A generalization of the Leeson effect
AA generalization of the Leeson effect
Enrico Rubiola and R´emi Brendel web page http://rubiola.org
FEMTO-ST InstituteCNRS UMR-6174, Besan¸con, FranceApril 30, 2010
Abstract
The oscillator, inherently, turns the phase noise of its internal com-ponents into frequency noise, which results into a multiplication by 1 /f in the phase-noise power spectral density. This phenomenon is known asthe Leeson effect . This report extends the Leeson effect to the analysisof amplitude noise. This is done by analyzing the slow-varying complexenvelope, after freezing the carrier. In the case of amplitude noise, theclassical analysis based on the frequency-domain transfer function is pos-sible only after solving and linearizing the complete differential equationthat describes the oscillator. The theory predicts that AM noise gives anadditional contribution to phase noise. Beside the detailed description ofthe traditional oscillator, based on the resonator governed by a second-order differential equation (microwave cavity, quartz oscillators etc.), thisreport is a theoretical framework for the analysis of other oscillators, likefor example the masers, lasers, and opto-electronic oscillators.This manuscript is intended as a standalone report, and also as com-plement to the book E. Rubiola,
Phase Noise and Frequency Stability inOscillators , Cambridge University Press, 2008. ISBN 978-0-521-88677-2(hardback), 978-0-521-15328-7 (paperback).
Revision history . First submission on arXiv.1 a r X i v : . [ phy s i c s . i n s - d e t ] A p r . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 Contents
Notation 31 Introduction 52 Basics 6
A Exotic issues 42
A.1 AM-PM coupling in the off-resonance resonator . . . . . . . . . . 42A.2 Parametric fluctuation of the S matrix . . . . . . . . . . . . . . . 43A.3 The Miller effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 References 46 . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 Notation
Symbol Meaning A amplifier voltage gain (thus, the power gain is A ) A ( s ) normalized-amplitude noise, see α ( t ) b i coefficients of the power-law approximation of S ϕ ( f ),b( t ) ↔ B( s ) resonator impulse response. Also b ϕ ( t ) ↔ B ϕ ( s ), etc. E ( s ) normalized-amplitude noise, see ε ( t ) f Fourier frequency, Hz f c amplifier corner frequency, Hz f L Leeson frequency, Hz F amplifier noise figure h ( t ) ↔ H ( s ) impulse responseh( t ) ↔ H( s ) phase or amplitude impulse response, takes subscript ϕ or αh i coefficients of the power-law model of S α ( f ) or S y ( f )(in case of ambiguity use h i for S y ( f ) and h (cid:48) i for S α ( f )) j imaginary unit, j = − k Boltzmann constant, k = 1 . × − J/K
L{ }
Laplace-transform operator L ( f ) single-sideband noise spectrum, dBc/Hz. L ( f ) = S ϕ ( f ), by definition N ( s ) gain fluctuation, see η ( t ) Q resonator quality factor R , R resistance, load resistance (often, R = 50 Ω) s Laplace complex variable, s = σ + jωs derivative operator S a ( f ) one-sided power spectral density (PSD) of the quantity at time T , T absolute temperature, reference temperature T = 290 K u ( t ) Heaviside (step) function, u ( t ) = (cid:82) δ ( t (cid:48) ) dt (cid:48) u ( t ) ↔ U ( s ) voltage (amplifier input) v ( t ) ↔ V ( s ) voltage. Also a dimensionless signal˜ v ( t ) complex-envelope associated to v ( t ) V , V dc or peak voltage V o , V o ( t ) phasor associated with an ac signal v ( t ) x ( t ) generic function y ( t ) generic function y ( t ) fractional frequency fluctuation, y ( t ) = [ ν ( t ) − ν ] /ν α ( t ) ↔ A ( s ) normalized-amplitude noise, may take subscript u or vβ ( s ) transfer function of the feedback path γ amplifier compression parameter (0 < γ < δ ( t ) Dirac delta function ε ( t ) ↔ E ( s ) normalized-amplitude noise η ( t ) ↔ N ( s ) amplifier gain fluctuation κ small phase or amplitude step ν frequency (Hz), used for carriers σ real part of the Laplace variable s = σ + jωτ resonator relaxation time . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ϕ ( t ) ↔ Φ( s ) phase noise χ dissonance, χ = ω/ω n − ω n /ωψ ( t ) ↔ Ψ( s ) phase noise (amplifier) ω imaginary part of the Laplace variable s = σ + jωω angular frequency, carrier or Fourier ω oscillator angular frequency ω n resonator natural angular frequency ω p resonator free-decay angular pseudo-frequencyΩ detuning angular frequency, Ω = ω − ω n Subscript Meaning ω , P , V , etc. i input, as in v i ( t ), ϕ i ( t ), Φ i ( s ) i electrical current, as in the shot noise S i ( ω ) = 2 qin resonator natural frequency ( ω n , ν n ) o output, as in v o ( t ), ϕ o ( t ), Φ o ( s ) p resonator free-decay pseudo-frequency ( ω p , ν p ) Symbol Meaning < > mean. Also < > N mean of N valuestime average, as in x ↔ transform inverse-transform pair, as in x ( t ) ↔ X ( s ) ∗ convolution, as in v ( t ) = h ( t ) ∗ u ↔ V ( s ) = H ( s ) U ( s ) (cid:16) asymptotically equal (cid:13) zero of a function (complex plane, Bode plot, or spectrum) × pole of a function (complex plane, Bode plot, or spectrum) Acronym Meaning
AM Amplitude Modulation (often ‘AM noise’)CAD Computer-Aid Design (software)FM Frequency Modulation (often ‘FM noise’)PM Phase Modulation (often ‘PM noise’)PSD (single-side) Power Spectral DensityRF Radio Frequency . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 The oscillator noise, which in the absence of environmental or aging effect iscyclostationary, is best described as a baseband process, after freezing the pe-riodic oscillation. The polar-coordinate representation of the limit cycle splitsthe model of the oscillator into two subsystems, in which all signals are theamplitude and the phase of the main system, respectively. Putting things sim-ply, these two subsystems are (almost) decoupled and all the nonlinearity goesto amplitude. This occurs because amplitude nonlinearity is necessary for theoscillation to be stationary. Conversely the phase, which ultimately is time,cannot be stretched.The baseband equivalent of a resonator, either for phase or amplitude is alowpass filter whose time constant is equal to the resonator’s relaxation time.Hence, the phase model of the oscillator consists of an amplifier of gain exactlyequal to one and the lowpass filter in the feedback path, as extensively discussedin [1]. The amplitude model is a nonlinear amplifier, whose gain is equal toone at the stationary amplitude and decreases with power, and the lowpassfilter in the feedback path. In the baseband representation both AM and PMperturbations map into additive noise, even in the case of flicker and otherparametric noises. This model gives a new perspective on the classical van derPol oscillator.The elementary theory of nonlinear differential equations tells us that nonlin-earity stretches the feedback time constant. Asymptotically, the time constantis split into two constants, one at the oscillator startup and one in stationaryconditions. If the gain varies linearly with amplitude, which is always true forsmall perturbations, the oscillator can be solved in closed form.After the pioneering work of D. B. Leeson [2], a number of different analyseshas been published. Sauvage derived a formula that has the same behavior ofthe Leeson formula using the autocorrelation functions [3]. Hajimiri and Lee[4, 5, 6] proposed a model based on the “impulse-sensitivity function” (ISF),which emphasizes that the impulse has the largest effect on phase noise if itoccurs at the zero-crossing of the carrier. This model, mainly oriented to thedescription of phase noise in CMOS circuits, is extended in [7]. Demir & al.proposed a theory based on the stochastic calculus [8], in which they introducea decomposition of phase and amplitude noise through a projection onto theperiodic time-varying eigenvectors (the Floquet eigenvectors), by which theyanalyze the oscillator phase noise as a stochastic-diffusion problem. This theorywas extended to the case of 1 /f noise [9]. Other articles are mainly orientedto the microwave oscillators [10, 11, 12]. Demir inspired a work on the phasenoise in opto-electronic oscillators [13]. Some of the articles cited make use ofsophisticated mathematics, as compared to our simple methods. All give littleor no attention to amplitude noise.It is well known that the instability of the resonator natural frequency con-tributes to the oscillator fluctuations, which in some cases turns out to be themost important source of frequency fluctuations. Almost nothing is knownabout the amplitude fluctuation of the resonator. That said, the resonatorinstability is not considered here. This article stands upon our earlier works[1] and [14]. The latter is mainly oriented to the ultra-stable quartz oscillator.Here, we present an unified approach to AM and PM noise in oscillators by ana-lyzing the mechanism with which the noise of the oscillator internal components . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 Phase noise is a well established subject, clearly explained in classical references,among which we prefer [17, 18, 19, 20] and [21, vol. 1, chap. 2]. The reader mayalso find useful [1, chap. 1].The quasi-perfect sinusoidal signal of angular frequency ω , of random am-plitude fluctuation α ( t ), and of random phase fluctuation ϕ ( t ) is v ( t ) = [1 + α ( t )] cos [ ω t + ϕ ( t )] (clock signal) . (1)We may need that | α ( t ) | (cid:28) | ϕ ( t ) | (cid:28) | ˙ ϕ ( t ) | (cid:28) S ϕ ( f ) = (cid:10) | Φ( jf ) | (cid:11) m (avg, m spectra) , (2)The uppercase denotes the Fourier transform, so ϕ ( t ) ↔ Φ( jf ) form a trans-form inverse-transform pair. In experimental science, the single-sided PSD ispreferred to the two-sided PSD because the negative frequencies are redundant.It has been found that the power-law model describes accurately the oscillatorphase noise S ϕ ( f ) = (cid:88) n = − b n f n (power law) (3)coefficient noise type b − frequency random walk b − flicker of frequency b − white frequency noise, or phase random walk b − flicker of phase b white phase noiseThe power law relies on the fact that white ( f ) and flicker (1 /f ) noises existper-se, and that phase integration is present in oscillators, which multiplies thespectrum × /f . This will be discussed extensively in Section 5. Additionalterms with n < − f − , otherwisethe input-output delay would diverge. Nonetheless, these steeper terms mayshow up in some regions of the spectrum, due to a variety of phenomena. We use interchangeably ω as a shorthand for 2 πν for the carrier frequency, and as a short-hand for 2 πf for the offset (Fourier) frequency, making the meaning clear with appropriatesubscripts when needed but omitting the word ‘angular.’ . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 Amplitude noise, far less studied than phase noise, can be described in verysimilar way. The reader may find useful the report [22], which describes indepth the experimental aspects.The amplitude noise is generally measured as the average PSD S α ( f ) = (cid:10) |A ( jf ) | (cid:11) m (avg, m spectra) , (4)where α ( t ) ↔ A ( jω ) form a transform-inverse-transform pair, and described interms of power law S α ( f ) = (cid:88) n = − h n f n (power law) (5)coefficient noise type h − (integrated flicker) h − amplitude random walk h − flicker of amplitude h white amplitude noiseIn the current literature the coefficients h i are used to model S y ( f ), i.e., thePSD of the fractional-frequency fluctuation y ( t ). We use the same coefficients forsimplicity, because the relationships between spectrum and two-sample (Allan)variance are the same.At very low frequency, the amplitude noise cannot be steeper than f − oth-erwise the amplitude would diverge. This applies to both two-port componentsand oscillators. Yet, steeper terms can show up in some regions of the spectrum. The simplest form of oscillator is a resonator with an amplifier of gain A in closedloop that compensates for the resonator loss /β . Stationary oscillation takesplace at the frequency ω that verify Aβ = 1. This is known as the Barkhausencondition. The actual oscillator can be represented with the scheme of Fig. 1,which includes a gain compression mechanism and noise. For our purposes thenoise is represented in polar coordinates as amplitude noise and phase noise.The gain compression is necessary for the amplitude not to decay or diverge.We assume that A is independent of frequency, at least in a range sufficientlylarger than the resonator bandwidth. For the sake of simplicity we normalizethe loop elements so that A = 1 and β = 1 at the oscillation frequency ω = ω and at the nominal output amplitude v = 1.The resonator has narrow bandwidth, hence it eliminates all the harmonicsmultiple of ω generated by the amplifier nonlinearity. Though the harmonicscan be present at the amplifier output, where the signal can be distorted, theydo not participate to the regeneration process that entertains the oscillation.Hence, the only practical effect of the nonlinearity on the loop dynamics is toreduce the gain at the fundamental frequency ω . Therefore, the quasi-sinusoidalapproximation can be used. Since the quantity β is the resonator gain, 1 /β is the loss. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-oscill-models [1+ α (t)] cos[ ω t+ φ (t)] AMPM η (t) ψ (t) A gaincompression β noise real amplifier resonator ψ (t) (cid:15511) Ψ (s) b(t) (cid:15511) B(s) low-pass Σ φ (t) (cid:15511) Φ (s) B -- near-dc PM noiseA -- full RF model of the oscillator file: ele-oscill-models ε (t) Σ α v (t) (cid:15511) A v (s) u A A b(t) (cid:15511) B(s) low-pass α u (t) (cid:15511) A u (s) gain fluctuat. η (t) (cid:15511) N (s) saturation C -- near-dc AM noise u v
Figure 1: Feedback oscillator and its decomposition in PM and AM models.Assuming that, as it occurs in practice, the resonator relaxation time τ islarger than 1 /ω by a factor of at least 10 , the oscillator behavior can bemathematically described in terms of the slow-varying complex envelope, asamplitude and phase were decoupled from the oscillation. In this representationthe oscillator splits into two subsystems, one for phase and one for amplitude,as shown in Fig. 1. Since phase represents time, which cannot be stretched , allthe non-linearity goes in the amplitude subsystem.The main advantage of the slow-varying envelope representation is that am-plitude noise and phase noise can be represented as additive noise phenomena,regardless of the physical origin. This eliminates the difficulty of flicker noise This is no longer true in extreme nonlinear oscillators, like the femtosecond laser, whichare out of our scope. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-clipping-types A u v A u van der Pol(quadratic) linear A = 1– γ ( u –1) hard clippingsoft clipping A =1 oscillator operation A A A file: ele-clipping-types Figure 2: Most common types of gain saturation. The quantities u and v arethe rms amplitude at the carrier frequency.and other parametric processes. The formalism is simple and tightly connectedto the experimentally observable quantities.We are interested in the mechanism that governs the noise propagation of theinternal components to the oscillator output. Virtually all oscillators are stableenough for the noise to be a small perturbation to the stationary oscillations,and consequently for a linear model to be accurate for any practical purpose.Linearization gives access to the Laplace-Heaviside formalism. The response y ( t ) to the input x ( t ) is therefore given by y ( t ) = x ( t ) ∗ h ( t ) ↔ Y ( s ) = X ( s ) H ( s )where h ( t ) is the impulse response , i.e., the response to the Dirac δ ( t ) function, H ( s ) is the transfer function , the symbol ‘ ∗ ’ is the convolution operator, thedouble arrow ‘ ↔ ’ stands for Laplace transform inverse-transform pair, and s = σ + jω is the Laplace complex variable. Given the input power spectral density S x ( f ), the output power spectral density is given by S y ( f ) = | H ( jf ) | S x ( f ) . The application of this idea to the oscillator rises some difficulties, which willbe solved in the next Sections.
In large signal conditions, all amplifiers have some kind of nonlinearity that lim-its the maximum output power. Neglecting the band limitation, when a sinu-soidal signal U rf ( t ) = U cos( ω t ) is present at the input of an amplifier, the sat-urated output can be written as the Fourier series V rf ( t ) = (cid:80) ∞ n =1 V n cos( nω t ). Here x ( t ) and y ( t ) are generic functions of time, thus not the phase time and the fractionalfrequency fluctuation commonly used in the oscillator literature. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-gamma u v = AuA = 1– γ ( u –1) oscillator operation file: ele-gamma v A u v γ γ γγ Figure 3: Most common types of gain saturation. The quantities u and v arethe rms amplitude at the carrier frequency.The n = 1 term is the fundamental and the n > V n and to introduce a phase in each sinusoidalterm. In a linear amplifier only the fundamental is present at the output, thus V n = 0, ∀ n ≥ u and v instead of theinstantaneous peak amplitudes U and V . Let us define the amplifier gain as A = vu (definition of A ) , which of course is equivalent to A = V /U . The gain A should not be mistakenfor the differential gain ∂v/∂u .Figure 2 shows the gain-saturation types most frequently encountered anddescribed underneath. The small-signal gain is denoted with A and the gainat the oscillator nominal amplitude u = 1 is denoted with A . Figure 2 isnormalized for A = 1. Around u = 1 the gain can be linearized as A = A [1 − γ ( u − ≤ γ < , which rewrites as A = 1 − γ ( u −
1) whith 0 ≤ γ < A = 1.The slope − γ deserves some comments. The condition γ > A ( u ) must decrease monotonically (Fig. 2). This is theamplitude-stabilization mechanism. A second obvious condition is that in theregular-operation region (i.e., around u = 1) the output v ( u ) must increasemonotonically. We show that this second condition is equivalent to γ < v = Auv = − γu + (1 − γ ) (Fig. 3) , The latter is the ‘cap’ parabola shown in Fig. 3. For v ( u ) to increase mono-tonically at u = 1 it is necessary that the point u = 1 is on the left-side of the . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 − γ γ > , whose solution is γ < v ( u ) increases monotonically holds for virtually all am-plifiers. Only a few exceptions are found, the most remarkable of which is achannel that includes a Mach Zehnder electro-optic modulator. In such casesthe Barkhausen Aβ = 1 may be met (at least) at two different amplitude levels,the first with γ < γ >
1. In one of such cases, it hasbeen mathematically proved and experimentally observed that the oscillationamplitude flips between these two levels, producing an amplitude oscillation athalf the frequency determined by the loop roundtrip time [23].
In the classical van der Pol oscillator [24], the amplifier input-output functionis defined as y = a x − a x , with a > a >
0. In mathematical treatisesthe coefficients a and a are sometimes set to one. Feeding the signal U rf ( t ) = U cos( ω t ) in such amplifier and taking only the fundamental frequency, theoutput is V rf ( t ) = U (cid:2) a − a U (cid:3) cos( ω t ). Accordingly, the gain becomes A = a − a U , which is a ‘cap’ parabola. In small-signal condition the gain is A , independent of the signal level. In-creasing the input level the output is clipped when it hits a threshold, wherethe sinusoid progressively turns into a square wave. The asymptotic amplitudeof the fundamental is 4 /π (2.1 dB) higher than the threshold. This behavior isoften encountered in amplifiers linearized by a strong feedback, as most circuitsbased on operational amplifiers. Of course the feedback is no longer effectivewhen the output is expected to exceed the supply voltage. With moderate feedback, the output clipping starts gradually when the outputapproaches the dynamic-range boundary. This behavior is typical of microwaveamplifiers. The knee of the gain curve occurs approximately at the 1 dB com-pression power.
The gain law A = 1 − γ ( u −
1) holds in the whole dynamic range. However thismodel may seem a mere academic exercise, it provides useful results in a simpleand compact form.
The contents of this Section is extensively discussed in [25], and briefly summa-rized here. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-addit-noise Σ v (t) u (t) U cos( ω t) file: ele-addit-noise RF noise v n (t) = v x (t) cos( ω t) – v y (t) sin( ω t) noise-freedevice Figure 4: Additive noise.
Let us consider a quasi-perfect device that adds a noise term v n ( t ) to the sinu-soidal input signal, as shown in Fig. 4. Assuming that the device gain is equalto one, the output signal is v ( t ) = V cos ω t + v n ( t ) , or equivalently v ( t ) = V cos ω t + v x ( t ) cos ω t − v y ( t ) sin ω t . (7)The random variables v x ( t ) and v y ( t ), called in-phase and quadrature compo-nent of noise, represent the noise v n ( t ) in the bandwidth of interest.Though the Cartesian representation (7) is the closest to the physics ofadditive noise, polar coordinates can also be used v ( t ) = V [1 + α ( t )] cos [ ω t + ϕ ( t )] , (8)where in low-noise conditions it holds that α ( t ) = v x ( t ) V and ϕ ( t ) = v y ( t ) V . The most relevant feature of the additive noise is that all the statisticalproperties of v n ( t ), thus of v x ( t ) and v y ( t ), are not affected by the input signal.There follow some relevant properties1. Referred to the input, there is an equal amount of AM and PM noise.Yet, AM/PM asymmetry can show up at the output if the amplitudenon-linearity compresses the AM noise.2. AM and PM noise are statistically independent.3. The shape of the noise spectrum is independent of the carrier frequency ω . Therefore the noise spectrum cannot have a term 1 /f , 1 /f etc.,centered at an arbitrary carrier frequency ω .4. The AM noise and the PM noise scale down with the carrier power.The additive noise is generally white, though it can have bumps due to deviceinternal structure and it rolls off out of the bandwidth. In the case of thermal . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-param-noise AMPM noise-freedevice u (t) v (t) φ (t) α (t) file:ele-param-noise correlatednoise Figure 5: Parametric noise.noise, including the noise figure F (defined only at the temperature T = 290K), the noise PSD of a generic white-noise process n ( t ) is S n ( f ) = F kT . (9)In polar coordinates the noise PSD is S α ( f ) = h and S ϕ ( f ) = b , with h = F kT P and b = F kT P , (10)where P is the carrier power.In the case of cascaded amplifiers, the Friis formula [26] applies, by which thenoise contribution of each stage is divided by the gain of the preceding stages h = kT P (cid:20) F + F A + F A A + . . . (cid:21) (Friis, AM noise) (11) b = kT P (cid:20) F + F A + F A A + . . . (cid:21) (Friis, PM noise) (12)where P is the carrier power. The parametric noise originates from a near-dc process that modulates the car-rier, as shown in Fig. 5. Accordingly, the polar-coordinate representation (8)is the closest to the physical mechanism. The most important parametric ran-dom phenomenon is flicker noise, whose PSD is proportional to 1 /f over sev-eral decades. Other types of parametric noise, with PSD proportional to 1 /f i , i = 2 , , . . . , can only exist in a limited frequency region. For example, 1 /f noise in the region between 1 mHz and 1 Hz has been observed as the phase noiseof radio-frequency amplifiers, and also as the offset fluctuation of operationalamplifiers. If these high-slope phenomena would be allowed to span over manydecades at low frequencies, the amplitude or the group delay would diverge inthe long run, which does not fit the experience about two-port devices.As a realistic approximation, one can assume that the near-dc process andthe modulation efficiency are independent of the carrier power P , hence thethe statistical properties of α ( t ) and ϕ ( t ) tend to be constant in a wide rangeof power h − = C and b − = C constant, independent of P . (13) . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 P has the amazing consequencethat the noise of a multistage amplifier is the sum of the individual contributions S α ( f ) = [ S α ( f )] + [ S α ( f )] + . . . (14) S ϕ ( f ) = [ S ϕ ( f )] + [ S ϕ ( f )] + . . . , (15)independently of the order of the single stages in the chain.Generally, a parametric process affects both amplitude and phase with sep-arate coefficients, as the dashed noise generator in Fig. 5 does. This introducessome correlation between AM and PM noise. Evidence of this statement isprovided by the following examples. • In a bipolar transistor, a noise source may affect the thickness of the baseregion. Such a process modulates simultaneously the gain (AM noise) andthe BE BC capacitances (PM noise), which turns in fully-correlated AMand PM noise. • In a laser medium, the pump power affects the partition between excitedatoms and ground-state atoms. Two noise phenomena are simultaneouslydriven by the power fluctuation of the pump. The first and more obviousphenomenon is the fluctuation of gain and of saturation power, whichshows up as AM noise — referred to as RIN in the jargon of laser optics.The second phenomenon results from the fact that the contribution of anatom to the refraction index changes if the atom is excited. This producesphase noise inside the loop, thus frequency noise in the laser beam. • The third example is provided by the fluctuation of cathodic emission invacuum tubes, like triodes, klystrons, magnetrons, TWTs, etc. Beside theobvious effect on gain, the electron emission impacts on the space charge,and in turn on the capacitance seen by the signal.In all the above examples, a single phenomenon yields fully correlated amplitudeand phase noise.
It has been seen that the AM noise and the PM noise spectra are S α ( f ) = h + h − f + . . . h = F kTP h − = C (constant) (16) S ϕ ( f ) = b + b − f + . . . b = F kTP b − = C (constant) (17)An example of phase noise spectrum is shown in Figure 6. This Figure em-phasizes the fact that the flicker noise is constant and that the white (additive)noise scales down as the power increases. The obvious consequence is that thecorner frequency f c also scales with power. A common mistake found in CADsoftware is that the flicker is described by a fixed corner frequency, independentof power. The reader is strongly encouraged to check before trusting a CADprogram. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-ampli-Sphi b P b , higher P f c = ( b –1 / FkT P
0 depends on P f" c f' c b –1 ≈ c on s t . vs . P b = FkT P S φ (f) , l og - l og sca l e f b –1 f –1 file: ele-ampli-Sphi Figure 6: Amplifier phase (or amplitude) noise power-spectrum density. ele-gain-fluctuation
A u ideal A = 1– γ ( u –1) oscillator operation fluctuating A = 1– γ ( u –1)+ η fluctuation η v A u file: ele-gain-fluctuation Figure 7: Parametric fluctuation of the amplifier gain.
Modeling the oscillator in the frequency domain, the gain is A = A [1 − γ ( u − u = 1. Introducing a slow fluctuation, A turns intothe slow varying function of time A ( t ) = A [1 − γ ( u − η ( t )] e jψ ( t ) approximated as A ( t ) (cid:39) A [1 − γ ( u −
1) + η ( t )] e jψ ( t ) (Fig. 7) , (18)where η ( t ) ↔ N ( s ) (amplitude fluctuation, i.e., AM noise) ψ ( t ) ↔ Ψ( s ) (phase fluctuation, i.e., PM noise)are the amplitude an phase gain fluctuations, respectively. Figure 7 shows thecombined effect of the gain amplitude fluctuation and compression.Flicker noise, which results from a parametric effect, impacts directly on thegain. It can be described by[ S η ( f )] flicker = h − f and [ S ψ ( f )] flicker = b − f (constant vs. P ) . . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-resonator-delta-method Q, ω n [1+ ε (t)] cos[ ω t+ ψ (t)] [1+ α (t)] cos[ ω t+ φ (t)] resonator Q , ω n δ (t) b φα (t)b αα (t) resonator cos( ω t+ κ ) t = 0 κ ∫ b αφ (t) dt set a small phase or amplitude step κ at t=0, and linearize for κ → cos( ω t ) [1+ α (t)]cos[ ω t+ φ (t)] [1+ ε (t)] cos[ ω t+ ψ (t)] [1+ α (t)] cos[ ω t+ φ (t)] δ (t) resonator Q , ω n b φφ (t)b αφ (t) file: ele-resonator-delta-method κ ∫ b φφ (t) dt Figure 8: AM and PM response of a resonator.Additive noise, albeit of quite different origin, can still be seen as a gainfluctuation because it affects the input/output relationship. Hence[ S η ( f )] additive = [ S ψ ( f )] additive = F kT P (constant vs. f ) . The resonator in actual load conditions is governed, or locally well approxi-mated by the differential equation¨ v o + ω n Q ˙ v o + ω n v o = L { v i ( t ) } , (19) Whoever has worked seriously in the field of oscillators, may have in mind three setsof parameters like ‘ ω n and ‘ Q .’ These sets refer (1) to the unloaded resonator, which isa mathematical abstraction not accessible to the physical experiment; (2) to the resonatorloaded by the measurement test set, from which the unloaded parameters are estimated; and(3) to the resonator loaded by the oscillator circuit. The external circuit, either the test set orthe resonator, increases the dissipation and affects the natural frequency. That said, it is tobe made clear that here the resonator is always loaded by the oscillator circuit , and thereforethat there is no point in discussing the other conditions . On the other hand, the process ofgetting ω n and Q from experimental data may be tricky or difficult. We skip this discussionbecause it depends on the specific resonator and oscillator, while we aim at a general theory. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ω n is the natural frequency, Q is the quality factor, v i ( t ) is the externalforce, and L is an operator. The most interesting form of the force term isL { v i ( t ) } = ω n Q ˙ v i ( t ) because it is homogeneous with the dissipative term. Thisoccurs with the series (parallel) RLC resonator driven by a voltage (current)source, and in other relevant cases. Accordingly, (19) becomes¨ v o + ω n Q ˙ v o + ω n v o = ω n Q ˙ v i ( t ) . (20)Using the Laplace transform, the resonator transfer function β ( s ) = V o /V i is β ( s ) = ω n Q ss + ω n Q s + ω n . (21)Equations (20) and (21) are normalized for the resonator to respond to a sinusoidat the exact resonant frequency ω n with a sinusoid of the same frequency, phaseand amplitude.We analyze the impulse response of the resonator phase and amplitude instationary-oscillation conditions. The phase response is the response to a pertur-bation δ ( t ) in the argument of the driving signal, as shown in Fig. 8. Similarly,the amplitude response is the response to a perturbation δ ( t ) in the amplitudeof the driving signal. In general literature the impulse response is denoted with h ( t ), and its Laplace transform with H ( s ). Since we use h ( t ) ↔ H ( s ) for theoscillator response, the phase or amplitude impulse response of the resonator isdenoted with b ( t ) ↔ B ( s ). It turns out that the resonator response is the samefor amplitude and phase.In our analysis we replace the impulse δ ( t ) with a small phase or amplitudestep κ u ( t ), where u ( t ) is the Heaviside function u ( t ) = (cid:90) ∞−∞ δ ( t ) dt = (cid:40) t < t > κ →
0. Then we use the general property of linear systemsthat the response to u ( t ) is (cid:82) b( t ) dt . Notice that u ( t ) can be seen as a switchthat changes state from off to on at t = 0; and that u ( − t ) switches from on tooff at t = 0. Let us consider the resonator driven by the signal v i ( t ) = 1 β cos( ω t − θ ) (probe) (22)where β and θ are chosen for the asymptotic output to be v o ( t ) = cos( ω t ) for t → ∞ , i.e., amplitude is equal one and phase equal zero in the general case ω (cid:54) = ω n . If the probe signal v i ( t ) is switched off at the time t = 0, the outputis v o ( t ) = cos( ω p t ) e − t/τ t > , (response) (23) . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 τ = 2 Qω n (relaxation time) (24)is the resonator relaxation time and ω p = ω n (cid:114) − Q (free-decay pseudo-frequency) (25)is the free-decay pseudo-frequency. For Q (cid:29)
1, we can approximate ω p (cid:39) ω n . Thisis justified by the fact that the phase error ζ accumulated during therelaxation time τ is ζ = ( ω n − ω p ) τ = 14 Q .
This is seen by replacing τ = Qω n and ω p = ω n (cid:112) − / Q in ζ , and by expand-ing in series truncated at the first order. The response to a sinusoid switched on at the time t = 0 takes the general form v o ( t ) = A cos( ω p t ) e − tτ + B sin( ω p t ) e − tτ + C cos( ω t ) + D sin( ω t ) t > , where A , B , C , and D are constants determined by the boundary conditions.We use the probe signal (22). This yields immediately C = 1 and D = 0.The constants A and B are found by assessing the continuity of v o ( t ) at t = 0,which gives A = − B = 0. Approximating ω p (cid:39) ω n for Q (cid:29)
1, theoutput is v o ( t ) = − cos( ω n t ) e − t/τ + cos( ω t ) t > . (26)Similarly, using a probe signal v i ( t ) = β sin( ω t − θ ), the output is v o ( t ) = − sin( ω n t ) e − t/τ + sin( ω t ) t > When the resonator is used at the exact natural frequency, it holds that ω = ω n , β = 1, and θ = 0.A phase step κ at t = 0 is described as the probe signal v i ( t ) = cos( ω t ) u ( − t ) (cid:124) (cid:123)(cid:122) (cid:125) switched off at t = 0 + cos( ω t + κ ) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) switched on at t = 0 , By virtue of linearity, the response is the sum of (23) plus (26), that is, v o ( t ) = cos( ω p t ) e − t/τ + cos( ω p t + κ ) (cid:2) − e − t/τ (cid:3) t > . (28)Expanding and using the approximations cos( κ ) (cid:39) κ ) (cid:39) κ for κ → ω p (cid:39) ω n for large Q , thus ω p (cid:39) ω , we get v o ( t ) = cos( ω t ) − κ sin( ω t ) (cid:2) − e − t/τ (cid:3) t > , . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 Resonator response in the phase space figure: 950 ! reson ! responsesource: oscill ! am ! noiseE. Rubiola, sep 2009 time, t/tau s t e p r e s p o n s e i m pu l s e r e s pon s e Figure 9: Resonator response to the step and to the impulse.This can be seen as a slowly varying phasor V o ( t ) = √ (cid:8) jκ (cid:2) − e − t/τ (cid:3)(cid:9) , whose angle arctan (cid:18) (cid:61){ V o ( t ) }(cid:60){ V o ( t ) } (cid:19) (cid:39) κ (cid:2) − e − t/τ (cid:3) t > κ u ( t ). After deleting κ and differentiating, we obtain theimpulse response b( t ) = τ e − t/τ .An amplitude step κ at t = 0 is described as the probe signal v i ( t ) = cos( ω t ) u ( − t ) (cid:124) (cid:123)(cid:122) (cid:125) switched off at t = 0 + (1 + κ ) cos( ω t ) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) switched on at t = 0 , Once again the response is the sum of (23) plus (26) v o ( t ) = cos( ω p t ) e − t/τ + (1 + κ ) cos( ω p t + κ ) (cid:2) − e − t/τ (cid:3) t > . (29)Expanding under the same approximations as above, i.e., cos( κ ) (cid:39) κ ) (cid:39) κ for κ →
0, and ω p (cid:39) ω n for large Q , and ω p (cid:39) ω , we get v o ( t ) = cos( ω t ) + κ (cid:2) − e − t/τ (cid:3) cos( ω t ) t > . This is a slowly varying phasor V o ( t ) = √ (cid:8) κ (cid:2) − e − t/τ (cid:3)(cid:9) , whose ampli-tude swing V o ( t ) − V o (0) (cid:39) κ (cid:2) − e − t/τ (cid:3) t > κ u ( t ). After deleting κ and differentiating, we obtain theimpulse response b( t ) = τ e − t/τ , the same already found for the phase impulse.In conclusion, the response to either a phase impulse or to an amplitudeimpulse is (Fig. 9) b( t ) = 1 τ e − sτ ↔ B( s ) = 1 /τs + 1 /τ . (30) . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-AM-PM-cpl-reson B αα B φα B αφ B φφ α v (t) (cid:15511) A v (s) ε (t) (cid:15511) E (s) ψ (t) (cid:15511) Ψ (s) φ (t) (cid:15511) Φ (s) file: ele-AM-PM-cpl-reson Figure 10: Detuning the resonator results in coupling AM to PM.Eq. (30) is that of a simple RC low-pass filter, which we will use in all blockdiagrams. The inverse of τ is known as the Leeson (cutoff) frequency of theresonator ω L = 1 τ = ω n Q or f L = 12 πτ = ν n Q .
Finally, it is to be remarked that (28) contains no amplitude terms at firstorder, and that (29) contains no phase terms at first order. This means that atthe exact resonant frequency there is no phase-amplitude coupling.
In this Section we analyze the impulse response of the resonator when the carrierfrequency is ω (cid:54) = ω n , with an offsetΩ = ω − ω n (detuning) . An amplitude perturbation ε ( t ) in the resonator driving signal results in anamplitude fluctuation α ( t ) = b αα ( t ) ∗ ε ( t ) plus a phase fluctuation ϕ ( t ) =b ϕα ( t ) ∗ ε ( t ). Similarly, the resonator responds to a phase perturbation ψ ( t )with a phase fluctuation ϕ ( t ) = b ϕϕ ( t ) ∗ ψ ( t ) plus an amplitude fluctuation α ( t ) = b αϕ ( t ) ∗ ψ ( t ). This is written in matrix form as (cid:20) αϕ (cid:21) = (cid:20) b αα b αϕ b ϕα b ϕϕ (cid:21) ∗ (cid:20) εψ (cid:21) ↔ (cid:20) A Φ (cid:21) = (cid:20) B αα B αϕ B ϕα B ϕϕ (cid:21) (cid:20) E Ψ (cid:21) , and shown in Fig. 10. In the following sections we will prove that the stepresponse is (Fig. 11) (cid:90) (cid:2) b (cid:3) ( t ) dt = (cid:34) sin(Ω t ) e − t/τ [1 − cos(Ω t )] e − t/τ [1 − cos(Ω t )] e − t/τ sin(Ω t ) e − t/τ (cid:35) (31)and that the impulse response is (Fig. 12) (cid:2) b (cid:3) ( t ) = (cid:34) (cid:0) Ω sin Ω t + τ cos Ω t (cid:1) e − t/τ (cid:0) − Ω cos Ω t + τ sin Ω t (cid:1) e − t/τ (cid:0) − Ω cos Ω t + τ sin Ω t (cid:1) e − t/τ (cid:0) Ω sin Ω t + τ cos Ω t (cid:1) e − t/τ (cid:35) (32) . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ! ! Step response of the detuned resonator
Parameter: detuning frequency F figure: 945 ! reson ! step ! responsesource: oscill ! am ! noiseE. Rubiola, jun 2009 time, t/tau F _ L / F _ L a l ph a / a l ph a , ph i/ ph i Figure 11: Step response of the resonator off the natural frequency. ! ! Impulse response of the detuned resonator
Parameter: detuning frequency F figure: 946 ! reson ! impulse ! responsesource: oscill ! am ! noiseE. Rubiola, jun 2009 time, t/tau F _ L / F _ L alpha/alpha, phi/phi0 F _ L / F _ L phi/alpha, alpha/phi Figure 12: Impulse response of the resonator off the natural frequency. (cid:2) B (cid:3) ( s ) = τ s + τ + Ω τs + τ s + τ + Ω − Ω ss + τ s + τ + Ω − Ω ss + τ s + τ + Ω τ s + τ + Ω τs + τ s + τ + Ω . (33)The resonator response has diagonal symmetryb αα ( t ) = b ϕϕ ( t ) ↔ B αα ( s ) = B ϕϕ ( s )b αϕ ( t ) = b ϕα ( t ) ↔ B αϕ ( s ) = B ϕα ( s ) . (34)The proof is given in Sections 4.3.1, 4.3.2, and 4.3.4 . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 A phase step κ at t = 0 is described as the probe signal v i ( t ) = 1 β cos( ω t − θ ) u ( − t ) (cid:124) (cid:123)(cid:122) (cid:125) switched off at t = 0 + 1 β cos( ω t − θ + κ ) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) switched on at t = 0 = 1 β cos( ω t − θ ) u ( − t ) + 1 β (cid:2) cos( ω t − θ ) cos κ − sin( ω t − θ ) sin κ (cid:3) u ( t ) (cid:39) β cos( ω t − θ ) u ( − t ) + 1 β (cid:2) cos( ω t − θ ) − κ sin( ω t − θ ) (cid:3) u ( t ) κ (cid:28) . Using (23), (26) and (27) under the large- Q approximation ( ω p = ω n ), the aboveyields the output v o ( t ) = cos( ω n t ) e − tτ + (cid:2) − cos( ω n t ) e − tτ + cos( ω t ) (cid:3) + κ (cid:2) sin( ω n t ) e − tτ − sin( ω t ) (cid:3) ( t > v o ( t ) = cos( ω t ) − κ sin( ω t ) + κ sin( ω n t ) e − t/τ t > . Introducing the detuning frequency Ω = ω − ω n , we get sin( ω n t ) = sin( ω t − Ω t ),thus sin( ω n t ) = sin( ω t ) cos(Ω t ) − cos( ω t ) sin(Ω t ). Hence, the output signal canbe rewritten as v o ( t ) = cos( ω t ) − κ sin( ω t ) + κ sin( ω t ) cos(Ω t ) e − tτ − κ cos( ω t ) sin(Ω t ) e − tτ , which simplifies to v o ( t ) = cos( ω t ) (cid:104) − κ sin(Ω t ) e − t/τ (cid:105) − κ sin( ω t ) (cid:104) − cos(Ω t ) e − t/τ (cid:105) . (35)Freezing the oscillation ω t , the above turns into the slow-varying phasor V o ( t ) = 1 √ (cid:110) − κ sin(Ω t ) e − t/τ + jκ (cid:2) − cos(Ω t ) e − t/τ (cid:3)(cid:111) κ (cid:28) (cid:61){ V o ( t ) }(cid:60){ V o ( t ) } = κ (cid:104) − cos(Ω t ) e − t/τ (cid:105) and amplitude | V o ( t ) | = | V o (0) | − κ sin(Ω t ) e − t/τ After deleting κ and differentiating, we obtain the impulse responseb ϕϕ ( t ) = (cid:104) Ω sin(Ω t ) + 1 τ cos(Ω t ) (cid:105) e − t/τ phase (36)b αϕ ( t ) = (cid:104) − Ω cos(Ω t ) + 1 τ sin(Ω t ) (cid:105) e − t/τ amplitude (37) . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 An amplitude step κ at t = 0 is described as the probe signal v i ( t ) = 1 β cos( ω t − θ ) u ( − t ) (cid:124) (cid:123)(cid:122) (cid:125) switched off at t = 0 + (1 + κ ) 1 β cos( ω t − θ ) u ( t ) (cid:124) (cid:123)(cid:122) (cid:125) switched on at t = 0 Using (23), (26) and (27), under the approximation ω p = ω n the above yieldsthe output v o ( t ) = cos( ω n t ) e − t/τ + (1 + κ ) (cid:2) − cos( ω n t ) e − t/τ + cos( ω t ) (cid:3) t >
0= cos( ω t ) + κ cos( ω t ) + κ cos( ω n t ) e − t/τ . Using cos( ω n t ) = cos( ω t ) cos(Ω t ) + sin( ω t ) sin(Ω t ), the output is v o ( t ) = cos( ω t ) (cid:110) κ (cid:104) − cos(Ω t ) e − t/τ (cid:105)(cid:111) − κ sin( ω t ) sin(Ω t ) e − t/τ (38)Freezing the oscillation ω t , the above turns into the slow-varying phasor V o ( t ) = 1 √ (cid:110) κ (cid:2) − cos(Ω t ) e − t/τ (cid:3) + jκ sin(Ω t ) e − t/τ (cid:111) of angle arctan (cid:61){ V o ( t ) }(cid:60){ V o ( t ) } = κ sin(Ω t ) e − t/τ κ (cid:28) | V o ( t ) − V o (0) | = κ (cid:104) − cos(Ω t ) e − t/τ (cid:105) κ (cid:28) κ and differentiating, we obtain the impulse responseb αα ( t ) = (cid:104) Ω sin(Ω t ) + 1 τ cos(Ω t ) (cid:105) e − t/τ amplitude (39)b ϕα ( t ) = (cid:104) − Ω cos(Ω t ) + 1 τ sin(Ω t ) (cid:105) e − t/τ phase (40) Interestingly, the phase noise bandwidth of the resonator increases when theresonator is detuned. This is related to the following facts.1. When the resonator is detuned, it holds that (cid:12)(cid:12)(cid:12)(cid:12) d arg[ β ( jω )] dω (cid:12)(cid:12)(cid:12)(cid:12) ω < (cid:12)(cid:12)(cid:12)(cid:12) d arg[ β ( jω )] dω (cid:12)(cid:12)(cid:12)(cid:12) ω n . (41)With lower slope, the oscillator phase noise is higher.2. Detuning the resonator, the symmetry of arg[ β ( jω )] around the oscillationfrequency is broken. This explains the frequency overshoot seen in Fig. 12for Ω (cid:54) = 0.3. The step response decays faster when the resonator is detuned. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 s )] from [b( t )] For the sake of completeness, we derive the full expression of [B( s )] from [b( t )],that is (33) from (32). Thanks to the symmetry properties (34), we only needto derive B αα ( s ) and B αϕ ( s ).Using the well known properties e − t/τ ↔ s + 1 /τe at f ( t ) ↔ F ( s − a ) , we notice that it holds e ± j Ω t e − t/τ ↔ s ∓ j Ω + τ . (42)B αα ( s ) = L { b αα ( t ) } is found using (42), after expanding (39) with the Eulerformulaecos(Ω t ) = 12 (cid:0) e j Ω t + e − j Ω t (cid:1) sin(Ω t ) = 1 j (cid:0) e j Ω t − e − j Ω t (cid:1) . (43)Thus,B αα ( s ) = L (cid:110)(cid:104) Ω 1 j (cid:0) e j Ω t − e − j Ω t (cid:1) + 1 τ (cid:0) e j Ω t + e − j Ω t (cid:1)(cid:105) e − t/τ (cid:111) = Ω j (cid:20) s − j Ω + τ − s + j Ω + τ (cid:21) + 12 τ (cid:20) s − j Ω + τ − s + j Ω + τ (cid:21) = Ω + s/τ + 1 /τ (cid:0) s + τ − j Ω (cid:1) (cid:0) s + τ + j Ω (cid:1) = 1 τ s + τ + Ω τ (cid:0) s + τ − j Ω (cid:1) (cid:0) s + τ + j Ω (cid:1) , and finally B αα ( s ) = 1 τ s + τ + Ω τs + τ s + τ + Ω qed . (44)Similarly, B αϕ ( s ) = L { b αϕ ( t ) } is found using (42), after expanding (37)with the the Euler formulae (43). Thus,B αϕ ( s ) = L (cid:110)(cid:104) − Ω2 (cid:0) e j Ω t + e − j Ω t (cid:1) + 1 j τ (cid:0) e j Ω t − e − j Ω t (cid:1)(cid:105) e − t/τ (cid:111) = − Ω2 (cid:20) s − j Ω + τ + 1 s + j Ω + τ (cid:21) + 1 j τ (cid:20) s − j Ω + τ − s + j Ω + τ (cid:21) = − Ω s − ω/τ + ω/τ (cid:0) s + τ − j Ω (cid:1) (cid:0) s + τ + j Ω (cid:1) , and finally B αϕ ( s ) = − Ω ss + τ s + τ + Ω qed . (45) . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-PM-scheme low-pass Σ relaxation time file: ele-PM-scheme ϕ ( t ) ↔ Φ( s ) ψ ( t ) ↔ Ψ( s ) τ = 2 Q/ω v v Figure 13: Phase-noise model of the feedback oscillator. ele-H-leeson H ϕ ( s ) | H ϕ ( f ) | ff L /f f f L = ν Q − /τ file: ele-H-leeson jωσ Figure 14: Phase-noise transfer function.
Figure 13 shows the phase-noise model of the oscillator. In this figure, all signalsare the phase fluctuation of the oscillator sinusoidal signal. Here, the resonatorturns into a lowpass filter of time constant τ , as explained in Section 4. A noise-free amplifier has a gain exactly equal to one because the amplifier repeats thephase of the input signal. The real amplifier introduces the random phase ψ ( t ),which in this representation is additive noise, regardless of the physical origin.For the sake of simplicity, we put in ψ ( t ) all the phase-noise sources.We define the phase-noise transfer function asH( s ) = Φ( s )Ψ( s ) . Applying the elementary feedback theory to the circuit of Fig. 13 we findH( s ) = 11 + B( s ) , where B( s ) is the resonator transfer function (30), and thereforeH( s ) = s + 1 /τs (Fig. 14) . (46)This is the Leeson effect, by which the oscillator integrates the slow phase fluc-tuation, turning it into frequency fluctuation. The phase-noise transfer functionis plotted in Fig. 14. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-AM-scheme low-pass Σ A relaxation time file: ele-AM-scheme τ = 2 Q/ω v v ε ( t ) ↔ E ( s ) u = 1 + α u α u ( t ) ↔ A u ( s ) α v ( t ) ↔ A v ( s ) v = 1 + α v Figure 15: Amplitude-noise model of the feedback oscillator.
Figure 15 shows the low-pass model that describes the oscillator amplitude.Since the gain A depends on amplitude, the Laplace/Heaviside formalism can-not be used directly. We first need to linearize the system in the appropriateconditions. Cutting the feedback loop at the amplifier input, we get u = ε + v , where v results from the lowpass filter v = 1 τ (cid:90) ( v − v ) dt . Combining the above equations and replacing v = Au and v = u − ε , we get u − τ (cid:90) ( A − u dt = ε + 1 τ (cid:90) ε dt (general IE) , (47)thus ˙ u − τ ( A − u = ˙ ε + 1 τ ε (general DE) . (48)Notice that (47)-(48) are general because A is still unspecified. Substituting A = 1 − γ ( u − u + γτ (cid:0) u − (cid:1) u = ˙ ε + 1 τ ε with A = 1 − γ ( u − . (49)The system free running is governed by the homogeneous equation˙ u + γτ (cid:0) u − (cid:1) u = 0 . (50)The solution is u ( t ) = 11 + Ce − γt/τ (solution of (50)) , (51)where C is a constant determined by the initial conditions. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-oscillator-start oscillator startup Parametersstartup Ao = 1.414saturation c = 0.414relax time tau = 0.03initial cond Vo(0) = 0.001 figure: ele (cid:239) oscillator (cid:239) startsource: oscill (cid:239) am (cid:239) noiseE. Rubiola, apr 2010 time s a t u r a t e d s m a ll (cid:239) s i gn a l Figure 16: Oscillator startup.
A simplified model for the oscillator is obtained by assuming that the linearapproximation A = 1 − γ ( u −
1) holds in the whole amplitude range. One canobject that this case is only of academic interest because in real amplifiers theparameter γ is constant only in a narrow region around u = 1, as shown inFig. 2. Nonetheless, the general description that follows can be easily adaptedto practical cases.Assuming that A = 1 − γ ( u − C is related to the initial u (0) by u (0) = 11 + C → C = 1 u (0) − . Thus, u ( t ) = 11 + (cid:16) u (0) − (cid:17) e − γt/τ . (52)In the absence of a switch-on transient, oscillation starts from noise. Thus, u (0)is a small positive quantity, hence 1 /u (0) − (cid:39) /u (0) and u ( t ) (cid:39)
11 + u (0) e − γt/τ < u (0) (cid:28) . (53)Accordingly the following asymptotic expression hold u ( t ) = u (0) e γt/τ t → u ( t ) = 1 t → ∞ (55)Figure 16 shows the complete oscillation start (52), which saturates to u = 1,and the small-signal approximation (54). Figure 17 shows a Spice simulation. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-Brendel-sim1 ! ! ! ! ! ! ! !
0V 4.0ms
V(u) !!!! = 0.1m ! !!!! = 10m ! SPICE simulation γ = 0.1, τ = 16 µsu(0) = 1E-4 ... 1E-2 file: ele-Brendel-sim1 Figure 17: Simulation of the startup of the model shown in Fig. 15, assumingthat the amplifier law A = 1 − γ ( u −
1) holds in the full range.
Actual oscillators differ from the above simplified model in that the small-signalgain follows the law A = 1 − γ ( u −
1) only in the vicinity of u = 1, as shownin Fig. 2. In the absence of a general model, we denote the small-signal gainwith A , which is a circuit-specific parameter that we assume to be constant for u →
0. Replacing A = A (constant), the homogeneous equation (50) becomes˙ u − τ ( A − u = 0 . The solution is u ( t ) = u (0) e ( A − t/τ , (56)where u (0) is the small initial condition set by noise. This solution is similar to(54), but for the different value of the time constant.A number of computer simulations were done independently by R. B. wellbefore the approach presented here was developed [27, 28]. This led to thepreliminary work published in [29]. Figure 18 shows the simulated startup.The left-hand side of the envelope, until t ≈ µ s, fits well the theoreticalprediction (56). It is interesting to study the amplitude free running when the initial conditionis set close to the steady state, thus at u (0) = 1 + κ with 0 < κ (cid:28)
1. Inthis conditions the approximation A = 1 − γ ( u −
1) holds, and therefore theamplitude is given by (52) with u (0) = 1 + κu ( t ) = 11 − κ κ e − γt/τ ( u (0) = 1 + κ ) , (57)For small κ , this is linearized as u ( t ) = 1 + κe − γt/τ , (58) . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-Brendel-startup time, μ s v o l t age , m V file: ele-brendel-startup envelope waveform Figure 18: Simulated oscillator startup.or equivalently α u ( t ) = κe − γt/τ (59)because α u = u − τ r = τγ (restoring time) (60)is the oscillator restoring time for amplitude perturbations. Since in virtuallyall amplifiers it holds that 0 < γ <
1, as widely discussed in Section 3.1, it holdsthat τ r > τ . Now, we study the oscillator response to the amplitude impulse ε ( t ) = δ ( t )occurring when the oscillator is in the steady state u = 1, v = 1. The impulseresponse is the derivative of the step response, linearized for small perturbation.Thus, referring to Fig. 15, we apply at the input the small step ε ( t ) = κ u ( t ) 0 < κ (cid:28) . We know from Section 6.4 that the small-signal response is a decaying expo-nential e − γt/τ . Hence the response is completely determined by the initial andfinal values u ( t ) = u ( ∞ ) + (cid:2) u (0 + ) − u ( ∞ ) (cid:3) e − γt/τ t > . (61)Since the perturbation takes time to propagate through the lowpass filter, itholds that u (0 + ) = 1 + κ . The final value is obtained by inspection on Fig. 15after bypassing the lowpass filter and setting (cid:15) ( t ) = κ . Thus u = v + κ (adder) v = [1 − γ ( u − u (amplifier and filter) , hence u = [1 − γ ( u − u + κ ( t → ∞ ) . . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 u = γ − (cid:112) γ + 4 γκ γ u = γ + (cid:112) γ + 4 γκ γ , and for small κu (cid:16) − κγ u (cid:16) κγ κ → . It is immediately seen that u < u >
0. Hence u is the physical solutionwhile u is discarded. Setting κ = 1 (unit step) and using (61), we find the stepresponse (cid:90) h ( ε ) u ( t ) dt = 1 + 1 γ u ( t ) − (cid:18) γ − (cid:19) e − γt/τ u ( t ) . (62)Notice that the term ‘1+’ is the steady state before the step is applied. Thesubscript u and the superscript ( ε ), which refer to the input ε and to the output u , are introduced to emphasize the difference versus other response functions.The impulse response is found by differentiating (62)h ( ε ) u ( t ) = δ ( t ) + γτ (cid:18) γ − (cid:19) e − γt/τ u ( t ) . (63)The Laplace transform H ( ε ) u ( s ) = L{ h ( ε ) u ( t ) } is found immediately using theproperties L{ δ ( t ) } = 1 and L{ e − at } = s + a H ( ε ) u ( s ) = 1 + γτ (cid:18) γ − (cid:19) s + γ/τ , which simplifies to H ( ε ) u ( s ) = s + 1 /τs + γ/τ . (64)So, the transfer function is completely determined by the roots(pole) s p = − γ/τ f p = γf L < f L (zero) s z = − /τ f z = f L . In this Section we study the effect of the parametric fluctuation of the gain byintroducing the random variable η ( t ), as anticipated in Section 3.3.4 and Fig. 7 A = 1 − γ ( u −
1) + η (65) η ( t ) ↔ N ( s ) . We linearize the system for low noise, and we search for the transfer functionsH ( η ) u ( s ) = A u ( s ) N ( s ) and H ( η ) v ( s ) = A v ( s ) N ( s ) , where α u ( t ) ↔ A u ( s ) and α v ( t ) ↔ A v ( s ) , are the amplitude fluctuations at the amplifier input and output, respectively. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-Hu-AM | H u ( f ) | /f f γf L /γ H u ( s ) − γ/τ file: ele-Hu-AM jωσ Figure 19: Amplitude-noise transfer function (amplifier input).
By replacing A = 1 − γ ( u −
1) + η in the homogeneous equation (50), we get˙ u + γτ ( u − u = ητ u . Since u = 1 + α u , it holds that ˙ u = ˙ α u and u − α u , thus˙ α u + γτ α u u = ητ u . For small fluctuations α u and η , we linearize the above using u (cid:39) α u + γτ α u = 1 τ η . The linearized system can now be described in using the Laplace transforms (cid:16) s + γτ (cid:17) A u ( s ) = 1 τ N ( s ) , (66)which gives the transfer function (Fig. 19)H ( η ) u ( s ) = 1 /τs + γ/τ . (67) We first need to relate α v to α u . This is done by replacing A = − γ ( u − η in v = Au v = [ − γ ( u −
1) + 1 + η ] u , and by expanding using v = 1 + α v and u = 1 + α u α v = 1 + η − γα u + α u − α u η − γα u . Neglecting the second-order noise terms α u η and α α v = (1 − γ ) α u + η , . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-Hv-AM f L γf L γf L γ <1 γ >1 | H α ( f ) | Leeson –1/ τ –1/ τ – γ / τ – γ / τ γ <1 γ >1 [not allowed] file: ele-Hv-AM jωjωσσ not allowed Figure 20: Amplitude-noise transfer function (oscillator output). ele-Brendel-sim2 ! Time ! V(u) file: ele-Brendel-sim2
SPICE simulation γ = 0.1, τ = 16 µs η =1E–3 step at t=4 ms Figure 21: Simulation of the step response of the AM model shown in Fig. 15.we get α u = α v − η − γ ↔ A u ( s ) = A v ( s ) − N ( s )1 − γ . Then, by replacing the above A u ( s ) in Eq. (66) we get (cid:16) s + γτ (cid:17) A v ( s ) = (cid:16) s + 1 τ (cid:17) N ( s ) , and finally H ( η ) v ( s ) = s + 1 /τs + γ/τ . (68)The transfer function H ( η ) v ( s ) is shown in Fig. 20 Notice that the case γ > < γ < η = 10 − , assuming that the gain-compression parameter is γ = 0 .
1. The risingexponential reaches the final value 1 + η/γ with a time constant τ /γ , whichconfirms Eq. (68).
We calculate the oscillator AM and PM spectra due to the Leeson effect alone.The fluctuation of the resonator natural frequency, not accounted in this Section, . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-spectrum-PM s u s t a i n . a m p l. f c S φ (f) f f c < f L f c f L f c f L Type 2: f L < f c Amplifier alone ou t pu t bu ff e r ou t . bu f . ou t pu t bu ff e r s u s t a i n i ng a m p li fi e r S φ (f) S φ (f) s u s t a i n . a m p l. file: ele-spectrum-PM Figure 22: Phase-noise spectrum in log-log scale. The cutoff frequency generatedby the zero of the transfer function at s = − /τ is shown on the horizontal axis.The pole at s = 0 cannot be shown on a logarithmic scale.may be added afterwards. With the remarkable exception of the laser, virtuallyall practical oscillators are followed by a buffer, which contributes with its ownnoise. Referring to first plot of Fig. 22 (amplifier PM noise), we notice thatthe output buffer has higher flicker and lower white noise than the sustainingamplifier. The buffer flicker is higher because the buffer has higher number ofstages, each of which adds its 1 /f phase noise independent of the carrier power[Eq. (13)]. Conversely, the buffer white noise is lower because this type of noiseis additive and the input power is higher at the buffer input [Equations (11)–(12)]. The same is seen on the first plot of Fig. 23, which refers to the amplifierAM noise. With reference to Fig. 22, the analysis starts from the sustaining-amplifier noise,which shows the flicker corner at f = f c . This noise is turned into the oscillatornoise by the transfer function H ϕ ( f ) [Eq. (46)], which is completely describedby a pole at f = 0 and a zero at f L = πτ on the Bode plot.With a low- Q resonator we get the spectrum of the Type 1, where f L > f c .At the oscillator output, before buffering, only the slopes 1 /f , 1 /f and f arepresent. The buffer noise is generally not visible because it rises with lower slope(1 /f ) on the right hand of the plot. Only in a few special cases, when noisespecial techniques are used to reduce the phase noise of the sustaining amplifier[30, 31], thus the gap between the flicker of the buffer and of the sustaining . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-spectrum-AM f c f fi l e : e l e - s pe c t r u m - A M f c < γ f L < f L Type 2: γ f L < f c < f L f c γ f L f L f c γ f L f L f c γ f L f L Type 3: γ f L < f L < f c Amplifier alone ou t pu t bu ff e r ou t pu t bu ff e r ou t pu t bu ff e r ou t pu t bu ff e r s u s t a i n i ng a m p li fi e r S α (f) S α (f) S α (f) S α (f) s u s t a i n . a m p l. s u s t a i n . a m p l. s u s t a i n . a m p l. Figure 23: Amplitude-noise spectrum in log-log scale. The cutoff frequenciesgenerated by the roots of the transfer function at s = − γ/τ and s = − /τ areshown on the horizontal axis.amplifier is large, some 1 /f noise shows up at the buffered output in the regionaround f L .If the resonator Q is higher we get the spectrum of the Type 2, where f L < f c .Before buffering, only the slopes 1 /f , 1 /f and f are present. The buffer 1 /f noise shows up because it is higher than the sustaining-amplifier noise and hasthe same slope. The amplitude noise (Figure 23) is more complex than the phase noise becausethe transfer function H v ( f ) [Eq. (68)] is described by two roots, a real pole at f = γ πτ and a real zero at f L = πτ . Increasing the resonator Q , these rootsmay occur both on the right-hand of f c in the Type-1 spectrum (low Q ), oneon the left hand and the other on the right hand of f c in the Type-2 spectrum(medium Q ), and both on the left-hand of f c in the Type-3 spectrum (high Q ). . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-AM-PM-cpl-param w ψ v = 1+ α v α v (t) (cid:15511) A v (s) u = 1+ α u α u (t) (cid:15511) A u (s) η (t) (cid:15511) N (s) ΣΣ A file: ele-AM-PM-cpl-param ψ ' (t) (cid:15511) Ψ ' (s) φ (t) (cid:15511) Φ (s) B αα B φφ Φ (s) AM loopPM loop w η Amplifiernoise H u ( s ) = 1 /τs + γ/τ H v ( s ) = s + 1 /τs + γ/τ A u (s) w γ H ϕ ( s ) = 1 + sτsτ AM → PMcoupling ψ " (t) (cid:15511) Ψ " (s) x(t) (cid:15511) X (s) Figure 24: Combined effect of the AM-PM coupling in amplifier and the loop,which forces the Barkhausen condition.Generally, the buffer 1 /f noise shows up only in the Type-3 spectrum, inthe 1 /f region between f L and f c . It may also show up in the Type-2 spectrumaround f L if the gap between the flicker of the buffer and of the sustainingamplifier is made large by the use of a noise-degeneration sustaining amplifier. We turn our attention to the AM-PM noise coupling mechanism shown inFig. 24. Noise modulates the gain. Yet the Barkhausen condition forces theloop gain to be equal to one through the gain-compression mechanism. Theconsequence is that the gain fluctuation is transformed into a fluctuation of theoscillation amplitude, and in turn into a fluctuation of the amplifier phase. Theconclusion is that the phase ψ seen by the Leeson effect is the sum of two con-tributions , the first comes straight from the amplifier, and the second resultsfrom the effect on the fluctuating amplitude. The detailed model that followsis shown in Figures 24 and 25, and discussed underneath.For the sake of simplicity, we assume that the oscillator is tuned at theexact natural frequency of the resonator, and we assume that the amplifier isperturbed by one dominant source of noise. These hypotheses give a realisticpicture of the oscillator.Denoting with x ( t ) ↔ X ( s ) the near-dc noise process, and introducing themodulation efficiency w η and w ψ , the amplifier gain is perturbed by a factor(1 + w η x ) e jw ψ x . Accounting for compression and neglecting the second-order . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-AM-PM-cpl-amp | A | u ideal A =1– γ ( u -1) fluctuating A = 1– γ ( u –1)+ η amplitudefluct η / γ a r g ( A ) u ideal fluctuating file: ele-AM-PM-cpl-amp Leeson effect η / γ ψ = ψ '+ ψ '' v A u gainfluct. η phasefluct. ψ 'near-dcrandom process amplifier noise γ f L f L fth(t)H(f) loop AM response η / γ ψ ' ψ " γ f L f L fth(t)H(f) loop AM response Figure 25: Combined effect of the AM-PM coupling in amplifier. TheBarkhausen condition turns the gain fluctuation η into amplitude fluctuation,and in turn to the phase fluctuation ψ (cid:48)(cid:48) . The latter adds to the phase fluctuation ψ (cid:48) .terms, the complete expression of the gain is A = (cid:2) − γ ( u −
1) + w η x (cid:3) e jw ψ x . The stationary oscillation is ruled by the Barkhausen condition Aβ = 1. Withthe normalization β ( ω ) = 1, this implies that | A | = 1. There follows that theinstantaneous gain fluctuation η ( t ) cannot increase | A | . Instead, η ( t ) causesthe oscillation amplitude u ( t ) to change from 1 to 1 + η ( t ) /γ , as shown in theupper plot of Fig. 25. In this condition the amplifier introduces a phase term ψ (cid:48)(cid:48) ∝ η/γ , which adds to the ‘simple’ phase noise ψ (cid:48) of the amplifier. Thesuperscripts ‘prime’ and ‘second’ are introduced in order to keep the symbols ψ and ϕ for the overall phase fluctuation. Therefore, the phase fluctuation seenby the Leeson effect (Sec. 5) is ψ = ψ (cid:48) + ψ (cid:48)(cid:48) . After (46), the first phase-noisecontribution is Φ (cid:48) ( s ) = w ψ s + 1 /τs X ( s ) . . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010
100 1000 10000 1e+05 1e+061101001000100001e+051e+06
Effect of AM (cid:239)
PM coupling
Parameterstau = 1.59e (cid:239)
6 F_L = 1.0e5gamma = 0.05 gamma*f_L = 5.0e3coupling = (cid:239) figure: 956 (cid:239) AM (cid:239) PM (cid:239) cplsource: oscill (cid:239) am (cid:239) noiseE. Rubiola, apr 2010 frequencyblack: Leeson effectblue: AM noisered: combined Figure 26: AM-PM coupling due to parametric noise. The ‘coupling’ parameteris equal to w γ w η /w ψ .By inspection on Fig. 24, the second phase noise contribution isΦ (cid:48)(cid:48) ( s ) = w γ w η s + 1 /τs + γ/τ s + 1 /τs X ( s ) . Since Φ (cid:48) ( s ) and Φ (cid:48)(cid:48) ( s ) depend deterministically on X ( s ), they must be added.Thus, the noise transfer functionH ( c ) ( s ) = Φ( s ) X ( s ) = Φ (cid:48) ( s ) + Φ (cid:48)(cid:48) ( s ) X ( s ) (69)is H ( c ) ( s ) = w ψ (cid:20) w γ w η w ψ s + 1 /τs + γ/τ (cid:21) s + 1 /τs X ( s ) . (70)The term ‘1’ is the simple Leeson effect as introduced in Section 5, but for thetrivial factor w ψ introduced because (70) refers to the near-dc process x ↔ X instead of to the phase ψ ↔ Ψ. The term w γ w η w ψ is the phase fluctuation inducedby AM noise.Figures 26 and 27 show the noise transfer function | H ( c ) ( f ) | in two cases.The signature of the AM-PM coupling shows up in the frequency range between γf L and f L . In this region, the plot is parallel to that of the simple Leeson effect.Interestingly, the AM-PM coupling can either increase or reduce the noise inthe region between γf L and f L . Of course, the phase-noise plot accounts forthe slope of the near-dc process X by which the transfer function is multiplied.Thus for example the same signature can be seen in the 1 /f region if X isflicker noise. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010
100 1000 10000 1e+05 1e+061101001000100001e+051e+06
Effect of AM (cid:239)
PM coupling
Parameterstau = 1.59e (cid:239)
6 F_L = 1.0e5gamma = 0.05 gamma*f_L = 5.0e3coupling = 0.4 figure: 957 (cid:239) AM (cid:239) PM (cid:239) cplsource: oscill (cid:239) am (cid:239) noiseE. Rubiola, apr 2010 frequencyblack: Leeson effectblue: AM noisered: combined Figure 27: AM-PM coupling due to parametric noise. The ‘coupling’ parameteris equal to w γ w η /w ψ . ele-OEO [1+ α (t)] cos[ ω t+ φ (t)] AMPM η (t) ψ (t) A gaincompression β f equivalent noise mode selector file: ele-OEO EOM β d delay line laser optics Figure 28: Opto-electronic oscillator. The equivalent noise originates in bothelectronic and optical path. Gain compression takes place either in the amplifier,in the photodetector or in the electro-optic intensity modulator.
The delay-line oscillator is a variant of the oscillator in which the resonatoris replaced by a delay line of delay τ d , so that the oscillation frequency is aninteger multiple of 1 /τ d . To the extent of the Leeson effect, the delay lineis equivalent to a resonator of quality factor Q = πν τ d because the slope d arg( β ) /dω is the same. Of course, longer delay gives access to lower phasenoise and higher frequency stability, provided the delay be stable. For thisreason the modern version of the delay-line oscillator, called OEO [15, 16] andshown in Fig. 28, makes use of an optical fiber as the delay unit. The optical fiberexhibits high thermal stability (6 . × − /K) and low loss (0.15 dB/km at 1.55 µ m wavelength, equivalent to 0.03 dB/ µ s), limited by the Rayleigh scattering.Other implementations are possible, based on a surface-wave devices and on . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 phase noise |H(jf)|^2optical (cid:239) fiber oscillator with selector frequency, Hz parameters: tau = 2E (cid:239) figure: 958 (cid:239) OEO (cid:239) xfer (cid:239) funcsource: oscill (cid:239) am (cid:239) noiseE. Rubiola, apr 2010 Figure 29: Noise transfer function of the opto-electronic oscillator, either forAM noise or PM noise. The parameter m is the mode order. Hence with 20 µ sdelay, the mode of order 2 × falls at ν = 10 GHz.electrical lines.The phase noise theory of the delay-line oscillator is widely discussed in [1,Chap 5]. Here we give some key element to extend the theory to AM noise andto the AM-PM noise coupling.Since the delay line is a wide-band device, the loop can sustain oscillationsat any frequency multiple of 1 /τ d . A mode-selector filter is therefore necessaryto choose one frequency by lowering the loop gain at the other frequencies. Forthis reason the feedback function of Fig. 1 is split into delay and filter, denotedwith the subscripts d and f , respectively. It is important to understand thatgroup delay of the mode selector must be orders of magnitude shorter than thedelay of the line because the sensitivity to environment parameters is weightedproportionally to the delay.There are practical reasons to use a resonator as the mode selector. Weassume that the delay-line attenuation is independent of frequency, moving theflatness defect to the resonator transfer function. Using the elementary theory ofthe Laplace transform and the material developed in Section 4, the slow-varyingenvelope representation of the feedback path isb f ( t ) = 1 τ f e − t/τ f ↔ B f ( s ) = 1 /τ f s + 1 /τ f (71)b d ( t ) = δ ( t − τ d ) ↔ B d ( s ) = e − sτ d (72)b( t ) = b f ( t ) ∗ b d ( t ) ↔ B( s ) = B f ( s ) B d ( s ) , (73)thus B( s ) = 1 /τ f s + 1 /τ f e − sτ d . (74) . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 s ) = 11 − B( s ) (75)and therefore H( s ) = s + 1 /τ f s + (1 − e − sτ d ) /τ f , (76)an example of which is shown in Fig. 29.The full extension to AM noise, as derived in Section 7 for the oscillatorbased on a simple resonator, takes cumbersome and tedious algebra. Yet at lowfrequencies, below the Leeson frequency, the asymptotic approximation of thedelay line is a resonator of quality factor Q = πν τ d . This simplification givesaccount for the low-frequency behavior, and at least a qualitative prediction forthe AM noise peaks. Let us start with the analysis of Figure 28 in open loop conditions. The pathfrom the amplifier to the EOM is broken. First, we observe that the bandpassfilter, however large, eliminates the harmonics at frequencies multiple of ω .Thus the light power at the photodetector input can be described by P λ ( t ) = P λ (cid:2) m cos( ω t ) (cid:3) , (77)where m = J ( z ) is the modulation index, J ( z ) is the first-order Bessel functionof the first kind, and z is proportional to the microwave voltage at the input ofthe intensity modulator [32]. Though the theoretical maximum is m (cid:39) . m = 1. The current at the photodetector input is I ( t ) = ρP λ ( t ) (78)= ρP λ (cid:2) m cos( ω t ) (cid:3) , (79)where ρ = ηq (cid:126) ν λ is the photodetector responsivity, η the quantum efficiency, and (cid:126) ω λ the photon energy. Assuming a quantum efficiency of 0.6, the responsivityis of 0.75 A/W at 1.55 µ m wavelength, and of 0.64 A/W at 1.31 µ m. Filteringout the dc component, the rms voltage across a resistor R at the photodetectoroutput is v = 1 √ R ρ m P λ (rms voltage at the photodetector out) (80)The path from the EOM to the microwave amplifier output of Fig. 28 can beseen as an ‘amplifier,’ plus a filter function. The ‘amplifier’ includes intensitymodulator, photodetector, and microwave amplifier. By virtue of (80) the laserpower affects the gain, thus the relative intensity noise (RIN) makes the gainfluctuate. Since in open loop η = δv/v , the laser RIN induces a gain fluctuation η = δP λ P λ S (RIN) η ( f ) = S RIN . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 − −
15 dB/dec appear in the spectrum. Atthe beginning of he process of collecting data from the literature, we suspectthat this is typical of the distributed-feedback laser. Anyway, regardless ofthe physical explanation beyond, the presence of − −
15 dB/decslopes in the RIN spectrum in conjunction with the Leeson effect could explainthe slope of −
35 dB/dec and −
25 dB/dec observed in the phase noise spectrumof some oscillators. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 A Exotic issues
This Appendix is not a finished work. We report on some facts intended tobe the seed for further analysis.
A.1 AM-PM coupling in the off-resonance resonator
It has been shown in Section 4 that the resonator operated at the exact natu-ral frequency responds to a phase perturbation with a decaying exponential ofphase, with no effect on the amplitude; and that it responds to an amplitudeperturbation with a decaying exponential of amplitude, with no effect on thephase. It has also been shown that cross terms appear off the resonance, for theresonator response is described by [b( t )] ↔ [B( s )] (32)–(33).The AM-PM coupling inside the resonator yields naturally to the oscillatormodel depicted in Fig. 30. In this figure, the symbol s (expressed or implied)cannot be the Laplace complex variable because the system is nonlinear. In-stead, s is to be interpreted as the derivative operator ddt , which is allowed.Hence, the simplest approach is to derive the resonator equations using theLaplace formalism, and then to convert these equations into regular differentialequations by replacing s → ddt .The lower loop of Fig. 30 yields the Leeson effect, as described in Sec. 5. Theupper loop models the amplitude noise as discussed in Sec. 6-7. The two loopsare coupled by the terms B αϕ and B ϕα , which are nonzero when the resonatoris off resonance. By inspection on Fig. 30 we get U α = E + AB αα U α + B ϕα Φ ele-AM-PM-cpl-osc v = 1+ α v α v (t) (cid:15511) A v (s) u = 1+ α u α u (t) (cid:15511) A u (s) ε (t) (cid:15511) E (s) ΣΣ A file: ele-AM-PM-cpl-osc ψ (t) (cid:15511) Ψ (s) φ (t) (cid:15511) Φ (s) B αα B φα B αφ B φφ U α (s) Φ (s) AM loopPM loopAM-PMcouplingresonator
Figure 30: Detuning the resonator results in coupling AM to PM. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010
43Φ = Ψ + B ϕϕ Φ + AB αϕ U α , which can be rewritten as U α = 11 − AB αα E + B ϕα − AB αα ΦΦ = 11 − B ϕϕ Ψ + AB αϕ − B ϕϕ U α . Combining the above equations, we get U α = 11 − AB αα E + B ϕα (1 − AB αα )(1 − B ϕϕ ) Ψ + AB ϕα B αϕ (1 − AB αα )(1 − B ϕϕ ) U α Φ = 11 − B ϕϕ Ψ + AB αϕ (1 − AB αα )(1 − B ϕϕ ) E + AB αϕ B ϕα (1 − AB αα )(1 − B ϕϕ ) Φ , hence U α = (1 − AB αα )(1 − B ϕϕ )(1 − AB αα )(1 − B ϕϕ ) − AB ϕα B αϕ (cid:20) E − AB αα + B ϕα Ψ(1 − AB αα )(1 − B ϕϕ ) (cid:21) Φ = (1 − AB αα )(1 − B ϕϕ )(1 − AB αα )(1 − B ϕϕ ) − AB αϕ B ϕα (cid:20) AB αϕ E (1 − AB αα )(1 − B ϕϕ ) + Ψ1 − B ϕϕ (cid:21) and finally (cid:20) U α Φ (cid:21) = 1(1 − AB αα )(1 − B ϕϕ ) − AB ϕα B αϕ (cid:20) − B ϕϕ B ϕα AB αϕ − AB αα (cid:21) (cid:20) E Ψ (cid:21) . (81)The above equation is let in closed form for further analysis. The followingformulae will be useful1 − AB αα = s + τ (2 − A ) s + (1 − A ) (cid:0) τ + Ω (cid:1) s + τ s + τ + Ω − B ϕϕ = s + τ ss + τ s + τ + Ω B αϕ = B ϕα = − Ω ss + τ s + τ + Ω . A.2 Parametric fluctuation of the S matrix All over this report, the oscillator loop is analyzed as a simple block diagram inwhich the signal flows in one direction only, and there is no interaction due toimpedances. Breaking this assumption, the amplifier and the resonator can bedescribed in terms of the scatter matrix S .The case of the traditional microwave oscillator, where amplifier and res-onator are described by a 2 × A and theresonator transfer function β ( s ) are the element S of the respective matrix.Hence, the amplifier AM and PM noise as introduced in the previous Sections, . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-scatter-2x2 S = Γ in S = A rev S = A S = Γ out Z = Γ +1Γ − Z amplifier port 1 port 2(input) (output) S S S = β S resonator port 1 port 2(input) (output) oscillatoroutput file: ele-scatter-2x2 Figure 31: In most oscillators, the amplifier and the resonator can be describedin terms of scatter matrix. ele-scatter-neg-r S port 1 resonator file: ele-scatter-neg-r S = Γ in S = A rev S = A S = Γ out amplifier port 1 port 2negative- R input (output) oscillatoroutput Figure 32: Scatter matrix representation of the negative-resistance (dipolar)oscillator.go in (cid:60){ S } and (cid:61){ S } , respectively. The finite isolation of the amplifier isrepresented as S (cid:54) = 0. This effect has little importance because the isolationratio of actual amplifier is high enough for the reverse signal not to circulate inthe loop. Another effect is due to the amplifier input and output impedances,related to the scatter matrix by Z = Γ+1Γ − Z , Γ in = S , and Γ out = S . Theamplifier input and output impedances interact with the resonator parameters.Thus, the fluctuations of S and S turn into frequency fluctuations.The quartz oscillator and other negative-resistance oscillators can also bedescribed with the scatter matrix formalism (Fig. 32). In this case, the resonatordegenerates into a single-value matrix. The amplifier S models the negativeresistance that makes the system oscillate. Strictly, only S is necessary. Yet,in most cases the amplifier also acts as a buffer of gain S = A buf , reverse gain(isolation) S = A rev , and output reflection coefficient S = Γ out .Of course, the scattering matrix formalism also apply to optics. The laser(oscillator) differs from the above analysis in that the laser amplifier is bidirec-tional and the signal may be a stationary wave. A.3 The Miller effect
The Miller theorem [33] states that an impedance Z in the feedback path of anamplifier of gain A can be replaced by two impedances, Z = Z − A (input) and Z = Z − /A (output) , . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 ele-miller A Z Z A Z Z = Z − A Z = Z − /A file: ele-miller Figure 33: Miller effect.connected at the amplifier input output, respectively (Fig. 33). For our pur-poses, the left-hand side of Fig. 33 is formally equivalent to the oscillator loop,for we can identify Z as the resonator in the feedback path, and A as the sus-taining amplifier.Unfortunately the Miller theorem cannot be inverted in the general casebecause it would be necessary to collapse three degrees of freedom ( A , Z and Z ) into two degrees of freedom ( A and Z ). The parameters of the specificcircuit are needed to get Z from Z and Z . Nonetheless, the Miller theoremprovides evidence that the gain fluctuations affect the impedances of the wholecircuit, and that a fluctuating impedance at the amplifier input or output canbe turned into a fluctuating impedance in parallel to Z , hence to the resonator.In turn, the oscillator frequency fluctuates. . Rubiola & R. Brendel, Generalized Leeson effect, April 30, 2010 References [1] E. Rubiola,
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