# Design a Crystal Oscillator to Match Your Application

Abstract: Quartz crystals are mechanical resonators with piezoelectric properties. The piezoelectric properties (electric potential across the crystal is proportional to mechanical deformation) allow their use as electrical circuit elements. Crystals are widely used as resonant elements in oscillators due to their high quality factor (QF), excellent frequency stability, tight tolerance, and relatively low cost. This tutorial explains the primary design considerations to be addressed in a design of a simple crystal oscillator using AT-cut crystals. The basic qualities of a crystal oscillator and factors that can affect their performance in a variety of applications are described. The topics discussed here are the compilation of issues encountered over a decade of design and applications for ISM-band radios. These topics include load capacitance, negative resistance, startup time, frequency stability versus temperature, drive-level dependency, crystal aging, frequency error, and spurious modes.

*Electronic Design*, September 07, 2012.

## Basics of a Crystal Model

**Figure 1**). The series RLC branch, often called the motional arm, models the piezoelectric coupling to the mechanical quartz resonator. The shunt capacitance represents the physical capacitance formed by both the parallel plate capacitance of the electrode metallization and the stray package capacitance.

*Figure 1. Simple electrical model of a fundamental-mode quartz crystal.*

R1 10Ω to 150Ω (equivalent series resistance, ESR)

L1 determined by C1 and the operating frequency (motional inductance)

C0 0.5pF to 5pF (shunt capacitance)

_{0}/QF) × dQ/dt + Q × ω

_{0}² = 0.

_{0}, is the square root of the inverse of the product of inductance and capacitance.

*Figure 2. Crystal mechanical model.*

**Figure 2**). The forces applied to the crystal, ignoring the fixed force and spatial offset due to gravity, result in an acceleration of the mass (Newton's Second Law of Motion). Two forces are assumed in the simple linear model, spring force and frictional force. The spring force is given by Hooke's Law, F = K × Y, where K is the spring modulus and Y is the displacement from equilibrium. The frictional loss is assumed to be proportional to the velocity of the plunger in the dashpot and the friction constant, D, of the dashpot. Equating these forces (with no external driving forces) gives:

_{0}/QF) × dY/dt + Y × ω

_{0}² = 0.

_{0}= √(1/(L × C) = √(K/M)

*Figure 3. Cubic quartz resonator. Electrodes on the top and bottom faces, A = L × W.*

**Figure 3**.

_{0}= √(K/M) ~ √(A/(T × A × T) = √(1/T²) = 1/T

**Figure 4**. In this mode of operation the center of gravity moves both vertically and horizontally. Thus, the preceding analysis is a one-dimensional approximation, useful for qualitative understanding of the mechanical resonance of an AT-cut crystal.

*Figure 4. AT-cut thickness, shear mode resonance.*

- Smaller crystal electrode areas are attractive for lower cost and perhaps smaller package size. However, this smaller area increases series resistance, which slows startup time (see following
**Startup Time**section) and can prevent oscillation. - Larger crystal electrode areas lower series resistance. However, this larger area increases the shunt capacitance which then lowers the active circuit negative resistance (see
**Negative Resistance**section below), which, in turn, also slows startup time and can prevent oscillation. The larger crystal electrode area increases the motional capacitance. With a larger motional capacitance comes greater sensitivity to frequency shift due to external capacitive loads, or frequency "pulling" (see**Load Capacitance**section below).

## Load Capacitance

**Figure 5**. Crystal manufacturers specify a given load capacitance along with a frequency of operation. Operation with a load capacitance that differs from the manufacturer's specified load capacitance results in an oscillation frequency error with respect to the manufacturer's specified frequency. The frequency error is due to capacitive "pulling" of the crystal. This can be demonstrated by combining the shunt and load capacitances in parallel, and then combining this summed shunt plus load capacitance in series with the motional capacitance to form the overall effective capacitance.

_{EFF}= C

_{MOTIONAL}× (C

_{LOAD}+ C

_{SHUNT})/(C

_{LOAD}+ C

_{SHUNT}+ C

_{MOTIONAL})

*Figure 5. Load capacitance.*

_{LOAD}+ C

_{SHUNT})/(C

_{LOAD}+ C

_{SHUNT}+ C

_{MOTIONAL}) is nearly unity, and the effective overall capacitance is very near the value of the motional capacitance. Note that as the load capacitance grows larger, (C

_{LOAD}+ C

_{SHUNT})/(C

_{LOAD}+ C

_{SHUNT}+ C

_{MOTIONAL}) approaches nearer to unity, and the effect of absolute changes in the load capacitance on the overall effective capacitance weakens (lower frequency pulling). In the same way, smaller motional capacitances also lower frequency pulling as (C

_{LOAD}+ C

_{SHUNT})/(C

_{LOAD}+ C

_{SHUNT}+ C

_{MOTIONAL}) approaches nearer to unity for any given load capacitance. See

**Figure 6**for the frequency versus load capacitance (pulling curve) of a typical crystal.

Figure 6. Typical pulling curve for 5fF C

_{MOTIONAL}, 3pF C

_{SHUNT}, 3pF specified C

_{LOAD}, 10MHz crystal.

## Negative Resistance

**Figures 7**and

**8**. Note that the three points A, B, and C are identical for both topologies, except for the AC-ground point.

*Figure 7. A Colpitts oscillator.*

*Figure 8. A Pierce oscillator.*

**Figure 9**). From this:

_{A}= -Z3 × I

_{C}= Z2 × I - Z2 × g

_{M}× V

_{A}= Z2 × I + Z2 × g

_{M}× Z3 × I = I × (Z2 + g

_{M}× Z3 × Z2)

_{M}is the small signal change in collector current per change in base to emitter voltage for a bipolar junction transistor (g

_{M}= ΔI

_{C}/ΔV

_{BE}), or the small signal change in drain current per change in gate to source voltage for a MOSFET (g

_{M}= ΔI

_{D}/ΔV

_{GS}).

_{CA}= V

_{C}- V

_{A}= I × (Z3 + Z2 + g

_{M}× Z3 × Z2)

_{IN}= V

_{CA}/I = Z3 + Z2 + g

_{M}/(C3 × C2 × (j × ω)²) = Z3 + Z2 - g

_{M}/(C3 × C2 × ω²)

*Figure 9. Determining the input impedance of the Pierce oscillator.*

_{IN}is the impedance presented to the crystal by two capacitors and the transconductor, then the impedance presented to the crystal is effectively the series combination of C3 and C2 in series with a negative resistance. Note that this allows for ease of setting the load capacitance of the crystal by appropriate choice of C3 and C2, independent of transconductance.

**Figure 10**.

*Figure 10. Equivalent circuit of a three-point oscillator with a crystal.*

_{IN}= Z3 + Z2 + g

_{M}× Z3 × Z2

_{IN}) in parallel with C

_{SHUNT}:

_{APPLIED}= [1/Z

_{SHUNT}+ 1/(Z3 + Z2 + g

_{M}× Z3 × Z2)]

^{-1}

_{APPLIED}= [(Z3 + Z2 +Z SHUNT + g

_{M}× Z3 × Z2)/(Z3 × Z

_{SHUNT}+ Z2 × Z

_{SHUNT}+ g

_{M}× Z3 × Z2 × Z

_{SHUNT})]

^{-1}

_{APPLIED}= (Z3 × Z

_{SHUNT}+ Z2 × Z

_{SHUNT}+ g

_{M}× Z3 × Z2 × Z

_{SHUNT})/(Z3 + Z2 +Z

_{SHUNT}+ g

_{M}× Z3 × Z2)

_{APPLIED}, the negative impedance presented by the three-point oscillator to the RLC motional branch of the crystal is:

_{APPLIED}} = -(g

_{M}× C3 × C2)/[ω² × (C3 × C2 + C3 × C

_{SHUNT}+ C2 × C

_{SHUNT})² + (g

_{M}× C

_{SHUNT})²]

_{APPLIED}} with respect to g

_{M}and setting the derivative equal to zero yields the transconductance g

_{M(MIN)R}, for which the minimum (largest magnitude) of negative resistance occurs:

_{M(MIN)R}= ω × [(C3 × C2)/C

_{SHUNT}+ C3 +C2]

_{M(MIN)R}the maximum magnitude of negative resistance occurs, yielding:

_{APPLIED}}|

_{MIN}= -1/{2 × ω × C

_{SHUNT}× [1 + C

_{SHUNT}× (C3 + C2)/(C3 × C2)]}

_{APPLIED}}, has the following characteristics:

- Is always negative.
- The absolute value of the negative resistance drops as C
_{SHUNT}increases. (See**Figures 11**and**12**.) - The maximum achievable absolute value of the negative resistance (at g
_{M(MIN)R}) drops as C_{SHUNT}increases. (See Figures 11 and 12.) - The absolute value of the negative resistance must be larger than the motional resistance of the crystal for oscillation to occur. Generally, a typical or nominal absolute value of the negative resistance should be greater than four times the motional resistance.

*Figure 11. Negative resistance versus load capacitance at 10MHz with transconductance of 5mA/V; load capacitance is due to the series combination of C3 and C2.*

*Figure 12. Negative resistance versus transconductance at 10MHz with load capacitance of 10pF; load capacitance is due to the series combination of C3 and C2 (each at 20pF).*

_{SHUNT}on both plots. Even a small increase in C

_{SHUNT}decreases the magnitude of the negative resistance in every possible configuration, especially near the peak of the negative resistance magnitude.

_{SHUNT}small and to increase C3 and C2 to apply the necessary load capacitance. As an example, consider the following cases where the crystal load capacitance is 8pF, the operating frequency is 10MHz, the crystal C

_{SHUNT}is 2pF, the parasitic values of C3 and C2 are 8pF (due to the IC package and PCB stray capacitances), and the transconductance is fixed (due to internal IC biasing and device size) at 1mA/V.

**Case 1**. Use 8pF ceramic capacitors in the in the positions of C3 and C2 to load the crystal. These 8pF capacitors are in parallel with the 8pF stray capacitances for total C3 and C2 values of 16pF. This will load the crystal with 8pF, as C3 and C2 appear in series with respect to the crystal. In this case the negative resistance calculated from preceding equation for Re{Z

_{APPLIED}} will be -627Ω.

**Case 2**. Use a 4pF ceramic capacitor in parallel with the crystal, as this saves the cost of one capacitor and the SMT placement of one capacitor versus Case 1. The C3 and C2 stray capacitances of 8pF each load the crystal with 4pF. The additional 4pF of shunt capacitance in parallel sum to a total of 8pF load capacitance. However, in this case the negative resistance will only be -466? due to the undesirable effects of increasing C

_{SHUNT}.

## Startup Time

^{|(R/2 × L)| × t}] × sin{2 × π × √[1/(L × C)] × t + Θ}

## Frequency Stability Versus Temperature

_{0}= A

_{0}+ A

_{1}(T - T

_{0}) + A

_{2}(T - T

_{0})² + A

_{3}(T - T

_{0})³

_{0}through A

_{3}are functions of the angle of the quartz cut.

**Figure 13**, this is impossible to achieve even with a perfect cut-angle, zero-tolerance, zero-aging crystal over the industrial temperature range of -40°C to +85°C. For this case a radio system with an internal temperature sensor and narrow-frequency-step fractional-N synthesizer, such as the MAX7049 transmitter, can be used to compensate for known crystal frequency temperature coefficients.

*Figure 13. Graph of relative frequency shift versus temperature for AT-cut crystal angles in minutes.*

## Aging

## Compilation of Frequency Error Sources

- Initial tolerance, which is the manufacturer's guaranteed frequency tolerance at +25°C and with the specified load capacitance applied to the crystal
- Frequency stability versus temperature
- Pulling due to load capacitance variations
- Aging

## Drive-Level Dependency

- Operate with large negative resistance, greater than four times the manufacturer's maximum specified series resistance. This will overcome nearly all DLD issues.
- Purchase from higher-quality crystal vendors.
- Pay the premium for DLD testing.

## Spurious Modes

## Conclusion

#### General References

- Vittoz, Eric A., Degrauwe, Marc G. R., and Bitz, Serge, "High-Performance Crystal Oscillator Circuits: Theory and Application,"
**IEEE Journal of Solid-State Circuits**, Vol. 23, No. 3, June 1988. - Kreyszig, Erwin,
**Advanced Engineering Mathematics**, Fifth Edition, John Wiley and Sons, 1983. - Bechmann, Rudolf, "Frequency-Temperature-Angle Characteristics of AT-type Resonators Made of Natural and Synthetic Quartz,"
**Proceedings of the IRE**, November 1956.