University of California Berkeley EECS 105 Fall 2003 Prof A Niknejad Problem Set 1 Due Monday September 1 2003 1 In the following problems perform the operating indicated and reduce the nal answer to the form of A iB a b c 1 i 1 i i 1 i 1 i i i d ii e i570 4i29 5i 2 Prove that for any number z 1 Re z z z 2 1 Im z z z 2i 3 Prove that conjugation distributes over sums di erences 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 0 and w 0 z arg arg z arg w w 7 Consider the high pass lter shown below C vs t R a Calculate the sinusoidal steady state frequency response by i directly solving the di erential equations and by ii the method of phasor analysis b Calculate the output voltage when the lter is driven by a signal Vs t 5V cos t 30 where 2 100 kHz Assume that the RC time constant 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 an application for such a circuit 8 A simple physical model for an inductor is shown in below The series resistance models the loss and the shunt capacitance models the self resonant frequency of the device a Find the impedance of the circuit as a function of frequency for this part only use L 1 nH R 3 and C 150fF b What s the magnitude of the impedance at DC What s the magnitude of the impedance at in nite frequency c Are there any frequencies when the phase of the impedance is zero d Plot the magnitude and phase response for the circuit L C R 2
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