1Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5Lecture 5: Resonance Prof. J. S. SmithDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithContextz In the last lecture, we discussed simple zeros and poles of a transfer function, and creating approximate Bode plots for hand plotting of phase and magnitude characteristics.z Is this lecture, we will discuss second order transfer functions, circuits which have resonances.2Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithSecond Order Circuitsz The series resonant circuit is one of the most important elementary circuits:z This model is not only useful for physical LCR circuits, but also approximates mechanical resonances, molecular resonance, microwave cavities, transmission lines, buildings, bridges, …Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithTime Domain analysisz The differential equations for this circuit are:dttdiLtvLL)()( =rtitvrr⋅=)()(dttdvCtiCc)()( =dttdvRCtvdtdttdvCdLvtvtvtvtvCCCsRCLs)()()()()()()(++⎟⎠⎞⎜⎝⎛=++=3Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. Smithz So the differential equation for the circuit is:z Let’s see how this circuit responds to a step input, zero before time t=0, and Vddfor t>0z First of all, note that the steady state solution is dttdvRCtvtvdtdLCtvCCCs)()()()(22++=ddsVtv=∞→ )(Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithTransient solutionz To find the transient solution, for t>0, let’s try a solution of the form:– Where s is a complex numberz Now we substitute this into our D. E.:z Giving us a second order equation for s:ddstCVAetv +=)(ststddstddCCCsRCsAeAeVAeLCsVdttdvRCtvtvdtdLCtv+++=++=222)()()()(RCsLCs ++= 1024Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. Smithz We can use the quadratic formula to find solutions for s: →z If s is real, then the circuit is overdamped, and the voltage will change exponentially to its steady state value.z If s is complex, the circuit is underdamped, and the solution will oscillate around the steady state value before settling down to it.RCsLCs ++= 102⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛±=LCLRLRs12222,1Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithUnderdamped casez For the underdamped case:z We will need to take sums of the complex exponentials to get real solutions, solutions are of the form:z Where: andand A and are determined by the boundary conditions22,1212⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛±=LRLCjLRs)sin()(1φωα++=−tAeVtvtddC221⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛=LRLCωLR2=αφ5Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithResponse of underdamped circuit to stepz If the circuit is moderately damped:0.5 1 1.5 2 2.5 30.250.50.7511.251.51.752Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithUnderdamped response to stepz And if very underdamped:0.5 1 1.5 2 2.5 30.250.50.7511.251.51.7526Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithUnderdamped Oscillations:z In the very underdamped case (R small), the ringing dies exponentially in a time:z And each oscillation takes a time:z So the number of oscillations of ringing that will occur is approximately:RL21=−απππω2212)2(121LCLRLC≈⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛=−−CLRLCRLNππ122=≈Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithFrequency domain analysisz With phasor analysis, this circuit is readily analyzed, for example, the input impedance: You can also write the expression for the voltage across any componentRCjLjZ ++=ωω1⎟⎠⎞⎜⎝⎛−+=++=LCLjRRCjLjZ2111ωωωωZ7Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithSeries LCR Impedancez For example, the voltage across the capacitor:SCCcVRCjLjCjZIV111−⎟⎟⎠⎞⎜⎜⎝⎛++==ωωωZDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithLow frequency behaviorz At low frequencies, the characteristic of this circuit is dominated by the capacitorThe inductor looks like a shortat low frequenciesThe ω in the denominator of theterm for the capacitor makes it the major contributionRCjLjZ ++=ωω18Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithHigh frequency behaviorz At high frequencies, the characteristics of this circuit are dominated by the inductorThe capacitor looks like a short at low frequenciesThe ω proportionality of theterm for the inductor makes it the major contributionRCjLjZ ++=ωω1Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithNear resonancez Near resonance, energy will oscillate between the capacitor and the inductorNotice that the terms for thecapacitor and the inductor haveopposite sign, so they can add upto zero impedance at one frequency At that frequency (ω0=[LC] -1/2) , energy is perfectlyoscillating between the inductor and the capacitor,→The only impedance left at that frequency is the resistor.RCjLjZ ++=ωω19Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. Smithz At resonanceLC12=ω01]Im[ =⎟⎟⎠⎞⎜⎜⎝⎛+=CjLjZωωDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithPower sidebarz The power into a circuit is just the voltage times the current, but remember this is not linear, so we’ll go back to complete notation:ZIVeIeItieVeVtvtjtjtjtj=+=+=−−ˆˆ)*ˆˆ(21)()*ˆˆ(21)(ωωωω10Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 5 Prof. J. S. SmithConvert from phasors, then multiplyz
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