Worksheet #12 - PHY102 (Spr. 2006)DC and AC circuitsDue Thursday April 13th 9pmIn earlier worksheets we have studied the behavior of damped, mass-spring systems. We also took a brief look at the linear and non-linear pen-dulum problems. The equations describing these problems also describe thebehavior of linear and non-linear circuits and there are many analogies be-tween mechanical and electrical problems that are often used in the physicsdiscussions. For example the concepts of underdamped, critically dampedand overdamped apply to electrical problems as well as to mechanical prob-lems. Similarly the concept of resonance applies to both mechanical andelectrical problems. In circuits, a resistor absorbs energy and acts as thedamping. In mechanical problems energy is transferred between potentialenergy and kinetic energy. In electrical problems energy is transferred be-tween energy stored in the magnetic field and energy stored in the electricalfield. In the case of circuits energy is transferred between capacitors andinductors.The relation between the magnitudes of the voltage and current in circuitelements L,R,C are as follows:VR= IR, VL= LdIdt, VC=1CZt0Idt (1)Kirchhoff’s voltage law states that the sum of the voltage drops around aseries LRC circuit must sum to zero, yielding the following equation,Vs− VR− VL− VC= 0 (2)Using the relations above between the current and voltages for L,R,C givenabove we find,Vs− IR − LdIdt−1CZt0Idt = 0 (3)Taking a derivative of this equation yields,dVsdt− Ld2Idt2− RdIdt−IC= 0 (4)Recall that a damped mass-spring system is described by Fs− bv − kx = ma,so that,−Fs+ md2xdt2+ bdxdt+ kx = 0 (5)1where Fsis the driving force on the mass-spring system. The analogies be-tween the mass-spring system and the electric circuit are evident from theseequations, that is, L acts like the mass, R like the damping and 1/C like thespring constant. The derivative of the source voltage acts like the drivingforce.Problems1. (Transients) Consider a series LRC circuit connected to DC voltage of1V. A switch in the circuit is initially open and the capacitor is initally un-charged. Consider closing the switch at time zero. Write a code to find andplot the current in the LRC circuit as a function of time. Run your code forvalues of L, R and C to illustrate the cases of underdamped and overdampedcircuits.2. (Resonance) Consider an LRC circuit driven by a steady sinusoidal ACsource with amplitude 1V. Find and plot the amplitude of the steady statecurrent in the circuit as a function of the ratio of the frequency of the ACsource to the natural frequency of the LRC circuit. The natural frequency isgiven by,ωn= (1/LC −
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