High Order Harmonic Oscillators in Microwave and Millimeter-wave ...

2 Oscillators
Sθ(ω)
k
∆ f
kT0F
P0
f0
fα
f
Fig. 2.9: Idealized power spectral density of amplifier noise including 1/f and thermal
components.
Using (2.38) in (2.37) gives the power spectral density of the output phase noise as
S φ (ω) = kT 0F
P 0
= kT 0F
P 0
( Kω
2
0 ω α
4Q 2 ∆ω + ω0
2
3 4Q 2 ∆ω + Kω )
α
2 ∆ω + 1
( Kωα ωh
2
∆ω 3
+ ω2 h
∆ω 2 + Kω α
∆ω + 1 )
. (2.39)
This result is sketched in Fig. 2.10. There are two cases, depending on which of the
middle two terms of (2.39) is more significant. In either case, for frequencies close to the
carrier at f 0 , the noise power decreases as 1/f 3 , or -18 dB/octave. If the resonator has
a relatively low Q, so that its 3 dB bandwidth f h > f α , then for frequencies between f α
and f h the noise power drops as 1/f 2 , or -12 dB/octave. If the resonator has a relatively
high Q, so that f h < f α , then for frequencies between f h and f α the noise power drops as
1/f, or -6 dB/octave.
At higher frequencies the noise is predominantly thermal, constant with frequency, and
proportional to the noise figure of the amplifier. A noiseless amplifier with F = 1 (0
dB) would produce a minimum noise floor of kT 0 = −174 dBm/Hz. In accordance with
Figure 2.6, the noise power is greatest frequencies closest to the carrier frequency, but
(2.39) shows that the 1/f 3 component is proportional to 1/Q 2 , so that better phase noise
characteristics close to the carrier are achieved with a high-Q resonator. Finally, recall
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