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Capacitance readout circuits based on weakly-coupled resonators Hafizi-Moori, Siamak 2016

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Capacitance Readout Circuits Based onWeakly-Coupled ResonatorsbySiamak Hafizi-MooriB.Sc., University of Tehran, 1991M.Sc., Tehran Polytechnic, 1995A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2016c© Siamak Hafizi-Moori 2016AbstractCapacitive sensors and their associated readout circuits are well known and havebeen used in many measurement applications in different industries. Improvingthe sensitivity, resolution and accuracy of measuring small capacitance changeshas always been one of the important research topics, especially in recent yearsthat sensors are becoming smaller in size with lower associated capacitance val-ues. This thesis focuses proposes a new method for implementing capacitancereadout circuits with higher sensitivity. This is the first time, to our knowledge,that this method has ever been applied directly in electrical domain for capacitancemeasurement applications.The proposed method, which is based on weakly-coupled-resonators (WCRs)concept, can achieve considerably (orders of magnitudes) higher sensitivity whilesimplifying the analog front end circuitry and reducing the cost. For compari-son, capacitance-to-frequency conversion readout circuits were chosen, which areone of the most reliable and best performing designs and also the closest to ourWCR method since both involve shift in natural modes due to capacitance changes.Analysis and SPICE simulations followed by experiments proved the concept. Theexperimental results have shown almost two orders of magnitude higher relativesensitivity for our two-degree-of-freedom (2DOF) WCR-based system. In the nextstep we proposed a novel (named hybrid) method to reduce the measurement er-iiAbstractror considerably (4 to 6 times lower). Hybrid method is robust and insensitiveto variations in excitation frequency, which is one of the main sources for errors.We have also analyzed the use of active inductors in our coupled resonators. Theanalyses and simulations proved the concept. This opens an avenue towards im-plementation of WCR-based readout in integrated circuits; specifically applicablefor micro-electro-mechanical systems (MEMS) devices, and even integrating bothMEMS sensors and the readout circuit in the same integrated circuit (IC) package.Another route on this research was to exploit the insensitivity and robustness ofthree-degree-of freedom (3DOF) weakly-coupled resonators to resonant frequencydeviations. Analyses, followed by simulations, proved that applying 3DOF WCRin sensing differential capacitance changes does not require frequency tracking, yethas the same sensitivity achieved in 2DOF-based readout circuits.iiiPrefaceI, Siamak Hafizi-Moori, am the principal contributor of all chapters. Dr. EdmondCretu, supervisor of the research, has provided guidelines, technical support andediting assistance on the manuscript.In the early stages of the project, as a reference for one of the conventionalreadout circuits, a capacitance-to-voltage readout circuit was designed and testedby Ahmed Sharkia and I, which is being presented in Appendix A and helped incompleting the experimental results of the following paper:E.H. Sarraf, A. Sharkia, S. Moori, M. Sharma and E. Cretu. “High SensitivityAccelerometer Operating on the Border of Stability with Digital Sliding ModeControl”, IEEE Sensors 2013.A version of chapter 4 has been published. S. Hafizi-Moori and E. Cretu,“Weakly-coupled resonators in capacitive readout circuits,” Circuits and SystemsI: Regular Papers, IEEE Transactions on, vol. 62, no. 2, pp. 337–346, 2015.A version of chapter 5 has been submitted to a journal and is under review.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 History of Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Readout Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Thesis Outlines . . . . . . . . . . . . . . . . . . . . . . . . . . . 8vTable of Contents2 Capacitive Sensors and Their Associated Readout Circuits . . . . . 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Capacitive Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Capacitance Readout Circuits . . . . . . . . . . . . . . . . . . . 172.3.1 Capacitance to Voltage Converter . . . . . . . . . . . . . 212.3.2 Capacitance to Duty Cycle Converter . . . . . . . . . . . 232.3.3 Capacitance to Phase Shift Converter . . . . . . . . . . . 262.3.4 Capacitance to Frequency Converter . . . . . . . . . . . 292.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Weakly-Coupled-Resonators as Capacitance Readout Circuits . . . 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Weakly Coupled Resonators . . . . . . . . . . . . . . . . . . . . 393.3 Reasons for Proposing WCRs as an Alternative for Readout Cir-cuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 WCR-Based Readout Circuit Analysis and Performance Estimation 474.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Theory of Operation . . . . . . . . . . . . . . . . . . . . . . . . 484.2.1 Analytical Solution . . . . . . . . . . . . . . . . . . . . 554.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 734.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77viTable of Contents5 Error Reduction in WCR-Based Capacitance Readout Circuits . . 795.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 Theory of Operation . . . . . . . . . . . . . . . . . . . . . . . . 835.2.1 Measurement Sensitivity . . . . . . . . . . . . . . . . . . 855.2.2 Measurement Error . . . . . . . . . . . . . . . . . . . . 955.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 1025.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076 3DOF WCRs in Capacitance Measurement . . . . . . . . . . . . . . 1086.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2.1 Differential Perturbation Detailed Analysis . . . . . . . . 1146.2.2 System Response to Common Mode Excitation . . . . . . 1236.2.3 Differential Perturbation Analysis in Common Mode Ex-citation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.3 Circuit Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 1326.3.1 Single-Sided Excitation, Differential Perturbation Case . 1326.3.2 Differential Excitation, Differential Perturbation Case . . 1376.3.3 Common-Mode Excitation, Differential Perturbation Case 1386.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427 Active Inductors in WCRs . . . . . . . . . . . . . . . . . . . . . . . 1447.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.2 Real (Nonideal) Inductors . . . . . . . . . . . . . . . . . . . . . 1447.3 Active Inductors . . . . . . . . . . . . . . . . . . . . . . . . . . 147viiTable of Contents7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538 Conclusions and Further Discussions . . . . . . . . . . . . . . . . . 1558.1 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . 1558.2 Prospects and Open Problems . . . . . . . . . . . . . . . . . . . 158Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Appendix A Circuit Simulations and Justification for Using CFC as theBenchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171A.2 CVC Simulation and Implementation Results . . . . . . . . . . . 171A.3 CDC Simulation Results . . . . . . . . . . . . . . . . . . . . . . 176A.4 CPC Simulation Results . . . . . . . . . . . . . . . . . . . . . . 179A.5 CFC Simulation Results . . . . . . . . . . . . . . . . . . . . . . 185viiiList of Tables2.1 Capacitance readout circuit methods, a brief comparison. . . . . . 363.1 Analogy between mass-spring-damper and RLC coupled oscillators. 464.1 Analytical values for 2DOF WCRs at out-of-phase resonance. . . 644.2 Comparison table between ∆∣∣∣ i2i1 ∣∣∣and ∆ ff methods of measurement. 674.3 Experimental results for both eigenvalue and eigenvector basedmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.1 Experimental results for all three methods. . . . . . . . . . . . . . 1066.1 Measured values at fixed excitation frequency (mode 2). . . . . . 1366.2 Simulation results for differential excitation at first mode. . . . . . 1376.3 Simulation results for differential excitation at third mode. . . . . 1376.4 Simulation results; common mode excitation at 2nd mode. . . . . 1387.1 Gyrator-based WCR simulation results (mode 1 excitation). . . . . 153A.1 CDC circuit simulation data. . . . . . . . . . . . . . . . . . . . . 178A.2 CPC simulation results, ratio of voltage magnitudes vs. perturbation.184A.3 CPC simulation results, ratio of voltage magnitudes vs. perturbation.186ixList of TablesA.4 CFC simulation results, output frequency vs. perturbation. . . . . 191xList of Figures1.1 Typical sensor and associated readout circuit. . . . . . . . . . . . 41.2 Loci of the dimensionless eigenvalues of the two-pendulum system. 61.3 An RLC series weakly coupled resonator system. . . . . . . . . . 82.1 Simple capacitor construction and schematic symbol. . . . . . . . 122.2 Different arrangements for capacitive sensors. . . . . . . . . . . . 132.3 Applications of capacitive sensors. . . . . . . . . . . . . . . . . . 152.4 Image of an accelerometer obtained with PolytecMSA−500r . . 162.5 Image of accelerometer designed at Georgia Institute of Technology. 172.6 Examples of capacitance-to-voltage (CVC) readout circuits. . . . 192.7 Differential CVC based on charge integration. . . . . . . . . . . . 222.8 An improved CVC readout, based on low duty cycle periodic reset. 232.9 Schematic of a CDC with direct configuration. . . . . . . . . . . . 242.10 Schematic representation of a CDC with the relaxation oscillator . 252.11 Phase shift generated using capacitance in an RC circuit. . . . . . 272.12 Phase shift plot for differential RC circuit. . . . . . . . . . . . . . 272.13 CPC using zero-crossing detection. . . . . . . . . . . . . . . . . . 292.14 CPC using analog multiplier. . . . . . . . . . . . . . . . . . . . . 292.15 CFC based on simple Hartley oscillator. . . . . . . . . . . . . . . 30xiList of Figures2.16 Switched-capacitor harmonic oscillator with AGC . . . . . . . . . 312.17 CFC based on CVC cascaded with VFC. . . . . . . . . . . . . . . 322.18 CFC based on integration and periodic reset. . . . . . . . . . . . . 343.1 Lumped-element model of a coupled 2DOF mechanical system . . 413.2 Loci of the dimensionless eigenvalues of the two coupled oscillators. 413.3 SEM image of a set of coupled gold-foil cantilevers. . . . . . . . . 424.1 Two weekly coupled mechanical resonators. . . . . . . . . . . . . 494.2 2DOF weekly-coupled series RLC resonators. . . . . . . . . . . . 504.3 Two weekly coupled resonators natural frequencies loci. . . . . . 524.4 Mode localization in two weekly-coupled-resonators. . . . . . . . 544.5 Effect of loss on sensitivity. Coefficient r in (4.25). . . . . . . . . 634.6 Relative shift in resonant frequency vs. eigenmode in 2DOF WCRs. 654.7 Circuit schematic of 2DOF WCRs for SPICE simulations. . . . . 664.8 AC analysis of 2DOF WCRs based on series RLC resonators. . . . 664.9 i1 plots, coupled RLC circuit AC analysis with sweeping C2. . . . 684.10 i2 plots, coupled RLC circuit AC analysis with sweeping C2 . . . . 684.11 Resonant frequency loci veering in 2DOF WCR. . . . . . . . . . 694.12 Sensitivity comparison between three different methods. . . . . . 704.13 LabVIEW-Multisim co-simulation for 2DOF WCRs. . . . . . . . 714.14 LabVIEW-Multisim co-simulation results. . . . . . . . . . . . . . 724.15 High-level-block-diagram of proposed capacitance readout. . . . . 734.16 Test setup for experimental measurements. . . . . . . . . . . . . . 744.17 Sensitivity comparison between simulations and experiments. . . . 764.18 Effect of parasitic parameters on frequency response. . . . . . . . 77xiiList of Figures5.1 Series RLC two weakly coupled resonators. . . . . . . . . . . . . 795.2 Relative shift in resonant frequency vs. eigenmode . . . . . . . . 805.3 Bode Plot for Series RLC Resonator . . . . . . . . . . . . . . . . 825.4 System high-level-block-diagram. . . . . . . . . . . . . . . . . . 835.5 Examples of conventional capacitance measurement methods. . . 845.6 Eigenvalue loci veering. . . . . . . . . . . . . . . . . . . . . . . . 855.7 Frequency response of the system for three values of perturbationδ =−0.1%, 0%and 0.1%. . . . . . . . . . . . . . . . . . . . . . 935.8 Error comparison and improvement by hybrid method. . . . . . . 965.9 Error comparison and improvement by hybrid method. . . . . . . 985.10 Amplitudes of I1 and I2 at out-of-phase resonance. . . . . . . . . . 1005.11 Analytical: linear approximation vs. exact for |I1|/|I2|. . . . . . . 1015.12 Analytical vs. simulation for |I1|/|I2|. . . . . . . . . . . . . . . . 1025.13 |I1|/|I2| plot around out-of-phase resonant frequencies. . . . . . . 1035.14 High-level-block-diagram of proposed capacitance readout. . . . . 1045.15 Magnitude of v1/v2 around out-of-phase resonance. . . . . . . . . 1055.16 Measurement error comparison. . . . . . . . . . . . . . . . . . . 1066.1 3DOF coupled spring-mass system with stiffness perturbation. . . 1096.2 3DOF weekly coupled series RLC resonators. . . . . . . . . . . . 1116.3 Frequency shift of all three modes in one-sided perturbation. . . . 1126.4 Frequency shift of all three modes in differential perturbation. . . 1136.5 3DOF WCR schematic with differential excitation. . . . . . . . . 1146.6 3DOF WCR schematic with common mode excitation. . . . . . . 1156.7 3DOF WCR schematic with differential excitation. . . . . . . . . 115xiiiList of Figures6.8 Frequency response; unperturbed 3DOF WCRs under differentialexcitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.9 3DOF WCR schematic with common mode excitation. . . . . . . 1206.10 Frequency response; unperturbed 3DOF WCRs under common modeexcitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.11 3DOF WCR, impact of loss on resonant frequencies. . . . . . . . 1236.12 Phase plot of ε1 and ε2 at ω2 vs. Q factor. . . . . . . . . . . . . . 1306.13 Magnitude plot of ε1 and ε2 at ω2 vs. Q factor. . . . . . . . . . . . 1326.14 Three WCR veering from SPICE simulation. . . . . . . . . . . . 1336.15 Three WCR, relative sensitivities, mode 1 excitation. . . . . . . . 1346.16 Three WCR, relative sensitivities, mode 2 excitation. . . . . . . . 1356.17 Three WCR, relative sensitivities, mode 3 excitation. . . . . . . . 1356.18 3WCR, normalized current I2at 2nd mode. . . . . . . . . . . . . . 1366.19 The effect of quality factor on f2- δ dependence. . . . . . . . . . 1396.20 I2 magnitude for mode 2, common mode excitation, differentialperturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.21 Effect of Q factor on the sensitivity. . . . . . . . . . . . . . . . . 1416.22 Effect of Q factor on the magnitude of ε2 (simulation). . . . . . . 1427.1 Equivalent circuits for a real inductor (from CoilCraft Inc.). . . . . 1457.2 An example of an active inductor. . . . . . . . . . . . . . . . . . 1477.3 Realization of a floating inductor using gyrators. . . . . . . . . . . 1497.4 Realization of a floating inductor using gyrators ,rearranged. . . . 1507.5 Circuit used in simulation of 2DOF WCR based on active inductors. 1517.6 Gyrator-based 2DOF WCR simulation at balance. . . . . . . . . . 152xivList of Figures7.7 Gyrator-based 2DOF WCR simulation for different perturbations. 1527.8 Relative sensitivity of gyrator-based 2DOF WCR. . . . . . . . . . 153A.1 CVC readout using charge integration, capacitor driving circuit. . 172A.2 CVC readout using charge integration, input stage differential am-plifier, filtration and demodulation. . . . . . . . . . . . . . . . . . 173A.3 CVC readout using charge integration, output buffer and LP filter. 173A.4 CVC based on differential charge amplifier, capacitance changesand output voltage plots. . . . . . . . . . . . . . . . . . . . . . . 174A.5 CVC based on differential charge amplifier, intermediate nodessimulation waveforms . . . . . . . . . . . . . . . . . . . . . . . . 175A.6 Varicap SPICE model. . . . . . . . . . . . . . . . . . . . . . . . 176A.7 Schematic representation of a CDC with the relaxation oscillator. . 177A.8 Simulation graph for the CDC readout circuit. . . . . . . . . . . . 178A.9 Simulation results for the CDC readout circuit. . . . . . . . . . . 179A.10 CPC readout circuit using charge amplifier. . . . . . . . . . . . . 181A.11 Improved CPC readout circuit. . . . . . . . . . . . . . . . . . . . 182A.12 Simulation results for the readout circuit. . . . . . . . . . . . . . . 183A.13 CPC parametric sweep simulation results. . . . . . . . . . . . . . 184A.14 CFC readout circuit based on Hartley oscillator. . . . . . . . . . . 185A.15 Simulation results for the readout circuit. . . . . . . . . . . . . . . 186A.16 CFC readout circuit based on switched-capacitors. . . . . . . . . . 189A.17 Simulation results for the readout circuit based on SC. . . . . . . . 190A.18 Simulation results for the readout circuit based on SC . . . . . . . 191xvList of AcronymsADC Analog-to-digital converterASIC Application specific integrated circuitsCDC Capacitance-to-duty cycle converterCFC Capacitance-to-frequency converterCPC Capacitance-to-phase shift converterCVC Capacitance-to-voltage converterDOF Degree-of-freedomESR Equivalent series resistanceIC Integrated circuitMEMS Micro-electro-mechanical systemsOPA Operational amplifierPCB Printed circuit boardPLL Phase-locked loopVARACTOR Variable-reactance diodexviList of AcronymsVARICAP Variable capacitor, variable-capacitance diodeVFC Voltage-to-frequency converterWCR Weakly-coupled-resonatorxviiAcknowledgmentsI would like to express my gratitude to my research supervisor Dr. Edmond Cretufor his supervision and his professional guidance. He is indeed much more than anacademic supervisor and I would always remember his support and advice. I amalso very thankful of Dr. Shahriar Mirabbasi and Dr. Robert Rohling who helpedme a lot, especially at the beginning and initiation of my research at UBC.I would like to thank all my Ph.D. examination committee members for theirtime and valuable comments and feedback.xviiiDedicationTo my wife,my mother andmemory of my fatherxixChapter 1Introduction1.1 History of SensorsHuman life is becoming more and more dependent on the measurement of physicalphenomena. The advancement in science owes a great deal to our ability to measurethe environment around us; as Lord Kelvin aptly puts it: “To measure is to know”.In the course of history, the methods for measurement have advanced alongsidethe advancements in science and engineering, resulted in the use of “sensors”. Asensor, in its crudest form, is a tool that yields a certain electrical output whenexposed to the a physical phenomenon. This broad definition includes everythingfrom the first electric thermostat patented by Warren S. Johnson in 1885, to themost advanced pressure sensors used in high performance cars today.Sensors have become the default tool for us to measure the properties of ourphysical surroundings, and with the growth in the number of their applications,they have reached a market share of $79.5 billion in 2013, and are expected toreach nearly $154.4 billion by 2020 [1]. Of course this explosive growth is helpedby the advancements in electronics and IC manufacturing capabilities, which beganfrom the invention of the transistor in 1947 at Bell Laboratories, and the first im-plementation of the monolithic IC at Fairchild Semiconductor in 1959. This trend11.1. History of Sensorshas continued to the present day by the introduction of increasingly sophisticatedand specialized sensors.The reasons for this fast growth in reliance on sensors could be summarized asfollows [2].• Sensors have an electrical output, which is the most versatile form of signalcarrier that can be used for processing and storing sensor related information.• Many different back-end circuitry options are available for use with the sen-sors, resulting in the ability to manufacture the sensor and the signal condi-tioning subsystem in the same package.• Given that the output of the sensors can be amplified, there is the possibilityof using active sensors, which do not absorb energy from the process beingmeasured.• Sensors can be designed to measure nonelectric quantities through the usageof appropriate material and techniques (changes in the properties of nonelec-tric material can be translated to electrical changes, which can be detected inthe electrical domain.• The sensors output can be displayed, recorded and further processed, to pro-vide more insight into the nature of the variations of the process being mea-sured.The need for measuring different types of physical quantities has led to the devel-opment of many different sensor types, each of which has its own unique character-istics. Sensors can be categorized based on their need for power (active or passive),their output signals (analog or digital), or their mode of operation, e.g. deflection21.1. History of Sensorstype or null type. However, in electronic engineering, it is preferred to categorizesensors based on the measured electrical quantity (e.g. resistance, capacitance, andinductance) [2].Resistive sensors are widely used in measurement applications since one of thesimplest way of mapping the measurand onto electrical variations is through theequivalent electrical resistance modulation. The outputs of the resistive sensorsare readily available for processing, hence these sensors have simple measurementcircuitry. Also, resistive sensors offer many options with regards to their size,resistance value, back-end circuitry and AC/DC operation [3]. Resistive sensorshave a high sensitivity in general; however, their resolution is affected by thermalnoise, which means that various environment related factors will influence theiroutput [3].Inductive sensors rely on the change of self or mutual inductance of a coil orset of coils for measurement. These sensors can be used in applications where thethickness of objects needs to be measured. The detection of change in inductivesensors can be done only by using AC readout circuits. Because of the effectof inductance on the neighboring circuitry, proper shielding is desirable for aircored inductors [3]. This aspect, coupled with the direct relationship between thephysical dimensions of the coil and the quality factor (i.e. higher quality factorcoil needs a lower equivalent series resistance, and consequently require the largercross section of winding wire for the same number of turns) , means that thesesensors are usually bulkier than other passive sensor types.Capacitive sensors are widely used for displacement measurement [3]. Becauseof their precise performance, low cost construction, simple structure and versatility,they are common solutions for measuring variables such as acceleration, humidity,31.2. Readout Circuitsliquid levels etc. Capacitive sensors work on the principle of measuring the capac-itance between two or more conductors in a dielectric environment [3, 4]. Becauseof their desirable characteristics, more emphasis has been placed on the capacitivesensors in this thesis.1.2 Readout CircuitsGenerally speaking, nearly all sensors are coupled to with an electrical subsystemin order to measure their respective electrical output. This electronic interface isknown as the readout circuit. The readout circuits are as diverse as the sensorsthemselves, but the primary task of all of them is signal conditioning, which in theIEEE standard 1451.4-2004 is defined as processing of a sensor output signal withoperations such as amplification, compensation, filtering and normalization [5].Figure 1.1: Block diagram of a typical sensor and the associated readout circuit.As can be seen in Figure 1.1, the readout circuit is the interface between thesensor and the rest of the system. It performs signal conditioning tasks with thepower received from the power supply. This thesis specifically considers readoutcircuits used for capacitive sensors. The existing capacitance readout circuits use41.3. Motivationvarious methods, such as capacitance-to-voltage (CVC), capacitance-to-frequency(CFC), capacitance-to-phase shift (CPC), capacitance-to-duty cycle (CDC) conver-sions etc. All of the aforementioned methods utilize analog circuitry for filtering,amplification and even switching.There are many challenges for the existing capacitance readout circuit tech-niques. Typically, factors such as intrinsic noise, switching noise (in circuits basedon switching elements), offset problems and temperature dependency of the cir-cuit components result in a relatively high number of passive and active circuitcomponents. In addition, measuring small variations in capacitance (the measuredparameter), is often disturbed by the inherent presence of parasitic capacitances,which could be even larger than the sensing capacitance. Capacitive MEMS sen-sors have a typical capacitance in the range of 0.2pF to 1pF, parasitic capacitanceof about 2pF and typical resolution range of 1aF to 10aF. Achieving a high sensitiv-ity/gain in measurement with a low signal to noise ratio has always been a seriouschallenge.In some applications the gain-bandwidth trade-off becomes another challenge;the higher the rate of the sensor capacitance changes, the lower the overall gain ofthe readout circuit. This tradeoff is not of a huge concern in this project since thecapacitance variation is considered to be quasi-static (that is, very slow relative tothe time constants of the readout circuits).1.3 MotivationIn order to address some of the aforementioned challenges, especially the high sen-sitivity and robustness, we searched for an alternate and innovative method with51.3. Motivationhigher sensitivity and inherent simplicity. There has been a very elegant methodfor measuring perturbations that has been used in mechanical domain for decades.This method is based on weakly-coupled resonators (WCRs), and has a long his-tory in mechanical and acoustic domain. WCRs exhibit an interesting feature re-lated to the mode localization (energy localization), which is the energy repartitionbetween the two resonators due to perturbation. Mode localization was examinedin solid state physics applications by P. W. Anderson for the first time [6, 7] whicheventually led him win the Noble prize in physics in 1977.The behavior of the resonant frequencies as functions of perturbation and thecoupling strengths between the resonators, when plotted, gives two sharply veer-ing traces with high local curvatures. This behavior, shown in Fig. 1.2, was firstinvestigated by Pierre [8] who named it loci veering.Figure 1.2: Loci of the dimensionless eigenvalues of the two-pendulum system interms of the disorder, ∆l; representative mode shapes are shown. (a) The stronginterpendulum coupling case, R = 0.5; neither eigenvalue loci veering nor modelocalization occur. (b) The weak interpendulum coupling case, R =0.025; bothcurve veering and strong localization occur.An alternative, more intuitive, representation of veering phenomenon is pre-61.3. Motivationsented in Fig 4.4. It is generally accepted that the eigenvalues of the WCRs systemrepresent the resonant frequencies of the system. As a result the term eigenvalueloci veering phenomena has been used to describe a range of similar behaviors indisordered structures in the mechanical and MEMS field [9, 10, 11, 12, 13, 14, 15].This thesis offers an innovative method for capacitance measurement basedon weakly-coupled resonators (WCRs), which is proven to have more sensitivityand circuit simplicity. It will be shown that WCR-based readout circuit can reachseveral orders of magnitude higher sensitivity than other state-of-the art methods(e.g. capacitance-to-frequency method). On the other hand, there is a challenge inmatching component values for both resonators. The higher the sensitivity of WCRmethod, the more the negative impact of mismatch on the correct reading. Anotherchallenge with WCRs is the bandwidth of the perturbation since the theory assumesimplicitly a quasi-static perturbation. In this thesis we also study the effect of thelosses on the sensitivity of the system, which has not been offered in the previousmechanical/MEMS researches.The simplest WCR in the electrical domain consists of two series/parallel RLCcircuits coupled through a capacitor /inductor. This thesis proposes to use theWCRs as an alternative for a capacitance readout circuit. Figure 2 shows the con-figuration of the WCR fundamental circuit examined in this thesis. Using such anarrangement for capacitance readout circuits results in lower number of compo-nents required, contributing to the low cost, low power and high reliability of thesecircuits. Moreover, as will be shown in the following chapters, the relative sensi-tivity of a WCR arrangement is much higher than the existing comparable readoutcircuit methodologies.71.4. Thesis OutlinesRLCCCVsR L CFigure 1.3: An RLC series weakly coupled resonator system.1.4 Thesis OutlinesWith the above mentioned information in mind, the main contributions of this the-sis include: the use of WCRs for capacitive measurement (the first applicationof WCRs principles in the electrical domain in this direction); achieving a muchhigher measurement sensitivity compared to the existing capacitance readout cir-cuit methodologies; proposing a method for minimizing the susceptibility of thereadout circuit to the excitation frequency errors; utilizing three WCRs for differ-ential capacitive measurements, yielding thus a lower dependance on the excitationfrequency and consequently a more robust readout circuit; and finally examiningthe possibility of using active inductors in a WCR arrangement for a capacitancereadout circuit (with the potential of future single-die integration of the capacitancereadout technique).The next chapters are structured as follows. Chapter 2 presents a representa-tive, but by no means exhaustive, literature review of the state-of-the-art in sensorreadout circuit technology. This will include overviews of various types of sensorsand different methodologies used for readout circuits, narrowing down to the read-out circuits used for capacitance measurement. A justification of choosing CFC81.4. Thesis Outlinesmethod as a reference for comparison with our proposed WCR-based method ispresented at the end of this chapter. The details of various state-of-the-art readoutcircuits mentioned in this chapter, along with simulation results, are presented inAppendix A.Chapter 3 begins by giving a more detailed and historical introduction to WCRs,and enumerating their various conventional uses. It then continues by formallyproposing the use of WCRs as the alternative method for capacitance readout cir-cuits. The justification for such a proposal is given and finally the reader is pre-sented with the research question.Chapter 4 presents the theoretical analysis and simulation results examiningthe use of the WCR methodology for the readout circuit. This is then followed bythe sensitivity analysis as well as the simulation and practical circuit implementa-tion results. These results are then compared with the conventional CFC method,showing the full extent of the sensitivity improvement.Chapter 5 examines the capacitance measurement error problem for the WCRmethodology. It then proposes a method to minimize the measurement error, byusing a combination of the CFC and WCR methods, resulting in a more robustreadout circuit.Chapter 6 proposes the use of three-degree-of-freedom WCRs in the readoutcircuit to perform differential capacitive measurement in a robust manner. Thischapter also studies the effect of losses (quality factor) on the sensitivity. It showsthe trade-offs between quality factor (Q), dynamic range of measurable perturba-tion and sensitivity. The analytical and simulation results are provided and com-pared for such an arrangement.Chapter 7 explores the use of an active inductor (in the form of an op-amp-91.4. Thesis Outlinesbased circuit) as alternatives for bulky passive inductors in implementing of WCRmethodology by theoretical analysis and simulation. This is helpful toward inte-grating a complete WCR-based readout circuit on a chip. This chapter is followedby final discussions, and outlining further avenues of research for the future inchapter 8.10Chapter 2Capacitive Sensors and TheirAssociated Readout Circuits2.1 IntroductionThis chapter introduces capacitive sensors and their associated readout circuits.The fundamentals of capacitive sensing are presented in §2.2, where various ca-pacitive sensor configurations are depicted together with different ways of cate-gorizing such. In addition , the benefits and limitations of capacitive sensors areexamined in detail.Section 2.3 begins by defining what readout circuits are and different categoriesthey fall into. Subsequent subsections are then devoted to examining each of themethods with more in-depth explanations for various configuration where neces-sary. The related simulations are presented in Appendix A. This chapter continueswith a justification for choosing one of the capacitance readout circuit methodolo-gies as a benchmark for comparison with our proposed WCR method. A summaryof this chapter is presented in the last section.112.2. Capacitive Sensors2.2 Capacitive SensorsThe past decades have seen a burgeoning attention to the use of capacitive sensorsfor sensing and detecting physical quantities such as pressure, rotational angles,linear displacement and acceleration [2]. As their name suggests, capacitive sen-sors rely on a capacitance change in order to measure the desired quantity.A capacitor in its simplest form consists of two conductive plates, separated bya dielectric, as shown in Fig 2.1.  d   d  yx(a) (b)Figure 2.1: Simple capacitor, (a) construction, (b) schematic symbol and electricfield.The distance between the plates, the plate overlapping area and the dielectricsubstance are the critical parameters in any capacitor, determining the capacitancevalue.If we consider the parallel-plate capacitor model and neglect the fringe fieldeffects, the capacitance can be calculated usingC = ε0εrAd, (2.1)122.2. Capacitive Sensorswhere C is the capacitance, ε0 = 8.85 pF/m is the dielectric constant for vacuum,εr is the relative dielectric constant, A = xy is the overlapping plate area, and d isthe distance between the plates.Various arrangements for capacitors used in capacitive sensors are shown inFig 2.2.(a) (b) (c)(d)zddC 1C 2(e)C 1 C 2dzz0(f)Figure 2.2: Different arrangements for capacitive sensors based on: (a,b) variationof area, (c) variation of gap between plates, (d) dielectric change, (e) differentialvariation in the gap, and ( f ) a differential variation in the area.Using variable capacitors as sensors poses some difficulties. One of the firstproblems with regards to such usage is the fringe effect present in parallel platecapacitors. Although fringe effects are considered negligible in many instances,this is only acceptable when the distance between the plates is far smaller than the132.2. Capacitive Sensorssize of the plates.Additionally there needs to be appropriate shielding for the capacitive sensorplates and the wires connected to them, to reduce capacitive interference. However,shielding wires to prevent capacitive interference introduces a new capacitance inparallel with the sensor (parasitic capacitance). This in turn results in a loss ofsensitivity, as the change in the sensor capacitance only changes a part of the overallcapacitance. Also, relative movement between the wires and the dielectric couldintroduce errors, caused by changes in the capacitor geometry.Another important matter is the quality of the dielectric used in the capacitor.There should be a constant and high electrical insulation between the plates. Ifthe insulation is poor, then there will be a leakage resistance in parallel with thecapacitor that affects the overall capacitance. This results in the impedance beingaffected by a factor other than the capacitance, which renders the measurementmethods ineffective and prone to errors. Dielectrics with high conductivity (suchas water) could be affected by thermal interference generated because of the powerpassing through their effective resistance and generating heat.In general, the capacitive sensors are categorized into variable capacitors anddifferential capacitors. In variable capacitive sensors one or more of the abovementioned parameters change based on the measured phenomenon, where as indifferential capacitive sensors, the values of two capacitors simultaneously changein opposite directions by the physical variable to be measured.Despite the limitations mentioned above, capacitive sensors enjoy several ad-vantages including: low power consumption, wide operating temperature range,value dependency mainly on the geometry and less on the material properties, highresolution and easy for fabrication.142.2. Capacitive SensorsAs a result, capacitive sensors have a wide variety of applications, includingbut not limited to the measurement of displacement, force, pressure, acceleration,angular velocity. Moreover, the recent rapid growth in human/machine interfacehas given rise to the application in touch screens in many personal communicationdevices, such as mobile phones and tablets. Another area of interest is in medicalinstrumentation, where accurate measurement of signals from the patients body isof great importance. Fig. 2.3 shows some of these applications, e.g. measuringinertia in aviation, tilt and inclination in dams, trains and off-shore platforms, andseismic and vibration in highrises and race cars [16].Figure 2.3: Applications of capacitive sensors (e.g. accelerometers and gyrosin navigation, aviation, race car data acquisition, oil and gas, seismic) and someMEMS based capacitive sensors fabrication [16].152.2. Capacitive SensorsThis figure also shows some samples of capacitive sensors designed by Coli-brys.One area of recent advancement in capacitive sensing is designing sensorsbased on micro-electro-mechanical systems (MEMS). Although MEMS are notinvestigated in detail in this thesis, a capacitive interface is the common configura-tion for them, due to better power efficiency and increased sensitivity [17]. Fig. 2.4shows an image of a MEMS capacitive accelerometer designed by Dr. Elie Sarrafat University of British Columbia (UBC) [18].Figure 2.4: Image of IMOMBCEHS0903 accelerometer obtained withPolytecMSA−500r.This capacitive sensor has two sets of differential capacitors, one gap varyingand one area varying. The typical value for the gap-varying capacitance in thisdesign is about 2 pF with a dynamic range of -10g to 10g, a noise floor of approx-imately 4 µg/√Hz, and a gain of 23.4 mV/g for the whole system including therelated readout circuit.162.3. Capacitance Readout CircuitsAnother example of such sensors is shown in Fig. 2.5. The plates that form thedifferential capacitors are also shown on the image [19].Figure 2.5: Image of accelerometer designed at Georgia Institute of Technology[19].No matter where the capacitive sensors are used, or the amount of capacitancechange they create, there is a need for a mechanism to measure this change andtranslate it into useable output. In literature this mechanism is known as a readoutcircuit. The next section introduces the readout circuits and their various types inmore detail.2.3 Capacitance Readout CircuitsAs mentioned in the previous section, all sensors, including capacitive ones, relyon a mechanism to measure a physical variable (measurand) and translate it to asuitable signal to be used for further processing through filtering and amplification.This mechanism, known as the readout circuit, is a broad topic by itself; as there areas many readout circuits as there are sensors themselves. This section of the thesisexamines capacitance readout circuits in more detail categorizing them based ontheir circuit configuration, time sampling and feedback.172.3. Capacitance Readout CircuitsFor either variable capacitance or differential capacitance types of sensors,there are many types of readout circuits available. Some of the most commonare presented in Fig. 2.6[2]. Generally, a readout circuit designed for a differen-tial capacitive sensor can also be used for a single variable capacitive sensor. Themost common way is to replace of the sensor capacitors with a fixed capacitor(usually called reference capacitor). Differential readout circuits typically utilizethe difference between the two amplifier outputs in their circuit configuration. Fig.2.6 presents examples of simple single-ended and differential readout circuits. Fig.2.6(a) is a simple charge amplifier. Cx is the sensor capacitor and C f is the feedbackcapacitor. R f provides the DC bias current for the operational amplifier (op-amp)input stage. Assuming Cx is an area-varying capacitive sensor with parameter x asthe ratio of the change in the area:Cx = ε0εrA0(1+ x)dandVo =−CxC f Vswhich shows the output voltage is proportional to the capacitance changes. Itwas assumed that R f is large enough to have a negligible effect in the output voltagefor the bandwidth of interest. In case a gap-varying capacitive sensor was used, itwould be helpful to swap Cx and C f to get a linear relationship between Vo and x,the gap-variation measurand.182.3. Capacitance Readout CircuitsVsCxCfRfVoVsCx1CfRfVo¡VsCx2VsCxCfRfVoR1R2Cx2C4RVoCx1C3RRRVsRVoR RRVsZ1Z3Z2Z4(a) (b) (c)(d) (e)Figure 2.6: Examples of capacitance-to-voltage converter (CVC) readout circuits.(a) single-ended charge amplifier for single variable capacitor (b) single-endedcharge amplifier for differential capacitive sensor. (c) bridge amplifier for sin-gle variable capacitive sensor. (d) differential amplifier for differential capacitivesensor. (e) instrumentation amplifier for differential capacitive sensor.The readout circuit shown in Fig. 2.6(b) is very similar to the circuit of Fig.2.6(a). It is rearranged to accommodate for a differential capacitive sensor. Figure2.6(c) is a pseudo-bridge version of Fig. 2.6(a). The output of the circuit can bewritten as [2]:Vo =VsR1/R2−Z3/Z11+R1/R2,where Z1and Z3 are the total impedances of Cx and C f || R f , respectively.192.3. Capacitance Readout CircuitsThe configurations of Fig. 2.6(d) and (e) are based on pseudo-bridges anddifferential amplifiers. Figure 2.6(e) has an additional stage to convert the outputto single-ended, a stage known as instrumentation amplifier. The output voltagefor these differential configurations is:Vo =Vs(Z3Z1− Z4Z2).Differential readout circuits are less susceptible to common mode and powersupply noises; they typically have larger input signal range, and generally a bet-ter resolution. Moreover, differential readout circuits have a larger common moderejection ratio (CMRR), which makes them more desirable. On the other hand,compared to single-ended readout circuits, differential readout circuits consumemore current. This is because single-ended readout circuits normally use mini-mum size and number of transistors. Also, due to the need for more components,differential readout circuits require a larger silicon footprint in application specificintegrated circuits (ASIC) or on the printed circuit boards (in case of discrete circuitimplementation).In terms of timing, there are two types of designs available for readout circuits,continuous and discrete time. Intrinsically, the continuous time design has a higherresolution since it does not suffer from sampling noise[20]. Discrete time readoutcircuits are a better option when dealing with larger resistances, for instance in thefeedback loops. Switched-Capacitor circuits are one way of implementation basedon discrete time operation.Examining the feedback structure of the sensor and corresponding readout cir-cuits, two types of structures are present, open-loop and closed-loop. In the open-202.3. Capacitance Readout Circuitsloop configuration the capacitance change from the sensor is amplified and turnedinto a usable signal by the readout circuit and the resulting output is then trans-ferred to subsequent data displays. The closed-loop configuration exploits a sec-ondary input to the sensor which mimics the magnitude of the primary capacitancechanged caused by the measured phenomenon. This secondary input is essentiallya negative feedback set by the readout circuit designed to keep the actual valueof capacitance in equilibrium. The true capacitance change can be measured bymonitoring the negative feedback line.As mentioned earlier, many different methods have been proposed and devel-oped for capacitance readout circuits. Each of these methods rely on measuringa parameter change with respect to the capacitance, e.g. voltage, phase shift, fre-quency etc.2.3.1 Capacitance to Voltage ConverterThe first method examined in this thesis is the capacitance to voltage converter(CVC), where the capacitance changes are translated into a change in voltage forfurther processing [20, 21, 22, 23, 24, 25]. The CVC method on its own has manydifferent configuration, which are briefly mentioned below.• CVC method using charge integration, which can be used for both singleand differential capacitance. In this configuration, the capacitance change isconverted to a corresponding electrical charge variation, that is afterwardsconverted into a change in voltage using an op-amp. A schematic diagramof this circuit can be seen in Fig. 2.7 below [21].212.3. Capacitance Readout CircuitsC 0fR0f VoCx0 +¢CxCfRfVcC 0x0¡¢C 0xVBVCVDHINADD0CDC 0DRDR0DCarrier Sensor CurrentDetectorAMDemodulatorInst:Amp:AM ¡ModulatorVAFigure 2.7: Differential CVC based on charge integration [21].This method is less susceptible to parasitic capacitance, however it needs alarge value feedback resistor which could be difficult to implement on an IC [20].The bandwidth of the capacitor in this configuration is from DC to 10 KHz and areported resolution of 24 aF is measured if a 12 pF capacitor is used [21].• CVC method using low duty cycle periodic reset configuration is anothersub category of the CVC method which enjoys low noise, linear capacitanceto voltage transfer function, and low susceptibility to system offset. Thisconfiguration has reached a 0.06% resolution for a 0.8 pF capacitor [20]. Aschematic diagram of this configuration is presented in Fig. Capacitance Readout CircuitsCx1CfS1VoCx2VDDS2S1S 02S 01CpReset SensingS1, S01S2, S02Figure 2.8: An improved CVC readout, based on low duty cycle periodic reset[20].• Other configurations designed to reduce offset and increase resolution, twoof which are chopper stabilized configuration, which is shown to reduce theinput offset effects [22] and a ratio-arm bridge which is a symmetrical andsensitive circuit, but requires transformer coils [23].2.3.2 Capacitance to Duty Cycle ConverterIn the capacitance to duty cycle converter (CDC) method, the changes in capaci-tance are translated into changes in the duty cycle of a pulse train. This method hastwo main configurations that are listed below.• CDC method using direct configuration is named as such because there is a232.3. Capacitance Readout Circuitsdirect relationship between the capacitance and the duty cycle. To explainthis further, a CDC readout circuit using direct configuration is presented inFig. 2.9.Vref VrefVEEVCCCrefCxQ1 Q2R2R1R3R4 R5R6R7R8VthresholdVcomp:VFigure 2.9: Schematic of a CDC with direct configuration.The timing of the the duty cycle can be expressed byT = R(Cx− (a−1)Cr) ln(Vre fVre f −Vth), (2.2)where T is the on time of the duty cycle, R is the load resistance, Cx is themeasured capacitance, a= 1+R5/R4 , Cr is the reference capacitance, Vre f isthe reference voltage, and Vth (Vthreshold) is the threshold voltage. It is evidentfrom (2.2) that there is a direct relationship between T and Cx. The directconfiguration enjoys simplicity, lower power consumption (because of the242.3. Capacitance Readout Circuitssmaller number of components), and easy linearization on the digital side.If a modern low voltage/power CMOS implementation is used, then thisconfiguration achieved a bandwidth of 1 KHz with a 13 bit resolution. Theresolution and bandwidth were limited by the speed of the op-amp, whichhas a 3 MHz maximum gain bandwidth and a 13 V/µs slew rate [26].• CDC method using relaxation-oscillator configuration uses two capacitorsfor sensing. These capacitors are multiplexed by diode switches to form anop-amp based integrator. A schematic of this configuration is presented inFig. 2.10.V0BACC1C2V5V3RtR1R2D1 D2V2R3A1 A2A3A4V4Figure 2.10: Schematic representation of a CDC with the relaxation oscillator .The interface presented in Fig. 2.10 detects the ratio of capacitances in theform of the duty ratio.D =THTH +TL=C2C1+C2,252.3. Capacitance Readout Circuitswhere D is the duty cycle of the signal at V5 port.This configuration allows for high speed measurements as reported in the lit-erature. In one test case, a resolution of 60 aF was achieved using a 30 MHzoscillation frequency with a reference capacitance of 3 pF [27].2.3.3 Capacitance to Phase Shift ConverterCapacitance as a reactive component creates a phase shift between voltage andcurrent in a circuit. Assuming the rest of the circuit parameters and values areconstant, the phase shift, in reference to the input voltage, is a function of thecapacitance. As an example in a simple RC circuit shown in Fig. 2.11(a), thephase is:∠vo = φ =−arctan( 1ωRCx ).Typically in capacitance measurement, there is a reference capacitor, Cr, which isthe reference for the changes of the sensor capacitance Cx. One of the commonchoices for Cr is the value of Cx at rest. An example of this configuration is shownin Fig. 2.11(b). The phase of the differential voltage at the output, vo, is representedby:∠vo = φ = arctan(1ωRCr)− arctan( 1ωRCx)≈− ∆C/CrωRCr + 1ωRCr,where ∆C =Cx−Cr. This approximation is true when Cx and Cr are close enoughi.e. φ < 6◦.262.3. Capacitance Readout CircuitsCxRCrCxRRA sin(!t) A sin(!t)A sin(!t)vovo = B sin(!t + Á)Á = ¡ arctan( 1!RCx)(a) (b)\vo = Á ¼ ¡¢C!RC2r+1!RFigure 2.11: Phase shift generated using capacitance in an RC circuit. (a) single-ended. (b) differential.This phase difference for the circuit in 2.11(b) is plotted in Fig. 2.12 for thevalues of Cr = 100nF, R = 10kΩ and five different values of Cx ∈ { 80nF, 90nF,100nF, 110nF, 120nF}.1 10 100 1k 10k 100k-6-4-20246P has e d if f er enc e (o )Frequency (Hz)Cx 80nF 90nF 100nF 110nF 120nFFigure 2.12: Phase shift plot for differential RC circuit.272.3. Capacitance Readout CircuitsThe phase difference is almost linear at around -5 dB to -15 dB frequencies (10Hz to 80 Hz); e.g. the phase shift is approximately 1◦ per 10 nF of capacitancechange (10% change in capacitance) at around 30 Hz. The common challenge incapacitance measurement based on the phase shift is the nonlinearity introducedby arctan function.There are many different methods for measuring the phase of a signal, or thephase difference between two signals [28], e.g. direct oscilloscope method, zero-crossing, three-voltmeter, phase-locked-loops (PLLs), Fourier transform etc. Anexample of high level schematic diagram for a capacitance-to-phase-shift converter(CPC) based on zero-crossing detection is shown in Fig. 2.13. In this circuit, zerocrossings of the signals passing through the sensor and reference capacitors, whichrepresent the phase of the signal, are detected using comparators. The square waveat the output of the comparators are used to set or reset the output of a R-S flip-flop.so the duty cycle of the flip-flop output is proportional to the phase shift betweentwo signals. Another circuit based on modulation and demodulation is shown inFig. 2.14. In this method, the capacitance changes are modulated by the carriersignal. A multiplier is made of two logarithmic amplifier and an analog summa-tion, followed by an anti-log amplifier. The output of the anti-log amplifier hastwo frequency components. The high frequency components is eliminated by thelow-pass filter (integrator) block. The low frequency component, which containsthe information regarding the phase difference, passes through the integrator. Thisphase difference has a one-to-one relationship with the difference in the capaci-tances (Cx−Cr).282.3. Capacitance Readout CircuitsNon¡ invertingZero¡ crossingdetectorRSIntegratorBufferQVsVoInvertingZero¡ crossingdetectorCrCxRRR1R1CCD1D1ABCDQFigure 2.13: CPC using zero-crossing detection.LogAmplifierIntegratorBufferVoLogAmplifierCrCxRRV0sin(!t)V0cos(!t)¡+Anti¡ LogAmplifierR1R1R2Figure 2.14: CPC using analog multiplier.2.3.4 Capacitance to Frequency ConverterThe last readout circuit design method introduced in this section converts the changein capacitance to frequency, and it is known as capacitance to frequency converter(CFC). The main distinguishing factor with regards to the CFC method is that itgenerally does not need an analog-to-digital converter (ADC), since a simple zerocrossing counter can be used. Fig. 2.15 shows a schematic of a simple CFC read-out circuit based on a Hartley oscillator. The oscillation frequency is a function ofthe capacitance CL with the equation fosc = 1/(2pi√C2LT), where LT = L1+L2 is292.3. Capacitance Readout Circuitsthe equivalent tank inductance [29].VCCR1 R2R3 R4R5R6L1 L2C2C1C3Q1Figure 2.15: CFC based on simple Hartley oscillator.Another example based on switched-capacitor oscillator is, illustrated in Fig.2.16. This CFC design, presented in 1985 [30] , enjoys low complexity, as it doesnot need an ADC. The design was based on implementation of a quadrature os-cillator (two integrated loop circuit) using switched capacitors. The relationshipbetween the frequency of the oscillation and the capacitor to be measured is:f0 =CmCfc2pi,where C is the integrating capacitor shown in Fig. 2.16, Cm , αC is the capacitorto be measured, and fc is the clock frequency of the SC circuit.302.3. Capacitance Readout Circuits¡+¡+fc fc fc fcControlPulseGeneratorZeroCrossingDetectorS/H§Vrefk²C®C ®CC CV1V21=f0V^+¡Figure 2.16: Switched-capacitor harmonic oscillator with AGC .CFC has been applied in many readout circuits for different application re-quirements and variety of implementation technologies. We are going to brieflypoint to some of these applications, without going to the details, to show the broadusage of CFC method in the literature.A design based on the relaxation oscillator was presented for a capacitive dig-ital hygrometer in 1995 [31] but no comparison with other contribution was pre-sented. The design presented in [32] has the advantage of making the frequencyindependent of the power supply in a wide dynamic range. The next contribution,presented in 1991, is also based on switched capacitors. It has two main stages ofCVC (based on switched-capacitor) followed by a voltage-to-frequency converter(VFC). It has low power, low cost, and linear capacitance to frequency transformcharacteristics [33]. A simplified schematic of the circuit is shown in Fig. 2.17.The CVC circuit based on SC is shown on the left. VR1 and VR2 are constant refer-ence voltages. CR and CX are reference and measurand capacitors, respectively. VCis the output of the CVC stage. The VFC circuit schematic is shown on the right312.3. Capacitance Readout Circuitsside. The input is VC and the output is VO.Figure 2.17: CFC based on CVC cascaded with VFC.The relationship between VC and CX , for the CVC section, is given by:VC =(CX −CR)CF(VR2−VR1)+VR1 (2.3)The relationship between the output frequency fo and VC is:fo =(C1/C2)(VC−VR1)VR2−VR1fC2(2.4)where fC is the clock, Φ1 or Φ2 frequency.The contribution presented in [34] improves the solutions presented in [31, 33,30] by introducing a digital compensation system. This also uses a CVC followedby a VFC . This solution boasts low complexity, eliminates the need for an ADC,and increases linearity. The same authors use the same solutions with some minorchanges in [35] and [36].A combination of CVC and CFC is used in a humidity and accelerometer sen-sor presented in the literature [29]. The CFC part uses a Hartley oscillator witha feedback loop. A comparative study presented in [37] improves the previous322.3. Capacitance Readout Circuitsworks [38, 33, 34] to propose a solution, which not only offers better performanceon frequency to code conversion, but has better electrical characteristics, wide in-put spectrum range, and a wide high frequency dynamic range. The proposeddesign is based on the relaxation oscillator. In another contribution presented in[39], repetitive charge integration and charge conservation is used to combine boththe CVC and VFC into a single CFC that requires only one op-amp. This designconverts the difference between the capacitance values to an output frequency bythe repeated charge integration method.The study presented in [40] improves on the methods presented in [36, 39, 41]to get a more accurate and wider frequency range by saving and accumulatingthe residual charges. A more recent study based on relaxation oscillator presentsan active bridge where the frequency is linearly related to capacitive imbalance[42]. A recent paper on CFCs only presents simulation results, which indicate hightemperature (up to 175◦C), excellent stability over a wide temperature range andgood accuracy and resolution while not using a complex ADC [43]. The simpleprinciple of the circuit is based on integration, comparison and periodic reset. Asimplified schematic of this circuit is shown in Fig. 2.18. TG is a transmission gatewhich discharges the sensing capacitor CS. The negative and positive inputs ofthe operational amplifier (OPA) are biased through constant current I and constantvoltage VWE , respectively.332.4. ComparisonFigure 2.18: CFC based on integration and periodic reset [43].When Vint , which is the integral of current I offsetted by VWE , becomes greaterthan threshold voltage Vth, a one-shot pulse gets generated which in turn dischargesthe capacitor CS and resets the output at the same time. The frequency of the one-shot output pulses are related to the capacitor value by:f ≈ ICS(Vth−VWE) (2.5)The common point about the studies presented above are that they are mainlygeared towards IC design; however, this thesis is focused on more fundamentalcircuit theory matters. As a result, the review of CFC designs presented above isperformed more for the purpose of completeness, not for a side by side comparison.2.4 ComparisonCapacitive sensors, and specifically capacitive-based micro-electro-mechanical sys-tems (MEMS), have more widespread use in comparison to their piezoelectricand piezoresistive counterparts, due to larger temperature operating ranges, lower342.4. Comparisonpower consumptions and good resolution [17]. Both single ended and differentialcapacitive sensing configurations are being commonly used. Nevertheless, design-ing a reliable and accurate capacitance readout circuit is challenging, especiallyfor capacitive sensing of the displacement in MEMS structures that require smallstructural size and hence very small capacitor values and their relative changes.For instance, present inertial MEMS sensors require small bandwidths (50 - 100Hz) with resolutions often reaching aF levels for nominal capacitance in the orderof 0.1 - 2 pF [44]. These small sensing capacitors in the presence of parasitic ca-pacitance, which is in pF range, along with the interconnect resistance will limitthe measurement resolution and bandwidth of the readout circuit.The more complex the readout circuit, the larger the risk of introducing para-sitic elements, leading to a deterioration of the overall sensing performance. This isvalid for custom system-in-a-package capacitive sensing solutions, but even morefor discrete readout circuit alternatives. The need for complex solutions appears inthe context of required added features, e.g. self-calibration, temperature compen-sation, self-testing and analog-to-digital conversion. Many different approachesand methods have been introduced for high sensitivity capacitance readout circuits:capacitance-to-voltage converters (CVC) [45, 41, 20, 21, 24, 25], capacitance-to-frequency converters (CFC) [39], capacitance-to-duty-cycle converters [26, 46]and capacitance-to-phase-shift converters (CPC) [47]. Each of these principlescan be implemented through multiple circuit techniques. For example, a CVCcan be implemented using charge integration, chopper stabilized, ratio-arm-bridge,low duty cycle periodic reset, AM based relaxation oscillator, etc., which are pre-sented in the above mentioned references. Comparisons between various capac-itance readout methods are detailed in [48, 49]. Table 2.1 shows a comparison352.4. Comparisonsummary between the common methods named above.Author / Manufacturer Method Performance Parameter(s)Ashrafi et al. [47] CPC Resolution: 0.7fF (32ppm)Zubair and Tang [38] CPC Resolution: 4.7fF (50ppm), 1.5˚/fFWolffenbuttel [50] CPC Resolution: 0.4fF, 1.5˚/fFHaider et al. [51] CVC Resolution: 1fF, Sensitivity: 1mV/fFIrvine Sensors [52] CVC Resolution: 4aF/√HzLotters et al. [21] CVC Resolution: 24aFSolidus [53] CFC Resolution: 20aFTable 2.1: Capacitance readout circuit methods, a brief comparison.The most commonly used capacitance readout circuits are CVCs based onswitched-capacitor charge amplifier and CFCs. The former is insensitive to par-asitic capacitance at the input of integrating amplifier [54]. The main concern re-lated to the SC method is the noise associated to the charge injection and clock feedthrough that occurs in MOS switches. CFCs are among the highest performancereadout circuits, due to their higher sensitivity and circuit simplicity. Althoughthey are susceptible to parasitic capacitances and resistances, temperature drifts,and other sources of variation in the nominal oscillation frequency, but then can bemade more robust by using a differential approach that compares the measured ca-pacitance to a reference capacitance. The only major drawback of this differentialapproach is the slower reaction since the circuit would have to switch between thesensor and the reference, taking twice as long [54].Since the focus of our project is not that much related to the most of the com-monly used capacitance readout circuits methodologies, their simulation detailsand analyses are left for Appendix A.362.5. Justification2.5 JustificationNow that all these readout circuit methods have been introduced, it is evident thatmany methods for designing readout circuits exist, each of which being suitablefor specific applications and measurement ranges. It is however important to findthe method most suitable to compare against our proposed WCR method. Histor-ically the measurement systems based on time or frequency are among the mostreliable methods of measuring systems. The output of these methods can be easilyconnected to a digital processing systems. They inherently are closer to digital im-plementations since they do not require analog-to-digital converters at their output.This thesis applies the WCR-based principles to the capacitance measurementproblem. While CFC methods exploit a shift in the resonant frequency with thecapacitance change, WCR-based circuits are related to the resonant modes (theresonant frequencies give the eigenvalues of the linear circuit), but focus rather onthe energy repartition between the existing eigenmodes, and the way this repar-tition is influenced by a change in capacitance that induces a symmetry-breakingphenomenon. Nevertheless, the nearest method to the proposed WCR is the CFC,since both methods rely on exploiting resonance related characteristics. Based onthis knowledge, the CFC method is chosen as the benchmark to which WCR-basedreadout circuits will be compared.2.6 SummaryThis chapter has presented an overview of capacitive sensors, and their varioustypes. Then readout circuits were introduced and various subcategories related toreadout circuits, namely CVC, CDC, CPC and CFC, were examined in some detail372.6. Summaryusing relevant literature. A summary and a brief comparison between these meth-ods was presented. Appendix A goes through more detailed simulation of thesereadout circuits. After examining all these methods in detail, the CFC was chosenas the benchmark for comparison with the WCR, both exploiting circuit resonancecharacteristics. The next chapter examines WCRs in general and considers theirusage as an alternative for conventional readout circuits.38Chapter 3Weakly-Coupled-Resonators asCapacitance Readout Circuits3.1 IntroductionAs mentioned in chapter 1, the physical principles which give WCRs their inter-esting characteristics, namely mode localization and eigenvalue local veering, havebeen analyzed and used in solid state physics, mechanics, acoustics, and for MEMSdevices. This chapter starts by introducing WCRs in more detail and reviewing theexisting literature concerning WCRs, which is mainly in the mechanical field. Thenthe chapter studies suitability of WCRs as capacitance readout circuits. To deter-mine its suitability, the WCR method is judged based on criteria such as sensitivity,robustness and simplicity.3.2 Weakly Coupled ResonatorsMany different fields exhibit the interesting interplay between resonant frequen-cies, coupling strength between coupled resonant systems, and perturbation. Thereis a phenomenon called mode/energy localization which happens in nearly iden-tical weakly-coupled resonators. Mode localization in its simplest form happens393.2. Weakly Coupled Resonatorsbetween two identical resonators that are weakly coupled and at least one of theelements of the resonators gets perturbed. For simplicity, we consider two loss-less spring-mass resonators in Fig. 3.1. At first assume there is no coupling be-tween the resonators i.e. kc = 0. We also assume that the resonators are identicali.e. m1 = m2 = m and ∆k = 0. In this case, both resonators have identical res-onant/natural frequencies (eigenvalues or normal modes) of ω0 =√k/m. Theseresonators under the same initial and excitation conditions, have the same dis-placements (eigenvectors or mode shapes). Now we assume that they are coupledthrough a weak coupling of spring kc. Once they are coupled, then the systembecomes a second order system and the identical natural frequencies split in twofrequencies (eigenvalues). The gap between these two modes is identified by thestrength of the coupling. The stronger the coupling, the farther apart the normalmodes. If this coupled resonators system is excited, e.g. by an initial condition,it starts oscillating and the energy will be exchanged between the two halves al-ternatively and evenly. In other words, the energy gets delocalized in the system.Now, if there is a perturbation introduced in the system, e.g. by changing the sec-ond spring constant from k to k+∆k, then the localization phenomenon happens,and one side will get more energy (magnitude of displacement) than the other side.This is also called mode localization. The relative change of this mode localizationdepends on the relative perturbation δ , ∆k/k. If the coupling is weak enough,the two eigenvalues (natural modes) are close to each other. In this case, anotherphenomenon, called normal mode veering, happens besides the mode localization.Normal mode veering, which is also called eigenvalue loci veering, is shownin Fig. 3.2 [12]. The vertical axis is the normalized eigenvalues and the horizon-tal axis is the relative perturbation δ . The higher the perturbation, the more gap403.2. Weakly Coupled Resonatorsk2 = k +¢km1 = m m2 = mk1 = kkCx1 x2Figure 3.1: Lumped-element model of a coupled two-degree-of-freedom (2DOF)spring–mass system.between the eigenvalues. The eigenvalues of the system are closest at δ = 0. Theloci is showing an abrupt change around δ = 0, which is called veering zone. Ifthe coupling is very week, these curve look like intercept lines, which is deceptive.This is known as eigenvalue veering or normal mode veering.Figure 3.2: Loci of the dimensionless eigenvalues of the two coupled oscillators interms of δ [12].These phenomena of mode localization and veering are well known in the fieldof mechanics, acoustics and MEMS. Mode localization have been used for detect-ing and measuring very small perturbations that are nearly impossible, or muchmore difficult, using other methods. As an example, ultrasensitive mass sensing413.2. Weakly Coupled Resonatorswas implemented by this method able to measure a 154pg mass, shown in Fig. 3.3with almost two orders of magnitude better sensitivity than conventional relativefrequency shift methods [9].Figure 3.3: Scanning electromicroscopy (SEM) image of the first set of coupledgold-foil cantilevers and SEM image of an attached microsphere (inset, circled)[9].These aspects were firstly analyzed in the solid state field by Anderson in 1978[6, 7], which was the fundamental for the energy localization in periodic disorderedstructures, which also is called mode localization. The same phenomenon createsanother effect which was named curve veering by Pierre in 1988 [8]. More workwith plates was performed in [55], proving that the natural frequencies of platesthat belong to the same symmetry family exhibit the veering phenomenon whenthe plates are subjected to geometrical changes. This led the authors to propose the423.2. Weakly Coupled Resonatorsconditions under which the eigenvalue loci veer do not cross. Another work pre-sented in [56], improved on the conditions proposed in [57] and expanded them toinclude general real-valued eigenvalues utilizing a perturbation method. The rela-tionship between the eigenvalue loci veering and mode localization was describedin [58]. This work was then continued in [59], and the criteria governing the oc-currence of veering as well as the relationship between the veering of eigenvalueloci and eigenvector sensitivity was detailed. The dependency of the loci veeringon one or two parameters in a system has been investigated by [60].Although the phenomenon of mode localization has been very well known andused in disciplines such as acoustic and structural dynamics [61, 8, 58, 62] for along time, few publications have investigated the application of this phenomenon insensing and measurements of perturbations. There are several recent reports show-ing the application of mode localization in detection and measurement of smallchanges that can be considered as perturbation in the system. Chen and Kareem[63] studied the curve veering of cable-stayed and suspension bridge frequencyloci. Spletzer et al. applied this concept as a method for ultrasensitive mass sens-ing in coupled microcantilevers [9, 15]. Thiruvenkatanathan et al. reported theuse of mode localization concept in designing higher sensitivity MEMS sensors[11, 12, 13]. The relative sensitivity of the mode localization method, which isbased on relative shift in eigenvectors of the system, is orders of magnitude higherthat the relative shift in the system resonant frequencies or eigenvalues [11]. This isshown in Fig. 3.1 and equations (3.1) and (3.2). Equation (3.1) shows the influenceof the stiffness perturbation on relative change in eigenvectors.433.3. Reasons for Proposing WCRs as an Alternative for Readout Circuits|un−u0n||u0n| ≈∣∣∣∣ ∆k4kc∣∣∣∣ , n = 1,2, (3.1)where un and u0n are magnitudes of normalized eigenvectors with and withoutperturbation respectively i.e. un = [a1 a2] where a1 and a2 are normalized mag-nitudes of x1 and x2. ∆k is the change in the stiffness and kcis the coupling stiffness.Equation (3.2) shows the influence of the same stiffness perturbation on therelative resonant frequency shift.| f − f0|| f0| ≈∣∣∣∣∆k2k∣∣∣∣ , (3.2)where f and f0 are resonant frequencies with and without perturbation respectivelyand k is the nonperturbed stiffness of the springs.Equations (3.1) and (3.2) show that the relative sensitivity based on the relativeshift in eigenvectors, is approximately k2kc times greater than of the relative shift inthe resonant frequency. For a week coupling, kc k, k2kc ratio could become ordersof magnitude larger.To the best of our knowledge, there has been no literature indicating the use ofmode localization to measure the perturbation, e.g. minute capacitance changes, inthe electrical domain, prior to this research.3.3 Reasons for Proposing WCRs as an Alternative forReadout CircuitsThe readout circuits and methods explained above, as well as many other state-of-the-art circuits for capacitance readouts, are valuable and indicate present research443.3. Reasons for Proposing WCRs as an Alternative for Readout Circuitsin terms of technology and implementation. Looking at this problem from a dif-ferent perspective opens a promising way for designing highly sensitive and lowcost capacitance readout circuits. As it has been proven in the mechanical/MEMSdomain, the mode localization is very sensitive to perturbation, and as capacitancechanges in the circuit can be considered a perturbation, this thesis aims to investi-gate whether WCRs are a suitable alternative to existing capacitance readout cir-cuits. By looking at the nature of this method and what has been achieved in themechanical field, it is very likely that using this method in electrical domain givesus a very good sensitivity with considerably less sophisticated analog circuitry. Interms of the physical implementation, as a digital signal processor is one of theessential units in almost all readout systems, it is much simpler, though not neces-sarily easier, to choose a method which is inherently closer or matches better with adigital domain implementation. The proposed method reduces the amount of ana-log circuitry and takes advantage of digital processors to measure the capacitance.Sensitivity and the resolution of this method in electrical domain should be studiedand verified. To apply the mode localization technique to the electrical systems, wecan use the analogy between the coupled resonators in Fig. 3.1 and two coupledLC circuits. The LC resonators could have either series or parallel configuration.These parallel and series LC circuits are dual of each other. In more general formin Fig. 3.1 there are dampers in parallel with each spring which have the sameroles as resistors in the electrical domain. This analogy between mechanical andelectrical domain is shown in Table 3.1. Note that to subscript index s , e.g. in Ls,stands for “series” and index p stands for “parallel” dual circuit.453.4. SummaryMechanical Electrical (Series RLC) Electrical (Parallel RLC)Mass (M) Inductance (Ls) Inductance (Cp = Ls)Damping (C) Resistance (Rs) Resistance (Rp = 1/Rs)Spring constant (1/K) Capacitance (Cs) Capacitance (Lp =Cs)Displacement (x) Electric Charge (qs) Electric Charge (qp)Force (Fs) Voltage source (Vs) Current source (Is)Table 3.1: Analogy between mass-spring-damper and RLC coupled oscillators.3.4 SummaryThis chapter first introduced the history and background of weakly coupled res-onators, veering and mode localization in mechanical and acoustic field. The resultof an application of WCRs applied to MEMS inertial sensors was also presented.The chapter was then concluded by briefly visiting the possibility of adopting WCRmethod in electrical domain to measure small capacitance changes. The followingchapter examines the details of two-degree-of-freedom WCR circuit in capacitancemeasurement.46Chapter 4WCR-Based Readout CircuitAnalysis and PerformanceEstimation4.1 IntroductionThis chapter presents an innovative and simple way of measuring small capaci-tance changes. The proposed method can be a breakthrough in capacitance readoutcircuits, commonly used for microsensors associated with physical quantities suchas pressure, rotational angles, linear displacement and acceleration [2].In this chapter we propose a new method, based on weakly-coupled resonators(WCR), which will be compared with the CFC method. It will be proven ana-lytically, and shown by both simulations and experimental tests, that this methodhas at least one order of magnitude, and ideally three orders of magnitude, betterrelative sensitivity in comparison to the resonant frequency shift method.The WCR-based readout circuit is based on the energy (or mode) localiza-tion phenomenon. If two identical resonators are weakly coupled and excited bya harmonic source, the oscillation energy is equally shared between them; mode474.2. Theory of Operationlocalization appears when one of them is perturbed relative to the other, leading toan unbalance of the energy repartition among the two individual resonators. Thissymmetry-breaking phenomenon is well known in acoustic resonators and othermechanical systems [64, 8], as described in chapter 3. Recently, more attentionhas been given to the use of WCRs in the mechanical aspect of resonating MEMSmass sensors. It has been proven that weakly-coupled resonators can achieve al-most three orders of magnitude higher sensitivity than conventional frequency shifttechniques for perturbation sensing [12, 11, 10, 13, 9]. We are hence transferringthis high sensitivity symmetry-breaking technique to the electrical domain, for thefirst time (to our knowledge). We will use it for a simple capacitance readout solu-tion. The measurement technique requires simple analog circuitry and is suitablefor integration with digital signal processing. The outcome of this chapter was pub-lished as a journal paper in IEEE transactions on circuits and systems I (TCAS-I)[65].4.2 Theory of OperationIn principle, mode-localized or WCRs consist of two, or more, nearly identical cou-pled resonators. As stated previously, the concept of WCRs has been used for thedetection of very fine changes (perturbations) in mechanical resonators. These typeof vibrating systems are analogous to quantum mechanical systems, e.g. the hy-drogen molecule [9]. The eigenstates of two identical weakly-coupled resonators,symmetric and antisymmetric, are very similar to "bonding orbitals" and "anti-bonding orbitals", respectively. The frequency (eigenvalue) of the antisymmetricmode is either higher or lower than that of the symmetric eigenmode, depending on484.2. Theory of Operationwhether the coupling factor is positive or negative, respectively. Starting from anidentical set of two resonators, a symmetry-breaking perturbation can be defined asa slight change in one of resonator’s parameters, e.g. the mass or spring constant inmechanical systems, or capacitance or inductance value in electrical systems. Theperturbation will lead to both a further separation between the resonant frequenciesassociated with the symmetric and antisymmetric modes, and to a redistribution ofthe energies (amplitudes) associated with these modes, for a given external exci-tation. The measurement of the ratio between their oscillation amplitudes (e.g.displacement or charges amplitudes) is therefore directly correlated with the mag-nitude of the perturbation. The energy is no longer divided evenly between tworesonators in such a case, and tends to localize more in one of them. An exampleof 2DOF WCRs in the mechanical domain is shown in Fig. 4.1 illustrating twospring-coupled mass-spring resonators.k2 = k +¢km1 = m m2 = mC1 = C CC C2 = Ck1 = k kCx1 x2Figure 4.1: Two weekly coupled mechanical resonators.The low stiffness spring connecting the two masses acts as the weak couplingin this system. The coupling coefficient for this system, neglecting the dampingeffect, is defined as kc/k. An analogous representation of this system, using seriesRLC circuits, is shown in Fig. 4.2, assuming that the excitation is applied through494.2. Theory of Operationa voltage source, and the damping in the coupling is negligible. The couplingcoefficient becomes C/Cc in this case.R1= R L1= L C1= C L2= L R2= RC2= C +¢CCCVSi1 i2++++¡¡¡¡q1q2Figure 4.2: 2DOF weekly-coupled series RLC resonators.The coupled resonant system can be represented by the set of differential equa-tions shown in 4.1. The most general case, damped and perturbed, is consideredhere as it covers all other cases, e.g. undamped or unperturbed.L1d2q1dt2 +R1dq1dt +1CC(q1−q2)+ 1C1 q1 = vsL2d2q2dt2 +R2dq2dt +1CC(q2−q1)+ 1C2 q2 = 0(4.1)It is assumed that the circuit is excited with a harmonic sinusoidal voltagesource. Applying a perturbation to this circuit was defined as inducing very smallchanges in the value of C2, from C to C+∆C. It was also assumed that the rate ofthe perturbation change is slow enough (quasi-stationary signal), is that the tran-sient parts of the responses, including the effect of initial conditions, do not play arole. These assumptions were equivalent with the ones already used in the literaturerelated to the weakly-coupled mechanical resonators. Therefore the transient partof the response was eliminated and the steady state part of the response was usedto detect the perturbation magnitude.With these assumptions, the initial conditionsand transient responses of the system are ignored. Hence (4.1) can be re-written in504.2. Theory of OperationLaplace domain as:L s2+ RL s+ 1LC (1+ k) − kLC− kLC s2+ RL s+ 1LC (1+ k−δ ) Q1Q2= Vs0 (4.2)δ , ∆C/C and k =C/CC are defined as the perturbation and coupling strengthof the system, respectively. Although in in circuits and systems it is more commonto use either node voltages or loop current as the circuit variables, here we usecharges on the capacitors in each resonators to follow the analogy given in Table3.1.The loci of the resonant frequencies of the system versus the relative changesin capacitance C2 (perturbations) are shown in Fig. 4.3 [12]. The plotted resultsare the outcome of SPICE simulations using National Instruments Multisim-12rwith the values of L1 = L2 = 10mH, C1 = 100nF, C2 = 100nF+∆C, with −4nF <∆C < 4nF as perturbation, and three different coupling capacitance, CC, values of5, 15 and 25µF. There are three pairs of curves shown, each corresponding todifferent coupling strengths (different CC values). It will be shown that, since thecoupling coefficient is positive in this case, for a given perturbation value, the upperbranch of the locus is related to the out-of-phase mode (higher natural frequency),while the lower branch corresponds to the in-phase resonant mode (lower naturalfrequency). The middle zone of the graph (around zero perturbation) is called thetransition zone. The shown trend, where the loci of these eigenvalues in weakly-coupled systems approach each other in the transition zone but do not intersect, iscalled loci veering.514.2. Theory of OperationFigure 4.3: Two weekly coupled resonators natural frequencies loci.As can be seen, the weaker the coupling is, the narrower the transition zoneand the higher the slopes in the curves around the transition zone are. This suddenveering and narrowing of the transition zone results in a higher degree of energylocalization, or larger changes in the relative amplitudes of oscillation in the leftand right resonators.This mode localization or shift in the eigenvector component values around thetransition zone is shown in Fig. 4.4. To show the eigenvectors abrupt changes in thetransition zone, the ratio between i2 and i1 (loop currents) is plotted in Fig. 4.4(a).For positive perturbations, amplitude of i2 becomes considerably larger than i1 atout-of-phase resonant excitation. Inversely, i1 gets quite larger than i2 at out-of-524.2. Theory of Operationphase mode excitation using the external voltage source. In other words, if thesystem is excited at in-phase resonant frequency, the energy will be more localizedin the left loop of the circuit for positive perturbations and in the right loop fornegative perturbations. The opposite is true for in-phase excitation - energy willbe localized in the left loop for negative perturbations, or in the right loop forpositive perturbations. This is also shown using mode shape vector orientationsalong with eigenvalue loci veering curve plot in Fig. 4.4(b). In this plot, normalizedeigenvalues are shown to be similar to the resonant frequencies loci in Fig. 4.3, asexpected.The key aspect in achieving high sensitivity to perturbation is the drastic changein the angle of the mode shape vector in the veering zone. In comparison, theeigenvalue changes are not as significant.In the series RLC-based WCRs shown in Fig. 4.2, we assume R1 = R2 = R,L1 = L2 = L and C1 = C2 = C, which means both resonators are identical in theabsence of any perturbation. The coupling is done via CC ,with CC C being thecondition for a weak coupling.534.2. Theory of Operation(a)(b)Figure 4.4: Mode localization in two weekly-coupled-resonators.544.2. Theory of OperationA small change in any of these values can be defined as a perturbation, whichslightly pushes the system away from the energy equidistribution. We try to exploitthe high sensitivity in the orientation of the eigenvector around the symmetry-breaking region, in order to use it as a measure of the applied perturbation (C2variations). We will show that this technique provides a higher sensitivity thanexisting state-of-the-art methods, e.g. monitoring the relative shift in the resonantfrequency. A detailed analytical solution to this second order differential equation,using Laplace transform, is presented in the next section.4.2.1 Analytical SolutionEquation 4.2, re-written below, representing the two weakly coupled RLC res-onator systems, is solved through Laplace-transform techniques.Ls2+ RL s+ 1LC (1+ k) −1LC−1LC s2+ RL s+1LC (1+ k−δ )Q1Q2=Vs0 (4.3)The solutions Q1 and Q2 are therefore given by:Q1 =1Ls2+ RL s+1LC (1+ k−δ )(s2+ RL s+1LC(1+2k− δ2))(s2+ RL s+1LC(1− δ2))Vs (4.4)Q2 =1L1LCC(s2+ RL s+1LC(1+2k− δ2))(s2+ RL s+1LC(1− δ2))Vs (4.5)554.2. Theory of OperationUsing Partial Fraction Expansion (PFE), the equations can be simplified toQ1 =1L(k1s2+ RL s+1LC (1+2k− δ2 )+k2s2+ RL s+1LC (1− δ2 ))Vs (4.6)Q2 =1L(k3s2+ RL s+1LC (1+2k− δ2 )+k4s2+ RL s+1LC (1− δ2 ))Vs (4.7)wherek1 =12(1+δ2k); k2 =12(1− δ2k)k3 =− 2k2(LC)2; k4 =2k2(LC)2The response of this WCRs system to a Dirac Delta input is given by the inverseLaplace transform of the above equations:q1 = 12L[1ωd1(1− δ2k)sin(ωd1t)+ 1ωd2(1+ δ2k)sin(ωd2t)]e−R2L tu(t)q2 = 12L[1ωd1 sin(ωd1t)− 1ωd2 sin(ωd2t)]e−R2L tu(t)(4.8)whereωd1 =1√LC√1− δ2,ωd2 =1√LC√1+2k− δ2564.2. Theory of OperationAs expected, the impulse response fades out with a time constant of R/2L.In order to deduce the response of the system to a harmonic voltage excitation,as in Fig. 4.2, it is useful to decouple the equations set in (4.3). We introduce thetransformed state variables:P1P2, 1√21− δ4k 1+ δ4k1+ δ4k −1+ δ4kQ1Q2 (4.9)orQ1Q2≈ 1√21− δ4k 1+ δ4k1+ δ4k −1+ δ4kP1P2 (4.10)The set of equations in the new state variables is given by:Ls2+Rs+ 1C(1− δ2)−A − δ 332Ck2− δ 332Ck2 Ls2+Rs+ 1C(1+2k− δ2)−BP1P2= 1√2(1− δ4k)Vs(1+ δ4k)Vs(4.11)whereA =132k2(2Lδ 2s2+2Rδ 2s+1C(4δ 2k+δ 3−2δ 2))andB =132k2(2Lδ 2s2+2Rδ 2s+1C(4δ 2k+δ 3+2δ 2)).574.2. Theory of OperationWith the assumption of very small relative perturbations, even in comparisonwith the coupling coefficient (i.e. δ  k), we can eliminate the terms containinghigher order terms of δ/k. The resulting simplified and completely decoupled setof equations is:(Ls2+Rs+ 1C(1− δ2))P1 = 1√2(1− δ4k)Vs(Ls2+Rs+ 1C(1+2k− δ2))P2 = 1√2(1+ δ4k)Vs(4.12)Each of these equations can now be solved independently of one other, andthe inverse transformation to the physical state variables Q1, Q2 will give theirexpression.Assume vs(t) = Asin(ωt)u(t) or Vs(s) = ω/(s2+ω2)The equations then become:P1(s) = ω√2L(1− δ4k)1s2+ RL s+1LC (1− δ2 )1s2+ω2P2(s) = ω√2L(1+ δ4k)1s2+ RL s+1LC (1+2k− δ2 )1s2+ω2(4.13)Using the inverse Laplace transform, the function x(t), in time domain, can beretrieved:x(t) = xt(t)+ xs(t) (4.14)where xt(t) and xs(t) are transient and steady state parts of x(t) respectively.Here, we focus on quasi-static perturbations, after the transient parts of theresponses have faded out. Therefore the transient part of the response can be elim-inated and the steady state part is used to detect the perturbation magnitude.584.2. Theory of OperationThe equations for P1 and P2 can be re-written asP1(s) =ω√2L(1− δ4k) k1s+ k2s2+ RL s+1LC(1− δ2) + −k1s+ k3s2+ω2 (4.15)whereω2n1 =1LC(1− δ2)and 2ζ1ωn1 =RLk1 =2ζ1ωn1(ω2n1−ω2)2+(2ζ1ωn1ω)2=RL(1LC(1− δ2)−ω2)2+(RLω)2k2 =(2ζ1ωn1)2−ω2(ω2n1−ω2)(ω2n1−ω2)2+(2ζ1ωn1ω)2=(RL)2−ω2( 1LC (1− δ2)−ω2)(1LC(1− δ2)−ω2)2+(RLω)2k3 =ω2n1−ω2(ω2n1−ω2)2+(2ζ1ωn1ω)2=1LC(1− δ2)−ω2(1LC(1− δ2)−ω2)2+(RLω)2P2(s) =ω√2L(1+δ4k) k′1s+ k′2s2+ RL s+1LC(1+2k− δ2) + −k′1s+ k′3s2+ω2 (4.16)where594.2. Theory of Operationω2n2 =1LC(1+2k− δ2)and 2ζ2ωn2 =RLk′1 =2ζ2ωn1(ω2n2−ω2)2+(2ζ2ωn2ω)2=RL(1LC(1+2k− δ2)−ω2)2+(RLω)2k′2 =(2ζ2ωn2)2−ω2(ω2n2−ω2)(ω2n2−ω2)2+(2ζ2ωn2ω)2=(RL)2−ω2( 1LC (1+2k− δ2)−ω2)(1LC(1+2k− δ2)−ω2)2+(RLω)2k′3 =ω2n2−ω2(ω2n2−ω2)2+(2ζ2ωn2ω)2=1LC(1+2k− δ2)−ω2(1LC(1+2k− δ2)−ω2)2+(RLω)2Assuming a quasi-static perturbation, P1 and P2 can be simplified and approxi-mated by the steady-state, harmonic, part of the response:P1(s) = ω√2L(1− δ4k)−k1s+k3s2+ω2P2(s) = ω√2L(1+ δ4k) −k′1s+k′3s2+ω2(4.17)orp1(t) = ω√2L(1− δ4k)(k3ω sin(ωt)− k1 cos(ωt))p2(t) = ω√2L(1+ δ4k)(k′3ω sin(ωt)− k′1 cos(ωt)) (4.18)Knowing p1 and p2, q1 and q2 can be solved.604.2. Theory of Operationq1(t) = 1√2[(1− δ4k)p1+(1+ δ4k)p2]q2(t) = 1√2[(1+ δ4k)p1+(−1+ δ4k)p2] (4.19)By substituting p1 and p2 in the above equationsq1(t) =12L[(1− δ2k)ω2n1−ω2(ω2n1−ω2)2+(RLω)2 +(1+ δ2k)ω2n2−ω2(ω2n2−ω2)2+(RLω)2]sin(ωt)− ω2LRL[(1− δ2k)1(ω2n1−ω2)2+(RLω)2 +(1+ δ2k)1(ω2n2−ω2)2+(RLω)2]cos(ωt)(4.20)q1(t) =12L[ω2n1−ω2(ω2n1−ω2)2+(RLω)2 − ω2n2−ω2(ω2n2−ω2)2+ (RLω)2]sin(ωt)+ω2LRL[− 1(ω2n1−ω2)2+(RLω)2 + 1(ω2n2−ω2)2+ (RLω)2]cos(ωt).(4.21)The amplitudes of q1 and q2 have peaks at ωn1 and ωn2, which are natu-ral/resonant angular frequencies of the system. The following approximation isused in estimating q1, q2:(1+δ2k)/(1− δ2k)≈ 1+ δkfor δ  k.The steady state solution for the ratio of the two capacitor charges will then be:614.2. Theory of Operation|q1||q2| =∣∣∣∣(1− δ2k)∣∣∣∣√√√√√√√√4k2ω20 +ω2n2Q2(1+(1+ δk)(1+ 4k2Q2(1+2k− δ2 )))24k2ω20 +ω2n2Q2(1+ 4k2Q2(1+2k− δ2 ))2 (4.22)with the following notations:ω0 =1√LC, Q =ω0LR, ωn2 =1√LC√1+2k− δ2(4.23)and defining the square root term as:r ,√√√√√√√√4k2L2C2 +R2L3C(1+2k− δ2)(1+(1+ δk)(1+ 4k2LR2C(1+2k− δ2 )))24k2L2C2 +R2L3C(1+2k− δ2)(4k2LR2C(1+2k− δ2 ))2 . (4.24)Using a rational function curve fit, with a least mean square (LMS) approxi-mation, r is almost independent of δ which is graphed in Fig. 4.5. In this approx-imation, with the values for L, C and k are 10 mH, 100 nF and 1/150 respectively,r becomes:r ∼=∣∣∣∣0.34171R+0.98−0.00252R+1∣∣∣∣ (4.25)Hence, the ratio between capacitor charges in (4.22) is well estimated, for re-sistor values 0 < R < 100Ω, by:624.2. Theory of Operation|q1||q2| ==∣∣∣∣0.34171R+0.98−0.00252R+1(1− δ2k)∣∣∣∣ (4.26)The parameter r indicates the effect of loss (resonator resistance) on sensitivity.Note that if loss is negligible then r gets close to unity. If loss is considerable, thenthe magnitude of q2 becomes smaller than the magnitude of q1, regardless of theamount of perturbation. In such cases, the value of r, which is |q1/q2| at balance(δ = 0) is greater than 1.Figure 4.5: Effect of loss on sensitivity. Coefficient r in (4.25).For the values of L = 10mH, R = 0.1Ω, CC = 15µF and C = 10nF (i.e. k =1/150), this ratio will be|q1||q2| = 1.014427 |(1−75δ )| (4.27)634.2. Theory of OperationThis is completely in-line with the values from the exact equation, as shown inTable 4.1 and Fig. 4.6. The following notations are used:ωn1 =1√LC√1− δ2, ωn2 =1√LC√1+2k− δ22ζiωni =RL, ωdi = ωni√1−ζi2 ; i = 1,2where ωn1 and ωn2 , or their corresponding frequencies, fn1 and fn2, are theresonant frequencies of the natural modes of the WCR system.δ ωn2(Rad/Sec)∣∣∣q1q2 ∣∣∣− ∣∣∣q1q2 ∣∣∣0 ωn2−ωn0ωn0-1.33E-4 31623.83 -0.0098896 3.333E-05-1.07E-4 31623.62 -0.0079273 2.667E-05-8E-5 31623.41 -0.0059573 1.999E-05-5.33E-05 31623.2 -0.0039795 1.333E-05-2.67E-05 31622.99 -0.0019937 6.667E-060 31622.78 0 02.667E-05 31622.57 0.0020017 -6.667E-065.333E-05 31622.35 0.0040114 -1.333E-058E-5 31622.14 0.0060293 -2.000E-051.07E-4 31621.93 0.0080552 -2.667E-051.33E-4 31621.72 0.0100894 -3.334E-05Table 4.1: Analytical values for 2DOF WCRs at out-of-phase resonance.In Table 4.1∣∣∣q1q2 ∣∣∣0is the value of ∣∣∣q1q2 ∣∣∣ when there is no perturbation (δ = 0).In Fig. 4.6, the relative shift in the resonant frequency of the out-of-phase mode(Table 4.1) is plotted with a magnified scale of 100, in order to make it noticeablein comparison to the relative shift of the ratio of capacitor charges. The sensitivitybased on relative capacitor charge measurement is approximately 300 times higher644.3. Simulationsthan the sensitivity based on the resonant frequency shift measurement. This datawas exported from circuit simulations using Multisim-12r.-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015-0.010-0.0050.0000.0050.010±(%)¯¯¯q1q2¯¯¯¡¯¯¯q1q2¯¯¯0   	   	      !n2¡!n0!n0100£Figure 4.6: Relative shift in resonant frequency vs. eigenmode in 2DOF WCRs.4.3 SimulationsA series of simulations were performed with sinusoidal input as forced excitation.Circuit simulations were conducted in National Instruments Multisim-12r for thecircuit shown in Fig. 4.7. The main reason for choosing Multisimr is the ca-pability of co-simulation with LabVIEWr, which integrates very well with theNI-PXI hardware platform used in our experiments. Perturbations of C2 valueswere simulated by using a voltage-dependent capacitor (∆C in parallel with C2).654.3. SimulationsWhen C+∆C = 100nF , the resonators are balanced.R10:1Ð 100mH 100nHL1 C1 C2 L2 R2¢CCC 15¹FVS0:1Vp100nH 100mH 0:1Ði1 i2Figure 4.7: Circuit schematic of 2DOF WCRs for SPICE simulations.The results of the AC analysis, shown in Fig. 4.8, illustrate the natural frequen-cies (modes) of the unperturbed circuit, at 5.033 kHz and 5.066 kHz, respectively.Figure 4.8: AC analysis of 2DOF WCRs based on series RLC resonators.From a practical perspective, the loop currents, i1 and i2, rather than chargesq1and q2, are taken as the eigenvectors components. The weak coupling is achievedthrough CC, which was chosen to be approximately 15 times larger than C1 and C2.If CC becomes too small, then the two resonators will become strongly coupled; if664.3. Simulationsit is too large, then the amount of energy transferred to the second resonator is notsufficient to be measured given the existing parasitics and noise sources.In the next step of the simulation, the circuit was excited using a sine waveinput at one of the resonant frequencies. Perturbations were introduced to C2 us-ing a voltage-controlled capacitor U1. The simulation results are aligned with thetheoretical analyses for both in-phase and out-of-phase modes, at 5.033 kHz and5.066kHz respectively. A parametric sweep analysis on C2, from 95nF to 105nF,shows the effect of perturbations on the natural frequencies and loop current values(mode shapes). The results are shown in Table 4.2.δ (%) fn2(Hz)∣∣∣∆uu0 ∣∣∣ ∣∣∣ i2i1 ∣∣∣ ∣∣∣ i2i1 ∣∣∣− ∣∣∣ i2i1 ∣∣∣0 100 ∣∣∣ fn2− fn0fn0 ∣∣∣-5 5183.38 0.8470 3.2496 2.2752 2.3116-4 5156.96 0.8088 3.0782 2.1037 1.7901-3 5131.13 0.7481 2.8134 1.8389 1.2802-2 5106.31 0.6407 2.3969 1.4225 0.7903-1 5083.48 0.4224 1.7465 0.7721 0.33970 5066.27 0.0000 0.9744 0.0000 0.00001 5058.06 0.4238 0.5012 0.4732 0.16212 5054.65 0.6362 0.3079 0.6666 0.22943 5053.25 0.7383 0.2181 0.7564 0.25704 5052.45 0.7937 0.1685 0.8059 0.27285 5051.85 0.8285 0.1374 0.8370 0.2846Table 4.2: Comparison table between ∆∣∣∣ i2i1 ∣∣∣and ∆ ff methods of measurement.Note that for the theoretical eigenvectors, the charges on the capacitors C1 andC2 were defined as vector components. Both analytical calculations and simula-tions show that the value of relative change in loop current ratio is very close tothe relative shift in eigenvectors (∆u/u0) [9]. In these simulations a weak couplingof C/CC = 1/150 and a maximum perturbation of δ = ∆C/C = 0.005 resulted in674.3. Simulationsa sensitivity of approximately 0.6, which is in agreement with the analytical cal-culations. Simulations show that the variation in the ratio between eigenvectorcomponents is about 30 times larger than the relative frequency shift. The resultsof these simulations are shown in Fig. 4.9 and Fig. 4.10.Figure 4.9: i1 plots, coupled RLC circuit AC analysis with sweeping C2 from 99%to 101% of nominal value.Figure 4.10: i2 plots, coupled RLC circuit AC analysis with sweeping C2 from 99%to 101% of nominal value.684.3. SimulationsThe veering of the resonant frequencies is plotted in Fig. 4.11, while a com-parison between three methods of indirectly measuring the perturbations shown inFig. 4.12.Figure 4.11: Resonant frequency loci veering in 2DOF WCR.CC = 15µF , R =0.1Ω, C = 100nF , L = 10mH.Note that the |∆ f/ f | method is up-scaled 100 times to be comparable in sensi-tivity magnitude with the other two methods.694.3. Simulations-5.0 -2.5 0.0 2.5 5.0-1.0- modeT he r es pons e of  th ree met hod st o q uas i-s tat i c p er t ur bat i ons   ¢u1= ju10j¢ jI2=I1j¡ 100£¢f=f0±(%)(a)-5.0 -2.5 0.0 2.5 5.0-1.0- mode±(%)¢u2= ju20j¢ jI2=I1j100£¢f=f0T he r es pons e of  th ree met hod st o q uas i-s tat i c p er t ur bat i ons   (b)Figure 4.12: Sensitivity comparison between three different methods; (a) compar-ison when exciting the in-phase (symmetric) mode (b) comparison when excitingthe antisymmetric mode.704.3. SimulationsAs it can be seen from the plots, for small perturbations the sensitivity of loopcurrents ratio is almost equal to the sensitivity of the relative change in modeshapes. The computation and hardware implementation of currents ratio estima-tion is easier and requires less processing and memory resources; it is therefore theadopted method. To further simplify the implementation, |i2/i1| ratio can be usedinstead of ∆ |i2/i1|, eliminating unnecessary previous state storage in the computa-tional algorithm.Figure 4.13: LabVIEW-Multisim co-simulation for 2DOF WCRs.The experiments, which will be described in the next section, were conductedusing National Instruments LabVIEWr software, together with a PXI data acqui-sition system. The above simulations were repeated using a LabVIEW-Multisimr714.3. Simulationsco-simulation framework (Fig. 4.13), to have a smoother transition from simula-tion to hardware implementation. The results have confirmed the previous resultsobtained using Multisim-only simulations.-1.0 -0.5 0.0 0.5 1.0-0.6-0.4-±(%)C omp ar i ng  r el at i ve s hi f ts  i n r es onant  f r eq uenc y, ei genmod e and  c ur rent  rat i o    100¢f=f0¢U= jU0j¢ jI2=I1jFigure 4.14: Relative shifts for different methods; LabVIEW-Multisim co-simulation result.The results are illustrated in Fig. 4.14, in a similar format as Fig. 4.12(b).There is a small difference in the co-simulation results, specifically around -3%perturbation. This comes from the fact that, for very high-Q (Q > 1000) circuits,the accuracy of integration in time domain is strongly dependent on the time-stepsettings in both LabVIEW and Multisim. Fine tuning this timing requires a vari-able time step of in the order of ns, which makes the simulation very slow andmemory-intensive (several hours on a system with 8GB of RAM). The AC analy-sis in Multisim simulations gives more reliable results without extensive simulation724.4. Experimental Resultstime.4.4 Experimental ResultsThe components used for the hardware implementation were R1 =R2 = 0.1Ω, C1 =C2 = 100nF, L1 = L2 = 10mH and CC = 15µF. C1 was regarded as a reference,while C2 corresponded to the sensing capacitance. The perturbation of C2 wasgenerated by adding extra capacitors in parallel to the initial 100nF value. Thetest-bench environment needed a voltage source with very low output impedancefor excitation, and an interface to measure the inductors or capacitors voltages.Measurement of the analog values was done using a National Instrument PXIe-1062Q system with a PXI-7854R analog interface module. A high level blockdiagram of the proposed instrumentation system is shown in Fig. 4.15.Figure 4.15: High-level-block-diagram of proposed capacitance readout.734.4. Experimental ResultsThe circuit was implemented on both a breadboard and a custom printed circuitboard (PCB), and connected to the PXI module from National Instruments. TheLabView-FPGA software toolkit was used for the data acquisition and processing.Images of the setup are shown in Fig. 4.16. Three 16-bit analog inputs of the PXIanalog interface were used for reading excitation input, as a reference, and bothinductor-capacitor junction node voltages.(a) (b)(c) (d)Figure 4.16: Test setup for experimental measurements. (a) Circuit connections(b) FPGA based data acquisition (PXI) (c) LabView GUI screen-shot (d) Completesystem connections.The inputs were connected to the PXI module via coaxial cables with BNCconnectors. The only additional equipment that is needed to set up the experiment744.4. Experimental Resultswas an external power supply, or alternatively a pair of 9V batteries, to power upthe input op-amp buffer.A single tone sine wave generator and a simple PLL algorithm were imple-mented using LabView-FPGA to keep the excitation frequency tracking the out-of-phase resonance frequency, when different perturbations were applied to C2.The system was also capable of manually controlling the sine wave generatorby breaking the PLL loop, in order to test the designed PLL. In our experiments,the capacitors voltage amplitudes were considered as components of the voltagevector (i.e. v = [v1 v2]T ). v0 corresponds to v at balance (i.e. δ = 0) whichrelates to the out-of-phase resonant frequency f0. Introducing a small perturbationleads to a resonant frequency shift by ∆ f and a change in v by ∆v.The values for ∆ f/ f0 and ∆v/v0 were taken as measures for the relative shiftin resonant frequency and the relative eigenvector shift methods, respectively. Theresults of these methods are shown in Table IV for different perturbations, δ , ofC2; both the simulated and the measured sensitivities are plotted in Fig. 4.17.δ (%) fn2 v1 v2∣∣∣∆vv0 ∣∣∣ 50 ∣∣∣∆ ff0 ∣∣∣0 5144 1.54059 1.30774 0 00.0295 5143.5 1.54481 1.30477 0.00502 0.004860.0586 5143 1.54792 1.30324 0.00823 0.009720.1119 5142.5 1.55792 1.30387 0.01425 0.014580.1948 5141 1.56675 1.30005 0.02300 0.029160.9787 5132 1.69866 1.29276 0.11538 0.11664Table 4.3: Experimental results for both eigenvalue and eigenvector based methodsThe experimentally obtained sensitivities were lower than in simulation due tonon-idealities of the circuit components, e.g. internal series resistance of the in-ductors and electronic component tolerances. These led to reduced quality factors754.4. Experimental Resultsof the resonators and higher tolerance in their coupling strengths. Nevertheless, asseen in Fig. 4.17 the capacitance-to-eigenvector shifting method proves to be atleast 50 times better than the capacitance-to-frequency shifting method.Figure 4.17: Sensitivity comparison between simulations and experiments.The experimental results using CC = 15µF did not show satisfactory results interms of distinguishing between the two resonant peaks (mainly due to the equiv-alent series resistance ESRs). There are other parasitic parameters and sources fornonlinearities, e.g. skin effect, core hysteresis etc. [66], which are not significantin our case due to the low frequency and narrow band of operation.In these lossy coupled resonators, there is a trade-off between the sensitivityand detectability of the two resonant modes. To increase the gap between the reso-764.5. Summarynant frequencies, coupling should increase (i.e. CC should decrease) which in turndecreases the sensitivity. In these experiments CC was changed to approximately4µF . Fig. 4.18 shows the effect of this change on the frequency response.(a) (b)Figure 4.18: Effect of parasitic parameters on frequency response. (a) CC = 15µF(b) CC = 4µF.4.5 SummaryThe work presented in this chapter expands upon a principle validated already inmechanical and acoustical engineering. The concepts of eigenvalues veering andmodes localization are fundamental in many physical problems; for instance themode localization aspect is reflected in the work of P. W. Anderson leading to theNobel prize in physics in 1977 (Anderson Localization) [6].We have applied this technique, for the first time (to our knowledge), to acapacitance-to-eigenvector shifting readout circuit and presented its superiorityover a capacitance-to- frequency shifting approach. The circuit is relatively sim-ple and easy to couple with digital processing circuitry. The theoretical analysiswas complemented by both numerical simulations (SPICE and LabVIEWr) andexperimental measurements on a custom-made PCB level circuit.774.5. SummaryDespite the presence of significant parasitic elements, e.g. ESR of inductorsand capacitors, the simple circuit was able to detect capacitive changes as lowas 30ppm and proved to be at least 50 times more sensitive than the equivalentfrequency-shifting method (Fig. 4.17). This opens up the new possibilities formore advanced integrated solutions relying on mode localization for readout cir-cuits of various electrical parameters. The next chapter focuses on the errors anduncertainties associated with the excitation frequency, which are important in bothCFC and WCR-based methods.78Chapter 5Error Reduction in WCR-BasedCapacitance Readout Circuits5.1 IntroductionThe superiority of the WCR over CFC method, in terms of the relative sensitivity,has been proven and shown in the previous chapter. A WCR-based readout circuitwith a coupling factor of as low as k = 0.02 can show a sensitivity of at least twoorders of magnitude higher than the equivalent CFC configuration. An exampleof such systems, based on coupled series RLC circuits, is shown in Fig. 5.1. Thecapacitor CC provides the weak coupling between the two series resonance circuits,with the coupling coefficient being defined as δ , ∆C/CC. The circuit will beanalyzed and discussed in more detail in the following sections.Figure 5.1: Series RLC two weakly coupled resonators.795.1. IntroductionAs stated in chapter 4, we have applied the WCR concept to capacitance changesmeasurement problem [65]. The sensitivities of CFC and WCR methods were an-alytically calculated. It also has been shown that the ratio between sensitivitiesof these methods (i.e. the slopes of the relative shifts in resonant frequency andeigenmode for CFC and WCR, respectively) is inversely proportional to the cou-pling strength between the two sections. For a weak coupling coefficient of 1/150,the relative sensitivity of the WCR method is 300 times more than of the CFCmethod. This is illustrated in Fig. 5.2. In this figure, ωn0 is the unperturbed res-onant angular frequency (either in-phase of out-of-phase excitation mode), whileωn2 is the resonant angular frequency shifted by the perturbation δ . q1 and q2 arecharges on the capacitance C1 and C2, respectively, and |q1/q2|0 is their ratio atunperturbed (δ = 0) condition.-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015-0.010-0.0050.0000.0050.010±(%)¯¯¯q1q2¯¯¯¡¯¯¯q1q2¯¯¯0   	   	      !n2¡!n0!n0100£Figure 5.2: Relative shifts in resonant frequency of CFC vs. eigenmodes of WCRmethodes.805.1. IntroductionThe main difference between these two methods is that CFC is based on eigen-values (natural frequencies) shifting of the system, while WCR scheme is based onsystem eigenmodes shifting, which is 1/k times, k , CCC  1, more sensitive thanCFC method. The previous chapter mainly focused on relative sensitivity improve-ment in capacitance change measurement but without analyzing its robustness toexcitation frequency errors. Therefore it is important to formally define expres-sions for measurement errors for both CFC-based and WCR-based readout circuits,to have a fair comparison between them. This chapter presents accurate and ana-lytically deduced error expressions for CFC and WCR methods. Furthermore itexploits these derivation for obtaining a new method with guaranteed lower mea-surement errors. Typically, such systems based on resonance monitoring require afeedback loop to keep the frequency of the excitation signal at resonance, e.g. byusing a PLL or some other locking mechanism [67]. There is a tradeoff betweensensitivity and phase noise. Higher sensitivity requires higher quality factors. Onthe other hand, a high quality factor resonator has a sharp slope of the phase (φ )vs. frequency ( f or ω) dependency around resonant frequency ( dφ/dω = 2Q/ω0,), which in turn makes the system more sensitive and prone to error at the lockingfrequency Fig. 5.3.There are several possible implementations for CFC readout circuits. In ourcase we have chosen the same double-resonator circuits for both methods, fre-quency shift and eigenmodes shift. A high level block diagram of the system isshown in Fig. 5.4. The linear combination of the CFC and WCR methods wascoined “Hybrid WCR” method. If the value of one of the circuit componentschanges, the resonant frequency of the system changes accordingly. The feed-back loop, including a phase detector and a loop filter, keeps the VCO tracking815.1. Introduction5026 5028 5030 5032 5034 5036 5038 5040−20−1001020Amplitude(dB)  Q = 500Q = 100Q = 205026 5028 5030 5032 5034 5036 5038 5040−50050Phaseφ(◦)f (Hz)  Q = 500Q = 100Q = 20Figure 5.3: Bode Plot for Series RLC Resonatorthe resonant frequency. There are different methods for tracking and locking tothe resonant frequency of such RLC tanks, either based on amplitude, phase ora combination of them. The phase information is used in this research for sim-plicity and ease of implementation. It is also assumed that all the uncertaintyand errors in the circuitry, e.g. thermal and electronic noises, ADC quantizationnoise, phase/frequency noise of the VCO etc, eventually show themselves in thefrequency/phase of the VCO output as a deviation from the exact resonant fre-quency of the resonator. This erroneous frequency locking has a direct impact onthe perturbation estimation using CFC method. It also indirectly affects the WCRmethod since all calculations are based on the assumption of the system being ex-cited at resonance.Our analysis shows that the errors in these two methods are of comparablemagnitude but of opposite sign. As a solution, we have linearly combined theresults of CFC and WCR methods to mitigate the estimation error.The rest of this chapter is structured as follows: theory is presented in sec-tion 5.2 followed by simulation results in section 5.3. Experimental results arepresented in section 5.4.825.2. Theory of Operationvsvci1 i2fs = f0(1 + ®vc)V COi1 i2vsAmplitudeDetectionLoopFilterI1 I2VsWCRMethodCFCMethodfsHybridWCR Method(this work)AnalogDigitalPhaseDetectorFigure 5.4: System high-level-block-diagram.5.2 Theory of OperationBefore discussing the details of the WCR system, it is useful to briefly discussthe related frequency dependent methods. A simple RLC resonator vs. Colpittsoscillator is shown in Fig. 5.5 as examples. Fig. 5.5a illustrates the capacitancemeasurement based on the RLC resonator by simply associating the resonant fre-quency and the capacitance C values, f0 =(2pi√LC)−1. The formula indicatesa one-to-one correspondence between capacitance and resonant frequencies mea-sured with the help of an excitation voltage Vs. An alternative method for a CFCbased capacitance measurement is to use an active oscillator, e.g. Colpitts oscilla-tor shown in Fig. 5.5b. The op-amp gain and its positive feedback fulfills both gain835.2. Theory of Operationand phase conditions for oscillation. The frequency and capacitance relationshipfollows f0 =(2pi√LCT)−1 equation, where CT =C1C2/(C1+C2) is the equivalentresonator tank capacitance. In capacitance measurements it is common to assumeeither C1 or C2 taken as reference capacitor while the other one serves as the sens-ing capacitor.VsR L C(a) RLC resonator.+-C1C2RfRiLA(b) Colpitts oscillator.Figure 5.5: Examples of conventional capacitance measurement methods. (a) RLCresonator and (b) Colpitts oscillator.In the simple 2DOF WCR circuit presented in Fig. 5.1, the sensing capacitor isone of the capacitors of the series resonators, which in our case is C2,C1+∆C. ∆Cis the sensor capacitance change to be measured. Its normalized value, normalizedperturbation, is defined as δ , ∆C/CC, where CC is the coupling capacitance. Therelative sensitivity of both CFC and WCR methods in detecting the perturbationis explained in [65]. The CFC method is based on the relative shift in resonantfrequency, while the WCR method is based on the relative shift in ratio of the loopcurrents (|I1/I2|).Both in-phase and out-of-phase resonant frequencies are directly related to thesystem eigenvalues, while loop currents (capacitance charges) are related to the845.2. Theory of Operationeigenvector components. Symmetry breaking phenomenon occurs around the pointof zero perturbation, where the orientation of the eigenvectors changes at a muchfaster rate than the complimentary changes in the eigenvalues magnitudes. This isshown in Fig. 5.6. The region around the zero perturbation is called the veeringzone.Figure 5.6: Eigenvalue loci veering.5.2.1 Measurement SensitivityFig. 5.1 shows a simple weakly coupled resonators circuit based on series RLCcircuits. The first goal is to carry out closed loop equations for magnitudes of loopcurrents, I1 and I2, and their ratio. This will be solved in two unperturbed andperturbed cases.First assume an unperturbed, ∆C = 0, case. The circuit is excited by a singleharmonic voltage signal. The phasor equation for the circuit can be interpreted by855.2. Theory of Operationequation (5.1). z11 z12z21 z22 I1I2= Vm∠00 , (5.1)wherez11 = z22 = R+ jωL+ 1jω(1C +1CC)andz12 = z21 =− 1jωCC .This can be simplified further by:1ωCM I1I2= Vm∠00 , (5.2)whereM = m11 m12m21 m22,m11 = m22 = RCω+ jLCω2+ 1j (1+ k),m12 = m21 = jk and k = CCC .Solving (5.2) for I1 and I2: I1I2= M−1 Vm∠00Cω, (5.3)|I1|=Vm Cω|det(M)|√(RCω)2+(LCω2−1− k)2, (5.4)and865.2. Theory of Operation|I2|= 1|det(M)| (kCω)Vm, (5.5)wheredet(M) =(RCω+ j(LCω2−1− k))2+ k2. (5.6)Hence ratio of loop currents will become:∣∣∣∣ I1I2∣∣∣∣= 1k√(RCω)2+(LCω2−1− k)2. (5.7)Both |I1/I2| and |I1| have minimum values at:ωz =√1+ k√LC(5.8)which is in the middle of the two resonant frequencies of ω01 and ω02 with thevalues of:ω01 =1√LC(5.9)andω02 =√1+2k√LC(5.10)The assumption is that the parameters δ , k and R are small enough so that allsecond and higher order terms of these parameters could be neglected.Equation (5.7) can be simplified by linearization around either of the resonantfrequencies, e.g. ω0 = 1/√LC:∣∣∣∣ I1I2∣∣∣∣= 2√LCk∣∣∣∣ω− 1√LC∣∣∣∣+ 1k√R2CL+ k2 (5.11)875.2. Theory of OperationNow we consider more general case by assuming the presence of perturbation∆C on capacitor C2 i.e. C2 =C+∆C. The equation (5.2) becomes:1ωC m11 m12m21 m22+ j δ1+δ I1I2= Vm∠00 (5.12)where δ = ∆C/C. Solving (5.12) for I1 and I2 leads to: I1I2= N−1 Vm∠00Cω, (5.13)whereN, (M+δM) = m11 m12m21 m22+ j δ1+δ . (5.14)If perturbation is small enough, δ  1 i.e. (1+δ )−1 ≈ 1−δ , then|I1|=√(RCω)2+(LCω2−1+δ − k)2|det(N)| CωVm, (5.15)and|I2|= kCω|det(N)|Vm, (5.16)hence∣∣∣∣ I1I2∣∣∣∣= 1k√(RCω)2+(LCω2−1+δ − k)2. (5.17)885.2. Theory of OperationSimilar to the unperturbed case, both |I1/I2| and |I1| have minimum values at:ωz =√1+ k−δ√LC, (5.18)which is slightly shifted in comparison with the unperturbed case. Unlike the un-perturbed case, this zero is not in the middle of the two resonant frequencies of:ω01 =√1−δ/2√LC(5.19)andω02 =√1+2k−δ/2√LC. (5.20)The approximation (1+ k−δ/2)1/2≈ 12[(1+2k−δ/2)1/2+(1−δ/2)1/2]is used,where k 1 and δ  1.Assuming circuit is symmetrical and all component values, except for C2, areconstant. The function |I1/I2| in (5.17), z(ω,δ ) , |I1/I2|, could be linearized inthe vicinity of δ = δ1 = 0 and ω = ω1 = ((1+2k)/LC)1/2 using Taylor series:z(ω,δ ) ' z(ω1,δ1)+∂∂ωz(ω1,δ1)(ω−ω1)+∂∂δz(ω1,δ1)(δ −δ1) . (5.21)The first order derivatives in (5.21) could be extracted from (5.17).∂∂ωz(ω1,δ1) =1kR2C2+2kLCd' 2k√LC, (5.22)895.2. Theory of Operation∂∂δz(ω1,δ1) =1d' 1k, (5.23)where d ,(R2CL (1+2k)+ k2)1/2.Approximations in equations (5.22) and (5.23) are done based on the R2C/Lk assumption which is true for this circuit.Using values of δ1 = 0, ω1 = ((1+2k)/LC)1/2 = 31832.897, k' 1/150, RC'10−9 and LC ' 10−9, (5.22) and (5.23) can be simplified further to:z(ω1,δ1) =1k√(RCω1)2+(LCω12−1+δ1− k)2=1k√R2CL(1+2k)+ k2 (5.24)' 1∂∂ωz(ω1,δ1)' 2k√LC = 0.0095 (5.25)∂∂δz(ω1,δ1)' 1k = 150 (5.26)Using these approximations, (5.21) can be re-written in simplified form of:∣∣∣∣ I1I2∣∣∣∣= 1+ 2√LCk(ω−√1+2kLC)+1kδ (5.27)Using component values for the circuit related to this work:905.2. Theory of Operation∣∣∣∣ I1I2∣∣∣∣= 1+0.0095(ω−31832.897)+150δ (5.28)or∣∣∣∣ I1I2∣∣∣∣= 1+0.0597( f −5066.363)+0.0015∆C (5.29)where f and ∆C are in Hz and pF respectively. Although the focus of this workis around out-of-phase mode but with a similar approach around in-phase mode,ω1 = (LC)−1/2 = 31622.777, the following linearization would be resulted.∣∣∣∣ I1I2∣∣∣∣= 1− 2√LCk(ω− 1√LC)− 1kδ , (5.30)∣∣∣∣ I1I2∣∣∣∣= 1+0.0597(5032.921− f )−0.0015∆C. (5.31)With approximation√1+2k ≈ 1+ k and √1+ k ≈ 1+ k/2, both in-phase andout-of-phase equations for |I1/I2| can be combined to a single equation:∣∣∣∣ I1I2∣∣∣∣= 2√LCk∣∣∣∣ω− 1√LC(√1+ k− δ2)∣∣∣∣+ ε. (5.32)Where ε is the error of linear approximation. This linearization should be veryprecise around resonant frequencies and less precise around the corner where twolines intersecting (ωz). To examine this lets calculate ε at these two frequenciesassuming there is no perturbation. At ω = ωz = ((1+ k−δ )/LC)1/2 :915.2. Theory of Operationε =1k√(RCωz)2+(LCωz2−1− k)2=RCk√LC√1+ k ' 0.047. (5.33)This value matches with the minimum of the curve in Fig. 5.11. The next pointof interest is at out-of-phase resonant frequency, ω = ω1 = ((1+2k)/LC)1/2 :ε =1k√(RCω1)2+(LCω12−1− k)2−1=RCk√LC√1+2k+L(k)2/R2C−1 (5.34)' 0.001139. (5.35)As it can be seen, the error at resonant frequency is negligible. This error is theequivalent of kε = 7.6×10−4% perturbation.At this point we are going back to 5.17, which calculates the ratio of the loopcurrent magnitudes as:∣∣∣∣ I1I2∣∣∣∣= 1k√(RCω)2+(LCω2−1+δ − k)2. (5.36)The frequency responses of (5.15), (5.16) and (5.36) are shown in Fig. 5.7.These parameters are shown for three different cases−one unperturbed and twoopposite perturbation values. For each perturbation value, the loop currents, aswell as any other physical parameters of the system, have two resonant frequen-cies. For the unperturbed case, δ = 0, loop currents |I1| and |I2| are overlapping925.2. Theory of Operationat either in-phase or out-of-phase resonant frequencies. In positive perturbationcases, e.g. ∆C = 100 pF or δ = 0.1% , the system has lower resonant frequen-cies, with |I1| higher than |I2| for in-phase excitation and lower for out-of-phaseexcitation. For negative perturbations, e.g. ∆C =−100 pF or δ =−0.1%, the sit-uation is reversed. The frequency band of interest is either around the in-phase orout-of-phase resonant frequencies.5020 5030 5040 5050 5060 5070 508005|I 1|and|I 2|(mA)012|I 1|/|I 2|f (Hz)  |I1| / |I2| for +100 pF|I1| / |I2| for no purturbation|I1| / |I2| for -100 pF∆C = -100 pFNo Perturbation (∆C = 0)∆C = +100 pFFigure 5.7: Frequency response of the system for three values of perturbation δ =−0.1%, 0%and 0.1%.As explained above and, in (5.32) and Fig. 5.7, |I1/I2| has a linear behavior ineither resonant frequencies neighborhood. This can be expressed by:|I1/I2|= 2k√LC∣∣∣ω− 1√LC (√1+ k− δ2)∣∣∣+1k(R2CL (1+2k)+ k2)0.5.(5.37)In the case of exciting the circuit at the neighborhood of the out-of-phase reso-nant frequencies, the (5.37) becomes:935.2. Theory of Operation∣∣∣∣ I1I2∣∣∣∣≈ 1+ 2k√LC(ω−(1+2kLC)0.5)+δ2k. (5.38)As an example for L = 10 mH, C = 100 nF and k = 1/150:∣∣∣∣ I1I2∣∣∣∣= 1+0.0597( f −5066.363)+0.0015∆C, (5.39)where f and ∆C are given in Hz and pF, respectively. Equation (5.38) is valid fora wide range of excitation frequencies. It has the maximum error around the zeroof the system, which is not a useful excitation frequency. It is very precise in theresonant frequency area, which is the focus of this work. For the high sensitivityWCR method, which is based on the veering phenomenon, the following equationcan be used to backtrack the perturbation δ from the measurement of the loopcurrents [65].∆∣∣∣∣ I1I2∣∣∣∣= δ2k (5.40)or∣∣∣∣ I1I2∣∣∣∣= 1+δ/2k, (5.41)where k ,C/CC 1 is the coupling factor of the circuit.Similarly, for the CFC method, the following equation is applicable [11, 65]:∣∣∣∣ ωω0 −1∣∣∣∣= δ4(1+2k) ≈ δ4 , (5.42)where ω0 = 1√LC or ω0 =1√LC√1+2k for in-phase or out-of-phase cases, respec-945.2. Theory of Operationtively. For the out-of-phase case, (5.42) can be written as:ω =√1+2k−δ/2√LC≈ ω0(1− δ4(1+2k)). (5.43)Equations (5.40) and (5.42) show that the WCR method has considerably ( 2ktimes) higher sensitivity than CFC method. The lower the coupling factor, thehigher the sensitivity of WCR. For instance, for k = 0.02, the WCR method isalmost two orders of magnitude more sensitive than the CFC method.While higher sensitivity is a desired performance metric, we are also interestedin improving the detection limit of the system. Two distinct type of noises limitthis factor. The thermal noise determined by the resistance in the circuit and themismatch between the excitation frequency and the actual resonant frequency. Thelow resistance of the RLC circuit makes the thermal noise to be negligible withrespect to the latter source of uncertainty.5.2.2 Measurement ErrorAs it is assumed in all the sensitivity calculations, to have a precise estimation ofthe perturbation, the circuit has to be excited at the resonant frequency. All equa-tions (5.37) to (5.43) are only valid at resonant frequencies, either the in-phase orout-of-phase. For both WCR and CFC based capacitance measurements, it is verycrucial to keep the excitation precisely at resonant frequency and to dynamicallycontrolling it according to the changes in perturbation. Any mismatch betweenthe excitation and actual resonance frequencies leads to an erroneous perturbationcalculation.The objective of this work is to introduce a method that is less sensitive to955.2. Theory of Operationexcitation frequency errors. Interestingly, this simple circuit shows a very uniquecharacteristic that helps us achieving our goal. Considering an out-of-phase sce-nario, if the excitation frequency is not exactly locked at resonant frequency, it willintroduce an error in estimating perturbation, δ , in either of these methods. Theerrors associated with CFC and WCR methods have almost the same ampli-tude, but opposite signs. This feature can be exploited to drastically reduce thereading error by combining the results of both methods. Analytical proof of thisfinding is presented below.There is a one-to-one relationship between δ and ω or, δ and |I1/I2| in CFCor WCR methods, respectively. Assume there is an error of ωerr , ω0−ω ( ferr ,f0− f ) in excitation frequency. This causes both CFC and WCR method to estimatethe perturbation with errors. This is illustrated in 5.8.j I1= I2jj I1= I2j±ff (Hz)±0 + "(±f)±0±0±0+"( ±i)f (Hz)±if0+ferrf0Figure 5.8: Error comparison and improvement by hybrid method.965.2. Theory of OperationIn WCR method, this error causes our system to compute a different |I1/I2|according to (5.38). This deviation in |I1/I2| propagates to the final estimation ofthe perturbation, given by (5.41). In summary:δ = 2k(∣∣∣∣ I1I2∣∣∣∣+2√LCk ωerr−1)(5.44)= 2k(∣∣∣∣ I1I2∣∣∣∣−1)+4√LCωerr (5.45)= 2k(∣∣∣∣ I1I2∣∣∣∣−1)+4√2k+1ω0 ωerr (5.46)where 2√LCk ωerr is the propagated error to |I1/I2|. Looking at (5.46), the ulti-mate error in estimating perturbation δ using WCR method will be defined as:ε(δi), 4√2k+1ω0ωerr. (5.47)Similarly,in the case of monitoring the resonant frequency shift, ωerr generatesa perturbation estimation error given by (5.43):δ = 4(1+2k)(1− ω+ωerrω0)(5.48)= 4(1+2k)(1− ωω0)−41+2kω0ωerr. (5.49)The final error in estimating perturbation using CFC method can be defined as:ε(δ f ),−41+2kω0 ωerr. (5.50)975.2. Theory of OperationThe key point here is that the final errors in perturbation estimation in thesetwo methods, ε(δi) and ε(δ f ), have opposite signs. By combining equations (5.46)and (5.49) to eliminate ωerr term, an unbiased actual perturbation estimation (δ )is achievable.δ =2√1+2k1+√1+2k(2(1− ωω0)+ k(∣∣∣∣ I1I2∣∣∣∣−1)) . (5.51)Figure 5.9 shows the effects of this hybrid method in correcting the error. Thehorizontal axis is the excitation-resonance frequency mismatch ( ferr , ωerr/2pi).As an example, an arbitrary and fixed perturbation of about 0.025 is consideredas the reference point. The ideal perturbation curve would be a horizontal line at0.025. The two lines crossing each other with opposite slopes are perturbation val-ues estimated using CFC and WCR methods. The hybrid method result is plottedand matches very closely with the ideal (no mismatch error) line.−0.2 −0.1 0 0.1δferr (Hz)  δaδfδiδhFigure 5.9: Error comparison and improvement by hybrid method.985.2. Theory of OperationTo illustrate this with an example, assume ∆Ca is the actual capacitance change,corresponding to actual perturbationδa. With ∆Ca = 25pF or δa = 2.5×10−4, theresonant frequency will be:fres =ω22pi=3.18309×1042pi= 5066.05Hz. (5.52)If the excitation frequency is off by +0.1 Hz ( f = 5066.15 Hz), then the esti-mated perturbations will be:δ f = 4√1+2k(1− ff 0)= 1.669×10−4, (5.53)andδi = 2k(∣∣∣∣ I1I2∣∣∣∣−1)= 2150 (1.025−1) = 3.299×10−4, (5.54)which correspond to ∆C f = 16.69pF (-33% error) and ∆Ci = 32.99pF (32% error)respectively.Using the hybrid method, the estimated ∆C, defined as ∆Ch, will become:∆Ch = (1.0133/2.0133)1/2 (16.69+32.99) = 25.005pF ,which has 0.02% error with respect to the actual value of 25pF.995.3. Simulations5.3 SimulationsSeveral SPICE simulations were performed to illustrate the circuit behavior andsupport the analytical results. Fig. 5.10 shows a more detailed view of the peaks offrequency responses of |I1| and |I2| in the region of out-of phase resonant frequen-cies for different values of δ . The middle peak is related to the unperturbed case,δ = 0, for which both loop currents have identical amplitudes at the resonant fre-quency. This is marked with the vertical arrow on Fig. 5.10. Increasing δ reducesthe resonant frequency, increases the amplitude of I1 and slightly decreases the am-plitude of I2 (thus increasing |I1|/ |I2|). Conversely, decreasing δ reduces both |I1|and |I2|. Note that the changes in |I2| are not significant at resonant frequencies fordifferent values of δ . The dotted lines in Fig. 5.10 are used as a guide to showthe trend of the loop current amplitude peaks As it can be seen in Fig. 5.10, forthe ±90 pF range for ∆C (0.18% change in δ ) the relative changes of resonant fre-quency and |I1|/ |I2| are 0.06% and 14%, respectively. The higher |I1|/ |I2| makesthe sensing easier for the analog front end.5065 5065.5 5066 5066.5 5067 5067.5 50684.|I 1|and|I 2|(mA)f (Hz)  |I1||I2|-δ ↑δ ↑δ = 0Figure 5.10: Amplitudes of I1 and I2 around out-of-phase resonant frequencies fordifferent perturbation values δ . From left to right ∆Cs are: -90, -60, -30, -10, 0, 10,30, 60 and 90 pF.1005.3. SimulationsAnother set of simulations show the precision of the linearization in equation(5.38). The circuit component values for this simulation were chosen to be R1 =R2 = 0.1Ω, C1 = 100 nF, C2 = 100 nF + ∆C, L1 = L2 = 10 mH and CC = 15µF.The frequency response of |I1|/ |I2| is calculated for both the exact and linearizedsolutions.The simulation was performed using MATLAB and the results are plotted inFig. 5.11. As it can be seen, both exact and linear approximation curves match verywell in most of the frequency domain, especially close to the resonant frequencies.The maximum error is in 5050 Hz neighborhood corresponding to the system zerobetween the two resonant frequencies, which is not the region of excitation.5030 5035 5040 5045 5050 5055 5060 5065 507000.|I 1|/|I 2|f (Hz)  exactlinear approx.∆C=-100 pF∆C=+100 pF∆C=0Figure 5.11: Analytical: linear approximation vs. exact for |I1|/|I2|.To validate the analytical expressions, SPICE simulator was used to provide theresults shown in Fig. 5.12. The values used for this simulation are the same as theones employed for the analytical estimation. The exact solution, the linearizationand the SPICE numerical simulations all provide results validating the technique1015.4. Experimental Resultsused.5030 5035 5040 5045 5050 5055 5060 5065 507000.|I 1|/|I 2|f (Hz)  Analytical with ∆C = +100 pFSimulation with ∆C = +100 pFAnalytical with ∆C = 0Simulation with ∆C = 0Analytical with ∆C = -100 pFSimulation with ∆C = -100 pFFigure 5.12: Analytical vs. simulation for |I1|/|I2|.For different δ values, |I1|/ |I2| will have corresponding shift in value main-taining their dependence on the excitation resonant frequency mismatch. This isillustrated in Fig. Experimental ResultsThe components used for the hardware implementation were R1 = R2 = 0.1Ω,C1 = C2 = 100nF, L1 = L2 = 10mH and CC = 4.7µF. C1 was considered as areference; C2 corresponded to the sensing capacitance. The perturbation of C2 wasgenerated by adding extra capacitors in parallel to the initial 100nF value. The test-bench environment needed a voltage source with very low output impedance forexcitation, and an interface to measure the inductors or capacitors voltages. Mea-surement of the analog values was done using a National Instrument PXIe-1062Q1025.4. Experimental Results5065 5065.5 5066 5066.5 5067 5067.5 50680.750.80.850.90.9511.|I 1|/|I 2|f (Hz)  δ = +0.09%δ = +0.06%δ = +0.03%δ = +0.01%δ = 0.00%δ = −0.01%δ = −0.03%δ = −0.06%δ = −0.09%Figure 5.13: |I1|/|I2| around out-of-phase resonant frequencies for different pertur-bation values.system with a PXI-7854R analog interface module. A high-level block diagram ofthe proposed instrumentation system is shown in Fig. 5.14.The circuit PCB was designed using Altium Designer 12 and connected to thePXIe-1062Q module from National Instruments. The LabView-FPGA softwaretoolkit was used for the data acquisition and processing. Three 16-bit analog in-puts of PXI analog interface were used for reading excitation input, as a reference,and both inductor-capacitor junction node voltages. The inputs were connectedto the PXI module via coaxial cables and BNC connectors. The only additionalequipment that is needed to set up the experiment was an external power supply, oralternatively a pair of 9V batteries, to power up the input op-amp buffer.A single tone sine wave generator and a simple PLL algorithm were imple-mented using LabView-FPGA to keep the excitation frequency tracking the out-of-phase resonance frequency, when different perturbations were applied to C2.1035.4. Experimental ResultsFigure 5.14: High-level-block-diagram of proposed capacitance readout.The system was also capable of manually controlling the sine wave generatorby breaking the PLL loop, in order to test the designed PLL. In our experiments,the capacitors voltage amplitudes were considered as components of the voltagevector (i.e. v = [v1 v2]T ). v0 corresponds to v at balance (i.e. δ = 0) whichrelates to the out-of-phase resonant frequency f0. The values for relative shift in|v1/v2|, in reference to the same ratio at balance, is plotted vs. excitation frequencyrange wide enough to include the second natural mode. These experimental resultsare plotted in Fig. 5.15 which are resembling the simulation and analytical results.As it is shown in Fig. 5.15a the plots are parallel lines that are vertically shiftedproportional to the perturbation value. To show the distinction between the parallellines clearer, a closer view of the results is plotted in Fig. 5.15b.1045.4. Experimental Results5100 5110 5120 5130 5140 51500.| V 1 /  V2|frequency (Hz) δ = 0 % δ = 0.03 % δ = 0.06 % δ = 0.115 % δ = 2.15 %δ increase(a) Wider perturbation range.5110 5120 5130 51400.870.900.930.960.991.| V 1 /  V2|frequency (Hz) δ = 0 % δ = 0.03 % δ = 0.06 % δ = 0.115 %δ increase(b) Zoomed in, narrower perturbation range.Figure 5.15: Magnitude of v1/v2 for excitation frequencies around out-of-phasemode for different perturbations.Another experiment that shows the effectiveness of the proposed method insystem robustness due to frequency error is performed. In this experiment, a fixed0.03% perturbation (∆C = 30pF) was applied on C2. Then the excitation frequencywas intentionally swept ±0.05% of the actual out-of-phase resonant frequency.The results for perturbation estimation based on all three methods of frequencyshift (δ f ), eigenmode shift (δi), and hybrid method (δh) along with the actual per-turbation (δa) which is a horizontal line at 0.03% are presented in table 5.1 andplotted in Fig. 5.16. Note that all the values are expressed as percentages.1055.4. Experimental Resultsωerr (%) δ f (%) δi (%) δh (%) δa (%)-0.049 0.240 -0.093 0.072 0.030-0.039 0.201 -0.068 0.065 0.030-0.029 0.160 -0.043 0.057 0.030-0.02 0.121 -0.019 0.050 0.030-0.01 0.081 0.006 0.043 0.0300 0.040 0.031 0.035 0.0300.01 0.001 0.055 0.028 0.0300.02 -0.039 0.080 0.021 0.0300.029 -0.079 0.104 0.014 0.0300.039 -0.120 0.129 0.006 0.0300.049 -0.159 0.154 -0.001 0.030Table 5.1: Experimental results for all three methods.It is clear that the proposed method is considerably closer to the actual pertur-bation value.-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06-0.2- δf δi δh δap er t ur bat i on ( %)ωerr (%)Figure 5.16: Measurement error comparison.1065.5. Summary5.5 SummaryBased on the proposal for using WCRs in measuring minute capacitance changesin this thesis, two sensitivity analyses have been performed - one using WCRs andthe other using CFC for comparison. The sensitivity analysis has focused on theerror in estimated δ due to the excitation-resonance mismatch. We have performedboth exact and linearized theoretical approach, complemented by SPICE numeri-cal simulations and experiments. The two estimation methods based on CFC andWCR have a dependence on the excitation-resonance mismatch of almost equalmagnitude but opposite sign. Combining the two measurement techniques to a hy-brid method allows us to obtain a capacitive sensing method that is both highlysensitive (due to the WCR method) and robust to excitation resonant mismatches.The estimated error with the new combined method is at least one order of mag-nitude smaller than either of the WCR or CFC schemes. In the next chapter wetry to examine the use of three-degree-of-freedom (3DOF) WCRs in capacitancemeasurement aiming more simplification in excitation method.107Chapter 63DOF WCRs in CapacitanceMeasurement6.1 IntroductionIn last two chapters we have seen the features and advantages of using 2DOFWCRs in capacitance readout design. This chapter explores the possibility andfeatures of exploiting higher (three)-degree-of-freedom WCR for capacitance read-outs. We have used mechano-electrical analogy to directly transfer the analysis per-formed on 3WCRs to the electrical domain, in terms of eigenvalues and eigenvec-tors sensitivities. We then run SPICE simulations to validate the theoretical insight.The majority of the capacitive sensors in MEMS, especially for accelerometers andgyroscopes, are based on sensing differential capacitors. The theoretical analysisproves that differential perturbations in 3DOF WCRs do not require resonance fre-quency tracking. We have shown this insensitivity to the excitation frequency inSPICE simulation.1086.2. Analysis6.2 AnalysisAs we have seen, a 2DOF coupled resonator has two normal modes. Similarly, a3DOF resonator, e.g. spring-mass, system shown in Fig. 6.1 with stiffness per-turbation, has three normal modes [68, 69]. k is the stiffness of each of the threemain spring, kc is the stiffness of the two coupling springs, m is the mass and ∆ki(i = 1,2, 3) is the perturbation of the spring of the ith resonator. Perturbation canact either as small mass asymmetries or be reflected in ∆k differences that breakthe symmetry of the system.m m mkc kck1 + ¢k1 k2 + ¢k2 k3 + ¢k3Figure 6.1: 3DOF coupled spring-mass system with stiffness perturbation.Since the system is initially symmetric, the analysis of a single perturbationmodifying the first or third mass will be the same. It has been shown that, if theperturbation modifies the spring stiffness of the first of the third resonator, theresonant frequencies ωi and mode shapes of the system φi are [69]:1096.2. Analysisω1 = ω0(1+δ6), φ1 =[1 1− δ3κ1− δκ]T,ω2 = ω0(√1+κ+δ4), φ2 =[1 − δκ−1− δ2κ]T,ω3 = ω0(√1+3κ+δ12), φ3 =[1 −2− δ6κ1+δ2κ]T, (6.1)where ω0 =√k/m is the natural frequency of each of the individual spring-mass resonators, κ , kc/k , and δ = ∆k/k.Another important case corresponds to a differential perturbation, where theperturbations applied to the first and third spring have the same magnitude butopposite signs +δ and −δ , respectively. The second (middle) resonator stays un-perturbed. In this case we have:ω1 = ω0, φ1 =[1 1+δ3κ1+δκ]T,ω2 = ω0√1+κ, φ2 =[1δ2κ−1+ δ2κ]T,ω3 = ω0√1+3κ, φ3 =[1 −2+ δ6κ1− δ2κ]T. (6.2)As the above equation shows, the resonant frequencies are independent of thedifferential perturbation δ (for δ  1). This is due to the fact that the loci ofresonant frequencies of the system with a varying differential perturbation haveextrema points in the veering zone (in the vicinity of δ = 0) for all three modes.This is very helpful in simplifying the excitation circuit design since there is no1106.2. Analysisneed for a sophisticated frequency tracking feedback loop to continuously adjustthe excitation frequency depending on the perturbation value. It is also shown thatin 3DOF WCR system, the sensitivity of the mode shapes in differential perturba-tion mode is almost twice as the mode shape sensitivity in the case of one-sidedperturbation [68, 69]. This feature makes the 3DOF WCR system attractive to beused as differential capacitance measurement system in the electrical domain.A 3DOF WCR circuit based on series RLC is shown in Fig. 6.2. R1 L1 C1 R2 L2 C2 R3 L3 C3CC1 CC1VsFigure 6.2: 3DOF weekly coupled series RLC resonators.The equivalent of the equation sets (6.1) (one-sided perturbation) for this RLCcircuit is:ω1 = ω0(1− δ6), φ1 =[1 1− δ3k1− δk]T,ω2 = ω0(√1+ k− δ4), φ2 =[1 − δk−1− δ2k]T,ω3 = ω0(√1+3k− δ12), φ3 =[1 −2− δ6k1+δ2k]T, (6.3)and for (6.2) (differential perturbation) this will be:1116.2. Analysisω1 = ω0, φ1 =[1 1+δ3k1+δk]T,ω2 = ω0√1+ k, φ2 =[1δ2k−1+ δ2k]T,ω3 = ω0√1+3k, φ3 =[1 −2+ δ6k1− δ2k]T, (6.4)where k,C/CC, δ ,∆C/C, andω0 = 1/√LC. As it can be seen the sensitivityof the resonant frequencies to perturbation δ , for all modes, is zero.We are expecting a shift in resonant frequency corresponding to the perturba-tion for single-ended perturbation and no shift in differential perturbation. SPICEsimulations, parameter sweep analysis, verifies this fact as shown in figures 6.3 and6.4.3DOF WCRs one-sided perturbationPrinting Time:Tuesday, December 15, 2015, 11:49:10 PMC3 sweepFrequency (Hz)5.02k 5.09k5.03k 5.08k5.05k 5.06kI 1 Mag ni t ud e0600m100m500m200m400m300mδ = − 0.1%δ = + 0.1%δ = + 0.05%δ = 0δ = − 0.05%Figure 6.3: Frequency shift of all three modes in one-sided perturbation of a 3DOFWCRs system.1126.2. Analysis3DOF WCRs differential perturbationPrinting Time:Tuesday, December 15, 2015, 11:50:50 PMC3 sweepFrequency (Hz)5.02k 5.09k5.03k 5.08k5.05k 5.06kI 1 Mag ni t ud e0600m100m500m200m400m300mδ = + 0.1%δ = + 0.1%δ = + 0.1%δ = + 0.1%δ = + 0.1%Figure 6.4: Frequency shift of all three modes in differential perturbation of a3DOF WCRs systemFor one-sided perturbation in Fig. 6.3, the frequency response of loop currentsshows the change in the resonant frequency, while Fig. 6.4 shows almost a fix res-onant frequencies. This is one of the major advantages of differential perturbationover one-sided perturbation in 3DOF WCRs.The sensitivity of mode shapes, defined as the relative changes in the angleof the mode shape in reference to δ . To define sensitivity, modal vectors willbe plotted in three-dimensional Cartesian space with standard basis and take theirprojection on the plane defined by the first and the third axes (akin to X-Z plane inXYZ coordinate system) [69]. The angle Θ j made by the projection of jth modalvector with the first axis can be used to define sensitivity of the mode. This way,the sensitivities for resonant frequencies and mode shapes are defined as:1136.2. AnalysisNω j ,∣∣∣∣ω j−ω j0ω j0∣∣∣∣ , NΘ j , ∣∣∣∣Θ j−Θ j0Θ j0∣∣∣∣ (6.5)where tanΘ j = a3 j and a3 j is the third element of φ j. Using the above defini-tion, the sensitivity for mode shapes can be calculated.NΘ1 =∣∣∣∣ 4pi δk∣∣∣∣ , NΘ2 = NΘ3 = ∣∣∣∣ 2pi δk∣∣∣∣ . (6.6)As we will see in the next section, we calculate the sensitivity simply by theratio of the magnitude of the change in mode shape vector over its magnitude priorto perturbation.6.2.1 Differential Perturbation Detailed AnalysisTo make the analysis of 3DOF WCRs circuit shown in Fig. 6.2 simpler, we can usesuperposition technique and split it to the circuits shown in Fig. 6.5 and Fig. 6.6. R1 L1 C1 R2 L2 C2 R3 L3 C3CC1 CC2Vd2¡Vd2Figure 6.5: 3DOF WCR schematic with differential excitation.1146.2. Analysis R1 L1 C1 R2 L2 C2 R3 L3 C3CC1 CC2Vc VcFigure 6.6: 3DOF WCR schematic with common mode excitation.Assuming Vd =Vs and VC =Vs/2, then the superposition of the sources on theleft hand side of the circuits results in Vs, and they cancel each other out on theright hand side. The circuit components are R1 = R2 = R3 = R, L1 = L2 = L3 = L,CC1 = CC2 = CC, C1 = C+∆C, C3 = C−∆C, and C2 = C. The perturbation δ is∆C/C and coupling value is k =C/CC.In differential case we can simplify the circuit as shown in Fig. 6.7. We firstconsider the unperturbed case i.e. δ = 0. Fig. 6.7(a) is the same as Fig. 6.5.The resonator elements are identical and henceZ1 = Z2 = Z3 = Z where Z( jω) =R+ j(Lω−1/(Cω)) or Z(s) = R+(Ls−1/(Cs)) in Laplace domain.CC1 CC2Vd2Z1 Z2 Z3 ¡Vd2(a) Differential Excitation.CCVd2Z Z=2(b) Differential excitation, half-circuit.Figure 6.7: 3DOF WCR schematic with differential excitation.By splitting Z2 in two series Z/2 impedances and using the symmetry of the1156.2. Analysiscircuit, the circuit can be simplified to half-circuit shown in Fig. 6.7(b).Vd2=(Z+1sCCZ21sCC+ Z2)I1 =(Z+Z2+ZCCs)I1 (6.7)= Z3+ZCCs2+ZCCsI1. (6.8)If we calculate the input admittance Yin , I1Vd/2 :Yin(s) =I1Vd/2=1Z2+ZCCs3+ZCCs, (6.9)orYin( jω) =1Z( jω)2+ jZ( jω)CCω3+ jZ( jω)CCω. (6.10)So, the resonant frequencies of the system are the zeros/minima of |Z( jω)| or|3+ jZ( jω)CCω|. These frequencies are:ω1 =1√LC= ω0 (6.11)ω3 =√1+3k√LC=√1+3kω0. (6.12)The magnitude of Yin( jω) is:1166.2. AnalysisYin( jω) =1|Z( jω)||2+ jZ( jω)CCω||3+ jZ( jω)CCω| (6.13)=√√√√√√ ωC(1− ω2ω20)2+(ωRC)2(1+2k− ω2ω20)2+(ωRC)2(1+3k− ω2ω20)2+(ωRC)2(6.14)=ωC√(1− ω2ω20)2+(ωRC)2√√√√√√1+ k2−2k(1+3k− ω2ω20)(1+3k− ω2ω20)2+(ωRC)2. (6.15)From the half-circuit in Fig. 6.7(b), I2 can be expressed versus I1 :I2I1=1jωCC1jωCC +Z=22+ jωCCZ(6.16)I2 =2k1+2k− ω2ω20 + jωRCCI1 (6.17)|I2| = 2k√(1+2k− ω2ω20)2+(ωRCC)2|I1| . (6.18)We are interested in the magnitudes of loop currents I1, I2 and I3 at resonant fre-quencies ω1 and ω3. Note that due to the circuit symmetry, I3 = I1.At ω = ω1 = ω0 = 1/√LC :Yin( jω0) =I1(Vd/2)=ω0Cω0RC√1+−5k29k2+(ω0RC)2(6.19)I1 = I3 =23R(Vd/2) , (6.20)1176.2. AnalysisThe approximation ofω0RCC k is used here, which is valid by a good margin(more than 4 orders of magnitude). I2 will be:_|I2| = 2k√(2k)2+(ω0RCC)2|I1| (6.21)≈ |I1|= 23R (Vd/2) . (6.22)At ω = ω3 =√1+3k/√LC we have:Yin( jω0) =I1(Vd/2)=ω0√1+3kC3k√1+k2(ω0RC)2(6.23)I1 = I3 ≈ 13R (Vd/2) , (6.24)and I2 will become:|I2|= 2k√(k)2+(ω0√1+3kRCC)2 |I1|= 2 |I1|= 23R (Vd/2) . (6.25)In summary, the differential excitation mode, stimulates mode 1 and mode 3.The resonant frequencies and magnitudes of the loop currents are:ω1 = ω0, φ1 =(Vd2)(23R)[1 1 1]T ,ω3 = ω0√1+3k, φ3 =(Vd2)(13R)[1 −2 1]T , (6.26)1186.2. AnalysisSPICE simulation shown in Fig. 6.8 verify the above analysis.Figure 6.8: Frequency response of unperturbed 3DOF WCRs circuit with differen-tial excitation.For common mode excitation using Vc = Vs/2, the circuit can be simplifiedas shown in Fig. 6.9. The only difference with the previous case is applyinga common mode voltage VC to both ends of the circuit. This symmetry of thecircuit results in no current passing through the middle resonator i.e. I2 = 0, whichsimplifies the circuit to the half-circuit shown in Fig. 6.9(b).1196.2. Analysisi = 1; 2; 3CC1 CC2VcZ1 Z2 Z3Zi = Ri + j(!Li ¡ 1=!Ci)(a) Common mode excitation.CCVcZ(b) common mode excitation,half-circuit.Figure 6.9: 3DOF WCR schematic with common mode excitation.Solving the circuit in 6.9(b) is simple.Yin( jω) =I1Vc=1Z+ 1jωCC=1R+ j(Lω− 1ω(1C +1CC))=CωRCω+ j(ωω0 − (1+ k)) . (6.27)The resonant frequency of the circuit with common mode excitation would beω2 =√1+ k√LC= ω0√1+ k. (6.28)The magnitude of loop current I1 can be calculated from:1206.2. Analysis∣∣∣∣ I1Vc∣∣∣∣ = Cω√(RCω)2+(ω2ω20− (1+ k))2 . (6.29)At ω = ω2 the magnitude of I1 becomes:|I1| = Cω√(RCω)2|Vc| (6.30)=1R|Vc| . (6.31)I1 has the same magnitude as I3 but it has opposite phase. As we stated above,I2 = 0. In summary:ω1 = ω0, φ1 =VcR[1 0 −1]T . (6.32)SPICE simulation shown in Fig. 6.10 verify the analysis the circuit in mode 2.1216.2. AnalysisFigure 6.10: Frequency response of unperturbed 3DOF WCRs circuit with com-mon mode excitation.As it can be seen from the solutions to the differential and common modeexcitation cases above, the differential excitation stimulates mode 1 and 3, andthe common mode excitation only stimulates mode 2. This could be very helpful,especially in cases of lossy circuits, where one of the challenges are detecting anddifferentiating between the two modes due to the overlapping of the resonancepeaks. This is shown in Fig. 6.11 for a 3DOF WCR circuit with higher resistancevalues, 2Ω (Q≈160), for each resonator. As it can be seen, mode 1 and mode 2frequencies are merging into each other in a way that are not detectable using aone-sided excitation. The peak frequency is also misleading, since mode 1 and 2peaks are merged and created only one peak at a frequency between mode 1 andmode 2 resonant frequencies.1226.2. AnalysisFigure 6.11: 3DOF WCR, impact of loss on resonant frequencies.The common mode excitation, which only excites mode 2, avoids the abovementioned difficulty. Hence we choose common mode excitation for the rest of theanalysis.6.2.2 System Response to Common Mode ExcitationWe have chosen excitation in mode 2, which can be achieved by common modecircuit excitation as explained above.We also chose differential perturbation sinceit is a common operating mode for most of the capacitive MEMS sensors. More-over, the resonant frequency in this configuration is independent of perturbationmagnitude.We start the analysis by introducing the quality factor of our 3DOF WCRscircuit in mode 2. The peak of I1 magnitude can be calculated from (6.29), whichis rewritten below.1236.2. Analysis∣∣∣∣ I1Vc∣∣∣∣ = Cω√(RCω)2+(ω2ω20− (1+ k))2 . (6.33)The peak happens at ω = ω2 = ω0√1+ k, which is:∣∣∣∣ I1Vc∣∣∣∣max=Cω2√(RCω2)2(6.34)=1R(6.35)The magnitude at low or high 3dB frequency, ω3dB, is 1/(√2R). The lowerω3dB is named ω3dBL and the high side is named ω3dBH .∣∣∣∣ I1Vc∣∣∣∣3dB=Cω3dB√(RCω3dB)2+(ω23dBω20− (1+ k))2 = 1√2R (6.36)Solving this equation for ω3dB give us:ω3dBL =12√(RL)2+4ω20 −R2L(6.37)ω3dBH =12√(RL)2+4ω20 +R2L. (6.38)The bandwidth of the frequency response is:1246.2. AnalysisB.W.= ω3dBH −ω3dBL = RL , (6.39)and the quality factor is:Q =ω2B.W.=ω0√1+ kR/L=1R√LC√1+ k. (6.40)Rearranging it for R results in:R =√1+ kQ√LC=ω2LQ. (6.41)We will use this equation later in this chapter to interpret the loss effect onsensitivity versus quality factor Q.6.2.3 Differential Perturbation Analysis in Common ModeExcitationAs we shown above, if our unperturbed 3DOF WCRs series RLC circuit is excitedin common mode configuration at mode 2 resonant frequency, it results in a zerovalue for I2 loop current and an I1 loop current with value represented in (6.27). I3will have the same magnitude as I1 does, with an opposite phase i.e. I3 = −I1. Inother word, we have an unperturbed system with the following state:1256.2. AnalysisI1 =CωRCω+ j(ω2ω20− (1+ k))Vc (6.42)I2 = 0 (6.43)I3 = −I1. (6.44)Applying a differential perturbation on C1 and C3 perturbs the system and conse-quently the loop currents. The differential perturbations can be expressed as:C1 = C+∆C =C (1+δ ) (6.45)C2 = C−∆C =C (1−δ ) , (6.46)where δ = ∆C/C.In order to compute the new system response we will use perturbation anal-ysis technique, assuming δ  1. In this analysis, we interpret all the perturbedloop currents as proportionally related to the unperturbed current I1 i.e. I1 will bechanged to I′1, I2 changes to I′2, and I3 becomes I′3. This could be formulated asshown below.I′1 = I1(1+ ε1) (6.47)I′2 = I1(ε2) (6.48)I′3 = −I1(1+ ε3), (6.49)where εi (i = 1,2,3) are small values representing the sensitivity of the changes1266.2. Analysisin loop currents to perturbation. We solve the circuit in Fig. 6.6 for loop currentsunder the perturbation condition. the main equations will be:(Z+ 1jωCC − δjωC)I′1− 1jωCC I′2 =Vc(2jωCC +Z)I′2− 1jωCC(I′1+ I′3)= 0(Z+ 1jωCC +δjωC)I′3− 1jωCC I′2 =−Vc,(6.50)where the following approximations are used.1C±∆C =1C(1∓δ ). (6.51)At resonant frequency, ω2 = ω0√1+ k, we will have:Z+1jω2CC= R, (6.52)orZ = R− 1jω2CC. (6.53)We assume δ and εi (i= 1,2,3) are small enough that we can eliminate any of theirsecond order or higher terms and their products. The loop currents with eliminatingterm δε1 will become:I1(1+ ε1)(R− δjω2C)− I1ε2 =VcI1ε2(2jω2CC − 1jω2CC +R)− 1jω2CC (I1(1+ ε1)− I1(1+ ε3)) = 0−I1(1+ ε3)(R+ δjω2C)− I1ε2 1jω2CC =−Vc(6.54)1276.2. Analysiswhich can be more simplified to:Rε1 = 1jω2(δC +ε2CC)ε2R+ 1jω2CC =1jω2CC (ε1− ε3)Rε3 =− 1jω2(δC +ε2CC) (6.55)Solving this equation for ε1, ε2 and ε3 , will result in:ε1 =−ε3 = ε22 (1+ jω2RCC) (6.56)ε2 =11+ 12 (Rω2CC)2+ j 12 Rω2CC(−δk). (6.57)Applying (6.41) to the above equations with the approximation belowRω2CC =√1+ kQk(6.58)=1Q√1k+1k2(6.59)≈ 1Qk(6.60)results in:ε2 =11+ 12Q2k2 + j12Qk(−δk)(6.61)ε1 =−ε3 = ε22(1+ j1Qk)(6.62)1286.2. AnalysisFor the ideal case, loss-less resonators, these values are:ε1 = −ε3 =− δ2kε2 = −δkIn this ideal case (Q→ ∞) the loop currents appear as spikes with infinite magni-tudes (Dirac impulses in the frequency domain). In real cases, with very low losses(very high Q), with the same assumptions for εi, the loop currents become:I′1 =VcR(1− δ2k)(6.63)I′2 =VcR(−δk)(6.64)I′3 =VcR(−1− δ2k)(6.65)In other words the mode shape vector for high Q system is:φ2 =VcR[1− δ2k− δk−(1+δ2k)]T. (6.66)Now we consider more realistic case that Q is not desirably high. In this case wecompute the magnitudes of εi in (6.61) and (6.62) are affected by Q. The magni-1296.2. Analysistudes are:|ε2| = δ/k√(1+ 12Q2k2)2+ 14Q2k2(6.67)|ε1|= |ε3| =√1+ 1Q2k22√(1+ 12Q2k2)2+ 14Q2k2(6.68)which can be simplified to:|ε2| = 1√1+ 54Q2k2 +14Q4k4(δk)(6.69)|ε1|= |ε3| =√1+ 1Q2k2√4+ 5Q2k2 +14Q4k4(δk)(6.70)In our circuit here, the argument of εi , ∠εi, is very small and hence |1+ εi| =1+ |εi|. This equation is not valid in general for complex numbers. Figure. 6.12shows the arguments ∠ε1 and ∠ε2 versus quality factor Q.0 100 200 300 400 500 600 700 800 900 1000−0.01−0.00500.0050.010.0150.020.0250.03Quality Factorε 1andε 2arguments(◦)Sensitivity in lossy resonators  ε1ε2Figure 6.12: Phase plot of ε1 and ε2 at ω2 vs. Q factor.1306.2. AnalysisAs it is graphed, the arguments are even less than 1◦ over the range shown forQ.In summary, when the circuit is excited in common mode, in the presence ofdifferential perturbation on C1 and C3 , the loop currents will change slightly to thebelow values:I1 =VcR(1+ |ε1|) (6.71)I2 = −VcR |ε2| (6.72)I3 = −VcR (1−|ε1|) (6.73)where εis are|ε2| = A(δk)(6.74)|ε1|= |ε3| = B(δk)(6.75)and coefficients A and B are functions of quality factor Q.A =1√1+ 54Q2k2 +14Q4k4(6.76)B =√1+ 1Q2k2√4+ 5Q2k2 +14Q4k4(6.77)The plot of A and B versus Q, Fig. 6.13, shows changes in A and B dependency toQ. the higher the Q, the higher the sensitivity i.e. the effect of δ/k is translated to1316.3. Circuit Simulationsthe loop current amplitudes with a higher coefficient.0 100 200 300 400 500 600 700 800 900 100000. FactorCoefficientsAandBEffect of loss on sensitivity  ABFigure 6.13: Magnitude plot of ε1 and ε2 at ω2 vs. Q factor.This plot suggest two approaches. If we have high enough quality factors, thesame input perturbation δ will reflect into an output perturbation ε2 magnitudelarger than ε1 and ε3 current perturbations. For lower quality factors, nevertheless,ε2 has a higher sensitivity to variations in the quality factors. In such cases, moni-toring I1 and I3 seems to be a better solution. If the operating conditions guaranteea stable Q factor, ε2 magnitude is larger than ε1 and ε3, and monitoring I2 resultsinto a higher sensitivity.6.3 Circuit Simulations6.3.1 Single-Sided Excitation, Differential Perturbation CaseIn the following simulations it is assumed that the resonators are identical; per-turbation is differential and occurs on C1 and C3, while the circuit is excited from1326.3. Circuit Simulationsone end by the harmonic voltage source Vs (single-sided excitation with differentialperturbation) i.e.C1 =C+∆C, C2 =C, C3 =C−∆C, CC1 =CC2 =Cc,R1 = R2 = R3 = R, ,and L1 = L2 = L3 = L,where k = CCC  1 and δ = ∆CC  1 are conditions for weak coupling and smallperturbations respectively.A SPICEr simulation of the circuit is done using Multisimr12 for Fig. 6.2and the loci of the normal modes are plotted in Fig. 6.14. The values R = 100mΩ,L = 10mH, C = 100nF, CC = 15µF are used in the simulation.-0.050 -0.025 0.000 0.025 0.0504900495050005050510051505200Nat ur al  fr eq uenc ies  ( Hz )δ f1 f2 f3Figure 6.14: Three WCR veering from SPICE simulation.1336.3. Circuit SimulationsBased on the data collected from the simulations, the following relative shiftfunctions are also calculated and then graphed vs. perturbation changes. the slopeof the graph represents the sensitivity.Sω1 =ω1−ω10ω10, Sω2 =ω2−ω20ω20, Sω3 =ω3−ω30ω30, (6.78)Sφ1 =∣∣∣∣φ1−φ10φ10∣∣∣∣ , Sφ2 = ∣∣∣∣φ2−φ20φ20∣∣∣∣ , Sφ3 = ∣∣∣∣φ3−φ30φ30∣∣∣∣ , (6.79)where φ j0 and φi ( j = 1,2,3) are the vectors with normalized loop currentsmagnitudes as the vector components for the ith normal mode without perturbationand with perturbation, respectively, e.g. φ1 = [I1 I2 I3] for mode 1 where I1 andI2 and I3 are normalized magnitudes of the loop currents at 1st normal mode. Sφ iis the relative shift in φi due to the perturbation δ .-0.0050 -0.0025 0.0000 0.0025 0.0050- at i ve s ens it i vi t yδ 100x|∆f / f0| |Φ - Φ0| / |Φ0|Figure 6.15: Three WCR, relative sensitivities, mode 1 excitation.1346.3. Circuit SimulationsThese two relative shift functions, Sωi and Sφ i are plotted in Fig. 6.15 to Fig.6.16 for all three modes of the circuit with the component values stated above.-0.0050 -0.0025 0.0000 0.0025 0.00500. at i ve s ens it i vi t yδ 100x|∆f / f0| |Φ - Φ0| / |Φ0|Figure 6.16: Three WCR, relative sensitivities, mode 2 excitation.-0.0050 -0.0025 0.0000 0.0025 0.00500. at i ve s ens it i vi t yδ 100x|∆f / f0| |Φ - Φ0| / |Φ0|Figure 6.17: Three WCR, relative sensitivities, mode 3 excitation.In these figures the Sωi curves are magnified by a factor of 100 to be bettercomparable with Sφ i curves. This eigenmode sensitivities are similar, in magni-1356.3. Circuit Simulationstudes, to 2DOF WCR configuration. The advantage is, nevertheless, the invarianceof the resonant frequencies on the differential perturbation magnitude, simplifyingthe readout circuit.We choose the 2nd normal mode as excitation frequency of our system andkeep it fixed regardless of the perturbation value (δ ). The circuit was simulatedand the values for Sφ2 were calculated and then plotted in Fig. 6.18.δ (%) i1(mA) i2(mA) i3(mA) I1 I2 I3 |φ2−φ02|/|φ02|-0.05 531.041 42.5 493.95 0.73096 0.05850 0.67991 0.04379-0.025 516.56 29.18 497.83 0.719455 0.0406 0.69336 0.01970 498.98 23.8 498.93 0.70674 0.03371 0.70667 00.025 479.22 31.21 497.83 0.69281 0.04512 0.71971 0.022240.05 456.96 45.27 493.95 0.67756 0.06712 0.7324 0.05129Table 6.1: Measured values at fixed excitation frequency (mode 2).-0.050 -0.025 0.000 0.025 0.0500. |φ2-φ02| / |φ02|Rel at i ve s hi f t i n mod e s hap eδ (%)Figure 6.18: Three WCR, relative shift in normalized loop current vector under fixexcitation at 2nd mode.1366.3. Circuit SimulationsThese simulation results show that, keeping the excitation frequency at unper-turbed resonant frequency gives almost the same results (slope) as if the excitationfrequency tracks and locks to the exact resonant frequency6.3.2 Differential Excitation, Differential Perturbation CaseA simulation based on a differential excitation and differential perturbations oncapacitors C1 and C3 is done. Two out of phase harmonic inputs excite the circuitfrom both ends as shown in Fig. 6.5.The results show that only mode 1 and mode 3 appear in this system. At mode2 excitation all the loop currents are almost zero. The results are shown in Table6.2 and 6.3.δ (%) i1(mA) i2(mA) i3(mA) I1 I2 I3 |φ2−φ02|/|φ02|-0.1 563.07 655.12 757.71 0.4900 0.5701 0.6594 0.12005-0.05 615.58 663.53 714.7 0.5338 0.5753 0.6197 0.060810 666.8 666.48 666.79 0.5774 0.5771 0.5774 00.05 714.72 663.64 615.56 0.6197 0.5754 0.5337 0.060820.1 757.78 655.33 563.02 0.6594 0.5702 0.4899 0.12009Table 6.2: Simulation results for differential excitation at first mode.δ (%) i1(mA) i2(mA) i3(mA) I1 I2 I3 |φ2−φ02|/|φ02|-0.1 359.94 665.55 309.86 0.4402 0.8140 0.3790 0.0434-0.05 346.61 665.83 321.6 0.4244 0.8153 0.3938 0.02170 333.81 665.91 333.84 0.4089 0.8158 0.4090 00.05 321.6 665.78 346.61 0.3938 0.8153 0.4245 0.02160.1 309.86 665.55 359.94 0.3790 0.8140 0.4402 0.0433Table 6.3: Simulation results for differential excitation at third mode.The results show a relative sensitivity of almost 150 ( 1k ) and 50 (12k ) for mode1 and mode 3, respectively.1376.3. Circuit Simulations6.3.3 Common-Mode Excitation, Differential Perturbation CaseAnother simulation based on common mode excitation and differential perturba-tion on C1 and C3 is done, as shown in Fig. 6.6. In this case, system only respondsto excitation mode 2. Loop currents for mode 1 and mode 3 are almost zero. theresults are shown in Table 6.4.δ (%) i1(mA) i2(mA) i3(mA) I1 I2 I3 |φ2−φ02|/|φ02|-0.1 1056.1 146.6 911.4 0.7529 0.1045 0.6498 0.1277-0.05 1032.9 74.5 958.7 0.7319 0.0528 0.6793 0.06460 1000.0 0.0 1000.0 0.7071 0 0.7071 00.05 958.8 74.6 1032.9 0.6794 0.0529 0.7319 0.06460.1 911.4 146.6 1056.1 0.6498 0.1045 0.7529 0.1277Table 6.4: Simulation results; common mode excitation at 2nd mode.These results show a relative sensitivity of about 127 for mode 2 with commonmode excitation. This complies with the sensitivity of the mode shape calculatedin (6.66). The analytical value for the relative sensitivity would be:Sφ2 =∣∣∣∣φ2−φ20φ20∣∣∣∣=√ε21 + ε22 + ε23√1+0+1=√32δk≈ 129.9δ (6.80)Repeating all these simulations for resonator series resistances of 1mΩ (Q =ω2LR ≈300,000) and 100mΩ (Q≈3,000) shows that the sensitivity is robust to Qvariations as predicted by theoretical analysis. There is a trade off between thequality factor Q, coupling strength k, and the desired dynamic range of measuredδ . The higher the quality factor and the weaker the coupling strength are, the lowerthe measurable δ range will be. Fig. 6.19 illustrates the effect of different qualityfactors on the dependence of the resonant frequency f2 (mode 2), on the differential1386.3. Circuit Simulationsperturbation magnitude.5049.4 5049.6 5049.8 5050.00204060i 1 ( mA )frequency (Hz) δ=−0.05% δ=−0.025% δ=0 δ=0.025% δ=0.05%R = 1mΩ(a) Q≈300,000 (R = 1 mΩ).5046 5048 5050 5052 50540. 1 ( mA )frequency (Hz) δ = −0.05% δ = −0.025% δ = 0 δ = 0.025% δ = 0.05%R = 100 mΩ(b) Q≈3,000 (R = 100 mΩ).Figure 6.19: The effect of quality factor on f2- δ dependence.1396.3. Circuit SimulationsIn the example above, with 1mΩ series resistance, the frequency response isso sharp, that keeping the excitation frequency fixed for a perturbation range of±0.05% does not work, Fig. 6.19a. The series resistances of the RLC resonatorsare increased by a factor of 100 to 0.1Ω to get measurable values for currents overthe full range of δ , Fig. 6.19b. On the other hand, if the resistance is high ( Q islow) the sensitivity will decrease. For high quality factors, as shown in Fig. 6.19a,f2 will slightly vary with δ , possibly requiring a tracking feedback loop for theproper common mode excitation if the dynamic range of δ is wide. This behaviorcontrasts with the lower quality factor as shown in Fig. 6.19b, where f2 is practi-cally insensitive to δ magnitude. Compared to the 2DOF WCR readout circuits, themeasurement setup is simplified not only because of a fixed excitation frequency(no need of resonant frequency tracking), but also because the measurement of thedifferential δ perturbation reduces to monitoring a single current (I2) amplitude.Fig 6.20 shows I2 magnitude vs. δ .Figure 6.20: I2 magnitude for mode 2, common mode excitation, differential per-turbation.In this figure the vertical axis is normalized by dividing to Vc/R. The slope1406.3. Circuit Simulations(sensitivity) compiles with the theory in (6.66). In this simulation Vc = 100 mV,R = 100 mΩ, and coupling strength is k = 1/150.Depending on the application, the amount of damping should be calculatedto fulfill both sensitivity and dynamic range trade-off. Moreover, as we have ob-served in 2DOF WCRs, lower quality factors makes the detection of the resonantfrequencies difficult. This happens due to the fact that both amplitudes and theresonant frequencies of the loop currents are affected by quality factors values. Inthese cases, the coupling strength modifications might be helpful. The strongerthe coupling is, the farther the normal modes are pulled apart, hence the easier thedetection of their resonant frequencies becomes. This on the other hand, reducesthe relative sensitivity to the perturbation to be measured.Another set of simulations shows the effect of Q on the sensitivity more clearly.A parameter sweep on Q factor for differential perturbation with common modeexcitation (focused on mode 2) gave us the sensitivity results plotted in Fig.  ( Vs /R)Q>1000Q=300Q=15000.0020.0040.006-0.015 -0.01 -0.005 0 0.005 0.01 0.015I 2/  (δ (%)Q=60Q=30Figure 6.21: Effect of Q factor on the sensitivity (slope of normalized current I2);common mode excitation.1416.4. SummaryIn this plot I2 is scaled in reference to the value of Vc/R.As it can be seen, the lower Q the less sensitive the circuit is. The simulationresults plot of magnitude plot of ε2, shown as A, defined in (6.76), on vertical axisin Fig. 6.22, is inline with analytical results estimated and plotted in Fig. 10 100 1000 10000 100000 1000000AQuality factor (Q)Figure 6.22: Effect of Q factor on the magnitude of ε2 (simulation).6.4 SummaryIn this chapter we explored the possibility and features of 3DOF WCRs in read-out circuits. We showed that differential perturbation makes the overall circuitrysimpler, by eliminating the need for frequency tracking of the excitation voltage,without sacrificing the relative sensitivity. We analytically showed that commonmode excitation at mode 2 resonant frequency simplifies the readout circuit, espe-cially in the real scenarios with lossy elements. We have used perturbation theoryto obtain analytical estimates for eigenmode sensitivities to differential perturba-tions. SPICE circuit simulations have validated the theoretical analysis. Practicaltrade-offs between coupling strength, quality factor and desired perturbation range1426.4. Summaryto be measured have been discussed. In the following chapter, we discuss one of thepossible ways of improving circuit realization for WCRs based on active inductors.143Chapter 7Active Inductors in WCRs7.1 IntroductionThis chapter examines the usage of active inductors (op-amp-based) for imple-mentation of WCRs. The advantages and disadvantages of this method are alsoexamined.The advantages are smaller size, with the possibility of die level integration ofWCR-based capacitance readout circuits.The trend for the capacitive sensors is towards smaller physical dimensions,and consequently lower capacitance values. In order to design coupled resonatorsfor smaller capacitance measurements and yet keep the resonant frequency lowenough to prevent ADC circuit complications, relatively large inductances are re-quired. Although the circuit is simple, large value inductors are considerably bulky,especially if a high quality factor is important (associated with a low DC series re-sistance).7.2 Real (Nonideal) InductorsAll real inductors have parasitic values causes their behavior deviates from theirideal constitutive equations. There are different approaches to model a real induc-1447.2. Real (Nonideal) Inductorstor using ideal components in the literature [70, 71, 72, 73]. One of the commonmodels, that is also used by CoilCraft Inc, one of the manufacturers of the induc-tors, is shown in Fig. 7.1[73, 72]. One of the main parasitic components is theequivalent series resistance (ESR) which is the DC resistance of the inductor (R2).There is an additional frequency-dependent resistance (RVAR1), due to the skin ef-fect, in series with the ideal inductor (LVAR). The parallel capacitance C causes aself-resonance. If the self-resonant frequency is f0 then the value of the capacitorC will be [71]:C =1(2pi f0)2L.(a) Model including the low power core losses(RVAR2).(b) Simplified model without core losses.Figure 7.1: Equivalent circuits for a real inductor (from CoilCraft Inc.).R1 is the equivalent series resistance (ESR) of capacitor C. For a typical induc-tor from CoilCraft the values of the equivalent circuits are:RVAR1 = k1√fRVAR2 = k2√f1457.2. Real (Nonideal) InductorsLVAR = k3− k4 log(k5 f )where k1 to k5 are empirical coefficients measured by the manufacturer. Asan example for CoilCraft LPS4018-323 inductor with nominal value of L = k3 =3.3uH these coefficients are:k1 = 1.8x10−4, k2 = 0.792, k3 = 3.3, k4 = 0.083, k5 = 9.8x10−6.In our WCR circuit implementation, we have tried many different inductorsand capacitors, to choose the ones with the least loss (or highest quality factors)and closest mach. In almost all types of inductors there is a trade-off between in-ductor loss, physical dimensions and inductance. In our case we wanted to pushthe capacitance value of the resonator as low as possible, to be closer to the realcases of capacitive sensors. On the other hand we had the bandwidth limitation ofthe National Instruments data acquisition system; PXI-7854R analog interface ofPXIe-1062Q system has a sampling rate of 750kHz. For practical implementationof the reconstruction filters, the sampling frequency should be more than theo-ritical Nyquist frequency; practically 5 to 10 times the excitation (resonant) fre-quency.To have enough samples in order to measure the peaks and zero-crossingprecise enough, sampling rate should be considerably lower, e.g. 5 to 10 times,than the highest frequency of the signal. This limitation leads us to pick a reso-nant frequencies below 10kHz. Unfortunately lower resonant frequencies requireinductors with larger inductance values, and if a low resistance (high quality fac-tor) inductor is required the inductor will become bulky. To allow for a die levelintegration of the WCR techniques, replacing physical coils with active inductorsis a promising solution. It even helps solving the circuit size issue, if we can use1467.3. Active Inductorsthem in our WCR system as a replacement for the main inductors.7.3 Active InductorsUsing active components in conjunction with passive RC circuits is a known so-lution in electronics. The concept has been used in active filter circuits for a longtime [74, 75, 76, 77] . One of the use of these active components and RC circuitsis to implement so called active inductors [78]. Realization of active inductors canbe gyrator gyrators [79] or non-gyrator based .In this section we would like to study the feasibility of using active inductorsin WCR-based readout circuit. We also try to keep the circuit as simple as possi-ble as one of the main advantages of our method. A simple gyrator-based activeinductor implementation is shown in Fig. 7.2. The circuit is shown in Fig. 7.2(a)its equivalent passive circuit is shown in Fig. Fig. 7.2(b).¡+RsRpCpViCpRp Leq = RpRsCpRsVi(a) (b)Figure 7.2: An example of an active inductor, (a) schematic, (b) equivalent circuit.The input impedance of the active circuit can be calculated easily assuming theop-amp is ideal i.e. infinite input impedance and gain, and zero output impedance.Since both input of the op-amp are at the same potential (and is equal to ICpRp):1477.3. Active InductorsICp1sCp= IRsRs. (7.1)With a KVL from the input voltage Vi through Rs and considering the voltageat the negative input of the op-amp is ICpRp:Vi = IRsRs+ ICpRp. (7.2)By eliminating ICp from 7.1 and 7.2:Vi = (1+ sRpCp)RsIRs , (7.3)orIRs =1Rs (1+ sRpCp)Vi (7.4)andICp =sCp(1+ sRpCp)Vi. (7.5)Knowing total input current is Ii = IRs + ICp , the input impedance Zincan bewritten as:Zin =ViIi=1+ sRpCp1Rs+ sCp, (7.6)Zin =ViIi=Rs+ sRsRpCpRs+ sRsCp. (7.7)1487.3. Active Inductorsor with a bit of rearrangement, it can be written as:Zin =ViIi=(1Rs+ sRsRpCp+1Rp+ 1sCp)−1, (7.8)which is direct representation of the equivalent circuit shown in Fig. 7.2(b)with the equivalent inductor value of Leq = RsRpCp. As it can be seen, Rs appearsas a resistance in series with the inductor Leq and Rp and Cp, which are in series,appear in parallel with Leq and Rs branch. This is very close to the real inductormodel discussed in the previous section.One fundamental limitation in realization of an inductor using simple gyra-tors (same as our case), is that the inductor has always a grounded terminal. Inour WCR circuit we need at least one of the inductors to be floating. There aresome solutions presented in the literature to make a floating inductor using twointerconnected gyrators. Usually the interconnection happens at the capacitanceconnections. Fig. 7.3 shows such a floating active inductor circuit based on oursingle-ended gyrator of Fig. 7.2.¡+RsRpCpVi1 Vi2U2R0pC 0pR0sU1Vo1 Vo2Figure 7.3: Realization of a floating inductor using gyrators.To simplify the analysis this circuit is rearranged as shown in Fig. 7.4. Theequations for the circuit are:1497.3. Active InductorsVi1−Vi2 = Rs (1+ sCpRp) IRs , ICp = sCpRsIRsVi2−Vi1 = R′s(1+ sC′pR′p)I′Rs , I′Cp = sC′pR′sI′Rs(7.9)Assume R′s = Rs, C′p =Cp and C′s =Cs, then I′Rs =−IRs and I′Cp = ICp , then:IRs =1Rs (1+ sCpRp)(Vi1−Vi2) , (7.10)andICp =sCpRsRs (1+ sCpRp)(Vi1−Vi2) . (7.11)¡+RpCpVi1Vi2U1U2Vo1Vo2RsR0pC0pR0sFigure 7.4: Realization of a floating inductor using gyrators ,rearranged.Eventually the differential input impedance of the circuit can be calculated as:1507.3. Active InductorsZin =Vi1−Vi2ICp + IRs=Rs (1+ sCpRp)1+ sCpRs, (7.12)which is the same as 7.7 except that this inductor is floating between Vi1 andVi2.To prove this realization works in a WCR system, the same 2DOF circuit in Fig.4.7 was simulated with the inductors L1 and L2 are replaced with a floating and asingle-ended active inductors respectively as shown in Fig. 7.5. This simulation isdone using Multisim 12r. The values considered in this design are:Rs = 8Ω, Rp = 100kΩ, Cp = 10nFU1LMV2011MA32476C1 10nFR1 8ΩR2100kΩV11 Vpk 5kHz 0° R3.1ΩV312 V V412 V VccVccVcc Vcc-Vcc-Vcc-Vcc -VccU2LMV2011MA32476C510nFR7 8ΩR8100kΩC6100nFU3LMV2011MA32476C410nFR9 8ΩR10100kΩC82µFC10100nFR13.1ΩViFigure 7.5: Circuit used in simulation of 2DOF WCR based on active inductors.In this circuit, C6 and C10 are considered as the main capacitors of the twocoupled resonator circuits. The parametric sweep of C10 was done in the range of99nF to 101nF i.e. a ±10% perturbation on C10. The AC simulation was done foreach value of C10 and the loop currents, IR3 and IR13, are plotted in Fig. 7.6 and7.7. The sensitivity factor is taken as the relative shift in u, I3/I13 at resonance foreach value of C10. The values of this relative shift ,|ui−u0i|/ |u0i| , are presented intable 7.1 and plotted in Fig. 7.8. The simulation results show that WCR method isalmost 30 times more sensitive than the relative resonance frequency shift method.1517.3. Active Inductors5000 5200 5400 5600 5800 6000 62000. oop  c ur rent s ( mA )frequency (Hz) IR3 IR13Figure 7.6: Gyrator-based 2DOF WCR simulation at balance, loop currents fre-quency response.5400 5600 58000. oop  c ur rent s ( mA )frequency (Hz)δ = -0.5%δ = -1%δ = 0.5%δ = 1%δ = 0IR3Figure 7.7: Gyrator-based 2DOF WCR simulation for different perturbations, loopcurrents frequency response.1527.4. SummaryC13 (nF) δ (%) i2(mA) i1(mA) i2(mA) |IR3/IR13| (u1−u01)/|u01|99 -1 5466.1 92.767 69.86 1.3279 0.0718699.5 -0.5 5458.9 89.556 69.809 1.2829 0.03551100 0 5451.7 86.341 69.693 1.2389 0100.5 0.5 5444.4 83.148 69.523 1.1960 -0.03463101 1 5437.2 79.997 69.303 1.1543 -0.06826Table 7.1: Gyrator-based WCR simulation results (mode 1 excitation).-1.0 -0.5 0.0 0.5 1.0-0.10- (u-u0)/u0 10 x ∆f/f0Rel at i ve s ens it i vi t yδ (%)Figure 7.8: Relative sensitivity of gyrator-based 2DOF WCR.7.4 SummaryIn this chapter we have introduced a typical model for real inductors, we haveexplained the reason behind the need for large inductors for our WCRs, and wehave suggested one alternative based on active inductors. One of the main chal-lenges related to the active inductors is the design of floating inductors. Our WCR1537.4. Summaryuses at least one floating inductor. This work offers a topology for realizationof floating inductors. A WCR circuit based on active inductors, one floating andone grounded, is suggested and simulated. Simulation results are validating thetheory. The relative sensitivity comparison between frequency method and modeshapes was presented and confirmed the higher sensitivity of the WCR method.The following chapter summarizes the contributions, challenges and prospects ofthis research.154Chapter 8Conclusions and FurtherDiscussions8.1 Research ContributionsThe focus of this thesis was on applying innovative methods for capacitance read-out circuits based on weakly-coupled-resonators, a technique applied in the elec-trical domain for the first time. Other than the novelty of this method in electricaldomain, WCR-based capacitance readout circuits provide higher sensitivity (theo-retically three orders of magnitude) over the method based on resonant frequencyshift method which is one of the state-of-the-art methods.Capacitive sensors are one of the most popular sensors in various industries andapplications. Higher sensitivity capacitance readout circuits are of high interest inrecent years since MEMS capacitive sensors, with smaller capacitance changes,have become common in the industry. This demand for high sensitivity and morereliable readout circuits was the main motivations for the present work. The re-search started with a literature survey of state-of-the art readout circuits in chapter(2). Some of these circuits were simulated along the path of this project and pre-sented in more detail in Appendix (A). One of the readout circuits, which is based1558.1. Research Contributionson capacitance-to-voltage conversion was made with discrete component at PCBlevel and used experimentally in another project conducted by Dr. Elie Sarraf [80].The application of 2DOF WCRs has been proposed in chapter (3), accompaniedby a theoretical analysis and proof for its higher sensitivity, followed by validat-ing simulations and experimental measurements presented in chapter (4). Trackingand exciting the the resonance frequency accurately is essential to the success ofthe 2DOF-WCR method. It is paramount that the resonant frequency is locked bythe excitation source with a high degree of precision. The accuracy of the reso-nant frequency manifests itself in the sharpness of the resonant frequency peaks.If circuit losses are present, the sharpness of the resonant frequency peaks is sig-nificantly reduced, to the effect that no distinct peaks would be distinguishable iflosses are higher than a certain threshold. This of course adversely affects the reso-nance detection in experiments. Moreover, the high sensitivity achievable in theory(orders of magnitude better than the CFC method), will be significantly reduced asshown in chapter 4.Along the way, we have found an improved, hybrid method, combining fre-quency shifting and energy localization measurements, as proposed in chapter 5.This technique is more robust, higher insensitivity to the excitation frequency devi-ations. In another step further, analytical and simulations studies of readout circuitbased on 3DOF-WCR have been presented in chapter 6. It has been proven that,similar to the mechanical and MEMS domains, the 3DOF-WCR method can beused to measure differential perturbations and has the advantages an invariant res-onant frequency over a certain range of perturbations. This is helpful in simplifyingthe circuit even more by eliminating the frequency tracking (e.g. PLL). This alsomakes it inherently insensitive to errors in excitation frequency. Another unique1568.1. Research Contributionsbehavior of 2DOF WCR circuits, under differential perturbation, is that only nat-ural mode 2 or modes 1 & 3 gets stimulated if they excited by common mode ordifferential source, respectively. This behavior becomes very helpful is real (lossy)systems, where the low quality factor impacts the frequency response by pushingthe resonant peaks toward each other and overlapping them to a point that the cur-rent peaks are not happening at any of the exact resonant frequencies. We haveproven that common mode excitation is insensitive to these impact of low qualityfactor for differential perturbations.One of the challenges of these WCRs, for relatively low resonant frequencies,is the bulkiness of the inductors. We have presented the use of active inductors (e.g.gyrator-based inductors) as an alternative way to reduce the circuit size and possi-bly implementing the entire readout circuit in as a single integrated circuit. Thiseliminates the need for physical inductors, as they are replaced by circuits basedon resistors, capacitors and op-amps which are much easier to be implemented inintegrated circuits. This was shown analytically in chapter 7 along with the relatedsimulations.Another advantage of readout circuits based on WCRs is insensitivity to thechanges in the ambient conditions, e.g. temperature and relative humidity, due tothe circuit symmetry. They also require simpler analog circuit. On the other, fullytuning and matching all resonators is a challenge which is needed to get a goodresult from WCR based circuits. There is also a limitation on the bandwidth of theperturbation relative to the resonant frequency since the perturbation assumed tobe quasi-static throughout this work. Another limitation of 2DOF WCRs, which iseliminated in 3DOF WCR with differential perturbation, is the need for frequencytracking system that adds to the system complexity.1578.2. Prospects and Open Problems8.2 Prospects and Open ProblemsSince the WCR-based readout circuits are being introduced in electrical domainfor the first time, there are several directions for expanding this research, some ofthem being outlined below.The circuit excitation can be done in different ways or with different wave-forms. Several questions remain: are there more efficient ways of excitation?Is differential excitation a good choice for differential capacitive sensors? Doessquare wave excitation, binary level, have any advantage over sine wave excita-tion? These questions are valid for both 2DOF and 3DOF WCR cases.Another topic for research is to evaluate the effects of higher rate of capac-itance changes on the circuit. There was a fundamental assumption throughoutthis research, in accordance with the other researches related to mode localiza-tion phenomena, namely the assumption of perturbations being quasi-static. Thisassumption simplifies the analyses and simulations. Now that the basics of theWCR-based readout circuit have been proven, there will be a good research topicto study the bandwidth of the perturbation δ , or ∆C. This research and all the otherrelated literature considered quasi-static perturbation. There is an open question ifwe go towards dynamic perturbation since it enters into nonlinear analysis due tothe fact that both capacitance and excitation signal are changing in time.On the application of active inductors, there is still a lot of potential for newresearch. Other types of active inductors could be designed, simulated and experi-mentally tested.There is a challenge about the inductor size and bulkiness, since the frequencyhas to stay relatively low (limitation of an ordinary data acquisition system) while1588.2. Prospects and Open Problemsthe sensor capacitance is low too. Finding solutions for this problem is anothertopic for future work. One solution that we have thought about is to use an en-velope detector before the data acquisition system. Knowing that in WCR-basedmethods, the voltages and/or currents are more important than the instantaneoussignal, an envelope detection technique could be helpful to obtain the magnitudes.The frequency of this envelope is equal to the rate of the capacitance changes,which is much lower rate of change than the excitation (resonant) frequency. Thisallows the use of smaller inductors.Applying active inductors or methods like envelope detection explained aboveor even combination of those could be another research topic. Considering newtechnologies which allow MEMS device and CMOS circuitry on the same die isvery promising in pushing this idea even further to implement a single-chip WCRbased capacitive MEMS sensor and associated readout circuit.There is also another potential to study a variation of WCR based capacitancereadout that keeps the system always at the balance (non-localized) by adding cali-brated capacitors from a capacitor bank and adding them to the reference capacitor.This should result in much larger dynamic range of perturbation.The last point that we want to mention here is that all these WCR based meth-ods and their variations can be used towards measuring small inductance too.159Bibliography[1] BBCResearch. Global Markets and Technologies for Sensors. 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High sensitivityaccelerometer operating on the border of stability with digital sliding modecontrol. In SENSORS, 2013 IEEE, pages 1–4, Nov 2013.[81] R.F. Wolffenbuttel, K.M. Mahmoud, and Paul P.L. Regtien. Compliant ca-pacitive wrist sensor for use in industrial robots. Instrumentation and Mea-surement, IEEE Transactions on, 39(6):991–997, Dec 1990.170Appendix ACircuit Simulations andJustification for Using CFC as theBenchmarkA.1 IntroductionThis chapter continues the discussion on the various methods for the readout cir-cuits introduced in §2.3 with more detail. Detailed simulation (and one implemen-tation) results for each of the readout circuits are presented.A.2 CVC Simulation and Implementation ResultsThis section presents the circuit design and simulation results for the CVC readoutcircuit. As is was mentioned before, there are various CVC readout circuits. Theone that we are simulating here is based on charge amplifier. This circuit wasimplemented using discrete components and was used experimentally in a researchrelated to the closed loop readout circuits using sliding-mode control [80]. Thecircuit schematic used for the simulation is shown in Fig. A.1 to A.3.171A.2. CVC Simulation and Implementation ResultsN1 N2V_REF_1VinCMNCMN CMNV_REF_2GND GNDV_CTRL+1V_CTRL-2REF3VCC1VARICAPV_CTRL+1V_CTRL-2REF3VCC2VARICAP1E-12C_REF_1CapGND1E-12C_REF_2CapGND10KR02Res2V_BIASVSRCGND1nF-10C01Cap5NEG5POSOUT 1IN-2IN+348 BUFFERAOPAMP1MEGR01Res2GNDCOM1NC2GND3V+4 VL 5IN 6V- 7NO 8S1DG419L5POS5NEGGND 5POS 5POS5NEGCLKFigure A.1: CVC readout using charge integration, capacitor driving circuit.172A.2. CVC Simulation and Implementation ResultsN1N210pFC02Cap10pFC03Cap5POS5POS5NEG10megR03Res210megR04Res2V1PV1NOUT+ 4IN-1IN+863OUT- 5Vocm2U4THS4131GND5NEGV2PV2N5POSGND100pFC05100pFC06120pFC07120pFC0820pFC0920pFC1036KR05120KR06120KR07OUT 1IN-2IN+348 U1AOPAMPOUT 7IN-6IN+548U1BOPAMP500R11500R13Res2500R12500R14GND5NEG5POSo1OUT+4IN-1IN+863OUT- 5Vocm2U5THS41315NEGV3PV3N5POSGND100pFC11120pFC13120pFC1420pFC1520pFC1624KR08220KR09220KR1016KR15GNDOUT 1IN-2IN+348 U6ATQWFMRAP 100pFC17 COM1NC2GND3V+4 VL 5IN 6V- 7NO 8S2DG419Lo2GND5POS5NEG5POSGNDCLK100pFC12V2PV2NFigure A.2: CVC readout using charge integration, input stage differential ampli-fier, filtration and demodulation.8.2KR181nFC193.9KR171.5nFC18GND1.8KR201nFC211.5KR1912nFC20GNDOUTo25NEG5NEG5POS5POS5POS5NEGOUT 7IN-6IN+548 BUFFERBBQHKHMFW1MEGR16GNDOUT 1IN-2IN+348U9AEAVJCLHDOUT 7IN-6IN+548U9BARLUJRXR1MEGR21GND12ANALOG Header 2GNDFigure A.3: CVC readout using charge integration, output buffer and LP filter.In Fig. A.1 there are two voltage-controlled capacitors used to simulate the173A.2. CVC Simulation and Implementation Resultsbehavior of a differential capacitive sensor. Two opposite phase voltage sourcesare used to control these variable capacitors. The common node of the differentialcapacitors, CMN node, is excited by a 1MHz square pulse train with a DC bias.This DC bias is needed in case of using a sensor such as a MEMS capacitive ac-celerometer to bias the circuit connected to proof-mass [80]. The signal injectedto common node passes through the differential capacitors and gets amplified byan op-amp based differential amplifier followed by two stages of second-ordermultiple-feedback (MFB) high-pass filters. The modulation with 1MHz carriermakes the effect of 1/f noise negligible by pushing the information signal awayfrom 1/f noise frequency region before amplification. The amplified differentialoutput translates to single-ended using an op-amp and then is demodulated by thesame 1MHz clock. After demodulation the signal goes through two low-pass fil-ters. The final output will be an amplified voltage proportional to the differentialcapacitance changes. The input and output of the circuit are plotted in Fig. A.4 andsimulation results for the middle stages are shown in Fig A.5. 0.000u 100.0u 200.0u 300.0u 400.0u 500.0uTime (Sec)Out put  ( V)-1.000 -0.750 -0.500 -0.250  0.000 0.250 0.500 0.750 1.000 1.250 1.500 De lt a_C  ( p F)-200.0m-150.0m-100.0m-50.00m 0.000m50.00m100.0m150.0m200.0m250.0m300.0m OutputDelta_CFigure A.4: CVC based on differential charge amplifier, capacitance changes andoutput voltage plots.174A.2. CVC Simulation and Implementation Results 0.000u 50.00u 100.0u 150.0u 200.0u 250.0u 300.0u 350.0u 400.0u 450.0u 500.0uTime (Sec)Di f fe re nt i al  Ca pa ci t anc e V ol t ag e-50.00m-40.00m-30.00m-20.00m-10.00m 0.000m10.000m20.00m30.00m40.00m50.00m (n1-n2)(a) Differential capacitor output voltage. 0.000u 50.00u 100.0u 150.0u 200.0u 250.0u 300.0u 350.0u 400.0u 450.0u 500.0uTime (Sec)Di f f. C ha rg e A mp . Out put  ( V)-400.0m-300.0m-200.0m-100.0m 0.000m100.0m200.0m300.0m400.0mv1p-v1n(b) Output of differential amplifier. 0.000u 50.00u 100.0u 150.0u 200.0u 250.0u 300.0u 350.0u 400.0u 450.0u 500.0uTime (Sec)F ir st  Fi l te r Di f f. Out put-750.0m-500.0m-250.0m 0.000m250.0m500.0m750.0mv2p-v2n(c) First high-pass filter differential output. 0.000u 50.00u 100.0u 150.0u 200.0u 250.0u 300.0u 350.0u 400.0u 450.0u 500.0uTime (Sec)2 nd  F il t er  D if f . Out put-4.000 -3.000 -2.000 -1.000  0.000 1.000 2.000 3.000 4.000 v3p-v3n(d) Second high-pass filter differential output. 0.000u 50.00u 100.0u 150.0u 200.0u 250.0u 300.0u 350.0u 400.0u 450.0u 500.0uTime (Sec)S ing le -e nd ed  Mod ul at ed  and  F il t er red  Out put Si gnal  ( V)-4.000 -3.000 -2.000 -1.000  0.000 1.000 2.000 3.000 4.000 o1(e) Output of differential to single-ended circuit prior to de-modulation. 0.000u 50.00u 100.0u 150.0u 200.0u 250.0u 300.0u 350.0u 400.0u 450.0u 500.0uTime (Sec)De mo dul at ed  S ig na l ( V)-3.000 -2.000 -1.000  0.000 1.000 2.000 3.000 4.000 5.000 o2(f) Demodulated signal prior to low-pass filtration.Figure A.5: CVC based on differential charge amplifier, intermediate nodes simu-lation waveforms175A.3. CDC Simulation ResultsThe SPICE model for variable capacitor (VARICAP) is given by:					 !	"	#		$	 	%"&!		 '(!"&)*+,-./0	+.1	+#-%#,2-%#/20%3333,	/0)),,-,4))313)56	+	 ,This model is shown in Fig. A.6.Figure A.6: Varicap SPICE model.Since a capacitance of 1pF is connected to the reference pin, the result of thismodel is C[pF] = 1[V/pF]Vctrl[V].A.3 CDC Simulation ResultsThis section presents the circuit design and simulation results for the CDC readoutcircuit. The circuit schematic used for the simulation is shown in Fig. A.7 [27]176A.3. CDC Simulation ResultsV0BACC1C2V5V3RtR1R2D1 D2V2R3A1 A2A3A4V4R4 R5Figure A.7: Schematic representation of a CDC with the relaxation oscillator..The differential capacitive sensor consist of is C1 = C0−∆C and C2 = C0 +∆C. The duty cycle of output (V5) is linearly related to the differential capacitancechanges ∆C:D =THTH +TL=C2C1+C2=12(1+∆CC0),where TL and TH are high and low time periods of output V5, respectively.The SPICE simulation were done using the following parameters for the circuit,and the results are shown in Fig XX.R1 = R2 = R3 =1kΩ, Rt =125MΩ, R4 =5kΩ, R5 =20kΩ,C0 =3pF, ∆C as a parameter varies between -0.45pF and 0.45pF.177A.3. CDC Simulation Results∆C/C0 D-0.15 0.4258-0.1 0.4515-0.05 0.47590 0.49930.05 0.52410.1 0.54730.15 0.5706Table A.1: CDC circuit simulation data.-0.2 -0.1 0.0 0.1 0.20.400.450.500.550.60Duty CycleC/C0Figure A.8: Simulation graph for the CDC readout circuit.178A.4. CPC Simulation Results		 	 	 	 	 	 	 	 	 	 	CDC simulationTime (s)0.0 3.0m500.0µ 2.5m1.0m 2.0m1.5mV ol t ag e ( V)-1414-99-550		Figure A.9: Simulation results for the CDC readout circuit.A.4 CPC Simulation ResultsThis section presents the circuit design and simulation results for the CPC readoutcircuit. In CPC, the change in capacitance is translated into a change in the phaseshift of a sinusoid. Fig. A.10 presents the schematic of a CPC readout circuit [81].It has two input voltages, Usin(ωt) as the main input and (U/a)sin(ωt +pi − φ)as compensating input. The main input goes through the sensor capacitance. Thevalue of the capacitance at rest (unperturbed) is C0 and the capacitance changes(perturbation) subject to measure is ∆C. The compensating input goes through acompensation capacitor CC. There are two conditions that are fundamental for thiscircuit to appropriately function as shown in Fig. A.10. First, the ratio of the main179A.4. CPC Simulation Resultsinput voltage amplitude to the compensating input voltage amplitude should beequal to the ratio of CCto C0, a,CC/C0. Second, the condition ωRtCt  1 shouldbe met. The op-amp output voltage, Uo, was calculated to be:U0 =− 1Ct√(∆C+12φ 2C0)2+(φC0)2.Ucos(ωt−θ)whereθ = arctan(0.5φ 2C0+∆CφC0).This method however is prone to non-linearity (in the form of tan−1), stray ca-pacitance1 sensitivity and additional frequency dependent phase shift. An improve-ment to this circuit was presented in [47] which mainly focused on linearization ofthe phase shift using a buffer amplifier. on nonlinear phase shift using a chargeamplifier and differential capacitance. This improved configuration is presented inFig. A.11 and alleviates the issues mentioned above using two buffers creating analmost frequency independent phase shift.1Stray capacitance is a form parasitic capacitance which affects the phase shift.180A.4. CPC Simulation ResultsCtC0 +¢CUoRtACCCC = aC0!RtCt >> 1Uasin(!t + ¼ ¡ Á)Usin(!t)Figure A.10: CPC readout circuit using charge amplifier.The improved circuit has an achievable theoretical resolution of 2 ppm usingideal components, however the experimented resolution of 0.7 fF in measuring a22 pF capacitor was achieved which translates to 32 parts per million (ppm). Thisis mainly due to practical limitations [47].181A.4. CPC Simulation ResultsCTRTADCComparatorCos(!t)ADCComparatorSin(!t)LPFLPFDSDCVSVCVO1CxC0FromSinusoidalOscillatorR RR RR1C1VB= Bsin(!t + ¼ ¡ Ã)VA= Asin(!t)VAVBVSSVCCFigure A.11: Improved CPC readout circuit.It is shown in this paper how to calculate the perturbation on Cx from measuredand digitized values related to voltages VS and VC (DS and DC).Cx1Cx0=tan(ψ)n.DCDS−1,where, Cx0 and Cx1 are the Cx values before and after perturbation, respectively.n is a constant related to the ADC reference voltages, and ψ is one of the elementsfor the phase (ωt +pi−ψ) determined by values of R1 and C1 [47]. The perturba-tion can be defined as δ ,Cx1/Cx0.182A.4. CPC Simulation ResultsThe simulation results are shown in Fig A.12.At balance, Cx1 = 0, with the condition of Cx0A =C0Bcos(ψ), the phase delayat VO1 node is about pi/2. This is one of the key points of tuning the circuit. Notethat ψ is an arbitrary and small value (3.5◦in this simulation).0 200 400-202V ol t ag e ( V)Time (µSec) VA VB(a) Input Signals.0 200 400-0.04- ol t ag e ( V)Time (µS) VO1 VCC VSS(b) VCC and VSS.0 50 100 150 200 250 300 350 400-12-10-8-6-4-202460. (V)Time ( Sec)CV(c) VC and VS.Figure A.12: Simulation results for the readout circuit.183A.4. CPC Simulation ResultsThe results of a SPICEr simulation with a sweep on parameter δ are presentedin the Table A.2δ VC/VS-0.03 0.508333333-0.02 0.341666667-0.01 0.1750 0.0083333330.01 -0.1583333330.02 -0.3416666670.03 -0.5Table A.2: CPC simulation results, ratio of voltage magnitudes vs. perturbation.The slope of the line (gain of the whole system) is cot(ψ) which is approxi-mately 16.68 and complies with the result of the simulation.-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03-0.6-0.4- 0.004760.999y xRRegression Line:Figure A.13: CPC parametric sweep simulation results.184A.5. CFC Simulation ResultsA.5 CFC Simulation ResultsThis section presents the simulation for two CFC based readout circuits. The firstcircuit schematic used for the simulation is the one that previously shown in Fig.2.15 which is repeated here in Fig. A.14 for ease of reading. The component valuesused for the simulation are R1 = 56kΩ, R2 = 112.2kΩ, R3 = 2.5kΩ, R4 = 500Ω,R5 = 500Ω, R6 = 1kΩ, C1 =C3 = 10µF, C2 = 100pF, L1 = 200µH, L2 = 50µHand the transistor Q1 is a typical NPN signal transistor, e.g. 2N2222 [29].VCCR1 R2R3 R4R5R6L1 L2C2C1C3Q1Figure A.14: CFC readout circuit based on Hartley oscillator.In this oscillator based readout circuit, C2 is assumed to be the sensor capaci-tance. The SPICE simulations are done with parameter sweep on C2 for the valuesbetween 100pF to 700pF. The simulation results are shown in table A.3 and areplotted in Fig A.15.185A.5. CFC Simulation ResultsC3(pF) fo(kHz)100 905150 758200 664250 600300 553350 512400 482450 455500 433550 415600 398650 381700 369Table A.3: CPC simulation results, ratio of voltage magnitudes vs. perturbation.0 100 200 300 400 500 600 700 8003004005006007008009001000fo (kHz)C3 (pF)Figure A.15: Simulation results for the readout circuit.The second CFC readout circuit that is chosen for simulation is based on186A.5. CFC Simulation Resultsswitched-mode capacitors [39]. This circuit schematic and the related waveformsare shown in Fig. A.16. CSEN and CREF are reference and sensing capacitorsrespectively. Both of these capacitors are connected to the negative input of an op-amp, the charge amplifier. These capacitors are driven by CLK and CLKB, whichpump electric charges to these capacitors differentially. CINT is the charge integra-tion capacitor. During periods of time where SWΦ1 is closed, the op-amp acts as aunity gain buffer and forces the voltage at the common node between all capacitorsto be at VCOM. When SWΦ2 is closed the op-amp and CINT function as charge inte-grator and adds electric charges to the previous charges on CINT . The output of thiscircuit is VOUT node. A third switch SWRST does the reset function and dischargesthe capacitor to make it ready for the next measuring cycle. The circuit has threemain phases:1. Reset phase: At the beginning of each new cycle CINT gets discharged whenSWRST and SWΦ1 are switched on. VOUT also goes to VCOM in this phase.2. Pump-in phase: When SWΦ2 is on the rising edge of CLK ( falling edge ofCLKB ) happens, the charge correspondence to the difference of CREF and CSEN isintegrated by CINT . The additional charge that is added to the COUT is:∆Q = ∆VOUTCINT = (CSEN−CREF)VDD = ∆CVDD (A.1)So the step increase in VOUT can be expressed as:∆VOUT =∆CCINTVDD. (A.2)3. Toggle phase: To have a proper integration functionality, the charge on CINTshould be conserved from the end of each pump-in phase to the beginning of the187A.5. CFC Simulation Resultsnext pump-in phase. This happens by turning on SWΦ1 and falling edge of CLK(rising edge of CLKB).Pump-in and toggle phases alternating and depending on the value of CSEN inreference to CREF the charge on CINT , or the voltage VOUT , increases or decreases.This continues until VOUT reaches a predefined upper limit VREF = VH or lowerlimit VREF =VL. The initial value for VOUT is VCOM after each reset phase. AssumeVOUT reaches VREF after n consecutive steps.VREF −VCOM = n ∆CCINT VDD. (A.3)On the other hand by definition of n:n =fclkfout. (A.4)So the final equation that shows the relationship between fout and ∆C can bewritten as:∆C =CINTVREF −VCOMVDDfoutfclk(A.5)188A.5. CFC Simulation Results¡+CSENCREF¡+¡+VoutSW©1SW©2SWRSTVHVLFFVDDVDDCLKCLKBCLKDCINTResetResetCLKCLKB©1©2VCOMFigure A.16: CFC readout circuit based on switched-capacitors.The simulation plot for the values of CSEN = 10.5pF, CREF =10pF, CINT =10pF, fclk =100kHz, VCOM = 0, VDD =3.3V and VREF =VDD/2 is shown in Fig. A.17.189A.5. CFC Simulation Results0.0 0.2 0.40240.0 0.2 0.40240.0 0.2 0.40120.0 0.2 0.40240.0 0.2 0.4024φ 1φ 2V outRes etC lkTime (mS)Figure A.17: Simulation results for the readout circuit based on SC.The results of the simulation with parameter sweep on CSEN for values between9.5pF to 10.5pF are captured in table A.4 and plotted in Fig. A.18. Note that thereis an error associated with the ratio of VREF to ∆VOUT not being a whole number,which is discussed in more details in literature [40].190A.5. CFC Simulation Resultsδ = ∆CCREF fout(kHz)0.05 7.1386677650.04 5.80150.03 4.35100.02 2.77630.01 1.2195-0.01 -1.2192-0.02 -2.8261-0.03 -4.3440-0.04 -5.5556-0.05 -7.1511Table A.4: CFC simulation results, output frequency vs. perturbation.-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06-10-50510f r eq uenc y ( k Hz )δ (=∆C/CREF)Figure A.18: Simulation results for the readout circuit based on SC191


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