Crystal Oscillator

A quartz crystal sliced from a piezoelectric material vibrates at a frequency so precise that it drifts by less than one second per year in a good oscillator. The crystal oscillator exploits this mechanical resonance to produce a frequency reference that no LC circuit can match. Every microcontroller clock, GPS receiver, and mobile phone baseband chip relies on a crystal oscillator for timing.

Core Concept

A quartz crystal exhibits the piezoelectric effect: mechanical stress creates an electrical voltage and vice versa. When an alternating voltage is applied at the crystal's natural mechanical resonant frequency, it vibrates with very high amplitude. The electrical equivalent circuit is an inductor Ls, capacitor Cs, and resistor Rs (the series branch) in parallel with Cp (package capacitance).

The crystal has two resonant frequencies. The series resonant frequency fs is where the series branch Ls-Cs resonates, giving minimum impedance. The parallel resonant frequency fp is slightly higher, where the total impedance is maximum. The difference between fs and fp is tiny, typically 0.1% to 1% of fs. Oscillators are designed to operate between these two frequencies, in the inductive region.

The Pierce oscillator is the most common crystal oscillator circuit for microcontroller clocks. It uses a CMOS inverter (such as the 74HC04 or the internal oscillator of an ATmega328) as the amplifier, the crystal as the feedback element, and two capacitors C1 and C2 (typically 18 to 22 pF) as load capacitors. The crystal operates slightly above fs, in its inductive region, so it behaves like an inductor and forms a Colpitts-like tank with C1 and C2.

Key Equations

Series resonant frequency:

fs = 1 / (2π × sqrt(Ls × Cs)) where Ls is the motional inductance in henries and Cs is the motional capacitance in farads (typically femtofarads).

Parallel resonant frequency:

fp = fs × sqrt(1 + Cs/Cp) where Cp is the parallel plate capacitance of the crystal package (typically 4 to 20 pF). Since Cs << Cp, fp is only slightly above fs.

Load capacitance for Pierce oscillator:

CL = (C1 × C2) / (C1 + C2) + Cstray The crystal is trimmed to oscillate at its marked frequency when CL matches the specified load capacitance (typically 12 pF or 18 pF).

Given:
  Crystal: 8 MHz, motional inductance Ls = 10 mH = 10 × 10^-3 H
  Motional capacitance Cs (to verify): find from fs formula
  Parallel capacitance Cp = 5 pF = 5 × 10^-12 F

Why this formula:
  Find Cs from the series resonant frequency, then find fp.

Formula:
  fs = 1 / (2π × sqrt(Ls × Cs))
  Rearranged: Cs = 1 / ((2π × fs)^2 × Ls)

Substitution:
  Cs = 1 / ((2π × 8 × 10^6)^2 × 10 × 10^-3)
     = 1 / ((5.027 × 10^7)^2 × 10^-2)
     = 1 / (2.527 × 10^15 × 10^-2)
     = 1 / (2.527 × 10^13)

Calculation:
  Cs = 3.96 × 10^-14 F = 0.0396 pF ≈ 40 fF

Now find fp:
  fp = fs × sqrt(1 + Cs/Cp)
     = 8 × 10^6 × sqrt(1 + 0.0396 × 10^-12 / 5 × 10^-12)
     = 8 × 10^6 × sqrt(1 + 0.00792)
     = 8 × 10^6 × 1.00395

Final Answer:
  fp ≈ 8.0316 MHz
  Separation between fs and fp = 31.6 kHz (only 0.395% above fs)

Exam Tip: GATE problems on crystal oscillators test two things most often. First, the relationship fp > fs and that oscillation occurs between fs and fp. The crystal is inductive in this region. Second, the load capacitance CL: if CL changes, the oscillation frequency changes. Increasing CL lowers the frequency slightly (called pulling). Manufacturers specify a nominal load capacitance (12 pF or 18 pF typically). Using the wrong load capacitors shifts the output frequency away from the marked value.

Key Properties

• Frequency stability of a crystal oscillator is typically 10 to 100 ppm, compared to 1000 to 10000 ppm for a comparable LC oscillator.
• The series resonant frequency fs depends only on Ls and Cs, which are determined by the crystal's physical dimensions and cut angle.
• The crystal operates in its inductive region between fs and fp. In this region, it acts as an inductor and replaces L in a Colpitts or Pierce oscillator.
• Q factor of a quartz crystal is 10,000 to 1,000,000, far higher than any LC tank. Higher Q means sharper frequency selectivity and better stability.
• The Pierce oscillator using a 74HC04 inverter with 22 pF load capacitors is the standard clock circuit for 3.3V and 5V microcontrollers.
• Temperature compensated crystal oscillators (TCXOs) hold frequency to within 0.5 ppm over a wide temperature range by compensating for the crystal's temperature coefficient.
• AT-cut crystals have near-zero temperature coefficient near 25°C and are used for most communication and timing applications.

Quick Revision

• Quartz crystal is a piezoelectric resonator. Its equivalent circuit is Ls, Cs, Rs (series) in parallel with Cp.
• Series resonance fs = 1/(2π sqrt(Ls Cs)). Parallel resonance fp = fs sqrt(1 + Cs/Cp).
• fp > fs always. The crystal is inductive between fs and fp.
• Crystal Q is 10,000 to 1,000,000. This gives outstanding frequency selectivity.
• Pierce oscillator: CMOS inverter + crystal + two 18-22 pF load capacitors + 1 MΩ bias resistor.
• Load capacitance CL = C1C2/(C1+C2) + Cstray. Wrong CL shifts oscillation frequency.
• Stability: crystal 10-100 ppm, LC oscillator 1000-10000 ppm, RC oscillator even worse.
• Exam trap: Saying the crystal oscillates exactly at fs. It oscillates slightly above fs, in the inductive region. The exact frequency depends on the external load capacitance CL.