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Application of feedback to electromagnetic seismometers 1965

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AN APPLICATION OF FEEDBACK TO - ELECTROMAGNETIC SEISMOMETERS by .ROBERT DAVID MELDRUM B.Sc, University, of B r i t i s h Columbia, 1963 A-THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIRMENTS FOR THE DEGREE OF MASTER OF SCIENCE in.the Department of GEOPHYSICS We accept t h i s thesis.as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1965 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 o f the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o lumbia, I a g r e e t h a t t h e 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 s t u d y . I f u r t h e r a g r e e 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s * I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i - c a t i o n o f 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 a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Geophysics The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date October, 1965 i i ABSTRACT . Negative feedback can be applied to an e l e c t r o - magnetic seismometer through, the co i l , terminals using a Max- well impedance bridge to bypass the c o i l impedance. The application.of feedback by t h i s method obviates the need f o r a second transducer, and permits d i r e c t c a l i b r a t i o n of the feedback seismometer. Feedback of a simulated ground acceleration pro- portional to^the acceleration of the suspended mass e f f e c t i v e - l y increases the mass, lengthening the natural period of the seismometer. Displacement•feedback e f f e c t i v e l y s t i f f e n s the spring, decreasing the resonant period, while the seismometer damping.can be increased with v e l o c i t y feedback. Experimentally determined s e n s i t i v i t y curves ob- tained with a Willmore Mk.I.seismometer demonstrate the c a p a b i l i t i e s of negative feedback i n a l t e r i n g the response of. the seismometer. A-resonant period.of 17 sec. correspond- ing to an equivalent suspended, mass of 123©'kg. i s r e a d i l y obtained. A discussion, of. instrument noise in.the feedback seismometer i s included. i i i AN APPLICATION OF FEEDBACK TO.ELECTROMAGNETIC SEISMOMETERS Table of Contents Abstract i i Table of Contents i i i L i s t of Tables v i L i s t of Figures v i i Acknowledgements x Chapter 1 Introduction.to Seismometry 1.1 Seismometers and seismographs 1 • 1.2 Seismometeroand seismograph response 3 1.3 Seismometer equivalent c i r c u i t s $ 1.4 Seismometer and.seismograph c a l i b r a t i o n ,10 Chapter 2 The Application of Feedback to Seismometers 2.1 Introduction 13 2.2 Review of previous work 13 Chapter 3 The Maxwell ; Bridge Feedback Seismometer 3.1 The Maxwell bridge as a means of applying feedback 16 3.2 The ef f e c t of. feedback on the seismometer 3.2.r Feedback seismometer transfer function .17 3.2.2 Acceleration s e n s i t i v i t y curves 20 3.2.3 Cal i b r a t i o n of. the feedback seismometer • 23 3-2.4 Root locus diagrams f o r the feedback seismometer 24 3.3 Instrument noise in.the feedback s e i s - mometer 3.3.1 Introduction 31 i v 3 - 3 - 2 Thermal noise and :noise i n . t h e feedback loop 32 3 - 3 - 3 D r i f t i n . t h e feedback seismometer 33 Chapter 4 Experimental V e r i f i c a t i o n of Feedback- Theory 4 - 1 The.Willmore Mk.I seismometer 3& 4 - 2 C a l i b r a t i o n and feedback c i r c u i t r y 41 4 - 3 Experimental r e s u l t s 44 4 - 4 I n t e g r a t o r , d i f f e r e n t i a t o r and summing a m p l i f i e r 50 4 - 5 P r a c t i c a l root locus diagrams 53 Chapter 5 Conclusion 55 B i b l i o g r a p h y 57 Appendix A Electromagnetic Transducer A . l Theory 59 A. 2'Electromagnetic damping 60 Appendix B The Laplace Transform and Transfer Functions B. l Laplace and inverse Laplace transforms 61 B. 2 Transfer f u n c t i o n s 6 l Appendix C Feedback C. l The t r a n s f e r f u n c t i o n of feedback systems 63 C .2 S t a b i l i t y of feedback systems 63 C. 3 System noise and feedback 64 Appendix D Root Locus Method D. l Theory 66 D .2 )s and 0J* of second order systems from root locus diagrams 67 Appendix E Operational Amplifiers E l l Transfer function of operational ampli- f i e r s E.2 Block symbols of operational amplifiers V I LIST. OF TABLES Table 4-1 Table 4-2 Information pertaining to acceleration s e n s i t i v i t y curves of Fig.4-3a Information pertaining to acceleration s e n s i t i v i t y curves of Fig.4-3-b 48 49 • V l l LIST .OF' FIGURES .Chapter-1 Introduction to Seismometry , 1-1 Simple seismometer 3 ,1*2 Electromagnetic seismometer acceleration s e n s i t i v i t y 5 ; 1-3 Electromagnetic seismograph acceleration s e n s i t i v i t y , 6 1-4 Acceleration s e n s i t i v i t i e s of long and short period seismographs i n . r e l a t i o n to the micro- seismic .region 7 1-5/Voltage-driven;seismometer equivalent c i r c u i t 9 1-6 Current-driven.seismometer equivalent c i r c u i t 9 1-7 Electromagnetic seismometer c a l i b r a t i o n . c i r c u i t 11 Chapter 3 The Maxwell Bridge Feedback•• Seismometer 3-1 Feedback seismometer c i r c u i t 17 3-2 Sim p l i f i e d feedback seismometer c i r c u i t .1$ 3-3 Feedback seismometer acceleration-sensi-< • t i v i t y 21 • 3-4 Seismometer a c c e l e r a t i o n . s e n s i t i v i t y . .21 - 3-5 Current-driven.seismometer equivalent c i r - c u it ' 22 3-6 Root locus f o r e f f e c t i v e mass increase 25 26 3-7 Root, locus f o r e f f e c t i v e damping constant .increase 3-3 Root locus f o r e f f e c t i v e spring constant increase 27 3-9 Root locus f o r e f f e c t i v e spring and damp- ing , constant., increase 27 3-10 Root,locus f o r e f f e c t i v e mass decrease 2& •3-11 Root.locus f o r e f f e c t i v e mass and.spring constant increases ' 29 V l l l 3-12 Root locus f o r e f f e c t i v e damping constant decrease 29 3-13 Root locus f o r e f f e c t i v e mass and damping constant increases 2 9 3-14 Root locus f o r e f f e c t i v e spring constant decrease 30 3-15 Root locus f o r e f f e c t i v e mass increase f o l - lowed by e f f e c t i v e damping constant i n - crease 30 3- l6 Feedback seismometer c i r c u i t f o r noise analysis 33 Chapter 4 Experimental V e r i f i c a t i o n of Feedback Theory 4- 1 Calibr a t i o n curve (ZJw)) f o r Willmore Mk.I seismometer 42 4-2 Experimental feedback and c a l i b r a t i o n schematic 44 4-3 Experimental,acceleration s e n s i t i v i t i e s of feedback-seismometer a. acceleration and displacement feedback 45 b. v e l o c i t y feedback applied, to 2 sec. period feedback s e i s - mometer 46 4-4 Integrator used i n experimental work 51 a. c i r c u i t - b. gain 4-5 D i f f e r e n t i a t o r used i n experimental work 51 a. c i r c u i t b. gain 4-6 Single amplifier producing combination of di r e c t and d i f f e r e n t i a t e d feedback 53 a. c i r c u i t b. gain 4-7 Acceleration feedback c o e f f i c i e n t versus v e l o c i t y feedback c o e f f i c i e n t f o r opera- t i o n a l amplifier of Fig.4-6a 53 4-8 Comparison of practical.and i d e a l root l o c i 54 a. i d e a l d i f f e r e n t i a t o r b. p r a c t i c a l d i f f e r e n t i a t o r c. ideal, integrator d. practical, integrator Appendix B The Laplace Transform and Transfer Functions B-l Block representation of a system with transfer function G-(s) 62 Appendix C .Feedback C-l Block diagram of feedback system 63 C-2 Block diagram, of system with:noise 64 C-3 Block diagram of noisy system with feed- back 65 Appendix D .Root-Locus Method D-l Block diagram.of second order•system with d i f f e r e n t i a t i n g feedbacks loop 67 D-2 ^s -and 6)'of second, order feedback systems from root.locus 68 Appendix E Operational Amplifiers E - l Operational amplifier 69 E -2 Operational, amplifier with..several,input •' signals 70 E-3 Series of standard, block.symbols 70 a. summing amplifier b. phase inverter c. multiplication:by a constant ACKNOWLEDGEMENTS . The concept of. the Maxwell bridge as a means of applying;feedback to a seismometer was conceived by Dr. R. D. Russell, who i s better known in.the f i e l d of mass spectrometry than seismology. It was. under his very able guidance that, t h i s study was conducted. His advice and en- couragement are truly.appreciated. Considerable p r o f i t was derived,from' discussions with'Oliver Jensen, a fellow graduate student who.is extend- ing t h i s research , with!.the construction of an operating seismograph employing a feedback seismometer based on the pri n c i p l e s described:herein. It.is.hoped that he too•has benefited. The inter e s t expressed.in.this project by Peter Michalow, electronics, technician. in,-the Geophysics Department, i s appreciated. His assistance.in overcoming va r i o u s , p r a c t i - cal d i f f i c u l t i e s was invaluable. The author i s g r a t e f u l to Dr. A. M. Crooker of the U.B.C. Physics-Department • whose analog,, computer was used, i n the experimental work, and.to,the Observatories Branch of the Department of Mines'and. Technical Surveys, Ottawa, who generously loaned.two Willmore Mk.I seismometers, to the Geophysics Department f o r t h i s and ensuing ;studies. F i n a n c i a l assistance was.received.from - National Research Council of Canada research grants to Drs. J. A. Jacobs and R. D. Russell. Chapter 1 INTRODUCTION.TO SEISMOMETRY 1.1 Seismometers and seismographs The seismograph,is an.instrument which.provides a record of the i n f i n i t e s i m a l movements of the ground caused by. e l a s t i c waves, t r a v e l l i n g through the earth,from distant earthquakes, and from other lesser sources. The detecting unit, of the seismograph, the seismometer-'-, produces a s i g - nal related, to the ground motion. This s i g n a l , which may be e l e c t r i c a l or mechanical (eg. the de f l e c t i o n of a pointT er.from.a zero mark), i s then.converted by a recording system into a continuous d i g i t a l or analog ;record. The seismometer generally, consists of a suspended mass which, because of i t s . i n e r t i a , tends to remain at rest while the frame from which i t i s suspended moves with the ground^. Ideally, the apparent motion of the mass i s 180° out of phase with,the motion of. the ground. In practice, however, the suspension.system applies a restoring;force to the mass.proportional t o , i t s displacement, with the resu l t that•it.tends to o s c i l l a t e with a period determined by the suspended mass and the stiffness.of. the suspension. It i s 1. The terms seismograph and seismometer have been differently, defined by various au t h o r i t i e s . The i n t e r - pretation given:here r e f l e c t s the opinion of.the author and i s based on d e f i n i t i o n s given by J. H. Hodgson [6] and by P. L. Willmore [15] . 2. The exception i s the strain.seismometer, which d i r e c t l y measures the compressions and d i l a t i o n s of the ground as the seismic waves pass. 2 necessary to apply a damping.force, proportional to,the v e l - o c i t y of the mass and opposing•its motion, to reduce t h i s tendency. The mass i s generally constrained to move i n one direction.only, and the seismograph.records one component of.. the ground motion. Seismographs have been b u i l t however, to produce records of a l l . three components simultaneously. The detection of. the d i f f e r e n t i a l motion between the mass and: the frame can be accomplished i n mahy ways. The o r i g i n a l method, involved, the magnification of. the-masses movements; by an,intricate system.of levers, and the record- ing, of the magnified, motion by a stylus writing on smoked paper. This method of recording;had several disadvantages including. ;the inconvenience of smoked paper records and the necessity of large suspended masses to overcome the f r i c t i o n between the-paper and stylus.and in.the lever system. With.the use of transducers^ and e l e c t r i c a l record- ing methods, higher magnifications with.smaller masses were achieved. Many seismographs use electromagnetic • transducers^ 4 - to drive sensitive galvanometers, the suspension movements of which are optically-magnified and recorded on photographic paper. A great v a r i e t y of transducer systems and recording methods have been employed.in seismographs. E l e c t r o s t a t i c 3. Devices which convert mechanical energy, into e l e c t r i c a l energy and vice versa. 4. See Appendix A - Electromagnetic transducer. transducers i n which one plate of a capacitor i s attached to the suspended mass are often used in.long period seismographs. These transducers allow external power sources and AC ampli- f i c a t i o n methods to be used. Chart recorders allowing the record,to be viewed as i t i s being written, and d i g i t a l rec- orders are i n use, while magnetic tape recording methods,are being developed. The method of suspending the mass varies consider- ably between seismometers, and usually depends on the purpose f o r which the instrument was designed [5,7] . A v e r t i c a l com- ponent- seismometer with a simple spring suspension i s d i s - cussed, i n the following section. 1.2 Seismometer and seismograph response Fig.1-1 shows a simple v e r t i c a l component seismometer consisting of a mass M , sus- pended by a spring of s t i f f n e s s U , and damped, by a f l u i d dash- pot with damping.constant D . If y i s the d i s - placement of the ground i n space, and " X . i s t h e displace- ment of the mass with respect Fig.1-1 Simple seismometer. to the case, then.the displacement of. the mass i n space i s (x-y) and i t s acceleration i s r-\(*-y) • The forces acting on the mass due to the spring and dashpot are respectively - U x and - Djf f o r displacement x and v e l o c i t y of the mass with respect to the case. 4 Applying-: Newton's second, law, M !!£____ _ _ ux - Ddx and the equation.of motion.is or _ _ -r 2 ^ __ + * V * _ f[_? where tt i s the damping r a t i o , and 6_ is.-the angular natural frequency. Since the seismometer s a t i s f i e s a second,order d i f f e r e n t i a l equation, i t . i s termed a second-order mechanical system. A Laplace transformation of. the equation of motion and rearrangement gives the transfer function of the s e i s - mometer, _ . _ _ J _ l l _ - s" where the upper case symbols represent the transforms of. the lower case quantities^. If the ground acceleration i s regarded as the input, the transfer function i s X(s) X(s) _ i The response to a sinusoidal ground acceleration %eJ,ji ( y. real) i s , by d e f i n i t i o n . of the transfer function 4 ^ f ( e _ x«e . ~ ~Y~0*̂  i s the frequency response 5. See Appendix B - Laplace transforms and transfer functions. of the seismometer to sinusoidal ground accelerations and I XOrt>l I Y 0 M M ̂ s "t^le a c c e l e r a " t i ° n s e n s i t i v i t y . The transfer function of an electromagnetic trans- ducer i s J^^= 3 where ^ i s the transducer constant, and and <sx(si are the transforms of, the output emf and r e l a t i v e v e l o c i t y of the transducer elements respectively. Therefore, regarding the ground acceleration as the input, the transfer function of a seismometer .with an electromagnetic transducer i s E(s) . ±M_.3£L . 3 9 Y O'w) i s then the acceleration s e n s i t i v i t y of the e l e c t r o - magnetic seismometer and i s plotted against the period of the ground movement i n Fig.1-2. \ =i£ i s the resonant period of T s (period) F i g . 1 - 2 Electromagnetic seismometer acceleration s e n s i t i v i t y . the seismometer. Now, the transfer function of a galvanometer i s where 0(5) i s the transformed angular d e f l e c t i o n of the sus- pension, Ite) i s the transformed c o i l current, \ is.the galvanometer damping r a t i o , &>3 i s the galvanometer angular natural frequency, h i s the galvanometer transducer constant. If the galvanometer i s connected d i r e c t l y to the electromagnetic seismometer, and.if. the galvanometer reaction (the e f f e c t ; o f the movements of the galvanometer suspension on the motion of the suspended mass of the seismometer) and the coil.inductances are n e g l i g i b l e , the transfer function of the r e s u l t i n g seismograph i s 9$ 9(ju)\ i s the accel-where .̂ i s an :attenuation constant, eration s e n s i t i v i t y ' of the seismograph and i s plotted i n Fig.1-3. Tg and 7s.are the resonant periods of the galvano- meter and seismometer respectively. Ta T* Y09 (period)—»- Fig.1-3 Electromagnetic seismograph acceleration s e n s i t i v i t y . I t - i s seen that.the passband of the electromagnetic 7 seismograph i s controlled.by the galvanometer on the short period side and by the seismometer on the long.period.side. If T 3 and T 5. were. interchanged, the response curve would have the same appearance, but a decreased s e n s i t i v i t y would r e s u l t . In the past, one of the aims of seismologists i n .the construction of seismographs was a h i g h . s e n s i t i v i t y over as broad a period range as possible. This practice resulted i n the discovery of microseisms, a continuous seismic back- ground noise with periods ranging from 1 to 7 seconds. Fur- ther increases i n s e n s i t i v i t y were possible only i n period ranges f o r which the microseismic noise l e v e l was low. This led:to the use of two seismographs f o r each component of ground motion recorded - one recording long:period signals, and;the other, short periods - with t h e i r response curves positioned to exclude the main microseismic region (Fig.1-4). log ( p e r i o d ) Fig.1-4 Acceleration s e n s i t i v i t i e s of long and short period .seismographs i n r e l a t i o n to the microseismic region. This attitude :has changed however, with the p o s s i b i l - ity, of using slow speed magnetic tape recording in.seismology. A slow speed tape recorder with f l a t response combined with a broadband seismometer would record.a wide spectrum of seismic signals. The tapes could l a t e r be played: back through various f i l t e r s to obtain the desired, r e s u l t s . In t h i s manner, no information i s l o s t t h r o u g h , f i l t e r i n g as long as.the system i s s t r i c t l y l i n e a r . The bandwidth of a - seismograph can be increased by - various methods, the simplest being the lengthening of the - seismometer, period and: the shortening of the galvanometer period. Long seismometer periods i n . v e r t i c a l instruments re- quire the use of weaker springs which are subject to thermo- e l a s t i c e f f e c t s and. mechanical fatigue. These ef f e c t s cause the mass to d r i f t from:its equilibrium,position, r e s t r i c t i n g the p r a c t i c a l length,of periods obtainable by t h i s method. Willmore [14] has shown.,that broad-band character- i s t i c s may be achieved by gross overdamping of both the seismometer and galvanometer. He has also investigated the application.of feedback to a seismograph as an alternate means of obtaining a wide passband [14] . 1.3'Seismometer equivalent c i r c u i t s . It i s well known that many, mechanical, systems may be more conveniently studied by considering t h e i r e l e c t r i c a l analogs: e l e c t r i c a l c i r c u i t s which satisfy, the same d i f f e r - e n t i a l equation as the mechanical system. An.electromagnetic seismometer i s an electromech- anical, system and. may. be studied,as such. It i s , however, much.simpler to replace the mechanical part of the system by i t s e l e c t r i c a l analog, leaving a completely equivalent e l e c t r i - 9 c a l system.which may be treated,with simple c i r c u i t theory. Such a procedure has long been followed:by seismol- ogists, using an equivalent c i r c u i t i n which a.voltage gen- erator represents the ground.velocity (Fig. 1 - 5 ). A simple derivation.of t h i s c i r c u i t may be found i n reference 3 . Fig. 1 - 5 Voltage-driven.seismometer equivalent c i r c u i t . K o l l a r and Russell [7]have recently, contended that the symmetry, inherent i n the mechanical resonant system.is preserved i f the voltage generator present i n . t h i s rather conventional equivalent c i r c u i t i s replaced by a current gen- erator ., The current driven equivalent c i r c u i t i s shown i n Fig. 1 - 6 and i s used throughout t h i s thesis^. Fig-.l - 6 Current-driven seismometer equivalent c i r c u i t . The values of the various components are shown and 7. This equivalent c i r c u i t was o r i g i n a l l y given.by S.A. Scherbatskoy and J. Neufeld, Equivalent e l e c t r i c a l . n e t - works of seismographs, Geophysics I I , 213-242, 1937. 10 a r e r e l a t e d t o t h e s e i s m o m e t e r - c o n s t a n t s , s p r i n g c o n s t a n t U, damping c o n s t a n t ]) , mass M , t r a n s d u c e r c o n s t a n t $ , and t o . t h e g r o u n d a c c e l e r a t i o n y . r?c and L c a r e t h e r e s i s t a n c e and i n d u c t a n c e o f t h e c o i l , and t h e s w i t c h i s a n a l a g o u s t o a c l a m p u s e d : t o p r e v e n t t h e mass f r o m s w i n g i n g . As K o l l a r and R u s s e l l o b s e r v e , t h e e l e c t r o m e c h a n i c a l s e i s m o m e t e r and t h e e q u i v a l e n t c i r c u i t a r e i n d i s t i n g u i s h a b l e by e l e c t r i c a l measurements made a t t h e o u t p u t t e r m i n a l s . 1.4 S e i s m o m e t e r and s e i s m o g r a p h c a l i b r a t i o n The c a l i b r a t i o n o f a s e i s m o m e t e r i n v o l v e s t h e d e t e r - m i n a t i o n o f i t s r e s p o n s e t o s i n u s o i d a l g r o u n d m o t i o n s o f any p e r i o d . W i l l m o r e [13] has d e v i s e d a c o m p l e t e l y e l e c t r i c a l method o f c a l i b r a t i n g e l e c t r o m a g n e t i c s e i s m o m e t e r s w h i c h d e - t e r m i n e s , t h e c o n s t a n t s U , D , $ , ( M. i s . assumed: known) , and d i r e c t l y p r o d u c e s c a l i b r a t i o n . c u r v e s . W h i l e W i l l m o r e has t r e a t e d , t h e s e i s m o m e t e r a s . a n . e l e c t r o m e c h a n i c a l s y s t e m , K o l l a r and R u s s e l l , u s i n g t h e c u r r e n t e q u i v a l e n t c i r c u i t , have e n l a r g e d o n , t h e W i l l m o r e method f r o m a c o m p l e t e l y e l e c t r i c a l p o i n t o f v i e w . They have a l s o c o n f i r m e d , t h e v a l i d i t y . - o f t h e a p p r o x i m a t i o n s u s e d i n . t h e a n a l y s i s . F o r t h e c a l i b r a t i o n , t h e c lamped s e i s m o m e t e r i s p l a c e d i n t h e " u n k n o w n " p o s i t i o n o f a M a x w e l l impedance b r i d g e ( F i g . 1 - 7 ) , and t h e b r i d g e i s b a l a n c e d , i n t h e u s u a l manner f o r " M A I N " i n p u t . The b a l a n c e c o n d i t i o n , i s i n d e p e n - d e n t of, t h e f r e q u e n c y and g i v e s t h e v a l u e s , o f "Rc and L t . K o l l a r and R u s s e l l have s h e w n . t h a t w i t h , t h e s e i s - mometer u n d a m p e d , t h e r a t i o o f d e t e c t o r o u t p u t s f o r " M A I N " 11 My 7 Fig.1-7 Electromagnetic seismometer c a l i b r a t i o n c i r c u i t . and "SUBSTITUTION" inputs i s '-gr^r from which 2/to) may be determined. The positions of the resonant peak and asymp- totes of a logarithmic plot of 2 - , ^ against CO , together with the known suspended mass, determine the values of U , 0 , and They also show that a po t e n t i a l vr applied to the "MAIN" input of the bridge produces the same r e s u l t as a current generator i n p a r a l l e l with 2, which, comparing with, the equivalent c i r c u i t , i s equivalent to a ground .accel- With t h i s knowledge, the acceleration sensi- t i v i t y of the seismometer with the bridge as an attenuator can be plotted, from which the acceleration s e n s i t i v i t y of the seismometer alone can-be deduced. The magnification and v e l o c i t y s e n s i t i v i t y are related to; the acceleration sens- i t i v i t y and may, also be determined. If, the galvanometer to be used with the seismometer i s the bridge detector, the s e n s i t i v i t y curves f o r the seismograph with the bridge as an attenuating network can:be determined. Willmore has placed a d i f f e r e n t i a t i n g c i r c u i t before eration MKB . 12 ~ the bridge, enabling.the v e l o c i t y s e n s i t i v i t y - t o • b e obtained d i r e c t l y . Chapter 2 THE APPLICATION OF FEEDBACK TO SEISMOMETERS 2.1 Introduction It was mentioned i n Chapter 1 that Willmore had investigated the application of feedback to a seismograph as a means of obtaining broad band response c h a r a c t e r i s t i c s . Feedback has also been.used:by others.to a l t e r the response c h a r a c t e r i s t i c s of a seismometer, p r i n c i p a l l y to reduce the d r i f t associated with long;period instruments. These investigators applied the feedback to the seismometer as opposed to Willmore, whose feedback loop enclosed both the seismometer and galvanometer. In a l l cases however, an electromagnetic transducer was used to apply the feedback s i g n a l to the suspended mass of the seismometer. In.this t h e s i s , feedback w i l l be used s o l e l y to a l t e r the o v e r a l l response of a seismometer, though the method of application w i l l be quite d i f f e r e n t . A b r i e f description of the work done by several researchers follows. 2.2 Review of previous work De Bremaecher et a l . [5] have used feedback to elim- inate d r i f t i n the long period Press and Ewing:seismometer in:the Rice University d i r e c t d i g i t i z i n g seismograph. The input si g n a l to the feedback c i r c u i t was a voltage propor- t i o n a l to'the ground displacement, derived from the heter- odyned frequency of two o s c i l l a t o r s tuned by a d i f f e r e n t i a l capacitor transducer. The feedback c i r c u i t consisted of a . 14 low... pass f i l t e r ( i n t e g r a t o r ) constructed w i t h an. o p e r a t i o n a l a m p l i f i e r , and was connected:to-the electromagnetic t r a n s - ducer c o i l such ..that the f o r c e exerted by/the transducer op- posed..the motion.of the mass. The low.pass, f i l t e r ensured /that. ©nly; very, long periods were a f f e c t e d , the r e s u l t - b e i n g t o e f f e c t i v e l y s t i f f e n : t h e s p r i n g f o r long.period s i g n a l s . Sutton ..and Latham [ l l ] ; have used.. e s s e n t i a l l y * the same method as . de Bremaecker to reduce d r i f t i n • a--, long . period seismograph designed f o r unattended operation. The output s i g n a l from a d i f f e r e n t i a l c a p a c i t o r transducer, p r o p o r t i o n a l to the mass displacement, was a m p l i f i e d and f e d through a low pass f i l t e r to the c o i l of a•coil•and.magnet damping assembly. A.centering;.motor engaged by a . t r i g g e r c i r c u i t was.also provided.to adjust the suspension,if. the d r i f t • e x - ceeded, the c a p a b i l i t i e s of. the feedback•current. In.both.of. the above cases.and.in.general where feedback has been.used.to reduce the long t e r m . d r i f t of a seismometer, the feed b a c k . s i g n a l can.be monitored.producing •a record of. t i d a l , t i l t s . and g r a v i t y changes ( t i d a l e f f e c t s i n - t h e v e r t i c a l component), provided the d r i f t • r e s u l t i n g ;from other s o u r c e s . i s n e g l i g i b l e . Tucker [l2] , on.••the other hand, has.used,feedback to shorten.the pe r i o d of a seismometer, which was o r i g i n a l l y chosen•long :to improve the s i g n a l to noise r a t i o of. the transducer and e l e c t r i c a l system, and w h i c h . l a y . i n the reg i o n of. i n t e r e s t . He describes . a: h o r i z o n t a l seismograph • designed f o r the study of microseisms, having n e a r l y f l a t • r e s p o n s e to • 1-5 acceleration, f'&v periods ranging ;f.i?©m; 1.5 sec. to > 12.0 sec. .. A variable coupling;transducer produced a l k c . signal which was fed through an amplifier and:bhase sensitive r e c t i f i e r producing a DC voltage proportional to - the mass displacement. This voltage was fed back through . a r e s i s t o r to a,.transducer c o i l , producing a restoring;force on-the mass proportional to i t s displacement. This e f f e c t i v e l y , increased,the spring s t i f f n e s s , shortening the seismometer period. The feedback r e s i s t o r was shunted by a capacitor which provided damping, since by the p r i n c i p l e of superposition, i t produced a current i n the c o i l r e s u l t i n g i n a force on ..the, mass proportional to i t s v e l o c i t y . The negative feedback also;improved the l i n - e a r i t y of the amplifier and seismometer, and s t a b i l i z e d sens- i t i v i t y . : In Willmore "s feedback seismograph [14] , the feed- back sig n a l was derived from a photoelectric amplifier which produced an output proportional to the galvanometer spot d e f l - ection. This output was fed through ,a resistance-capacitance differentiating;network.into a secondary transducer c o i l , producing a force on the"mass proportional, to the galvanometer spot velocity.. Such feedback r e s u l t e d : i n a changeain,the galvanometer reaction.unobtainable by, other methods, which, with. large amounts. of. damping:, produced, wide band response c h a r a c t e r i s t i c s . Chapter 3 THE MAXWELL,BRIDGE FEEDBACK.SEISMOMETER 3 • 1 The Maxwell bridge as a means .-of applying feedback a In applying feedback, to a system, i t • i s Asimple fact.that, the feedback signal must be added.to the input. This was accomplished by means of electromagnetic.trans- ducers, in,, the feedback seismometers just discussed: a cur- rent i flowing in,.the transducer c o i l causes a mass motion equivalent to that which. would, be produced by a ground ac- celeration -pp . Another method of e l e c t r i c a l l y simulating a ground movement was discussed,in § 1.4: a voltage impressed ,across the "MAIN" input.terminals of the Maxwell bridge used i n calibration,produces the same ef f e c t on.the seismometer as., a ground acceleration j-̂j=r • Russell (private communication) ; has suggested i t would be possible to. add feedback,, to a seismometer simply by applying;the feedback.signal, to the "MAIN" input•terminals of a Maxwell bridge. u t i l i z e d as a., permanent attenuating net? work, between, .the seismometer and. the recording ;system. This method eliminates, in,the case of electromagnetic seismo- meters, the need.for a second.transducer. Also, when c a l - ibrated by Willmore vs Maxwell bridge method, corrections u s u a l l y required.when,the bridge i s removed, or replaced.by another, attenuator are not • necessary. 17 3.2 The ef f e c t of feedback on the seismometer 3.2.L Feedback seismometer transfer function^-. . A c i r c u i t diagram showing the application of feed- back to a^seismometer using the Maxwell bridge i s i l l u s t r a t e d using transform notation i n Fig.3-1. The input ground accel- Fig.3-1 Feedback seismometer c i r c u i t . eration, i s represented by the. current generator i n para- l l e l , with TsLs) . The output s i g n a l Vte)= . i s fed,-through a d i f f e r e n t i a l amplifier with transfer function FYs")., to the "MAIN" input terminals of the bridge. . Since t h i s i s equiva- lent- to placing a current generator i n ; p a r a l l e l with ~ZS , the c i r c u i t of Fig.3-1 may be replaced with the simp l i - f i e d version, shown,in Fig.3-2. and ( £j, and Q. i n . para- l l e l ) are required to be large f o r successful operation of the 1. See Appendix C - Feedback. 1 _ Fig.3-2 S i m p l i f i e d seismometer c i r c u i t . bridge i n t h i s manner and have been neglected. The f u n c t i o n of the bridge to change a voltage source Vf?) i n t o a current source .is emphasized by show- in g the block ^ independant of F(s) . Assuming the input impedance of the a m p l i f i e r to be i n f i n i t e , there w i l l be no currents f l o w i n g through 2 C or ^ , and the t r a n s f e r f u n c t i o n of the seismometer i n the bridge without feedback i s I (s) M s l t P s - i -U which i s p r o p o r t i o n a l to the a c c e l e r a t i o n s e n s i t i v i t y of the seismometer since ll^J - — 5 — v The t r a n s f e r f u n c t i o n of the feedback loop i s and the closed loop t r a n s f e r f u n c t i o n i s then _ _ _ u ^(5) Rearranging, Since ? 5 /s) i s an impedance f u n c t i o n , the t r a n s f e r + ____ , -1 19 function.of the feedback seismometer indicates, the addition ,of. feedback has e f f e c t i v e l y added an impedance + in.par- a l l e l with z?s . If Ffe)oc s', S" , S , the added impedance i s an,inductance, resistance, and.dapacitance respectively, the magnitude of which depends on.the gain i n the feedback-loop and the sign.of which depends on the sign.of the feedback. Since the r e s i s t o r , capacitor, and. inductor comprising 25 are respectively related to the damping fa c t o r , mass, and.spring constant of. the seismometer, the e f f e c t of adding additional impedances i n p a r a l l e l with _ s . i s to a l t e r the values of these parameters, which:in turn a f f e c t s the resonant f r e - quency and damping r a t i o . .Setting F(s)-iR6(4s'+b+cs) , the transfer function of. the feedback seismometer i s ' , V/<r) „ ,// » and Yr^- —___1__E where M* , 0', U', are the e f f e c t i v e mass, damping and,spring .constants. The damping.ratio and.resonant frequency are then gjj/uf and ^7 and can be varied... at w i l l , by adjust- ing -the • nature and. amount, of the - feedback. Due to the i n s t a b i l i t y of gain and d i s t o r t i o n gen- e r a l l y associated with positive feedback, i t w i l l not be d i s - cussed. F(s) w i l l henceforth:be assigned positive values and "negative feedback w i l l , be assumed. This does not, .however, r e s t r i c t , t h e c a p a b i l i t i e s of the system, as both the damping. 20 r a t i o and resonant frequency can be either increased or de- creased, by-negative feedback. While an objective of t h i s thesis i s to demonstrate the f e a s i b i l i t y of using ,a Maxwell, bridge to apply feedback to an electromagnetic seismometer, the effects produced by feedback are associated with any, feedback seismometer. 3.2.2 Acceleration.sensitivity, curves. The a c c e l e r a t i o n . s e n s i t i v i t y of the feedback s e i s - mometer to sinusoidal ground motions of angular frequency OJ i s At•low frequencies, at high.frequencies, ro") VLJUI) and at the resonant frequency, ^ = /Mr > a n d ~w~^x~ Since M and ^ are f i x e d , the height of the res- onant peak i s altered only by an e f f e c t i v e increase in.the damping constant. The r e l a t i v e positions of the asymptotes on arlogarithmic p l o t - f o r d i f f e r e n t values of M/ , 0' ^ U ' , can;be determined from these expressions, allowing a series of acceleration s e n s i t i v i t y , curves to;be sketched as i n F i g . 3-3. A s i m i l a r set. of s e n s i t i v i t y , curves r e s u l t i n g from actual changes i n the mass, spring and,damping constants i s given.in Fig.3-4. The corresponding asymptotic and peak expressions f o r are derivable from, those given above f o r the feedback seismometer by dropping the primes. 21 lo^( period) —»• Fig.3-3 Feedback seismometer a c c e l e r a t i o n . s e n s i t i v i t y . loj ̂period)-—>» Fig.3-4 Seismometer acceleration s e n s i t i v i t y . It i s apparent from a comparison of Figs.3-3 and 3-4 that while actual and e f f e c t i v e increases i n both the spring and damping.constants have the same e f f e c t on the ac- celeration s e n s i t i v i t y of the seismometer, actual and e f f e c t - ive increases" .in'the suspended mass do not. An explanation 22 of. t h i s r e s u l t i s f a c i l i t a t e d with the aid of the seismometer equivalent c i r c u i t i n Fig. 3 - 5 . An.actual, increase i n the Fig.3 - 5 Current-driven seismometer equivalent c i r c u i t . spring constant appears.in,the equivalent c i r c u i t as a decrease i n the inductance — . An e f f e c t i v e increase i n the spring constant by the application of feedback was shown i n the preceeding section to e f f e c t i v e l y add;an inductance in•• p a r a l l e l with 2, , thereby reducing;the t o t a l inductance i n j?s . These two means of. increasing the spring; constant are thus equivalent. and. an increase by -the same f a c t o r with either procedure w i l l produce i d e n t i c a l acceleration s e n s i t i v i t y curves^ as.indicated i n Figs.3 - 3 and 3 - 4 . The equivalence of actual and ef f e c t i v e increases i n the damping,constant can s i m i l a r i l y be shown. Now, both actual and e f f e c t i v e increases i n mass increase -the capacitance i n -2S , but an ;actual mass increase M j j also increases the input current source -—- , resulting;m,a greater s e n s i t i v i t y , to.a given ground acceleration.as seen i n Fig. 3 - 4 . The acceleration response of a seismometer with 2. In the mechanical system, t h i s i s true provided the s h i f t i n the equilibrium.position of the mass f o r actual increases does not - introduce n o n - l i n e a r i t i e s into the opera- t i o n of. the transducer. 23 a suspended mass fMI effectively, increased by feedback to Kl 7 M i s lower by a factor than that of a seismometer with an actual, mass of M' : the energy, extracted from the ground; by the instrument depends on the true and not,the effective sus- pended mass. Since the feedback signal is derived from.the seis- mometer open circuit voltage which is proportional to the velocity of the suspended mass.relative ;to the case, the effective increases in.the mass, spring and damping constants are observed only when the mass i s in.motion. Consequently, the rest•position of the suspended mass does not change with effective increases in the mass and spring constant as i t would were the changes real. That the damping.constant i s effectively, increased only when.the mass i s in,motion is of no consequence since the damping force is zero when the mass is at rest. Other operational differences between actual and effective changes in the spring and damping/constants-arise in noise considerations discussed, in S3 . 2 . 2 . (1 ) and §3 .-3 .3 . 3 . 2 . 3 : Calibration of the feedback seismometer. The feedback seismometer can be calibrated directly since the bridge used to apply, the feedback is capable of con verting any number of voltage sources into current sources providing, their driving circuits do not interact. A.voltage ir , so applied to the "MAIN" input of the bridge i s equiva- lent to a ground acceleration • ^ 9 i s known, the acceleration sensitivity- of, the feedback seismometer (or 24 seismograph, i f the seismograph recording system i s the bridge detector) can be determined. The transducer constant may be determined; by disconnecting;the feedback c i r c u i t and,calibrat- ing the seismometer i n the bridge as explained,in § 1.4. In,practice, to ensure t h e i r respective c i r c u i t s do not i n t e r a c t , the feedback and c a l i b r a t i o n signals may be added by a summing amplifier-' before being applied to the bridge. 3.2.4 Root locus- diagrams f o r the feedback seismometer. The root locus method^- i s often employed i n . inves- t i g a t i n g the s t a b i l i t y of feedback systems^. Its sole pur- pose here, however, w i l l be to provide graphical analysis of the ef f e c t of. feedback on.the natural frequency and.damping r a t i o of the seismometer since the concept- of s t a b i l i t y usually, associated with feedback control systems and feedback amplifiers has l i t t l e meaning-here. It was shown.in §3.2 . 1 that the application.of ce r t a i n types of feedback.to a s e i s - mometer resulted only- i n e f f e c t i v e l y * a l t e r i n g the suspended mass, damping and spring constants. Large loop gains.lead only to large deviations in;the e f f e c t i v e values of these mechanical constants from the o r i g i n a l values and not to i n - s t a b i l i t i e s as might- be expected with ,a servosystem. The 3. See Appendix E, - Transfer functions of operation- al., amplifiers. 4. See Appendix D - Root locus method. 5. See Appendix' Cv2 - S t a b i l i t y of feedback.systems. 25 root locus diagrams, despite t h e i r basis of. feedback stabi l i t y - c r i t e r i a , simply provide information on the natural frequency and damping r a t i o of a seismometer as. the various constants, M , P., W , are altered by feedback or by other means. Root l o c i , are given.for the three basic types of feedback discussed, and. f o r l i n e a r combinations. of these. t. The feedback transfer function, the mechanical constant.al- tered, and the open:loop transfer function from which the root*, locus plots are constructed, are included with each d i a - gram. The seismometer i s considered to be i n i t i a l l y under- damped ( ), with the r e s u l t that i t s poles (which are. also the poles of the open loop transfer function) l i e at complex conjugate positions i n the s-plane. i ) F^)= A S , e f f e c t i v e mass.increase, open loop transfer function - *-8" F i g . 3 - 6 Root locus f o r e f f e c t i v e mass increase. The root locus i s the s o l i d arc tangent to the J<*>-axis at the o r i g i n . As the gain (B ), increases from zero at the poles to °o at the zeros (the e f f e c t i v e mass increases from M to 00 ) , the damp- ing r a t i o V/ and. the angular nat- u r a l frequency of the feedback seismometer decrease from f r and CJS to zero. The poles of seismometers with the same o r i g i n a l 26 resonant frequency, but d i f f e r e n t damping;ratios than the seis- mometer represented above w i l l , l i e at complex conjugate pos- i t i o n s .along;the dashed arc. i i ) ?(s)~ci , e f f e c t i v e damping constant increase, open loop transfer function = The root, locus l i e s on.the nega- t i v e r e a l axis, branching out into the s-plane i n , an arc of radius cs*. centred on the zero at the o r i g i n . As the gain increases from O , 6_s= constant =- c_»s , while ^ i n - creases to 1. at the break-away point, where the seismometer i s c r i t i c a l l y damped, and.to values greater than,1 on the axis where the seismometer becomes overdamped. The poles of, the over- damped seismometer l i e at - $*CJS±*yfa, j T r 1 - i ' . , e f f e c t i v e spring constant increase, open B Fig.3-7 Root locus, f o r e f f e c t i v e damping constant increase i i i ) F M = o s " loop transfer function _ s*+g? ̂ s + ^ The root locus i s a l i n e p a r a l l e l to the J w - a x i s , i n d i c a t i n g Us increase and decrease with increasing gain or spring : constant. 27 Fig.3-8 Root locus f o r e f f e c t i v e spring constant increase. iv) F(s) * (&+ tsT') s bs " ' ( i + Ts) } e f f e c t i v e spring and, damp- ing ; constant increases, open, loop: transfer function _ B([+Ts) 1 T The root locus i s an arc centred on the zero at - ̂ — — — — . T a Fig.3-9 .Root locus f o r e f f e c t i v e spring and damping;constant increases. As i n the o r i g i n a l seismometer, where an increase by the same fac t o r of two of M , D , U , could.be interpreted 28 as a decrease by ;the same f a c t o r of the t h i r d , an e f f e c t i v e increase i n two of M , D , U , by-feedback produces s i m i l a r e f f e c t s . An e f f e c t i v e increase by the same f a c t o r i n the spring;and damping^constants i s , from the point of view of seismometer response completely/equivalent to an actual mass decrease. i e . , M 9 S M g s However, an e f f e c t i v e increase by the same factor of the mass and either of U or D i s equivalent to a decrease i n the t h i r d constant ( D or U ) accompanied by a decrease fey -̂ p- i n the s e n s i t i v i t y , which was shown i n § 3 .3 .2 to a c - company, an e f f e c t i v e mass increase. Thus, i n Fig.3-9, i f T * ' . J , then and the root locus diagram has the appearance of Fig.3-10. Comparison, with ithe root locus given.for e f f e c t i v e mass i n - crease (Fig.3-6) confirms the pro- p o s i t i o n that t h i s case represents an.effective mass decrease. Fig.3-10 Root locus f o r e f f e c t i v e mass decrease. v) FCs)= (as + bs") =. bs~'(l +T a s l ) , e f f e c t i v e mass and spring con- stant increases, open loop,transfer.function 29 g O + T V ) i f . 4 f Fig.3-11 Root locus f o r e f f e c t i v e mass and spring constant increases, Fig.3-12 Root locus.for ef- fe c t i v e damping constant de- crease. If I = ̂  , the root, locus takes on the appearance .shown i n Fig 3-12, i n d i c a t i n g a damping constant decrease (Compare with F i g 3-4) • vi) F(s) = (as + b)= b ( l+Ts) , • e f f e c t i v e mass and damping constant increases••, open, loop. transfer function ( I + T S ) Fig.3-13 Root,, locus f o r e f f e c t i v e mass and damping •: constant increases. If T . * 30 , the root locus has the appearance shown-in Fig.3-14 which comparing with Fig.3-8 indicates a spring constant decrease. H i *-<r Fig.3-14 Root.locus for ef f ectiyeosppingf; constant decrease;. JO) While the root locus diagrams given are.essent- i a l l y .qualitative i n nature, accurate plots can be construct- ed f o r a p a r t i c u l a r seismometer f o r quantitative analysis. It i s also i n s t r u c t i v e to consider the pole pos- i t i o n s of the feedback seismometer f o r the feedback combina- 1 tions discussed above as r e s u l t i n g ,from a series•of changes, f i r s t of one mechanical constant, then of another. For example, i f . the mass were effectively, increased length- ening the period, Vs and <*)* are deter- mined from the s o l i d arc on the s- plane i n Fig.3-15, based on the root, locus diagram of Fig.3-6. The gain could be increased u n t i l the required natural frequency was ob- Fig.3-15 Root, locus f o r e f f e c t i v e mass i n - crease followed by ef- fe c t i v e damping con- stant increase. 31 tained. If. the damping :constant were then e f f e c t i v e l y , i n - creased, the poles of the. feedback seismometer would;move along the dashed, l i n e , after the root locus diagram of Fig . 3-7, and the damping ; r a t i o could; be adjusted to.-the desired value. 3.3 Instrument noise i n the feedback seismometer 3 . 3 . 1 Introduction. Noise in.seismometers i s generally divided,into two broad.classes; instrument noise and,seismic noise. Seismic noise i s any undesirable s i g n a l of either a r t i f i c i a l or natural o r i g i n , which enters.-the seismometer from,the ground. Instrument noise, on'the.other:hand, r e f e r s . t o : s i g - nals produced i n : t h e detector i t s e l f , and, may/be attributed to -such things as mechanical fatigue and thermo-elastic ef- f e c t s in,the suspension elements, and to thermal noise i n the associated c i r c u i t r y , a n d damping system. Barometric changes, a i r currents and interactions of the seismometer with ambient magnetic f i e l d s must also be classed as sources of instrument noise, but w i l l not be discussed since they may-be minimized by proper design. . . Byrne has extensively dealt with thermal noise i n seismic detectors by considering the application to seismo- meters of two theorems by Nyquist on thermal noise i n elec- tromechanical systems. It was shown,that the thermal noise present i n the seismometer i s determined by associating Johnson noise generators with the r e s i s t i v e elements i n the equivalent c i r c u i t . 32 While Byrne has used,voltage generators with the conventional seismometer equivalent c i r c u i t , K o l l a r and Russell have shown that,the analysis i s greatly, s i m p l i f i e d by the use of current generators.in the current-driven equivalent c i r c u i t . They, have also introduced.a means of determining the equivalent input noise of. thermal noise sources i n com- pl i c a t e d networks. This method,involves the cancellation of the e f f e c t of the noise by an unknown input which is.then determined from, the requirement that.the output be zero by simple c i r c u i t theory. They point out that t h i s procedure r e s u l t s i n "a phase error of 7T radians but that the phase of a noise source i s not s i g n i f i c a n t . This procedure w i l l be used ex c l u s i v e l y i n the following paragraphs dealing with>thermal noise i n the feed- back seismometer. The value of the noise current generator i n p a r a l l e l with ? s , which, produces the same e f f e c t at the seismometer output as :the noise source i n the c i r c u i t w i l l ; be calculated. This input noise current and; hence the elec- t r i c a l noise in,the seismometer, i s then completely equiva- lent to a ground acceleration -^a-• 3 . 3 • 2 Thermal.noise and noise i n the feedback.loop. Since the inductive reactance of the transducer c o i l at frequencies near the seismometer resonant frequency i s often,negligible compared to.the c o i l resistance, i t w i l l be neglected i n the. following analysis. The capacitor C B 33 can then be omitted from.the bridge. The external feedback c i r c u i t r y ^ represented by the d i f f e r e n t i a l amplifier with transfer function F can;be replaced by a voltage generator FV with i n t e r n a l resistance and the r e s u l t i n g feedback seismometer c i r c u i t i s shown i n Fig.3-16. Fig.3-16 Feedback seismometer c i r c u i t f o r noise analysis. The thermal noise associated with each r e s i s t o r i n the c i r c u i t of Fig.3-13 and the noise inherent i n the external feedback c i r c u i t w i l l be i n d i v i d u a l l y considered. F„ w i l l be assumed to be n e g l i g i b l e . i ) Thermal noise i n "damping r e s i s t o r " ^- i n Z5 . An RMS current generator ^ ^ f y ^ f where h = Boltz- mann's constant, and T i s the absolute temperature, placed i n p a r a l l e l with 2 S (and hence w i t h ^ ) represents the thermal noise present i n the bandwidth of the resistance ̂  . A current generator i n p a r a l l e l with ? s i s con- sidered to be of such a value t h a t . i t cancels the e f f e c t of 6. Excluding the bridge. 34 the noise source. Since i t . i s . then required that V = 0 , the voltage generator FV may be replaced by a short c i r - cuit , and no current may flow through T?£ and ft (Fig.3-17) . Fig.3-17 Thus L«- ' V gVE> , and the thermal noise due to damping remains unchanged by- feedback,, irregardless of whether the feedback: has e f f e c t - i v e l y altered the damping,constant. i i ) Thermal noise i n VA . A voltage generator V WT? cdf' i n series with J?e represents.. the thermal noise i n the bandwidth o f f present i n (Fig. 3-18). Fig.3-18 '' • If i t i s balanced by the input noise current , 35 then V - O and the generator FV maybe replaced.by a short c i r c u i t . Also, no current can flow/through , with the resu l t that i i i ) Thermal noise i n T?B. Representing the thermal:noise by a voltage generator /fikTRndf i n series with 1?, , and requiring V to be zero leads to.the c i r c u i t shown i n Fig.3 - 1 9 and the requirement = >AfkP»dr" f W W - Fig.3 - 1 9 T Kirchhoff*s. law gives + ^r) + U + U) 2. = VrT?.af These two equations can be solved f o r t» s Ko 2, iv) Thermal noise i n Rp. The thermal noise associated, with Pb i s represented by the voltage generator fiWr^Af i n series with R"D . IfV=0 then i n Fig.3-20 36 c, (eQ + ft) •+ u f t - ^kTe0df ^ o F i g . 3-20 Kirchhoff^s law. gives, two. more equations, These three equations give v) Thermal noise i n A voltage generator *j4kTR*Jf i n series with RB represents the thermal noise associated,with RR . The c i r - c u i t shown i n Fig.3-21 and.the requirement r e s u l t i f V i s to be zero. -AAAAA- L _ J •AAAA- F i g . 3 - 2 1 K i r c h h o f f T s l a w s g i v e 6', 4. __)C KV*. = tfPPF&d? S o l v i n g t h e s e t h r e e e q u a t i o n s g i v e s v i ) N o i s e i n e x t e r n a l f e e d b a c k c i r c u i t r y . C o n s i d e r an RMS n o i s e v o l t a g e £ t o be p r e s e n t i n , - t h e i n p u t o f t h e v o l t a g e g e n e r a t o r o f F i g . 3 - l 6 . T h i s i s e q u i v a l e n t t o a n o i s e £ a c r o s s t h e i n p u t o f t h e d i f f e r - e n t i a l a m p l i f i e r r e p r e s e n t i n g t h e f e e d b a c k - c i r c u i t . I f V i s z e r o , t h e n t h i s n o i s e c a n . b e ' r e p r e s e n t e d by a v o l t a g e g e n e r a t o r €F as shown i n F i g . 3 - 2 2 . 3a r • •AA/vV* t,<* V eF 4 v » ' V V W 1 1- Fig.3-22 • Also, i f V =0 , Ki r c h h o f f f s l a w g i v e s " (, l?0+/?c + ?J+ ((?rt4-reB)+ l H? T= o U,+ tJ(P*i-1?») « - £ F These equations, may be solved, f o r •£* giving 2 « ' IP* + J?b) The t o t a l equivalent input noise current, i n the bandwidth due to the above mentioned sources i s found by taking the square root of the sum of the squares of the i n d i v i d u a l RMS noise currents. The corresponding input ground acceleration i n the bandwidth df i s then — times the t o t a l input noise current. The equivalent ground acceleration i n a s p e c i f i e d bandwidth due to these sources can be calculated by. i n t e - grating the corresponding expression.for bandwidth over the required frequency band. 3 . 3 . 3 D r i f t - i n the feedback seismometer. It was shown i n § 3 . 5 . 2 that the equivalent input •noise associated with the thermal noise in.the seismometer due to damping was unchanged by the application of feedback. In general, the equivalent input noise of a system .remains the same before and,after feedback i s applied?. Thus, d r i f t s due to thermo-elastic e f f e c t s and mechanical, fatigue i n the spring which can be represented as long period input ground accelerations, are unchanged by-the application of feedback. Increasing the period of the s e i s - mometer, by.feedback w i l l , however, increase i t s s e n s i t i v i t y to long term.drift, b u t - i f the d r i f t i s o r i g i n a l l y n e g l i - gible with.respect to long period signals, i t w i l l remain so. 7. Proof i n Appendix C. Chapter 4 EXPERIMENTAL VERIFICATION.OF FEEDBACK- THEORY 4.1 .The Willmore Mk.I seismometer Experimental v e r i f i c a t i o n . of, the theory;presented in. CJhapter 4 was carried out using a Willmore Mk.I seismometer, formerly a standard,in;the.Canadian seismograph network 1. The Willmore Mk.I i s a portable, short - period seismometer, capable of both v e r t i c a l and: horizontal operation. Its suspended mass of 4 . 2 6 :kg i s a c y l i n d r i c a l permanent magnet, constrained to move a x i a l l y along the seismometer axis by f i v e spokes, which also provide the restoring force i n hor- i z o n t a l operation. Two t r i a n g u l a r leaf springs support the mass f o r v e r t i c a l operation. Minor period adjustments are r e a d i l y made to obtain.a resonant•period ;of 1 sec. i n both the horizontal and v e r t i c a l positions. The electromagnetic transducer consists, of a c o i l , fastened.to the base plate of the seismometer, Which enters a c i r c u l a r gap between.the suspended magnet*s concentric pole pieces. In.the instrument used, the c o i l was wound, on.a s p e c i a l .aluminum.former which provided some degree of eddy current damping. Additional damping i s . u s u a l l y obtained.by shunting the c o i l with.a suitable resistance. An external thumb screw enables the magnet to be clamped.against the bottom stop, during trans- portation. 1. Now.replaced, by the Willmore Mk.II. 41 The seismometer was operated solely, i n the hor- i z o n t a l p osition f o r these experiments. It-was calibrated using a Maxwell bridge by the method explained in|l./+. The r e s u l t i n g c a l i b r a t i o n curve i s shown i n Fig.4-1j with the ordinate values adjusted to give the value of is in.ohms. The various seismometer con- stants were M = suspended, mass = 4.26: kg D.= damping constant =14.2 Nsec/m U = spring constant = 16S N/m % = transducer constant = l6l.O v o l t sec/m & - c o i l resistance = 1020X_ Lc. = c o l l inductance2 = 1.1_'h X = damping r a t i o = 0.266 The values of. the r e s i s t o r , capacitor, and i n - ductor representing.the mechanical part of the seismometer, is, in.the current-driven equivalent c i r c u i t (Fig.1-6) were respectively l_"25fl, l65yUF, and 154 h. 4.2 C a l i b r a t i o n and feedback c i r c u i t r y The bridge used to apply feedback to the s e i s - mometer was of course, the same bridge used to calibrate i t . The resistances and capacitance comprising the bridge were measured on a General Radio impedance bridge and, were 2. Ls was found to vary .with frequency. The value given, was determined at., the seismometer r:'e_3op;a£_fcy, frequency, l c p s . 42 I /OO Or 0./ ,710 /0.0 period (sec.)—•— Fig.4-1 C a l i b r a t i o n Curve ( ̂ / w J J . f o r Willmore Mk.I Seismometer. 43 i?B= 102.2k, r?R = 105/}, Rj> = 10.5k, c» = O . l l ^ F . The substitution r e s i s t o r = 10.0k was used i n c a l i b r a t i o n only.. The output from:the bridge was fed.through a switch to an Astrodata nanovoltmeter operated as a f l o a t i n g preamplifier to a Tektronix #502 oscilloscope. The output of a KROHN-HITE-model 440AR: ultra-low frequency o s c i l l a t o r which.provided the c a l i b r a t i n g voltage, was connected through a 40db attenuator to-the switch leading to the preamplifier and oscilloscope where i t could be measured. The preamplifier gain was f i x e d at 100 to avoid bandwidth differences, between.ranges. Since a l l data re- quired, were r a t i o s of two signals, the corner frequency of the amplifier at~ 3 . 2 cps did not affect the r e s u l t s . A block diagram i l l u s t r a t i n g , the application of feedback.to.the seismometer i s given,in Fig. 4 - 2 . It also i l l u s t r a t e s the method-of determining -the acceleration s e n s i t i v i t y of the feedback seismometer explained i n S 3 . 2 . 3 . Vacuum tube operational amplifiers and c o e f f i c i e n t potentiometers of a Heathkit Model C Analog Computer were used;to provide the gain.and. to perform.the required math- ematical operations i n the feedback.loop. 10$ capacitors and.5% r e s i s t o r s were used.with the amplifiers. I t was found.that.these amplifiers acquired a dc offset when confronted with a capacitive load such as the Maxwell bridge. The closing of. the feedback, loop-further enhanced, the problem. This e f f e c t could be overcome if. the 44 feedback c i r c u i t consisted of an integrator, by placing a 330H.resistance i n series with C8 ; The addition, of t h i s resistance did.not .affect.the operation of the bridge. This procedure failed, however, when a d i f f e r e n - t i a t o r comprised the feedback c i r c u i t , and i t was necessary •to-remove Q from the bridge. This did affect.the operation .of the bridge at higher frequencies (above 3 cps) but as the purpose of the d i f f e r e n t i a t o r in-the feedback loop was .to decrease the natural frequency, the absence of C% was not .important. Fig.4-2- Experimental feedback and.calibration schematic. 4.3 Experimental r e s u l t s Several experimentally, determined acceleration s e n s i t i v i t y curves f o r the feedback seismometer with d i f - ferent e f f e c t i v e masses, spring and damping constants are compared with.the o r i g i n a l a c c e l e r a t i o n . s e n s i t i v i t y curve of the seismometer i n Figs.4-3a and 4-3b. The curves of Fig.4~3a .perio d;|sec.J 100.0 a) acceleration (periods greater than 1 sec.) and.displacement (periods less than.1 sec.) feedback Fig.4-3 Experimental acceleration s e n s i t i v i t i e s of feedback seismometer. 4 6 1.0 2 0 IO.O period; [seel — b. Velocity feedback applied t o 2 sec. p e r i o d f e e d - back- seismometer. Fig.4 - 3 Experimental acceleration s e n s i t i v i t i e s of feedback seismometer. 47 withiresonant periods greater than 1 sec. were obtained with negative acceleration.feedback (ef f e c t i v e spring constant increase). The curves of Fig.4-3b display, the c a p a b i l i t i e s of feedback, in.damping the seismometer and were obtained by .adding-negative v e l o c i t y feedback.to the 2 sec. resonant period feedback seismometer. Tables 4-1 and,4-2 contain,various.information about.the feedback seismometers, which.the curves represent. .All, information: present in-the tables, was derived from, the curves and from,the o r i g i n a l , values of the various constants. T.§ i s the resonant period of the feedback seismometer, and i s determined from, the position of the peak.of the accel- eration , s e n s i t i v i t y , curves. While the curves of Fig.4-3a indicate an,increase in,the damping;constant (see Table 4-1 and Fig.3^3) , v e l o c i t y feedback was not applied, to the seismometer and:the cause of. the additional damping i s unknown. One p o s s i b i l i t y i n - vestigated, was that of. an unwanted phase shift, i n the feed- back loop. A phase lag i n the differentiating.network and a phase lead i n the integrating network could introduce a component of v e l o c i t y fe-edback. As w i l l be explained, l a t e r , the d i f f e r e n t i a t o r and integrator used i n the feedback loop have high. and. low frequency, cutoffs respectively, r e s u l t i n g ;in phase differences, of the required sign,to introduce damp- ing. In Table 4-1, <t> i s the magnitude of the phase angle required to produce the damping observed i n the various curves. The values of (f> were calculated from one of ; 0.25 0.5 1. 0* 2 .0 4 . 0 8 .0 17.2 f u[n] 1920 5530 4800- 758 325 258 added , impedance ' L[h.] 10.1 51.4 c|>] - 492 2460 10,400 48,600 M[k»] 4.26* .•4.26* 4 . 2 6 * 17.0 68.2 - 273 1265 e f f e c t i v e ( D [ ^ ] 27.7 - 19.1 -.14.2* ,:19-6 . 4 8 . 4 94.0 •11*5.5 2690 672 • 168* -168* . 168* 168* 168* C 0.129 . 0.178 O .285* 0.184 0.226 . 0.219 ,0 . 125 '[')*• 0.3 0.55 22.84 28.15 16.5 4 . 6 * O r i g i n a l value, f Defined i n text on page -47. Table 4-1 Information pertaining;to acceleration s e n s i t i v i t y , curves of F i g . 4 - 3 a . 49 where AM , AD > AU represent the e f f e c t i v e increases i n each of the parameters M , D , ^ . was found to be too large i n a l l cases except f o r T 0 = 0.5 sec. and 0.25 s e c , f o r which, i t - was too small. <j> i n general was also too large to be associated with phase s h i f t s in.the operational amplifiers used. curve . no ̂  1 , 2 3 4 . 4300 1215 44© 197 added i impedance L[H-] ' — — C \ff]. > MM 492 17.0 ' 492 17.0 . 492 17.0 492 17.0 e f f e c t i v e ( D 19.6 35.5 73.0 146 163* 163* . 163* 16S* <• 0.134 . 0.336 0 .691 . 1.33 t Curves are numbered" * O r i g i n a l value. consecutively from least damped. Table 4-2 Information, pertaining to acceleration s e n s i t i v i t y curves of Fig . 4 ~ 3 b . 50 Since 4> is. small, for. T Q = 0.5 sec. and. 0.25 sec. f o r which CB. was. retained., in. .the. bridge ,. and. large f o r the other cases, f o r which Q, was removed, from., the bridge, i t < i s possible that, the c o i l inductance of the. seismometer com- bined, with-the above p o s s i b i l i t i e s to produce the required ..phase s h i f t s . The d i f f e r e n t i a t i n g • c i r c u i t used, to obtain .the curve.for T Q = 2.0 sec. had.different c h a r a c t e r i s t i c s from..:the others ..which-may/explain, the deviation from..the tendency, of <f> to. increase towards the higher resonant frequencies. 4•4:Integrator, d i f f e r e n t i a t o r and.summing amplifier^ The summing amplifiers...were constructed, with.meg- .ohm. resistances, and the m u l t i p l i c a t i o n s by 10 (recommended ,l i m i t with these.amplifiers) were accomplished with 100k and .. 1 megohm, resistances. Although operational.amplifiers.are t h e o r e t i c a l l y •capable of i d e a l integrations and d i f f e r e n t i a t i o n s , the presence of noise requires.the use of' p r a c t i c a l d i f f e r e n t i a - tors and.integrators. The integrating c i r c u i t used,is shown .in Fig.4~4a.and a- logarithmic:.plot. of the gain i n Fig .4 -4b . The transfer function.of. t h i s integrator i s where X.=x[Cx • While t h i s , reduces to an, i d e a l integrator • only, i f Vi.- <*> , i t . approximates . an: i d e a l integrator f o r •3. See Appendix E.- Transfer functions of operation- a l : a m p l i f i e r s . 51 angular frequencies a) such, that oSTL» \ . ^ l/TJ log u > a) c i r c u i t b) gain Fig.4-4 Integrator used i n experimental work. The values of the resistances and capacitance used were = 100k, *Rv = 1 meg., and •= 2.0/<F. , with : the corner period of 12.4 sec. The d i f f e r e n t i a t i n g c i r c u i t used i s shown,in F i g . 4-5a and a logarithmic:plot of the gain i n Fig.4~5b. a) c i r c u i t . b) gain Fig.4-5 D i f f e r e n t i a t o r used, i n experimental work. The transfer function.Of t h i s d i f f e r e n t i a t o r i s 3> C, S (I +T.s)(l+T s S) where T7 = p;C, , T; » withTf>T1-. This .reduces to an i d e a l d i f f e r e n t i a t o r only, i f = 0 and 1?, - 0, but. approxi- mates .an i d e a l d i f f e r e n t i a t o r f o r angular frequencies such = that 6JT;« I . The values of the resistances and capacitances used were: F| = 100k, R\. = lmeg. , 0.Ol^uF, and C,. was f o r T Q = 2.0 sec. 0.143̂ F,.and f o r the other three curves, l.O^F. The corner period at was 0.09 sec. (11.1 cps) and 0.63 sec respectively f o r the two values of C, , and at-^p-^ was O.O63 sec. (15.9 cps). It was necessary, to keep the corner frequencies of the low frequency cutoff of the integrator and the high frequency, cutoff of the d i f f e r e n t i a t o r out of the frequency ranges of i n t e r e s t . If t h i s was not done, the order of the system, increased and the e f f e c t i v e spring constant and mass became functions of frequency, d i s t o r t i n g the s e n s i t i v i t y curves. While i t . i s possible to obtain d i r e c t feedback with ,a separate operational,amplifier and add i t with a summing amplifier to the other feedback signals to produce damping, combinations of dir e c t feedback, with d i f f e r e n t i a - ted feedback, are possible with one amplifier by shunting R and C,, i n the d i f f e r e n t i a t o r by a r e s i s t o r as shown i n F i g . 4-6a. A logarithmic plot.of the gain^is given i n Fig . 4 -6b . The transfer function i s ft. (> +~r s) P'(n-T,sXi + T 1s) where T = T", + ^'c, . For angular frequencies CJ< ^ , the d i f f e r e n t i a t o r t ransfer function,.is ^ — ^ ^ which gives v e l - ocity,and acceleration,feedback (remembering that the s e i s - mometer output i s proportional to the v e l o c i t y of the mass). 53 log fg* 1"! a) c i r c u i t Fig.4-6 S i n g l e amplifier•producing.combination.of d i r e c t . a n d d i f f e r e n t i a t e d feedback. p The v e l o c i t y feedback c o e f f i c i e n t i s •£/ and v a r i e s w i t h P / , while the acceleration.feedback c o e f f i c i e n t , i s fi-^i^'^) and a l s o v a r i e s w i t h " P , ' . However, 'if. as i l l u s t r a t e d i n F i g . 4-7,^1 i s appropriately, chosen, the damping;may. be v a r i e d through .a wide .range without a f f e c t i n g the a c c e l e r a t i o n feed- back c o e f f i c i e n t .and. Tience • the resonant-period, of. the feed- back .seismometer. I ° 9 ( a c c e l e r a t i o n f e e d - back c o e f f i c i e n t ) 1?,'=!?, log (ve/ocity -feedfcacfe coefficient)- F i g . 4 - 7 ' A c c e l e r a t i o n feedback c o e f f i c i e n t . v e r s u s v e l o c i t y feed- back c o e f f i c i e n t f o r o p e r a t i o n a l a m p l i f i e r of Fig . 4 - 6 a . 4-5 P r a c t i c a l r o o t - l o c u s diagrams Since the d i f f e r e n t i a t o r and i n t e g r a t o r a c t u a l l y used d i f f e r from the i d e a l cases d e a l t - w i t h , i n . t h e theory 54 i n Chapter 3, i t i s reasonable to expect the root locus diagrams to take on a d i f f e r e n t appearance. However, as pointed out i n §4.3, i f the time constants were correctly- chosen, the p r a c t i c a l circuits.behave ideally, f o r the f r e - quency ranges of i n t e r e s t . Thus, while the root locus plots, contain extra poles, zeros, and.branches, the f i n a l r esults are e s s e n t i a l l y the same as shown in.the following f i g u r e s . a) i d e a l d i f f . b) p r a c t i c a l d i f f . * f \ c) i d e a l i n t . d) p r a c t i c a l i n t . Fig.4-3 Comparison of i d e a l and p r a c t i c a l root l o c i . C h a p t e r 5 .CONCLUSION ; I t has been s h o w n . t h a t by a p p l i c a t i o n o f n e g a - t i v e f e e d b a c k , t h e shape and. p o s i t i o n o f t h e r e s p o n s e c u r v e s o f a s e i s m o m e t e r c a n be a l t e r e d a t w i l l , w i t h , r e g a r d ; t o t h e damping and n a t u r a l p e r i o d . N e g a t i v e a c c e l e r a t i o n , f e e d b a c k a p p l i e d t o t h e s u s p e n d e d mass o f a s e i s m o m e t e r e f f e c t i v e l y i n c r e a s e s t h e m a s s , l e n g t h e n i n g t h e p e r i o d , w h i l e n e g a t i v e d i s p l a c e m e n t f e e d b a c k s h o r t e n s t h e p e r i o d by e f f e c t i v e l y s t i f f e n i n g t h e s p r i n g . The damping f a c t o r i s e f f e c t i v e l y i n c r e a s e d by n e g a t i v e v e l o c i t y f e e d b a c k . S i m i l a r r e s u l t s f o l l o w t h e a p p l i c a t i o n of. n e g a t i v e f e e d b a c k t o pendulum s e i s m o m e t e r s i n w h i c h , "moments o f i n e r t i a : a r e c o n s i d e r e d . i n . p l a c e o f s u s p e n d e d m a s s e s . The a b i l i t y t o l e n g t h e n , o r s h o r t e n ; t h e r e s o n a n t p e r i o d , a n d ; hence s h i f t t h e r e s p o n s e band o f a s e i s m o m e t e r b y - t h e a p p l i c a t i o n o f f e e d b a c k w i l l a l l o w a p a r t i c u l a r i n - s t r u m e n t t o be a d a p t e d , t o v a r i o u s r e s e a r c h : p r o j e c t s . The W i l l m o r e Mk. I I , s e i s m o m e t e r w i t h a b u i l t i n . p e r i o d a d j u s t m e n t , has a - p e r i o d r a n g e o f f r o m 0.6 s e c . t o 3.0 s e c . b u t . t h i s . i s f a r , l e s s t h a n t h e c a p a b i l i t i e s o f a s e i s m o m e t e r w i t h . f e e d - b a c k . W h i l e t h e amount o f e l e c t r o m a g n e t i c damping w h i c h . c a n . b e a p p l i e d t o a s e i s m o m e t e r i s r e s t r i c t e d b y - t h e f i e l d s t r e n g t h . o f t h e t r a n s d u c e r m a g n e t , t h e use o f n e g a t i v e f e e d - b a c k e n a b l e s any d e s i r e d d e g r e e o f damping t o be o b t a i n e d . I t a p p e a r s . t h e n , t h a t . t h e f e e d b a c k m e t h o d . m i g h t be an e x c e l l - e n t means o f g r o s s l y o v e r d a m p i n g s e i s m o m e t e r s . t o a c h i e v e t h e 56 broad, band c h a r a c t e r i s t i c s desired, f o r magnetic tape recording. The actual shape of the response curves, aside from ..differences in.damping, can. be altered with feedback i f the r e s t r i c t i o n that the unloaded,feedback seismometer be a sec- ond order system.is removed. The application of displace- ment feedback, through . a., low ' pass. f i l t e r to reduce d r i f t , i n long;period seismometers, as described, i n Chapter 2, re- sul t s , i n a third: order system. In these seismometers, the spring . constant•becomes.a function.of period, increasing with longer periods. The acceleration, response of. the feedback seismometer at., long periods, determined, by the spring con- stant, drops off at a rate greater than.the normal 6db/octave. The use of the Maxwell bridge i n applying feed- back to an electromagnetic seismometer through the trans- ducer c o i l terminals obviates -the. need of a. second, trans- ducer and. i t s necessary c a l i b r a t i o n . The Maxwell bridge also permits d i r e c t c a l i b r a - t i o n . of. the the feedback.seismometer and.seismograph based on. the method of Willmore. The corrections usually.: required . when the bridge u s e d , i n . c a l i b r a t i o n . i s replaced;by another attenuator are not• required, since the bridge.is.a permanent feature of. the feedback seismometer. The f i n a l step in, the theory, presented, here i s being taken by. 0 . G. Jensen, of. the Dept. of Geophysics at U.B.C. in,the building of a seismograph incorporating.the feedback methods presented. BIBLIOGRAPHY •1. Benioff, H., Earthquake seismographs and associated instruments, Advances, i n Geophysics 2, 219-275, 1955. 2. Bohn, E.V., The Transform.Analysis of Linear Systems, Addison-Wesley Publishing Co. I n c R e a d i n g , Mass., 1963. 3. B r u n e l l i , B.E., and V.V. Alekseev, Seismograph system with feedback, Bull.(Izv.),Acad. S c i . USSR, Geophys. Ser., No-1, 524-527, 1959. 4. Byrne, C.J., Instrument noise i n seismometers, B u l l . Seismol. Soc. Am. 51, 69-84, 196I. 5. De Bremaecker, J.C1., P. Donoho, and J.G. Michel, A dir e c t d i g i t i z i n g seismograph, B u l l . Seismol. Soc. Am. 52, 661-672, 1962. 6. Hodgson, J.H.., Earthquakes and Earth Structure, 60-69, Prentice Hall- Inc., Englewood C l i f f s , New Jersey, 1964. 7. Kollar, F., and R.D. Russell, Seismometer analysis using an.electric:current .analog, (Submitted.for publica- tion, i n B u l l . Seismol. Soc. Am.). 8. Richter, C.F., Elementary Seismology, 210-231, W.H. Free- man and Co. Inc., San: Fransisco, 1958. 9. Rykov, A.V., The e f f e c t of feedback on the parameters of a pendulum, Bull.(Izvv)'Acad. S c i . USSR, Geophysl Ser., No.2, 636-64O, 1963. 10. Savant, C.J., Basic Feedback Control System Design, 81- 123, McGraw-Hill-Book Company Inc., New-York, 1958. 11. Sutton, G.H., and G.V. Latham, Analysis of a feedback- controlled .seismometer, J. Geophys.. Res. 69, 3865-3882, 1964. 12. Tucker, M.J.,, An el e c t r o n i c feedback seismometer, J. S c i . Instr. 35, 167-179, 1958. 13J Willmore, P.L., The application.of the Maxwell; impedance bridge to the c a l i b r a t i o n of electromagnetic s e i s - mographs, B u l l . Seismol. Soc. Am. 49, 99-114, 1959• 53 14. Willmore, P.L., Some properties of heavily damped electromagnetic seismographs, Roy. Ast. Soc. Geophys. J. 4 , 339-404, 1961. 15. Willmore, P.L., The detection of earth, movements, Runcorn, S.K., (ed..) , Methods and Techniques . i n ' Geophysics, 230-2,76, Interscience Pub. Ltd., London, 1959. A p p e n d i x A ELECTROMAGNETIC TRANSDUCER • A . l T h e o r y The e l e c t r o m a g n e t i c " t r a n s d u c e r c o n v e r t s ; m e c h a n i c a l energy, i n t o e l e c t r i c a l : e n e r g y by a p p l i c a t i o n o f F a r a d a y » s l a w o f e l e c t r o m a g n e t i c i i n d u c t i o n . A c o i l , o f t e n a t t a c h e d t o t h e s u s p e n d e d m a s s , l i e s i n . t h e f i e l d . o f a permanent magnet . a t t a c h e d , t o t h e s e i s m o m e t e r frame-'-. I f t h e mass and c o i l . .move w i t h a: v e l o c i t y \r" r e l a t i v e t o - t h e f r a m e , an emf 9 t r i s i n d u c e d , a c r o s s , the , open: t e r m i n a l s o f t h e . c o i l where J i s t h e t r a n s d u c e r c o n s t a n t . , and, i s r e l a t e d , t o t h e m a g n e t i c f l u x t h r e a d i n g t h e c o i l and t o t h e number o f t u r n s i n t h e c o i l . I f t h e c o i l i s n o t open c i r c u i t e d , b u t i s c o n n e c t e d t o a r e s i s t i v e a t t e n u a t i n g . n e t w o r k o r i s s h u n t e d by a r e s i s - t o r t o p r o d u c e e l e c t r o m a g n e t i c damping (Av2) t h e o u t p u t v o l t - age i s no l o n g e r gur b u t i s . m o d i f i e d by, t h e c o i l and s h u n t i m p e d a n c e s . I n t h e s e c a s e s , i f t h e i n d u c t i v e r e a c t a n c e o f t h e c o i l i s . n o t n e g l i g i b l e a t t h e f r e q u e n c i e s o f s i g n a l s . b e i n g i n d u c e d . by-- t h e g r o u n d m o t i o n , t h e s e i s m o m e t e r becomes a . t h i r d o r d e r s y s t e m , c h a n g i n g t h e r e s p o n s e c h a r a c t e r i s t i c s d e r i v e d , i n C h a p t e r 1. The e l e c t r o m a g n e t i c t r a n s d u c e r a l s o c o n v e r t s e l e c t - r i c a l e n e r g y i n t o m e c h a n i c a l e n e r g y : a c u r r e n t I f l o w i n g i n . t h e c o i l c a u s e s a f o r c e 9<- t o be e x e r t e d on t h e s u s p e n d e d 1. An a r r a n g e m e n t e x a c t l y - o p p o s i t e t o t h i s i s f o u n d , i n . t h e W i l l m o r e S e i s m o m e t e r . 60 mass of the seismometer. The mass responds to t h i s f o r c e - i n M as. i f the seismometer were subject to a ground acceleration..^. A.2 Electromagnetic damping Damping req u i r e s . t h e a p p l i c a t i o n of a f o r c e to the pendulum, which i s p r o p o r t i o n a l and opposite to the pendulum v e l o c i t y . I t may be obtained e l e c t r o m a g n e t i c a l l y • b y shunt- i n g the c o i l with,a s u i t a b l e r e s i s t a n c e . If. the c o i l of r e - s i s t a n c e I?c i s .shunted, by a r e s i s t o r ^ s then, n e g l e c t i n g the c o i l inductance, a current $+@ w i l l f l o w in;-the . c i r c u i t and.a f o r c e p p w i l l be a p p l i e d to the suspended mass op- posing i t s . m o t i o n 2 . The p o t e n t i a l d i f f e r e n c e across the shunt r e s i s t o r may, then be the seismometer output and, i s Iv^t The electromagnetic damping constant i s $ lf>t and i s maximum, i f the c o i l c i s ' ~ s h o r t . c i r c u i t e d ' . ( f?r = O ) . The maximum amount of electomagnetic damping which can.be a p p l i e d i s p r o p o r t i o n a l to..,the square of the transducer constant, and hence dependant on the c h a r a c t e r i s t i c s ..of, the transducer elements. 2. Note however, i f the c o i l inductance i s c l a r g g o enough, i t can.change t h i s behaviour s u b s t a n t i a l l y . Appendix' B THE LAPLACE TRANSFORM AND,TRANSFER FUNCTIONS B.l Laplace and inverse .la-place transforms - The Laplace transform of a function of time i s defined as oo where s=<r+]u> i s the complex frequency. The inverse Laplace transform of F(s) i s defined as In.practice these inte g r a l s may. be evaluated using,complex variable theory; or the re s u l t s may simply'be extracted, from tables of Laplace transforms. B.2 Transfer, functions The transfer function of a l i n e a r system i s a function of the complex frequency S , and i s defined to be the r a t i o of. the exponential response to an exponential ex- c i t a t i o n . Thus, i f a' sign a l ••ir<e**' i s e x c i t i n g a system of transfer function G(s) the response .will, be •b,(*)vie =u-0e and ^ More generally, consider a si g n a l [-of or t <r O] which can be resolved into a series of exponential spectral terms by the Laplace transform: 62 J*» « w - a n / * * 6 * * where Vifsjs/fy-^ The e f f e c t of each spectral term on the .system.response can be i n d i v i d u a l l y determined. The input spectral term at complex frequency ^ i s VŜ sAjjL̂ fib; and consequently, from, the d e f i n i t i o n . of, the transfer function, the corresponding output term i s ^ ^ ^ f d s - ^dllS-d*. . W S , = Vi(s> ~ T p j i s the general d e f i n i t i o n of the transfer function In words, the transfer function of a l i n e a r system;is.the r a t i o of the Laplace transform of a response to the Laplace transform of the ex c i t a t i o n . The t r a n s f e r function of a system may be deter- mined, by;taking the Laplace transform of the system d i f - f e r e n t i a l equation. A system, with, transfer function (r(s) i s usually represented by a- block in.which the transfer function i s printed. ( F i g . B - l ) . V,(s) G(s) F i g . B - l Block representation of a system with tr a n s f e r function Appendix C FEEDBACK C L The transfer function of feedback systems Consider the system shown i n the block diagram, of F i g . C I . (rfc) and are transfer functions and ViCs). Wo(s) , and E(s) are Laplace transforms of signals at the positions shown. The EG) F(s) F i g . C - l Block diagram of feedback system c i r c u l a r symbol i s a sub- tra c t o r , and indicates that part of the output ( F(s)V+(s) i n transform notation), i s being added negatively to the input. Such a system:is c a l l e d a negative feedback system. From F i g . C - l , V». fs) - E(s) G-(s) and E(^) = V;(s) - F(S)V*(%) Elimination of Ets) gives H(s)» - (^ F(S)6-M where His) i s the tra n s f e r function.of the feedback system and. i s known as the closed, loop transfer function. F(^)^s) i s known as the open loop transfer function.and also as the loop gain. C 2 S t a b i l i t y of feedback systems From a phsical point of view, i f . the feedback sys- tem experienced phase s h i f t s at some value of the constant part of the loop gain such that the feedback s i g n a l would be added to rather than subtracted from..the input, the system. 64 would saturate or become unstable. Such possible phase s h i f t s must be considered in.the design of feedback systems. From a mathematical point of view, the response of a general system with, transfer function F/(s) ;to applied i n i t i a l conditions i s of the form Z.C«e where the s K = +ju>H kit are the poles of H(s) f i e . , H(%)= oo , and n i s the num- ber of such.poles. I t i s seen that associated with each pole i s a c h a r a c t e r i s t i c exponential response € . For the system.to be stable, the c h a r a c t e r i s t i c exponential's must be damped: < 0 . Hence the poles, of Nfc) must-lie i n the l e f t half s-plane. If H(s) - J^TT^SJ f or a feedback system, i t i s re- quired that the roots of \+F(s)6(s)*0 l i e , i n the l e f t half s-plane. This c r i t e r i o n , i s the basis of the various methods of s t a b i l i t y analysis which are i n use. The root locus method i s discussed i n Appendix D and other methods may, be found i n references 2 and 10. C.3 System noise and feedback A system with transfer function C-(s)- Gi(s)<3^(%) N(S) G;(S) Gil?) has a noise source at the position shown.in Fig.C-2, With zero input, the system output i s GrCs)N(s) . This output can be considered the re s u l t of, a si g n a l applied N(s) at-the system, input. This equivalent input noise i s then "ĝ j" Fig.C-2 Block diagram of system with noise. .65 If. a feedback path ...with transfer function F/s) enclosed, the system ..(.Figv C-3), we have, f o r zero input, N(s) E(s» Vfs) F(s) and tfs)- Gvfc)Ms) + E(S)&(S) Fig.C-3 Block diagram,of noisy.system/with.feedback. which i s the output noise due to the noise source IN/(s) , The equivalent•input noise is.now G-Js) N(s) \+ F(s) G-(s) _ N|-(s) which was the r e s u l t f o r the system,without feedback. There- fore, the equivalent input noise of a system, i s unaltered,by the application of feedback, provided.the feedback path.con- tains no-noise sources. It then.follows that.the signal, to noise r a t i o of a? system, i s the same before and. aft e r feedback-" i s applied. Appendix D ROOT LOCUS. METHOD D. 1: Theory The s t a b i l i t y , of: a feedback system with transfer function His;=- / + depends on the pos i t i o n of the poles-^- of Hfe) i n the complex;frequency* plane. (See Appendix C.2). The root.locus method plots the positions of the roots of I + F/s)6V«s) (poles of H(sV ) on the s-plane with the loop gain as a parameter. The equation I + F(s)&(s)- O defining the roots re- duces to. two equations: |F(s)6-(s)| = I o^[F(s)G(s)J= T + awn n= 0,11,12,... The root, locus i s the locus of a l l points. i n ; the s-plane s a t i s f y i n g the phase condition, with the magnitude condition•being used, to determine the value of the loop gain at .any, point on.the locus. As the loop gain,increases from 0 to oc , the locus i n general, originates.at,the 1 poles of F(s)<^r(s) (marked.by X ), and. terminates onthe zeros^ (marked, by .0 ), with the d i r e c t i o n of gain, increase . being given:by arrowheads on:the•locus. The value of the loop gain where the locus.crosses into the rig h t h a l f of the s-plane ( <r = O). i s the largest value f o r which, the system/, w i l l remain ..stable. This value w i l l , r e s u l t in.a steady undamped o s c i l l a t i o n . if, l i n e a r i t y i s maintained. 1. Poles of X^s) are those S f o r which X(s) = 00. 2. Zeros of X (s) are those s f o r which X(s)= O . 67 D .2 £ and 6)« of second order systems from root locus diagrams Consider a system with transfer function a s E(s) _ which may, represent ;the acceleration s e n s i t i v i t y of an :un- loaded electromagnetic seismometer where E"Cs) i s the open c i r c u i t output voltage and YCs) i s the input ground acceler- ation. In.this case, &>s=^pj y £ $ = 2tf\c0s a n c i ^""Mf • Also consider the application of negative acceleration feed- back, f o r example,, to the seismometer as shown i n - F i g . D - l . The open loop transfer function i s As1* Vfs> b s with two zeros at the o r i g i n of the s-plane, and two poles at. F i g . D - l . Block diagram of second order system.with Pythagarus 1-theorem and e l e - d i f f e r e n t i a t i n g feedback loop. mentary trigonometry show the length of a radius vector between the origin.and one of these poles to be C0T , and the angle between.the radius vector and the negative r e a l axis to be cos" 1 ^ . The root locus f o r t h i s feedback system i s i l l u s t r a t e d i n Fig.D-2. As A increases, the poles of the feedback seismometer move along the c i r c u l a r arc comprising the locus from, the poles of. the open .loop transfer function to the zeros. Since the feedback s y s t e m i s of second order, 63 i t s poles are determined;by.an.equation of. the form ..and .are at Ss.and <Ws' , are . the damping ; r a t i o and.angular n a t u r a l f r e - quency of. the feedback, s e i s - mometer and. may; be determined g r a p h i c a l l y , a s . described a- bove once the ga i n A.and hence t h e . p o s i t i o n s of the Fig.D-2 V and < o f sec- ond order system from root l o c u s . poles on.the locus are s p e c i f i e d . (Fig.D-2). For A -O , the poles of. the feedback seismometer are c o i n c i d e n t .with those of, the seismometer before feedback-was applied.and to4 - 60s and = }s Root locus diagrams can:thus be used,to study, the e f f e c t - o f v a r i o u s types.of feedback on.the n a t u r a l , frequency and damping ; r a t i o of. the seismometer. Appendix E OPERATIONAL AMPLIFIERS E. L Transfer function of operational amplifiers An operational amplifier, i s a high gain dc amp- l i f i e r . When a vacuum.tube operational amplifier i s en- closed by a feedback path as i n F i g . E - l , the input to the amplifier becomes a v i r t u a l ground. Since the currents flowing.into t h i s junction must sum to zero, i n trans- \/ v form notation, Js. + _L = 0 if 2, and i s the V,(s) ?,(s) transfer function of the system. If Z, and ? f are resistances "Pi and T?^ , then F i g . E - l Operational amplifier. V, (si r?, which i s the Laplace transform; of u ; ( f ) = - ^ <r,(t) An operational amplifier operated, i n t h i s manner, i s a phase inverter and,multiplier. If 2, and E^.are respectively a resistance and. a capacitance, then, f o r no i n i t i a l charge on the capacitor, Vote) _ I V (s) which i s the Laplace transform of v,(4)* - ̂ Jvitod't , and the system, performs an inte g r a t i o n with a gain of j ^ - . S i m i l a r i l y , i f ?,. i s a capacitance and ? f . a r e s i s tance then r - l ^ C s ..and " o-.tt)--P fC, f^ i f i . Several signals can be combined as shown i n Fig.E 7 0 Fig.E-2 Operational amplifier with several input signals. . with ' • ̂ (a ) . - * v,^)-|^>4 . ^ - | - V , W While t r a n s i s t o r operational amplifiers operate i n a d i f f e r e n t manner, the re s u l t i s the same. E.2 Block symbols of operational amplifiers Three standard, block symbols which appear i n the thes i s are i l l u s t r a t e d i n Fig.E-3. - i r o- CL cur a) summing amplifier b) phase inverter Fig.E-3 Series of standard block symbols. c) m u l t i p l i c a - t i o n by a con- stant

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