What is the difference between unloaded and loaded Q?

Unloaded Q (Q_U) is the Q of the resonator by itself, with no external connections. It represents only the internal losses of the resonator (conductor loss, dielectric loss, radiation). This is the intrinsic property of the resonator. Loaded Q (Q_L) is the Q of the resonator when connected to external circuits (source and load). The external connections add additional energy loss paths (the desired signal extraction), which always reduce Q. The relationship is: 1/Q_L = 1/Q_U + 1/Q_ext, where Q_ext represents the energy coupling to the external circuit. For a filter, Q_L determines the bandwidth: BW = f_0/Q_L. But the insertion loss depends on Q_U: IL = 20 log(1 minus Q_L/Q_U) per resonator. If Q_L approaches Q_U, nearly all the stored energy is lost internally (high insertion loss). Good filter design requires Q_U much greater than Q_L, typically Q_U/Q_L greater than 5 for acceptable insertion loss. Example: a microstrip resonator with Q_U = 200 used in a filter requiring Q_L = 50. IL per resonator = 20 log(1 minus 50/200) = 20 log(0.75) = 2.5 dB. For a 4-pole filter: 10 dB total IL (unacceptable for many applications, which is why high-Q resonator technologies are so important).

What determines component Q?

Capacitor Q: Q = 1/(omega times C times ESR), where ESR is the equivalent series resistance. High-quality RF capacitors (porcelain, ultra-low-loss PTFE): Q greater than 1000 at 1 GHz. Standard MLCC (X7R): Q of 50 to 200 at 1 GHz. ESR increases with frequency due to skin effect and electrode resistance. Capacitor Q dominates at high frequencies. Inductor Q: Q = omega times L / R_series. The series resistance includes DC resistance, skin effect loss, proximity effect loss, and core loss (for wound inductors). Air-core SMD (0402): Q of 30 to 80 at 1 GHz. Wire-wound: Q of 50 to 150 at lower frequencies. On-chip spiral: Q of 5 to 20 (substrate loss dominates). Inductor Q is typically the limiting factor in integrated circuits. Transmission line resonators: microstrip quarter-wave: Q of 100 to 300. Stripline: Q of 200 to 500. Substrate integrated waveguide: Q of 200 to 500. Machined waveguide cavity: Q of 5000 to 50000. Dielectric resonator: Q of 5000 to 50000 (depends on material). Crystal (quartz): Q of 10000 to 1000000. Superconducting cavity: Q greater than 10 to the 10.

How does Q affect filter and oscillator performance?

Filters: the Q of the resonators determines the filter's insertion loss and the achievable selectivity. A Chebyshev bandpass filter with n poles, bandwidth BW, center frequency f_0, and resonator Q_U has an approximate insertion loss of: IL approximately equals 4.343 times f_0 times sum of g_i / (BW times Q_U) dB, where g_i are the prototype element values. For a 5-pole, 2 percent bandwidth filter at 2 GHz with Q_U = 200: IL approximately equals 4.343 times 2000 times 8.8 / (40 times 200) = 9.5 dB. With Q_U = 5000 (cavity): IL approximately equals 0.38 dB. This is why narrow-band filters require high-Q resonators. A 0.1 percent bandwidth channel filter is practically impossible with microstrip (Q = 200) but straightforward with a cavity (Q = 10000). Oscillators: the Leeson model shows that phase noise improves as Q squared: L(f_m) is proportional to 1/Q_L squared. Doubling Q improves phase noise by 6 dB. This is why low-phase-noise oscillators use the highest-Q resonators available: quartz crystals for HF, sapphire-loaded cavities for microwave, and superconducting cavities for the ultimate performance.

Resonance Metric

Quality Factor

/kyoo fak-ter/ — Q

Q = ω₀W/P_loss = f₀/BW_3dB. Higher Q: sharper, lower loss. 1/Q_L = 1/Q_U + 1/Q_ext. Capacitor: Q=1/(ωC×ESR). Inductor: Q=ωL/R. Microstrip: 100-300. Cavity: 10k-100k. Crystal: 10k-1M. Filter IL: ∝1/Q_U. Oscillator PN: ∝1/Q_L². 2×Q = −6 dB phase noise. Q is everything in RF.

Cavity: 10k-100k

Crystal: 10k-1M

On-chip: 5-20

Understanding Quality Factor

Q is the single most important figure of merit in RF resonant circuit design. It determines filter insertion loss, oscillator phase noise, and the achievable selectivity of any frequency-selective circuit. Every RF engineer's career is a continuous quest for higher Q.

The challenge is that Q varies enormously across technologies: an on-chip spiral inductor has Q of 5-20, while a superconducting cavity exceeds 10 billion. Choosing the right resonator technology for a given application is one of the most impactful design decisions in RF engineering.

Q Factor Equations

Definitions:
Q = 2π × energy stored / energy lost per cycle
Q = ω₀W_stored/P_dissipated
Q = f₀/BW_3dB

Loading:
1/Q_L = 1/Q_U + 1/Q_ext
Critical coupling: Q_ext = Q_U, Q_L = Q_U/2

Filter insertion loss (per resonator):
IL = 20log(1 − Q_L/Q_U) dB
Q_U/Q_L > 5: IL < 2 dB/resonator

Resonator Q Comparison

Technology	Q	Freq Range	Size	Application
On-chip spiral	5-20	1-100 GHz	μm	IC VCO/filter
MLCC/SMD	50-200	DC-6 GHz	0402-0805	Matching/bias
Microstrip λ/4	100-300	1-30 GHz	mm-cm	PCB filter
Dielectric res.	5k-50k	1-30 GHz	mm-cm	DRO, BTS filter
Machined cavity	10k-100k	0.5-100 GHz	cm	BTS duplexer

Common Questions

Frequently Asked Questions

Loaded vs unloaded?

Q_U: resonator alone (internal losses). Q_L: with external coupling (always lower). 1/Q_L=1/Q_U+1/Q_ext. Filter BW=f₀/Q_L. IL depends on Q_U/Q_L ratio: if Q_L approaches Q_U, most energy lost internally. Need Q_U>5×Q_L for low IL. Critical coupling: Q_ext=Q_U.

Component Q?

Cap: Q=1/(ωC×ESR). Porcelain: Q>1000. MLCC X7R: 50-200. Inductor: Q=ωL/R. SMD air-core: 30-80. On-chip: 5-20 (substrate loss). Transmission line: microstrip 100-300, stripline 200-500. Cavity: 10k-100k. Crystal: 10k-1M. SC cavity: >10¹⁰. Inductor Q = IC bottleneck.

Impact?

Filter IL ∝ Σg_i/(BW×Q_U). 2% BW, 5-pole, Q=200: IL~10dB. Q=5000: IL~0.4dB. Narrow filters need high Q. Oscillator: PN ∝ 1/Q². 2×Q = −6dB PN. Crystal OCXO (Q=100k): −175 dBc/Hz. On-chip VCO (Q=10): −90 dBc/Hz. Q = RF performance ceiling.

Related Terms

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RF Design

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