Measuring Reactive-Load Impedance with Transmission-Line Resonators Beyond the Perturbative Limit

TL;DR

This work develops a closed-form analytic framework to extract reactive-load impedance and dielectric loss from transmission-line resonators terminated by reactive loads, valid beyond perturbative limits. Central to the method are the resonator frequency shift, energy-participation ratio $p$, and loss tangent $\tan\delta$, with exact relations $f_r/f_{open} = (1/π)\arctan(Z_0/X) + n$, $p = |\sin\phi|/(\phi + |\sin\phi|)$, and $Q_i^{-1} = [2|\sin\phi|\tan\delta + 2π Q_{open}^{-1}]/[\phi + |\sin\phi|]$, where $\phi = 2π f_r/f_{open}$. The authors show maximal parameter sensitivity when $|X|$ is comparable to the resonator impedance $Z_0$ and introduce multimode self-calibration to extract both $f_{open}$ and device capacitance/inductance without a separate reference. Experimental validation with Nb CPW resonators terminated by vdW hBN capacitors yields dielectric constants $\kappa_{hBN}$ around 3.0 and loss tangents in the few $\times 10^{-6}$ range, consistent with literature, while multimode measurements substantially reduce uncertainty. Overall, the framework enables efficient, simulation-free microwave metrology of nanoscale materials and quantum devices, with broad applicability to resonator-based material characterization.

Abstract

We develop an analytic framework to extract circuit parameters and loss tangent from superconducting transmission-line resonators terminated by reactive loads, extending analysis beyond the perturbative regime. The formulation yields closed-form relations between resonant frequency, participation ratio, and internal quality factor, removing the need for full-wave simulations. We validate the framework through circuit simulations, finite-element modeling, and experimental measurements of van der Waals parallel-plate capacitors, using it to extract the dielectric constant and loss tangent of hexagonal boron nitride. Statistical analysis across multiple reference resonators, together with multimode self-calibration, demonstrates consistent and reproducible extraction of both capacitance and loss tangent in close agreement with literature values. In addition to parameter extraction, the analytic relations provide practical design guidelines for maximizing energy participation ratio in the load and improving the precision of resonator-based material metrology.

Figures (7)

• Figure 1: (a) Schematic of a waveguide resonator with characteristic impedance $Z_0$ and length $\ell$, terminated by a DUT characterized by $Z_L = r + jX$. The resonator is coupled to the busline through a coupling capacitor $C_c$. (b) The reactive termination shifts the resonant frequency from $f_\text{open}$ to $f_r$, and changes the internal quality factor from $Q_\text{open}$ to $Q_i$. The two corresponding ports for $S$-parameter measurement are labeled in (a). (c) Standing wave voltage (solid) and current (dashed) profiles along the resonator length for three representative reactive terminations: $\lvert X\rvert \ll Z_0$, $\lvert X\rvert = Z_0$, and $\lvert X\rvert \gg Z_0$. The termination is modeled as a purely reactive load ($jX$), shown in the schematic. The plots illustrate the capacitive terminations, as current lags the voltage by $\pi/2$ at the DUT.
• Figure 2: (a–b) Normalized resonant frequencies $f_r/f_\text{open}$ (a) and corresponding DUT participation ratios $p$ (b) as functions of normalized capacitive reactance $\lvert X\rvert /Z_0$ ($X<0$) for modes $n=1$ and $n=2$. Here, $Z_0=50~\Omega$. The short-ended ($\lambda/4$) and open-ended ($\lambda/2$) limits are indicated for reference. (c–d) Corresponding results for inductive reactance ($X>0$), shown for modes $n=0$ and $n=1$. In panels (a–d), dashed curves show the frequency-dependent reactance of example DUT (capacitors in brown, inductors in teal). For a 7 GHz resonator, the capacitors are chosen as 47 fF (Case I) and 606 fF (Case II), and the inductors as 77 pH (Case III) and 909 pH (Case IV). At resonance, Cases I and III correspond to $X = -500~\Omega$ and $X = +5~\Omega$, representing the perturbative regime. In contrast, Cases II and IV yield $\lvert X\rvert \approx 50~\Omega$, corresponding to conditions of maximal DUT energy participation ratio. The resonant frequency for each DUT is obtained from the intersection of its dashed reactance curve with the solid resonance curve, indicated by open circles. The corresponding participation ratios are shown in (b) and (d). (e) Relative uncertainty in extracted capacitance $\tilde{\sigma}_C = \sigma_C/C$ and inductance $\tilde{\sigma}_L = \sigma_L/L$ as a function of DUT energy participation ratio $p$. Uncertainties are normalized to the baseline frequency uncertainty of a reference resonator, $\tilde{\sigma}_{f_\text{open}} = \sigma_{f_\text{open}}/f_\text{open}$. Results are shown for Cases I–IV, defined consistently across panels (a)–(d).
• Figure 3: (a) Simulated results (circles) for capacitive DUT at fundamental mode $n=1$ obtained using Qucs for selected values of the loss tangent closely match the analytic predictions (dashed lines), computed by substituting the same loss tangent values into the model. Lower panel: DUT energy participation ratio $p$ versus $\lvert X\rvert /Z_0$ for a capacitive DUT configuration. The maximum achievable participation in this setup is approximately $17.85\%$. The two open circles reproduce the capacitive Cases I and II from Fig. \ref{['fig2']}(a). (b) Extracted DUT loss tangent for the two representative Cases in (a). Solid lines show $\tan\delta~Q_\text{open}$ as a function of $Q_i/Q_\text{open}$ (left axis), and dashed lines show the corresponding relative uncertainties $\sigma_{\tan\delta}/\tan\delta$, normalized to the baseline quality factor uncertainty of a reference resonator (right axis).
• Figure 4: (a) Optical image of a hanger-type Nb CPW resonator terminated by a PPC. The zoomed region highlights the PPC, with colored outlines marking the boundaries of the vdW flakes and the PPC area labeled in pink. An Al patch is included schematically to indicate the electrical connection. The schematic below shows the capacitor cross-section. (b) Normalized capacitance, $C/\varepsilon_0$, versus the geometric ratio $A/d$ (overlap area $A$ divided by hBN thickness $d$). The gray dashed line represents the PPC model with $\kappa_{\mathrm{hBN}} = 3.03$. Linear fits yields $\kappa_{\mathrm{hBN}} = 3.33 \pm 0.06$ from the single-mode analysis (circles) and $\kappa_{\mathrm{hBN}} = 3.06 \pm 0.08$ after multimode self-calibration (diamonds), demonstrating improved capacitance accuracy after correcting for $f_0$. (c) Histogram of inverse quality factor $Q_\text{open}^{-1}$ for reference resonators (gray bars). The data can be described by a log-normal distribution (solid line). Vertical dashed lines indicate the single-photon $Q_i^{-1}$ values at fundamental modes of DUTs A–C. (d) Extracted loss tangent $\tan\delta$ for DUTs A-C. Box-and-whisker plots show the single-mode results, which rely on comparison to reference resonators. Diamonds with error bars show the multimode results, which eliminate the need for any reference resonator. The gray dashed line and shaded band indicate representative literature values for hBN. The low-$Q$ tail observed in (c) leads to apparent negative outliers in the single-mode $\tan\delta$ extraction; these are removed entirely after multimode calibration, yielding more accurate and statistically tighter results.
• Figure 5: Experimental setup used for resonator characterization. The total attenuation of the input microwave line from room temperature to the mixing chamber is approximately 65 dB. A vector network analyzer (VNA) is used as both the microwave source and detector.