10LABO Ganago Student Lab10

7/27/2019 10LABO Ganago Student Lab10

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Lab 10: Laplace Transform Analysis Technique

LAB EXPERIMENTS USINGNI ELVIS II

AND NI MULTISIM

Alexander GanagoJason Lee Sleight

University of Michigan

Ann Arbor

Lab 10

Laplace Transform Analysis Technique

2010 A. Ganago Introduction Page 1 of 8


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Goals for Lab 10

Learn about:

The Laplace transforma powerful tool for calculations of time-dependentresponses of linear circuits to arbitrary input signals Multisim Arbitrary Waveform Generatora tool to create any type of input

signal, which can be used for both theoretical simulations (in pre-lab) and real

experiments on real circuits (in the lab)

In the pre-lab:

Use Multisim Arbitrary Waveform Generator to create an input signal thatincorporates a rectangular pulse, a triangular pulse, and a ramp

Simulate the responses of first-order circuits and second-order circuits to the inputsignal you created, at various frequencies

In the lab:

Build several first-order circuits and second-order circuits with the same nominalvalues of components as you used in the pre-lab simulations

Measure the responses to the input waveform, which you created in the pre-lab, atvarious frequencies, of the following circuits:

o First-order low-pass (LP) filtero First-order high-pass (HP) filtero Second-order low-pass filter at critical damping

In the post-lab:

Compare the simulated and measured responses, and explain their distinctions:o Of the same circuit to input signals at various frequencieso Of the low-pass versus high-pass filtero Of the first-order low-pass filter versus second-order low-pass filter

Optional goals:

Measure under-damped responses of a second-order low-pass filter to the inputwaveform, which you created in the pre-lab, at various frequencies

Explain the dependence of the responses on the damping ratio0

of the circuit

and on the frequency of the input signal.



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Introduction

Laplace transform is a powerful technique for calculations of the time-dependent

response of linear circuits to arbitrary input signals. In lectures and homework, you will

learn about the mathematical foundation of this technique and perform calculations toapply it to several circuits.

In this lab, we will not ask you to do calculations: instead, we rely on Multisim toperform them for you in pre-lab simulations. The main focus of the lab is to do the same

thing twice: first, you will simulate the circuit response in the pre-lab, then you will

build real circuits and measure their actual responses in the lab; eventually, you willcompare your theoretical results and your lab data in the post-lab. As usual, similarity

between theoretical predictions and lab data proves that the theory is good; disagreement

between the two deserves special attention and requires an explanation.

In the pre-lab, you will use the Multisim Arbitrary Waveform Generator to create aninput signal that incorporates elements of 3 standard waveformsa rectangular pulse, a

triangular pulse, and one segment of a negative ramp (see Figure 10-1). The duration ofthese 3 elements (and the blank space between them) makes up one period of your input

waveform. This waveform will serve as the input signal both in your pre-lab simulation

of circuit responses and in your real measurements in the lab. The actual period T (in

sec) of your input signal depends on the frequency, with which the function generator

plays your waveform. You will use different settings of this frequency, depending on theparticular assignment.

Figure 10-1. The input waveform incorporates a rectangular pulse, a triangular pulse, andone period of a negative ramp.

You will apply this special input signal to first-order and second-order circuits, whichyou will simulate in the pre-lab and build in the lab. All these circuits are familiar to you



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from Lab 9, where you used them as filters; for your convenience, we will keep using the

notations of Lab 9 such as shown in Figures 10-2 through 10-4.

The distinction between these two labs is not in the circuits but in the signals: in Lab 9,

you used sinusoidal inputs, which is typical in studies of filters; in Lab 10, you will use

the waveform of Figure 10-1 as the input. Therefore, instead of using phasors, whichwork only under sinusoidal steady-state conditions, we must employ the much more

powerful technique of Laplace transform, which works for any type of input signals.

Specifically, you will simulate and build:

1. A first-order LP filter shown in Figure 10-2,2. A first-order HP filter shown in Figure 10-3, and3. A second-order LP filter shown in Figure 10-4.

Figure 10-2. A first-order LP filter.



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Figure 10-3. A first-order HP filter.

Figure 10-4. A second-order LP filter.

The response of your circuit depends on the ratio of the period of the input

waveform to the characteristic time

r T

of the circuit

Tr

=

This ratio is varied as you change the frequency setting on your function generator and/or

when you change the component values of your circuit.For the first-order LP and HP filters, the characteristic time equals



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RC=

For the second-order LP filter, the characteristic time can be estimated as

1LC

The effects of the frequency are easy to explain for the first-order LP and HP filters; 3

particular cases belong to the 3 ranges of the ratioT

r

= .

(1)If the ratio is very large, such as Tr ~ 10

= 0 , then the capacitor is given enough

time to charge, discharge, and recharge to a negative voltage during each part of

the input signal (see Figure 10-1). In other words, the capacitor voltage willclosely follow the input voltage, except right after the moments when the input

voltage has changed abruptlyat the vertical edges of the rectangular pulse andof the ramp, because the capacitor voltage must be continuous, avoiding abrupt

changes. On the contrary, the resistor voltage will remain near zero, except for the

time intervals when the capacitor voltage cannot follow the abrupt changes of theinput voltage, because, according to KVL, the sum of resistor voltage and

capacitor voltage must match the input voltage.

(2) At the other extreme, when the ratio is very small, such as Tr ~ 0.01

= , the

capacitor is not given any time to charge thus its voltage is very small. By the

same token, the resistor voltage closely follows the input.

(3)At intermediary values of the ratio, such as Tr ~

= 1 , the capacitor is given some

time to charge, discharge, and recharge, but this time is too short for it to charge

fully: recall that , that is 37% of the charge still remains on the

capacitor after it discharges during the time equal to . It means that the capacitor

partly charges during the rectangular pulse (see Figure 10-1); then it partlydischarges before the triangular pulse begins, and so on. In other words, the

response to all parts of the input signal, measured as the capacitor voltage, will be

convoluted. The same is true for the resistor voltage.

10.37e

For qualitative analysis of the circuit response, it may also be helpful to remember that a

low-pass filter acts (approximately) as an integrator at high frequencies, and a high-passfilter works (also, approximately) as a differentiator at low frequencies. Therefore, the

triangular and ramp parts of your input waveform will be rounded off by a low-pass filter,

while a high-pass filter will produce nearly constant (negative and positive) outputs

corresponding to the linear slopes of the triangular and ramp.

The responses of a second-order circuit may (recall Lab 6).



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Recall that a second-order circuit can exhibit 3 types of responses, depending on the ratio

of the damping factor and the undamped natural frequency 0 :

If0

1

< , the response is under-damped, and includes oscillations;

If0

1

= , the response is critically damped: it does not include oscillations and is

relatively fast;

If0

1

> , the response is over-damped, similar to the exponential response of a

first-order circuit, and slower than the critically damped response.

Review Lab 6, where you studied all these types of responses of second-order circuits to

abrupt changes of input voltage, similar to the beginning (rectangular pulse) part of theinput waveform, which you will use in this Lab (Figure 10-1).

In our experiments with NI ELVIS II, it proved a bit of a challenge to observeoscillations in the second-order LP filter (Figure 10-4), because we needed a small

resistance of the circuit, which significantly loaded the function generators output. To

overcome this difficulty, we used a buffer (voltage follower) based on an op amp such asyou studied in Lab 4: see Figure 10-5.

Figure 10-5. Voltage follower prevents loading of the function generator output when thecircuit resistance is very low.

The required part of lab work is focused on the responses of a second-order LP filter

(Figure 10-5) near critical damping to the input waveform (Figure 10-1) at variousfrequencies. Although the critically damped response does not involve oscillations, it is

distinct from the response observed in a first-order LP filter (Figure 10-2).



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As an Exploration for extra credit, we offer measurements of the under-damped

responses of a second-order LP filter to the same input waveform at various frequencies.The set of data with strong oscillations is indeed distinct from anything you can observe

in first-order circuits.

As you will see, the presence of oscillations in the output signal is bad because they maskthe form of the input signal. At the same time, comparison between the responses of two

typesunder-damped and critically dampedproves that, in order to recover the shape

of the input signal, one simply has to increase the energy losses in the circuit, forexample, to change damping from very low to critical.



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Pre-Lab:

0.Arbitrary Waveform GeneratorAs explained in the introduction, this lab will make use of the Arbitrary WaveformGenerator, in order to create a non-trivial input signal. The following tutorial describes a

step-by-step process for creating the waveform, which you will be using in the pre-lab,

in-lab, and post-lab.

It is essential that you begin your work on this lab with the creation of the waveform,

because you will need this waveform for doing the Pre-Lab simulations, In-Lab

experiments, and Post-Lab data analysis. Lacking this waveform at the beginning of

your In-Lab work will lead to a significant loss of points and even more significant loss

of valuable lab time.

Use Multisim to create a new NI ELVIS II Schematic.

Double click on the Arbitrary Waveform Generator Icon (located above the function

generator).

Click the button to launch the waveform editor.

Click the new segment button three times.

Change the duration of the first three segments to be 0.01 seconds, and the last (4 Pth

P)

segment to be 0.1 seconds.

Select the first segment and click new component once.

Select the new component and change the function library to pulse.

Set the rise and fall to be 5% each. Set the Amplitude to be 2.00. Leave the other settings

unchanged.

Select the second segment and click new component once.

Select the new component and change the function library to triangle.

Set the frequency to be 50 Hz. Set the Amplitude to be 1.00 (note the negative sign).

Leave the other settings unchanged.

Select the third segment and click new component once.

Select the new component and change the function library to ramp.

Set the start to be 1.00 and the end to be 1.00 (note the negative sign).

A. Ganago Pre-Lab Page 1 of 6


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Leave the final segment as a constant 0 V DC.

Your final waveform should look like the following figure:

Select FileSave in order to save the waveform as a .wfm. This will allow you edit the

waveform at a later time, in the event that you made a mistake. Pre-Lab File #1.

Select FileSave As to save the file as a .wdt. This is the file that the Arbitrary Waveform

Generator will use to create the signal, which will be used in pre-lab simulations and also

fed to your real circuits in the lab as the input. Pre-Lab File #2.



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1.First Order Low Pass FilterUse Multisim to simulate the following circuit:

Use: R = 1 k and C = 1 F

Use the Arbitrary Waveform Generator to supply the input signal, which you created in part 0.

The signal is produced on the respective analog output port, #31,32. Set the gain to be 5.

Set the update rate of the Arbitrary Waveform Generator to be 1,000 S/s.

Run the simulation.

Use the OSCOPE to view the input and output waveforms. Create a printout of the plot.

(Pre-Lab Printout #1).

Set the update rate of the Arbitrary Waveform Generator to be 10,000 S/s.Run the simulation.




Run the simulation.



Set the update rate of the Arbitrary Waveform Generator to be 500,000 S/s.Run the simulation.



Make sure you save a copy of your file (for use in the lab). Pre-Lab File #3.



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2.First Order High Pass FilterUse Multisim to simulate the following circuit:

Use R = 1 k and C = 1 F, as before.

Use the Arbitrary Waveform Generator to supply the input signal which you created in part

0. (The signal is produced on the respective analog output port, #31,32). Set the gain to be 5.


Run the simulation.Use the OSCOPE to view the input and output waveforms. Create a printout of the plot.









Run the simulation.

Use the OSCOPE to view the input and output waveforms. Create a printout of the plot.(Pre-Lab Printout #8).




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3.Second Order Low Pass Filter, Near Critical DampingUse Multisim to simulate the following circuit:

Use R = 200 (not the different resistance value)C = 1 F

L = 10 mH

Use a 741 Op Amp to create the buffer. Dont forget to wire the Op Amp supply lines up to

the VPS VI (use +10 V DC and 10 V DC as the supply voltages).

Use the Arbitrary Waveform Generator to supply the input signal, which you created in part0 on analog output 0. (The signal is produced on the respective analog output port, #31). Set

the gain to be 5.





Run the simulation.

Use the OSCOPE to view the input and output waveforms. Create a printout of the plot.(Pre-Lab Printout #10).






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Discussion of your results: Simulated responses

Write up brief comparisons of the Pre-lab printouts that you obtained, and provide brief

explanations of the observed distinctions/similarities:

o For the same circuit (for each of the 3 circuits) in response to input signals atvarious frequencies

o For the first-order Low-Pass versus first-order High-Pass filter circuits (for eachof the 4 frequencies)

o For the first-order Low-Pass filter versus second-order Low-Pass filter nearcritical damping (for each of the 4 frequencies)



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In-Lab Work

Part 1: First-Order Low-Pass Filter

Open your Multisim file from the pre-lab for the first-order low-pass filter. Build the following circuit:

Use R = 1 kC = 1 F

Connect VBINB as the AO 0 port on the NI ELVIS II (terminal #31).

Measure VBINB and VBOUTB on AI channels 0 and 1 respectively.

Double click the icon for the Arbitrary Waveform Generator (AWG) on your Multisim

circuit. Ensure that the waveform which you created in the pre-lab is loaded into outputchannel 0.

Ensure the Update rate to be 1,000 S/s with an immediate trigger, and that the device is

Multisim.

Open the OSCOPE VI.

Run the simulation.

Set the device of the AWG to be NI ELVIS II and run continuously.

Power on the PB.

Run the AWG VI.

Use the Oscilloscope to view the experimental and theoretical waveforms. Note: you

will have to first set the device to be NI ELVIS II and adjust the input channels.

2010 A. Ganago In-Lab Page 1 of 8


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Create a printout which clearly displays both pairs of the input and output waveforms.

(In-Lab Printput #1).

The following procedures are very similar to the above; the focus is on varying the

frequency of the input signal, or the Update rate.

Switch the device of the AWG and OSCOPE to be Multisim.

Set the Update rate to be 10,000 S/s with an immediate trigger.

Run the simulation.

Switch the device of the AWG and OSCOPE to be NI ELVIS II, and run the VIs.

Use the Oscilloscope to view the experimental and theoretical waveforms.


(In-Lab Printout #2).



Run the simulation.







Run the simulation.







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Part 2: First-Order High-Pass Filter Open your Multisim file from the pre-lab for the first-order high-pass filter.

Build the following circuit:

Use R = 1 kC = 1 F

Connect VBINB as the AO_0 port on the NI ELVIS II (terminal #31).


Double click the icon for the AWG on your Multisim circuit. Ensure that the

waveform which you created in the pre-lab is loaded into output channel 0.


Multisim.

Open the OSCOPE VI.

Run the simulation.

Set the device of the AWG to be NI ELVIS II and run continuously.

Power on the PB.

Run the AWG VI.







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Part 3: Second-Order Low-Pass Filter near Critical

Damping Open your Multisim file from the pre-lab for the second-order low-pass filter.

Build the following circuit:

Use R = 200 C = 1 FUse a 741 Op Amp for the buffer.

Connect VBINB as the AO_0 port on the NI ELVIS II (terminal #31).


Double click the icon for the AWG on your Multisim circuit. Ensure that the

waveform which you created in the pre-lab is loaded into output channel 0.


Multisim.

Open the VPS VI, and set the SUPPLY to be 10V. Ensure the device is Multisim.

Open the OSCOPE VI.

Run the simulation.

Set the device of the AWG and VPS to be NI ELVIS II and run continuously.

Power on the PB.

Run the AWG and VPS VIs.





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The following procedures are very similar to the above; the focus is on varying thefrequency of the input signal, or the Update rate.

Switch the device of the AWG, VPS, and OSCOPE to be Multisim.


Run the simulation.

Switch the device of the AWG, VPS, and OSCOPE to be NI ELVIS II, and run the VIs.


Create a printout which clearly displays both pairs of the input and output waveforms.(In-Lab Printout #10).

Switch the device of the AWG, VPS and OSCOPE to be Multisim.


Run the simulation.

Switch the device of the AWG, VPS and OSCOPE to be NI ELVIS II, and run the VIs.



Switch the device of the AWG, VPS and OSCOPE to be Multisim.


Run the simulation.






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Part 4: Explorations for Extra Credit

Under-damped Responses of a Second-Order Low-

Pass Filter Continue to use the circuit from Part 3. However, replace the 200 resistor with a10 resistor. Leave all other components unchanged.

Ensure the Update rate to be 1,000 S/s with an immediate trigger, and that the device isMultisim.

Open the VPS VI, and set the SUPPLY to be 10V. Ensure the device is Multisim.

Open the OSCOPE VI.

Run the simulation.

Set the device of the AWG and VPS to be NI ELVIS II and run continuously.

Power on the PB.

Run the AWG and VPS VIs.







Run the simulation.







Run the simulation.







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Run the simulation.







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Post-Lab:

1.First-Order Low-PassA.For each of your first-order low-pass plots (In-Lab Printouts #1-4), discuss theagreement/disagreement between the pre-lab simulation results and the in-lab

experimental results.

B.Explain how the circuits response is dependent on the input signal frequency. Isthis consistent with your knowledge of low-pass filters?

C.A low-pass filter acts approximately as an integrator (at high frequencies). Doyour findings from In-Lab Printout #1-4 support or disprove this statement?

2.First-Order High-PassA.For each of your first-order high-pass plots (In-Lab Printouts #5-8), discuss the

agreement/disagreement between the pre-lab simulation results and the in-lab


B.Explain how the circuits response is dependent on the input signal frequency. Isthis consistent with your knowledge of high-pass filters?

C.A high-pass filter acts approximately as a differentiator (at low frequencies). Doyour findings from In-Lab Printout #5-8 support or disprove this statement?

3.Second-Order Low-Pass (Near Critical Dampening)A.For each of your second-order low-pass plots (In-Lab Printouts #9-12), discuss

the agreement/disagreement between the pre-lab simulation results and the in-lab

experimental results.B.Explain how the circuits response is dependent on the input signal frequency. Is

this consistent with your knowledge of low-pass filters?

C.Compare your second-order low-pass filter results (In-Lab Printouts #9-12) toyour first-order low-pass filter results (In-Lab Printouts #1-4). Discuss the

similarities/differences.

2010 A. Ganago Post-Lab Page 1 of 2


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4.Explorations: Second-Order Low-Pass (Under Damped)A.For each of your second-order low-pass plots (In-Lab Printouts #13-16), discussthe agreement/disagreement between the pre-lab simulation results and the in-lab


B.Explain how the circuits response is dependent on the input signal frequency. Isthis consistent with your knowledge of low-pass filters?

C.Compare your under damped second-order low-pass filter results (In-LabPrintouts #13-16) to your critically damped second-order low-pass filter results

(In-Lab Printouts #9-12). Discuss how the damping ratio0

affects the circuits

response.

10LABO Ganago Student Lab10

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