Crystal Oscillator\ Resonance and Start-Up Visualization

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Objective

This ANITEX document explains the operating principle of a quartz crystal oscillator. The interactive Science Viewer illustrates the crystal equivalent circuit, series and parallel resonance, oscillation start-up, amplitude stabilization, and the output waveform.

Quartz Crystal Equivalent Circuit

A quartz crystal can be represented by a motional branch containing R_m, L_m, and C_m in series, with a shunt capacitance C_0 in parallel: Z_m()=R_m+j( L_m-1 C_m). The total crystal impedance is obtained by placing this motional branch in parallel with C_0.

The series resonant frequency is approximately f_s=12L_mC_m.

The parallel resonant frequency is approximately f_p f_s1+C_mC_0.

Because C_m is normally much smaller than C_0, f_p lies only slightly above f_s.

Oscillation Condition

An oscillator requires positive feedback satisfying the Barkhausen conditions near the operating frequency: |A(j)(j)| 1 during start-up, and A(j)(j)=2 n.

Small electrical noise initially excites the resonator. Near the crystal resonance, the desired frequency is strongly selected because of the crystal's high quality factor Q. A simplified educational start-up envelope is A(t)=A_(1-e^-t/_g), and the output is represented by v_o(t)=A(t)(2 f_0 t). As the output grows, amplifier nonlinearity reduces the effective loop gain toward unity and the oscillation settles to a stable amplitude.

Interactive ANITEX Animation

The following block is interpreted by the ANITEX / PHP Science Viewer.

Graph Interpretation

The impedance graph shows the narrow region between series resonance f_s and parallel resonance f_p. The start-up graph shows how a tiny noise-derived signal grows toward a stable amplitude. The loop-gain graph begins above unity and approaches unity as nonlinear amplitude limiting occurs. The output graph shows the corresponding sinusoidal oscillation.

Important Note

This ANITEX model is an educational representation. A practical crystal oscillator should be analyzed using the actual crystal equivalent parameters, load capacitance, amplifier transfer function, parasitic elements, and device nonlinearities.