EECS 105 Fall 2003 Lecture 4 Lecture 4 Resonance Prof Niknejad Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Lecture Outline Some comments questions about Bode plots Second order circuits Series impedance and resonance Voltage transfer function bandpass filter Bode plots for second order circuits Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Good 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 j Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Bode Plot Question 1 Note that the slope of a pole or zero is independent of the break point 20 log 1 j 20 log 20 log 20 log This term is a constant On a log scale this term has a fixed slope of 20 dB decade On a log log scale all straight lines have the same slope the slope gets translated into an intercept shift Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Bode Plot Question 2 Why do you sometimes use a positive sign and other times a negative sign in the transfer function 1 j z1 1 j z 2 1 j zn H G0 j 1 j p 2 1 j p 2 1 j pm K Which one is right The plus sign is right For passive circuits the poles are all in the LHP left half plane A simple RC circuit has RC 0 Otherwise the circuit has a negative resistor We can synthesize negative resistance with active circuits Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Bode Plot Question 3 I know what an arctan looks like and it looks nothing like what you showed us Linear Scale Department of EECS Log Scale University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Log Scales Log scales move forward non uniformly Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Why j When we factor our transfer functions why do we always like to put things in terms of j as opposed to say Recall that we are trying to find the response of a system to an exponential with imaginary argument e j t LTI System H H j e j t 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 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Second Order Circuits The 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 Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Series LCR Impedance With phasor analysis this circuit is readily analyzed Z 1 Z j L R j C 1 1 Z j L R R j L 1 2 j C LC 1 Im Z L 1 2 0 LC Department of EECS 1 LC 2 University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Resonance Resonance occurs when the circuit impedance is purely real Imaginary components of impedance cancel out For a series resonant circuit the current is maximum at resonance VL V C VL Vs VR VC VR Vs 0 Department of EECS VL VL VR Vs Vs VC VR VC 0 0 University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Series Resonance Voltage Gain Note that at resonance the voltage across the inductor and capacitor can be larger than the voltage in the resistor VL Vs Vs VL I j 0 L j 0 L j 0 L Z 0 R VC VR jQ Vs Vs 0 L Vs 1 VC I j 0 L j 0C Z 0 j R jQ Vs Department of EECS 0 L 1 1 LC 1 L 1 Z0 Q R 0C R C R C R R University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Second Order Transfer Function So we have Vo V0 R H j Vs j L 1 R j C To find the poles zeros let s put the H in canonical form V0 j CR H j Vs 1 2 LC j RC One zero at DC frequency can t conduct DC due to capacitor Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Poles of 2nd Order Transfer Function Denominator is a quadratic polynomial R j L V0 j CR H j 2 1 R Vs 1 LC j RC 2 j j LC L R j 1 2 L H j 0 R LC 02 j 2 j L j H j Department of EECS 0 Q 0 j j Q 2 0 2 0 L Q R University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Finding the poles Let s factor the denominator 0 j j 02 0 Q 2 0 02 0 1 2 0 j 0 1 2Q 4Q 2Q 4Q Poles are complex conjugate frequencies The Q parameter is called the quality factor or Q factor This is an important parameter Im Re Q R 0 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Resonance without Loss The transfer function can be parameterized in terms Im of loss First take the lossless case R 0 2 2 0 0 0 j 0 2Q 4 Q Q Re When the circuit is lossless the poles are at real frequencies so the transfer function blows up At this resonance frequency the circuit has zero imaginary impedance and thus zero total impedance Even if we set the source equal to zero the circuit can have a steady state response Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad Magnitude Response The response peakiness depends on Q 0 R j 0 0 L Q H j R 02 2 j 0 02 2 j 0 0 L Q j H j 0 1 Q 1 H 0 0 Q 10 H j 0 02 j Q 02 02 j 0 0 Q 1 Q 100 Department of EECS 0 University of California Berkeley EECS 105 Fall 2003 Lecture 4 Prof A Niknejad How Peaky is it Let s find the points when the transfer function squared has dropped in half 2 0 Q 1 2 H j 2 2 0 2 2 2 0 Q 1 1 2 H j 2 2 …
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