University of California, Berkeley Fall 2003EECS 105 Prof. A. NiknejadProblem Set 1Due Monday September 1, 20031. In the following problems, perform the operating indicated and reduce the finalanswer to the form of A + iB(a)1+i1−i(b)i1−i+1−ii(c)√i(d) ii(e) i570− 4i29− 5i2. Prove that for any number z,Re(z)=12(z +¯z)Im(z)=12i(z − ¯z)3. Prove that conjugation distributes over sums, differences, and quotients.4. Write each of the following numbers in polar form.(a) −1(b) 3(c) −4i(d) 2 − i(e)√2(1 + i)5. Find the six sixth roots of unity and plot them on the complex plane.6. Prove that for any z =0andw =0arg(zw)=arg(z) − arg(w)7. Consider the high-pass filter shown below.vs(t) CR(a) Calculate the sinusoidal steady-state frequency response by (i) directly solvingthe differential equations and by (ii) the method of phasor analysis.(b) Calculate the output voltage when the filter is driven by a signalVs(t)=5Vcos(ωt +30◦)whereω =2π 100 kHz. Assume that the RC timeconstant of the circuit is ten microseconds. Use phasors for your calculations.What’s the magnitude response in dB?(c) Describe the operation of the circuit as C →∞. Can you think of anapplication for such a circuit?8. A simple physical model for an inductor is shown in below. The series resistancemodels the loss and the shunt capacitance models the self-resonant frequency of thedevice. (a) Find the impedance of the circuit as a function of frequency (for this partonly, use L =1nH,R = 3Ω, and C = 150fF). (b) What’s the magnitude of theimpedance at DC? What’s the magnitude of the impedance at infinite frequency? (c)Are there any frequencies when the phase of the impedance is zero? (d) Plot themagnitude and phase response for the
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