44 4444 44362010-02-22 19:17:00 / rev f84c9915f679+2.3 Low-pass filtersThe next example is an analysis that originated in the study of circuits(Section 2.3.1). After those ontological bonds are snipped – once thesubject is “considered independently of its original associations” – thecore idea (the abstraction) will be useful in understanding diverse naturalphenomena including temperature fluctuations (Section 2.3.2).2.3.1 RC circuitsRCV0V1Linear circuits are composed of resistors, capaci-tors, and inductors. Resistors are the only time-independent circuit element. To get time-dependentbehavior – in other words, to get any interestingbehavior – requires inductors or capacitors. Here,as one of the simplest and most widely applicablecircuits, we will analyze the behavior of an RC circuit.The input signal is the voltage V0, a function of time t. The input signalpasses through the RC system and produces the output signal V1(t). Thedifferential equation that describes the relation between V0and V1is (from8.02)dV1dt+V1RC=V0RC. (2.11)This equation contains R and C only as the product RC. Therefore, itdoesn’t matter what R and C individually are; only their product RCmatters. Let’s make an abstraction and define a quantity τ as τ ≡ RC.This time constant has a physical meaning. To see what it is, give thesystem the simplest nontrivial input: V0, the input voltage, has been zerosince forever; it suddenly becomes a constant V at t = 0; and it remainsat that value forever (t > 0). What is the output voltage V1? Until t = 0,the output is also zero. By inspection, you can check that the solution fort > 0 isV1= V1 − e−t/τ. (2.12)In other words, the output voltage exponentially approaches the inputvoltage. The rate of approach is determined by the time constant τ. Inparticular, after one time constant, the gap between the output and input44 4444 44362010-02-22 19:17:00 / rev f84c9915f679+2.3 Low-pass filtersThe next example is an analysis that originated in the study of circuits(Section 2.3.1). After those ontological bonds are snipped – once thesubject is “considered independently of its original associations” – thecore idea (the abstraction) will be useful in understanding diverse naturalphenomena including temperature fluctuations (Section 2.3.2).2.3.1 RC circuitsRCV0V1Linear circuits are composed of resistors, capaci-tors, and inductors. Resistors are the only time-independent circuit element. To get time-dependentbehavior – in other words, to get any interestingbehavior – requires inductors or capacitors. Here,as one of the simplest and most widely applicablecircuits, we will analyze the behavior of an RC circuit.The input signal is the voltage V0, a function of time t. The input signalpasses through the RC system and produces the output signal V1(t). Thedifferential equation that describes the relation between V0and V1is (from8.02)dV1dt+V1RC=V0RC. (2.11)This equation contains R and C only as the product RC. Therefore, itdoesn’t matter what R and C individually are; only their product RCmatters. Let’s make an abstraction and define a quantity τ as τ ≡ RC.This time constant has a physical meaning. To see what it is, give thesystem the simplest nontrivial input: V0, the input voltage, has been zerosince forever; it suddenly becomes a constant V at t = 0; and it remainsat that value forever (t > 0). What is the output voltage V1? Until t = 0,the output is also zero. By inspection, you can check that the solution fort > 0 isV1= V1 − e−t/τ. (2.12)In other words, the output voltage exponentially approaches the inputvoltage. The rate of approach is determined by the time constant τ. Inparticular, after one time constant, the gap between the output and inputGlobal comments 1Global commentsare input/output signals always going to be so readily recognizable?I think a tree would be useful here to show all the factors that can affect resistance todaily temp variationsI agree, by the time you reach the end of this section you realize that these two situationsparallel nicely.Now we can do the last 2 problems of the pset! woohoo!I enjoyed this chapter because I understood all of it after taking 2.005 and 6.002, which iscool!It seems like this chapter combines abstraction with divide and conquer. It seems like areally practical application based on the example.WHy is this a transfer function? Is B a Laplace transform?Thanks for incorporating course 2 material (ie. 2.005) and course 6 material. I think itmakes it easier for both majors in the class to understand.Are these approximations taken from your book of constants? Or like your usual generalestimates?44 4444 44362010-02-22 19:17:00 / rev f84c9915f679+2.3 Low-pass filtersThe next example is an analysis that originated in the study of circuits(Section 2.3.1). After those ontological bonds are snipped – once thesubject is “considered independently of its original associations” – thecore idea (the abstraction) will be useful in understanding diverse naturalphenomena including temperature fluctuations (Section 2.3.2).2.3.1 RC circuitsRCV0V1Linear circuits are composed of resistors, capaci-tors, and inductors. Resistors are the only time-independent circuit element. To get time-dependentbehavior – in other words, to get any interestingbehavior – requires inductors or capacitors. Here,as one of the simplest and most widely applicablecircuits, we will analyze the behavior of an RC circuit.The input signal is the voltage V0, a function of time t. The input signalpasses through the RC system and produces the output signal V1(t). Thedifferential equation that describes the relation between V0and V1is (from8.02)dV1dt+V1RC=V0RC. (2.11)This equation contains R and C only as the product RC. Therefore, itdoesn’t matter what R and C individually are; only their product RCmatters. Let’s make an abstraction and define a quantity τ as τ ≡ RC.This time constant has a physical meaning. To see what it is, give thesystem the simplest nontrivial input: V0, the input voltage, has been zerosince forever; it suddenly becomes a constant V at t = 0; and it remainsat that value forever (t > 0). What is the output voltage V1? Until t = 0,the output is also zero. By inspection, you can check that the solution fort > 0 isV1= V1 − e−t/τ. (2.12)In other words, the output voltage exponentially approaches the inputvoltage. The rate of approach is determined by the time constant τ.