Introduction to Complex Impedance. Passive Low Pass Filter (1 st order) Passive High Pass Filter. Active Low Pass Filter. Active High Pass Filter

Transcription

1 Introduction to Complex Impedance for Passive Low Pass Filter ( st order) Passive High Pass Filter ( st order) Active Low Pass Filter ( st order) Active High Pass Filter ( st order) Active Low Pass Filter (2 nd order)

2 Lecture Schedule (Najera and Quinones) Date Najera Date Quinones 4/0 Complex Impedance AND Gate 4/7 Complex Impedance Ultrasonic Sensor 4/24 Quiz 5 Complex Impedance Review 4/3 Glucose Circuit con t Complex Impedance AND Gate 4/20 Quiz 5 Complex Impedance Ultrasonic Sensor Review 4/27 Exam 5/ Exam 5/4 No Lecture

3 Lab Schedule (Najera and Quinones) Date Najera Date Quinones 4/2 Ballistocardiograph 4/9-4/3 Ballistocardiograph 4/9 Ultrasonic 4/6-4/20 Ultrasonic 4/26 Ultrasonic 4/23-4/27 Ultrasonic 5/3 Ultrasonic-report due by end of lab Scorecard due 4/30-5/3 Ultrasonic-report due by end of lab Scorecard due

4 Homeworkdue 4/3 (Quinones), due 4/7 (Najera) P23, P24 Due 4/3 (Quinones) P23, P24 Due 4/7 (Najera)

5 Modeling Filters with Complex xx Numbers or Complex Impedance BRIEF EXPLANATION OF COMPLEX IMPEDANCE

6 Modeling Filters with Complex xx Numbers or Complex Impedance XY PLANE COMPLEX PLANE SINGLE SIGNAL

7 Modeling Filters with Complex xx Numbers or Complex Impedance XY PLANE COMPLEX PLANE MUTIPLE SIGNALS DIFFERENT AMPLITUDES AND PHASE SHIFTS

8 Modeling Filters with Complex Numbers or xx Complex Impedance If V and I are complex amplitudes of Voltage and Current, respectively. Complex impedance Z substitutes resistance R in the Ohm s Law Equation and has units of ohms ( ) then V t = Ve jwt i t = Ie jwt and complex impedance is defined as Z = R for resistors Z = V I Z = jwc for capacitors

9 Modeling Filters with Complex xx Numbers or Complex Impedance HOW TO USE KCL AND COMPLEX IMPEDANCE TO MODEL AN OP-AMP CIRCUIT

10 The type of filters examined xx in this course OP-AMP FILTERS EXAMINED IN THIS COURSE FIRST ORDER LOW PASS * HIGH PASS * BAND PASS * SECOND ORDER LOW PASS # *Active and Passive Filters # Only Active Filters

11 FIRST ORDER COMPLEX xx IMPEDANCE MODELS LOW PASS HIGH PASS PASSIVE = jwcr + = jwcr jwcr + ACTIVE = R 2 R (jwr 2 C + ) = R 2 R jwr C + jwr C SECOND ORDER BAND PASS LOW PASS = R 2 R (jwc 2 R 2 + ) = (jwrc + ) ACTIVE 2 jwr C + jwr C PASSIVE **Notice the common components for all types of low pass filters and for all types of high pass filters**

12 Modeling Filters with Complex xx Numbers or Complex Impedance PASSIVE FILTERS LOW AND HIGH PASS

13 Modeling Filters with Complex xx Numbers or Complex Impedance Vin R Vout C Vin Vout Z = R Z2 = jwc = jwcr + low pass filter Vin Vout Vin Vout Z = jwc Z 2 = R = jwcr jwcr + high pass filter

14 Modeling Filters with Complex xx Numbers or Complex Impedance ACTIVE FILTERS LOW AND HIGH PASS FILTERS

15 Modeling Filters with Complex xx Numbers or Complex Impedance C=nF LOW PASS FILTER R=25k R2=M Vout = R 2 R (jwcr 2 + ) HIGH PASS FILTER Vin R=25k C=nF R2=M Vout = R 2 /R jwcr + jwr C **BOTH ARE ACTIVE FILTERS**

16 Modeling Filters with Complex xx Numbers or Complex Impedance ACTIVE FILTERS LOW PASS FILTERS

17 First Order: Low Pass xxfilter w/gain FIRST ORDER: FOR EVERY DECADE OF FREQUENCY, THE AMPLITUDE DROPS 20 db = R 2 R jwcr 2 + WHAT IS THE VALUE OF fc FOR THIS FILTER? WHAT IS THE GAIN? The gain of the Inverting Op-Amp is equal to Gain = R 2 R Low Pass Filter Vin R C R2 0 F 0 k w = angular frequency = 2 f k Vout Vref

18 First Order: Low Pass xxfilter w/gain FIRST ORDER: FOR EVERY DECADE OF FREQUENCY, THE AMPLITUDE DROPS 20 db Where the gain of the Op-Amp is equal to Wave Generator: f = Hz A = 0. V Offset = 2.5 V Low Pass Filter fc =.59 Hz C R2 = R 2 R jwcr 2 + Gain = R 2 20 db R decade Vin R Vout w = angular frequency = 2 f NETWORK ANALYZER Vref

19 First Order: Low Pass xxfilter w/gain FIRST ORDER: FOR EVERY DECADE OF FREQUENCY, THE AMPLITUDE DROPS 20 db fc =.59 Hz = R 2 R jwcr 2 + w = angular frequency = 2 f MATLAB Code: >> w=logspace(-3,5,00); >> j=sqrt(-); >> R=E3; >> R2=0E3; >> C=0E-6; >> Vout=(-R2/R)*./(+j*w*R2*C); >> loglog(w/2/pi,abs(vout)) Y = logspace(a,b,n) generates n poi nts between decades 0^a and 0^b Converts w in radians/sec to f in Hz since w = 2 f or f = w/2 Creates a loglog plot of Vout vs f **NOTICE THE MATLAB CODE FOR MATHMATICAL OPERATIONS INVOLVING MATRICES**

20 First Order: Low Pass xxfilter / Plot FIRST ORDER: FOR EVERY DECADE OF FREQUENCY, THE AMPLITUDE DROPS 20 db >> w=logspace(-3,3,00); >> j=sqrt(-); >> R=E3; >> R2=0E3; >> C=0E-6; >> Vout=(-R2/R)*./(+j*w*R2*C); >> loglog(w/2/pi,abs(vout)) = R 2 R jwcr 2 + w = angular frequency = 2 f fc =.59 Hz MATLAB Y = decadelogspace(a,b,n) generates n poi nts between decades 0^a and 0^b decade Converts w in radians/sec to f in Hz since w = 2 f or f = w/2 Creates a loglog plot of w vs Vout

21 Modeling Filters with Complex xx Numbers or Complex Impedance ACTIVE FILTERS SECOND ORDER LOW PASS FILTERS

22 Second Order: Low Pass xx Filter SECOND ORDER: FOR EVERY DECADE OF FREQUENCY, THE AMPLITUDE DROPS 40 db 0 k 0 k 0 F = jwcr fc =? 0 F w = angular frequency = 2 f fc =.59 Hz

23 Second Order: Low Pass xx Filter SECOND ORDER: FOR EVERY DECADE OF FREQUENCY, THE AMPLITUDE DROPS 40 db f =200 mhz A = V Offset = 2.5 V 0 k 0 k 0 F = jwcr f =.59 Hz A = V Offset = 2.5 V fc =? 0 F w = angular frequency = 2 f

24 Second Order: Low Pass xx Filter SECOND ORDER: FOR EVERY DECADE OF FREQUENCY, THE AMPLITUDE DROPS 40 db 0 k 0 k 0 F = jwcr db decade fc =? 0 F w = angular frequency = 2 f NETWORK ANALYZER