EE40 Lec 13 Filters and Resonance Prof Nathan Cheung 10 13 2009 Reading Hambley Chapter 6 6 6 8 Chapter 14 10 14 5 EE40 Fall 2009 Slide 1 Prof Cheung Common Filter Transfer Function vs Freq H f H f Low Pass High Pass Frequency Frequency H f H f B dR Band Reject j t B d Pass Band P Frequency EE40 Fall 2009 Frequency Slide 2 Prof Cheung Filters Passive resonant circuits that contain only passive elements namely resistors R capacitors C and inductors L Active resonant circuits that contain op amps transistors amps transistors and or other active devices in addition to the passive elements EE40 Fall 2009 Slide 3 Prof Cheung Filter Order The order of a filter is equal to the absolute value of the highest power of in its transfer function EE40 Fall 2009 Slide 4 Prof Cheung First Order Filter Circuits High Pass VS R C Low Pass Low Pass VS R L High Pass HR R R 1 j C HR R R j L HC 1 j C R 1 j C HL j L R j L EE40 Fall 2009 Slide 5 Prof Cheung First Order Low Pass Filter EE40 Fall 2009 Slide 6 Prof Cheung Second Order RLC Filter Circuits VS EE40 Fall 2009 Band Pass Z R 1 j C j L R HBP R Z Low Pass High Pass C Band HLP 1 jj C Z Reject HHP j L Z L HBR HLP HHP Slide 7 Prof Cheung Second Order BandPass Filter Write the expression for VR Now find HBP VR j RC H BP 2 VS 1 LC j RC EE40 Fall 2009 Slide 8 Prof Cheung Passive Filter Bandpass Linear Linear plots VR j RC H BP VS 1 2 LC j RC EE40 Fall 2009 Slide 9 Prof Cheung Resonance Frequency Defined as frequency when the total impedance is purely l resistive i ti i e i zero imaginary i i component t j Z R j L C Therefore EE40 Fall 2009 Slide 10 Prof Cheung The Quality Factor Q The Quality Factor Q characterizes the degree of selectivity of the circuit circuit It is determined by dB For a bandpass filter Notice that Q depends on the resonant frequency Linear scale EE40 Fall 2009 Slide 11 Prof Cheung Resonance Bandwidth fo 1 2 This is a Linear Linear plot EE40 Fall 2009 Slide 12 At resonance with large Q VL and VC VR Prof Cheung Another Way to write H f See Hambley text Hc f Hc f R R 1 j L j C 1 1 1 1 L j j RC R o L 1 QS R o RC 0 1 jQs 0 2 Hc f EE40 Fall 2009 1 f0 f 1 Qs f0 f 1 o LC 2 2 Slide 13 Prof Cheung To Generate the Bode Plots Low f f0 1 1 2 dominates H c f Hc f 2 f0 f f0 jQs Qs f f 2 y 10 log H c f K 20 log f Slope 20dB dec H c f 90 High f f 1 2 dominates H c f Hc f f f0 jQs f0 f0 Q f s 2 2 y 10 log H c f K 20 log f Slope 20dB dec H c f 90 2 at f f 0 H c f 1 H c f 1 y 0dB H c f 0 EE40 Fall 2009 Slide 14 Prof Cheung Bode Plots 20dB dec EE40 Fall 2009 Slide 15 20dB dec Prof Cheung Second Order LowPass Filter VC 1 H LP VS 1 2 LC j RC M LP 1 1 Q 2 2 2 1 2 0 1 o LC 2 EE40 Fall 2009 Q 0 L 1 o o RC R Slide 16 Prof Cheung Another Way to write H f See Hambley 1 1 j C j RC Hc f 1 1 L j L 1 j R j C j RC R L 1 1 1 Hence 0 L define 0 Qs Let 0 2 LC R R 0C 0 C j j 1 1 L j 0 L j 0 0 Qs Qs 0 R 0 R j RC j 0 RC 0 Hc f 1 jQs 0 0 jQs 2 Hc f EE40 Fall 2009 f0 Q s f 2 f f 1 Qs 0 f0 f 2 Slide 17 Prof Cheung Second Order HighPass Filter VL 2 LC H HP VS 1 2 LC j RC M HP EE40 Fall 2009 0 2 1 2 2 2 2 1 0 Q 0 Slide 18 Prof Cheung Second Order Bandreject Filter VL VC H BR 1 H BP VS EE40 Fall 2009 Slide 19 Prof Cheung Parallel LC Circuit See Hambley 6 7 1 Zp 1 1 j C R j L Iin I Z 1 1 H f 0 p R I0 R 1 1 1 j RC R j L j C R j L R j L R L j 0 R j j 0 Q p j RC j 0 RC Qp 0 L 0 0 Hc f 1 1 jQ p f f 0 f0 f R RC L 1 2 Hc f Qp f f 1 Q p 0 f0 f EE40 Fall 2009 2 Slide 20 Prof Cheung C Active Filters Contain less components no inductors Transfer function that is insensitive to component tolerance or load variations Easily E il adjusted dj t d Allow a wide range of useful transfer functions EE40 Fall 2009 Slide 21 Prof Cheung Single Pole Lowpass Filter 1 R f j C Vout f H LP Vs Rs Rf Rs 1 1 j R C f f 1 Rf 1 H LP GLP GLP LP 1 j Rs Rf C f LP EE40 Fall 2009 Slide 22 Prof Cheung Single Pole Highpass Filter Vout Zf Rf H HP Vs Zs R s j Cs j HP H HP G HP 1 j HP EE40 Fall 2009 GHP Slide 23 Rf Rs HP 1 Rs Cs Prof Cheung Cascaded Active Filters EE40 Fall 2009 Slide 24 Prof Cheung Cascaded Active Filters EE40 Fall 2009 Slide 25 Prof Cheung Cascaded Active Filters EE40 Fall 2009 Slide 26 Prof Cheung Cascaded Active Filters EE40 Fall 2009 Slide 27 Prof Cheung Cascaded Active Filters EE40 Fall 2009 Slide 28 Prof Cheung Cascaded Active Filters EE40 Fall 2009 Slide 29 Prof Cheung Appendix For reference only See Hambley 14 5 The open loop gain of an Op Amp decreases with frequency The unity gain bandwidth ft several MHz for typical op amps EE40 Fall 2009 Slide 30 Prof Cheung Appendix For reference only Circuit supposed to have a closed lop gain of 10 20db Given ft 4MHz G circuit can operate with f 400kHz EE40 Fall 2009 Slide 31 Prof Cheung
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