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

2 Oscillators
voltage transfer function of a parallel RLC resonator.
H(ω) =
1 + jQ
1
(
ω
ω 0
− ω 0
ω
) =
1
1 + 2jQ∆ω/ω 0
, (2.36)
where ω 0 is the resonant frequency of the oscillator, and ∆ω = ω − ω 0 is the frequency
offset relative to the resonant frequency.
Noise-free
amplifier
Vi(ω)
Sθ(ω)
A=1
V0(ω)
Sφ(ω)
H(ω)
Fig. 2.7: Feedback amplifier model for characterizing oscillator phase noise.
Since the input and output power spectral densities are related by the square of the
magnitude of the voltage transfer function [13], we can use (2.35)-(2.36) to write
S φ =
=
1
2
= 1 + 4Q2 ∆ω 2 /ω0
2 S
∣1 − H(ω) ∣ 4Q 2 ∆ω 2 /ω0
2 θ (ω)
(
ω 2 ) ( )
0
1 + S
4Q 2 ∆ω0
2 θ (ω) = 1 + ω2 h
S
∆ω0
2 θ (ω) (2.37)
where S θ (ω) is the input power spectral density, and S φ (ω) is the output power spectral
density. In (2.37) we have also defined ω h = ω 0 /2Q as the half-power (3 dB) bandwidth
of the resonator.
The noise spectrum of typical transistor amplifier with an applied sinusoidal signal at
f 0 is shown in Fig. 2.8. Besides kTB thermal noise, transistors generate additional noise
that varies as 1/f at frequencies below the frequency f α . This 1/f, or flicker, noise is
likely caused by random fluctuations of the carrier density in the active device. Due to
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