RF/Microwave Oscillator Phase Noise Calculation

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RIGOROUS RF AND MICROWAVE OSCILLATOR PHASE NOISE CALCULATION BY ENVELOPE TRANSIENT TECHNIQUE

E. Ngoya, J. Rousset, and D. Argollo

IRCOM –UMR CNRS 6615– University of Limoges– 123, Av. A.Thomas, 87060 Limoges Cédex (France) [email protected]

derivation of oscillator (free running or forced) phase noise ABSTRACT analysis is done. A simulation of MESFET oscillator is A novel method is presented which allows accurate presented with a comparison against the conventional conversion calculation of RF and microwave oscillator phase noise. A matrix approach showing the effectiveness of the new method. thorough frequency domain amplitude and phase noise equation is derived for arbitrary circuit topology. The method COMPUTING RESPONSE OF CIRCUITS DRIVEN BY FM is based on envelope transient simulation technique. It also CARRIERS speeds up analysis of oscillator transients and circuits driven by chirp and frequency modulated signals. The commonly adopted form of RF and microwave circuit equation in frequency domain is: INTRODUCTION A(ω)X (ω) + B(ω)Y (ω) + D(ω)G(ω) = 0 Oscillator phase noise characterization is of prime importance in  −ωmax < ω < ωmax (1) RF and microwave communication systems design. If rigorous  and general purpose CAD techniques for noise calculation in  y(t) =ψ (x(t)) forced circuits like mixers and amplifiers are well established today [1-2], this is not the case for free running oscillators. It has where G(ω) , Y (ω) and X (ω) are respectively Fourier been observed by several authors that the classical conversion coefficient vectors of the driving sources g(t) , nonlinear matrix approach used to compute noise in forced circuits is not ψ sufficient for computing phase noise of free running oscillator sources y(t) and state variables of the circuit x(t) . (x(t)) is a because the system is ill-conditioned. Important works has been static characteristic defining the nonlinear sources. In the most done by authors in last years with the aim of resolving this general case x(t) contains all node voltages and non admittance problem, through both time domain integration (TDI) and element currents, plus their derivatives and time shifted versions: harmonic balance (HB) principles[3-6]. Time domain techniques di = dv z −τ −τ unfortunately have the inconvenient of being hardly applicable to x(t) [v(t),iz (t), , ,v(t v ), iz (t z )] . dt dt distributed element circuits and high Q oscillators. On the other For the following, let us consider that any electrical variable hand solutions carried in frequency domain (FD) around HB technique suffer from its inability to handle oscillator frequency w(t) can be decomposed as a sum of finite bandwidth modulation (FM) induced by noise perturbation. For this reason, modulations around harmonically related carriers ω k as below: in [5] for example, noise equilibrium equation has been separated N K = jωk t in two non related terms, namely conversion equation and w(t) ∑ Wk (t)e (2) modulation equation, which in reality are intimately linked. In [6] k=−N an alternate technique is presented which claims simultaneous BW / 2 K 1 K Ω computation of conversion and modulation noise effects, W (t) = ∫W (Ω)e j tdΩ k 2π unfortunately the equilibrium equation is likely to be singular as −BW / 2 the system matrix involves terms in 1/Vk , where Vk are circuit T ωk = K Λ node voltage harmonics of the noiseless steady state. Hence under the assumption that the transfer functions In this paper, a rigorous approach based on Envelope transient ω ω ω (ET) [7] analysis technique is presented. Because ET handles in A( ) , B( ) and D( ) can be closely approximated by a a natural way oscillator transients in the form of time varying first order polynomial around allω k , within the modulation phasors (amplitude, phase and frequency), oscillator phase noise bandwidth BW , we can derive a first order differential can be readily computed by considering noise signals just as any ↑ equation [7-8] from (1), which describes the dynamics of other modulation signal. We have a direct access to noise induced frequency jitter, amplitude and phase modulation in time time varying phasors (or modulations) of the various domain, from which the overall output noise spectrum is variables. computed by Fourier transform. In the following we are presenting ET equation for analyzing circuits driven by FM ↑ carrier signals and oscillator transients, which has not been yet The above assumption is always true for all lumped element considered in the literature. Based on this a straightforward circuits, using modified nodal formulation and is valid for practical modulation bandwidths as regards distributed elements

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D D > = − A(ω )X (t) + A dX (t)/ dt + (4) k H,..., H k k D k k D > Eq (6) is very suitable for all systems driven by chirp type B(ω )Y (t) + B dY (t) / dt + k k D k k D signals like radars and especially for oscillator transient ω + > = D( k )Gk (t) DkdGk (t) / dt 0 and phase noise simulation as it will be shown. D H D jω τ Y (t) = F [ψ ( ∑ X (t)e p c )]⊗δ ( f − f ) k τc p k OSCILLATOR TRANSIENTS SIMULATION p=−H

k = −H ,..., H C C C During transient states, oscillator frequency and In the above equation Ak ,Bk , Dk stand for derivatives of amplitude are moving continuously so that its dynamics is ω ω ω A( ) , B( ) , D( ) at frequency ω k and Fτc[] for Fourier well described by eq (6), except that the frequency δω transform along carrier time axis τ c modulation term 0 (t) is also an unknown of the system. Equation (4) can be solved using trapezoidal or An elegant way to adress this problem is to use an Gear integration rule from time origin to a desired time oscillator probe [9]. An oscillator probe is an active point or until the transient dies out. At this point it is voltmeter whose equivalent circuit is shown in Fig.1. important to notice that signals are sampled at the rate of i (t) 0, ω − ω ≤ BW / 2 p ω = 0 the time varying complex envelopes (amplitude and phase), Z p ( )  ∞, ω − ω > BW / 2 not the carrier waveform. This is why ET is superior to 0 oscillator e (t) conventional brute force TDI. Nevertheless in the case the v p (t) p modulation contains an important frequency modulation index, the resulting amplitude and phase can have a large rate as compared to modulation duration or period, and this still necessitates a large number of envelope samples. Fig.1 Oscillator probe

To resolve this problem, it is necessary to keep track The probe contains an ideal voltage source supplying a explicitly of any frequency modulation term present in the signal of arbitrary amplitude and time varying frequency driving signal. For simplicity of the notation, we will t K jω t +∫ δω (τ )dτ hereafter consider only the case of a single fundamental = ℜ 0 0 0 ω e p (t) e[Ep1(t)e ] in series with an ideal carrier 0 . The derivations can be easily extended to the filter which has zero impedance in a bandwidth BW multiple fundamental carrier case. around the reference frequency ω and infinite impedance Let us consider a driving signal g(t) with a time varying 0 outside. The bandwidth is large enough to pass all the carrier frequency term (frequency modulation) δω0 (t) and K energy spectrum delivered by e p (t) . The aim of the probe complex envelope G (t) : 1 is to supply the unknown time varying frequency term t δω ω + δω τ τ 0 (t) and to fix a phase reference to oscillator, setting j 0t ∫ 0 ( )d K instantaneous phase of probe stimulus to some arbitrary g(t) = ℜe[G (t)e 0 ] (4) 1 value. When connected to a circuit node, the probe must When applied to the nonlinear circuit all variables defined satisfy the constraint of zero load in order not to disturb the by eq (2) now take the form: circuit equilibrium, i.e no current flowing into probe: H ~ jkω0t Ip (t) =η ( Ep (t) , δω (t) )= 0 (7) w(t) = ∑ Wk (t)e (5) 1 p 1 0 k=−H Eq (7) introduces two supplementary equations (real and t K K jk∫ δω (τ )dτ ~ 0 0 imaginary of Ip1(t) ) with one more unknown E p1(t) to be where Wk (t) = Wk (t)e added to (6). Simultaneous solution ofK (6) and (7) gives Expressed in this form we Kcan sample separately the the time varying complex amplitude X k (t) of the state AM&PM modulation term W (t) and the FM term k variables along with the time varying frequency term δω 0 (t) instead of the total compound modulation term δω (t) of the oscillator. Because we are directly sampling ~ 0 Wk (t) which has a much higher rate. Reconsidering (4) frequency, TDI of equilibrium equation can be carried with with signal expression defined by (5) we readily find the even very coarse time steps. following equation. K K D A()kω + kδω (t) X (t) + A dX (t)/ dt + (6) OSCILLATOR PHASE NOISE ANALYSIS 0 0 k K k k K ()D It is today well admitted [10] that the basic random B kω0 + kδω0 (t) Yk (t) + BkdYk (t) / dt + K K fluctuations (noise sources) taking place in electronic ()ω + δω + D = D k 0 k 0(t) Gk (t) DkdGk (t) / dt 0 devices can be well described by a summation of distinct

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sinusoids (pseudo sinusoid) with random amplitude and δ + dXk + δω θ Ak Xk (t) Ak jkXk0 Ak 0 (t) + (12) phase ( Ni , i ) dt = jθi jΩit = jΩit ∂Y ∂Y dX n(t) ∑ Ni e e ∑ Nie (8) B ∑ k0 δX (t) + B ∑ k0 k + jkB Y δω (t) k ∂ k k ∂ k k0 0 i i p Xp0 p Xp0 dt Since energy level of noise can be assumed sufficiently dNsk (t) small, superposition theorem holds and response of the = Fk Nsk (t) + Fk circuit as a result of noise perturbation can be computed on dt a frequency by frequency basis. However in a nonlinear Eq (12) is a linear equation that can be efficiently solved in circuit pumped by a quasi-periodic drive at a set of frequency domain. In fact since we are considering a {} sinusoidal noise perturbation, the noise response and frequency ωk , k = −H,..., H , there will be an exchange of frequency jitter are also sinusoidal so that we may write: energy between noise sinusoids at all frequencies of the set K + Ω − − Ω {} δX (t) = ∆X e j st + ∆X e j st (13) Θs = ωk ± Ωs ,k = −H,..., H Consequently it is k k k indispensable to carry analysis simultaneously for all δω = ℜ ∆ω jΩst 0 (t) e[ 0e ] sinusoids at frequencies defined by the set Θ . The noise s Considering (13) into (12) and taking account of (7), we Ω frequency sample s is called noise sideband frequency readily find oscillator frequency domain noise equilibrium Considering that oscillator circuit described by eq equation, where we can see that conversion and modulation (6)-(7) is perturbed by a spot noise sample effects are intimately linked in a single equation of the K H ω form. n (t) = ∑ N (t)e jk 0t (9) s sk C(Ω ) M ∆  F  =− s X k H  t   =  N (14) K Ω − Ω  up 0 ∆ω  0  = + j st + − j st   0  N sk (t) N e N k e k where we need to add (9) as a new driving source into eq (6)-(7). ∆ = ∆ + ∆ + T ∆ = ∆ + ∆ + T For this purpose it is necessary to accommodate (9) in the X [ X −H ,..., X H ] , N [ N−H ,..., N H ] , form of a FM signal by writing: C(Ωs ) is the conversion matrix, M is the frequency K K t jk∫ ∆ω0 (τ )dτ modulation gradient and up is the extraction vector of the N (t) = N (t)e 0 (10) sk k oscillator probe current, stating the oscillator arbitrary K K − t ∆ω τ τ jk∫ 0 ( )d phase reference condition ∆Vp1 =Vp−1 = ∆Ep1 . N (t) = N (t)e 0 k sk Considering the signal definitions (5) and (13) and Eq (6) may then be rewritten as below to account for the assumption that frequency jitter due to noise is small contribution of n (t) Ω s compared to s the total noise complex amplitude at A()kω + kδω (t) X (t) + A dX (t) / dt + (11) Ω 0 0 k k k positive sideband frequency s , around harmonic k is ↑ B()kω + kδω (t) Y (t) + B dY (t) / dt + found to be: 0 0 k k k ()ω + δω + = ~ + kX + D k 0 k 0 (t) Gk (t) DkdGk (t) / dt ∆X = k0 ∆ω + ∆X (15) k Ω 0 () 2 s k F kω0 + kδω0 (t) Nk (t) + FkdNk (t) / dt From (14) and (15) the complete output noise correlation Suppose that the oscillator has no other RF drive except is fully determined by inspection technique. noise, and that we have chosen reference frequency ω0 to ∆ ~ + ∆ ~ + * be exactly the steady state oscillation frequency. In this Given the three correlation terms Xk ( X k ) , case when we reach the steady state, δω0 (t) is identically ~ + ~ + * ~ + ~ + * ∆X− (∆X− ) and ∆X− (∆X ) oscillator PM noise at equal to the frequency jitter induced by noise and the state k k k k K K K K K some node voltage is computed by the traditional formula. variable X (t) writes X (t) = X +δX (t) where δX (t) k k k0 k k   K ~+ 2 ~+ 2  ~+ ~+ * j2θ  ∆ + ∆ − ℜ ∆ ()∆ Vk0  is the noise response and Xk0 the noiseless steady state. Vk V-k 2 e V-k Vk e   At this stage eq (11) can be solved in a direct TDI δφ 2 (Ω ) = s 2 k V approachK as outlined before. Alternatively, since k0 δ Xk (t) and δω0 (t) are small, it is numerically more efficient to derive a noise perturbation equation from (11) by a linear expansion around the noiseless equilibrium.

Doing so and eliminating the steady state terms, we get ↑ oscillator noise envelope transient equation Otherwise total noise spectrum needs to be computed by DFT of overall modulation term in eq (5). This occurs as Ωs tends to zero

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APPLICATION CONCLUSION We have used the above outlined method to compute A new method for simulation of systems driven by FM the output frequency and power of the MESFET oscillator signals and phase noise analysis of arbitrary topology described in reference [9], when it is forced by a chirp oscillators has been presented. It is shown that frequency pulse starting at 100ns and ending at 900ns. Fig.2 shows conversion and modulation effects taking place in a free the time varying input and output frequencies. We may see running oscillator as a result of noise perturbation are that the oscillator gets locked from 400ns to 560ns, that is intimately linked within a single equation. A general for input frequency in the range 3 to 3.1GHz. purpose and self contained oscillator phase noise equation is derived in frequency domain. 3.35 Oscillator Output Freq Chirp input Freq 3.25 REFERENCES [1] R. Q. Twiss, “Nyquist’s and Thevenin’s theorems generalized for 3.15 nonreciprocal linear networks”, J. Appl. Phys., vol. 26, pp. 599- 602, May 1955. 3.05 [2] V. Rizzoli, F. Mastri and C. Cecchetti, "Computer aided noise

Freq (GHz) 2.95 analysis of MESFET and HEMT mixers", IEEE trans. Microwave Theory Tech , vol. 37, No.9, pp. 1401-1410, Sept.89 2.85 [3] F. X. Kaertner, "Determination of the correlation spectrum of oscillators with low noise"., IEEE trans. Microwave Theory Tech , 2.75 vol. 37, No.1, pp. 90-101, Jan. 1989 0 200 400 600 800 1000 Time (ns) [4] V. Rizzoli, F. Mastri and D. Masotti, "General noise analysis of nonlinear microwave circuits by the piecewise harmonic balance technique", IEEE trans. Microwave Theory Tech , vol. 42, No.5, Fig.2 Time varying oscillator input and output frequency pp. 807-819, May 1994 [5] W. Anzill, Marion Fillebock and P. Russer, "Low phase noise design of microwave oscillators", Int. J. Microwave and Millimeter 0 Wave CAE , vol. 6, No.1, pp. 5-25, Jan. 1996 Envelope transient method Conversion matrix method [6] F. Farzaneh, E. Mehrshahi, "A novel noise analysis method in -30 microwave oscillators based on harmonic balance", Proceedings of the 28th European Microwave Conference, pp. 255-260, 1998 -60 [7] E. Ngoya and R. Larchevèque, “Envelope Transient Analysis: A -90 New Method for the Transient And Steady State Analysis of Microwave Communication Circuits and Systems”, IEEE MTT-S -120 Digest, pp. 1365-1368, 1996 PM Noise (dBc/Hz) Noise PM -150 [8] J. Roychowdhury, “Efficient Methods for Simulating Highly Nonlinear Multi-rate Circuits”, Proc. DAC, 1997 -180 [9] E. Ngoya, A. Suarez, R. Sommet, and R. Quéré, "Steady state 0.1 1 10 100 1000 10000 100000 analysis of free or forced oscillators by harmonic balance and Noise sideband frequency (KHz) stability investigation of periodic and quasi-periodic regimes", Int. J. Microwave and Millimeter Wave CAE , vol. 5, No.3, pp. 210- 223, March. 1995 Fig.3 Oscillator phase noise: comparisonof new method [10] P. Penfield, , "Circuit theory of periodically driven nonlinear and conversion matrix approach systems", Proceedings of the IEEE , vol. 54, No.2, pp. 266-280, Feb. 1966

Fig.3 shows the simulated phase noise spectrum of the oscillator using the new method and the conventional conversion matrix approach. We can observe important differences between the two simulations enforcing the observation that plain frequency conversion approach can be largely inaccurate for oscillator phase noise analysis. For this simulation MESFET noise was modeled by 1/f flicker current noise, along with channel shot noise. The new method predicts very well the practically observed 30dB/decade slope of phase noise figure near the carrier and naturally coincides to the conversion matrix figure as noise sideband frequency is large.

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