Operational Amplifier - Outerspace-Time-Continuum · PDF fileWebster Lab 2: Operational...

Operational Amplifier

Joshua Webster

Partners: Billy Day & Josh Kendrick

PHY 3802L

10/16/2013

Webster Lab 2: Operational Amplifier

1

Abstract:

The purpose of this lab is to provide insight about operational amplifiers and to

understand the role they play in circuitry. In this lab, a circuit is constructed with an operational

amplifier being used as a summing amplifier. Data is recorded for different individual

experiments with the circuit. The first experiment is to find what effect grounding the inputs has

on the output voltage and gain. The second experiment shows the effects of solely increasing the

input voltage, and allows us to calculate the theoretical (nominal) values for the output voltage.

The third experiment is to show that changing the supply voltage affects the saturation value of

the output. The fourth experiment shows the effect that the frequency has on gain. The fifth

experiment is to show how the slew rate affects the output signal wave form. The sixth and final

part of the experiment is to measure the rise time and calculate the slew rate. The slew rate of the

operational amplifier was determined to be 0.536 V/μs.


2

Table of Contents

Abstract: .......................................................................................................................................... 1

Introduction ..................................................................................................................................... 3

Background ..................................................................................................................................... 4

Experimental Techniques................................................................................................................ 7

Diagrams and Images .................................................................................................................. 7

Data ............................................................................................................................................... 11

Part 1: ........................................................................................................................................ 11

Parts 2 & 3: ................................................................................................................................ 11

Part 4: ........................................................................................................................................ 12

Part 5: ........................................................................................................................................ 13

Part 6: ........................................................................................................................................ 14

Part 7: ........................................................................................................................................ 15

Part 8: ........................................................................................................................................ 18

Analysis......................................................................................................................................... 19

Discussion ..................................................................................................................................... 22

Conclusion .................................................................................................................................... 23

Appendix ....................................................................................................................................... 24

References ..................................................................................................................................... 25


3

Introduction

An operational amplifier or op amp is a device that is used extensively in analog electric

circuitry. Op amps can be utilized for a number of mathematical tasks including addition,

subtraction, multiplication, division, differentiation, and integration. It has dual-inputs, a single-

output, and functions as a linear amplifier with a high open-loop gain, high input resistances, and

low output resistance.1 Since the resistances of the inputs are very high, the current running

through an op amp is very small, which allows us to take the input current to be zero. This

experiment deals with the normal operation of an op amp, in which we will be supplying

symmetric voltages, and using it as a differential amplifier. A differential amplifier amplifies the

voltage difference between two input signals. The op amp chip being utilized is a 741A op amp,

which comes in an 8-pin dual-inline package (DIP).2

The following main sections of this report will consist of the background, experimental

techniques, data, analysis, discussion, and a conclusion. Also included will be an appendix for

any extra information that doesn’t necessarily fit into the other sections, and a section for any

references made in the text.

1 (Operational Amplifiers (Op Amps), 2001)

2 (PHY 3802L Experiments)


4

Background

Operating in normal mode, we will apply symmetric supply voltages across the

operational amplifier. The “golden op amp rules” state that the input and output currents are

zero, and the input and output voltage difference is zero. These rules, however, are used as a

close approximation to reality and are only entirely true for an ideal op amp.

The output voltage can be represented by,

( ) ( )

In the above equation: Vout is the output voltage, A0 is the open-loop gain, and V+ and V- are the

positive and negative input voltages respectively.

Since op amps are used in circuits with negative feedback, the effective input voltage can be

represented as:

( )

In the above equation, B is the feedback factor, which is determined by the feedback circuit.

The amplification with feedback, also known as closed-loop gain is described using the

following equation:

( )

In the above equation: Af is the amplification with feedback or gain, Vout is the output voltage,

and Vin is the input voltage.

From the property of the op amp, we find:

( ) ( )

Arranging terms,

( )

Where A0 is the open loop gain and B is the feedback factor. This expression shows that the

closed loop gain is smaller than the open loop gain. If then ⁄ . This means that

the gain of the amplifier is only dependent on the feedback factor B, and not on the open loop

gain A0. The variation of A0 is insignificant.

The uncertainty in the gain can be calculated using the following equation with the average

uncertainties in the measurements:


5

|

| |

|

|

| |

| ( )

Using the fact that the circuit is a scaling summer circuit3 and Ohm’s Law, an equation for Vout

can be obtained.

( )

Vout is the output voltage, VA and VB are the input voltages, and Rf, R1, and R2 are the resistances

of the resistors.

To find the uncertainty in Vout we must use the error propagation formula found in the Appendix

(A.1):

√(

)

(

)

(

)

( )

From this, we find:

√(

)

(

)

(

)

(

)

(

)

√(

)

(

)

(

)

(

)

(

)

( )

The uncertainties in the resistor values are given by the individual resistors.

The slew rate of an op amp can be defined as the slope of the output voltage versus time:

( )

An equation can be formulated that relates the slew rate to the frequency:

3 (Carter & Brown, 2001)


6

( )

( ) 4

For the equation above: is the frequency, one cycle is ⁄ seconds with a period of 2π, and Vout

is the output voltage.

Solving equation 10 for the frequency allows us to estimate the frequency at which gain begins

to decrease. Keep in mind that this equation is for a slew rate in units of V/s, and that this

equation is used as an estimate for the frequency.

( )

4 (Research Solutions and Resources LLC, 2011)


7

Experimental Techniques

Diagrams and Images

Diagram 1: This diagram shows the color coding for resistors. It was used to determine which

resistors we needed for our circuit. This image is from the lab website.

Figures 1 & 2: Shown below is the operational amplifier setup that was used in this lab.

Amplifiers are often represented as a triangle in diagrams. Using the op amp as a differential

amplifier; pins 2 and 3 are for the input signals, pin 4 is for the negative supply voltage, pin 7 is

for the positive supply voltage, and pin 6 is for the output. These images are from the lab

manual.


8

Figure 3: Shown below is the circuit constructed for this lab. vB and vA are inputs 1 and 2

respectively, R1 and R2 are input resistors, Rf is a feedback resistor. This image is from the lab

manual.

Image 1: This image shows the actual circuit that was constructed for this experiment.

Rf 741A Op Amp

R2 R1


9

Image 2: The image below shows the oscilloscope (top, Tektronix TDS2012B), and function

generator/power supply (bottom, FG 501A 2MHz Function Generator, DM 502A Autoranging

DMM, PS503A Dual Power Supply).

The summing amplifier shown in Figure 3 was constructed as shown in Image 1. A single

2 kΩ resistor was not available, so two 1 kΩ resistors were used in its place. Part 1 in the Data

section shows the values of the individual resistors. A 741A op amp with a dual-inline package

was used. The circuitry was connected to the supply voltage and function generator device, and

the supply voltage was set to +12 V and -12 V.

Using the function generator, a 1 kHz sinusoidal signal of about 100 mV was applied to

one input with the other grounded. The sign and magnitude of the amplification factor were

recorded. This was repeated, with the signal and ground connected to their respective opposite

inputs as before.

Next, the same signal was connected to both inputs. The output magnitude was measured

and compared to predicted values. The amplitude of the input signal was to be varied starting

from 100 mV going up to 1.2 V, however, a systematic error due to human error in recording the

measurements of the device caused the readings to be off by one-half of their intended value. At

each increment, the output voltage amplitude and gain were determined. A plot was then

constructed after the data was obtained and the nominal output amplitude was determined using

equation 6. Then supply voltage was changed to ±10 V, and the same sets of measurements were

recorded at the same input signal amplitudes.


10

The supply voltage was set back to ±12 V. The gain as a function of frequency was

measured for an input amplitude of around 500 mV (actually 250 mV, because of a systematic

error). The data for 3 frequency values per decade was recorded. For example, the three

frequency values for the first decade would be at intervals around 100 Hz, 300 Hz, and 500 Hz.

The sinusoidal input signal was then changed to a square input signal. For low, middle,

and high frequency values, the output signals were observed and recorded in the form of images.

The input signal was then changed to a triangular wave, and the same data was recorded.

Finally, the frequency was set to 1 kHz with an amplitude of 500 mV (250 mV). The rise

time of the square wave signal was determined using the oscilloscope for both the input to the

amplifier and the output from the amplifier. The rise time can only accurately be found by

“zooming in” with the oscilloscope until the trough and peak of the input or output takes up the

entire screen (graph). This also allows for the rise time to be estimated by eye using the graph

itself.


11

Data

Part 1:

Resistor values are R1 =R1a + R1b = (1052 ± 1 Ω) + (1063 ± 1 Ω) = 2115 ± 2 Ω, R2 = 1056

± 1 Ω, and Rf = 10120 ± 10 Ω. R1 is the sum of two resistors.

Parts 2 & 3:

Table 1: With the circuit constructed, a 1.014 kHz sinusoidal signal was applied at 102 mV with

input one (VA or R1) grounded. The resultant output, input 1 grounded Vout, was a sinusoidal

wave at 980 mV. A signal of 1.018 kHz at 103 mV with input 2 (VB or R2) grounded produced

an output signal, input 2 grounded Vout, of 1.016 kHz at 488 mV. With the same signal of 1.016

kHz at 100 mV applied to both inputs the output was 1.021 kHz at 1.420 V.

Freq.

(Hz)

Vin

(mV)

∆V in

(mV)

Vout

(mV)

∆V out

(mV)

Gain

(Af)

Input 1

Grounded 1014 102 1 980 10 9.61

Input 2

Grounded 1018 103 1 488 1 4.74

Both Inputs

Same Signal 1016 100 1 1420 10 14.20


12

Part 4:

Table 2: This table shows the values obtained through the measurements taken with the supply

voltage set to ±12 V and their uncertainties due to the accuracy of the device. Vin and Vout are the

input and output voltages respectively, ∆V in and ∆V out are the uncertainties in the input and

output voltages respectively. The nominal output was calculated using equation 7 and the

uncertainty in the nominal output, σNominal, is calculated using equation 8. The nominal voltage

was taken to be the absolute value, for means of comparison and graphing.

Vin

(V)

∆V in

(mV)

Vout

(mV)

∆V out

(mV)

Freq.

(Hz)

Gain ∆Gain Nominal Output

(mV)

σNominal

(mV)

51 1 720 10 990.1 14.1 0.20 733 10.74

101 1 1440 10 990.1 14.3 0.10 1451 10.81

148 1 2140 10 990.1 14.5 0.07 2126 10.92

200 1 2820 10 989.1 14.1 0.05 2874 11.08

250 1 3540 10 990.1 14.2 0.04 3592 11.28

302 1 4320 10 990.1 14.3 0.03 4339 11.54

344 1 4920 10 990.1 14.3 0.03 4943 11.77

400 1 5650 100 989.1 14.1 0.25 5747 12.12

452 1 6400 100 990.1 14.2 0.22 6494 12.49

515 10 7350 100 990.1 14.3 0.19 7400 107.36

550 10 7900 100 989.1 14.4 0.18 7903 107.40

600 10 8600 100 988.1 14.3 0.17 8621 107.45

Graph 1: A plot of the output amplitude as a function of the nominal output amplitude. Errors

bars were included, but are very small compared to the scale of the graph. The errors are listed in

Table 2. The absolute value of the nominal output amplitude is shown in this graph.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 100 200 300 400 500 600 700

Vo

ut (

mV

)

Vin (mV)

Vout vs Vin (Part 4)

Series1


13

Part 5:

Table 3: This table shows the values obtained through the measurements taken when the supply

voltage was set to ±10 V. The data headings are the same as in Table 2 and are explained there.

Vin

(mV)

∆V in

(mV)

Vout

(mV)

∆V out

(mV)

Freq.

(Hz)

Gain Nominal Output

(mV)

51 1 710 10 987.5 13.9 733

103 1 1460 10 990.1 14.2 1480

150 1 2160 10 990.1 14.4 2155

200 1 2860 10 990.1 14.3 2874

250 1 3540 10 988.1 14.2 3592

302 1 4320 10 988.1 14.3 4339

352 1 5100 100 989.1 14.5 5058

400 1 5650 100 989.1 14.1 5747

452 1 6400 100 987.2 14.2 6494

500 10 7150 100 988.1 14.3 7184

550 10 7900 100 988.1 14.4 7903

595 10 8600 100 990.1 14.5 8549

Graph 2: A plot of the experimentally determined output voltage versus the theoretically

determined, “nominal”, output voltage. Data for this graph is listed in Table 3.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 2000 4000 6000 8000 10000

No

min

al O

utp

ut

Vo

ltag

e (

mV

)

True Output Voltage: Vout (mV)

Vout vs. Nominal (Part 5)

Series1


14

Part 6:

Table 4: This table shows the values measured and calculated for the frequency decades. The

data headings are the same as in Table 2 and are explained there. The nominal output voltage is

dependent on Vin (which varies marginally) and the resistances (which do not change).

Vin

(mV)

∆Vin

(mV)

Vout

(mV)

∆Vout

(mV)

Freq.

(Hz)

∆Freq.

(Hz)

Gain Nominal Output

(mV)

246 1 3480 10 99.4 0.005 14.15 3534.58

233 1 3280 10 299 0.05 14.11 3347.79

232 1 3280 10 500 0.05 14.14 3333.42

246 1 3560 10 987.2 0.05 14.47 3534.58

236 1 3360 10 2987 0.5 14.27 3390.90

234 1 3340 10 4995 0.5 14.27 3362.16

236 1 3320 10 9750 0.5 14.10 3390.90

244 1 3000 10 29170 5 12.30 3505.84

244 1 2320 10 48760 5 9.51 3505.84

251 1 1260 10 101000 50 5.03 3606.42

255 1 416 1 304500 50 1.63 3663.89

252 1 254 1 507600 50 1.01 3620.79

252 1 126 1 1008000 500 0.50 3620.79

232 1 82 1 1503000 500 0.35 3333.42

244 1 62 1 2000000 500 0.25 3505.84

Graph 3: A plot of gain versus the logarithmic frequency for the values listed in Table 4.

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

0 1 2 3 4 5 6 7

Gai

n

Log(Frequency)

Gain vs. Log(Frequency)

Series1


15

Part 7:

Image 3: This image of a square input signal at 1.002 kHz was taken during the lab. It shows the

output signal being a square wave as well.

Image 4: This image of a square input signal at 100.0 kHz was taken during the lab. It shows a

change in the output signal wave form.


16

Image 5: This image of a square wave input signal at 1.008 MHz was taken during the lab. It

shows the output as triangular wave.

Image 6: This image of a triangular wave input signal at 1.000 kHz was taken during the lab. It

shows the output signal as having the same triangular waveform.


17

Image 7: This image is of a triangular wave at 99.80 kHz. It shows the output signal as being

sinusoidal.

Image 8: This image is of a triangular wave at 1.008 MHz. It shows a sinusoidal output wave.


18

Part 8:

Image 9: This image shows the square wave input signal at 1 kHz frequency and 466 mV. From

the image the rise time of the input signal can be determined to be 29.76 ns.

Image 10: This image shows the output signal of the square wave at 1 kHz frequency and 466

mV. The rise time of the output signal can be determined from this image to be 8.960 μs.


19

Analysis

The gain or amplification with feedback can be calculated using equation 3 as follows:

( )

In Parts 2 & 3 the results shown in Table 1 were as expected. The values in the table are sensible,

because of the two different resistances on inputs 1 and 2. When both of the inputs are connected

with the same signal the output voltage should increase. To calculate the uncertainty associated

with the gain we will use equation 6. The uncertainty in the input and output voltages are

associated with the device, and are accepted to be 1 digit of the last readable place.

|

| |

| ( )

|

| ( ) |

( ) | ( )

The nominal output amplitude is calculated using equation 7. It is said to be nominal because it is

the theoretical value for the output amplitude. The oscilloscope gives the true value. VA and VB

are the input voltages and are equal for this case.

( )

( )

( )

The uncertainty in the nominal output amplitude is given by equation 8. The uncertainties on the

resistors are given by the color coding on the resistors themselves. Since two resistors were used

for R1, the uncertainty was estimated to be twice the value of the uncertainty of each individual

resistor as determined from the resistance value recorded using a voltmeter.

√(

)

(

)

(

)

(

)

(

)

( )

√(

)

( ) (

( ) )

( ) (

( ) )

( ) (

)

( ) (

)

( )


20

For the above equation, VA and VB are in units of volts, and resistances are in Ohms.

In Part 5 the output voltage was changed to ±10V, to make the saturation value more

pronounced. There was a systematic error in our experiment due to incorrect reading of the

voltages. The values actually being recorded were the peak to peak values (on the oscilloscope)

instead of trough to peak, resulting in values that are half of what needed to be recorded for

saturation to take effect. The recorded values in the data tables were corrected for this error, but

since the voltage was not at a high enough level saturation of the signal did not occur.

The slew rate can be calculated using equation 9, where the ∆Vout is just the change in the

output voltage cutting out the top and bottom 10% (just measuring the middle 80%).

Specifically, in Image 10 the blue channel 2 line is the output, so starting at the first integer box

(not measuring the half boxes on top and bottom) measuring upwards there are 4 boxes each with

a value of 1.20 V. Therefore, ∆Vout = 4.80 V.

( )

( )

Table 5: For both the input and output, the values for rise time and slew rate are listed.

Vin

(V)

Vout

(V)

∆Vout

(V) Gain

Rise Time

(μs)

Slew Rate

(V/μs)

Input Channel 0.466 3.5 0.1 7.5 0.02976 117.6075269

Output Channel 0.466 3.5 0.1 7.5 8.960 0.390625

In Graph 3, the change in gain can be seen versus the change in frequency. The gain is

experimentally determined to be fairly steady (fluctuating slightly around 14) until a frequency

of about 10 kHz is applied. The gain begins to rapidly decrease as the frequency is raised even

higher. This drop in gain is the effect of the amplifier being incapable of amplifying the signal at

the frequency at which it is arriving. This effect is visible when using an oscilloscope, as shown

in Images 3-10. The frequency value at which the signal can no longer be amplified can be

determined by:

( )


21

(

) (

)

This analytically determined value for the frequency at which the op amp can no longer

amplify the signal is fairly consistent with the values found in Table 4. Some variation is to be

expected due to equation 11 being only a rough estimate of the true frequency.


22

Discussion

In Parts 2 & 3 the data is consistent with the values being within the range of

acceptability. In Part 4 all of the data seems to be reliable. As the input voltage is increased, the

output voltage jumps up for every 50 mV added to the input. The output voltage then

successively increases by about 1000 mV for every 50 mV added to the input until the amount it

increases by goes down to 100 mV in the last data point. The nominal output voltage seems to be

close to the true recorded values. The uncertainties in the nominal don’t quite make up the

difference between the nominal and the true values. This could be due to slight voltage losses

through the circuit that are, for the most part, negligible as the equation is based on an ideal

amplifier.

During Part 5, the saturation value was intended to be noticed by lowering the supply

voltage, and going through the same steps as in part 4. Due to a systematic error, in which the

voltages being applied were actually half of the intended values, this result did not occur. The

saturation value for typical op amps is around one volt below the supply voltage5, which is a

reasonable estimate due to slight voltage loss across the circuit.

In Part 6, Graph 3 shows that as the frequency increases the gain decreases. A gain of

greater than one means that amplification is occurring, whereas a gain of less than one means

that it is a passive circuit resulting in voltage loss instead of amplification. The data recorded in

Table 4 is a testament to this.

In Part 7, Images 3-8 show that as the frequency of the input wave is increased the output

wave signal changes. Specifically for this op amp, a square wave input signal will begin to have

a triangular wave output signal at around 100 kHz. A triangular wave input signal will begin to

have a sinusoidal wave output signal at around 100 kHz. This is due to the slew rate of the op

amp. The voltage can only be raised at a specific rate. Increased input frequency results in a

stretched wave form that is even more pronounced at higher input frequencies. This is why it

isn’t noticeable at lower frequencies.

At high frequencies the slew rate has a greater affect, which results in smaller gain,

because the op amp cannot amplify the signal at the rate at which it is being received. The

determined value for the slew rate (0.536 V/μs) is perfectly within the acceptable range. For the

741A op amp, the typical slew rate is 0.7 V/μs. The minimum is slew rate is listed to be 0.3

V/μs.6

5 (Oliveira Sannibale, 2012)

6 (National Semiconductor Corporation, 2000)


23

Conclusion

The data collected and calculations performed in the experiment give good insight into

operational amplifiers and general circuitry. The first set of data shows the effects that grounding

the different inputs has on the output voltage and gain. The second set of data shows the effects

of solely increasing the input voltage, and allows us to calculate the theoretical (nominal) values

for the output voltage. The third set of data was intended to show that changing the supply

voltage affects the saturation value of the output, but no visible effect was present in the case of

this experiment due to a systematic error. The fourth set of data shows that as the frequency is

increased to a significant amount, the gain decreases, and the op amp no longer amplifies. The

fifth set of data shows that the slew rate has a visible effect on the output wave signal. The final

set of data shows the rise time from which the slew rate of the op amp was determined to be

0.536 V/μs. It can also be concluded that the slew rate has a relation to the frequency dependence

of the gain.


24

Appendix

A.1 Formula for the propagation of errors:

√(

)

(

)

(

)

( )


25

References

Operational Amplifiers (Op Amps). (2001). Retrieved October 1, 2013, from siliconfareast.com:

http://www.siliconfareast.com/opamps.htm

Carter, B., & Brown, T. R. (2001, October). Handbook of Operational Amplifier Applications.

Retrieved October 5, 2013, from Texas Instruments:

http://www.physics.fsu.edu/courses/Fall13/phy3802L/exp3802/mech_em/sboa092a.pdf

National Semiconductor Corporation. (2000, August). LM741 Operational Amplifier. Retrieved

October 6, 2013, from www.national.com:

http://www.physics.fsu.edu/courses/Fall13/phy3802L/exp3802/mech_em/lm741.pdf

Oliveira Sannibale, V. d. (2012, December). Analog Electronics Basic Op-Amp Applications.

Retrieved October 16, 2013, from California Institute of Technology:

http://www.ligo.caltech.edu/~vsanni/ph5/pdf/Ph5.Chapter.BasicOpAmpCircuits.pdf

PHY 3802L Experiments. (n.d.). Retrieved October 1, 2013, from FSU Physics:

http://www.physics.fsu.edu/courses/Fall13/phy3802L/exp3802/mech_em/opampexp1.pdf

Research Solutions and Resources LLC. (2011, March 6). Control Amplifier Bandwidth and

Slew Rate. Retrieved October 8, 2013, from Resources for Electrochemistry:

http://www.consultrsr.com/resources/pstats/bwidth.htm

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