Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The precise measurement of mass spectrometer ion currents Whittles, Arthur Brice LeRoy 1960

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1960_A6_7 W4 P7.pdf [ 3.33MB ]
Metadata
JSON: 831-1.0103737.json
JSON-LD: 831-1.0103737-ld.json
RDF/XML (Pretty): 831-1.0103737-rdf.xml
RDF/JSON: 831-1.0103737-rdf.json
Turtle: 831-1.0103737-turtle.txt
N-Triples: 831-1.0103737-rdf-ntriples.txt
Original Record: 831-1.0103737-source.json
Full Text
831-1.0103737-fulltext.txt
Citation
831-1.0103737.ris

Full Text

THE PRECISE MEASUREMENT OF MASS SPECTROMETER ION CURRENTS by ARTHUR BRICE LEROY WHITTLES B. Sc., University of B r i t i s h Columbia, 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1960 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f ^ ^ A C A The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 3 , Canada. Date fot • \l s 13k 0  ( i ) TABLE OF CONTENTS ABSTRACT ( i i ) LIST OF ILLUSTRATIONS (iv) ACKNOWLEDGEMENTS (vi) CHAPTER 1 The Input Properties of Mass Spectrometer Measuring Systems 1.1 Introduction 1 1.2 Early Methods of Po s i t i v e Ion ^ Detection and Measurement 2 1.3 Electrometer Vacuum Tubes 4'* CHAPTER 2 Input Resistors 2.1 Introduction 9 2.2 The Voltage C o e f f i c i e n t of Victoreen Hi-Meg Resistors 11 2.3 The Non-Linearities of Film Resistors 21 CHAPTER 3 The Response of a Type 1 System 3.1 Introduction 24 3.2 Simple Feedback System 27 3.3 Response of a Type 1 System 37 CONCLUSIONS 42 BIBLIOGRAPHY 46 APPENDIX The Analogue Computer i n Servo System Design A.1 Introduction 49 A.2 The Response of a System 52 A.3 Equalization 56 A.4 Conclusions 64 ( i i ) ABSTRACT This thesis i s primarily concerned with the problems involved i n making precise mass spectrometer ion current measurements. A survey of the l i t e r a t u r e of the past decade shows that u n t i l the new type of system discussed i n Chapter 3 was presented by R.D. Russell and F. K o l l a r , remarkably few developments were made i n improving the precision of ion current measurements for the isotopes of the heavier elements. This s i t u a t i o n i s probably p a r t l y due to a f a i l u r e to apply the modern methods of analysis which are presently used i n the design of feedback control systems. Chapters 1 and 2 deal with the main requirements of ion current measuring systems. It has been assumed in previous systems that the r e s i s t o r , through which the ion current i s passed, has an approximately constant value f o r a l l values of the current. A method has been developed to measure the degree of t h i s non-linearity, and r e s u l t s are given f o r the Victoreen Hi-Meg r e s i s t o r . The theory necessary f o r the measurement i s developed i n d e t a i l . The f i n a l chapter deals with a general study of mass spectrometer measuring systems. A comparison i s made of the two general types of measuring systems now i n use, with an attempt to determine t h e i r r e l a t i v e advantages and disadvantages. The new system noted above i s also ( i i i ) discussed with the objectives of determining i t s t h e o r e t i c a l performance, and of i l l u s t r a t i n g the methods of analysis. An appendix has been included to show the usefulness of the analogue computer i n the design of an ion current measuring system. (iv) LIST OF ILLUSTRATIONS F i g . 1: Typical Grid C h a r a c t e r i s t i c s of an Electrometer Tube 6 Fig . 2: Bridge and Electrometer Detector C i r c u i t 13 F i g . 3: A Mass Spectrometer D.C. Measuring System 13 F i g . 4: Voltage C o e f f i c i e n t of Victoreen Hi-Meg Resistors 17 F i g . 5: Simple Feedback System 28 F i g . 6: Simple Feedback System 28 F i g . 7: Current Pulse I(t) => I sin 2wt 32 F i g . 8: Superposition of Single Peaks of I ( t ) - I sin 2wt and I(t) - I cos 2wt 32 F i g . 9: Idealized Current Pulse 32 Fig.10: A Type 1 Mass Spectrometer Measuring System 36 Fig.11: Generalized Form of a Mass Spectrometer Measuring System 36 Fig.12: Generalized Operational Amplifier 51 Fig.13: Simple Two Pole System 51 Fig.14: Some Network Impedance Transfer Functions 51 Fig.15: Simulated Amplifier-Motor Transfer Functions 53 Fig.16: Summing Amplifier 53 Fig.17: Response of an Amplifier-Motor System to a Step Input Voltage 55 Fig.18: Simulated Amplifier-Motor System 55 o Fig.19: Amplifier-Motor Root-Locus Plot 57 Fig.20: Input Capacities 58 Fig.21: The Transfer Function Diagram of a Motor-Amplifier System with Input Capacities 58 (v) Fig.22: Root Locus Diagrams for Amplifier-Motor System with Input Capacities 60 Fig.23: Some D.C. Equalization Networks 61 Fig.24: Amplifier-Motor System with Veloc i t y Feedback 63 Fig.25: V e l o c i t y Feedback on the Analogue Computer 63 (vi) ACKNOWLEDGEMENTS This research was conducted under the supervision of Professor R.D. Russell, whose help and advice i s greatly appreciated. The writer i s also indebted to F. K o l l a r , and to the other members of the Geophysics Staff for th e i r many help f u l suggestions. The Heathkit E l e c t r o n i c Analogue Computer was provided by Dr. A.M. Grooker of the Physics Department. During t h i s i nvestigation f i n a n c i a l assistance was received from the National Research Council of Canada. -1-CHAPTER 1 The Input Properties of Mass Spectrometer Measuring Systems. 1.1 Introduction The design of a suitable ion current measuring system i s a primary inter e s t i n the construction of a mass spectrometer that i s used to determine the precise abundances of the isotopes of an element. Consider the measuring system i l l u s -trated i n F i g . 3, page 13. The ion current being measured i s c o l l e c t e d by c o l l e c t o r C , and the exchange current passes through the r e s i s t o r s R and R^ to ground. Since the voltage drop developed across R i s E = IR , E i s used to determine I . Thus for a precise measurement of the ion current i t i s necessary that the detector input resistance be much larger than R , so that a l l the exchange current flows through R . Although R i s generally i n the order of 1 0 1 1 ohms the e x i s t i n g techniques discussed i n the remainder of Chapter 1 allow t h i s condition to be met. It i s also necessary that R remains constant for d i f f e r e n t ion current valuesj that i s , the voltage co-e f f i c i e n t of the r e s i s t o r must be n e g l i g i b l e . This condition i s discussed i n Chapter 2. -2-1.2 Early Methods of Po s i t i v e Ion Detection and Measurement Pos i t i v e ion currents were f i r s t detected by the use of fluorescent screens (Goldstein, 1886; Thomson, 1913), but t h i s method was both d i f f i c u l t and tedious, and was supplanted with the introduction of the photographic plate. Thomson (1913), Aston (1925), and others applied the photo-graphic method to the study of mass spectra, but found that r e l a t i v e isotope abundances could not be obtained with any degree of precision. E l e c t r i c a l methods were introduced to overcome t h i s objection. The f i r s t arrangements for the e l e c t r i c a l detection of p o s i t i v e rays employed electrometers. A good summary of these instruments may be found i n Strong (1956) and only t h e i r general features are of interest here. The Dalezalek and Compton electrometers are representative types (Strong, 1956; Suydam, 1948). A vane i s suspended in a p i l l b o x divided into four equal and insulated quadrants, and the e l e c t r o s t a t i c i n t e r a c t i o n of the vane and quadrants causes the vane to rotate. A small mirror on the suspension f i b e r i s used to measure the ro t a t i o n , which i s proportional to the charge on the electrometer. Other types of electrometers include the Wulf String, the Perucca, and the Lindemann electrometers. The operation of these instruments depends p r i n c i p a l l y on the deflection of a -3-quartz f i b e r between charged plates. While cert a i n electrometers are capable of measuring currents i n the order of lO"" 1^ amperes, they are generally quite d e l i c a t e , and the measurements are d i f f i c u l t to make even with extensive c a l i b r a t i o n . The more rugged electrometer vacuum tubes and the v i b r a t i n g reed electrometers have largely replaced electrometers i n the measurement of ion currents. The v i b r a t i n g reed electrometer i s widely used today i n ion current measurements. The main features of t h i s i n s t r u -ment are summarized by Duckworth (1958), and the theory of operation may be found i n Gunn (1932), and Palevsky et a l . (1947). The ion current passes through a r e s i s t o r R and c o l l e c t s on an input capacitor. This capacitor i s mechanically vibrated and a periodic voltage i s produced across i t s termi-nals. The amplitude of t h i s voltage i s proportional to the charge on the input capacitor. The D.C. voltage i s thereby converted to an A.C. voltage, which can be amplified by con-ventional vacuum tube amplifiers. The input impedance of these 15 electrometers i s i n the order of 10 ohms, and the input ca-pacity i s approximately 40 micro-microfarads, without feedback. If the instrument i s used i n a feedback arrangement t h i s ca-pacity w i l l be reduced somewhat, but w i l l not be eliminated, and a large time constant w i l l s t i l l be present. The main ad-vantage of t h i s instrument i s based on the f a c t that ampli-f i c a t i o n i s much more e a s i l y accomplished with A.C, than with D.C. The disadvantages include the large time constant, lower -4-accuracy, and the s i z e and cost of the device. A v i b r a t i n g reed electrometer would s a t i s f y the conditions necessary f o r a suitable ion detector input. The choice bet-ween t h i s and an electrometer tube input does not concern us here,as i t i s largely dependent upon the s p e c i f i c application for which the system i s to be used. 1.3 Electrometer Vacuum Tubes Generally, electrometer tubes are s p e c i a l l y designed vacuum tubes having extremely low g r i d currents; however, certain conventionally constructed vacuum tubes may be used under s p e c i a l conditions. Nottingham (1930) conducted an extensive study into electrometer tubes and c i r c u i t s , i n which he analysed the causes of g r i d current. The g r i d current i s a measure of the D.C. input resistance of the g r i d , and sets a l i m i t on the magnitude of current that can be detected. Nottingham measured three probable components of the g r i d current, and was able to f i n d the r e l a t i o n s h i p between the g r i d current and voltage. His r e s u l t s were si m i l a r to those i l l u s t r a t e d i n Fi g . 1, page 6. He was able to show that, while there must be a f i n i t e (but large) D.C. g r i d input resistance, the incre-mental input impedance (dE / d l ) may be v i r t u a l l y i n f i n i t e at o g the proper operating point. In the same year several s p e c i a l l y designed tubes (Nelson, 1930; Metcalf and Thompson, 1930) were described. The l a t t e r -5-tube (FP-54) had a g r i d current as low as 10 amperes and an input capacity of 2.5 micro-microfarads, which was a con-siderable improvement over previous designs. This tube was the r e s u l t of a careful study of the causes of g r i d current. Metcalf and Thompson showed that the g r i d current resulted from i n s u l a t i o n leakage, gas ions, thermionic g r i d emission, cathode p o s i t i v e ion emission, and photo-electrons. By c a r e f u l electrode construction and placement, low i n t e r -electrode voltages, and the introduction of a p o s i t i v e space charge g r i d between g r i d and cathode, these sources of g r i d current were eliminated or greatly reduced. The Western E l e c t r i c D-96475 was of s i m i l a r design. Conventional vacuum tubes have also been adapted to serve as electrometer tubes (Bennet, 1930; Huntoon, 1935; Gabus and Pool, 1937). Although the input resistance i s generally not greater than 1 0 1 3 ohms, these tubes, u n t i l recently, were more r e a d i l y available and cheaper than the s p e c i a l l y designed electrometer tubes. The receiving type 954 acorn tube (Gabus and Pool, 1937) i s a t y p i c a l example. By interchanging the screen and control grids of t h i s pentode, and by operating with a low plate pot e n t i a l (6 v o l t s ) , the input resistance was increased from 10 8 to 10*3 ohms. The input capacity, however, was 10 micro-microfarads, without feedback, which gave a considerable time constant. These modified tubes are suitable f o r many ion current measurements but more c r i t i c a l applications require the s p e c i a l l y constructed electrometer tubes. - 6 -F i g . 1: Typical Grid C h a r a c t e r i s t i c of an Electrometer Tube - 7 -The Raytheon CK5886 i s an example of a presently used electrometer tube. It i s of inte r e s t to discuss t h i s tube rather f u l l y as i t has been used i n the voltage c o e f f i c i e n t measurement of the Victoreen Hi-rMeg r e s i s t o r (Chapter 2), and has c h a r a c t e r i s t i c s which approach those desired for an i d e a l detector system (high input resistance, and low input capacity). The CK5886 i s a subminiature electrometer pentode with f l e x i b l e terminal leads. Low plate voltage (6 v o l t s ) , low filament voltage (1.25 v o l t s ) , and low filament current (10 mi1Hamperes) may be u t i l i z e d . The g r i d current may be de-termined i n the manner suggested by Strong (1956). The plate current i s plotted as a function of the g r i d voltage i n the usual .manner, f i r s t with no resistance in-the g r i d c i r c u i t and then with a high resistance i n s e r i e s . The horizontal difference between the two curves represents the p o t e n t i a l drop i n the resistance due to the g r i d current, and by d i -viding t h i s voltage difference by the value of the r e s i s t o r , the g r i d current i s found. A shielded box i s necessary for t h i s measurement. A curve of g r i d voltage vs. g r i d current i s i l l u s t r a t e d i n F i g . 1. This curve immediately suggests the following points. If the tube i s given a g r i d bias of -2.1 v o l t s , then the D.C. input resistance i s reported to 14 be i n the order of 3x10 ohms, and the writer found t h i s to be correct. The incremental impedance w i l l be greater than 17 10 ohms. The high D.C. input resistance i s of considerable importance i n the bridge c i r c u i t used i n Chapter 2, since - 8 -under these conditions the detector does not load the bridge to any extent. The very high incremental impedance i s im-portant also. In a mass spectrometer the ion current i s usually i n the form of wide current pulses f o r the d i f f e r e n t masses, so a high incremental impedance w i l l ensure that a l l the current change w i l l pass through the g r i d r e s i s t o r R , and none into the electrometer preamplifier (Fig. 3). The input capacity of the tube and c i r c u i t i s generally quite small (few micro-microfarads). K o l l a r (1960) has suggested an i n t e r e s t i n g c i r c u i t that might v i r t u a l l y e l i m i -nate the capacity due to the tube. A t r a n s i s t o r feedback loop may be used to drive the interconnected plate and screen grids with the same voltage v a r i a t i o n s that appear on the cathode of the tube. Since a l l tube electrodes follow the g r i d the e f f e c t s of the tube capacities w i l l be eliminated, and unity gain achieved. The present writer was able to v e r i f y these r e s u l t s using an equivalent vacuum tube c i r c u i t . In summary, then, the techniques for obtaining a large preamplifier input impedance already e x i s t , and one i s j u s t i -f i e d i n assuming that the ion current I (Fig. 3) flows e n t i r e l y through the r e s i s t o r R The assumption that R has a n e g l i g i b l e voltage co-e f f i c i e n t w i l l be dealt with i n the following chapter. -9-CHAPTER 2 Input Resistors. 2.1 Introduction The detection systems of most modern mass spectrometers are of the form i l l u s t r a t e d i n F i g . 3. The ion current I , a r r i v i n g at the c o l l e c t o r cup C , passes d i r e c t l y through the 1 0 1 1 ohm r e s i s t o r R , and gives r i s e to a voltage E across R . This voltage i s impressed on the g r i d of an electrometer tube i n the preamplifier stage, and i s amplified i n the following stages. The electrometer tube, provided i t i s at a s u i t a b l e operating point w i l l have a very large input resistance compared to R The measurement of the isotope r a t i o s , then, i s the measurement of the r e l a t i v e voltages E=RI , where I w i l l be the current r e s u l t i n g from the ion currents at the d i f f e r -ent masses. In a l l previous measuring systems R has been assumed constant, within a small error, for a l l voltages E . It was the purpose of t h i s part of the investigation to de-termine to what extent t h i s i s true. A summary of the r e s u l t s has been published (Whittles, 1960), a r e p r i n t of which i s enclosed at the back of t h i s thesis. Very l i t t l e information appears to be presently available -10-on the voltage c o e f f i c i e n t s of large r e s i s t o r s (10** to 1 0 1 4 ohms), although considerable work has been Carried out by Williams and Thomas (1959), and others, on current noise and non-linearity of deposited carbon r e s i s t o r s i n the 10 4 ohm range. For the purpose of t h i s discussion, the present writer proposes to use as the d e f i n i t i o n of the voltage c o e f f i c i e n t of R : - - i ^ (2-1) R dE where E = the voltage across R j and, dR = the incremental change i n R for an incremental change,in E , dE Thus as the value of E across R i s increased the value of R w i l l change, and the d i r e c t i o n of change w i l l depend upon the r e s i s t o r used. For the r e s i s t o r s i n -vestigated t h i s change i s negative, and the r e l a t i o n s h i p between R and E was found to be of an exponential form. In the f i r s t measurements ca r r i e d out by the writer R was determined f o r various applied voltages by a convention-a l Wheatstone bridge. In the second, the bridge was modified so that the voltage c o e f f i c i e n t could be determined d i r e c t l y . In the Wheatstone bridge, R , and a p r e c i s e l y measured standard 10 megohm r e s i s t o r formed one arm, and two 0.1% pre c i s i o n r e s i s t o r boxes the other. The n u l l detector was a Hewlett Packard D.C. meter, used on the 10 microvolt range. -11-The input resistance i s approximately one megohm. Despite extensive precautions e l e c t r o s t a t i c e f f e c t s were quite serious, and the r e s u l t s almost e n t i r e l y unsatisfactory. The data indicated that R was constant within +0.06% over the voltage range of 0 to 22 v o l t s , and decreased thereafter approximately exponentially with respect to E It was decided that a di r e c t measurement of the voltage c o e f f i c i e n t was required to obtain the desired accuracy. 2.2 The Voltage C o e f f i c i e n t of Victoreen Hi-Meg Resistors (i ) Introduction The Victoreen Hi-Meg* r e s i s t o r discussed here i s a car-bon coated glass rod, the ends of which are banded with s i l -ver to obtain the best e l e c t r i c a l contact. The r e s i s t o r proper i s vacuum sealed i n a glass envelope, and t h i s enve-lope i s treated with a s i l i c o n e varnish to keep the r e s i s t o r free from humidity e f f e c t s . The maximum permissible voltage i s 1000 v o l t s , and the nominal voltage c o e f f i c i e n t , defined by the manufacturer as: ( R at 1 vo l t - R at 100 vo l t s ) 1 % ^ R at 1 v o l t 100 v o l t i s -0.03% / v o l t for 1 0 1 1 ohms, and -0.02% / vo l t f or 10 9 and l O 1 ^ ohms. J.W. Weigl (private communication) has i n -dicated that the maximum permissible voltage can be ex-ceeded s u b s t a n t i a l l y without hysteresis or detectable •Tradename -12-damage. The measurements were carr i e d out by connecting two Vic-toreen Hi-Meg r e s i s t o r s ( R and R* ) i n s e r i e s to form one arm of a Wheatstone bridge, and two precision r e s i s t o r s to form the other arm. One of the Victoreen r e s i s t o r s ( R ), was chosen to have approximately ten times the resistance of the other, so that the voltage across the bridge appeared predominantly across the larger r e s i s t o r . Thus the voltage c o e f f i c i e n t of the smaller of the two can be neglected, or else included as a minor correction. A s p e c i a l detector was constructed, and i s i l l u s t r a t e d i n F i g . 2. The difference amplifier i s constructed of two high input impedance Raython CK5886 electrometer pentodes connected as cathode followers. As discussed i n Chapter 1, i f the tubes are at the proper operating point the input resistance i s very large compared to that of the Victoreen Hi-Meg r e s i s t o r being tested, and the detector does not load the bridge. Sligh t differences i n the tube c h a r a c t e r i s t i c s and i n the filament voltages resulted i n a bias difference between the tubes amounting to 100 m i l l i v o l t s . This was n u l l i f i e d by adjusting the screen g r i d voltages through the 100 kilohm potentiometer, and allowed operation of the c i r c u i t with i d e n t i c a l g r i d biases, and i d e n t i c a l cathode currents. Under these conditions, any d r i f t should have been nearly the same i n the two tubes, and thus cancel i n the d i f f e r e n t i a l output. The use of compensating c i r c u i t s -13-F i g . 2: Bridge and Electrometer Detector C i r c u i t K t ) Electrometer Preamplifier High Gain D.C. Amplifier 1 | Q(t) R < >R i. E x(t) To Chart Recorder F i g . 3: A Mass Spectrometer D.C. Measuring System -14-was unnecessary as the measurements were completed within a short time. The voltage gain of the cathode follower was 0.61. The output impedance was approximately 30 kilohms. The detector and a l l r e s i s t o r s were enclosed i n a l i g h t -proof grounded-metal chassis, with the electrometer tubes and the Victoreen Hi-Meg r e s i s t o r s inside on a l u c i t e chassis. Hence the leakage paths were at a minimum. Common supply voltages were used f o r the plates, the screen grids, and common cathode r e s i s t o r s , but separate mercury c e l l s were used f o r the filaments. A l l supply and filament b a t t e r i e s , and switches, were mounted on another l u c i t e chassis. This chassis was also placed i n the ground-ed-metal box. Transients due to mechanical vibrations were reduced by placing the enclosed chassis i n a cradle of rubber bands, which i n turn, was placed on a heavy s t e e l sheet on a concrete f l o o r . The cathode ray oscilloscope was a Tektronix type 502, with the d i f f e r e n t i a l input of one trace used as a vacuum tube voltmeter to measure the voltage difference between the cathodes of the CK5886 tubes. This was s u f f i c i e n t l y s e n s i t i v e to detect an unbalance voltage of about 10 micro-v o l t s without loading the bridge. The input impedance of the 502 i s approximately 1 megohm, compared to the cathode follower output impedance of 30 kilohms. -15-( i i ) Method and Results In carrying out the measurements the bridge was balanced fo r an a r b i t r a r y bridge voltage V . This voltage was then reduced by 6 v o l t s , and the r e s u l t i n g error voltage e measured on the oscilloscope. As w i l l be shown i n part ( i i i ) , i f one assumes that the voltage c o e f f i c i e n t i s small, that the bridge i s always near balance, and that the detector does not load the bridge, the average value of the voltage c o e f f i c i e n t over the 6 v o l t range i s given by: 1 dR (Ri + Ro) 3 e 100 % = — ± - ± (2-3) R dE R i 2 R 2 G 6(V-6) v o l t where, at balance: V = the i n i t i a l bridge voltage; E = the voltage across R ; R l > "*2 = v a l u e s °f the two low resistance arms; and where: «i — = 10 approximately i n t h i s experiment; R 2 G = gain of the detector =0.61 . Since R and R 1 are also i n the r a t i o 10:1, R 1 although a Hi-Meg r e s i s t o r has only 0.1 V across i t compared to 0.9 V across R . Hence the bridge voltage appears pre-dominantly across the larger r e s i s t o r , and the voltage co-e f f i c i e n t of the smaller can be neglected, or else included -16-as a minor correction. This correction i s given i n part ( i i i ) . The r e s u l t s are i l l u s t r a t e d in Fig. 4. In p l o t t i n g the graph the measured voltage c o e f f i c i e n t was assigned to the middle of each 6 vo l t range. The voltage c o e f f i c i e n t , f o r small voltages, i s found to vary i n a li n e a r fashion from -0.003% per vo l t to -0.016% per vo l t for the 1 0 1 0 ohm re-s i s t o r , and s l i g h t l y higher values f o r the 10 1* ohm r e s i s t o r . In t h i s voltage range the voltage c o e f f i c i e n t i s much below the manufacturer's nominal value for the 1 to 100 v o l t range; however, a value approximately equal to the manufacturer's value was found f o r the resistance change from 1 to 100 volts. This l a t t e r r e s u l t suggests that the voltage c o e f f i c i e n t vs. E r e l a t i o n s h i p i s not actually a straight l i n e f o r higher voltages, but rather some curve with a slowly decreasing slope f o r increasing E . By extrapolation of the straight l i n e back to E = 0 vo l t s , 1^ and Rg , for balance, may be computed and the value of R obtained from R = RjR 1 / R 2 An i n t e r e s t i n g comparison of the r e s u l t s of t h i s i n -vestigation with those of Williams and Thomas (1959) may be made i f we assume a str a i g h t l i n e r e l a t i o n s h i p between R and E . Then: -1 dR A. = AE + B , (2-4) R dE -17-Volts (E) F i g . 4: Voltage C o e f f i c i e n t of Victoreen Hi-Meg Resistors -18-where A and B are pos i t i v e constants that may be evalu-ated from the graph. If (2-4) i s integrated we f i n d : R = R Q exp(-AE 2/2 - BE) (2-5) If we put dR = R Q - R , and note that RQ/R - 1 : l-exp(-AE 2/2 - BE) Expanding the exponential, and noting that the exponent i s small: In ( — } - ln(AE 2/2 + BE) (R0> If AE 2/2 << BE , then: ln(dR) - InE + In(BR Q) A plot of ln(dR) vs. InE would have a slope of 1. On the other hand, i f BE << AE 2/2 : ln(dR) = 2• In E + ln(AR Q/2) which has a slope of 2. These r e s u l t s are s u b s t a n t i a l l y i n agreement with those of Williams and Thomas (1959), who showed that the r e l a t i o n of IndR vs. InE was a straight l i n e of slope 1.2 to .1.9, depending on the r e s i s t o r used. Although the r e s i s t o r s are very d i f f e r e n t i n s i z e and con-s t r u c t i o n , the non-linearities found are s i m i l a r . i n £5> (R Q) In -19-( i i i ) Voltage C o e f f i c i e n t Formula (a) If the assumptions outlined i n 2.2 ( i i ) are made, equation (2-3) may be derived as follows: at balance, with a voltage V across the bridge: R 1V RoV (2-6) R + R 1 Rj + R 2 If V i s changed by the 6 v o l t step to V 1 , a small error voltage e 1 appears across the input of the difference amplifier: R^-V1 RoV1 e 1 =• - (2-7) R + dR + R 1 Rj + R 2 R 1 i s assumed to have zero voltage c o e f f i c i e n t . Substitute (2-6) into (2-7) to obtain: ! R1V1dR^ e - (R + dR + R 1)(R + R 1) neglecting the sign. But dR << R , and R1/R = R 2/Ri > s ° : e 1 R^dR V 1 (R + Rl)2 and: 1 dR = (R + R 1 ) ^ ! R V-V 1 R^V^V-V 1) -20-or: 1 dR (R, + R o ) ^ 1 100% = * , (2-8) R dV R-jRgVMV-V1) v o l t As equation (2-8) stands, — — re f e r s to the percent change R dV of R with respect to the change of bridge voltage V . To change t h i s dependence to the voltage E across R , we note that: R]V E= — , at balance, R ^  + Rg or: R-idV dE R l + R2 Additionally, since the oscilloscope measures an error voltage e across the output of the difference amplifier, e 1 may be replaced by e/G . Equation (2-8) becomes: 1 dR (Ri + Ro) 3 e 100% = — ± — — £ (2-9) R dE Ri 2R2 GV^V-V 1) v o l t If V-V* = 6 , we arr i v e at the f i n a l form, equation (2-3). (b) If R1 does not have a zero voltage c o e f f i c i e n t , s i m i l a r arguments show that (2-9) becomes: 1 dR R dE (Ri + R 2 ) 3 e (R 2) 1 dR] R X 2R 2 GV^V-V 1) (R 2) R 1 d E ^ v o l t 100% -21-where E i s the voltage across R . If R2^Ri = 1 / 1 0 > and i f the voltage c o e f f i c i e n t s of the two r e s i s t o r s are nearly the same: (Ro) 1 dR 1 (Ro) 1 dR 1 1 dR (R x) R 1 dE 1 (R x) R dE 10 R dE This correction may be neglected, (iv) Temperature C o e f f i c i e n t The nominal temperature c o e f f i c i e n t of these r e s i s t o r s i s quite large, -0.1% / degree centigrade for 1 0 1 1 ohms. However, i f the r e s i s t o r i s mounted inside the vacuum with the c o l l e c t o r and. electrometer tube any sudden thermal d i s -turbances w i l l have l i t t l e or no e f f e c t , and long term d i s -turbances w i l l only introduce an e a s i l y corrected-for d r i f t . For these reasons the temperature c o e f f i c i e n t may be neglected. 2.3 The Non-Linearities of Film Resistors The probable causes of the n o n - l i n e a r i t i e s of f i l m re-s i s t o r s has been outlined by G r i s d a l et a l . (1951). An i d e a l r e s i s t o r would possess a resistance that i s constant with time, temperature, voltage, and frequency under a l l conditions of application. Wire-wound r e s i s t o r s approach t h i s i d e a l , but t h i s type of construction may only be used fo r low resistances. Metal f i l m r e s i s t o r s may also be used, but only up to values i n the order of a few -22-thousand ohms. Above t h i s value they become unstable. Higher resistance requirements have necessitated non-metallic f i l m r e s i s t o r s ' of which carbon appears to be the most s a t i s f a c t o r y , both because i t possesses a r e l a t i v e l y high s p e c i f i c resistance, and because i t can be r e a d i l y produced i n f i l m form. Unfortu-nately, such non-metallic r e s i s t i v e materials possess r e l a t i v e -l y less stable e l e c t r i c a l and mechanical properties. To avoid large temperature c o e f f i c i e n t s , poor s t a b i l i t y i n time and humidity, and appreciable voltage c o e f f i c i e n t s , i t has been found necessary to produce the carbon f i l m by the pyro l y s i s of hydrocarbon vapors. This process involves the thermal decomposition of hydrocarbons, with a r e s u l t i n g depo-s i t i o n of carbon onto the surfaces of suit a b l e r e f r a c t o r y and chemically stable objects, such as glass or ceramic cores. This process allows an increase i n the resistance, homoge-neity, and s t a b i l i t y of the f i l m and improves the high frequency c h a r a c t e r i s t i c s and power d i s s i p a t i o n . Additionally, the s p e c i f i c resistance of carbon even i n the thinnest films i s e s s e n t i a l l y the same as i n the bulk, so purely geometrical or mechanical factors determine the resistance of the f i l m . A further development has been the introduction of a few per-cent of boron into p y r o l y t i c carbon films. This gives a f i l m which possesses a resistance s t a b i l i t y comparable to that of the wire-wound type. The films are deposited i n the form of very small —7 c r y s t a l packets (3x10 cm.) which are quite anisotropic, so -23-the resistance w i l l probably depend both upon the orientation of the c r y s t a l s , and the properties of the c r y s t a l boundaries. It i s found that the s p e c i f i c resistance decreases with an increase i n the degree of c r y s t a l o r ientation, which i n -dicates that the voltage c o e f f i c i e n t i s not a thermal e f f e c t . It seems quite l i k e l y that the increase i n the conductivity with voltage i s p a r t l y due to the increase i n the i n t e r a c t i o n of the molecules of two adjoining c r y s t a l faces. Additionally, since carbon i s i n the same chemical family as s i l i c o n and germanium we should expect some semi-conductor behaviour. Thus, although the outer electrons are bound more c l o s e l y to the nucleus i n carbon than i n the other two elements, one would expect an approximately exponential increase i n current with respect to the applied voltage. This was observed i n equation ( 2 - 5 ) . F i n a l l y , i t i s noted that the supporting surfaces for the films have both chemical and physical influences on the properties of the f i l m . Thus n o n - l i n e a r i t i e s are to be ex-pected as a r e s u l t of mechanical imperfections on the sup-porting surfaces, and as a r e s u l t of variations i n thick-ness of the f i l m . These variati o n s i n thickness are mainly due to the chemical properties of the supporting material and impurities i n t h i s material. -24-CHAPTER III The Response of a Type 1 System. 3.1 Introduction One of the e a r l i e s t arrangements f o r the e l e c t r i c a l measurement of mass spectrometer ion currents i s that re-ported by Taylor (1935). This measuring system employed an electrometer tube D.C. amplifier with a d i r e c t reading galvanometer i n the plate c i r c u i t . While the system had the advantage of being very simple, i t was quite suscepti-ble to d r i f t and not capable of very precise measurements, p a r t i c u l a r l y i f the isotope abundances d i f f e r e d considerably. Wide variations i n the isotope abundances may be measured by the introduction of a shunt selector as sug-gested by Smith et a l . (1937). This selector switched the large input r e s i s t o r R . Since the value of the re-s i s t o r used depended upon the isotope being measured, the input voltage to the amplifier was always of the same order of magnitude. This arrangement has the disadvantage of introducing additional leakage paths, and of requiring precise values f o r the input r e s i s t o r s . Hippie et a l . (1945) were the f i r s t to introduce the c i r c u i t which forms the basis of most modern mass spectrome--25-ter measuring systems. This system i s i l l u s t r a t e d i n Fig . 3. As pointed out e a r l i e r , i n the steady state, the voltage E across R i s nearly equal to . A n a l y t i c a l l y , the error voltage Q(t). = E(t) - E j ( t ) may be made a r b i t r a r i l y small by increasing the gain G ; however, t h i s i s d i f f i c u l t to accomplish i n practice, as c i r c u i t and input capacities tend to make the system less stable for high gains. A f i n i t e error voltage w i l l generally be present, and the magnitude of th i s voltage w i l l be proportional to that of the input si g n a l . Russell and Ko l l a r (Kollar, 1960) have suggested a system that provides an e s s e n t i a l l y zero error voltage even i f the input capacities are substantial. This system i s i l l u s t r a t e d i n F i g . 10 and w i l l be discussed more f u l l y i n sections 3.2 and 3.3. Instead of taking s i n g l e measurements, i t i s possible to compare, simultaneously, two ion currents as a r a t i o . This procedure was suggested by Aston (1930) but was not used i n h i is own experiments. Straus (1941) was the f i r s t to apply the method i n his comparison of two nickle isotope abundances. The two ion currents at the d i f f e r e n t masses charged up two condensors, and the charges were then compared i n a n u l l measurement. This development was further improved by Nier et a l . (1947a,b). These workers made simultaneous comparisons of - 2 6 -two ion currents with a c i r c u i t s i m i l a r to that used by Hippie et a l . was replaced by a manually operated potentiometer, and the output was connected through a high value r e s i s t o r to a second c o l l e c t o r and a second measuring system. The ion current r a t i o was then determined by a s e n s i t i v e galvanometer in the plate c i r c u i t of t h i s second measuring system. Urey et a l . (1950) employed a v i b r a t i n g reed electrome-ter and a chart recorder i n place of the electrometer-galva-nometer measuring system of Nier. This arrangement increased the pr e c i s i o n over that of previous measurements by a factor of f i v e . Wanless and Thode (1953) have also suggested modifi-cations. Russell and K o l l a r have pointed out that t h e i r system may also be used i n a s i m i l a r manner. However, instead of comparing the ion currents at two d i f f e r e n t masses, i t may be possible to use one system to measure the t o t a l current of a l l the isotopes, and the second to measure that of a given mass. The output of the f i r s t system would supply the voltage , (Fig. 10) f o r the second. Hence any f l u c t u -ations i n the t o t a l ion current w i l l cause s i m i l a r f l u c t u -ation i n , and the r e l a t i v e abundance of the given mass current being measured should remain constant. The cor-r e c t i o n i s e n t i r e l y automatic. -27-3.2 Simple Feedback System (i) Advantage of a Feedback System The simple system used by Taylor (1935) i s known as an open loop system. This term describes a system (a four terminal network) i n which the input drives the output d i r e c t l y through intermediate components, and one i n which the output does not a f f e c t the input. In a closed loop system the output i s connected back to the input as i n Fig. 5. In the open loop system the gain depends d i r e c t l y upon the state of the intermediate components, so variations of temperature, humidity, etc., may cause changes i n the output that are independent of the input s i g n a l . This type of system i s therefore unsuitable for precise measurements. The very precise determinations of isotope abundances would re-quire the use of a feedback or closed loop system, i n which the output i s "fed back" to, and subtracted from, the input. To demonstrate how the feedback system la r g e l y removes the gain dependence on the intermediate components we may consider the system i l l u s t r a t e d i n F i g . 5. By the d e f i n i t i o n of the gain G : E 1 GQ But: E Thus: G E = G 1E 1 + G (3-1) G 1 i s the e f f e c t i v e gain of the closed loop system. - 2 8 -A A J £ Q 1 / B B i t F i g . 5 : Simple Feedback System E(s) F i g . 6: Simple Feedback System -29-A change i n the intermediate components w i l l cause a change i n the gain: dG 1 1 dG (1+G) 2 If G i s i n the order of 500, a 10% change i n G w i l l produce only a 0.02% change i n G 1 . A feedback system, then, causes the output to be l a r g e l y independent of changes i n the intermediate components. It should be noted that the foregoing discussion does not apply to the noise c h a r a c t e r i s t i c s of a system. The noise present i n a system may be either attenuated, or accentuated, depending upon the conditions of feedback. To f a c i l i t a t e the analysis of feedback systems i t i s useful to express the system, and a l l voltages, i n terms of Laplace transformations (Savant, 1958). ( i i ) D e f i n i t i o n s The Laplace transform of the function E(t) i s : E(s) - E ( t ) e " s t d t The "transfer function" of a system (a four terminal network) may then be defined as the r a t i o of the Laplace transformed output to the Laplace transformed input. The "open loop transfer function" of a system i s that transfer function for which there i s no feedback loop; for example: E i ( s ) ~~T = G(s) E (s) -30 would be the open loop transfer function of F i g . 5, provided the feedback loop from A 1 to B were broken, and B were grounded. On the other hand, the "closed loop transfer function" of F i g . 5 i s ; (s) G(s) E (s) " 1 + G(s) In t h i s terminology F i g . 5 may be redrawn as F i g . 6. The c i r c u l a r symbol denotes that Q(s) = E(s) - E-^Cs) . In general G(s) w i l l hot be a constant, but rather of the form: G(sT-,+l) (sTo+1) ... G(s) - ; I (3-2) s" ( s T 2 + l ) ( s T 4 + l ) . . . G i s some constant that may conform to the D.C. gain, and T^ , etc. are the time constants of the system. A system having t h i s transfer function i s known as a type n system. The values of s=0, -1/T 2, -I/T4, ... , i n the denominator, are c a l l e d the "poles" of the system, and the values s^-l/T^, 1/T3, ... , are c a l l e d the "zeros". Whether or not a system w i l l o s c i l l a t e i n a feedback arrangement depends upon the location of the system poles and zeros i n the s-plane, and not upon the form of the d r i v i n g function E(s) . The damped transient terms, however, depend upon both the system and the d r i v i n g function; the system determines the frequency of these terms, and d i s c o n t i n u i t i e s i n the d r i v i n g s i g n a l , or i n one of i t s derivatives, de--31-terraines t h e i r magnitude. The transient caused by a d i s -continuity i n the function w i l l be of larger magnitude than that caused by a discontinuity i n the f i r s t derivative. The voltage output response to any given input voltage may be found, as a function of time, by taking the inverse Laplace transformation of the product of the closed loop transfer function, and the transform of the input d r i v i n g s i g n a l . ( i i i ) The Choice of a Feedback System The ion current present i n a mass spectrometer that has e l e c t r o s t a t i c or magnetic f i e l d scanning may be ap-proximated reasonably well by a current pulse of the form 2 I(t) = I s i n wt , perhaps with a f l a t portion added at the peak (Fig. 7, and F i g . 8). While s i n wt i s actually continuous, the single pulse form may be obtained by sub-2 t r a c t i n g a s i n (wt-ir) wave; however, the treatment can be s i m p l i f i e d by considering only the sin^wt i n the region 0 s t < 7 r / w . As w i l l be shown l a t e r , the Laplace transform of I ( t ) i s : I (s) = j ! 1 * 2 . , (3-5) s(s z+4w z) If the system i s a type 0 or non-integrating system, the open loop transfer function may be written: G t e ^ + 1) (sT 3 + 1) , ° ( S ) = (sT 2 + l ) ( s T 4 + 1) F i g . 7; Current Pulse I(t) = I sin 2wt F i g . 9s Idealized Current Pulse - 3 3 -The system w i l l be of the general form i l l u s t r a t e d i n Fig..11. The error voltage i s : Q(s) = I(s) R - E ^ s ) and since: E ^ s ) = G(s) Q(s) t h e n : Q ( S ) , | I ^ 2 _ ( 3 . 3 ) It i s desirable, p a r t i c u l a r l y f o r absolute isotope abundance measurements, that the error voltage Q(t) be as small as possible, and zero at the peak where the measurement E^Ct) i s taken. Q(t) may be found by s p l i t t i n g Q(s) i n -to p a r t i a l f r a c t i o n s and evaluating the inverse Laplace trans-formations: R(sT 2+l)(sT 4+l) 2lw 2 Q(s) • ' ( s T 2 + D (sT 4+l) + G^ T j + l ) (sT 3+l) s(s^+4w2) 2IRw 2(sT 2+l)(sT 4+l) (s+A) (s+B) s(s 2+4w 2) v O T T >„2 ( a b cs+d e) Q(s) = 2IRw N 1 ^ ^ — (s+A s+B s 2+4w 2 s) A and B w i l l i n general be complex expressions, and a, b,c,d,e, are constants. The f i r s t two terms w i l l be damped transient terms, and the t h i r d i s a term of the form: cos 2wt + <L_ s i n 2wt . The fourth term w i l l be the constant 2w -34-e , which i s proportional to 1/G . Since e i s inde-pendent of the time, Q(t) w i l l not be zero even at the peak of the current pulse. The gain G may be increased to de-crease e , but not i n d e f i n i t e l y as the s t a b i l i t y of a system i s generally inversely proportional to the gain,, and the transient terms may become s i g n i f i c a n t for large gains. The transient terms are damped with respect to time, so i f the scanning rate were slow these terms would die out before the peak of the current pulse were reached even with a type 0 system at high gain. However, fo r very precise determinations of isotope abundances, measurements should be taken i n the shortest possible time to present constant operating con-d i t i o n s . It i s more d i f f i c u l t to provide a type 0 system that has both a small error voltage, and a fast scanning rate. With a type 1 or integrating system i t i s easier to re-solve the preceding d i f f i c u l t y . The type 1 system may also have the advantage of a shaft output that can be either used i n a d i g i t a l conversion arrangement, or read d i r e c t l y , thus removing the measuring dependence upon the more inaccurate chart recorders which most type 0 systems use. Furthermore, r a t i o recording, as mentioned i n section 3.1, may be made e n t i r e l y automatic. The system suggested by Russell and K o l l a r (Kollar, 1960) i s a type 1 system (Fig. 10). The electrometer-t r a n s i s t o r preamplifier i s designed to present a low output impedance to the chopper amplifier that follows, and the -35-chopper amplifier i s used to convert the D.C. to A.C. to drive the servo motor M . This arrangement might be re-placed by a suitable vibrating reed electrometer. The motor drives a 10 turn h e l i c a l potentiometer P that i s known to be l i n e a r at least within 0.025%. A p r e c i s e l y graduated c i r c u l a r d i a l i s fastened upon the end of t h i s potentiometer shaft and measurements may be taken d i r e c t l y from t h i s d i a l . A chart recorder may be used as an indicator by allowing the motor to drive a second potentiometer P 1 . The motor moves the potentiometer P wiper to decrease any error voltage. (iv) Remarks The l i m i t s of the gain of t h i s system w i l l be f i x e d by the following considerations. The minimum value of the gain i s that required for a given si g n a l to overcome the f r i c t i o n i n the motor and potentiometers. The maximum value w i l l depend upon the desired minimum transient response, and thus upon the scanning speed. The foregoing discussion has ignored the e f f e c t s of the capacity across R and the interelectrode capacities of the electrometer tube. These capacities add other poles and zeros to the system open loop transfer function, and may thus a f f e c t the s t a b i l i t y of the system. An example of t h i s w i l l be discussed i n the appendix. Electrometer-Transistor Preamplifier Chopper Amplifier AAA/WWW Tp Chart Recorder Fig . 10s A Type 1 Mass Spectrometer Measuring System I(s) AAAMA/VW For type 1 system of F i g . 10s G G(s) s(sT+l) K T K AE b " " I F " Motor Constant Motor Time Constant E b/B*- Voltage/Radian on Potentiometer P A •=> Amplifier Gain Figo l i s Generalized Form of a Mass Spectrometer Measuring System -37-3,3 Response of a Type 1 System (Fig. 10). (i) Response to I(t) =1 sin 2wt. It i s of interest to discuss the response only i n the region 0 S t £ TT/W, as the measurement i s confined to the ion current peak at wt = TT/2 The motor transfer function (Savant, 1958) i s of the form: K(s) K s(sT+l) so the open loop transfer function i s : G ( s ) = T^Ty <3"4) where G i s a constant (Fig. 11). In the region defined: I(t) = I sin 2wt = I(l-Cos2wt) then: (1 _ s ) (s s 2+4w 2) I(s) = I ; o—r2 K s ) - ff2l9 (3-5) s(s z+4w^) The error voltage Q(s) w i l l be: -38-Rs(sT+l) 2w2I s(sT+l)+G s(s 2+4w 2) 2w 2IR(sT+l) (s 2T+s+G)(s 2+4w 2) This may be solved by the use of p a r t i a l f r a c t i o n s or by the Laplace transformation inversion theorem. The so-lu t i o n i s : O i l ) = 2w2 ( c - B i n wit f d-cos w*t) IR wA(b^+4w2a^) where: ( g . s i n 2wt + h.cos 2wt) 2w(e 2 + f 2 ) a = -1/2T b - 4w2 + — - -2T T 2T" ( r p 2 i p ) / >J| 4 T 2 e = — - 4w2 T f = -2w/T (3-7) = -1 <<l - 2w 2 ) T (2T ) -39-h 8w 3 + w QL_ ( T 2 2G) T ) T G 4T2 1 I n g e n e r a l , Q ( t ) / I R w i l l be i n t h e o r d e r o f a few p e r c e n t . ( i i ) Remarks. W h i l e t h e a c t u a l r e s p o n s e depends upon t h e c i r c u i t p a r a m e t e r s , e q u a t i o n (3-7) p r o v i d e s some g e n e r a l c o n s i d e r -a t i o n s , (a) The e r r o r v o l t a g e Q ( t ) i s i n t h e f o r m o f a damped h i g h f r e q u e n c y t r a n s i e n t s u p e r i m p o s e d on a s i n e wave o f p e r i o d 2w. (b) The damping o f t h e t r a n s i e n t d e pends m a i n l y upon t h e motor c o n s t a n t T w h i c h s h o u l d be as s m a l l as p o s s i b l e . The g a i n w i l l , i n p a r t , d e t e r m i n e t h e magni-t u d e o f t h e t r a n s i e n t . I f G becomes l a r g e e q u a t i o n (3-7) r e d u c e s t o : T h i s e r r o r r e s p o n s e i s i l l u s t r a t e d i n F i g u r e 7. S e v e r a l a d d i t i o n a l p o i n t s become a p p a r e n t . Under t h e above a s s u m p t i o n : (a) t h i s s y s t e m , n e g l e c t i n g f r i c t i o n and w i t h a r e a s o n a b l e s c a n r a t e , g i v e s a z e r o e r r o r v o l t a g e a t t h e c u r r e n t peak where t h e measurements a r e t a k e n : (b) t h e e r r o r v o l t a g e i s i n v e r s e l y p r o p o r t i o n a l t o t h e peak s c a n n i n g t i m e j and, ( c ) t h e e r r o r v o l t a g e w i l l d e c r e a s e w i t h i n -c r e a s i n g g a i n G Q ( t ) = — s i n 2wt G w (3-8) IR -40-Measurements c o n d u c t e d by R.D. R u s s e l l and F. K o l l a r have i n d i c a t e d t h a t t h e e r r o r v o l t a g e o f t h e i r s y s t e m i s s i m i l a r t o t h a t o f e q u a t i o n ( 3 - 8 ) . T h e s e measurements a l s o i n d i c a t e t h a t t h e e r r o r v o l t a g e a t t h e peak i s n o t e x a c t l y z e r o , b e c a u s e o f t h e f r i c t i o n i n t h e motor and p o t e n t i o m e t e r . ( i i i ) R e s p o n s e t o I ( t ) = I s i n 2 w t w i t h a f l a t t e n e d p e a k . S i n c e t h e measurements w h i c h d e t e r m i n e t h e i s o t o p e a b u n d a n c e s a r e t a k e n a t t h e t o p o f t h e c u r r e n t p u l s e , an i d e a l i z e d p u l s e s h a p e w o u l d be t h a t w h i c h p o s s e s s e s a f l a t t o p and s i n ^ w t s i d e s . The a n a l y t i c e x p r e s s i o n w h i c h g i v e s t h i s s h a p e i s t h e sum o f a s i n ^ w t p u l s e , and a c o s wt p u l s e t h a t b e g i n s a t wt = T T/2 . The e r r o r v o l t a g e Q ( t ) w i l l be t h e s u p e r p o s i t i o n o f t h e e r r o r s due t o e a c h p u l s e , and i s o f t h e f o r m i l l u s t r a t e d i n F i g u r e 8. A g a i n , n e g l e c t i n g f r i c t i o n , t h e e r r o r v o l t a g e w i l l be z e r o a l o n g t h e p u l s e t o p . A w i d e p u l s e t o p i s d e s i r a b l e f r o m t h e p o i n t o f v i e w o f p r e c i s e abundance measurements., ( i v ) R e s p o n s e t o a t r a p e x o i d a l c u r r e n t p u l s e . A c u r r e n t p u l s e o f t h i s f o r m i s i l l u s t r a t e d i n F i g u r e 9. The e r r o r v o l t a g e w i l l c o n s i s t o f damped t r a n s i e n t s o f p e r i o d w 1 , b e g i n n i n g a t wt = 0, A, B, and C, s u p e r -i m p o s e d on s t e a d y s t a t e e r r o r s a l o n g t h e l e a d i n g and t r a i l i n g s i d e s . I f t h e t r a n s i e n t s a r e h i g h l y damped t h e e r r o r v o l t a g e w i l l be z e r o , n e g l e c t i n g f r i c t i o n , a l o n g t h e p u l s e t o p . -41-(v) Conclusions. A current pulse of the form shown i n Figure 7 i s probably more common for higher mass measurements, while Figure 8 i s probably representative of the low masses. From the point of view of precise measurements the l a t t e r pulse shape i s more desirable. This shape might be obtained either through an improvement i n the ion beam optics, or by varying the scanning rate for the d i f f e r e n t parts of the pulse. While Figure 9 represents the assumed i d e a l pulse shape, t h i s possesses no apparent advantage over that shown i n Figure 8. The response c h a r a c t e r i s t i c s of a type 1 system appear to be better, generally, than that of a type 0 system. A greater precision of measurement, and a f a s t e r scanning rate, also appear to be possible with the type 1 system. Typical examples of systems that include servo motors are worked out i n considerable d e t a i l i n Savant (1958), and Ahrendt and Savant (1960). 42-CONCLUSIONS Most modern mass spectrometer measuring systems are of the general form i l l u s t r a t e d i n Figure 11. C i s the Faraday cup which c o l l e c t s the ion current I(s) , G(s) i s the detector and measuring system, R-^  a low value r e s i s t o r , and R a dropping r e s i s t o r whose value i s i n the order of lO** ohms. It i s shown i n Chapter 1 that the very large input resistances, both D.C. and incre-mental, of either electrometer tubes or v i b r a t i n g reed electrometers ensure that I(s) flows e n t i r e l y through R . The voltage drop I(s)R across t h i s r e s i s t o r i s thus a measure of the ion current. In a l l previous measuring systems R has been assumed approximately constant for a l l voltages E developed across i t s termi-nals; that i s , the voltage c o e f f i c i e n t dR/RdE was con-sidered to be n e g l i g i b l e . However, i n making a very precise measurement of the ion current, t h i s change of resistance with voltage should be known so that, i f necessary, a correction may be applied. The method described i n Chapter 2 was developed to determine t h i s voltage c o e f f i c i e n t , and a summary of the r e s u l t s may be found i n the r e p r i n t which i s enclosed at the back of t h i s thesis (Whittles, 1960). The r e s i s t o r , whose voltage c o e f f i c i e n t i s to be measured, i s placed -43-i n a Wheatstone bridge and the output of the bridge measured by a s p e c i a l l y constructed electrometer tube detector (Fig. 2). For small voltages, the voltage c o e f f i c i e n t i s found to vary in a l i n e a r fashion (Fig. 4) from -0.003% per v o l t to -0.016% per v o l t for the 1 0 1 0 ohm r e s i s t o r , and s l i g h t l y higher values for the 1 0 1 1 ohm r e s i s t o r . These values are con-siderably below the manufacturer's nominal values of -0.02% per v o l t f o r the 1 0 1 0 ohms, and -0.03% per v o l t f o r 1 0 1 1 ohms. The voltage c o e f f i c i e n t i s probably a r e s u l t of the known anisotropic c r y s t a l packet structure of the p y r o l y t i c carbon f i l m of the r e s i s t o r , and the semi-conductor behaviour of carbon suggests i t s l i n e a r form. At the present stage of development of mass spectrometer measuring systems, the correction necessitated by the v o l t -age c o e f f i c i e n t i s probably n e g l i g i b l e ; however, the method of i t s determination could be applied more widely to a study of the voltage c o e f f i c i e n t s and other n o n - l i n e a r i t i e s of high value r e s i s t o r s . From the discussions of section 3.1 and 3.2, i t becomes evident that a feedback system i s necessary for very precise ion current measurements. A general feedback system i s i l l u s t r a t e d i n Figure 11. I(s) i s the Laplace transform of the input ion current, and I(s)R i s the input s i g n a l . E ^ s ) i s the Laplace transformation of the output voltage across Rj , Q(s) i s the difference of I(s)R and E (s) . G(s) i s c a l l e d the open loop transfer function -44-and i s t h e r a t i o o f E-^Cs) t o Q ( s ) . The c l o s e d l o o p t r a n s f e r f u n c t i o n i s G ( s ) / 1 + G ( s ) and t h e r a t i o E-^(s) t o I ( s ) R , Two g e n e r a l t y p e s o f f e e d b a c k s y s t e m s a r e p r e s e n t -l y i n u s e as mass s p e c t r o m e t e r m e a s u r i n g s y s t e m s , t h e t y p e 0 o r n o n - i n t e g r a t i n g s y s t e m , and t h e t y p e 1 o r i n t e g r a t i n g s y s t e m . T h e s e s y s t e m s a r e a n a l y s e d i n s e c t i o n s 3.2 and 3.3 and t h e e r r o r r e s p o n s e i s f o u n d , as a f u n c t i o n o f t i m e , by d e t e r m i n i n g t h e i n v e r s e L a p l a c e t r a n s f o r m o f Q(s) S i n c e i n t h e a c t u a l measurements E-^(s) i s t a k e n t o be t h e i o n c u r r e n t v o l t a g e I ( s ) R , t h e n Q ( s ) = I ( s ) R - E 1 ( s ) s h o u l d be as n e a r t o z e r o as p o s s i b l e , p a r t i c u l a r l y f o r a b s o l u t e i s o t o p e abundance measurements. T h r e e p o s s i b l e i n p u t i o n c u r r e n t p u l s e s a r e g i v e n on page 32, and t h e e r r o r v o l t a g e s f o r t h e t y p e 1 s y s t e m g i v e n i n F i g u r e 10 have been computed. T h e s e e r r o r v o l t a g e s a r e f o u n d t o be z e r o , neg-l e c t i n g f r i c t i o n , a l o n g t h e p u l s e t o p where t h e measure-? ments a r e t a k e n . The e r r o r i s u s u a l l y i n t h e o r d e r o f a few p e r c e n t o f t h e i n p u t . The t y p e 0 s y s t e m i s u n d e s i r a b l e f o r a b s o l u t e measurements as i t w i l l have a f i n i t e e r r o r v o l t a g e a t t h e p u l s e t o p . F o r r e l a t i v e i s o t o p e abundance measurements, on t h e o t h e r hand, e i t h e r s y s t e m i s s u i t a b l e ; however, t h e t y p e 1 s y s t e m has t h e a d d i t i o n a l a d v a n t a g e o f a s h a f t o u t p u t t h a t c a n e i t h e r be u s e d i n a d i g i t a l c o n v e r s i o n a r r a n g e m e n t , o r r e a d d i r e c t l y , t h u s a v o i d i n g t h e i n h e r e n t l i m i t a t i o n s o f a c h a r t r e c o r d e r . Most t y p e 0 s y s t e m s u s e a c h a r t r e c o r d e r - 4 5 -to determine E-^Cs) (Fig. 3). Finally, the type 1 system may be used with a faster scanning rate than the type 0, and, as mentioned in section 3.1, ratio recording may be made entirely automatic. In general, the analytical calculations and answers are rather complex, as equation 3-7, page 38, suggests. The analogue computer can be used to avoid these lengthy calculations as on this instrument the answer wi l l be provided in the form of a voltage; this second approach is discussed in the Appendix. -46-BIBLIOGRAPHY Ahrendt W.R., and C.J. Savant Jr. (1960); Servomechanism  Practice. McGraw-Hill Book Company, Inc. Aston F.W. (1925); Mass Spectra and Isotopes. Edward Arnold and Co., London (1933). Aston F.W. (1930); The Photometry of Mass Spectra and the Atomic Weight of Krypton, Xenon, and Mercury, Proc. Roy. Soc., V126, 511. Bennet P.D. (1930); An Electrometer Tube, Rev. S c i . Inst., VI, 466. Bradley F.R., and R.McCoy (1952); D r i f t l e s s D.C. Amplifier, E l e c t r o n i c s , V25, 144-8. Duckworth H.E. (1958); Mass Spectroscopy, p. 52, Cambridge University Press. Gabus G.H., and M.L, Pool (1937); A Portable Phototube Using an R.C.A. 954 Tube, Rev. S c i . Inst., V8, 196. Goldstein (1886), B e r l . Ber., V39, 691. Grisd a l R.O., A.C. P f i s t e r , and W. van Roosbroeck (1951); P y r o l y t i c Film Resistors: Carbon and Borocarbon, B e l l System Technical Journal, V30, 271-314. Gunn R. (1932); P r i n c i p l e s of a New Portable Electrometer, Phy. Rev., V40, 307. Herman P.J., K.H. Starks, and J.A. Rudolph (1956); Basic Applications of Analog Computers, Instruments and Automation, V29, 464. Hippie J.A., D.J. Grove, and W.M. Hickam (1945); E l e c t r o n i c Problems Involved i n the P r a c t i c a l Application of the Mass Spectrometer, Rev. S c i . Inst., V16, 69-75. Huntoon R.D. (1935); An Inexpensive D-C. Amplifier, Rev. S c i . Inst., V6, 322. Johnson C.L. (1956); Analog Computer Technique, McGraw-Hill Book Company, Inc. -47-K o l l a r F. ( I 9 6 0 ) : T h e P r e c i s e I n t e r c o m p a r i s o n o f L e a d I s o t o p e R a t i o s , Ph.D. T h e s i s , D e p a r t m e n t o f P h y s i c s , U n i v e r s i t y o f B r i t i s h C o l u m b i a . K o r n G.A., and T.M. K o r n ( 1 9 5 2 ) ; E l e c t r o n i c A n a l o g C omputers , M c G r a w - H i l l Book Company, I n c . M e t c a l f G.F., and B . J . Thompson ( 1 9 3 0 ) ; A Low G r i d C u r r e n t Vacuum Tube, P h y s . Rev., V36, 1489. N e l s o n H. ( 1 9 3 0 ) ; An E l e c t r o m e t e r Tube, Rev. S c i . I n s t . , V I , 281. N i e r A.O., E.P. Ney, and M.G. Ingham ( 1 9 4 7 a ) : A N u l l Method f o r t h e C o m p a r i s o n o f Two I o n C u r r e n t s i n a Mass S p e c t r o m e t e r , Rev. S c i . I n s t . , V18, 294-297. N i e r A.O. ( 1 9 4 7 b ) j A Mass S p e c t r o m e t e r f o r I s o t o p e and Gas A n a l y s i s , Rev. S c i . I n s t . , V18, 398-411. N o t t i n g h a m W.B. ( 1 9 3 0 ) ; Measurement o f S m a l l D.C. P o t e n t i a l s and C u r r e n t s i n H i g h R e s i s t a n c e C i r c u i t s by U s i n g Vacuum T u b e s , J o u r , o f t h e F r a n k l i n I n s t i t u t e , V209, 287-348. ~ ~ ~ P a l e v s k y H., R.K. Swank, and R. G r e n c h i k ( 1 9 4 7 ) ; D e s i g n o f Dynamic C o n d e n s o r E l e c t r o m e t e r s , Rev. S c i . I n s t . , V18, 298. ~ ~ S a v a n t C . J . ( 1 9 5 8 ) ; B a s i c F e e d b a c k C o n t r o l S y s t e m D e s i g n , M c G r a w - H i l l Book Company, I n c . S m i t h P.T. , W.W. L o z i e r , L.G. S m i t h , and W. B l e a k n e y ( 1 9 3 7 ) ; A H i g h S e n s i t i v i t y Mass S p e c t r o m e t e r w i t h an A u t o m a t i c R e c o r d e r , Rev. S c i . I n s t . , V8, 51-55. S o r o k a W.W. ( 1 9 5 4 ) ; A n a l o g Methods i n C o m p u t a t i o n a n d  S i m u l a t i o n . M c G r a w - H i l l Book Company, I n c . S t r a u s H.A. ( 1 9 4 1 ) ; A New Mass S p e c t r o m e t e r and t h e I s o t o p i c C o m p o s i t i o n o f N i c k l e , P h y s . Rev., V59, 430-3. S t r o n g J . ( 1 9 5 6 ) ; P r o c e d u r e s i n E x p e r i m e n t a l P h y s i c s , P r e n t i c e - H a l l , I n c . , ( 1 9 3 8 ) . Suydam V.A. (1948) ; F u n d a m e n t a l s o f E l e c t r i c i t y and Magnetism, D. V a n N o s t r a n d Company, I n c . T a y l o r D.D. ( 1 9 3 5 ) ; A M o d i f i e d A s t o n - T y p e S p e c t r o m e t e r and Some P r e l i m i n a r y R e s u l t s , P h y s . Rev., V47, 666-71. -48-Thomson J.J. (1913); Rays of Po s i t i v e E l e c t r i c i t y , Longmans Green and Co., London. Urey H.C., C.R. McKinney, J.M. McCrea, S. Epstein, and H.A. Al l e n (1950); Improvements i n Mass Spectrometers for the Measurement of Small Differences i n Isotope Abundance Ratios, Rev. S c i . Inst., V21, 724-30. Wanless R.K., and H.G. Thode (1953); A Mass Spectrometer for High Precision Isotope Ratio Determinations, Jour, S c i . Inst. (London), V30, 395-8. Wass C.A.A. (1955); Introduction to E l e c t r o n i c Analog Computers, McGraw-Hill Book Company, Inc. Whittles A.B.L. (1960); Voltage C o e f f i c i e n t of Victoreen High-Meg Resistors, Rev. S c i . Inst., V31, 208-9. Williams T.R., and J.B. Thomas (1959); Current Noise and Non-linearity i n P y r o l y t i c Carbon Films, Rev. S c i . Inst., V30, 586-90. -49-APPENDIX The Analogue Computer i n Servo System Design A«l. Introduction. Although the analogue computer may be used to solve many types of e l e c t r i c a l , mechanical, and mathematical problems, i t i s probably most widely used i n the design of servo systems. A high gain (G= 50,000) D.C. amplifier, with R-C input and feedback networks, constitutes the basic element of an analogue computer. This arrangement i s c a l l e d an operational amplifier, and i s i l l u s t r a t e d i n Figure 12. If one assumes that the D.C. amplifier has.a very high input impedance so that a l l the input current flows through Z i and Zj- , and that the gain G i s large, then i t can be shown that the transfer function of the system i s : G(s) - ~ Z f ( s ) (a) Z i ( s ) By the use of various simple input and feedback net-works i t i s possible to simulate the transfer functions of many d i f f e r e n t systems. Suppose, for example, that we have a system whose transfer function has two poles. -50-S = -1/Tj , - 1 / T 2 • T o simulate t h i s system on the analogue computer one may set up the operational amplifier shown i n Figure 13. The transfer function can then be found by determining the impedances of the networks; Z ± ( t ) - R x + -J: so; Zi(s) = R X + _L = 1 + s T l and: Z f (s) - R2^/sC2) _ R 2 R 2+l/sC 2 1+sT so; 2 G (s) - E ° < S > " E ± ( s ) (l+sT-jMl+sTg) where: T j = R i ^ ; T 2 = R 2C 2 j and, T 3 = R 2 c i The transfer impedance functions of many d i f f e r e n t networks have been tabulated by Bradley and McCoy (1952), and a few of these are i l l u s t r a t e d i n Figure 14. As an example of how these tables may be used, consider an amplifier-motor transfer function: G(s) s(sT+l) If the network i l l u s t r a t e d i n Figure 14 (d) i s used f o r -51-G(s) = E o ( s ) - - M f > E ^ s ) Z ±(s) Fig . 12: Generalized Operational Amplifier i 1C2 E H S A A A A / W W V ) R 2 •AAAAAMAAr E, F i g . 13: Simple Two Pole System A / V W W W v ° R (a) R (b) 1 sC R (c) R s(RC)+l (d) 2R 1 + SRC R 0 - A A A A A A / V -(e) sRC + 1 sG Network Transfer Function Fi g . 14: Some Network Impedance Transfer Functions -52-Z-^  , and that i n Figure 14 (b) f o r Z^ , the operational amplifier may be set up as i n Figure 15 (a). The transfer function of t h i s arrangement may be found from equation (a): - Z f ( s ) _ - l / s C 2 G ( S ) " Z ±(s) ~ 2R 1(l+sT 1) or: G ( S ) = s ( s T 1 + l ) where: T =» ^ l ^ l j G = — 1 2 For a given motor time constant T^ , and motor constant K , one may choose appropriate values for R^ , , and G 2 • C 2 may be used to change the "gain" of the "amplifier". The system i l l u s t r a t e d in Figure 15 (b) w i l l also give an amplifier-motor transfer function, but the various constants w i l l be d i f f e r e n t . One other operational amplifier should be mentioned here (Fig. 16). The summing amplifier i s used for the c i r c u l a r symbol described on page 30. A.2. The Response of a System. With most feedback or servo systems one i s usually interested i n determining the response to various input functions. As an example of how the analogue computer may -53-R l R l I (a) o A A A / W W R< w w v £ (b) F i g . 15: S i m u l a t e d A m p l i f i e r - M o t o r T r a n s f e r F u n c t i o n s R l A / W W W R 2 A / W W W R 3 A W W W W E o = -R 3 E 1 + R 3 E 3 Ri F i g . 16: Summing A m p l i f i e r -54-be u s e d t o q u i c k l y o b t a i n t h i s answer, c o n s i d e r t h e a m p l i -f i e r - m o t o r s y s t e m i n F i g u r e 10. I f t h e i n p u t v o l t a g e i s a s t e p f u n c t i o n i t i s p o s s i b l e t o show a n a l y t i c a l l y t h a t t h e o u t p u t i s a damped t r a n s i e n t t e r m s u p e r i m p o s e d upon a s t e p f u n c t i o n ( F i g . 1 7 ) . The s y s t e m i s s i m u l a t e d on t h e computer as i l l u s t r a t e d i n F i g u r e 18. i s t h e summing a m p l i f i e r and s e r v e s t o i s o l a t e t h e f e e d b a c k s i g n a l f r o m t h e s i g n a l g e n e r a t o r o u t -p u t . T he i n v e r t i n g a m p l i f i e r G 2 i s i n c l u d e d i n t h e l o o p , as i t i s n e c e s s a r y t o i n v e r t t h e s i g n a l t o p r o v i d e n e g a t i v e f e e d b a c k . G3 and n e t w o r k s r e p r e s e n t t h e a m p l i f i e r , m otor, and p o t e n t i o m e t e r . The s i g n a l g e n e r a t o r p r o v i d e s a s e r i e s o f p o s i t i v e and n e g a t i v e s t e p s ( a s q u a r e wave) a b o u t 50 msec, a p a r t . and E Q may be d i s p l a y e d on a d u a l beam o s c i l l o s c o p e . T he r e s u l t s a r e t h o s e i l l u s t r a t e d i n F i g u r e 17. T h e v a r i o u s s p e c i f i c a t i o n s o f t h e s e r v o s y s t e m ( o v e r -s h o o t , s t e a d y s t a t e e r r o r s , e t c . ) may be o p t i m u m i z e d b y v a r y i n g t h e c i r c u i t p a r a m e t e r s . I f one w i s h e s t o t e s t t h e r e s p o n s e t o o t h e r t y p e s o f i n p u t f u n c t i o n s , s u c h a s t h e c u r r e n t p u l s e s d e s c r i b e d i n s e c t i o n 3.3, i t i s n e c e s s a r y t o c o n s t r u c t a f u n c t i o n g e n e r a t o r . E ± ( t ) / \ ! \ -55-X7 • * < — / F i g . 17: R e s p o n s e o f an A m p l i f i e r - M o t o r S y s t e m t o a S t e p I n p u t V o l t a g e 1 m rAAAAAAAAAn 1 m E . <f W\AAAM/V 1 m rAAAAAAAAAn 1 m AAAAAA/VW-U^)  S i g n a l V J G e n e r a t o r 1 m 0.0005 mfd 100 K 100 K >AAAAA/WW\AAAAJ 7LZ 0.01 mfd 7 E, F i g . 18: S i m u l a t e d A m p l i f i e r - M o t o r S y s t e m -56-A.3. Equalization. (i) Introductions Root Locus Diagrams. As mentioned previously, the input capacities w i l l a f f e c t the s t a b i l i t y of the system. The degree of i n -s t a b i l i t y may be controlled to some extent by decreasing the gainj however, the gain must be s u f f i c i e n t l y high to meet the steady state error requirements, or to overcome the f r i c t i o n in the motor and potentiometers. To e f f e c t t h i s compromise i t i s usually necessary to change the system's open loop transfer function either by the i n t r o -duction of an equalization network or by providing additional feedback paths. Before either of these possi-b i l i t i e s can be discussed, i t i s necessary to introduce the concept of the "root-locus diagram". Consider a system with the open loop transfer function G(s) . If the roots of the expression G(s)+1 = 0 are determined for G £0 , and plotted i n the s-plane, the r e s u l t i s a root locus diagram. The roots are generally complex: s = a+jw . For example, the open loop transfer function of the amplifier-motor system i l l u s t r a t e d i n Figure 10 i s : G(s) s(sT+l) This system has poles at s = 0 , - l / T , and no zeros. A plot of the roots of G(s)+1 - 0 , f o r G>0 , w i l l be of the form (Savant, 1958, page 83): - 5 7 -X = poles / / / / 0 = zeros / / / / F i g . 19: A m p l i f i e r - M o t o r Root L o c u s P l o t The a r r o w s i n d i c a t e t h e d i r e c t i o n o f i n c r e a s i n g g a i n G The p o l e s o f t h e s y s t e m a r e t h e r o o t s o f G ( s ) + 1 = 0 , f o r G = 0 . I f a r o o t l o c u s p l o t e n t e r s t h e s h a d e d a r e a t o t h e r i g h t o f s = 0 ( a > 0) , t h e s y s t e m , f o r t h a t g a i n , w i l l o s c i l l a t e when t h e f e e d b a c k l o o p i s c o n n e c t e d . I n t h i s r e g i o n t h e t r a n s i e n t t e r m s a r e n o t damped, b u t i n c r e a s e e x p o n e n t i a l l y w i t h t i m e . P o i n t s t o t h e l e f t o f s =» 0 ( a < 0) a r e s t a b l e o p e r a t i n g p o i n t s , and t h e t r a n s i e n t s i n t h i s r e g i o n w i l l be e x p o n e n t i a l l y damped w i t h t i m e . The m a g n i t u d e o f t h e s e damped t r a n s i e n t t e r m s w i l l i n c r e a s e w i t h t h e g a i n . I f t h e i n p u t c a p a c i t i e s a r e n o t s i g n i f i c a n t t h e a m p l i f i e r - m o t o r s y s t e m i s a l w a y s s t a b l e ( F i g . 1 9 ) . The i n p u t c a p a c i t i e s o c c u r a c r o s s R , and f r o m t h e e l e c t r o m e t e r g r i d t o g r o u n d ( F i g . 2 0 ) . The t r a n s f e r f u n c t i o n d i a g r a m o f t h i s s y s t e m i s g i v e n i n F i g u r e 21. The p o s s i b l e r o o t l o c u s p l o t s h a v e b e e n c o n s t r u c t e d by F. K o l l a r and R.D. R u s s e l l , and a r e i l l u s t r a t e d i n -58-K s ) . Q(s) s(sT+l) E Q ( s ) R '1 C2 Fig. 20: Input Capacities Ei(s)=I(s)R i Q(s) G s ( s T m + l ) ( s T 3 + l ) j Eo<s) sTj+l T l " R 1 C 1 T2 " R 2 C 2 T 3 " T l + T 2 F i g . 21: The Transfer Function Diagram of a Motor-Amplifier System with Input Capacities -59-F i g u r e 22 (a) and ( b ) . S i m i l a r p l o t s may be f o u n d i n S a v a n t (1958, page 9 8 ) . Suppose t h a t t h e i n p u t c a p a c i t i e s have c a u s e d t h e s y s t e m t o r e s e m b l e t h a t r e p r e s e n t e d by F i g u r e 22 ( a ) , w h i c h i s u n s t a b l e f o r h i g h e r g a i n s . I t w o u l d be d e s i r a b l e t o change t h i s s y s t e m i n t o e i t h e r o f t h o s e r e p r e s e n t e d b y F i g u r e 19, o r F i g u r e 22 ( b ) , as t h e s e s y s t e m s w o u l d be s t a b l e f o r a l l g a i n s . T h i s c hange may be a c c o m p l i s h e d by t h e i n t r o d u c t i o n o f an e q u a l i z a t i o n n e t w o r k , o r by t h e u s e o f v e l o c i t y f e e d b a c k . ( i i ) E q u a l i z a t i o n Network. To p r o v i d e s t a b l e o p e r a t i n g c h a r a c t e r i s t i c s one may i n s e r t i n t o t h e s y s t e m a n e t w o r k w h i c h adds a d d i t i o n a l p o l e s a n d z e r o s t h a t c a n be a d j u s t e d t o c a n c e l T± and T 3 . A g r o u p o f p o s s i b l e D.C. n e t w o r k s i s g i v e n i n F i g u r e 23. I f t h e e q u a l i z a t i o n n e t w o r k i s t o be i n s e r t e d i n t o an A.C. p a r t o f t h e f e e d b a c k l o o p an e q u i v a l e n t A.C. n e t w o r k must be u s e d ( S a v a n t , 1958, page 2 2 4 ) . The l a t t e r two n e t w o r k s , F i g u r e 23 ( c ) and ( d ) , w o u l d n o r m a l l y be u s e d o n l y t o s t u d y t h e p r o b l e m on an a n a l o g u e c o m p u t e r . ( i i i ) V e l o c i t y F e e d b a c k . I n v e l o c i t y f e e d b a c k a s i g n a l , p r o p o r t i o n a l t o t h e s h a f t v e l o c i t y o f t h e m o t o r , i s f e d b a c k t h r o u g h an a m p l i -f i e r t o t h e i n p u t o f t h e m otor ( F i g . 2 4 ) . T h i s s i g n a l may be o b t a i n e d e i t h e r by t h e u s e o f a t a c h o m e t e r , w h i c h i s e s s e n t i a l l y a v o l t a g e g e n e r a t o r , o r by u t i l i z i n g t h e b a c k E.M.F. o f t h e m otor ( K o l l a r , 1 960), - 6 0 -F i g . 2 2 ; R o o t - L o c u s D i a g r a m s f o r M o t o r - A m p l i f i e r S y s t e m w i t h I n p u t C a p a c i t i e s -61-•AAAAAA/WV-G(s) -a C s T ^ l ) (sT^a+l) Rr a = R l + R 2 Rl o V W v V v W v (a) (b) T l ~ R 1 C 1 G(s) = To = (sT 2+l) a *2 R 1 + R 2 R 2 C 2 C l Ri » * W v W W V V R 2 A A A A A A A A A H ( C ) Ro (sTn+1) G( S ) = - 2 1 R x (ST2+1) T l ~ R 1 C 1 T 2 . - R 2 C 2 R 2 rAAAAAAA-9 ^ / v W W W V -G(s) -C x (sT 2+l) ( s T ^ l ) -o TT = R 1 C 1 R 2 C 2 (d) F i g . 23; Some D . C . Equalization Networks -62-This problem may be set up on the computer as i n Figure 25. G-^  and G 3 are summing amplifiers, G 2 re-presents the amplifier and the single pole, G 4 the motor, Gg the v e l o c i t y feedback amplifier, and G g the single zero i n the feedback loop. R^ controls the amplifier gain A , and Cg the gain B of the v e l o c i t y feedback ampli-f i e r . The values chosen were: » 0.5 mmfd., C 2 = 4 mmfd., R •=» 1 0 1 1 ohms, T - 0.065 seconds, and K = 10 v o l t sec." 1 radians" 1. Since the system: s(sT + 1) may be replaced by: Bs K BK + 1 s sT 4. 1 BK + T -L 1 then as B i s increased the new motor time constant T/BK + 1 w i l l decrease, and the system w i l l become that represented by Figure 22 (b). • • « • 0 « • - 6 3 -(s) R Y STQ+1 K s(sT+l) Bs E Q ( s ) _L sTj+l Fi g . 24: Amplifier-Motor System with V e l o c i t y Feedback 0.5 mfd 1 m r - A A A A A / V i lm p w w w 100 K - w v w v w v -100 K r V V v N A A A A / V i 100 K 100 K 0.5 mfd rVWWWf 100 K v w w w 0.05 mfd E Q ( s ) 1 m 1 m A A / W W W f A A A A / W V 1 0.12 mfd Fi g . 25: V e l o c i t y Feedback on the Analogue Computer 6 4 -The analogue computer may be put to many other uses the solutions of mechanical problems, d i f f e r e n t i a l , d i f f e r e n t i a l - i n t e g r a l , l i n e a r and non-linear equations, etc. These uses have been described by Herman et a L (1956), Wass (1955), Soroka (1954), Korn (1952), Johnson (1956), and many others. A*4 GohclusiOns Probably the chief advantage in using the analogue computer i n the design and study of a servo system i s the readiness with Which one may come to understand the o v e r a l l properties of the system, without being d i s -tracted by the large amount of technical d e t a i l necessary i n i t s construction* The response of the system to a given input si g n a l may be obtained without the long c a l c u l a t i o n an a n a l y t i c a l answer often requires. A rough root-locus diagram may be made very quickly• f o r a second order system (two poles) the exact rbOt-locus plot may be determined. Equalization networks may be tested. Various changes i n the system's open loop transfer function may be studied and developed* A l l these p o s s i b i l i t i e s , and many others, can be effected by simply changing or adding simple R=d networks* The aid provided by the analogue computer i n the design of a servo system amply repays the small e f f o r t required to learn i t s Use. -65-The p r e c e d i n g e x p e r i m e n t s were c a r r i e d o u t on a H e a t h k i t E l e c t r o n i c A n a l o g Computer w h i c h was k i n d l y p r o v i d e d by Dr. A.M. C r o o k e r o f t h e P h y s i c s D e p a r t m e n t . Reprinted f rom T H E REVIEW OF SCIENTIFIC INSTRUMENTS, V o l . 31, N o . 2, 208-209, February, 1960 Printed in U. S. A. Voltage Coefficient of Victoreen High-Meg Resistors A. B. L. WHITTLES Geophysics Laboratory, Department of Physics, University of British Columbia, Vancouver, British Columbia (Received September 14, 1959; and in final form, December 7,1959) TH E Geophysics Laboratory at this University has been constructing mass spectrometer facilities with the object of measuring natural variations in the isotope ratios of lead with a higher precision than has been achieved previously. In the course of this work, we under-took to measure the voltage coefficient of the 1010- and 10n-ohm Victoreen resistors which were used in one of our instruments, since the nominal voltage coefficients for the range 1-100 v given in the manufacturer's literature (0.02% and 0.03%/v) were high enough so that they could have been a significant factor in our measurements. Since information about the voltage coefficients of these resistors does not seem to be readily available in the litera-ture, the results are reported here. It should be noted, however, that only one of each of the resistors was tested, and the results may not be typical of this class of resistors. Considerable work has been carried out by Williams and Thomas 1 , 2 on current noise and nonlinearity of deposited carbon resistors in the 104-ohm range.-Although the re-sistors are very different in size and construction, the non-linearity found was, similar to that found in -the present experiment. The measurements were carried out by connecting two Victoreen high-meg resistors (R and Rl) in series to form two arms of a Wheatstone bridge, and two precision re-sistors to form the other two arms. One of the Victoreen resistors was chosen to have approximately ten times the resistance of the other, so that the voltage across the bridge appeared predominantly across the larger resistor; thus, the voltage coefficient of the smaller of the two could be neglected or else included in a minor correction. Although our interests were in the range less than 5 v, we made meas-urements up to 34.5 v, in order to obtain some information about the behavior of these resistors at higher voltages. For the detector we used the circuit shown in Fig. 1. The CRO was a Tektronix type 502, with the differential input of one trace used as a vacuum tube voltmeter. The sensi-tivity was high enough to detect a lO^tv signal. A l l neces-sary precautions were taken to minimize the effect of drift, leakage, microphonics, and photoelectric currents. In carrying out the measurements the bridge was balanced for an arbitrary bridge voltage V. Then the bridge voltage was reduced by 6 v and the resulting error voltage e measured on the oscilloscope. If one assumes that FIG. 1. Circuit of bridge and detector.. 2 N O T E S O 6 10 15 20 25 30 35 VOLTS F I G . 2. Voltage coefficient of sample 1010- and 10 I l-ohm resistors. the voltage coefficient is small , that the bridge is always near balance, and that the detector does not load the bridge, i t can be shown that the average value of the vol t -age coefficient [here denned as 1/R (dR/dE)~] over the 6-v range is given by 1 dR (Ri+RiY e 100 RdE J?! 2i? 2 G 6 ( 7 - 6 ) A t balance: F = t h e ini t ia l bridge voltage, E=voltage across R, i?ii?2 = value of the two low resistance arms, wi th 2?i/l?2=10 approximately i n this experiment, G = g a i n of the detector, G=0.61 w i t h the circuit used. The results obtained are il lustrated i n F i g . 2. I n plott ing the graph, the measured voltage coefficient was assigned to the middle of each 6-v range. The voltage coefficient was found to vary in a linear fashion from — 0.0025%/v to —0.016%/v for the 10 1 0-ohm resistor over the voltage range 0-32 v ; i t varied between slightly larger values for the 10 u -ohm resistor. I n this voltage range, the voltage coefficient is much below the manufacturer's nominal value for the 1- to 100-v range; however, we d id obtain approxi-mately the manufacturer's value for the resistance change from 1 to 100 v . The advice of Professor R . D . Russell and M r . F . K o l l a r in making this measurement is acknowledged w i t h thanks. M r . D. O. W a r d , Victoreen Instrument Corporat ion, read this manuscript and contributed helpful suggestions. The mass spectrometer project is financed b y the N a t i o n a l Research Counci l of Canada and the Geological Survey of Canada. 1 T . R . Williams and J . B . Thomas, Rev. Sci. Instr. 30, 586 (1959). 2 T . R . Williams and J . B . Thomas, I R E Trans, on Componet Parts 5, 151 (1958). 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0103737/manifest

Comment

Related Items