Analytical formulation

A set of input tones will be introduced into the oscillator circuit, at the frequencies ω ₁, ω ₂,…, ω_K, assumed to be closely spaced about the original free-running oscillation frequency ω_o. When varying a suitably chosen tuning parameter η, the oscillator should be able to lock to each of the different input frequencies ω_k. To get analytical insight, the oscillator will be described at its fundamental frequency, in terms of its current-to-voltage ratio Y(V,ω,η) at a particular observation node [Reference Suarez, Ramirez and Sancho22–Reference Fernández, Ramírez, Suárez and Sancho26], where V and ω are the excitation amplitude and frequency. This admittance function will be the current-to-voltage ration of an AG connected at the observation node in the HB simulation of the oscillator circuit. In the absence of the input signals, at its free-running solution, the oscillator fulfils the following two-tier equation system [Reference Suárez and Quéré11]:

(1)$$\eqalign{Y\lpar V_o\comma \;\omega _o\comma \;\eta _o\rpar = \; & 0 \cr \bar{H}\lpar \bar{X}^{\prime}\comma \;V\comma \;\omega \comma \;\eta _o\rpar = \; & 0 \comma} $$

where $\bar{H}$ represents the HB system that constitutes the inner tier and V_o and ω_o, are the free-running amplitude and frequency at η_o. Note that the phase origin has been arbitrarily taken at the observation node Ve^{j 0} and the vector $\bar{X}^{\prime}$ contains all the state variables except Ve^{j 0}. The above system can be solved through optimization of V and ω in order to fulfil the goal Y = 0.

Now, the effect of the input source will be considered. Injection-locking to the particular input tone ω_q, belonging to the set of input tones ω ₁, ω ₂,…,ω_K, will be assumed. The oscillator will be formulated in the envelope domain at the carrier frequency ω_q. Thus, the node voltage will be expressed as $\lpar {V_o + {\rm \Delta }V\lpar t\rpar } \rpar e^{j\phi \lpar t\rpar }e^{j\omega _qt}$, where ΔV(t) is the time-varying increment of the voltage amplitude with respect to the free-running value V_o and ϕ(t) is the time-varying phase, in the presence of the input sources. Applying Kirchoff's laws at the observation node one obtains the following system:

(2)$$\!\!\!\eqalignb{& Y\lpar {V_o + {\rm \Delta }V\lpar t\rpar \comma \;\eta_o + \Delta \eta \comma \;\omega_o + \Delta \omega_q + s/j} \rpar \lpar {V_o + {\rm \Delta }V\lpar t\rpar } \rpar e^{\,j\phi \lpar t\rpar } \cr &\quad= \sum\limits_k {I_k\exp j\lpar {\Delta \omega_{k\comma q}t} \rpar } \comma \;\quad {\rm \Delta }\omega _{k\comma q} = \omega _k-\omega _q\comma} $$

where Δω _q = ω _q − ω _oand s is a complex frequency increment and I_k is the equivalent complex current associated with each input signal. Note that Δω _k,q is the offset frequency of each input tone k ≠ q with respect to ω _q. Under a small input power and small frequency spacing between the input tones, the function Y can be expressed in a first-order Taylor series expansion about V_o, ω_o, and η_o [Reference Ramirez, Ponton, Sancho and Suarez21,Reference Suarez, Ramirez and Sancho22]. Taking into account that s acts as a time differentiator, one obtains:

(3)$$Y_V{\rm \Delta }V\lpar t\rpar + Y_\eta {\rm \Delta }\eta + Y_\omega \lpar {\Delta \omega_q + \dot{\phi }\lpar t\rpar -j{\rm \Delta }\dot{V}\lpar t\rpar /V_o} \rpar = \sum\limits_k {\displaystyle{{I_k} \over {V_o}}\exp j\lpar {-\phi \lpar t\rpar + \Delta \omega_{k\comma q}t} \rpar } \comma \;$$

where Y_V, Y_η, and Y_ω are the derivatives of Y with respect to V, η, and ω, calculated at the free-running steady-state oscillation V_o, ω_o, and η_o. These derivatives are obtained by means of the same AG used for the calculation of the free-running oscillation [Reference Suárez10]. In that simulation, the AG operates at the unknown oscillation frequency ω _o, with the amplitude V _o, agreeing with the first harmonic of the node voltage. The AG must fulfill the oscillation condition Y(V _o,ω _o) = 0, which is solved through optimization in commercial HB software. Once the free-running solution has been determined, the derivatives of the admittance function are calculated through finite differences [Reference Ramirez, de Cos and Suarez13] (Fig. 1). The derivative Y_V with respect to the amplitude is calculated by considering the increments V_o ± ΔV, while the frequency is kept at ω _o. The frequency derivative Y_ω is calculated by considering the increments ω_o ± Δω, while the amplitude is kept at V_o. The derivative Y_η with respect to the parameter is calculated by considering the increments η_o ± Δη, while the AG amplitude and frequency are kept at V_o, ω_o.

Fig. 1. Calculation of the derivatives of the oscillator nonlinear-admittance function through finite differences in harmonic balance, using an auxiliary generator.

Solving (3) for $\Delta \omega _q + \dot{\phi }\lpar t\rpar$ one obtains:

(4)$$ \eqalign{& \Delta \omega _q + \dot{\phi }\lpar t\rpar = \cr & \displaystyle{{-\vert {Y_\eta } \vert \sin \alpha _{v\eta }{\rm \Delta }\eta + \sum\nolimits_k {\displaystyle{{\vert {I_k} \vert } \over {V_o}}\sin \lpar {-\phi \lpar t\rpar + \Delta \omega_{k\comma q}t + \phi_k-\alpha_v} \rpar } } \over {\vert {Y_\omega } \vert \sin \alpha _{v\omega }}}\comma \;}$$

where |I_k| and ϕ_k are the magnitude and phase of the input tones, α_vω = ang(Y_ω)−ang(Y_V), α_vη = ang(Y_η)−ang(Y_V), and α_v = ang(Y_V). Note that (4) describes the circuit response in both locked and unlocked conditions. When the oscillator is locked to ω_q, the phase ϕ(t) in (4) can be represented in a Fourier series at K−1 frequencies Δω _1,q, …, Δω _K,q, so one can express ϕ(t) = ϕ_o + Ф_mix(t), where ϕ_o is a constant value and Ф_mix(t) has zero average: <Ф_mix(t) = 0>. Taking this into account, to detect the locking bands, one will introduce the expression ϕ(t) = ϕ_o + Ф_mix(t) into (4) and perform an averaging:

(5)$$\eqalign{& \left\langle {\Delta \omega_q + {\dot{\Phi }}_{mix}\lpar t\rpar } \right\rangle \ = \ \displaystyle{{\vert {I_q} \vert } \over {V_o}}\ \ \displaystyle{{\left\langle {\sin \lpar -\phi_o-\Phi_{mix}\lpar t\rpar -\alpha_v\rpar } \right\rangle } \over {\vert {Y_\omega } \vert \sin \alpha _{v\omega }}} \cr & \quad-\displaystyle{{\vert {Y_\eta } \vert \sin \alpha _{v\eta }} \over {\vert {Y_\omega } \vert \sin \alpha _{v\omega }}}{\rm \Delta }\eta + \displaystyle{{\left\langle {\sum\limits_{k\ne q} {\displaystyle{{\vert {I_k} \vert } \over {V_o}}\sin \lpar {-\phi_o-\Phi_{mix}\lpar t\rpar + \Delta \omega_{k\comma q}t + \phi_k-\alpha_v} \rpar } } \right\rangle } \over {\vert {Y_\omega } \vert \sin \alpha _{v\omega }}}.} $$

Since Δω _q is kept at a constant value, the average phase shift ϕ_o will vary with the tuning parameter Δη. Under small amplitudes |I _k|, Ф_mix(t) can be neglected in the first term on the right-hand side of (3). As the amplitudes |I _k| decrease, the above equation approaches:

(6)$$\Delta \omega _q = \displaystyle{{\vert {I_q} \vert } \over {V_o}}\displaystyle{{\sin \lpar -\phi _o-\alpha _v\rpar } \over {\vert {Y_\omega } \vert \sin \alpha _{v\omega }}}-\displaystyle{{\vert {Y_\eta } \vert \sin \alpha _{v\eta }} \over {\vert {Y_\omega } \vert \sin \alpha _{v\omega }}}{\rm \Delta }\eta . $$

One can also solve (3) in terms of the increment of the voltage amplitude ΔV(t). Because the frequency reference is ω_q, the voltage amplitude at this frequency is V_o + ΔV(t). Thus, the amplitude at the tone ω_q at which the oscillator is injection-locked is enhanced by the self-oscillation and will be higher than that of the rest of the spectral lines. Solving for the increment ΔV(t) and averaging, one obtains:

(7)$$\eqalign{\left\langle {{\rm \Delta }V\lpar t\rpar } \right\rangle = & \displaystyle{{\displaystyle{{\vert {I_q} \vert } \over {V_o}}\left\langle {\sin \lpar \alpha_\omega + \phi_o + \Phi_{mix}\lpar t\rpar \rpar } \right\rangle } \over {\vert {Y_V} \vert \sin \alpha _{v\omega }}}-\displaystyle{{\vert {Y_\eta } \vert \sin \alpha _{\eta \omega }} \over {\vert {Y_V} \vert \sin \alpha _{v\omega }}}{\rm \Delta }\eta \cr & \quad + \displaystyle{{\left\langle {\sum\limits_k {\displaystyle{{\vert {I_k} \vert } \over {V_o}}\sin \lpar {\alpha_\omega -\phi_o-\Phi_{mix}\lpar t\rpar + \phi_k-\Delta \omega_{k\comma q}t} \rpar } } \right\rangle } \over {\vert {Y_V} \vert \sin \alpha _{v\omega }}}\comma} $$

where α_ω = ang(Y_ω), α_ηω = ang(Y_ω)−ang(Y_η), and $\Delta \dot{V}\lpar t\rpar$ has been neglected. Taking into account the relationship between the average phase shift ϕ_o and Δη, the average of the amplitude increment 〈ΔV(t)〉should approach an ellipsoidal curve versus Δη in the locking band. However, due to the presence of two turning points in the closed curve, only one section of this curve will be stable. Note that the turning points are local/global bifurcations [Reference Guckenheimer and Holmes27, Reference Thompson and Stewart28] at which a quasi-periodic solution is generated from a zero-frequency shift, in a manner analogous to the generation of a limit cycle in a saddle-node bifurcation [Reference Guckenheimer and Holmes27, Reference Thompson and Stewart28].

Application to a FET-based oscillator

The above analysis has been applied to the voltage-controlled oscillator in Fig. 2. It is based on the transistor ATF-34143 and is tuned with the varactor SMV1235. It is able to oscillate in the frequency band 2.7–2.78 GHz. The input tones will be generated with the Agilent E4438C ESG Vector Signal Generator and are equally spaced by default. In order to generate as many tones as possible, the spacing Δf = 16 MHz is chosen, enabling six input tones, between 2.7 and 2.78 GHz (case (i) in Fig. 2). Initially, synchronization at ω_q/(2π) = 2.7 GHz is assumed. First, the results obtained through the integration of (3) and with circuit-level envelope transient simulations [Reference Cos, Suarez and Sancho20] (using ω_q as the only fundamental frequency, as in (3)) have been compared. The spectrum obtained under injection-locked conditions for a multitone input signal in the two cases is shown in Fig. 3. As can be seen, there is an excellent agreement that demonstrates the validity of (3).

Fig. 2. FET-based (ATF-34143) oscillator at f_o = 2.7 GHz. The frequency is tuned with the varactor diode SMV1235. (a) Schematic, showing the two different kinds of input signal considered in this work: (i) six input tones equally spaced between 2.7 and 2.78 GHz and (ii) a rectangular signal with 1 Vpp, 25% duty cycle and frequency 90 MHz. The dc-block capacitance is suppressed in this second case. (b) Photograph.

Fig. 3. Spectrum obtained through (3) and with envelope transient for the spectral-line power P_in = −33 dBm and frequency spacing Δf = 16 MHz.

Figure 4(a) presents the phase shift −ϕ(t) (without averaging) for three different values of the tuning voltage η = V_D, within the synchronization band. In agreement with (4), after a transient, the phase exhibits a periodic variation about a constant value −ϕ _o, which changes with the tuning voltage V_D. Note that the oscillator will get locked to a different input tone when varying V_D. To obtain the synchronization bandwidth about each input tone ω ₁, ω ₂,…,ω_K, one should change the reference frequency of the analysis. Otherwise there will be an additional ramp function (ω_q−ω_k)t in Ф_mix(t). The synchronization band is obtained representing the average of −ϕ(t) versus V_D, as shown in Fig. 4(b). This analysis of the synchronization band requires a sufficiently large time offset to ensure that the circuit is in steady state at each V_D step. The apparent discontinuities in the diagram of Fig. 4(b) are because the fundamental frequency of the envelope-domain analysis, ω_q, is, as stated, alternatively set to each of the input frequencies. The synchronization bands are easily distinguished (in solid line), since, in agreement with (6),〈 − ϕ(t)〉 = −ϕ _o varies continuously in each band, with an excursion of slightly less than 180°, corresponding to the stable solutions. This excursion is not centered about ϕ_o = 0° because α_v is different from zero. The three bands in Fig. 4(b) correspond to synchronization at f ₁ = 2.7 GHz, f ₂ = 2.716 GHz, and f ₃ = 2.732 GHz.

Fig. 4. Behavior under six input lines, between 2.7 and 2.78 GHz, with Δf = 16 MHz. (a) Time variation of the phase shift for three V_D values, within the first synchronization band (f ₁ = 2.7 GHz). (b) Average value of ‒ϕ(t) versus V_D.

Figure 5 presents the variation average amplitude 〈ΔV(t)〉 in the same intervals of tuning voltage V_D considered in Fig. 4. The same three locking bands are detected, distinguished by the ellipsoidal variation of 〈ΔV(t)〉 versus V_D. Only the stable upper half of the ellipsoidal synchronization curves is obtained in the envelope-domain analysis. Note that the circuit becomes unlocked at the turning points of each curve.

Fig. 5. Behavior under six input lines, between 2.7 and 2.78 GHz, with Δf = 16 MHz. Average value of ΔV(t) versus V_D, which approaches the upper half of an ellipse in each synchronization band.

The capability of the oscillator to discriminate a particular tone can be evaluated from the spectrum of the oscillation signal, without changing the fundamental frequency, kept at ω_q. The frequencies the spectral line with larger output power and its neighboring spectral lines are identified and traced versus V_D (Fig. 6(a)). The circuit is synchronized where these frequency components remain constant when varying this parameter. Note that outside the synchronization band, mixing effects give rise to a significant variation of the dominant spectral component and its neighbors. In Fig. 6(b) the output power at the frequencies of the input tones is traced versus V_D, which allows evaluating the frequency-selection capability. The boundaries of the synchronization bands are easily detected by the fast variation of the spectrum frequencies, after the occurrence of each local/global bifurcation [Reference Suárez10, Reference Guckenheimer and Holmes27, Reference Thompson and Stewart28]. The bandwidths slowly decrease when moving away from the oscillator free-running frequency (2.7 GHz).

Fig. 6. Behavior under six input lines, between 2.7 and 2.78 GHz, with Δf = 16 MHz. Detection of synchronization bands from the analysis of the solution spectrum. (a) Frequencies of the spectral line with larger output power and its neighboring spectral lines, traced versus V_D. (b) Output power at the frequencies of the input tones.

Figure 7 presents the measurement results. Figure 7(a) shows the six input tones, with Δf = 16 MHz. Figure 7(b) presents an unlocked spectrum for V_D = 0.28 V. Figures 7(c) and 7(d) present the spectrum for V_D = 0.8 V (synchronization to f ₂), and V_D = 2.85 V (to f ₅). We attribute the differences to inaccuracies and dispersion in the lumped-component models.

Fig. 7. Measurement results. (a) Input tones. (b) Unlocked spectrum for V_D = 0.28 V. (c) Synchronization to f ₂ for V_D = 1.69 V. (d) f ₄ for V_D = 2.85 V.

Case P = N

For P = N the sub-harmonic injection-locked oscillator will be described with an outer-tier admittance function Y extracted from HB simulations. This admittance function is similar to the one considered in the section “Oscillator injected by multiple input tones”. However, the circuit is now in injection-locked conditions with respect to the sub-harmonic source, at the high sub-harmonic order N. Thus, the function Y depends on the observation node amplitude V and phase ϕ, since the oscillation frequency is determined by the input source frequency and fulfils f_o = Nf_in. Thus, the synchronized solution will fulfil the following two-tier equation system:

(9)$$\eqalign{Y\lpar V_s\comma \;\phi _s\rpar = \; & 0 \cr \bar{H}\lpar \bar{X}^{\prime}\comma \;V_s\comma \;\phi _s\rpar = \; & 0 \cr \omega _o = \; & N\omega _{in}}\comma $$

where $\bar{H}$ represents the HB system that constitutes the inner tier, V_s and ω_s are, respectively, the amplitude and phase of the steady-state solution and the vector $\bar{X}^{\prime}$contains all the state variables except the voltage at the observation node Ve ^jϕ and its complex conjugate. The above system can be solved through the optimization of V and ϕ in order to fulfil the goal Y = 0

For the noise analysis, two different contributions will be considered: the noise from the input source and the circuit own noise sources. Regarding the input source, the phase noise will be much larger than the amplitude noise, so the latter will be neglected. On the other hand, the circuit noise will be modeled with an equivalent noise-current generator at the observation node i _n(t). The spectral density of this equivalent noise-current source |I _N(Ω)|²is fitted with the oscillator in free-running conditions, in order to obtain the same phase-noise spectral density as with the entire set of circuit noise sources. The fitting is carried out in circuit-level HB, using the conversion-matrix approach [Reference Paillot, Nallatamby, Hessane, Quere, Prigent and Rousset30–Reference Nallatamby, Prigent, Sarkissian, Quéré and Obregón32]. The result is shown in Fig. 14. Note that in injection-locked conditions, one can expect the oscillator to follow the phase noise of the input source (usually growing as Ω³ when approaching this carrier) up to a certain (relatively high) offset frequency, above which the impact of flicker noise will usually be negligible. Thus, the fitting can be limited to the effect of the white noise sources.

Fig. 14. Fitting the spectral density of the original free-running oscillator with a single equivalent current source of spectral density |I _N(Ω)|². The fitting is carried out in circuit-level HB, using the conversion-matrix approach.

The phase noise contribution of the input source (with arbitrary waveform) will be obtained from the jitter τ(t) of this source [Reference Kaertner33, Reference Takahashi, Ogawa and Kundert34]. One must take into account that a time translation τ of the input signal gives rise to an identical time translation of the oscillator solution, which does not affect the value the admittance function Y(V _s, ϕ _s), depending only on the phase shift with respect to the input source. However, the time translation will modify the absolute node phase, which is increased as ϕ + Nω_inτ. Thus, the jitter of the input source will give rise to the phase perturbation Nψ(t) = Nω _inτ(t) at the oscillation frequency. Because there are also noise contributions coming from the oscillator circuit [modeled with the equivalent source i _n(t)], the total phase perturbation of the voltage at the observation node will be Δϕ _T(t) = Nω _inτ(t) + Δϕ(t)where Δϕ(t) is a small increment with respect to the steady-state phase ϕ_o due to the noise perturbations. On the other hand, the node amplitude, under the influence of τ(t) and i _n(t) becomes V _s + ΔV(t).

In the presence of the noise perturbations the outer-tier system at Nω_in can be obtained by performing a first-order Taylor series expansion of the total node admittance Y about the particular steady-state synchronized solution, V_s, ϕ_s, and ω_s, which provides:

(10)$$\lsqb {Y_V \Delta V\lpar t \rpar + Y_\phi \Delta \phi \lpar t\rpar -jY_\omega s} \rsqb \lpar {V_s + \Delta V\lpar t\rpar } \rpar e^{\,j\lpar {N\omega_{in}\tau \lpar t \rpar + \Delta \phi \lpar t\rpar } \rpar } = I_N\lpar t \rpar \comma \;$$

where s acts like a time differentiator and the subindexes stand for derivatives of Y with respect to the corresponding variables V, ϕ and ω, and I _N(t)is the envelope of the current-noise source i _n(t) about Nω _s. After the time differentiation, one obtains the following equation:

(11)$$ \eqalignb{& Y_V \Delta V\lpar t \rpar + Y_\omega \left({\lpar {N\omega_{in}\dot{\tau }\lpar t\rpar + \Delta \dot{\phi }\lpar t\rpar } \rpar -j\displaystyle{{\Delta \dot{V}\lpar t\rpar } \over {V_s + \Delta V\lpar t\rpar }}} \right) \cr & + Y_\phi \Delta \phi \lpar t\rpar = \displaystyle{{I_N\lpar t \rpar } \over {V_s}}\comma \;$$

where the subindexes indicate the quantity with respect to which the admittance function is differentiated. The derivatives $Y_V\comma \;Y_\omega \comma \;Y_\phi$ are calculated at the steady-state synchronized solution through finite differences in HB, following the procedure described in [Reference Ramirez, de Cos and Suarez13, Reference Ramirez, Ponton, Sancho and Suarez21].

Next, the complex equation (8) is split into real and imaginary parts and the Fourier transform is applied in the slowly varying time scale of the noise perturbations, with associated frequency Ω. Solving for Δϕ _T(Ω) = Δϕ(Ω) + Nψ(Ω) and multiplying by the adjoint, the phase-noise spectral density [Reference Ramirez, Ponton, Sancho and Suarez21] is given by:

(12)$$\eqalignb{\vert {\Delta \phi_T\lpar \Omega \rpar } \vert ^2 = & \displaystyle{{{\vert {Y_{\rm V} \times Y_\phi } \vert }^ 2 N^2{\vert {\psi \lpar \Omega \rpar } \vert }^2 + 2{\vert {Y_V} \vert }^2\lpar {{\vert {I_N} \vert }^2/V_s^2 } \rpar } \over {{\vert {Y_V \times Y_\phi } \vert }^2 + {\vert {Y_V \times Y_\omega } \vert }^2\Omega ^2 }} \cr = & \displaystyle{{{\vert {Y_\phi } \vert }^ 2{\rm si}{\rm n}^ 2\alpha _{V\phi } N^2{\vert {\psi \lpar \Omega \rpar } \vert }^2 + 2\lpar {{\vert {I_N} \vert }^2/V_s^2 } \rpar } \over {{\vert {Y_\phi } \vert }^ 2{\rm si}{\rm n}^ 2\alpha _{V\phi } + {\vert {Y_\omega } \vert }^ 2{\rm si}{\rm n}^ 2\alpha _{V\omega } \Omega ^2 }}\comma} $$

where higher-order terms in Ω have been neglected and the complex-number products of the form a × bare real and are defined as: $a \times b = {\mathop{\rm Re}\nolimits} \lpar a\rpar {\mathop{\rm Im}\nolimits} \lpar b\rpar -{\mathop{\rm Re}\nolimits} \lpar b\rpar {\mathop{\rm Im}\nolimits} \lpar a\rpar = \vert a \vert \vert b \vert \sin \lpar \angle b-\angle a\rpar$. In turn, the angles in the third term are defined as $\alpha _{ab} = \angle b-\angle a$. To derive the above expression, it has been taken into account that the real and imaginary parts of the equivalent noise source $\bar{I}_N$ are uncorrelated and the input noise is uncorrelated with the oscillator noise. Dividing the numerator and denominator by $\vert {Y_{\rm V} \times Y_\phi } \vert ^ 2$, one obtains:

(13)$$\vert {\Delta \phi_T\lpar \Omega \rpar } \vert ^2 = \displaystyle{{ N^2{\vert {\psi \lpar \Omega \rpar } \vert }^2 + \lpar {1/\lpar {{\vert {Y_\phi } \vert }^ 2{\rm si}{\rm n}^ 2\alpha_{V\phi }} \rpar } \rpar 2\lpar {{\vert {I_N} \vert }^2/V_s^2 } \rpar } \over {1 + \lpar {\lpar {{\vert {Y_\omega } \vert }^ 2{\rm si}{\rm n}^ 2\alpha_{V\omega }} \rpar /\lpar {{\vert {Y_\phi } \vert }^ 2{\rm si}{\rm n}^ 2\alpha_{V\phi }} \rpar } \rpar \Omega ^2 }}\comma \;$$

which can be rewritten as:

(14)$$\vert {\Delta \phi_T\lpar \Omega \rpar } \vert ^2 = \displaystyle{{ N^2{\vert {\psi \lpar \Omega \rpar } \vert }^2 + A_p\lpar {{\vert {I_N} \vert }^2/V_s^2 } \rpar } \over {1 + \lpar \Omega ^2/\omega _{3dB}^2 \rpar }}\comma \;$$

where the following parameters have been introduced:

(15)$$A_p = \displaystyle{2 \over {{\vert {Y_\phi } \vert }^ 2{\rm si}{\rm n}^ 2\alpha _{V\phi }}}\comma \;\quad \omega _{3dB} = \displaystyle{{\vert {Y_\phi } \vert {\rm sin\lpar }\alpha _{V\phi }\rpar } \over {\vert {Y_\omega } \vert {\rm \lpar sin}\alpha _{V\omega }\rpar }}. $$

As gathered from (14), the injection-locked oscillator exhibits a low-pass response with respect to the input phase noise and its own noise sources, with a 3 dB cut-off frequency ω _3dB. As shown in (15), the corner frequency ω_3dB will be, in principle, smaller for a higher quality factor of the oscillator (larger $\vert {Y_\omega } \vert$) and larger for a higher $\vert {Y_\phi } \vert$, implying a higher sensitivity to the input source and thus a larger locking range. Note, however, that the angles α _Vϕ, α _Vω can be very relevant.

For small offset frequency, the phase noise follows N ²|ψ(Ω)|², as derived from the jitter τ(t) of the input signal. The effect of the circuit noise sources is determined by $A_p\vert {I_N} \vert ^2/V_s^2$. If $A_p\vert {I_N} \vert ^2/V_s^2 \gt N^2\vert {\psi \lpar \Omega \rpar } \vert ^2$ from a certain Ω < Ω_3B, before decaying −20 dB/dec.

The predictions of the semi-analytical expression (14) have been compared with a phase-noise analysis of the sub-harmonic injection-locked oscillator at N = 30, performed in HB, with the conversion-matrix approach [Reference Paillot, Nallatamby, Hessane, Quere, Prigent and Rousset30–Reference Nallatamby, Prigent, Sarkissian, Quéré and Obregón32]. The results of this comparison, in the presence of the equivalent white-noise source i _n(t), are shown in Fig. 15. The curves providing the phase-noise spectral density obtained with the two methods are overlapped. Remember that the spectral density of the white-noise source |I _N|² was fitted in the standalone free-running oscillator. The corner frequency is f _3 dB ≅ 1.5 MHz.

Fig. 15. Comparison of the results of the semi-analytical expression (14) and the conversion-matrix approach when only white-noise sources are considered. Note that the spectral density of the white-noise source |I _N|² was fitted in the standalone free-running oscillation.

For the prediction of the whole spectrum, the phase noise of the rectangular input source at the fundamental frequency (90 MHz) has been experimentally characterized, using the R&S FSWP8 – Phase Noise Analyzer, and introduced in (14). Figure 16 presents a comparison of the predictions from (14) with the experimental measurement of the oscillator phase at Nf_in. Note that there is an excellent agreement since the two curves are nearly superimposed. At the lower offset frequencies, the phase noise is 20log(30) higher than the one at the lower harmonic frequency of the input source, included in the figure. The phase-noise spectrum in free-running conditions is also shown for comparison. The spectrum of the injection-locked oscillator follows the input-noise source up to its corner frequency f _3 dB ≅ 1.5 MHz. Up to this frequency, there is a negligible effect of the circuit own noise sources. At low-frequency offset, there is an improvement of more than 30 dB. To obtain a lower spectral density at higher offset frequencies an input source with lower phase noise should be used.

Fig. 16. Comparison of the predictions from (14) with the experimental measurement of the oscillator phase at Nf_in. At the lower offset frequencies, the phase noise is 20log(30) higher than the one at the lower harmonic frequency of the input source, included in the figure. The phase-noise spectrum in free-running conditions is also shown for comparison.

Case P ≤ N

For P < N the quantities $A_p/V_s^2$ and Ω_3 dB, will be identified from envelope-domain simulations through a numerical procedure. To obtain the frequency response of the oscillator phase with respect to the equivalent noise source |I _N|², this noise source will be replaced with a deterministic tone at the frequency Ω. Then, a two-tone envelope-domain analysis will be carried out, using the frequency basis (ω ₁ = ω _o/P, ω ₂ = Ω). Then, for each Ω, the phase shift will be calculated from the voltage spectrum at the observation node [Reference Ramirez, Ponton, Sancho and Suarez21], doing:

(16)$$\!\!\!\!\! \eqalign{\Delta \phi \lpar N\omega _{in} + \Omega \rpar = \, \displaystyle{{V^\ast \lpar N\omega _{in}-\Omega \rpar e^{\,j\phi _s}-V\lpar N\omega _{in} + \Omega \rpar e^{{-}j\phi _s}} \over {2V\lpar N\omega _{in}\rpar }}. }$$

To obtain the voltage components in the above expression, the time-varying harmonic components resulting from the envelope-domain analysis can be processed to provide:

(17)$$\eqalign{V\lpar N\omega _{in} + \Omega \rpar = \; & \left\langle {X_{P\comma 1}\lpar t\rpar } \right\rangle \comma \;\;V\lpar N\omega _{in}-\Omega \rpar \cr = \; & \left\langle {X_{P\comma -1}\lpar t\rpar } \right\rangle \comma \;\;V\lpar N\omega _{in}\rpar = \left\langle {X_{P\comma 0}\lpar t\rpar } \right\rangle } \comma \;$$

where the operator 〈 〉 applies an averaging in the period T _in = 1/f _in and X _k,l(t) is the harmonic component corresponding to the frequency kω ₁ + lω ₂. Note that the time-varying nature of the harmonics X _k,l(t) is due to the presence of frequency components at multiples of the injection frequency f _in, which are removed by the operator 〈 〉. Sweeping Ω and tracing |Δϕ(Nω _in + Ω)|² versus Ω one obtains a frequency response from which one can identify the two quantities $A_p/V_s^2$ and Ω_3 dB. Once these quantities are available, one can obtain the phase-noise spectrum through (17).