Transmission-Line Resonators & Q Factor

Fig 1 — Common microstrip resonators. Each is a line section whose ends (open, short, or joined into a ring) trap a standing wave; the feed lines couple to it through gaps or parallel edges

What Is Resonance?

A resonator stores energy that oscillates between two forms — electric and magnetic — at a natural frequency. In a lumped LC circuit the energy moves between the capacitor and the inductor. In a transmission-line resonator it moves between the voltage and current maxima of a standing wave. At resonance the average stored electric and magnetic energies are equal, W_e = W_m, so the input impedance is purely resistive.

Every practical resonator also loses energy, and the quality factor measures how slowly it does so:

Definition of Q

Q = ω_{0} \frac{average energy stored}{power lost} = ω_{0} \frac{W_{e} + W_{m}}{P_{loss}}

Q is 2π times the ratio of the energy stored to the energy lost per cycle. A high-Q resonator rings for many cycles after it is excited and has a narrow bandwidth.

The series and parallel RLC models

Derivation — impedance of a series RLC near resonance

A series RLC has Z_in = R + jωL + 1/(jωC). With ω₀² = 1/(LC) and ω = ω₀ + Δω,

Z_{in} = R + j ω L (1 - \frac{ω_{0}^{2}}{ω^{2}}) = R + j ω L \frac{(ω - ω_{0}) (ω + ω_{0})}{ω^{2}} \approx R + j 2 L Δ ω

Power stored is W_e + W_m = ½L|I|² at resonance, and P_loss = ½R|I|², so Q = ω₀L/R. Writing L = QR/ω₀,

Series and parallel RLC near resonance

Z_{in} \approx R (1 + j 2 Q \frac{Δ ω}{ω_{0}})

Y_{in} \approx \frac{1}{R} (1 + j 2 Q \frac{Δ ω}{ω_{0}})

BW = \frac{ω_{0}}{Q}

The half-power points occur where the reactive part equals the resistive part, giving a fractional bandwidth of 1/Q. The parallel case is the dual, with Q = ω₀RC.

Transmission-line resonators behave exactly like these lumped circuits near each resonance, which is why filter and oscillator theory written for LC circuits carries over directly to microwave lines.

Open-Circuit and Short-Circuit Resonators

A lossless line terminated in a short or open has a purely reactive input impedance (Part 12), Z_sc = jZ₀ tan βℓ and Z_oc = −jZ₀ cot βℓ. Resonances occur where these pass through zero (series resonance) or infinity (parallel resonance). Four combinations arise:

Resonator	Length at f₀	Behaves like	Higher resonances
Short-circuited λ/2	λ_g/2	Series RLC	2f₀, 3f₀, 4f₀ …
Short-circuited λ/4	λ_g/4	Parallel RLC	3f₀, 5f₀ … (odd only)
Open-circuited λ/2	λ_g/2	Parallel RLC	2f₀, 3f₀, 4f₀ …
Open-circuited λ/4	λ_g/4	Series RLC	3f₀, 5f₀ … (odd only)

The half-wave resonator and Q = β/2α

Derivation — RLC equivalent of a short-circuited λ/2 line

A lossy short-circuited line has Z_in = Z₀ tanh(αℓ + jβℓ) = Z₀(tanh αℓ + j tan βℓ)/(1 + j tan βℓ tanh αℓ). At ω = ω₀ + Δω the electrical length is βℓ = π + πΔω/ω₀, so tan βℓ ≈ πΔω/ω₀; for small loss tanh αℓ ≈ αℓ. The product of the two small terms in the denominator is negligible:

Z_{in} \approx Z_{0} (α ℓ + j π \frac{Δ ω}{ω_{0}})

Comparing with the series RLC, R + j2LΔω, gives the equivalent elements and the Q:

Short-circuited λ/2 resonator

R = Z_{0} α ℓ

L = \frac{π Z_{0}}{2 ω_{0}}

C = \frac{1}{ω_{0}^{2} L}

Q = \frac{ω_{0} L}{R} = \frac{π}{2 α ℓ} = \frac{β}{2 α}

Because ℓ = π/β at resonance, Q depends only on the line's own loss — not on Z₀ or the length. The same result holds for every line resonator.

The open-circuited λ/2 line is the dual: near resonance it is a parallel RLC with R = Z₀/(αℓ), and the same Q = β/2α. On microstrip, the open-ended half-wave line is the most common resonator of all — it needs no via, and it is the basis of end-coupled, parallel-coupled and hairpin filters, and of the rectangular patch antenna (antenna types).

The quarter-wave resonator

Derivation — RLC equivalent of a short-circuited λ/4 line

Now βℓ = π/2 + πΔω/(2ω₀), so tan βℓ = −cot(πΔω/2ω₀) ≈ −2ω₀/(πΔω), a large number. Dividing numerator and denominator of Z₀ tanh(γℓ) by j tan βℓ:

Z_{in} = Z_{0} \frac{1 + \tanh α ℓ / (j \tan β ℓ)}{\tanh α ℓ + 1 / (j \tan β ℓ)} \approx \frac{Z_{0}}{α ℓ + j π Δ ω / (2 ω_{0})}

Short-circuited λ/4 resonator (parallel RLC)

R = \frac{Z_{0}}{α ℓ}

C = \frac{π}{4 ω_{0} Z_{0}}

L = \frac{1}{ω_{0}^{2} C}

Q = ω_{0} R C = \frac{π}{4 α ℓ} = \frac{β}{2 α}

The quarter-wave resonator is half the length of the half-wave one and has the same Q. Its next resonance is at 3f₀, which gives filters built from it a wider spurious-free stopband.

Fig 2 — Input impedance of a short-circuited λ/4 resonator against frequency, for unloaded Q = 50 and 200 at f₀. Parallel resonances (peaks) at f₀ and 3f₀; series resonances (nulls) at 2f₀ and 4f₀. Higher Q gives taller, narrower peaks

Q₀ = 200Q₀ = 50

Microstrip Resonators

On microstrip the guided wavelength replaces the free-space one, λ_g = λ₀/√ε_eff, and every open end is electrically longer than drawn by Δℓ (Part 13). The common forms (Fig 1) are:

Open-ended λ/2 line — simplest, no via; spurious response at 2f₀.
Shorted λ/4 line — half the size and the next response at 3f₀, but the via adds inductance and loss, and its position must be controlled precisely.
Ring resonator — a closed loop resonant when its mean circumference is nλ_g. With no open ends there are no end corrections, so it is the standard test structure for measuring ε_eff, dispersion and loss of a substrate.
Hairpin and folded resonators — a λ/2 line bent into a U to save area; the arms couple to each other, so EM simulation is needed.
Stepped-impedance resonators — a low-impedance centre with high-impedance ends; shorter than λ/2 and with spurious resonances pushed away from 2f₀.

Coupled-line resonators

When two resonators sit close together their fields overlap and energy passes between them. Two identical coupled resonators no longer have one resonance but two, split either side of f₀. The split measures the coupling coefficient:

Coupling coefficient from the split resonances

k = \frac{f_{2}^{2} - f_{1}^{2}}{f_{2}^{2} + f_{1}^{2}} \approx \frac{f_{2} - f_{1}}{f_{0}}

Electric (edge-to-edge, open ends facing) and magnetic (shorted ends facing) coupling add or subtract; the formula gives the net magnitude. Larger spacing → smaller k.

Parallel-coupled (edge-coupled) half-wave resonators overlap by λ_g/4. Each coupled section is a pair of coupled lines described by even- and odd-mode impedances, just like the differential pair in Part 11; the gap sets Z₀e and Z₀o and therefore k.

Quality Factor — Unloaded, Loaded and External

Unloaded Q

The unloaded Q, Q_u (or Q₀), counts only losses inside the resonator itself. On microstrip there are three mechanisms, and their reciprocals add:

Unloaded Q of a microstrip resonator

\frac{1}{Q_{u}} = \frac{1}{Q_{c}} + \frac{1}{Q_{d}} + \frac{1}{Q_{r}}

Q_{c} = \frac{β}{2 α_{c}}

Q_{d} = \frac{1}{\tan δ_{eff}}

\tan δ_{eff} = \frac{ε_{r} (ε_{eff} - 1)}{ε_{eff} (ε_{r} - 1)} \tan δ

Qc from conductor loss α_c (Part 8), Qd from dielectric loss, Qr from radiation at open ends and bends. The dielectric term depends only on the effective loss tangent.

The conductor term grows with the cross-section: wider strips on thicker substrates carry the same current with less resistance, so Q_c rises roughly in proportion to h. The radiation term falls with h and εr⁻¹: thick, low-εr substrates radiate more from open ends (that is why patch antennas use them). The dielectric term is fixed by the material. Typical microstrip resonators reach Q_u ≈ 50 on FR-4, 100–250 on low-loss laminates, and a few hundred on alumina; coaxial and dielectric resonators reach thousands, waveguide cavities ten thousand or more.

Loaded and external Q

A resonator is useful only when it is coupled to something — a feed line, a filter port, a transistor. The coupled circuit draws energy from the resonator just as its internal losses do. Its effect is described by the external Q, Q_e, and the loaded Q, Q_L, combines both:

Loaded Q

\frac{1}{Q_{L}} = \frac{1}{Q_{u}} + \frac{1}{Q_{e}} g = \frac{Q_{u}}{Q_{e}}

The coupling coefficient g = Q_u/Q_e classifies the coupling: g < 1 under-coupled, g = 1 critically coupled (matched for a one-port), g > 1 over-coupled.

Derivation — measuring Qu from a transmission (two-port) response

For a resonator coupled to two ports, the transmission near resonance is a Lorentzian whose half-power bandwidth gives Q_L directly. At resonance the power delivered to the external circuit (ports) is the fraction Q_L/Q_e of all the power dissipated, and for a symmetric resonator the transmission coefficient is

Two-port resonator response

S_{21} (f) \approx \frac{S_{21} (f_{0})}{1 + j 2 Q_{L} \frac{f - f_{0}}{f_{0}}}

| S_{21} (f_{0}) | = \frac{Q_{L}}{Q_{e}} = 1 - \frac{Q_{L}}{Q_{u}}

Q_{L} = \frac{f_{0}}{Δ f_{3 dB}}

Measure f₀, the 3 dB bandwidth and |S21(f₀)| on a VNA, then solve for Qu. The method works best for weak coupling (|S21| below about −10 dB), where QL ≈ Qu.

Fig 3 — Transmission through a two-port resonator with the same external Q (Qe = 30) and three values of unloaded Q. Lower Qu gives more insertion loss at resonance and a wider, flatter peak

Qu = 400 (−0.6 dB)Qu = 130 (−1.8 dB)Qu = 50 (−4.1 dB)

Applications in Filters

Almost every microwave bandpass filter is a chain of coupled resonators. Filter synthesis starts from the lowpass prototype values g₀, g₁ … g_N+1 (Butterworth, Chebyshev …) and converts them to the two quantities a resonator layout can control:

Coupled-resonator filter design equations

k_{i, i + 1} = \frac{FBW}{\sqrt{g_{i} g_{i + 1}}}

Q_{e, in} = \frac{g_{0} g_{1}}{FBW}

Q_{e, out} = \frac{g_{N} g_{N + 1}}{FBW}

FBW is the fractional bandwidth. The coupling k between each adjacent pair is set by its spacing; the external Q of the first and last resonators is set by the feed (tap position or input gap).

The designer then simulates two resonators at a time to find the spacing that gives each k, and one resonator with its feed to find the tap that gives Q_e. The unloaded Q decides the filter's midband loss:

Midband insertion loss from finite Qu (Cohn)

IL \approx 4.343 \sum_{i = 1}^{N} \frac{g_{i}}{FBW Q_{u}} dB

Narrow filters need high-Q resonators: the loss is inversely proportional to FBW·Qu.

Applications in Oscillators

An oscillator is an amplifier with a resonator in its feedback path; the resonator sets the frequency and, through its loaded Q, the phase noise. In Leeson's model the close-in phase noise falls as 1/Q_L² — doubling Q_L improves it by 6 dB. Microstrip λ/4 and λ/2 resonators are used where cost and size matter; dielectric resonators coupled to a microstrip line (DROs) where low phase noise matters; and ring or hairpin resonators where tunability with a varactor is needed. The oscillators guide covers the circuits; here the important points are the frequency stability and the Q of the line resonator itself (Examples 9 and 10).

Worked Examples

FR-4 values: 50 Ω, h = 1.6 mm, w = 3.07 mm, ε_eff = 3.24, tan δ = 0.02. RO4003C values: εr = 3.55, tan δ = 0.0027, 50 Ω on 0.508 mm (w = 1.14 mm) or 0.813 mm (w = 1.82 mm), ε_eff ≈ 2.79. Copper σ = 5.8 × 10⁷ S/m.

Length of an Open-Ended Half-Wave Resonator

Problem: Design an open-ended λ/2 microstrip resonator on FR-4 for 2.45 GHz. The open-end extension is 0.63 mm per end (Part 13, Example 2). At what frequency is the first spurious resonance?

Half a guided wavelength

\frac{λ_{g}}{2} = \frac{299.79}{2 \times 2.45 \times \sqrt{3.24}} = \frac{299.79}{8.82} = 33.99 mm

Correct for both open ends

ℓ_{drawn} = 33.99 - 2 \times 0.63 = 32.73 mm

Answer: draw the line 32.7 mm long; next resonance near 4.9 GHz

The next resonance is at 2f₀ (slightly below, because dispersion raises ε_eff at higher frequency). A filter built from these resonators has a spurious passband near 4.9 GHz — a problem for a 2.45 GHz transmitter's second harmonic. A λ/4 resonator or a stepped-impedance resonator moves it further away.

Measuring εr with a Ring Resonator

Problem: A ring resonator on 0.508 mm RO4003C has a 50 Ω (1.14 mm) strip and a mean radius of 15.0 mm. The first resonance (n = 1) is measured at 1.89 GHz. Find ε_eff, and the substrate εr that reproduces it in the microstrip formulas.

Mean circumference = λg at n = 1

L = 2 π \times 15.0 = 94.25 mm

ε_{eff} = {(\frac{c}{L f_{1}})}^{2} = {(\frac{299.79}{94.25 \times 1.89})}^{2} = {1.683}^{2} = 2.83

Invert the Hammerstad–Jensen formula (w/h = 2.24)

ε_{eff} (ε_{r}) = 2.83 \Rightarrow ε_{r} \approx 3.62

Answer: ε_eff = 2.83 and εr ≈ 3.62 — higher than the 3.55 used for design

With εr = 3.55 the ring would resonate at 1.906 GHz; the measurement is 0.8% lower. This is exactly how laminate makers arrive at a "design Dk" for microstrip that differs from the clamped-stripline "process Dk" (for RO4003C, 3.55 versus 3.38): the microstrip value includes the effects of copper roughness and the real field distribution. Measuring the higher-order resonances (n = 2, 3 …) gives ε_eff against frequency, i.e. dispersion.

Unloaded Q of a Microstrip Resonator on FR-4

Problem: Estimate Q_c, Q_d and Q_u at 2.45 GHz for the FR-4 line, ignoring radiation. Use the wide-strip estimate α_c ≈ R_s/(Z₀w).

Conductor term

R_{s} = \sqrt{\frac{π f μ_{0}}{σ}} = 0.0129 Ω

α_{c} = \frac{0.0129}{50 \times 3.07 \times 10^{- 3}} = 0.084 Np/m

β = \frac{2 π f}{c} \sqrt{3.24} = 92.4 rad/m

Q_{c} = \frac{92.4}{2 \times 0.084} = 549

Dielectric term

\tan δ_{eff} = \frac{4.3 \times 2.24}{3.24 \times 3.3} \times 0.02 = 0.0180 Q_{d} = 55.5

Combined

Q_{u} = {(\frac{1}{549} + \frac{1}{55.5})}^{- 1} = 50.4

Answer: Qu ≈ 50, almost entirely limited by FR-4's dielectric loss

Improving the copper would hardly help; only a lower-loss dielectric raises Q. Real FR-4 resonators often measure Q_u ≈ 40–50 once radiation and roughness are included.

Unloaded Q on RO4003C — Effect of Substrate Thickness

Problem: Repeat Example 3 for 50 Ω resonators on 0.508 mm and 0.813 mm RO4003C.

Same Qd for both (β = 85.7 rad/m)

\tan δ_{eff} = \frac{3.55 \times 1.79}{2.79 \times 2.55} \times 0.0027 = 0.00241 Q_{d} = 415

Conductor term

0.508 mm : α_{c} = 0.227 Np/m, Q_{c} = 189

0.813 mm : α_{c} = 0.142 Np/m, Q_{c} = 302

Combined

Q_{u} (0.508) = {(\frac{1}{189} + \frac{1}{415})}^{- 1} = 130

Q_{u} (0.813) = {(\frac{1}{302} + \frac{1}{415})}^{- 1} = 175

Answer: Qu ≈ 130 on 0.508 mm and ≈ 175 on 0.813 mm — now conductor loss dominates

On a low-loss laminate the copper sets the Q, and thicker substrates give higher Q because the strip is wider. Surface roughness (Part 8) typically lowers these figures by 20–30%. The trade-off is radiation: thicker substrates radiate more from open ends and bends.

RLC Equivalent of a Shorted Quarter-Wave Resonator

Problem: Find R, L and C of a short-circuited λ/4 50 Ω resonator on FR-4 at 2.45 GHz, with total attenuation α = α_c + α_d = 0.084 + 0.832 = 0.916 Np/m and ℓ = 17.0 mm.

Equivalent elements (ω₀ = 1.539 × 10¹⁰ rad/s)

C = \frac{π}{4 ω_{0} Z_{0}} = 1.02 pF

L = \frac{1}{ω_{0}^{2} C} = 4.14 nH

R = \frac{Z_{0}}{α ℓ} = \frac{50}{0.0156} = 3.21 k Ω

Check the Q

Q = ω_{0} R C = 1.539 \times 10^{10} \times 3210 \times 1.02 \times 10^{- 12} = 50.4

Answer: R = 3.2 kΩ, L = 4.1 nH, C = 1.0 pF; Q = 50, as in Example 3

These values let the line resonator be dropped into a lumped oscillator or filter design. A lower-impedance line gives more C and less L, which reduces the effect of a transistor's parasitic capacitance on the resonant frequency.

Extracting Qu, QL and Qe from a Measurement

Problem: A gap-coupled resonator measures f₀ = 2.45 GHz, 3 dB bandwidth 70 MHz and |S21(f₀)| = −6.0 dB. Find Q_L, Q_u and Q_e.

Loaded Q and peak transmission

Q_{L} = \frac{2450}{70} = 35.0 | S_{21} | = 10^{- 6 / 20} = 0.501

Unloaded and external Q

Q_{u} = \frac{Q_{L}}{1 - | S_{21} |} = \frac{35.0}{0.499} = 70.1 Q_{e} = \frac{Q_{L}}{| S_{21} |} = 69.9

Answer: Q_L = 35, Q_u ≈ 70, Q_e ≈ 70 (each port about 140)

The coupling is strong (g ≈ 1), so Q_L is half of Q_u and the result is sensitive to the measured |S21|. For an accurate Q_u the gaps would be widened until |S21(f₀)| ≈ −20 dB, where Q_L is within 10% of Q_u.

Designing the Coupling of a Two-Pole Filter

Problem: A two-resonator 0.1 dB Chebyshev filter (g₁ = 0.8431, g₂ = 0.6220, g₀ = 1) is required at 2.45 GHz with 5% bandwidth. Find the required k₁₂ and Q_e. A trial layout shows split resonances at 2.38 and 2.52 GHz — is the spacing right?

Required coupling and external Q

k_{12} = \frac{0.05}{\sqrt{0.8431 \times 0.6220}} = 0.0690 Q_{e} = \frac{0.8431}{0.05} = 16.9

Trial layout

k = \frac{{2.52}^{2} - {2.38}^{2}}{{2.52}^{2} + {2.38}^{2}} = \frac{0.686}{12.01} = 0.0571

Answer: k₁₂ = 0.069 and Qe = 16.9 needed; the trial coupling (0.057) is too weak — reduce the gap

The resonance split must widen to about 2.37/2.54 GHz (about 170 MHz apart). Q_e = 16.9 is very strong coupling — a gap would be far too narrow, so the feed is usually tapped directly onto the resonator at a point where its voltage gives the required Q_e.

Insertion Loss of a Narrowband Filter versus Qu

Problem: A three-pole 0.1 dB Chebyshev filter (g = 1.0316, 1.1474, 1.0316) has a 5% bandwidth. Estimate its midband loss with resonators of Q_u = 50 (FR-4), 130 (RO4003C) and 400.

Σg = 3.2106; IL = 13.94/(FBW·Qu)

Q_{u} = 50 : \frac{13.94}{2.5} = 5.6 dB

Q_{u} = 130 : \frac{13.94}{6.5} = 2.1 dB

Q_{u} = 400 : \frac{13.94}{20} = 0.70 dB

Answer: 5.6 dB on FR-4, 2.1 dB on RO4003C, 0.7 dB with Qu = 400

A 5% filter on FR-4 loses over 70% of the power — and the loss doubles if the bandwidth is halved. Narrowband filters below 1 dB loss need Q_u in the hundreds: low-loss substrates, suspended stripline, coaxial or dielectric resonators.

Resonator Q and Oscillator Phase Noise

Problem: An oscillator using a microstrip resonator has Q_L = 25. By how much would the close-in phase noise improve with a dielectric resonator of loaded Q 1000, other things being equal?

Leeson: close-in phase noise ∝ 1/QL²

Δ ℒ = 20 \log_{10} \frac{Q_{L 2}}{Q_{L 1}} = 20 \log_{10} \frac{1000}{25} = 32 dB

Answer: about 32 dB lower close-in phase noise

This is why fixed-frequency microwave sources (LNB local oscillators, radar references) use dielectric resonators, while wideband VCOs accept the noise of low-Q line resonators in exchange for tuning range. Loaded Q matters, so the resonator is coupled as lightly as the loop gain allows.

Temperature Drift of a Line Resonator

Problem: RO4003C has a temperature coefficient of dielectric constant of +40 ppm/°C and an in-plane expansion of about 11 ppm/°C. Estimate the drift of a 2.45 GHz microstrip resonator from −20 °C to +80 °C. The fraction of ε_eff that follows εr is about 0.89 (the same factor as in tan δ_eff).

f ∝ 1/(ℓ√εeff)

\frac{Δ f}{f} \approx - \frac{1}{2} \frac{Δ ε_{eff}}{ε_{eff}} - \frac{Δ ℓ}{ℓ} = - \frac{1}{2} (0.89 \times 40) - 11 = - 29 ppm/°C

Over 100 °C

Δ f = - 29 \times 10^{- 6} \times 100 \times 2.45 GHz = - 7.1 MHz

Answer: about −29 ppm/°C, a 7 MHz drift over the range

For a 5% filter (122 MHz wide) this is a 6% shift of the passband — tolerable, but it must be included in the design margin. Oscillators need far better stability: dielectric resonators are made with a temperature coefficient chosen to cancel the circuit's drift, and synthesizers lock the oscillator to a crystal reference.

Summary

A line section with open or short ends is a resonator; near resonance it is exactly a series or parallel RLC.
λ/2 resonators repeat at every multiple of f₀; λ/4 resonators only at odd multiples.
Every line resonator has Q = β/2α; on microstrip 1/Q_u = 1/Q_c + 1/Q_d + 1/Q_r, with Q_d = 1/tan δ_eff.
Coupling adds external Q: 1/Q_L = 1/Q_u + 1/Q_e; measure Q_L from the 3 dB bandwidth and Q_u = Q_L/(1 − |S21|).
Coupled-resonator filters are designed with k = FBW/√(g_ig_i+1) and Q_e = g₀g₁/FBW; their loss scales as 1/(FBW·Q_u).
Oscillator phase noise scales as 1/Q_L²; ring resonators measure ε_eff, dispersion and loss.

The final part brings the whole series together. Part 15: Practical RF Transmission-Line Design walks through a complete design flow: choosing a structure and substrate, calculating dimensions, designing connector launches and via fences, simulating, fabricating and verifying with a VNA.

📘 Go deeper: Resonators in oscillators

How the resonator's loaded Q sets the frequency stability and phase noise of RF oscillators. Open Oscillators and VCOs →

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