EECS 105 Spring 2004 Lecture 5 Lecture 5 Resonance Prof J S Smith Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 5 Prof J S Smith Context z z 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 Is this lecture we will discuss second order transfer functions circuits which have resonances Department of EECS University of California Berkeley 1 EECS 105 Spring 2004 Lecture 5 Prof J S Smith Second Order Circuits z z The series resonant circuit is one of the most important elementary circuits 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 Berkeley EECS 105 Spring 2004 Lecture 5 Prof J S Smith Time Domain analysis z The differential equations for this circuit are vr t ir t r ic t C vs t vL t vC t vR t dvC t dt vL t L diL t dt dv t d C C dv t dt vs L vC t RC C dt dt Department of EECS University of California Berkeley 2 EECS 105 Spring 2004 Lecture 5 z Prof J S Smith So the differential equation for the circuit is d2 dv t vs t LC 2 vC t vC t RC C dt dt z z Let s see how this circuit responds to a step input zero before time t 0 and Vdd for t 0 First of all note that the steady state solution is vs t Vdd Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 5 Prof J S Smith Transient solution z To find the transient solution for t 0 let s try a solution of the form vC t Ae st Vdd z Where s is a complex number Now we substitute this into our D E dv t d2 v t vC t RC C 2 C dt dt 2 st st Vdd LCs Ae Vdd Ae RCsAe st vs t LC z Giving us a second order equation for s 0 LCs 2 1 RCs Department of EECS University of California Berkeley 3 EECS 105 Spring 2004 Lecture 5 z Prof J S Smith We can use the quadratic formula to find solutions for s 0 LCs 2 1 RCs 2 s1 2 z z R R 1 2L 2 L LC If s is real then the circuit is overdamped and the voltage will change exponentially to its steady state value If s is complex the circuit is underdamped and the solution will oscillate around the steady state value before settling down to it Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 5 Prof J S Smith Underdamped case z For the underdamped case s1 2 z R 1 R j 2L LC 2 L 2 We will need to take sums of the complex exponentials to get real solutions solutions are of the form vC t Vdd Ae t sin 1t z 2 and Where 1 R LC 2 L R 2L and A and are determined by the boundary conditions Department of EECS University of California Berkeley 4 Response of underdamped circuit to step EECS 105 Spring 2004 Lecture 5 z Prof J S Smith If the circuit is moderately damped 2 1 75 1 5 1 25 1 0 75 0 5 0 25 0 5 1 1 5 2 2 5 Department of EECS 3 University of California Berkeley EECS 105 Spring 2004 Lecture 5 Prof J S Smith Underdamped response to step z And if very underdamped 2 1 75 1 5 1 25 1 0 75 0 5 0 25 0 5 Department of EECS 1 1 5 2 2 5 3 University of California Berkeley 5 EECS 105 Spring 2004 Lecture 5 Prof J S Smith Underdamped Oscillations z In the very underdamped case R small the ringing dies exponentially in a time 2L R And each oscillation takes a time 1 z 1 2 LC 1 R 2 1 2 2 LC 2 L z So the number of oscillations of ringing that will occur is approximately 1 L LC 2L N R 2 R C Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 5 Prof J S Smith Frequency domain analysis z With phasor analysis this circuit is readily analyzed for example the input impedance Z Z j L Z j L 1 j C R 1 R R j L 1 2 j C LC 1 You can also write the expression for the voltage across any component Department of EECS University of California Berkeley 6 EECS 105 Spring 2004 Lecture 5 Prof J S Smith Series LCR Impedance z For example the voltage across the capacitor Z 1 1 1 j L Vc I C Z C R VS j C j C Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 5 Prof J S Smith Low frequency behavior z At low frequencies the characteristic of this circuit is dominated by the capacitor The inductor looks like a short at low frequencies The in the denominator of the term for the capacitor makes it Z j L 1 R j C the major contribution Department of EECS University of California Berkeley 7 EECS 105 Spring 2004 Lecture 5 Prof J S Smith High frequency behavior z At high frequencies the characteristics of this circuit are dominated by the inductor The capacitor looks like a short at low frequencies The proportionality of the term for the inductor makes it Z j L 1 R j C the major contribution Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 5 Prof J S Smith Near resonance z Near resonance energy will oscillate between the capacitor and the inductor Notice that the terms for the capacitor and the inductor have 1 R opposite sign so they can add up Z j L j C to zero impedance at one frequency At that frequency 0 LC 1 2 energy is perfectly oscillating between the inductor and the capacitor The only impedance left at that frequency is the resistor Department of EECS University of California Berkeley 8 EECS 105 Spring 2004 Lecture 5 z Prof J S Smith At resonance 1 0 Im Z j L j C 2 1 LC Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 5 Prof J S Smith Power sidebar z 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 1 j t j t Ve V e 2 1 i t I e j t I e j t 2 V Z I v t Department of EECS University of California Berkeley 9 EECS 105 Spring 2004 Lecture 5 Prof J S Smith Convert from phasors then multiply z Power P v t i t 1 j t j t …
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