ABSTRACT A RESONANT CAVITY HIGH FREQUENCY OSCILLOMETER by Dante Anthony Costanzo A high frequency oscillometer operating at 100 megacycles per second ‘was developed. A reentrant cavity was employed as the resonant circuit element. High frequency voltage was induced in the cavity by inductive coupling of the cavity to a constant voltage and constant frequency generator. The current flowing in the cavity was indicated by a tuned detector circuit which was capacitatively coupled to the cavity. The high frequency energy received by the detector probe is rectified and applied to the grid circuit of a dc amplifier tube which is part of a modified Wheatstone Bridge circuit. Bridge unbalance current is linearily related to signal intensity. One electrode of a precision variable condenser was directly connected to the inner cylinder and the other electrode was connected to the outer cylinder of the cavity. Connected in parallel with the electrodes of the precision variable condenser was a condenser-type cell. When the oscillometer was initially tuned to resonance, a change in conductivity and/or dielectric constant in the solution contained in the cell detuned the oscillometer. Resonance was reestablished by adjustment of the precision condenser as indicated by maximum detector current. Dante Anthony Costanzo The changes in instrument responses, effective cavity current, cell capacitance at resonance, and cell capacitance at half power points are related to changes in admittance of the cell-solution network. Equivalent circuits are pr0posed to represent cavity, detector and cell-solution networks. A qualitative interpretation of instrument responses in terms of these circuits is presented. Instrument performance was tested with a simulated titration of an aqueous solution of hydrochloric acid with sodium hydroxide and a simulated titration of a glacial acetic acid solution of sodium acetate with perchloric acid. Dielectric constants of some pure organic solvents were determined by means of direct capacitance measurements. A RESONANT CAVITY HIGH FREQUENCY OSCILLOMETER by Dante Anthony Costanzo A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1967 ACKNOWLEDGEMENTS The author wishes to express his appreciation to his major professor, Dr. Andrew Timnick, for his encouragement and guidance given throughout this investigation and in the preparation of this thesis. Acknowledgement is also extended to Mr. Arthur H. Johnson for his helpful suggestion pertaining to instrument design and to the late Mr. Frank Betts for his cooperation and supervision in the shop. A debt of gratitude is due to the author's wife, Donna Lee, for her patience, understanding and encouragement through his graduate studies. A special acknowledgement is extended to the Analytical Chemistry Division of the Oak Ridge National Laboratory for the preparation of the drawings contained herein. ii VITA The author, Dante Anthony Costanzo, was born April 26, 1932, in Cleveland, Ohio. He attended elementary, grade and high school in Lakewood, Ohio. He graduated from Lakewood High School in 1950. He attended Michigan State University from 1950 to 1954 and received a Bachelor of Science degree in 1954 with a major in chemistry. He received a Forest Acker's scholarship (1951-52) and a Superior Student scholarship (1952-54) during his undergraduate studies. He continued in attendance at Michigan State University from 1954 until 1959 in pursuit of an advanced degree. He was a graduate teaching assistant during the school periods 1954 through 1959. The author is a member of the Society of Sigma Xi, the Society of Sigma Pi Sigma, RESA, the American Association for the Advancement of Science, the American Chemical Society, and the American Institute of Chemists. iii TABLE OF CONTENTS page ACKNOWLEDGMENTS........................ ...... ..... ................. ii VITA............ ...... ......... ......... ...... ..... ....... ......... iii LIST OF TABLES..................... ................................ vii LIST OF FIGURES... ....... . ....... . ......................... .. ...... viii LIST OF APPENDICES....... ......... . ................................ xi INTRODUCTION.............. ......................................... 1 Statement of Problem .......................... . ............... 4 THEORY............ ........ ...... ..... . ........................ , ..... 6 Series Resonance......... ..................................... 6 Parallel Resonance.......................... ........ ... ....... lO Cavity Resonator.............................................. 20 Electrical PrOperties of the Condenser-Type Cell and Solution. 32 EXPERIMENTAL....................................................... 41 Preparation of Reagents......................... ....... ....... 41 a. Aqueous Solutions..................................... 41 b. Acetous Solutions.............. ...... ................. 41 c. Methanolic Solutions.................................. 42 Purification of Solvents used in Dielectric Constant Measurements............................................... 43 Resonant Cavity Oscillometer............................. ..... 44 Signal Generator.............................................. 44 Cavity Resonator. ...... . ...................................... 49 Cell Assembly..... ..... ....... ......... .. ..................... 53 Detector Circuit .......... . ................................... 60 iv page InCidental Instr‘mentation. O I O O I O O O O O O O O O O O O O O O O O O O O O O 0 O O O O O O O O 63 Oscillometer Response Measurements............................. 64 Effect of Detector Probe Capacitance upon the Oscillometer Response.............. .................. ............. ....... 67 Evaluation of Cell Capacitance Measurements .................... 68 Evaluation of Cell Parameters C? and Co ........................ 69 Dielectric Constant Measurements ............................... 71 Oscillometer Response Curves.......... ....... ............. ..... 71 Low Frequency Conductance Measurements.................... ..... 72 Simulated Titrations........................................... 74 RESULTS AND DISCUSSION...................... ..... ................... 75 Equivalent Circuit of the Cavity Resonator Oscillometer. ....... 76 Condition of Resonance......................................... 90 Effect of the Detector Probe Capacitance upon Oscillometer Response................................. ..... .............. 99 Cell Capacitance Measurements — Reliability and Precision Data.......................................... ..... ........ 108 Evaluation of Cell Parameters, Cg and 60........ ....... ....... 114 Effect of Solution Parameters and Resonant Frequency upon the Oscillometer Response..... ............... . .............. 114 Effect of Solution Conductivity upon Oscillometer Response.... 116 Effect of Solvent Dielectric Constant upon the Oscillometer Response............. ...................... . ............. .. 123 Effect of Resonant Frequency upon the Oscillometer Response... 128 Ill III page Oscillometer Response Curves for Perchloric Acid, Sodium Acetate and p-Nitroaniline in Glacial Acetic Acid ....... ... 133 APPLICATION..... ........ . .......................................... 136 Oscillometric Titrations.... ....... ..... ..... . ..... . .......... 137 Dielectric Constant Measurements .......... . ................... 142 CONCLUSIONS. .. ........ . ..................... . ...................... 147 LITERATURE CITED.. ..... ..... ............... ..... .. ................. 150 APPENDIX....... ....... ........................... . .............. 154 vi TABLE II. III. IV. VI. VII. VIII. IX. XI. LIST OF TABLES page Vernier Capacitor Calibration: Dielectric Sample Holder, Type No. 1690-A, Serial No. 438 .......... u ............... 58 Relationship Between the Resonant Frequency, Cell Capacitance, and Detector Probe Capacitance at Optimum Detector Current.... ..... .................. .............. 109 Reliability of the Cell Capacitance Measurements: Resonant Frequency - 100 Mc/sec ................................... 110 Reliability of the Cell Capacitance Measurements: Resonant Frequency - 120 Mc/sec ................................... 112 Evaluation of the Cell Parameters C6 and C? ................. 115 Oscillometer Response Data for Aqueous Solutions of NaCl at a Resonant Frequency of 100 Mc/sec.......................121 Oscillometer Response Data for Methanolic Solutions of NaCl at a Resonant Frequency of 100 Mc/sec....................126 Effect of Dielectric Constant on the Oscillometer Response (Solvents: Methanol and Water - Solute: NaCl - Frequency: 100 Mc/sec). ............................................. 127 Oscillometer Response Data for Aqueous Solutions of NaCl at a Resonant Frequency of 75 Mc/sec.... ................. ...131 Effect of Frequency on the Oscillometer Response (Solvent: Water - Solute: NaCl - Frequencies: 75 and 100 Mc/sec)...132 Dielectric Constant Measurements of Pure Solvents at'a Resonant Frequency of 100 Mc/sec and 25° ..... .. ......... ..145 vii LIST OF FIGURES FIGURE page 1. Magnitude and Phase Angle of the Impedance of a Series Resonant Circuit as a Function of Capacitance ...... ..... ...... 7 2. Current Flowing in a Series Resonant Circuit as a Function of Capacitance at Constant Resistance.....;...;.,.............. 11 3. Current Flowing in a Series Resonant Circuit as a Function of Capacitance at Constant Inductive Reactance... .............. 12 4. Magnitude and Phase Angle of the Impedance of a Parallel Resonant Circuit as a Function of Capacitance ................ l8 5. Electromagnetic Field Configuration in a Cylindrical Reentrant Cavity.......................... ........ ...... ...... .. ....... 21 6. Application of Kirchhoff's Law to Simplify the Electrical Behavior of the Coupled Signal Generator—Resonant Cavity Network..................................................... 28 7. Equivalent Circuit of the Condenser-Type Cell and Solution: C? and R9, Capacitance and Resistance of Glass Walls: C8 and R8, Capacitance and Resistance of the Solution (A) Fundamental Equivalent Circuit, (B) Parallel Equivalent Circuit of A.......................................... ...... 34 8. High Frequency Parallel Equivalent Capacitance and Conductance as a Function of the Fundamental Equivalent Circuit Low Frequency Conductance ...... ... ............. . ........... ..... 37 9. Resonant Cavity High Frequency Oscillometer and Instrumental Set Up... ......... . ........ ...... ......... . ..... . ........... 45 10. Resonant Cavity High Frequency Oscillometer ............ u ....... 46 viii 11. 12. 13. 14. 15. 16. l7. 18. 19. 20. 21. 22. Circuit Diagram of the Resonant Cavity High Frequency Oscillometer. ............... .......... ..................... Longitudinal Cross Section Views of the Cavity Resonator ...... . Cross-Section View of the Cell-Dielectric Sample Holder—Cavity Resonator Assembly................. ........................ High Frequency Condenser—Type Cell............................. Dielectric Sample Holder Micrometer Capacitance Calibration (Type l690—A, Serial No. 438).................. ............. The Fundamental Equivalent Circuit of the Resonant Cavity Oscillometer................................................ Simplified Equivalent Circuit of the Resonant Cavity Oscillometer .............. .................................. An Equivalent Circuit as Represented by the EXpression I I.) .. ...LZ . V Z ; I = ,7 “2 , ‘7 .................................... 2 UM Zia" Z _ -.. 7 __ 1.12.31. 2 Z Z ’ 2 Detector Current as a Function of: (A) Cell Capacitance; (B) Detector Probe Capacitance (at 100 Mc/sec) .............. Simplified Equivalent Circuit Representation of the Oscillometer Fundamental Equivalent Circuit (Refer Fig. 16). Effect of Detector Probe Capacitance upon Oscillometer Response .................................................... Oscillometer Responses versus Low Frequency Conductivity of Aqueous Solutions of l«1, 2—1, and 3-1 Electrolytes (Freq.: lOO Mc/sec) ................................................. ix page 47 50 52 54 57 77 79 92 93 100 117 23. 24. 25. 26. 27. 28. 29. 30. page Effect of Solvent Dielectric Constant upon Oscillometer Response: Instrument Response versus Low Frequency Conductivity of Aqueous and Methanolic Solutions of Sodium Chloride (Freq. : 100 Mc/sec).... ........ ., .......... . ....... ....124 Effect of Resonant Frequency upon Oscillometer Response: . Instrument Response versus Low Frequency Conductivity of Aqueous Solutions of Sodium Chloride at 75 and 100 Mc/sec.....129 Oscillometer Response versus Low Frequency Specific Conductivity of Glacial Acetic Acid Solutions of Perchloric Acid, Sodium Acetate, and p-Nitroaniline (Freq. : 100 Mc/sec).134 Oscillometer Response Curves for Aqueous Solutions of Sodium Chloride (Freq. : 100 Mc/sec) ................................. 138 Simulated Oscillometric Titration of Sodium Hydroxide with Hydrochloric Acid in Water (Freq. : 100 Mc/sec) ............... 140 Simulated Low Frequency Conductometric Titration of Sodium Hydroxide with Hydrochloric Acid in Water. .................... 141 Simulated Oscillometric Titration of Sodium Acetate with Perchloric Acid in Glacial Acetic Acid (Freq. : 100 Mc/sec)...l43 Low Frequency Conductometric Titration of Sodium Acetate with Perchloric Acid in Glacial Acetic Acid ........................ 144 Equivalent Series Resonant Circuit to Represent the Resonant Cavity Oscillometer and Equivalent Shunt Resistance of the Cell-Solution Circuit.. ............... . ....... ..... ........... 170 Fundamental Equivalent Circuit to Represent the Resonant Cavity Oscillometer (Detector Coupled Capacitively to the Cavity)....l76 LIST OF APPENDICES APPENDIX page 1. Sample Derivation for Universal Resonance Curves............ 154 II. Parts List for the Resonant Cavity Oscillometer. ........... 157 III. Operating Procedure for the Resonant Cavity Oscillometer.... 159 IV. Hewlett-Packard Model 608A VHF Signal Generator Specifications....................... ...... ... ......... 161 V. Operating Procedure for the Hewlett-Packard Model 608A VHF Signal Generator .............................. .......... 163 IV. Sample Derivation for the Cell Parameters Equations (C0 and C )....................................................h. 165 9 VII. Susceptance Variance Method and Derivation of Equations..... 169 VIII: Fundamental Equivalent Circuit Representation of the Resonant Cavity Oscillometer and Derivation of the Circuit Equations (for the Capacitive Coupling) .................. 175 xi INTRODUCTION Oscillometry is a term applied to a class of physico—chemical measurements which employs high frequency oscillators as a power source and reactive coupling of such circuits with test solutions by means of electrodes isolated from the sample. Instruments which are suitable for making high frequency measurements are named oscillometers and the methods of the class are characterized as oscillometric methods.1 In the oscillometric method of analysis, the vessel containing the solution to be analyzed is placed between the plates of a capacitor (condenser-type cell) or in the field of a coil (coil-type cell). The cell may be part of a resonance circuit of a high frequency oscillator, or may be part of a resonance circuit excited by high frequency energy from an external oscillator. The oscillators operate in the megacycle per second (Me/sec) frequency region at either a fixed frequency or at variant frequencies depending upon specific circuitry. A change in composition of test solution contained within the cell is indicated by a change in the oscillometer response. The response is associated with the electrochemical properties of the test sample and is effected by the entire chemical system between the electrodes. The instrument is responsive to the dielectric constant and/or to the conductivity of the test solution, depending upon the Specific circuitry. Oscillometer reSponses which may be observed to follow the course of a titration, to evaluate dielectric constants, to monitor flowing streams or chromatographic effluents, etc., are: effective tank circuit current or voltage, cell capacitance, oscillator plate current or grid current and bias, and frequen:y. The oscillometer response measured is related to either the total admittance current or to the real or imaginary part of the admittance current flowing through the high frequency cell containing the test sample. Those os:illometer responses related to the real part of the admittance current are proportional to the Q-factor of the circuit, and those responses related to the imaginary part are proportional to the susceptance. Several excellent reviews are available which describe the present state of instrumentation and applications of the oscillometric method of analysis.1-15 The principles and theoretical foundation of the oscillometric methods of analysis have been discussed by several authors.16-25 Reilly and McCurdyl3’2A quantitatively developed in a perspicuous manner, the theory underlying the high frequency measurements when employing a condenser type cell. The authors sucessfully employed a twin—T impedance bridge to obtain absolute values of electrical quantities for a condenser- type cell to test their theory. Reilly and McCurdy13’24 have shown that the optimum working concen— tration, that concentration in which the instrument responds in a nearly linear manner with changes in solution composition, is directly propor- tional to the operating frequency of the oscillometer and to the geometry of the cell. To extend the instrument response to higher concentrations, higher operating frequencies must be employed. Many instruments have been described which operate in the fre- 21,26,28 quencies between 1-30 Mc/sec. These instruments may be employed for concentration measurements of solutions equivalent in conductance to an aqueous 0.06 M NaCl solution. In instruments operating in this frequency range, ordinary inductances and capacitances are employed in resonant circuit construction. Above 30-40 Mc/sec, the distributed capacitances and inductances of circuit connectors and components become critical and ordinary circuits are no longer usable. To circumvent this difficulty, coaxial lines, transmission lines and cavity resonators are employed to construct stable oscillators. Johnson and Timnick,29 and Pungor,3O constructed half—wavelength coaxial-line oscillators operating at approximately 125—130 Mc/sec. Lane31 and Fisher, Kelley, Stelzner and Wagner32 designed instruments operating at 200 to 250 Mc/sec which include parallel-transmission-line oscillators. Pancek,33 Stelzner,34 Blaedel and Malmstadt,35 and Huber 19 utilized concentric line oscillators operating at and Cruse frequencies between 350—500 Mc/sec. Works, Daken, and Baggs,36 in 1945, adapted a cavity resonator to the measurement of dielectric constants and power factors of solid dielectric materials at 200 Mc/sec. A signal generator was employed as a source of high frequency energy to excite two cavities, one, a measur— ing cavity into which the sample was inserted, and the other, a calibrated wavemeter. Parallel crystal diodes in conjunction with a high sensi— tivity galvanometer were employed to indicate a current proportional to the voltage in the cavity. Voltage, wavemeter, and heterodyning methods were employed to determine dielectric constants and power factors. An oscilloscope replaced the high sensitivity galvanometer in the frequency‘ modulation method to determine dielectric constants and power factors. Johnson37 adapted a cavity resonatcr obtained from a U.S. Navy Model OAO-2 radar test equipment frequency meter to the measurement of dielectric constants and power factors of solid dielectric materials. Johnson loosely coupled an oscillator operating at a fixed frequency Of 100 Mc/sec to the cavity resonator. A General Radio Type 1690«A di- electric sample holder, into which the sample was inserted, was connected to the cavity. A crystal diode and vibrating reed electrometer with current sensitivity of 10'15 amperes, was employed in the detector circuit. Blaedel and Malmstadt35 and Huber and Cruse19 described oscillo- meters operating at 350 and 415 Mc/sec reapectively. Each oscillometer contained two oscillators, a reference oscillator and a working oscill~ ator. The reference oscillator operated at a fixed frequency and employed a cavity resonator. The working oscillator, employed a concen- tric line as the resonant circuit element. The sample cell was coupled to the working oscillator, and as the composition of the sample varied, the resonant frequency of the working oscillator varied. Changes in beat frequency between these two oscillators, reflected changes in sample composition. A reentrant cylindrical cavity resonator similar to that described by Johnson37 was adopted as the resonant circuit element of the oscillo— meter developed and employed in this investigation. Statement of Problem. This study was undertaken to; (1) construct a prototype oscillometer to Operate in the 100 Mc/sec frequency range which employs a cavity resonator as the resonant circuit element, (2) to elucidate the nature of the instrument response for an instrument operating in this frequency range, (3) to test the performance of the oscillometer by means of practical physicc-chemical measurements an aqueous and nonaqueous systems. THEORY Series Resonance If a sinusoidal voltage is applied to a series combination of resistance, inductance and capacitance as shown in Fig. 1,38.4O the current flowing in the circuit is given by (1) I — E - ' 7 R + J!\Ufi£ - {DC} where I is the current in amperes, E is the applied voltage in volts, 8 is the resistance in ohms, L is the inductance in henries, C is the capacitance in farads, w is the angular frequency equal to wa, where f is the frequency in cycles per second, and j is equal to (~l)l/2. Effective impedance Z, the retarding effect on current flow, can be described by the complex quantity (2) z = R + jiX, - X?! where XZ is the inductive reactance in ohms, equal to wL and X0 is the 1 capacitive reactance equal to 3% . The effective impedance is the vector sum of resistance and reactance and has both magnitude and phase. The magnitude of the impedance is given by (3) Z = ‘I 25 q. 1 1 T 1 I 1 1 \ \x 20 —- \ r A \ “>J \\ " ‘5 ” E \\XO, whereas, for the series circuit, R~iEl1rallel resonant circuit as a function of capacitance. The impedance j-EB broken up into the equivalent resistive and reactive components. At resonance, the reactance and phase angle curves pass through =5€3ro and the total impedance and resistance curves are a maximum, the Characteris tic circuit impedance (31) Z0 = R. Below resonance the circuit is capacitive. The current leads the resultant voltage. The phase angle is negative. IM PEDANCE AT RESONANCE IMPEDANCE (0R RESISTANCE on REACTANCE) PHASE ANGLE , degrees 18 1.0 r FUNDAMENTAL ~ _ ‘I ‘ PARALLEL CIRCUIT . 0.8— II \\ ‘ — ' I3 mgébANCE I' ‘\ LéRié #0 ‘ _.>’ .2 C16 i: \\\\1 \ t , - o ._\ - : :"-.\ 0.4 —-g RESISTIVE — ; COMPONENT - 0.2 — — \“ - 0"" _ _____§.__.__ ___ ................... . EQUIVALENT _ A ................. SERIES CIRCUIT ~ 0.2 -l;l Requiv. fl _ ; REACTIVE _ — COMPONENT 5 0 4 _ 2 _: xequiv. "* a_ . _ 5 - 0-5 ‘V I I I I I I I ‘ +100 - - +80_ A .4I'" ............... .2 (D I" +60 - :5 x T +40 — V I! —1 +20 - i a o——————————.-1lI—————————— -20 _ A I. .. -40 _ 2 I. a -60 - w ./ — -so 5% ”ma-I" "00' I I I I I I . I 0.80 0.85 0.90 00.95 1.00 1.05 ‘ 1.10 1.15 1.20 CA PAC I'TANCE CAPACITANCE AT RESONANCE Figure 4. Magnitude and Phase Angle of the Impedance of a Parallel Resonant Circuit as a Function of Capacitance 19 Above resonance the circuit is inductive. The current lags the resultant voltage. The phase angle is positive. The reactance curve exhibits a minimum at a capacitance value corresponding to f 2 (32) C(min) = co]? _ 5] earlci a maximum at a capacitance value corresponding to 1 (33) C(max) '— OI Q! The variation of voltage across the parallel circuit with capaci— t ive tuning is identical to the variation of the equivalent impedance (3:15. 1:he parallel network (equation (25). The width of the voltage (equivalent to the impedance) resonance (1‘;. 3:?”‘ve, at 0.707 of the voltage at resonance, is related to the circuit r [C - 0 (sq ( _ 7 2 10.707 -2; 3 ‘Y - Y - C - 2 1_ Q I ,I0.707 0 Val-1 ‘TEEE=-‘re 72 and yl, represent the values of the fractional tuning and C? :1 and C1 correspond to the absolute values of the capacitances on e1 1t::l her side of resonance at 0. 707 of the maximum voltage. The variation of voltage with capacitance for the parallel reson- a.“ 1.:21 circuit is similar to the variation of current with capacitance for 20 the series resonant circuit. At resonance, the actual voltage of the parallel circuit varies with the circuit Q in the exact manner as does the current for the series resonant circuit (Fig. 3). The only difference between the series resonant circuit, Fig. l, and the parallel resonant circuit, Fig. 4, is that for series resonance, tzhe inductance L and capacitance C are in series as seen from the source ()f electromagnetic oscillation (the generator) while for parallel Iresonance, L and C are in parallel as seen from the generator. The circuit as shown in Fig. 1, would still be a series resonance cziqrcuit if resistance B were placed in parallel with either L or C, eri which case the electrical behavior of the circuit, represented by t: Pie resonance curves, would hardly be changed provided that the circuit 4;? ‘was not altered. Also, the circuit as shown in Fig. 4 would still be a parallel resonance circuit if resistance B were placed in series with either L or C. Cavity Resonator Cavity resonators are characterized by any closed 8 urface having conducting walls which can sustain or support oscillating TEE‘:3LEectromagnetic fields within them when excited by external generators. I”:tr“'EEE=sonators of this type are considered as standard resonant circuit <52: :Jlaements at high frequencies because of their perfect shielding, low D Qwer loss, high shunt impedance and adaptability to various electronic Q Sstems. Fig. 5 shows schematically a reentrant type hybrid coaxial cavity : Qsonator.19’41 The cavity consists of two concentric cylindrical ‘::="'=Dnductors shorted at one end, and each closed at the other end by 1E;“ 1:1ane conducting disks at right angles to their axis to form a gap. 21 ELECTRIC FIELD II; I :31 , /, ,___._.__ vl'llu'll|l I'll-Ill 'lllllll ’_._:__.=\\ \\ w /I 11 I I I, \ \‘ I \ IIIIII .\IIII...I.. II II“ WHIIIIIII TIIIII .IIIIIII. <4 ~ \ I x \ ,__:_:__: \ \ \ \ 'l'lll ‘Illal'l 1|l|lll|l Il'll MAGNETIC FIELD O 000 0000 O 000 ooooL 0009 0000 o 000 o 0.09 o 000 o 000 0 OOO Electromagnetic Field Configuration in a Figure 5. Cylindrical Reentrant Cavity ...v. I... o. ‘ (I) .._ .‘,‘ 10' h, -) (I. ”u, a.‘ 22 The hybrid type cavity is a transition between the nonentrant "perfect" "perfect" coaxial cavity. cylindrical cavity and the In this investigation, the end closure of the inner conductor, which forms the gap, was removed to decrease the capacitance associated Barrow and Mieher have shown that a hybrid cylindrical with the gap. cxaaxial cavity modified in this manner behaves electrically like the 1137brid type. When the cavity is excited by electrical oscillations, oscillating electrical and magnetic fields are induced and sustained in the radial 19, 41 Eitlci axial planes within the cavity as shown in Fig. 5. The cavity may have an infinite number of natural frequencies of C>So::.:i.llation, and may oscillate in more than one mode at any given frequency. ’ 42 However, any cavity will have certain allowed resonance The value of these frequencies and the mode of oscillation f requencies. are determined by the shape and dimensions of the cavity resonator. EP<:>71=‘ any given type or shape of a cavity, the resonant frequency will h) €53 inversely proportional to its mechanical dimensions. The simple types of cavity resonators have been analyzed in detail use of Maxwell's equations and pertinent engineering information, Although the electrical behavior and response of by 43,44 E145: €553 been derived. "hybrid" type cavity have been determined, adequate analysis or com- tug 1):: “Eahensive electrical measurements for this type have not been report— eq 41 The probable configuration of the electrical and magnetic fields the lowest mode of oscillation existing in a hybrid coaxial cavity Q33 19,41 1:.‘E Eonator is schematically shown in Fig. 5. 23 The electrical lines of forces are most dense in the axial position between the gap formed by the conducting end closure disks of the concentric cylindrical conductors. The electric lines of force are equally distributed in a radial plane and decrease in density toward the shorted end of the coaxial section. For the modified hybrid type, in which the conducting disk of the inner conductor is removed, the electric lines of force are most dense in a radial plane nearest the (Open end of the inner conductor. The magnetic lines of force are orthagonal to the electric lines caf force and are most dense at the shorted end of the coaxial section. UFtIe magnetic lines of force decrease in density in the axial direction t:<3ward the gap, and decrease in density in a radial direction away from the axis. Between the components of the electric and magnetic fields existing ‘PO'ZIthin an excited cavity, there exist linear relationships which corres- lE>wc3nd to relationships between the voltage and current in ”lumped” RLC ‘ITI—ietworks. It is possible therefore, to synthesize RLC networks which lbjK-‘ave an impedance function similar to that of a cavity. Accordingly, :jL~ ‘t is possible to represent the cavity by an equivalent RLC network wiich is independent of the type of cavity.44 The values of circuit parameters, inductance L, and capacitance C ‘EEits re functions of the reactance and depend on shape and dimensions of 1t::-‘he cavityf"O If the modified cavity may be considered analogous in electrical 1t:=> ehavior to a coaxial line, the capacitance, inductance, and impedance TEEiL.ssociated with the cavity will depend on the ratio of the radii of the ...- uvv' o... - ... fll 24 concentric cylinders. Also, the capacitance, C, will depend on the length of the gap and the dielectric constant of the medium enclosed by the cavity. The inductance L, will depend on the permeability of the medium. The impedance Z, will depend on the resistivity, the dielectric constant and permeability of the medium. The shunt resistance, R, for a reentrant cavity is a function of the loss in the circuit and is approximately given by40 (35) R = r 2 1,06583 Zn-S— . b 3 2p + %J+ Zn where shunt resistance, ohms length of coaxial section, cms wave length, cms outside radius of inner cylindrical conductor inside radius of outer cylindrical conductor resistivity of the_conductors, abohms (e m u) per cm Vp/ww = ”skin depth" = 2nf = frequency, cycles per second 3 kREQDU‘QVHSU The high frequency resistance depends therefore not only on the “=flimensions of the cavity and resistivity of the conducting surfaces 1.=::Iut also upon the frequency. The current in the cavity flows only in ‘EEa thin surface layer of the conducting walls and depends on the total =EEsurface area of the conducting walls. The circuit Q of the cavity can be conveniently defined by equation '(13) and has the same significance as for the "lumped” resonance cir— <=:uits. In terms of the magnetic field H within the cavity, the circuit 25 Q is given by40 2/§_’ . fH dv p/nw fH da (36) Q where magnetic intensity resistivity of conducting walls, abohms per cm 2nf = frequency, cycles/sec = volume of cavity, cm3 = area of the inner surface of the cavity, cm 2 2 SCH-flint: The Q of a cavity therefore is determined by the volume to surface ratio of the cavity, the resistivity of the conductors, and the mode and frequency of oscillation. The resonant wave length of a reentrant cavity resonator as shown in Fig. 5 is given by40 2 . (37) A0 = 2n[x a ° Zn QJ where wavelength for resonance, cm length of cavity radius of inner cylindrical conductor radius of outer cylindrical conductor distance between end closures (length of gap) A 0 x a b d In practice, the circuit parameters of a cavity resonator are not only a function of the characteristic impedance of a cavity but also a function of the impedance coupled into the cavity by external circuits (i.e., the signal generator, detector, and load). Accordingly, the circuit Q, and the resonant frequency of the cavity resonator are a function of the impedance coupled into the cavity from the voltage 26 source, the detector, and the load. Cavity resonators are coupled to external circuits by means of small loop or rod coupling probes, inserted through holes into the metal enclosure, which are connected to external circuits by coaxial lines.38’44 The loop probe which provides coupling with the magnetic field, is oriented within the cavity so as to enclose magnetic flux lines corres— ponding to the desired mode of operation. If an alternating current is passed through the loop, the excited cavity will support oscillation of the desired mode. Conversely, oscillations existing in an excited cavity will induce a voltage in the loop. The rod probe, which provides coupling with the electric field, is oriented to coincide with the electric lines of force. Maximum coupling is obtained in the axial position within the gap of a "hybrid“ resonator. For the case of the modified "hybrid" resonator as employed in this investigation, maximum coupling is obtained in radial position nearest the open end of the inner conductor. A voltage applied to the rod probe will also excite oscillations within the cavity, and conversely, oscillations in an excited cavity will induce a voltage in the rod probe. The size and shape of the probe, and the location and orientation of the probe within the cavity, will determine not only the degree of coupling (energy transfer), but will also determine the mode of oscilla- tion employed. By suitable location and orientation of the probe, definite information about field configuration may also be obtained. {... ‘vn >- ...»..- h ...“: kud (In ‘v-x DIV U) . 1. I.) () o9. a G 27 The modified hybrid cavity and cavities in general may be conven- iently represented by either the equivalent admittance RLC network or the equivalent impedance RLC network shown in Fig. 6A.46 As will be shown, either choice of cavity representation is equivalent when one considers the mode of operation and the observed electrical behavior of the cavity. In Fig. 6B is shown the equivalent circuit drawn to represent a signal generator inductively coupled to a cavity by means of a loop probe.43 In this representation, V1 is the applied voltage of the signal generator, R1 is the equivalent series resistance of the loop, L1 is equivalent inductance of the loop, L2 is the total equivalent inductance of the cavity, R2 is the total equivalent series resistance of the cavity and C2 is the total equivalent series capacitance of the cavity. M12 is the mutual inductance between the inductors L1 and L2. The impedance representation was selected for convenience only. The electrical behavior for the cavity network would not be appreciably altered if the admittance form of the cavity were selected in which R2, would be in parallel with C2. The electrical behavior of the inductively coupled circuits shown in Fig. 6B can be calculated with the aid of the following rules:38 Rule 1. As far as the signal generator circuit is concerned, the effect that the presence of the coupled cavity circuit has is exactly as though an impedance (liZJZ/Z2 had been added in series with the signal generator resistance and inductance, (where Z is 2 equal to series impedance of the cavity circuit when considered by itself) 28 C2 A. Equivalent Resonant Cavity Circuits ADMITTANCE FORM IMPEDANCE FORM R2 L-2 C2 8. Fundamental Circuits Of The Coupled Networks R1 R2 m KI}, . Cc L4 L2 -T- M12 C. Equivalent Circuit Referred To Signal Generator RI ALLA 'VVV L ________ J COUPLED IMPEDANCE = (”M'2)2 Z2 0. Equivalent Circuit Referred to Resonant Cavity COUPLED IMPEDANCE (lama)2 F-‘E‘l Figure 6. L2 R2 3: C2 Application of Kirchhoff's Law to Simplify the Electrical Behavior of the Coupled Signal Generator-Resonant Cavity Network 29 Rule 2. The voltage induced in the cavity circuit by the signal generator current, I1, has a magnitude of “M1211 and lags the current that produces it by 90°. In complex quantity notation, the induced voltage is —jwM I . 121 Rule 3. The cavity current is exactly the same current that would flow if the induced voltage were applied in series with the cavity inductance, L2, and if the signal generator circuit were absent. These rules may be demonstrated by writing down the circuit equations to represent the voltage relationship for the signal generator and cavity 47,48 circuits. According to Kirchhoff's law, these equations are (38) V =[Z )I + jwM I 1 'L 1 1 12 2 and (39) 0 =[Z ]I + jail! I 2 2 12 1 where Z is the series impedance of the signal generator loop, equal to 1 R1 + ijl and Z2 is the series impedance of the cavity equal to R2 + j mL - 6%7). Solving the pair of equations to eliminate L , 2 2 2 yields NM 2 12 (40) V = Z + I 1 1 22 1 As referred to the signal generator the equivalent series impedance Z . (equiv) of the coupled circuits is the ratio of the signal generator 30 voltage to the signal generator current, thus / “. K]. W I2 = = J__JJL__ (41) Z(equiv) I1 Zl + z - 2 [wM. '2 The term 4—242+—-represents the coupled impedance arising from the presence of the cavity (refer to Fig. 6C). It is evident from equation (40) that the current flowing in the signal generator circuit is given by V (42) I = 1 I z + [MM 12/2 1 k 12,! 2 A second current relationship which may be derived by simultaneous solution of equation (38) and (39), is for the value of the current flowing in the cavity as given by -ij V (43) I _ 12 1 Y 2-ZZ+wM 1 2 12 Rearrangement of the terms of equation (43) yields .mM _‘ __;UL_ . J 2 V1 V = 1 V = ____2____. (44) 12 “M 1 z Z2 + 212 2(equiv) The significance of equation (44) is that the actual circuit shown in Fig. 6B may now be represented by the series equivalent circuit shown 31 in Fig. 6D, in which the current and voltage relationships are referred to the cavity. This equivalent circuit representation permits one to explain the electrical behavior of the cavity in terms of the cavity circuit parameters, L , R , and C , at constant applied voltage V . 2 2 2 ‘ A This equivalent circuit may also be derived by use of Thévenin theorem which states that:48 Any linear network containing one or more sources of voltage and having two terminals behaves, in so far as a load impedance connected across the terminals is concerned, as though the network and its generators were equivalent to a simple gener- ator having an internal impedance Z and a generator voltage E, where E is the voltage that appears across the terminals when no load impedance is connected and Z is the impedance that is measured between the terminals when all sources of voltage in the network are short circuited. In the equivalent circuit representation shown in Fig. 6D, the equivalent cavity circuit is represented as R , L , C in series with an 2 2 2 mM 2 impedance Z which represents the equivalent impedance coupled into 1 the cavity by the signal generator. The equivalent signal generator voltage, as given by —ij (45) V =—-J-2-V, 2 21 1 behaves as if it were placed in series with an equivalent series imped- ance given by f \ IwM l2 (46) z :2 +4—242— 2(equiv) 2 1 an. au- »... . n.- .- '5‘.» “~‘ ‘,_~ ... “H ‘I._ H "‘A. ‘.~ 32 Thus, as seen from the signal generator, the cavity capacitance, 02, and inductance L2 are in series. Therefore, a cavity resonator will be analagous in electrical behavior to a series resonant circuit (Fig. 1, 2 and 3) and may be represented by the idealized series reson- ant circuit. Cavity resonators may be tuned to resonance by varying the fre- quency of the applied voltage or by coupling reactance into the cavity. At resonance, the current flowing in the cavity is a maximum. A probe which couples the cavity to a detector, is used to determine the relative magnitude of the current flowing in the cavity. A more detailed discussion of the electrical behavior of the cavity resonator is contained in the section ”Results and Discussion“. Electrical Properties of the Condenser-Type Cell and Solution. To interpret and evaluate changes in oscillometer response to changes in solution parameters, it is most convenient to represent the cell-solution combination as an equivalent circuit in which the electrical prOperties of the cell and solution are represented by lumped values of capacitance and resistance (or conductance). In this manner, changes in electrical pr0perties of the equivalent circuit may be related to changes in con— ductivity and dielectric constant of the test solution. In turn, changes in oscillometer reSponse, which result from the change in the electrical properties of the cell-solution load, may be related to the changes in the electrical properties of the equivalent cell-solution circuit. A summary of the treatment of the electrical properties of the condenser-type cell and solution accomplished by Reilly and McCurdy follows.13’24 33 A condenser-type cell was employed in the oscillometer. The vessel containing solution to be analyzed is placed between the plates of a coaxial condenser. In essence, the cell is a parallel plate condenser whose electrodes surfaces are separated by three units of dielectric two thicknesses of the vessel wall and the test solution as schematically shown in Fig. 7. The fundamental equivalent circuit to represent the cell—solution combination is given in Fig. 7, in which the electrical properties are represented as lumped values of capacitance and resistance; C9 is the capacitance due to the dielectric properties of the glass walls of the vessel, CS is the total capacitance due to the dielectric properties of the solution; Hg is the resistance due to the glass vessel walls, and R8 is the total resistance due to the test solution. If the resistance [Hg] due to walls of the vessel is assumed to be so large that it makes a negligible contribution to the cell-solution high frequency response, the cell-solution network may be reduced to the fundamental equivalent circuit shown in Fig. 7A. The net admittance of the fundamental equivalent circuit is given by 12 2 f; 3 I I ——-w C + m C C C + C R, 9 ‘5’, 96M; 61 (47) y: 1 2f #2 +3 1 2’ l2 ——z-+w C +CJ ——2-+mlC +C R3 [9 8 Rs 9 8; where Y is the admittance in mhos; Rs equals the total solution resis- tance in ohms; C9 is the capacitance due to the dielectric properties of the walls of the vessel in farads; Cs is the total capacitance due to the solution, in farads; w is the angular frequency, equal to an, where 34 ELECTRODES SOLUTION R9 Rs R9 "NJNI1'AVAV1 —ll—H l——ll— Cg C,5 Cg T Co Cs Rs CP RP A 8 Figure 7. Equivalent Circuit of Condenser—Type Cell and Solution: C and R9, Capacitance and Resistance of Glass Walls: C8 and Rs’ Capacitance and Resistance of the Solution (A) Fundamental Equivalent Circuit, (B) Parallel Equivalent Circuit of A 35 f is the frequency in cycles per second; and j is the operator equal to (_1)1/2. The net admittance is a complex quantity, equal to the sum of a real and imaginary term, and may be conveniently represented by the general equation (48 Y = G + ° B ) pJp The real part of the admittance, GP, is the conductance term and is equal to fiL-, the reciprocal of the parallel equivalent resistance. The P imaginary part of the admittance, Bp is the susceptance term and is equal to wCfi, where C? is the parallel equivalent capacitance. Thus, it is possible to reduce the fundamental equivalent circuit of Fig. 7A to the simplified equivalent circuit of Fig. 7B. The parallel equivalent conductance,Cp,is given by (49) GP=§L= 2‘ 2 K + C + C s p 2 w I 9 0 ] szC 2 g and the parallel equivalent capacitance, Cp, is given by \ I K20 + w2C C e[C + c e] g g 0 ca. 0 (50) C = , p K2 + wzlc + c e]2 k 9 0 I where K (equal to 2;) is equal to the low frequency conductivity of the 8 36 solution (in mhos); C is the capacitance of the sample space of the vessel containing air; and e is the dielectric constant of the medium contained in the cell. The Cp term and the Cp terms, given by equation (49) and (50) respectively, are identical to the real and imaginary terms of the admit— tance (equation (47) with the exception of the quantities K (equal to £7) and Cos (equal CS). 8 The high frequency conductance (GP) and the susceptance (m0?) terms of the admittance of the parallel equivalent circuit (Fig. 7B) operate independently of each other. Consequently, it is possible to evaluate each quantity individually in terms of the electrochemical properties of the solution. The high frequency conductance term Cp is predominately governed by the electrochemical behavior of ions in which electrical energy is con— verted to heat energy when ions migrate through a voltage gradient. The susceptance term (pr) is governed primarily by the electro- chemical behavior of molecules in which intra-molecular displacement of charges take place under the influence of an applied voltage gradient. Equations (49) and (50) state that the parallel equivalent high frequency conductance and the parallel equivalent capacitance are functions of the frequency f, the solution parameters K and e, and the cell parameters C? and Co' In this investigation, we are concerned only with the effects of f, K, and 6 upon the oscillometer response. A single cell was constructed and employed throughout the investigation. Fig. 8 shows the expected relationship between Cp and C? and K. Reasonable values of f, Cg, Cv’ and s, were substituted into equations 6p) , HIGH FREQUENCE CONDUCTANCE ( micromhos 37 § IIII'fIIIIIIIIlIIII'IIIIITIIIlf'IU __q Cp 100 7m I I 1 I IIIII I I I TIIII] I I I I ITITT I I I I IIIII I I I IIIII‘ FUNDAMENTAL 3 600 EQUIV. CIRCUIT ‘2 K- Ill? 2 Cg ‘‘‘‘‘‘‘‘‘‘ “ 500 _{ [— j 400 55' 50‘E - PARALLEL - EQUIV. CIRCUIT : 300 6p = Wp 1 7. O 1 1 11111] 1 1 1 111111 1 1 1 111111 1 1 1 1111 T T TTTTn T T Trnn r T TTTTTT T T TTTTrr T T TTTT l l ‘ 24 L FUNDAMENTAL EQUIV. ’1 : CIRCUIT PARAMETERS ; g : c, - 24.02 ppfarads j In m T c, - 3.182 ppfarads . g1: : e - 78.54 1 <1 9 - ~ 1:8 23 _— FREQUENCY -. 2 9 - f ' I00 Mc/sec . O '- . a.-— _ . €565 _ - 2 ’ 3 4.2 E . 11.1 E . . j 22; .1 <1 . . E b d a. - _‘ l' .. 24 I. l 1 1111111 1 l Lllllll l Jillllll l l lllllll l l I 1111: 0.000I 0.001 0.01 0.1 1.0 10 LOW FREQUENCY CONDUCTANCE (K) , mhos’ Figure 8. High Frequency Parallel Equivalent Capacitance and Conductance as a Function of the Fundamental Equivalent Circuit Low Frequency Conductance 38 (49) and (50) to obtain the theoretical curves. The dielectric constant of the solution, E, is assumed to be constant and independent of the solution conductivity, K. At very small or very large values of the low frequency solution conductivity, K, the limit of the high frequency conductance approaches zero, (51) Zim G = 0 P w + K + 0 At a value of the low frequency conductance equal to (52) f \ KI’peak) = clog + COEJ the high frequency conductance is a maximum and is equal to (53) G = g 4 + The value of K for which Cp is a maximum is obtained by differentiating the expression for Cp, given by equation (49), with respect to K, equating dG -Z%-equal to zero, and solving the resulting expression for K. When the value of the low frequency conductivity approaches zero, the parallel equivalent capacitance approaches a limit given by 39 C Cos (54) Zcm C? = C + COS K + 0 When the value of K becomes very large, C? approaches a limit given by (55 Zim C = C ) P The difference in the extreme values of the equivalent parallel capaci- tance ACb, as expressed by equation (54) and (55), is equal to C 2 =.___JL___ (56) AC? Cb +-Cbc The value of K at the mid—point,where the parallel equivalent capacitance is half way between the extremes,is found by setting K20 + wZC C e|c + C e C'Z/ZAC’; g lqOIQT OJ 9 K2 + azic + C 2:2 \g 0} (57) CpImid-point) = Thus, the value of K at the mid-point is equal to (58) KImid—point) = w4gg + Cog} A comparison of equations (52) and (58) show that the maximum value of the parallel high frequency conductance, G , and the mid-point p/max) parallel equivalent capacitance value,C p(mid p0int)’occur at the same 40 value of the low frequency conductance, or (59) KIpeak) = KImid—point) = “ICQ + 655} From a comparison of equation (53) and (56), it follows that the relationship between G and AC is given by p(max) p wAC (6O) Gp(max) = 2 Equations (52), (54) and (53) show that the magnitudes of Kipeak)’ KImid-point)’ and prmax)’ respectively, are directly proportional to the frequency, where as, AC? (equation 56) is independent of the frequency. A decrease in the frequency will result in a prOportional decrease in the magnitude of G and K ('7 p(max)’ K(peak) The magnitude of Asp (mid—pointJ' will remain constant with a change in frequency. Equations (53) and (56) show that a decrease in the dielectric constants of the medium will result in an increase in the magnitude of and ACp. The magnitudes of K( and K )’ will de- Gp(max) peak) (mid-point crease with a decrease in the dielectric constant of the medium. EXPERIMENTAL Preparation of Reagents. a. Aqueous Solutions, Laboratory distilled water was passed through a "Deminizer" ion exchange column and distilled from alkaline potassium permanganate in an all glass apparatus. This double distilled water was used for all aqueous solution preparations. Reagent grade hydrochloric acid was used to prepare a stock solution which was compared to a standard sodium hydroxide solution. A dilute sodium hydroxide solution, free of carbonate, was prepared from a filtered, saturated sodium hydroxide solution. The dilute sodium hydroxide solution was standardized against primary grade potassium acid phthalate to the phenolphthalein end point. Reagent grade sodium chloride and barium chloride dihydrate, dried at 105°, were used to prepare stock solutions of known concentrations. Optical grade lanthanum sesquioxide, ignited to constant weight, was dissolved in the minimum of hydrochloric acid to prepare a stock solution of lanthanum chloride. Dilute solutions of sodium chloride, barium chloride, lanthanum chloride, and hydrochloric acid were prepared by diluting desired aliquots of the respective stock solutions. b. Acetous Solutions. Glacial acetic acid, acetic anhydride, and perchloric acid, 70-73Z, conforming to A.C.S. specifications were used in the acetous solution preparations. Acetic anhydride was added to glacial acetic acid to remove water. Glacial acetic acid thus treated was used in all acetous solution pre- parations. 41 42 An acetous stock solution of perchloric acid was prepared and standardized against primary grade potassium acid phthalate to the crystal violet endpoint.4 Eastman Kodak p—nitroaniline was recrystallized from a mixture of equal volumes of water and ethyl alcohol until the uncorrected melting point of the solid remained constant to within 0.5°C of the accepted value of 148°C.50 An acetous stock solution was prepared and its concen- tration was determined by a potentiometric titration with standard acetous perchloric acid as titrant. Reagent grade anhydrous sodium acetate was dissolved in the treated glacial acetic acid and the concentration of this stock solution was determined by potentiometric and low-frequency conductometric titration with standard acetous perchloric acid as titrant. Dilute solutions of perchloric acid, sodium acetate and p-nitr0* aniline were prepared by diluting desired aliquots of the respective stock solutions with the treated glacial acetic acid. c. Methanolic Solutions. Commercial methanol was refluxed over grignard alloy and fractionally distilled and the middle fraction of the anhydrous methanol collected.‘49 Reagent grade sodium chloride was used to prepare a saturated methanolic stock solution. The concentration of the stock solution was determined by evaporating an aliquot of the stock solution at 105°C and weighing the residue. Dilute solutions of methanolic sodium chloride were prepared by diluting a desired aliquotof the stock solution with the anhydrous methanol. 43 Weights calibrated against Bureau of Standards calibrated weights, N.B.S. Test No. 87925, were used in all weighing. All solutions were prepared at 25°C. Calibrated pipets, bursts, and volumetric flasks were used for all solution preparation and the titrations. Purification of Solvents used in Dielectric Constant Measurements. Double distilled water was redistilled and collected in an all quartz apparatus. Mallinckrodt, thiophene free, benzene was recrystallized three times from its own melt.51 The benzene was refluxed over phosphorous pentoxide and fractionally distilled. The purified benzene was stored over metallic sodium ribbons. Eastman Kodak chlorobenzene was repeatedly shaken with concentrated sulfuric acid in a separatory funnel until no brown coloration of the sulfuric acid was observed.52 The solvent was washed successively with water, 2.M.K2003 and water. The material was dried over calcium chloride for several days and fractionally distilled. Baker Chemical purified nitrobenzene was recrystallized five times from its own melt.52 The material was dried over phosphorous pentoxide and fractionally distilled under reduced pressure. Eastman Kodak (White Label) 1, 2—dichloroethane was washed with 4 ygNaOH and then with water.53 The material was dried for several weeks over calcium chloride and fractionally distilled, while protected from sunlight, under reduced pressure. Baker Chemical acetone was purified by the Shipsey-Werner method which involves the salvation of sodium iodide by acetone to form sodium 44 iodide triacetonate.54 Acetone was saturated with anhydrous sodium iodide at ambient temperature and the excess solid removed by decant- ation. The saturated solution was cooled to -10°C and the solid sodium iodide triacetonate was separated on a filter, transferred to a dis- tillation apparatus and the acetone was distilled and collected in fractions. Resonant Cavity Oscillometer. The oscillometer and auxillary instrumentation employed in this investigation is shown in Fig. 9. The oscillometer consists of a signal generator, reentrant cavity resonator, cell assembly, and detector. The cell assembly includes the condenser-type cell, thermostated by a circulating water bath, and the dielectric sample holder. The detector components include the detector probe capacitor and high sensitivity galvanometer. Auxilliary equipment, used to obtain classical conductance measure- ments, are the conductivity bridge and cell, and the capacitance decade box. In Fig. 10 is shown a top view of the chassis upon which the cavity resonator and some associated electrical circuit components are mounted. A schematic diagram of the electrical circuit of the oscillometer is shown in Fig. 11. The parts list for the oscillometer (Fig. 11) is given in Appendix II. The operating procedure for the cavity resonator oscillometer is given in Appendix III. Signal Generator. A Hewlett-Packard Model 608 A VHF signal generator was used as a source of high frequency energy to excite the cavity resona- tor. The pertinent electrical specifications and operating procedures 45 a: umm Hauamguude can woumaoadnomo kunmsvmum swam .mufirmo uamdomom .m mus—mam I .225an 3860533 . xom 2.380 83:33. £30 ., .33 :3: ., oocuxooqoo ..otooaoo . . «coca to.oo.oo 1...... __~o 522230 320: 0383.. 5; 333:8 .255 ...:> 2:835 £8 :83 05.0.3.8 ...o on: Low 5.230 3323.505. 46 Hmumfioaafiumo koamavmum :me Anuw>mo uamaommm .OH muswfim 3.483%. ...-U. . on. Q x 2.225 5369.626 5352.330 scam news... 8323.3 .8588 $950.52.? 47 koamavmum swam >ua>mo udmdomam anu mo amuwman uwaoufiu n N . Hoe m £2:sz w40> l'al Hammaoaaauwo .HH shaman u Om .> 0: .a40 wmomd Shop—Mo R .U I «m .. >._._>Jm2umm< ..on ,o J TAOY «94523 1.4205 uI> O... 48 for the signal generator are given in Appendices IV and V respectively. The Type N RF output connector of the signal generator was replaced with a more conventional Amphenol 83-1H (Series UHF) connector. The signal generator was connected to the cavity resonator and other equip- ment by means of an RG8/U coaxial cable terminated at both ends with Amphenol 83-lSP (Series UHF) connectors. The operating frequencies of the signal generator, as indicated on the frequency dial, were compared and agreed very closely with known frequencies originating from a U.S. Army Signal Corps Model l-222—A signal generator which contained an internal quartz crystal calibrating oscillator. Since RF voltage measuring equipment was not available, a high fre- quency voltage detector37’55g56 employing a lN-34A crystal diode was constructed and used to determine the stability of the signal generator output voltage at 5 Mc/sec frequency intervals between 75 and 150 Mc/sec. The voltage stability tests were performed in the following manner. The signal generator was tuned to the desired frequency and the attenuator adjusted to give a voltmeter indication at the "Set Level”. At 100 Mc/sec the relative output voltage as indicated by the crystal diode detector was measured at 10 minute intervals over a two hour period and every hour thereafter for 10 hours. At all other frequencies, the output voltage was measured at 10 minute intervals over a one hour period. For all practical purposes the output voltage of the signal generator was constant at a given frequency. However, it was necessary to retune the signal generator to the desired frequency and readjust the voltage control before every measurement to insure that the values corresponded to the desired 49 frequency and output voltage. Due to the nonlinear frequency response of the crystal diode detector, it was impossible to compare the relative magnitude of output voltage at the various frequencies even though the signal generator output voltage was adjusted to the ”Set Level” meter indication. Consequently, the signal generator output voltmeter indication was employed as a measure of the relative output voltage at all frequencies. The exact value of the output voltage could not be determined because the impedance terminating at the output terminal of the signal generator was not known. The output voltage of the signal generator was adjusted to the "Set Level" meter indication for all high frequency measurements obtained with the oscillometer. Necessary adjustments of the signal generator were made prior to each individual measurement to insure that values obtained corresponded to the desired frequency and output voltage. Cavity Resonator. A silver plated brass reentrant cavity resonator (Fig. 12) was obtained from a U.S. Navy Model DAD—2 radar test equipment frequency meter (manufactured by The Liebel—Florsheim Co., Cincinnati, Ohio). As the resonant circuit element of the radar test equipment, the cavity provided for frequency measurement between 105 to 127 Mc/sec. The cavity was capacitatively tuned to resonance by inserting an iron rod plunger into a flared open end of the inner conductor. The plunger was driven by a vernier screw. The cavity resonator was modified to be operable in the 100 Mc/sec frequency region as the resonant circuit element of the oscillometer by removing the capacitance associated with the iron plunger and flared end 50 LOOP PROBE (To VHF Signal Generator) ,. Efirs ’/ // ’1 ’1 _IL :1. 7 1/‘///1’/I b M- ‘ I I ’/ ’//’/ ’1 ’1 ///// 7/32'—'l F11. T32 r7/ ’1 '1 I // ’/ / ’1 ’1 11 11 ROD PROBE (To DetectOr) / r; 491. R INSULATOR IL. WTTOM VIEW ' [f GLASS VESSEL THERMOSTATABLE (DNDENSER-TYPE CELL 11 --——-————OIELECTRIC SAMPLE HOLDER MOUNTING BLOCK CONCENTRIC /CYLINDRICAL CONDUCTORS :-—CLOSURE 0st SIQE VIEW Figure 12. Longitudinal Cross-Section Views of the Cavity Resonator 51 of the inner conductor. The conducting disk to which the iron plunger and vernier screw were attached, was replaced by a solid machined aluminum disk. The inside surface of the conducting disk was polished to insure minimum resistance to the current path of the conducting surface. The flared end of the inner concentric conductor was removed and the length of the inner conductor selected so that the cavity could be employed in the 100 Mc/sec frequency region as the resonant circuit element of the oscillometer, and to obtain maximum circuit Q. 7 The cavity was rigidly mounted on the chassis provided in the Model DAD-2 frequency meter and contained in a metal enclosure. The signal generator was inductively coupled to the cavity resonator by terminating the coaxial cable, connected between the output terminal of the signal generator and the cavity resonator in a loop as shown in Figs. 12 and 13. Silver was used as the loop material, in preference to other metals, to insure minimum resistance to current flow. The loop was located in the same position as provided in the Model GAO-2 frequency meter. The plane of the loop was oriented in the axial and radial direction within the cavity (orthagonal to the magnetic flux lines existing in the excited cavity). The dimensions of the loop were selected to give a maximum degree of coupling (energy transfer) and minimum power loss as experimentally indicated by a maximum of the detector current and sharpness of response curve when the trail loops were employed in the oscillometer at 100 Mc/sec. The length of the loops were experimentally observed to be more critical than the width, in increasing the degree of coupling. 52 manammm¢ uoumnommm mufi>m01umvaom oaaamm oHuuomHmHQIHHao anu mo 3afi> doauoomlmmouu .MH auswfim mmobono 200 105‘ zommm >t><0 _ .c: _ \IIIIIIIII umoE oom «Samba I mOh mmdqo 53 Cell Assembly. In Fig. 13 is shown a cross-sectional view of the cell assembly employed in the oscillometer. The cell assembly consists of a thermostatable cell and a General Radio Type l690-A dielectric sample holder (manufactured by General Radio Co., Cambridge, Mass.). The cell assembly is schematically represented in Fig. 11 by capacitors Cs’ Ct and Cb, where Cs represents the capacitance contributed by the condenser-type cell and test solution, and C and CD, the variable capacitors of the t dielectric sample holder. The condenser—type cell was experimentally designed to meet two requirements: (1) the cell could be employed for high frequency measure— ments between 75 to 150 Mc/sec with air as the dielectric contained in the cell to determine instrument Operating characteristics; and (2) the cell could be employed for high frequency measurements at 100 Mc/sec of pure and conducting liquids whose dielectric constants were between 1 and 80. The dielectric sample holder was used to measure changes in cell capacitance because: (1) it contained calibrated variable capacitors (Ct and CU); (2) the unit was provided with a vernier capacitor which could be employed for susceptance variance measurements;57’58 (3) the unit could be connected to the cavity and cell with a minimum of dis— tributed capacitance; and (4) the unit could be rigidly attached to the cavity and cell. The high frequency condenser-type cell is shown in Fig. 14. The inside wall of the jacket of the vessel holder serves as one electrode of the coaxial condenser and the right angle rod, which protrudes into the finger of the borosilicate vessel served as the other plate. The 54 24nvnfl Zrnnl #- '1? _G_LA§$. 105mm VESSEL 75mm 1mmjlt— TOP VIEW e——7n"n fiat . k? r—4 -% Hi 7F— . AF'P‘ 'fl -=4kfif 3%" :x9, I % l 1r *5" -_- ' TIE—1%"fl I 4%“ 3" ' 3%" —i J". Likingfi 4t*“%”‘* Slog VIEW FRONT VIEW Figure 14. High Frequency Condenser—Type Cell 55 cell holder was machined to very close tolerance to insure a constant resistance and capacitance to the current path between the vessel wall and electrode. The borosilicate vessel, constructed to represent an annular sample Space, was reproducibly positioned in the holder by alignment of markings placed on the surface of the vessel and on the lip of the holder. The condenser—type cell was attached to the dielectric sample holder as shown in Fig. 13. The plates of the cell are connected in parallel to the capacitors C and Cu of the dielectric sample holder. The parallel t capacitance network formed by the cell and sample holder is represented in Fig. 11 by Cs’ Ct’ and CU. It was assumed that in connecting the dielectric sample holder cell assembly to the resonant cavity, the net capacitive contribution of the assembly, C + CU + C8, is in parallel to t the total equivalent capacitance of the resonant cavity circuit. A cross sectional view of the General Radio Type 1690-A dielectric sample holder, employed as measuring capacitors C and Cv is shown in t Fig. 13. The main micrometer capacitor (Ct) is formed by two electrodes, 2.000 1 0.0025 inches in diameter. The surfaces are ground optically flat within a few wavelengths. The lower electrode is positioned by a Vycor insulator. The upper electrode is positioned by the micrometer type screw. The screw is surrounded by a flexible copper tube, or metal bellow, to insure low and constant resistance and inductance in the current path to the movable electrode which is at ground potential. The spacing in mils between the electrodes is read directly on the drum and barrel. The air capacitance as a function of spacing is given by 56 (61) C 706.4 where Ct is the calculated air capacitance in picofarads, and t is the spacing between the electrode surfaces in mils. A correction curve (Fig. 15) was provided with the dielectric sample holder, giving the measured deviations from the calculated values over the range 300 mils to 10 mils Spacing. The corrections in picofarads are added algebrai- cally to the capacitance corresponding to readings of the drum and barrel. A vernier capacitor (CU) is formed by the micrometer screw projecting into a hole in the ungrounded electrode. Micrometer screw travel is measured in terms of 500 divisions, each corresponding very closely to 0.01 picofarads. In Table I is given the calibration data for the vernier screw capacitor. The correction chart was provided by the Standardizing Labora- tory, General Radio Company. The figures in the body of the table are corrections in picofarads to be added algebraically to the readings of the vernier capacitor dial and drum for capacitance differences. When these corrections are applied, the resultant calibration is internally consistant to 0.002 picofarads or 0.1%, whichever is the greater. Thus, capacitance differences may be measured to 0.004 picofarads. The calibration data was obtained at 1000 cycles per second with the low potential or shield terminal grounded. Each setting was approached in the direction of increasing scale reading to eliminate any possible 57 Anna .02 Hmaumm .mo uamnommm osu mo uwoouflo unmam>wsvm Housmaqusm mnw .oa muawam fl IIIIIIIIIIIII J _ mOHUMHmo _ " mm H _ 7%me _. 3.3%-.“ ... ...... a.» u .238- .38. _ misam _ "mop/28mm _ n 1254230? “0&5de “ i > 340 n m4 m0 _ _ _ _ _ _ u u u _ n u _ _ _ _ _||||l|||||lL “a c a u _> n .P u "N N4 _ _m.v UlH__Ul U ..Uur _ IIIIIII n _ n . _ u. u _ _ _ _ _ __ . _ . m u _ b4??? ._ I— u _ u _ _ “ NI)— _ _lu llllll L 'Ilulllllb ..l ||||||||| L_ _ _ .m “ Eozmmzmo " 4 F I ...: :2 IE? um I | I— 78 probe into the cavity enclosure. This form of coupling may be repre- sented by several different equivalent representation circuits. Since the probe interacts with the electrostatic as well as the electromagnetic fields existing in the excited cavity, an inductive coupling representa— tion has been selected rather than a capacitative coupling representation for convenience and to facilitate circuit analysis.46 Similar qualita— tive circuit relationships as develOped herein may be obtained by use of other coupling networks representations. The distributed inductance of the detector probe at angular frequency w is represented by L3. The variable detector probe capacitance (Fig. 11) and distributed capacitance of the detector circuit are lumped together and represented by a variable capacitor, C3. The equivalent series resistance of the detector network is R2. The mutual inductance between the cavity and detector inductor is M23. To facilitate the analysis of oscillometer response and more con— veniently show the relationships between the voltage, current, and impedance for the network, the fundamental circuit shown in Fig. 16 may be reduced to the fundamental equivalent circuit shown in Fig. 17. The only significant difference between the two fundamental circuit repre- sentations is that in Fig. 17, an equivalent series capacitance CZ' and equivalent series resistance R ' have been substituted for the network 2 consisting of R9, R , CZ, C Ct and Cp as shown in Fig. 16. p v’ The equivalent series impedance, Z', for the netwOrk consisting of R2, RP, CZ, CU, Ct and C? is given by 79 “‘Eficfiflfif“-'" GENERATOR . _ . . Y. n. V. A. C. _ _ . . .3 L2 “I IIIII Milli.“ “ 3 . . L R_ _ . Lo 3 m. _ R . C. _ E. _ Tl. _ 3 W: . u C _ . _ r lllllllllllll IL Simplified Equivalent Circuit of the Resonant Cavity Figure 17. Oscillometer 80 (69) 2' = 32+ Hp 2 + w2 C + C + C + 0 ‘2R 2 . 2 v P} P t ' \ ’ \ m c + c + c + c IR 2 L" t 1) p11? 1 + w2 c + c + c + C 2R 2 L . 2 t v P: P . -J Equation (69) shows that this network may be represented by an equiva- lent resistance R ', which is equal to 2 R (70) R ' = R + r 4P 7 2 1 + wZiC + C + c + c 2 t ‘0 p“. 2R 2 P and in series with an equivalent capacitive, given by f 1 + wZIC + C + C + CJZR 2 (71) C I = F41 U t P l) 2 w2lC + C + C l 2 v Y t + CkJR 2 P The use of R2' and C2' in the development of the circuit theory per- mits one to use terms in the mathematical expression which are much less cumbersome and less difficult to employ. When a sinusoidal voltage, V,, is applied to the cavity by means of i the signal generator, a voltage will be induced in the cavity via the coupling between inductors L1, and L2 by the current, I‘, flowing in the signal generator circuit. The complex notation of the voltage, V , induced in the cavity, is 2 81 (72) V = ij I 2 121 The induced voltage has a magnitude of lezI' and lags behind the signal generator current by 90°. O The current 12 flowing in the cavity circuit is exactly the same current that would flow in the cavity, if the induced voltage V2 were applied in series with the inductor L2 and the signal generator circuit were absent. The current I2 flowing in the energized cavity will in turn induce a voltage, V3, in the detector. The complex notation of the induced de— tector circuit voltage is given by (73) V = -ij I 3 232 The current I3 flowing in the detector circuit is the same current that would flow in circuit 3 if the induced voltage, VQ, were applied in series with the inductor L3 and the cavity circuit were absent. If the output V1, of the signal generator is a sinusoidal voltage and all circuit parameters are constant, the following relationships may be written. (74) [R + ij JI + ij I = V 1 11 122 1 r 1 (75 R'+'wL-———]]I+°MI+‘MI=0 ) 232w0'2'7233'7121 2 (76) 82 For the sake of simplicity in writing, the following abbreviations are adopted: (77) (78) (79) (80) (81) Equations (82) (83) (84) Z = R + ij , signal generator impedance 1 1 1 \ Z ' = R ' + j wL —-——£T , cavity resonator impedance 2 2 2 wC , 2 ’ 1 Z = R + j wL - -——-, detector circuit impedance 3 a 3 wC 3 Z = ij , mutual impedance between the generator M 12 12 and cavity inductors ZM = ij , mutual impedance between the cavity and 23 23 detector inductors (74), (75), and (76) are reduced to Z I + Z I = V .M 1 1 12 2 1 Z I+ZI+Z I: (4 1 2 2 M23 2 12 = 0 Z I + Z I 23 2 3 3 83 The simultaneous solution of equations (82), (83), and (84) yield { 2 z z — z V 2 3 M71) 1 (85) Il=zzz —zzM7[—Z3ZM l 12 3 l 23 -2 —2M 23V: _ ‘2 - (86) 12"zzz—z.zMi—zsz‘Z ‘ 2 3 l 23 12 2M zM Vl _ 12 23 (87) Iq_ZZZ-ZZ 1.2 T “ 12 3 1M 3M 23 12 As is evident by equations (85), (86), and (87) the nature and magni- tude of the current flow in the various circuits are complex and depend upon the nature and magnitude of the impedances of the individual circuits. However, by means of equations (85), (86), and (87), it is possible to synthesize equivalent networks which are very useful in demonstrating the nature of the interaction and coupling of impedance in the various circuits. For example, the terms in equation (86) may be rearranged to yield an equivalent expression given by aw _ __12.V Z 1 2 ZM ZM z I _ __12. _ ..JUL. 2 Zl Z3 Equation (88) may be satisfied by an equivalent circuit shown in Fig. 18. This is an equivalent circuit referred to the cavity. In this simplified equivalent circuit, the equivalent voltage 84 N EIN| H H cowmmmumxm as“. .3 wouamwmuamm mm 9.2.5.30 ...—513.33“ a4 .3” muswwm mN 3233.2. MI: 1 u ombmfimm 2N Ema/$58 — N num> .Nu nu AU .>~...._N . mo ..1 ..... 4 554238 . N mozanvm Housmamvsam umumEOHHflomo mnu mo coaumuammmummm ufisuuwo unmam>flsvm wowwwamfiwm .ON munwflm 94 subscript 2 refers to the equivalent cavity circuit and the subscript 3 refers to the detector circuit. Furthermore, the circuit parameters, V ', 2 C ', L , M , R and C denote the same quantities as shown or expressed 2 3 23 3 3 in preceding equations and circuit representations. R ” and L ' repre- 2 sent the total equivalent series resistance and inductance, respectively, of the cavity as given by w2M 23 n: r 2 12, ‘1 (105) R2 R2 + R 2 + wZL 2 1 1 and 1112M 2L I __ _ 1 ‘7 l (106) L2 — L2 R 2 + 'wL .2 1 | 1| \ J w?M R The quantity R " includes the resistive term R 2 +Li2i 2 which represents 2 1 I the resistive impedance reflected into the cavity by inductively coupling the signal generator to the cavity. The term R ' is givgn by equation 2 w M L (70). The quantity representing L2’ includes a term R7+IinWYWhiCh ‘l I represents the equivalent reactance reflected into the cavity by the sig- nal generator, via the inductive coupling of the cavity to the generator. The equation to represent the detector current, I3, may be derived in a manner similar to the derivations of equations (85), (86) and (87), and is given by (107) I = - 3 95 where: r 1 (108) zll=RN+ij'__—]_'. 2 2 2 wC \ 2 J M 23 23 (110) 2 =11 +ji111L — ‘15— 3 1 3 w 3 \ ) -ij VZ' _ 121 , (111) I — R " + jX "'R + 3X ‘ + X 3 3 1| ' M 2 9 1 3 3 :I J 23 where: (112) X = wM M23 23 I I- '____ (113) X2 — wL2 wC , 2 (114) X =wL -—1—- 3 3 w03 When the terms of equation (111) are rationalized and grouped into the real and imaginary parts, one obtains an expression for the detector cur- rent given by 96 M J 3 R "‘1’; 'R “R — X 'X + X 1 Z 32J |23 23 \ 1-4 ’X 'R + X R "‘ + j’R UR - X 'X + X 2"V 'X J 2 3 3 I I 2 3 2 3 2 NM (115) I3 = ** IX 'R + ‘ 2 3 \ The magnitude of the detector current is given by I V2 XM — V1 7 7: ‘T- (116) I3 - ffX 1R + X R n ._ .1- R HR _ X IX + X, ._ l7/2 l 2 3 3 2 l 2 3 2 3 M I K , \ 23 25 J or ' I V2 XM21 (117) 13 = ,X :23 7 + X 23 "1 + R "ZR 2 + 2R "X Z 2/2 2 3 3 2 2 3 2 M 23 + X '2X 2 - 2X 'X X 3 + X ’ 2 3 2 3 M M , 23 23 k 1 Partial resonance in the coupled circuits, as shown in Fig. 20, is obtained when any one circuit parameter is so varied as to cause maximum effective detector current, I3, under conditions of applied voltage, VZ’. From equation (117), it is evident that in theory, partial reson- ance may be obtained by adjusting any one of the five circuit parameters [R2', 83, XZ', X3 or.&M J, which appear only in the positive terms of the 23 denominator. However, in this discussion we are concerned primarily with the values of 02' and C3 [or X2' and qu which will produce partial resonance. J The value of X2' or X3 which will produce partial resonance may, in general, be found by differentiating the expression for I3 with respect 97 to the proper X, equating dI/dX equal to zero and solving for X in terms of the other parameters. For example, the value of X ' which will produce partial resonance is given by dI ———3- = = _ ' {q ,; 2 2; _ J 21 (118) dX , 0 V2 XM 1/2‘2X :R? + X3 ! PX3XM l 2 2? I I 1 1 ’23 P k Therefore, 2 2 X3XM ngM ' - = ———-—2-3._ (119) X (res) - ifs—272%? Z 3 - 3 If one substitutes for XM , X ' and X , the value given by equations 2 '2 23 (112), (113) and (114) respectively, J _ wL3 - wé‘ MM 2 1 \ ‘ '2 2'3 (120 [NL ' "' = k ‘1 3* ) - 2 wC ' 2 + [wL - -l—wz ‘ 2 free) 3 I 3 wca‘ From equation (120), the value of C ' to produce partial resonance is 2 obtained and is given by 2 P88) TwL _ (”é w2M 2 (1) (UL ' " L 3 7 L 2: 2 R 2 + ‘wL --l— 3 (DC . k \ 3) I The significance of equation (121) is that for a given value of 03, 02 must have the value staUuito produce maximum detector current, I3. 98 If the reactance of the detector circuit is equal to zero ,by virtue of IwL3 - Z£71 = 0, then C ' should be so adjusted so that the reactance 2 of the cavity, X2 ', is equal to zero. Thus, the value of C ' is given by 2 In a similar manner, it may be shown that the value of X3 to produce partial resonance is given by r 2 X2 XM = . 23 3(res) R '3 + X '2 2 2 (123) X By substituting into equation (123) the values of XM , X1 and X2' given 23 by equations (112), (113) and (114) respectively, one obtains the value of 03 which will produce partial resonance, 1 (124) 03(res) - It is evident from equation (116) that if both the cavity reactance, X2' and the detector reactance X3 are equal to zero, the detector cur- rent will be an optimum and will be equal to M 2 (HM 3V2: _. 23 _ 23 (125) Iglxa = 0, X ' _ OJ R "133+ XM Z R "133 + (oi-M 221 k 2 2 e 99 The significance of equation (125) is that, if the reactance of the cavity, X2', and the reactance of the detector, X“, are adjusted to equal zero, 5 and R and KM are fixed, then the detector current will be a function of the total equiSalent series resistance, R9', of the cavity network. Effect of the Detector Probe Capacitance upon Oscillometer Response. In Fig. 21 is shown the relationship between the detector probe capaci- tance and the oscillometer responses at 75, 80, 90 and 100 Mc/sec. The data were obtained in the manner described on page 67. Each data point on the cell capacitance and detector current curves correspond to reso— nance values for the given value of the detector probe capacitance. In order to conveniently show the relationship between the cell capacitance and the detector probe capacitance at the various frequenc- ies, the difference between the measured cell capacitance, (for a given detector probe capacitance value) and the cell capacitance corresponding to optimum detector current were plotted. For example, at 100 Mc/sec, the detector probe capacitance was arbitrarily adjusted to 39.5 pico— farads and the cell capacitance {Ct and Cu] were adjusted to obtain maximum effective detector current. The cell capacitance at resonance was equal to 27.843 picofarads. This value corresponds to the minimum of the cell capacitance curve (at 100 Mc/sec) and is equal to -O.109 picofarads (27.843 - 27.942). Therefore, at any given value of the detector probe capacitance, the measured cell capacitance is the algebraic sum of the value obtained from the curve and the cell capacitance at Optimum detector current. The cell capacitance at optimum detector current obtained at 75, 80, 90 and 100 Mc/sec are given in Fig. 21. The relationship between the detector probe capacitance and the lOO +OA5 E; '+OJO- 7 — Z J < r; 1: +0.05— — 2 E _ (<1: .9 o, . -----l _____ O 3. Cell. Capocntonce at d. Ophmum Detector _] _ 005 Curreng _ _J . HMc/sec) cuwf) “J 75 62.589 0 _ 010 80 53.173 _ ' M) same 100 21942 -O.15 ' 1 '5' 1.0 "' T m 5 ”.3. o 8 i a) 332: ' . fl :2 ._ g t E 0.7 — , — u) 5 c: u: urg (16—- . . __ . _ n: g g a -. O - . +— NoCI a I- G’ 80012 _J h _.| : . 1.0013 3 : 6 :— o t 1 1 1111111 1 1 1111111 1 1 1111111 1 1 1 111111 1 t3 , 3.0 — a zxn : I u: un— _ _ :3 “-8I 2.5-—- 3 a .1 so: I 31 ‘3uf° : I O: m E 2 O - .— CD .2 I I E Z a _ a “I 3“ :1 I 5 ' ‘ 2 5-1 - :— I 3 <1 _ - m E I I— —I 1" 4... - .. U) 04 1.0 — —_ E .1 : WW _ -I l l l l ‘ w 1 1 111111 1 1 1L1111 J J_111111 1 1 111111 1 1 111111 c) 120—— -— _ - 1... I .5 ~ — m: 80-— - g; _ _ c>a. __ ._ 5 6° . g o _ - :1. p- __ __ (J 40 t3 _ - 13 20-— - - . 1 O 1 1 1111111 1 11111111 1 11111111 1 11111111 1 1 111 4 40 100 1000 10.000 100.000 SPECIFIC CONDUCTANCE, umhos/cm Figure 22. Oscillometer Responses versus Low Frequency Conductivity of Aqueous Solutions of 1-1, 2-1, and 3-1 Electrolytes (Freq. : 100 Mc/sec) 118 and 25°. As shown in Fig. 22, for an aqueous system the instrument response is independent of the nature of the electrolyte and dependent upon the solution conductivity. A general interpretation of the oscillo- meter responses is based upon equation (47 ) which relates the electrical properties of the cell solution network, as shown in Fig. 8, to the solution conductivity. The observed changes in oscillometer response are directly attributed to changes in electrical behavior of the cell- solution network. 75";muupv. As predicted by equation (49), the equivalent high frequency con— roportional to the total equivalent conductance of the oscillometer (equation (29), a change in Cépwill result in a proportional change in AC? 0.70.7 The maximum change in AC’ is related to the equivalent parallel 0.707 (lonductance of the oscillometer and is given by 00 I 30 [mam - AC ("77'7“ qu ’rrax‘ 7 , '7 'I, , ' . l .4,’ (144) G = l (l, 07 0‘ 0.. .=———"‘ . 2 . 2 (max) ‘WhereZMIWY , is equal to the difference between the minimum value, AC ,1), where Rk‘is equal to zero and the maximum value, and 0.70707??? 119 AC which corresponds to K0 / ‘ . '7 max ..' . 0.70, (peak) As would be expected, a change in the total equivalent conductance of the oscillometer results in a change in the detector current,-I, at resonance. As shown in Fig. 22, the detector current decreases with an increase in solution conductivity, reaches a minimum, and then increases thereafter with increasing solution conductivity. The value of the solution conductivity, K , where-I is a minimum, corresponds exactly 0, . - mtnj tC)K(peakI for the AC” 707response curve. V.’ Theory predicts that the equivalent parallel capacitance, qp, of the «cell-solution network should remain essentially constant at very small \Lalues of K, then increase with increasing.K and finally reach a limiting \ralue at very large value of.K (refer to Fig. 8). The difference be— tween the extreme values of Cp is equal to LICp and is given by equation (56). As previously stated, a change in solution parameters (conductivity c>r dielectric constant) will detune the oscillometer from resonance. TPhe change in cell capacitance necessary to reestablish resonance will 1>e exactly equal, but opposite in sign to the change in fip' Therefore, an increase in C as shown in Fig. 8 (for an increase in X) will result 31n a decrease in the oscillometer cell capacitance as shown in Fig. 22. .At very large values of the solution conductivity,Cz9 should approach a tIlaximum limiting value as given by equation (55). Therefore, the cell Capacitance should approach a minimum limiting value at large values of the solution conductivity. In Fig. 22, the cell capacitance response curve has not reached a limiting value at the highest solution conductivity tested but appears 120 to approach it. Specific conductivity data beyond approximately 50,000 micromhos --cm—l could not be obtained with any degree of certainty with the conductivity bridge employed. A comparison of the observed oscillometer response to that predicted by theory can be accomplished most conveniently by use of the data given in Table VI. The experimental value for AC , ,, K , AC , K , «j 707 max; 0/ p C, .4 - - 'peak) :mta-poznt) I’min) and K” were obtained from the reaponse data given in Fig. 22. ”mth The experimental values for K. ., F peak» . . . and K . were cal- .mtd-petan .mznl culated by use of the expression (145) K = KC - a/7. where K is equal to the solution conductance (mhos);.K0 is equal to the specific solution conductance; a/Z is equal to the effective cell constant of the condenser-type cell (cm);<2 is equal to the effective area of the smaller of the two electrodes (cm2) and 1 is the effective distance be- tween the electrodes (cm). The cell constant,<1/Z was evaluated for the condenser-type cell by use of the eXpression61 .._E .51 (146) C - 4w Z or (147) c = 0. 0885 e - a— Z 121 Table VI. Oscillometer Response Data for Aqueous Solutions of NaCl at 100 Mc/sec I Response . I I Quantity Experimental *Theoretical Measured I I | I I Cap. Diff. I AC , , picofarads 2.02 2.11 0.707777%) '~ 662 at Half Gp( ), micromhos 635 .max _ I Power K , micromhos ~cm 1,450 4,780 0, I I .peak) I P01nts K, , micromhos I 52,100 172,000 IpeakI ACp, picofarads >3 2.11 K0 , micromhos -cm_ I 1,450 4,780 { . _ . Cell Cap. mtd patnt) K(mid—p0int)’ micromhos 52,100 I 172,000 I Ifmin)’ microamp. I 17.0 ----- Detector KO , micromhos -cm- 1,500 4,780 (min) Current KI . , micromhos 53,900 172,000 .mtn) *Values used in calculations C9 = 24.02 picofarads (Table V) Co = 3.182 picofarads (Table V) EH 8 78.54 (25°C.) 20 122 where(? is equal to the effective capacitance of the cell and s is equal to the dielectric constant of the medium contained in the cell. When the cell contains air, the effective capacitance of the cell is given by (148) CO 0.0985 N|D where Co is equal to the capacitance due to the sample space of the vessel and is therefore equal to the experimental cell parameter, CO, given in Table V. The effective cell constant based upon the experimental value for CO and equation (145) is equal to 36.0 cm. As indicated in Table VI, the experimental values of the cell para— meters 00 and Cg were used in the calculations for the theoretical data. The theoretical value for Ifmfn‘ could not be evaluated due to the fact . I 1' that the resistive and reactive impedances of the complex oscillometer network necessary to evaluate I/m’nl were not known. As predicted by theory, the experimental values for K , C/pggk) K , and K0 are approximately equal to each other. How- 0(mid-p9int) ’min) ever, there is a significant difference between the eXperimental conduc-~ tivity values for K , Km and K- , and the correspond- 0 peak) “(mid-point} diminI ing theoretical values. The difference may be due, in part or in whole, to the erroneous experimental values of Cg and Cc (Table 111) used in the calculations of the theoretical values of K0 and Kn . fpeakl “/mid-point) The experimental value calculated for G is in good agreement r’maxI with the theoretical value. The agreement in the data can be explained on the basis of the measurements involved in the calculation. For the 123 measurement of AC , the main dial capacitor, C , was fixed, and the (l 707 t precision vernier screw capacitor, Cv’ was varied. Since A00.” 7 is the difference between two CU values, the error associated with the individual CU measurements are partially cancelled when the difference is taken. As a consequence, the absolute error in the value of AC 7 will be less than the absolute error of either of the two CU values. As shown in Table VI, the experimental value of GOP is approximately 1.5 times greater than the theoretical value. The discrepancy between the experimental and theoretical values is probably due, in part or whole, to erroneous values of Cg and 00 used in the calculations of the theo- retical value for ACp, and to the errors in the cell capacitance measure- ments. It is interesting to note that if the experimental value of 0 (max) is employed to calculate ACP (refer to equation (60), one obtains a value for A0p equal to 2.02 picofarads which is in good agreement with theory. This agreement would further indicate that the error involved in the cell capacitance is primarily due to error involved in the measurement of Ct' Effect of Solvent Dielectric Constant upon the Oscillometer Response. In Fig. 23 is shown the relationship between the oscillometer response and the solution specific conductance for aqueous and methanolic solutions of sodium chloride. The response data for the aqueous solutions of sodium chloride are the same data as shown in Fig. 22 and was included in Fig. 23 for comparative purposes. Due to difficulties encountered in balancing the conductivity bridge 1 circuit at conductivity values greater than 12,000 micromhos -cm— for the methanolic solutions, sufficient response data was not obtained to 124 I2_ I I IIIIII I I IIIII I I IIIIII I I IIIIII I IIIIII C I 'I I I : H :— ‘5 m 10 E- -3 o _ 1 3 I— d t W 9 1‘ ‘2 g 3 : : b '- -4 3, .2 8 :' —. o :1 : 1 .J :1 7 :- J ..1 r S “J .4 o E . 6 :- '3 t . 5 ’— l l lllllJJ l 1 1111111 1 1 111111] 1 1 1111111 1 1 111111 LL] 0 L. s 6 T (n a: f.) I Z {1" z 5 T O -— .— 0. ‘5 8 : w m 4 C‘ LIJ m (KB I- 0: o “J 2 I 5.52 3:- ; +_— CL =3. 2 Lu On. :5 2 L 2 at _, : D t; _ - o _ — a an 2.5_ _ (D on - ‘ m 338 20.— 1 '— 40*- _ -I Z I—CL 3. .- uJ — q _ 1 4.1 - 3 (L4 “' - I! <1 ' 7 I— 0 I 7 (D 1- _ _ 8 h 1 1 111111 1 1 1111111 1 1 1 111111 1 1 1 111111 1 1 11111? q 140 - 'r 5 420 — — u: ~ . E 100 — a 8 a I- -I a: 5 so — — .9 =1 L - 3 60 t- o 100 Mc/sec “ I: " A 75 Mc/sec ‘ o 40 — — 20- ‘— 1 1 1 1 11111 1 1 1 111111 1 1 1 111111 1 1 1 111111 1 1 11111 I 10 100 1000 10,000 I00,000 SPECIFIC .CONDUCTANCE . #mhos/cm Figure 24. Effect of Resonant Frequency upon Oscillometer Response versus Low Frequency Conductivity of Aqueous Solutions of Sodium Chloride at 75 and 100 Mc/sec 129 9 :— T I TIIIII' I I IIIIIII I I IIIIII' I I IIIIIII I I IIII ‘47 u: = 5 o '— ~46 E 8 i : I: m :_ 8 o .‘J o g "’ U1 3 d g a I 1‘, \ I .. O .1 4" ‘7—2 2‘45 0 § : 8 m 2 _J I .. N 1 _l — .4 1:1 I : O I- : 6 E— .: 44 L J 1 1111111 1 11111111 1 11111111 1 11111111 1 1 111.-I m ‘z’ 3 I m 0'). 3.0: 1 g, 3:3 I I o — — a 5 a. 2.5_ _ m m I— -I a: w as - — o s L .2 '- z 32 2.0_ .. z Sci=1 I I w — :1 _ :I d ..l - a 2 < = : k £>II I I (n +- 4 L0-— E .14 C 8 1 1 111111 1 1 1111111 1 1 1111111 L 1 1111111 1 L1_11111 140— _I l- -1 l— .. z 420 -n m L 1 a: a . - . a: S 80 — a 2 =1 - -I 8 60 — o 100 Mc/sec ‘ If: ’ A 75 Mc/ssc ‘ o 40-' -‘ I— .. 20 h— _ 1 1 1111111 1 1 1111111 1 LULILII 1 JJ L11111 1 1 11111 1 10 100 1000 10,000 100,000 SPECIFIC .CONDUCTANCE . umhos/cm Figure 24. Effect of Resonant Frequency upon Oscillometer Response versus Low Frequency Conductivity of Aqueous Solutions of Sodium Chloride at 75 and 100 Mc/sec 130 maximum of the reflective cell capacitance difference at half-power point response curves. As shown in Fig. 24, the cell capacitance curves at ?5 and 100 Mc/sec have not reached limiting values at the highest conductivity solu— tion conductivity tested but appear to approach them: As a result, it is not possible to evaluate the effect of frequency upon ACE; However, it can be shown that at 75 and 100 Mc, the difference in cell capacitance between the limiting value at small values of solution conductance and the cell capacitance corresponding to maxima of the 300.70_ response curves %n: K(peaijare equal, One may assume, therefore, that:f}) would be independent of the resonant frequency as indicated by equation (56). In Table IX are given the experimental and theoretical oscillometer response data for a resonant frequency of 75 Mc/sec; The experimental data necessary to evaluate C? and (M7 were obtained from the ’max) response curves shown in Fig. 24. The experimentally derived values of Ca, and (aagiven in Table V were employed in the calculations of the theoretical data. In all cases, the agreement between the experimental and theoretical data is not exceptionally goode The large differences observed between the experimental and theoretical data is due, in fact or whole, to the erroneous values of Cé,and€%) employed in the evaluation of the theo- retical data and the errors associated with the cell capacitance measure— ments. A comparison of the experimental and theoretical response data for resonant frequencies of 75 and 100 Mc/sec is given in Table X. The experimental data necessary to evaluate the various ratio are given in 131 Table IX. Oscillometer Response Data for Aqueous Solutions of NaCl at a Resonant Frequency of 75 Mc/sec Response Quantity Measured or Evaluated EXperimental *Theoretical Measured AC0 70', picofarads 1.85 1.57 Cap. Diff. .' I G , micromhos 436 369 at Half Power p(max) Points K0 , micromhos -cm-l 1,200 3,540 ’peak) K, , micromhos 43,700 129,000 :peak) AC , picofarads :2,7 1.57 P -1 Cell Cap K0, . . , micromhos -cm I 1,200 3.540 :mtd—poznt) 7 K(mid—point)’ micromhos 43, 00 129,000 I{min)’ microamps 38 —--- Detector -1 K0 , micromhos -cm 1,250 3,540 ( I Current peak, Kerak)’ micromhos 45,500 129,000 *Values used in calculations. Cg - 20.72 picofarads (Table V) 00 - 3.224 picofarads (Table V) EH - 78.54 (25¢C.) 20 132 Table X. Effect of Resonant Frequency on Oscillometer Response (Solvent: Water - Solute: NaCl - Frequencies: 75 and 100 Mc/sec) ITIa) I b) Ratio :I Theoretical E Experimental I L : T .i K (Peak)(75) 0.75 0.75 :1 0.84 K (peak)(100) " ll 1 i 0p I ! (max)(75) I: 0.75 i 0.56 ! 0.69 Gp ' l ‘5 (max)r700) II | I l ! 1 | AC | p(75) 1.00 0.74 » 1.2 AC pf100) | | L L a) Values of CO, Cg, and EH assumed to be constant and independent 20 of frequency. b) Experimental values of C9 and C0 (Table III) used in theoretical calculations. The dielectric constant of water assumed to be independent of frequency, equal to 78.54 at 25°C. 133 Table VII and IX. Under the columns labeled "Theoretical”, two values are given for each of the ratios evaluated. In column (a), the ratios were evaluated on the basis of C0 and C9 being constant and independent of the fre- quency. In column (b), the ratios were evaluated on the basis of the values of Cg and C0 experimentally derived at 75 and 100 Mc/sec and given in Table III. A comparison of column (a) to column (b) values of 0 AC p(ma:c,’75 and p75 Cp ACP / evaTfigégQOat 75 andOYOO Mc/sec are in error. It is furtuitous that the Kerak775 {peak)100 The agreement between the experimental and theoretical quantities indicate that the experimental values of C9 and C0 column (a) and column (b) values for are equal. given in Table X is poor and is due to errors in the cell capacitance measurements used to evaluate the experimental quantities. Oscillometer Response Curves for Perchloric Acid,gSodium Acetate and peNitroaniline in Glacial Acetic Acid. In Fig. 25 is shown the re- lationship between the oscillometer response obtained at 100 Mc/sec and the low frequency specific conductance of perchloric acid, sodium acetate and p-nitroaniline in glacial acetic acid at 25°. The data were obtained as described on page 71. Only in the case of perchloric acid was sufficient data obtained to clearly define the shapes of the response curve and permit a comparison to the predicted response (Fig. 8). Due to the limiting conductivity of p-nitroaniline and sodium acetate in glacial acetic acid63, specific conductivity data beyond 1 micromho -cm-1 and 100 micromhos -cm-1, respectively, could not be obtained. The largest conductivity values indicated in Fig. 25 for p-nitroaniline and sodium acetate correspond to INSTRUMENT RESPONSE 134 Q __ -- 11,- --. 2 WWI—IEWWTTT‘ T‘T I ITF‘ITT 1“? Y'Y‘H 11‘ 'L O. . ‘v‘ _. 181— ' — A 16 P- _ LLI U ‘— .4 2 g g 14 -— - 0 o 1' ‘ 4 E 0.. .,__ 1 _. <1 i 2 1— o i _ 7 j 10 ._ _ tn 0 + . 8- _ >- \ —1 6 1 11111111 1 1 1111111 4 11111111 L L1111111 ‘10] 1111111 0.: ’- q 9 . 1o — LU m . {51— 0.2 8 1 9:0 J Ga?) 8:: E 6 a JZUJE 4 3 :1 ... CO :1. 00. 4 .— 355 _ “13 c): 2 _ ..1 .Je I “J“ 1 1 1 1 0 O 1 1111111 1 1111111 1 1 111111 1111111 1 1111111 100 m - 0 H0104 '— 80 e N c H 0 I 5 0 2 3 2 q 3; . Noe-©1012 D 60 — c>§ . m c>° 4O - 1— =1 0 - “J L] 20 a O .1 \‘0-§_-.o- ’0 O 1 1 1111111 1 1 1441111 1 11111111 1 1 111111 1 J 111111 0.1 1 10 100 1000 10,000 SPECIFIC CONDUCTANCE , #mhos/cm Figure 25. Oscillometer Response versus Low Frequency Specific Conductivity of Glacial Acetic Acid Solutions of Perchloric Acid, Sodium Acetate, and p-Nitroaniline (Freq. : 100 Mc/sec) 135 saturated solutions. In general, it can be said that with an increase in solution con- ductance, the decrease in cell capacitance and detector current is greatest for p-nitroaniline and least for perchloric acid. Also, with an increase in solution conductance, the increase in the cell capacitance difference at half power points is greatest for p-nitroaniline and least for perchloric acid. The relatively small differences observed between the cell capacitance difference at half power point response curves for three solutes are not readily apparent as shown in Fig. 25. As expected, the general shapes of the response curves for perchloric acid in glacial acetic acid are comparable to the response curves ob— tained for aqueous solutions of electrolytes. However, due to the low dielectric constant of glacial acetic acid, equal to approximately 6 as compared to 80 for water’KQpeak) observed for the glacial acetic acid solution is less than K(peak) observed for the aqueous solutions. Also, AC and G P p(max) is greater for the glacial acetic solution as compared to the respective values for water. The difference between the various reaponse curves for p-nitroaniline, sodium acetate and perchloric acid is readily apparent even though sufficient data were not obtained to clearly define the curves. The differences observed in response for these solutes is probably due to ion-pair formation and dipole interaction63. APPLICATION To test the performance of the oscillometer, a simulated aqueous titration of sodium hydroxide with hydrochloric acid and a nonaqueous (glacial acetic acid) titration of sodium acetate with perchloric acid were performed and the dielectric constants of some pure solvents were evaluated at a resonant frequency of 100 Mc/sec and 25°C. In the oscillometric titration method, it is common practice to first obtain oscillometer response curves for the titration system to be studied, which show the relationship between the instrument response and the solution conductivity (or concentration) of the substance to be titrated, titrant, or product. Having obtained a response curve it is possible to select the proper conductivity (or concentration) range in which the substance titrated, titrant or product must fall to obtain straight line titration curves. A quantitative interpretation of the oscillometric titration curves can be accomplished by means of the oscillometer response curves and the low frequency conductance titration curves obtained for the titration system.l3’24 To evaluate the dielectric constant of a test solvent, two methods may be employed. The first method11 involves the use of a calibration curve which relates cell capacitance at resonance to the dielectric constant of the solvent. To evaluate the dielectric constant of an unknown, the test sample is introduced into the cell assembly vessel and the cell capaci- tance at resonance measured. The dielectric constant of the test sample is determined by use of the calibration curve° 136 137 In the second method,13’24 the cell parameters, CO and C9, are evaluated as described on page 69. Having obtained C0 and Cg, it is then possible to evaluate the dielectric constant of an unknown by use of equation (68) as described on page 71. In this investigation, the second method was employed to evaluate the dielectric constants of some pure solvent in order to demonstrate the use of equation (68) and to observe the error in the derived dielectric constant measurement which result from errors in the cell capacitance and cell parameter measurements. Oscillometric Titrations. To illustrate the application of the response curves for the simulated aqueous titration of sodium hydroxide with hydrochloric acid, the oscillometric response data presented for sodium chloride in Fig. 22 was selected and is shown in Fig. 26. The only difference between the response curves shown in Fig. 22 and 26 is that in Fig. 22, the abscissa is scaled in logarithmic units of specific conductivity, whereas, in Fig. 26, the abscissa is linear. In general, a linear relationship between the instrument response and solution con- ductivity is preferred to the semi-log relationship when the response curves are employed to interpret oscillometric titrations curves. As shown in Fig. 26, the oscillometer responds with maximum sensi— tivity in a nearly linear manner through the conductivity range A to B. Therefore, in an oscillometric titration, the concentration of substance to be titrated, titrant or product, should be selected to fall in this region of solution conductivity in order to obtain straight line titra- tion curves. Through the interval A to B, the cell capacitance and detector current decrease and the cell capacitance difference at half 138 91'7 x T 1117fi711 11*111r ITITWTI1 :, TI .1 J V, .. U —4 o a ... . =1 ._ i _ LLJ o L Z <[ .. ’2 _ o 1". <1 ‘1 »- 513 to £39 141 m. E N. o. m w v o _ _ _ i _ __ _ _ _ _ _ _ . . e . l _ Ono. . . . l coo. _ . 1 1 Cu 0 d 34 l . l cow. 0 H I 1 do . . m 1 1 00: N nu w . 1 m o m T l 08. m 3 1 fl 1 . 08. w 0 v s 1 _e and. ram. .momzh w l 1002 283.0 .285 l 88 w _ O. h — _ — _ _ _ _ _ — ? _ 142 conductance. The low frequency conductivity titration curve shown in Fig. 28 was obtained with an identical series of solutions as used to obtain the oscillometric titration curves. Through the course of the neutralization reaction, the solution low frequency specific conductivity decreases and the oscillometer cell capacitance and detector current increase, whereas, the cell capacitance difference at half power points decreases. Beyond the equivalence point, the solution specific conductivity increases and the cell capacitance and detector current decrease, whereas, the cell capacitance difference at half power points increases. It follows that oscillometric titration curves relating cell capacitance or detector current to volume of titrant should be inverted V-shaped curves and that the oscillometric titration curve relating cell capaci- tance difference to volume of titrant should be V-shaped as shown in Fig. 27. In Fig. 29 are shown the oscillometric titration curves obtained for the simulated titration of sodium acetate with perchloric acid in glacial acetic at 100 Mc/sec and 25‘. A general interpretation of these curves are based upon Fig. 30, the simulated low frequency conductometric titration and Fig. 23, the oscillo— metric response curves and may be accomplished in the manner as discussed for the aqueous titration. Dielectric Constant Measurements. in Table XI are given the di- electric constants of some pure solvents evaluated at 100 Mc/sec and 25°. The method employed to evaluate the dielectric constants is described on page 71. 143 19.80— _ L1J " -. E; 19.60— _ E » 1 - 6319.40— : 28.60ml _ cit-E » | . S §19.20F _ -J 1 d 1900— j 0 ~ 1 1 1 1 1 1 1 1 1 1 1 1 l 1 l 1 1 8 2.21— j Lu 2 _ U) L1J - 3 $12 2.o~— _. a u.z: _ 4 U) 9:55 lfl Cid. L8—— .~ (I LUCK-o 1- .1 1— 0L1Jo 2 52516— 1 _ LlJ 1.0“- 1. | .4 E gai 128.62ml E 35 1.4— I a) c»? _ g _H_ 1.2— _. :1“ . 1 O 1.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 70" j 60— ' '— | 50* l 28.58m1 — l 24.96 ml 0.3673N 1108311302 4 _ O THEOR. 5.13., 28.77 ml DETECTOR CURRENT, #omp 30 1 111191111 111 11; 12 16 2O 24 28 32 36 4O VOLUME OF 0.3186N HCIO4 , ml Figure 29. Simulated Oscillometric Titration of Sodium Acetate with Perchloric Acid in Glacial Acetic Acid (Freq. : 100 Mc/sec) 144 vfio< ofiumo< ngomao a“ vHo< uauoanoumm nufiB oumumo< adwvom mo coflumuufiH ofiuumEOuoavcoo madmavoum 304 .om Guzman .8 $0.0: 2005.0 “.0 m230> mm mm mm 2m 0m 0. N. m #—___—fld—a__—_dq—.__%dq__+_ co N _E O®.m~ CD 0d :1 L0 wax souwfi ‘aonvmncmoo OIdIOEdS .8 Emu tam .momxp N08.8082 22.0.0.0 .8 00.2w ~—--—_-P—_—_F_-_—bpb—pF_—_—_r mN IIITIIITIIIIIIIIIIIIFII 1.11111111111111111114111- 145 Table XI. Dielectric Constants of some Purified Solvents Evaluated at 100 McIsec and 25’ l I Relative 1 Dielectric Constant l ’ i Error ' Literature 'Experimental , Per Cent 1 l l Benzene I 2.275 1 2.278 I +0.1 l | ' Chloroform I 4.806 2 4.790 1 -0.3 l i i Chlorobenzene 1 5.621 1 5.586 1 —0.6 l l l 1,2 Dichloroethane 10.36 ' 10.21 I -1.5 l l Acetone 20 71 I 20.?9 1 +0.6 I l ! Nitrobenzene 34.82 I 35.21 1 +1.0 ‘ l | Water 78.54 79.80 1 +1.6 1 a) b) Reference 62. Average of three determinations. 146 Although the cell capacitance measurements and the cell parameter values C9 and C0 employed in the calculation of the dielectric constants (equation (68) are thought to be in error, the experimentally derived dielectric constants are in good agreement with literature values. The relative error of the measurements is shown to be largest for the sol- vents of high dielectric constant. This apparent increase in the relative error of the measurement with increasing dielectric constant is primarily due to the errors associated with the quantity 10Ca - ACS1 in equation (68). The greater the difference between ACE and cCa, the larger the absolute error in [35a - ACQI. As discussed on pages 108—1l3, the error g’J in the cell capacitance measure associated with the main dial capacitor, Ct’ appears to increase with increasing separation of the electrodes. When a material with a high dielectric conscant is contained in the cell assembly vessel, the separation of the electrodes of Ct at resonance will be large as compared to the separation of the electrodes at resonance for the empty vessel. As a consequence, the error in 068, will be large as compared to ACa. If the main dial capacitor, Ct’ and vernier screw capacitor, C”, were calibrated, it would be possible to accurately measure the cell capaci- tance values, ACa, ACS, and AC and then evaluate precisely cell H 9 parameters, C6 and Cg. In turn, the dielectric constant of a test material could be determined precisely. CONCLUSIONS A prototype high frequency oscillometer Operating at 100 megacycles per second was developed. A cylindrical reentrant cavity was employed as the resonant circuit element. High frequency voltage was induced in the cavity by inductive coupling of the cavity to a constant voltage and constant frequency generator. The current flowing in the cavity was indicated by a tuned detector circuit which was capacitively coupled to the cavity. The high frequency energy received by the detector probe is rectified and applied to the grid circuit of a dc amplifier tube which is part of a modified Wheatstone Bridge circuit. Bridge unbalance is linearly related to signal intensity. One electrode of a precision variable capacitor was directly con- nected to the inner cylinder and the other electrode was connected to the outer cylinder of the cavity. Connected in parallel with the pre— cision variable condenser was a condenser—type cell. When the oscillo- meter was initially tuned to resonance, a change in conductivity and/or dielectric constant of the solution contained in the cell detuned the oscillometer. Resonance was reestablished by adjustment of the pre- cision condenser as indicated by maximum detector current. Equivalent circuits are pr0posed to represent the signal generator, cavity, detector and cell-solution networks. A qualitative interpretation of the instrument response in terms of these circuits is presented. The net effect of coupling the tuned detector circuit to the resonant cavity is to lower the Q-factor of the resonant cavity (or to increase the equivalent shunt conductance of the cavity). Optimum detector 147 148 current at resonance is obtained when the reactance of the cavity and detector circuits are equal to zero (i.e., when both circuits are tuned to resonance). The oscillometer response curves obtained indicate that the changes in instrument response, effective detector current at resonance, cell capacitance at resonance and cell capacitance at half power points are related to changes in admittance of the cell-solution network as described by Reilly and McCurdy. The design of the instrument is such that changes in susceptance and high frequency conductance of the cell-solution network can be measured directly. Equations were derived and a method described and used to evaluate the cell parameters Cb, the capacitance due to the walls of the vessel, and, Co, the capacitance due to the annular sample spaces containing air. Also, an equation was derived and used to evaluate the dielectric con- stant of some pure solvents. In the present study, no attempt was made to calibrate the main dial and vernier screw capacitors of the dielectric sample holder at 100 Mc/sec. The calibration data supplied by the manufacturer and obtained at 1000 cycles per second was employed in all capacitance measurements. In order to obtain precise capacitance measurements, it is imperative that the dielectric sample holder be calibrated at 100 Mc/sec. In turn, precise dielectric constant measurements would be possible. Instrument performance was satisfactorily tested with a simulated titration of an aqueous solution of hydrochloric acid with sodium hy- droxide and a simulated titration of a glacial acetic acid solution of sodium acetate with perchloric acid. Interpretation of the oscillometric 149 titration curves were accomplished by use of the oscillometer response curves and low frequency conductance titration curves. It is without question that certain improvements in the mechanical arrangements and design can be made to improve the performance of the resonant cavity oscillometer, such as: (l) A practical titration cell be designed to permit stirring of the solutions. (2) For an instrument to be used primarily for titrations, the resonant cavity can be positioned vertically and the titration vessel inserted into the end of the cavity so that the test solution is located in the gap formed by the end clos- ures of the inner and outer cylinders. In this position, the electro- static field existing within the cavity is most dense. Consequently, the interaction between the electrostatic field and the test solution should be greatest in this position. (3) To increase the Q—factor of the oscillometer, a crystal diode detector and vibrating reed electro— meter can be used. By proper selection of the resistive and capacitive components in the crystal diode circuit, it should be possible to mini- mize the resistive impedance coupled into the cavity as compared to the tuned detector circuit employed in the present study. (4) To extend the concentration ranges in which the oscillometer reaponds with maximum sensitivity in a linear manner, a smaller resonant cavity can be employed. The resonant frequency of the cavity will be inversely proportional to its mechanical dimensions. 10. ll. 12. 13. 14. 15. 16. 17. 18. 19. 20. LITERATURE CITED P.H. Sherrick, D.A. Dawe, R. Karr and F. Ewen, "Manual of Chemical Oscillometry", E.R. Sargeant and Co., Chicago, Illinois (1954). G.A. Harlow and D.H. Morman, Anal. Chem. 28, 485R (1966)° A. Bellomo and O. Klug, Chem. Anal. (Warsaw) 2, 879 (1966). M.F.C. Iadd and W.H. Lee, Talanta 13, 941 (1965). M. Szczepanik, Przemyls Chem.449 542 (1965). V.A. Zarinski and I.A. Ger'es, Zavodskaya Lab. 29, 1157 (1963). E. Pungor, J. Electranal Chem._;, 189 (1962). M.F.C. Ladd and W.H. Lee, Talanta 4, 274 (1960). A. Timnick, L.L. Fleck and E.R. Hooser, Chemistry in Canada, March, 23. (1960). A. Bellomo and G. D'Amore, Atti Soc. Peloritana Sc. fis, mat nat., _5, 119 (1959). E.R. Sargeant and Company, Scientific Apparatus and Methods 23 l (1957). V.A. Zarinski and I.R. Mandelberg, Zavodskaya Lab. 22, 262 (1956). C.N. Reilly in "New Instrumental Methods in Electrochemistry" by P. Delahay, Interscience Publishers, New York, N.Y., (1954), pp. 319- 345. G.G. Blake, "Conductometric Analyses at Radio Frequency”, Chemical Publishing Co., Inc., New York, N.Y. (1952). E. Pungor and L. Balazs, Magy. Kem Folyocrant ng 257 (l966). M. Syczepanik, Przemyal Chem. 33, 471 (1965). K. Nakano and H. Tadano, Nippon Kagaku Zasshi 82, 485 (1964). v.1. Ermakov. P.A. Zagorets, and N.I. Smernov, Russ. J. Phys. Chem. .86, 625 (1962). R. Huber and K. Cruse, Z. phys. Chem. 123 273 (1957). R. Huber and K. Cruse, Angew. Chem. 66, 625 (1954). 150 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 151 R. Huber and K. Cruse, Angew. Chem. 96, 625 (1954). J.L. Hall, J.A. Gibson, H.O. Phillips and F.E. Gritchfield, Anal. Chem. 26, 1539 (1954). S. Fujiwara and S. Hayaski, Anal. Chem. 263 239 (1954). C.N. Reilly and W.H. McCurdy, Jr., Anal. Chem. 25, 86 (1953). W.J. Blaedel and H.V. Malmstadt, Anal. Chem. 24, 455 (1952). M.L. Mittal and 8.8. Dube, Bull. Chem. Soc. Japan 39, 1064 (1966). A. Bellomo and G'Amore, Chem. Tech. (Berlin) 143 340 (1962). 1.0. Graniskaya and B.A. Lopatin, Zavadskaya Lab. 22) 9 (1960). A.H. Johnson and A. Timnick, Anal. Chem. 283 889 (1956). E. Pungor and K. Huber, A. Anal. Chem. 154, l (1957). 8.8. Lane, Analyst 82, 406 (1957). D.J. Fisher, M.T. Kelley, R.W. Stelzner, and E.B. Wagner, I.S.A. Journ. of Instr., Auto Control and Automation 43 474 (1957). M. Pancek, Chem. Iistylég, 1367 (1958). R.W. Stelzner: Private communication, Oak Ridge National Laboratory, Oak Ridge, Tenn., Sept., 1959. W.J. Blaedel and H.V. Malmstadt, Anal. Chem. 223 1413 (1950). C.N. Works, T.W. Dakin, and F.W. Baggs, Proceedings of the I.R.E., April, 245 (1945). A.H. Johnson: Private communication, Michigan State University, East Lansing, Michigan, Sept., 1956. F.E. Terman, "Electronic and Radio Engineering", 4th Ed. McGraw-Hill Book Co., New York, N.Y. (1955). C.G. Cannon, "Electronics for Spectroscopists”, Hilger and Watts Ltd., London (1960). T.E. Terman, "Radio Engineers Handbook" lst Ed., McGraw—Hill Book Co., New York, N.Y. (1943). W.L. Barrow and W.W. Mieher, Proceedings I.R.E. 283 184 (1940). W.W. Hensen, J. Appl.Phys. 2, 654 (1938). 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 152 R.I. Sarbacher and W.A. Edson, "Hyper and Ultrahigh Frequency Engineering", John Wiley and Sons, Inc., New York, N.Y. (1946), p. 389. J.C. Slater, "Forced Oscillations in Cavity Resonators”, Rad. Lab. (MIT), Report No. 188, Dec. 31, 1942. G.B. Collins, "Microwave Magnetrons", McGraw-Hill Book Company, Inc., (1948), p. 310. C.G. Montgomery, "Technique of Microwave Measurements”, MIT Radiation Laboratory Series, McGraw—Hill Book Company, Inc., New York, N.Y. (1947) pp. 285-319. 8.1. Pearson and G.J. Maler, "Introduction to Circuit Analyses", Jonh Wiley and Sons, Inc. (1965. R.N. Kerchner and G.F. Corcoran, "Alternating-Current Circuits", 4th Ed., John Wiley and Sons, Inc., New York, N.Y. (1960). K.G. Stone, "Determination of Organic Compounds", McGraw-Hill Book Co., Inc. , New York, N.Y., p. 92. A.I. Vogel, "Practical Organic Chemistry", Longmans, Green and Co., Inc., New York, N.Y. (1948), p. 558. N.J. Leonard and L.E. Sutton, J. Am. Chem. Soc. 19, 1564 (1948). K.B. McAlpine and C.P. Smyth, J. Chem. Phys. 2» 55 (1935). C.P. Smyth, R.W. Dornte, and E.B. Wilson, J. Am. Chem. Soc. 53, 4242 (1931). R.J. Livingston, J. Am. Chem. Soc. 99, 1220 (I947). "The Radio Amateur's Handbook", 33rd Ed. American Relay League, West Hartford, Conn. (1956), p. 84. W.I. Orr, "The Radio Handbook", 16th Ed., Editors and Engineers, Ltd., Summerland, California (1962), pp. 232-33. ASTM Standards, "Electric Insulation, Plastics, Rubber", Part 6, ASTM Designation: D150—47T, American Society for Testing Materials, Philadelphia, Pa., (1949). L. Hartshorn and W.H. Ward, Proceedings, I.E.E. (London) 12) 597 (1936). "Operating Instructions for Type l690-A, Dielectric Sample Holder", General Radio Company, Cambridge, Mass. (1954). 60. 61. 62. 63. 153 1966 Book of ASTM Standards "Electrical Insulating Materials”, Part 29, ASTM Designation: D—lSO-65T, American Society for Testing Materials, Philadelphia, Pa. (1966). A.E. Knowton "Standard Handbook for Electrical Engineers", 8th Ed., McGraw-Hill Book Company, New York, N.Y. (1949), p. 64. A.A. Maryott and E.A. Smith, ”Table of Dielectric Constants of Pure Liquids“, National Bureau of Standards Circular 514, (1951). S. Glasstone, "Physical Chemistry", 2nd Ed., D. Van Nostrand Co., Inc., New York, N.Y. (1946). APPENDIX I Sample Derivation for Universal Resonance Curves* The effective impedance of a series resonance circuit as shown in Fig. l is given by (3) Z = IR‘ + {mi - If? ' (8) wLO = Egr- O and (10) Z0 = R where the subscript zero denotes resonance. If the inductance, L, and resistance, R, of the series circuit are constant and independent of the frequency, the circuit may be tuned to resonance by adjusting the series circuit capacitance, C. Equation (3) may be rearranged to express the impedance of the circuit in terms of the circuit Q and the fractional capacitative tuning which is given by (15) Y = Eq- 0 *The equation numbers appearing in the Appendices correspond to equation numbers appearing in the body of the text with the exception of equations prefaced by A which are new expressions and appear only in the Appendices. 155 where C is the actual capacitance and CO is the capacitance at resonance. ‘- Rearrangement of equation (1) in terms of the fraction tuning yields " 1112‘. (A-l) Z: R2 + wL— 111—ll l2/2 wC yv I K 0‘ .J I or I 7 "’J 3/2 (A’Z) Z = R Z + I3£-- 1111 Since the inductance L is constant and is equal to L0, the above equation may be rearranged in terms of Q, as given by . ’ 7“;'- (A-3) Z = R I + IQ - Qlill 1/2 1 1Y2 ' 1 . j , where wLo 2 Thus, the ratio of the actual impedance to the impedance at resonance is 1‘211/2 R11 * @211 ‘ 711 R 01-4) Z—Z = 0 Factoring out the R terms, the ratio of the actual impedance to the impedance of the RLC series resonant circuit is given by “1 + Q2 (16) EZ— = 0 156 The universal resonance curves for the series and parallel RLC circuit, obtained from equations (17), (18), and (19) and equations (26), (27), (28), and (29) may all be derived in a similar way by rearrangement of the appropriate expression for the quantities invclved, in terms of the circuit Q and fractional capacitative tuning. . 0 APPENDIX II Parts list for the Resonant Cavity Oscillometer Cavity resonator, Fig. 12 Condenser-Type Cell, Fig. 13 Coupling Probes, Fig. 12 Type 1960-A Dielectric Sample Holder, General Radio Co., Fig. 14 250 pfd, 21 plate air condenser 500 pfd, 500 V. 0.1 mfd, 400 V. 0.5 mfd, 400 V. 0.1 mfd, 400 V. 4.7 megohms, 0.5 watt 3.3 megohms, 0.5 watt 20,000 ohms, 0.5 watt potentiometer 10,000 ohms, 0.5 watt 47,000 ohms, 0 5 watt 390 ohms, 0.5 watt 4700 ohms, 0.5 watt 25,000 ohms, 0.5 watt potentiometer 6800 ohms, 0.5 watt R. F. choke, 16 turns, No- 22 enamel Primary 115 V., 60 cycles: secondary No. l: 180-0—180V; No. 2: 6.3 V., 1.2 amps 0.30 microammeter 0-200 microammeter, 0-0051 microamp div 157 Section 955 6807 6 x 5—GT DPST toggle switch DPST toggle switch 158 fi - " E APPENDIX III Operating;Procedure for the Resonant Cavity Oscillometer Connect the input terminal of the cavity resonator to the source of high frequency oscillation (Refer Appendix V, Operating Procedure for the Hewlett Packard Model 608A VHF Signal Generator). Set the ammeter toggle switch to HIGH SEN. and the high sensitivity galvanometer switch to OFF. Connect the power cable to a source of 115 volt 60 cycle current, and turn the power switch ON. Allow 10 to 15 minutes warm up for the 6SQ7 tube to stabilize. Rotate the detector SENSITIVITY control to the extreme counter :lock- wise position (lowest detector sensitivity). Set the ammeter toggle switch to the LOW SENS. position. The de- tector current is indicated by the 0—30 microammeter. Turn the detector capacitance dial to 250. Rotate the main dial of the dielectric sample holder until a maximum meter indication is obtained. If the meter indication initially is less than zero, rotate the ZERO SET control until the meter indica- tion is greater than zero. Set the high sensitivity galvanometer, 0.200 microamps, to the X0.001 sensitivity position and rotate the ZERO ADJ. control until the meter reads zero. Set the meter toggle switch to HIGH SENS. 159 ‘1"— 10. ll. 12. 13. 14. 160 Adjust the detector SENSITIVITY control to give a desired meter indication with the oscillometer tuned to resonance. It may be necessary to rotate the vernier screw of the dielectric sample holder to tune precisely the oscillometer to resonance as indicated by a maximum meter indication (refer to Fig. 19). Detune the cavity from resonance by rotating the main dial of the dielectric sample holder until a minimum meter indication is observed. Increase the galvanometer sensitivity by step switching, as the meter indications decrease, to determine precisely the minimum meter indication. Rotate the ZERO SET control until the meter indication is zero (clockwise rotation of the ZERO SET control will increase the meter indication). The ZERO SET control most be readjusted for every change in the setting of the SENSITIVITY control. Set the galvanometer sensitivity switch to X0.001. Turn the detector capacitance dial to a new setting. Then, rotate the main dial of the dielectric sample holder until a maximum meter indication is observed. Rotate the vernier screw to precisely de- termine the maximum. Repeat this procedure until a maximum meter indication is observed for a given setting of the detector capaci- tance dial when the oscillometer is tuned to resonance (refer to Fig. 20). After having determined the setting of the detector capacitance to give maximum oscillometer response, as described above, high fre— quency measurements can be made. APPENDIX IV Hewlett-Packard Model 608A VHF Signal Generator Specifications Frequency Range 10 to 500 megacycles in 5 ranges Ranges: 10-21 Mc, 21-45 Mc, 45-100 Mc, 100—230 MC, 230-500 Mc Calibration Accuracy Within t 1%. Frequency settings can be duplicated within 0.2%. Ougput Voltage 0.1 microvolt to 1.0 volt, continuously variable, Direct Reading controls calibrated in voltage and dbm. Rated Load Impedance 50 ohms resistive Internal Impedance 50 ohms; maximum VSWR 1.2 Output Voltage Accuragx Within * 1 db rated load over entire frequency range. Leakage Negligible Power Supply Rating Voltage — 115/230 volts 1 10% Frequency - 50—60 cycles/sec Wattage - 150 watts 161 162 RF Output Connector Amphenol No. SO—239 CPR-49194, 83-1H Series UHF APPENDIX V Operating Procedure for the Hewlett-Packard Model 608A VHF Signal Generator Connect the OUTPUT terminal of the Model 608A generator to the cavity resonator power input terminal by means of a RG8/U coaxial cable terminated with Amphenol 83-lSP (PL-259 CPH-b9l90) connectors. Connect the power cable to a source of 110 volts - 60 cycles current, and turn the POWER switch ON. Allow 10 to 15 minutes warmup for the output frequency and level to stabilize. Set the RANGE control to the desired frequency range. Rotate the frequency control until the desired frequency is indicated by the frequency dial. Set the CW, PULSE 400 =, 1000 =, Ext. MOD switch to the CW position (output without modulation). Adjust the OUTPUT lEVEL control until a reading is obtained on the output voltmeter. Do not bring this level up to a full scale reading on the meter. Adjust the TRIMMER knob for a maximum reading on the output voltmeter. Set the OUTPUT LEVEL control to SET LEVEL point on the output volt- meter and adjust the ATTENUATOR control so its dial reads 500 millivolts. a. Periodically readjust the TRIMMER knob for a maximum reading on the output voltmeter to 163 164 insure output frequency stability. Readjust the OUTPUT LEVEL control to the SET LEVEL whenever a change in output is indicated by the output voltmeter. APPENDIX VI Sample Derivation for the Cell Parameters Equations ’00 and qu 1 A l At resonance, the total equivalent capacitance of the resonant cavity oscillometer for any given frequency is given by (A'S) C(total) = where Cx is the total equivalent capacitance of the oscillometer excluding PF; z ' aux-me“; the capacitance associated with the cell-solution load, Cs’ and the I ct - owls. capacitance associated with the dielectric sample holder, \ The capacitance term Cw includes the distributed capacitance of the resonant cavity and also includes the equivalent capacitance as a result of capacitive reactance coupled into the cavity by the signal generator and the detector circuits. The magnitude of the capacitance terms C(total) and Cx are unknown. However, at a given frequency, they are assumed to be constant and operate independently of the capacitance terms I C and [C - C J . s t v s The capacitance term [Ct - Cv's is the difference in capacitance I between the reading of the main micrometer dial, C , and the vernier t screw, CU, of the dielectric sample holder and represents the capacitance value measured for the test sample contained in the cell. The capacitance term, 08’ represents the capacitance of the condenser- type cell and is given by C C068 (A-6) Cs = C + C e g 0 s 165 166 where C9 is the capacitance due to glass walls of the cell, C0 is the capacitanCe of the cell containing air and as is the dielectric constant of the test sample contained in the cell. Equation (A—6) is identical in form to equation (54) and assumes that the resistance of dielectric contained is so large as to be negligible in the equivalent parallel capacitance term for the condenser—type cell (Refer to equation (50). In general, when a dielectric is contained within the cell, the total capacitance is thus given by C C068 = C (A 7) C(total) x + C + C c s g 0 s where ACS is equal to the capacitance difference between Ct and CD, [Ct - Cu 3’ which is measured for the particular dielectric. I When air is contained in the cell, the total equivalent oscillo- meter capacitance at resonance is given by C C = __£L£L_. C(total) cm + Co + Cg + ACa (A-8) ( where AC& is equal to the capacitance difference C+ - Cv’a measured for \V ,‘ air. When mercury is contained in the cell, the equivalent parallel capacitance of the cell (equation (51) is reduced to Cé’ and the total equivalent oscillometer capacitance is given by (A—9) = C + C + AC x 9 H g C(total) 167 r x where ACH is equal to the capacitance difference, {Cf - Cv[H 9 ’9 for mercury. measured By use of equations (A-7), (A-8) and (A—9) one may derive the cell constants CE and C? in the following manner: Equation (67), l 1 9} is obtained by equating equations (A—8) and (A-9) to eliminate the cons— tant terms C(totaZ) and Cm. The resulting expression is solved for Co in terms of C , AC , and AC . g a H Equation (66), I lee - ZHAC8 - ACH i C: ) ‘91 g AC8 - ACE? e -1 3 AC& - ACH 1 91 is obtained by equating equations (A-?) and (A-8) in order to eliminate the constant C and Cr. The resulting expression is solved for (total) C? in terms of CO, A08, ACHQ’ and as. The value of CO, given by equation (67) is substituted into the expression obtained for C? and C? reevaluated in terms of A08, AC H , ACfi, and £3. The value of C9 may be calculated by measurement of AC& for air, AC for mercury, and A08 for a material of known dielectric constant, es. Having determined the value of C? by use of equation (67), it is then possible to substitute the value of C9, ACa, ACH into equation (66) 9 and calculate the value of C0. 168 Equation (68), \ 000 C [ACa-ACJ+C+C e=—9- 9 0 8 00 \ CCO 0 -[AC —ACJ-—fl— g a s C + C r 9 0) is obtained by equating equations (A-7) and (A—8) to eliminate the cons- tants C of C , 9 (total) C o , AC , and AC . a 8 and Cx. The resulting eXpression is solved for es in terms I... APPENDIX VII Susceptance Variance MethodS7-58and Derivation of Equations When a constant alternating voltage, V2, is applied between the terminals, a and b, of a series resonance circuit as shown in Fig. A-l, the current flowing in the circuit is given by 3;. V _ . = = 2 (A 10) I2 Zt RP R + ij + . 1 + wC R 2 2 J 2 p ; 4 In this equivalent circuit representation the subscript 2 denotes the g; resonant cavity and therefore, is consistant with previous circuit re— presentations. However, the voltage source V2, the resistance R2, the inductance L2, and the capacitance CZ, all represent the total equivalent series parameters of the oscillometer. In this representation the parallel resistance, RP, would be analogous to the equivalent parallel cell solu« tion load (equation (47) THEORY). The voltage drOp, V’, across the capacitance 62 (or across points c-d) is given by \ ,1 R _ r = P (All) V I21+ijR 2 P or r I? tip 1 + ij R (A-12) V' = 7 PR V2 R + ij + .p 1 + wC R L 2 2 J 2 PJ 169 170 $.14 ufloouwo nowuoaomlaaou mzu mo moamumfimom.u:=:m ucoam>Hovm cam umuoEOHHHomo >ua>mu uamcommm mcu uaomoumom ou ufiooufiu uamcomom mowuom unmam>wvum .HI¢ ouawwm 171 Rearrangement of the terms in equation (A-12) yields V ____JL?___ R + JwL (A-l3) V' = 2 1 ... . no . __1_._.. R p R + JwL P 2 2 If the quantity §T_713;Z"19 rationalized and the‘terms of the-equation 2 2 (A-14) are grouped into the real and imaginary parts, the voltage V' is ‘.\ ‘ .1.’ equal to -’ R L ‘ l o i - —Z—~2—V ' [RZT-I- ML: 3 32 + on L2] 2 (A—15) V = 1 31L . L9 'fi-+RT+ AZLZ*J“’02 'RY+AZLY P 2 2 2 2 I Since the quantity fiL-is equal to GP, the parallel shunt conductance, and P R the quantity R 2 +£ZL z is equal to the equivalent shunt conductance, 2 Ge’ of the R - L2 network, equation (A—lS) is reduced to 2 ( ij - v 2w as R Z + wZL V2 (A-16) V' = l L2 o - 2 Y GtIJ‘” CZ RY+m1LZ 1, 2 2 J where Gt is equal to the total equivalent shunt conductance, and R 1 (A—17) Gt=—+R2 [ZAYLY‘GP +08 P 2 2 Resonance is obtained when the capacitance C is adjusted to yield 2 maximum effective voltage V'lres) under condition of constant applied 172 voltage, V2, at angular frequency, m. It is evident from equation (A-l7) that the value of the capacitance, C , that will yield maximum effect- zfres) ive voltage, V is equal to I (res)’ L _ , 12 gfres) R Z + m7L47 2 2 (A-18) C The maximum effective voltage at resonance is then equal to r wL - 2 6 R22 + wZLOZ ’ 2 (A-lg) V'l ._ = l 1* 1‘ tree, Gt The ratio of the voltage for any given value of C2, to the voltage at resonance is given by v! 0+ (A—20) V.——-— = . L ‘ lres) . I 9 F - A at + J“ 2 R 1 + wZL 1; 2 J or ’3 r G ‘ V' * t (A-21) V, = I L ', _1 (res) G 2 + W2|n _ 2. f 7! t u ’02 R Z + sz Z! i 2 2 i The value of C , for any given value of-VTm———-zis given by 2 ”res; L Gt 7 vJ/2 - = I l a t—. .. . .- (A 22) 02 R t + wYE Z m V' :27 1 2 2 Iv! , l1 (res/l } L 41' v - ... Since the quantity R 2 + sz z is equal to 02(res;’ equation (A 22) is 2 2 reduced to 173 Gt V] 1/2 (A-23) C = C , , ——- 2 gfres u) . V' If for a given value of , Ca and Cb are chosen to represent (res) the two capacitance values of C to satisfy the conditions of equation 2 (A—23), their values may be given by G ’ 2/2 (A—24) c = c + J?- r—i—a —z' a (res) m V' ’ ' V. I I 5‘ (”88)} J and G _[ 51./'2 (A-ZS) c = c t ,-——1——-? b 2(res) IV, ' -1 wli V'{res) I It is evident from equations (A-24) and (A—25) that the capacitanre at resonance C 2(res)’ is equal to ( - = ‘A 26 C'2(res) and that the total equivalent shunt conductance G t’ is equal to t *1 “-1/2 = 2 VI ‘2 '1 2(f-__%fi__]2 ’1\1/2 _7______ “T““- IV (res)J 1V (res); l 1 1‘ (A—27) 0 mica - Cb] 174 If the capacitance values Ca and Cb are the two capacitance values I, of 02 (on either side of resonance) which correSpond tO'VTL‘“"‘= 6.707, (res) (the half power points), the value of Gt is equal to wAC (A-28) at =- —-g~191 The Q-factor for the cavity resonance circuit (Fig. l) is related £1 fl to the capacitance difference at the half power points by E wC ( 2C (A-29) Q = 2 res) _ 2(res) Gt — AC0.707 F" APPENDIX VIII Fundamental Equivalent Circuit Representation of the Resonant Cavipy Oscillometer and Derivation of the Circuit Equations (for the Capacitive Coupling) The fundamental equivalent circuit drawn to represent the resonant cavity oscillometer is shown in Fig. A—2. The subscript 1 denotes the signal generator circuit. The constant applied signal generator voltage is given by V . R. and L1 represent the total equivalent resistance and inductance of the loop probe used to couple the signal generator to the resonant cavity. The mutual induc- T' tance between the inductor L3 and inductor U2 is represented by M.2 The subscript 2 refers to the resonant cavity circuit. L2 repre- sents the total distributed inductance of the resonant cavity. R represents the total equivalent series resistance of the resonant cavity and includes not only the distributed resistance of the cavity but also includes the equivalent series resistance of the condenser—type cell— solution load (refer to Fig. 11). The variable capacitance, C2, repre- sents the total equivalent series capacitance of the cavity network and includes the total distributed capacitance of the cavity, the parallel f capacitance network of the dielectric sample holder 1C \ t and CD1 and the equivalent capacitance associated with the condenser—type cell—solution load (refer to Fig. 11). The subscript 3 refers only to that portion of the detector circuit (Fig. 11) which preceeds the detector tube V?. The variable capacitance C , represents the detector probe capacitance which is connected to the 3 175 176 Ahua>mo onu ou 5Ho>HuaommmU vmamsoo Houomumnv umuoaoaaaomo hufi>mo ucmnommm mnu ucomoummm ou ufiooufio ucmHm>fioum Houaoamvaom .NI< muowwm J WI llllllllll a; H. IIIIIIIIIII .4 mOHUMHmo _. m0._.._._>