Analog Computing Technique


 Laurence Barrett
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1 Analog Computing Technique by obert Paz Chapter Programming Principles and Techniques. Analog Computers and Simulation An analog computer can be used to solve various types o problems. It solves them in an analogous way (pun intended). Two problems or systems are considered analogous i certain or all o their respective measurable quantities obey the same mathematical equations. Most general purpose analog computers use an active electrical circuit as the analogous system because it has no moving parts, a high speed o operation, good accuracy and a high degree o versatility. Active electrical networks consisting o resistors, capacitors, and op amps connected together are capable o simulating any linear system since the orward voltage
2 transer characteristics o these networks are analogous to the basic linear mathematical operations encountered in the system s mathematical model. By using diode unction generators and special circuits which have nonlinear voltage transer characteristics, it is also possible to simulate nonlinear systems. The mathematical model o an analog computer programmed to simulate a speciic physical system is identical to the mathematical model o the system. The voltage transer characteristics o the electrical networks are analogous to the desired mathematical operations. The input and output voltages (computer variables) are analogous to the corresponding mathematical variables (problem variables) o the problem. Because o limitations o the computer or its associated input/output equipment, it is usually necessary to change the scale o the computer variables, thus orcing the values o a computer variable to dier rom the corresponding problem variable values. It is important to understand that an analog computer solution is simply a voltage wave orm whose time dependency is the same as that o the desired variable. The normal procedure or simulating a system starts with determining the mathematical model describing the physical quantities o interest. An analog block diagram is made to relate the sequence o mathematical operations and to aid in scaling the variables. From the analog block diagram the electrical components are connected together (patched). The computer is operated and the computer variables observed on a recorder or oscilloscope. Since the output is a computer variable (voltage wave orm) it is necessary to convert the output variable back to the original problem variable.. Solving Dierential Equations with an Analog Computer. A typical simulation o a physical system involves a mathematical model consisting o a set o one or more dierential equations and initial conditions on the variables. I the system is linear, the dierential equations are linear and the operations required are ) summation, ) sign inversion, 3) multiplication by a constant, 4) integration and 5) dierentiation. For practical reasons, the integration operation is easier to implement than the dierentiation operation. The reason lies in the act that computer signals are real voltages and, thereore, are corrupted by noise to some etent. Since integration has a tendency to average out the eects o noise (while dierentiation will accentuate it), a more precise solution can be obtained using integration techniques. Each o these operations may be represented as shown in Figure. Actual realization o these will be discussed in the net section. Analog Computer Principles
3 (t) (t) y(t) = = (t) (t) 3 (t) (t) a a(t) (a) 3 (t) Summation (b) Multipication by a Constant (c) (t) y(0) Integration y(t) = t = y(0) 0 (t)dt (d) (t) Sign Inversion (t) Figure.: Basic Linear Operations As an eample, consider the computer solution o the dierential equation dy dt = y, y( 0) = (.) Since the analog computer solves the equivalent integral equation, we integrate both sides. t y( t ) = y( 0) y( t ) dt. (.) The deinition o integration given in Figure. would represent equation (.) i the input were the same as the output. This condition can be easily implemented by connecting the input to the output as shown in Figure.. 0 y(0) y(t) = t = y(0) 0 y(t)dt Figure.: Block Diagram ealization o the Integral Equation. It is important to note that it is not necessary to know the input o the integrator in order to solve equation (), but only that the input must be equal to the output at all times. The idea o eeding an unknown output back to the input to generate a solution is basic to analog Chapter. Programming Principles and Techniques 3
4 computer solution o dierential equations. This is not unreasonable since the dierential equation determines the class o solutions, while the initial conditions determine the speciic solution. A higher order linear dierential equation may be handled by reducing it to a set o irst order equations and ollowing a similar procedure. For eample, d y dy y = 0, y( 0) =, ( y 0) = 0, (.3) dt dt may be turned into the set o equations, ( ) (.4a), ( ) (.4b) where = y, and = dy dt. The equivalent integral orms are t ( t ) = ( 0) ( t ) dt, (.5a) 0 t ( t ) = ( 0) ( t ) ( t ) dt (.5b) 0 [ ] Equations (.5a) and (.5b) may be implemented using the circuit shown in Figure.3. (0) (0) (t) (t) (Equation.5b) (Equation.5a) Figure.3: Circuit ealization o the Second Order Equation. 3. Physical ealization o Linear Operations on an Analog Computer. In section, the dierential equation was epressed in terms o a set o general mathematical operation. No attempt was made to discuss how these operations were realized with physical 4 Analog Computer Principles
5 components. In this section, the basic linear operations o summation, multiplication by a constant, and integration will be discussed. The operation o sign inversion will be inherent in the summation and integration operations as a result o construction convenience and versatility. (a) The Operational Ampliier (Op Amp). The operational ampliier is a high gain ampliier with a wide variety o applications. The ampliier is usually described in terms o its gain, input impedance, output impedance, bandwidth, and oset characteristics. An operational ampliier usually has two input terminals. The two input terminals are marked with a () to indicate the noninverting input and a ( ) to indicate the inverting input. An equivalent circuit or an op amp and a standard symbol are shown in Figure.4. V V V V (a) (b) in out V g V g = A v (V V ) Figure.4: (a) The Circuit Symbol or an Op Amp, (b) An Equivalent Op Amp Circuit. (b) Summers and Inverters. The electrical circuit whose transer characteristics are analogous to the mathematical operation o summation is shown in Figure.5 ( or an n input summer). Applying Kircho s current law at the summing junction gives or, in terms o voltages Note that since V V V i... in i = ia (.6) V V V V V... o =. (.7) n = A V, then equation (.7) can be rewritten o v n in Chapter. Programming Principles and Techniques 5
6 V V V V... n o = o, (.8) A n v where = n in L. (.9) i V i V n V o in out n i n V g Figure.5: Summing Ampliier By isolating, we obtain, V o = V... Vn. (.0) n A A v v Since the op amp has a very high voltage gain (usually > 0 5 ), we assume that A v. Thus, equation (.) reduces to V o V =... V n. (.0) n Usually, analog diagrams are given in terms o symbols which represent the electrical circuit. For this weighted summation, the analog symbol is shown in Figure.6, and we have the output equation Vo = KV... K nvn, (.) and 6 Analog Computer Principles
7 K i =, i =, K, n. (.) i V V n K K n Figure.6: Weighted Summing Ampliier Since the circuit in Figure.5 is to be implemented on the analog computer, it is essential that the summing operation indicated by Figure.6 and equations (.) and (.) be understood thoroughly. Note that the inherent sign inversion is a result o the negative voltage gain o the op amp. Thus, the inverter which inverts the sign o its input is a special case o the summer with only one input and with K =. The transer characteristic is given by Vo = V. (.3) V V V 3 V Figure.7: Analog Diagram or Eample.. Eample.: Determine the circuit to produce the output voltage given by Vo = V 0V V3 0 V4. (.4) Solution: An analog diagram (i.e. in terms o symbols) or equation (.4) is given in Figure.7. The circuit diagram which must be patched on the analog computer to realize equation (.4) is shown in Figure.8. Chapter. Programming Principles and Techniques 7
8 V 50kΩ 50kΩ V 5kΩ 50kΩ 50kΩ V 3 V 4 5kΩ 50kΩ Figure.8: Electronic Circuit or Eample.. In an actual ampliier, the output voltage will not be zero when V i and V n are zero. This eect is called the oset and is usually measured as is shown in the drawing below. The value o s required to reduce to zero under the conditions shown is called the oset voltage. Under the conditions shown ( = 0), the oset current, I os, is deined as I I. Note the termination attached to the noninverting input in Figure.9. y i s I I y n n = i i Figure.9: Op Amp Circuit Showing the Oset Voltage In an actual ampliier, the output voltage is not normally zero when V = V. This eect is oten described in terms o the common mode gain (CMG) and the common mode rejection ratio (CM). Measurement o the CMG and CM are illustrated in Figure.0. y CM = kv V o CM (.5) 8 Analog Computer Principles
9 s Figure.0: Op Amp Coniguration or Determining CM Operational ampliiers come in many orms. As an eample the characteristics o the µa 709 ( a simple integrated circuit ampliier) are given below. A v 45,000 s mv (typical or i < 0k Ω) I os in out CM 0.µA 00kΩ 50Ω 90dB (c) Multiplication by a Constant ; eerence Voltages. The op amps on the Comdyna GP6 computer are constructed with 5K and 50k resistors only. Thus, in order to realize gains (constant multipliers) other than /0, and 0, another technique must be used. The coeicient potentiometer (pot) is a voltage divider which allows the output voltage to be some raction o the input voltage. A pot thus has a gain o less than unity. The electrical circuit diagram and analog computer diagram are shown in Figure.. The transer characteristic is given by V o = av, 0 a. (.6) in V in a = av in V in a (a) (b) Chapter. Programming Principles and Techniques 9
10 Figure.: (a) Circuit Symbol, (b) Analog Computer Symbol or Potentiometer. The gain a is always placed outside the pot with the inside reserved or a pot identiication number. A constant term in a dierential equation is obtained by using a dc reerence voltage which is supplied on the computer. Its value is usually ± the range o the op amps. For the GP6, this is ± 0v. Note that the actual dials or the potentiometer are illustrated in Appendi A. Eample.: Determine the analog diagram and circuit to implement the equation Vo = V 5. 4 V. 6. (.7) Solution: The analog diagram is given in Figure. and the electrical circuit in Figure.3. A GP6 wiring diagram is supplied in Figure.4 to illustrate the actual connections needed. V V Figure.: Analog Diagram or Eample.. V kΩ 5kΩ 50kΩ V.54 5kΩ 5kΩ kΩ Figure.3: Circuit Diagram or Eample.. 0 Analog Computer Principles
11 V V SJ IC SJ IC COEFFICIENT POT SJ SJ 3. B. B..B..B Figure.4: GP6 Wiring Diagram or Eample.. (d) Integrators. Integration is the most important operation available on the analog computer. In act, analog computers owe their eistence to their ability to integrate rapidly. Integration is dierent rom inversion and summation because it is time dependent. Integration can be accomplished by replacing the eedback resistor o the summer with a capacitor. The resulting electrical circuit or an integrator is shown in Figure.5. i C i V n i n V V n Figure.5: Electrical Circuit or an n Input Integrator. Using the same procedure and approimations as used or the summer, the transer characteristic or the circuit o Figure.5 becomes Chapter. Programming Principles and Techniques
12 t V ( ) V V t V n( ) o( ) = o( 0) t t... C C d t (.7) 0 n Again, there is a sign inversion in the integration operation as a result o the negative voltage gain o the op amp. Note that summation and integration can be perormed with a single ampliier. The analog computer symbol or the integrator is shown in Figure.6. (0) V V n K K n Electronic Switch Figure.6: Analog Computer Diagram or an Integrator. The electronic switch terminal (ES) is used to control the operating modes o the integrator. The normal modes o operation are the initial condition (IC) mode and the operate (OP) mode. The IC mode allows the integrator capacitors to be charged to the initial values, while the OP mode causes the solution to occur. This sequence is shown or the eample in Figure.7. The switches or the IC, HD, OP, and O are illustrated in Appendi A. 0V 0v 0V t (a) E.S. t = 0 (b) 0v Figure.7: (a) Analog Computer Diagram or Eample.3, (b) Voltage Output or Eample.3. The electronic switch is nothing more than a SPDT switch with one pole grounded. Control o the electronic switch is made through a switch driver and logic circuits which are usually controlled by the computer mode control switches on the ront panel. It is important to note, concerning the analog symbol or the integrator, that i) ( 0 ) must be applied to the IC Analog Computer Principles
13 terminal to get ( 0 ) at the output and ii) the electronic switch, ES, is usually implied and not shown unless it is used in an unconventional manner. Circuit diagrams or the integrator are shown in Figure.8. In the initial condition state, illustrated by Figure.8a, the capacitor charges to V ic which can be used to represent an initial condition. When in the OP mode, the output o the system is the negative o the integral o the input starting rom the initial condition. The equivalent circuit under these conditions is shown in Figure.8b. Note that the initial condition part o the circuit is grounded. V ic V ic V i i (a) C V ic V i i (b) C Figure.8. (a) Integrator Circuit in IC Mode, (b) Integrator Circuit in OP Mode. 4. A Systematic Procedure or Programming Dierential Equations Although the procedure presented in Section may be used to determine the analog diagram o a linear dierential equation, a more systematic version will be developed. This procedure does not require that a irst order integral equation be considered or each integrator, but instead requires an equation or the highest order derivative. This equation will correspond to Equation (.4) in Section. There is no essential dierence in the theory or results o the two techniques. Only the steps required dier. Consider, or purposes o illustration, the second order linear dierential equation and initial conditions, d dt d 0. 5 = 4, ( 0) = 0, ( 0) =. (.8) dt I any derivative o a variable is known, then it may be integrated to obtain the variable. In the case o Equation (8) i the second order derivative (d /dt ) were known, then it could be Chapter. Programming Principles and Techniques 3
14 integrated once to obtain the irst order derivative (d/dt) and a second time to obtain the variable (). Note that since the integrator has a sign inversion associated with it, the output o every odd integration is negative. But Equation (.8) gives the second order derivative in terms o the lower ordered derivative. Thus, () I d dt is known, d dt and may be obtained; () I d dt and are known, d dt may be obtained. This circular argument states that only the relationship between the inputs and outputs o the integrators are known, not the actual values o the inputs and outputs. This should not seem too strange since a dierential equation represents only the class o solutions. The boundary values or initial conditions are necessary to determine a particular solution. The initial conditions have not yet been considered. The programming procedure is then as ollows: (a) Assume that the highest order derivative ( d dt in this case) is known and generate all lower order derivatives as shown in Figure.9. Note that the output should always be, not , since is the desired solution. Figure.9: Analog Circuit or a Double Integrator (b) Solve the dierential equation or the input to the integrator string and orm the indicated sum. d dt d = (.9) dt (Don t orget that the summer also inverts!). See Figure Figure.0: Generating the Second Derivative. 4 Analog Computer Principles
15 (c) Combine the results o (a) and (b) using pots, summers and inverters where required. See Figure.. (d) Add all the initial conditions. ecall that the applied initial condition should be the negative o the integrator output at t = 0. See Figure.. 0v.4.5 [ (0)] [(0)] Figure.: Analog Diagram or the SecondOrder System. Although Figure. is the inal diagram, it is possible to remove the summer by recalling that the integrator can also be a summer. Inverters must then be inserted or removed rom each eedback loop to take care o the sign inversion. Usually, the orm which uses the least number o elements (pots, amps, etc.) is the preerred orm since the probability o a bad lead or element in the patched problem is lessened. Figure. represents the same problem as Figure., but with the summer removed. Figure.3 shows how Figure. would actually be wired up on the GP6. 0v 0v.4..5 Figure.: Alternate Diagram, with educed Number o OpAmps Chapter. Programming Principles and Techniques 5
16 SJ IC SJ IC SJ SJ COEFFICIENT POT 3 SJ 7 5. B. B..B..B. SW OP SW OP Figure.3:. GP6 Wiring Diagram or Figure. As another eample, consider a general third order linear dierential equation. The procedure may be etended or a system o any order. We have with initial conditions Using the procedure outlined above: (a) Form the integrator string 3 d a d a d a t 3 o = ( ), (.0) dt dt dt ( 0) =, ( 0) =, ( 0) =. (.) o o o Figure.4: A ThirdOrder String o Integrators. (b) Solve or the input to the integrator string and orm the indicated sum. 6 Analog Computer Principles
17 3 d = a d a d a t 3 o ( ), (.0) dt dt dt (t) a 0 a a Figure.5: Generating the Third Derivative. (c) Combine the results o (a) and (b). (d) Add the initial conditions and remove the summer since one less ampliier is required. The result is shown in Figure.6. (t) [(0)] a a [ (0)] [(0)] a 0 Figure.6: The General ThirdOrder System. 5. Tricks with Transer Functions Suppose we have the irstorder transer unction G( s) Y ( s) a = =, 0 < a 0, 0 < b 0 U ( s) s b This can be drawn in either o the allintegrator diagrams shown in Figure.7. u(t) a y(t) u(t) a y(t) b b Chapter. Programming Principles and Techniques 7
18 Figure.7: Equivalent AllIntegrator Diagrams Because the gains are o values less than 0, then nothing more than a 0 input is needed. Eample analog computer diagrams are shown or the two block diagrams as shown in Figure.8. We note that the analog computer does very well at implementing a irstorder system. u(t) a/0 0 0 y(t) u(t) 0 0 a/0 y(t) b/0 b/0 (a) (b) Figure.8: Analog Computer First Order Diagrams O the two circuits, both have advantages and disadvantages. The circuit (a) has the advantage that the output comes directly rom an opamp, and can thus be used to drive another circuit. This circuit is also good i an initial condition, y( 0 ), is given. It has the disadvantage that its input has a pot beore entering the op amp. Thus, this circuit will tend to load whatever signal is driving it. The circuit (b) has the advantage that it does not load the circuit that is driving it, but has a pot on the output, and is thus not very useul in driving another circuit. Both o these circuits, however may be used in a higherorder transer unction realization. For eample, i a transer unction has simple real poles and the cascade actorization a a G( s) = s p s p a n L s p where all the coeicients are o magnitude less than 0, then these circuits may be simply cascaded (put in series) to orm the overall transer unction. Some suggestions to this approach include i) Using circuit (b) to implement the irst (input) stage, and (a) as the output stage. ii) Alternating circuits to combine coeicients and thus reduce the number o required potentiometers. iii) Using an additional inverter on the output i there are an odd number o poles. I our transer unction has the partial raction epansion n 8 Analog Computer Principles
19 c c c G( s) = d n L s p s p s p then we may implement this transer unction using the irstorder circuits in parallel, with the output o each entering into a summer. In this case, circuit (b) would be preerable to prevent loading the input. n Chapter. Programming Principles and Techniques 9
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