Using a Split-Ring Resonator to Measure Electromagnetic Properties of Liquids

Methods of Experimental Physics II. Spring 2015

Melissa Bosch and David Bosch

Abstract

We measured the resonance frequency and quality factor of a split-ring resonator when it was suspended in air, water, and aqueous solutions of !NaCl of various concentrations. From this data data, we determined the real and complex permittivity of water and observed the qualitative effects of !NaCl concentration on the quality factor and resonance frequency of the split-ring resonator. The real part of permittivity was found to be 77.74 ± 0.01, agreeing within 1

of the predicted 77.71. The imaginary permittivity was over 100

from a literature extrapolation. The dependance of quality factor on saline concentration followed our prediction qualitatively, but not quantitatively.

Introduction

Split ring resonators (SRRs) consist of a hollow, metallic object with a split along its length; the SRR used in this report is cylindrical. A SRR can be modeled as an LRC circuit, the inductance arising from the single “turn” of the SRR, capacitance from the two opposing sides of the slit, and resistance arising from the material of the SRR itself. [1]

For this experiment, the quality factor and resonant frequency of the SRR was measured when the SRR was suspended in air, submerged in water, and submerged in a !NaCl solution. The SRR was driven with a coupling loop and function generator, and another coupling loop and spectrum analyzer were used to record the signal response. From the resultant quality factor and resonant frequency measurements, values for the relative permittivity of water were found.

Theory

If a time-varying magnetic field of angular frequency ω is applied along the axis of the SRR, Faraday's Law of induction implies that a counteracting electromotive force will be generated on the surface of the SRR. The induced current will travel along the inner radius the SRR to skin depth δ, circling transverse to the axis. The effective resistance, capacitance, and inductance were written in terms of the geometric properties and skin depth of the SRR. Capacitance from the two faces of the SRR's slit were modeled as a parallel-plate capacitor and the inductance was approximated by a single-turn solenoid. Using well-known equations relating quality factor and resonance frequency to these parameters in a series LRC circuit, quality factor and resonance frequency were predicted in air.

To account for differences in measured and predicted quality factor, a resistive "fudge factor" can be introduced. was assumed to be frequency-dependant to simplify computations. It was necessary to introduce such a term due to the fact that distance between the emitter and reciever - thus coupling strength - directly affected the measred quality factor.

In water, capacitance is increased by a factor of complex permitivity. Complex analysis yields the prediction:

where the upper equation relates the real and imaginary parts of relative permitivity to the resonant frequencies in air and water, and the second equation relates the determinable quality factor in saline to the quality factor component arising from Rex, the predicted quality factor in air, and the ratio of real and imaginary relative permitivity.

In saline, the conducting effects of the medium form a paralell current path across the SRR gap. Analysis yielded:

where is the quality factor of the saline solution, σ is the conductivity of the saline solution, and is the resonant frequency (theoretically constant with σ). Quantitatively, quality factor should approximate the in-water resonance at low conductivities, but become inverselty proprortional to σ at high conductivities.

Apparatus and Method

Figure 1: Schematic of the SRR and coupling loops mounted within a cylindrical waveguide (a.k.a sewer pipe). The waveguide was shielded by multiple layers of aluminum foil to reduce radiative losses. (Original Figure)

The SRR was positioned coaxially within a cylindrical waveguide. One coupling loop was connected to a function generator and supplied a time-variant B-field along SRR axis. A second loop on the opposite face of the SRR detected the response and transmitted it to a Spectrum Analyzer.

The data acquisition process was automated through !LabVIEW. Our program measured the SRR output voltage read by Spectrum Analyzer for input signal frequencies in the vicinity of the resonance peaks.

Results and Analysis

Origin was used to perform a least-squares fit on each dataset to a Lorentzian.

Fig. 2. The normalized resonance measured when the SRR was submerged in air (pink circles squares: = 366.80 ± 0.01 MHz, Q = 2821 ± 5) and air (blue squares: = 41.620 ± 0.007 MHz, = 187 ± 5).

Using the in-air results, and the equations (4) and (5), was estimated to be 4.41 ± 0.02 m. Using this new factor, it was determined = 77.76 ± 0.01 and = 0.261 ± 0.001. These values differ 1σ and 114σ, respectively, from preditions based off of extrapoloation of Buchner et. al.'s data.This discrepancy may be due to numerous possibilities; for one, it is possible that

is frequency dependent. In addition the assumed value for the resistivity of the SRR copper or the extrapolation method itself could be invalid. Lastly, we speculate the discrepancy between the measured and predicted

is partly be due to too strong coupling between receiver and emitter, which could make the input resistance of the spectrum analyzer non-negligible. Supporting this notion, we noted that increasing coupling loop spacing raised Q dramatically.

Figure 3: Measurement of Saline-solution resonance. Left: Four resonance peaks acquired at four different saline concentrations. Right: The measured SRR quality factor vs. conductivity. The blue data points are experimental data, and the green trend is the prediction.

As predicted, at low σ, the peak plateaus, dominated by , and at high σ, it transitions to a regime inversely proportional to σ. However, quantitatively, results differ from prediction.

We reason that this could be due to an inhomogeneous saline solution, or the aforementioned coupling concern.

Acknowledgements

Special thanks to Kurt Wick, Peter Martin and Dr. Jake Bobowski for their invaluable help and advice! Without their help, this project might not have been finished.

References

[1] Bobowski, J. S. “Using Split-ring resonators to measure the electromagnetic properties of materials: An experiment for senior physics undergraduates.” Department of Physics, University of British Columbia Okanagan, Kelowna, British Columbia, Canada VIV 1V7. (2013)

[2] Buchner, Richard, Josef Barthel, and J. Stauber. "The dielectric relaxation of water between 0 C and 35 C." Chemical Physics Letters 306.1 (1999): 57-63.