Crystal Oscillator Waveform Analysis

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Introduction

A crystal oscillator uses the mechanical resonance of a quartz crystal to generate a highly stable periodic electrical signal.

Quartz exhibits the piezoelectric effect. When an electric field is applied to the crystal, mechanical deformation occurs. Conversely, mechanical vibration generates an electrical voltage.

The crystal therefore behaves as a very high-Q resonant circuit.

Equivalent Circuit of a Quartz Crystal

A quartz crystal can be represented by the equivalent electrical circuit

R_m, L_m, C_m

connected in series, with a parallel capacitance C_0.

The impedance of the motional branch is

Z_m() = R_m + j( L_m - 1 C_m ).

The total crystal impedance is

Z_X = Z_m 1j C_0.

Series Resonance

Series resonance occurs when

L_m = 1 C_m.

Therefore

_s = 1L_mC_m

and

f_s = 12L_mC_m.

At series resonance, the reactive components cancel.

Thus

Z_m R_m.

The impedance becomes very small and the crystal strongly passes the resonant frequency.

Parallel Resonance

Because of the parallel capacitance C_0, another resonance occurs slightly above the series resonant frequency.

The approximate parallel resonant frequency is

f_p = f_s 1+C_mC_0 .

Usually

f_p > f_s.

The oscillator circuit operates very close to one of these resonant frequencies.

Oscillation Condition

For sustained oscillation, the Barkhausen criterion must be satisfied.

|A(j)(j)| = 1

and

A(j)(j)=360^ n.

The amplifier compensates for losses in the crystal while the crystal determines the oscillation frequency.

Time-Domain Waveform

The ideal steady-state oscillator voltage can be represented as

v_o(t) = V_p(2 f_0 t+).

For example,

V_p=2.5\ V, f_0=10\ MHz.

Therefore

v_o(t) = 2.5 ( 2(1010^6)t ).

The period is

T=1f_0.

For f_0=10\ MHz,

T=100\ ns.

Oscillation Start-Up

Immediately after power is applied, the oscillator does not reach its final amplitude instantly.

Noise inside the circuit provides a small initial signal.

The resonant component is repeatedly amplified.

The approximate build-up waveform can be represented as

v(t) = V_ ( 1-e^-t/ ) (2 f_0t).

Here

represents the effective start-up time constant.

As time increases,

1-e^-t/ 1.

Therefore the waveform gradually approaches

v(t) V_(2 f_0t).

Waveform Analysis

During start-up:

A(t) = V_ ( 1-e^-t/ ).

The amplitude increases with time.

At steady state:

A(t) V_.

The oscillator frequency remains approximately

f(t) f_0.

The crystal's high quality factor

Q = _0 L_mR_m

produces a very narrow resonance bandwidth.

Approximately,

BW = f_0Q.

A high Q therefore results in excellent frequency selectivity and frequency stability.

Interactive ANITEX Simulation

The following simulation visualizes:

crystal oscillator start-up, sinusoidal output waveform, amplitude build-up, crystal vibration, resonance response, series resonant frequency, frequency-domain resonance peak.

Interpretation

When the PLAY button is pressed, the simulation begins with a very small oscillation.

The amplitude increases according to

A(t) = V_ ( 1-e^-t/ ).

At the same time, the quartz crystal vibrates mechanically.

The frequency response graph shows a sharp resonance peak near

f_0.

Increasing Q makes the resonance peak narrower.

Therefore,

Q BW frequency selectivity.

This is the fundamental reason why quartz crystal oscillators provide much better frequency stability than ordinary RC oscillators.

Summary

A crystal oscillator converts electrical energy into mechanical vibration and back into electrical energy through the piezoelectric effect.

Its high-Q resonance determines the oscillation frequency.

The output waveform evolves from noise into a stable sinusoidal signal:

v_o(t) = V_ ( 1-e^-t/ ) (2 f_0t)

and finally approaches

v_o(t) = V_(2 f_0t) .