What is the Barkhausen criterion for oscillation?

For a circuit to sustain steady-state oscillation, two conditions must be met simultaneously at the oscillation frequency: the loop gain must be exactly unity (magnitude condition) and the total phase shift around the loop must be exactly 0 or 360 degrees (phase condition). At startup, the loop gain must exceed unity to allow oscillation to build from noise. As the amplitude grows, the gain-limiting mechanism (transistor saturation, AGC, or an amplitude limiter) reduces the gain to exactly unity, stabilizing the amplitude. If the gain falls below unity, oscillation dies. If the phase condition is met at multiple frequencies (multiple modes), the highest-gain mode dominates. The resonator's function is to ensure the phase condition is met at only one frequency with sufficient selectivity (high Q) to suppress competing modes.

Why does resonator Q determine phase noise?

Phase noise is the short-term random fluctuation of the oscillator frequency, measured in dBc/Hz at an offset from the carrier. Leeson's model shows that close-in phase noise improves as Q squared: L(fm) is proportional to (f0/2Qfm)^2, where f0 is the oscillator frequency, Q is the loaded resonator Q, and fm is the offset frequency. A quartz crystal with Q = 100,000 at 10 MHz produces phase noise below minus 150 dBc/Hz at 10 kHz offset. An LC VCO with Q = 50 at the same frequency produces minus 100 dBc/Hz at 10 kHz. The 66 dB difference comes entirely from the Q^2 relationship: (100000/50)^2 = 4 million = 66 dB. This is why every low-phase-noise oscillator uses the highest-Q resonator available at its operating frequency.

When should I use a TCXO vs. an OCXO?

A TCXO (temperature-compensated crystal oscillator) uses a varactor and a temperature sensor to correct for the crystal's frequency drift with temperature. Stability: plus or minus 0.5 to 2.5 ppm over minus 40 to plus 85 degrees C. Power: 10 to 50 mW. Warmup: seconds. Cost: 2 to 20 dollars. An OCXO (oven-controlled crystal oscillator) holds the crystal at a constant temperature (typically 80 to 90 degrees C) inside a thermally insulated enclosure. Stability: plus or minus 0.01 to 0.1 ppb. Power: 1 to 5 W (oven heating). Warmup: 2 to 10 minutes. Cost: 50 to 500 dollars. Use a TCXO for GPS, cellular, and IoT where ppm stability is sufficient. Use an OCXO for instrumentation, radar, and telecom where ppb stability is required.

Active Components

Oscillator

Every radio system needs at least one oscillator. The GPS satellite's rubidium atomic clock oscillator drifts less than 1 nanosecond per day, enabling position accuracy to 3 meters. The WiFi router's PLL-locked VCO tunes across 5.15 to 5.85 GHz in microseconds to hop between channels. The spectrum analyzer's YIG-tuned oscillator sweeps 26 GHz of bandwidth in a single smooth sweep. Each is an oscillator, but the resonator technology, Q factor, tuning mechanism, and resulting phase noise span five orders of magnitude. The resonator determines everything: quartz crystals for stability, varactor-tuned LC tanks for speed, dielectric resonators for microwave purity, and atomic references for ultimate accuracy.

Category: Active Components

Condition: Barkhausen: |Aβ| = 1, ∠Aβ = 0°

Key Spec: Phase noise, stability, tuning

The Resonator Determines the Oscillator

Type	Resonator Q	Phase Noise (10 kHz)	Tuning Range	Stability	Application
OCXO	100,000+	−160 dBc/Hz	None (fixed)	±0.01 ppb	Instrumentation, radar
TCXO	50,000 to 100,000	−150 dBc/Hz	None (fixed)	±0.5 to 2.5 ppm	GPS, cellular, IoT
DRO	5,000 to 15,000	−120 dBc/Hz	None (fixed)	±5 ppm	Microwave LO, PtP
LC VCO	20 to 50	−100 dBc/Hz	Octave (2:1)	Unlocked: ±500 ppm	PLL, synthesizer
YIG oscillator	1,000 to 5,000	−115 dBc/Hz	Multi-octave	±50 ppm	Spectrum analyzers, EW
Ring oscillator	5 to 10	−80 dBc/Hz	Wide	Unlocked: poor	CMOS PLL, clock gen

Barkhausen criterion:
|Aβ(f₀)| = 1, ∠Aβ(f₀) = n × 360°

Leeson's phase noise model:
L(f_m) = 10·log[(1 + (f₀/(2Qf_m))²) × (1 + f_c/f_m) × FkT/(2P_s)]

Q impact on phase noise:
L ∝ 1/Q² (at close-in offsets)
Q from 50 to 100,000: phase noise improves by 66 dB

Common Questions

Frequently Asked Questions

What is the Barkhausen criterion?

Loop gain = 1 and loop phase = 0° at the oscillation frequency. At startup, gain exceeds 1 to build oscillation from noise. Limiting mechanism (saturation/AGC) settles gain to exactly 1. The resonator's Q ensures the phase condition is met at only one frequency.

Why does Q determine phase noise?

Leeson: phase noise ∝ 1/Q². Crystal (Q = 100k) at 10 MHz: −150 dBc/Hz at 10 kHz. LC VCO (Q = 50): −100 dBc/Hz. The 66 dB difference = (100k/50)² = 4 million. Every low-noise oscillator uses the highest Q resonator available.

TCXO vs. OCXO?

TCXO: varactor compensation, ±0.5 to 2.5 ppm, 50 mW, seconds warmup, $2 to $20. OCXO: oven-stabilized crystal, ±0.01 ppb, 1 to 5 W, 2 to 10 min warmup, $50 to $500. TCXO for GPS/cellular; OCXO for radar/instrumentation.

Related Terms

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Signal Source

Oscillator Phase Noise Estimator

Enter resonator type, Q factor, operating frequency, and output power. Estimate close-in phase noise using Leeson's model and compare against system requirements.

Estimate Phase Noise