# Today. 1/25/11 Physics 262 Lecture 2 Filters. Active Components and Filters. Homework. Lab 2 this week

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1 /5/ Physics 6 Lecture Filters Today Basics: Analog versus Digital; Passive versus Active Basic concepts and types of filters Passband, Stopband, Cut-off, Slope, Knee, Decibels, and Bode plots Active Components and Filters Review basics of OpAmps First Order Active Filters OpAmps with complex analysis Transfer Functions Bode Plot of Active Filters Homework Reading (up to 4.07) and by //. (p. 38),. (p. 39),.4(p.40, don t need phasors), and.6 (page 4) and problems on next slide Lab this week Lab and Lab a notebooks due Friday /4/ at 0am. Work on lab a experiment (not PSpice) and Lab a meeting Do Lab a pre-lab BEFORE lab meeting on Thursday /7/ Physics 6: Laboratory Electronics II Spring 0 Lect Page

2 Homework Passive RC Filter Write an expression for the complex transfer function (Vout(f)/Vin(f)) in terms of R, C, and f. Write an expression for the magnitude of the transfer function in terms of R and C. Draw a sketch of it with axis label with numbers for key values. Write an expression for the phase of the transfer function. What are values of the magnitude and phase of the transfer function at the cutoff frequency? Select a value for C so the cutoff frequency is 3.0kHz. Passive RL Filter Repeat but substitute an inductor L for the capacitor C C Find value of L for a cutoff frequency of 600 Hz. First-Order Active RC filter Repeat steps for RC Filter above, R Vin but for active filter circuit to the right. 5.0k Write expressions in terms of C, R, and R. Select a value for C so the cutoff frequency is 6.0kHz. Physics 6: Laboratory Electronics II Spring 0 Lect Page Vin 7Vac 0Vdc V k 0 R 0k R C Vout Vout

3 Filters Filters: An electrical filter is a device designed to pass a certain group of signals or suppress other groups of signals from a collection of signals. Analog versus Digital Filters Analog Filters: Process real continuous, analog signals. Digital Filters: Numerically process signals that have been discretely sampled and digitized. Passive versus Active Analog Filters Passive Filters: Filters implemented with resistors, capacitors, and inductors. Gain is less than or equal to one. Active Filters: Filters implemented with resistors, capacitors, inductors, and active devices such as operational amplifiers or transistors. Can have gain greater than one. Physics 6: Laboratory Electronics II Spring 0 Lect Page 3

4 Basic Filter Concepts Frequency dependent impedances R C R Low Frequencies: C-> infinity circuit => R High Frequencies: C->0 circuit => R R ( Z < R ) R L R Low Frequencies: L-> 0 circuit => R R ( Z < R ) High Frequencies: L->infinity circuit =>R Physics 6: Laboratory Electronics II Spring 0 Lect Page 4

5 Types of Ideal Filters Transfer Functions of Ideal Filters T(f) T(f) T(f) T(f) Passband(s): Frequency that pass weakly attenuated or have gain. Stopband(s): Frequencies that are strongly attenuated. Real World Filters: No sharp cutoffs; gain rolls off, stopband G>0. Physics 6: Laboratory Electronics II Spring 0 Lect Page 5

6 Passive Filter: Generalized Voltage Divider V ( ) s V s Z ( ) 0 Z ( ) Transfer Function V out ( ) V T f V G ( ) ( ) out( ) Vs( ) Z( ) Z( ) V Gain (or attenuation) Physics 6: Laboratory Electronics II Spring 0 Lect Page 6 Z out V s( ) Z( ) Z( ) Z ( ) Z ( ) ( f) ( ) f /( ) Z( f) Z( f)

7 Passive RC Filter Analysis Vin R Z 0 f C Z Vout T ( f) ( / ( jc)) R( / ( jc)) jrc RC exp j tan ( RC) T ( f) exp( j ( f)) Mag T ( f) Phase ( f / f ) ( f) tan f / f c c Cut-off frequency f C RC Physics 6: Laboratory Electronics II Spring 0 Lect Page 7

8 Filter Gain Characterization Pass Band Transition 40 db Stop Band Ripple PBG Knee Band Width 3 order = slope = Vac 0Vdc V R 0k C 0.u f 0 c 3 Gain-Frequency plot (log-log) of the Bode Diagram Physics 6: Laboratory Electronics II Spring 0 Lect Page 8

9 Logarithmic Nomenclature Define Decibel: db T(f) V P 0log 0log 0 0 V P Factor db Multipliers on a Log Axis Name Ratio Octave : Decade 0: Physics 6: Laboratory Electronics II Spring 0 Lect Page 9

10 Bode Diagrams Bode gain diagram: Log-Log plot of T(f) vs f Log(f) = x-axis Log( T(f) ) = y-axis Bode phase diagram: Log-Lin Plots of Phase(f) vs f Log(f) = x-axis Phase(f) = y-axis (Linear) Quick analysis of filter behavior: Filter type Pass band gain Cut-off frequency, aka; Corner or Knee frequency 3dB down point ( T(f) =/sqrt() Stop band ( skirt ) slope (order) Phase shift R 0k Vs 4.5Vpp 0 C 0.05u Physics 6: Laboratory Electronics II Spring 0 Lect Page 0

11 R Bode Diagram: Phase Vs 0k C v out As 0, Z C 4.5Vpp 0.05u v out v S (in phase) 0 As, Z 0 C Since v i R v, R R S i i and is in phase with v But, v is -90 out of phase with i C C R S So, v is -90 out of phase with v. out Physics 6: Laboratory Electronics II Spring 0 Lect Page C S

12 Multiple-Order (Passive) Filter.0V 00mV 3dB down R k R m=3 k R3 m= k 0mV Vac 0Vdc.0mV V R C 0.u R R3 V C 0.u V C3 0.u V 00uV Vac 0Vdc V k C 0.u V k C 0.u k V C3 0 0.u V 0uV 0Hz 00Hz.0KHz 0KHz 00KHz.0MHz 0 V(R:) V(R:) V(R3:) Frequency Physics 6: Laboratory Electronics II Spring 0 Lect Page

13 Cascaded Passive Filters Cascaded passive filters increase the order of the total filter. Order = number of cascaded first order filters Slope in stopband = order of filter Each section s input impedance will load previous section. Degrades response and load capacity No gain, only loss Gain (voltage or current) has to included separately Could introduce buffers (op-amps) as integral part of filter. Physics 6: Laboratory Electronics II Spring 0 Lect Page 3

14 Passive and Active Filters Passive Filter R k V AC = 0V V C 0.u Equivalent OpAmp Filter/ First order Active Filter C 0.u n R k -Vcc 0 Vin R k LM74-4 V- OS OUT 6 Vout U OS V Vcc Physics 6: Laboratory Electronics II Spring 0 Lect Page 4

15 + - Operational Amplifiers Review OpAmps Basics Open Loop: Closed Loop: OpAmps in common feedback circuits Physics 6: Laboratory Electronics II Spring 0 Lect Page 5

16 First Order Active Filter Analysis V i i i V 0 V Z out in 0, so V i Z ( f) out C Z0.u n R k -Vcc ZR LM74 4 V- Vin - OS k 6 OUT OS U V+ Z( f) V ( f ) Z 0 +Vcc Vout in Z f ( f) T f Z f Physics 6: Laboratory Electronics II Spring 0 Lect Page 6 ( ) ( ) ( )

17 Z First Order Active Filter Analysis T ( f) R jcr jc R R jc R V ( f) Z jr C R ( ) ( / ) out V in f Z R R j c C RC Physics 6: Laboratory Electronics II Spring 0 Lect Page 7

18 First Order Active Filter Analysis - II V ( f) R V T f j f V in( f) R j/ c V in out out ( ) exp( ( )) T ( f) R R ( f / f ) c f C RC 0 ( ) 80 tan ( / ) f f f c Physics 6: Laboratory Electronics II Spring 0 Lect Page 8

19 Gain-Frequency for First Order Active Filter 0 Knee Slope= Gain 0. PBG f C k 0k 00k M Frequency(Hz) PBG = Gain Pass Band Physics 6: Laboratory Electronics II Spring 0 Lect Page 9

20 Impedance Converters Optional for fun slides Active Circuits can be used to Invert Impedance An applied voltage must sink current Convert capacitor into an inductor Make hard to manufacture inductors out of cheap capacitors Physics 6: Laboratory Electronics II Spring 0 Lect Page 0

21 Negative-Impedance Converter Z in = -Z + R - Z R Z in = -Z NIC Z Physics 6: Laboratory Electronics II Spring 0 Lect Page

22 Gyrator Z in = R /Z R NIC R NIC R Z Z in = R /Z Gyrator Z If Z=/(jC), Then Z in = jcr A capacitor (plus op amps) can act as an inductor L=CR Physics 6: Laboratory Electronics II Spring 0 Lect Page

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