Worksheet #11 – PHY102 (Spring 2010)DC and AC circuitsIn earlier worksheets we have studied the behavior of a damped, mass+springsystem. We also took a brief look at the linear and non-linear pendulumproblems. The equations describing these problems also describe the behav-ior of linear and non-linear electrical circuits, and there are many analogiesbetween mechanical and electrical problems that are often used in physics dis-cussions. For example the concepts of underdamped, critically damped andoverdamped apply to electrical problems as well as to mechanical problems.Similarly the concept of resonance applies to both mechanical and electricalproblems. In circuits, a resistor absorbs energy and acts as the damping.In mechanical problems, energy is transferred between potential energy andkinetic energy. In electrical problems, energy is transferred between energystored in the magnetic field and energy stored in the electrical field. In thecase of electrical circuits, energy is transferred between capacitors and in-ductors.The relation between the magnitudes of the voltage (V) and current (I)in circuit elements 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 with respect to time yields,dVsdt− Ld2Idt2− RdIdt−IC= 0 (4)1Recall that a damped mass-spring system is described by Fs− bv − kx = ma,so that,−Fs+ md2xdt2+ bdxdt+ kx = 0 (5)where Fsis the driving force on the mass-spring system. The analogy be-tween the mass+spring system and the electric circuit are evident from theseequations: L acts like the mass, R like the damping and 1/C like the springconstant. The derivative of the source voltage acts like the driving force.Problem 1 – TransientsConsider a series LRC circuit connected to DC voltage of 1 Volt at time0. (A resistor with resistance R, an initially-uncharged capacitor with ca-pacitance C, an inductor with inductance L, an ideal battery of with voltage1V, and a switch are connected in series; the switch is open for t < 0 andclosed for t > 0.) Write a Mathematica code to find and plot the currentin the LRC circuit as a function of time. Run your code for various valuesof L, R and C to illustrate the cases of underdamped and overdamped circuits.Hint: in the underdamped (damped oscillation) case, Mathematica maygive you the solution in a form that contains complex exponentials insteadof a nicer form of trig function times exponential with no imaginary terms.To convert it, you may need to useFullSimplify[ExpToTrig[ ... ]]Problem 2 – ResonanceConsider an LRC circuit driven by a steady sinusoidal AC source with rmsamplitude 1 Volt. Find and plot the amplitude of the steady state currentin the circuit as a function of the ratio of the frequency ω of the AC sourceto the natural frequency ωnof the LRC circuit. That natural frequency isgiven byωn=q1/LC − (R/2L)2. (6)(Perhaps you are familiar with a simpler form ωn=q1/LC for the resonantfrequency; but the resonant frequency actually shifts somewhat in the pres-2ence of damping, according to the more accurate formula given here. Alsonote that the frequencies used here are the “angular frequency” measured inradians per second, as opposed to f = ω/(2 π), which is measured in cyclesper second (Hz).)Hint # 1: In principle, you can solve this using DSolve. That is somewhattricky, however, because you have to get rid of the transient terms in thesolution, which decay exponentially. Easier is to just assume that the answerwill have a current of the form i = A cos ω t and find the value for the am-plitude A that satisfies Eq. (2).Actually, instead of assuming i = A cos ω t, it might be better to assumei = A cos(ω t + φ). Allowing the additional phase φ here will let you assumethat the corresponding phase in Vsis equal to zero.Hint #2: You will need to get Mathematica to transform something of theform c1cos(ωt) + c2sin(ωt) into the form c3cos(ωt + c4). To help with that,see what happens when you runy = Cos[omega t + phi]z = TrigExpand[y]Coefficient[z,Cos[omega t]]Also plot the voltage across the capacitor and the voltage across the inductoron the same graph. Make separate graphs for situations where the appliedfrequency is above, below, and at the resonant frequency. Choose the otherparameters such that the damping is significant but not too
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