What is the Q factor?

The quality factor (Q) is the fundamental metric for resonator performance. It has three equivalent definitions: Q = 2 pi times (energy stored) / (energy dissipated per cycle). Q = f_0 / BW_3dB (the resonant frequency divided by the 3 dB bandwidth). Q = omega_0 L / R for a series RLC circuit, or Q = R / (omega_0 L) for a parallel RLC circuit. Higher Q means: sharper frequency selectivity (narrower bandwidth), lower loss in filters (insertion loss is inversely proportional to Q), lower phase noise in oscillators (phase noise improves as 20 log Q), more precise frequency references, and longer ring-down time (energy decays as exp(minus omega_0 t / 2Q)). Unloaded Q (Q_u) is the inherent Q of the resonator without external loading. Loaded Q (Q_L) includes the effect of external coupling: 1/Q_L = 1/Q_u + 1/Q_ext. A resonator with Q_u = 10000 coupled to give Q_L = 5000 has 50 percent of its energy dissipated in external loads (by design, for a filter) and 50 percent in internal losses.

What resonator types are used at RF?

Lumped LC: inductor and capacitor. Q limited by inductor losses (Q_L = 30 to 100 at RF frequencies). Used below 1 GHz where distributed structures are too large. Size: millimeters. Microstrip: half-wave or quarter-wave transmission line sections. Q = 100 to 300, limited by conductor and radiation losses. Used in planar filters up to 30 GHz. Dielectric resonator (DR): ceramic puck with very high permittivity (epsilon_r = 20 to 90) and very low loss (Q = 5000 to 50000). Used in base station filters and stable oscillators (DRO). Size: much smaller than a cavity at the same frequency. Cavity: enclosed metal structure (rectangular or cylindrical). Q = 5000 to 100000 depending on mode, frequency, and surface finish. Used in high-performance filters and oscillators. Size: approximately lambda/2 in each dimension. Quartz crystal: piezoelectric resonator. Q = 10000 to 1000000. The standard for precision frequency references. Used in oscillators (TCXO, OCXO) and filters. Frequency range: 1 kHz to 200 MHz. BAW/FBAR: bulk acoustic wave resonator. Q = 500 to 2000. Used in miniature RF filters for phones (duplexers). Frequency: 500 MHz to 6 GHz.

How does Q affect oscillator phase noise?

Leeson's equation for oscillator phase noise shows a direct relationship with Q: L(f_m) = 10 log((2FkT/P_s) times (1 + (f_0 / (2 Q_L f_m)) squared) times (1 + f_c / f_m)). At offset frequencies within the half-bandwidth (f_m less than f_0/(2Q)): phase noise improves as 20 log Q. Doubling Q reduces phase noise by 6 dB. This is why high-Q resonators are essential for low-phase-noise oscillators. Examples: a lumped LC VCO with Q = 20 at 2 GHz might achieve minus 100 dBc/Hz at 100 kHz offset. A dielectric resonator oscillator (DRO) with Q = 5000 at 10 GHz achieves minus 130 dBc/Hz. A sapphire-loaded cavity oscillator with Q = 200000 at 10 GHz achieves minus 170 dBc/Hz. A quartz crystal oscillator (OCXO) with Q = 100000 at 10 MHz achieves minus 175 dBc/Hz at 10 kHz offset, which is the gold standard for frequency stability. The practical limit is that higher-Q resonators are larger, more expensive, and have narrower tuning range, which is why VCOs (wide tuning, low Q) are combined with PLLs referenced to crystals (narrow tuning, high Q).

Energy Storage

Resonator

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EM energy storage at f₀. Q = f₀/BW = 2π×E_stored/E_lost. LC: Q 50-200. Microstrip: Q 100-300. Dielectric: Q 5000-50000. Cavity: Q 5000-100000. Crystal: Q 10⁴-10⁶. Higher Q = sharper selectivity, lower filter IL, lower oscillator phase noise (20logQ). 1/Q_L = 1/Q_u + 1/Q_ext. Leeson: L(f_m) ∝ 1/Q².

Crystal: Q 10⁶

DR: Q 50k

LC: Q 50-200

Understanding Resonators

Resonators are the frequency-selective building blocks of RF systems. Every filter is built from coupled resonators. Every oscillator contains a resonator that determines its frequency and spectral purity. The quality factor (Q) of the resonator fundamentally limits the performance of both: higher Q means sharper filters and cleaner oscillators.

The quest for higher Q has driven resonator technology from simple LC circuits (Q of 50) to superconducting cavities (Q exceeding 10¹¹). Each resonator technology occupies a niche defined by its Q, frequency range, size, cost, and tunability.

Resonator Equations

Q factor:
Q = f₀/BW_3dB
Q = 2π × W_stored/W_dissipated
Series RLC: Q = ω₀L/R
Parallel RLC: Q = R/(ω₀L)

Loaded Q:
1/Q_L = 1/Q_u + 1/Q_ext

Leeson (oscillator PN):
L(f_m) = 10log((2FkT/P_s)×
(1+(f₀/(2Q_Lf_m))²))
2×Q: −6 dB phase noise

Resonator Technology Comparison

Type	Q	Freq Range	Size	Application
Lumped LC	50-200	DC-1 GHz	mm	VCO, matching
Microstrip	100-300	1-30 GHz	mm-cm	PCB filter
Dielectric	5k-50k	1-50 GHz	cm	BTS filter, DRO
Cavity	5k-100k	1-100 GHz	cm-m	Hi-perf filter
Quartz crystal	10k-1M	1kHz-200MHz	mm	TCXO, OCXO

Common Questions

Frequently Asked Questions

Q factor?

f₀/BW = energy stored/energy lost. Higher Q = sharper filter, lower IL, lower oscillator phase noise (20logQ improvement). Q_u (unloaded): intrinsic. Q_L (loaded): includes external coupling. 1/Q_L = 1/Q_u + 1/Q_ext. Crystal Q=10⁶ = gold standard.

Types?

LC: Q 50-200, <1GHz, smallest. Microstrip: Q 100-300, planar, radiation-limited. DR: ceramic puck, ε_r=20-90, Q=5k-50k, BTS filters. Cavity: Q=5k-100k, λ/2 size, hi-perf. Crystal: piezoelectric, Q=10⁶, OCXO standard. BAW/FBAR: Q=500-2k, phone duplexers.

Phase noise?

Leeson: L(f_m) ∝ 1/Q². 2×Q = −6 dB PN. LC VCO Q=20: −100 dBc/Hz @100kHz. DRO Q=5000: −130. Sapphire Q=200k: −170. Crystal OCXO Q=100k: −175 @10kHz. Higher Q = larger, costlier, less tunable. VCO+PLL: combines wide tune (VCO) with crystal stability.

Related Terms

← Resistance Return Loss →

RF Design

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