Online proceedings - EDA Publishing Association

The simulation output phase noise has been compared with the
theoretical phase noise from Leeson effect.
11-13 May 2011, Aix-en-Provence, France
One can calculate the phase noise of such an oscillator with
the Leeson effect [5], and examine the phase noise variations
with the resonator physical parameters. The Leeson equation
between the oscillator and the feedback amplifier [6] phase
spectral densities follows relation (8):
Leeson frequency
17Hz
1 ν
0 2
Sϕ
( f ) = [1 + ( ) ] S ( f )
2 φ
(8)
f 2Q
S ϕ
( f ) is the phase spectral density of the oscillator
S φ
( f ) is the phase spectral density of the amplifier
ν
0 is the resonance frequency and Q is the quality factor of
the resonator.
f is the offset frequency from the carrier.
The cut-off frequency, called Leeson’s frequency, is:
ω
f L
= 0
(9)
2Q
Simulations are run taking the physical quantities of the VIA
(table 1) into account. The quality factor of the characteristic
equation (1) is:
Fig. 8. phase noise output
This Leeson's model theoretical value matches the verilog-A
model simulation described above. Leeson’s frequency varies
only with the damping coefficient and the mass.
In the next simulations, only the mass of the
resonator is changed. The theoretical Leeson's frequency is
proportional to m -1 . We have previously shown that the
Leeson's frequency was 15.5 Hz with a mass of 1.10 -8 kg. For
1.10 -9 kg and 1.10 -10 kg, we have respectively frequencies of
155Hz and 1.55 kHz, as verified by the following simulation
on figure 9:
ω m
Q = 0
(10)
ρ
x
The Leeson’s frequency is:
ρ
x
f
L
= (11)
2m
f L
= 15. 5Hz
Phase noise analysis is performed with Virtuoso simulator and
the output signal is selected to create the phase noise output. It
becomes possible to read the Leeson’s frequency (figure 8).
Fig. 9. phase noise output depends on the mass
C. Influence of the transconductance amplifier noise
The white noise and flicker noise can be added to this
model and particularly to the transconductance amplifier
block.
1. White noise
White noise function is added to the the transconductance
amplifier plus pin. Therefore, we add a VerilogA block. A
Cadence special function is used : white_noise().
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