Chapter 6 Oscillators with Multi-Band Resonators 103
6.2 Phase-Noise of Oscillators with a Generalized Resonator Structure
6.2.2 Phase Noise
Since at the resonant frequency or at a very close frequency to it, the derivative of impedance magnitude and its real part is zero35, the expression in (6.55) can be further simplified to
0
) ( 2
0 ω ω
ω ω
d
Qenergy ≅ ⋅dΦ (6.56)
where Φ(ω) is the phase response of the resonator impedance function. The form of (6.56) was suggested before [118] in the context of feedback model of second-order oscillators.
Here, we show its generality for all passive resonator structures using energy arguments.
Note that (6.55) and (6.56) are general expressions for all passive structures at resonance and should be used instead of the widely used definition for one-port networks where the quality factor is defined as the ratio of imaginary to the real part of impedance function.
Figure 6.16: Illustration of phase-noise definition per unit bandwidth
Hence oscillators with a smaller phase-noise number will have a purer power spectrum density around the oscillation frequency. Despite its deceptively simple look, calculation of oscillator output spectrum in the presence of noise is not always easy and many efforts has been dedicated to this task for over half a century [39]-[112]. The linear time-invariant analysis of the problem given by Leeson in [39] provides a simple expression for phase- noise that has been used extensively by designers for decades. Probably, it is due to the simplicity of the expressions in [39] that the more concrete analysis by Lax in [109] has not been given the same attention by the circuit design community. Since oscillators are a major building block in most communication circuits and their phase-noise affects the sensitivity of the radio system, attention has been given to accurate analysis of phase-noise in integrated oscillators recently [110]-[113]. Approaches vary quite a bit, with [110]
emphasizing on the time-varying aspect of the nonlinear problem, deriving a phase-noise expression that can help in designing oscillators with a lower phase-noise number. The exact solution of the stochastic nonlinear problem is possible and the results can be integrated into a circuit simulator [111]. More recently the problem of noise in integrated oscillators has been once again traced back in its physical origin of diffusion processes via the concept of virtual damping [112].
In this section, we will first argue that many resonator structures can be modeled with a second-order resonator for frequencies close to resonance and the conditions that allow us to do so. Then, using the well-known models to calculate phase-noise for second-order systems, we will show a general link between the phase-noise of an oscillator and Qenergy of its resonator.
ω ω ω ω
dBc
∆∆∆∆ωωωω
1 Hz
ωωω ω
power spectral density
From the early works of Foster [119], we know that for any lossless network, the reactance function can be represented as
= ∞
+
− −
=
∑
a C LX
n n
n ω
ω ω ω ω ω
1 0
2 2
2 1 )
( (6.58)
where ωn are the poles of the network and an is the residue of the pole ωn. The capacitor C0
models that pole at zero frequency while the inductor L∞ model the pole at infinity. If ωns are not very close, for frequencies close to any of the poles, say ωk, the kth term in the sum dominates and (6.58) simplifies to
2 2
) 2 (
k k to
close
X a
k ω ω
ω ω ω ω
≈ − (6.59)
Differentiating the susceptance (inverse of (6.59)) and use of (6.54) gives
( )
storedk E
V d
a dB
k =− ⋅
−
=
= 4
1 2
ω
ω ω (6.60)
From comparing (6.58)-(6.60) to the results for a second order parallel LC resonator, we can derive the equivalent capacitor and inductor for a complex network in resonance
2 2
2
2 2
, 2 2
2 1
k stored k
k eq
stored k
eq E
a V L
V E C a
ω ω =− ⋅ ⋅
= −
− ⋅
− =
= (6.61)
If the poles of the network are low-loss36, the effect of loss can be lumped into a single resistor in parallel with the equivalent second order resonator
)
2 ( 0
2
ω
P RR V
dissipated
eq =
= ⋅ (6.62)
The quality factor of this equivalent second order system is the same as one expects from (6.55). The equivalent model derived above can be used to calculate the phase-noise in oscillators using the previously developed theories for second order systems.
First, we want to show how the results obtained in this section can be applied to the phase-noise model in [110]. This model starts with calculating the effect of injected noise to
36 This assumption is along the same line that led us to deriving (6.56). Both assumptions are to assure that in the neighborhood of resonance that we are interested in, the dominant effect comes from the corresponding pole and not from other poles of the resonator. If these conditions are not satisfied, the null in the impedance transfer function becomes shallower due to loss as the poles get closer.
phase displacements for a resonator at oscillation. According to [110] the transfer function of injected impulse current at different times to excess phase in the oscillating output is a periodic function of time, called the impulse sensitivity function (the ISF is shown with Γ(ω0t) hereafter) and directly affects the phase-noise expression of oscillators. Although direct calculation of ISF in general is possible, it gets more difficult for higher-order resonators. An example of the direct calculation of ISF for a fourth-order resonator is given in the Appendix E.
For a general resonator with impedance function Z(s) that is already oscillating at ω=ω0, the effect of impulse current at t=t0 on its output can be calculated using
} )
( { )
( )
( t V
unperturbedt L
1Z s e
st0V = +
−⋅
− (6.63)In (6.63) Vunperturbed(t) is resonator’s oscillating output before injecting the impulse current and L-1 is the inverse Laplace transform. Clearly the effect of the second part of (6.63) is also oscillation at ω=ω0 which combined with the pre-injection term yields an oscillation with a different phase shift37. Since the effect of passive resonator is only to shape the ISF through (6.63), the derivations and discussions in [110] will remain valid for a complicated resonator structure by merely using the equivalent values of (6.61)-(6.62). The link to the physical resonator stored energy is still preserved through those expressions.
The phase-noise model of [109] starts off by the Taylor expansion of the impedance function around the resonance frequency of a second-order system. Similarly we can write
) ) (
) ( ( ) ( ) ) (
) (
( 02 0 0 0
0 0
0 ω ω
ω ω ω
ω ω ω ω ω
ω ω ≅ ω ω=ω − + + + ω=ω −
d jX dR
d R jdX j
Z closeto (6.64)
The standard procedure for second-order systems in [109] can be followed to derive the Langevin equation for the phase of the oscillator. In the standard second-order LC system dX(ω)/dω at resonant frequency is simply the value of inductor, L. The result will be the familiar phase-noise dependency on ω0L/R that is called the tank quality factor. However in the general case of (6.64) the phase-noise will depend on ( ) ( )
2 0
0
0 ω
ωω ω
ω R
d
⋅dX which is the
familiar energy-based definition of quality factor (expression (6.55)). This result agrees favorably with [112] where the phase-noise treatment is based on the physical phase diffusion model. There, it is showed that for the second-order LC system, the phase-noise is directly related to the total stored energy in the tank and inversely proportional to the thermal energy stored in the system.
In summary, we first argued that any complex resonator structure with multiple poles (resonant frequencies) that are sufficiently far apart, behaves similar to a second order system close to its resonant frequency. This link has been made based on energy arguments.
Hence, phase-noise models that are expressed for second order systems can be further expanded to higher-order systems for frequencies close to the oscillation frequency38.