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A theoretical and experimental investigation of sinusoidal and relaxation oscillations in thermistor… Morley, Gordon Arthur 1957

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A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF SINUSOIDAL AND RELAXATION OSCILLATIONS IN THERMISTOR - CAPACITOR SYSTEMS by • GORDON ARTHUR MORLEY B.A.Sc, University^of British Columbia, 1955 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of PHYSICS We accept this thesis as conforming to the standard required for candidates for the degree of MASTER OF APPLIED SCIENCE THE UNIVERSITY OF BRITISH. COLUMBIA April, 1957 ABSTRACT It is known that a thermistor displays inductive- reactance and negative resistance at low frequencies when biased with a current greater than turnover current. Thus when shunted by a capacitance in this condition sustained oscillations are possible. These oscillations range from a sinusoidal small - amplitude character, when the system is just above the threshold for oscillation, to a strongly.relaxational type when a large shunt capacitance is employed. The investigation described in this thesis involved a study of the dynamic properties of these oscillations and their relation to the static properties of the thermistor. The relevant thermistor parameters which were measured included the dependence of resistance on temperature, the thermal conductance and the thermal time constant. For the determination of the latter, special very low frequency techniques were developed. Thermistors of high resistance and low thermal time constant were, selected in order to avoid the necessity for very large capacitances in the study of relaxation oscillations. The oscillations were studied over a wide range of capacitances covering the transition from sinusoidal to relaxation type. The voltage extrema and period were measured as functions of capacitance, voltage supply resistance and operating point. Also, in order to elucidate the dynamic processes involved, the transient phenomena produced by abrupt pertubations were investigated. Li parallel with the experimental program, an investigation was.made of the features of the differential equations describing the behaviour of the system, especially for the limit cycles corresponding to relaxation oscillations. The.asymptotic form-of the cycles were derived for the case where the circuital time constant greatly exceeds the thermal time constant of the thermistor. The complicated nature of the equations precluded a solution in a closed form and approximational methods were found to be necessary. However, in the case of sinusoidal oscillations of small amplitude, the period can be exactly expressed in terms of the system parameters (Burgess, Nov. 1955) and this result was confirmed experimentally. In presenting t h i s t h e s i s In 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 at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 reference and study. I f u r t h e r agree that permission f o r extensive 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 . I t i s understood that 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 gain s h a l l not be allowed without my w r i t t e n permission. The U n i v e r s i t y of B r i t i s h Columbia, Vancouver S, Canada. Department ACKNOWLEDGEMENTS I should like to express my gratitude, to my thesis director, Professor R. E . Burgess, for his help in the preparation of the material in this report. This thesis was facilitated by .Defence Research Board Grant 9512-22 while I was on educational leave with financial assistance from Canadian Armament Research and.Development Establishment. iv. TABLE OF CONTENTS Page PART 1 INTRODUCTION 1 PART 2 STATIC PROPERTIES OF THERMISTORS 3 1. Physical Properties 3 2. Static Voltage -current Characteristics 5 2.a. General 5 2.b. Definition of Thermal Conductivity 6 2.c. Properties of Static Characteristic 6 3. Experimental 10 PART 3 SMALL - AMPLITUDE TIME VARYING PHENOMENA IN THERMISTORS 13 1. . Definition of Thermal Time Constant 13 2. Sm all Signal Differential E quation 14 3. Response to Sinusoidal Input 15 4. Response to Step Input 17 5. Small-amplitude Sinusoidal Oscillations 19 ,6. Experimental 20 PART 4 THEORY OF NON LINEAR OSCILLATIONS 24 .1. General 24 2. Phase Planes 24 Page 3. . Singular Points 25 3.a. General 25 3.b. Special Cases 27 3.c. Stability and Classifications of Singular Points 28 3. d. Examples of Singular Points 28 4. Oscillations 30 4. a. General 30 4.b. Relaxation Oscillations 31 PART 5 STUDY OF OSCILLATIONS IN THERMISTORS 32 1. Experimental Techniques 32 2. Limited Applicability of Linear Theory of Oscillations 33 3. Phase Planes for Thermistor Oscillations 34 3.a. General 34 3.b. The V - I Phase Plane 35 3. c. The V-»9< Phase.Plane 38 4. Division of the Period of Oscillations into Epochs 40 4. a. General 40 4 .b. Charging Epoch 40 4 .C . Discharging Epoch 41 BIBLIOGRAPHY 46 vi. LIST OF FIGURES Facing Page 2.1 Photographs of bead-type thermistors 3 2.2 .Dependence of specific resistance upon inverse temperature of typical thermistor material 3 2.3 Application of current to thermistor 5 2.4 Turnover of static characteristic 8 2.5 Static chaxacteristic for ideal thermistor 9 2.6 Dependence of resistance upon inverse temperature 10 2.7 Circuit used for static measurements 10 2.8 Static current - voltage characteristic 11 2.9 Dependence of thermal function upon current 11 2.10 Dependence of excess temperature upon applied power 11 3.1 Current - voltage loci with applied sinusoid 16 3.2 Small amplitude a c equivalent circuits 16 3.3 Loci of impedance function 17 3.4 Circuit for step input 17 3.5 Intersections of load line and static characteristic 18 3.6 Circuit for oscillations 19 3.7 Bridged T circuit for measuring impedance 21 3.8 Circuit for observing current - voltage loci 21 3.9 Photographs of current voltage loci 21 v i i . Facing Page 3.10 Response to step input 22 5.1 Circuit for observing oscillations 32 5.2 Dependence of period upon capacitance at an operating point 34 5.3 Dependence of amplitude of oscillation upon capacitance 34 5.4 Deduced form of closed trajectory 36 5.5 Photographs of I - V plane showing spirals 38 5.6 Hypothetical trajectory during extreme relaxation oscillations 39 5.7 Photographs of voltage and current as functions of time during oscillations 40 5.8 Period of oscillation as a function of capacitance for different load resistors 41 5.9 Period of oscillation as a function of load resistance 41 5.10 Drawings of voltage - excess temperature phase plane 43 5.11 Drawings; of excess temperature - rate -of - change -of - excess - temperature phase plane for condenser discharging into a thermistor 44 5.12 Photographs of current - voltage plane during condenser discharge through thermistor 45 PART 1 INTRODUCTION The thermistor is a two terminal device possessing a high negative temperature coefficient of resistance and a large thermal inertia. Burgess (Oct. 1955) has shown that under certain conditions of d - c bias, the thermistor will exhibit an incremental admittance consisting of a negative conductance and negative susceptance. In a later paper, Burgess (Nov. 1955) also has shown that if a positive susceptance of equal magnitude, in the form of a capacitance, be connected in parallel to the thermistor, small-amplitude sinusoidal.oscillations will be sustained if the d - c source conductance is only slightly less in magnitude than the negative incremental conductance of the thermistor. If the magnitude of the parallel capacitance is increased, the oscillations become relaxational and do not appear to bear treatment in terms of the Van der Pol equation. This paper presents the results of a theoretical and experimental investigation of these relaxation oscillations. Preliminary to the investigation, existing theory of the thermistor and some general remarks on non linear vibration theory are given. In. Part 2 the form of the static current - voltage characteristic is deduced and the results of measurements of its parameters for certain thermistors are quoted. Part 3 concerns small - amplitude time varying phenomena in the thermistor. It essentially reproduces the two papers.of Burgess and relates how the small - amplitude a - e theory was used to measure the thermal tine constant of the thermistor. Part 4 gives some definitions and theorems of non-linear vibration theory which are employed in the investigation. Part 5 presents the investigation of relaxation - type oscillations over a wide range of the value of the parallel capacitance. F I G U R E 2-1 P H O T O G R A P H S O F B E A D - T Y P E T H E R M I S T O R S H G U R E 2,2 DEPENDENCE OF SPECIFIC Rfc5>\STKNCE UPON «KVSftSE INVERSE TEMPtRATOSE DESCENT 3. PART 2 STATIC PROPERTIES OF THERMISTORS 1. Physical Properties Thermistors are thermally sensitive resistors whose variation in resistance is characterized by a high negative temperature coefficient. They are generally made by heating compressed powders of semiconducting materials to a temperature at which they sinter into a compact mass and then firing them on metal - powder contacts. Semiconductors are substances whose electrical resistivity at or near room temperature is much more than that of typical conductors and much less than that of typical insulators. In general, around room temperature these materials when pure, have negative temperature coefficients of resistivity of about 5% per degree centigrade compared to platinum and copper which are typical conductors and have positive temperature coefficients of about 0.4% per degree centigrade. Figure 2.2 is a graph of the logarithm of specific resistance versus reciprocal absolute temperature of a typical thermistor material. This graph indicates that the logarithm of the thermistor resistance R varies approximately linearly as the reciprocal of the absolute temperature of the thermistor T. Thus for a given thermistor one may write lOg R f t / - J r , or In R = J l constant, T 4 . or finally b R : Roje T (2.1) where, R = thermistor resistance T = absolute temperature of thermistor b =• constant equal to the slope of In R - vs - -rp- graph e = naperian base R,o = value of R where projected curve crosses the. line 4p- = 0. It is apparent that b is a characteristic of a given material since a plot of log R versus -ip has the same slope.as.a plot of the logarithm of specific resistance versus ~ . The dimension of b is temperature and it is generally specified in degrees kelvin or degrees centigrade. It plays the role of activation energy in (2.1) similar to the work function in the equation for thermionic emission. It is apparent that R^ is dependent in general upon the type of material and the physical construction such as the size of the element and area of the contacts of a given thermistor. If the dependence of In R upon —^- be.carefully examined, it may be found ( Becker et al 1947) that the slope increases as the temperature increases, thus a more precise expression may be d R = A e "T^ (2.2) T c where A , c and d are empirical constants. The constant c is generally a small positive or negative number at zero. Equation (2.1) will be used for the model in this paper. FIGURE 2-3 A P P L I C A T I O N OF C U R R E N T TC> THERMISTOR T I M E - H Q R 1 2 - O N T A L V O U . T A & E - V E R T I C A L 5. 2. Static Voltage - Current Characteristic (2.a) General It is convenient here to introduce the variable t& which is defined as the excess of the temperature T of the element above the ambient  temperature Ta, or & = T - Ta (2.3) If a current I is passed through a thermistor and Ta is held constant, then after sufficient time the voltage at the terminals of the thermistor V will reach a steady value (see Figure 2.3). If the current is increased with Ta held constant then the applied electric power will increase causing an inaease in the excess temperature of the thermistor. Thus a series of points (V, I) may be obtained which define a curve called the static characteristic. If a current I is passed through the thermistor and the electric power is prevented from increasing the temperature above the ambient Ta, then since T = Ta+10* and fO* = 0, the relation X = R e -Ta— I is valid for all the values of V and.I. The points (V, I) again define a characteristic curve. This curve is called the isothermal characteristic or simply isothermal . It is apparent that in the V - I plane an isothermal is a straight line the slope.of which depends.upon Ta. Thus, the thermistor may be considered a linear circuit element device if its temperature is held constant. In practice the isothermal condition may be observed by increasing the cooling apparatus or by applying the power in pulses . 6. (2 .b) Definition of Thermal Conductivity It will now be assumed that the dissipated power in the steady state is a function of the excess temperature •©* The. balance of the heat supplied and the rate of cooling gives VI = f (T - Ta) . f (td» ) . (2.4) The term on the right f (1©I ) is the rate at which heat is lost from the thermistor. The form of f (|£l) has been given (Bollman and Kreer, 1950) as the sum of a thermal conductance term and a radiation - loss term, resulting in, f (m ) = W 4- k R £.(Ta + »©• ) 4 - T a 4 j , (2.5) where k is the thermal conductance of the thermistor and k R is the radiation coefficient of the thermistor. The. dimensions of k are power per unit temperature and the units are generally watts per degree centigrade. Over most measurable temperature ranges either the second term of (2.5) is generally negligible with respect to the first term or at least the dominant factor of the second term is 3 k Ta iQt . In the model in this paper the relation VI = Jcie», (2.6) expressing Newton's law of cooling will be assumed. Combining equations (2.1) and (2.6), one obtains the following static characteristic -f •= R - e s > • <2-7> k (2 .c) Properties of the Static Characteristic There are several interesting properties which can be obtained from (2.7). To facilitate this, several new functions-.will be.defined. 7. The temperature coefficient of resistance jx, will be defined by the relation ^ R ( d * ) _ ' (2.8) which upon calculation from (2.7) gives T 2 ( T a W p (2.9) The thermal function <^  will be defined by t£ = bt& (2.1Q) (Tatie<)2 and is a dimensionless quantity which can never exceed b . It is noted 4 T a that since VI = kid* both ^  and <|> may be written as functions of or as functions of VI. The slope r of the static characteristic is r = dV = = T 2 - b(T - Ta) R (2.11) dl 1 +4> T - 2 b(T - Ta) and is the ac resistance at zero frequency for any operating point. The first part of (2.11) may be rearranged so that <(> = R j j t (2.12) R+r At the origin of the V - I plane, V = I = >^ •= 0 and b r = R^© e Ta =Ra the zero-current resistance of the thermistor at the ambient temperature T a . Thus the static characteristic at the origin is tangent to the isothermal corresponding to T = Ta. .As. Lis. increased ^ becomes larger until finally it may equal unity. At the point where <j> = 1, . r = 0 and T = T t , the F I G U R E 2 4 T U R N O V E R OF STATIC C H P\R ACT E R 1ST IC VOLTAGE V 8. phenomena of turnover is observed since this corresponds to the voltage maximum in the static characteristic (see Figure 2.4). The value T = T t is the thermistor temperature at turnover T t = b - (b2 - 4b-Ta)V (2.13) It is noted that turnover will only occur if b >4 Ta. If the ambient temperature is 300 °K and b has a typical value for thermistor materials of 4000°K, turnover will always occur. If b » 4Ta, then T t = Ta( l+Ta_) b is valid. The.voltage minimum at Tat b is of mathematical interest only since this temperature is well beyond the melting.point of materials used in Ta 2 thermistor production. The turnover condition T t - Ta = —— -2T 20°C is physically realizeable and is of great interest in many properties.and applications of thermistors. T a 2 b The following relations are valid for T t = Ta + VI = (VI)tGTi T a 2 b R = R t 2f R a e " 1 , The point of inflexion of the static characteristic where the slope^  r reaches its maximum negative value is at the point where d / d V V ^ 0 (2.14) / d V ] * 0 dT V dl / Putting x = T - Ta and c = _b_ and T = Ti at the point dr_ = 0, it is found Ta Ta dT FIGURE2-5 S T A T > C CHKR^TER»»TIC FOR I D E A L T H E R M I S T O R F O L L 0 / / i N c S V O L T A G E 9. that ( X i+ l ) 3 (xi - 3 ) + c 2 x i 2 = 0 , If X i « A then 8 and resulting in or and also c ~ TJ , 4 „ — J — T _ _ •) X i 43 J6»i = 3Ta: J3b - 4 Ta J3*- 1 R t ST R i e^ = 2.16 R i RiCT Rae * = .18Ra Ii Cl It ( e ^ " " ' 1 ^ = 1-93 I t <. These relations occuring for an ideal thermistor following — - = R «0 exp b k under the condition Ta >^ b may be summarized in the following table and in Figure 2.5. Summary of the Properties of Ideal Thermistor Characteristics R r 4* Comment « T a Ra(l+£) Ra i l ? 1 Origin T a 2 T a 2 Rae _ 1 0 1 Turnover .2 ^— Rae"''3 - Ra SL, J T Inflexion b (2+^3) b b b Ta Rae 3~Ta - Rae 4 T f l — — Maximum * A c l 4Ta FIGURE 2-6 DEPENDENCE O F RESISTANCE UPON INVERSE T E M P E R A T U R E FOR A V E G D T Y P E £>5M T H E f c * 4 ( S T O R - i i 1. XNVERSE T E M P E R A T U R E X J O ~ 3 r FIGURE ZH CIRCUIT USED FOR STATIC MEASUREMENTS THERMISTOR ANMETB* VOLTMCTfiR 3. Experimental Data were obtained for the temperature variation of resistance of several different thermistors. The thermistor under measurement was placed in a bath of hot transformer oil in a vacuum flask. Simultaneous readings of resistance R from a Wheatstone bridge connected to the thermistor and of the temperature T from a thermistor immersed in the oil were taken while the oil cooled to room temperature. The Wheatstone bridge was operated so as to pass negligible power into the thermistor and the thermometer was immersed so that its bulb was at the same depth in the oil as the thermistor bead. Figure 2.6 shows the graph of log R vs T "1 from a set of readings. Over the temperature range measured no variation from a straight line was observed. Thus, the equation b R = R ^ e T (2.1) is adequate. Measurements of the static characteristic were made using the circuit shown in Figure 2.7. In the V - I plane the equation E = I R i f V gives the load line. Its intersection with the static characteristic is the operating point ( V 0 Jo )• • R l w a s composed of a variable resistance.and a fixed safety resistance. E was a variable - voltage power supply. Figure 2.8 shows a graph of the static characteristic for a Victory Engineering Corporation (VECO) type 65A1 thermistor insair. Values of R were calculated at various points on the curve and corresponding values of were calculated from the log R - versus . - T * graph. Values of <b were calculated from the values of Id* by the relation ZOO ISO' FIGURE 2-8 STAT/C CURRENT-VOLTKGfc\ CHARACTERISTIC FOk V t C O 45M TUERMlvro* < H h Z UJ c£ cc Z> O XNFLSCTION So-o -r VOLTAGE V THERMAL FUNCTION df> FIGURE2.10 DEPENDENCE OF EXCESS TEMPERATURE VPON EXCESS TEMPERATURE OEG C £ * T 4 • b (Ta+ie*)2 (2.10) Figure 2.9 shows the dependence of <j> upon the thermistor current I. From the values of (V , I) at various points along the static curve, values of the power VI were, calculated and a A graph was made of the power - excess temperature relation along the static curve (see Figure 2.10). For the equation VI = kie« (2.6) to be valid, this curve must be straight line. Deviations from a straight line might be due to: 1. Uneven heating of the thermistor bead resulting in the non-applicability of the log R vs T _ 1 relation measured with T = Ta for T ^ Ta. 2. Heat losses due to radiation at high temperatures of the bead necessitating consideration of second or higher order terms in of the relation VI. = k»6i+ k R £ (Ta«M9l) 4 - T a 4 J (2.5) 3. Heating of the surroundings of the bead to such an extent that changes in the effective ambient temperature.are not proportional to the excess temperature. 4. Heating of the surroundings of the bead resulting in a change of the thermal conductivity k. It was noted that VI - vs - curve had an increasing slope with increasingf^l . Items 2 and 4 could have caused this. Values of b, and average slopes of the VI vs Id* curves for several thermistors were: 12. milliwatts Thermistor R ohms b uK k oK Surroundings Servotherm 1317 3.15 4160 1.1 Still air VECO 65A3 3.11 4320 0.13 Still air VECO 65A1 2.45 4310 \ .0094 Vacuum - as supplied by, manufacturer ( ( ( 0.12 Still air - with bulb opened to admit atmosphere It was noted that the manufacturer's specifications give a value of 0.1 milliwatts per degree centigrade for k for both VECO type 65A3 and 65Al. This agrees with the measured value for the type 65A3 in still air. However, in the case of the type 65Al which is in a sealed evacuated glass bulb there is a discrepancy unless the glass bulb is opened and the bead comes in contact with still air. PART 3 SMALL AMPLITUDE TIME. VARYING PHENOMENA IN THERMISTORS 1. Definition of Thermal Time Constant If the non-steady-state.condition is, considered, it is evident that the balance of power can no longer be represented by the equation VI = k»©» but rather the input electrical power must equal the sum of the dissipated power plus the rate at which thermal energy is being supplied to the thermistor. If the_rate at which heat is lost is determined only by the instantaneous excess temperature ^ , the balance condition may be written VI = k e i + H 4^ ? (3-1) dt where H is the heat capacity of the thermistor at temperature Ta+l©* „ .Several cases where the rate of heat loss is not dependent only upon l& are considered by Burgess (Oct 1955). However, in the development here, the validity of (3.1) and the independence of H upon t& will be assumed. If a power V G I 0 = k *©* 0 n a s been.applied to the thermistor for such a time that all transients have died out and then the thermistor is open circuited, the excess temperature will be given by \Oi = t&0 exp ( - k t ) H (3.2) The quantity H_ is called the thermal time constant of the thermistor and will k be denoted by*G . Equation (3.1) may be rewritten as VI = k (te« +^dH&«) . dt 2 . Small Signal Differential Equation The quantities v, i and 6 will be defined by .v = V - V 0 ) ) i = I - I o ) ) 9 = i8»-»e»o ) (3.3) (3.4) where Vo. I 0 o are the values of V, I and at some operating point. The quantities fa o, JUo and R-o are the respective values of <fc , and R at the operating point so that (3.5) Vo = Ro •= RoO exp Tafl©J Io Vo Io = kfdfe fa = lAoW> -- b r&Q (Ta+ieb) 2 If the first equation of (3.5) is expanded in a Taylor's series in 0 , it is seen that It the condition I o + i vi«C V 0 I 0 R 0 + e M + g £ £ : holds, only the first order terms.of the expansion need be considered. Solving thus for © , it is found that 6 1 , v _ i ( To 17 ) (3.6) 15. and d t M » ( T„ * h % u (3'7) Substituting (3.6) and (3.7) in (3.3) and expanding VI in terms of v and i arid keeping only first order terms it is found that A* Vo dt ^ dt • or since V 0Io = kt**©.and. <^0= / i <» V ° ^ k (3.8) _y_ (l + <^o)+ X_ dv =J^ (l-4b)+^ di_ # Vo Vo dt IQ Io dt This is the general differential equation governing small changes in V and.I. 3. Response to Sinusoidal Input If v = Vi.e and i = Ii e Jw* where and Ii may be complex and where w » 2 7t x frequency f and if Z » — , it is found from (3,8) that l l Z = R 0 f i r ± o ± ^ * l (3.9) is the relation for the small - signal driving - point impedance of the thermistor at any operating defined by R 0 and 4>o* If Z = R (w)+ jX (w) it is found that R(w) = 1 - 4>o 2 * w 2 X 2 Ro (1+<^«) 2 + W 2 T * 2 (3.10) and X(w) = 2 ^ w t Ro ) 2 t w ^ 2 " .(3.11) <r=Z-Seps T - - 5 ^ p S V--.\CHPS Ti.«Z3°C H O R I Z O N T A L t o n / ' I O V VERTICAL lDlV = I O F I G U R E 3-1 CURRENT VOLTAGE LOCI WITH APPLIED SINUSOID FIGURE 3.2 SMALL AMPLITUDE AC EQUIVALENT CIRCUITS OF THERMISTOR If w >> 1 it is clear that R (w) = R 0 and X (w) = 0. This is the condition that the applied power is varying so rapidly that the thermal inertia of the thermistor causes the excess temperature to remain essentially constant over a whole cycle at the value if vi << Vo Io • The impedance then corresponds to Ro, the reciprocal of the slope of the isothermal passing through the operating point. If w t « l it is clear that R(w) = Ro (I -<fro) = r (i+<fc>) and X (w) = 0. This is the condition that the applied power VI is varying so slowly that the temperature essentially reaches its equilibrium value at each point of the cycle, thus the current - voltage locus follows the static characteristic and the impedance is equal to the slope of the static characteristic at the operating point. At medium frequencies where w is neither much larger nor much smaller than , the current will lag the .applied voltage and the current - voltage locus becomes an ellipse. Photographs of current - voltage loci at different operating points.and frequencies are shown in Figure 3.1. Several equivalent circuits are immediately apparent from the form of the impedance function and are shown in Figure 3.2 . Regarding the impedance equation Re FIGURE 33 LOCI or FIGURE 3-4 C I R C U I T F O R S T E P I N P U T #1 THERMISTOR it is seen that its representation in the Z plane is a series of semicircles Ro with <|>0 as a parameter (see Figure 3.3). The infinite frequency value of Z is real and has the magnitude unity. The zero frequency value is rp_ Ro Ro and is also real. If ^>©>l and hence ro < 0; i.e. the thermistor is biased beyond turnover, it is noted that R (w) becomes negative for all frequencies below some critical frequency f G = JVo_ .= (4>o - 1) » (3 ..12) 2 Tt "^Kt At this frequencyJEQ the real part of the impedance is zero and the thermistor behaves like a pure, positive reactance given by J- J. Z(w Q) : j R Q /4>ft- lV"= j ( - r 0 R 0 ) x . (3.13) 4. Response to Step Input The response of the thermistor to a small step function.of voltage will now be considered. Since linear theory will be used, the value of the change of voltage e s applied to the thermistor and load resistor Rj in series must be small enough that e s v o holds. The circuit will be.of the form shown in Figure 3.4. The relation e s .-. i R 2 + v (3.14) will hold for this circuit. Combining this with the general equation dv + v ( l + & ) = R 0 f»CdL4 i ( l " <fro)"l (3-8) dt L dt J V O L T A G E V V O L T A G E V V O U T A G E V FIGURE3.5 INTERSECTIONS OF LOAD LINE A N D STATIC CHARACTERISTIC a first order differential equation in i is obtained which may be easily integrated to give for the pertubation of the current i •= es,(l44>o)  Ro (l-$o)+R* (If 4b.) constant exp - PR0 ( 1 - )+ R l (If (fro ) 1 t. L R 0+ R l J If X is defined by t ' = X (RQ-» R i ) R 0 ( i - 4W-)+ R i (i+4tf and the initial condition i = e s Ri+Ro at i = o is applied, the equation for the current i becomes e c / i x i , o A_ n —A. l « (l+<fc> -2<t>oR0 « * % ) , Ro ( 1 - ^ o H R l ( i H o ) R 0 + R l (3.15) It is interesting to note that if R i < - R o a -••) •= - r 0 (i+4.) then and the exponent in (3.15) becomes positive so that the circuit becomes unstable. If the thermistor is biased at a point on the static characteristic where R^< - r Q , the load line will cross the static characteristic in at least one other place (see Figure 3.5) and any pertubation e s will cause the system to jump to one of the other intersections of the load line and the static characteristics For certain values of E and Rj, the upper intersection exists only theoretically since the hypothetical value of the corresponding temperature is above the melting point FIGURE 3.6 C I R C U I T FOR O S C I L L A T I O N S —r~i — i — 12 ^ = C A C T U A L CIRCUIT A C EQUIVALENT of the thermistor material. If the thermistor tends to jump to such a point it of course burns out. If = - r 0 , the denominator of (3.15) vanishes indicating that under this condition the linear terms.are insufficient and that higher order terms must be considered in the expansion of V, I and l©< . A similar equation to (3.15) involving the change of voltage V may be developed and is v = Ro (1 - 4>0 + 2 <boRi \ R! (l+ 4> 8)+R 0 (l . -f lb) R o T R 2 (3.16) 5. Small Amplitude Sinusoidal Oscillations The circuit to be considered is shown in Figure 3.6. The development will follow that of Burgess (Nov 1955). The thermistor will be assume d to be biased beyond turnover and the amplitude of oscillation will be assume d to be small so that the linear equations will be valid. The admittance of the circuit is of the form Y T = - i + jwC + _*_ ( 1+ $o+ jwTf \ ^1 Ro V 1 -4>e+jwt: / (3.17) or if then G r(w)+ j B T(w), 2 . 2 . 2 w t 2*. 2 G T ( W ) = RT+i- f 1 4 •» % ¥ Bp (w) = wC - -L ( 2w % <fto R o 1 (1 -4>o)W<t2 The conditions for oscillations to occur are G j (w) £ 0 Brp (W) X 0 (3.18) For this development to be valid the oscillations must be small so that non-linearities are small. Thus the first relation of (3.18) should approach the equality as closely as possible. Applying equations (3.18) to the admittance in (3.17) it is seen that C m i n = r / 1 + 1 \ (3.19) 4> e-lUl Ro/ l and w An upper limit of oscillation frequency corresponding to w m a x occurs when the d - c supply is-a perfect constant - current source; i.e., — = 0, and Cmjn R o has its lowest possible value C ~* mrn (T4V i) Ro and X ™max - ( <^Q2 - 1 ) X It is noted that w m a x corresponds to the w Q defined in (3.12). 6. Experimental Early in the programme of research, attempts were made to verify the semi-circular locus of the impedance function and to measure the thermal time constant of the thermistor flakes in several Servotherm type 1317 bolometer units. Several different types of bridges were set up but found to be unsuccessful because of the difficulties associated with the high impedance levels and high voltages involved in these thermistors. Finally,. a bridged - T C I C\ I D C % n BRID&ED T CIRCUIT FOR r l b U K L ^ - / INSURING- I M P E D A N C E FIGURE 3-8 CIRCUIT FOR OBSERVING CURRENT-VOLTAGE LOCI ft. L o F«e<? f OJJ ^ S A M P L I N G <ieU««ATo«S""^ ^ R6SISTAWC6 — -3 i . . . . FIGURE 3.9 V4- 6 C v P H O T O G R A P H S O F C UR-^EMT-MOLTAGe L O C I T O SWOW E F F E C T C F CHkK'&E 0 £ F R E Q U E N C Y O F SIG-NAL T H E R M I S T O R IS 8lA^£t> B E Y O N D T U R N O V E R filter of the type shown in Figure 3.7 was chosen. In this bridge: the resistors Rg, Rj^ and the supply voltage E determine the operating point ( V 0 , IQ ) . None of E , Rg and Rj__ need be varied during a - c measurements. Measurements of the impedance were made over a range, of frequency from ,5 to 100 cps. At: several different operating points the measured impedance was essentially the value of the reciprocal of the slope of the isothermal through the operating point; i.e., Z (w) = R 0» X = 0. This led to the conclusion that manufacturer's value of 0.01 sec for X was much smaller than the actual which must have been at least 0.1 sec. Later in the research programme the thermal time constant was measured of a V E C O type 65A1 (with its glass tube opened), using two different methods. The first method consisted of applying a variable-frequency, small-amplitude, a - c signal to the thermistor biased beyond turnover and observing the voltage - current locus on an oscilloscope as the frequency of the generator was varied (see Figure 3.8). The loci were ellipses whose axe! s 1 direction varied with frequency (see Figure. 3.9). At the frequency where the axes were, vertical and horizontal the thermistor behaved like, a pure reactance and the relation i fo ( W " I ) * (3-12) or t = j _ ( W - n 2TCfo was known to hold. The value of *£ at different operating points were FIGURE 3-10 R E S P O N S E TO S T E P I N P U T W H E N T H E R M I S T O R IS B I A S E D B S V O N D T U R N O V E R Io (^Amps) TT (see) .90 .52 100 .61 125 .56 150 .63 200 .63 250 .62 300 .64 The other method consisted of biasing the thermistor beyond turnover and applying a small and instantaneous pertubation e s to the supply voltage (see Figure 3.4). The change of voltage v across the thermistor follows e s R Q .(!.-»»+"2 ^a.R] e f ) T V©  2 j o Ri e ~ £ R l (!+•<» )+Ro(l -•©) R o + R l (3.16) At time t = 0, v will jump from v = 0 to v - e s R Q Rl+Ro As.t increases, v decreases (see Figure 3.10) to a negative value if <^©^ !• Then the condition v = 0 will occur at t = to given by 2 <feoRi Ro + R l 1 -  4> e ^ = 0 , or t - f R i ( l+ * » ) 4 R„(l -•tf ) 1 t o t R l -r R o J In In 2 (Ro+ R i ) ( 4 - i ) (3.21) Using (3.21) me value of t was calculated at several operating points 23. Io (y^Amps) *X (sec) 50 .67 100 .64 200 .58 Averaging the values of % measured by both these methods gives % - .61 sec compared;to the manufacturer's value of *t - 1 sec. PART 4 THEORY OF NON LINEAR OSCILLATIONS 1. General In this section, the various methods of examination of systems which can be described by two first - order differential equations of the form dx - P (x, y) dt dy. = Q (x, y) dt (4.1) will be discussed. This is an autonomous dynamical system since P and Q are not explicit functions of time. It is noted.that the general second order differential equation d2x ^ dx f (x, d x ) + g (x, dx} s © (4.2) dt 2" dt dt dt can be transformed to (4.1) thus dx = y dt dy = f (x, y) - g (x, y) 4 dt 2. Phase,Planes Suppose a solution of (4.1) exists of the form x = x(t) y r y(t) Then since this is a dynamical system with two degrees of freedom and since the. state of the system at time t is fixed by the values (x, y), these two quantities may be considered as coordinates of a phase plane . To each state of the system there corresponds a point M (x, y) which is known as.a representative point. As t varies M will describe a curve called.a trajectory or integral curve which is a description of the history of the system. A complete trajectory, of course, represents the history of system throughout all time . The totality of all paths of M represents all possible histories of the system, any one of which is determined by a single point. Thus with the possible exception of the intersection of the curves P = 0 and Q - 0, only one path may pass through any one representative point. The velocity of the point M along a trajectory is the phase velocity of the system. It is.a plane vector with P and Q its components in the y and x directions. Its direction is given by arc tan P(x, y) = arc tan dy Q(x, y) dx at every point where. P and Q do not vanish simultaneously. The.locus of points where dy is constant is called an isocline. The points where P and dx Q vanish simultaneously are called singular points. 3. Singular Points 3. a General Since singular points are at the intersection of the.curves P = 0 and Q = 0, all velocities of the system vanish and the system is in a rest or equilibrium position there. The nature of a singular point reveals much qualitative information regarding the solution of the differential equation, thus some general theorems regarding singular points will be quoted. Poincare (1892) has shown that the differential equation dy e ax+ by+ P,2 («,y) dx cx + ;dy + Q2 («> y) in which A = ad - be ^ 0 and.in which P 2 and Q 2 approach zero like x 2 .4. y 2 n a s ^ o n i y singularity at the origin and the behaviour at the origin is similar to that of the linear differential equation dy = ax +• by dx cx+dy * (4.3) If a differential equation is of the form dj/ 1 = A + ax-1+..by1 dxl B+cxl + dyl ' (4.4) the only singularity is at the point (x 0 , y o) where I-A b I la - A I x° " J-B d j yo = /c - B1 and A = If the transformations | c d I . x = x 1 - x 0 y - y 1 - y 0 are made (4.4) is reduced to the form of (4.3). Since Poincare (1892) has shown that the nature of a singularity is preserved in undergoing linear transformations, there is no loss in generality in considering (4.3) instead of (4.4). The general solution of (4.3) is (x -cC) y)^' (x - y ) ^ * -constant (4.5) 27. where b+c ± J(b - c) 2 - 4 ad (4.6) and o(%± - b - c ±4(b-c) 2+4ad 3 .b Special Cases of - ax+by  dx cxHh dy (4.7) It can be shown (Stoker 1950 Chap. Ill) that any form of (4.3) can be reduced to certain special cases by linear transformations of x and y. These special cases and the names given to their singularities are listed in the following table. 1. 2. Condition a n d - 0 bc>0 m = b c a - d - 0 bc<0 Differential Equation = m I dx x dy r -n • Z. dx X Solution T - Yo m yx11 = y 0 x o n Type of Singularity at Origin node saddle 3. b = c = 0 ad<0 -<ua 5p2 d a d = 0 b - c = 0 b = qa q>0 a .= b = c d - 0 dy = -p2 x dx y dr = qr de If x = r cos 9 y = r sin & dy_ = xfcy_ dn y y^+p^x2 •- y Q+p 2x 2 center r « r Q exp q(0-£|i y = x spiral node 3.c Stability and Classifications of Singular Points A singular point is stable if there exists some neighbourhood around the singularity inside of which all representative points approach the singularity as t increases. A singular point is unstable if there exists some neighbourhood around the singularity inside of which all representative tend to leave the neighbourhood as t increases. Using these criteria for stability, the following classifications can be made of the types of singularities of (4.3). (i) Stable if b+c < 0 !) Node if A<.0 (ii) Unstable if b+c >0 I D > 0 (B) Saddle if A> 0 II D < 0 (A) Center if bfc= 0 (B) Spiral if b+c ^-0 (i) Stable.if b*c < 0 (ii) Unstable if b+c>0 IH D = 0 (A) Node (i) Stable if b+c <0 (ii) Unstable if b+c>0 where D = (b - c) 2+ 4 ad ; jL\ = ad - be 3.d Examples of Singular Points It is instructive to note that the singularity of the equation for the linear harmonic oscillator exhibits all the possible types of singularities, except a saddle, if negative damping is allowed. The equatiunifor such a system are and d2x 4. 2r dx_ + s 2x = 0 dt2" dt dy = - s 2x - 2ry dt dx z y dt 29. which are of the form of (4.3) if a = - s 2 b• • - 2r c = 0 d = 1 It is seen that b+c = 2r D = 4r 2 8* « The following table illustrates the nature of the singularities for different values of r and s. . Singularity Stable Node Stable Spiral Point Center Unstable Spiral Point Unstable Node Condition r > s 0>r>s r = 0 0>-r>s - r > s Damping Solution Positive, greater of* _*vt or equal to critical Ci cos £ t + c O Positive less than critical None Negative, less than critical Negative, greater C|« T ^ i C -or equal to critical o r An example of a saddle point is the singularity of the equation governing the behaviour of a simple, rigid pendulum near its uppermost position. This equation is d 2 x - q 2 sin x = 0 dtZ or d 2x - - 2 dt2 q z x = 0 keeping only linear terms. It is apparent A = - q 2 < 0 and D = 4q 2 > 0 which are the conditions for a saddle. 4. Oscillations 4.a General A closed trajectory in a phase plane corresponds to a periodic phenomena in the system represented by the phase plane. This statement follows from the fact that a representative point on the. closed curve returns to the same position after some time T; i.e., The symbol T will be used to denote period in the sequel and will not be confused with T the absolute temperature. The period T of an oscillatory process where the path of integration is to be taken around the closed trajectory in the direction of increasing time. trajectory the number of saddle points must be one less than the sum of the number of nodes, centers and spiral points. Thus it follows that inside a closed trajectory there must exist one node, center or spiral point. x (t * T) = x (t) y(t+T) = y(t) may in principle be calculated from the line integral T It can be shown (Stoker 1950 Chap. IE) that inside a closed If all the representative points in a region of a phase plane tend to a single closed trajectory free of singular points as t increases that trajectory is called a limit cycle. 4 .b Relaxation Oscillations In most oscillatory systems of (4.1), P and Q involve parameters of the system which may be varied; e.g., the factor £ in the Van der Pol equation 3 dy r £ (y " I ) " x dt 3 dx - y dt For certain values of the parameters; e.g., €L^^l., the oscillations are very nearly sinusoidal. As the parameters change; e.g., £ is increased, the oscillations may ultimately become characterized by, two distinct epochs;  e-E*» 10» o n e m which energy is stored up and one in which energy is discharged nearly instantaneously. Oscillations.of this nature are called relaxation oscillations. x A W - ro O S C I L L O S C O P E PoR. voi.TA,« SUPPLY , SAMPLING T O OSOLUOSCOpg Pee cocftswT FIGURE SI CIRCUIT FOR OBSERVING O S C I L L A T I O N S PART 5 STUDY OF OSCILLATIONS IN THERMISTORS 1. Experimental Techniques The basic circuit employed was that of Figure 5.1 which has been shown in 5 of Part 3 to oscillate if E and Ri-are such that the thermistor is biased beyond turnover and C is greater than some minimum value. Great difficulty was experienced in obtaining thermistors both sufficiently robust to withstand the extremely high temperatures during large -amplitude oscillations and having sufficiently low value of the. ratio t was easily, obtainable using oil and paper condensers. Also difficulties were met because of the apparent errors in the manufacturer's specifications of t and k in several thermistors (see .3 of Part 2 and 6 of Part 3). The final choice of a thermistor type for experimental investigation was made by biasing a number of thermistors beyond turnover and increasing the parallel capacitance in each case until sustained oscillations were observed and then choosing the one requiring the least capacitance. The type chosen was a VECO type 65A3 which was later substituted by a VECO type 65A1 which has, as explained in 3 of Part 1, after opening its glass tube, similar electrical and thermal properties to the 65A3, but is somewhat better protected from mechanical damage. The measured values of its parameters were so that 33. k = 0.13 m W deg ' i t = 0.61 sec R r 3.45 ohms b = 4310°K Oscillations were observed on an oscilloscope.in the V - I plane and in the time domain, i.e., V (t) and I (t). .A current sampling resistor of 1 Kohm was inserted in series with the thermistor for current measurements and a large resistor was used in series with the oscilloscope for voltage measurements. The effect of the 1 Kohm resistance was found to be negligible except at very large instantaneous currents occuring when the thermistor experienced its peak of temperature rise. An arbitrary current maximum of 1.5 mA, corresponding to — 150°C on the static curve, was set on the current during oscillations to avoid risk of burn - out. The effect of loading of the shunt resistance.of the oscilloscope circuit for voltage measurements was calculated, and the appropriate adjustments to R^ and E were made to preserve a known operating point. 2. Limited Applicability of Linear Theory of Oscillations In Part 3 an equation was derived relating the frequency f of small - amplitude sinusoidal oscillations to the parallel capacitance C, (3.20) when (3.19) If the period of oscillation T =• i f T = 2 % % (5.1.) T^ZI^C F I G U R E D E P E N D E N C E OF PERIOD UPON CAPACITANCE AT AN OPERATING POINT i i j » CAPACITANCE C JULWs FIGURE 5-3 DEPENDENCE O F A M P L I T U D E OF OSCILLATION UPON CAPACITANCE PARALLEL CAPACITANCE C - Mfd Data, were taken to show the dependence of T upon C. at a fixed operating point > for several different values of Rj and was compared to (5.1) (see Figure 5.2). The curves nearly coincided for C - CxninJ i.e., where the amplitude of oscillation was small. Naturally a marked divergence was noted as the value of C was increased since^^O) and (3.19) are valid only for infitesimal oscillations; i.e., there is only one value of C and f for a given operating point and R^. Data were also taken to show the relation of the amplitude of the voltage waveform and the value of C. The difference between the voltage maximum V m a x and the voltage minimum V m i n indicates that for only small increases of C beyond Cmin the voltage amplitude becomes very large and the voltage variation is very unsymmetrical relative to the value of the voltage at the operating point Vo (see Figure 5.3). These two results point out the very limited range of C in which the linear theory is applicable and also the inadequacy of applying pertubation methods to the linear theory. 3. Phase Planes for Thermistor Oscillations  3.a. General The general problem of oscillation consists of the simultaneous solution of the two thermistor equations V = exp b = R (t&) = J_ (2.1) I Ta+t6» G (te*) P = VI = k(te* + X, (3-3) dt 35. and the circuital relation E = V t IRx +• CRi d V dt (5.2) It is seen that there are many pairs of variables for which phase planes exist. Among them are those listed in the following table. Variables I, V ie», v dt Static Characteristic V R-^exp b T" Ta+VI V 2 = k«e»R(®«) P = k»d* diet =• 0 d t Load Line E = IRj+V E =v[l4-RiG(ie»)] E -(R(*») P]* xU+RiGW] Comments Variables easily measured. Load line is a straight line. Simplest form of differential equation Static characteristic is straight line. Differential equation is complicated. E = [k(»e<+'Cdj6»)R(lW)J* Static characteristic dt is one of axes. x [l+RiG^e»)] Differential equation is complicated. 3-.b. The V - I Phase Plane The first to be investigated was the V - I plane. Eliminating all other variables the system may be reduced to the following first order differential equation where dl_ dV CRi_bI Qvi - k>6* (I, V)3 V ~£ffa+«ef(l, vg z k (B-y -IRlJ (5.3) •©• ( I , V) = Ta. In V IRoa *>TATtC CHARACTERISTIC VJERTlCAL TANG-EKT VOLTAGE v/ Oscillations correspond to closed trajectories around the singular point (V 0, I Q) at the intersection, of E = IRi + V and VI = k»6»(I, V) where ~ = ^ = 0 • The trajectories of (5.3) have vertical tan|jents on the locus of ^  = 0, thus the load line E - JR\ - V = 0 is the isocline for SJL =oo>. The trajectories are tangent to the isothermal corresponding to maximum and minimum temperatures where = or VI = klGt(I, V) which are the points where the trajectories cross the static characteristic. The isocline for = 0 corresponding to dV current maximum and minimum is not easily soluble. It is to the right of the static characteristic above the load line and to the left of the static characteristic below the load line (see Figure 5.4). Since — > 0 below the load line and dt d ¥ < 0 above the load line it is apparent that the motion of any, representative dt point is in a counterclockwise direction around the operating point (1f0, Lj). The nature of the singular point (V G, IQ) was next examined. The transformations v = V - V 0 i = I-Io were made and and — were solved with second and higher order terms dt dt in v and i discarded to give the pair of linear equations dv - - _y_ - i_ dt CR, C di dt v /0b+ 1 - 1 V 1 +i/4b-l-T \ \ « C R i / ' R 0 V T ~ CRo' The quantities D, A and b +care D = (b - c ) 2 + 4 ad sfop- 1 -jL^ / j _ - l \ ) 2 - 4/fc+l - l j _ V *e G A R R i y ( t CRI)CR 0 = ad - be = _1_ f 6o- 1 - <t>0+ 1 \ b + c = 6ft - 1 - J _ / . J_ \ % C \ R 0 T R i / * A condition of great interest is that for the operating point to be a center. For b + c = 0 one has the condition that the system is just oscillating with infitesimal amplitude. For b+ c = 0 , D is less than zero and - I U o R l / (5-4) This; value of C is identical to the value Cjnjn in (3.19). For values of greater than Cmin, the singularity is. an unstable spiral point provided D <. 0 and for values of C smaller than Cmjn, the singularity is a stable spiral point also provided that D * 0. The conditions at a typical operating point are $ » = 2.0 R Q = 0.60 Mohms Rl = 4.0 Mohms X a 0.61 sec *o = -0.20 Mohms for which the singularity is Unstable Node C>13.8 ufd Unstable Spiral 13.8> C > 1.17 ufd Center C = 1.17 ufd VERTICAL I0W--1OV H O K l Z O N T M 1 D*f* 2 0 ^ ^ FIGURE S 5 P H O T O G R A P H S O F I - V P L A N E S T A B L E A M D U N S T A B L E S P I R A L Stable Spiral ^Stable Node .104 <-C £ 1.17 ufd C *.104 ufd Figure 5.5 shows some photographs.illustrating the nature of the singularity in the V - I plane for several values of C. 3.c. The V -»9» Plane The differential-equations in the V -Id4 plane are d>at . . l f y2G(lBl) d t * V k ' ) ) (5.5) dV = _1_ (E - V^+R^ie*)) ) d t CRi ) where G(|60 = exp - b Ta+lft* and their first order approximations valid near the operating point are dt « *C V 0 dv = -(froVoGo Q - (14-RiG n ) v dt C »6»o R l which have the same values of D, A and b+c as the equations for v and i since to first order they are linear transformations from the equations in v and i . The equations of (5.4) may be rewritten as the single first-order differential equation diet = CRi V 2G(W) x - (5.6) d V t E. - V (l+RiG(i»t)) V O L T A G E V It is noted that this equation (5.6) is much simpler than(5.3). It is apparent that the isocline for d .= 0 is the static characteristic and dV the isocline for = e& is the load line. d V dl©^ Also, it is seen that.if C 0, then - ^ 0 everywhere in d V the plane except at the load line. Thus in the limit as C becomes, zero one would expect all representative points to jump horizontally to the load line and then to follow the load line to the operating point. If C -^oo , the case for extreme relaxation oscillations, ^ST^aio everywhere in the plane except at the static characteristic. Thus one would expect all representative points to jump vertically to the static characteristic and once there, to follow it. In particular, a representative point starting from the origin would.be expected to follow the static curve to the turnover point. At this point it would tend to jump to the hypothetical branch of the ideal static characteristic and follow it until the voltage minimum was reached and then jump to the lower branch of the static characteristic (see Figure 5 .6 ) . Since a trajectory such as this is not physically realizable, as the upper branch of the ideal static characteristic is at a hypothetical temperature greater than bCT 4000°K it was not possible to set up extreme relaxation oscillations in the thermistor. For values of C •<• 0 0 the trajectories would be expected to have a value of 16* somewhat lower than the value of )04 corresponding to the static characteristic for a given V. TIME l S E C M b l Y VOLT/V*£ ZOV= I DIV Q^ZOpte O f 5 u & CH.^juW r l b U K - v * / $ CURRENT *S FUNCTlCMs OF 40. 4. Division of the Period of Oscillations into Epochs  4.a. General Figure 5.7 shows photographs of thermistor voltage V and current I as functions of time during oscillation. It is seen that as the value of C increases, in both the currentand voltage functions two distinct epochs may be recognized. These were given the names charging and discharging epochs corresponding to the time regions in which is respectively positive dt and negative. 4 .b. Charging Epoch For the purposes of analysis the charging epoch T^ is defined as that portion of the period for which the closed trajectory is below the load line. If i s sufficiently large it has been shown that " X ~ dt is very small compared with — over the charging portion of the cycle. Thus the equation for the balance of power V 2 G O C Q = r&i + n diet k dt becomes approximated by the equation for the static characteristic &t = V2G(iei) which may in principle be solved fori©* and substituted in the equation for ^ giving for T c Vm<t* \ d.V J E - V(l+ Rr< T c = CR X _V^ (5.7) iG(»©i)) IO -o Iii a. F I G U R E 5-8 PERIOD OF O S C I L L A T I O N AS A F U N C T I O N O F CAPACITANCE f=0R D I F F E R E N T L O A D R E S I S T A N C E S E =540 VDLTb Wo- 7 2 V 0 L T S > I L 2. 3 ^ CAPACITANCE C - ^ W s —L F I G U R E S 9 PERIOD OF OSCILLATION! AS A FUNCTION OF I i i i l i t > O z. 4 ^ s 1 0 1 2 LOAD RESISTANCER. ftSi. where V m a x and V,^^ are the maximum and minimum values of V over a cycle. A lower bound on T c is given by the inequality T c > CRi In E - Vmin E - V m a x or for sufficiently large values of C since Vmin becomes small (see Figure 5.3) T c > CR X Vmax - Vmin E If the discharging epoch can be assumed small enough, T c should be a first order approximation to the period T. The form of (5.7) indicates that the period of oscillation should be proportional to the parallel capacitance. C. This was true over a wide range of the value of C and the slope of the T versus C line increases with increasing Rx. Figure 5.8 shows the period of oscillation T as a function of capacitance C. Figure 5.9 shows the period of oscillation T as a function of load resistance R^ for several different values of G. The .relation between Rj and T is linear for values of R\ such that the operating point is well above turnover. The form of these functional relations between T, C and T, R^ suggest the following empirical relation T - ao+ai C+a2 R].+a3 CRi would be valid over a range of operating points. 4. c. Discharging Epoch It was decided that a study of the discharging of a condenser through a thermistor would possibly yield information regarding the discharging epoch of the oscillation. For the circuit; of a capacitor discharging into :a;tiiermistor:the following 42. equations hold i G#> exp - b  V Ta + e « p = i v -k (je* + ^_dje») dt C ^ + I..= 0 dt If TQIS is the duration, of the discharging epoch, T DIS = C \ d V -(5.8) J V G (ieo and C In Vmax < T D I S < C In Vmax G ( ^U Vmin G(0) ^min Several phase planes were considered for regarding this problem among which were & - V and the Id* - d l ^ > planes. dt The differential equations were dl©» = -C J V - j O * R | (5 .9) dV c (" v ie<  1 *r L k v -1 where and where R - V_ = exp b I Ta + 101 dx = (te*+*x Xitx - 2 -q)- x (5.10) dim x t x = diet dt b (Ta-flftt ) 2 D 10 o,0 to 8° TO*1 2^-° VOLTAGE, V FIGURE S 1 0 DRAW INGS OF V 0 L T A & E -EYCESS TEMPERATURE 0 20 40 fco go /&c ii© VOLTAGE V and G - G QO exp —b . Ta + ie< The -V plane was first considered. The origin is a stable node for all real values, of the parameters C, t andk. Isoclines corresponding di&t to slope ^ = d y are given by V = - k a : ^ + 1 ^  k 2 + 4kie« R In particular for |5 = 0 V = (k l©! R)^ which is the static characteristic. For the locus of points where the current I is maximum, d l = o dt which gives and dV = - »LV = b V ov = p. dlO« ' ( T a + e O 2 V : /k(tei R + t (Tai-HOP2) V C b For sufficiently large values of C and sufficiently small values this locus very nearly,coincides with the static characteristic. For large values of 161 this locus is given by k (Ta+iei) • Figure 5.10 is a drawing of the V plane for two different values of C with several isoclines.and the. locus for — = 0 drawn in. dt Regarding the differential equation in the , x = J^gi dt plane (5.9) it is seen that the static characteristic is given by x = 0, S T A T I C C H A R A C T E R I S T I C # = 0 > i B<y IO© £ 5 S TEMP °C —+ C=I0M W F I G U P R 5.11 DRIWIWO OF EXCESS - T&MPt RATUf?£ -RATE OF CHAU&-& OF EXCESS TEMP£RATUR£ PHASic PLANE FOR Cort&£NS£R CHSCH^R^Mr 'NT© A THEKWisroR STATIC CHARACTERS TIC fco 80 id-o EXCESS TEMP °' t"ZO dx which is the isocline for d|0l = ©° . The isocline for gjgj - °» W rtLL + x {Um T 1 - 2 G « ) - 21d» G 4 • r C C or for the maximum gp» i s given by the equation Y2 dx The isoclines with slope — = - 1 are the locus where P is a maximum d*e« 7^ in which case x = 2G (Ta - H O * ) 2 C b and the locus where P = 0 in which case x = The locus for maximum current I where — = 0 is d t x = _G_ which is not an isocline. The trajectories along mis locus have.a slope given by . dx .-. -Oi diet ' >e» * x x) - 1 Figure 5 .11, shows drawings of the I©*, x = plane for / values of C = 1 dt and C = 10 ufd. . It is seen that trajectories starting above a certain minimum value of x have in succession dldt 2 maximum of —— if GaVi >"(VI)t dt L 2 maximum of P if , 5GVi ^ (VI)t XL 2 2 maximum of I if CVj ^ (VI)t 2 2 O3-0xi*d CMtyU VERTICAL IDlVHOyixA HOR120RTAL [D(V=l5v F I G U R E S l Z P H O T O G R A P H S OF C U R R E N T - V O L T A G E P L & M F P U R ING-C O M D t MS£R DISCHARGE THERNUSTfcR maximum of i&i minimum of dt where Vi is the initial voltage of the condenser. Attempts were made to approximate the discharge by several different functions such as and ^ These functions were found to be greatly inadequate due to their lack- of sufficient number of adjustable parameters preventing an adequate approximation to the solution of the differential equation for d t L V Goeo /J c Figure 5.12 shows photos of the V- I plane during discharging of a condenser. BIBLIOGRAPHY Becker, J. A., Green, C. B., Pearson, G.L., Properties and Uses of Thermistors, B.S.T.J., vol. 26, pp 170 - 212; 1947. Bollman, J.H., Kreer, J. G., The Application of Thermistors to Control Networks. Proc. I.R.E., vol. 39, pp 20 - 26; January, 1950. Burgess, R. E., The A. C. Admittance of Temperature - Dependent Circuit Elements. Proc. Phys. Soc, vol. 68, pp 766 - 774; October 1,1955. Burgess, R . E ., Electrical Oscillations in Thermistors and Germanium Point - Contact Rectifiers. J. Electronics, vol. 1, pp 297 - 3€2 November, 1955. Poincare, H., Sur les Courbes Etefinie par une Equation Differentialle. Oeuvres, Gauthier - Villars, Paris, vol. 1; 1892. Stoker, J.J., Nonlinear Vibrations. Interscience Publishers Inc.;, 1950 

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