Computer aided static and transient design of a class of multivibrators: the parallel Schmitt circuit.

OF A CLA$S1OF 1ULTIVIBRATORS: THE PARALLEL SCHMITT CIRCUIT

by

VICTOR KWOK-KING FUNG

S.B., massachusetts Institute of Technology

(1966)

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY September, 1966

Signature of Author

Signature Redacted

Departmdnt of Electrical 4igifieerin, August 22, 166

Signature Redacted

Certified by A Thesis SuDervisor

Signature Redacted

Accepted by

Chairman, DepartmentalCoinmittee on Graduate

DISCLAIMER NOTICE

Due to the condition of the original material, there are unavoidable flaws in this reproduction. We have made every effort possible to provide you with the best copy available.

Thank you.

Some pages in the original document contain text that runs off the edge of the page.

by

VICTOR KWOK-KING FUNG

Submitted to the Department of. Electrical Engineering on August 22, 1966 in partial fulfillment of the requirements for the degree of Master of Science.

ABSTRACT

This thesis is concerned with the static and tran-sient design of the Parallel Schmitt circuit. Particular emphasis is given to methods which can be developed into a systematic procudures for the analysis of all classes of multivibrators and which can be mechanized on the digital computer.

The static design calcul.tes the value of the four unknown resistances subject tb four input constraints. The first two constraints are supplied by analytic exp-ressions for the circuit at its breakpoints, while the latter two are chosen arbitrarily for their convenience and usefulness in the transient analysis and the optimi-zation. The correlation between the VF and VK is almost prefect, showing deviations less than 5% and oftentimes less than 1%.

The incremental D.C. loop gain is derived and used to find the conditions for a regenerative circuit. These are found to be consistent with the conditions obtained by treating the circuit as a device with a particular V-I characteristic. The power consumption of the circuit

i investigated and the conditions and proof given for its minimization.

The system of non-linear first order simultaneous eqlations used to describe the transient behaviour was obtained by substituting simple charge control models for the transistors. Their coefficients, which changed with

current levels, were assumed to be constants in a linear approximation to the actual solution. Instead of the usual

"IS" shaped curve of output collector current versus time,

mostly within 50% of the measured value.

The final part presents a method for the numerical integration of the non-linear transient equations. Using this programtwo ways are suggested for the minimazation of the power consumption and the hysteresis.

Paul E. Gray, and his supervisors at General Radio Company, Mr. James K. Skilling and Mr. Robert E. Owen for their constant guidance and

CHAPTER T

I. INTRODUCTION...1

1.1 Multivibrators,...,... ..

1

1.2 A Summary of Each Chapter...4

1.3 The Parallel Schmitt Circuit ... S II. STATIC ANALYSIS*... ... .. ... ... 12

2.1 Breakpoint analyses of the Parallel...12

Schmitt Circuit 2.11 Analytis Expression for VF ... 16

2.12 Analytic Expression for VH ... 16

2.2 iq the Allowable error in firing voltage for maximium variations in transistor current gains ...

.19

2.3 The fourth constraint: emitter current at firing ... 2.4 Output Requirements ... 22

2.5 Solution for Unknowns ... 24

III. THE TERMINAL V-I CHARACTERISTIC...27

3.1 The Different oiodes of Operation...27

3.2 Waveforms of non-zero rise time...,...29

3.21 Single lumped reactance outside'....30 the device

3.22 Analysis of the Active Elements

V P

4.1 Deriation of the Loop Gain 4.2 -Properties of the Loop Gain

4.3 Relationship of Loop Gain to Other

Circuit Functions ...

OWER CONSUMPTION ... 0 ...

5.1 Analytic Expressions for the Power

at the Breakpoints... 5.2 Power Dissipation in the Circuit

5.3

Minimum Power Conditions ... PART II

.35

.35

.42 .44 .44

.46

.46

CHAPTER VI TRANSIENT ANALYSIS ... ,...50

6.1 The Charge Control Theory ...

50

6.11 Relationships with other transistor

parameters . ...

.51

6.12 Changes in time constahts with

colletor currents ...

52

6.2 Derivation of a Set of Simultaneous

Differential Equions...60

6.3 The Two Methods of Transient Analysis ... 63

6.31 Analysis involving total variables .. 63 6.32 Analysis involving incremental

6.61 Conditions for imaginary roots.74

6.62 Condition for two positive roots....75 6.63 Condition for one positive and

one negative root...76

6.64 Condition for two negative roots....76

6.7 Numerical Example...79 CHAPTER VII THE 7.1 7.2 SPEED UP CAPACITOR. ...

83

New Equations including CX**---.-*-'.. 83 Effects of the Speed Up Capacitor...86

PART III CHAPTER VIIILITHEB 8.1 8.2 8.: IX THE 9.1 9.2 STATIC SYSTEM... The Static Program... The Subroutines... _{. .. 0}... Verification of D.C. Progr with actual resultsa... TRANSIENT SYSTEN...

General Remarks.. 0.0 0..0.

The Transient Program...

000000000 anuspredi ct 00 . 0. ... .. .. ... .. ... g.. .88 .88 .88 ions .93 .95 .95 .95

XI CONCLUSION.. ... ... ... .... . . ... * . e 102 11.1 The Conclusion..**. ... .. .. ... 102 11.2 Suggestions for further study... 104

APPENDIX APPENDIX APPENDIX APPENDIX APPENDIX 1 2

3

4

5 Fairchild 2N708 Specifications... .... 105 Comparison of Errors for . 112

D.C. solution neglecting junction drop..113 D.C. program Results...114

Computer Programs...123

VS (Volts) VO (Volts) VF (Volts) VH (Volts) XIL (Ma) Supply voltage Ou

jatVoltage

Firing voltage specifieddby user Holding voltage specified by user Current drawn by output load

INPUT CONSTANTS BETAX Betan VBEI (Volts) VBE2 (Volts) VG1 (Volts) VG2 (Volts) Taul (nsec) Tau2 (nsec) TAUB1 (nsec) TAUB2 (nsec) Maximum C .M* Minimum C

Base-emitter junction voltage at Cutoff of _Q

Base emitter junction voltage at Cutoff of Q2

Base emitter junction voltage at Cutin of Qi

Base emitter junction voltage at Cutin of Q2

Base diffusion time constant of Q1 Base diffusion time constant of '2 Base recombination time constant of Q

BETAT VF1 (Volts) VH1 (Volts) DELTA ALPHA SIGMA

Q

ETA XIEF (mA) XIEH (mA) XIE Zs Z Zi SYSTEM VARIABLES Typical

Voltage on base of Q2 at firing (VF

used by system)

Voltage on base of Q2 at holding (VH used by system)

Normalized Parameter

Normalized Parameter

Normalized Parameter

Beta quotient

G2-

v

Allowable variation in Vf as hfe of Q changes from max to (min

Emitter current at firing Emitter current at holding

Change in emitter current

Transient normalized parameter Transient normalized parameter Transient normalized parameter

GMAX PF (mW) PH (mW) PS (mW) VFA (Volts) VHA (Voh)i) XI1 (mA) X12 (mA) VFNAX (Volts) VFMIN (Volts) VHMAX ( Volts) VHMIN (Volts)

Maximum incremental D.C. loop gain

Power dissipated at firing

Power dissipated at holdingri.r Power dissipated at saturation

Actual Vf after resistance standardized Actual Vh after resistances standardized

Collector current of Q2 at which D.C. open loop gain just becomes greater than one

Collector current of Q at which D.C. open loop gain just begomes less than one

Maximum possible Vf for extreme,. toler-ance on circuit variables

Minimum possible Vf for extreme toler-ance on circuit variables

Maximum possible Vh for extreme toler-ances on circuit variables

Minimum possible Vh for extreme toler-ances on circuit variables

UNKNOWS

Emitter resistance RE (k Rt)

RO (k.S)

XII (JC)

XIO (J

)

(mA)

T ( J ) (nsec)

Output resistance - Collector

re-sistance of Q2

Input transistor current

Matrix-collector current of Q as a function of time

Output transistor current Matrix -

Col-lector current of Q2 as a function of

time

Time matrix - times at which XII and

PART 1

CHAPTER I

INTRODUCTION

1.1 MULTIVIBRATORS

Multivibrators .are two stage amplifiers in which the output of the second stage supplies regenerative

feed-1

back to the input of the first amplifier. The circuit,

with resistive and reactive elements connecting the amplifier stages, possesses a number of active and passive states,

between which switching behaviour is possible.

An alternate approach to analyzing the circuit~s op-eration is to regard it as a two terminal device with a non-linear V-I characteristics in which there is always a negative resistance (active) region bounded bygresistance

(passive) regions.

Using transistors as the amplifier device, there are four possible configurations in which transistors can be connected in regenerative feedback. They are shown in Figs. 1 through 4 together with the resistive circuit representative of each class. The V-I characteristics at the terminals

CLASS A Identical pair C - b, -b CLASS C Complementary pair SC - b , c-b CLASS B Identical pair c -b, e- e CLASS D Compldmentary pair c - b , e - e

SA'

E LES -JORDAN Ckt.

(

CLASS A

)

A

A'

T _{AALEL SCHMITT} Ckt.A( CLASS B

)

A-SERIEA S CHMIT T Ckt.

(

CLASS C)

A'c

AA' are either single valued in current (current controlled devices, Fig.

5)

or are single valued in voltage (voltage controlled devices, Fig. 6).

The object of this thesis is to develop a general method of analysis for these multivibrators by considering the

Parallel Schmitt circuit of class B as a typical example. A procedure involving first the static and then the dynamic

design and finally optimization of important circuit functions is developed and mechanized on the digital computer. The

two complementary approaches of regarding the multivibrator as two active devices connected in regenerative feedback

('active device approach') and as a device with a negative resistance V-I characteristic ('V-I characteristics approach') is presented and reconciled at each point. In general, the former method is employed in the transient analysis of

Part II and the latter is mainly used in the static analysis of Part I.

1.2 A SUMMARY OF EACH CHAPTER

Part I contains the full static design of the Parallel

Schmitt circuit. The details of the circuit are presented in the last section of Chapter I. In Chapter II four con-straints for the static design are introduced with the reasons for their choice. The unknowns are related to these constraints by analytic expressions, and finally, a

i'D

N ec~c~1\v~

I

Re

toS c~vec

CURRENT CONTROLLED DEVICE

t

~c~ciVive

c,~wce~

/

U-I

1~

V

VOLTAGE CONTROLLED DEVICE

solution. Chapter III treats the Parallel Schmitt circuit simply as a device with the V-I characteristics of Fig. 7 and considers the behaviour of the circuit under different biasing conditions. The second section presents the

different ways in which the "V-I characteristics approach" and the "active devices approach" account for the 'inertial of the circuit. In Chapter IV, the expression for the loop gain is derived and some of its properties discussed. Chapter V presents analytic expressions for the power consumption

and states the conditions and a proof for minimum power consumption.

Part II deals with the transient response of the cir-cuit. Chapter VI starts off with a summary of the Charge Control Theory, presenting the basic charge relationships used in the analysis. With this model, a set of non-linear simultaneous differential equations, with coefficients which vary with the collector current levels in the transistors,

is found to describe the transient behaviour. In thefext setion, an approximate linear solution to these equations

-the

is obtained using boundary conditions derived from eee4

roi~~ ~cavo Wnus cre~ ck't t=0 n cf" lop n in 4halto iV. Finally, the accuracyt

of the prediction for the rise time is correlated with empirical results. Chapter VII introduces a speed up capacitor and analyses its effects on the rise time.

In Part III, computer programs are written for the static and transient methods developed in the previous

:lj

4

31L Ci,

N

,

V-I CHARACTERISTICS OF THE PARALLEL SCHMITT CIRCUIT

xI.

duc

Ac

e-1V

0A, ie

1c.)o

)

three subroutines, STAND, for standardizing resistance values, VLMIT, for calculating the limits on VF and V and GAIN, for calculating the collector currents at which thegain exceeds one. The transient design program is presented in Chapter IX, where numerical methods are used to solve exactly the set of non-linear differential equations derived in Chapter VI.

Using these programs, Chapter X develops an overall system for minimizing the power or the hysteresis subject to a set of static requirements and a minimum rise time. Finally, the conclusion is presented in Chapter XI.

1.3 THE PARALLEL SCHMITT CIRCUIT

The Parallel Schmitt circuit (Fig. 8) is represen-tative of the multivibrators of Class B. Since its in-vention by O.H. Schmitt in 1938,2 it has gained widespread use in many pulse circuit design applications. As shown in

its V-I characteristics of Fig. 7,3 the Parallel Schmitt circuit has four clearly defined regions. There are three positive resistance regions where Q₁ or-Q₂* is either active

2. See Reference (2)

Vs

1E

SOutf't

66

THE PARALLEL SCHMITT CIRCUIT

'F %

C Q V- c.

or saturated and an active negative resistance region where both Q and Q are in the active state. The usual

opera-1 2

tion of the circuit is confined to regions I, II, and III. An alternative mode of operation with V-I characteristics

shown in Fig. 9, contains a region in which the second transistor is s&turated. It is rarely used and so only the first mode of operation shown in Fig. 7 is considered

in this paper.

It may be appropriate to note here that whenever experimental verification is done in the paper, the transistors used are Fairchild 2N70P npn silicon tran-sistors. Their complete characteristics are shown in Appendix I.

I

9

AN ALTERNATIVE MODE OF OPERATION

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--CHAPTER II

STATIC ANALYSIS

2.1 BREAKPOINT ANALYSIS OF THE PARALLEL SCHMITT CIRCUIT

Based on a plot of the input V-I characteristics of the Parallel Schmitt circuit (see Fig. 7), breakpoint analyses can be done at the holding and firing voltages, VH and VF. A simple transistor model, (see Fig. 10) which includes only the current gain characteristic

( 1

c

-

IB

) and a base-emitter junction voltage drop, is used. The junction voltage is assumed to be that of a typical diode, varying exponentially with collector current as shown in Fig. 11. VG denotes the base-emitter junction voltage drop at Cutin, where the transistor is on the

threshold of conduction and V denotes the junction voltage BE

well into the active region. In the examples chosen in this paper, the Cutin current (IG) is approximately 0.125 ma. and the active region current (IA) is 10 ma. with a de-viation of not more t~hn .02 volts along the whole range of current as shown in Graph I. Because of the steepness of the slopes of these curves, these two values are used for VG and VBE throughoutregardless of the collector current level. The variation of D.C. current gain, A, with collector

current for 2N708's is shown in Graph II. (Compare with Fairchild 2N708 Specifications, Appendix 1).

C

Flqur'ee

10

--

I-fIla

TRANSISTOR MODEL - -

-I

- - --- - I

-

A I

Vc,

BASE-EMITTER JUNCTION CHARATERISTICS

C--~

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2111 EXPRESSIONS FOR VF

.I At the firing point, F , Q is on the threshold of conduction with a junction voltage of VG and negligible collector current flow, IG; Q_{2 is active with VBE and IEF}

(See Fig. 12). Using the equivalent circuit of Fig. 13 the following expressions are derived.

VF'- VF -VG

-+VBEz

Z)

(

Rc +.x)

VF

=VS

+VJ

E

2

Ae

(g)

C-Y

g

R8~ 1E

(I?

+/

Where VBE2 and are the values of these variables at the operating colleqtor current level, IEF. The analysis is done in terms of VF the voltage at the base of Q2, after it has been transformed from the input firing voltage, VFV because of its convenience.

2.11 ANALYTIC EXPRESSION FOR VH

Similartly, at the holding point, H, by assuming Q

is active and Q2 is at cutin, the following expressions are found (See Fig. 14 and 15). Again the analysis is done in terms of VH the voltage at the base of Q2 at holding.

VH' ZVH - VBI +VGZ

~

3

'=V

+

( c/E

)

z4

Rec+ r'x+Re

+

RC(4

a

PARALLEL SCHMITT CIRCUIT AT FIRING

-f

VFVsr

1

1131

EQUIVALENT CIRCUIT AT FIRING

7.

1 *1~

VfI

I

go

VF

a

V&l

V k 0 a v, e.

E

I.

PARALLEL SCHMITT CICUIT AT HOLDING

ai

cu~

z

1441

I

EQUIVALENT CIRCUIT AT HOLDING

xI4

zc_

iz

VBE,

The Input terminal current at holding,IH, is given by

.1

(4,+)4)

2.2 THE ALLOWABLE ERROR IN FIRING VOLTAGE FOR MAXIMUM VARIATIONS IN TRANSISTOR CURRENT GAINS

Detailed examination of the Schmitt circuit (Fig. 8) shows that. R0 is completely decoupled (except that Q2 cannot

be saturated)from the rest of the network and is simply determined by the desired output voltage swing. The exact expression for RO is given in Section 2.4 Hence there are only four unknowns RE, RC, RX and RB involved in the circuit. By specifying the nominal firing and holding voltages and through the use of Equations (2) and (4), two constraints are specified. _{The two additional constraints required to solve} for the four unknowns are given by , the error in VF for maximum variations in the transistor current'gain of Q2 and IEF , _{the emitter current at the firing point.}

It should be made clear at this point that the choice of constraints is completely arbitrary. The above constraints were selected because they were felt to be the ones most useful to the designer and because thoy ;crz folt te bo the

91qcs mest k~ef-lto 7 h to e s ' gg r ant b c Ethey can be

be very suitable for the optimization in Part III.

is defined as the allowable percentage variation in VF as the D.C. current gain of Q2 varies from 6"/1Y7Ito /My.

It is obtained by taking the partial derivative of V with respect toPin Equation (2) and setting VBE to zero.

- Re-R

In principle a method of partial derivatives is not valid for finding I because there are as much as 400%

changes in .

1

should really be found by considering the differences between VF at,/nAxand at/>"v. However,

in practize, the difference between the two methods is so small as* to make the simplicity of taking partial derivatives desirable.-' A comparison of the two methods and the errors

involved in choosing the derivative is shown in Appendix 2.

* This method yields an n' as follows

Furthermore, the1 in Equation (5) is defined in terms of

a typical Ot which is the geometric mean of -fl7na and '1flAx.

Graph II shows that for any reasonable value of collector

current (: Ze 7//'n4) #zL-E vO- t' with a variation of no more

than 25%. Henceforth, for simplicity, all

Se

will be equal to - regardless of collector current level.

The two limits onq are also derived by imposing the condition that R and RC be non-negative.

VIC-I-Viv~

₍₇₎

2.3 THE FOURTH CONSTRAINT: THE EMITTER CURRENT AT FIRING

The set of necessary constraints is completed by assuming a certain value of IEF' the current in the emitter branch

(RE branch) at the firing point.

7yc

_ ___V' _-_V __.. (

9)

e

The selection of IEF is governed by four conditions. First, the transistors must be operating in a region of appreciable current gain ( _{=40 or more), which means thd, typically,IEF} has to be greater than a milliamp. Second, the size of IEF

determines the current available at the collector of Q2 and hence the load that can be connected to that point. Third, the power consummed in the circuit is inversely proportional to IF This property is treated in greater detail in Part III.

EF a.VL S.3)

in which IEF is adjusted for a minimum power optimization. Finally, as will be apparent in the transient analysis of Part II, the cutoff frequency, fT, decreases with collector current. (See Graph V) Thus a low current level, IEF, will slow the response of the circuit.

2.4 OUTPUT REQUIREMENTS

The Parallel Schmitt circuit is commonly used to drive a device e.g. a diode gate, which draws current only when the ciruit is in one state. The two possible situations are

illustrated in Fig. 16 and 17. They show in Case 1, a load which draws a current, IL, when the output is high (circuit

in state (10) and in Case 2 one which delivers a current, _IL' when the output is low (circuit in state (01) ). Thus the

output voltage at the collector of Q2 varies from VS to V₀ for holding and firing respectively where V0 is the

minimum output voltage. In the following discussion, Case 2 is chosen for the output load condition.

The output resistance is given by

2:,CF -g

in order not to saturate _Q2,

C

0,

sr

-& u tomae4 '4-- vooaeck

OF

kjvej(2

11

tL

I

P\O

'*Fl uve. t C

) 0 Q A-V \Jt L 0 Ck C

The minimum value for IEF is given by

2.5 SOLUTION FOR UNKNOWNS

With the four Equations (2), (4),

(5),

and (8), a unique set of values for the unknowns RE , RC , RX and RB can be found. To facilitate calculations a set of normalized

parameters is introduced:

eE(qf

No physical significance can be attached to

S

but c,. is the inverse of the value of the resistive divider ratio on the base of Q2 and _{is the unloaded voltage gain of Q}

2 '

The static design procedure is as follows:

i) Pick the desired values of VF' VH and V .

ii) Find the value of the circuit constants 44/A Et. VG and VBE from data sheets (See Appendix 1)

or by actual measurement.

iii) Pick the value of IEH governed by the con-siderations in Section 2.3 ,

v) Pick the desired value ofaIbetween these limits. vi) Determine the value of VO and IL as dictated by

the output requirements.

vii) Calculate the required output resistance by:

.74F - IL.

checking to see that Q2 is not saturated (Equation 11).

viii) Solve for the normalized parameters,

V4's

- V91

Z I

-'.-..

4

(

x) Solve for unknowns.

V -- VSE, (782

4

(z.,')

tee

=

te /9

The above is an outline of a procedure for the sequen-tial solution of the resistance unknowns, .At no point is a simultaneous solution necessary if the unknowns are cal-culated in this order. An approximate set of equations which neglects the effects of base emitter junction voltage

drops are given in Appendix 3. While they are quite in-accurate, their great simplicity makes them useful in applications where only rough solutions are required.

CHAPTER III

THE TERMINAL V-I CHARACTERISTICS

3.1 THE DIFFERENT MODES OF OPERATION

The Schmitt circuit can be considered simply as a device with the V-I characteristics shown in Fig. 18.

Boundary conditions, imposed in the form of a source voltage and a source resistance connected at the input terminals, appear as a load line on the V-I characteristics curve

(Fig. 18, 19). The most important factor which affects the operation of the circuit is whether the magnitude of the source resistance is greater or smaller than the magnitude of the negative resistance of the device., From expressions for VF, VH and IH the absolute value of the negative resis-tance is found to be,

feN

(-f

&e

(%

The three possible modes.-of operation are considered below. The first is the astable mode with the value of the

RS greater than the value of the negative resistance (RS >RNI)

the load line intersecting the V-I characteristic at only one point in the (11) region. With purely resistive elements connected across its terminals, the circuit is stable

in this condition. However,any energy storage elements, whether they are introduced purposely or as a result of

rt

)I

)I

604)

k1l Astable operation (2) Bistable Operation (3) Monostable Operation

L-I

strays, appearing in the form of parallel capacitance across the terminalsAwill cause the circuit to oscillate.

It is the potential for this behavior which allows this mode of operation to be called astable. The circuit 'free runs' with its own characteristic frequency through the limit

cycle F-F1 -H-H1 (shown in Fig. 18) Generating what is known

as relaxation oscillations at it terminals.

The second is the bistable mode with Rs less than the absolute value of the negative resistance (Rs <IRD. 'The load line intersects the V-I characteris ic at three points, two of which are unstable with a stable point in the middle. This mode is characterized by a switching behaviour between the two

stable states even in the absence of any energy storage. The Circuit changes state- whenever the input voltage increases to the firing or decreases to the holding voltages regard-less of the input wave shape.

The third way ;is the monostable mode which is equi-valent to the bistable mode ( Rs.<N except that the load line intersects the V-I characteristics at only one stable

point. With a R-C timing circuit at its input terminals, the circuit can be made to traverse a full limit cycle and return to its single point of equilibrium. Every input pulse regard-less of shape produces an output pulse of constant shape and duration as preset by the circuit and the R-C time constant.

3.2 WAVEFORMS OF NON-ZERO RISE TIME

shown in Fig. 18, biased in ;the bistable mode, the circuit switches instantaneously between the two stable states. In an ideal circuit where no reactive elements are present, the terminal variables can change instantaneously. However, in reality, each transition from F to F₁ and from H to H₁ has associated with it a finite rise time, which is accounted

for by postulating some energy storage. This can be done in two ways.

3.21 ANALYSIS WITH A LUMPED REACTANCE OUTSIDE THE DEVICE

In a device such as the Parallel Schmitt circuit, this "inertia" can be introduced in the form of a series

induct-4

ance at the input terminals (Note: in a device with a voltage controlled V-I characteristic, it can be done by introducing a parallel capacitance). The method assumes that during transistions, the circuit actually follows the V-I characteristic in the negative resistance region. The

discrepancy between the voltage on the load line and the device terminal voltage, is taken up by the energy storage element. Based on this assumption, one can arrive at a

prediction for the rise time.5 However, the main disadvantage

4. See Refernce (4)

of this approach is that the value of this lumped series in-ductance reflects the inertia of the device as well as the stray reactances in the contacts and the leads. In the absence of any theoretical basis for picking the value of

this energy storage, an appropriate value is usually chosen from experience or by trial and error.

3.22 ANALYSIS OF THE DEVICE AS A REGERNERATIVE FEEDBACK AMPLIFIER

This second method treats the device as a regernative feedback network containing.,two blements, the transistors. The "inertia" of the device is manifested in the diffusion and recombination time consants in the base of these

transistors. Hence, by assuming a 'Charge Control model' for the transistors, the rise time is found in terms of physically measurable parameters of the circuit.

While the first method is by far the/simpler, its difficulty lies in the fact that it trIds to pick a reactance which will give the correct rise time without

first knowing that rise time. The accuracy of the solntion is entirely dependent on the value of this energy storage element. The second method has the advantage of being based on physically measurabIe quantities. However, the analysis, even in its simplest form is formidable and at

computer. The latter method is chosen for Part II of this project.

3.3 THE HYSTERESIS

Assuming again that the circuit is biased in the bistable mode, a phenomenon is observed, as the circuit

is switched from one state to the other by changing

Vg. (Fig. 19) UsingEFigi 18, consider a case in which the circuit starts off in (01), it switches to (10) as V

reaches VF and continues along the load line in the (10) state for increasing V-. However, as Vi is decreased, it is found that the circuit no longer triggers, i.e. changes state, at VF, but is returned to the (01) state only when VH is reached. If the output waveform at the collector of Q₂ ( See Fig. 7) is plotted for a sine wave input, the above phenomenon is immediately apparent, (see Fig. 20). This difference between the trigger voltages

for the positive and negative going transitions is known as the hysteresis, H , of the circuit. From the V-I

characteristic curve it is equal to VF - VH.

It should be made clear that a hysteresis is inherent in any device with a negative resistance region. In fact, the value of the negative resistance is directly proport-ional to the hysteresis as. shown in Equation (22). In some applications in which a distinct trigger level is

Il -'_ __ I. 61I

Z7A'PcT

4ki4'

6C77/cJ1-rf1

.s

2

4

_{r4 6~ee}

i/f

N

VF

-- 1-

V.

H

I

a'

I'

I

I

I

I

I

I

I

I

I

I

01

of.

vlll, j

required for both the upward and downward transitions, it may be desirable to minimize the hysteresis. However, as will be seen in the following chapters, the switching

speed is faster for a larger value of hysteresis and so, a choice must be made to decide between a fast transient response or a smaIl hysteresis. In any case the hysteresis is both an important static characteristic and a measure of the dynamic behaviour of the circuit.

CHAPTER IV

LOOP GAIN :ANALYSIS

4.1 DERIVATION OF THE LOOP GAIN

The transistor model used to derive the incremental D.C. loop gain is shown in Fig. 21. The effects of the base resistanceW which is inversely proportional to the collector current, is included because it can vary from 0.1 to 10 -ohms over a range of IC from 10 to 0.1 ma. The frequency dependence of

p

is not accounted for and so the expressions derived are only valid for D.C. conditions. Substituting the model into the circuit of Fig. 7 (see also Fig. 22), and conveniently opening the loop at the collector of Q, (no equivalent loading is necessary as the resistance of the collector current

source of Q, appears to be infinite) the equivalent circuit shown in Fig. 23 results.

Introducing the current I into the node where the loop is opened4i the gain is given by,

e9-

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _

tAA* p

f~B

r

C. go I' 0

E

TRANSISTOR MODEL USED IN LOOP GAIN ANALYSIS

(fl*~

~ ~ow%

'zC-Y.~

a

j

P.

THE EQUIVALENT CIRCUIT FOR LOOP GIAN ANALYSIS

( LOOP OPENED AT COLLECTOR OF QZ )

Li.

t~

16

_L

Isr

1.

~J'c.

p

I

I

L

EQUIVALENT CIRCUIT FOR CALCULATING THE GAIN

liz

~~

2

;i

e

3

I

*

mxtl--kE

R~X

4c 7P ,

I)e

where the and

r4lf

are dependent on the total collector current levels IC, and IC2 of the circuit.

4.2 PROPERTIES OF THE LOOP GAIN

The D.C. loop gain of the circuit is intimately related to the transient performance. Although the exact relation can be found by putting the frequency dependence of into the model for the transistors, it is not done because a more direct approach can be used

(see Part III). The exact relationship can be determined by empirical methods, if desired, but, in any case, one can be certain that a circuit with a higher loop gain provides a faster rise time. In fact, the circuit is not regenerative and will not show switching behaviour if the gain is at no point greater than one. Hence the loop gain provides a figure of merit, for transient

performance, from purely static considerations. Also, the gain expression is important in calculating the range of collector current over which the gain exceeds one. The collector current I,, at which the gain just exceeds one indicates the point where the circuit is going into

switch regeneratively in the transient sense.

By assuming the following relationships, the gain as a function of IC2, _{the collector current of Q2 is}

obtained.

II

et

(

le.-+

-C.=

F

(--7,)

The exact dependence of on IC is given in Graph II, but for most purposes can be assumed to be equal to a constant. Relation (27) implies that RE is large enough so that the emitter branch is assumed to be a current source. Setting R = 0 and substituting in Equation (24),

which is the maximum gain at each collector current level. It can be seen from Equation (24) that any positive value of Rs will tend to decrease the gain.

A plot of loop gain versus IC2 is shown in Graph III for some typical examples. It is observed that

in-creasing hysteresis implies higher gain and larger ranges of collector current for which the circuit is

124

r

+r

T

4.3. RELATIONSHIP OF LOOP GAIN TO OTHER CIRCUIT FUNCTIONS

To show that the loop gain is consistent with the other static circuit functions, conditions for a regen-erative circuit are rederived using that expression. The upper limit on RS for the circuit to be regenerative from the"V-I characteristic Approach" in Chapter III, is given by Equation (22).

Setting the loop gain greater than one in Equatidnh (24) results in the same expression.

(The slight discrepancy between Equations (22) and (29)

is because some terms were neglected in the gain expressA

ionbut since in typical circuits the difference

can be disregarded.) Applying this result to the discussion in Chapter III, the gain is found to be greater than one for circuits biased in the bistabl6 or monostable mode and less than one for those in the astable mode.

Setting the junction voltages to zero in Equations (2) and (4) and using the above condition,

Z> ex

(p1)3

Again an identical constraint is obtained by setting all base resistance to zero and the gain equal to br greater

than one in Equation (23).

(3z)

Hence it can be seen that conditions for a regenerative circuit derived from the expression for the loop gain is completely consistent with those derived from the "V-I Characteristics Approach" and the hysteresis. Further-more, the above shows that when the circuit is regenerative in the bistable and monostable modes, it's loop gain is greater than one and when it is non-regeneratile-lin, the astable mode, it's loop gain is less than one.

CHAPTER V

POWER CONSUMPTION

5.1 ANALYTIC EXPRESSIONS FOR THE POWER AT THE BREAKPOINTS

From a consideration of the equivalent circuits at the break points F and G of fitgures 12 and 14 and neg-lecting all voltages, the following analytic expressions for the power consumption at firing, PF , at holding PH are obtained.

-

2

(x'*'

)~

(3)

PCe

+

Using the equivalent circuit as shown in Fig. 25 at saturation, S , (defined in Fig. 7) the power at saturation,

5.2 POWER DISSIPATION IN THE CIRCUIT

For a typical case in the bistable mode of operation the load line sweeps out a limit cycle as marked in Fig. 25. The power dissipated in the (01) region is given by P F However, PH provides only the minimum and PS the maximum power consumped in the (10) state. The power distipated in the active region (11), is negligible because of the

comparatively short times spent in that region. (of the order of 50 ns.) Hence, the total power consumed in the circuit is given by a linear combination PF PH and P ,

the values of al , a₂ and a3 being determined by the driving function (e.g. duty cycle of a pulsed input.)

5.3

MINIMUM POWER CONDITIONS

Careful observation of Equations (33) and (34) shows that the PF and PH can each be separated into two compon-ents. Considering Fig. 26, the power at firing can be rewritten:

_

'

'8&

(

a

'4.

S

Limit cycle : 0 H - F - F - F1 -H -H1 - 0

LIMIT CYCLE FOR BISTABLE MODE OF OPERATION

p()

tE

I

I

+

Vs

PC,6)

THE TWO COMPONENTS OF POWER FLOW IN THE PARALLEL SCHMITT CIRCUIT

--. ow1.

I Li

DEVICE

A

a

Therefore, to minimize each component of PF , the values of the resistances RE and RB must be at a maximum. From Equation (18),

Picking a maximum value of RE corresponds to a minimum value

of IEF. Alsofrom Equation (5), a partial derivative of/It

with respect to RB is taken,

d ie (c + (A'c,'L 'x)

The fact that it is always positive implies that a maximum RB corresponds to a maximum q . The arguments are entirely analagous for P Hence, a minimum power consumption at. holding and at firing results for a minimum IEF and

max-imum . Finally, considering P S as a function of RB _and

RE in Equation (35), partial derivatives are obtained which show that P is also minimized for maximum R and R .

S B E2

_ ' _ _ _ _ _ _ _

[

~

Ae

X~6

?'~J

-

VL

( ~ 7

Thus, to minimized each component PF f H and P and hence the total power consumption, IEF must be a minimim and

4

a maximum.

The implications of making IEF small, but within the limits set by Equations (Ii) and (10), is that the current

gain of Q, and Q2 drop off as the currents become too

small. It will be seen in Part 3 that this lowers the value of the cutoff frequency, fT and hence slows the rise

time. The effect of increasing t is to increase the error in VF caused by variations in of and so, in this case there is a tradeoff between the accuracy of VF and the power consumption.

PART II

CHAPTER VI

TRANSIENT ANALYSIS

6.1 THE CHARGE CONTROL THEORY

Observing the one-to-one correspondence- between the dperating state, as given by the terminal currents, and the interqal charge distributions in a transistor, Beaufoy and Sparks proposed, in 1957, 6,7 to study the transistor as a charge-control device. The dynamic behaviour of a

transistor can be conveniently explained in terms of the transient charges in the minority carrier distributions in the base.

Based on the following assumptions:

I) uniform base region-disregarding the

ii) iii)

iv)

v)

properties of the space chwt e layer, Low-level injection in the base

a one dimensional model

lifetimes in the emitter and collector regions negligible as compared to those in the base because of high impurity concentrations, recombination takes place in the base only,

See Reference (6) See Reference (7)

6.

relations are derived:

Tg

dt

where 1B = Base current

I

= Collector current C

Q, = stored base charge

= Recombination time constant in the base = Diffusion time constant in the base

The first term in Equation (42) models the component of base current necessary to replenish minority carriers lost by recombination in the base. The second reflects any transient changes in the current levels. In Equation

t43), the collector current is assummed to be proportional

to the stored base charge. These two equations form the basis for the simple charge control model used in the transient analysis.

6A'11 Relationships with other transistor parameters

The current gain-bandwidth product 6.;r is given by:

6.12 Changes in Time Constants with Collector Currents

The diffusion and recombination time constants are directly related to the D.C. current gain and the gain band-width product f . From Equation (45).

I

(467

fT and as a function of Collector current is readily available in most transistor specification sheets (See

Appendix 1) or can easily be measured. An experimental gOT

can be used

7

setup for measuring high frequency current gains and 8

hence fT ( fT = f The results of both methods of deriving

Z

and Tg are also included. Graphs V i through VII show curves derived from Fairchild 2N708

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1111

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--I

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- ----I liII 2- -~ ~ I T --- --7J

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111li

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i TL1 -r __ _ rf~- --444 H~f~j+ -ii-tt ti-i-Tm- 'ri--rrrrr-'i-rr-ttrnrt ni-tv '-r-rrrr-rrtw -rn-rn r ri-i-i ii I' Ti t i--ii-i--tiui-r1--ti-ri- 1

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Olt

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{2; g'~j L.Z

LAIR

t T I I i i i -

--Specification sheets (Appendix I). Graphs VIII through XI are derived by actual measurement with the above method.

Alternatively r andN can be measured directly

by methods well covered in the literature.9

6.2 DERIVATION OF A SET OF SIMULTANEOUS DIFFERENTIAL EQUAT IONS

Substituting the above model into the Parallel

Schmitt Circuit, the following equivalent circuit results.

(See Fig. 27). Below is a set of simultaneous

differ-ential equations derived by using Equations (42) and (43) for both _{Q and Q2}

29

d

W

e4

c-FIGURE 27

Where:

*___R

_(4?)

-r7

(') )(53

The two equations (48) and (49) model the

tran-sition process through the active region (11). (See Fig. 8). They correspond physically to a case in which

the input voltage is taken up to either VF or VH and

then the regenerative action of the circuit is allowed to take over to switch the circuit into the other state. It will be seen that a very small incrementadrive, .-s e-cr noe 4oth auren ih ,ecry to a is

this transition.

The difficulty of solution of Equation (48) and (49) lies in the fact that

77

and

T

₈ change with operating

conditions.

2.=fk),

yhese functions of the

life times can be derived from plots of fT and versus IC or can be measured directly.

6.3 THE TWO METHODS OF TRANSIENT ANALYSIS

Most analyses of time varying linear or piecewise linear systems can be perfoemed in terms of total quan-tities or incremental quanquan-tities which are perturbations from some fixed operating level. The Parallel Schmitt circuit is no exception and is amenable to both these types of analyses if we are to assume that the diffusion and recombination time constants do not depend on coll-ector current levels. If the non-linear system is

analysed, with these depedences included, then the

total variables approach is found to be more convenient.

6.31 Analysis involving total variables

The transient behaviour can best be described in terms of total collector current levels ICl and IC2

of Q, and Q2. Turning back to Equations (48) and (49),

it is observed that these equations are derived from Fig. 27 (which has source and input voltages) and hence

involve the total quantities IC1 and IC₂. These equations, are characterized by a driving function on the right

hand side, to account for the fact that switching always occurs at some finite operating level i.e. at VH or VF, and non-linear coefficients 2a and 7B which are func-tions of IC1 and IC2* However, for a linear approximation

7"

and ?₈ are assumed to be constant with current.

A solution of Equations (48) and (49) will yield I cit) and IC₂(t), which contain a homogeneous and a particular solution.

This method, when the full non-linearity of the and ?- are accounted for, is particularly adaptable to numerical integration on the computer, because

2'

and ?'S are given in terms of changes in total currents.

6.32 Analysis involving incremental quantities

The transient solution can also be obtained by

considering the incremental changes in collector current levels iC1 and iC2. Again, for a linear solution the

and ?'g can be assumed to be constant with small perturbations in total ICIs. If the equations below

are substituted into Equations (48) and (49),

Ze: = a C(

where Io and I' are E.C. current levels when the cir-cuit is excited by VH and VS, )the following equations are obtained to describe the transient behaviour, in terms of incremental quantities,

Z

7/

1

_.

It is seen that in the quiescent state the incremental currents are zero as expected.

The "incremental method" of analysis is inherently linear as it implies that a linearization (of

2

, f

)

has been performed around the operating points (I0 and

I.'). When the non-linearities are introduced, the

currents iC1 and iC2 can be obtained by transforming the origin of the coordinates to the operating points. These methods are at best troublesome and are not preferred to

the "total variables approach" in non-linear analysis.

6.4 BOUNDARY CONDITIONS

The boundary condition in the transient analysis is 3provided by the constraint that the collector currents

-e.~ wme, ontq6, have to be continuous across t=O.

For the analysis involving total variables, the exbitation needed for switching is provided by a change in VH, defined as A . However, notice that the excitation is applied in the form of a negative change in VHrA., rather than an"intuitively obvious" positive excitation. This is immediately obvious if one considers a case in which the load line sweeps down towards VH (See Fig. 25 )

At holding, any positive change in VH will drive the circuit rapidly back into the (10) state while only a decrease in VH can accomplish the transition.

Using the consttaint of continuous collector current at t=O, the currents at t=0- are obtained by solving,

c 1 _2.,

k CS I k~k

and using the continuity condition,

Resubstituting into Equation (48) and (49) with the excitation -A,

Hence the boundary conditions, the slopes of the collector current at t=0, are obtained.

In a linear approximation method, which will be developed fully in Section 6.5, there are well defined

complex frequencies and the currents as functions of time are in the form of growing and decaying exponentials. The tetminating condition, for switching from (10) to the (01) state at VH, is reached when either IC,

or IC₂ becomes i&e-t zero or equal to IEF respectively. (See Fig. 28). The rise time is given by the time th6A10% to the time it takes t6 reach the 90% of the final value. The boundary conditions for the incremental analyses

C II

0I

~'o4

"F%

\4ckm

e\Jr

kov

ATV

c62..R

I.-

v e

o,

s)

I

r

kz

I%~* %~k~L Vh~

e~~A ~

AppOXA.cC~t

I

A

1 1

to 4.1 ,,

N

Uk

\Q

are exactly the same as those derived for the total quan-tities, except that the total quantities are replaced by incremental ones. Using the fact that before and just after the excitation,-A, is applied at t=0, the

currents iC1 and iC2 are zero, the equivalents of Equations (58) and (59) are rederived.

It may be well to point out that the magnitude of

A

in this and the "total variables analysis", is immaterial as long as it is finite. Although the rise time on the absolute time scale depends critically on A , the value of the rise time is independent of A if it is taken as the time difference between the "90% and 10% points".

(See Fig. 30).

Unlike the total variable analysis, a plot of the incremental currents versus time starts off at the origin as shown in Fig. 29. One exponential growing in the positive direction and one going in the negative direction. The tetminating condition is reached when the

total incremental change i.e. the sum of iC1 and i C2' has reached IEF*

The method involving'total quantities are used throughathe rest of the paper because, in the opinion of the author, it is conceptually easier and has more relationship with the J.C. conditions.

6.5 TRANSIENT SOLUTION BY LINEAR APPROXIMATION

The fact that the diffusion and recombination time constants, 27 and

Z8

, vary with collector current, introduces non-linearities int Equations (48) and (49). Their exact solution, which will be completed in Part III, is only possible by humerical techniques on the digital computer. When completed, the solution to the output waveform at the collector of Q2, will show an "S"

shaped curve, in which the rise time is given by the 10% and 90% points (Fig. 30). However, for the average

designer who desires only a rough prediction of the

rise time, a linear approximation can be readily performed by hand.

The method assums 27 and to be constant with

current and for convenience and

'b

: .

go

10 I0

IWIJJ

;.7~ma-

/,-,/~et

-4

4/Q/

a

frequency, S, can be defined. The associated homogeneous

equations of Equations (48) and (49) are:

A,

$1f

4

6.s

c16[6s?/

-~7#k

(T3)

The characteristic equation is given by:

[(s

Setting

.+-L~

..

-:-and letting

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[ .5 1

,A ----

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i

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(a6)

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