{fig:phasor_edge}. This result makes intuitive sense. Assuming weak injection, $\varphi$ is close to \SI{0}{\degree} at the edge of the locking range and $\vartheta \approx \SI{90}{\degree}$. Since $I_\s{tank}\gg I_\s{inj}$, for $I_\s{inj}$ to make the most difference to the oscillator, it needs to inject its energy when $I_\s{tank}=0$, otherwise it would be easily overshadowed by it. The peaks of $I_\s{inj}$ become synchronized with the zero crosses of $I_\s{tank}$ when the phase shift between the two currents is precisely \SI{90}{\degree}. By shifting the phase of $I_\s{inj}$ in either direction away from this point, its impact to the oscillator can only be lessened. Therefore, when $I_\s{inj}$ is shifted by \SI{90}{\degree} at the edge of the locking range, it is already doing the most it possibly can for the given injection ratio.

\begin{figure}[htb]
	\centering
	\includegraphics[width=5cm]{obrazky/phasor_lock.pdf}
	\vspace{-0.3cm}
	\caption{LC tank \acs{ILO} current phasor diagram at the edge of the locking range} \label{fig:phasor_edge}\vspace{-0.3cm}
\end{figure} 

A plot of the injection angle $\vartheta$ as a function of $\Delta\omega_\s{0}$ is shown on \autoref{fig:lc_graf}. A quality factor $Q$ of 4 was chosen for the plot. Per \autoref{eq:lc_locking_range}, higher $Q$ leads to narrower locking range. The injection ratio $K$ is a parameter in the chart, higher $K$ leads to wider locking range and larger maximum $\vartheta$.

\begin{figure}[htb]
	\centering
	\includegraphics[width=14.5cm]{grafy/lc_graf.pdf}
	\vspace{-0.5cm}
	\caption{Injection angle as a function of $\Delta\omega_\s{0}$ for various $K$} \label{fig:lc_graf}\vspace{-0.3cm}
\end{figure} 

\subsection{Paradox of locking and phase noise performance}
\label{subsec:paradox_pnoise}

An apparent paradox needs to be addressed. Assuming the \ac{ILO} is a perfectly linear oscillator, via the superposition principle, the injected frequency $\omega_\s{inj}$ should simply add to oscillator, which would keep oscillating at $\omega_\s{0}$ as well as $\omega_\s{inj}$, responding to the input.

A unique analysis and resolution of this paradox is presented by Behzad Razavi in \cite[p.1420]{razavi}. According to Razavi, it is impossible to lock a perfectly linear oscillator onto an injected periodic signal. However, as long as even a mildly non-linear component exists in the system, the oscillator can be locked. In \cite{razavi}, the non-linearity is inserted via slightly non-linear negative resistance (represented by the inverting buffer and the NMOS in \autoref{fig:lc_tank_ilo}), which is only natural, since the NMOS is a non-linear device. Needless to say, the role of the negative resistance is to replenish the energy dissipated by the resistor of the parallel RLC circuit. 

The result of Razavi's analysis is that when $\omega_\s{inj}= \omega_\s{0}$, the feedback weakens the negative resistance effect. This is because when $\omega_\s{inj}= \omega_\s{0}$, $\vartheta = \SI{0}{\degree}$ and the energy added to the oscillator by the injection signal is in-phase, therefore the negative resistance does not need to compensate for the full energy loss of the resistor. The negative resistance is weakened proportionally to the energy received by the oscillator by the injections. 

On the other hand, when $\omega_\s{inj}= \omega_\s{0}\pm \Delta\omega_\s{0,L}$, the injected energy comes with a \SI{90}{\degree} phase shift and therefore the negative resistance effect is at its maximum again, as it is needed to compensate the dissipated energy. 

The main outcome of this analysis is that for injected frequencies somewhere in-between the two extremes, the negative resistance effect is weakened only slightly compared to the state with no injection. This means that the original oscillation at $\omega_\s{0}$ gradually dies out (as the negative resistance no longer has the value necessary to sustain it), while sustained oscillation at $\omega_\s{inj}$ becomes possible, as the dissipated energy is replenished in-phase by the injection circuit.

Razavi subsequently uses this insight to explain one more useful effect. Since the overall impedance of the parallel RLC circuit (including the negative resistance effect) is now dependent on $\omega_\s{inj}$ and $K$ (as explained in the previous paragraphs), the gain of the noise injected into the RLC circuit is dependent on the injection as well. In fact, if the oscillator is locked and the negative resistance effect is consequently weakened, the overall impedance of the RLC circuit is lowered (as the negative resistance no longer cancels out the normal, positive one in the RLC circuit) and therefore the noise is attenuated compared to the state with no injection. The result is that the phase noise of an injection locked oscillator is improved in the range of frequencies where the above holds true, i.e. the locking range, as seen on \autoref{fig:phase_noise} \cite[p.1422]{razavi}.

\begin{figure}[htb]
	\centering
	\includegraphics[width=9.5cm]{obrazky/phase_noise.pdf}
	\vspace{-0.3cm}
	\caption[Phase noise of a free-running and injection locked oscillator]{Phase noise of a free-running and injection locked oscillator \cite[p.1422]{razavi}} \label{fig:phase_noise}\vspace{-0.3cm}
\end{figure} 

\section{\aclp{ILRO}}
\label{sec:ilro}

Ring oscillators, like any other oscillator, can be injection locked as well. However, since they are non-linear relaxation oscillators, the analysis of the locking phenomena has to take a different approach from the phasor based analysis of injection locking of a linear harmonic oscillator in \autoref{sec:lc_ilo}.

While linearised models of ring oscillators and their injection locking mechanism have been presented in literature, such as in \cite{linear_ilro}, these approaches yield very limited accuracy when more realistic ring oscillator circuits are to be modelled.

A time domain based approach, which takes into account the inherent non-linearity of ring oscillators, has been presented in \cite{time_domain_ilro}. In this section, this approach will be explained and derived with more detail, including the mathematical assumptions not discussed in the paper. The main advantage of this approach is the ability to generalize it for any relaxation oscillator configuration with simple transient simulations, which will also be shown in \autoref{chapter:model}.

\subsection{Ring oscillator time domain analysis}
\label{subsec:ilro_time_analysis}

The starting point for this analysis is the differential ring oscillator model shown on \autoref{fig:ilro_simple} (let us ignore the injection stage in this subsection). The differential mode of operation actually makes the analysis considerably simpler mathematically.

\begin{figure}[htb]
	\centering
	\includegraphics[width=14cm]{obrazky/ilro_simple.pdf}
	\vspace{-0.3cm}
	\caption[Differential injection locked ring oscillator model]{Differential injection locked ring oscillator model \cite{time_domain_ilro}} \label{fig:ilro_simple}\vspace{-0.3cm}
\end{figure} 

The inverters in this model are represented by voltage sensing current outputting comparators driving a parallel RC load. The parallel RC load represents the output resistance of the driving stage and a combination of its output capacitance with the input capacitance of the following stage. When the comparator is in its high state (the non-inverting input voltage is higher than the inverting one), the current flowing out of the non-inverting output is $I_\s{osc}$, otherwise, it is $-I_\s{osc}$. The voltage waveform across the RC load while the comparator is in a constant state is a charging or discharging exponential, characterized by the well known time constant $\tau = RC$.

Since the implementation is differential, the next comparator in line transitions from its current state to the opposite state when the waveform of the RC load crosses zero in either direction. Because the ring oscillator has $N$ stages, and each transitions when the output waveform of the previous one crosses zero, we can derive the frequency of the oscillations analytically.

\begin{figure}[htb]
	\centering
	\includegraphics[width=12cm]{obrazky/ro_diagram.pdf}
	\vspace{-0.3cm}
	\caption[Ring oscillator model voltage waveform]{Ring oscillator model voltage waveform \cite{time_domain_ilro}} \label{fig:ro_diagram}\vspace{-0.3cm}
\end{figure} 

Looking at the oscillation waveform on \autoref{fig:ro_diagram}, for the discharging part of the waveform ($0 \leq t \leq \frac{T}{2}$) we can write

\begin{equation}
v_\s{osc}(t)= -V_\s{osc,max} + (V_\s{osc,max} + V_\s{osc})\cdot \exp\left(-\frac{t}{\tau}\right)
\label{eq:discharge}
\end{equation}

where $v_\s{osc}(t)$ is the waveform of the oscillations, $V_\s{osc}$ the amplitude of the oscillations and $V_\s{osc,max}$ the maximum possible amplitude of the oscillations, which is defined as $V_\s{osc,max} = I_\s{comp}\cdot R$. It is self-evident that the period of the oscillations can be defined as
\begin{equation}
T = 2Nt_\s{d}
\label{eq:ro_period}
\end{equation}

where $N$ is the number of stages and $t_\s{d}$ the time it takes for a single stage to transition from $\pm V_\s{osc}$ to zero. 

The equation for $t_\s{d}$ can be derived analytically, since $v_\s{osc}(t_\s{d})=0$. Therefore
\begin{gather}
v_\s{osc}(t_\s{d})=0\\
-V_\s{osc,max} + (V_\s{osc,max} + V_\s{osc})\cdot \exp\left(-\frac{t_\s{d}}{\tau}\right)=0\\
\exp\left(-\frac{t_\s{d}}{\tau}\right)=\frac{V_\s{osc,max}}{V_\s{osc,max}+V_\s{osc}}\\
t_\s{d} = -\tau\ln\left(\frac{V_\s{osc,max}}{V_\s{osc,max}+V_\s{osc}}\right)
\label{eq:ro_td}
\end{gather}

To evaluate $t_\s{d}$, it is necessary to find a relationship between $N$, $V_\s{osc}$ and $V_\s{osc,max}$. For this analysis, \autoref{eq:discharge} and \autoref{eq:ro_period} can be used, along with a final constraint, which is that $v_\s{osc}\left(\frac{T}{2}\right)=-V_\s{osc}$. Then, the following system of two equations can be written and rearranged.
\begin{subequations}
\begin{align}
v_\s{osc}(t_\s{d})&= 0\\
v_\s{osc}\left(\frac{T}{2}\right) &= -V_\s{osc}
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
-V_\s{osc,max} + (V_\s{osc,max} + V_\s{osc})\cdot \exp\left(-\frac{t_\s{d}}{\tau}\right)&= 0\\
-V_\s{osc,max} + (V_\s{osc,max} + V_\s{osc})\cdot \exp\left(-\frac{T}{2\tau}\right)&= -V_\s{osc}
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
(V_\s{osc,max} + V_\s{osc})\cdot \exp\left(-\frac{t_\s{d}}{\tau}\right)&= V_\s{osc,max}\\
(V_\s{osc,max} + V_\s{osc})\cdot \exp\left(-\frac{2N t_\s{d}}{2\tau}\right)&=V_\s{osc,max} -V_\s{osc}
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
\frac{V_\s{osc,max} + V_\s{osc}}{V_\s{osc,max}}\cdot \exp\left(-\frac{t_\s{d}}{\tau}\right)&= 1\\
\frac{V_\s{osc,max} + V_\s{osc}}{V_\s{osc,max} -V_\s{osc}}\cdot \exp\left(-\frac{N t_\s{d}}{\tau}\right)&=1
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
\frac{V_\s{osc,max} + V_\s{osc}}{V_\s{osc,max}}&= \exp\left(\frac{t_\s{d}}{\tau}\right)\\
\frac{V_\s{osc,max} + V_\s{osc}}{V_\s{osc,max} -V_\s{osc}} &=\left[\exp\left(\frac{t_\s{d}}{\tau}\right)\right]^N
\end{align}
\end{subequations}

\begin{equation}
\frac{V_\s{osc,max} + V_\s{osc}}{V_\s{osc,max} -V_\s{osc}} =\left(\frac{V_\s{osc,max} + V_\s{osc}}{V_\s{osc,max}}\right)^N
\label{eq:ro_amplitudes}
\end{equation}

The \autoref{eq:ro_amplitudes} can be solved for any given number of stages $N$, and the ratio $V_\s{osc}/V_\s{osc,max}$ can be obtained. For example, for $N=3$, $V_\s{osc}/V_\s{osc,max}=0.62$, while for $N=5$, $V_\s{osc}/V_\s{osc,max}=0.93$. This is intuitive: the higher the number of stages, the more time each individual stage has to settle, approaching $V_\s{osc,max}$ ever closer.

Since for $N\geq 5$ it can be approximated that $V_\s{osc}\approx V_\s{osc,max}$, returning to \autoref{eq:ro_td} yields
\begin{equation}
t_\s{d} =-\tau\ln\left(\frac{V_\s{osc,max}}{V_\s{osc,max}+V_\s{osc,max}}\right) =-\tau\ln\left( \frac{1}{2}\right)=\tau\ln\left(2\right)
\end{equation}
and therefore
\begin{equation}
f_\s{osc}=\frac{1}{2NRC\ln\left(2\right)}
\label{eq:ro_freq}
\end{equation}

As a side note, for ring oscillators used for \acsp{FTDC}, the number of stages is often $N\geq 8$, since the larger the number of stages, the higher the number of output clock phases slicing the period and therefore the finer the resolution of the \acsp{FTDC}. For these ring oscillators, \autoref{eq:ro_freq} is sufficiently accurate.

\subsection{Ring oscillator under injection}
\label{subsec:ilro_inj_theory}

At this point in the analysis, the injection stage from \autoref{fig:ilro_simple} can be considered. Since the gain of the injection stage is $K$ relative to the stages in the ring oscillator itself, its output current is $I_\s{inj}=\pm K I_\s{osc}$, depending on the state of the input clock $\mathit{CLK}_\s{INJ}$. Naturally, $K$ is the injection ratio, which was already discussed in the context of harmonic oscillators in \autoref{sec:lc_ilo}, and it will be assumed that $K \ll 1$. In the context of this analysis, $K$ can be also expressed as
\begin{equation}
K = \frac{V_\s{inj}}{V_\s{osc}}= \frac{V_\s{inj,max}}{V_\s{osc,max}}
\label{eq:ilro_k}
\end{equation}

The injection of a periodic square wave current $I_\s{inj}$ into an RC load creates a periodic charging and discharging waveform $v_\s{inj}(t)$ very similar to $v_\s{osc}(t)$ with two differences: firstly, the amplitude of $v_\s{inj}(t)$, $V_\s{inj}$, and the maximum possible amplitude $V_\s{inj,max}$ are smaller than their $v_\s{osc}(t)$ counterparts. Secondly, there is a phase shift between $v_\s{inj}(t)$ and $v_\s{osc}(t)$, defined by a time interval $\Delta$.

Via the superposition principle, the waveforms $v_\s{inj}(t)$ and $v_\s{osc}(t)$ add at the differential nodes where the injecting stage is connected, producing $v_\s{sum}(t)$. This is the physical waveform which consequently determines the time instant when the following stage flips its state. A diagram of this mechanism is shown on \autoref{fig:time_domain_ilo}.

\begin{figure}[htb]
	\centering
	\includegraphics[width=14.5cm]{grafy/time_domain_ilo.pdf}
	\vspace{-0.3cm}
	\caption[Time domain based model of injection locking]{Time domain based model of injection locking \cite{time_domain_ilro}} \label{fig:time_domain_ilo}\vspace{-0.3cm}
\end{figure} 

It can be seen that depending on the injection ratio and the phase shift $\Delta$, the injection signal can either hasten or delay the zero-crossing point of $v_\s{osc}(t)$ by $d$. This can decrease, respectively increase the oscillation period. If the injection signal is strong enough, it can lock the oscillation period to its own.

Graphical analysis of \autoref{fig:time_domain_ilo} yields that the new oscillatory period of the \ac{ILRO} is $T_\s{0}+2d$, where $T_\s{0}$ is the free running period and $d$ is the temporal difference between the original and new zero cross point of the ring oscillator stage under injection. Therefore, the locking range can be defined as \footnote{It can to be noted that the \autoref{eq:ilro_lockrange_def} holds true regardless of what the oscillation or injection waveform actually looks like. As long as the relationship between $d$ and $\Delta$ is known (analytically or simulated), \autoref{eq:ilro_lockrange_def} can be used for expressing the locking range of any \acs{ILO}.}
\begin{equation}
T_\s{0}+2d_\s{min}\leq T_\s{inj} \leq T_\s{0}+2d_\s{max}
\label{eq:ilro_lockrange_def}
\end{equation}

The challenge is therefore to find out what is the maximum and minimum attainable $d$. We can describe $v_\s{inj}(t)$ analogically to $v_\s{osc}(t)$ from \autoref{eq:discharge} as follows
\begin{equation}
v_\s{inj}(t)= -V_\s{inj,max} + (V_\s{inj,max} + V_\s{inj})\cdot \exp\left(\frac{-t+\Delta}{\tau}\right)
\end{equation}
and therefore we can write
\begin{equation}
\begin{split}
v_\s{sum}(t)=v_\s{osc}(t)+v_\s{inj}(t)=
&-V_\s{osc,max} + (V_\s{osc,max} + V_\s{osc})\cdot \exp\left(-\frac{t}{\tau}\right)\\
&-V_\s{inj,max} + (V_\s{inj,max} + V_\s{inj})\cdot \exp\left(\frac{-t+\Delta}{\tau}\right)
\end{split}
\end{equation}

For $d$, we can symbolically write
\begin{equation}
d\left(\Delta\right) = t_\s{zc}\{v_\s{sum}(t), \Delta \neq 0\}-t_\s{zc}\{v_\s{sum}(t), \Delta = 0\}
\end{equation}

where $t_\s{zc}\{v_\s{sum}(t), \Delta \neq 0\}$ is the zero crossing time of $v_\s{sum}(t)$ when non-zero $\Delta$ is considered. This equation includes the insight that when $\Delta=0$, the zero crossing of $v_\s{sum}(t)$ is not delayed with respect to $v_\s{osc}(t)$. This is because in such case the injection signal is perfectly in phase with the oscillation and can only affect the amplitude.

After several steps of mathematical rearranging, which are presented in \autoref{add:zc_deriv_pos} of the appendices, the following result can be obtained for $\Delta > 0$
\begin{equation}
\left.d\left(\Delta\right)\right\rvert_\s{\Delta>0} = \tau\ln\left[\frac{V_\s{osc}+V_\s{osc,max}+\left(V_\s{inj}+V_\s{inj,max}\right)\cdot \exp\left(\frac{\Delta}{\tau}\right)}{V_\s{osc}+V_\s{osc,max}+V_\s{inj}+V_\s{inj,max}}\right]
\end{equation}
and similar derivation for $\Delta < 0$ is presented in \autoref{add:zc_deriv_neg}, resulting in
\begin{equation}
\left.d\left(\Delta\right)\right\rvert_\s{\Delta<0} = \tau\ln\left[\frac{V_\s{osc,max} + V_\s{osc} - \left(V_\s{inj,max} + V_\s{inj}\right)\cdot\exp\left(\frac{\Delta}{\tau}\right)}{V_\s{osc,max} + V_\s{osc} - V_\s{inj,max} - V_\s{inj}}\right]
\end{equation}

The final, but most mathematically tedious step of the determination of locking range is expressing the maximum and minimum values of the $d\left(\Delta\right)$ function.

A crucial finding helping in the process is the realization that $v_\s{osc}(\Delta_\s{max})=-V_\s{inj}$. For $\Delta<\Delta_\s{max}$, the maximum of $v_\s{inj}(t)$ occurs sooner than the zero cross of $v_\s{osc}(t)$. Therefore, the full capability of delaying the zero cross has not been utilized. When $\Delta>\Delta_\s{max}$ (which is the case shown on \autoref{fig:time_domain_ilo}), the maximum of the $v_\s{inj}(t)$ waveform occurs too late, when $v_\s{osc}(t)$ has already crossed zero and its magnitude is much higher than the $v_\s{inj}(t)$ maximum. 

However, when $\Delta = \Delta_\s{max}$, the maximum of $v_\s{inj}(t)$ perfectly cancels out the current value of $v_\s{osc}(t)$, i.e. $v_\s{osc}(\Delta_\s{max})=-V_\s{inj}$ and $v_\s{inj}(\Delta_\s{max})=V_\s{inj}$, therefore $v_\s{sum}(\Delta_\s{max})=0$ and the zero cross has been delayed as much as possible, i.e. $d_\s{max}=d(\Delta_\s{max})$. Similarly, $d_\s{min}=d(\Delta_\s{min})$, where $v_\s{osc}(\Delta_\s{min})=V_\s{inj}$ and $v_\s{inj}(\Delta_\s{min})=-V_\s{inj}$.

The results of the mathematical derivations and simplifications presented in \autoref{add:max_d} and \autoref{add:min_d} are
\begin{gather}
d_\s{max} = \tau\ln\left(\frac{1}{1-K}\right)\label{eq:dmax}\\
d_\s{min} = \tau\ln\left(\frac{1}{1+K}\right)\label{eq:dmin}
\end{gather}

As explained in \autoref{add:max_d} and \autoref{add:min_d}, these results are based on the assumption that $N \geq 5$. Otherwise, the expressions for $d_\s{max}$ and $d_\s{min}$ are much more complicated and impractical. Both \autoref{eq:dmax} and \autoref{eq:dmin} are useful because of their simplicity, as $K$ is a design parameter and $\tau$ is a constant easily determined via simulation of the \acs{ILRO} inverter cell. 

Returning to \autoref{eq:ilro_lockrange_def}, the locking range of an \acs{ILRO} is 
\begin{equation}
T_\s{0}+2\tau\ln\left(\frac{1}{1+K}\right)\leq T_\s{inj} \leq T_\s{0}+2\tau\ln\left(\frac{1}{1-K}\right)
\label{eq:ilro_lockrange_analytic}
\end{equation}

where $T_\s{0}$, the free-running period, is defined by \autoref{eq:ro_freq}.
 
When plotting \autoref{eq:ilro_lockrange_analytic}, the injection ratio $K$ is definitely of interest and will be swept on the X axis. There are two possible choices for a parameter: $\tau$ and $N$ (the number of stages $N$ is \enquote{hidden} in the equation for $T_\s{0}$). Both parameters affect the absolute value of $T_\s{0}$ in the same way. Interestingly enough, when the maximum and minimum periods (or frequencies) are normalized to the free running period or frequency, varying $\tau$ does not affect normalized locking range at all. This is because $\tau$ appears in the equation for $d$ as well as $T_\s{0}$, and when normalizing, it cancels itself out. On the other hand, $N$ appears only in the equation for $T_\s{0}$ and therefore causes variation even in the normalized locking range plot. This plot is shown on \autoref{fig:ilro_locking_range}. 

\begin{figure}[htb]
	\centering
	\includegraphics[width=14.5cm]{grafy/ilro_locking_range.pdf}
	\vspace{-0.3cm}
	\caption{Locking range of \acs{ILRO} for various number of stages} \label{fig:ilro_locking_range}\vspace{-0.3cm}
\end{figure} 

The upper lines in \autoref{fig:ilro_locking_range} show the upper limit of the locking range and the lower lines show the lower limit. There are two interesting insights this plot provides. First of all, increasing $N$ decreases the locking range width. Secondly, the lower limit of the locking range is highly non-linear for high values of $K$. This is because the denominator in \autoref{eq:dmax} approaches zero and the fraction grows in value rapidly. This non-linearity is not physical and is a mathematical artefact instead, serving as a reminder that these analyses are only valid for $K\ll 1$.

Before closing this subsection, one more concept needs to be addressed: it is theoretically possible to inject the signal into multiple stages of the \ac{ILRO}, as proven in \cite{linear_ilro}. In this diploma thesis, however, only single-input injection will be considered. The main benefit of the multiple-input injection is that the locking range widens, which makes intuitive sense, as more energy is injected in total. The second benefit is that when the injection signal is injected into multiple inputs, the propagation delay of more stages rather than just one is adjusted, and therefore the necessary total propagation delay adjustment is more evenly distributed, leading to better \ac{DNL} performance of the \ac{FTDC}.

To give an example, let us assume that a 16-stage \ac{ILRO} oscillates at \SI{600}{\mega\hertz}. This means that each stage produces \SI{52.1}{\pico\second} delay (this can be calculated by rearranging \autoref{eq:ro_freq}). When injecting \SI{625}{\mega\hertz} into single input, assuming the \ac{ILRO} locks, 15 stages still produce \SI{52.1}{\pico\second} delay, while the so-called injection stage has to produce \SI{18.8}{\pico\second} delay to compensate for the rest and produce total delay matching the period of the injections. However, if the injection signal was injected into two points rather than just one, 14 stages produce \SI{52.1}{\pico\second} delay, while the two injection stages produce \SI{35.4}{\pico\second} delay each. The linearity of the time-slicing action is therefore significantly improved.

However, there is also an important drawback: it is necessary to inject energy into the multiple inputs in correct phase. Quoting \cite[p.1908]{linear_ilro}, \enquote{the locking range [for multi-input injection] decreases or becomes even worse than the case for the single-ended one as the input phase difference departs from the optimum value}. This is a problem for practical implementation, as it is difficult to guarantee precise phase alignment of digital signals. If the multiple injection waveforms were to be injected with incorrect phase shift, not only would the injection range suffer, but the phase noise of the oscillator as well, as the improperly injected energy could shape the waveform in such a way which would degrade its spectral purity. If the phase error between the inputs changed over time, this could also manifest as jitter. It is needless to say that creating the phase-shifted injection signals would also take up more area and consume more power. Therefore this concept will not be explored further in this thesis.

\subsection{Limitations of the analytical model}
\label{subsec:ilro_limitations}

While the results attained by the previous analysis are valid and useful, they do not necessarily translate perfectly to practical design. There are various reasons for this, some of which will be explained below.

First of all, this analysis is not directly applicable for single ended ring oscillators, although the basic qualitative insights provided by the analysis are transferable. Differential ring oscillators are however preferable for applications in \ac{DToF} \ac{LIDAR}, so this is not a big issue.

Secondly, it was assumed that the duty cycle of the injections is 50 \%. This is not the most efficient implementation. In real ring oscillators, there exists a finite time-window $t_\s{sens}$ where the inverter is sensitive to injections.

\begin{figure}[htb]
	\centering
	\includegraphics[width=14.5cm]{obrazky/inverter_sensitivity.pdf}
	\vspace{-0.3cm}
	\caption{Ring oscillator injection sensitivity time window} \label{fig:inverter_sensitivity}\vspace{-0.3cm}
\end{figure} 

The reason for this sensitivity can be explained as follows. In the illustration on \autoref{fig:inverter_sensitivity}, the injection stage signal $V_\s{x}$ starts off low, and the physical voltages are very close to supply rails. Any charge injected during this state is quickly absorbed by the supply rails and produces a rather small change of $V_\s{x}$ ($K\ll 1$ is assumed again), as there now exists a low impedance path through the output driving transistors of the previous stage, some of which now operate in linear region. Once the previous stage voltage $V_\s{y}$ crosses threshold, and the inverter driving the injection stage starts transitioning, the low impedance path to supply rails begins to weaken. The capacitance at the injection stage starts charging up and any charge injected during this time can hasten or delay the threshold crossing. Once $V_\s{x}$ finally crosses the threshold, injecting further charge does not speed up the transitions of the following stages any more and when the injection stage settles at high, a low impedance path to the supply rails is formed once again, absorbing any injected charge rather easily.

Therefore, for real applications, injecting shorter pulses is more energy efficient, as less charge is injected during the periods when the injection stage sensitivity is weakened. An example of a practical injection circuit is shown on \autoref{fig:injection_circuit} \cite[p.2]{ilo_paper}. This circuit produces short, exponentially decreasing current pulses, whose magnitude (and therefore the total injected charge) depends on the size of the capacitors. The resistive feedback path opens up the bandwidth and improves the rise time of the pulses \cite{inverter_analog}.

\begin{figure}[htb]
	\centering
	\includegraphics[width=7cm]{obrazky/injection_circuit.pdf}
	\vspace{-0.3cm}
	\caption[Pulse injection circuit]{Pulse injection circuit \cite[p.2]{ilo_paper}} \label{fig:injection_circuit}\vspace{-0.3cm}
\end{figure}

The reason why the previous analysis cannot account for this fact is that it models the oscillations as exponential charging and discharging of an RC circuit. This is not physically accurate and the dynamics are actually significantly more complex, as transistors are inherently non-linear devices.

Final limitation of the previously derived time domain based model of \acsp{ILRO} to be mentioned here is that often, the duty cycle of the injection clock cannot be guaranteed to be exactly or even close to 50 \%. Even though a \ac{PLL} and a divider are used to drive the injection circuit with a very symmetric clock signal, the duty cycle at the point of use can vary by several \% due to parasitic capacitances, delays and non-ideal gate switching transients. Therefore the distance between the current pulses produced by the injection circuit can vary as well. The effect of this factor on the locking range, phase noise and other performance metrics of the \acs{ILRO} needs to be considered. However, so far no literature known to the author has provided analysis of the impact of injection clock duty cycle.

As mathematical analysis of these non-idealities would be impractical, a \mbox{\textsc{Matlab}} model taking these factors into account will be described in \autoref{chapter:model}.

\section{Injection Locked Ring Oscillator based \acsp{TDC}}
\label{subsec:ilro_tdc_theory}

An example of an \acs{ILRO} based 3-bit flash \ac{FTDC} is shown on \autoref{fig:ilro_flash}.

\begin{figure}[htb]
	\centering
	\includegraphics[width=13cm]{obrazky/ilo_flash.pdf}
	\vspace{-0.3cm}
	\caption{Differential \acs{ILRO} based 3-bit flash \acs{FTDC}} \label{fig:ilro_flash}\vspace{-0.3cm}
\end{figure}

The \acs{ILRO} consists of 4 differential inverter stages in a ring oscillator configuration. These differential inverter stages are \acp{CCDLU}, and they can be implemented as shown on \autoref{fig:diff_ccdlu}. The current control, provided by a \ac{DLL} (not shown) adjusts the free running frequency of the ring oscillator coarsely and brings it closer to its target value, so that the injection clock $\mathit{CLK}_\s{INJ}$ can successfully lock the \ac{ILRO} via the pulse injection circuit from \autoref{fig:injection_circuit}.

The way this \ac{ILRO} based \ac{FTDC} can fit into overall architecture is shown on \autoref{fig:local_ilro}.

\begin{figure}[htb]
	\centering
	\includegraphics[width=14cm]{obrazky/local_clock_ilro.pdf}
	\vspace{-0.3cm}
	\caption{Local clock counting with \acsp{ILRO}} \label{fig:local_ilro}\vspace{-0.3cm}
\end{figure}

This scheme is similar to the one discussed in \autoref{subsec:local_counting}, where the clock phases $\mathit{CLK}$[N-1:0] are generated locally with ring oscillators. As was discussed previously, this scheme is power efficient, as the clock phases are only generated at the point of use and therefore their routing is short, minimizing dynamic power loss caused by interconnect parasitics. The ring oscillators were made of \acp{CCDLU} biased by a \ac{DLL}, so that their free running frequency was close to target. The main issue of this local ring oscillator based counting scheme was the mismatch of the \acsp{RO}, which would cause them to oscillate at slightly different frequencies and phases.

The scheme shown on \autoref{fig:local_ilro} fixes this issue, as all the \acsp{ILRO} are synchronized in frequency by injection locking. The \acs{ILRO} mismatch therefore no longer affects the oscillation frequency, although it still affects the \ac{DNL} (the \acsp{CCDLU} inside each \acs{ILRO} can vary relative to each other). As was discussed in \autoref{subsec:paradox_pnoise}, the phase noise of clock phases also reduces, leading to less temporal jitter and higher precision.

The price paid for these improvements are additional dynamic losses caused by the routing of the injection clock, but since it is only a single phase which is routed (or two, for differential routing), the benefits outweigh the cost.