Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6 002 Circuits and Electronics Spring 2003 Handout S03 042 Homework 8 Issued Wed Apr 2 Due Fri Apr 11 Problem 8 1 The inductor in the LC circuit below has an initial current i I Amperes At t 0 the switch opens i v C 0 1 F C L SW L 40 mH A Determine the natural frequency and the period of the oscillation which occurs for t 0 Specify units B Write a differential equation for the current i or the voltage v which applies for t 0 C Solve this equation apply the indicated initial conditions and write expressions for i t and v t for t 0 Problem 8 2 The circuit of Problem 8 1 is modified by adding a high value parallel resistor of conductance G The initial inductor current is i I and the switch opens at t 0 i v G C SW C 0 1 F L L 40 mH G 5x10 4 Mhos A Write a differential equation for v which applies for t 0 B Determine the characteristic equation for the circuit the roots of which are the natural frequencies C Determine the damping factor the natural frequency and the Q of the lightly damped oscillator Problem 8 3 The circuits shown below are driven by sinusoids In each case express the indicated variables as functions of time i e f t A cos t and write expressions for the magnitudes and the phases Hint Use a complex exponential as the driving function A R Vcos t L vL B i t Icos t Problem 8 4 responses R C vC The circuits show below which are driven by sinusoids have the indicated I iS t Icos t R L vL vL t I q L 1 L 2 R cos t 2 tan 1 L R R II vS t Vcos t i t C vC 1 1 RC 2 vC t V C 1 RC 2 i t V cos t tan 1 RC cos t 2 tan 1 RC A Focus first on circuit I Recall that the magnitude of the impedance of an inductor varies as L and note that the circuit has the form of a current divider At the limit of very low frequencies R L reason from the circuit alone to determine an approximate value for the magnitude of vL Verify your answer by using the given response B At the limit of very high frequencies R L use circuit reasoning to determine an approximate value for the magnitude of vL C At what frequency does the current iS t divide equally in magnitude between R and L D At this frequency what are the magnitude and phase of the response The following questions are similar to A D above but apply to circuit II Note that it has the form of a voltage divider and recall that the magnitude of the impedance of a capacitor varies as 1 C E For repeat of Part A what is the condition for the low frequency limit that corresponds to the R L Reason from the circuit alone to determine approximate values for the magnitudes of vC and i at very low frequencies Verify your answers by using the given responses F For repeat of Part B what is the condition for the high frequency limit that corresponds to the R L Reason from the circuit along to determine approximate values for the magnitudes of vC and i at very high frequencies Verify your results by using the given responses G At what frequency does the voltage vS t divide equally in magnitude between R and C H At this frequency what are the magnitudes and phases of the responses
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