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SIMULTANEOUS ALGEBRAIC EQUATIONS

A different type of problem is one involving simultaneous algebraic equations. A system of two

equations in two unknowns may be represented by the equations

a, x + b,y = C,

ajX +

— C

These equations may be rewritten in the form

2L_ |

y - c 2

b2

a 2

a 2 b2

The computer circu it for these equations is shown in Figure 32.

— wwv

2L

a ,

Figure 32

CIRCUIT FOR SIMULTANEOUS ALGEBRAIC EQUATIONS

As an example, consider the equations

5

X

+ 3 y

=

210

2

X +

5 y

=

160

These equations becom e

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Содержание Heathkit EC-1

Страница 1: ...EDUCATIONAL ELECTRONIC ANALOG COMPUTER MODEL EC 1 ...

Страница 2: ... in Heath kits are Y iwatt Higher wattage rated resistors when specified are progressively larger in physical size Small wire wound resistors x i watt 1 or 2 watt may be color coded but the first band will be double width CURRENT STANDARD CODE W hile RM A 2nd fSisnifitanf f i9ure Black JAN C la ss M ultiplier Tolerance JA N 1948 RM A CODE RMA 3 DOT OBSOLETE RATED 500 W V D C 20 TOL M ultiplier Sig...

Страница 3: ...and third to show some illustrative examples of problems which can be solved on the Computer This manual is not intended to be exhaustive but rather to be a guide in the operation of the computer For this reason frequent references are made to the available literature Several excellent books as well as many articles are available Some of these are listed in the references at the end of the manual ...

Страница 4: ...c u it 15 Repetitive O s c illa to r 15 General Operating In stru ction s 16 Basic Mathematical Operations A ddition 17 M ultiplication 19 Integration 19 Illustrative Problems Falling Body 21 Spring Mass S y s t e m 24 Simultaneous Algebraic E quations 26 P r o je c tile 27 Bouncing B a ll 28 R eferen ces 31 S p ecifica tion s 32 Parts L i s t 36 S ch em a tic 40 Page 2 ...

Страница 5: ...computer operates continuously makes it very useful in such operations as integration for this reason computers used this way are sometimes known as Differential Analyzers One of the most powerful applications of analog computers is simulation in which physical properties not easily varied are represented by voltages which are easily varied Thus the knee action of an automobile front wheel suspens...

Страница 6: ...cients and initial conditions The general procedure in solving a problem is to 1 Set the machine variables voltages to the correct initial conditions 2 Make the computing elements operative and force the voltages to vary in the manner prescribed by the differential equations 3 Observe and or record the voltage variations with respect to time which constitute the solution of the given problem 4 Sto...

Страница 7: ...hes zero as A approaches infinity In practice A is made large with respect to eQ by using high gain amplifiers so that eg becomes very small and for practical purposes eg can be considered to be at ground potential Since the input to the amplifier is the grid of a tube the current through the amplifier from the input can be considered to be zero with the result that the current if through the resi...

Страница 8: ...omputer The equation eo 5 l _ e may be written e_ where o K K which is in effect multiplication by a constant Since in most cases the output voltage is of the opposite sign to the input voltage the amplifier also acts as an inverter or sign changer To change the value of the constant K it is necessary only to change Rf or Rf Generally Rf is kept at 1 megohm and Rj is changed This may be done by us...

Страница 9: ... are used as shown in Figure 5 the operational amplifier becomes an adder e VWW e o J W M r e JWvW M W e0 i Figure 5 AMPLIFIER AS ADDER Again making use of eg 0 the sum of the currents in the input resistors equals the current through the feedback resistor ThUS i 2 i 3 f But since i _ r it follows that L e 2 e3 D R R3 L1 2 3 f Multiplication of both sides of the equation by Rf gives the result S t...

Страница 10: ...IER CIRCUIT USED FOR SUBTRACTION which may be written where i t and R f K2 r7 Division can be performed by multiplying by the reciprocal of the number but since this som e times requires a ratio of Rf Ri of less than unity which may cause unstable operation of the amplifier this method is generally undesirable The circuit shown in Figure 7 shows a more desirable method DIVISION BY A CONSTANT The v...

Страница 11: ... of 10 than the true value of the answer so the indicated answer must be divided by the same value as used in re ducing the division to a value of less than one in this case 10 Thus for this example o 0 25 110 1 Rj i since 2 5 0 25 x 10 choosing Rj r i m e g this becomes 1 1 x eO 0 25 10 ei which becomes 10 0 q 25 If the divisor had been 25 a factor of 100 would have been chosen yielding 100 o o 2...

Страница 12: ... similar analysis that the operational amplifier can be used to differentiate The amplifier is used very seldom for this purpose however since noise in the input tends to be magnified by differentiation whereas it tends to cancel out in integration Such circuits also tend to be unstable In practice the value of the feedback resistor Rf when used is generally 1 megohm and the value of the feedback ...

Страница 13: ...eleration which the body experiences is constant near the surface of the earth and due to the force exerted on the object by the gravitational field of the earth This may be written as an equation d y _ dt 2 where y is the distance the object falls in time t and g is the acceleration given the object by the earth s gravitational field By integrating twice it is possible to obtain an expression for...

Страница 14: ...antage is that machine error is decreased especially the error due to the leakage resistance of the feedback capacitors In general the solution of a problem on the computer should not require more than l to5 minutes Longer solution times require special precautions REPETITIVE OPERATION It is desirable for many problems to repeat the solution and observe the effect on the solution of changing the v...

Страница 15: ...ly true when repetitive operation is used In this case one of the computer amplifiers is used to provide the sweep The oscilloscope must be a DC scope If a permanent record of the solution is desired a photograph of the oscilloscope trace may be made or a recording galvanometer may be used Examples of both methods are shown in the illustrative problems NON LINEAR OPERATION A discussion of non line...

Страница 16: ...k and input resistors thus simplifying the balancing operation The triode section of the 6U8 is connected as a cathode follower output stage The two NE 2H neon lamps are used as a reference voltage dropping element by which the output signal level can be dropped from approximately 112 volts to zero volts without loss in gain as would occur if a purely resistive element were used The input and outp...

Страница 17: ...s by means of a meter and a FUNCTION switch on the panel By use of the FUNC TION switch and meter the output of any of the operational amplifiers may also be measured An OPERATION switch turns the computer on starting solution of the problem After the prob lem is solved it is necessary to reset the computer to the starting conditions This is accom plished by setting the OPERATION switch to the RES...

Страница 18: ...his operation for each of the other eight amplifiers With the METER RANGE switch set at 10 V repeat the above operation for each amplifier Once again repeat the above opera tion for each amplifier with the METER RANGE switch set at 1 V The amplifiers should be checked for balance before each problem run It is not necessary to remove computing compo nents from the problem board when balancing ampli...

Страница 19: ...nd terminal red binding post will be connected to the IC supply voltage of the required potential A 4PST relay is used for inserting initial conditions and for removing residual voltages from the amplifiers For convenience the connections to the relay contacts are brought to binding posts on the panel where they may be connected across problem components as required The relay contacts are normally...

Страница 20: ...R FUNCTION switch at 10 V and the number 1 INITIAL CONDITION control to extreme counterclockwise position Turn on the HIGH VOLTAGE switch and turn the IC 1 control clockwise until the meter reads 3 6 volts Now unplug the end of the patch cord at the red METER INPUT binding post and plug it into either of the INPUT binding posts of amplifier 1 In the same manner connect IC 2 to the meter connect th...

Страница 21: ...ift in the amplifier I M t o e0 Bf g I OOP OOP C3 7 VOLTS 10X3 7 VOLTS R I 100 000 Figure 19 AMPLIFIER USED FOR MULTIPLICATION INTEGRATION It was shown on Page 10 that by replacing the feedback resistor of an amplifier with a capacitor the output of the amplifier is proportional to the integral of the input voltage or more specific ally e J I e d t e j c Rjtf J 1 c Plug a 1 megohm precision resist...

Страница 22: ...read ing should increase rather uniformly from zero to approximately 100 V but should be stopped at 60 volts by turning the OPERATION switch to RESET This will stop integration and remove the charge from the capacitor leaving it ready for operation again Turning the OPERATION switch to REPETITIVE will start the computer operating but now the oscillator will automati cally reset the initial conditi...

Страница 23: ...e some of the types of problems the solution of which can readily be obtained on the computer Components are furnished for use in solving some of these problems For other problems additional components are required These may be standard radio components Ordinary 5 or 10 resistors may be used where accuracy of results is not important such as when only the general nature of the problem solution is ...

Страница 24: ...er 2 Plug mounted 1 pfd capacitors into the feedback sockets of amplifiers 1 and 2 Connect a patch cord short from the output of amplifier 1 to the input of amplifier 2 the one with the 1 megohm resistor Connect patch cords from the AMPLIFIER INPUT binding post and AMPLIFIER OUTPUT binding posts to relay contacts 1 for amplifier 1 and relay contacts 2 for amplifier 2 Set IC 1 control to the first ...

Страница 25: ... DC oscilloscope such as the Heath DC Oscilloscope by con necting the vertical input of the oscilloscope to the red AMPLIFIER OUTPUT binding post and the ground to the black AMPLIFIER OUTPUT binding post No sweep is needed to show the falling body but if one wants to show the path when an initial horizontal velocity is given the body a sweep voltage is necessary To insure synchronization one of th...

Страница 26: ...T FRICTIONAL FORCE ADDED This circuit is set up on the computer as before with the addition of a feedback resistor across the first amplifier For simplicity therelayand ground connections are omitted as well as the initial condition power supply connections These are the same as in the previous problem The solution is shown in Figure 28 Figure 28 A Figure 28 B 191 SOLUTION OF FALLING BODY PROBLEM ...

Страница 27: ...e both zero The differential equation may be written M s T T F s 7 r v The simplified computer diagram is shown in Figure 30 Figure 30 SPRING MASS SYSTEM CIRCUIT In this diagram all resistor values are megohms and all capacitor values are microfarads Ground connections are omitted for purpose of simplicity F t is supplied by one of the initial condition power supplies Various solutions are shown i...

Страница 28: ...unknowns may be represented by the equations a x b y C ajX C These equations may be rewritten in the form 2L_ y c 2 b2 a2 a 2 b2 The computer circuit for these equations is shown in Figure 32 wwv 2L a Figure 32 CIRCUIT FOR SIMULTANEOUS ALGEBRAIC EQUATIONS As an example consider the equations 5 X 3 y 210 2 X 5y 160 These equations become Page 26 ...

Страница 29: ... total voltage respectively PROJECTILE PROBLEM A variation of the falling body problem is to fire a projectile upward at an angle from the hori zontal and see the path on the oscilloscope Air resistance may be introduced also Thus the effects of the initial velocity gravity and air resistance may be observed In the X direction the equation of motion is with V at T 0 d 2 M d t d t oxi and the equat...

Страница 30: ... in Figure 35 Figure 35 A Figure 35 B SOLUTION OF PROJECTILE PROBLEM a As recorded on oscilloscope b As recorded on pen recorder BOUNCING BALL A problem which is interesting to watch as well as one which illustrates the more complex type of problem that the computer is capable of handling is the Bouncing Ball problem For solv ing this problem a few additional components are required which are read...

Страница 31: ...portant but must have center tap 2 10 Kf2 1 2 watt resistors 2 25 Kf2 controls 1 0 1 M fd 150 volt capacitor 1 1 0 jufd 150 volt capacitor 2 100 K 2 controls 105 125V 5 0 6 0 C Y C L E B A L L S H A P E s 2 5 K 2 5 K Figure 36 PHASE SHIFTER FOR BOUNCING BALL PROBLEM These components are standard radio components The capacitors may be paper or plastic tubular The complete diagram for the problem in...

Страница 32: ...a a m v i A g VW Ar l I 1 AM M AMA V tr 2 Q C o ro L D Z I O DC V Page 30 Figure 38 ...

Страница 33: ...RONIC COMPUTERS PRINCIPLES AND APPLICATIONS Philosophical Library New York 1956 Proceedings of the NATIONAL ELECTRONICS CONFERENCE Volume V 1949 Chicago 1950 Stibitz George R and Larrivee Jules A MATHEMATICS AND COMPUTERS McGraw Hill 1957 PAMPHLETS Howe C E and Howe R M A TABLETOP ELECTRONIC DIFFERENTIAL ANALYZER University of Michigan Department of Aeronautical Engineering Publication AIR 6 1953 ...

Страница 34: ...s each using a high gain circuit with a pentode driving a cathode follower output Each amplifier has the following characteristics GAIN Open loop gain approximately 1000 OUTPUT Any value between 60 and 60 volts with load current up to 0 7 ma HUM AND NOISE LEVEL Approximately 4 mv referred to input under normal conditions DRIFT Short term drift less than 5 mv referred to input FREQUENCY RESPONSE Do...

Страница 35: ...ven in the Operational Manual DIMENSIONS Computer Cabinet 19 5 8 wide x 111 2 high x 15 deep Panel 19 wide x 101 2 high May be mounted in standard rack if desired WEIGHT Net Weight 37 1 2 lbs Shipping Weight 43 lbs POWER REQUIREMENTS 1050125 volts 50 60 cps 100 watts ASSEMBLY TIME Average about 25 30 hours May require more or less time depending on experience of builder and quality of workmanship ...

Страница 36: ...S The following graphs show the response of atypical DC amplifier of the EC 1 Individual ampli fiers will show some variation from these curves LOAD RESISTANCE IN OHMS FREQUENCY IN CYCLES PER SECOND Page 34 ...

Страница 37: ...AMPLIFIER OUTPUT I N DB FREQUENCY IN CYCLES PER SECOND Page 35 ...

Страница 38: ... Wound Resistor 10 Watt Wire Wound Resistor Precision Resistors 2 145 1 19 Kft 1 2 54 1 200 Kf2 1 2 14 2 1 meg 1 2 55 1 2 meg 1 2 17 1 10 meg 1 Controls Switches 10 91 9 3 Kfi linear screwdriver adjust 10 12 8 100 K 2 linear 10 14 1 250 Kf2 linear 10 94 1 7 5 meg linear 60 3 9 DPDT slide switch 61 9 2 SPST toggle switch with hardware 63 190 1 3 position rotary switch 63 191 1 4 position rotary swi...

Страница 39: ...50 silicon rectifier M eter Tubes Lamps 407 58 1 50 0 50 microampere meter 1 411 46 4 OB2 tube 411 59 1 OA2 tube 411 60 1 6AQ5 tube 411 73 1 12BH7 tube 411 80 9 6U8 tube 411 95 1 6BH6 tube 412 1 1 47 pilot lamp 412 15 18 NE 2H neon lamp 412 3 1 NE 51 neon lamp Grommets Bushings Clamps 73 4 18 3 16 rubber grommet 73 2 4 3 4 rubber grommet 75 17 146 Nylon bushing 75 24 1 Plastic strain relief bushin...

Страница 40: ...rts 200 M201 1 203 M167F245 1 204 M219 1 204 M220 1 204 M221 1 90 97 1 3 48 x 1 8 3 48 x 1 4 4 40 x 1 2 6 32 x 3 8 6 32 x 3 8 8 32 x 3 8 10 32 x 1 2 3 48 x 3 16 4 40 x 1 4 hex nut 6 32 x 1 4 hex nut 8 32 x 5 16 hex nut 3 8 hex control nut 3 8 flat control washer 1 2 flat switch washer 6 lockwasher 8 lockwasher 3 8 control lockwasher 6 solder lug Chassis Panel Right panel support bracket Left panel...

Страница 41: ...PTION Per Kit _______________ 9 100 KS2 1 precision resistor 9 1 meg 1 precision resistor 3 0 1 jufd 200 volt 5 mylar capacitor 3 1 0 jufd 200 volt 5 mylar capacitor 2 Silicon diode 27 Plastic double plug 48 Banana plug 48 Red banana plug cap 1 Roll red test lead wire SILICON DIODE PLASTIC DOUBLE PLUG Page 39 ...

Страница 42: ...i 5 j u Q o s ...

Страница 43: ...parts and wiring When removing insulation from the end of hookup wire it is seldom necessary to expose more than a quarter inch of the wire Excessive insulation removal may cause a short circuit condition in respect to nearby wiring or terminals In some instances transformer leads of solid copper will have a brown baked enamel coating After the trans former leads have been trimmed to a suitable le...

Страница 44: ...HEATH COMPANY A S u b s i d i a r y o f D a y s t r o m Inc TH E W O R L D S F I N E S T E L E C T R O N I C E Q U I P M E N T I N K I T B E N T O N H A R B O R M I C H I G A N F O R M Litho in U S A ...

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