Lecture 4: ResonanceLecture OutlineGood Questions…Bode Plot Question #1Bode Plot Question #2Bode Plot Question #3Log ScalesWhy jω?Second Order CircuitsSeries LCR ImpedanceResonanceSeries Resonance Voltage GainSecond Order Transfer FunctionPoles of 2nd Order Transfer FunctionFinding the poles…Resonance without LossMagnitude ResponseHow Peaky is it?Half Power Frequencies (Bandwidth)More “Notation”Second Order Circuit Bode PlotBode Plot: Damped CaseUnderdamped CaseUnderdamped Mag PlotUnderdamped PhasePhase Bode PlotBode Plot GuidelinesEnergy Storage in “Tank”Energy Dissipation in TankPhysical Interpretation of Q-Factorthin-Film Bulk Acoustic Resonator (FBAR) RF MEMSDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4Lecture 4: ResonanceProf. NiknejadDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadLecture OutlineSome comments/questions about Bode plotsSecond order circuits:–Series impedance and resonance–Voltage transfer function (bandpass filter)–Bode plots for second order circuitsDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadGood Questions…Why does the Bode plot of a simple pole or zero always have a slope of 20 dB/dec regardless of the break frequency?You’ve been sloppy with signs, what’s the deal?Why does the arctangent plot look so funny?Why do we factor the transfer function into terms involving jω?Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadBode Plot Question #1Note that the slope of a pole or zero is independent of the break point:On a log-log scale, all straight lines have the same slope … the slope gets translated into an intercept shift!log20log20log201log20 jOn a log scale, this term has a fixed slope of 20 dB/decadeThis term is a constantDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadBode Plot Question #2Why do you sometimes use a positive sign and other times a negative sign in the transfer function?The plus sign is right! For “passive” circuits, the poles are all in the LHP (left-half plane). A simple RC circuit has:Otherwise the circuit has a negative resistor!We can synthesize negative resistance with active circuits…)1()1)(1()1()1)(1()()(22210pmppznzzKjjjjjjjGHWhich one is right?0RCDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadBode Plot Question #3I know what an arctan looks like and it looks nothing like what you showed us!Linear Scale Log ScaleDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadLog ScalesLog scales move forward non-uniformly…Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadWhen we factor our transfer functions, why do we always like to put things in terms of jω, as opposed to say ω? Recall that we are trying to find the response of a system to an exponential with imaginary argument:Real sinusoidal steady-state requires the argument to be imaginary. We must therefore only consider the transfer functions for such values …If this doesn’t make sense, hang in there!Why jω?LTI SystemHtje)()(tjejHDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadSecond Order CircuitsThe series resonant circuit is one of the most important elementary circuits:The physics describes not only physical LCR circuits, but also approximates mechanical resonance (mass-spring, pendulum, molecular resonance, microwave cavities, transmission lines, buildings, bridges, …)Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadSeries LCR ImpedanceWith phasor analysis, this circuit is readily analyzed RCjLjZ 1LCLjRRCjLjZ2111011]Im[2LCLZLC12ZDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadResonanceResonance occurs when the circuit impedance is purely real Imaginary components of impedance cancel outFor a series resonant circuit, the current is maximum at resonance+VR−+ VL –+ VC –+Vs−VLVCVRVs0VLVCVRVs0VCVLVRVs0Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadSeries Resonance Voltage GainNote that at resonance, the voltage across the inductor and capacitor can be larger than the voltage in the resistor:+VR−+ VL – + VC –sssLVjQLjRVLjZVLjIV0000)(sssCVjQLjRVjLZVCjIV0000)(1RZRCLRCLCRCRLQ0001111Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadSecond Order Transfer FunctionSo we have:To find the poles/zeros, let’s put the H in canonical form:One zero at DC frequency can’t conduct DC due to capacitorRCjLjRVVjHs1)(0+Vo−RCjLCCRjVVjHs201)(Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadPoles of 2nd Order Transfer FunctionDenominator is a quadratic polynomial:LRjjLCLRjRCjLCCRjVVjHs220)(11)(LRjjLRjjH220)()(LC120QjjQjjH02200)()(RLQ0Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadFinding the poles…Let’s factor the denominator:Poles are complex conjugate frequenciesThe Q parameter is called the “quality-factor” or Q-factorThis is an important parameter:ReIm0)(2002QjjQjQQQ 4112420020200 0RQDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 4 Prof. A. NiknejadResonance without LossThe transfer function can be parameterized in terms of loss. First, take the lossless case, R=0:When the circuit is lossless, the poles are at real frequencies, so the transfer function blows up!At this
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