Lecture Week 10. Quiz #7 Op-Amps II Homework P17 and P18 Active Filters Active Filters Analysis Active Filters Bode Plots Filter Design Homework

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1 Lecture Week 10 Quiz #7 Op-Amps II Homework P17 and P18 Active Filters Active Filters Analysis Active Filters Bode Plots Filter Design Homework

2 Quiz 7 Op-Amp II(20 pts.) Please clear desks and turn off phones and put them in back packs You need a pencil, straight edge and calculator 20 minutes Keep eyes on your own paper Follow the same format as for homework

3 Homework - P17 Assume that the circuit below has reached equilibrium and that the capacitor is fully charged. (a) Determine the value of V A, (b) the current flowing through R 1, R 2, and R 3, and (c) the power consumed by R 3. Show all calculations required to solve this problem, all units, and all unit conversions for full credit.

4 Homework P18 (a) Write the expression for KVL for the circuit below, and (b) determine the value of the current (I).

5 Active Filters An active filter not only gets rid of unwanted frequencies, but it also amplifies those signals that we want A passive filter will only get rid of unwanted frequencies The higher the order of the filter, the more aggressive the roll off will be (the range of frequencies where the amplitude of the signal decreases will be shorter) VS

6 Active Filters Cutoff Frequencies An active filter will still amplify signals after the cutoff frequency is reached DO NOT assume that the filter will jump straight to zero after it has hit its cutoff frequency STILL AMPLYFING

7 Making Sense of a Bode Plot Everything above 0 db means amplification or > 0 db means no change or = Everything below 0 db means diminishment or <

8 Analysis of Active Low-Pass C Vin R1 R2 Vout = R 2 R 1 1 (jwcr 2 + 1) Which R contributes to the filtering part, and which contribute to the amplification part?

9 Exercise: Analysis of Active High-Pass Filter using KCL at node A and complex impedance. R2 Vin R1 C A Vout = R 2 /R 1 jwcr jwr 1 C

10 FIRST ORDER First Order COMPLEX IMPEDANCE MODELS LOW PASS HIGH PASS PASSIVE = 1 jwcr + 1 = jwcr jwcr + 1 ACTIVE = R 2 R 1 1 (jwr 2 C + 1) = R 2 R 1 jwr 1 C 1 + jwr 1 C SECOND ORDER Second Order BAND PASS LOW PASS = R 2 R 1 1 (jwc 2 R 2 + 1) = 1 (jwrc + 1) 2 PASSIVE jwr 1 C jwr 1 C 1 **Notice the common components for all types of low pass filters and for all types of high pass filters** ACTIVE

11 COMPLEX IMPEDANCE MODELS FIRST ORDER All low pass filters have this expression: = 1 jwcr + 1 SECOND All high ORDER pass filters have this expression: = jwcr jwcr + 1

12 Band-Pass Filter Identify the low pass filter fc value and the high pass filter fc value Active Band-Pass Filter Bode Plot

13 First Order: Low Pass Filter w/gain FIRST ORDER: FOR EVERY DECADE OF FREQUENCY, THE AMPLITUDE DROPS 20 db = R 2 R 1 1 jwcr Low Pass Filter WHAT IS THE VALUE OF fc FOR THIS FILTER? WHAT IS THE GAIN? C1 R2 10 F 10 k w = angular frequency = 2 f Vin R1 The gain of the Inverting Op- Amp is equal to 1 k Vout Gain = R 2 R 1 Vref

14 First Order: Low Pass Filter w/gain Wave Generator: f = 1Hz A = 0.1 V Offset = 2.5 V Gain = R 2 R 1 = 10 NETWORK ANALYZER 20 db 1 decade Vref

15 First Order: Low Pass Filter w/gain Wave Generator: f = 1Hz A = 0.1 V Offset = 2.5 V Image Source:

16 First Order: Low Pass Filter w/gain FIRST ORDER: FOR EVERY DECADE OF FREQUENCY, THE AMPLITUDE DROPS 20 db fc = 1.59 Hz = R 2 R 1 1 jwcr MATLAB Code: >> w=logspace(-3,5,100); >> j=sqrt(-1); >> R1=1E3; >> R2=10E3; >> C=10E-6; >> Vout=(-R2/R1)*1./(1+j*w*R2*C); Y = logspace(a,b,n) generates n points between decades 10^a and 10^b >> loglog(w/2/pi,abs(vout)) 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 w = angular frequency = 2 f **NOTICE THE MATLAB CODE FOR MATHMATICAL OPERATIONS INVOLVING MATRICES**

17 First Order: Low Pass r / Plot FIRST ORDER: FOR EVERY DECADE OF FREQUENCY, THE AMPLITUDE DROPS 20 db >> w=logspace(-3,3,100); >> j=sqrt(-1); >> R1=1E3; fc = 1.59 Hz >> R2=10E3; >> C=10E-6; >> Vout=(-R2/R1)*1./(1+j*w*R2*C); >> loglog(w/2/pi,abs(vout)) Converts w in radians/sec to f in Hz = R 2 R 1 1 jwcr w = angular frequency = 2 f MATLAB Y = 1 decadelogspace(a,b,n) generates n poi nts between decades 10^a and 10^b 1 decade since w = 2 f or f = w/2 Creates a loglog plot of w vs Vout

18 Active Filter Design What are the frequencies of the signals you are trying to capture? Is the signal that you are trying to capture small in amplitude? How much noise are you willing to have in your output?

19 Design Example Design Example List of Requirements: 1. Pass frequencies above 1Hz (HIGH PASS FILTER) and diminish or eliminate frequencies above 15 Hz (LOW PASS FILTER) 2. Amplitude signal will be in the range of millivolts 3. We need a *fairly clean signal * By fair we mean that the data displayed to the user is usable, if not, we expect feedback from the user

20 Bandwidth Design We need a bandpass to pass a certain range of frequencies Calculate the necessary cutoff frequencies for the low-pass and high-pass F c1 and F c2 filters. DO NOT calculate the cutoff frequency exactly at the specified frequencies, remember that the filter does not suddenly jump to zero amplitude! F c1 = 0.1 Hz = F c2 = 10 Hz = 1 2πR 1 C 1 1 2πR 2 C 2 OR

21 Amplification Design Always look at the amplitude of the raw signal! We are in the millivolts range, and we want to amplify our signal to the volts range. How can we do it? Calculate the gain to be 10 or more! Inverting Amplifier G = R f R in Non-Inverting Amplifier G = R R 2 OR

22 Filter Noise Are we happy with the signal to noise ratio? High frequencies are an issue? Add more low-pass filters Low frequencies are an issue? Add more high-pass filters Yes we are happy with our filter!

23 Homework- Due 11/6 (Najera), Due 11/9 (Martinez- Acosta) P19 P20

24 What s Next in Week 11? Will introduce LAB Module VII Blood Pressure Sensor (Design your own circuit!) LECTURE Quiz #8 Op-Amps III More active filters Build, Test, and Model a 2 nd order Low Pass Filter Please bring laptops to all lectures and labs.

25 Questions?