Chapter 2N Class 2: RC Circuits Contents 2N Class 2: RC Circuits 1 2N.1Capacitors........................................................... 2 2N.1.1Why?......................................................... 2 2N.1.2CapacitorStructure................................................... 2 2N.1.3 A static description of cap behavior... ......................................... 2 2N.1.4 ...A dynamic description of cap behavior ....................................... 3 2N.2Time-domainViewofRC’s.................................................. 3 2N.2.1 An easy case: constant I ................................................ 3 2N.2.2 A harder case but more common: constant voltage source in series with a resistor (“exponential” charging) ..... 4 2N.2.3IntegratorsandDifferentiators............................................. 6 2N.3FrequencyDomainViewofRC’s................................................ 9 2N.3.1 The impedance or reactance ofacap.......................................... 9 2N.3.2 Digression: Deriving ZC ............................................... 10 2N.3.3 RC Filters ....................................................... 11 2N.3.4Decibels........................................................ 14 2N.3.5 Estimating a Filter’s Attenuation ............................................ 17 2N.3.6 Input and output impedance of an RC Circuit..................................... 19 2N.3.7PhaseShift....................................................... 20 2N.3.8PhasorDiagrams.................................................... 21 2N.4TwoUn-glamorousbutImportantCapApplications:“blocking”and“decoupling”...................... 25 2N.4.1 Blocking capacitor................................................... 25 2N.4.2“Decoupling”or“Bypass”Capacitor.......................................... 26 2N.5 A Somewhat Mathy View of RC Filters ............................................ 27 2N.6ReadingsinAoE........................................................ 28 REV 21; January 26, 2015. 1Revisions: insert Ray redrawings (10/14); correct Xc to Zc in phasor diag figure, add subsection explaining how we get away with ignoring phase shift, and note 25-degree shift at 1/2 lowpass f3dB (9/14); revise per Paul’s comments, add headerfile (7/14); fixrefto text filter treatment (4/14); add “Why?”, use external header file(1/14); add short mathy section re RC frequency response, at very end (nov13); corrected log-log fall figure, which had show rise (1/13); add index (6/12; add details of passband atten, from Tek note (1/12); 1 2 Class 2: RC Circuits 2N.1 Capacitors Today things get a little more complicated, and more interesting, as we meet frequency-dependent circuits. We rely on the capacitor to implement this new trick, which depends on the capacitor’s ability to “remember” its recent history. That ability allows us to make timing circuits (circuits that let something happen a predetermined time after something else occurs); the most important of such circuits are oscillators—circuits that do this timing oper- ation over and over, endlessly, in order to set the frequency of an output waveform. The capacitor’s memory also lets us make circuits that respond mostly to changes (differentiators) or mostly to averages (integrators). And the capacitor’s memory implements the RC circuit that is by far the most important to us: a circuit that favors one frequency range over another (a filter).2. All of these circuit fragments will be useful within later, more complicated circuits. The filters, above all others, will be with us constantly as we meet other analog circuits. They are nearly as ubiquitous as the (resistive-) voltage divider that we met in the first class. 2N.1.1 Why? We have suggested a collection of applications for RC circuits. Here is a shorter answer to what we mean to do with the circuits we meet today: • generate an output voltage transition that occurs a particular delay time after an input voltage transition; • design a circuit that will treat inputs of different frequencies differently: it will pass more in one range of frequencies than in another (this circuit we call a “filter”). 2N.1.2 Capacitor Structure Figure 1: The simplest capacitor configuration: sandwich This capacitor is drawn to look like a ham sandwich: metal plates are the bread, some dielectric is the ham (ceramic capacitors really are about as simple as this). More often, capacitors achieve large area (thus large capacitance) by doing something tricky, such as putting the dielectric between two thin layers of metal foil, then rolling the whole thing up like a roll of paper towel (mylar capacitors are built this way). 2N.1.3 A static description of cap behavior... AoE §1.4.1 A static description of the way a capacitor behaves would say add note on decibels, on bypass and blocking caps.; many errata, sept 10, esp frequency-extreme figure (9/10). 2Incidentally, in case you need to be persuaded that remembering is the essence of the service that capacitors provide, note that much later in this course, partway into the digital material, we will meet large arrays of capacitors used simply and explicitly to remember: several sorts of digital memory (dynamic RAM, Flash, EEPROM and EPROM) use thousands or millions of tiny capacitors to store their information, holding that data in some cases for many years Class 2: RC Circuits 3 Q = CV where Q is total charge, C is the measure of how big the cap is (how much charge it can store at a given voltage: C = Q/V), and V is the voltage across the cap. This statement just defines the notion of capacitance. It is the way a Physicist’s might describe how a cap behaves, and rarely will we use it again. 2N.1.4 ...A dynamic description of cap behavior Instead, we use a dynamic description—a statement of how things change with time: I=C dV/dt This is just the time derivative of the “static” description. C is constant with time; I is defined as the rate at which charge flows. This equation isn’t hard to grasp. It says ‘The bigger the current, the faster the cap’s voltage changes.’ A Hydraulic Analogy Again, flowing water helps intuition: think of the cap (with one end grounded) as a tub that can hold charge: Figure 2: A cap with one end grounded works a lot like a tub of water A tub of large diameter (cap) holds a lot of water (charge), for a given height or depth (V). If you fill the tub through a thin straw (small I), the water level—V—will rise slowly; if you fill or drain through a fire hose (big I) the tub will fill (“charge”) or drain (“discharge”) quickly. A tub of large diameter (large capacitor) takes longer to fill or drain than a small tub. Self-evident, isn’t it? 2N.2 Time-domain View of RC’s Now let’s leave tubs of water, and anticipate what we will see when we watch the voltage on a cap change with time: when we look on a scope screen, as you will do in Lab 2. 2N.2.1 An easy case: constant I AoE §1.4.4.1 Figure 3: Easy case: constant I −→ constant dV/dt 4 Class 2: RC Circuits This tidy waveform, called a ramp, is useful, and you will come to recognize it as the signature of this circuit fragment: capacitor driven by constant current (or “current source”). This arrangement is used to generate a triangle waveform, for example: Figure 4: How to use a cap to generate a triangle waveform: ramp up, ramp down But the ramp waveform is relatively rare, because current sources are relatively rare. Much more common is the next case. 2N.2.2 A harder case but more common: constant voltage source in series with a resistor (“exponential” charging) AoE §1.4.2.1 Figure 5: The more usual case: cap charged and discharged from a voltage source, through a series resistor Here, the voltage on the cap approaches the applied voltage—but at a rate that diminishes toward zero as Vcap approaches its destination. It starts out bravely, moving fast toward its Vin (charging at 10 mA, in the example above, thus at 10V/ms); but as it gets nearer to its goal, it loses its nerve. By the time it is 1 volt away, it has slowed to 1/10 its starting rate. (The cap behaves a lot like the hare in Xeno’s paradox: remember him? Xeno teased his fellow-Athenians by asking a question something like this: ‘If a hare keeps going halfway to the wall, then again halfway to the wall, does he ever get there?’ (Xeno really had a hare chase a tortoise; but the electronic analog to the tortoise escapes us, so we’ll simplify his problem.) Hares do bump their noses; capacitors don’t: Vcap never does reach Vapplied,inanRC circuit. But it will come as close as you want. 2N.2.2.1 “Exponential” Charge and Discharge The behavior of RC charging and discharging is called “exponential” because it shows the quality common to members of that large class of functions. A function whose slope is proportional (or inversely proportional) to its value is called “exponential:” the function ex behaves as the discharging RC circuit does. Its slope is equal to its value. Class 2: RC Circuits 5 Familiar exponential functions are those describing population growth (the more bacteria, the faster the colony grows) and a draining bathtub (the shallower the water, the more slowly it drains). The capacitor’s discharge curve follows this rule more obviously than the charging curve, so let’s consider that case: as the capacitor discharges, the voltage across the R diminishes, reducing the rate at which it discharges. This is evident in the discharge curve of fig. 6. The RC discharges the way a bathtub drains. Here’s a fuller account: Figure 6: RC charge, discharge curves Two numbers, in the plot of fig. 6 are worth remembering: • in one RC (called “one time-constant”) Vcap goes 63% of the way toward its destination • in five RCs, 99% of the way If you need an exact solution to such a timing problem: −t/RC Vcap = Vapplied · (1 − e ) In case you can’t see at a glance what this equation is trying to tell you, look at e−t/RC by itself: • when t = RC, this expression is e−1 =1/e, or 0.37.