Chapter 2 Design Basics of CMOS VCO
2.10 Phase Noise
Noise injected into an oscillator by its constituent devices or by external means may influence both the frequency and the amplitude of the output signal. In most cases, the disturbance in the amplitude is negligible or unimportant, and only the random deviation of the frequency is considered.
For a nominally periodic sinusoidal signal, we can write x(t)=Acos[fct+φn(t)], where φn(t) is a small random excess phase representing variations in the period. The function φn(t) is called “phase noise”. Note that for∣φn(t)∣<<1 rad, we have x(t)≒Acosfct-Aφn(t) sinfct; that is, the spectrum of φn(t) is translated to ±fc.
In RF applications, phase noise is usually characterized in the frequency domain. For an ideal sinusoidal oscillator operating at fc, the spectrum assumes the shape of an impulse, whereas for an actual oscillator, the spectrum exhibits “skits” around the carrier frequency (Fig. 2.10). The frequency fluctuations correspond to jitter in the time domain, which is a random perturbation of zero crossings of a periodic signal (Fig. 2.11).
Figure 2.10 Frequency spectrum of ideal and real oscillators
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Figure 2.11 Jitter in the time domain relates to phase noise in the frequency domain
Frequency fluctuations are usually characterized by the single sideband noise spectral density normalized to the carrier signal power (Fig. 2.10). It is defined as
( ,1
( ,c ) 10 log sideband c
carrier
P f f Hz
L f f
P
⎡ + Δ )⎤
Δ = ⎢ ⎥
⎣ ⎦ (11)
and has units of decibels below the carrier per hertz (dBc/Hz). Pcarrier is the carrier signal power at the carrier frequency fC and Psideband(fc+Δf, 1 Hz) denotes the single sideband power at the offset Δf from the carrier fC at a measurement bandwidth of 1 Hz.
Figure 2.12 Oscillator output power spectrum
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The typical oscillator output power spectrum is shown in Fig. 2.12. The noise distribution on each side of the oscillator signal is subdivided into a larger number of strips of width Δf located at the distance fm away from the single. It should be noted that, generally, the spectrum of the output single consists of the phase noise components. Hence, to measure the phase noise close to the carrier frequency, one needs to make sure that any contributions of parasitic amplitude modulation to the oscillator output noise spectrum are negligible compared with those from frequency modulation. The single sideband phase noise L(fm) usually given logarithmically is defined as the ratio of signal power PssΔf in one phase modulation sideband per bandwidth Δf=1 Hz, at an offset fm away from the carrier, to the total signal power Ps.
Time invariant model
In this section, phase noise analysis is described by using time invariant model. Time invariant means whenever noise sources injection, the phase noise in VCO is the same. In the other words, phase shift of VCO caused by noise is the same in any time. Therefore, it’s no need to consider when the noise is coming. Suppose oscillator is consists of amplifier and resonator. The transfer function of a band-pass resonator is written as
( )
2The transfer function of a common band-pass is written as
( )
Compare equation (12) with (13). Thus,
0
1
ω = LC and Q=ω0RC (14)
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The frequency ω ω= 0+ Δω which is near oscillator output frequency. If ω0 Δω, we can use Taylor expansion for only first and second terms. Hence
( )
The close-loop response of oscillator is expressed by
( ) ( )
The above equation is double sideband noise. The phase noise faraway center frequency ω
Where F is empirical parameter (“often called the device excess noise number”), k is Boltzman’ s constant, T is the absolute temperature, PS is the average power dissipated in the resistive part of the tank, ω0 is the oscillation frequency, and Q is the effective quality factor of the tank with all the loading in place(also known as loaded Q). From equation (18), increasing power consumption and higher Q factor can get better phase noise. Increasing power consumption means increasing the power of amplifier. This method will decrease noise figure (NF) and improve phase noise.
From, equation (18), we can briefly understand phase noise. But the equation and actual measured results are different. The VCO spectrum is shown as Fig. 2.12. The phase noise equation can be modified as the same as equation (31) that is called Lesson’s model [5].
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These modifications, due to Leeson, consist of a factor to account for the increased noise in the 1/(Δω)2 region, an additive factor of unity (inside the braces) to account for the noise floor, and a multiplicative factor (the term in the second set of parentheses) to provide a 1/ Δω3 behavior at sufficiently small offset frequencies. With these modifications, the phase-noise spectrum appears as in Fig. 2.13.
It is important to note that the factor F is an empirical fitting parameter and therefore must be determined from measurements, diminishing the predictive power of the phase-noise equation. Furthermore, the model asserts that Δω1/ f3, the boundary between the 1/(Δω)2
and 1/ Δω3 regions, is precisely equal to the 1/f corner of device noise. However, measurements frequently show no such equality, and thus one must generally treat Δω1/ f3 as an empirical fitting parameter as well. Also, it is not clear what the corner frequency will be in the presence of more than one noise source with 1/f noise contribution. Last, the frequency at which the noise flattens out is not always equal to half the resonator bandwidth, ω0/ 2Q. Both the ideal oscillator model and the Leeson model suggest that increasing resonator Q and signal amplitude are ways to reduce phase noise. The Leeson model additionally introduces the factor F, but without knowing precisely what it depends on, it is difficult to identify specific ways to reduce it. The same problem exists with Δω1/ f3 as well. Last, blind application of these models has periodically led to earnest but misguided attempts to use active circuits to boost Q. Sadly, increases in Q through such means are necessarily accompanied by increases in F as well, preventing the anticipated improvements in phase noise. Again, the lack of analytical expressions for F can obscure this conclusion, and one
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continues to encounter various doomed oscillator designs based on the notion of active Q boosting.
Figure 2.13 Phase noise: Leeson versus (18).
Time Variant
In the general case, multiple noise sources affect the phase and amplitude of an oscillator.
This chapter begins by investigating the effect of a single noise source on the amplitude and phase of the oscillator.
Figure 2.14 Equivalent systems for phase and amplitude
Since each input source generally affects both amplitude and phase, a pair of equivalent systems, one each for amplitude and phase, can be defined. Each system can be viewed as a single-input, single-output system as shown in Fig. 2.14. The input of each system in Fig.
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2.14 is a perturbation current (or voltage) and the outputs are the excess phase, φ(t), and amplitude, A(t). Both systems shown in Fig. 2.14 are time-variant as shown by the following examples.
The first example is an ideal parallel LC tank oscillating with voltage amplitude, as shown in Fig. 2.15. If one injects an impulse of current at the voltage maximum, only the voltage across the capacitor changes; there is no effect on the current through the inductor. Therefore, the tank voltage changes instantaneously, as shown in Fig. 2.15. Assuming a voltage- and time-invariant capacitor, the instantaneous voltage change ΔV is given by
total
V q C
Δ = Δ (19)
where Δq is the total charge injected by the current impulse and Ctotal is the total capacitance in parallel with the current source. It can be seen from Fig. 2.15 that the resultant change in A(t) and φ(t) is time dependent. In particular, if the impulse is applied at the peak of the voltage across the capacitor, there will be no phase shift and only an amplitude change will result, as shown in Fig. 2.15(a). On the other hand, if this impulse is applied at the zero crossing, it has the maximum effect on the excess phase, φ(t), and the minimum effect on the amplitude, as depicted in Fig. 2.15(b).
Figure 2.15 Impulse response of an ideal LC oscillator
To emphasize the generality of this time-variance, consider two more examples. The
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relaxation oscillator known as the Bose oscillator is shown in Fig. 2.16. It consists of a Schmitt-trigger inverter and an RC circuit. The hysteresis in the transfer function of the inverter and the RC time constant determine the frequency of oscillation. The resulting capacitor voltage waveform is shown with a solid line in Fig. 2.17.
As before, imagine an impulsive current source in parallel with the capacitor, injecting charge at t=τ, as shown in Fig. 2.16. All of the injected charge goes into the capacitor and changes the voltage across it instantaneously. This voltage change, ΔV, results in a phase shift, Δφ, as shown in Fig. 2.17. As can be seen from Fig. 2.17, for a small area of the current impulse (injected charge), the resultant phase shift is proportional to the voltage change, ΔV, and hence to the injected charge, Δq. Therefore, Δφ can be written as
0 0
max max
( ) V ( ) q
V q
φ ω τ Δ ω τ Δ
Δ = Γ = Γ Δ q qmax (20)
Figure 2.16 Bose oscillator with parallel perturbation current source
Figure 2.17 The waveform of the Bose oscillator shown in Figure 3
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where Vmax is the voltage swing across the capacitor and qmax=CnodeVmax is the maximum charge swing. The function, Γ(x) is the time-varying “proportionality factor”. It is called the impulse sensitivity function (ISF), since it determines the sensitivity of the oscillator to an impulsive input. It is a dimensionless, frequency- and amplitude-independent function periodic in 2π that describes how much phase shift results from applying a unit impulse at any point in time.
In any event, to develop a feel for typical shapes of ISF’s, consider two representative examples, first for an LC and a ring oscillator in Fig. 2.18(a) and (b).
Figure 2.18 Example ISF for (a) LC oscillator and (b) ring oscillator.
It is critical to note that the current-to-phase transfer function is linear for small injected charge, even though the active elements may have strongly nonlinear voltage current behavior.
It should also be noted that the linearity and time-variance of a system depends on both the characteristics of the system and its input and output variables. The linearization of the current-to-phase system of Fig. 2.14 does not imply linearization of the nonlinearity of the voltage-current characteristics of the active devices. In fact, this nonlinearity affects the shape of the ISF and therefore has an important influence on phase noise, as will be seen shortly.
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Noting that the introduced phase shift persists indefinitely, the unity phase impulse response can be easily calculated from (20) to be
0
Thanks to linearity, the output excess phase, φ(t), can be calculated for small charge injections using the superposition integral
0
where i(t) represents the input noise current injected into the node of interest. Equation (22) is one of the most important results of this section and will be referred to frequently.
The output voltage, V(t), is related to the phase, φ(t), through a phase modulation process.
Thus the complete process by which a noise input becomes an output perturbation in V(t) can be summarized in the block diagram of Fig. 2.19. The essential features of the block diagram of Fig. 2.19 are a modulation by a periodic function, an ideal integration and a nonlinear phase modulation. The complete process thus can be viewed as a cascade of an LTV system that converts current (or voltage) to phase, with a nonlinear system that converts phase to voltage.
Figure 2.19 The equivalent block diagram of the process.
Since the ISF is periodic, it can be expanded in a Fourier series
0 0 0
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where the coefficients cn are real-valued, and θn is the phase of the nth harmonic. As will be seen later, θn is not important for random input noise and is thus neglected here. Using the expansion in (23) for Γ(ω0τ) in the superposition integral and exchanging the order of summation and integration, the following is obtained:
0
Equation (24) identifies individual contributions to the total φ(t) for an arbitrary input current i(t) injected into any circuit node, in terms of the various Fourier coefficients of the ISF. The decomposition implicit in (24) can be better understood with the equivalent block diagram shown in Fig. 2.20.
Each branch of the equivalent system in Fig. 2.20 acts as a bandpass filter and a downconverter in the vicinity of an integer multiple of the oscillation frequency. For example, the second branch weights the input by c1, multiplies it with a tone at ω0 and integrates the product. Hence, it passes the frequency components around ω0 and downconverts the output to the baseband. As can be seen, components of perturbations in the vicinity of integer multiples of the oscillation frequency play the most important role in determining φ(t).
Figure 2.20 The equivalent system for ISF decomposition
To investigate the effect of low frequency perturbations on the oscillator phase, a low frequency sinusoidal perturbation current, i(t), is injected into the oscillator at a frequency of Δω ω0 :
24 ( ) 0cos( )
i t =I Δωt (25)
Figure 2.21 Conversion of a low frequency sinusoidal current to phase
where I0 is the amplitude of i(t). The arguments of all the integrals in (24) are at frequencies higher than Δω and are significantly attenuated by the averaging nature of the integration, except the term arising from the first integral, which involves c0. Therefore, the only significant term in φ(t) will be
0 0 0 0
As a result, there will be two impulses at ±Δω in the power spectral density of φ(t), denoted as Sφ(ω) as shown in Fig. 2.21.
As another important special case, consider a current at a frequency close to the oscillation frequency given by
1 0
( ) cos ( )
i t =I ⎡⎣ω + Δωt⎤⎦ (27)
A process similar to that of the previous case occurs except that the spectrum of i(t) consists of two impulses at ±(ω0+Δω) as shown in Fig. 2.22. This time the dominant term in (24) will be the second integral corresponding to n=l. Therefore, φ(t) is given
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Figure 2.22 Conversion of a tone in the vicinity of ω0
by integer multiple of the oscillation frequency will result in two equal sidebands at ±Δω in Sφ(ω). Hence, in the general case φ(t) is given by
Unfortunately, we are not quite done: (29) allows us to figure out the spectrum of φ(t), but we ultimately want to find the spectrum of the output voltage of the oscillator, which is not quite the same thing. The two quantities are linked through the actual output waveform, however. To illustrate what we mean by this linkage, consider a specific case where the output may be approximated as a sinusoid, so that vout( ) cos[t = ω0t+φ( )]t . This equation may be considered a phase-to-voltage converter; it takes phase as an input, and produces from it the output voltage. This conversion is fundamentally nonlinear because it involves the phase
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modulation of a sinusoid.
Performing this phase-to-voltage conversion, and assuming “small” amplitude disturbances, we find that the single-tone injection leading to (29) results in two equal-power sidebands symmetrically disposed about the carrier
2
The foregoing result may be extended to the general case of a white noise source
(30)
Equation (30) implies both upward and downward frequency translations of noise into the noise nea
, weighted by coefficient , so 1/f device noise ultimately becom
r the carrier, as illustrated in Fig. 2.23. This figure summarizes what the foregoing equations tell us: components of noise near integer multiples of the carrier frequency all fold into noise near the carrier itself.
Noise near dc gets upconverted
es 1/f3 noise near the carrier; noise near the carrier stays there, weighted by ; and white noise near higher integer multiples of the carrier undergoes downconversion, turning into noise in the 1/f2 region. Note that the 1/f2 shape results from the integration implied by the step change in phase caused by an impulsive noise input. Since an integration (even a time-varying one) gives a white voltage or current spectrum a 1/f character, the power spectral density will have a 1/f2 shape.
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Figure 2.23 Evolution of circuit noise into phase noise.
It is clear from Fig. 2.23 that minimizing the various coefficients cn (by minimizing the ISF) will minimize the phase noise. To underscore this point quantitatively, we may use Parseval’s theorem to write
2 2
where is the rms value of the ISF. All other factors held equal, reducing will reduce the phase noise at all frequencies. Equation (33) is the rigorous equation for the 1/f2 region and is one key result of the LTV model. Note that no empirical curve-fitting parameters are present in (33).
Γrms Γrms
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Calculation of the 1/f3 Noise Corner
Many active and passive devices exhibit low frequency noise with a power spectrum that is approximately inversely proportional to the frequency. It is for this reason that noise sources with this behavior are referred to as 1/f noise.
Noting that device noise in the 1/f region can be described by
2 2 1/
where ω1/f is the corner frequency of device 1/f noise, (30) and (34) result in the following expression for phase noise in the 1/f 3 portion of the phase noise spectrum:
2 2
which describes the phase noise in the 1/f 3 region. The 1/f 3 corner frequency is then
3
it will generally be lower. In fact, sinceΓdcis the dc value of the ISF, there is a possibility of
the control of the designer, usually through adjust
reducing by large factors the 1/f 3 phase-noise corner.
The ISF is a function of the waveform, and hence potentially under
ment of the rise- and fall-time symmetry. This result is not anticipated by LTI approaches, and is one of the most powerful insights conferred by this LTV model. This result has particular significance for technologies with notoriously poor 1/f noise performance, such as CMOS and GaAs MESFET’s.
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Figure 2.24 (a) Waveform and ISF for the symmetric waveform (b) the asymmetric waveform
To u in Fig
2.2
symmetric rising and falling edge is much larger than that in t
nderstand what affects c0, consider two ring oscillators, with waveforms shown
4. The first waveform has symmetric rising and falling edges, i.e., its rise time is the same as its fall-time. Assuming a time-invariant node capacitor1, the sensitivity of this oscillator to a perturbation during the rising edge is the same as its sensitivity during the falling edge, except for a sign. Therefore, the ISF has a small dc value. The second case corresponds to an asymmetric waveform with slow rising edge and a fast falling edge. In this case, the phase is more sensitive during the rising edge, and is also is sensitive for a longer time; therefore, the positive lobe of the ISF will be taller and wider as opposed to its negative lobe which is short and thinner, as shown in Fig. 2.24.
The dc value of the ISF for the a
he symmetric case, and hence a low frequency noise source injecting into it shows a stronger upconversion of low frequency noise. A limited case of the effect of odd-symmetric waveforms on phase noise. However minimizing (37) is more a general criterion because although odd-symmetric waveforms may have small c0 coefficients, the class of waveforms with small c0 is not limited to those with odd symmetry.
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