Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Circuits and ElectronicsSpring 2003Handout S03-042 - Homework #8Issued: Wed. Apr 2Due: Fri. Apr 11Problem 8.1: The inductor in the LC circuit below has an initial current i = I Amperes. Att = 0 the switch opens.SWL+-vCiC = 0.1µFL = 40mH(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) andv(t) for t > 0.Problem 8.2: The circuit of Problem 8.1 is modified by adding a high-value parallel resistorof conductance G. The initial inductor current is i = I and the switch opens at t = 0.SWL+-vCiC = 0.1µFL = 40mHG = 5x10-4MhosG(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 fre-quencies.(C) Determine the damping factor α, the natural frequency ω, and the Q of the lightly dampedoscillator.Problem 8.3: The circuits shown below are driven by sinusoids. In each case express theindicated variables as functions of time, i.e. f(t) = A cos(ωt + φ) and write expressions for themagnitudes and the phases.Hint: Use a complex exponential as the driving function.(A)+−Vcos(ωt)LRvL=?+-(B)Icos(ωt)CvC=?+-Ri(t)=?Problem 8.4: The circuits show below, which are driven by sinusoids, have the indicatedresponses.I)iS(t)=Icos(ωt)LvL+-RvL(t) = IωLq1+(ωLR)2cos¡ωt +π2− tan−1¡ωLR¢¢II)vS(t)=Vcos(ωt)CvC+-Ri(t)+−vC(t) = V1√1+(ωRC)2cos¡ωt − tan−1(ωRC)¢i(t) = VωC√1+(ωRC)2cos¡ωt +π2− tan−1(ωRC)¢(A) Focus first on circuit I). Recall that the magnitude of the impedance of an inductor varies asωL, and note that the circuit has the form of a current divider.At the limit of very low frequencies¡ω <<RL¢reason from the circuit alone to determine anapproximate value for the magnitudeof vL. Verify your answer by using the given response.(B) At the limit of very high frequencies¡ω >>RL¢use circuit reasoning to determine an approx-imate value for the magnitudeof 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 theform of a voltage divider and recall that the magnitude of the impedance of a capacitor varies as1ωC.(E) For the repeat of Part A), what is the condition for the low frequency limit that corresponds to¡ω <<RL¢? Reason from the circuit alone to determine approximate values for the magnitudesof vCand i at very low frequencies. Verify your answers by using the given responses.(F) For the repeat of Part B), what is the condition for the high frequency limit that corresponds to¡ω >>RL¢? Reason from the circuit along to determine approximate values for the magnitudesof vCand 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
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