Department of EECS EECS 105 Fall 2003 Lecture 4 Prof Niknejad University of California Berkeley Lecture 4 Resonance Lecture Outline Series impedance and resonance Voltage transfer function bandpass filter Bode plots for second order circuits Prof A Niknejad University of California Berkeley Some comments questions about Bode plots Second order circuits Department of EECS z z EECS 105 Fall 2003 Lecture 4 Department of EECS University of California Berkeley Why do we factor the transfer function into terms involving j Why does the arctangent plot look so funny z z You ve been sloppy with signs what s the deal Why does the Bode plot of a simple pole or zero always have a slope of 20 dB dec regardless of the break frequency Good Questions Prof A Niknejad z z EECS 105 Fall 2003 Lecture 4 Bode Plot Question 1 Prof A Niknejad On a log scale this term has a fixed slope of 20 dB decade University of California Berkeley On a log log scale all straight lines have the same slope the slope gets translated into an intercept shift This term is a constant 20 log 1 j 20 log 20 log 20 log Note that the slope of a pole or zero is independent of the break point Department of EECS z z EECS 105 Fall 2003 Lecture 4 Bode Plot Question 2 Prof A Niknejad Which one is right University of California Berkeley 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 K 1 j z1 1 j z 2 L 1 j zn H G0 j 1 j p 2 1 j p 2 L 1 j pm Why do you sometimes use a positive sign and other times a negative sign in the transfer function Department of EECS z z z z EECS 105 Fall 2003 Lecture 4 Bode Plot Question 3 Linear Scale Prof A Niknejad University of California Berkeley Log Scale I know what an arctan looks like and it looks nothing like what you showed us Department of EECS z EECS 105 Fall 2003 Lecture 4 Log Scales Prof A Niknejad University of California Berkeley Log scales move forward non uniformly Department of EECS z EECS 105 Fall 2003 Lecture 4 Why j Prof A Niknejad LTI System H H j e j t University of California Berkeley 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 e j t 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 Department of EECS z z z z EECS 105 Fall 2003 Lecture 4 Second Order Circuits Prof A Niknejad University of California Berkeley The physics describes not only physical LCR circuits but also approximates mechanical resonance mass spring pendulum molecular resonance microwave cavities transmission lines buildings bridges The series resonant circuit is one of the most important elementary circuits Department of EECS z z EECS 105 Fall 2003 Lecture 4 Series LCR Impedance j C 1 R 1 Im Z L 1 2 0 LC 1 1 LC 2 1 Z j L R R j L 1 2 j C LC Z j L Z Prof A Niknejad University of California Berkeley With phasor analysis this circuit is readily analyzed Department of EECS z EECS 105 Fall 2003 Lecture 4 Resonance Prof A Niknejad VR Vs VL VC Vs VR 0 VC VL VR Vs 0 VC VL VR Vs University of California Berkeley 0 VC VL 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 Department of EECS z z z EECS 105 Fall 2003 Lecture 4 0 L VC VR jQ Vs 1 University of California Berkeley Vs 0 L Vs VC I j 0 L j 0C Z 0 j R jQ Vs LC 1 L 1 Z0 1 1 Q R 0C R C R C R R VL Vs Vs VL I j 0 L j 0 L j 0 L Z 0 R Note that at resonance the voltage across the inductor and capacitor can be larger than the voltage in the resistor Department of EECS z Prof A Niknejad Series Resonance Voltage Gain EECS 105 Fall 2003 Lecture 4 Vo V0 R H j Vs j L 1 R j C University of California Berkeley One zero at DC frequency can t conduct DC due to capacitor V0 j CR H j Vs 1 2 LC j RC To find the poles zeros let s put the H in canonical form So we have Department of EECS z z z Prof A Niknejad Second Order Transfer Function EECS 105 Fall 2003 Lecture 4 H j Q 0 2 j j 2 0 j Q 0 R University of California Berkeley Q 0 L V0 j CR H j 2 R 1 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 R j L Denominator is a quadratic polynomial Department of EECS z Prof A Niknejad Poles of 2nd Order Transfer Function EECS 105 Fall 2003 Lecture 4 Finding the poles 02 2 0 Q 0 02 0 0 Q R Poles are complex conjugate frequencies The Q parameter is called the quality factor or Q factor This is an important parameter Im Re Prof A Niknejad University of California Berkeley 1 j 0 1 2 4Q 2 2Q 4Q 2Q 0 j j 2 0 Let s factor the denominator Department of EECS z z z z EECS 105 Fall 2003 Lecture 4 University of California Berkeley 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 2 2 0 0 0 j 0 2 2Q 4 Q Q Re Prof A Niknejad The transfer function can be parameterized in terms Im of loss First take the lossless case R 0 Resonance without Loss Department of EECS z z z z EECS 105 Fall 2003 Lecture 4 Magnitude Response H j 0 1 0 Q 100 Q 10 j Q 1 H j 0 Q Q 0 1 Prof A Niknejad University of California Berkeley 02 02 j 0 j 02 0 R j 0 0 L Q H j R 02 2 j 0 02 2 j 0 0 L Q The response peakiness depends on Q Department of EECS H 0 0 z EECS 105 Fall 2003 Lecture 4 How Peaky is it 2 0 2 …
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