Microstrip Ring Resonator Metrology Calculator

Q: Why is a ring resonator preferred over a straight half-wavelength resonator for substrate characterization?

Straight half-wavelength (λ/2) resonators have two open ends that radiate energy into free space and store fringing electric fields in the air. These parasitic end effects cause large measurement errors. A ring resonator has no open ends, eliminating radiation and end-capacitance errors.

Q: What causes split resonance peaks in a ring resonator?

Any structural asymmetry in the ring, such as non-uniform trace etching, localized dielectric inhomogeneity, or asymmetrical feed coupling gaps, excites orthogonal degenerate modes that split the single resonance peak into two distinct adjacent peaks.

Extract dielectric permittivity (ε_r) and loss tangent (tanδ) from measured S21 ring resonance frequencies without open-end radiation loss errors.

Calculation Mode

Ring Mean Radius r₀ (mm)

Resonant Mode n

Measured Resonant Freq f₀ (GHz)

Measured -3dB Bandwidth Δf (MHz)

Peak Insertion Loss S21 (dB)

Physical Substrate Parameters

Substrate Height h (mm):

Ring Trace Width W (mm):

Copper Thickness t (μm)

Coupling Gap S_gap (mm)

Extracted Dielectric Constant ε_r

3.66

ε_eff = 2.87

Extracted Loss Tangent (tanδ)

0.0038

Unloaded Q₀ = 134

Ring Resonator Metrology Parameters

Ring Circumference (2πr₀):

94.25 mm

Guided Wavelength λ_g:

47.12 mm

Loaded Quality Factor (Q_L): 123.5

Coupling Coefficient: Loose (Weak Coupling OK)

Harmonic Resonance Predictions

Mode n = 1: 1.76 GHz

Mode n = 2: 3.52 GHz (Current)

Mode n = 3: 5.28 GHz

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Microstrip Ring Resonator Permittivity Metrology

Determining the complex dielectric properties ((\varepsilon_r) and (\tan\delta)) of high-frequency PCB laminates (such as Rogers RO4350B, PTFE, or FR-4) requires measurement structures that do not suffer from open-end fringing capacitance or radiation losses. The microstrip ring resonator is the industry standard metrology device because its circular geometry is closed and boundary-free.

1. Resonance Condition & Effective Permittivity

Resonance occurs when the mean circumference of the ring is an integer multiple of the guided wavelength (\lambda_g):

$$2\pi r_0 = n \cdot \lambda_g = \frac{n \cdot c}{f_n \sqrt{\varepsilon_{\text{eff}}}}$$

From the measured resonance frequency (f_n) of mode (n), the effective dielectric constant is extracted directly:

$$\varepsilon_{\text{eff}} = \left( \frac{n \cdot c}{2\pi r_0 \cdot f_n} \right)^2$$

2. Quality Factors & Loss Tangent Extraction

From the loaded Q-factor (Q_L = f_0 / \Delta f) and insertion loss (S_{21}), the unloaded Q-factor (Q_0) is separated from external port loading:

$$Q_0 = \frac{Q_L}{1 - 10^{S_{21}/20}}$$

Total unloaded loss is the sum of conductor loss ((1/Q_c)) and dielectric loss ((1/Q_d = \tan\delta)):

$$\frac{1}{Q_0} = \frac{1}{Q_c} + \frac{1}{Q_d} \implies \tan\delta = \frac{1}{Q_0} - \frac{1}{Q_c}$$

3. Weak Coupling Rule

The gap (S_{\text{gap}}) between feed lines and the ring must be wide enough to maintain (S_{21} < -20\text{ dB}). Loose coupling ensures the feed lines do not perturb the ring's natural resonant frequency.

Frequently Asked Questions

Why is a ring resonator preferred over a straight half-wavelength resonator for substrate characterization?

Straight half-wavelength (λ/2) resonators have two open ends that radiate energy into free space and store fringing electric fields in the air. These parasitic end effects cause large measurement errors. A ring resonator has no open ends, eliminating radiation and end-capacitance errors.

What causes split resonance peaks in a ring resonator?

Any structural asymmetry in the ring, such as non-uniform trace etching, localized dielectric inhomogeneity, or asymmetrical feed coupling gaps, excites orthogonal degenerate modes that split the single resonance peak into two distinct adjacent peaks.