8LABO Ganago Student Lab9

7/28/2019 8LABO Ganago Student Lab9

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Lab 9: Filters and Transfer Functions

LAB EXPERIMENTS USINGNI ELVIS II

AND NI MULTISIM

Alexander Ganago

Jason Lee Sleight

University of Michigan

Ann Arbor

Lab 9Filters and Transfer Functions

2010 A. Ganago Introduction Page 1 of 16


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Introduction

Filters are circuits that pass or amplify signals at certain frequencies and block or

suppress signals at other frequencies. Essential for their performance are circuit elements

whose impedances are frequency-dependentcapacitors and inductors, as well aseffective or parasitic capacitances and inductances. Due to the parasitic capacitances and

inductances, any circuit can act as a filter, especially at high frequencies.

In order to study a circuit as a filter, we must consider its input and output voltages and

currents, as shown in Figure 9-1.

Figure 9-1. Input (1) and output (2) currents and voltages of a circuit.

Here comes an important novelty: when we look at a circuit as a filter, most often, we are

interested not in a particular value of the output voltage or current but in the ratio of the

output to the input. This ratio has a special name of transfer function.

Combining two inputs (voltage and current) and two outputs (voltage and current), we

can consider 4 types of transfer functionsvoltage gain, current gain, transferimpedance, and transfer admittance [see your textbook for more detail]. Here we focus on

the voltage gain:

( )( )

( )OUT

IN

VH

V

=

Note that the voltage gain is defined for sinusoidal signals; in a linear circuit, a sinusoidal

input produces a sinusoidal output at the same frequency, but (in general) with a different



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amplitude and phase. In order to keep track of both the amplitude and phase angle, we

use phasors; thus, at each frequency, the transfer function is a complex number.

The transfer function magnitude

( ) ( )( )

OUT

IN

VHV

=

determines the ratio of amplitudes and is used to describe the four types of filters, as

shown in Figures 9-2 through 9-5:

Low-Pass (LP), High-Pass (HP), Band-Pass (BP), and BR (BR).

In this lab, you will study circuits that serve as examples of these 4 types of filters.

Figure 9-2. Transfer function magnitude of a Low-Pass (LP) filter.



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Figure 9-3. Transfer function magnitude of a High-Pass (HP) filter.

Figure 9-4. Transfer function magnitude of a Band-Pass (BP) filter.



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Figure 9-5. Transfer function magnitude of a BR (BR) filter.

Of course, we need to determine what frequencies each filter passes and whichfrequencies it blocks. By conventional definition, the pass-band of a filter is the

frequency range, where the power of the output signal exceeds 50% of the maximal

output power. With a very good accuracy, the 50% power corresponds to the drop of thetransfer function magnitude by 3 dB from the maximum. The half-power frequency is

often called the cutoff, or corner frequency and denoted C (see Figures 9-2 to 9-5).

Refer to your textbook for discussion and calculations.

Frequencies outside the pass-band are called the stop-band (see Figures 9-2 to 9-5).

The simplest filter circuit is a voltage divider built of two circuit elements with

impedances and as shown in Figure 9-6.1Z 2Z



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Figure 9-6. A simple filter built as a voltage divider of two circuit elements.

The transfer function of the filter shown in Figure 9-6 equals:

2

1 2

ZH( )

Z Z =

+

In this lab, you will use a resistor and a capacitor; depending on which connections youchoose for the input and the output, you will build an LP filter (Figure 9-7) and an HP

(Figure 9-8).

Figure 9-7. A first-order LP RC filter.



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The transfer function of the filter shown in Figure 9-7 is easy to calculate with phasors:

( )

C

1

1 1j CH

1 1 j RC

R 1 jj C

= = =

+ ++

Here, is the frequency of the sinusoidal signal (which you can vary on your functiongenerator) and

C

1

RC =

is the characteristic frequency of the circuit (which you can vary by choosing another

resistor or capacitor). The term first-order filter comes from the fact that, in the equation

above, the terms with frequencies are linear (not quadratic).

The magnitude of this transfer function is:

( )2

CC

1 1H

1 j 1

= = + +

At very low frequencies C > this transfer function magnitude is approximately



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( ) CC2

C

1H

1

>> =

+

In other words, it is inversely proportional to the signal frequency. For example, if thesignal frequency increases by a factor of 10 or by one decade (see below), the transferfunction magnitude drops by a factor of 10.

Recall that decibels (dB) are used as a logarithmic measure of the transfer functionmagnitude:

( ) ( )10dBH 20 log H =

The slope of the transfer function magnitude, such as observed at high frequencies for anLP filter, is often expressed in decibels per decade or dB/dec, where a decade is any

interval of frequencies between 1 and 2 such that2

1

10

= , for example, from 10 Hz

to 100 Hz, or from 42 MHz to 420 MHz.

On a plot that has a logarithmic scale of frequencies and shows the transfer function

magnitude in decibels, such decrease corresponds to a straight line with the slope of

20 dB/dec for the first-order LP filter (Figure 9-2).

Using the same resistor and capacitor, you can change their connections to the input and

output signals and obtain a HP filter, as shown in Figure 9-8.

Figure 9-8. A first-order HP filter.



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The filter in Figure 9-8 has the same cutoff frequency C1

RC = ; its transfer function is

sketched in Figure 9-3.

At high frequencies C >> , its transfer function magnitude is constant, and at low

frequencies C


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A series RLC circuit can be used to obtain all 4 types of second-order filters, as shown in

Figures 9-11 through 9-14.

Figure 9-11. A second-order LP filter.

Figure 9-12. A second-order HP filter.



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Figure 9-13. A second-order BP filter.

Figure 9-14. A second-order BR filter.

Algebraic expressions for the transfer functions of the filters shown in Figures 9-11 to 9-

14 contain quadratic, or second-order terms2

,2

C , etc., therefore such filters arecalled second-order. See the calculations and discussions in your textbook.

Here we note that, the transfer function magnitude of a second-order LP filter (see Figure

9-11), is expected to have the slope of 40 dB/dec, because its transfer function

magnitude drops by a factor of 100 over each decade at high frequencies C >> .



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We also note that the minimal transfer function magnitude of the second-order BR filter

(Figure 9-14) theoretically reaches zero, or (negative infinity) on the dB scale, at the

resonant frequency 01

LC = but in practice it does not happen because of the parasitic

resistance of the inductor and other imperfections of the circuit. Therefore, in your pre-

lab simulations you can obtain very low magnitudes of the transfer function for this BRfilter (the exact value depends on the frequencies, at which the software calculates

( )H , which depends on the number of points per decade, and so on) but in the lab youwill probably observe a much shallower notch in the transfer function magnitude plot.

If losses in the second-order circuit are not high, both the LP and HP transfer function

magnitude plots reveal the resonant peaks near 01

LC = as shown in Figures 9-15 and

9-16.

Figure 9-15. The transfer function magnitude plot for a second-order LP filter.



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Figure 9-16. The transfer function magnitude plot for a second-order HP filter.

The frequency, at which the transfer function magnitude is maximal, is usually close to

the resonant frequency 01

LC = ; however, in the case of a weak resonance (broad

peak), the maximum on the curve is shifted toward lower frequencies for LP filters andtoward high frequencies for HP filters.

The filters discussed above are called passive, because they do not contain sources of

power. Active filters include, for example, op amps, which require power supplies.

Figure 9-17 shows an active filter based on an inverting amplifier.

Figure 9-17. An inverting amplifier can act as an active filter.



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The transfer function of this active filter equals:

( ) 21

ZH

Z =

The negative sign is important when you calculate and measure the phase shift between

the input and the output signals.

In this lab, you will build a first-order LP filter based on an inverting amplifier; itsdiagram is shown in Figure 9-18.

Figure 9-18. An active first-order LP filter based on an inverting amplifier.

The output of this filter can be used as an input for another filter thus several filters can

be used as stages of a larger filter circuit. For example, Figure 9-19 shows a filter of two

identical stages, and Figure 9-20 shows a filter of three identical stages.

In this lab, you will build and study active circuits of one stage (Figure 9-18), and you

will have an option to explore for extra credit an active circuit of two and three identical

stages (see Figures 9-19 and 9-20).



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Figure 9-19. An active LP filter built of 2 identical stages (for extra credit exploration).

Figure 9-20. An active LP filter built of 3 identical stages (for extra credit exploration).



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Pre-Lab:

1.First-Order FiltersPart 1.1. First-Order Low-Pass Filter

Use Multisim to simulate the following circuit:

Use R = 10

C = 470 FUse the Bode Plotter tool to create a frequency analysis from 11,000 Hz.

A.Using the given circuit parameters, calculate the theoretical cutoff frequency of thefilter.

B.Use the cursors on the Bode Plotter to measure and record the cutoff frequency,fBCUTOFF,1B, of the filter. Discuss its agreement/disagreement with the theoretical cutoff

frequency from Part 1.1.A.

C.Create a printout of the simulation results (Pre-Lab Printout #1).D.On the printout, show a linear approximation to the transfer function (constant at lowfrequency and with the slope of 20 dB/dec at high frequency). Determine whether the

crossing point of the two linear functions matches the cutoff frequency. Briefly discuss

whether the 20 dB/dec linear approximation matches the actual transfer function.

2010 A. Ganago Pre-Lab Page 1 of 7


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Part 1.2. First-Order High-Pass Filter


Use R = 10 C = 470 F

Use the Bode Plotter tool to create a frequency analysis from 11,000 Hz.

E. Using the given circuit parameters, calculate the theoretical cutoff frequency of thefilter.

F. Use the cursors on the Bode Plotter to measure and record the cutoff frequency,fBCUTOFF,2B, of the filter. Discuss its agreement/disagreement with the theoretical cutofffrequency from Part 1.2.A.

G.Create a printout of the simulation results (Pre-Lab Printout #2).H.On the printout, show the linear approximation to the transfer function (with the slope

of +20 dB/dec at low frequency and a constant at high frequency). Determine whether

the crossing point of the two linear functions matches the cutoff frequency. Briefly

discuss whether the +20 dB/dec linear approximation matches the actual transfer

function.



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2.Second-Order FiltersPart 2.1. Second-Order Low-Pass Filter


Use R = 10

C = 4.7 F

L = 10 mHUse the Bode Plotter tool to create a frequency analysis from 1010,000 Hz.

A.Using the given circuit parameters, calculate the theoretical maximal gain frequency ofthe filter.

B.Use the cursors on the Bode Plotter to measure and record the maximal gain frequency,fBMAX,1B, of the filter. Discuss its agreement/disagreement with the theoretical maximal

gain frequency from Part 2.1.A.

C.Create a printout of the simulation results (Pre-Lab Printout #3).D.On the printout, show the linear approximation to the transfer function (constant at low

frequency and with the slope of 40 dB/dec at high frequency). Determine whether the

crossing point of the two linear functions matches the maximal gain frequency. Briefly

discuss whether the 40 dB/dec linear approximation matches the actual transfer

function.



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Part 2.2. Second-Order High-Pass Filter


Use R = 10

C = 4.7 F


E. Using the given circuit parameters, calculate the theoretical maximal gain frequency ofthe filter.

F. Use the cursors on the Bode Plotter to measure and record the maximal gain frequency,fBMAX,2B, of the filter. Discuss its agreement/disagreement with the theoretical maximal

frequency from Part 2.2.A.

G.Create a printout of the simulation results (Pre-Lab Printout #4).H.On the printout, show the linear approximation to the transfer function (with the slope

of +40 dB/dec at low frequency and constant at high frequency). Determine whether

the crossing point of the two linear functions matches the maximal gain frequency.

Briefly discuss whether the +40 dB/dec approximation matches the actual transfer

function.



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Part 2.3. Second-Order Band-Pass Filter


Use R = 100

C = 4.7 F


I.Using the given circuit parameters, calculate the theoretical frequency of the maximalgain of the filter, as well as the bandwidth of the filter.

J.Use the cursors on the Bode Plotter to measure and record the frequency of maximalgain, fBMAX,3B, of the filter, as well as the bandwidth of the filter. Discuss its

agreement/disagreement with the respective theoretical values from Part 2.3.A.

K.Create a printout of the simulation results (Pre-Lab Printout #5).L. On the printout, show the linear approximation to the transfer function (with the slope

of +20 dB/dec at low frequency and with the slope of 20 dB/dec at high frequency).

Determine whether the crossing point of the two linear functions matches the frequency

of maximal gain. Briefly discuss whether the +20 dB/dec and 20 dB/dec linear

approximations match the actual transfer function.



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Part 2.4. Second-Order Band-Reject Filter


Use R = 100

C = 4.7 F


M.Using the given circuit parameters, calculate the theoretical frequency of the minimalgain of the filter, as well as the bandwidth of the filter.

N.Use the cursors on the Bode Plotter to measure and record the frequency of minimalgain, fBMIN,1B, of the filter, as well as the bandwidth of the filter. Discuss its

agreement/disagreement with the respective theoretical values from Part 2.4.A.

O.Create a printout of the simulation results (Pre-Lab Printout #6).



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3. Active Filters


Use RB1B = 1 k

RB2B = RBLB = 5 k

C = 1 FUse a Virtual Op Amp.

Use the Bode Plotter tool to create a frequency analysis from 11,000 Hz.

P. Use the cursors on the Bode Plotter to measure and record the cutoff frequency,fBCUTOFF,3 B, of the filter

Q.Create a printout of the simulation results (Pre-Lab Printout #7).



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In-Lab Work

Part 1: First-Order Filters

Part 1.1. First-Order Low-Pass Filter Power on the ELVIS II.

Build the following circuit:

Use: R = 10

C = 470F

Measure VBINB on AI1 and VBOUTB on AI0.

Power on the PB.

From the NI ELVISmx Instrument Launcher, launch the Bode Analyzer.

Set the frequency range to be 11,000 Hz, 15 steps per decade, V BPPKB = 5V, normal

polarity.

Run the Bode Analyzer.

Use the cursor to measure the cutoff frequency:

fBCUTOFF,1B = _________ Hz

Create a printout of the Bode plot. (In-Lab Printout #1)

Power off the PB.

2010 A. Ganago In-Lab Page 1 of 12


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Part 1.2. First-Order High-Pass Filter

Build the following circuit (Hint: components are the same as in Part 1.1 but their

connections are different):

Use R = 10

C = 470F


Power on the PB.

Run the Bode Analyzer with the same settings as Part 1.1.




Power off the PB.



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Part 2: Second-Order FiltersPart 2.1. Second-Order Low-Pass Filter


Use R = 10

C = 4.7F

L = 10 mH


Power on the PB.

Set the frequency range of the Bode Analyzer to be 1010,000 Hz, 15 steps per decade,VBPPKB = 5V, normal polarity.


Use the cursor to measure the maximal gain frequency:

fBMAX,1B = _________ Hz


Power off the PB.



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Part 2.2 Exploration: Second-Order High-Pass Filter

If you do not want to perform the explorations, skip to Part 2.3.

Build the following circuit (Hint: components are the same as in Part 2.1 but theirconnections are different):

Use R = 10

C = 4.7F

L = 10 mH


Power on the PB.



fBMAX,2B = _________ Hz


Power off the PB.



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Use R = 100 (Note: this is different from Parts 2.1 and 2.2)

C = 4.7F

L = 10 mH


Power on the PB.

Set the frequency range of the Bode Analyzer to be 1010,000 Hz, 15 steps per decade,

VBPPKB = 5V, normal polarity.



fBMAX,3B = _________ Hz



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Use the cursor to measure the frequencies and , which determine thebandwidth:

C1, Part 3f C2, Part 3f

C1, Part 3f = ________ Hz

C2, Part 3f = ________ Hz

Note that the sketch above shows frequencies in rad/sec, while your measurements

produce frequencies in Hz.


Power off the PB.



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Build the following circuit (Hint: components are the same as in Part 2.3 but their

connections are different):

Use R = 100

C = 4.7F

L = 10 mH


Power on the PB.


Use the cursor to measure the minimal gain frequency:

fBMIN,4B = _________ Hz



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Use the cursor to measure the frequencies and , which determine the bandwidth:C1f C2f

C1, Part 4f = ________ Hz

C2, Part 4f = ________ Hz

Note that the sketch above shows frequencies in rad/sec, while your measurementsproduce frequencies in Hz.


Power off the PB.



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Part 2.5 Explorations Band Reject Filter

Change your circuit from Part 2.4 so that R = 1 k.

Power on the PB.


Use the cursor to measure the minimal gain frequency, as well as the frequencies

and which determine the bandwidth:C1, Part 5f C2, Part 5f

fBMIN,5B = _________ Hz

C1, Part 5f = ________ Hz

C2, Part 5f = ________ Hz


Power off the PB.



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Part 3: Active FiltersBuild the following circuit:

Where RB1B = 1 k

RB2B = RBLB = 5 k

C = 1 F


Power on the PB.

Set the frequency range of the Bode Analyzer to be 11,000 Hz, 15 steps per decade,

VBPPKB = 0.1V, normal polarity.





Power off the PB.

This is the end of the required lab. If you are not going to continue with the

explorations, then power off the PB and ELVIS II and clean up your workstation.



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Part 4: Explorations Active Filters


Use: RB1B = 1 k

RBFB = RBLB = 5 k

C = 1 F


Power on the PB.

Run the Bode Analyzer with the same settings as Part 4.




Power off the PB.

Continued on the next page.



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Use: RB1B = 1 k

RBFB = RBLB = 5 k

C = 1 F


Power on the PB.

Run the Bode Analyzer with the same settings as Part 4.




This is the end of the lab. Power off the PB and ELVIS II and clean up your

workstation.



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Post-Lab:

1.First-Order FiltersPart 1.1. First-Order Low-Pass Filter

A.Discuss (quantitatively) the agreement/disagreement between the cutoff frequencyfound in your simulation (from Pre-Lab Part 1.1) and measured in the lab (from In-Lab

Part 1.1).

B.Discuss (qualitatively) the agreement/disagreement between the phase shift of yoursimulated circuit (from Pre-Lab Part 1.1) and your experimental circuit (from In-Lab

Part 1.1) at various frequencies.

Part 1.2. First-Order High-Pass FilterC.Discuss (quantitatively) the agreement/disagreement between the cutoff frequency

found in your simulation (from Pre-Lab Part 1.2) and measured in the lab (from In-Lab

Part 1.2).

D.Discuss (qualitatively) the agreement/disagreement between the phase shift of yoursimulated circuit (from Pre-Lab Part 1.2) and your experimental circuit (from In-Lab


Continued on the next page

2010 A. Ganago Post-Lab Page 1 of 4


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2.Second-Order FiltersPart 2.1. Second-Order Low-Pass Filter

A.Discuss (quantitatively) the agreement/disagreement between the maximal gainfrequency found in your simulation (from Pre-Lab Part 2.1) and measured in the lab

(from In-Lab Part 2.1).

B.Discuss (qualitatively) the agreement/disagreement between the phase shift of yoursimulated circuit (from Pre-Lab Part 2.1) and your experimental circuit (from In-Lab


C.Compare your results (qualitatively) of your second-order low pass filter to those ofyour first-order low pass filter. Discuss issues such as overshoot, fallout rate (slope of

the transfer function magnitude plot in dB/dec), and phase shift. What are the

advantages/disadvantages of each circuit?

Part 2.2 Exploration: Second-Order High-Pass Filter

D.Discuss (quantitatively) the agreement/disagreement between the maximal gainfrequency found in your simulation (from Pre-Lab Part 2.2) and measured in the lab


E. Discuss (qualitatively) the agreement/disagreement between the phase shift of yoursimulated circuit (from Pre-Lab Part 2.2) and your experimental circuit (from In-Lab


F. Discuss (qualitatively) the agreement/disagreement between the slope of the transferfunction magnitude plot (in dB/dec) within the stop-band of you simulated circuit (from

Pre-Lab Part 2.2) and your experimental circuit (from In-Lab Part 2.2). Explain your

results.

G.Compare your results (qualitatively) of your second-order high pass filter to those ofyour first-order high pass filter. Discuss issues such as overshoot, fallout rate, andphase shift. What are the advantages/disadvantages of each circuit?

2010 A. Ganago Post-Lab Page 2 of 4


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H.Discuss (quantitatively) the agreement/disagreement between your simulated maximalgain frequency (from Pre-Lab Part 2.3) and your experimental maximal gain frequency


I.Discuss (quantitatively) the agreement/disagreement between your simulated bandwidth(from Pre-Lab Part 2.3) and your experimental bandwidth (from In-Lab Part 2.3).

J.Discuss (qualitatively) the agreement/disagreement between the phase shift of yoursimulated circuit (from Pre-Lab Part 2.3) and your experimental circuit (from In-Lab

Part 2.3).


K.Discuss (quantitatively) the agreement/disagreement between your simulated minimalgain frequency (from Pre-Lab Part 2.4) and your experimental minimal gain frequency


L. Discuss (quantitatively) the agreement/disagreement between your simulatedbandwidth (from Pre-Lab Part 2.4) and your experimental bandwidth (from In-Lab Part

2.4).

M.Discuss (qualitatively) the agreement/disagreement between the phase shift of yoursimulated circuit (from Pre-Lab Part 2.4) and your experimental circuit (from In-Lab

Part 2.4).

Part 2.5 Explorations Band Reject Filter

N.Calculate the theoretical minimal-gain frequency and the bandwidth for the circuit usedin In-Lab Part 2.5. (series RLC with R = 1 k, C = 4.7F, L = 10 mH).

O.Discuss (quantitatively) the agreement/disagreement between the theoretical minimalgain frequency above and your experimental minimal gain frequency (from In-Lab Part

2.5).

P. Discuss (quantitatively) the agreement/disagreement between the theoretical bandwidthabove and your experimental bandwidth (from In-Lab Part 2.5).

3.Active Filters 2010 A. Ganago Post-Lab Page 3 of 4


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A.Discuss (quantitatively) the agreement/disagreement between your simulated cutofffrequency (from Pre-Lab Part 3) and your experimental cutoff frequency (from In-Lab

Part 3).

B.Compare your results (qualitatively) of your active low pass filter to those of your first-order and second-order low pass filter. Discuss issues such as overshoot, fallout rate,

and phase shift. What are the advantages/disadvantages of the active circuit?

4.Explorations Active FiltersA.Explain your results. Discuss issues such as overshoot, fallout rate, cutoff frequency,

and phase shift for each circuit. Give a qualitative comparison between each of the

filters (1-stage active from Part 3, 2-stage active from Part 4, and 3-stage active from

Part 4.). Explain the advantages/disadvantages of using more stages in your active

filter.

8LABO Ganago Student Lab9

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