Amplitude Noise of Electrical Oscillators Based on the LTV Model

In this paper, using the Linear Time Variant (LTV) noise model of oscillators, an analytical expression for Amplitude Impulse Sensitivity Function (A-ISF) based on the limit cycle of the oscillator is derived for the first time. Also Considering the existing expression for Phase Impulse Sensitivity Function (P-ISF), it is shown that P-ISF and A-ISF are correlated. The derived A-ISF is used to simulate the amplitude noise of the oscillators in the 1/f 2 region of the noise spectrum by the LTV model. Simulation and experimental results show the validity of the derived formula. DOI: http://dx.doi.org/10.5755/j01.eee.18.8.2607


I. INTRODUCTION
For many years, phase and amplitude noises of electrical oscillators have been the subject of study, and many models have been introduced.The Linear Time Variant Model (LTV) [1], [2] is a formulated and simple model that offers closed form formulas for phase and amplitude noises which include the effect of circuit elements in the output noises.In the LTV model, Phase and Amplitude Impulse Sensitivity Functions (P-ISF and A-ISF respectively) are defined which relate circuit noise sources to the output phase and amplitude noises respectively.To obtain P-ISF and A-ISF, many simulations of the oscillator circuit at certain points and with certain parameters are needed [1], [2] which makes the method difficult to be performed.Some closed form expressions for P-ISF have been presented in [1], [3]- [5] which enable us to calculate the P-ISF only by one simulation.In [1], this analytical expression has been extracted from the limit cycle of the oscillator.In [3]- [5], based on the structure of the LC oscillators, some analytical expressions have been presented for P-ISF.However, the method in [1] is more general since it has been extracted from the limit cycle and therefore does not rely on the topology of the oscillator.While the phase noise has been calculated by the LTV model, there is no closed formula for A-ISF and no amplitude noise has been calculated by this model [6].
Recently, numerical models for the phase noise have been introduced which solve nonlinear differential equations of the oscillator circuit numerically by Harmonic Balance (HB) technique and calculate phase and amplitude noises and Manuscript received January 20, 2012; accepted March 18, 2012.
their correlation [7]- [11].While these methods are accurate, some special and complicated computations are needed to evaluate the phase and amplitude noises which have not always been included in the commercial software.
In this paper, the amplitude and phase noises of electrical oscillators using the LTV model are investigated.For the first time, an analytical formula is derived for A-ISF based on the output limit cycle of the oscillator.So the calculation of this function can be easily performed only by one simulation.Since the extractions of P-ISF and A-ISF are derived from the limit cycle, they have the same origin and are correlated which is discussed in this paper.The behavior of the derived formula is investigated and the amplitude noise of electrical oscillators is calculated.Simulation and experimental results show the validity of the derived expressions and the evaluated amplitude and phase noises.

II. BRIEF REVIEW OF THE LTV MODEL
In a practical oscillator the output signal is [1]  , where ) (t

 and )
(t A are instantaneous phase and amplitude of the output respectively.Consider an ideal and simplified model of a parallel LC oscillator shown in Fig. 1.Based on the LTV model, if a current impulse is injected at time  to the system, the instantaneous voltage change in the capacitor at the node i is [1] , , where i q  is the total injected charge corresponding to the current impulse source and i C is the total equivalent capacitance of the node i .For small injected charges, the phase change is is the phase impulse sensitivity function (P-ISF) which is a dimensionless, frequency-and amplitudeindependent periodic function and describes the phase deviations for a unit impulse input at time  .This phase change will be transformed into the output signal by the oscillator circuit and makes the phase noise.
In the 1/f 2 region of the phase noise spectrum, the resultant phase noise based on the LTV model is [1] , 2 where is the RMS value of P-ISF, max q is the maximum charge on the node capacitor, f i noise  2 is the white noise source spectral density in the oscillator circuit and   is the offset frequency from the carrier.The conventional method to calculate P-ISF is that an impulse current source is injected to the oscillator circuit at different phases of the oscillation period.Then the oscillator is simulated for a few cycles to reach the stable condition.This impulse causes a time shift T  in the output waveform.The excess phase for this time shift is where T is the period of oscillation.By sweeping the time in which the impulse is injected, different Similar to the phase deviation, for small injected charges, the amplitude change is [2] where is the amplitude impulse sensitivity function (A-ISF) which is a periodic function and describes the amplitude deviations for a unit impulse input at time .This amplitude change will be transformed into the output signal by oscillator circuit and makes the amplitude noise.Similar method to that of P-ISF is employed to calculate A-ISF [2].
In the 1/f 2 region of the spectrum, the resultant amplitude noise is [ where is the RMS value of A-ISF, 0  is the angular oscillation frequency and Q is the oscillator quality factor.

III. ANALYTICAL EXPRESSION FOR P-ISF
In [1], an analytical formula for P-ISF has been derived based on the output limit cycle.The extraction is brought here for simplicity.
For a stable oscillator, the limit cycle is defined as a closed trajectory in n-dimensional state space, where the state vector X traverses it once in every period of the oscillation, as shown in Fig. 2. Suppose that a perturbation vector X  , caused by circuit noise sources, changes the state of the system from X to X X   .This change in the state vector causes an equivalent displacement along the trajectory of the limit cycle.This displacement in turn makes a time shift which is related to the final phase shift.The unit tangential vector at the point of perturbation is .

|| ||
The equivalent phase shift due to the perturbation vector is where X  is the first derivative of the state vector with respect to time and  denotes the norm function.If the state variables are node voltages and an impulse is applied to the node i , there will be a change in i V  given by (2).So (8) reduces to Considering (9) with the normalized waveform function where and f  is the derivative of the normalized waveform with respect to time on node i .Comparing (10)  with (3) results in the following P-ISF In the special case of the second order systems (which is a usual condition in practical oscillator circuits), (11) leads to the following expression for P-ISF [1] , ) ( IV. ANALYTICAL EXPRESSION FOR A-ISF Here, we use the limit cycle of Fig. 2 and derive an analytical formula for the amplitude impulse sensitivity function (A-ISF).Since this formula is extracted from the limit cycle, any nonlinear behavior of the oscillator circuit will be included in A-ISF.Also in contrast to the conventional method which needs many simulations, using the formula only needs one simulation over one cycle of the oscillation to calculate A-ISF.For the normal vector, at the point of the perturbation we can write where i X   is equal to the amplitude deviation due to perturbations, i.e.
i A  at the node i .Substituting (7) in (13) If we consider node voltages as the state variables, for node i , we have Normalizing (1) to max , i V and considering (15) gives Comparing ( 16) with ( 5), we get For second order systems, (17 which represents the analytical expression for A-ISF.Now, we show that P-ISF and A-ISF are correlated. Consider Fig. 2. The components || X  and  X of the perturbation vector X  have the same origin and are correlated.Extraction of (12) shows that P-ISF corresponds to || X  .Also extraction of (18) shows that A-ISF corresponds to  X .Therefore P-ISF and A-ISF is correlated.Using ( 12) and ( 18), the relationship between P-ISF and A-ISF will be .
To investigate the amplitude noise, two oscillator structures are considered.First, a Colpitts oscillator topology presented in [1] is proposed as in Fig. 3 with the given parameters for the transistor BFR520.With R=10kΩ, L=200nH, C1=40pF and C2=200pF, the oscillation frequency is 62MHz.For this oscillator, the output current, the output ac voltage (with dc value omitted) versus Radian and the A-ISF calculated by (18) are shown in Fig. 4a, Fig. 4b and Fig. 4c respectively.To investigate the behavior of the calculated A-ISF, consider Fig. 1a.If an impulse is injected at the peak of the output signal, it makes the maximum amplitude deviation.If an impulse is injected at the zero crossing of the output, no amplitude deviation occurs as in Fig. 1b.
Considering Fig. 4b and Fig. 4c, applying an impulse at the positive or negative peak of the output voltage of the Colpitts oscillator (point A in Fig. 4b) causes maximum amplitude deviation at the output and consequently, at this point A-ISF must have maximum magnitude which is point A in Fig. 4c.Also, applying an impulse at zero crossing point (point B in Fig. 4b) causes no amplitude deviation at the output and consequently A-ISF must be zero at this point which is point B in Fig. 4c.This discussion shows that the behavior of A-ISF calculated by (18) is correct.Similar to discussion for A-ISF behavior done for the Colpitts oscillator, applying an impulse at the positive or negative peak of the output voltage of the ring oscillator (point A in Fig. 6b) causes a maximum amplitude deviation at the output and consequently A-ISF at this point must have maximum magnitude which is point A in Fig. 6c).
Also applying an impulse at the zero crossing point B in Fig. 4b causes no amplitude deviation at the output and therefore A-ISF is zero at point B in Fig. 4c).This discussion shows that the behavior of A-ISF calculated by ( 18) is valid for this kind of oscillators.The transistor current of the Colpitts oscillator in Fig. 4a has a pulse shape and its magnitude is nearly zero during a part of the period.It modulates any stationary noise sources in the oscillator circuit (thermal noise of the resistor and shot noise of the transistor) and makes cyclostationary noise sources.In this case, new effective A-ISF will be introduced as [1]      , where   is a periodic function that describes the noise source modulation and can be calculated from the transistor current and is normalized to one [1].The same situation exists for the current of the ring oscillator as in Fig. 6a.Now we calculate the amplitude noise of the Colpitts oscillator.In the 1/f 2 region of the output noise spectrum, the total white noise power spectral density of the oscillator circuit is the shot noise of the transistor and the thermal noise of the resistor [1] , where e q is the unit electron charge, T is the absolute temperature, k is the Boltzmann constant and C I is the transistor dc current.Also Q can be approximated by

C R L
for the LC oscillators where the variables are the resonator components.With R=10 kΩ, L=200nH, C1=40pF and C2=200pF, I C =8mA and C total =33.33pF.The maximum voltage swing is 8.14V.Therefore q max =271.47 pC.Also Q=129, Λ eff,RMS =0.22, and the noise power spectral density is 2.56×10 -21 A 2 /Hz.Fig. 9 shows the amplitude noises calculated by (6) and by the Harmonic Balance (HB) method.Very good agreement can be inferred from the figure which can validate the derived expression of A-ISF.
Table I shows the amplitude noise for a Colpitts oscillator measured in [14] and calculated by (6).Good agreement for the results can be achieved from the table.The differences are for the colored noise sources.Now we calculate the amplitude noise of the ring oscillator.For this oscillator, the quality factor is [12] where N is the numbers of stages and dt dV / is the derivative of the output voltage of the ring oscillator with respect to the time and 0  is the oscillation frequency.In the 1/f 2 region of the output noise spectrum, the total white noise power spectral density for the ring oscillator circuit is the channel thermal noise which is [13] , 4 where T is the absolute temperature, k is the Boltzmann constant,  is the thermal noise coefficient and is typically 2/3 for the MOSFETs.m g is the transconductance of the transistor.m g is measured at the middle of the output transition in which the output voltage changes from zero to maximum.
For the 19-stage ring oscillator, C total =1.256pF and the maximum voltage swing is 1.8V.Therefore q max =2.261pC.Also Q=8.1964, Λ eff,RMS =0.0304.For PMOS m g is 28.1 mS and for NMOS is 62.4 mS.So the noise power spectral density is 1.01×10 -21 A 2 /Hz.Fig. 10 shows amplitude noises calculated by (9) and the HB method.Very good agreement can be inferred from the figure which can validate the derived expression of A-ISF.

Fig. 1 .
Fig. 1.Time domain response of an oscillator (a) impulse is injected at the peak (b) impulse is injected at the zero crossing point.

According to Fig 2 ,
the perturbation vector X  can be decomposed into its components: the tangential vector to the trajectory || X  and the normal vector to the trajectory  X .

Fig. 5 .
Fig. 5. Ring oscillator.As another example a 19-stage ring oscillator (in TSMC 0.18μm technology) with oscillation frequency of 943 MHz is considered as in Fig. 5.The output current, the output ac voltage (with dc value omitted) versus Radian and the A-ISF calculated by (18) are shown in Fig. 6a, Fig. 6b and Fig. 6c respectively.Similar to discussion for A-ISF behavior done for the Colpitts oscillator, applying an impulse at the positive or negative peak of the output voltage of the ring oscillator (point A in Fig.6b) causes a maximum amplitude deviation at the output and consequently A-ISF at this point must have maximum magnitude which is point A in Fig.6c).Also applying an impulse at the zero crossing point B in Fig.4bcauses no amplitude deviation at the output and therefore A-ISF is zero at point B in Fig.4c).This discussion shows that the behavior of A-ISF calculated by (18) is valid for this kind of oscillators.