Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6 002 Electronic Circuits Fall 2004 Homework 10 Handout F04 50 Issued 11 12 2004 Due 11 19 2004 Helpful Readings for this Homework Chapter 13 Chapter 14 Sections 15 1 15 3 Exercise 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 e s0 t that satisfies the homogenous differential equation For example if dV V 0 dt then e s0 t 1 is a homogenous solution when the natural frequency s 0 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 v IN t V SI cos S t Note that vOUT t will take the form v OUT t V SO S cos S t S S Signal Generator L RIN vIN t C R vOUT t Figure 1 The 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 and electronic crossovers Amanda tells him All you need is a resistor and an inductor in series Take the high frequencies 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 frequency tweeter Crossover Amp tweeter high frequencies Amp 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 v i t V i cos t where V i is real The outputs of the network are the voltages across the resistor and inductor v ar t V ar cos t V ar and v al t V al cos t V al where V ar and V al denote the amplitude and V ar and V al denote the phase of complex numbers V ar and V al Find the amplitude and phase as functions of for both V ar and V al as follows a jx Using the Taylor series expansions for e cos x and sin x show that e jx cos x j sin x Fol jx lowing this recognize that cos x Re e 1 b Find the magnitude and phase of A Bj Express A Bj and in polar form A Bj c Find differential equations which can be solved for v ar t and v al t d Following Part a let v i t Re V i e j t Also let v ar t Re V ar e j t and v al t Re V al e j t where V ar and V al are complex functions of With these substitutions use the corresponding differential equation to find V ar and V al 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 V ar and V al as functions of and V i f ar al ar and al and and log with Sketch and clearly label the dependence of log V Vi Vi Vi i V V V V L respect to log Identify the low and high frequency asymptotes on each sketch R 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 frequency 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 R var t vi t L val t Problem 10 4 optional but recommended Chapter 14 Problem 16 parts a b and c only page 1183
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