Crystal Oscillators (1): Resonant Frequencies Derived

This lecture by Vincent Chang continues a series on crystal oscillators, focusing on deriving the resonant frequencies and explaining the frequency response of a quartz crystal. The instructor begins by reviewing the equivalent circuit of a crystal, which includes a series RLC branch in parallel with a capacitor. From this circuit, he derives the crystal impedance and identifies two key frequencies: the series resonant frequency (ωs) and the parallel resonant frequency (ωp). He then simplifies the impedance to a purely reactive expression and plots the reactance as a function of frequency. The plot shows that the crystal behaves capacitively at low and high frequencies, but inductively in a very narrow band between ωs and ωp. He emphasizes that ωp is only slightly higher than ωs because the parallel capacitance is much larger than the series capacitance. The lecture then applies this understanding to a Colpitts-type oscillator where the crystal replaces the inductor. By applying the Barkhausen criterion, which requires the sum of reactances in the loop to be zero, he shows that the crystal must be inductive, forcing the oscillation frequency to lie within the narrow inductive band. Since this band is extremely narrow, the oscillation frequency is essentially equal to the series resonant frequency of the crystal. This explains why crystal oscillators are highly stable: the frequency is determined almost entirely by the crystal’s resonant frequency, which is insensitive to temperature and time. The lecture concludes with a summary of the key takeaway: the crystal’s near-delta-function inductive region allows the oscillator frequency to be set precisely by the crystal itself.