6 December 2002 Physics 217, Fall 2002 1Today in Physics 217: simple active AC circuits Operational amplifiers (opamps) Amplification with opamps Arithmetic with opamps Integration and differentiation with opamps Analog computersCircuit diagram for the venerable 741 opamp(National Semiconductor Corp.).6 December 2002 Physics 217, Fall 2002 2Passive and active circuit elementsThe circuit elements we’ve seen so far – capacitors, resistors and inductors – are called passive. These devices always obey the same relation between their current and voltage, independent of how large or small the current or voltage is.  That is, they have the same impedanceat all voltages. Activeelectronic devices are those whose current-voltage properties (impedance) change with voltage.  Semiconductor diodes or vacuum tubes, for instance.Some active devices can amplify: they allow one to control and change a large electrical power by changing a small electrical power, or vice versa.  Examples: transistors, vacuum-tube triodes, and circuits made from these devices.6 December 2002 Physics 217, Fall 2002 3Operational amplifiersThis is not a course in electronics; we will not discuss the details of transistors. But one particular class of transistor circuit that can provide amplification is simple enough to learn the basics of in a few minutes: the operational amplifier, or opamp.  Internally, they’re not simple, and the finest details of their performance is way beyond the scope of our discussion, but most of what they do can be described in only two concise statements. Let’s learn enough about opamps to use them in simple active AC circuits, thereby enriching substantially the number of AC-circuit situations in which you will be able to solve problems.6 December 2002 Physics 217, Fall 2002 4Opamps (continued)Single opamp in eight-pin dual-inline package (DIP).Connection diagram. +V and –V are DC power supplies, usually +15 and -15 volts.-+Circuit-diagram symbol: leave off DC power-supply and offset-null connections.OutputInvertinginputNoninvertinginput6 December 2002 Physics 217, Fall 2002 5Opamps (continued)An opamp has two inputs and one output.  An increase in the potential at the noninverting input leads to an increase in the potential at the output. An increase in the potential at the inverting input leads to a decrease in the potential at the output.  Thus the output change is proportional to the difference between the input voltage change. The proportionality constant (the open-loop gain, or amplification) is huge, usually larger than 100000. -+OutputInvertinginputNoninvertinginput6 December 2002 Physics 217, Fall 2002 6Opamps (continued)The two rules of opamps: If a current path is provided between the output and the inverting input, the output voltage will adjust automatically to produce zero voltage difference between the two inputs. The inputs draw no current.Connection of an amplifier’s output to its inverting input is called negative feedback.6 December 2002 Physics 217, Fall 2002 7Amplification with opampsThe two rules make it easy to find the relation between input and output voltages, DC or AC. Here are a few examples involving simply amplification of a single voltage. (Here, and throughout, voltage means potential relative to ground.) Inverting amplifier:voltage at + is zero, so thevoltage at – is too (rule#1). Thusflows in But the samecurrent must flow in (rule #2), so-+1R2RinVoutVIIin 1IV R=1.R2R2out 2 in1.RVIR VR=− =−6 December 2002 Physics 217, Fall 2002 8+-Amplification with opamps Noninverting amplifier.Here the voltage at – must be(rule #1), and the wire connecting – to the point between the resistors carriesno current (rule #2), so thatpoint is also at voltage 1R2RinVoutVIIinVin:V()()in1in 2out 1 2 1 2 in11,1.VIRVRVIRR RR VRR==+= +=+6 December 2002 Physics 217, Fall 2002 9Arithmetic with opamps Summing amplifier:Easily extended to as many inputs as one wants.-+RRRI1I2I1V2VoutV()()()121212out 1 20rule #1,.rule #2.VVVIIRRII IVIRVV−=⇒= ==+⇒=−=−+6 December 2002 Physics 217, Fall 2002 10Arithmetic with opamps (continued) Differential amplifier:-+RRR2I1I2V1VoutV()()()22221121out 122121rule #22rule #12112.22VIRVVIR VIVVRVVRVVIRVVVVV+−−−=====−=−=−=− − =−6 December 2002 Physics 217, Fall 2002 11Calculus with opampsSo far, these circuits would operate equally well on constant or time-varying voltages. Here we begin purely AC cases. Differentiator:-+RCoutVinVI()()inoutinoutrule #2 ,rule #1 ..dqdVICdt dtVRdVVRCdt===−⇒=−I6 December 2002 Physics 217, Fall 2002 12Calculus with opamps (continued) Integrator:The integration would go on forever. If one doesn’t want it to – for instance, if one wanted to change functions at the input – one closes the reset switch briefly. A new integration begins as soon as it’s opened.-+RCoutVinVIIReset switch()()()inout inrule #1 ,rule #2 .1.outVIRdqdVCdt dtVVtdtRC===−=−∫6 December 2002 Physics 217, Fall 2002 13One can make differentiators and integrators with inductors, too, but in general it works better with capacitors; inductors can’t be manufactured quite as precisely as capacitors, and don’t behave as close to ideally as capacitors do.-+-+Calculus with opamps (continued)out inLVVdtR=−∫inoutdVLVRdt=−inVinVRRLL6 December 2002 Physics 217, Fall 2002 14-+-+Analog computersSince we have circuit modules that integrate and differentiate voltages, and circuit modules that add and subtract voltages, we can now “write” linear differential equations as circuits. Take, for example, the differential equation we used on Wednesday:We can generate terms that look like these with two integrators:220020.dq dqqQdtdtωω++=()222dVRCdtdVRCdt−VRCCR6 December 2002 Physics 217, Fall 2002 15Analog computers (continued)Let’s take the linear and first-derivative terms and add them together:-+-+-+-+()222dVRCdtdVRCdt−VRCCRRR’RRRdVRCdt′dVRC Vdt′−−To set these equal, simply connect them with a wire!6 December 2002 Physics 217, Fall 2002 16Analog computers (continued)Thus-+-+-+-+()222dVRCdtdVRCdt−VRCCRRR’RRRdVRCdt′dVRC Vdt′−−()222 210.dV R dVVdtdt R CRC′++ =6 December 2002 Physics 217, Fall 2002 17Analog computers (continued)So to simulatewe just need to choose the resistors and capacitors so that and the numbers we’d measure for