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1 Physics 364, Fall 2012, reading due your answers to by 11pm on Thursday Course materials and schedule are at Assignment: This week s textbook reading is only about 8 pages and is pretty clearly written, so there s no need to skim. (a) First read carefully Eggleston s sections (b) Then read through my notes (starting on next page), which this week aren t so different from the book s approach. (c) Then me your answers to the questions below. Please look over this assignment early enough that you have time to ask me questions if anything is unclear. 1. What is the below-left circuit intended to do? In other words, if this circuit is performing its usual function, how does V out relate to V in? C V in R + V out 2. (a) If I choose R 1 = 2 kω and R f = 20 kω in the above-right circuit, what is the relationship between V out and V in? (Try to get both the magnitude and the sign right.) (b) Assuming that the golden rules correctly describe this circuit s operation, what potential difference (voltage) is measured between point A and ground? (c) Again using the golden rules, what relationship can you write between I 1 and I f? 3. Is there anything from this reading assignment that you found confusing and would like me to try to clarify? If you didn t find anything confusing, what topic did you find most interesting? 4. How much time did it take you to complete this assignment? Also, how many total hours did you spend on Physics 364 in the past week? If you feel that some of these hours were a poor use of your time (for most learning per hour invested), please suggest ways in which I might better align the course with your learning style. phys364/reading03.tex page 1 of :00

2 We will spend the next two weeks on opamps ( operational amplifiers ). An opamp is a high-gain differential amplifier with very high input resistance. It is nearly always used with negative feedback. Negative feedback means, for example, that if my car starts to veer toward the right, I turn the steering wheel to the left; and if the car starts to go off-course to the left, I turn the wheel to the right. Negative feedback has widespread use for making a desired outcome stable, such that a deviation from the desired outcome causes a correction of opposite sign to be applied. What is the point of an amplifier, anyway? Circuits having only passive components have no power gain: the power in the signal coming out of the circuit cannot exceed the power in the signal going in. If you want to drive your massive stereo speakers from your wimpy MP3 player, you need an intermediate circuit that will arrange for the power going into the large speakers to exceed the power coming out of the small MP3 player. This can t be done with the light bulbs, resistors, capacitors, inductors, and diodes that we have seen so far. So one common use of amplifiers is e.g. to turn a voltage signal V in into a signal V out that is e.g. ten times as large. While a transformer (another passive device) with turns-ratio n 2 /n 1 > 1 could produce V out = (n 2 /n 1 )V in > 1 for a time-varying signal, it could do so only at the cost of lower current: I out = (n 1 /n 2 )I in. To boost the power in a signal, we need an active device such as an amplifier. An amplifier can also turn a weak voltage source into a strong voltage source for example serving as a go-between for a source whose relatively large R thevenin would preclude it from driving a relatively small R load without drooping. We ll see how this works when we study the opamp follower (a.k.a. buffer amplifier) circuit below. Perhaps the most remarkable use of opamps is that they allow you without too much effort to perform all kinds of mathematical operations. You can make a weighted sum: V out = i w iv in,i. You can integrate (without last week s annoying V out V in restriction): V out V in dt. You can even compute logarithms: V out log(v in ). Opamps are building blocks that allow you to build useful circuits whose performance depends more on the chosen values of a few passive components than on the details of the opamp itself. They will become one of your favorite LEGO bricks able to fit into your circuit in many different configurations, which you will soon learn to recognize and know how to analyze. phys364/reading03.tex page 2 of :00

3 An opamp (shown above) has two inputs (labeled + and ) and one output. The input is called the inverting input, and the + input is called the non-inverting input. The + and do not refer to the polarity of the input signal, but rather to the fact that the output is proportional to V + V. An opamp also needs an external source of power, which we label V S± (letter S for supply ): V S± are typically something like ±15 V, though the exact choice depends on the opamp model and what power-supply voltages you have available. Most of the time, V S± are omitted from the schematic diagram their presence is implicit. The basic function of an opamp is to produce an output proportional to the difference between its inputs, with a very large constant of proportionality: V out = A (V + V ) with A 1 (typically A 10 6 or more). V out cannot go past the power supply rails, i.e. V S < V out < V S+. In fact, most opamps do not permit V out to get closer than 1 or 2 V from V S±, so for V S± = ±15 V, typically V out cannot exceed ±13 V. 1 Most of what you need to know about analyzing opamp circuits can be summed up in two Golden Rules. We will see next week why the rules work. We will also see later some limitations of real-world opamps that cause deviations from the ideal behavior described by the golden rules. But for this week, we ll just try to get comfortable with using the golden rules as stated. Rule #1. V out takes on whatever value is needed to arrange that V + V = 0. Rule #2. The inputs draw negligible current. Rule #0 (the fine print). (a) There must be negative feedback: i.e. a fraction of V out must be fed back into V, the inverting input. (b) V out must not be saturated (i.e. V out can t be stuck at or near the V S± rails). (c) Negative feedback is especially important at DC, to prevent the opamp from going into saturation. Rule #2 is a consequence of the opamp s very high input resistance, and does not depend on Rule #0. We will see next week that Rule #1 is just a consequence of the opamp s very high gain, as long as Rule #0 is satisfied. Let s draw out the schematic diagrams for several classic opamp configurations. As we go along, we will use the golden rules to analyze the operation of each circuit. The first opamp circuit we ll study is called a follower (shown below, left), because V out = V in, i.e. V out follows V in. It is also called a buffer, because it can be placed in between two circuit fragments that normally would prefer not to talk directly to 1 Opamps that allow V out to span the full V S± range will boast rail-to-rail operation on their data sheets. phys364/reading03.tex page 3 of :00

4 one another. 2 To analyze this circuit, notice that Rule #1 implies that V out = V in (so that V + = V ). Then notice that Rule #2 implies that the input resistance of this circuit is R in = (for an ideal opamp, anyway), since a change in V in will not cause any change in current at the + terminal. V in + V out What use is a circuit for which V out = V in? Well, do you remember from Lab 1 what happened when we used the output of one voltage divider as the input for a second voltage divider, as shown above (right)? For the first voltage divider, the Thevenin resistance (a.k.a. source resistance, a.k.a. output resistance) is R thev,1 = 667 Ω (using V B and ground as the two output terminals). Meanwhile, the input resistance of the second voltage divider is R in,2 = R + 2R = 3R. As long as 3R 667 Ω, the analysis is simple: V B = 2V 3 A, and V C = 2V 3 B = 4V 9 A. But if for example R = 1 kω, then V B droops about 18% below its unloaded (or open-circuit) value, and the analysis is more complicated. By inserting an opamp follower between the output of the first voltage divider and the input of the second voltage divider (see figure below), we restore the simple V C = 4V 9 A result, independent of the value of R. You can think of the opamp buffer circuit as rose-colored glasses for each of the circuits that it separates: the upstream circuit sees what it prefers to see (an ideal R in = for its downstream circuit); and the downstream circuit sees what it prefers to see (an ideal R thev = 0 for its upstream circuit). 3 This can be very handy for interfacing two circuits that normally would not be so compatible. The next circuit (figure below, left) is the inverting amplifier configuration. Rule #2 implies I 1 + I 2 = 0, and Rule #1 implies V = 0. Using Ohm s law for R 1 gives I 1 = V in /R 1, and for R 2 gives I 2 = V out /R 2. Then I 2 = I 1 (because the input draws negligible current) gives V out = R 2 R 1 V in. 2 Think of a buffer state that keeps the peace between two hostile nations. 3 Remember that an ideal voltage source has R thev = 0, and that the ideal load for a voltage source is an open circuit (R load = ). One more thing: I didn t make this up I borrowed the metaphor directly from the Harvard course s notes! phys364/reading03.tex page 4 of :00

5 Since the upstream circuit (i.e. whatever is driving V in ) sees what looks like a resistor R 1 connected to ground (we call V a virtual ground here), the inverting amplifier s input resistance is R in = R 1. The next configuration (above, right) is the non-inverting amplifier. Rule #1 implies V = V in, and Rule #2 (no current drawn by input) lets us use the voltagedivider equation to relate V to V out : we find V 2 out R 1 +R 2 R = V in. Solving for V out, ( V out = 1 + R ) 1 V in. R 2 In this case, R in = (for an ideal opamp), i.e. the input resistance is very high, because the + input draws negligible current. Next, we have the summing amplifier configuration (below, left). Since V = 0 ( virtual ground ), the current flowing to the right through R A, R B, and R C is, respectively, V A /R A, V B /R B, and V C /R C. By Rule #2, the sum of these three currents must flow to the right through R. So V out is (minus) the weighted sum of the inputs: ( VA V out = I total R = + V B + V ) ( ) C R R R R = V A + V B + V C. R A R B R C R A R B R C Can you think of a way to include a coefficient of opposite sign? Next is the differential amplifier configuration (above, right). By Rule #2, the current I A that flows through the upper R 1 continues through the upper R 2. Similarly, the current I B that flows through the lower R 1 continues through the lower R 2. Ohm s law gives V out I A R 2 = V and V I A R 1 = V A, which combine (eliminating I A ) to give Vout V R 2 = V V A R 1, which simplifies to R 1 V out = (R 1 +R 2 )[V ] R 2 V A. Meanwhile, the voltage-divider equation gives V + = R 2 R 1 +R 2 V B, and Rule #1 gives V = V +. R Substituting for [V ], we get R 1 V out = (R 1 + R 2 )[ 2 R 1 +R 2 V B ] R 2 V A, which gives R 1 V out = R 2 V B R 2 V A, which simplifies to V out = R 2 R 1 (V B V A ). phys364/reading03.tex page 5 of :00

6 Amplifying the difference between two inputs is often a helpful technique for eliminating interference from unwanted signals. Another useful opamp circuit is the current-to-voltage amplifier (below, left). Sometimes your incoming signal is most easily described as a weak current source. For instance, many devices for photon detection emit an electric current in proportion to the incoming light intensity. Whereas the easiest load for a voltage source to drive is an open circuit (R load = ), the easiest load for a current source to drive is a short circuit (R load = 0): connecting too large a resistance to a weak current source will suppress the flow of current. 4 Because the opamp s + input is grounded, the input is held at virtual ground by feedback (Rule #1). So the current flowing into the opamp current-to-voltage amplifier sees what looks like a short-circuit to ground, which is ideal for a current source. By Rule #2, I in must flow to the right through resistor R, since the opamp inputs draw negligible current. So I in +V out /R = 0, which yields V out = RI in. The next application (above, right) is called the opamp integrator. By Rule #1, V = 0, so the current flowing to the right through R is just I = V in /R. By Rule #2, current I must continue to the right through C, since negligible current flows into the opamp inputs. With the left side of the capacitor fixed at V = 0, the result of current I flowing through C from left to right is C dv out /dt = I = V in /R, so V out (t) = 1 RC t dt V in (t ). The opamp circuit makes a very nice integrator there is no longer the annoying V out V in restriction that we had with the passive RC integrator of Lab 2. One serious problem with this circuit, however, is that if the time-averaged value of V in differs at all from zero, V out will saturate near V S±. We have violated Rule #0c, that there should be feedback at DC. There are two common solutions to this problem. The first (shown below, left) is to use a switch to zero the charge on the capacitor at t = 0 (i.e. whenever the integration should begin). Then V out (t) = 1 RC t 0 dt V in (t ) 4 For current sources, there is a theorem called Norton s theorem that is analogous to Thevenin s theorem. Calculating the Norton equivalent circuit allows you to quantify this loading effect for current sources. phys364/reading03.tex page 6 of :00

7 and you can choose RC such that V out does not saturate until well after the desired integration time has elapsed. I remember finding this circuit in the analogue of Physics 414 that I took in college: to observe changes in magnetic flux during phase transitions of a superconductor, the experiment reported Φ B by integrating the emf E induced in a wire coiled around the sample. 5 The second solution (shown above, right) is to include a bleeder resistor R bleed to drain the capacitor s charge over some long time scale R bleed C. Now there is a feedback path at DC, and the circuit is doing a weighted integration, emphasizing the recent time interval R bleed C. I didn t try to work this out explicitly, but by intuition I think the result is something like V out (t) = 1 RC t dt V in (t ) e (t t )/(R bleed C). Another way to look at this is as a generalization (using a complex impedance in place of the feedback resistor) of the inverting amplifier circuit. Without R bleed the amplifier s gain at f = 0 is ; with R bleed, the DC gain is reduced to R bleed /R. One final application (above) is a logarithmic amplifier. Using the Shockley diode equation, I diode = I 0 e V diode/(25 mv), we use Rule #1 (V = 0 since + input is at ground), and then Rule #2 (the currents flowing into the input sum to zero) to find I 0 e Vout/(25 mv) + V in /R = 0. Rearranging, ( ) Vin V out = 25 mv log. I 0 R The quantity 25 mv is really kt/e, as we ll see once we study diodes and transistors in detail. 6 And the constant I 0 (usually written I sat ) is a very small current (e.g. 5 Alas, when we did the experiment, the opamp chip was dead, and I had not yet taken an electronics course, so I didn t know how to fix it. We wound up collecting the E(t) data directly and then doing the integration numerically in a spreadsheet. 6 You might remember that kt at room temperature is about 1 40 ev. phys364/reading03.tex page 7 of :00

8 picoamps) that depends on the physical characteristics of the diode. It s quite a neat trick that you can get a circuit to take a logarithm! Notice that V out figures out how to undo whatever is in the feedback loop. This technique can be used more generally. LTspice! One more thing that I hope to put into these notes by Wednesday is a set of instructions for installing LTspice on your own computer. It is actually quite handy to be able to make a computer simulation of a circuit before you try to build it. (Example shown below.) phys364/reading03.tex page 8 of :00

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