E40M. Op Amps. M. Horowitz, J. Plummer, R. Howe 1


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1 E40M Op Amps M. Horowitz, J. Plummer, R. Howe 1
2 Reading A&L: Chapter 15, pp Reader, Chapter 8 Noninverting Amp Inverting Amp Summing Amp M. Horowitz, J. Plummer, R. Howe 2
3 How to Measure Small Voltages? Arduino input has fullscale around 5V It produces a 10 bit answer (1024) This means a LSB (least significant bit) is 5mV Need to make the signal bigger before input to Arduino So, we will use an amplifier Many ways to build amplifiers One often uses a standard building block for amplifiers Called an Operational Amplifier, or OpAmp A circuit with very high gain at low frequencies (< 10 khz) M. Horowitz, J. Plummer, R. Howe 3
4 Electrical Picture Signal amplitude 1 mv Noise level will be significant will need to amplify and filter We ll use filtering ideas from the last two lectures M. Horowitz, J. Plummer, R. Howe 4
5 OP AMPS M. Horowitz, J. Plummer, R. Howe 5
6 Op Amp Is a common building block It is a highgain amplifier Output voltage is A (V+ V) Gain, A, is 10 K to 1 M LM741 Output voltage can be + or Often can swing between +Vdd and Vdd supplies Huh? M. Horowitz, J. Plummer, R. Howe 6
7 OpAmp Power Supply Up to now we had one supply voltage, Vdd All voltages were between Vdd and Gnd Generally measured relative to Gnd So all voltages were positive. A sinewave goes positive and negative And most input signals do that too It is convenient to have a reference where The output can be positive and negative Can do that by changing what we call the reference M. Horowitz, J. Plummer, R. Howe 7
8 Moving the Reference V dd = 5 V V dd = 2.5 V v out v out 2.5 V  The voltages are all the same, only the reference voltage has moved M. Horowitz, J. Plummer, R. Howe 8
9 What You Will Actually Do Use the USB supply Just change the reference voltage 5 V +  R R +  M. Horowitz, J. Plummer, R. Howe 9
10 Op Amp Behavior Relationship between output voltage and input voltage: v o = A( v + v ) = A( v p v ) n A is the opamp gain (or openloop gain), and is huge 10K1M The input currents are very, very small so i p 0 and i n 0. M. Horowitz, J. Plummer, R. Howe 10
11 Since the Output Swing is Limited The high gain only exists for a small range of input voltages If the input difference is too large, the output saturates Goes to the max positive or negative value possible Close to supply voltages M. Horowitz, J. Plummer, R. Howe 11
12 What Does This Do? V cc = 5 V v out v in v out V M. Horowitz, J. Plummer, R. Howe 12
13 Same Circuit Different Reference V dd = 2.5 V v out + 2 v in v out V M. Horowitz, J. Plummer, R. Howe 13
14 How To Get A Useful Amplifier The gain of the op amp is too high to make a useful amplifier We need to do something to make it useful We will use analog feedback to fix this problem Feedback makes the input the error between the value of the output, and the value you want the output to have. Let s see how to do this M. Horowitz, J. Plummer, R. Howe 14
15 Connect V out to V in v out = A(V + V ) = A(v in v out ) + V cc = 5 V v in V cc = 5 V v out ( A +1)v out = Av in v out = A ( A +1) v in v in  M. Horowitz, J. Plummer, R. Howe 15
16 What Is Going On We solved the equation to find the answer But how does the opamp get this answer? Think about what happens when the input increases in voltage From 0 V to 0.1 V Initially the output can t change There is capacitance at every node The opamp thinks it needs to create a huge output voltage So it drives current into the output Which charges the capacitor Causing the output to increase This then decreases the input difference M. Horowitz, J. Plummer, R. Howe 16
17 Feedback in an Opamp Circuit As the output rises The input difference decreases So A* DV in also decreases The system is stable when A* DV in is exactly equal to V out If A is large (10 6 ) for any V out Say in the range of ± 10V Dv in will be very, very small Can approximate that by saying Dv in will be driven to 0 Output will be set so v in+ v in M. Horowitz, J. Plummer, R. Howe 17
18 BUT This is only true if you connect the output feedback To the negative terminal of the amplifier What happens if you connect it to the positive terminal? M. Horowitz, J. Plummer, R. Howe 18
19 Ideal Op Amps The Two Golden Rules for circuits with ideal opamps* No voltage difference between opamp input terminals No current into opamp inputs * when used in negative feedback amplifiers M. Horowitz, J. Plummer, R. Howe 19
20 USEFUL OP AMPS CIRCUITS M. Horowitz, J. Plummer, R. Howe 20
21 Approach To Solve All Opamp Circuits First check to make sure the feedback is negative If not, STOP! Find the output voltage that makes the input difference 0 Assume V + = V  Find V out such that KCL holds We ll do some examples M. Horowitz, J. Plummer, R. Howe 21 E40M Lecture 19
22 Noninverting Amplifier i p = 0 so v p = v s V + = V  so v n = v p = v s i 1 i 1 = i 2 so v o v s R 1 = v s R 2 i 2 v o 1 = v R s R 1 R 2 R v o = v 1 +R 2 s R 2 M. Horowitz, J. Plummer, R. Howe 22
23 Inverting Amplifier At node v n v n v s R s + v n v o R f + i n = 0 But v n =v p = 0 and i n = 0, so v s R s v o R f = 0 or v o = v s R f R s M. Horowitz, J. Plummer, R. Howe 23
24 CurrenttoVoltage Converter i 2 i p = i n = 0 i 1 v n = v p = 0 i R So i R = 0 as well KCL at the v n node: i 1 = i s = i 2 = v o R f so v o = i s R f M. Horowitz, J. Plummer, R. Howe 24
25 OP AMP FILTERS M. Horowitz, J. Plummer, R. Howe 25
26 Adding Capacitors Sinusoidal voltage v o = v s Z f Z s = C f Suppose we add a capacitor in the feedback j 2πFC R f f R s We can treat this exactly as we did the earlier circuits by using impedances. Our earlier analysis showed vo = v s R f R s 1 Z s = R Z s f = 1 + j 2πFC R f f R = v f 1 s R s 1+ j 2πFR f C f M. Horowitz, J. Plummer, R. Howe 26
27 Sketching the Bode Plot 20 log 10 V o /V s 60 v o R f 1 = v s R V s 1+ j 2πFR f C s f V o F [Hz] 20 R s = 1 kω, R f = 100 kω, C f = 160 nf F c = 1/(2pR f C f ) = 10 Hz M. Horowitz, J. Plummer, R. Howe 27
28 Learning Objectives Understand how living things use electricity Understand what an op amp is: The inputs take no current The output is 10 6 times larger than the difference in input voltages The two Golden Rules of op amps in negative feedback Input currents are 0; V in = V in+ Be able to use feedback to control the gain of the op amp For inverting and noninverting amplifiers Understand op amp filters and differential amplifiers M. Horowitz, J. Plummer, R. Howe 28
29 More Examples M. Horowitz, J. Plummer, R. Howe 29
30 Summing Amplifier i 1 i 2 i 3 i p = i n = 0 v n = v p = 0 KCL at the summing point (or summing node): v i 1 + i 2 = i 3 so 1 + v 2 = v o R 1 R 2 R Output voltage is a scaled sum of the input voltages: v o = R f v R 1 + R f v 1 R 2 2 M. Horowitz, J. Plummer, R. Howe 30
31 A Subtracting (Difference) Amplifier? Take an inverting amplifier and put a 2 nd voltage on the other input? i 1 + i 2 = 0 so v n v 1 R s + v n v o R f = 0 v 1 v2 v n = v 2 so v 2 v 1 R s = v o v 2 R f v o R f = v 2 v 1 R s + v 2 R f v o = v 1 R f R s + v 2 R f +R s R s Not quite what we wanted. We d like v o a (v 1 v 2 ). M. Horowitz, J. Plummer, R. Howe 31
32 Differential Amplifier 1.0 v 1 v n R 1 = v n v o R 2 v 1 v 2 R 4 R 3 +R 4 R 1 = v 2 R 4 R 3 +R 4 v o R 2 v o = v 1 + v 2 R 4 + R 1 R 4 R 2 R 1 R 1 R 3 +R 4 R 2 R 3 +R 4 But if R 3 = R 1 and R 4 = R 2 v o = v 2 v 1 ( ) R 2 R 1 M. Horowitz, J. Plummer, R. Howe 32
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