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

2 Oscillators
Random phase
variation
Amplitude
Discrete spurious
signal
f0
f
Fig. 2.6: Output power spectrum of a typical RF oscillator.
or other random noise sources). Note (2.28) that an instantaneous phase variation is
indistinguishable from a variation in frequency.
Small changes in the oscillator frequency can be represented as a frequency modulation
of the carrier by letting
θ(t) = ∆f
f m
sin ω m t = θ p sin ω m t, (2.29)
where ∆f is -3 dB band width, f m = ω m /2π is the modulating frequency. The peak
phase deviation is θ p = ∆f/f m (also called the modulation index). Substituting (2.29)
into (2.28) and expanding gives
v 0 (t) = V 0 [cos ω 0 t cos(θ p sin ω m t) − sin ω 0 t sin(θ p sin ω m t)] (2.30)
where we set A(t) = 0 to ignore amplitude fluctuations. Assuming the phase deviations
are small, so that θ p ≪ 1, the small argument expressions that sinx ∼ = x and cos x ∼ = 1
can be used to simplify (2.30) to
v 0 (t) = V 0 [cos ω 0 t − θ p sin ω m t sin ω 0 t]
= V 0 {cos ω 0 t − θ p
2 [cos(ω 0 + ω m )t − cos(ω 0 − ω m )t]} (2.31)
This expression shows that small phase or frequency deviations in the output of an oscillator
result in modulation sidebands at ω 0 ± ω m , located on either side of the carrier
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