Frequency Correlation of Quartz Crystal Oscillators

By: Bruce R. Long
Piezo Crystal Company

first presented at the RF Expo EAST, 1990

INTRODUCTION

Imagine a hard-pressed design engineer, up against a deadline to deliver several RF systems, each needing a stable, accurate frequency source. A quick look through a popular circuit handbook for a crystal oscillator circuit yields a likely winner, and the crystals are ordered. But the handbook doesn't mention the effect of oscillator load reactance. So, the crystals are specified without a load. When the crystals are put in the circuit, they appear to be off frequency. What happened?

Quartz crystal resonators are widely used in electronics as highly stable and accurate frequency sources, but the oscillation frequency depends as much on the oscillator circuit as it does on the crystal. This presentation will clarify resonator correlation and show how correlation problems are avoided. Correlation here means correspondence between the crystal resonator frequency and oscillation frequency in a circuit.

RESONATOR PHYSICAL DESCRIPTION

A quartz crystal resonator is simply a circular piece of quartz with electrodes plated on both sides, mounted inside an evacuated enclosure. But resonator design demands knowledge of mechanics, acoustics, wave motion, piezoelectricity and electronic circuit theory. Quartz is ideal for resonators because it is hard, yet flexible, dimensionally stable, non-conductive, and most important, piezoelectric.

The piezoelectric effect links the mechanical and electrical properties of the resonator. Electrode voltage causes mechanical movement. Likewise, mechanical displacement generates an electrode voltage. Therefore, a crystal mechanical resonance can be treated as if it were electrical in nature. Physical displacement of an operating resonator is minuscule, generally only a few atomic diameters, but tremendous mechanical forces are in play. For example, the surface acceleration of a 50 MHz third overtone crystal exceeds five million gravities.

Making a resonator starts with slabs of quartz sawed from raw quartz crystal stock. Both natural quartz mined from the earth and cultured quartz grown artificially in high pressure autoclaves are used. The slabs are cut into squares, rounded, and ground flat. These thin pieces of unfinished quartz are called blanks. Thin blanks have high resonant frequencies, but they must remain thick enough to withstand grinding without excessive breakage. For this reason, the upper limit for mechanically ground fundamental mode resonators is approximately 35 MHz. These paper thin quartz blanks are too fragile to withstand further grinding.

High frequency resonators are made using overtone responses. This extends the upper frequency limit beyond 200 MHz.

The frequency tolerance of a finished resonator can be as precise as plus or minus one part per million, making exact control of blank thickness critical. Grinding must stop the moment the correct thickness is reached. Fortunately, the piezoelectric effect makes possible continuous measurement of the blank thickness. Vibration occurs as abrasive slides across the crystal. That, in turn, excites the crystal resonance generating radio frequency energy-energy that is detected and measured by a simple short-wave receiver.

Polishing removes small scratches left by grinding and leaves the surface ready for plating. The electrode mass loads the resonated active region and that lowers the crystal resonant frequency. To compensate, blanks are ground for resonance slightly above the desired frequency and set exactly on frequency by adjusting the thickness of the electrode. A 10 MHz resonator shifts 5 to 7 kHz during the plating process and the amount of shift is roughly proportional to the resonator frequency. That means a 100 MHz resonator moves about 50 to 70 kHz. During the plating process, the resonator is connected to a test oscillator and the resonant frequency is monitored with a frequency counter. Plating stops the instant the correct frequency is reached.

RESONATOR ELECTRICAL DESCRIPTION

The crystal's mechanical resonance appears as a very high Q electrical resonance with the equivalent circuit shown in Figure 1.

Figure 1: Crystal Resonator Equivalent Circuit

Components C, L and R make up a series resonant circuit representing the blank's mechanical resonance. Resistor R represents energy lost in the crystal and its mounting structure. The capacitance C₁ varies with crystal elasticity and the crystal mass determines the inductance L. These parameters are sometimes called the motional inductance and capacitance, highlighting their mechanical origins. The electrodes also form the plates of a capacitor with a quartz dielectric. This capacitance, represented by C₀, varies with the size of the electrodes. Resonator electrical parameters are measured with a crystal impedance bridge, a vector voltmeter, or a network analyzer. The instrument's frequency resolution must be good since circuit impedance varies sharply with frequency.

Figure 2 shows how resonator reactance changes with frequency. At Fs, the series resonant frequency, the motional capacitance and inductance cancel leaving a small capacitive component from C₀. Resonator reactance goes to zero at a slightly higher frequency. Here the effect of C₀ is cancelled by a small inductive component coming from the resonator motional branch and the impedance is purely resistive. In practice, the series resonant frequency and the resonator zero reactance point are quite close together. Confusion comes when the distinction between the two is ignored.

Figure 2: Resonator Reactance Frequency Function

Above series resonance, the motional inductance and capacitance partially cancel, leaving a high Q equivalent inductor which, when combined with the electrode capacitance C₀, create the parallel resonant response at Rp. The parallel resonant frequency is strongly affected by circuit loading. External load capacitance adds to C₀ lowering the parallel resonant frequency. The following formulas give the resonant frequencies:

Resonator impedance changes drastically between series and parallel resonance. At series resonance the crystal impedance is very low, limited only by the loss resistor R. But at parallel resonance-only a few parts per million higher in frequency-the impedance is quite high. The steep phase slope that accompanies the rapid impedance transition accounts for the frequency stability of quartz crystal oscillators.

Which resonance point, series or parallel, controls the oscillation frequency? That depends on the oscillator.

CONDITIONS FOR OSCILLATION

Resonators are passive. They do not oscillate without additional circuitry and a source of energy. A resonator, placed as a feedback element around an amplifier, makes an oscillator. The amplifier replaces energy lost in the resonator. The resonator controls the frequency of oscillation.

Two conditions must exist to start and sustain oscillations. First, the phase shift around the loop must be an integer multiple of 360 degrees. This guarantees that successive cycles add constructively. Second, the sum of gain and loss around the loop must equal or exceed one. During start-up, the loop gain is more than one and the oscillation amplitude grows exponentially. However, no system can sustain unlimited exponential growth. Ultimately, the amplifier is driven into gain compression, which stabilizes the gain at a value just sufficient to overcome resonator losses. Clipping limits gain in most oscillators, but some precision oscillators use separate automatic gain control (AGC) circuitry or clipping diodes. Oscillation conditions, known as the Barkhausen criteria, are named for the German scientist who first presented them.

RESONATOR OPERATION POINT

Crystal oscillator circuits traditionally are labeled as parallel or series resonant, with the crystal specified accordingly. But a close look at oscillator theory shows that this bit of conventional wisdom is too simple. Oscillators operate on the point of the crystal reactance slope that satisfies the Barkhausen criteria. This point coincides with the crystal series resonance point only if the phase shift in the rest of the circuit sums to a multiple of 360 degrees. In most oscillator circuits, the crystal operates in the inductive region between series and parallel resonance.

CRYSTAL OSCILLATOR CIRCUITS

Figure 3 shows the basic equivalent circuit common to the Pierce, the Colpitts and the Clapp oscillators. The only difference between them is the location of the grounded node. Performance differences are due to changes in the amplifier input and output impedance and to the varying effect of stray capacitance as the ground node is moved. The Colpitts oscillator is used at low to medium frequencies, and is replaced by the Pierce at higher frequencies. All three circuits are parallel mode oscillators with the crystal in each acting as an equivalent inductor. These oscillators operate significantly above the crystal series resonant frequency. Therefore, crystals used in these circuits are specified for the load capacitance of the oscillator.

Figure 3: Common Oscillator Configurations

In the Pierce circuit, the feedback path around the common emitter amplifier is obvious, starting at the transistor collector and returning to the amplifier input through the crystal. The Colpitts oscillator, in contrast, is a little obscure. Gain comes from an emitter follower amplifier. Capacitors C₁ and C₂, along with the crystal equivalent inductor, make up a tapped tuned circuit, which matches the low impedance amplifier output to the high impedance amplifier input. The Clapp circuit is very similar in this respect, except a common base amplifier is used.

The input impedance of a parallel resonant oscillator has two parts - a negative resistance coming from combined amplification and positive feedback, and a capacitive reactance coming mostly from feedback capacitors C₁ and C₂. The former cancels resonator losses, allowing sustained oscillation. The latter makes up the oscillator load capacitance. Figure 4 shows a formula giving the input impedance of an ideal Colpitts oscillator, with the transistor represented as an ideal voltage controlled current generator. Capacitors C₁ and C₂ dominate the imaginary part of the expression but also appear in the real portion. This equation should be used with caution because it assumes linear transistor operation. It is inaccurate for oscillators that saturate the transistor during part of each cycle. It also fails to account for stray capacitance and transistor phase shift.

Figure 4: Colpitts Load Impedance

SERIES MODE OSCILLATORS

In a series mode oscillator the crystal resonator operates at or near its zero reactance point. Figure 5, from page 161 of reference 3, shows a series mode oscillator consisting of two CMOS logic inverters. Bias resistors place the gates in the linear region. Load resistors RL sets the resonator loaded Q. This circuit works best at low frequencies, where the gate propagation delay can be safely neglected. At low frequencies the phase shift through the gates totals 360 degrees ­ 180 degrees per inverter. Zero phase shift across the feedback path meets the Barkhausen criteria so this circuit oscillates near the resonator zero reactance point.

At higher frequencies gate propagation delays start showing an effect. As the phase shift in the forward path moves past 360 degrees, the oscillation frequency drops below series resonance into the capacitive region of the crystal reactance curve. Here the crystal acts as an equivalent capacitor, forming an RC phase lead network with resistor RL, which compensates for the gate delay. This hurts frequency stability because the resonator reactance slope is not as steep here as in the inductive region and because gate propagation delay changes with temperature. Operation below series resonance does not follow good engineering practice and such oscillators are uncommon.

Figure 5: CMOS Series Mode Crystal Oscillator

Another series mode oscillator, the common collector Butler, is shown in Figure 7. It strongly resembles the Colpitts parallel resonant oscillator, but the resonator is replaced with an inductor. The crystal goes between the junction of C₁ and C₂ and the transistor emitter.

Figure 7: Butler Test Oscillator