Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 - Electronic CircuitsFall 2004Homework #10Handout F04-50Issued 11/12/2004 - Due 11/19/2004Helpful Readings for this Homework: Chapter 13, Chapter 14, Sections 15.1-15.3Exercise 10-1: Chapter 14, Exercise 7, parts b, c, and d only; (page 1169-1170).Note the terminology “natural frequency” in part (d). The definition of natural frequency is the value of s0, in a waveform of the form es0t, that satisfies the homogenous differential equation. For example, ifdVdt-------Vτ----+ 0=then es0t is a homogenous solution when the natural frequency s01τ--–=.Exercise 10-2: Chapter 15, Exercise 2 (page 1241).Problem 10-1:Assume that the network in Figure 1 is in sinusoidal steady state. Determine the response vOUT(t) to the input vINt() VSIωSt()cos≡ . Note that vOUT(t) will take the form vOUTt() VSOωS() ωSt φSωS()+()cos= . +-RINvIN(t)LRC-+vOUT (t)Signal GeneratorFigure 1The results of this problem will be used in the Pre-Lab of Lab 3.Problem 10-2: Chapter 15, Problem 3 (page 1255).Problem 10.3: James is building a stereo system but needs some help. He needs to take the output of his 8-track tape player and somehow split it into high and low frequencies before sending the signals to the MOSFET amplifiers he built last month, and then to his loudspeakers, so he asks Amanda and Bo for help. A quick web search leads them to http://www.eatel.net/~amptech/elecdisc, where they learn more than they ever wanted to know about car audio andelectronic crossovers. Amanda tells him, “All you need is a resistor and an inductor in series. Take the high frequen-cies off one and the low frequencies off the other.” She then leaves to work on her 6.002 problem set. Unfortunately, James forgot to ask which element to connect to his low-frequency woofer and which to connect to his high-fre-quency tweeter.CrossoverAmpAmptweeter (high frequencies)woofer (low frequencies)The network suggested by Amanda is shown below. For the purpose of our analysis, it is driven in steady state by the sinusoidal input voltage vit() Viωt()cos= , where Vi is real. The outputs of the network are the voltages across the resistor and inductor, vart() VarωtVar∠+()cos= and valt() ValωtVal∠+()cos= , where Var and Val denote the amplitude and Var∠ and Val∠ denote the phase of complex numbers Var and Val. Find the amplitude and phase, as functions of ω , for both Var and Val as follows:a) Using the Taylor series expansions for ejx, x()cos , and x()sin , show that ejxx()cos jx()sin+= . Fol-lowing this, recognize that x()cos Re ejx{}= .b) Find the magnitude and phase of ABj+ . Express ABj+ and 1ABj+--------------- in polar form.c) Find differential equations which can be solved for vart() and valt().d) Following Part a, let vit() Re Viejωt{}= . Also, let vart() Re Varejωt{}= and valt() Re Valejωt{}= , where Var and Val are complex functions of ω . With these substitutions, use the corresponding differ-ential equation to find Var and Val. Hint: recall that in the differential equations we can drop the Re{} notation until the end.e) Following Parts a and b, find the magnitude and phase for both Var and Val, as functions of ω and Vi.f) Sketch and clearly label the dependence of VarVi---------⎝⎠⎛⎞log and VarVi---------∠, and ValVi--------⎝⎠⎛⎞log and ValVi--------∠, with respect to ωLR-------⎝⎠⎛⎞log. Identify the low- and high-frequency asymptotes on each sketch.g) The breakpoint frequency for a plot is that frequency at which its low- and high- frequency asymptotes cross. Find the breakpoint frequency for the plots in Part f. (For a low-pass filter, its output is nearly constant below this frequency, while above this frequency, its output decreases with increasing fre-quency. The situation is reversed for the case of the high-pass filter.)h) Which terminals should be connected to his high-frequency tweeter and which to his low-frequency woofer? Discuss qualitatively, but physically, how the outputs act as low-pass or high-pass filters.RLvar(t)val(t)vi(t)Problem 10-4 (optional, but recommended): Chapter 14, Problem 16, parts a,b, and c only; (page
View Full Document
Unlocking...