ELTR 135 (Operational Amplifiers 2), section 2

Transcription

1 ELTR 135 (Operational Amplifiers 2), section 2 Recommended schedule Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Topics: Operational amplifier oscillators Questions: 1 through 10 Lab Exercise: Opamp relaxation oscillator (question 46) Topics: Calculus explained through active integrator and differentiator circuits Questions: 11 through 20 Lab Exercise: Opamp triangle wave generator (question 47) Topics: Logarithm review Questions: 21 through 35 Lab Exercise: Opamp LC resonant oscillator (question 48) Topics: Log/antilog circuits (optional) Questions: 36 through 45 Lab Exercise: Work on project Topics: Review Lab Exercise: Work on project Exam 2: includes Oscillator circuit performance assessment Project due Question 49: Sample project grading criteria Troubleshooting practice problems Questions: 50 through 59 General concept practice and challenge problems Questions: 60 through the end of the worksheet 1

2 ELTR 135 (Operational Amplifiers 2), section 2 Skill standards addressed by this course section EIA Raising the Standard; Electronics Technician Skills for Today and Tomorrow, June 1994 E Technical Skills Analog Circuits E.10 Understand principles and operations of operational amplifier circuits. E.11 Fabricate and demonstrate operational amplifier circuits. E.12 Troubleshoot and repair operational amplifier circuits. E.20 Understand principles and operations of sinusoidal and non-sinusoidal oscillator circuits. E.21 Troubleshoot and repair sinusoidal and non-sinusoidal oscillator circuits. B Basic and Practical Skills Communicating on the Job B.01 Use effective written and other communication skills. Met by group discussion and completion of labwork. B.03 Employ appropriate skills for gathering and retaining information. Met by research and preparation prior to group discussion. B.04 Interpret written, graphic, and oral instructions. Met by completion of labwork. B.06 Use language appropriate to the situation. Met by group discussion and in explaining completed labwork. B.07 Participate in meetings in a positive and constructive manner. Met by group discussion. B.08 Use job-related terminology. Met by group discussion and in explaining completed labwork. B.10 Document work projects, procedures, tests, and equipment failures. Met by project construction and/or troubleshooting assessments. C Basic and Practical Skills Solving Problems and Critical Thinking C.01 Identify the problem. Met by research and preparation prior to group discussion. C.03 Identify available solutions and their impact including evaluating credibility of information, and locating information. Met by research and preparation prior to group discussion. C.07 Organize personal workloads. Met by daily labwork, preparatory research, and project management. C.08 Participate in brainstorming sessions to generate new ideas and solve problems. Met by group discussion. D Basic and Practical Skills Reading D.01 Read and apply various sources of technical information (e.g. manufacturer literature, codes, and regulations). Met by research and preparation prior to group discussion. E Basic and Practical Skills Proficiency in Mathematics E.01 Determine if a solution is reasonable. E.02 Demonstrate ability to use a simple electronic calculator. E.05 Solve problems and [sic] make applications involving integers, fractions, decimals, percentages, and ratios using order of operations. E.06 Translate written and/or verbal statements into mathematical expressions. E.09 Read scale on measurement device(s) and make interpolations where appropriate. Met by oscilloscope usage. E.12 Interpret and use tables, charts, maps, and/or graphs. E.13 Identify patterns, note trends, and/or draw conclusions from tables, charts, maps, and/or graphs. E.15 Simplify and solve algebraic expressions and formulas. E.16 Select and use formulas appropriately. E.17 Understand and use scientific notation. E.18 Use properties of exponents and logarithms. 2

3 ELTR 135 (Operational Amplifiers 2), section 2 Common areas of confusion for students Difficult concept: Opamp relaxation oscillator circuit. This circuit can be difficult to grasp, because there is a tendency to immediately apply one of the canonical rules of opamp circuits: that there is negligible voltage between the inverting and noninverting inputs when negative feedback is present. It is wrong to apply this rule here, though, because this circuit s behavior is dominated by positive feedback. This positive feedback comes through a plain resistor network, while the negative feedback comes through a first-order lag RC time constant network, which means the positive feedback is immediate while the negative feedback is delayed. Because of this, the opamp output swings back and forth between saturated states, and never settles to an equilibrium position. Thus, it does not act like a simple amplifier and there will be substantial voltage between the two opamp inputs. A better way to consider this circuit is to think of it as a comparator with hysteresis, the inverting input chasing the output with a time lag. Difficult concept: Rates of change. When studying integrator and differentiator circuits, one must think in terms of how fast a variable is changing. This is the first hurdle in calculus: to comprehend what a rate of change is, and it is not obvious. One thing I really like about teaching electronics is that capacitor and inductors naturally exhibit the calculus principles of integration and differentiation (with respect to time), and so provide an excellent context in which the electronics student may explore basic principles of calculus. Integrator and differentiator circuits exploit these properties, so that the output voltage is approximately either the time-integral or timederivative (respectively) of the input voltage signal. It is helpful, though, to relate these principles to more ordinary contexts, which is why I often describe rates of change in terms of velocity and acceleration. Velocity is nothing more than a rate of change of position: how quickly one s position is changing over time. Therefore, if the variable x describes position, then the derivative dx dt (rate of change of x over time t) must describe velocity. Likewise, acceleration is nothing more than the rate of change of velocity: how quickly velocity changes over time. If the variable v describes velocity, then the derivative dv dt must describe velocity. Or, since we know that velocity is itself the d derivative of position, we could describe acceleration as the second derivative of position: 2 x dt 2 Difficult concept: Derivative versus integral. The two foundational concepts of calculus are inversely related: differentiation and integration are flipsides of the same coin. That is to say, one un-does the other. If you can grasp what one of these operations is, then the other is simply the reverse. One of the better ways to illustrate the inverse nature of these two operations is to consider them in the context of motion analysis, relating position (x), velocity (v), and acceleration (a). Differentiating with respect to time, the derivative of position is velocity (v = dx dt ), and the derivative of velocity is acceleration (a = dv dt ). Integrating with respect to time, the integral of acceleration is velocity (v = a dt) and the integral of velocity is position (x = v dt). Fortunately, electronics provides a ready context in which to understand differentiation and integration. It is very easy to build differentiator and integrator circuits, which take a voltage signal input and differentiate or integrate (respectively) that signal with respect to time. This means if we have a voltage signal from a velocity sensor measuring the velocity of an object (such as a robotic arm, for example), we may send that signal through a differentiator circuit to obtain a voltage signal representing the robotic arm s acceleration, or we may send the velocity signal through a integrator circuit to obtain a voltage signal representing the robotic arm s position. 3

4 Questions Question 1 This is a very common opamp oscillator circuit, technically of the relaxation type: A V B -V Explain how this circuit works, and what waveforms will be measured at points A and B. Be sure to make reference to RC time constants in your explanation. file Question 2 A variation on the common opamp relaxation oscillator design is this, which gives it variable duty cycle capability: V -V Explain how this circuit works, and which direction the potentiometer wiper must be moved to increase the duty cycle (more time spent with the opamp output saturated at V and less time spent saturated at -V). file

5 Question 3 Dual, or split, power supplies are very useful in opamp circuits because they allow the output voltage to rise above as well as sink below ground potential, for true AC operation. In some applications, though, it may not be practical or affordable to have a dual power supply to power your opamp circuit. In this event, you need to be able to figure out how to adapt your dual-supply circuit to single-supply operation. A good example of such a challenge is the familiar opamp relaxation oscillator, shown here: V -V First, determine what would happen if we were to simply eliminate the negative portion of the dual power supply and try to run the circuit on a single supply (V and Ground only): V Then, modify the schematic so that the circuit will run as well as it did before with the dual supply. file

6 Question 4 How many degrees of phase shift must the feedback circuit (the square box in this schematic) introduce to the signal in order for this inverting amplifier circuit to oscillate? Power source Feedback network Inverting amplifier file Question 5 How many degrees of phase shift must the feedback circuit (the square box in this schematic) introduce to the signal in order for this noninverting amplifier circuit to oscillate? Power source Feedback network Noninverting amplifier file Question 6 Explain what the Barkhausen criterion is for an oscillator circuit. How will the oscillator circuit s performance be affected if the Barkhausen criterion falls below 1, or goes much above 1? file

7 Question 7 Identify what type of oscillator circuit this is, and write an equation describing its operating frequency: R pot R 2 U 1 R 1 C 1 C 2 file Question 8 Explain the purpose of the tank circuit (L 1 and C 1 ) in the following oscillator circuit, and write an equation describing its operating frequency: R pot C 1 U 1 R 1 L 1 file

8 Question 9 This Wien bridge oscillator circuit is very sensitive to changes in the gain. Note how the potentiometer used in this circuit is the trimmer variety, adjustable with a screwdriver rather than by a knob or other hand control: R pot R 2 U 1 R 1 C 1 C 2 The reason for this choice in potentiometers is to make accidental changes in circuit gain less probable. If you build this circuit, you will see that tiny changes in this potentiometer s setting make a huge difference in the quality of the output sine wave. A little too much gain, and the sine wave becomes noticeably distorted. Too little gain, and the circuit stops oscillating altogether! Obviously, it is not good to have such sensitivity to minor changes in any practical circuit expected to reliably perform day after day. One solution to this problem is to add a limiting network to the circuit comprised of two diodes and two resistors: R pot D 1 R 3 U 1 D 2 R 4 R 2 R 1 C 1 C 2 With this network in place, the circuit gain may be adjusted well above the threshold for oscillation (Barkhausen criterion) without exhibiting excessive distortion as it would have without the limiting network. Explain why the limiting network makes this possible. file

9 Question 10 This interesting opamp circuit produces true three-phase sinusoidal voltage waveforms, three of them to be exact: C C C R/2 R R/2 R R/2 R With all the resistors and capacitors, you might have guessed this to be a phase-shift type of oscillator circuit, and you would be correct. Here, each parallel RC network provides 60 degrees of lagging phase shift to combine with the 180 degrees of phase shift inherent to the inverting amplifier configurations, yielding 120 degrees of shift per opamp stage. Derive a formula solving for the operating frequency of this oscillator circuit, knowing that the impedance of each parallel RC network will have a phase angle of -60 o. Also, determine where on this circuit you would obtain the three promised sine waves. file Question 11 How much current will go through the load in this op-amp circuit? Load 2.2 kω 4 V What impact will a change in load resistance have on the operation of this circuit? What will change, if anything, supposing the load resistance were to increase? file

10 Question 12 f(x) dx Calculus alert! How much current will go through the capacitor in this op-amp circuit, and what effect does this have on the output voltage? 10 µf 2.2 kω 4 V file Question 13 What will the output voltage of this integrator circuit do when the DPDT ( Double-Pole, Double- Throw ) switch is flipped back and forth? C R Be as specific as you can in your answer, explaining what happens in the switch s up position as well as in its down position. file

11 Question 14 f(x) dx Calculus alert! Integrator circuits may be understood in terms of their response to DC input signals: if an integrator receives a steady, unchanging DC input voltage signal, it will output a voltage that changes with a steady rate over time. The rate of the changing output voltage is directly proportional to the magnitude of the input voltage: Typical integrator response V in V in Time Time A symbolic way of expressing this input/output relationship is by using the concept of the derivative in calculus (a rate of change of one variable compared to another). For an integrator circuit, the rate of output voltage change over time is proportional to the input voltage: d dt V in A more sophisticated way of saying this is, The time-derivative of output voltage is proportional to the input voltage in an integrator circuit. However, in calculus there is a special symbol used to express this same relationship in reverse terms: expressing the output voltage as a function of the input. For an integrator circuit, this special symbol is called the integration symbol, and it looks like an elongated letter S : T 0 V in dt Here, we would say that output voltage is proportional to the time-integral of the input voltage, accumulated over a period of time from time=0 to some point in time we call T. This is all very interesting, you say, but what does this have to do with anything in real life? Well, there are actually a great deal of applications where physical quantities are related to each other by time-derivatives and time-integrals. Take this water tank, for example: 11

12 Flow Height One of these variables (either height H or flow F, I m not saying yet!) is the time-integral of the other, just as is the time-integral of V in in an integrator circuit. What this means is that we could electrically measure one of these two variables in the water tank system (either height or flow) so that it becomes represented as a voltage, then send that voltage signal to an integrator and have the output of the integrator derive the other variable in the system without having to measure it! Your task is to determine which variable in the water tank scenario would have to be measured so we could electronically predict the other variable using an integrator circuit. file Question 15 How much current (I) would have to be forced through the resistor in order to generate an output voltage of 5 volts? 2.2 kω Load I =??? file

13 Question 16 f(x) dx Calculus alert! How much current (I) would have to be forced through the resistor in order to generate an output voltage of 5 volts? 2.2 kω 10 µf V in At what rate would V in have to increase in order to cause this amount of current to go through the capacitor, and thereby cause 5 volts to appear at the terminal? What does this tell us about the behavior of this circuit? file

14 Question 17 f(x) dx Calculus alert! If an object moves in a straight line, such as an automobile traveling down a straight road, there are three common measurements we may apply to it: position (x), velocity (v), and acceleration (a). Position, of course, is nothing more than a measure of how far the object has traveled from its starting point. Velocity is a measure of how fast its position is changing over time. Acceleration is a measure of how fast the velocity is changing over time. These three measurements are excellent illustrations of calculus in action. Whenever we speak of rates of change, we are really referring to what mathematicians call derivatives. Thus, when we say that velocity (v) is a measure of how fast the object s position (x) is changing over time, what we are really saying is that velocity is the time-derivative of position. Symbolically, we would express this using the following notation: v = dx dt Likewise, if acceleration (a) is a measure of how fast the object s velocity (v) is changing over time, we could use the same notation and say that acceleration is the time-derivative of velocity: a = dv dt Since it took two differentiations to get from position to acceleration, we could also say that acceleration is the second time-derivative of position: a = d2 x dt 2 What has this got to do with electronics, you ask? Quite a bit! Suppose we were to measure the velocity of an automobile using a tachogenerator sensor connected to one of the wheels: the faster the wheel turns, the more DC voltage is output by the generator, so that voltage becomes a direct representation of velocity. Now we send this voltage signal to the input of a differentiator circuit, which performs the timedifferentiation function on that signal. What would the output of this differentiator circuit then represent with respect to the automobile, position or acceleration? What practical use do you see for such a circuit? Now suppose we send the same tachogenerator voltage signal (representing the automobile s velocity) to the input of an integrator circuit, which performs the time-integration function on that signal (which is the mathematical inverse of differentiation, just as multiplication is the mathematical inverse of division). What would the output of this integrator then represent with respect to the automobile, position or acceleration? What practical use do you see for such a circuit? file

15 Question 18 f(x) dx Calculus alert! A familiar context in which to apply and understand basic principles of calculus is the motion of an object, in terms of position (x), velocity (v), and acceleration (a). We know that velocity is the time-derivative of position (v = dx dv dt ) and that acceleration is the time-derivative of velocity (a = dt ). Another way of saying this is that velocity is the rate of position change over time, and that acceleration is the rate of velocity change over time. It is easy to construct circuits which input a voltage signal and output either the time-derivative or the time-integral (the opposite of the derivative) of that input signal. We call these circuits differentiators and integrators, respectively. Differentiator Integrator V in x dx dt Input Output V in x x dt Input Output Integrator and differentiator circuits are highly useful for motion signal processing, because they allow us to take voltage signals from motion sensors and convert them into signals representing other motion variables. For each of the following cases, determine whether we would need to use an integrator circuit or a differentiator circuit to convert the first type of motion signal into the second: Converting velocity signal to position signal: (integrator or differentiator?) Converting acceleration signal to velocity signal: (integrator or differentiator?) Converting position signal to velocity signal: (integrator or differentiator?) Converting velocity signal to acceleration signal: (integrator or differentiator?) Converting acceleration signal to position signal: (integrator or differentiator?) Also, draw the schematic diagrams for these two different circuits. file Question 19 f(x) dx Calculus alert! You are part of a team building a rocket to carry research instruments into the high atmosphere. One of the variables needed by the on-board flight-control computer is velocity, so it can throttle engine power and achieve maximum fuel efficiency. The problem is, none of the electronic sensors on board the rocket has the ability to directly measure velocity. What is available is an altimeter, which infers the rocket s altitude (it position away from ground) by measuring ambient air pressure; and also an accelerometer, which infers acceleration (rate-of-change of velocity) by measuring the inertial force exerted by a small mass. The lack of a speedometer for the rocket may have been an engineering design oversight, but it is still your responsibility as a development technician to figure out a workable solution to the dilemma. How do you propose we obtain the electronic velocity measurement the rocket s flight-control computer needs? file

16 Question 20 f(x) dx Calculus alert! A Rogowski Coil is essentially an air-core current transformer that may be used to measure DC currents as well as AC currents. Like all current transformers, it measures the current going through whatever conductor(s) it encircles. Normally transformers are considered AC-only devices, because electromagnetic induction requires a changing magnetic field ( dφ dt ) to induce voltage in a conductor. The same is true for a Rogowski coil: it produces a voltage only when there is a change in the measured current. However, we may measure any current (DC or AC) using a Rogowski coil if its output signal feeds into an integrator circuit as shown: Rogowski coil (air-core current transformer) i power conductor v out i Integrator Connected as such, the output of the integrator circuit will be a direct representation of the amount of current going through the wire. Explain why an integrator circuit is necessary to condition the Rogowski coil s output so that output voltage truly represents conductor current. file

17 Question 21 The concept of a mathematical power is familiar to most students of algebra. For instance, ten to the third power means this:... and eight to the seventh power means this: 10 3 = = = = 2,097,152 Just as subtraction is the inverse function of addition, and division is the inverse function of multiplication (because with inverse functions, one undoes the other), there is also an inverse function for a power and we call it the logarithm. Re-write the expression 10 3 = 1000 so that it uses the same quantities (10, 3, and 1000) in the context of a logarithm instead of a power, just as the subtraction is shown here to be the inverse of addition, and division is shown to be the inverse of multiplication in the following examples: 3 8 = 11 ( and - are inverse functions) 11 3 = = 14 ( and are inverse functions) 14 2 = 7 file Question = 1000 (powers and logs are inverse functions) log 10??? =??? Given the following mathematical expression, write another one defining a logarithm using the same variables: file Question 23 If: x y = z Then: log?? =? Electronic calculators with logarithm capability have at least two different types of logarithms: common logarithm and natural logarithm, symbolized as log and ln, respectively. Explain what the difference is between these two types of logarithms. file

18 Question 24 Note the following logarithmic identities, using the common (base 10) logarithm: log 10 = 1 log 100 = 2 log 1000 = 3 log = 4 In the first equation, the numbers 10 and 1 were related together by the log function. In the second equation, the numbers 100 and 2 were related together by the same log function, and so on. Rewrite the four equations together in such a way that the same numbers are related to each other, but without writing log. In other words, represent the same mathematical relationships using some mathematical function other than the common logarithm function. file Question 25 Note the following logarithmic identities, using the common (base 10) logarithm: log 0.1 = 1 log 0.01 = 2 log = 3 log = 4 In the first equation, the numbers 0.1 and 1 were related together by the log function. In the second equation, the numbers 0.01 and 2 were related together by the same log function, and so on. Rewrite the four equations together in such a way that the same numbers are related to each other, but without writing log. In other words, represent the same mathematical relationships using some mathematical function other than the common logarithm function. file

19 Question 26 Examine the following progression of mathematical statements: (10 2 )(10 3 ) = = = What does this pattern indicate? What principle of algebra is illustrated by these three equations? Next, examine this progression of mathematical statements: log 10 5 = log = 5 log = log = 5 log 10 2 log 10 3 = log = 5 What does this pattern indicate? What principle of algebra is illustrated by these three equations? file Question 27 Examine this progression of mathematical statements: (100)(1000) = (100)(1000) = 10 5 log[(100)(1000)] = log 10 5 log 100 log 1000 = log 10 5 log 10 2 log 10 3 = log = 5 What began as a multiplication problem ended up as an addition problem, through the application of logarithms. What does this tell you about the utility of logarithms as an arithmetic tool? file Question 28 Suppose you owned a scientific calculator with two broken buttons: the multiply ( ) and divide ( ). Demonstrate how you could solve this simple multiplication problem using only logarithms, addition, and antilogarithms (powers): 7 5 =??? The answer to this problem was easy enough for you to figure out without a calculator at all, so here are some more practice problems for you to try: 19

20 23 35 = = = = file Question 29 Examine this progression of mathematical statements: log = = 101 ( ) = log 10 1 log 1000 log 100 = log 10 1 log 10 3 log 10 2 = log = 1 What began as a division problem ended up as a subtraction problem, through the application of logarithms. What does this tell you about the utility of logarithms as an arithmetic tool? file Question 30 Suppose you owned a scientific calculator with two broken buttons: the multiply ( ) and divide ( ). Demonstrate how you could solve this simple multiplication problem using only logarithms, addition, and antilogarithms (powers): 12 3 =??? The answer to this problem was easy enough for you to figure out without a calculator at all, so here are some more practice problems for you to try: = = = = file

21 Question 31 Examine this progression of mathematical statements: (1000) 2 = (1000) 2 = 10 6 log[(1000) 2 ] = log 10 6 (2)(log 1000) = log 10 6 (2)(log 10 3 ) = log 10 6 (2)(3) = 6 What began as an exponential problem ended up as a multiplication problem, through the application of logarithms. What does this tell you about the utility of logarithms as an arithmetic tool? file Question 32 Suppose you owned a scientific calculator with two broken buttons: the power (y x ) and root ( x y). Demonstrate how you could solve this simple power problem using only logarithms, multiplication, and antilogarithms (powers): 3 4 =??? The answer to this problem was easy enough for you to figure out without a calculator at all, so here are some more practice problems for you to try: 25 6 = = = = file

22 Question 33 Examine this progression of mathematical statements: 1000 = log 1000 = log ( ) ( ) log = log ( ) 1 2 (log 1000) = log ( ) 1 2 (log 103 ) = log ( ) 3 2 (log 10) = log ( ) 3 2 (1) = log ( ) 3 2 = log ( ) 3 2 = 1.5 What began as a fractional exponent problem ended up as a simple fraction, through the application of logarithms. What does this tell you about the utility of logarithms as an arithmetic tool? file Question 34 Suppose you owned a scientific calculator with two broken buttons: the power (y x ) and root ( x y). Demonstrate how you could solve this simple root problem using only logarithms, division, and antilogarithms (powers): 3 8 =??? The answer to this problem was easy enough for you to figure out without a calculator at all, so here are some more practice problems for you to try: 4 13 = = = = file

23 Question 35 You may be wondering why anyone would bother using logarithms to solve arithmetic problems for which we have perfectly good and effective digital electronic calculator functions at our disposal. For example, why would anyone do this: log 7log when they could just do the following on the same calculator? 7 5 The quick answer to this very good question is, when it is more difficult to directly multiply two numbers. The trouble is, most people have a difficult time imagining when it would ever be easier to take two logarithms, add them together, and raise ten to that power than it would be to simply multiply the original two numbers together. The answer to that mystery is found in operational amplifier circuitry. As it turns out, it is much easier to build single opamp circuits that add, subtract, exponentiate, or take logarithms than it is to build one that directly multiplies or divides two quantities (analog voltages) together. We may think of these opamp functions as blocks which may be interconnected to perform composite arithmetic functions: a c = a - b Subtractor b c a c = a b Summer b c b = ln a b = e a a Log b a Antilog b Using this model of specific math-function blocks, show how the following set of analog math function blocks may be connected together to multiply two analog voltages together: b = ln a a Log b V 1 V 2 a c = a - b Subtractor b c a c = a b Summer b c (V 1 )(V 2 ) b = ln a b = e a a Log b a Antilog b 23

24 file Question 36 The relationship between voltage and current for a PN junction is described by this equation, sometimes referred to as the diode equation, or Shockley s diode equation after its discoverer: I D = I S (e qv D NkT 1) Where, I D = Current through the PN junction, in amps I S = PN junction saturation current, in amps (typically 1 picoamp) e = Euler s number q = Electron unit charge, coulombs V D = Voltage across the PN junction, in volts N = Nonideality coefficient, or emission coefficient (typically between 1 and 2) k = Boltzmann s constant, T = Junction temperature, degrees Kelvin At first this equation may seem very daunting, until you realize that there are really only three variables in it: I D, V D, and T. All the other terms are constants. Since in most cases we assume temperature is fairly constant as well, we are really only dealing with two variables: diode current and diode voltage. Based on this realization, re-write the equation as a proportionality rather than an equality, showing how the two variables of diode current and voltage relate: I D... Based on this simplified equation, what would an I/V graph for a PN junction look like? How does this graph compare against the I/V graph for a resistor? I D V D file

25 Question 37 Plot the transfer function ( versus V in ) for this opamp circuit, and explain how the circuit works: V in V in What type of mathematical function is represented by this circuit? file

26 Question 38 We know that an opamp connected to a voltage divider with a voltage division ratio of 1 2 will have an overall voltage gain of 2, and that the same circuit with a voltage division ratio of 2 3 will have an overall voltage gain of 1.5, or 3 2 : V in = 2 (V in) V in V 2 ( = 3 in) 1 2 Voltage divider circuit 2 Vout 3 Voltage divider circuit There is definitely a mathematical pattern at work in these noninverting opamp circuits: the overall voltage gain of the circuit is the mathematical inverse of the feedback network s voltage gain. Building on this concept, what do you think would be the overall function of the following opamp circuits? V in V in =??? =??? y x Feedback network y x Feedback network y = x 4 y = x 2 file

27 Question 39 Plot the transfer function ( versus V in ) for this opamp circuit, and explain how the circuit works: V in V in What type of mathematical function is represented by this circuit? file

28 Question 40 Plot the transfer function ( versus V in ) for this opamp circuit: V in V in What type of mathematical function is represented by this circuit? file

29 Question 41 Identify the mathematical function of this circuit (if you look closely, you ll notice that the transistors are connected in such a way that they act very similar to diodes): x R y R Note: the two resistors labeled R are equal in value. file Question 42 Identify the mathematical function of this circuit (if you look closely, you ll notice that the transistors are connected in such a way that they act very similar to diodes): x R R y Note: the two resistors labeled R are equal in value. file

30 Question 43 Suppose that in the course of building this exponential circuit you encounter severe inaccuracies: the circuit seems to work some of the time, but often its output deviates substantially (as much as /- 10%) from what it ought to be: Intended transfer function: y = x 0.5 x R y R Based on what you know of the components in this circuit, what could be varying so much as to cause these errors? What do you recommend as a solution to the problem? file

31 Question 44 Design an op-amp circuit that divides one quantity (x) by another quantity (y) using logarithms. To give you a start on this circuit, I ll provide the initial logarithmic op-amp modules in this diagram: x - ln x y - ln y Note: it will be helpful for your analysis to write the mathematical expression at each op-amp output in your circuit, so you may readily see how the overall math function is constructed from individual steps. file Question 45 Find a datasheet for the AD538, an integrated circuit manufactured by Analog Devices. Then, read it carefully and explain how it is able to perform arithmetic functions such as multiplication, division, powers, and roots. file

32 Question 46 Competency: Opamp relaxation oscillator Schematic Version: C 1 R 1 U 1 R 2 R 3 Given conditions V = R 1 = R 3 = -V = R 2 = C 1 = Parameters (pk-pk) Predicted Measured f out Fault analysis Suppose component fails What will happen in the circuit? open shorted other file

33 Question 47 Competency: Opamp triangle wave generator Schematic Version: C 1 R 1 R 5 R 4 C 2 U 1 R 2 R 3 U 2 Given conditions V = R 1 = R 3 = R 5 = C 2 = -V = R 2 = R 4 = C 1 = Parameters (pk-pk) Predicted Measured f out Fault analysis Suppose component fails What will happen in the circuit? open shorted other file

34 Question 48 Competency: Opamp LC resonant oscillator w/limiting Schematic R pot Version: D 1 R 2 U 1 C 1 D 2 R 3 R 1 L 1 Given conditions V = R 1 = L 1 = R 2 = R 3 = -V = C 1 = R pot = Parameters Predicted Measured f out Calculations file

35 Question 49 NAME: Project Grading Criteria PROJECT: You will receive the highest score for which all criteria are met. 100 % (Must meet or exceed all criteria listed) A. Impeccable craftsmanship, comparable to that of a professional assembly B. No spelling or grammatical errors anywhere in any document, upon first submission to instructor 95 % (Must meet or exceed these criteria in addition to all criteria for 90% and below) A. Technical explanation sufficiently detailed to teach from, inclusive of every component (supersedes 75.B) B. Itemized parts list complete with part numbers, manufacturers, and (equivalent) prices for all components, including recycled components and parts kit components (supersedes 90.A) 90 % (Must meet or exceed these criteria in addition to all criteria for 85% and below) A. Itemized parts list complete with prices of components purchased for the project, plus total price B. No spelling or grammatical errors anywhere in any document upon final submission 85 % (Must meet or exceed these criteria in addition to all criteria for 80% and below) A. User s guide to project function (in addition to 75.B) B. Troubleshooting log describing all obstacles overcome during development and construction 80 % (Must meet or exceed these criteria in addition to all criteria for 75% and below) A. All controls (switches, knobs, etc.) clearly and neatly labeled B. All documentation created on computer, not hand-written (including the schematic diagram) 75 % (Must meet or exceed these criteria in addition to all criteria for 70% and below) A. Stranded wire used wherever wires are subject to vibration or bending B. Basic technical explanation of all major circuit sections C. Deadline met for working prototype of circuit (Date/Time = / ) 70 % (Must meet or exceed these criteria in addition to all criteria for 65%) A. All wire connections sound (solder joints, wire-wrap, terminal strips, and lugs are all connected properly) B. No use of glue where a fastener would be more appropriate C. Deadline met for submission of fully-functional project (Date/Time = / ) supersedes 75.C if final project submitted by that (earlier) deadline 65 % (Must meet or exceed these criteria in addition to all criteria for 60%) A. Project fully functional B. All components securely fastened so nothing is loose inside the enclosure C. Schematic diagram of circuit 60 % (Must meet or exceed these criteria in addition to being safe and legal) A. Project minimally functional, with all components located inside an enclosure (if applicable) B. Passes final safety inspection (proper case grounding, line power fusing, power cords strain-relieved) 0 % (If any of the following conditions are true) A. Fails final safety inspection (improper grounding, fusing, and/or power cord strain relieving) B. Intended project function poses a safety hazard C. Project function violates any law, ordinance, or school policy file

36 Question 50 Predict how the operation of this relaxation oscillator circuit will be affected as a result of the following faults. Consider each fault independently (i.e. one at a time, no multiple faults): C 1 R 1 V -V R 2 R 3 Resistor R 1 fails open: Solder bridge (short) across resistor R 1 : Capacitor C 1 fails shorted: Solder bridge (short) across resistor R 2 : Resistor R 3 fails open: For each of these conditions, explain why the resulting effects will occur. file

37 Question 51 Identify at least two different component faults that would result in a change in duty cycle for this oscillator circuit: D 1 C 1 R 1 V D 2 -V R 2 R 3 file Question 52 Suppose this LC oscillator stopped working, and you suspected either the capacitor or the inductor as having failed. How could you check these two components without the use of an LCR meter? R pot C 1 U 1 R 1 L 1 file

38 Question 53 The purpose of this circuit is to provide a pushbutton-adjustable voltage. Pressing one button causes the output voltage to increase, while pressing the other button causes the output voltage to decrease. When neither button is pressed, the voltage remains stable. Essentially, it is an active integrator circuit, with resistors R 1 and R 2 and capacitor C 1 setting the rates of integration: V Increase V R 1 C 1 CA3130 R 2 Decrease -V -V After working just fine for quite a long while, the circuit suddenly fails: now it only outputs zero volts DC all the time. Identify at least one component or wiring failure that would account for the circuit s faulty behavior, and explain why that fault would cause the circuit to act like this. file

39 Question 54 This active integrator circuit processes the voltage signal from an accelerometer, a device that outputs a DC voltage proportional to its physical acceleration. The accelerometer is being used to measure the acceleration of an athlete s foot as he kicks a ball, and the job of the integrator is to convert that acceleration signal into a velocity signal so the researchers can record the velocity of the athlete s foot: Active integrator circuit R 2 (Attached to the athlete s shoe) R 1 C 1 Accel. cable U 1 To recording computer R 3 During the set-up for this test, a radar gun is used to check the velocity of the athlete s foot as he does come practice kicks, and compare against the output of the integrator circuit. What the researchers find is that the integrator s output is reading a bit low. In other words, the integrator circuit is not integrating fast enough to provide an accurate representation of foot velocity. Determine which component(s) in the integrator circuit may have been improperly sized to cause this calibration problem. Be as specific as you can in your answer(s). file

40 Question 55 Predict how the operation of this exponentiator circuit will be affected as a result of the following faults. Consider each fault independently (i.e. one at a time, no multiple faults): D 1 R 1 V in U 1 Resistor R 1 fails open: Solder bridge (short) across resistor R 1 : Diode D 1 fails open: Diode D 1 fails shorted: For each of these conditions, explain why the resulting effects will occur. file Question 56 Predict how the operation of this logarithm extractor circuit will be affected as a result of the following faults. Consider each fault independently (i.e. one at a time, no multiple faults): R 1 D 1 V in U 1 Resistor R 1 fails open: Solder bridge (short) across resistor R 1 : Diode D 1 fails open: Diode D 1 fails shorted: For each of these conditions, explain why the resulting effects will occur. file Question 57 Explain why carefully matched resistors and transistors are necessary in log/antilog circuits, using your own words. Also, explain why the operational amplifiers themselves need not be matched as precisely as the discrete components. file

41 Question 58 This square root extraction circuit used to work fine, but then one day it stopped outputting the square root of the input signal, and instead simply reproduced the input signal with a gain of 1: R 1 Q 1 x U 1 U 2 R 2 Q 2 R 4 U 3 y R 3 What could have gone wrong with this circuit to cause it to stop calculating the square root of the input? Be as detailed as possible in your answer. file Question 59 Identify at least two independent component faults which could cause this square root extraction circuit to always output 0 volts instead of the square root of input voltage x as it should: R 1 Q 1 x U 1 U 2 R 2 Q 2 R 4 U 3 y R 3 Explain why each of your proposed faults would cause the output to stay at 0 volts. file

42 Question 60 Most operational amplifiers do not have the ability to swing their output voltages rail-to-rail. Most of those do not swing their output voltages symmetrically. That is, a typical non-rail-to-rail opamp may be able to approach one power supply rail voltage closer than the other; e.g. when powered by a 15/-15 volt split supply, the output saturates positive at 14 volts and saturates negative at volts. What effect do you suppose this non-symmetrical output range will have on a typical relaxation oscillator circuit such as the following, and how might you suggest we fix the problem? V -V file

43 Question 61 This resonant LC oscillator circuit is very sensitive to changes in the gain. Note how the potentiometer used in this circuit is the trimmer variety, adjustable with a screwdriver rather than by a knob or other hand control: R pot C 1 U 1 R 1 L 1 The reason for this choice in potentiometers is to make accidental changes in circuit gain less probable. If you build this circuit, you will see that tiny changes in this potentiometer s setting make a huge difference in the quality of the output sine wave. A little too much gain, and the sine wave becomes noticeably distorted. Too little gain, and the circuit stops oscillating altogether! Obviously, it is not good to have such sensitivity to minor changes in any practical circuit expected to reliably perform day after day. One solution to this problem is to add a limiting network to the circuit comprised of two diodes and two resistors: R pot D 1 R 2 D 2 R 3 C 1 U 1 R 1 L 1 With this network in place, the circuit gain may be adjusted well above the threshold for oscillation (Barkhausen criterion) without exhibiting excessive distortion as it would have without the limiting network. Explain why the limiting network makes this possible. file

44 Question 62 f(x) dx Calculus alert! Calculus is a branch of mathematics that originated with scientific questions concerning rates of change. The easiest rates of change for most people to understand are those dealing with time. For example, a student watching their savings account dwindle over time as they pay for tuition and other expenses is very concerned with rates of change (dollars per year being spent). In calculus, we have a special word to describe rates of change: derivative. One of the notations used to express a derivative (rate of change) appears as a fraction. For example, if the variable S represents the amount of money in the student s savings account and t represents time, the rate of change of dollars over time would be written like this: The following set of figures puts actual numbers to this hypothetical scenario: Date: November 20 Saving account balance (S) = $12, Rate of spending ( ) ds dt = -5, per year ds dt List some of the equations you have seen in your study of electronics containing derivatives, and explain how rate of change relates to the real-life phenomena described by those equations. file Question 63 f(x) dx Calculus alert! Define what derivative means when applied to the graph of a function. For instance, examine this graph: y x y = f(x) "y is a function of x" Label all the points where the derivative of the function ( dy dx ) is positive, where it is negative, and where it is equal to zero. file

45 Question 64 f(x) dx Calculus alert! Shown here is the graph for the function y = x 2 : y y = x 2 x Sketch an approximate plot for the derivative of this function. file

46 Question 65 f(x) dx Calculus alert! According to the Ohm s Law formula for a capacitor, capacitor current is proportional to the timederivative of capacitor voltage: i = C dv dt Another way of saying this is to state that the capacitors differentiate voltage with respect to time, and express this time-derivative of voltage as a current. Suppose we had an oscilloscope capable of directly measuring current, or at least a current-to-voltage converter that we could attach to one of the probe inputs to allow direct measurement of current on one channel. With such an instrument set-up, we could directly plot capacitor voltage and capacitor current together on the same display: Hz FUNCTION GENERATOR k 10k 100k 1M Volts/Div A m 2 20 m Position 5 10 m Sec/Div 250 µ 1 m 50 µ10 5 m µ 25 m 2.5 µ 100 m 0.5 µ 10 5 m 500 m 0.1 µ coarse fine DC output 20 2 m DC Gnd AC A B Alt Chop Add Volts/Div B m Position 2 20 m 5 10 m Invert 10 5 m Intensity Focus Beam find 20 2 m DC Gnd AC Off Cal 1 V Gnd Trace rot. Norm Auto Single Reset µ 2.5 off X-Y Position Triggering Level A B Alt Holdoff Line Ext. Ext. input AC DC LF Rej Slope HF Rej I/V converter For each of the following voltage waveforms (channel B), plot the corresponding capacitor current waveform (channel A) as it would appear on the oscilloscope screen: 46

47 Note: the amplitude of your current plots is arbitrary. What I m interested in here is the shape of each current waveform! file

48 Question 66 f(x) dx Calculus alert! According to the Ohm s Law formula for a capacitor, capacitor current is proportional to the timederivative of capacitor voltage: i = C dv dt Another way of saying this is to state that the capacitors differentiate voltage with respect to time, and express this time-derivative of voltage as a current. We may build a simple circuit to produce an output voltage proportional to the current through a capacitor, like this: Passive differentiator circuit C R shunt The resistor is called a shunt because it is designed to produce a voltage proportional to current, for the purpose of a parallel ( shunt )-connected voltmeter or oscilloscope to measure that current. Ideally, the shunt resistor is there only to help us measure current, and not to impede current through the capacitor. In other words, its value in ohms should be very small compared to the reactance of the capacitor (R shunt << X C ). Suppose that we connect AC voltage sources with the following wave-shapes to the input of this passive differentiator circuit. Sketch the ideal (time-derivative) output waveform shape on each oscilloscope screen, as well as the shape of the actual circuit s output voltage (which will be non-ideal, of course): 48

49 Note: the amplitude of your plots is arbitrary. What I m interested in here is the shape of the ideal and actual output voltage waveforms! Hint: I strongly recommend building this circuit and testing it with triangle, sine, and square-wave input voltage signals to obtain the corresponding actual output voltage wave-shapes! file

50 Question 67 f(x) dx Calculus alert! Calculus is widely (and falsely!) believed to be too complicated for the average person to understand. Yet, anyone who has ever driven a car has an intuitive grasp of calculus most basic concepts: differentiation and integration. These two complementary operations may be seen at work on the instrument panel of every automobile: Miles per hour Miles On this one instrument, two measurements are given: speed in miles per hour, and distance traveled in miles. In areas where metric units are used, the units would be kilometers per hour and kilometers, respectively. Regardless of units, the two variables of speed and distance are related to each other over time by the calculus operations of integration and differentiation. My question for you is which operation goes which way? We know that speed is the rate of change of distance over time. This much is apparent simply by examining the units (miles per hour indicates a rate of change over time). Of these two variables, speed and distance, which is the derivative of the other, and which is the integral of the other? Also, determine what happens to the value of each one as the other maintains a constant (non-zero) value. file